Representation Theory

CONTEMPORARY MATHEMATICS 478

Fourth International Conference on Representation Theory July 16 –20, 2007 Lhasa, China

Zongzhu Lin Jianpan Wang Editors

American Mathematical Society

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CONTEMPORARY MATHEMATICS

478

Representation Theory

Fourth International Conference on Representation Theory July 16–20, 2007 Lhasa, China

Zongzhu Lin Jianpan Wang Editors

American Mathematical Society Providence, Rhode Island

Editorial Board Dennis DeTurck, managing editor George Andrews Abel Klein Martin J. Strauss

2000 Mathematics Subject Classiﬁcation. Primary 16Gxx, 17Bxx, 20Cxx, 20Gxx; Secondary 17B10, 17B20, 17B37, 17B45, 17B56, 20G05, 20G10, 20G42, 20C05, 20C08, 20C30.

Library of Congress Cataloging-in-Publication Data Representation theory / Zongzhu Lin, Jianpan Wang, editors. p. cm. — (Contemporary mathematics ; v. 478) Includes bibliographical references. ISBN 978-0-8218-4555-4 (alk. paper) 1. Representations of algebras—Congresses. 2. Representations of groups—Congresses. I. Lin, Zongzhu. II. Wang, Jianpan, 1949– III. Title. QA150 .R46 2008 515.7223—dc22 2008034291

Copying and reprinting. Material in this book may be reproduced by any means for edu- cational and scientiﬁc purposes without fee or permission with the exception of reproduction by services that collect fees for delivery of documents and provided that the customary acknowledgment of the source is given. This consent does not extend to other kinds of copying for general distribution, for advertising or promotional purposes, or for resale. Requests for permission for commercial use of material should be addressed to the Acquisitions Department, American Math- ematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. Requests can also be made by e-mail to [email protected]. Excluded from these provisions is material in articles for which the author holds copyright. In such cases, requests for permission to use or reprint should be addressed directly to the author(s). (Copyright ownership is indicated in the notice in the lower right-hand corner of the ﬁrst page of each article.)

c 2009 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Copyright of individual articles may revert to the public domain 28 years after publication. Contact the AMS for copyright status of individual articles. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10987654321 141312111009

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Contents

Preface vii List of Talks ix List of Participants xiii Sum formulas and Ext-groups Henning Haahr Andersen 1 Schur-Weyl duality in positive characteristic Stephen Doty 15 The centers of Iwahori-Hecke algebras are filtered Andrew Francis and Weiqiang Wang 29 On Kostant’s theorem for Lie algebra cohomology University of Georgia VIGRE Algebra Group 39 G-stable pieces and partial flag varieties Xuhua He 61 Steinberg representations and duality properties of arithmetic groups, mapping class groups, and outer automorphism groups of free groups Lizhen Ji 71 Characters of simplylaced nonconnected groups versus characters of nonsimplylaced connected groups Shrawan Kumar, George Lusztig, and Dipendra Prasad 99 Classification of finite-dimensional basic Hopf algebras according to their representation type Gongxiang Liu 103 Twelve bridges from a reductive group to its Langlands dual G. Lusztig 125 Some new highest weight categories Brian J. Parshall and Leonard L. Scott 145 Classification of quasi-trigonometric solutions of the classical Yang–Baxter equation Iulia Pop and Alexander Stolin 155 The relevance and the ubiquity of Prüfer modules Claus Michael Ringel 163

vi CONTENTS

Quivers and the euclidean group Alistair Savage 177 eu2-Lie admissible algebras and Steinberg unitary Lie algebras Shikui Shang and Yun Gao 189 Lusztig’s conjecture for ﬁnite classical groups with even characteristic Toshiaki Shoji 207 A survey on quasiﬁnite representations of Weyl type Lie algebras Yucai Su 237

Maximal and primitive elements in baby Verma modules for type B2 Nanhua Xi 257 Irreducible representations of the special algebras in prime characteristic Yu-feng Yao and Bin Shu 273

Preface

The fourth International Conference on Representation Theory (ICRT- IV) was held in Lhasa, China, during July 16–20, 2007. The first three conferences were held in Shanghai (1998), Kunming (2001), and Chengdu (2004). The main goal of the ICRT is to bring together representation theorists of various subjects such as algebraic groups, quantum groups, finite groups, Lie algebras, vertex operator algebras, Hecke algebras and complex reflection groups, quivers, finite dimensional algebras, Hall algebras, and other related topics. The representation theory has evolved to connect many different fields of mathematics and the ICRT intends to communicate the common ideas and approaches to common questions. During the five day programs, participants have endured both mathematical and physical challenges. There was also a two day sub-conference on Mathematical History which covered many interesting historical topics such as early mathematics text books in Tibet and the first Chinese national who ever obtained formal Ph.D. in mathematics. This volume contains eighteen papers that more or less reflect the topics of the conference. All papers were carefully refereed. Several referees read several versions of the manuscripts and made critical comments to ensure the high quality of the papers. The editors want to express their sincere appreciation to all referees for their anonymous contribution. Due to page limitation, the editors regret that not all submitted papers are included and thank all authors for their contribution to this proceedings. It was a challenge from many respects to bring the conference to this ”high level”. The conference would not be possible without many people’s hardwork. Jianpan Wang and Hebing Rui had traveled to Lhasa several times to arrange hotel and lecture facilities. Both of them have been working for more than a year raising fund for the conference and making travel arrangements for international participants. Many participants took trains through the newly built railroad at 5 km altitude and the train ride lasts for more than two days. Making such arrangement was impossible without the help of many local mathematicians at various cities. The mathematicians in Tibet University, Da Luosang Langjie and Yutian Fei, provided invaluable help making local arrangement including the post-conference tours. The volunteer staff member, Xin Wang (a graduate student at University of Virginia), provided an impeccable professional-type services during conference. The financial support from the following organizations, projects, and individu- als are greatly appreciated: — Tianyuan Mathematical Fund, National Natural Science Foundation of China — “985” fund, East China Normal University

vii

viii PREFACE

— “111 Project”, Department of Mathematics, East China Normal Univer- sity — Changjiang Scholars and Innovative Research Team, Department of Math- ematics, East China Normal University — Jun Hu, “New Century Excellent Talents in University”, Beijing Institute of Technology — Fang Li, “New Century Excellent Talents in University”, Zhejiang Uni- versity — Hebing Rui, “New Century Excellent Talents in University”, East China Normal University — Yucai Su, “One-Hundred Talent Program”, University of Science and Technology of China — Jianpan Wang, Fund of Science and Technology Commission of Shanghai Municipality, No. 06JC 14024 — Jianpan Wang, fund of National Natural Science Foundation of China, No. 10631010 The ICRT-IV was co-hosted by East China Normal University and the Tibet University at Lhasa. The organizing committee consists of Jianpan Wang (Chair, East China Normal University), Da Luosang Langjie (Co-chair, Tibet University at Lhasa), Bangming Deng (Beijing Normal University), Yutian Fei (Tibet Uni- versity), Jun Hu (Beijing Institute of Technology), Ruyun Ma (Northwest Normal University), Liangang Peng (Sichuan University), Hebing Rui (East China Normal University). The program committee consists of Jianpan Wang (Chair, East China Nor- mal University), Jiping Zhang (Co-chair, Peking University), Chongying Dong (UC Santa Cruz), Jie Du (University of New South Wales), Zongzhu Lin (Kansas State University), Yucai Su (The University of Science and Technology of Chi- nese), Weiqiang Wang (University of Virginia), Nanhua Xi (Academy Sinica), Jie Xiao (Tsinghua University). Last but not least, the editors want to thank Christine M. Thivierge, Associate Editor for Proceedings at American Mathematical Society who worked so closely and patiently with the editors to ensure the timely publication of this proceedings.

Zongzhu Lin Jianpan Wang

June 2008

List of Talks

Principal Speakers Henning H. Andersen Sum formulas and Ext-groups Susumu Ariki Crystal theory and hecke algebras of type Bn Michel Broué Families of unipotent characters and cyclotomic algebras Jonathan Brundan Cyclotomic Hecke algebras and parabolic category O Jon Carlson Modules of constant Jordan type Shun-Jen Cheng Kostants homology formula for infinite-dimensional lie superalgebras Chongying Dong Representation theory for vertex operator algebras Yun Gao Irreducible Wakimoto-like modules for lie algebras of type A Shrawan Kumar Special isogenies and tensor product multiplicities George Lusztig Unipotent elements in small characteristic Brian Parshall Some results on quantum and algebraic group cohomology and applications Liangang Peng Conical extensions of derived categories Claus M. Ringel The relevance and the ubiquity of prüfer modules Olivier Schiffmann Geometric construction of macdonald polynomials Leonard Scott Some Z/2-graded representation theory Toshiaki Shoji, Lusztigs conjecture for finite classical groups Yucai Su Quasifinite representations of some lie algebras containing the Virasoro algebra

xLISTOFTALKS

Toshiyuki Tanisaki Differential operators on quantized flag manifolds at roots of 1 Nanhau Xi Some maximal elements in a baby Verma module Contributed 30 Minute Talks Thomas Brüstle Gentle algebras given by surface triangulations Vlastimil Dlab Standardly stratified approximations Jie Du Quantum gl∞, infinite q-Schur algebras and their representations Ming Fang Dominant dimensions and double centralizer properties Andrew Francis Symmetric polynomials of Jucys-Murphy elements and the centre of the Iwahori-Hecke algebra of type A Xuhua He Minimal length elements and G-stable pieces Jun Hu Morita equivalences of cyclotomic hecke algebras of type G(r, p, n) Lizhen Ji Duality of arithmetic groups, mapping class groups and outer automorphisms groups of free groups Otto Kerner Cluster tilted algebras of rank three Yanan Lin, Generic sheaves over elliptic curves Zongzhu Lin Canonical bases for algebras arising from quivers with loops Gongxiang Liu Classification of finite dimensional basic hopf algebras according to their representation type Alistair Savage A geometric construction of a crystal graph commutor Bin Shu Representations and cohomology of Jacobson-Witt algebras Mei Si, Discriminants of Brauer algebras Alexander Stolin Classification of quantum groups Liping Wang Leading coefficients of the Kazhdan-Lusztig polynomials of the affine Weyl group of type B˜2 Adrian Leonard Williams Minimal dimensions of some irreducible representations of the symmetric group Jiping Zhang The block union of finite groups

LIST OF TALKS xi

Mathematical History Talks Tom Archibald Chinese mathematicians and the international research community: the case of Hu Mingfu and integral equations Elena Ausejo Commercial arithmetic in the Spanish renaissance Jose A. Cervera The Chou Suan by Giacomo Rho: An example of mathematical adaptation in China Joseph W. Dauben Zhu Shijie and the Jade Mirror of the Four Unknowns Qi Han Antoine Thomas (1644C1709) and the ﬁrst introduction of western algebra into china Da Luosang Langjie Duchung Zurtsi : A mathematics textbook for secular and oﬃcial schools of Tibet Karen H. Parshall 4000 Years of algebra: an historical tour from BM 13901 to Moderne Algebra Yongdong Peng The early communication of cybernetics in China (1929C1966) Yibao Xu Chinese gougu theory versus Euclidean geometry: views of a seventeenth- century Chinese mathematician David E. Zitarelli Miss Mullikin and the internationalization of topology

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List of Participants

Henning H. Andersen Bangming Deng Aarhus University, DENMARK Beijing Normal University, Beijing Tom Archibald Vlastimil Dlab Simon Fraser University, CANADA Carleton University, CANADA Susumu Ariki Chongying Dong Kyoto University, JAPAN University of California at Santa Cruz, USA Elena Ausejo University of Zaragoza, SPAIN Jie Du University of New South Wales, Xiaotang Bai AUSTRALIA Nankai University, Tianjing Zhoutian Fan Stephen Berman Beijing Normal University, Beijing University of Saskatchewan, CANADA Ming Fang Michel Broué Beijing University of Technology, Institut Henri-Poincaré Paris, Beijing FRANCE Yutian Fei Jonathan Brundan Tibet University, Lhasa University of Oregon, USA Andrew Francis University of West Sydney, Thomas Brüstle AUSTRALIA Université de Sherbrooke, CANADA Qiang Fu Jon F. Carlson Tongji University, Shanghai University of Georgia, USA Yun Gao Bintao Cao York University, CANADA Institute of Mathematics, Academia Sinica, Beijing Jingyun Guo Xiangtan University, Xiangtan Jose A. Cervera ITESM, Campus Monterrey, MEXICO Qi Han Institute for History of Natural Science, Shun-Jen Cheng Academia Sinica, Beijing Academia Sinica (Taipei), Taipei Yang Han Joseph W. Dauben, Institute of Systems Science, Academia The City University of New York, USA Sinica, Beijing

xiii

xiv LIST OF PARTICIPANTS

Xuhua He Frantiˇsek Marko Stony Brook University, USA Penn State University, Hazleton, USA Terrell L. Hodge Qingnian Pan Western Michigan University, USA Huizhou University, Huizhou Masaharu Kaneda Brian J. Parshall Osaka City University, JAPAN University of Virginia, USA Shrawan Kumar Karen H. Parshall University of North Carolina, USA University of Virginia, USA Jiuzu Hong Lianggang Peng Institute of Mathematics, Academia Sichuan University, Chengdu Sinica, Beijing Yongdong Peng Jun Hu Hubei Education Press, Wuhan Beijing Institute of Technology, Beijing Claus M. Ringel Lizhen Ji University of Bielefeld, GERMANY University of Michigan, USA Hebing Rui Otto T. Kerner East China Normal University, Heinrich-Heine-Universität, Shanghai GERMANY Alistair Savage Jonathan Kujawa University of Ottawa, CANADA University of Georgia, USA Olivier Schiffmann Da Luosang Langjie Ecole´ Normale Supérieure, FRANCE Tibet University, Lhasa Leonard Scott Fang Li University of Virginia, USA Zhejiang University, Hangzhou Jianyi Shi Lei Lin East China Normal University, East China Normal University, Shanghai Shanghai Toshiaki Shoji Yanan Lin Nagoya University, JAPAN Xiamen University, Xiamen Bin Shu Zongzhu Lin East China Normal University, Kansas State University, USA Shanghai Gongxiang Liu Mei Si Institute of Mathematics, Academia East China Normal University, Sinica, Beijing Shanghai Li Luo Alexander Stolin Institute of Mathematics, Academia University of Gothenburg, SWEDEN Sinica, Beijing Yucai Su George Lusztig University of Science and Technology of MIT, USA China, Hefei

LIST OF PARTICIPANTS xv

Toshiyuki Tanisaki Osaka City University, JAPAN Jianpan Wang East China Normal University, Shanghai Li Wang Shanghai Normal University, Shanghai Liping Wang Institute of Mathematics, Academia Sinica, Beijing Weiqiang Wang University of Virginia, USA Xiaoming Wang East China Normal University, Shanghai Xin Wang University of Virginia, USA Adrian Williams Impare College, University of London, UK Nanhua Xi Academia Sinica, Beijing Yibao Xu The City University of New York, USA Rong Yan Institute of Mathematics, Academia Sinica, Beijing Ziting Zeng Beijing Normal University, Beijing Jiping Zhang Peking University, Beijing Pu Zhang Shanghai Jiaotong University, Shanghai Qinhai Zhang Shanxi Normal University, Xi’an Shizhuo Zhang Beijing University of Technology, Beijing David E. Zitarelli, Temple University, USA

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Contemporary Mathematics Volume 478, 2009

Sum formulas and Ext-groups

Henning Haahr Andersen

1. Introduction

Let G be a reductive algebraic group over a field k of characteristic p>0. In [4] we gave a unified proof for the sum formulas for the terms in the Jantzen filtrations of Weyl modules on the one hand side and on the terms of the corresponding filtrations involving tilting modules on the other hand. At the same time our approach in loc. cit. applies to the quantum group Uq associated with G when q is a root of unity. One advantage of these sum formulas is that they do not require any restrictions on the prime p or on the order of the root of unity in question. In contrast, many other methods and results on Weyl modules and tilting modules are only valid in the range p ≥ h, the Coxeter number of G. Sometimes we even require p 0 with no specified bound, see e.g. [6].

The Jantzen filtration for Weyl modules as well as their counterparts involving tilting modules are constructed by passing to the Chevalley groups GZ for G, respectively the Lusztig form Uk[v,v−1] of the quantum group Uq. Our approach in [4] is based on an Euler type formula for Ext-groups between certain GZ-modules, respectively Uk[v,v−1]-modules. In this paper we shall briefly recall the key ingredients of this method. In particular, we shall focus on the Ext-groups between (dual) Weyl modules for GZ. We shall illustrate how these groups contain a lot of information relevant for the representation theory of G. At the same time we will discover that the individual groups are very hard to compute. As an explicit example we make a detailed study of the case where G = GLn and where the highest weights of the Weyl modules involved differ by a single root. In this situation we manage to calculate all Ext-groups (they all vanish except the first). This case was handled by Kulkarni in [11] by rather long direct computations. Our approach avoids most of these and have also the advantage that it works without modifications in the corresponding quantum case.

Math Subject Classiﬁcation numbers: 20G05, 17B37.

c 2009 Americanc 0000 Mathematical (copyright Societyholder) 1

2 HENNING HAAHR ANDERSEN

As a further illustration of how to obtain information about individual such Ext-groups we show how the translation arguments in [2] can be applied to give similar calculations for adjacent Weyl modules for GZp in the general case. This however requires p ≥ h.

Just like the approach in [4] our calculations here are carried out in such a way that they immediately generalize to quantum groups at roots of 1. So if Uq is the quantum group corresponding to a root system R of type A then we determine i + + Ext (∇k[v,v−1](λ), ∇k[v,v−1](λ − β)) for all i ∈ N,λ ∈ X and β ∈ R . Uk[v,v−1] Likewise for a general quantum group we handle adjacent dual Weyl modules at a root of unity q ∈ k of order at least the Coxeter number.

The results in this note go somewhat further in the direction of explicit determination of Ext-groups than what I presented in my lecture at ICRT-IV in Lhasa. I would like again to thank the organizers of this extremely nice and stimulating conference.

2. Euler type formulas

2.1. Notation. As in the introduction we let G denote a reductive algebraic group over a field k.Wesetp =char(k) and assume throughout that p>0. We choose a maximal torus T in G and a Borel subgroup B containing T .ThenR will be the root system for (G, T ). We fix a set of simple roots S in R by requiring that the roots of B are the corresponding negative roots −R+.Thenumberofpositive roots is called N. This is also the dimension of the flag variety G/B.

The character group for T is denoted X.WeletX+ be the set of dominant characters, i.e., X+ = {λ ∈ X |λ, α∨≥ 0 for all α ∈ R+}.Notethat characters of T extend to B.Soforλ ∈ X we often denote by the same symbol the 1-dimensional B-module obtained from this character.

The Weyl group W = NG(T )/ZG(T )forG acts naturally on X.Ifα ∈ R then ∨ the reﬂection sα ∈ W corresponding to α is given by sα(λ)=λ −λ, α α for all λ ∈ X. We shall also use the ‘dot-action’ deﬁned by w · λ = w(λ + ρ) − ρ, w ∈ W, λ ∈ X.Hereρ is the half sum of the positive roots.

Each element w ∈ W is a product of simple reflections (reflections for simple roots) and we have the corresponding length function l on W taking w into the minimal number of such simple reflections needed to express w. The unique longest element in W is denoted w0.Ithaslengthl(w0)=N.

If M is a ﬁnite dimensional T -module and λ ∈ X then the weight space Mλ is { ∈ | ∈ } deﬁned by Mλ = m M tm = λ(t)m for all t T .Wesaythatλ is a weight λ ∈ Z of M if Mλ = 0. The character ch M is ch M = λ∈X (dim Mλ)e [X].

SUM FORMULAS AND EXT-GROUPS 3

For each λ ∈ X+ we have a Weyl module ∆(λ)forG with highest weight ∗ λ. Its contragredient dual ∆(λ) is denoted ∇(−w0λ). Note that then the dual + Weyl module ∇(µ) attached to µ ∈ X has highest weight µ (because w0(λ)isthe smallest weight of ∆(λ)).

Each ∇(µ) contains a unique simple submodule which we denote L(µ). Then L(µ) is also the unique simple quotient of ∆(µ). The collection {L(µ)}µ∈X+ is up to isomorphisms a complete list of ﬁnite dimensional simple G-modules.

2.2. Cohomology modules. Let M be a ﬁnite dimensional B-module. Then 0 G we will write H (M)fortheG-module IndB M induced by M. This is also the 0-th cohomology (i.e., the set of global sections) for the vector bundle on G/B associated with M. More generally, we denote by Hi(M)thei-th cohomology of i G this bundle, or alternatively the value of the i-th right derived functor R IndB on M.Itiswellknown(asG/B is a projective variety) that the cohomology H•(M) is ﬁnite dimensional, and that Hi(M)=0fori>N.

The Euler character of a B-module M is given by χ(M)= (−1)i ch(Hi(M)). µ∈X

Note that χ is additive, i.e., if 0 → M1 → M → M2 → 0 is a short exact sequence of ﬁnite dimensional B-modules then χ(M)=χ(M1)+χ(M2).

Recall that the Weyl modules in Section 2.1 are special instances of cohomology N 0 modules. To be precise, we have ∆(λ) H (w0 · λ)and∇(λ) H (λ) for all λ ∈ X+. Kempf’s vanishing theorem [10] implies that χ(λ)=ch∆(λ)=ch∇(λ).

2.3. Chevalley groups. Let GZ be a split and connected reductive algebraic group scheme over Z corresponding to G.InotherwordsGZ is the associated Chevalley group. Then G is obtained from GZ by extending scalars to k.More generally, we write GA for the group scheme over an arbitrary commutative ring A obtained via the base change Z → A.(ThecaseA = Zp, the ring of p-adic integers, will be needed in Chapter 5.) We use similar notation relative to the subgroups T and B.Inparticular,TZ is a split maximal torus in GZ with Tk = T .

Note that for a GA-module V that is free of ﬁnite rank as an A-module, ch(V ) makes sense by considering ranks of weight spaces. If our ﬁeld k is an A-algebra then we have for such a module ch(V )=ch(V ⊗A k).

i For any commutative ring A and any BA-module M we write HA(M)forthe G -module Ri IndGA M.See[9], I.5 for the general properties of these modules. A BA In particular, we recall that if A is noetherian and M is ﬁnitely generated over A, i then HA(M) is also ﬁnitely generated over A,see[9], Proposition I.5.12 c).

4 HENNING HAAHR ANDERSEN

Given any commutative ring A,foreachλ ∈ X+ we have the following two GA-modules: the Weyl module ∆A(λ) and the dual Weyl module ∇A(λ). These modules are characteristic-free, i.e., as A-modules both are free of rank equal to dim ∆(λ)andwehaveGA-module isomorphisms ∆A(λ) ∆Z(λ)⊗A and ∇A(λ) ∇ ⊗ ∇ 0 N · Z(λ) A.JustasforG,wehave A(λ)=HA(λ)and∆A(λ) HA (w0 λ).

2.4. Ext groups. Consider ﬁnitely generated GZ-modules M and N.By[9], i II.B, the groups ExtGZ (M,N) are ﬁnitely generated and vanish for large enough i. We will also need the following special cases of some vanishing results which can also be found in loc. cit. Proposition 2.1. For λ, µ ∈ X+,

i ∇ a) ExtGZ (∆Z(µ), Z(λ)) = 0 unless µ = λ and i =0. ∇ Z HomGZ (∆Z(λ), Z(λ)) = . i ∇ ∇ ≥ − ≥ b) ExtGZ ( Z(µ), Z(λ)) = 0 unless µ λ and ht(µ λ) i. ∇ ∇ Z HomGZ ( Z(λ), Z(λ)) = .

The universal coeﬃcient theorem [9], Proposition I.4.18 a) gives analogous results over GA for other commutative rings A. In particular the proposition stays valid after replacing each Z by the ring of p-adic integers Zp.

± a1 a2 ··· ar ∈ Z 2.5. Euler coefficients. If n = p1 p2 pr is the decomposition of n into different prime powers then we set r div(n)= ai[pi] ∈D(Z) i=1 where D(Z) is the divisor group for Z.WhenM is a finite Z-module of order |M| then we set div(M)=div(|M|). We use this to define the following Euler coefficients of GZ and BZ-modules.

Z i Let V and V be GZ-modules, both finitely generated over .ThenExtGZ (V,V ) is finite for all i>0. This follows from Section 2.4 and the universal coefficient i theorem [9], Proposition I.4.18 a), because ExtGC (A, B) = 0 for all i>0andfor any two rational GC-modules A and B (GC being reductive). If the GC-modules VC = V ⊗Z C and V ⊗Z C do not have an isomorphic simple summand, then HomGZ (V,V ) is finite. This happens in particular when V or V is finite. By Sec- i ∈ + tion 2.4 we have in any case ExtGZ (V,V )=0wheni 0. So whenever λ X and V is a GZ-module such that VC does not contain the irreducible module with highest weight λ (this is always the case when V is a finite GZ-module because then VC = 0), then the following expression gives a well defined element in D(Z) G − i i eλ (V )= ( 1) div(ExtGZ (∆Z(λ),V)). i≥0 G Clearly, eλ is additive on exact sequences of such GZ-modules (in particular finite GZ-modules). For more details see [4], Section 4.1.

SUM FORMULAS AND EXT-GROUPS 5

2.6. The alternating formulas. In [4] we computed the Euler coeﬃcients G eλ (V ) from Section 2.5 in the case where V is the cokernel Q(µ) of the natural homomorphism ∆Z(µ) →∇Z(µ) (These formulas were the key that made us able to deduce the sum formulas). Note that because of Proposition 2.1 a) if λ = µ then G − G we have eλ (Q(µ)) = eλ (∆Z(µ)). This means that the formulas in loc .cit. give an i expression for the alternating sum of div(ExtGZ (∆Z(λ), ∆Z(µ))). It is convenient for our purposes here to have the corresponding statements for dual Weyl modules. This goes as follows:

Let λ, µ ∈ X+ with λ = µ.Forβ,γ ∈ R+ we set ∨ Vβ(λ, µ)={(x, m) | x ∈ W, 0 λ + ρ, γ ,w· µ = λ − nγ}. Replacing Weyl modules by dual Weyl modules in [4] we can now state the following two formulas for the alternating sum of the Ext-groups between dual Weyl modules Theorem 2.2. If λ and µ are two diﬀerent dominant weights then − i i ∇ ∇ − l(x) ( 1) div(ExtGZ ( Z(µ), Z(λ))) = ( 1) div(m) + i≥0 β∈R (x,m)∈Vβ (λ,µ) and − i i ∇ ∇ − l(w) ( 1) div(ExtGZ ( Z(µ), Z(λ))) = ( 1) div(n). + i≥0 γ∈R (w,n)∈Uγ (µ,λ)

Remark 2.3. In [4]wecomputedthesizeofVβ(λ, µ) (and equivalently of Uγ (λ, µ)) for most root systems. They turn out to be quite small. In particular if R is of type A then there is for a given pair (λ, µ)atmostoneβ ∈ R+ for which Vβ(λ, µ) is non-empty and if non-empty Vβ(λ, µ) has just two elements. This makes the right hand sides of the formulas in this theorem easy to compute. We shall take advantage of this in the next section.

3. The general linear group

3.1. The natural module. In this section we consider the case G = GL(V ) where V is an (n+1)-dimensional vector space over k.WeidentifyG with GLn+1(k) by choosing a basis {e1,e2, ··· ,en+1} for V .ThenweletT , respectively B denote the subgroup consisting of all diagonal, respectively lower triangular matrices in G. We let i ∈ X denote the character of T (and of B) which projects an element onto its i-th diagonal entry. Then 1,2, ··· ,n+1 are the weights of V and we choose the enumeration of the basis above such that ei has weight i. The root system for (G, T ) consists of the roots i − j where 1 ≤ i = j ≤ n + 1 and the simple roots are i − i+1,i=1, ··· ,n. ∈ n+1 ∈ + Whenever λ X we write λ = i=1 λii.Thenλ X if and only if λ1 ≥ λ2 ≥···≥λn+1.

6 HENNING HAAHR ANDERSEN

We choose a Z-form VZ of V by setting VZ = Ze1 ⊕···⊕Zen+1.ThenGZ = GL(VZ )istheZ-group functor whose value at a ring A is GZ(A)=GL(VA)= GLn+1(A)whereVA = VZ ⊗Z A.AlsoTZ and BZ are the corresponding subgroup functors.

3.2. The GL2 case. Let n = 1, i.e. G = GL2(k). + Lemma 3.1. If λ, µ ∈ X satisfy µ ≤ λ + 1 then

i ∇Z ∇Z ⊗ Z a) ExtGZ ( (µ), (λ) V )=0for all i>0, Z − Z i /(λ1 λ2 +1) for i =1, b) Ext (∇Z(λ + 1), ∇Z(λ + 2)) GZ 0 otherwise.

0 Proof: The tensor identity tells us that ∇Z(λ) ⊗Z VZ HZ(λ ⊗Z VZ). The BZ-sequence

0 → 2 → VZ → 1 → 0 gives a short exact sequence of GZ-modules

0 →∇Z(λ + 2) →∇Z(λ) ⊗Z VZ →∇Z(λ + 1) → 0 1 ∨ − − ≥− because HZ(λ + 2) = 0 (note that λ + 2,α1 = λ1 λ2 1 1).

i ∇ ∇ Now Proposition 2.1 ensures the vanishing of both ExtGZ ( Z(µ), Z(λ + 2)) i ∇ ∇ ≥ and ExtGZ ( Z(µ), Z(λ + 1)) for all i 1whenµ<λ+ 1. Hence we have proved a) except when µ = λ + 1. In that case our arguments above still gives the ∇ ∇ ⊗ Z vanishing for i>1 and by observing that HomGZ ( Z(λ + 1), Z(λ) VZ) ∇ ∇ HomGZ ( Z(λ + 1), Z(λ + 1)) we also get the following exact sequence → Z →φ Z → 1 ∇ ∇ → 1 ∇ ∇ ⊗ → 0 ExtGZ ( Z(λ+1), Z(λ+2)) ExtGZ ( Z(λ1+1), Z(λ) Z VZ) 0.

Since λ + 1 − (λ + 2)=α1 has height 1 we get from Proposition 2.1 b) the i ∇ ∇ vanishing of ExtGZ ( Z(λ + 1), Z(λ + 2)) for i>1. Therefore the left hand side in the alternating formula (Theorem 2.2) consists in our case just of 1 term and | 1 ∇ ∇ | − hence tells us that ExtGZ ( Z(λ + 1), Z(λ + 2)) = λ1 λ2 +1.Inviewofthe above exact sequence we may ﬁnish the proof of both a) and b) by checking that the map φ is multiplication by λ1 − λ2 +1.

Set r = λ1 − λ2. Recall that ∇Z(λ) may be identiﬁed with the r-th symmetric r r−i i ··· power S (VZ). We set vi = e1 e2,i=0, 1, ,r. These elements form then a Z ∇ ··· -basis for Z (λ). We denote by vi,i=0, 1, ,r + 1 the analogously deﬁned ∇ ∇ basis for Z(λ + 1). It is then an easy exercise to check that HomGZ ( Z(λ + 1), ∇Z(λ) ⊗ VZ) is generated by the following map → − ⊗ ⊗ ··· vi (r +1 i)vi e1 + ivi−1 e2,i=0, 1, ,r+1.

Here we have set v−1 = vr+1 = 0. On the other hand, the natural map vi ⊗ e1 → ⊗ → ∇ ⊗ ∇ vi,vi e2 vi+1 generates HomGZ ( Z(λ) VZ), Z(λ + 1)). It follows that φ is indeed multiplication by r +1.

SUM FORMULAS AND EXT-GROUPS 7

Remark 3.2. Note that the second part of this lemma could maybe more appropriately be stated as follows Z ∨Z i / λ, α1 for i =1, Ext (∇Z(λ), ∇Z(λ − α1)) GZ 0otherwise.

+ for all λ with λ−α1 ∈ X . This is the n = 1 case of our main result in this section.

3.3. Restrictions to parabolic subgroups. We return to the case of a general n. Here we set P1 equal to the standard maximal parabolic subgroup corresponding to {α2,α3, ··· ,αn}.Then1 is a character of P1 and we have 0 V H (G/P1,1). Moreover, we have a short exact sequence of P1-modules 0 (1) 0 → H (P1/B, 2) → V → 1 → 0.

Analogously we have the standard maximal parabolic subgroup P2 corresponding to the subset {α1,α2, ··· ,αn−1} of simple roots. This time n+1 is a character of P2 and we have an exact P2-sequence 0 (2) 0 → n+1 → V → H (P2/B, 1) → 0.

Let L1 ≤ P1 be the standard Levi subgroup of P1.ThenL1 T1 × G1 where T1 is the 1-dimensional torus (embedded in G as the first diagonal entry) and G1 is the subgroup of G consisting of those matrices which have 0 in all entries of the first row and column except for the first entry which is 1. Clearly, G1 GLn(k).

We let V1 denote the natural module for G1. This extends to a module for P1 0 + and in fact V = H (P /B, )| . More generally, if for λ ∈ X we set ∇ (λ)= 1 1 2 G1 L1 0 0 H (L /L ∩B,λ) then this L -module extends to P and we have H (P /B, λ)| 1 1 1 1 1 L1 ∇ L1 (λ).

This implies that if also µ ∈ X+ then i H0 P /B, µ ,H0 P /B, λ i ∇ µ , ∇ λ (3) ExtP1 ( ( 1 ) ( 1 )) ExtL1 ( L1 ( ) L1 ( )) × ∇ for all i. On the other hand, the decomposition L1 = T1 G1 gives L1 (λ) ⊗∇ ∇ λ11 1(λ )whereλ = i≥2 λii and 1(λ ) is the dual Weyl module for G1 with highest weight λ. This implies i ∇ µ , ∇ λ µ λ , i ExtG1 ( 1( ) 1( )) if 1 = 1 (4) Ext (∇L (µ), ∇L (λ)) L1 1 1 0otherwise.

Analogously, we have for the Levi subgroup L2 ≤ P2 that L2 G2 × T1 where G2 GLn(k)andT1 this time is embedded in G as the last diagonal entry. The nat- 0 ural module V for G extends to a P -module and we have V = H (P /B, )| . 2 2 2 2 2 1 G2 + Also if for λ ∈ X we write λ = λ + λn+1n+1 then in notation analogous to the 0 ∇ above we have H (P2/B, λ)|L L2 (λ) and we get just as above 2 i ∇ µ , ∇ λ µ λ , i ExtG2 ( 2( ) 2( )) if n+1 = n+1 (5) Ext (∇L (µ), ∇L (λ)) L2 2 2 0otherwise.

8 HENNING HAAHR ANDERSEN

More generally, associated to any subset I ⊂{α1,α2, ··· ,αn} we have a standard maximal parabolic subgroup PI with Levi subgroup LI .Ifλ ∈ X we set 0 0 0 ∨≥ ∈ HI (λ)=H (PI /B, λ). Then HI (λ) =0ifandonlyif λ, αi 0 for all i I. ∇ 0 ∩ 0 ∈ + We set I (λ)=H (LI /B LI ,λ)=HI (λ)|LI .Whenλ X the natural PI - homomorphism ∇(λ) →∇I (λ) is surjective, i.e. we have a short exact sequence of PI -modules 0 → N (λ) →∇(λ) →∇ (λ) → 0. I I − ∈ N Here all weights µ of NI (λ)satisfyµ = λ i aiαi with ai and ai =0for some i ∈ I. I For λ ∈ X we set λ = λjj with the sum extending over those j for which αj or αj−1 belong to I. In analogy with (4) and (5) we get then (in the obvious notation) i ∇ µI , ∇ λI µ − µI λ − λI , i ExtGI ( I ( ) I ( )) if = (6) Ext (∇I (µ), ∇I (λ)) LI 0otherwise; ∈ ∨ ∨≥ ∈ for all µ, λ X for which µ, αi , λ, αi 0 for all αi I.

For all this see [9] Chapter II.4.

When restricted to PI the G-module ∇(λ) has a good ﬁltration [8]. By the above ∇I (λ) occurs exactly once as a quotient in such a ﬁltration and all other factors ∇I (µ)haveµ of the above form. Remark 3.3. All the results in this subsection have Z-versions. This will be important in the following computations. It is left to the reader to place the relevant subscripts Z in the statements to get their formulations over Z.

3.4. A key vanishing result. We remain in the case G = GL(V )withV having dimension n + 1. Now we extend the result in Lemma 3.1 a) to the case of general n. + Proposition 3.4. Let µ, λ ∈ X with µ ≤ λ + 1.Then i ∇ ∇ ⊗ ExtGZ ( Z(µ), Z(λ) Z VZ)=0 for all i>0.

Proof: To simplify notation we omit all subscripts Z in this proof. We use the notation from Section 3.3 above.

By the tensor identity we have V ⊗∇(λ) H0(V ⊗ λ). Hence by Kempf’s vanishing theorem we get i ∇ ⊗∇ i ∇ ⊗ ExtG( (µ),V (λ)) ExtB( (µ),V λ) for all i.TheshortexactP1-sequence (1) therefore leads to the long exact sequence of Z-modules ···→ i ∇ 0 ⊗ → i ∇ ⊗∇ → i ∇ →··· ExtB( (µ),H (P1/B, 2) λ) ExtG( (µ),V (λ)) ExtB( (µ),λ+1) .

SUM FORMULAS AND EXT-GROUPS 9

i ∇ Since µ >λ+ 1 we have ExtB( (µ),λ + 1) = 0 for all i>0. Moreover, i ∇ µ ,H0 P /B, ⊗ λ i ∇ µ ,H0 P /B, ⊗ H0 P /B, λ ExtB( ( ) ( 1 2) ) ExtP1 ( ( ) ( 1 2) ( 1 )) i ∇ µ | , ∇ ⊗∇ λ ExtL1 ( ( ) L1 L1 ( 2) L1 ( )). Here the ﬁrst isomorphism again relies on Kempf’s vanishing theorem whereas the last one comes from (3).

As recalled in Section 3.3 ∇(µ) has a good ﬁltration as a P1-module. The ∇ ≤ factors occurring have the form L1 (ν)withν µ. Hence we are done if we prove i ∇ ν , ∇ ⊗∇ λ i> ExtL1 ( L1 ( ) L1 ( 2) L1 ( )) = 0 for all 0 whenever ν ≤ λ + .Butby(4)wehave 1 i ∇ ν , ∇ ⊗∇ λ ν λ , i ExtG1 ( 1( ) 1( 2) 1( )) if 1 = 1 Ext ((∇L (ν), ∇L (2)⊗∇L (λ)) L1 1 1 1 0otherwise.

Recall also from the previous section that ∇1(2) V1. Hence the desired vanishing follows via our induction on n (the induction start is provided by Lemma 3.1) by ≤ ≤ noting that if ν1 = λ1 then ν λ + 2 for all ν λ + 1.Infact,ifν = − n ∈ N − λ + 1 i=1 aiαi with ai then ν1 = λ1 means λ1 = λ1 +1 a1, i.e. a1 =1. − n ≤ But then ν = λ + a12 i=2 aiαi λ + 2.

3.5. The main theorem. We can now state the main result of this section. Theorem 3.5. Let β be an arbitrary positive root. Then for any λ ∈ X+ for which λ − β is also in X+ we have Z ∨− Z i /( λ + ρ, β 1) if i =1, Ext (∇Z(λ), ∇Z(λ − β)) GZ 0 otherwise.

Proof: Again in this proof we omit all Z subscripts.

There exist 1 ≤ r ≤ s ≤ n such that β = αr + αr+1 + ···+ αs = r − s+1.We ﬁrst reduce to the case r =1,s= n:

Set I = {αr,αr+1, ··· ,αs}. As in Section 3.3 we let PI , respectively LI denote the corresponding standard parabolic subgroup, respectively its Levi part. Note that we have LI Tr−1 × GLs+2−r × Tn−s where the tori Tr−1 and Tn−s have dimensions given by their index. Then we have via the results in Section 3.3 i ∇ λ , ∇ λ − β i ∇ λ , ∇ λ − β ExtG( ( ) ( )) ExtPI ( I ( ) I ( )) i ∇ λ , ∇ λ − β i ∇ λI , ∇ λ − β I . (7) ExtLI ( I ( ) I ( )) ExtGI ( I ( ) I (( ) ))

Here GI = GLs+2−r and the last isomorphism follows from (6) by noting that λj =(λ − β)j for all j with js+1.

This means that we may replace G by GI . Hence we may indeed assume r =1 and s = n.

We proceed now by induction on ht(β)=n.Ifn =1thenweareinthe GL2-case and we may appeal to Lemma 3.1 for the result. So assume n>1and

10 HENNING HAAHR ANDERSEN observe that since n+1 is the lowest weight of V the module V ⊗ (λ − 1)contains λ − 1 + n+1 = λ − β as a B-submodule. We therefore have an exact sequence of G-modules

(8) 0 →∇(λ − β) → V ⊗∇(λ − 1) → Q → 0 ∇ − n−1 ∇ − n−2 ··· ∇ where Q has a good G-ﬁltration with factors (λ i=1 αi), (λ i=1 αi), , (λ). ∇ Z i ∇ Hence HomG( (λ),Q) whereas ExtG( (λ),Q)=0fori>1 by our induction on ht(β).

1 ∇ We claim that also ExtG( (λ),Q) is zero. To see this we use the sequence (2) tensored by λ − 1. When we induce the resulting sequence to G we get the exact 0 0 sequence 0 →∇(λ−β) → V ⊗∇(λ−1) → H (G/B, H (P2/B, 1)⊗(λ−1)) → 0. 0 0 This means that Q H (G/B, H (P2/B, 1) ⊗ (λ − 1)) and we get (using the same line of arguments as above) 1 ∇ 1 ∇ 0 ⊗ − ExtG( (λ),Q) ExtB( (λ),H (P2/B, 1) (λ 1)) 1 ∇ λ ,H0 P /B, ⊗ H0 P /B, λ − ExtP2 ( ( ) ( 2 1) ( 2 1)) 1 ∇ λ , ∇ ⊗∇ λ − 1 ∇ λ ,V ⊗∇ λ − . ExtL2 ( L2 ( ) ( 1) L2 ( 1)) ExtG2 ( 2( ) 2 2( 1)) The last term vanish by Proposition 3.4 applied to G2 GLn.

i ∇ ∇ − This means in view of (8) and Proposition 3.4 that ExtG( (λ), (λ β)) i−1 ∇ ExtG ( (λ),Q)=0fori>1 and that we have an exact sequence → Z → Z → 1 ∇ ∇ − → 0 ExtG( (λ), (λ β)) 0. 1 ∇ ∇ − It follows that ExtG( (λ), (λ β)) is cyclic. Now we apply the alternating formula Theorem 2.2 to conclude that its order is λ + ρ, β∨−1 as desired. Remark 3.6. The result in (7) holds more generally. The same arguments as i ∇ λ , ∇ µ i ∇ λI , ∇ µI we used above show that ExtG( ( ) ( )) ExtGI ( I ( ) I ( )) whenever µ ∈ λ − NI. Stated in ’partition language’ this says that when computing Ext- groups between two (dual) Weyl modules given by the two partitions λ and µ we may strip off the first rows in λ and µ whenever their lengths coincide. The same goes for the last rows. Similarly we can also strip off the first columns if they are identical. In fact, the character δ = 1 + 2 + ··· + n+1 is the determinant on T . It extends to G and hence for any λ ∈ X+ we have ∇(δ + λ)=δ ⊗∇(λ). It follows that i ∇ ∇ i ∇ ∇ ExtG( (δ + µ), (δ + λ)) ExtG( (µ), (λ)) for all i.

4. Adjacent Weyl modules

4.1. In this section we discuss a result from [3] concerning extensions between adjacent Weyl modules. We consider here the case of an arbitrary connected reductive algebraic group G and we assume that the deﬁning ﬁeld k has char(k)=p ≥ h where h is the Coxeter number for the root system of G.

If β is a positive root and n ∈ Z then we denote by Hβ,n = {λ ∈ X |λ + ρ, β∨ = np} the aﬃne hyperplane associated to (β,n). We say that two alcoves C

SUM FORMULAS AND EXT-GROUPS 11

and C in X are adjacent if there is a unique pair (β,n) such that Hβ,n separates C and C .Ifλ ∈ C has mirror image λ ∈ C , i.e. λ = sβ,n · λ then we also say that (λ, λ) are adjacent p-regular weights. Equivalently, λ is maximal among all weights strongly linked to the p-regular weight λ,cf.[1].

Denote by Zp the ring of p-adic integers. Then we can write any m ∈ Zp in ν (m) the form m = up p with u ∈ Zp a unit and νp : Zp → N the p-adic valuation. In this notation we have Theorem 4.1. Let (λ, λ) be a pair of adjacent p-regular dominant weights separated by the hyperplane Hβ,n.Ifλ>λ then Z/pνp(np)Z i , i ∇ ∇ if =1 ExtGZ ( Zp (λ), Zp (λ )) p 0 otherwise.

In [3] we proved this result by exploiting properties of translation functors over Zp. Here we shall give an alternative proof which deduces it from our alternating formula Theorem 2.2.

4.2. The field case. We first treat the corresponding problem for Ext-groups for G. The arguments in this case were first given in [2] Proposition 4.2. If (λ, λ) is a pair of adjacent p-regular dominant weights with λ>λ then k if i =0, 1, Exti (∇(λ), ∇(λ)) G 0 otherwise.

Proof: Let C and C denote the alcoves containing λ and λ respectively. Choose µ ∈ C¯ ∩ C¯ in such a way that Hβ,n is the only hyperplane containing µ.Denote µ λ by Tλ , respectively Tµ the translation functor from the λ-block to the µ-block, respectively vice versa (see [9] for the deﬁnition as well as the following properties of translation functors). Then we have µ∇ ∇ µ∇ (9) Tλ (λ) (µ) Tλ (λ ), and there is an exact sequence →∇ → λ∇ →∇ → (10) 0 (λ ) Tµ (µ) (λ) 0. ∇ ∇ ⊂ ∇ λ∇ µ∇ ∇ This gives HomG( (λ), (λ )) HomG( (λ),Tµ (µ)) HomG(Tλ (λ), (µ)) EndG(∇(µ)) k.IfHomG(∇(λ), ∇(λ )) = 0 then the sequence (10) splits and the λ∇ µ simple module L(λ) would be a submodule of Tµ (µ). But Tλ L(λ) = 0 and hence λ∇ µ ∇ HomG(L(λ),Tµ (µ)) HomG(Tλ L(λ), (µ)) = 0.

These considerations show that the long exact sequence obtained from (10) by applying HomG(∇(λ), −) starts out as follows → → → → 1 ∇ ∇ → 1 ∇ λ∇ →··· (11) 0 k k k ExtG( (λ), (λ )) ExtG( (λ),Tµ (λ ))

12 HENNING HAAHR ANDERSEN

i ∇ λ∇ i µ∇ ∇ i ∇ ∇ As ExtG( (λ),Tµ (λ )) ExtG(Tλ (λ), (µ)) ExtG( (µ), (µ)) = 0 for all i ∇ ∇ i−1 ∇ ∇ i>0 we get from (11) ExtG( (λ), (λ )) ExtG ( (λ), (λ)) = 0 for i>1. The proposition follows.

4.3. Proof of Theorem 4.1. Recall that the universal coeﬃcient theorem gives for arbitrary η, ν ∈ X+ and i ∈ N i i 0 → Ext (∇Z (η), ∇Z (ν)) ⊗Z k → Ext (∇(η), ∇(ν)) → GZp p p p G

Zp i+1 Tor (Ext (∇Z (η), ∇Z (ν)),k) → 0. 1 GZp p p Taking η = λ and ν = λ we get from this and Proposition 4.2 i Ext (∇Z (λ), ∇Z (λ )) = 0 for all i ≥ 2. GZp p p ∇Z λ , ∇Z λ Observe that also HomGZp ( p ( ) p ( )) = 0. So Theorem 2.2 gives that the 1 ν (np) order of Ext (∇Z (λ), ∇Z (λ )) is p p .Infact,sinceλ and λ are adjacent we GZp p p ∨ have Vα(λ ,λ)=0forα = β and Vβ(λ ,λ)={d, m} where d = λ + ρ, β −np < p 1 and m = np. Hence we are done once we check that Ext (∇Z (λ), ∇Z (λ )) GZp p p 1 ∇ ∇ ∇ ∇ is cyclic. However, if not then ExtG( (λ), (λ )) (as well as HomG( (λ), (λ )) would have dimension higher than 1. Proposition 4.2 says that this is not so. Remark . i ∇ ∇ ∈ 4.3 The general question of determining ExtG( (µ), (λ)) for i N,λ,µ∈ X+ is wide open. Even the case i = 0 is very far from being solved. Our results in this section says that HomG(∇(λ), ∇(λ )) is 1-dimensional whenever the two weights λ and λ are adjacent. It follows from our results in Section 3 that when G = GL(V )andβ is a positive root then HomG(∇(λ), ∇(λ − β)) k if p divides λ + ρ, β∨−1 and 0 in all other cases. Some further results for GL(V )can be found in [7].

+ Question 4.4. Does there exist λ, µ ∈ X such that HomG(∇(λ), ∇(µ)) has dimension bigger than 1? Equivalently, are there examples where the ﬁnite abelian 1 ∇ ∇ group ExtGZ ( Z(µ), Z(λ)) is non-cyclic?

5. The quantum case

In this section k will still denote a ﬁeld but we allow its characteristic p to be 0. On the other hand we exclude p = 2 and if the root system R contains a copy of G2 we also exclude p = 3. We ﬁx a root of unity q ∈ k of order l or 2l with l odd. Let v be an indeterminate and denote by Uv the quantum group over Q(v) −1 associated to the root system R.SetA = k[v, v ]andletUA be the A-form of Uv (via the Lusztig divided power construction, see [5]). We make k into an A-algebra by sending v to q and set then Uq = UA ⊗A k. More generally, for any A-algebra A we set UA = UA ⊗A A .

Without going into details (for those we refer to [5]) we just mention the categories CA and Cq consisting of integrable UA-, respectively Uq-modules. If V ∈CA

SUM FORMULAS AND EXT-GROUPS 13 is a finitely generated torsion A-module then we set in analogy with the definition of eG in Section 2.5 λ eU V − i i λ ,V . λ ( )= ( 1) div(ExtCA (∆A( ) )) i≥0 + Here λ ∈ X ,∆A(λ) denotes the Weyl module for UA with highest weight λ and div : A →D(A) is defined as in [4].

As in Section 2 this leads to the following alternating formula Theorem 5.1. Let λ, µ ∈ X+ be two diﬀerent dominant weights. Then

i i l(x) a) ≥ (−1) div(ExtC (∇A(µ), ∇A(λ))) = ∈ + ∈ (−1) div([m]d ). i 0 A β R (x,m) Vβ (λ,µ) β i i l(w) b) (−1) div(Ext (∇ (µ), ∇ (λ))) = + (−1) div([n] ). i≥0 CA A A γ∈R (w,n)∈Uγ (µ,λ) dγ

5.1. Type A. In this subsection we assume that R is a root system of type A. In particular all roots have the same length and all dβ are 1. Arguing as is Section 3 we then obtain Theorem 5.2. Let β be an arbitrary positive root. Then for any λ ∈ X+ for which λ − β is also in X+ we have ∨− i A/([ λ + ρ, β 1])A if i =1, ExtC (∇A(λ), ∇A(λ − β)) A 0 otherwise.

5.2. Adjacent Weyl modules. Let R again be arbitrary. Denote by A(v−q) the localization of A at the maximal ideal generated by v − q.Wethenhavethe following direct analogue of Theorem 4.1. Theorem 5.3. Suppose l ≥ h.Let(λ, λ) be a pair of adjacent l-regular dominant weights separated by the hyperplane Hβ,n and with λ>λ.

a) If char k =0then k i , i ∇ ∇ if =1 ExtC ( A(v−q) (λ), A(v−q) (λ )) A(v−q) 0 otherwise. b) If char k = p>0 then − pνp(n) i ∇ ∇ A/(v q) A if i =1, ExtC ( A(v−q) (λ), A(v−q) (λ )) A(v−q) 0 otherwise.

References

[1] Henning Haahr Andersen, The strong linkage principle, J. Reine Ang. Math. 315 (1980), 53–59. [2] Henning Haahr Andersen, On the structure of the cohomology of line bundles on G/B, J. Algebra 71 (1981), 245–258. [3] Henning Haahr Andersen, Filtrations and tilting modules, Ann. scient. Ec.` Norm. Sup. 30 (1997), 353–366.

14 HENNING HAAHR ANDERSEN

[4] Henning Haahr Andersen and Upendra Kulkarni, Sum formulas for reductive algebraic groups, Advances in Math. 217 (2008), 419–447. [5] Henning Haahr Andersen, Patrick Polo and Wen Kexin, Representations of quantum algebras, Invent. math. 104 (1991), 1–59. [6] Henning Haahr Andersen, Jens Carsten Jantzen and Wolfgang Soergel, Representations of quantum groups at a p−th root of unity and of semisimple groups in characteristic p: Indepen- dence of p, Asterisque 220 (1994), pp. 1–321. [7] Roger Carter and M. T. J. Payne, On homomorphisms between Weyl modules and Specht modules, Math. Proc. Camb. Phil. Soc. 87 (1980), 419–425. [8] Stephen Donkin, Rational representations of algebraic groups, Lecture Notes in Mathematics 1140 (Springer 1985). [9] Jens Carsten Jantzen, Representations of algebraic groups. Second edition. Mathematical Sur- veys and Monographs, 107. American Mathematical Society, Providence, RI, 2003. [10] George Kempf, Linear systems on homogeneous spaces, Ann. Math. (2) 103 (1976), 557-591. [11] Upendra Kulkarni, On the Ext groups between Weyl modules for GLn.J.Algebra304 (2006), 510–542.

Department of Mathematics, University of Aarhus, Building 530, Ny Munkegade,8000 Aarhus C, Denmark

E-mail address: [email protected]

Contemporary Mathematics Volume 478, 2009

Schur–Weyl duality in positive characteristic

Stephen Doty

Abstract. Complete proofs of Schur–Weyl duality in positive characteristic are scarce in the literature. The purpose of this survey is to write out the details of such a proof, deriving the result in positive characteristic from the classical result in characteristic zero, using only known facts from representation theory.

1. Introduction

Given a set A write SA for the symmetric group on A, i.e., the group of bijections of A.Forσ ∈ SA and a ∈ A we always write aσ for the image of a under σ. In other words, we choose to write maps in SA on the right of their argument. This means that στ (for σ, τ ∈ SA) is deﬁned by a(στ)=(aσ)τ.

We will write Sr as a shorthand for S{1,...,r}. Consider the group Γ = GL(V ) of linear automorphisms on an n-dimensional vector space V over a ﬁeld K. We write elements g ∈ Γ on the left of their argument. (Indeed, maps are generally written on the left in this article, except when they belong to a symmetric group.) The given action (g, v) → g(v)ofΓonV induces a corresponding action on a tensor power V ⊗r, with Γ acting the same in each tensor position: g(u1 ⊗···⊗ur)=(g(u1))⊗···⊗(g(ur)), for g ∈ Γ, ui ∈ V .Evidentlythe ⊗r action of Γ commutes with the “place permutation” action of Sr,actingonV on the right via the rule (u1 ⊗···⊗ur)σ = u1σ−1 ⊗···⊗urσ−1 . In this action, a vector that started in tensor position iσ−1 ends up in tensor position i,thusa vector that started in tensor position i ends up in tensor position iσ. We write KG for the group algebra of a group G. The fact that the two actions commute means that the corresponding representations ⊗r ⊗r (1.1) Ψ : KΓ → EndK (V ); Φ : KSr → EndK (V ) induce inclusions ⊆ ⊗r ⊆ ⊗r (1.2) Ψ(KΓ) EndSr (V ); Φ(KSr) EndΓ(V ) ⊗r ⊗r where EndSr (V ) (respectively, EndΓ(V )) is deﬁned to be the algebra of linear ⊗r operators on V commuting with all operators in Φ(Sr) (respectively, Ψ(Γ)).

The author is grateful to Jun Hu for bringing reference [12] to his attention, and to the referee for useful suggestions.

c 2009 Americanc 0000 Mathematical (copyright Societyholder) 151

216 STEPHEN DOTY

Equivalently, the commutativity of the two actions says that the representations in (1.1) induce algebra homomorphisms → ⊗r → ⊗r (1.3) Ψ:KΓ EndSr (V ); Φ:KSr EndΓ(V ). The statement that has come to be known as “Schur–Weyl duality” is the following. Theorem 1 (Schur–Weyl duality). For any inﬁnite ﬁeld K, the inclusions in (1.2) are actually equalities. Equivalently, the induced maps in (1.3) are surjective.

In case K = C this goes back to a classic paper of Schur [21].1 The main purpose of this survey is to write out a complete proof of the theorem for an arbitrary infinite field, assuming the truth of the result in case K = C.The strategy, suggested by S. Koenig, is to argue that the dimension of each of the four algebras in the inclusions (1.2) is independent of the characteristic of the infinite field K. The claim for a general infinite field K then follows immediately from the classical result over C, by dimension comparison. We make no claim that this strategy is “best” in any sense; it is merely one possible approach. For a completely different recent approach, see [16].

2. Surjectivity of Ψ Let us first establish half of Theorem 1, namely the surjectivity of the induced map → ⊗r Ψ:KΓ EndSr (V ) in (1.3). For a very direct (and shorter) approach to this result, see the argument on page 210 of [1]. As already stated, the strategy followed ⊗r here is to argue that the algebras Ψ(KΓ), EndSr (V ) have dimension (as vector spaces over K) which is independent of the characteristic of the infinite field K.

We first establish that dimK Ψ(KΓ) is independent of K (so long as K is infinite). For this we need a general principal, which states that the “envelope” and “coefficient space” of a representation are dual to one another. To formulate the principle, let Γ be any semigroup and K any field (not necessarily infinite). Denote by KΓ the K-algebra of K-valued functions on Γ, with the usual product and sum of elements f,f of KΓ given by (ff)(g)=f(g)f (g), (f + f )(g)=f(g)+f (g), for g ∈ Γ.

Given a representation τ :Γ→ EndK (M)inaK-vector space M,thecoefficient Γ space of the representation is by definition the subspace cfΓ M of K spanned by the Γ coefficients {rab} of the representation. The coefficients rab ∈ K are determined relative to a choice of basis va (a ∈ I)forM by the equations (2.1) τ(g) vb = rab(g) va a∈I for g ∈ Γ, b ∈ I. Let KΓ be the semigroup algebra of Γ. Elements of KΓ are sums of the form ∈ g∈Γ agg (ag K) with finitely many ag = 0. The group multiplication extends by linearity to KΓ. The given representation τ :Γ→ EndK (M) extends by linearity to an algebra homomorphism KΓ → EndK (M); by abuse of notation we denote this

1A proof of Schur–Weyl duality over C can be extracted from Weyl’s book [25]. A detailed and accessible proof is written out in [11, Theorem 3.3.8].

SCHUR–WEYL DUALITY IN POSITIVE CHARACTERISTIC 173 extended map also by τ.Theenvelope2 of the representation τ is by deﬁnition the subalgebra τ(KΓ) of EndK (M). The representation τ factors through its envelope; that is, we have a commutative diagram τ / KΓFF End99K (M) FF rrr FF rr (2.2) FF rr F### + rrr τ(KΓ) in which the leftmost and rightmost diagonal arrows are a surjection and injection, respectively. Taking linear duals, the above commutative diagram induces another one ∗ (KΓ)∗ o τ End (M)∗ JdJd q K JJ qq JJ qq (2.3) JJ qq J2R xxqxq q ∗ τ(KΓ) in which the leftmost and rightmost diagonal arrows are now an injection and surjection, respectively. There is a natural isomorphism of vector spaces (KΓ)∗ KΓ, given by restricting a linear K-valued map on KΓ to Γ; its inverse is given by the process of linearly extending a K-valued map on Γ to KΓ.

Lemma 2([4, Lemma 1.2]). The coeﬃcient space cfΓ(M) may be identiﬁed with ∗ ∗ the image of τ , so there is an isomorphism of vector spaces (τ(KΓ)) cfΓ M.

Proof. Relative to the basis va (a ∈ I) the algebra EndK (M)hasbasiseab (a, b ∈ I), where eab is the linear endomorphism of M taking vb to va and taking all other vc,forc = b, to 0. In terms of this notation, equation (2.1) is equivalent with the equality (2.4) τ(g)= rab(g) eab. a,b∈I ∗ Let eab be the basis of EndK (M) dual to the basis eab,sothateab is the linear functional on EndK (M) taking the value 1 on eab and taking the value 0 on all ∗ other ecd. Then one checks that τ carries eab onto rab. This proves that cfΓ(M) may be identiﬁed with the image of τ ∗, as desired.

We apply the preceding lemma to the representation M = V ⊗r of Γ = GL(V ), ⊗r to conclude that dimK Ψ(KΓ) is equal to dimK cfΓ(V ). Now the reader may easily check that coeﬃcient spaces are multiplicative, i.e., cfΓ(M ⊗ N)=cfΓ(M) · Γ cfΓ(N). Here the multiplication takes place in K . We will apply this fact to ⊗r r compute the dimension of cfΓ(V )=(cfΓ(V )) .

From now on we choose (and ﬁx) a basis {v1,...,vn} of V and identify V with n K and Γ with GLn(K), by means of the chosen basis. Then the action of Γ on V is by matrix multiplication.

2This terminology is adapted from [25], where Weyl writes about the “enveloping algebra” of a group representation as the algebra generated by the endomorphisms on the representing space coming from the action of all group elements. In modern terminology, this is just the image of the representation’s linear extension to the group algebra.

418 STEPHEN DOTY

⊗r Lemma 3. For Γ=GLn(K) and K any inﬁnite ﬁeld, cfΓ(V ) is the vector space AK (n, r) consisting of all homogeneous polynomial functions on Γ of degree n2+r−1 r. We have dimK AK (n, r)= r =dimK Ψ(KΓ).

Γ Proof. Let cij ∈ K be the function which maps a matrix g ∈ Γontoits (i, j)th matrix entry. By definition, a function f ∈ KΓ is polynomial3 if it belongs to the polynomial algebra K[cij :1 i, j n]. The cij are algebraically independent since K is infinite. Note that the cij are the coefficients of Γ on V , i.e., cfΓ V = 1i,jn Kcij. An element f ∈ K[cij :1 i, j n] is homogeneous of degree r if f(ag)= arf(g) for all a ∈ K and all g ∈ Γ. Here we define ag to be the matrix obtained from g by multiplying each entry by the scalar a. Now from the equality cfΓ V = 1i,jn Kcij and the multiplicativity of ⊗r coefficient spaces, it follows that cfΓ(V ) is the vector space AK (n, r) consisting of all homogeneous polynomial functions on Γ of degree r. The equality n2+r−1 dimK AK (n, r)= r , now follows by an easy dimension count (or one can look at [10, §2.1]), and this is the same as dimK Ψ(KΓ) by Lemma 2.

⊗r The preceding lemma establishes the fact that dimK cfΓ(V ) is independent of the characteristic of K (so long as K is infinite). So we turn now to the task of ⊗r establishing a similar independence statement for dimK EndSr (V ). Let us restrict the action of Γ to the “maximal torus” T ⊂ Γgivenbyall diagonal matrices in Γ = GLn(K). The abelian group T is isomorphic to the direct product (K×) ×···×(K×)ofn copies of the multiplicative group K× of the field K, so its irreducible representations are one-dimensional, given on a basis element λ1 ··· λn ∈ N z by the rule diag(a1,...,an)(z)=a1 an z, for various λi . For convenience λ λ1 ··· λn of notation, write t =diag(a1,...,an), λ =(λ1,...,λn), and t = a1 an .Now T acts semisimply on V ⊗r, and we have a “weight space decomposition” ⊗r ⊗r (2.5) V = λ∈Nn (V )λ ⊗r ⊗r λ where (V )λ = {m ∈ V : tm = t m, for all t ∈ T }. Since the action of T on V ⊗r commutes with the place permutation action ⊗r of Sr, it follows that each weight space (V )λ is a KSr-module. It is easy to ⊗r write out a basis for (V )λ in terms of the given basis {v1,...,vn} of V . Clearly ⊗r ⊗···⊗ V has a basis consisting of simple tensors of the form vi1 vir for various multi-indices (i1,...,ir) satisfying the condition ij ∈{1,...,n} for each 1 j r. ⊗···⊗ Each simple tensor vi1 vir has weight λ =(λ1,...,λn)whereλi counts the number of indices j such that ij = i. Thus it follows that i λi = r.Letuswrite ∈ Nn ⊗r Λ(n, r) for the set of all λ such that i λi = r. Then each summand (V )λ is zero unless λ ∈ Λ(n, r), so we may replace Nn by Λ(n, r) in the decomposition (2.5). ⊗r From the above it follows that a basis of (V )λ, for any λ ∈ Λ(n, r), is given ⊗···⊗ by the set of all vi1 vir of weight λ. ⊗r As a KSr-module, the weight space (V )λ may be identified with a “permuta- λ λ ⊗ tion” module M . Typically, M is defined as the induced module 1 (KSλ) (KSr),

3The notion of “polynomial” functions on general linear groups goes back (at least) to Schur’s 1901 dissertation.

SCHUR–WEYL DUALITY IN POSITIVE CHARACTERISTIC 195 where by 1 we mean the one dimensional module K with trivial action, and where Sλ is the Young subgroup × ×···× S{1,...,λ1} S{λ1+1,...,λ1+λ2} S{λn−1+1,...,λn−1+λn} of Sr determined by λ =(λ1,...,λn). By [2, §12D] this has a basis (over K) 4 indexed by any set of right coset representatives of Sλ in Sr. Lemma . ⊗r 4 For any ﬁeld K, dimK EndSr (V ) is independent of K. Proof. From the decomposition (2.5) it follows that we have a direct sum decomposition of End (V ⊗r)=Hom (V ⊗r,V⊗r)oftheform Sr Sr ⊗r ⊗r ⊗r EndSr (V )= λ,µ∈Λ(n,r) HomSr ((V )λ, (V )µ). By Lemma 7(b) in the next section, we may identify ⊗r ⊗r λ µ HomSr ((V )λ, (V )µ) HomSr (M ,M ) for any λ, µ ∈ Λ(n, r). By Mackey’s theorem (see [2, §44] or combine [22,Proposi- λ µ tion 22] with Frobenius reciprocity), it follows that dimK HomSr (M ,M )isequal to the number of (Sλ, Sµ)-double cosets in Sr, which is independent of K.This proves the claim. Alternatively, one can avoid the Mackey theorem by applying James [13, Theorem 13.19] directly (see also [7, Proposition 3.5]).

Now we can obtain the main result of this section, which proves half of Schur– Weyl duality in positive characteristic. We remind the reader that the validity of C C Cn ⊗r Theorem 1 for K = is assumed, so in particular Ψ( Γ) = EndSr (( ) ). Proposition 5. For any inﬁnite ﬁeld K, the image Ψ(KΓ) of the represen- ⊗r tation Ψ is equal to the centralizer algebra EndSr (V ),sothemapΨ in (1.3) is surjective.

Proof. By Lemmas 3 and 4 we have equalities

dimK Ψ(KΓ) = dimC Ψ(CΓ), n ⊗r Cn ⊗r dimK EndSr ((K ) )=dimC EndSr (( ) ) C Cn ⊗r for any infinite field K.SinceΨ( Γ) = EndSr (( ) ) it follows that dimK Ψ(KΓ) = n ⊗r dimK EndSr ((K ) ) for any infinite field K, and thus by comparison of dimensions the first inclusion in (1.2) must be an equality. Equivalently, the map Ψin (1.3) is surjective.

3. Surjectivity of Φ

It remains to establish the surjectivity of the induced map Φ in (1.3). This surjectivity was first established in positive characteristic in [3, Theorem 4.1].5 We will outline an alternative proof here, following our avowed strategy of showing that ⊗r the dimensions of Φ(KSr), EndΓ(V ) are independent of the characteristic of the infinite field K.

In order to establish the independence statement for Φ(KSr) we apply results of Murphy and H¨arterich in order to compute the annihilator of the action of Sr

4Reference [2] works with left modules instead of right ones, so for our purposes left and right need to be interchanged there. 5The statement of Theorem 4.1 in [3] is actually much more general.

620 STEPHEN DOTY on V ⊗r. Note that Murphy and Härterich worked with the Iwahori–Hecke algebra (with parameter q)intypeA, so one needs to take q = 1 in their formulas in order to get corresponding results for the group algebra KSr. The results of Murphy and Härterich hold over an arbitrary commutative integral domain, so K does not need to be an infinite field in this part. So we assume from now on, until the paragraph after Corollary 12, that K is a commutative integral domain. Let λ be a composition of r. We regard λ as an infinite sequence (λ1,λ2,...) of nonnegative integers such that λi = r. The individual λi are the parts of λ, and the largest index such that λ =0andλj =0forallj>is the length, or number of parts, of λ. Any composition λ may be sorted into a partition λ+, in which the parts are non-strictly decreasing. When writing compositions or partitions, trailing zero parts are usually omitted. If λ is a partition, we generally write λ for the transposed (or conjugate) partition, corresponding to writing the rows of the Young diagram as columns.

Given a composition λ =(λ1,...,λ )ofr, a Young diagram of shape λ is an arrangement of boxes into rows with λi boxes in the ith row. A λ-tableau T is a numbering of the boxes in the Young diagram of shape λ by the numbers 1,...,r so that each number appears just once. In other words, it is a bijection between the boxes in the Young diagram and the set {1,...,r}.SuchaT is row standard if the numbers in each row are increasing when read from left to right, and standard if row standard and the numbers in each column are increasing when read from top to bottom.

The group Sr acts naturally on tableaux, on the right, by permuting the entries. Given a tableau T , we define the row stabilizer of T to be the subgroup R(T )of Sr consisting of those permutations that permute entries in each row of T amongst themselves, similarly the column stabilizer is the subgroup C(T ) consisting of those permutations that permute entries in each column of T amongst themselves. Let λ be a composition of r.LetT λ be the λ-tableau in which the numbers 1,...,r have been inserted in the boxes in order from left to right along rows, read λ from top to bottom. Set Sλ = R(T ). This is the same as the Young subgroup × ×··· S{1,...,λ1} S{λ1+1,...,λ1+λ2} of Sr defined by the composition λ. Given a row standard λ-tableau T , we define λ d(T ) to be the unique element of Sr such that T = T d(T ). Given any pair S, T of row standard λ-tableaux, following Murphy [19]weset −1 −1 (3.1) xST = d(S) xλd(T ); yST = d(S) yλd(T ). where x = w and y = (sgn w) w. λ w∈Sλ λ w∈Sλ Theorem 6(Murphy). Let K be a commutative integral domain. Each of the sets {xST } and {yST },as(S, T ) ranges over the set of all ordered pairs of standard λ-tableaux for all partitions λ of r,isaK-basis of the group algebra A = KSr.

Note that xST and yST are interchanged by the K-linear ring involution of KSr which sends w to (sgn w)w,forw ∈ Sr. This gives a trivial way of converting results about one basis into results about the other. We will need several equivalent descriptions of the permutation modules M λ, λ ⊗ which we now formulate. Let λ be a composition of r. Recall that M = 1 (KSλ) (KSr), where 1 is the one dimensional module K with trivial action. In [13,

SCHUR–WEYL DUALITY IN POSITIVE CHARACTERISTIC 217

Deﬁnition 4.1], an alternative combinatorial description of M λ is given in terms of “tabloids” (certain equivalence classes of tableaux), and in [5, (1.3)] the authors write out an explicit isomorphism between these two descriptions. The following gives two additional descriptions of M λ, the second of which was used already in the previous section. Lemma 7. For any composition λ of r, the permutation module M λ is isomorphic (as a right KSr-module) with either of

(a) the right ideal xλ(KSr) of KSr; ⊗r ⊗r (b) the weight space (V )λ in V ,whereV is free over K of rank at least as large as the number of parts of λ.

Proof. Let Dλ = {d(T )} as T varies over the set of row standard tableaux of shape λ. This is a set of right coset representatives of Sλ in Sr.Themapd → xλd gives the isomorphism (a), in light of Lemma 3.2(i) of [5]. The isomorphism (b) works as follows. Given d ∈Dλ,writed = d(T ) for some (unique) row standard ⊗···⊗ tableau T of shape λ.UseT to construct a simple tensor vi1 vir of weight λ, by letting ij be the (unique) row number in T in which j is found. This map is well deﬁned, and is a bijection since there is an obvious inverse map.

We recall that compositions are partially ordered by dominance, deﬁned as follows. Given two compositions λ, µ of r,writeλ µ (λ dominates µ)if ij λi ij µi for all j.Onewrites λ µ (λ strictly dominates µ)ifλ µ and the inequality ij λi ij µi is strict for at least one j. The dominance order on compositions extends to the set of row standard tableaux, as follows. Let T be a row standard λ-tableau, where λ is a composition of r. For any s

S T if for each s r,[S↓s] [T↓s]; (3.2) S T if for each s r,[S↓s] [T↓s]. Note that if S, T are standard tableaux, respectively of shape λ, µ where λ and µ are partitions of r,thenS T if and only if T S.HereT denotes the transposed tableau of T , obtained from T by writing its rows as columns.

Let ∗ be the K-linear anti-involution on A = KSr given by ( b w)∗ → b w−1 w∈Sr w w∈Sr w for any bw ∈ K. An easy calculation with the deﬁnitions shows that ∗ ∗ (3.3) xST = xTS; yST = yTS for any pair S, T of row standard λ-tableaux.

We write c ∈{x, y} in order to describe the cell structure of A = KSr relative to both bases simultaneously.

Theorem 8(Murphy,[19, Theorem 4.18]). Let c ∈{x, y}.Letλ be a partition of r.TheK-module A[λ]= KcST , the sum taken over all pairs (S, T ) of standard µ-tableaux such that µ λ, is a two-sided ideal of A,asisA[λ]=

822 STEPHEN DOTY KcST , the sum taken over all pairs (S, T ) of standard µ-tableaux such that µλ. For any a ∈ A and any pair (S, T ) of λ-tableaux, we have (3.4) cST a = U ra(T,U) cSU mod A[ λ] where ra(T,U) ∈ K is independent of S,andinthesumU varies over the set of standard λ-tableaux.

In the language of cellular algebras, introduced by Graham and Lehrer [9], for c ∈{x, y} the basis {cST } is a cellular basis of A. Note that by applying the anti-involution ∗ to (3.4) we obtain by (3.3) the equivalent condition ∗ (3.5) a cTS = U ra(T,U) cUS mod A[ λ] for any a ∈ A and any pair (S, T )ofλ-tableaux. Now ﬁx n and r,andletP be the set of partitions λ of r such that λ1 >n.Note that P is empty if n r.SetA[P ]= KyST , where the sum is taken over the set of pairs (S, T ) of standard tableaux of shape λ, for all λ ∈ P . It follows from (3.4), (3.5) that A[P ] is a two-sided ideal of A because P satisﬁes the property: λ ∈ P , µ λ =⇒ µ ∈ P for any partition µ of r.NotethatA[P ] is the zero ideal if n r. Lemma 9. The kernel of Φ contains A[P ].

Proof. If n r then P is empty and there is nothing to prove, so we may assume that nn.) The alternating sum α is a factor of yλ, i.e., we have yλ = αβ for ⊗r some β ∈ KSr,soyλ acts as zero as well. Since V is spanned by such simple ⊗r tensors, it follows that yλ acts as zero on V . −1 It follows immediately that every yST = d(S) yλd(T ), for λ ∈ P , acts as zero on V ⊗r, for any λ-tableaux S, T ,sinced(S)−1 simply permutes the entries in the tensor, and then yλ annihilates it. Since A[P ] is spanned by such yST , it follows that A[P ] is contained in the kernel of Φ.

We will use a lemma of Murphy to establish the opposite inclusion. Let (S, T ) be a pair of λ-tableaux, where λ is a composition of r. The pair is row standard if both S, T are row standard; similarly the pair is standard if both S, T are standard. The dominance order on tableaux defined in (3.2) extends naturally to pairs of tableaux, by defining: (3.6) (S, T ) (U, V )ifS U and T V. For a, b ∈ A let (a, b) denote the coefficient of 1 in the expression ab∗ = c w, w∈Sr w where cw ∈ K.Then(, ) is a non-degenerate symmetric bilinear form on A = KSr. It is straightforward to check that this bilinear form satisfies the properties (3.7) (a, bd)=(ad∗,b); (a, db)=(d∗a, b) for any a, b, d ∈ A. Lemma 10 (Murphy, [20, Lemma 4.16]). Let (S, T ) be a row standard pair of µ-tableaux and (U, V ) a standard pair of λ-tableaux, where µ is a given composition of r and λ a partition of r. Then:

SCHUR–WEYL DUALITY IN POSITIVE CHARACTERISTIC 239

(a) (xST ,yU V )=0unless (U, V ) (S, T );

(b) (xUV ,yU V )=±1 where T denotes the transpose of a tableau T . This is used in proving the following result, which in particular shows that the rank (over K) of the annihilator of the symmetric group action on V ⊗r is independent of the characteristic of K. Proposition 11 (H¨arterich, [12, Lemma 3]). The kernel of Φ, i.e., the annihilator ann V ⊗r,isthecellidealA[P ]. KSr Proof. By Lemma 9, the kernel of Φ contains A[P ], so we only need to prove the reverse containment. Let ∈ a = (S,T ) aST yST ker Φ where aST ∈ K, and the sum over all pairs (S, T ) of standard tableaux of shape λ, where λ is a partition of r. It suﬃces to prove: (∗) aST = 0 for all pairs (S, T )of standard tableaux of shape µ ∈ P c,whereP c is the complement of P in the set of all partitions of r. We note that P c is the set of conjugates λ of partitions λ in Λ(n, r). Write Λ+(n, r) for the set of partitions in Λ(n, r); this is the set of partitions of r into not more than n parts. We proceed by contradiction. Suppose (∗)isnottrue.SincebyLemma9we have ∈ aST yST ∈ ker(Φ), it follows that shape(S,T ) P b = shape(S,T )∈P c aST yST is also in the kernel of Φ; i.e., the element b annihilates V ⊗r. Under the assumption we have b =0.Let( S0,T0) be a minimal pair (with respect to ) ∈ c with shape(S0,T0) P such that aS0T0 =0.SoaST = 0 for all pairs (S, T ) with (S0,T0) (S, T ). Let λ0 be the shape of T0 (same as shape of S0). Then λ ∈ Λ+(n, r), and we have 0 (xλ0S b, d(T0)) = (xλ0S aST yST ,d(T0)) 0 0 − ∗ 1 = aST (d(T0) xλ0S ,yST ) 0 = a (x ,y ) ST T0S0 TS where all sums are taken over the set of (S, T ) of shape some member of P c. Here, µ we write xµT shorthand for xT µT , where (as before) T is the µ-tableau in which the numbers 1,...,r have been inserted in the boxes in order from left to right along rows, read from top to bottom.

By Lemma 10(a) all the terms in the last sum are zero unless (S0,T0) (S, T ), in other words (x ,y ) = 0 for all pairs (S, T ) which are strictly more dominant T0S0 TS than (S0,T0). By assumption, aST = 0 for all pairs (S, T ) strictly less dominant than (S ,T ). Thus, the above sum collapses to a single term a (x ,y ), 0 0 S0T0 T0S0 T0S0 and by our assumption and Lemma 10(b) this is nonzero. This proves that x b =0.Thus b does not annihilate the permutation λ0S0 λ0 ∈ + λ0 module M xλ0 A.Sinceλ0 Λ (n, r) as noted above, and thus M is isomorphic to a direct summand of V ⊗r, we have arrived at a contradiction. This proves the result.

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Corollary . 12 For any commutative integral domain K,theK-module Φ(KSr) − 2 is free over K,ofrankr! λ∈P N(λ) ,whereN(λ) is the number of standard tableaux of shape λ. In particular, the K-rank of Φ(KSr) is independent of K.

Proof. By the preceding proposition, Φ(KSr) A/A[P ]. This is free over K ⊗r because it is a submodule of the free K-module EndK (V ). By deﬁnition, A[P ] 2 is free over K of rank λ∈P N(λ) , so the result follows.

Now we return to the assumption that K is an inﬁnite ﬁeld, and consider why ⊗r dimK EndΓ(V ) is independent of K. This involves facts about the representation theory of algebraic groups that are less elementary than facts used so far. We identify the group Γ = GLn(K), the group of K-rational points in the algebraic group GLn(K), where K is an algebraic closure of K, with the group scheme GLn over K.

For Γ = GLn(K)weletT be the maximal torus consisting of all diagonal ele- n ments of Γ. Regard an element λ ∈ Z as a character on T (via diag(a1,...,an) → × λ1 ··· λn ∈ a1 an for ai K ). Consider the Borel subgroup B consisting of the lower triangular matrices in Γ, and let ∇(λ) be the induced module (see [14,PartI, §3.3]): Γ { ∈ −1 ∈ ∈ } indB(Kλ)= f K[Γ] : f(gb)=b f(g), all b B,g G n for any λ ∈ Z ,whereKλ is the one dimensional T -module with character λ, regarded as a B-module by making the unipotent radical of B act trivially. ∗ The dual space M =HomK (M,K) of a given rational KΓ-module M is again arationalKΓ-module, in two different ways: (i) (g · f)(m)=f(g−1m); (ii) (g · f)(m)=f(gtm)(gT is the matrix transpose of g) for g ∈ Γ, f ∈ M ∗, m ∈ M. Denote the first dual by M ∗ and the second by M T. T ∗ Let ∆(λ)=∇(λ) . Itisknownthat∆(λ) ∇(−w0λ) where w0 is the longest element in the Weyl group W . The modules ∇(λ), ∆(λ) are known as “dual Weyl modules” and “Weyl modules”, respectively.6 The most important property these modules satisfy, for our purposes, is the following K if j =0andλ = µ (3.8) Extj (∆(λ), ∇(µ)) Γ 0otherwise. This is a special case of [14, Part II, Proposition 4.13]. Say that a Γ-module M has a ∇-filtration (respectively, ∆-filtration) if it has an ascending chain of submodules

0=M0 ⊆ M1 ⊆···⊆Mt−1 ⊆ Mt = M i such that each successive quotient Mi/Mi−1 is isomorphic with ∇(λ ) (respectively, ∆(λi)) for some λi ∈ Zn. Another fact we need goes back to [24, Theorem B, page 164]: (3.9) ∆(λ) ⊗ ∆(µ) has a ∆-ﬁltration

6 Weyl and dual Weyl modules for GLn(K) are studied in [10, Chapters 4, 5], where they are respectively denoted by Dλ,K and Vλ,K .

SCHUR–WEYL DUALITY IN POSITIVE CHARACTERISTIC 2511 for any λ, µ ∈ Zn. (Note that this fundamental result has been extended in [6], which in turn was extended in [18].) From (3.9) it follows immediately by taking duals that (3.10) ∇(λ) ⊗∇(µ)hasa∇-ﬁltration for any λ, µ ∈ Zn. The following result, which says that V ⊗r is a “tilting” module for Γ, is now easy to prove. Lemma 13. V ⊗r has both ∇-and∆-ﬁltrations.

Proof. One has V = ∇(ε1)=∆(ε1)whereε1 =(1, 0,...,0). The result then follows from (3.9) and (3.10) by induction on r.

For the next argument we will need the notion of formal characters. Any rational KΓ-module M has a weight space decomposition M = λ∈Zn Mλ where λ Mλ = {m ∈ M : tm = t m, for all t ∈ T }.

λ λ1 ··· λn Here t = a1 an where t = diag(a1,...an) as previously defined, just before (2.5). Set X = Zn and let Z[X]bethefreeZ-module on X with basis consisting of all symbols e(λ)forλ ∈ X, with a multiplication given by e(λ)e(µ)=e(λ + µ), for λ, µ ∈ X.IfM is finite dimensional, the formal character ch M ∈ Z[X]ofM is defined by ch M = λ∈X (dimK Mλ) e(λ). The formal character of ∆(λ), which is the same as ch ∇(λ) since the maximal torus T is fixed pointwise by the matrix transpose, is given by Weyl’s character formula [14, Part II, Proposition 5.10].7 ⊗r Proposition 14. For any infinite field K, dimK EndΓ(V ) is independent of K.

⊗r Proof. Let 0 = N0 ⊆ N1 ⊆ ··· ⊆ Ns−1 ⊆ Ns = V be a ∇-filtration ⊗r ⊗r and 0 = M0 ⊆ M1 ⊆ ··· ⊆ Mt−1 ⊆ Mt = V a ∆-filtration. Write (V : ∇(λ)) for the number of successive subquotients Ni/Ni−1 which are isomorphic to ∇(λ), and similarly write (V ⊗r :∆(λ)) for the number of successive subquotients Mi/Mi−1 which are isomorphic to ∆(λ). Since characters are additive on short exact sequences, we have ch V ⊗r = (V ⊗r : ∇(λ)) ch ∇(λ)= (V ⊗r :∆(λ)) ch ∆(λ). λ∈X λ∈X ⊗r Since V is self-dual (under the transpose dual) we may choose the filtration (N∗) ⊗r ⊗r to be dual to the filtration (M∗). It follows that s = t and (V : ∇(λ)) = (V : ∆(λ)) for all λ. Now one applies (3.8) and a double induction through the filtrations. The argument is standard homological algebra, safely left at this point as an exercise for the reader. At the end one finds that ⊗r ⊗r ∇ 2 dimK EndΓ(V )= λ∈Zn (V : (λ)) where the number of nonzero terms in the sum is finite. The result follows.

7 The computation of the ch ∆(λ)forGLn(C) goes back to Schur’s 1901 dissertation. Thus, these characters are sometimes called Schur functions. See [17]or[23, Chapter 7] for exhaustive accounts of their many properties.

1226 STEPHEN DOTY

Now we are ready to prove the second half of Schur–Weyl duality in positive characteristic. We remind the reader that we assume the validity of Theorem 1 in case K = C.

Proposition 15. For any inﬁnite ﬁeld K, the image Φ(KSr) of the represen- ⊗r tation Φ is equal to the centralizer algebra EndΓ(V ),sothemapΦ in (1.3) is surjective.

Proof. The argument is essentially the same as the proof of Proposition 5. By Corollary 12 and Proposition 14 we have equalities

dimK Φ(KSr)=dimC Φ(CSr), n ⊗r Cn ⊗r dimK EndGLn(K)((K ) )=dimC EndGL(n(C)(( ) ) C Cn ⊗r for any infinite field K.SinceΦ( Sr) = EndGLn(C)(( ) ) it follows that n ⊗r dimK Φ(KSr)=dimK EndGLn(K)((K ) ) for any infinite field K, and thus by comparison of dimensions the second inclusion in (1.2) must be an equality. Equivalently, the map Φ in (1.3) is surjective.

By putting together Propositions 5 and 15 we have now established Theorem 1 in positive characteristic, assuming its validity for K = C. Remark . 16 (a) Let K be an arbitrary infinite field. Lemma 3 gives the n2+r−1 equality dimK (KΓ) = r , and the proof of Lemma 4 in light of [13,Theo- ⊗r + + rem 13.19] gives the equality dimK EndSr (V )= λ,µ∈Λ(n,r) N(λ ,µ ), where + + + N(λ ,µ ) counts the number of “semistandard” tableaux of shape λ and weight + − 2 µ . Corollary 12 says that dimK Φ(KSr)=r! λ∈P N(λ) ,whereN(λ)isthe number of standard tableaux of shape λ, and the proof of Proposition 14 shows that ⊗r ⊗r ∇ 2 dimK EndΓ(V )= λ∈Λ+(n,r)(V : (λ)) . Thus, in order to obtain a proof of Theorem 1 in full generality (without assuming its validity for K = C)fromthe methods of this paper, one only needs to demonstrate the combinatorial identities n2 + r − 1 (3.11) = N(λ+,µ+); r λ,µ∈Λ(n,r) (3.12) r! − N(λ)2 = (V ⊗r : ∇(λ))2. λ∈P λ∈Λ+(n,r) The author has not attempted to construct a combinatorial proof of these identities. If one assumes the validity of Theorem 1 in the case K = C, then these identities follow from the results in this paper. Alternatively, if one can find an independent proof of the identities, then one would have a new proof of Theorem 1 in full generality, including the case K = C. (b) There is a variant of Theorem 1 worth noting. One may twist the action ⊗r of Sr on V by letting w ∈ Sr act as (sgn w)w (so Sr acts by “signed” place permutations). This action also commutes with the action of Γ = GL(V ), and Theorem 1 holds for this action as well. This may be proved the same way. In the course of carrying out the argument, one needs to replace permutation modules by “signed” permutation modules, and interchange the role of Murphy’s two bases {xST }, {yST }.

SCHUR–WEYL DUALITY IN POSITIVE CHARACTERISTIC 2713

(c) There is also a q-analogue of Theorem 1, in which one replaces GLn(K)by the quantized enveloping algebra corresponding to the Lie algebra gln, and replaces KSr by the Iwahori–Hecke algebra H(Sr). The generic case (q not a root of unity) of this theorem was ﬁrst observed in Jimbo [15], and the root of unity case was treated in Du, Parshall, and Scott [8]. Alternatively, one may derive the result in the root of unity case from Jimbo’s generic version, using arguments along the lines of those sketched here.

References [1] R. Carter and G. Lusztig, On the modular representations of general linear and symmetric groups, Math. Zeit. 136 (1974), 193–242. [2] C. Curtis and I. Reiner, Representation theory of finite groups and associative algebras, Interscience (Wiley) 1962; reprinted by AMS Chelsea Publishing, Providence, RI, 2006. [3] C. de Concini and C. Procesi, A characteristic free approach to invariant theory, Advances in Math. 21 (1976), 330–354. [4] R. Dipper and S. Doty, The rational Schur algebra, Represent. Theory 12 (2008), 58–82 (electronic). [5] R. Dipper and G. James, Representations of Hecke algebras of general linear groups, Proc. London Math. Soc. (3) 52 (1986), 20–52. [6] S. Donkin, Rational Representations of Algebraic Groups: Tensor Products and Filtrations, Lecture Notes in Math. 1140, Springer-Verlag, Berlin 1985. [7] S. Doty, K. Erdmann, and A. Henke, A generic algebra associated to certain Hecke algebras, J. Algebra 278 (2004), 502–531. [8] J. Du, B. Parshall, and L. Scott, Quantum Weyl reciprocity and tilting modules, Comm. Math. Phys. 195 (1998), 321–352. [9] J.J. Graham and G.I. Lehrer, Cellular algebras, Invent. Math. 123 (1996), 1–34. [10] J.A. Green, Polynomial Representations of GLn, (Lecture Notes in Math. 830), Springer-Verlag, Berlin 1980; Second edition 2007. [11] R. Goodman and N.R. Wallach, Representations and invariants of the classical groups, Encyclopedia of Mathematics and its Applications, 68, Cambridge University Press, Cambridge, 1998. [12] M. Härterich, Murphy bases of generalized Temperley–Lieb algebras, Archiv Math. 72 (1999), 337–345. [13] G.D. James, The Representation Theory of the Symmetric Groups, (Lecture Notes in Math. 682), Springer-Verlag, Berlin 1978. [14] J.C. Jantzen, Representations of algebraic groups, Second edition. Mathematical Surveys and Monographs, 107, American Mathematical Society, Providence, RI, 2003. [15] M. Jimbo, A q-analogue of U(gl(N + 1)), Hecke algebras, and the Yang-Baxter equation, Lett. Math. Phys. 11 (1986), 247–252. [16] S. König,I.H. Slung˚ard, and C.C. Xi, Double centralizer properties, dominant dimension, and tilting modules, J. Algebra 240 (2001), 393–412. [17] I.G. Macdonald, Symmetric functions and Hall polynomials, Second edition, Oxford Mathematical Monographs, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1995. [18] O. Mathieu, Filtrations of G-modules, Ann. Sci. Ecole` Norm. Sup. (4) 23 (1990), 625–644. [19] G.E. Murphy, On the representation theory of the symmetric groups and associated Hecke algebras, J. Algebra 152 (1992), 492–513. [20] G.E. Murphy, The representations of Hecke algebras in type An, J. Algebra 173 (1995), 97–121.

1428 STEPHEN DOTY

[21] I. Schur, Uber¨ die rationalen Darstellungen der allgemeinen linearen Gruppe (1927); in I. Schur, Gesammelte Abhandlungen III, 65–85, Springer Berlin 1973. [22] J.-P. Serre, Linear representations of ﬁnite groups, Graduate Texts in Mathematics, Vol. 42, Springer-Verlag, New York-Heidelberg, 1977. [23] R.P. Stanley, Enumerative combinatorics, Vol. 2, Cambridge Studies in Advanced Mathematics, 62, Cambridge University Press, Cambridge, 1999. [24] J.P. Wang, Sheaf cohomology on G/B and tensor products of Weyl modules, J. Algebra 77 (1982), 162–185. [25] H. Weyl, The Classical Groups, Their Invariants and Representations, 2nd ed., Princeton Univ. Press, 1946.

Mathematics and Statistics, Loyola University Chicago, Chicago, Illinois 60626 U.S.A. E-mail address: [email protected]

Contemporary Mathematics Volume 478, 2009

The centers of Iwahori-Hecke algebras are ﬁltered

Andrew Francis and Weiqiang Wang

Abstract. We show that the center of the Iwahori–Hecke algebra of the symmetric group Sn carries a natural ﬁltered algebra structure, and that the structure constants of the associated graded algebra are independent of n.Aseries of conjectures and open problems are also included.

1. Introduction 1.1. The main results. The class elements introduced by Geck-Rouquier [GR] form a basis for the center Z(Hn) of the Iwahori-Hecke algebra Hn of type A over the ring Z[ξ], where the indeterminate ξ is related to the familiar one q by −1 ξ = q − q . In this paper, we shall parameterize these class elements Γλ(n)by partitions λ satisfying |λ| + (λ) ≤ n which are the so-called modiﬁed cycle types, just as Macdonald [Mac, pp.131] does for the usual class sums of the symmetric group Sn. Write the multiplication in Z(Hn)as ν (1.1) Γλ(n)Γµ(n)= kλµ(n)Γν (n). ν The main result of this Note is the following theorem on these structure con- ν stants kλµ(n). Theorem . ν 1.1 (1) For any n, kλµ(n) is a polynomial in ξ with non- ν negative integral coeﬃcients. Moreover, kλµ(n) is an even (resp., odd) polynomial in ξ if and only if |λ| + |µ|−|ν| is even (resp., odd). ν | |≤| | | | (2) We have kλµ(n)=0unless ν λ + µ . | | | | | | ν (3) If ν = λ + µ ,thenkλµ(n) is independent of n.

Itfollowsfrom(2)and(3)thatthecenterZ(Hn) is naturally a ﬁltered algebra and the structure constants of the associated graded algebra are independent of n. We in addition formulate several conjectures, including Conjecture 3.1 which ν simply states that kλµ(n)arepolynomialsinn, and further implications on the algebra generators of Z(Hn).

2000 Mathematics Subject Classiﬁcation. Primary: 20C08; Secondary: 16W70. W.W. is partially supported by an NSF grant.

c 2009 Americanc 0000 Mathematical (copyright Societyholder) 291

230 ANDREW FRANCIS AND WEIQIANG WANG

1.2. Motivations and connections. Our motivation comes from the original work of Farahat-Higman [FH] on the structures of the centers of the integral symmetric group algebras ZSn. Indeed, Theorem 1.1 specializes at ξ =0tosome classical results of loc. cit.. In addition, Conjecture 3.1 in the specialization at ξ = 0 (which is a theorem in loc. cit.) when combined with the specialization of Theorem 1.1 allowed them to define a universal algebra K governing the structures of the centers Z(ZSn) for all n simultaneously. Farahat-Higman further developed this approach to establish a distinguished set of generators for Z(ZSn), which is now identified as the first n elementary symmetric polynomials in the Jucys-Murphy elements. This has applications to blocks of modular representations of Sn. This Note arose from the hope that the results of Farahat–Higman might be generalized to the Iwahori-Hecke algebra setup and in particular it would provide a new conceptual proof of the Dipper-James conjecture. Recently, built on the earlier work of Mathas [Mat], the first author and Graham [FrG] obtained a first complete combinatorial proof of the Dipper-James conjecture that the center of Hn is the set of symmetric polynomials in Jucys–Murphy elements of Hn (cf. [DJ]for an earlier proof of a weaker version). A basic difficulty in completing the Farahat- Higman approach for Iwahori-Hecke algebras is that no compact explicit expression is known for the Geck-Rouquier elements Γλ(n) (see however [Fra]forauseful characterization). Our Theorem 1.1 is a first positive step along the new line. In another direction, the results of Farahat-Higman have been partially generalized by the second author [W] to the centers of the group algebras of wreath products G Sn for an arbitrary finite group G (e.g. a cyclic group Zr), and these centers are closely related to the cohomology ring structures of Hilbert schemes of points on the minimal resolutions. It will be interesting to develop the Farahat- Higman type results for the centers of the Iwahori-Hecke algebra of type B or more generally of the cyclotomic Hecke algebras which are q-deformation of the group algebra Z(Zr Sn).

1.3. This Note is organized as follows. In Section 2, we prove the three parts of our main Theorem 1.1 in Propositions 2.2, 2.4 and 2.6, respectively. In Section 3, we formulate several conjectures, which are Iwahori-Hecke algebra analogues of some results of Farahat-Higman. We conclude this Note with a list of open questions.

2. Proof of the main theorem

2.1. The preliminaries. Let Sn be the symmetric group in n letters generated by the simple transpositions si =(i, i +1),i =1,...,n− 1. Let ξ be an indeterminate. The Iwahori-Hecke algebra Hn is the unital Z[ξ]- algebra generated by Ti for i =1,...,n− 1, satisfying the relations

TiTi+1Ti = Ti+1TiTi+1

TiTj = TjTi, |i − j| > 1 2 (2.1) Ti =1+ξTi. −1 The order relation (2.1) comes from the more familiar (Ti − q)(Ti + q )=0via − −1 ··· ∈ the identification ξ = q q . If w = si1 sir Sn is a reduced expression (where ··· r will be referred to as the length of w in this case), then define Tw := Ti1 Tir . This definition is independent of the reduced expression for w. Itiswellknown

CENTERS OF IWAHORI-HECKE ALGEBRAS ARE FILTERED 313 that the Iwahori-Hecke algebra is a free Z[ξ]-module with basis {Tw | w ∈ Sn},and it is a deformation of the integral group algebra ZSn. The Jucys-Murphy elements Li (1 ≤ i ≤ n)ofHn are defined to be L1 =0and Li = T(k,i), 1≤k 1, we define the modified cycle-type of w to beρ ˜ =(ρ1 − 1,...,ρt − 1), following Macdonald [Mac, pp.131]. Given a partition λ,letCλ(n) denote the conjugacy class of Sn containing all elements of modified type λ if |λ| + (λ) ≤ n. Accordingly, let cλ(n)denotethe class sum of Cλ(n)if|λ| + (λ) ≤ n, and denote cλ(n)=0otherwise. The center Z(Hn) of the Iwahori-Hecke algebra is free over Z[ξ] of rank equal to the number of partitions of n; that is, it has a basis indexed by the conjugacy classes of Sn (see [GR]). In this paper, we shall parameterize these Geck-Rouquier class elements Γλ(n) by the modified cycle types λ. The elements Γλ(n)for|λ|+(λ) ≤ n are characterized by the following two properties among the central elements of Hn [Fra]: (i) The Γλ(n) specializes atξ = 0 to the class sum cλ(n); (ii) The difference Γ (n) − T contains no minimal length elements λ w∈Cλ(n) w of any conjugacy class.

In addition, we set Γλ(n)=0if|λ| + (λ) >n. 2.2. The structure constants as positive integral polynomials. By in- spection of the deﬁning relations, Hn as a Z-algebra is Z2-graded by declaring that ξ and Ti (1 ≤ i ≤ n−1) have Z2-degree (or parity) 1 and each integer has Z2-degree 0.

Lemma 2.1. Every Γλ(n) is homogeneous in the above Z2-grading with Z2- degree equal to |λ| mod 2. Proof. There is a constructive algorithm [Fra, pp.14] for producing the elements Γλ(n). This ﬁnite algorithm begins with the sum of Tw with minimal length elements w from the conjugacy class Cλ(n), then at each repeat of this algorithm, → → the only additions are of form (i) Tw Tw+Tsiwsi , (ii) Tsiw Twsi +Tsiw+ξTsiwsi , → or (iii) Twsi Twsi + Tsiw + ξTsiwsi , and the algorithm eventually ends up with the element Γλ(n). Each of the three type of additions clearly preserves the Z2- degree. As the minimal length elements have the same parity as |λ|, this proves the lemma. Denote by N the set of non-negative integers. Proposition . ν 2.2 For any given n, kλµ(n) is a polynomial in ξ with non- ν negative integral coeﬃcients. Moreover, kλµ(n) is an even (respectively, odd) polynomial in ξ if and only if |λ| + |µ|−|ν| is even (respectively, odd).

Proof. As is seen in the proof of Lemma 2.1, the class elements are in the positive cone Z(H )+ := Z(H ) ∩ N[ξ]T . Because of the positive coeﬃ- n n w∈Sn w cients in the order relation (2.1), the positive cone is closed under additions and products. Since the class elements contain minimal length elements from exactly one conjugacy class and contains those minimal length elements with coeﬃcient 1

432 ANDREW FRANCIS AND WEIQIANG WANG

[Fra] (see Sect. 2.1 above), the coefficient of a class element in a central element C is precisely the coefficient of the corresponding minimal length element in the Tw ν ∈ N expansion of C. This shows that kλµ(n) [ξ]. ν The more refined statement on when kλµ(n)isanevenoroddpolynomialinξ follows now from Lemma 2.1.

2.3. The ﬁltered algebra structure on the center. Let mµ(n)bethe monomial symmetric polynomial in the (commutative) Jucys-Murphy elements L1,...,Ln, parameterized by a partition µ.Itisknownthatmµ(n) ∈Z(Hn). Some relations between the class elements and the monomial symmetric polynomials in Jucys–Murphy elements are summarized as follows (see [Mat, FrG, FrJ]).

Lemma 2.3. (1) For any partition λ, we can express mλ(n) in terms of the class elements Γµ(n) as mλ(n)= bλµ(n)Γµ(n) for bλµ(n) ∈ Z[ξ]. |µ|≤|λ|

(We set bλµ(n)=0if |µ| + (µ) >n, or equivalently if Γµ(n)=0.) (2) For any λ, the coefficients bλµ(n) with |µ| = |λ| are independent of n. (3) Let λ be a partition with |λ| + (λ) ≤ n. Then, each Γλ(n) is equal to mλ(n) plus a Z[ξ]-linear combination of mµ(n) with |µ| < |λ|. Proof. Part (1) is a consequence of [Mat, Theorems 2.7 and 2.26]. By [FrG, Lemma 5.2], the coefficient of Tw (for an increasing w ∈ Sn of the right length) in a so-called quasi-symmetric polynomial in Jucys–Murphy elements is independent of n. Monomial symmetric polynomials are just sums of the corresponding quasi-symmetric polynomials, independently of n.Thisproves(2). Part (3) can be read off from the proof of [FrJ, Theorem 4.1].

Proof. By Lemma 2.3 (3), the product Γλ(n)Γµ(n)isequaltomλ(n)mµ(n) plus a linear combination of products mλ (n)mµ (n) satisfying |λ |+|µ | < |λ|+|µ|. A product of monomial symmetric polynomials mα(n)mβ(n) is a sum of monomial symmetric polynomials mγ (n) satisfying |α| + |β| = |γ|.Consequently,Γλ(n)Γµ(n) is a sum of mγ (n) with partitions γ of size at most |λ|+|µ|. The proposition follows now by Lemma 2.3 (1).

Assign degree |λ| to the basis element Γλ(n)andletZ(Hn)m be the Z[ξ]-span of Γλ(n)ofdegreeatmostm,form ≥ 0. Then, Proposition 2.4 provides a ﬁltered algebra structure on the center Z(Hn)=∪mZ(Hn)m.

Remark 2.5. Denote by An the (commutative) Z[ξ]-subalgebra of Hn generated by the Jucys-Murphy elements L1,...,Ln. The algebra An is filtered by the (m) ≥ ··· subspaces An (m 0) spanned by all products Li1 Lim of m Jucys-Murphy elements. The filtrations on An and Z(Hn) are compatible with each other by the inclusion Z(Hn) ⊂ An. However the algebra Hn does not seem to admit a natural filtration which is compatible with the one on An by inclusion An ⊂Hn.Thisis very different from the symmetric group algebra ZSn, which admits such a filtration by assigning degree 1 to every transposition (i, j).

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2.4. The graded algebra grZ(Hn). Proposition . | | | | | | ν 2.6 If ν = λ + µ ,thenkλµ(n) is independent of n. ν ν (In this case, we shall write kλµ(n) as kλµ.) Proof. ν By the definition of kλµ(n), we can assume without loss of generality that |λ| + (λ) ≤ n and |µ| + (µ) ≤ n. By Lemma 2.3 (3), Γλ(n)Γµ(n)=mλ(n)mµ(n)+X,whereX is a linear combination of products of monomials whose combined partition size is less than |λ| + |µ|. The monomials appearing in X correspond to partitions of size less than | | | | | | ν ν = λ + µ , and thus will not contribute to kλµ(n) by Lemma 2.3 (1). The product mλ(n)mµ(n) is a sum of monomials mα(n) satisfying |α| = |λ| + |µ| with ν coefficients independent of n; the contribution of each such mα(n)tokλµ(n)is independent of n by Lemma 2.3 (2). Summing all these contributions produces ν kλµ(n) which is independent of n. Proposition 2.6 is equivalent to the statement that all the structure constants of the graded algebra grZ(Hn) associated to the filtered algebra Z(Hn) are independent of n.

2.5. Examples. We provide some explicit calculations of the structure con- ν stants kλµ(n) for the multiplication between Γλ(n), with n =3, 4, 5. For the sake of notational simplicity, we will write Γλ(n)asΓλ with n dropped in the following examples. We also drop parentheses in the subscripts of class elements, denoting

Γ(λ1,...,λk) by Γλ1,...,λk . The square brackets denote the top-degree parts of each product. The compatibility of these examples with Theorem 1.1 is manifest.

(1) Let n =3.InZ(H3), we have 2 Γ1Γ1 = (ξ +3)Γ2 +2ξΓ1 +3Γ∅. (2) Let n =4.InZ(H ), we have 4 2 2 Γ Γ = (ξ +3)Γ +(ξ +2)Γ +3ξΓ +6Γ∅, 1 1 2 1,1 1 4 2 3 3 Γ1Γ2 = (ξ +4ξ +4)Γ3 +(2ξ +6ξ)Γ2 +(2ξ +4ξ)Γ1,1 2 +(3ξ +4)Γ +4ξΓ∅, 1 2 Γ1Γ1,1 = (ξ +2)Γ3 +2ξΓ2 + ξΓ1,1 +Γ1. (3) Let n =5.InZ(H ), we have 5 2 2 Γ Γ = (ξ +3)Γ +(ξ +2)Γ +4ξΓ + 10Γ∅ 1 1 2 1,1 1 4 2 4 2 Γ1Γ2 = (ξ +4ξ +4)Γ3 +(ξ +2ξ +1)Γ2,1 3 3 2 +(3ξ +8ξ)Γ +(3ξ +4ξ)Γ +(6ξ +6)Γ +10ξΓ∅ 2 1,1 1 Γ Γ = (ξ2 +2)Γ +(2ξ2 +3)Γ +2ξΓ +4ξΓ +3Γ 1 1,1 3 2,1 2 1,1 1 6 4 2 Γ1Γ3 = (ξ +6ξ +10ξ +5)Γ4 5 3 5 3 +(2ξ +10ξ +13ξ)Γ3 +(2ξ +8ξ +7ξ)Γ2,1 4 2 4 2 3 2 +(3ξ +10ξ +6)Γ +(3ξ +8ξ +4)Γ +(4ξ +6ξ)Γ +5ξ Γ∅ 2 1,1 1 8 6 4 2 Γ2Γ2 = (ξ +7ξ +16ξ +15ξ +5)Γ4 7 5 3 7 5 3 +(2ξ +14ξ +29ξ +19ξ)Γ3 +(2ξ +13ξ +22ξ +11ξ)Γ2,1

634 ANDREW FRANCIS AND WEIQIANG WANG

6 4 2 6 4 2 +(3ξ +20ξ +32ξ +7)Γ2 +(3ξ +19ξ +26ξ +8)Γ1,1 5 3 4 2 +(4ξ +25ξ +27ξ)Γ +(5ξ +30ξ + 20)Γ∅ 1 6 4 2 Γ2Γ1,1 = (ξ +6ξ +10ξ +5)Γ4 5 3 5 3 +(2ξ +10ξ +11ξ)Γ3 +(2ξ +9ξ +9ξ)Γ2,1 4 2 4 2 3 2 +(3ξ +11ξ +6)Γ2 +(3ξ +9ξ +4)Γ1,1 +(4ξ +7ξ)Γ1 +5ξ Γ∅.

2.6. A universal graded algebra. Introduce a graded Z[ξ]-algebra G with a basis given by the symbols Γλ,whereλ runs over all partitions, and with multiplication given by ν ΓλΓµ = kλµΓν . |ν|=|λ|+|µ| ν By Propositions 2.2 and 2.6, the structure constants kλµ are independent of n and 2 actually lie in N[ξ ]. Furthermore, we have surjective homomorphisms G→grZ[Hn] for all n, which send each Γλ to Γλ(n). The following proposition is immediate.

Proposition 2.7. The Z[ξ]-algebra G is commutative and associative.

Below for the one-row partition (m), we shall write Γ(m) simply as Γm.

Theorem 2.8. The Q(ξ)-algebra Q(ξ) ⊗Z[ξ] G is a polynomial algebra with generators Γm, m =1, 2,....

Proof. Given a partition λ =(λ1,λ2,...), the product is of the form ··· Γλ1 Γλ2 = dλµ(ξ)Γµ µ for dλµ(ξ) ∈ N[ξ]. As ξ goes to 0, Γµ goes to the class sum cµ and dλµ(ξ) speciﬁes to the structure constant dλµ as deﬁned in [Mac, pp.132] which we recall: ··· cλ1 cλ2 = dλµcµ. µ

It is known therein that the (integral) matrix [dλµ]for|λ| = |µ| = k with any k is triangular with respect to the dominance ordering of partitions and all its diagonal entries are nonzero, thus the matrix [dλµ]isinvertibleoverQ. This forces the matrix [dλµ(ξ)] invertible over the ﬁeld Q(ξ). Thus each Γµ is generated by Γ1, Γ2,...over Q(ξ). By deﬁnition, the elements Γµ for all partitions µ are linearly independent. Thus the theorem follows by comparing the graded dimensions of the algebra Q(ξ) ⊗Z[ξ] G and the polynomial algebra in Γm, m =1, 2, ···.

It is not clear whether the matrix [dλµ(ξ)] remains triangular with respect to the dominance order when ξ = 0. A similar phenomenon with a negative answer appears with the matrix M3 in Mathas [Mat, p.310] (Mk gives the coeﬃcients of class elements Γλ in monomial symmetric polynomials in Jucys-Murphy elements mµ,forλ, µ k).

CENTERS OF IWAHORI-HECKE ALGEBRAS ARE FILTERED 357

3. Conjectures and discussions 3.1. Several conjectures. We expect the following conjecture to hold. Conjecture . ν 3.1 Given partitions λ, µ and ν, there exists a polynomial fλµ Q ν ν in one variable with coeﬃcients in [ξ], such that fλµ(n)=kλµ(n) for all n.

Recall bλµ(n) from Lemma 2.3 (1). Similarly, we conjecture that there exists polynomials gλµ(x) with coefficients in Q[ξ], such that gλµ(n)=bλµ(n) for all n. The specialization at ξ = 0 of Conjecture 3.1 is a result in [FH]. Below we shall assume that Conjecture 3.1 holds. Set B to be the ring of polynomials in Q[x] which take integer values at integers. We can define a B[ξ]-algebra K with a basis given by the symbols Γλ,whereλ runs over all partitions, and the multiplication given by ν ΓλΓµ = fλµΓν . ν ν | |≤| | | | K K Since fλµ = 0 unless ν λ + µ , the algebra is an algebra filtered by m (m ≥ 0) which is the B[ξ]-span of Γλ with |λ|≤m. Wehaveanaturalsurjective homomorphism of filtered algebras

pn : K−→Z(Hn) given by pn fλΓλ = fλ(n)Γλ(n). λ The algebra G introduced earlier becomes the associated graded algebra for the ﬁltered algebra K up to a base ring change, i.e., grK = B[ξ] ⊗Z[ξ] G.Forr ≥ 1, set Er := Γλ. |λ|=r

Conjecture 3.2. The B[ξ]-algebra K is generated by Er, r =1, 2, ···. Conjecture 3.2 in the specialization with ξ = 0 is a main theorem of Farahat- Higman [FH]. Note (cf. e.g. [FrG]) that Er(n):= Γλ(n) ∈Z(Hn) |λ|=r can be interpreted as the r-th elementary symmetric function in the n Jucys– Murphy elements. If Conjecture 3.2 holds, then the surjectivity of the homomorphism pn : K→Z(Hn) implies that the center Z(Hn) is generated by Er(n), 1 ≤ r ≤ n − 1. That would provide a new and conceptual proof of the Dipper- James conjecture [DJ, FrG] along with additional results of independent interest. 3.2. Open questions and discussions. A fundamental diﬃculty in pursuing the approach of [FH] for the Iwahori-Hecke algebras is present in the following.

Question 3.3. Find an explicit expression for the elements Γλ(n) for all λ. The more challenging Conjecture 3.2 is likely to follow from an aﬃrmative answer to Question 3.4 below, as a similar calculation in the case of symmetric groups plays a key role in the original approach of [FH].

836 ANDREW FRANCIS AND WEIQIANG WANG

Question . ν | | | | 3.4 Calculate the structure constants kλ (m) with ν = λ + m. Recall the positivity and integrality from Theorem 1.1 (1). Question 3.5. Find a combinatorial or geometric interpretation of the posi- ν tivity and integrality of the structure constants kλµ(n) as polynomials in ξ. With the connections between results of Farahat-Higman and cohomology rings of Hilbert schemes of n points on the aﬃne plane in mind (cf. [W]), we post the following.

Question 3.6. Are there any connections between the center Z(Hn)andthe equivariant K-group of Hilbert schemes of n points on the affine plane? Let W be an arbitrary finite Coxeter group. The group algebra ZW is naturally filtered by assigning degree 1 to each reflection (not just simple reflection) and degree r to any element w ∈ W with a reduced expression in terms of reflections of minimal length r. This induces a filtered algebra structure on the center Z(ZW )as elements of a conjugacy class have the same degree. In the cases of types A and B, this definition agrees with the notion of degree for general wreath products introduced in [W]. (In the case of symmetric groups, the degree of cλ(n)coincideswith |λ|.) The Geck-Rouquier basis has been defined for centers of the integral Iwahori- Hecke algebras HW associated to any such W [GR], and its characterization (as in Sect. 2.1) holds in this generality [Fra]. Note that the generalization of Theo- rem 1.1 (1) to all such HW holds with the same proof. We ask for a generalization of Theorem 1.1 (2) as follows. Question 3.7. Let W be an arbitrary finite Coxeter group. If we apply the notion of degree above to the Geck-Rouquier elements in the center Z(HW ), does it provide an algebra filtration on Z(HW )? We expect that the answer to Question 3.7, at least for Iwahori-Hecke algebras of type B, is positive. More generally, we ask the following. Question 3.8. Establish and characterize an appropriate basis of class elements for the centers of the integral cyclotomic Hecke algebras associated to the complex reflection groups Zr Sn. Furthermore, generalize the results of this Note and [W] to the cyclotomic Hecke algebra setup. Are there any connections between these centers and equivariant K-groups of Hilbert schemes of points on the minimal 2 resolution C /Zr? It will be already nontrivial and of considerable interest to answer the question for the Iwahori-Hecke algebras of type B corresponding to r =2.

Acknowledgment. The authors thank the organizers for the high level In- ternational Conference on Representation Theory in Lhasa, Tibet, China in July 2007, where this collaboration was initiated.

References [DJ] R. Dipper and G. James, Blocks and idempotents of Hecke algebras of general linear groups, Proc. London Math. Soc. (3) (1) 54 (1987), 57–82. [FH] H. Farahat and G. Higman, The centres of symmetric group rings, Proc. Roy. Soc. (A) 250 (1959), 212–221.

CENTERS OF IWAHORI-HECKE ALGEBRAS ARE FILTERED 379

[Fra] A. Francis, The minimal basis for the centre of an Iwahori-Hecke algebra, J. Algebra, 221 (1999), 1-28. [FrJ] A. Francis and L. Jones, A new integral basis for the centre of the Hecke algebra of type A, Preprint 2007, arXiv:0705.1581. [FrG] A. Francis and J. J. Graham, Centres of Hecke algebras: the Dipper-James conjecture,J. Algebra 306 (2006), 244–267. [GR] M. Geck and R. Rouquier, Centers and simple modules for Iwahori-Hecke algebras,In: M. Cabanes (Eds.), Finite reductive groups (Luminy, 1994), 251–272, Progr. Math., 141, Birkh¨auser Boston, 1997. [Mac] I. G. Macdonald, Symmetric functions and Hall polynomials,SecondEd.,ClarendonPress, Oxford, 1995. [Mat] A. Mathas, Murphy operators and the centre of Iwahori-Hecke algebras of type A,J.Algebr. Combin. 9 (1999), 295–313. [W] W. Wang, The Farahat-Higman ring of wreath products and Hilbert schemes,Adv.in Math. 187 (2004), 417–446.

School of Computing and Mathematics, University of Western Sydney, NSW 1797, Australia E-mail address: [email protected] Department of Mathematics, University of Virginia, Charlottesville, VA 22904, U.S.A. E-mail address: [email protected]

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Contemporary Mathematics Volume 478, 2009

On Kostant’s Theorem for Lie Algebra Cohomology

UNIVERSITY OF GEORGIA VIGRE ALGEBRA GROUP1

1. Introduction 1.1. In 1961, Kostant proved a celebrated result which computes the ordinary Lie algebra cohomology for the nilradical of the Borel subalgebra of a complex simple Lie algebra g with coefficients in a finite-dimensional simple g-module. Over the last forty years other proofs have been discovered. One such proof uses the properties of the Casimir operator on cohomology described by the Casselman- Osborne theorem (cf. [GW, §7.3] for details). Another proof uses the construction of BGG resolutions for simple finite-dimensional g-modules [Ro]. Recently, Polo and Tilouine [PT] constructed BGG resolutions over Z(p) for finite-dimensional irreducible G-modules where G is a semisimple algebraic group with high weights in the bottom alcove as long as p ≥ h−1(h is the Coxeter number for the underlying root system). One can then use a base change argument to show that Kostant’s theorem holds for these modules over algebraically closed fields of characteristic p when p ≥ h − 1. It should be noted that Friedlander and Parshall had earlier obtained a slightly weaker formulation of this result (cf. [FP1, §2]) The aim of this paper is to investigate and compare the cohomology of the unipotent radical of parabolic subalgebras over C and Fp.Wepresentanewproof of Kostant’s theorem and Polo-Tilouine’s extension in Sections 2–4. Our proof employs known linkage results in Category OJ and the graded G1T category for the first Frobenius kernel G1. There are several advantages to our approach. Our proofs of these cohomology formulas are self-contained and our approach is presented in a conceptual manner. This enables us to identify key issues in attempting to compute these cohomology groups for small primes. In Section 5, we prove that when p

1The members of the UGA VIGRE Algebra Group are Irfan Bagci, Brian D. Boe, Leonard Chastkofsky, Benjamin Connell, Bobbe J. Cooper, Mee Seong Im, Tyler Kelly, Jonathan R. Ku- jawa, Wenjing Li, Daniel K. Nakano, Kenyon J. Platt, Emilie Wiesner, Caroline B. Wright and Benjamin Wyser.

391

240 UNIVERSITY OF GEORGIA VIGRE ALGEBRA GROUP

MAGMA. These examples suggest interesting phenomena which lead us to pose several open questions in Section 7.

1.2. Notation. The notation and conventions of this paper will follow those given in [Jan]. Let k be an algebraically closed field, and G a simple algebraic group defined over k with T a maximal torus of G. The root system associated to the pair (G, T ) is denoted by Φ. Let Φ+ be a set of positive roots and Φ− be the corresponding set of negative roots. The set of simple roots determined by + Φ is ∆ = {α1,...,αl}. We will use throughout this paper the ordering of simple roots given in [Hum1] following Bourbaki. Given a subalgebra a ⊂ g which is a sum of root spaces, let Φ(a) denote the corresponding set of roots. Let B be the Borel subgroup relative to (G, T ) given by the set of negative roots and let U be the unipotent radical of B. More generally, if J ⊆ ∆, let PJ be the parabolic subgroup relative to −J and let UJ be the unipotent radical and LJ the Levi factor of PJ .LetΦJ be the root subsystem in Φ generated by the simple roots in J,with + ∩ + positive subset ΦJ =ΦJ Φ .Setg =LieG, b =LieB, u =LieU, pJ =LiePJ , lJ =LieLJ ,anduJ =LieUJ . Let E be the Euclidean space associated with Φ, and denote the inner product on E by , .Letˇα be the coroot corresponding to α ∈ Φ. Set α0 to be the highest short root. Let ρ be the half sum of positive roots. The Coxeter number associated to Φ is h = ρ, αˇ0 +1. Let X := X(T ) be the integral weight lattice spanned by the fundamental weights {ω1,...,ωl}.LetM be a finite-dimensional T -module and M = ⊕ λ∈X Mλ be its weight space decomposition. The character of M, denoted by λ ∈ Z ch M = λ∈X (dim Mλ)e [X(T )]. If M and N are T -modules such that ≤ ≤ dim Mλ dim Nλ for all λ then we say that ch M ch N.ThesetX has a partial ≥ − ∈ Z ordering defined as follows: λ µ if and only if λ µ α∈∆ ≥0 α.Thesetof + r dominant integral weights is denoted by X = X(T )+ and the set of p -restricted weights is Xr = Xr(T ). For J ⊆ ∆, the set of J-dominant weights is + { ∈ | ∈Z ∈ + } XJ := µ X µ, αˇ ≥0 for all α ΦJ . and denote the p-restricted J-weights by (XJ )1. The bottom alcove CZ is defined as

CZ := {λ ∈ X | 0 ≤λ + ρ, αˇ0≤p}. 0 G Set H (λ)=indBλ where λ is the one-dimensional B-module obtained from the character λ ∈ X+ by letting U act trivially. The Weyl group corresponding to ΦisW and acts on X via the dot action w · λ = w(λ + ρ) − ρ where w ∈ W , λ ∈ X.

2. Cohomology and Composition Factors

2.1. For this section, let R = Z, C or Fp,andletJ ⊆ ∆. Then uJ has a basis consisting of root vectors where the structure constants are in R.Inorderto construct such a basis one can take an appropriate subset of the Chevalley basis for • ∗ g. The standard complex on Λ (uJ ) has differentials which are R-linear maps and • we will denote the cohomology of this complex by H (uJ ,R). Moreover, the torus • ∗ T acts on the standard complex Λ (uJ ). The differentials respect the T -action so • ∗ it suffices to look at the smaller complexes (Λ (uJ ))λ. The cohomology of this • n ∗ complex will be denoted by H (uJ ,R)λ.Foreachn,(Λ (uJ ))λ is a free R-module n of finite rank, so the cohomology H (uJ ,R)λ is a finitely generated R-module.

ON KOSTANT’S THEOREM FOR LIE ALGEBRA COHOMOLOGY 413

One can use the arguments given in Knapp [Kna, Theorem 6.10] to show that the cohomology groups when R = C or Fp satisfy Poincar´e Duality: n ∼ N−n ∗ ⊗ N ∗ (2.1.1) H (uJ ,R) = H (uJ ,R) Λ (uJ ) as T -modules where N =dimuJ . The Universal Coeﬃcient Theorem (UCT) (cf. [R, Theorem 8.26]) can be used to relate the cohomology over Z to the cohomology over C and Fp.TheZ-module C is divisible, so from the UCT (cf. [R, Corollary 8.28]) we have n ∼ n (2.1.2) H (uJ , C)λ = H (uJ , Z)λ ⊗Z C.

On the other hand, when k = Fp, the UCT shows that n ∼ n 1 n−1 (2.1.3) H (uJ , Fp)λ = (H (uJ , Z)λ ⊗Z Fp) ⊕ ExtZ(Fp, H (uJ , Z)λ). For every n, the formulas (2.1.2) and (2.1.3) demonstrate that n n dim H (uJ , C)λ ≤ dim H (uJ , Fp)λ. n n In particular, ch H (uJ , C) ≤ ch H (uJ , Fp). One should observe that additional n cohomology classes in H (uJ , Fp)λ can arise from either the ﬁrst or second summand • in (2.1.3) because of p-torsion in H (uJ , Z)λ. n • ∗ ⊗ For a uJ -module, one can deﬁne H (uJ ,M) using a complex involving Λ (uJ ) n M [Jan, I 9.17]. If M is a pJ -module then H (uJ ,M)isalJ -module. If M, N are n n ∗ ⊗ ≥ arbitrary uJ -modules then Extu (M,N)=H (uJ ,M N)forn 0.

2.2. Category OJ . For this section, k = C.FixJ ⊆ ∆. Denote the Weyl group of ΦJ by WJ , viewed as a subgroup of W .LetU(g) denote the universal enveloping algebra of g.

Definition 2.2.1. Let OJ be the full subcategory of the category of U(g)- modules consisting of modules V which satisfy the following conditions: (i) The module V is a ﬁnitely generated U(g)-module. (ii) As a U(lJ )-module, V is the direct sum of ﬁnite-dimensional U(lJ )-modules. (iii) If v ∈ V ,thendimC U(uJ )v<∞. Let Z be the center of U(g) and denote the set of algebra homomorphisms

Z → C by Z .Wesaythatχ ∈ Z is a central character of V ∈OJ if zv = χ(z)v ∈ ∈ ∈ Oχ O for all z Z and all v V .Foreachχ Z ,let J be the full subcategory of J consisting of modules V ∈OJ such that for all z ∈ Z and v ∈ V , v is annihilated by some power of z − χ(z). We have the decomposition O Oχ J = J . χ∈Z Oχ O We call J an infinitesimal block of category J . For the purpose of this paper we will only need to apply information about the integral blocks so we can assume that the weights which arise are in X.Thekey objects in integral blocks of OJ are the parabolic Verma modules, which are defined as follows. For a finite-dimensional irreducible lJ -module LJ (µ) with highest weight ∈ + + µ XJ extend LJ (µ)toapJ -module by letting uJ act trivially. The induced module U ⊗ ZJ (µ)= (g) U(pJ ) LJ (µ) is a parabolic Verma module, which we will abbreviate as PVM.

442 UNIVERSITY OF GEORGIA VIGRE ALGEBRA GROUP

The module ZJ (µ) has a unique maximal submodule and hence a unique simple quotient module, which we denote by L(µ); L(µ) is also the unique simple quotient of the ordinary Verma module Z(µ):=U(g) ⊗U(b) µ. All simple modules in the integral blocks of OJ are isomorphic to some L(µ). For each µ ∈ X, the ordinary ∈ + Verma module Z(µ) (and any quotient thereof, such as ZJ (µ)orL(µ)ifµ XJ )has ∈ Oµ Oχµ a central character which we will denote by χµ Z .Ifχ = χµ,write J := J . The Harish-Chandra linkage principle yields

χµ = χν ⇔ ν ∈ W · µ. This implies that the simple modules (and hence the PVM’s and projective inde- Oµ { ∈ | · ∈ +} composable modules) in J canbeindexedby w W w µ XJ (by identifying repetitions). For µ ∈ X,let Φµ = {α ∈ Φ |µ + ρ, αˇ =0}.

If Φµ = ∅,thenwesaythatµ is a regular weight;otherwise,itisasingular Oµ Oν weight.Ifµ and ν are both regular weights, then J is equivalent to J by the Jantzen-Zuckerman translation principle. For each α ∈ Φ, let sα ∈ W denote the reﬂection in E about the hyperplane { ∈ | · ∈ +} orthogonal to α.Ifµ is a regular dominant weight, then w W w µ XJ is the set (2.2.1) J { ∈ | ∈ } { ∈ | −1 + ⊆ +} W = w W l(sαw)=l(w) + 1 for all α J = w W w (ΦJ ) Φ which is the set of minimal length right coset representatives of WJ in W .Letw0 J J J (resp. wJ , w) denote the longest element in W (resp. WJ , W ). Then w0 = wJ w.

2.3. The following theorem provides information about the LJ composition fac- • tors in H (uJ ,L(µ)) when k = C.ForV a ﬁnite dimensional semisimple LJ -module, write [V : LJ (σ)]LJ for the multiplicity of LJ (σ)asanLJ -composition factor of V .

Theorem 2.3.1. Let k = C, V ∈OJ and λ ∈ X. i Z λ ,V ∼ L λ , i ,V (a) ExtOJ ( J ( ) ) = HomlJ ( J ( ) H (uJ )) i ∈ · (b) If [H (uJ ,L(µ)) : LJ (σ)]LJ =0where µ X+ then σ = w µ where w ∈ J W . Proof. i Z λ ,V ∼ i Z λ ,V (a) First observe that ExtOJ ( J ( ) ) = Ext(g,lJ )( J ( ) )(relative Lie algebra cohomology) and by Frobenius reciprocity we have i Z λ ,V ∼ i L λ ,V ∼ i , L λ ∗ ⊗ V . Ext(g,lJ )( J ( ) ) = Ext(pJ ,lJ )( J ( ) ) = H (pJ lJ ; J ( ) )

Since uJ pJ , one can use the Grothendieck spectral sequence construction given in [Jan, I Proposition 4.1] to obtain a spectral sequence, i,j i ∩ j ∗ ⊗ ⇒ i+j ∗ ⊗ E2 =H(pJ /uJ , lJ /(lJ uJ )); H (uJ , 0; LJ (λ) V ) H (pJ , lJ ; LJ (λ) V ). i,j ∼ i j ∗ ⊗ However, E2 = H (lJ , lJ ;H (uJ , 0; LJ (λ) V )) = 0 for i>0, so the spectral sequence collapses and yields j ∼ 0 j ∗⊗ ∼ j ∗⊗ HomlJ (LJ (λ), H (uJ ,V)) = H (lJ , lJ ;H (uJ ,LJ (λ) V )) = H (pJ , lJ ; LJ (λ) V ). i (b) Suppose that [H (uJ ,L(µ)) : LJ (σ)]LJ = 0. Then from part (a), i i [H (uJ ,L(µ)) : LJ (σ)]LJ =dimHomlJ (LJ (σ), H (uJ ,L(µ))) i Z σ ,L µ . = dim ExtOJ ( J ( ) ( ))

ON KOSTANT’S THEOREM FOR LIE ALGEBRA COHOMOLOGY 435

i J Z σ ,L µ σ w · µ w ∈ W But, ExtOJ ( J ( ) ( )) = 0 implies by linkage that = where .

2.4. Now let us assume that k = Fp.LetWp be the affine Weyl group and Wp be the extended affine Weyl group. In this setting we regard G as an affine reductive group scheme with F : G → G denoting the Frobenius morphism. Let r −1 F be this morphism composed with itself r times and set GrT =(Fr) (T ). The category of GrT -modules has a well developed representation theory (cf. [Jan,II Chapter 9]). Group schemes analogous to GrT can be defined similarly using the Frobenius morphism for LJ , PJ , B, U,etc. The following theorem provides information about the composition factors in the uJ -cohomology for p ≥ 3.

Theorem 2.4.1. Let k = Fp with p ≥ 3. i ∈ + · (a) If [H (uJ ,L(µ)) : LJ (σ)]LJ =0where µ X then µ = w σ where w ∈ Wp. i ∈ ∈ · (b) If [H (uJ ,L(µ)) : LJ (σ)]LJ =0where µ X1 and σ (XJ )1 then µ = w σ where w ∈ Wp. Proof. i (a) Suppose that [H (uJ ,L(µ)) : LJ (σ)]LJ = 0. From the Steinberg (1) tensor product theorem, we can write LJ (σ)=LJ (σ0)⊗LJ (σ1) where σ0 ∈ (XJ )1 ∈ + i ⊗ ∈ and σ1 XJ . Therefore, [H (uJ ,L(µ)) : LJ (σ0) pγ1](LJ )1T =0forsomeγ1 X. + One can also express µ = µ0 + pµ1 where µ0 ∈ X1 and µ1 ∈ X so that i ∼ i (1) H (uJ ,L(µ)) = H (uJ ,L(µ0)) ⊗ L(µ1) . i ⊗ i ⊗ Therefore, [H (uJ ,L(µ)) : LJ (σ0) pγ1](LJ )1T = 0 implies that [H (uJ ,L(µ0)) pγ2 : ⊗ ∈ i ⊗ LJ (σ0) pγ1](LJ )1T =0forsomeγ2 X,thus[H(uJ ,L(µ0)) : LJ (σ0) pγ](LJ )1T = 0forsomeγ ∈ X (where γ = γ1 − γ2). Observe that (2.4.1) i ⊗ ⊗ i [H (uJ ,L(µ0)) : LJ (σ0) pγ](LJ )1T =dimHom(LJ )1T (PJ (σ0) pγ, H (uJ ,L(µ0))). where PJ (σ0) ⊗ pγ is the (LJ )1T projective cover of LJ (σ0) ⊗ pγ. Next consider the composition factor multiplicities for the cohomology of L(µ0) over the Frobenius kernel (UJ )1, i ⊗ [H ((UJ )1,L(µ0)) : LJ (σ0) pγ](LJ )1T ⊗ i =dimHom(LJ )1T (PJ (σ0) pγ, H ((UJ )1,L(µ0))). We can also give another interpretation of this composition factor multiplicity. First, let us apply the Lyndon-Hochschild-Serre spectral sequence for (UJ )1 ∼ (PJ )1T ,(PJ )1T/(UJ )1 = (LJ )1T : (2.4.2) i,j i j i+j E =Ext (PJ (σ0) ⊗ pγ, H ((UJ )1,L(µ0))) ⇒ Ext (PJ (σ0) ⊗ pγ, L(µ0)). 2 (LJ )1T (PJ )1T

Since P := PJ (σ0) ⊗ pγ is projective as an (LJ )1T -module, the spectral sequence collapses and we have P, i U ,L µ ∼ i P, L µ Hom(LJ )1T ( H (( J )1 ( 0)) = Ext(PJ )1T ( ( 0))

∼ i G1T = Ext (coind P, L(µ0)). G1T (PJ )1T

644 UNIVERSITY OF GEORGIA VIGRE ALGEBRA GROUP

For p ≥ 3, there exists another ﬁrst quadrant spectral sequence which can be used to relate these two diﬀerent composition factor multiplicities [FP2, (1.3) Proposition]: 2i,j i ∗ (1) ⊗ j ⇒ 2i+j E2 = S (uJ ) H (uJ ,L(µ0)) H ((UJ )1,L(µ0)). − Since the functor Hom(LJ )1T (P, ) is exact, we can compose it with the spectral sequence above to get another spectral sequence: 2i,j i ∗ (1) ⊗ j (2.4.3) E2 = S (uJ ) Hom(LJ )1T (P, H (uJ ,L(µ0))) ⇒ 2i+j Hom(LJ )1T (P, H ((UJ )1,L(µ0))).

Suppose that σ0 + pγ∈ / Wp · µ0. Then by the linkage principle for G1T : ⊗ i Hom(LJ )1T (PJ (σ0) pγ, H ((UJ )1,L(µ0)))

∼ i G1T = Ext (coind PJ (σ0) ⊗ pγ, L(µ0)) = 0 G1T (PJ )1T for all i ≥ 0. Therefore, the spectral sequence (2.4.3) abuts to zero. The differential 0,j 2,j−1 2i,j i ∗ (1) ⊗ 0,j d2 in the spectral sequence maps E2 to E2 .NotethatE2 = S (uJ ) E2 ≥ 0,0 2i,0 ≥ for all i, j 0. Since 0 = E0 = E2 , it follows that E2 =0fori 0. Therefore, 0,1 2i,1 ≥ 2i,j E2 =0,thusE2 =0fori 0. Continuing in this fashion, we have E2 =0for j ⊗ all i, j. In particular, using (2.4.1) and (2.4.3), [H (uJ ,L(µ0)) : LJ (σ0) pγ](LJ )1T = j 0,j dim Hom(LJ )1T (P, H (uJ ,L(µ0))) = dim E2 =0forallj which is a contradiction. This implies that µ0 and σ0 are in the same orbit under Wp,thusµ = w · σ where w ∈ Wp. (b) Under the hypotheses, we can apply the above argument with 0 = γ1 = γ2 = γ. Therefore, µ = w · σ where w ∈ Wp. 2.5. We present the following proposition which allows one to compare composition factors of the cohomology with coefficients in a module to the cohomology with trivial coefficients. Note that this proposition is independent of the characteristic of the field k.

Proposition 2.5.1. Let J ⊆ ∆ and V be a ﬁnite-dimensional PJ -module. If i ∈ + i ⊗ [H (uJ ,V):LJ (σ)]LJ =0for σ XJ then [H (uJ ,k) V : LJ (σ)]LJ =0.

Proof. The simple finite-dimensional PJ -modules are the simple finite-dimensional LJ -modules inflated to PJ by making UJ act trivially. We will prove the proposition by induction on the composition length n of V .Forn = 1, this is clear because V is simple and UJ acts trivially so i ∼ i H (uJ ,V) = H (uJ ,k) ⊗ V. Now assume that the proposition holds for modules of composition length n, and let V have composition length n + 1. There exists a short exact sequence 0 → V → V → L → 0 where V has composition length n and L is a simple PJ -module. We have a long i exact sequence in cohomology which shows that if [H (uJ ,V):LJ (σ)]LJ =0then i i either [H (uJ ,V ):LJ (σ)]LJ =0or[H(uJ ,L):LJ (σ)]LJ = 0. By the induction i ⊗ i ⊗ hypothesis, this implies [H (uJ ,k) V : LJ (σ)]LJ =0or[H(uJ ,k) L : LJ (σ)]LJ = 0.

ON KOSTANT’S THEOREM FOR LIE ALGEBRA COHOMOLOGY 457

i The short exact sequence above can be tensored by H (uJ ,k) to obtain a short exact sequence: i i i 0 → H (uJ ,k) ⊗ V → H (uJ ,k) ⊗ V → H (uJ ,k) ⊗ L → 0.

The result now follows because one of the terms on the end has an LJ composition factor of the form LJ (σ) by the induction hypothesis, so the middle term has to have a composition factor of this form.

3. Parabolic Computations In this section we prove several elementary results which will be ingredients in our proof of Kostant’s Theorem and its generalization to prime characteristic in Section 4. 3.1. Given Ψ ⊂ Φ+,write Ψ = β. β∈Ψ For w ∈ W put (3.1.1) Φ(w)=−(wΦ+ ∩ Φ−)=wΦ− ∩ Φ+ ⊂ Φ+. We recall some basic facts about Φ(w). Lemma 3.1.1. Let w ∈ W . (a) |Φ(w)| = l(w). (b) w · 0=−Φ(w).

(c) If w = sj1 ...sjt is a reduced expression, then −1 { } Φ(w )= αjt ,sjt αjt−1 ,sjt sjt−1 αjt−2 ,...,sjt ...sj2 αj1 . Proof. (a) [Hum1, Lemma 10.3A], (b) [Kna, Proposition 3.19], (c) [Hum2, Exercise 5.6.1] Lemma 3.1.2. Let J ⊆ ∆ and w ∈ W . ⊂ + + ∈ J (a) Φ(w) Φ ΦJ =Φ(uJ ) if and only if w W . (b) If w · 0=−Ψ for some Ψ ⊂ Φ+ then Ψ=Φ(w). Proof. (a) Assume w ∈ J W .Letβ ∈ Φ(w). Then β ∈ Φ+,andβ ∈ wΦ− −1 ∈ − ∈ + J whence w β Φ .Thusβ/ΦJ by the second characterization of W in (2.2.1). Conversely, assume w/∈ J W . Then by the ﬁrst characterization of J W in (2.2.1), w has a reduced expression beginning with sα for some α ∈ J (by the ∈ ∈ + Exchange Condition, for instance). Then by Lemma 3.1.1(c), α Φ(w); but α ΦJ ⊂ + + so Φ(w) Φ ΦJ . (b) We prove this by induction on l(w). If l(w)=0thenw =1andw · 0=0, so clearly the only possible Ψ is Ψ = ∅ = Φ(1). Given w with l(w) > 0, write w = sαw with α ∈ ∆andl(w )=l(w)−1. Then α ∈ Φ(w)andα/∈ Φ(w )=sα(Φ(w) {α}); cf. the proof of [Hum2, Lemma 1.6]. + Suppose w · 0=−(γ1 + ···+ γm)fordistinctγ1,...,γm ∈ Φ .Then w · 0=sα · (w · 0) = sα(w · 0) + sαρ − ρ = −(sαγ1 + ···+ sαγm + α). There are two cases.

Case 1:Noγi = α.Thensαγ1,...,sαγm,αare distinct positive roots: sα permutes the positive roots other than α,andnosαγi = α because sα(−α)=α.Butthen by induction, {sαγ1,...,sαγm,α} =Φ(w ), and this contradicts α/∈ Φ(w ).

846 UNIVERSITY OF GEORGIA VIGRE ALGEBRA GROUP

Case 2:Someγi = α.Sayγm = α.Thensα(γm)=−α,sow · 0=−(sαγ1 + ···+ sαγm−1). By induction, Φ(w )={sαγ1,...,sαγm−1}. Hence Φ(w)=sαΦ(w ) ∪ {α} = {γ1,...,γm} as required. 3.2. Saturation. Lemma 3.1.2 guarantees that, for w ∈ J W , w · 0=−Φ(w) n ∗ { } is a weight in Λ (uJ ), where n = l(w). Specifically, if Φ(w)= β1,...,βn then ∧···∧ { | ∈ } the vector fΦ(w) := fβ1 fβn has the desired weight, where fβ β Φ(uJ ) ∗ { | ∈ } is the basis for uJ dual to a fixed basis of weight vectors xβ β Φ(uJ ) for uJ . Lemmas 3.1.1 and 3.1.2 guarantee that the weight w ·0 occurs with multiplicity one • ∗ → • ∗ in Λ (uJ ). In particular, since the differentials in the complex 0 Λ (uJ ) preserve n weights, we see that fΦ(w) descends to an element of H (uJ ,k)ofweightw · 0, and • n is the only degree in which this weight occurs in H (uJ ,k)(wherek = C or Fp). • In order to prove that fΦ(w) generates an LJ -submodule of H (uJ ,k)ofhighest weight w · 0, we need the following condition, which could be described by saying + that Φ(w) is “saturated” with respect to ΦJ . Proposition . ∈ J ∈ ∈ + − ∈ 3.2.1 Let w W .Ifβ Φ(w),γ ΦJ ,andδ = β γ Φ, then δ ∈ Φ(w). Proof. We prove this by induction on l(w). If w =1thenΦ(w)=∅ and the statement is vacuously true. So assume l(w) > 0. Write w = w sα with α ∈ ∆ and l(w)=l(w) − 1; then necessarily w ∈ J W .Toseethis,notethatwα < 0, so −1 + −1 + ⊂ + { } ⊂ + (w ) (ΦJ )=sαw (ΦJ ) sα(Φ α ) Φ .Now + − Φ(w)=Φ ∩ w sαΦ =Φ+ ∩ w(Φ− {−α}∪{α}) =(Φ+ ∩ wΦ−) ∪{wα} =Φ(w) ∪{wα}, whereinthethirdequalitywehaveusedthefactthatwα>0. By induction, + Φ(w ) is saturated with respect to ΦJ . So it remains to check the condition of the lemma when β = wα. − ∈ ∈ + ∈ Let β = w α and suppose δ = β γ Φforsomeγ ΦJ .Sinceβ + + ∈ + ∈ + + Φ ΦJ by Lemma 3.1.2, and γ ΦJ , necessarily δ Φ ΦJ . Consider the −1 −1 − − −1 ∈ J ∈ + root (w ) δ =(w ) (w α γ)=α (w ) (γ). Since w W and γ ΦJ ,we know (w)−1(γ) > 0. Since α is simple, (w)−1δ<0. That is, δ ∈ w(Φ−). Thus, δ ∈ Φ(w ) ⊂ Φ(w), as required. 3.3. Prime characteristic. In the prime characteristic setting we will need to work harder than in characteristic zero, because our control over the composition factors in cohomology in Theorem 2.4.1 is much weaker than in Theorem 2.3.1. We begin by recording two simple technical facts which will be needed later. Proposition 3.3.1. (a) Let λ, µ ∈ X and suppose λ = wµ where w = ≤ ≤ − sj1 ...sjt with t minimal. Then αjr ,sjr+1 ...sjt µ =0for 1 r t 1. (b) Suppose α ∈ Φ+ has maximal height in its W -orbit. Then β,α≥0 for all β ∈ Φ+. Proof. (a) Since t is minimal, − sjr+1 ...sjt µ = sjr ...sjt µ = sjr+1 ...sjt µ sjr+1 ...sjt µ, αˇjr αjr . This implies the desired inequality.

ON KOSTANT’S THEOREM FOR LIE ALGEBRA COHOMOLOGY 479

(b) Otherwise, sβ(α)=α −α, βˇβ would be a root of the same length as α, but higher, contradicting the hypothesis. We will be able to cut down the possible weights in cohomology when p ≥ h−1. The proof will make use of certain special sums of positive roots. For 1 ≤ i ≤ l set Φ = { α ∈ Φ+ |ω ,α > 0 } i i + = { α ∈ Φ | α = rjαj with ri > 0 } (3.3.1) + + =Φ ΦJ , where

J = Ji =∆ {αi}, { ∈ | ≥ } Φi = α Φi α, αi 0 , and deﬁne (3.3.2) δi = Φi ,δi = Φi .

We begin by collecting some elementary properties of δi.

Proposition 3.3.2. (a) w(Φi)=Φi for all w ∈ WJ . (b) δi = cωi for some c ∈ Z. J J J (c) −δi = w·0 where J =∆{αi} (recall w is the longest element of W ).

Proof. (a) For j = i and α a positive root involving αi, sj(α) is again a positive root involving αi.Thussj permutes Φi.Sincethesj with j = i generate WJ , the result follows. (b) From (a), for j = i, sj(δi)=δi.Thuswhenδi is written as a linear combination of fundamental dominant weights, the coeﬃcient of ωj is 0. That is, δi = cωi for some scalar c.Sinceδi ∈ ZΦ, c ∈ Z. (c) Write 2ρ = α + α = δi +2ρJ . α∈Φ+ α∈Φ+ ωi,α >0 ωi,α =0

Apply the longest element wJ of WJ , and use the ﬁrst computation in (a):

2wJ ρ = wJ δi − 2ρJ = δi − 2ρJ . Thus 1 − 1 − − 1 − wJ ρ = 2 δi ρJ = 2 δi (ρ 2 δi)=δi ρ, and so J −δi = −wJ ρ − ρ = wJ w0ρ − ρ = w · ρ. ≤ 3.4. The crucial property of δi is that δi, αˇi h. The proof will require a few steps. First, put J =∆ {αi} as before, and recall that wJ denotes the longest element of the parabolic subgroup WJ .Letwi ∈ W be an element of shortest possible length such that

(3.4.1) wiwJ αi = α, the highest root in Wαi.

Proposition 3.4.1. Let i, J, wi and α be as above. −1 (a) wJ (Φi Φi)=Φ(wi ). − − −1 (b) wJ (δi δi)=ρ wi ρ. ∨ (c) δi, αˇi =1+ ρ, α .

1048 UNIVERSITY OF GEORGIA VIGRE ALGEBRA GROUP

≤ (d) δi, αˇi h. Proof. ∈ ∈ (a) Observe that β wJ (Φi Φi) if and only if β = wJ α with α Φi and α, αi < 0; equivalently (using Proposition 3.3.2(a)), β,wJ αi < 0andβ ∈ Φi.Thus ∈ ⇐⇒ ∈ (3.4.2) β wJ (Φi Φi) β Φi and wiβ,α < 0. ∈ ∈ + ∈ − Assuming β wJ (Φi Φi), then β Φ and wiβ Φ (by Proposition ∈ −1 3.3.1(b)); equivalently β Φ(wi ) (by (3.1.1)). ∈ −1 ∈ + To prove the reverse inclusion, assume that β Φ(wi ); i.e., β Φ and − wiβ ∈ Φ . We claim it is enough to show that wiβ,α < 0 (the second condition ∈ ∈ + ∈ − of (3.4.2)). For if β/Φi then β ΦJ , hence wJ β ΦJ ,andthus wiβ,α = β,wJ αi = wJ β,αi≥0, since αj,αi≤0forj = i. − It remains to show wiβ,α < 0, or, equivalently, wiβ,α=0,since wiβ ∈ Φ

(recall Proposition 3.3.1(b)). Write wi = sj1 ...sjt with t minimal. By Lemma ≤ ≤ 3.1.1(c) we have β = sjt ...sjr+1 αjr for some 1 r t.Putµ = wJ αi and λ = α in Proposition 3.3.1(a) to obtain wiβ,α = sj1 ...sjr αjr ,sj1 ...sjt wJ αi = αjr ,sjr+1 ...sjt wJ αi =0.

(b) Using (a) and Lemma 3.1.1(b), − −1 − −1 · − −1 wJ (δi δi)= wJ (Φi Φi) = Φ(wi ) = wi 0=ρ wi ρ.

(c) Using (b) and the idea of the proof of Proposition 3.3.2(c), − − −1 − 1 1 − −1 δi δi = wJ (ρ wi ρ)=wJ [(ρ 2 δi)+ 2 δi] wJ wi ρ − − 1 1 − −1 − − −1 = (ρ 2 δi)+ 2 δi wJ wi ρ = δi ρ wJ wi ρ. −1 Thus δi = ρ + wJ wi ρ and so −1 ∨ δi, αˇi = ρ + wJ wi ρ, αˇi =1+ ρ, wiwJ αˇi =1+ ρ, α .

∨ (d) Combine (c) with the inequality ρ, α ≤ρ, αˇ0 = h − 1. 3.5. The next proposition is the key to our proof of Kostant’s Theorem in characteristic p ≥ h − 1. Proposition 3.5.1. Assume p ≥ h − 1.Supposeσ = w · 0+pµ is a weight of Λ•(u∗) where w ∈ W and µ ∈ X.Thenσ = x · 0 for some x ∈ W . Proof. The proof is again by induction on l(w). Assume w =1sothatpµ is a sum of distinct negative roots. Set ν = −µ so that pν = Ψ for some Ψ ⊂ Φ+. For any 1 ≤ i ≤ l, ≤ ≤ Ψ , αˇi δi, αˇi δi, αˇi .

The ﬁrst inequality follows because αj, αˇi≤0ifj = i whereas αi, αˇi =2,so including only positive roots that involve αi can only make the inner product bigger. The second inequality follows similarly: including only those positive roots α with c α, αˇi≥0 obviously can only increase the inner product. Writing Ψ =2ρ−Ψ , where Ψc =Φ+ Ψ, applying the same inequality for Ψc, and using the fact that ρ, αˇi =1,weobtain − ≤ ≤ 2 δi, αˇi Ψ , αˇi δi, αˇi .

ON KOSTANT’S THEOREM FOR LIE ALGEBRA COHOMOLOGY 4911

 ≤ But we also have δi, αˇi h by Proposition 3.4.1(d). Thus

(3.5.1) 2 − h ≤ pν, αˇi≤h.

Since p ≥ h − 1andν, αˇi∈Z, the ﬁrst inequality implies ν, αˇi≥0 for all i. That is, ν is dominant. If p>h, the second inequality implies that ν, αˇi =0for all i,andthusν = 0. This completes the proof in the case w =1whenp>h. From Proposition 3.3.1(b), it follows that

pν, αˇ0 = Ψ, αˇ0≤2ρ, αˇ0 =2(h − 1).

Since p ≥ h − 1, we deduce that ν, αˇ0 =0, 1 or 2. Suppose for the moment that we handle the case ν, αˇ0 = 2; this case does not arise if p = h. Recall also that we know ν is dominant. If ν, αˇ0 =0thenν = 0; this can be seen sinceα ˇ0 is the highest root of the dual root system, and thus involves every dual simple rootα ˇi with positive coeﬃcient [Hum1, Lemma 10.4A]. So the coeﬃcient of ωi in ν must be 0 for every i. Suppose ν, αˇ0 =1.Thenν is a minuscule dominant weight. Also pν = Ψ must belong to the root lattice. When p = h − 1, one can check for each irreducible root system that p does not divide the index of connection f (the index of the weight lattice in the root lattice); cf. [Hum1, p. 68]. Thus ν itself must lie in the root lattice. However, a case-by-case check using the list of minuscule weights (e.g., [Hum1, Exercise 13.13 and Table 13.1]) shows that this never happens. Assume p = h. The Coxeter number is prime only in type Al.Inthiscase every fundamental dominant weight ωi is minuscule, and h = f = l +1so pωi is in the root lattice. Suppose ν = ωi. Recall from Proposition 3.3.2(b) that δi = cωi; we compute c = cωi, αˇi = α, αˇi =2+(l − 1) = l +1=h, α∈Φ+ ωi,α >0 wherewehaveusedthefactthatαi, αˇi =2,αj +···+αi, αˇi = αi+···+αk, αˇi = 1for1≤ j 0 } and γ = Ψ0. We claim − − ≥ that γ =(h 1)α0.Toseethis,notethatsα0 Ψ0 = Ψ0 (recall that α, αˇ0 0 ∈ + − for α Φ ). So sα0 γ = γ. Substituting this into the formula for sα0 γ gives 1 − γ = 2 γ,αˇ0 α0.But γ,αˇ0 = 2ρ, αˇ0 =2(h 1), and this proves the claim. + Now assume p = h − 1, ν, αˇ0 =2,and(h − 1)ν = Ψ for some Ψ ⊂ Φ . Then

2(h − 1) = (h − 1)ν, αˇ0 = Ψ, αˇ0≤2ρ, αˇ0 =2(h − 1), so we must have equality at the third step. It follows from the deﬁnition of Ψ0 above, and the fact that γ,αˇ0 =2(h−1), that Ψ0 ⊂ Ψ. But then Ψ0 Ψ =(h−1)(α0 − ν), so α0 − ν is a dominant weight (by the argument given for Ψ at the beginning of this proof), and α0 − ν, αˇ0 = 0 by the deﬁnition of Ψ0. As mentioned earlier, − − − − − · this implies α0 ν =0.Thusσ = pµ = pν = (h 1)α0 = ρ, αˇ0 α0 = sα0 0. This completes the case w =1.

1250 UNIVERSITY OF GEORGIA VIGRE ALGEBRA GROUP

The induction step is almost identical to that in Lemma 3.1.2(b). Write w = sαw as in that proof, and suppose as before that w · 0+pµ = −(γ1 + ···+ γm)for + distinct γ1,...,γm ∈ Φ .Then

 w · 0+psαµ = −(sαγ1 + ···+ sαγm + α).

This is a sum of m ± 1 distinct negative roots (according to whether or not some γi = α). By induction, w · 0+psαµ = x · 0forsomex ∈ W . Apply sα· to get the result.

3.6. In this section we prove results about complete reducibility of modules that will be later used in our cohomology calculations.

J + Proposition 3.6.1. Let p ≥ h − 1, w ∈ W ,andλ ∈ CZ ∩ X .Then

(a) LJ (w · 0) is in the bottom alcove for LJ ; (b) LJ (w · 0) ⊗ L(λ) is completely reducible as an LJ -module.

Proof. (a) First decompose J := J1 ∪ J2 ∪···∪Jt into indecomposable components, and let β0 be the highest short root of one of the components Ji =: K. Observe that for w ∈ J W ,

−1 w · 0+ρK , βˇ0 = wρ − ρ + ρK , βˇ0 = wρ, βˇ0 = ρ, w βˇ0 where in the second equality we have used that both ρ and ρK have inner product 1 J with each simple coroot appearing in the decomposition of βˇ0.Nowsincew ∈ W ∈ + −1 ∈ + ≤ −1 ˇ ≤ − ≤ · and β0 ΦJ , w β0 Φ ,andthus0 ρ, w β0 h 1 p. Hence, w 0 belongs to the closure of the bottom LJ alcove. (b) Suppose that LJ (ν + µ)isanLJ composition factor of LJ (w · 0) ⊗ L(λ) where ν + µ is J-dominant and ν is a weight of LJ (w · 0) and µ is a weight of L(λ). We will show that ν + µ belongs to the closure of the bottom LJ alcove. First observe that µ, αˇ≤λ, αˇ0 for all α ∈ Φ. Indeed, we can choose w ∈ W such that wµ is dominant and since µ is a weight of L(λ), wµ ≤ λ. Therefore,

−1 µ, w βˇ = wµ, βˇ≤wµ, αˇ0≤λ, αˇ0 for all β ∈ Φ. Using the notation and results in (a), in addition to the fact that λ ∈ CZ,we have

ν + µ + ρK , βˇ0 = ν + ρK , βˇ0 + µ, βˇ0

≤w · 0+ρK , βˇ0 + µ, βˇ0

≤ (h − 1) + λ, αˇ0

= ρ, αˇ0 + λ, αˇ0

= λ + ρ, αˇ0 ≤ p.

The complete reducibility assertion follows by the Strong Linkage Principle [Jan, Proposition 6.13] because all the composition factors of LJ (w · 0) ⊗ L(λ)areinthe bottom LJ alcove.

ON KOSTANT’S THEOREM FOR LIE ALGEBRA COHOMOLOGY 5113

4. Kostant’s Theorem and Generalizations 4.1. In this section we will prove Kostant’s theorem, and its extension to characteristic p by Friedlander-Parshall (p ≥ h)[FP1] and by Polo-Tilouine (p ≥ h − 1) [PT], for dominant highest weights in the closure of the bottom alcove. We begin by proving the result for trivial coeﬃcients, and then use our tensor product results to prove it in the more general setting.

Theorem 4.1.1. Let J ⊆ ∆. Assume k = C or k = Fp with p ≥ h − 1.Then as an LJ -module n ∼ H (uJ ,k) = LJ (w · 0). w∈J W l(w)=n Proof. First observe that when p = 2 the condition that p ≥ h − 1 implies that Φ = A1 or A2. For these cases the theorem can easily be verified directly. So assume that p ≥ 3. We first prove that every irreducible LJ -module in the sum on the right side is a composition factor of the left side. By the remarks at the beginning of Section 3.2, J n we have for each w ∈ W with l(w)=n the vector fΦ(w) ∈ H (uJ ,k), where Φ(w)= {β1,...,βn}. To show that fΦ(w) is a maximal vector for the Levi subalgebra lJ , ∈ + fix γ ΦJ .Then m ∧···∧ ∧···∧ (4.1.1) xγ fΦ(w) = fβ1 xγ fβi fβm . i=1

Fix β = βi for some 1 ≤ i ≤ m. For any root vector xδ,

(xγ fβ)(xδ)=−fβ([xγ,xδ]) is nonzero if and only if 0 =[ xγ ,xδ] ∈ gβ, if and only if β = γ + δ (since root spaces are one-dimensional). Assume xγ fβ is nonzero; then it is a scalar multiple of fδ where δ = β − γ is a root. Since β ∈ Φ(w), Proposition 3.2.1 implies that ∈ δ Φ(w); that is, δ = βj for some j = i.Thusxγ fβ = fβj already occurs in the wedge product in (4.1.1). So every term on the right hand side of (4.1.1) is 0, proving that fΦ(w) is the highest weight vector of a LJ (resp. (LJ )1) composition n factor of H (uJ ,k)whenk = C (resp. k = Fp). But, the high weight is in the bottom LJ -alcove so we can conclude in general that this high weight corresponds to a LJ composition factor isomorphic to LJ (w · 0). We now prove that all composition factors in cohomology appear in Kostant’s formula. By Theorem 2.3.1 when k = C, and by Theorem 2.4.1, Proposition 3.5.1, n and Lemma 3.1.2 when k = Fp,anyLJ composition factor of H (uJ ,k)isan J LJ (w · 0) for w ∈ W . By Lemmas 3.1.1(a) and 3.1.2(b), l(w)=n and LJ (w · 0) occurs with multiplicity one in cohomology. Moreover, when k = Fp, by Proposition 3.6.1 all the composition factors LJ (w · 0) lie in the bottom LJ alcove. By the Strong Linkage Principle, there are no nontrivial extensions between these irreducible LJ modules. So in either case, n H (uJ ,k) is completely reducible and given by Kostant’s formula. • ∗ We remark that the largest weight in Λ (u )is2ρ.Moreover,2ρ + ρ, αˇ0 = 3ρ, αˇ0 =3(h − 1). This weight is not in the bottom alcove unless p ≥ 3(h − 1). This necessitates a more delicate argument for the complete reducibility of the cohomology when p ≥ h − 1.

1452 UNIVERSITY OF GEORGIA VIGRE ALGEBRA GROUP

4.2. We can now use the previous theorem to compute the cohomology of uJ with coeﬃcients in a ﬁnite-dimensional simple g-module.

+ Theorem 4.2.1. Let J ⊆ ∆ and µ ∈ X . Assume that either k = C,ork = Fp + with µ + ρ, βˇ≤p for all β ∈ Φ .ThenasanLJ -module, n ∼ H (uJ ,L(µ)) = LJ (w · µ). w∈J W l(w)=n Proof. Observe that the conditions on µ imply p ≥ h − 1. Namely, we have

(4.2.1) p ≥µ + ρ, αˇ0 = h − 1+µ, αˇ0≥h − 1.

For p = 2, the only case that remains to be checked is the case when Φ = A1 and L(µ)=L(1) is the two dimensional natural representation. This can be easily verified using the definition of cocycles and differentials in Lie algebra cohomology. So assume that p ≥ 3. First consider the case k = Fp with p = h − 1. Then the inequalities in (4.2.1) + must all be equalities, whence µ, αˇ0 =0.Sinceµ ∈ X , it follows that µ =0. But now we are back to the setting of Theorem 4.1.1, where the result is proved. Thus for the rest of this proof we may assume k = C or k = Fp with p ≥ h. We first prove that every LJ composition factor of the cohomology occurs in the n direct sum on the right side. Let LJ (σ)beanLJ composition factor of H (uJ ,L(µ)). By Proposition 2.5.1 and Theorem 4.1.1, we have that LJ (σ)isanLJ composition J factor of LJ (w · 0) ⊗ L(µ)forsomew ∈ W with l(w)=n. Moreover, by definition µ ∈ X1(T ) and by the proof of Proposition 3.6.1(b), σ ∈ (XJ )1. Hence, by Theorem 2.3.1 or 2.4.1, σ = y · µ for some y ∈ Wp (when k = C we set Wp = W ). · 0 · · ⊗ According to Proposition 3.6.1, LJ (w 0) = HJ (w 0), and LJ (w 0) L(µ)is completely reducible. Therefore, by using Frobenius reciprocity ∼ 0 =Hom (L (σ),L (w · 0) ⊗ L(µ)) = Hom (L (σ),w· 0 ⊗ L(µ)). LJ J J BLJ J From this statement, one can see that σ = y · µ = w · 0+ν for some weight ν of L(µ). Choose x ∈ W such that ν = xν with ν dominant. Note that ν is still a weight of L(µ), so in particular ν ≤ µ. Rewriting the previous equation gives (w−1y) · µ = w−1xν.

Applying [Jan, Lemma II.7.7(a)] with λ =0,ν1 = µ ∈ X(T )+ ∩ W (µ − λ), we conclude that ν = µ. Now apply [Jan, Lemma II.7.7(b)] to conclude that there exists w1 ∈ Wp such that −1 w1 · 0=0 and w1 · µ = w xµ. But since p ≥ h, ρ lies in the interior of the bottom alcove, so the stabilizer of 0 −1 under the dot action of Wp is trivial; i.e., w1 =1.Thusµ = w xµ,orequivalently, w · µ = w · 0+xµ = w · 0+ν = y · µ = σ.Sincew ∈ J W and l(w)=n,thisproves that every composition factor in cohomology occurs in Kostant’s formula (possibly with multiplicity greater than one). We now prove that every LJ irreducible on the right side occurs as a composition factor in cohomology, with multiplicity one. Let σ = w · µ for w ∈ J W with

ON KOSTANT’S THEOREM FOR LIE ALGEBRA COHOMOLOGY 5315

• • ∗ ⊗ l(w)=n.Theσ weight space of C =Λ(uJ ) L(µ)containsatleasttheone dimensional space n ∗ ⊗ Λ (uJ )w·0 L(µ)wµ since w · µ = w · 0+wµ = −Φ(w) + wµ. To see that this is the entire σ weight space of C• we use a simple argument of Cartier [Cart], which we reproduce here for the reader’s convenience. Note first that there is a bijection between subsets Ψ ⊂ Φ+ and subsets Ψ ⊂ Φ satisfying Φ=Ψ −Ψ, namely Ψ=Ψ ∩ Φ+ and Ψ=Ψ ∪−(Φ+ Ψ). Note that the collection of sets of the form Ψ is invariant under the ordinary action of W . It is easy to check that for such pairs, − − 1 (4.2.2) ρ Ψ = 2 Ψ . Suppose σ = −Ψ + ν for some Ψ ⊂ Φ+ and some weight ν of L(µ). It suffices − − 1 toshowΨ=Φ(w)andν = wµ.Wehaveσ + ρ = ρ Ψ + ν = 2 Ψ + ν. Thus −1 −1 − 1 −1 −1 − µ + ρ = w (σ + ρ)=w ν 2 w Ψ = w ν Γ + ρ, −1 −1 + where we have applied (4.2.2) to w ΨandsetΓ=w Ψ ∩ Φ . −1 −1 − But since w ν is a weight of L(µ)wecanwritew ν = µ i miαi with ∈ Z mi ≥0.So µ = µ − miαi −Γ. i −1 We conclude that all mi =0,sow ν = µ and ν = wµ.Also, Γ=∅ =⇒ w−1Ψ=Φ − =⇒ Ψ= wΦ− =⇒ Ψ=wΦ− ∩ Φ+ =Φ(w). This is what we wanted to show. Since the w · µ weight space in the chain complex C• is one dimensional and occurs in Cn, we conclude, as in the case of trivial coefficients, that w ·µ is a weight n n in the cohomology H (uJ ,L(µ)). A corresponding weight vector in C is

v = fΦ(w) ⊗ vwµ, ∈ ∈ + where fΦ(w) is as in the proof of Theorem 4.1.1 and 0 = vwµ L(µ)wµ.Fixγ ΦJ ; then xγ v = xγ fΦ(w) ⊗ vwµ + fΦ(w) ⊗ xγ vwµ. We know from the proof of Theorem 4.1.1 that xγ fΦ(w) = 0. Suppose xγ vwµ were not zero. Then it would be a weight vector in L(µ)ofweightwµ + γ.ByW -invariance, µ + w−1γ would be a weight of ∈ J ∈ + −1 ∈ + L(µ). But w W and γ ΦJ imply w γ Φ , and this contradicts that µ is the highest weight of L(µ). Therefore v is annihilated by the nilradical of the Levi subalgebra, and hence its image in cohomology generates an LJ composition factor n of H (uJ ,L(µ)) isomorphic to LJ (w · µ). Note also that our argument proves that this composition factor occurs with multiplicity one. The LJ highest weights are in the closure of the bottom LJ alcove by Propo- sitions 2.5.1 and 3.6.1, and thus the cohomology is completely reducible as an LJ -module.

1654 UNIVERSITY OF GEORGIA VIGRE ALGEBRA GROUP

5. The Converse of Kostant’s Theorem 5.1. Existence of extra cohomology. The following theorem shows that there are extra cohomology classes (beyond those given by Kostant’s formula) that arise in H•(u,k) when char k = p and p

n ∼ n u/uJ 1 n−1 (5.1.1) H (u,k) = H (uJ ,k) ⊕ H (u/uJ , H (uJ ,k)). By the remarks at the beginning of Section 3.2, we can find explicit cocycles such that, as a T -module, w · 0 → H•(u,k) w∈W • whereas by Lemmas 3.1.1 and 3.1.2, the only weights in H (uJ ,k)(orevenin • ∗ · ∈ ∈ J Λ (uJ )) of the form w 0withw W occur when w W .Sowemusthave 1 • w · 0 → H (u/uJ , H (uJ ,k)). w∈W J W Thus it suffices to find “extra” cohomology in the first term on the right hand side of (5.1.1), meaning a cohomology class in characteristic p which does not have an analog in characteristic zero. Since u/uJ is isomorphic to the nilradical of the Levi subalgebra lJ , the first part of the proof of Theorem 4.1.1 shows that for w ∈ J W with l(w)=n,we n u/u have an explicit invariant vector of weight w · 0inH (uJ ,k) J .Thuswegetan inclusion n u/uJ n w · 0 → H (uJ ,k) ⊂ H (uJ ,k). w∈J W l(w)=n

By [Jan, Lemma 2.13] this induces an LJ -homomorphism from a sum of Weyl modules (for LJ ) n (5.1.2) φ: S = VJ (w · 0) → H (uJ ,k) w∈J W l(w)=n which is injective on the direct sum of the highest weight spaces. Next we claim that · (5.1.3) HdLJ φ(S)= LJ (w 0). w∈J W l(w)=n

ON KOSTANT’S THEOREM FOR LIE ALGEBRA COHOMOLOGY 5517

∼ ⊂ To see this, note ﬁrst that φ(S) = S/Ker φ,andKerφ RadLJ S because of the injectivity of φ on the highest weight spaces of the indecomposable direct summands VJ (w · 0) of S. This means that ∼ RadLJ φ(S) = RadLJ (S/Ker φ)=(RadLJ S)/Ker φ. Thus

HdLJ φ(S)=φ(S)/ RadLJ φ(S) ∼ = (S/Ker φ)/((RadLJ S)/Ker φ) ∼ = S/ Rad S LJ ∼ = LJ (w · 0) w∈J W l(w)=n as claimed. −1 Now choose α to be a short simple root, and ﬁx w ∈ W such that w α = α0, the highest short root. Then w ∈ J W and −1 w · 0+ρ, αˇ = wρ, αˇ = ρ, w αˇ = ρ, αˇ0 = h − 1 >p.

Thus λ := w · 0 is not in the restricted region for LJ .Writeλ = λ0 + pλ1 with ∈ ∈ + λ0 (XJ )1 and 0 = λ1 XJ . There are two cases, according to whether or not φ(VJ (λ)) is a simple LJ -module. ∼ ∼ Case 1: φ(VJ (λ)) = LJ (λ). By Steinberg’s tensor product theorem, LJ (λ) = (1) (1) LJ (λ0) ⊗ LJ (λ1) .Sinceλ1 =0(on J), LJ (λ1) has dimension at least two, and u/uJ acts trivially on it. So this produces at least a two-dimensional space of n u/u vectors in H (uJ ,k) J arising from LJ (λ) which produces “extra” cohomology. ⊂ Case 2: N := RadLJ φ(VJ (λ)) =0.ThenN RadLJ φ(S)and

u/uJ u/uJ n u/uJ 0 = N ⊂ φ(S) ⊂ H (uJ ,k) . n u/u Since by (5.1.3) all the “characteristic zero” cohomology in H (uJ ,k) J has al- u/uJ ⊂ ready been accounted for in HdLJ φ(S), the vectors in N RadLJ φ(S)must be “extra” cohomology in characteristic p.

5.2. Explicit extra cohomology. In this section we exhibit additional coho- • mology that arises in H (u,k)wherek = Fp in case Φ = An.

Theorem 5.2.1. Let p be prime and Φ be of type An where n = p +1.Then the vector p ∧ ∧ ∧···∧ ∧···∧ f−α0 γ1 γ2 γi γp i=1 2p−1 ∧ appears as extra cohomology in H (u,k),whereγi = f−(α1+···+αi) f−(αi+1+···+αn). Proof. p ∧ ∧ ∧···∧ ∧···∧ Let E = i=1 f−α0 γ1 γ2 γi γp. Consider the vector p−1 f− ∧ [d(f− )] := f− ∧ d(f− ) ∧ d(f− ) ∧···∧d(f− ), α0 α0 α0 α0 α0 α0 p−1 times ··· ∈ 2 ∗ with d(f−α0 )=γ1 + γ2 + + γp Λ (u ). First note by direct calculation one has γi ∧ γj = γj ∧ γi and γi ∧ γi = 0. We can now apply the multinomial theorem

1856 UNIVERSITY OF GEORGIA VIGRE ALGEBRA GROUP

m ··· m ≥ for [d(f−α0 )] =[γ1 + γ2 + + γp] for m 2: m m [d(f− )] =[γ + γ + ···+ γ ] α0 1 2 p m r1 r2 rp = [γ1] ∧ [γ2] ∧···∧[γp] r1,...,rp r1,...,rp p m m! where ri = m and = . Consider the case when m = p − 1. i=1 r1,...,rp r1!r2!...rp! r Since [γi] i =0forri ≥ 2, the only nonzero terms occur where ri =0forsomei, and rj =1forallj = i.Wehave p − p−1 p 1 [d(f− )] = (γ ∧···∧γ ∧···∧γ ) α0 0, 1,...,1 1 i p i=1 p =(p − 1)! (γ1 ∧···∧γi ∧···∧γp). i=1 ∧ p−1 − So we have that f−α0 [d(f−α0 )] =(p 1)! E. Since the terms in the above sum are linearly independent, this shows that E =0.Toprovethat E ∈ Ker d,welook ∧ p−1 at f−α0 [d(f−α0 )] .Sinced is a diﬀerential, d(d(f−α0 )) = 0. Also note we can p ∧ ∧···∧ ∧···∧ apply the multinomial theorem again to get [d(f−α0 )] = p!(γ1 γ2 γi γp). Consequently, p−1 p−1 p−1 d f−α ∧ [d(f−α )] = d(f−α ) ∧ [d(f−α )] − f−α ∧ d [d(f−α )] 0 0 0 0 0 0 p−1 p − ∧ ∧ p−2 =[d(f−α0 )] f−α0 d(d(f−α0 )) [d(f−α0 )] i=1

= p!(γ1 ∧ γ2 ∧···∧γp).

It now follows that 1 p−1 d(E)=d (f− ∧ [d(f− )] ) (p − 1)! α0 α0 p! = (γ ∧ γ ∧···∧γ ) (p − 1)! 1 2 p

= p (γ1 ∧ γ2 ∧···∧γp). Thus d(E) = 0 in characteristic p (but not in characteristic 0). We need to verify that E is not in the image of the previous diﬀerential. This 2p−2 ∗ will follow by demonstrating that Λ (u )−pα0 = 0 because the diﬀerentials re- 2p−2 ∗ spect weight spaces. Any weight in Λ (u )isoftheformβ1 + β2 + ···+ β2p−2 where the βi are distinct negative roots. Observe that β1 + β2 + ···+ β2p−2, αˇ0≥ −2p + 1. One can deduce this because for each i, βi, αˇ0 =0, ±1, ±2andis equal to −2 if and only if βi = −α0. On the other hand, −pα0, αˇ0 = −2p,thus 2p−2 ∗ Λ (u )−pα0 =0.

• 6. Examples for H (uJ , Fp) The following low rank examples were calculated using our computer package developed in MAGMA [BC, BCP]. Recall that the cohomology has a palindromic behavior so in the A4 table the degrees are only listed up to half the dimension of n n uJ .SetH =dimH (uJ ,k).

ON KOSTANT’S THEOREM FOR LIE ALGEBRA COHOMOLOGY 5719

Type A3, h − 1=3

J p H0 H1 H2 H3 H4 H5 H6 ∅ 0 1 3 5 6 5 3 1 2 1 3 6 8 6 3 1 {1} or {3} 0, 2 1 3 6 6 3 1 {2} 0, 2 1 4 5 5 4 1 {1, 3} or {2, 4} 0, 2 1 4 6 4 1 {1, 2} or {2, 3} 0, 2 1 3 3 1

Type A4, h − 1=4

J p H0 H1 H2 H3 H4 H5 H6 H7 ∅ 0 1 4 9 15 20 22 ...... 2 1 4 11 25 38 42 ...... 3 1 4 9 17 25 28 ...... {1} or {4} 0 1 4 10 19 26 ...... 2 1 4 12 25 32 ...... 3 1 4 10 20 27 ...... {2} or {3} 0 1 5 12 19 23 ...... 2 1 5 12 23 33 ...... 3 1 5 12 20 24 ...... {1, 3} or {2, 4} 0, 2, 3 1 6 13 23 30 ...... {1, 4} 0, 2, 3 1 4 14 25 28 ...... {1, 2} or {3, 4} 0, 3 1 4 12 18 ...... 2 1 4 12 19 ...... {2, 3} 0, 3 1 6 14 14 ...... 2 1 6 14 15 ...... {1, 3, 4} or {1, 2, 4} 0, 2, 3 1 6 15 20 ...... {1, 2, 3} or {2, 3, 4} 0, 2, 3 1 4 6 ......

2058 UNIVERSITY OF GEORGIA VIGRE ALGEBRA GROUP

Type G2, h − 1=5

J p H0 H1 H2 H3 H4 H5 H6 ∅ 0 1 2 2 2 2 2 1 2, 3 1 3 6 8 6 3 1 {1} 0, 2 1 4 5 5 4 1 3 1 4 7 7 4 1 {2} 0 1 2 3 3 2 1 2 1 3 6 6 3 1 3 1 4 7 7 4 1

7. Further Questions The results in the preceding sections and our low rank examples naturally suggest the following open questions which are worthy of further study.

(7.1) Let G be a simple algebraic group over Fp and g =LieG. Determine a maximal c(J, p) > 0 such that

n n ch H (uJ , C)=chH (uJ , Fp) for 0 ≤ n ≤ c(J, p).

(7.2) Let Φ = An with |∆| = n. • • a) Does |∆ − J| >pimply that ch H (uJ , C) =chH (uJ , Fp)? • • b) Does |∆ − J|

(7.3) Let G be a simple algebraic group over Fp and g =LieG. Assume that p is [p] a good prime. Let N1(g)={x ∈ g : x =0} (restricted nullcone). From work of Nakano, Parshall and Vella [NPV], there exists J ⊆ ∆ such that N1(g)=G · uJ (i.e., closure of a Richardson orbit).

Does there exist J ⊆ ∆withN1(g)=G · uJ such that • • ch H (uJ , C)=chH (uJ , Fp)?

(7.4) Let G be a simple algebraic group over Fp and g =LieG. Compute

n ch H (uJ , Fp) for all p. It would be even better to describe the LJ -module structure. Solving (7.4) would complete the analog of Kostant’s theorem for the trivial module for all characteristics. One might be able to use (7.3) as a stepping stone

ON KOSTANT’S THEOREM FOR LIE ALGEBRA COHOMOLOGY 5921 to perform this computation. Moreover, this calculation would have major implications in determining cohomology for Frobenius kernels and algebraic groups (cf. [BNP]).

8. VIGRE Algebra Group at the University of Georgia This project was initiated during Fall Semester 2006 under the Vertical Inte- gration of Research and Education (VIGRE) Program sponsored by the National Science Foundation (NSF) at the Department of Mathematics at the University of Georgia (UGA). We would like to acknowledge the NSF grant DMS-0089927 for its ﬁnancial support of this project. The VIGRE Algebra Group consists of 3 faculty members, 2 postdoctoral fellows, 8 graduate students, and 1 undergraduate. The group is led by Brian D. Boe, Leonard Chastkofsky and Daniel K. Nakano. The email addresses of the members of the group are given below. Faculty: Brian D. Boe [email protected] Leonard Chastkofsky [email protected] Daniel K. Nakano [email protected] Postdoctoral Fellows: Jonathan R. Kujawa [email protected] Emilie Wiesner [email protected] Graduate Students: Irfan Bagci [email protected] Benjamin Connell [email protected] Bobbe J. Cooper [email protected] Mee Seong Im [email protected] Wenjing Li [email protected] Kenyon J. Platt [email protected] Caroline B. Wright [email protected] Benjamin Wyser [email protected] Undergraduate Student: Tyler Kelly [email protected]

9. Acknowledgements We would like to thank the referee for providing useful suggestions.

References [BNP] C.P. Bendel, D.K. Nakano, C. Pillen, Second cohomology for Frobenius kernels and related structures, Advances in Math. 209 (2007), 162–197. [BC] W. Bosma, J. Cannon, Handbook on Magma Functions, Sydney University, 1996. [BCP] W. Bosma, J. Cannon, C. Playoust, The Magma Algebra System I: The User Language, J. Symbolic Computation 24 (1997), 235–265. [Cart] P. Cartier, Remarks on “Lie algebra cohomology and the generalized Borel-Weil theorem”, by B. Kostant, Ann. of Math. (2) 74 (1961), 388–390. [FP1] E.M. Friedlander and B.J. Parshall, Cohomology of inﬁnitesimal and discrete groups, Math. Ann. 273 (1986), no. 3, 353–374. [FP2] , Geometry of p-unipotent Lie algebras, J. Algebra 109 (1986), 25–45.

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[GW] R. Goodman, N. R. Wallach, Representations and Invariants of the Classical Groups, Encyclopedia of Mathematics and its Applications 68, Cambridge University Press, 1998. [Hum1] J.E. Humphreys, Introduction to Lie Algebras and Representation Theory, Springer- Verlag, New York, 1972. [Hum2] , Conjugacy Classes in Semisimple Algebraic Groups, Math. Surveys and Mono- graphs, AMS, vol. 43, 1995. [Jan] J.C. Jantzen, Representations of Algebraic Groups, Academic Press, 1987. [Kna] Anthony W. Knapp, Lie Groups, Lie Algebras, and Cohomology,MathematicalNotes 34, Princeton University Press, 1988. [NPV] D.K. Nakano, B.J. Parshall, D.C. Vella, Support varieties for algebraic groups, J. Reine Angew. Math. 547 (2002), 15–49. [PT] P. Polo and J. Tilouine, Bernstein-Gelfand-Gelfand complexes and cohomology of nilpotent groups over Z(p) for representations with p-small weights, Ast´erisque (2002), no. 280, 97–135, Cohomology of Siegel varieties. [Ro] A. Rocha-Caridi, Splitting criteria for g-modules induced form a parabolic and the Bernstein-Gelfand-Gelfand resolution of a ﬁnite dimensional irreducible g-module, Trans. AMS 262 (1980), 335–366. [R] J. Rotman, An Introduction to Homological Algebra, Academic Press, 1979.

Department of Mathematics, University of Georgia, Athens, Georgia 30602

Contemporary Mathematics Volume 478, 2009

G-stable pieces and partial ﬂag varieties

Xuhua He

Abstract. We will use the combinatorics of the G-stable pieces to describe the closure relation of the partition of partial ﬂag varieties in [L3, section 4].

Introduction In 1977, Lusztig introduced a finite partition of a (partial) flag variety Y . In the case where Y is the full flag variety, this partition is the partition into Deligne-Lusztig varieties (see [DL]). In this case, it follows easily from the Bruhat decomposition that the closure of a Deligne-Lusztig variety is the union of some other Deligne-Lusztig varieties and the closure relation is given by the Bruhat order on the Weyl group. In this paper, we will use some combinatorial technique in [H4] to study the partition on a partial flag variety. We show that the partition is a stratification and the closure relation is given by the partial order introduced in [H2,5.4]and[H3, 3.8 & 3.9]. We also study some other properties of the locally closed subvarieties that appear in the partition.

1. Some combinatorics

1.1. Let k be an algebraic closure of the finite field Fq and G be a connected reductive algebraic group defined over Fq with Frobenius map F : G → G.Wefix an F -stable Borel subgroup B of G and an F -stable maximal torus T ⊂ B.LetI be the set of simple roots determined by B and T .ThenF induces an automorphism on the Weyl group W whichwedeontebyδ. The autmorphism restricts to a bijection on the set I of simple roots. By abusion notations, we also denote the bijection by δ. For any J ⊂ I,letPJ be the standard parabolic subgroup corresponding to J and PJ be the set of parabolic subgroups that are G-conjugate to PJ .Wesimply write P∅ as B.LetLJ be the Levi subgroup of PJ that contains T . For any parabolic subgroup P ,letUP be the unipotent radical of P .Wesimply write U for UB.

2000 Mathematics Subject Classiﬁcation. 14M15, 20G40. The author is partially supported by NSF grant DMS-0700589.

611

262 XUHUA HE

For J ⊂ I,wedenotebyWJ the standard parabolic subgroup of W generated J J by J and by W (resp. W ) the set of minimal coset representatives in W/WJ J K K J (resp. WJ \W ). For J, K ⊂ I,wesimplywriteW ∩ W as W . J K For P ∈PJ and Q ∈PK ,wewritepos(P, Q)=w if w ∈ W and there exists −1 −1 −1 g ∈ G such that P = gPJ g , Q = gwP˙ K w˙ g ,where˙w is a representative of w in N(T ). For g ∈ G and H ⊂ G,wewritegH for gHg−1.

We ﬁrst recall some combinatorial results.

1.2. For J ⊂ I,letT (J, δ) be the set of sequences (Jn,wn)n≥0 such that (a) J0 = J, (b) Jn = Jn−1 ∩ Ad(wn−1)δ(Jn−1)forn ≥ 1, J δ(J ) (c) wn ∈ n W n for n ≥ 0, ∈ ≥ (d) wn WJn wn−1Wδ(Jn−1) for n 1. Then for any sequence (Jn,wn)n≥0 ∈T(J, δ), we have that wn = wn+1 = ··· ··· → −1 and Jn = Jn+1 = for n 0. By [Be], the assignment (Jn,wn)n≥0 wm for m 0 deﬁnes a bijection T (J, δ) → W J . Now we prove some result that will be used in the proof of Lemma 2.5.

Lemma 1.1. Let (Jn,wn)n≥0 ∈T(J, δ) be the element that corresponds to w. Then (1) w(L ∩ U )w−1 ⊂ U . J PJ1 Pδ(J) ∩ −1 ⊂ ∩ ≥ (2) w(LJ UP )w L − UP for i 1. i Ji+1 δ(Ji 1) δ(Ji) Proof. We only prove part (1). Part (2) can be proved in the same way. Assume that part (1) is not true. Then there exists α ∈ Φ+ − Φ+ such that J J1 ∈ + ∈ − ≤ ∈ J ∈ + wa Φδ(J).Leti J J1 with αi α.Sincew W ,wehavethatwαi Φδ(J). −1 ∈ ∈ −1 + + By deﬁnition, w = w1v for some v Wδ(J .Thenαi w Φδ(J) = w1vΦδ(J) = δ(J) w1Φδ(J).Sincew1 ∈ W ,wemusthaveαi = w1αj for some j ∈ δ(J). Hence i ∈ J1, which is a contradiction. Part (1) is proved. −1 J 1.3. Deﬁne a WJ -action on W by x · y = δ(x)yx .Forw ∈ W ,set I(J, δ; w)=max{K ⊂ J;Ad(w)(K)=δ(K)} and [w]J = WJ · (wWI(J,δ;w)). Then W = w∈W J [w]J .See[H4, Corollary 2.6]. sj Given w, w ∈ W and j ∈ J,wewritew −→δ w if w = sδ(j)wsj and l(w ) ≤ l(w). If w = w0,w1, ··· ,wn = w is a sequence of elements in W such that for all sj k,wehavewk−1 −→δ wk for some j ∈ J,thenwewritew →J,δ w . We call w, w ∈ W elementarily strongly (J, δ)-conjugate if l(w)=l(w)and −1 there exists x ∈ WJ such that w = δ(x)wx and either l(δ(x)w)=l(x)+l(w)or l(wx−1)=l(x)+l(w). We call w, w strongly (J, δ)-conjugate if there is a sequence w = w0,w1, ··· ,wn = w such that wi−1 is elementarily strongly (J, δ)-conjugate to wi for all i. We will write w ∼J,δ w if w and w are strongly (J, δ)-conjugate. If w ∼J,δ w and w →J,δ w ,thenwesaythatw and w are in the same (J, δ)-cyclic shift and write w ≈J,δ w . Then it is easy to see that w ≈J,δ w if and only if w →J,δ w and w →J,δ w. By [H4, Proposition 3.4], we have the following properties: ∈ ∈ J ∈ → (a) for any w W ,thereexistsw1 W and v WI(J,δ;w1) such that w J,δ w1v.

G-STABLE PIECES AND PARTIAL FLAG VARIETIES63 3

 (b) if w, w are in the same WJ -orbit O of W and w, w are of minimal length J in O,thenw ∼J,δ w .Ifmoreover,O∩W = ∅,thenw ≈J,δ w .

1.4. By [H4, Corollary 4.5], for any WJ -orbit O and v ∈O, the following conditions are equivalent: (1) v is a minimal element in O with respect to the restriction to O of the Bruhat order on W . (2) v is an element of minimal length in O. We denote by Omin the set of elements in O satisfy the above conditions. The J elements in (WJ · w)min for some w ∈ W are called distinguished elements (with respect to J and δ). J As in [H4, 4.7], we have a natural partial order ≤J,δ on W deﬁned as follows: J Let w, w ∈ W .Thenw ≤J,δ w if for some (or equivalently, any) v ∈ (WJ · w )min,thereexistsv ∈ (WJ · w)min such that v ≤ v . J In general, for w ∈ W and w ∈ W ,wewritew ≤J,δ w if there exists v ∈ (WJ · w)min such that v ≤ w .

2. GF -stable pieces

2.1. For J ⊂ I,setZJ = {(P, gUP ); P ∈PJ ,g ∈ G} with the G × G-action deﬁned by − · g2 1 (g1,g2) (P, gUP )=( P, g1gUP g2 ). { ∈ Set hJ =(PJ ,UPJ ). Then the isotropic subgroup RJ of hJ is (lu1,lu2); l ∈ } LJ ,u1,u2 UPJ . It is easy to see that ∼ ZJ = (G × G)/RJ . J Set GF = {(g, F(g)); g ∈ G}⊂G × G.Forw ∈ W ,set

ZJ,F ;w = GF (B,BwB) · hJ .

We call ZJ,F ;w a GF -stable piece of ZJ . Lemma 2.1. Let w, w ∈ W . (1) If w →J,δ w ,then GF (B,BwB) · hJ ⊂ GF (B,Bw B) · hJ ∪∪v

sj Proof. It suﬃces to prove the case where w −→δ w for some j ∈ J. Notice that F (BsiB)=Bsδ(i)B for i ∈ I. If wsj

(B,BwB) · hJ =(B,BwsjB)(B,BsjB) · hJ

=(BsjB,Bwsj B) · hJ

⊂ GF (B,Bsδ(j)BwsjB,B) · hJ ⊂ GF (B,Bw B) · hJ ∪ GF (B,BwsjB) · hJ . If moreover, l(w )=l(w), then Bsδ(j)Bwsj B = Bw B and GF (B,BwB)·hJ = GF (B,Bw B) · hJ .

464 XUHUA HE

If sδ(j)w

(B,BwB) · hJ =(B,Bsδ(j)B)(B,Bsδ(j)wB) · hJ

⊂ GF (BsjB,Bsδ(j)wB) · hJ

= GF (B,Bsδ(j)wBsj B) · hJ ⊂ GF (B,Bw B) · hJ ∪ GF (B,Bsδ(j)wB) · hJ . If moreover, l(w )=l(w), then Bsδ(j)wBsj B = Bw B and GF (B,BwB)·hJ = GF (B,Bw B) · hJ . If wsδ(j) >wand sjw>w,thenl(w )=l(w). By [L1, Proposition 1.10], w = w. The statements automatically hold in this case.

Lemma 2.2. We have that ZJ = ∪w∈W J ZJ,F ;w.

Remark. We will see in subsection 2.3 that ZJ is the disjoint union of ZJ,F ;w for w ∈ W J .

Proof. Let z ∈ ZJ .SinceG × G acts transitively on ZJ , z is contained in the G-orbit of an element (1,g) · hJ for some g ∈ G. By the Bruhat decomposition of G,wehavethatz ∈ GF (1,Bw1B) · hJ for some w1 ∈ W . We may assume furthermore that w1 is of minimal length among all the Weyl group elements w1 ∈ · with z GF (1,Bw1B) hJ . By part (1) of the previous lemma and 1.3 (a), ∈ · ∪∪ · z GF (B,BwvB) hJ l(w )

Q 2.2. For any parabolic subgroups P and Q of G,wesetP =(P ∩ Q)UP .It is known that P Q is a parabolic subgroup of G. The following properties are easy to check. g (1) For any g ∈ G,(gP )( Q) = g(P Q). (2) If P ∈B,thenP Q = P for any parabolic subgroup Q.

J Lemma 2.3. Let J, K ⊂ I and w ∈ W .SetJ1 = J ∩ Ad(w1)K,where g ∈ ( PK ) w1 =min(wWK ). Then for g BwB, we have that PJ = PJ1 . Proof. By 2.2 (1), it suﬃces to prove the case where g =˙w.Now

w˙ w˙ 1 ( PK ) ( PK ) ∩ w˙ 1 ∩ w˙ 1 ∩ w˙ 1 (PJ ) =(PJ ) =(LJ LK )(LJ UPK )(UPJ PK )UPJ ∩ w˙ 1 ∩ ∩ w˙ 1 ∩ w˙ 1 ∩ w˙ 1 =(LJ LK )((LJ U) LK )(LJ UPK )(UPJ PK )UPJ .

G-STABLE PIECES AND PARTIAL FLAG VARIETIES65 5

J J K Since w ∈ W and w1 =min(wWK ), we have that w1 ∈ W . Therefore ∩ w˙ 1 LJ LK = LJ1 and

∩ ∩ w˙ 1 ∩ w˙ 1 ((LJ U) LK )(LJ UPK ) ∩ ∩ w˙ 1 ∩ ∩ w˙ 1 =((LJ U) LK )((LJ U) UPK )

w˙ 1 =(LJ ∩ U) ∩ PK = LJ ∩ U.

w˙ ( PK ) ∩ So (PJ ) = LJ1 (LJ U)UPJ = LJ1 U = PJ1 .

n Lemma 2.4. To each (P, gUP ) ∈ ZJ , we associate a sequence (P ,Jn,wn)n≥0 as follows

g n−1 P 0 = P, P n =(P n−1)F ( P ) for n ≥ 1, ⊂ n ∈P n g n ≥ Jn I with P Jn ,wn =pos(P ,F( P )) for n 0.

J n Let w ∈ W .Let(P, gUP ) ∈ ZJ,F ;w and (P ,Jn,wn)n≥0 be the sequence ∈T −1 associated to (P, gUP ).Then(Jn,wn)n≥0 (J, δ) and wm = w for m 0. Proof. Using 2.2 (1), it is easy to see by induction on n that the sequence F (h) −1 F (h) n associated to ( P, hgUP F (h) )is( P ,Jn,wn)n≥0. Then it suﬃces to prove ∈ −1 −1 the case where (P, gUP )=(PJ ,kUPJ )forsomek Bδ (w) B. ∈T Let (Jn,wn)n≥0 (J, δ) be the element that corresponds to w.Thenwn = −1 ≥ min(w Wδ(Jn))forn 0. By the previous lemma, we can show by induction n n P P n ≥ J J n ≥ w on that = Jn for all 0. Then n = n for 0. Moreover, n = n k n F (k) ∈ −1 −1 pos(P ,F( P )) = pos(PJn , Pδ(Jn))=wn since k Bδ (w) B.

(A similar result with a similar proof appears in [H1, Lemma 2.3].)

→ J −1 2.3. We can now deﬁne a map β : ZJ W by β(P, gUP )=wm for n m 0, where (P ,Jn,wn)n≥0 is the sequence associated to (P, gUP ). Then −1 ZJ = w∈W J β (w) is a partition of ZJ into locally closed subvarieties. Since −1 −1 ZJ,F ;w ⊂ β (w)andZJ = ∪w∈W J ZJ,F ;w,wehavethatZJ,F ;w = β (w)and

ZJ = w∈W J ZJ,F ;w.

J Fix w ∈ W and let (Jn,wn)n≥0 be the element in T (J, δ) that corresponds m to w. Clearly, the map (P, gUP ) → P for m 0 is a morphism ϑ : ZJ,F ;w → PI(J,δ;w). Lemma 2.5. Let w ∈ W J .Setx = δ−1(w)−1 and K = δ−1I(J, δ; w).Then · · (UPK )F (x, 1) hJ =(UPK x, UPδ(K) ) hJ . Proof. Notice that · · (UPK x, UPδ(K) ) hJ =(UPK )F (x, UPδ(K) ) hJ ∩ · =(UPK )F (x, UPδ(K) LJ ) hJ ∩ · =(UPK )F (x(UPδ(K) LJ ), 1) hJ .

∈ ∩ ∈ ∩ −1 So it suﬃces to show that for any v UPδ(K) LJ ,thereexistsu UPK Lδ (J) −1 −1 ∈ such that x uxF (u) vUPJ .

666 XUHUA HE

Let (Jn,wn)n≥0 ∈T(J, δ) be the element that corresponds to w. By Lemma 1.1, −1 −1 ∩ ⊂ x (Lδ (J) UP −1 )x UPJ , δ (J1) −1 −1 ∩ ⊂ ∩ ≥ x (Lδ (Ji) UP −1 )x LJi−1 UPJ for i 1. δ (Ji+1) i

We have that δ(K)=Jm for some m ∈ N.Nowv = vmvm−1 ···v0 for some ∈ ∩ ∈ −1 ∩ vi LJi UPJ . We deﬁne ui Lδ (Ji) UP −1 as follows: i+1 δ (Ji+1) ∈ −1 ∩ Let um = 1. Assume that k

Let uk be the element with −1 −1 F (uk) =(x (umum−1 ···uk+1)x)vmvm−1 ···vkF (umum−1 ···uk+1) −1 −1 =(x uk+1x)F (umum−1 ···uk+1) vkF (umum−1 ···uk+1) ∈ L ∩ U . JK PJk+1

∈ −1 ∩ Thus uk+1 Lδ (Jk) UP −1 and that δ (Jk+1) −1 −1 (x (umum−1 ···uk+1)x)F (umum−1 ···uk) = vmvm−1 ···vk. This completes the inductive deﬁnition. Now set u = umum−1 ···u0.Then (x−1ux)F (u)−1 −1 −1 −1 −1 =(x (umum−1 ···u1)x)F (u) (F (u)(x u0x)F (u) ) ∈ vUPδ(J) . The lemma is proved. By the proof of Lemma 2.2, Z = G (U δ−1(w)−1U , 1) · h . J,F ;w F Pδ−1I(J,δ;w) Pδ(I(J,δ;w)) J Then we have the following consequence.

J Corollary 2.6. Let w ∈ W .ThenGF acts transitively on ZJ,F ;w.

Remark. Therefore there are only ﬁnitely many GF -orbits on ZJ and they J are indexed by W . This is quite diﬀerent from the set of G∆-orbits on ZJ . Proposition 2.7. Let w ∈ W .Then

· J GF (B,Bw) hJ = w ∈W ,w ≤J,δ wZJ,F ;w . Remark. Similar results appear in [H3, Proposition 4.6], [H2, Corollary 5.5] and [H4, Proposition 5.8]. The following proof is similar to the proof of [H4, Proposition 5.8]. Proof. We prove by induction on l(w). × → → · Using the proper map p : GF BF ZJ ZJ deﬁned by (g, z) (g, F(g)) z, one can prove that

GF (B,Bw) · hJ = GF (B,Bw) · hJ = ∪v≤wGF (B,Bv) · hJ .

G-STABLE PIECES AND PARTIAL FLAG VARIETIES67 7

J By 1.3 (a), w →J,δ w1v for some w1 ∈ W and v ∈ WI(J,δ;w). By Lemma 2.1, GF (B,Bw) · hJ = GF (B,Bw) · hJ ∪∪w

By the proof of Lemma 2.2, GF (B,Bw1v) · hJ ⊂ GF (B,Bw1) · hJ .Thusby induction hypothesis,

· ⊂∪ J GF (B,Bw) hJ w ∈W ,w ≤J,δ wZJ,F ;w . J On the other hand, if w ∈ W with w ≤J,δ w, then there exists w ≈J,δ w with w ≤ w. Then by Lemma 2.1, ZJ,F ;w = GF (B,Bw ) · hJ ⊂ GF (B,Bw) · hJ .

· ∪ J ∩ Therefore GF (Bw,B) hJ = w ∈W ,w ≤J,δ wZJ,F ;w .By2.3,ZJ,F ;w1 ZJ,F ;w2 = J ∅ ∈ · J if w1,w2 W and w1 = w2.ThusGF (Bw,B) hJ = w ∈W ,w ≤J,δ wZJ,F ;w . The proposition is proved.

3. A stratiﬁcation of partial ﬂag varieties

3.1. It is easy to see that there is a canonical bijection between the GF - orbits on ZJ and the RJ -orbits on (G × G)/GF which sends GF (1,w) · hJ to −1 −1 RJ (1,w )GF /GF . Notice that the map (g1,g2) → g2F (g1) gives an isomor- ∼ phism of RJ -varieties (G × G)/GF = G,wheretheRJ -action on G is deﬁned by −1 (lu1,lu2) · g = lu2gF(lu1) .

Using the results of GF -orbits on ZJ above, we have the following results. J (1) For w ∈ W , RJ · w = RJ · (BwB). If moreover, w ∈ (WJ · w)min,then RJ · (Bw B)=RJ · (BwB). (2) G = w∈J W RJ · w. ∈ · J −1 −1 · (3) For w W , RJ w = w ∈ W,(w ) ≤J,δ w RJ w . Notice that if J = ∅, part (2) above follows easily from Bruhat decomposition. One may regard (2) as an extension of Bruhat’s Lemma. We will also discuss a variation of (2) in section 4.

3.2. Now we review the partition on PJ introduced by Lusztig in [L3, section 4]. n To each P ∈PJ , we associate a sequence (P ,Jn,wn)n≥0 as follows n−1 P 0 = P, P n =(P n−1)F (P ) for n ≥ 1, ⊂ n ∈P n n ≥ Jn I with P Jn ,wn =pos(P ,F(P )) for n 0. J By [L3, 4.2], (Jn,wn)n≥0 ∈T(J). Thus we have a map i : PJ → W .For w ∈ J W ,let PJ,w = {P ∈PJ ; wm = w for m 0}.

Then PJ = w∈J W PJ,w . It is easy to see that PJ,w = {P ∈PJ ;(P, UP ) ∈ ZJ,F ;w−1 }. Notice that Lie (G∆)+Lie(GF )=Lie(G) ⊕ Lie (G). Then for any x ∈ ZJ , G∆ ·x and GF ·x intersects transversally at x.Inparticular,PJ,w is the transversal intersection of G∆ · hJ and ZJ,F ;w−1 . We simply write P∅,w as Bw. By 3.2 (3), g −1 Bw = {B1 ∈B;pos(B1,F(B1)) = w} = { B; g F (g) ∈ BwB˙ }.

868 XUHUA HE

Since the Lang isogeny g−1F (g) is an isomorphism GF \G → G,wehavethat

(a) Bw = v≤wBv.

Now we can prove our main theorem.

Theorem 3.1. Let p : B→PJ be the morphism which sends a Borel subgroup B to the unique parabolic subgroup in PJ that contains B .Then J (1) For w ∈ W , p(Bw)=PJ,w . If moreover, v ∈ (WJ · w)min,thenp(Bv)= p(Bw)=PJ,w .

∈ P B J −1 −1 P (2) For w W , J,w = p( w)= w ∈ W,(w ) ≤J,δ w J,w .

Remark. The closure relation of PJ,w was conjectured by G. Lusztig in private conversation. J g −1 Proof. (1) Let w ∈ W and g ∈ G with B ∈Bw.Theng F (g) ∈ BwB. Thus g · −1 · ( PJ ,UPJ )=(g, g) hJ =(g, F(g))(1,F(g) g) hJ −1 ∈ GF (B,Bw ) · hJ = ZJ,F ;w−1 and p(Bw) ⊂PJ,w . −1 By 3.1, for any g ∈ G,thereexistsl ∈ LJ such that (gl) F (gl) ∈ Bw˙ B for some w ∈ J W . Hence g g (a) p(B)=∪g∈G p( B)=∪w∈J W ∪g−1F (g)∈Bw˙ B p( B)

= ∪w∈J W p(Bw ) ⊂w∈J W PJ,w = PJ .

Since p is proper, we have that p(B)=PJ . Thus the inequality in (a) is actually J an equality and p(Bw )=PJ,w for all w ∈ W . If moreover, v ∈ (WJ · w)min, then by 3.1, there exists l ∈ LJ such that −1 −1 −1 g g gl (gl) F (gl)=l g F (g)F (l) ∈ BwB˙ .Thusp( B)= PJ = PJ ∈ p(Bw)and p(Bv) ⊂ p(Bw). Similarly, we have that p(Bw) ⊂ p(Bv). Then p(Bv)=p(Bw). Part (1) is proved. (2) Since p is proper, we have that PJ,w = p(Bw). By 3.2 (a), Bw = v≤wBv B ∪ B ∪ g and p( w)= v≤wp( v)= g∈G,g−1F (g)∈BvB˙ p( B). By 3.1 (3), g B ∪ J −1 −1 ∪ −1 p( w)= w ∈ W,(w ) ≤J,δ w g F (g)∈Bw˙ B p( B) ∪ J −1 −1 B = w ∈ W,(w ) ≤J,δ w p( w)

∪ J −1 −1 P = w ∈ W,(w ) ≤J,δ w J,w . Part (2) is proved.

Let us discuss some other properties of PJ,w .

Proposition 3.2. Assume that G is quasi-simple and J = I.ThenPJ,w is irreducible if and only if suppδ(w)=I.

Proof. By [L3, 4.2 (d)], PJ,w is isomorphic to PK,w,whereK = I(J, δ; w). By [BR, Theorem 2], PK,w is irreducible if and only if wWK is not contained in WJ for any δ-stable proper subset J of I. Let J be the minimal δ-stable subset of I with wWK ⊂ WJ .Itiseasytosee that if suppδ(w)=I,thenJ = I. On the other hand, suppose that suppδ(w) = I ∈ − ∈ and J = I. Then for any i K suppδ(w), we have that wαi δ(K). Since

G-STABLE PIECES AND PARTIAL FLAG VARIETIES69 9 ∈ Z ∈ wαi αi + j∈supp(w) αj, we must have that wαi = αi and i δ(K). In − ∈ − particular, K suppδ(w)isδ-stable, wαi = αi for all i K suppδ(w)and − − ∈ − K suppδ(w)=I suppδ(w). Since G is quasi-simple, there exists i K suppδ(w) ∨ ∈ ··· such that (αi,αj ) < 0forsomej supp(w). Now assume that w = sj1 sj2 sjm { ∨ } ··· is a reduced expression and m =max n;(αi,αj ) =0 .Thensj1 sj2 sjm αi = ··· n sj1 sj2 sjm αi = αi.Thus ∨ ∨ ∨ 0 > (α ,α )=(s ···s α ,s ···s α )=(α ,s ···s α ). i jm j1 jm i j1 jm jm i j1 jm jm ∨ However, sj ···sj α is a negative coroot. Thus 1 m jm ∨ (α ,s ···s α ) ≥ 0, i j1 jm jm which is a contradiction. Therefore if suppδ(w) = I,thenJ = I. The proposition is proved.

3.3. By [L3, 4.2 (d)], PJ,w is isomorphic to PK,w,whereK = I(J, δ; w). Similar to [DL, 1.11], we have that

−1 PK,w = {g ∈ G; g F (g) ∈ PK wP˙ K }/PK −1 w˙ = {g ∈ G; g F (g) ∈ wP˙ K }/PK ∩ PK { ∈ −1 ∈ } Ad(w ˙ )◦F ∩ w˙ = g G; g F (g) wU˙ PK /LK (UPK UPK ).

Let P ∈PK,w such that there exists a F -stable Levi subgroup L of P .Then similar to [DL, 1.17], we have that

−1 PK,w = {g ∈ G; g F (g) ∈ PF(P )}/P = {g ∈ G; g−1F (G) ∈ F (P )}/P ∩ F (P ) −1 F = {g ∈ G; g F (g) ∈ F (UP )}/L (UP ∩ F (UP )).

4. An extension of Bruhat decomposition After the paper was submitted, I learned from A. Vasiu about his conjecture in [Va, 2.2.1]. We state it in the following slightly stronger version.

Corollary 4.1. Let P be a parabolic subgroup of G of type J with a Levi subgroup L.LetR = {(lu, lu ); l ∈ L, u, u ∈ UP } and deﬁne the action of R on G by (lu, lu) · g = lugF(lu)−1.Then (1) There are only ﬁnitely many R-orbits on G, indexed by J W . (2) If moreover, there exists a maximal torus T ⊂ P such that F (T )=T , then each R-orbit contains an element in NG(T ).

g g J Proof. We may assume that P = PJ and L = LJ . For any w ∈ W ,set ∗ −1 ∗ −1 w = gwF(g) .ThenitistoseethatR · w = g(RJ · w)F (g) .Nowpart(1) follows from 3.1 (2). g −1 If moreover, T = T ⊂ P is F -stable, then we have that g F (g) ∈ NG(T ). ∗ −1 −1 −1 −1 ∗ Thus w = gwF(g) = g(wF(g) g)g and wF(g) g ∈ NG(T ). So w ∈ NG(T ) and part (2) is proved.

1070 XUHUA HE

Acknowledgements We thank G. Lusztig for suggesting the problem and some helpful discussions. We also thank Z. Lin for explaining to me a conjecture by A. Vasiu and some helpful discussions. After the paper was submitted, I learned from T. A. Springer that he also obtained some similar results about the GF -stable pieces in a diﬀerent way using the approach in [Sp].

References [Be] R. Bédard, On the Brauer liftings for modular representations, J. Algebra 93 (1985), no. 2, 332-353. [BR] C. Bonnafé and R. Rouquier, On the irreducibility of Deligne-Lusztig varieties,Comptes Rendus Math. Acad. Sci. Paris, 343 (2006), 37–39. [DL] P. Deligne and G. Lusztig, Representations of reductive groups over finite fields, Ann. of Math. (2) 103 (1976), no. 1, 103–161. [H1] X. He, Unipotent variety in the group compactification, Adv. in Math. 203 (2006), 109-131. [H2] X. He, The character sheaves on the group compactification, Adv. in Math., 207 (2006), 805-827. [H3] X. He, The G-stable pieces of the wonderful compactification, Trans. Amer. Math. Soc., 359 (2007), 3005-3024. [H4] X. He, Minimal length elements in some double cosets of Coxeter groups, Adv. in Math. 215 (2007), 469-503. [L1] G. Lusztig, Hecke algebras with unequal parameters, CRM Monograph Series 18, American Mathematical Society, 2003. [L2] G. Lusztig, Parabolic character sheaves,I,II,Mosc.Math.J.4 (2004), no. 1, 153–179; no. 4, 869–896. [L3] G. Lusztig, A class of perverse sheaves on a partial flag manifold, Represent. Theory 11 (2007), 122-171. [Sp] T. A. Springer, An extension of Bruhat’s Lemma,J.Alg.313 (2007), 417–427. [Va] A. Vasiu, Mod p classification of Shimura F-crystals, arXiv:math/0304030.

Department of Mathematics, Stony Brook University, Stony Brook, NY 11794 E-mail address: [email protected]

Contemporary Mathematics Volume 478, 200 9

Steinberg representations and duality properties of arithmetic groups, mapping class groups, and outer automorphism groups of free groups

Lizhen Ji

Abstract. An important cohomological property of a countable group Γ concerns its duality property of its homology and cohomology groups with coefficients. In this paper, we review the duality property of arithmetic subgroups of linear algebraic groups and the identification of the dualizing module with the Steinberg representation (or module) of the arithmetic subgroups, discuss duality property of the mapping class groups of surfaces (for example, they are duality groups but not Poincaré duality groups) and also the identification of the dualizing module with a generalized Steinberg module, and prove that the outer automorphism group Out(Fn) of free groups Fn is also a duality group but not a Poincaré duality group. Based on the discussions in this paper, we also propose a conjecture that the outer automorphism group of a duality group is a virtual duality group. Hope that the discussions in this paper will add to the list of amazing applications to many different fields of the Steinberg representations.

Contents

1. Introduction 72 2. Poincaré duality of closed manifolds and generalizations 74 3. Homology and cohomology of groups 76 4. Duality groups, Poincaré duality groups 87 5. Algebraic groups, Tits building, Steinberg representations, arithmetic subgroups 81 6. Borel-Serre compactification and duality properties of arithmetic subgroups 83 7. Mapping class groups and Teichmüller spaces 89 8. Curve complex, duality properties of mapping class groups 90 9. Outer automorphism groups of free groups, outer space 92 10. Duality properties of outer automorphism groups 94

1991 Mathematics Subject Classification. Primary 57P10, 11F75, 30F60, 20F65. Key words and phrases. Steinberg representation, arithmetic group, Tits building, mapping class group, Teichmüller space, outer automorphism group, duality group, Poincaré duality. Partially supported by NSF grant DMS 0604878.

c 2009 Americanc 0000 Mathematical (copyright Societyholder) 711

2L72 IZHENJI

11. Comments and conjectures 95 References 97

1. Introduction There are many similarities between finite groups of Lie type (or finite Chevalley groups) and arithmetic subgroups of linear semisimple algebraic groups defined over Q (or over other global fields). For example, they are both naturally associated with algebraic groups over Z as in the case of SL(n, Z/pZ) (a finite group) and SL(n, Z) (an arithmetic subgroup of SL(n, Q)) arising from the algebraic group SL(n)={g ∈ GL(n, C) | det g =1}, which is defined over Z. In the study of representation theory of finite groups of Lie type, the Steinberg representation and its variants have played a fundamental role. In fact, it has been used crucially in both the ordinary representations over C and modular representations of such finite groups, representations of algebraic groups over algebraically closed fields of positive characteristic, and representations of Lie algebras, and determination of the number of unipotent elements of finite Chevalley groups. See the comments and notes [St, pp. 580-586], the survey paper [Hu] and the book [Ca] for extended discussions about the Steinberg representations and their applications mentioned above. See also the book [Be] for the generalized Steinberg module for finites groups at primes p which divide the order of the groups, which is the Lefschetz module of the canonical action of the groups on the simplicial complex associated with the poset of p-subgroups. There are several constructions of the Steinberg representation for finite Cheval- ley groups. One method is to obtain it as the top dimensional non-vanishing homology group of the Tits building of the associated algebraic group (or a BN-pair) in [So]. This method is the most useful one for the purpose of defining the generalized Steinberg modules (or representations) and for studying duality properties of groups considered in this paper. One reason is that the geometry at infinity of Lie groups and symmetric spaces is often described by the Tits building (or rather parabolic subgroups) of the Lie groups. In fact, in the study of cohomology groups of arithmetic subgroups Γ of linear semisimple algebraic groups G defined over the field Q of rational numbers (or more generally over number fields), the Tits building ∆Q(G)ofG also occurs naturally and an analogue of the Steinberg representation (or module) of G(Q)wasused by Borel and Serre [BS1] to show that arithmetic subgroups Γ are virtual duality groups, but not virtual Poincaréduality groups if they are non-uniform, i.e., the quotient Γ\G(R) is non-compact, which is equivalent to that the the Q-rank r of G is positive (Theorem 6.1). In fact, the symmetric space X associated with its BS real locus G = G(R) admits the Borel-Serre partial compactification X whose boundary is homotopy equivalent to the Tits building ∆Q(G). Various cohomological finiteness properties of torsion-free arithmetic subgroups Γ can be derived from BS the action of Γ on X and X , and the duality properties of Γ follow from the BS Poincaré duality for manifolds with boundary applied to X and the Solomon- Tits theorem on the homotopy type of ∆Q(G) (or rather the induced Steinberg representation of Γ on Hr−1(∆Q(G)). A particularly important connection with

STEINBERG REPRESENTATIONS AND DUALITY PROPERTIES OF GROUPS73 3 the previous case of finite Chevalley groups is that the dualizing module of arithmetic subgroups is equal to the generalized Steinberg module. In this paper, we also give a proof that non-uniform arithmetic subgroups are not Poincaré duality groups without using the Borel-Serre compactification and related results on Tits building, in the case the symmetric space X = G(R)/K associated with G is linear, i.e., a homothety section of a symmetric cone. A natural generalization of arithmetic subgroups is the class of S-arithmetic 1 1 subgroups such as SL(n, Z[ , ··· , ]), where p1, ··· ,pm are prime numbers. The p1 pm same results were proved for S-arithmetic subgroups of G(Q) in [BS2] (see Theorem Q 6.6 below). For this purpose, we need to use Bruhat-Tits buildings of G( pi )as Q replacement of symmetric spaces for the group G( pi ) to act on properly. A similar result holds for S-arithmetic subgroups of reductive algebraic groups of rank 0 over function fields (Theorem 6.7). Another important class of groups closely related to the class of arithmetic subgroups consists of the mapping class group Γg,p of surfaces of genus g with p punctures. In [Ha1] (see also [Ha2] [I2]), Harer proved that Γg,p is a virtual duality group. In a joint work with Ivanov [IJ], it was proved that Γg,p is not a virtual Poincaré duality group. In this case, the dualizing module is also given by a generalized Steinberg module, which is the homology group of a natural simplicial complex, the curve complex C(Sg,p). In the proofs, actions of Γg,p on the Teichmüller space BS Tg,p, the Borel-Serre partial compactification Tg,p and the curve complex C(Sg,p), and a weak analogue of the Solomon-Tits theorem on the homotopy type of C(Sg,p) (Theorem 8.4) are all used. Once the non-Poincaré duality of Γg,p is proved, a stronger analogue of the Solomon-Tits theorem on the homotopy type of C(Sg,p) can be derived (Theorem 8.5). Yet another class of groups closely related to arithmetic groups consists of the outer automorphism group Out(Fn) of free groups Fn on n generators, n ≥ 2. The analogue of the symmetric spaces and Teichmüller spaces for Out(Fn)isthe outer space Xn consisting of suitable marked metric graphs where Out(Fn)acts by changing the marking, introduced by Culler and Vogtmann in [CV]. In [BF], Bestvina and Feighn proved that Out(Fn) is a virtual duality group (Theorem 10.1) by using an analogue of the Borel-Serre partial compactification of Xn.In this paper, we show that Out(Fn) is not a virtual Poincaré duality group (Theorem 10.2). In this case, the dualizing module has not been identified, though there is some candidate for an analogue of the Steinberg module [HV]. The plan of the rest of this paper is as follows. In §2, we explain different Poincaré dualities of homology and cohomology groups of various kinds of manifolds: compact closed orientable manifolds, noncompact manifolds, and non- orientable manifolds, and also homology and cohomology with local coefficient systems. In §3, we introduce the homology and cohomology of groups with coefficients, and establish their relations with homology and cohomology with local coefficient systems of manifolds. In §4, motivated by the Poincaré dualities of manifolds, we define duality groups and Poincaré duality groups. We also discuss several criterions for duality groups and Poincaré duality groups. In §5, we introduce the spherical Tits building of a reductive algebraic group G over the field Q, the Solomon-Tits theorem, and the Steinberg module of G(Q). We also remark that similar constructions and results hold for reductive algebraic groups over general fields k,in particular, finite fields and p-adic fields. In §6, we construct a finite CW-complex

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BΓ-space for arithmetic subgroups Γ, explain the relation between its boundary and the spherical Tits building, and show that arithmetic subgroups are virtual duality groups whose dualizing module is given by the Steinberg module. We also prove that non-uniform arithmetic subgroups acting on linear symmetric spaces are not virtual Poincaré duality groups without using the Borel-Serre compactification and the Tits building. We conclude this section by explaining that S-arithmetic subgroups are also duality groups but not Poincaré duality groups in general. In §7, we introduce mapping class groups Mod(S)ofasurfaceS and the Teichmüller space T (S) associated with S, and a realization of the Borel-Serre type compactification of T (S) in terms of truncated submanifolds. In §8, we show that the curve complex C(S) plays the same role in describing the geometry at infinity of Teichmüllerspace as the Tits building in describing that of symmetric spaces, and explain how it can be used to prove that Mod(S) is a duality group, and the non-Poincaré duality of Mod(S) implies a stronger analogue of the Solomon-Tits theorem for C(S). In §9, we introduce the outer automorphism group Out(Fn) and the associated outer space Xn.In§10, we first recall the result of Bestvina-Feighn that Out(Fn)isa virtual duality group, and then prove that Out(Fn) is not a virtual Poincaré duality group. In §11, we raise a conjecture that if Γ is a duality group, then Out(Γ) is a virtual duality group and discuss cases where this conjecture is true. By studying side-by-side duality properties of these three classes of groups: arithmetic groups, mapping class groups, and outer automorphism groups, we hope that we could bring out further their similarities and emphasize the role of the generalized Steinberg modules in studying them. Acknowledgments. I would like to thank K.Brown and M.Goresky for very helpful and informative correspondences about duality properties of homology and cohomology groups, and references. This paper is based on a talk at The 4th International Conference on Representation Theory held in Lhasa, Tibet, from July 16–20, 2007. I would like to thank the organizers for providing a stimulating environment and their hospitality during the conference. I would also like to thank an anonymous referee for reading the paper carefully and for simplifying the proof of Theorem 10.2.

2. Poincaré duality of closed manifolds and generalizations In this section, we introduce many kinds of Poincaré dualities for manifolds, which could be noncompact, have nonempty boundary, or could be non-orientable. These versions will be needed in studying homology of groups and their duality properties in later sections. Besides the first standard version of the Poincaré duality, other versions do not seem to be so well-known to non-experts. In this paper, all manifolds are connected. The basic Poincaré duality for oriented manifolds. An important property of the cohomology and homology groups of a compact oriented manifold M n is the well-known Poincaréduality: for every i =0, ··· ,n, there is a canonical isomorphism

i ∼ (2.1) H (M,Z) = Hn−i(M,Z), which is obtained by the cap product with the fundamental class [M]ofM.

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As a consequence, the Betti numbers bi of M, which is equal to the rank of Hi(M,Z), or the dimension of the real vector space Hi(M,R), satisfy the equality: for every i, bi = bn−i. This is the original form of the duality as stated by Poincaré. Another form of the Poincaré duality states that there is a well-defined intersection pairing

∼ (2.2) Hi(M,Z) × Hn−i(M,Z) → H0(M,Z) = Z and that the pairing becomes nondegenerate when it is tensored with Q: ∼ (2.3) Hi(M,Q) × Hn−i(M,Q) → H0(M,Q) = Q. (Note that the tensor product with Q is important. Otherwise, the pairing in Equation (2.2) is not nondegenerate in general if the homology groups contain torsion elements.) Poincaré-Lefschetz duality for manifolds with boundary. The Poincaré duality in Equation 2.1 does not hold for noncompact manifolds or manifolds with nonempty boundary. On the other hand, a modified version holds. Suppose that M n is an oriented manifold with possible nonempty boundary ∂M. M is not necessarily compact. i Z Let Hc(M, ) be the cohomology group of X with compact support. Then the Poincarédualityfor noncompact M is the following isomorphism: for every i ≤ n, i Z ∼ Z (2.4) Hc(M, ) = Hn−i(M,∂M, ). This duality is often called the Poincaré-Lefschetz duality. See [Ma, p. 343] [BS1, p. p. 473] [Br1, p. 211] for some discussions and references for this version of Poincaréduality. For later applications, we note that this Poincaré duality also holds for manifolds with corners, since the corners of the manifold can be smoothed out without changing its topology. We will apply it to EΓ, the universal covering space of a classifying space BΓofΓwhenBΓ is a compact manifold with boundary or corners. Poincaré duality for closed non-orientable manifolds. If M n is a non-orientable closed manifold, then the Poincaré duality for M is usually stated in terms of homology and cohomology groups with the coefficient Z/2Z. For example, if M n is a closed connected non-orientable manifold, then for every i =0, ··· ,n, there is a canonical isomorphism i ∼ (2.5) H (M,Z/2Z) = Hn−i(M,Z/2Z). See [Mu, p. 383] for example. On the other hand, in order to connect better with the non-orientable Poincaré duality groups defined in the next section, it is helpful to state a stronger version of this Poincaré duality in Equation (2.5) in terms of the orientation sheaf. Specifically, let ωM be the orientation sheaf of M. It can be defined in several different but equivalent ways. First, for any open subset U ⊂ M, assign the abelian n Z Z n Z group HomZ(Hc (U, ), ), where Hc (U, ) is the cohomology with compact support. Then ωM is the associated sheaf and is locally constant [Ive]. Alternatively, let M˜ be the universal covering space of M.ThenM˜ is orientable. Let Ω be the

6L76 IZHENJI orientation module of M˜ , which is an inﬁnite cyclic group whose two generators correspond to the two orientations of M˜ . The fundamental group π1(M)actson M˜ and also on Ω such that γ ∈ π1(M)actsonΩas±1 depending on whether γ is ˜ × orientation preserving or not. Then the associated locally system M π1(M) Ωon M is equal to the locally constant orientation sheaf ω. In terms of the orientation sheaf ω, the manifold M is orientable if and only if ω is a constant sheaf with the stalk equal to Z. Then the general Poincar´e duality for a closed manifold M n, independent of whether it orientable or not, states that for every i =0, ··· ,n, there is a canonical isomorphism i ∼ (2.6) H (M,Z) = Hn−i(M,ω ⊗ Z).

In the above equation, the group on the right hand, Hn−i(M,ω⊗Z), is the homology of M with local coefficient system ω ⊗ Z. ∼ Since π1(M) acts trivially on Ω/2Ω = Z/2Z, the above duality in Equation (2.6) implies the Poincaré duality in Equation (2.5) for non-orientable closed manifolds. Poincaré duality for closed manifolds with local coefficient systems. More generally, for any local coefficient system E on M,andforeveryi = 0, ··· ,n, there is a canonical isomorphism i ∼ (2.7) H (M,E) = Hn−i(M,ω ⊗ E). Suppose that M n is a closed oriented manifold. Then the orientation sheaf is trivial, and for any local coefficient system E on M,wehavethePoincaré duality: i ∼ (2.8) H (M,E) = Hn−i(M,E). See [S] and [Ei] for definition and other details on homology and cohomology with local coefficient systems, and intersection theory of cycles and Poincaré duality for non-orientable manifolds. There are also other versions of Poincaréduality for not necessarily orientable noncompact manifolds with boundary and with local coefficient systems, but they are not needed in this paper. Remark 2.1. If M is a compact topological space that is not a smooth manifold, then the Poincaré duality, in particular the version in Equations (2.1) or (2.3) does not hold for M. For some stratified spaces, there is a family of intersection homology groups, depending on some parameters, called perversities, which satisfy an analogue of the duality in Equation (2.3). See [GM] for details.

3. Homology and cohomology of groups Given a group Γ with discrete topology, there exists a topological space (or rather a CW-complex) BΓ which satisﬁes the conditions:

π1(BΓ) = Γ,πi(BΓ) = 0,i≥ 2.

The latter vanishing conditions πi(BΓ) = 0, i ≥ 2, are equivalent to that the universal covering space EΓ=BΓ is contractible. The space BΓ is unique up to homotopy equivalence and is called the classifying space of Γ. (It is also the K(Γ, 1)-space in the family of K(Γ,n)-spaces, n ≥ 1.) The space EΓ admits a proper and ﬁxed point free action of Γ and is a universal

STEINBERG REPRESENTATIONS AND DUALITY PROPERTIES OF GROUPS77 7 space for proper and fixed-point free actions of Γ. In fact, it is characterized by the following properties: (1) EΓ is contractible. (2) EΓ is a Γ-CW-complex such that Γ acts properly and fixed point freely. In practice, an effective method to find good models of BΓ is to find good models of EΓ first, and then take the quotients Γ\EΓasBΓ-spaces. Since BΓ depends on Γ up to homotopy equivalence, its homology groups i Hi(BΓ, Z) and cohomology groups H (BΓ, Z) only depend on Γ. They give rise to homology and cohomology groups of Γ with the trivial coefficient Z: i i (3.1) Hi(Γ, Z)=Hi(BΓ, Z),H(Γ, Z)=H (BΓ, Z). For many applications to groups Γ, we also need to consider homology and i cohomology groups Hi(Γ,E), H (Γ,E) of Γ with coefficients in ZΓ-modules E (or more generally RΓ-modules, where R is a ring). Usually they are defined in terms of a projective resolution F : ··· → F2 → F1 → F0 → E. The homology is defined by

(3.2) Hi(Γ,E)=Hi(F ⊗Γ E), where F ⊗Γ E is the complex given by Fi ⊗Γ E = Fi ⊗ E/ ∼,andtherelation∼ is defined by f ⊗ m ∼ γf ⊗ γm,forf ∈ Fi,m∈ E,γ ∈ Γ. The cohomology is defined by i i (3.3) H (Γ,E)=H (HomΓ(F, E)). See [Br1, Chap. III] for more detail. If E = Z, the trivial ZΓ-module, then the chain complex of EΓ=BΓgivesa free and hence projective resolution of Z, and the identification in Equation (3.1) follows. (Note that BΓ has a CW-complex structure, and its lift to EΓ is equivariant with respect to Γ). If E is a nontrivial ZΓ-module, then it gives a nontrivial local system on BΓ: EΓ×Γ E, still denoted by E, and similar geometric interpretations of the homology and cohomology groups of Γ hold:

i i (3.4) H (Γ,E)=H (BΓ,E),Hi(Γ,E)=Hi(BΓ,E). See [Ei, Theorem 28.1] for details. An important cohomological invariant of a group Γ is its cohomological dimension, denoted by cd Γ, which is deﬁned by: (3.5) cd Γ = sup{i ∈ Z | Hi(Γ,E) =0 , for some ZΓ-module E}. It is known that if cd Γ < +∞, then Γ is torsion-free. If a group Γ contains nontrivial torsion elements but contains some torsion-free subgroups Γ of ﬁnite index, then cd (Γ) is independent of the choice of Γ and is called the virtual cohomological dimension of Γ, denoted by vcd Γ. Then the geometric interpretations in Equation 3.4 immediately implies the following. Proposition 3.1. For any group Γ and any BΓ-space, cd Γ ≤ dim BΓ.In particular, if there exists a BΓ-space with dim BΓ < +∞,thencd Γ < +∞.

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AgroupΓissaidtobeoftype FP if the trivial ZΓ-module Z admits a finite length resolution by finitely generated projective modules. It follows immediately from the definition that if Γ is of type FP,thencdΓ< +∞, and the homology and i cohomology groups Hi(Γ, Z)andH (Γ, Z) are finitely generated in every degree i. AgroupΓissaidtobeoftype FP∞ if the trivial ZΓ-module Z admits a resolution by finitely generated projective modules. It also follows immediately from the definition that for every group Γ of type FP∞, the homology and cohomology i groups Hi(Γ, Z)andH (Γ, Z) are finitely generated in every degree i. An effective method to prove that a group Γ is of type FP or FP∞ is to construct good models of BΓ-spaces. Proposition 3.2. If Γ admits a finite CW-complex BΓ-space, then Γ is of type FP.IfΓ contains a subgroup Γ of finite index which is of type FP, for example, if it admits a finite CW-complex BΓ -space, then Γ is of type FP∞. Proof. The first statement follows from the projective resolution of Γ induced from the chain complex of EΓ, and the second statement follows from the first and [Br1, Proposition 5.1, p. 197].

4. Duality groups, Poincaré duality groups In this section, we first introduce the notion of duality groups, Poincaré duality groups, virtual duality groups, and virtual Poincaré duality groups. Then we give a geometric interpretation of the dualizing module (Proposition 4.4), and several useful criterions for duality properties (Propositions 4.5 and 4.8, and Corollary 4.9). Let M be an oriented closed manifold. Suppose that M is aspherical, i.e., πi(M)={e} for i ≥ 2. Denote the fundamental group π1(M)byΓ.Then the Poincaré duality for M and the identifications Hi(M,Z)=Hi(Γ, Z)and Hi(M,Z)=Hi(Γ, Z) in Equation (3.1) implies the following duality property: i ∼ (4.1) H (Γ, Z) = Hn−i(Γ, Z). By the Poincaré duality for homology groups with local coefficient systems in Equa- tion (2.8) and the realization in Equation (3.4), it follows that for any Γ- (or ZΓ-) module E, and every i ≥ 0, there is a canonical isomorphism: i ∼ (4.2) H (Γ,E) = Hn−i(Γ,E). Now let Γ be any group with discrete topology. If the group Γ satisfies the above condition in Equation (4.2) for all ZΓ-modules E, then Γ is called an orientable Poincarédualitygroupof dimension n. Assume that M is a closed non-orientable aspherical manifold. Its universal covering space M˜ is clearly orientable, and its orientation sheaf is trivial. Identify its orientation module Ω with Z after a choice of an orientation of M.ThenΓ= π1(M)actsonZ nontrivially. The Poincaré duality for non-orientable manifolds in Equation (2.7) and the realization in Equation (3.4) imply that for every ZΓ-module E, and every i ≥ 1, i ∼ (4.3) H (Γ,E) = Hn−i(Γ, Z ⊗ E). In general, if a group Γ acts on Z, i.e., Z is a ZΓ-module, and satisfies the above condition in Equation (4.3) for all ZΓ-module E, then Γ is called a Poincaréduality ∼ group of dimension n. If the action of Γ on Z is trivial, then Z ⊗ E = E,andΓ

STEINBERG REPRESENTATIONS AND DUALITY PROPERTIES OF GROUPS79 9 is an orientable Poincaré duality group. Otherwise, Γ is a non-orientable Poincaré duality group. More generally, a group Γ is called adualitygroup(or a generalized Poincaré duality group)ofdimensionn if there exists a ZΓ-module D such that for every i ≥ 0 and every ZΓ-module E, there exists an isomorphism i ∼ (4.4) H (Γ,E) = Hn−i(Γ,D⊗ E). In this case, the module D is called the dualizing module of Γ. It is known [Fa, Theorem 3] that under the assumption that Γ is of type FP,ifD is finitely generated abelian group, then D must be isomorphic to Z which is regarded as a Γ-module, and hence Γ is a Poincaré duality group of dimension n. A group Γ is called a virtual Poincarédualitygroupif there exists a subgroup of finite index Γ which is a Poincaréduality group. The notion of virtual duality groups can be defined similarly. Proposition 4.1. If Γ is a duality group of dimension n,thenΓ is of type FP, and the cohomological dimension of Γ is equal to n;inparticular,Γ is torsion- free. If Γ is a virtual duality group of dimension n, then the virtual cohomological dimension of Γ is equal to n. For the first statement, see [Bi, Theorem 9.2], and for the second, see [Br1, p. 220, Theorem 10.1(iv)]. Therefore, duality groups enjoy various cohomological finiteness properties. A useful algebraic criterion for duality is the following one [Br1, p. 220] [Fa, Theorem 3] (see also [Bi]). Proposition 4.2. Let Γ be a group of type FP.ThenΓ isadualitygroupof dimension n if and only if one of the following equivalent conditions holds: (1) Hi(Γ, ZΓ) = 0 for i = n,andHn(Γ, ZΓ) is a torsion-free abelian group. (2) for every abelian group E, the cohomology group Hi(Γ, ZΓ ⊗ E)=0for i = n. If the above conditions are satisfied, then Hn(Γ, ZΓ) is the dualizing module D of Γ. As an abelian group, D is either isomorphic to Z or infinitely generated. Remark 4.3. In the above proposition, if Γ is a nontrivial duality group, then the module Hn(Γ, ZΓ) is automatically nontrivial. In fact, by [Br, Proposition 6.7, p. 202], under the assumption that Γ is of type FP, the equality (4.5) cd Γ = max{i | Hi(Γ, ZΓ) =0 } holds. Then the conditions in the proposition imply that cd Γ is finite. This implies that Γ is torsion-free and hence the non-triviality of Γ implies that Γ ⊇ Z.Bythe monotonicity of the cohomological dimension, it follows that cd Γ ≥ cd Z =1. Then Equation 4.5 again implies that Hn(Γ, ZΓ) is nontrivial. Examples of Poincaré duality groups. The existence of a good model of BΓ is often the key to show that Γ is a duality group. As in the situation at the beginning of this section, if a group Γ admits a BΓ-space given by a compact aspherical manifold M, then Γ is a Poincaré duality group. For example, it is clear that the free abelian group Zn admits a BΓ-space given by the closed orientable aspherical manifold Zn\Rn and is hence an orientable Poincaré duality group of dimension n.

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More generally, every torsion-free, finitely generated nilpotent group Γ is also an orientable Poincaré duality group. In fact, by a known result of Malcev [M], such a group Γ can be embedded into a simply connected nilpotent Lie group N as a lattice. Since every lattice Γ in N is automatically uniform, i.e., Γ\N is compact, and N is diffeomorphic to Rn, n =dimN, the closed orientable manifold Γ\N gives a desired BΓ-space. If G is a simply connected solvable Lie group and Γ is a lattice subgroup of G. Then Γ is torsion-free, and Γ\G is a closed manifold and gives a BΓ-space (see [Ra1]). In particular, Γ is a Poincaré duality group of dimension n,where n =dimG. BΓ-space given by manifolds with nonempty boundary. If the compactness condition on BΓisdropped,orBΓ is given by a compact manifold with nonempty boundary, then it is not obvious if Γ is a Poincaré duality group or even a duality group. For example, if Γ is a torsion-free non-uniform arithmetic subgroup of a semisimple linear algebraic group, such as a torsion-free subgroup of SL(2, Z) of finite index, then there are BΓ-spaces given by compact manifold with boundary but Γ is not a Poincaré duality group. On the other hand, the following result is very useful in studying the duality of Γ. Proposition 4.4. Suppose that a BΓ-space is given by a finite CW-complex. Let EΓ=BΓ be the universal covering space of BΓ.Then i Z ∼ i Z H (Γ, Γ) = Hc(EΓ, ). In particular, if Γ is a Poincare duality group of dimension n, then its dualizing n Z module is equal to Hc (EΓ, ). Suppose that a BΓ-space is given by an n-dimensional smooth compact manifold with nonempty boundary. Then its universal covering space EΓisacon- tractible manifold with boundary. Fix an orientation of EΓ. Proposition 4.5. Assume that Γ admits a BΓ-space given by an n-dimensional smooth compact manifold with nonempty boundary as above. If the boundary ∂EΓ is homotopy equivalent to a bouquet of spheres Sr−1 of dimension r − 1 (i.e., a wedge product of Sr−1), then Γ is a duality group of dimension n−r.Furthermore, if the bouquet contains at least one but only finitely many spheres Sr−1,thenit must contain exactly one sphere, and Γ is a Poincaré duality group in this case; otherwise, the bouquet contains infinitely many spheres and Γ is not a Poincaré duality group. Proof. By the Poincaré duality for manifolds with boundary in Equation (2.4), i Z ∼ Z Hc(EΓ, ) = Hn−i(EΓ,∂EΓ, ). By the long exact sequence for the pair EΓ,∂EΓ, we obtain that i Z ∼ ˜ Z Hc(EΓ, ) = Hn−i−1(∂EΓ, ). i Z Then the assumption on the homotopy type of EΓ implies that Hc(EΓ, )isequal to 0 for i = n − r and is free for n − r, and hence Γ is a duality group of dimension n − r by Proposition 4.2.

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In the above proposition, if BΓ is a closed manifold, the boundary ∂EΓis empty and hence of dimension −1, i.e., r = 0. This implies that Γ is a duality group of dimension n. Remark 4.6. Note that every smooth compact manifold with corners is home- omorphic to a smooth manifold with boundary by smoothing out the corners, and we can also assume the BΓ-space in the above proposition is a compact manifold with corners. A related result on cohomological dimension is the following [Br1, Proposition 8.1, p. 210]. Proposition 4.7. Suppose that a BΓ-space is a compact manifold with nonempty boundary. Then cd Γ ≤ dim BΓ − 1.

If M is a closed aspherical manifold with π1(M) = Γ, then it is a BΓ-space. On the other hand, the product M ×[−1, 1] is a manifold with nonempty boundary and gives another BΓ-space. In general, if a group Γ admits a BΓ-space given by a manifold which could be noncompact or with boundary, it seems difficult to determine the minimal dimension in which a manifold exists, and it seems also difficult to decide whether a duality group Γ is a Poincaré duality group or not by using the condition whether such a BΓ-manifold has non-empty boundary or not. On the other hand, the following algebraic criterion for non-Poincaré duality is useful for some groups [Str]. Proposition 4.8. Suppose Γ is of type FP and its cohomological dimension, cd Γ,isequalton.IfHn(Γ, ZΓ) is finitely generated as an abelian group, then for every subgroup Γ of Γ of infinite index, cd Γ < cd Γ. Corollary 4.9. If Γ is a group of type FP and there exists a subgroup Γ of infinite index such that cd Γ =cdΓ,thenΓ is not a Poincaré duality group. Proof. If not, then by definition, Hn(Γ, ZΓ) is isomorphic to Z.Thenthe above proposition implies that cd Γ < cd Γ. This is a contradiction.

5. Algebraic groups, Tits building, Steinberg representations, arithmetic subgroups

In this section, we first introduce the notion of spherical Tits building ∆Q(G) of a reducitive algebraic group G defined over Q. Then we recall the Solomon-Tits theorem on the homotopy type of the building ∆Q(G), and obtain the Steinberg module of G(Q). In Remark 5.4, we introduce the Steinberg module of reductive algebraic groups over general fields, which will be used in studying duality properties of S-arithmetic subgroups over both number fields and function fields of curves over finite fields. Then we define arithmetic subgroups and point out some references on applications of the Steinberg module StQ(G) of arithmetic subgroups in number theory, in particular, modular symbols. Let G ⊂ GL(n, C) be a connected linear algebraic group defined over Q.It is often obtained as subgroups preserving some structures on Cn, for example, a volume form, which leads to SL(n), or a skew-symmetric form, which leads to Sp(n). For simplicity, we assume in this section that G is a reductive algebraic group. An important object associated with G is the spherical Tits building ∆Q(G), which

1282 LIZHEN JI is a simplicial complex whose simplexes are parametrized by Q-parabolic subgroups of G. Recall that a subgroup P of G is called a parabolic subgroup if the quotient P\G is a projective variety. If P is deﬁned over Q,thenP is called a Q-parabolic subgroup. For each proper Q-parabolic subgroup P of G, denote the corresponding simplex of ∆Q(G)byσP. Then these simplexes satisfy the following compatibility and normalizing conditions:

(1) If P is a proper maximal Q-parabolic subgroup of G,thenitssimplexσP is zero dimensional, i.e., a point. Q (2) For every pair of -parabolic subgroups P1, P2, σP1 is a face of σP2 if and only if P1 contains P2. In particular, for every simplex σP, its vertices correspond to the set of all proper maximal Q-parabolic subgroups that contain P. Let r be the Q-rank of G.Thenr>0 if and only if G contains a proper Q-parabolic subgroup. Assume that r>0. From the above description and the structure of Q-parabolic subgroups (in particular, a minimal Q-parabolic subgroup is contained in exactly r diﬀerent proper maximal Q-parabolic subgroups), it follows that the dimension of ∆Q(G)isequaltor − 1. Clearly, the Q-locus G(Q)ofG acts on the set of Q-parabolic subgroups of G by conjugation and hence acts on ∆Q(G) by simplicial automorphisms. A known result of Solomon and Tits is the following [So] [Br2].

Proposition 5.1. The Tits building ∆Q(G) is homotopy equivalent to a bou- r−1 quet of countably inﬁnitely many spheres S .Inparticular,Hr−1(∆Q(G), Z) is a free abelian group on inﬁnitely many generators.

The action of G(Q)on∆Q(G) induces an action on Hr−1(∆Q(G), Z), and hence Hr−1(∆Q(G), Z) becomes a G(Q)-module. This is called the Steinberg module of G(Q) (or rather of ZG(Q)), and denoted by StQ(G)orSt(G). If the rank r =0, we call the trivial ZG(Q)-module Z the Steinberg module of G(Q). Remark 5.2. The spheres Sr−1 in the bouquet can be parametrized as follows. The building ∆Q(G) contains sub-complexes, which are all isomorphic to the Cox- eter complex associated with the Weyl group of G over Q. These sub-complexes are called apartments. Each apartment gives a triangulation of the sphere Sr−1 and contains ﬁnitely many simplexes, and the top dimensional simplexes of apartments are called chambers of the building. The building ∆Q(G) is the union of such apartments. In fact, for any ﬁxed chamber C of ∆Q(G), the union of the collection A of the apartments which contain this chamber is equal to ∆Q(G). Then each apartment in A corresponds to a sphere in the bouquet in the above proposition. Remark 5.3. Since G(Q) acts transitively on the set of minimal Q-parabolic subgroups, it follows that as a ZG(Q)-module, the Steinberg module StQ(G)is generated by one element. Since the action is not simply transitive, it is not a free ZG(Q)-module of rank 1. Let P be a minimal Q-parabolic subgroup G,andUP be the unipotent radical of P.ThenUP(Q) acts simply transitively on Σ. This implies that as a ZUP(Q)- module, Hr−1(∆Q(G), Z) is free of rank 1. See [BS1, Remark 8.6.8] for more detail.

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Remark 5.4. Instead of the field Q, we can consider any field k,suchasa finite field Fq (for example, Z/pZ,wherep is a prime number), the field Qp of p-adic numbers, or the function field of a curve over a finite field Fq, for example, Fq(t), where t is a free variable. Then for any reductive algebraic group G over k, there is also a spherical Tits building ∆k(G) whose simplexes are parametrized by k-parabolic subgroups of G. There is also a Solomon-Tits theorem stating that r−1 ∆k(G) is homotopy equivalent to a bouquet of sphere S ,wherer is the k-rank of G. The non-vanishing homology group Hr−1(∆k(G), Z)isaZG(k)-module, called the Steinberg module of G(k), and denoted by Stk(G). The building ∆k(G)isa finite simplicial complex if and only if k is a finite field, and hence Stk(G) is a finite abelian group if and only if k is a finite field. When G is a Chevalley group and k is finite field, the module Stk(G)isthe original Steinberg module [So]. As pointed out in the introduction, this module and generalizations have played a fundamental role in representation theories of algebraic groups and many other applications (see [St, pp. 580-586] [Ca] [Hu] [Be]).

Remark 5.5. Since Stk(G) is generated over ZG(k) by one element, there is a surjective map ZG(k) → Stk(G). See [To] for descriptions of the kernel of this map.

Definition 5.6. A subgroup Γ of G(Q) is called an arithmetic subgroup if it is commensurable with G(Z)=G(Q) ∩ GL(n, Z), i.e., Γ ∩ G(Z) is of ﬁnite index in both Γ and G(Z).

Clearly, an arithmetic subgroup Γ acts on the set of Q-parabolic subgroups and also on the Tits building ∆Q(G). The action of Γ on Hr−1(∆Q(G), Z)givesthe latter a ZΓ-module structure. The module is also called the Steinberg module of Γ (or ZΓ). Besides the application in the next section that the Steinberg module of a torsion-free arithmetic subgroup Γ is its dualizing module, this Steinberg module is also important in number theory. For example, when G = SL(n)andΓ=SL(n, Z), then Γ acts transitively on ∆Q(G), and it is reasonable to expect that the Steinberg module is generated over ZΓ by one element [AP, Theorem 4.1]. This has important applications in number theory through the so-called modular symbols. See the references of [To] and [GuM] for other related results.

6. Borel-Serre compactification and duality properties of arithmetic subgroups In this section, we first use the Borel-Serre compactification of locally symmetric spaces to construct finite CW-complex BΓ-spaces for torsion-free arithmetic subgroups of reductive algebraic groups G, and hence show that arithmetic subgroups are of type FP∞ and of type FP if they are torsion-free. Then we use the relation between the boundary of the Borel-Serre partial compactification of the symmetric space and the spherical Tits building to show that arithmetic subgroups are duality groups. For non-uniform arithmetic groups acting on linear symmetric spaces, we give a proof without using the Borel-Serre compactification that they are not Poincaré duality groups (Theorem 6.4). We conclude this section with duality properties of S-arithmetic subgroups.

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Let G be a linear algebraic group as in the previous section. Let G = G(R) be the real locus of G. Then every arithmetic subgroup Γ ⊂ G(Q) is a discrete subgroup of G. In this section, we assume that Γ is a lattice of G, i.e., Γ\G has finite volume. This is not always true, for example, if G = GL(n)andΓ=GL(n, Z). On the other hand, if G is semisimple or G has no nontrivial character defined over Q, then this condition on Γ is satisfied. Let K ⊂ G be a maximal compact subgroup of G.ThenX = G/K is diffeomorphic to Rn,wheren =dimX.IfG is reductive, then with respect to an invariant metric, X is a symmetric space not containing any compact factors and hence is of non-positive sectional curvature. Let r be the Q-rank of G as above. Then it is known that Γ\X is compact if and only if r =0. A basic and important result of [BS1] is the following. Theorem 6.1. Let Γ ⊂ G(Q) be an arithmetic subgroup as above. Then Γ is a virtual duality group of dimension n − r,wheren =dimX, and the dualizing module is given by the Steinberg module; and Γ is a virtual Poincarédualitygroup if and only if the Q-rank r =0, i.e., Γ is a uniform lattice. To prove this theorem, we need some preparations, in particular, the Borel-Serre compactification in [BS1]. Since Γ is a discrete subgroup of G, it acts properly on X. If Γ is torsion-free, then Γ acts fixed point freely. This implies that Γ\X is a BΓ-space and X is a EΓ-space. When r = 0 and Γ is torsion-free, then Γ\X is a closed aspherical manifold with π1 = Γ. This implies that Γ is a Poincaréduality group, and Theorem 6.1 is proved in this case. On the other hand, when r>0, Γ\X is not finite CW-complex, and Proposition 4.5 can not be applied. One way to obtain a BΓ-space given by a finite CW-complex is to construct a compactification Γ\X of Γ\X such that the inclusion Γ\X→ Γ\X is a homotopy equivalence. BS This is achieved by the Borel-Serre compactification Γ\X of Γ\X in [BS1]. Briefly, it is constructed in two steps: BS (1) Construct a partial compactification X by attaching a boundary component eP for every proper Q-parabolic subgroup P of G: BS (6.1) X = X ∪ eP. P BS Each boundary component eP is contractible and the X is a real analytic manifold with corners with the interior equal to X. BS (2) Show that the Γ-action on X extends to a proper action on X with a BS compact quotient, which gives Γ\X . Since a manifold with corners is clearly homotopy equivalent to its interior, BS the inclusion Γ\X→ Γ\X is a homotopy equivalence when Γ is torsion-free. BS Consequently, Γ\X is a compact BΓ-space when Γ is torsion-free. A simple example to keep in mind is the case when X is the Poincaré upper half-plane, and Γ\X is a hyperbolic (Riemann) surface. In this case, the Borel-Serre partial

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BS compactification X is obtained by adding a copy of R at every rational boundary point, i.e., a point in Q ∪{i∞} ⊂ R ∪{i∞} = ∂X, and every cusp of Γ\X is compactified by adding a circle. An immediate corollary is the following. Proposition 6.2. If Γ is a torsion-free arithmetic subgroup, then Γ is of type FP. In general, every arithmetic subgroup Γ, which might contain nontrivial torsion elements, is of type FP∞. In particular, in every degree i, Hi(Γ, Z) and Hi(Γ, Z) are finitely generated. This follows from Proposition 3.2 and the fact that every arithmetic subgroup contains a torsion-free subgroup of finite index. BS In the Borel-Serre partial compactification Γ\X , the boundary components eP are contractible and satisfy the incidence relation: ⊂ P1 P2 if and only if eP1 is contained in the closure of eP2 . Together with the fact that every boundary component eP is contractible and the nerve-cover principle, this implies the following result [BS1]. BS Proposition 6.3. The boundary ∂X is homotopy equivariant to the Tits building ∆Q(G), and hence is homotopy equivalent to a bouquet of countably infinitely many spheres Sr−1, when the Q-rank r of G is positive. Proof of Theorem 6.1. The case r = 0 was proved earlier. Assume that r>0. When Γ is a torsion- BS free, X is a EΓ-space with a compact quotient. Together with Proposition 4.5, Proposition 6.3 implies that Γ is a duality group of dimension dim X − r with ∼ the dualizing module equal to the Steinberg module StQ(G) = Hr−1(∆Q(G)) = BS Hr−1(∂X ).

BS To motivate results in the later section, we describe a realization of Γ\X in terms of a submanifold Γ\XT of Γ\X such that the inclusion Γ\XT → Γ\X is a homotopy equivalence. When Γ\X is a hyperbolic Riemann surface, it is clear that Γ\XT can be obtained by cutting off each cusp end. Since each cusp neighborhood is a topological cylinder, the whole space Γ\X can be deformed retracted to Γ\XT . This is equivalent to the following steps: (1) Remove Γ-equivariant horoballs at rational boundary points from the Poincaré upper half-plane to get a truncated submanifold XT . (2) Show that X can be deformation retracted to XT . (3) The Γ-action preserves XT , and the quotient Γ\XT is the desired submanifold Γ\XT . For a general symmetric space X and arithmetic subgroup Γ, the construction is similar. But the notion of neighborhoods of points at infinity, i.e., the analogues of the removed horoballs here, is more complicated. See the paper [Sa] for details and references. A natural question is whether XT can be further deformed so that there exists a subspace Y of X satisfying the following conditions: (1) Y is an equivariant deformation retract of X with respect to an arithmetic subgroup Γ, (2) the quotient Γ\Y is compact,

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(3) and the dimension of Y is equal to cd Γ when Γ is torsion-free. Certainly, the dimension on Y is the minimal one allowed. It turns out that when X is a linear symmetric space, i.e., a homothety section of a symmetric cone, such a subspace Y and deformation retraction exist [As]. Using this, we can give a different proof of that fact that Γ is not a Poincaré duality group when Γ is a torsion-free arithmetic subgroup. The most basic example of linear symmetric spaces is X = SL(n, R)/SO(n), which is a homothety section of the symmetric cone Πn(R) of positive definite matrices of rank n over R. Besides this family of symmetric cones, there are three other families:

(1) Πn(C), the space of positive definite matrices over the complex numbers C, (2) Πn(H), the space of positive definite matrices over the quaternions H, (3) Ln,theLorentzcone, (4) an exceptional symmetric cone Π3(O), the cone of all 3×3 positive definite matrices over the algebra O of octonions. See the book [FK] for more details. Theorem 6.4. Assume that X = G/K is a linear symmetric space and Γ ⊂ G(Q) is a torsion-free non-uniform arithmetic subgroup, for example, when Γ ⊂ SL(n, Z) is a torsion-free subgroup of finite index. Then the following results hold: (1) There exists a Γ-equivariant deformation retraction of X to a simplicial complex Y which is of dimension equal to dim X − r such that Γ\Y is compact, i.e, a finite CW-complex. In particular, cd Γ ≤ dim X − r. (2) There exists a subgroup Γ of Γ such that cd Γ =dimX −r.Thisimplies that cd Γ = dim X − r. (3) The index [Γ : Γ] is equal to infinity, and hence Γ is not a Poincaré duality group. Proof. The existence of Y and an equivariant deformation of X were proved in [As]. To prove (2), let P be a minimal Q-parabolic subgroup of G,andNP its unipotent radical. Then its real locus by P = P(R)admitsaQ-Langlands decomposition ∼ P = NPAPMP = NP(R) × AP × MP, where NP = NP(R), AP is the identity component of the real locus of a maximal Q-split torus of P,andMP is a reductive Lie group. Then ΓP =Γ∩ P is a discrete subgroup and is contained in the subgroup NPMP with a compact quotient. Let KM be a maximal compact subgroup of MP.ThenNPMP/KM is diffeomorphic to Rdim X−r. In fact, the Langlands decomposition of P induces a horospherical decomposition of X: X = NP × AP × MP/KM , where MP/KM is a symmetric space not containing any compact factors, and hence ∼ ∼ dim X−r NPMP/KM = NP × MP/KM = R and dim AP = r.SinceP is a minimal Q-parabolic subgroup, ΓP \NPMP/KM is a closed aspherical manifold, and consequently ΓP is a Poincaré duality group of dimension dim X − r. This implies that cd ΓP =dimX − r.Thisproves(2).Since

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∼ the volume of ΓP \X = ΓP \NPMP/KM × AP is inﬁnite (think of the case when X is the Poincar´e upper half plane and ΓP \X is a cylinder with an exponentially growing end), the index [Γ : ΓP ]=∞. Then (3) follows from Corollary 4.9 and the equality cd ΓP =cd Γ.

Remark 6.5. If X is not a linear symmetric space but the Q-rank r of G is equal to 1, we can also show directly, without using the Borel-Serre compactification, that every torsion-free arithmetic subgroup Γ is not a Poincaré duality group. The argument goes as follows. Each end of Γ\X is a topological cylinder and truncating off these ends gives a compact submanifold Γ\XT with boundary which is homotopy equivalent to Γ\X. By Proposition 4.7, cd Γ ≤ dim X−1. On the other hand, as in the proof of the above theorem, for any proper Q-parabolic subgroup P, the intersection ΓP =Γ∩ P ,whereP = P(R), is a subgroup of infinite index with cd Γ =dimX − 1. This implies that

cd Γ = dim X − 1= cd ΓP .

Since [Γ : ΓP ]=∞, Corollary 4.9 implies that Γ is not a Poincaré duality group. (For truncating a general locally symmetric space to obtain a compact submanifold with boundary which is a deformation retraction of the whole space, see [Ra2].) The statement that Γ is a duality group but not a Poincare duality group can also be proved using the truncated submanifold Γ\XT as follows. It is clear that the universal covering XT of Γ\XT has a nonempty boundary ∂XT which is homotopy equivalent to a bouquet of infinitely many spheres S0 (note S0 consists of two points and ∂XT has infinitely contractible connected components). By Proposition 4.5, Γ is a duality group but not a Poincaréduality group. S-arithmetic subgroups and duality properties. An important generalization of arithmetic arithmetic subgroups is the class of S-arithmetic subgroups. Let p1, ··· ,pk be a finite set of prime numbers and 1 1 S = {∞,p1, ··· ,pk}. The ring ZS = Z[ , ··· , ]ofS-integers consist of rational p1 pk numbers whose denominators are only divisible by primes p1, ··· ,pk. Then a subgroup Γ of G(Q) is called an S-arithmetic subgroup if it is commensurable with G(Q) ∩ GL(n, ZS). An important example of S-arithmetic subgroups is SL(n, ZS). Q For each prime pi,denotethe pi -rank of G by ri. Denote the Steinberg Q module of G( pi )(seebelow)bySti. Then an important result of Borel and Serre in [BS2] is the following. Theorem 6.6. Let G be a reductive algebraic group defined over Q,andΓ ⊂ Q G( ) be an S-arithmetic subgroup. Then Γ is a virtual duality group of dimension X k r − r r Q dim + i=1 i ,where is the -rank of G, and the dualizing module is the ⊗ k tensor product of local Steinberg modules StQ(G) i=1 Sti.Ifeitherr>0 or S contains at least one prime number, then Γ is not a virtual Poincaré duality group.

The idea is as follows. For each prime pi, there is a Bruhat-Tits building Xi Q associated with G( pi ), which is an inﬁnite simplicial complex of dimension ri and Q on which G( pi ) acts simplicially and properly. Endowed with the Tits metric, Xi is also a so-called CAT(0) and hence contractible (see [Br2] and [Ji]). It can Q be compactiﬁed by adding the spherical Tits building ∆(G( pi )) of the algebraic

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Q group G over pi . Then it follows that i Z ∼ i ∪ Q Q Z ∼ i−1 Q Z Hc(Xi, ) = H (Xi ∆(G( pi )), ∆(G( pi )), ) = H (∆(G( pi )), ) Q is non-zero only in dimension ri (note that ∆(G( pi )) is a simplicial complex of − Q ri Z dimension ri 1). The action of G( pi )onHc (Xi, )istheSteinberg module ri Z Sti. As an abelian group, Hc (Xi, ) is inﬁnitely generated. (Note that Xi is not a manifold and hence there is no version of Poincar´eduality which can be used to i Z compute Hc(Xi, ) as in the previous cases.) × k Q Since an S-arithmetic subgroup Γ is embedded in the product G i=1 G( pi ) as a discrete subgroup, it follows that Γ acts properly on the space k XS = X × Xi. i=1

If Γ is torsion-free, then Γ\XS is a BΓ-space. If r>0, Γ\XS is noncompact and is not a finite CW-compex. On the other BS hand, replacing X by the Borel-Serre partial compactification X gives a finite BS CW-complex BΓ-space Γ\XS ,where k BS BS XS = X × Xi. i=1 k − The Kunneth formula implies that when j =dimX + i=1 ri r, k k BS − BS j Z dim X r Z ⊗ ri Z ⊗ Hc (XS , )=Hc (X , ) Hc (Xi, )=StQ(G) Sti, i=1 i=1 and vanishes for other values of j. This implies that if Γ is torsion-free, then it is a duality group of dimension k − equal to dimX + i=1 ri r with the dualizing module given by the tensor product ⊗ k StQ(G) i=1 Sti. In general, Γ is a virtual duality group. As pointed out before, for each i, Sti is not finitely generated as an abelian group. This implies that if either r>0orS contains at least one prime pi,then the dualizing module of Γ is not finitely generated and hence Γ is not a Poincaré duality group. S-arithmetic subgroups over function fields. Global fields consist of two types: number fields, which are finite extensions of Q, and function fields of curves over finite fields Fq, which are finite separable extensions of Fq(t), where t is a free variable. For a function field k and a reductive algebraic group G defined over k,wecan also define S-arithmetic subgroups. By methods similar to those described in the previous paragraphs, in particular, the Bruhat-Tits building associated with G(kp) for any place p of k, the following result is also true [BS2, Theorem 6.2]. Theorem 6.7. Suppose G is a reductive algebraic group defined over a function field k and the k-rank of G is equal to 0. Then every S-arithmetic subgroup of G(k) is a virtual duality group, but not a virtual Poincaré duality group unless the S- arithmetic subgroup is finite.

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The assumption that the k-rank of G is equal to 0 in the above theorem is crucial. Otherwise, S-arithmetic subgroups of algebraic groups of positive rank over function ﬁelds always contain nontrivial torsion elements and hence their virtual cohomology dimension is equal to inﬁnity, which implies that they are not virtual duality groups. In fact, a stronger result holds: they are not of type FP∞.See [BuW] and references there.

7. Mapping class groups and Teichmüller spaces Let S be an orientable surface. Let Diff(S) be the group of all diffeomorphisms 0 0 of S,andDiff(S) its identity component. Then ModS =Diff(S)/Diff (S)isthe extended mapping class group (or also called the extended Teichmüller modular group)ofS.LetDiff+(S) be the subgroup of orientation preserving diffeomorphisms of S.ThenMod+(S)=Diff+(S)/Diff0(S)isthemapping class group of S. When S = Z2\R2 is the torus, i.e., S is a closed surface of genus 1, then Mod+(S) can be identified with SL(2, Z), and Mod(S) can be identified with GL(2, Z). Therefore, if S is a closed surface of genus g ≥ 2, or more generally an oriented surface of negative Euler characteristic χ(S), then Mod(S)andMod+(S) are natural generalizations of GL(2, Z)andSL(2, Z). For simplicity, in the following we assume that S is a closed orientable surface of genus g ≥ 2, and ModS is also denoted by Modg. The analogue of the symmetric spaces for the mapping class groups is the Te- ichmüller space Tg which consists of equivalence classes marked hyperbolic metrics on S, where a marked a hyperbolic metric on S is a hyperbolic surface Σ together with a homotopy class of diffeomorphisms ϕ : S → Σ, and denoted by (Σ, [ϕ]). Two marked hyperbolic surfaces (Σ1, [ϕ1]) and (Σ2, [ϕ2]) are equivalent if there is an isometry between Σ1 and Σ2 which commutes with the two markings. Therefore,

Tg = {(Σ, [ϕ]) | ϕ : S → Σ is a diﬀeomorphism}/ ∼ .

(As abstracted hyperbolic surfaces, if there exists an isometry between Σ1 and Σ2, then they are the same. Two equivalent marked hyperbolic surfaces (Σ1, [ϕ1]) and (Σ2, [ϕ2]) are also the same marked abstract hyperbolic surface. Therefore, Tg is the space of all marked hyperbolic structures on S. Because of this, we also write

Tg = {(Σ, [ϕ]) | ϕ : S → Σ is a diﬀeomorphism}.

Another way to define Tg is as follows. Let H(S) be the space of all hyperbolic metrics on S. Then the group of all diffeomorphisms of S,Diff(S), acts on H(S), and the quotient Diff0(S)\H(S) under the identity component Diff0(S)canbe identified with Tg.) 6g−6 It is known that Tg is a manifold diffeomorphic to R and Modg acts on Tg by changing the marking: for any [ψ] ∈ Modg,whereψ ∈ Diff(S), and a marked hyperbolic surface (Σ, [ϕ]), then [ψ] · (Σ, [ϕ]) = (Σ, [ϕ ◦ ψ]).

It is also known that this action of Modg on Tg is proper. Therefore, for any torsion-free subgroup Γ of Modg of ﬁnite index, Γ\Tg is a BΓ-space. +\ By deﬁnition, the quotient Modg Tg is the moduli space of Riemann surfaces (or algebraic curves) of genus g and is hence noncompact. The reason for the non- compactness is that starting with any hyperbolic surface, we can pinch a closed

2090 LIZHEN JI geodesic and get a sequence of hyperbolic surfaces which has no convergent subse- quences. To get a finite CW-complex BΓ-space for a torsion-free subgroup Γ ⊂ Modg of BS finite index, we can use with a Borel-Serre type compactification Γ\Tg oratrun- cated subspace Γ\Tg(ε) such that the inclusions below are homotopy equivalence: BS Γ\Tg(ε) → Γ\Tg → Γ\Tg . BS The Borel-Serre type compactification Γ\Tg has been constructed (or out- lined) by Harvey [Hav] and finished by Ivanov [I1] (see also the references of [IJ]). On the other hand, it is easier to construct the truncated subspace Γ\Tg(ε). For every point x =(Σ, [ϕ]) of Tg, and closed geodesic c in Σ, denote the length of c with respect to the hyperbolic metric by x(c). Then it is known that there exists a positive constant εg depending only on g such that every two closed geodesics c1 and c2 of Σ with x(c1),x(c2) ≤ εg are disjoint. For any ε ≤ εg, define a truncated subspace Tg(ε)ofTg by

Tg(ε)={x =(Σ, [ϕ]) ∈ Tg | for every closed geodesic c ⊂ Σ,x(c) ≥ ε}.

Then Tg(ε) is a real analytic manifold with corners. By the Mumford criterion, the quotient Γ\Tg(ε)iscompactif[Modg :Γ]< +∞ and is a real analytic with corners if Γ is also torsion-free. It is also known that Tg(ε) is a deformation retract of Tg and hence is contractible. Together with Proposition 3.2, this implies the following result (see [I1] for detail and references).

Proposition 7.1. For every torsion-free subgroup Γ of ﬁnite index of Modg, Γ\Tg(ε) is a ﬁnite CW-complex BΓ-space, and hence Γ is of type FP. Consequently, Modg is of type FP∞. 8. Curve complex, duality properties of mapping class groups

As pointed in the previous section, Modg is an analogue of arithmetic subgroups. It is naturally expected that they share many properties. One important result in [Ha2, Theorem 4.1] is the following.

Theorem 8.1. For every torsion-free subgroup Γ ⊂ Modg of ﬁnite index, Γ is a duality group of dimension 4g − 5. Therefore, Modg is a virtual duality group of virtual cohomological dimension 4g − 5. The dualizing module is given by an analogue of the Steinberg module and will be described later. Since Modg\Tg is noncompact, Modg is an analogue of non-uniform arithmetic subgroups, and the following result is naturally expected [IJ, Theorem 1.5].

Theorem 8.2. For every torsion-free subgroup Γ ⊂ Modg of finite index, Γ is not a Poincaré duality group. Therefore, Modg is not a virtual Poincaréduality group. A natural method to prove the above results is to understand the homotopy type of the boundary ∂Tg(ε) and apply the criterion in Proposition 4.5. For this purpose, we need to introduce the curve complex C(S)ofthesurfaceS,whichis an analogue of the spherical Tits building ∆Q(G) associated with algebraic groups earlier and was introduced by Harvey in [Hav].

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By definition, the vertices of the curve complex C(S) are free homotopy classes [c] of simple closed curves c in S. Vertices [c1],...,[ck+1] form the vertices of a k-simplex if and only if they are all different and every two curves ci1 and ci2 for 1 ≤ i1 0. Then for ε ≤ εg, the boundary faces of Tg(ε) are defined by these length functions: (c)=ε. Clearly, only when the geodesics c1, ··· ,ck+1 are disjoint, there are hyperbolic surfaces x =(Σ, [ϕ]) ∈ Tg such that x(ci)=ε,fori =1, ··· ,k+1. From this, it is clear that the boundary faces of Tg(ε) are parametrized the simplexes of C(S). In fact, the following result is true.

Proposition 8.3. The space Tg(ε) is a contractible manifold with corners. Its boundary faces when Tg(ε) is considered as a manifold with corners are contractible and parametrized by the simplexes of the curve complex C(S), and the whole boundary ∂Tg(ε) is homotopy equivalent to C(S).

Each boundary face of ∂Tg(ε) is the product of a truncated Teichmüller space with some Rm, and hence is contractible [I1], and the inclusion relation between these faces is the opposite of the inclusion of the simplexes of C(S). Therefore, C(S) plays the role of spherical Tits building. A natural and important problem is to prove an analogue of the Solomon-Tits theorem. In [Ha2](seealso[Ha1, Chap. 4, §1], and [I4, §3], or [I1, Theorem 3.3.A] for a different proof), Harer proved the following weak analogue of the Solomon-Tits theorem. Theorem 8.4. The curve complex C(S) is homotopy equivalent to a bouquet of spheres ∨Sn, where the dimension n =2g − 2. On the other hand, it is not obvious that the bouquet contains at least one sphere, i.e., C(S) is not contractible. In fact, this question has been raised by several people (see the introduction of [IJ] for more details about this question), and can be answered positively as follows: The above Theorem 8.4, and Propositions 4.5 and 4.2 together with a fact that the cohomological dimension of a group is greater than or equal to the cohomological dimension of any subgroup imply that Modg is a virtual duality group of dimension 4g −5, and then the bouquet for the homotopy type of C(S) contains at least one sphere. The dualizing module of Modg (or rather its torsion-free subgroups of finite index) is H2g−2(C(S), Z) with the natural action of Modg. See [IJ] for more details. In [IJ, Theorem 1.4], the following result was proved and gives a stronger analogue of the Solomon-Tits theorem. Theorem 8.5. The curve complex C(S) is homotopy equivalent to the bouquet of countably infinitely many spheres Sn,wheren =2g − 2. Unfortunately, the proof is not direct as in the case of spherical Tits buildings. Indeed, in [IJ], the idea was to show that Modg contains a subgroup, a so-called Mess subgroup Γ, of infinite index whose virtual cohomological dimension is equal

2292 LIZHEN JI to that of Modg, i.e., 4g −5. Then by Corollary 4.9, Modg is not a virtual Poincar´e duality group. Applying Proposition 4.5 to EΓ=Tg(ε) implies that the homotopy type of the boundary ∂Tg(ε), or equivalently the curve complex C(S), contains inﬁnitely many spheres. It will be desirable to give a more direct proof of Theorem 8.5. See [Bro] for results on H2g−2(C(S)) as a ZModg-module.

9. Outer automorphism groups of free groups, outer space

In this section, we ﬁrst recall the reduced outer space Xn and its spine Kn, then the determination of the virtual cohomological dimension of Out(Fn). Let Fn, n ≥ 2, be the free group on n generators, and

Out(Fn)=Aut(Fn)/Inn(Fn) be the group of outer aumorphisms of Fn.Whenn =2,Out(Fn)=GL(2, Z). It is known that there is a surjective homomorphism

π : Out(Fn) → GL(n, Z) induced by the map Fn → Zn = Z ×···×Z.Inparticular,Out(Fn)containsa −1 distinguished subgroup π (SL(n, Z)), which is usually denoted by SOut(Fn). The outer automorphism group Out(Fn) plays an important role in the theory of combinatorial group theory. See [V1-2] [Be] for surveys on various aspects of this group. One important reason for considering Out(Fn) together with arithmetic subgroups of linear algebraic groups and mapping class groups is that they share many similar properties. The results in this section provide more evidence for this point. In view of all these, it is natural that Out(Fn) can be considered as a a natural variation of arithmetic subgroups of semisimple Lie groups and mapping class groups of surfaces. As discussed in the previous section, symmetric spaces and actions of arithmetic subgroups on them are fundamental to understand arithmetic subgroups and Teichmüller spaces are also crucial in studying the mapping class groups. The analogous space for Out(Fn) is the so-called reduced outer space Xn. Briefly, the outer space was introduced by Culler and Vogtmann in [CV] and was defined to be the space of marked metric graphs with the fundamental group equal to Fn and the total length of all edges equal to 1. Specifically, let Rn be the rose with n petals, i.e., the wedge product of n circles 1 1 S (or the bouquet of n circles S ). Then π1(Rn)=Fn,andRn is a BΓ-space for Γ=Fn.(Wecanidentifyπ1(Rn)withFn by sending the homotopy class of each petal to a generator of Fn). We consider only graphs G that do not contain vertices of valency 1 or 2. Then a marked graph G with π1(G)=Fn is a graph together with a homotopy class of homotopy equivalence ϕ : Rn → G. A metric graph is a graph G with an assignment of nonnegative edge lengths (e)toedgese of G such that the sum of edges in every nontrivial loop in G is is strictly positive (but some edge lengths could be zero). We scale the edge lengths so that the total sum of all edge lengths is equal to 1. A metric graph is denoted by (G, ), and a marked metric graph is denoted by (G, , [ϕ]). Two marked metric graphs (G1,1, [ϕ1]) and (G2,2, [ϕ2]) are defined equivalent if there is an isometry between (G1,1)and(G2,2) that commutes with the markings [ϕ1]and[ϕ2]. (In other words, an equivalence class of marked metric graphs is one marked abstract

STEINBERG REPRESENTATIONS AND DUALITY PROPERTIES OF GROUPS93 23 metric graph, or an abstract metric graph together with a specified isomorphism ∼ π1 = Fn.) Then the set of equivalence classes of such marked metric graphs with π1 = Fn is called the outer space associated with Fn. If we only consider the sub- collection of graphs which do not contain any separating edge (i.e., its complement is disconnected), we get the reduced outer space Xn. 0 Let Htp(Rn) be the group of all homotopy equivalences of Rn,andHtp(Rn) 0 the identity component of Htp(Rn). Then the quotient Htp(Rn)/Htp(Rn) is canonically isomorphic to Out(Fn). Specifically, every automorphism ϕ : Fn → Fn corresponds to an homotopy equivalence between Rn and Rn, which maps a petal representing a generator x of Fn to the loop representing the ϕ(x), and the automorphism of Fn is inner if and only if the homotopy class contains the identity map. It follows that Out(Fn)actsonXn by changing the marking of the marked metric graphs in Xn as in the case of the action of Modg on the Teichmüller space. It is known that the outer space Xn is an infinite simplicial complex of dimension 3n − 4, and the action of the group Out(Fn)onXn is proper. The quotient Out(Fn)\Xn is the moduli space of normalized metric graphs without containing any separating edges. When n =2,Xn can be canonically realized with an ideal triangulation of the upper half plane H2 = {x + iy | x ∈ R,y >0} (or the unit disc in C)[V1-2]. An important result in [CV, Theorem, p. 93] is the following.

Theorem 9.1. The reduced outer space Xn is contractible.

This implies that for any torsion-free subgroup Γ ⊂ Out(Fn), Γ\Xn is a BΓ- space, and hence cd Γ ≤ 3n − 4. Consequently, we have an upper bound on the virtual cohomological dimension of Outn(Fn):

vcd Outn(Fn) ≤ dim Xn =3n − 4. Since the total sum of some nontrivial loops could become arbitrarily small and go to 0, the quotient Out(Fn)\Xn is noncompact. Therefore, Out(Fn)isan analogue of non-uniform arithmetic subgroups, and the above upper bound on vcd Out(Fn) is unlikely to be sharp. Indeed, let Kn be the spine of Xn, i.e., the geometric realization of the partially ordered set of open simplices of Xn, which can be canonically realized as a subset of Xn. It turns out to be a simplicial complex of dimension 2n − 3, and Out(Fn) leaves Kn stable and acts properly on it with the compact quotient Out(Fn)\Kn [CV]. Another important result in [CV, Theorem 6.1.1] is the following.

Theorem 9.2. The spine Kn is an equivariant deformation retract of Xn and hence is contractible. Together with Proposition 3.2, it implies the following corollary.

Proposition 9.3. For every torsion-free subgroup Γ ⊂ Out(Fn), Γ\Kn is a ﬁnite CW-complex BΓ-space, and hence Γ is of type FP and cd Γ ≤ 2n − 3. Consequently, Out(Fn) is of type FP∞,andvcd Out(Fn) ≤ 2n − 3. The following observation of Gersten [CV, p. 93] shows that the upper bound on vcd Out(Fn) in the above theorem is sharp.

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Proposition 9.4. For every n ≥ 2, Out(Fn) contains a subgroup isomorphic 2n−3 2n−3 to Z , which implies vcd Out(Fn) ≥ vcd Z =2n − 3. Consequently,

vcd Out(Fn)=2n − 3. This subgroup can be written down explicitly [CV, p. 93].

10. Duality properties of outer automorphism groups Given the duality results for arithmetic subgroups and mapping class groups, it is natural to expect that similar duality results hold for Out(Fn). In [BF, Theorems 1.1 and 1.4], by applying a version of Morse theory to a partial compactiﬁcation of the outer space Xn, which is an analogue of the Borel- BS Serre partial compactiﬁcation X of symmetric spaces X, Bestvina and Feighn proved the following result, using the algebraic criterion in Proposition 4.2.

Theorem 10.1. The outer group Out(Fn) is (2n−5)-connected at inﬁnity, and i hence for any torsion-free subgroup Γ of Out(Fn) of ﬁnite index, H (Γ, ZΓ) = 0 for i<2n − 3;fori =2n − 3, H2n−3(Γ, Z) is a free abelian group. Consequently, Out(Fn) is a virtual duality group of dimension 2n − 3.

Since Out(Fn)\Xn is noncompact and Out(Fn) is similar to a non-uniform arithmetic subgroup, the following result is naturally expected.

Theorem 10.2. For every n ≥ 2, the group Out(Fn) is not a virtual Poincar´e duality group.

Proof. Suppose that the opposite is true, i.e., that Out(Fn)isavirtual Poincaré duality group. Then there exists a torsion-free subgroup Γ of Out(Fn)of − ∼ finite index which is a Poincaré duality group, i.e., H2n 3(Γ, ZΓ) = Z. By Corollary 4.9, it suffices to find an infinite index subgroup Γ of Γ such that cd Γ =2n − 3=cdΓ. Indeed, this will lead to a contradiction with the assumption that Γ is a Poincaré duality group. 2n−3 By Proposition 9.4, Out(Fn) contains a subgroup Λ isomorphic to Z .We claim that Γ also contains a free abelian subgroup of rank 2n − 3, which clearly has the cohomological dimension equal to 2n − 3. To prove this, consider the left- multiplication of Λ on the finite set Out(Fn)/Γ. The stabilizer in Λ of the identity coset Γ is equal to Γ ∩ Λ and clearly has finite index in Λ (note that the Λ-orbit through the coset Γ is finite). This implies that Γ ∩ Λ is also a free abelian group of rank 2n − 3.

Remark 10.3. In the previous cases of arithmetic subgroups of linear algebraic groups and mapping class groups, the dualizing modules can be identified with some Steinberg modules, which are defined in terms of the top non-vanishing homology group of natural simplicial complexes, and the duality property follows from the Poincaré duality for manifolds with boundary. In the case of Out(Fn), the dualizing module does not have such a concrete interpretation yet, and the duality is also proved using the algebraic criterion in Proposition 4.2. Since Xn is not a manifold, probably one could not appeal to the Poincaré duality for manifolds. On the other hand, it is an important and natural problem to find an analogue of the spherical Tits building ∆Q(G) and the curve complex C(S) which describes the geometry at

STEINBERG REPRESENTATIONS AND DUALITY PROPERTIES OF GROUPS95 25 inﬁnity of Xn and whose homology group gives the dualizing module. One such candidate is the factor complex in [HV]. Remark 10.4. There is no doubt that Theorem 10.2 was expected by many people and should be known to many experts. On the other hand, it has not been written down explicitly before. It was observed by the author while trying to list similar properties of the three classes of groups discussed in this paper. From such listing, this result is natural.

Let x1, ··· ,xn be a free basis of Fn. The symmetric automorphism subgroup Σn of Aut(Fn) consists of those automorphisms that send each generator xi to a conjugate of some xj. The pure symmetric automorphism group, denoted by P Σn, is the subgroup of symmetric automorphisms that send each xi to a conjugate of xi. The image of P Σn in Out(Fn) is called the outer pure symmetric automorphism group and denoted by OPΣn. After a preliminary version of this paper was submitted, we learnt the following result [BMM].

Theorem 10.5. The group OPΣn is a duality group of dimension n − 2,and the group P Σn is a duality group of dimension n − 1.

By [Co, Theorem 5.1], the group OPΣn contains a free abelian subgroup of rank n − 2, and the group P Σn contains a free abelian subgroup of rank n − 1. Then by the same argument as in the proof of Theorem 10.2, we can prove the following result.

Theorem 10.6. For n ≥ 3, the groups OPΣn and P Σn are not Poincaré duality groups. Remark 10.7. The above arguments show that if a virtually non-abelian group ΓisoftypeFP and of cohomological dimension d but contains a free abelian subgroup of rank d, then Γ is not a virtual Poincaré duality group. Besides the examples discussed above, other groups satisfying this condition includes the partial symmetric outer automorphism group P Σ(n, k), which consists of automorphisms which send the first k generators of {x1, ··· ,xn} to conjugates of themselves (see [BCV] for details). Therefore, it follows that P Σ(n, k)isnota virtual Poincare duality group either.

11. Comments and conjectures In the above sections, we have shown that three important classes of groups are duality groups. It is natural to consider which group operations preserve the class of duality groups. Clearly, if Γ1 and Γ2 are both duality groups which admit EΓ1 and EΓ2 that are manifolds satisfying the conditions in Propisition 4.5, then EΓ1 × EΓ2 is a E(Γ1 × Γ2)-space satisfying the conditions of Proposition 4.5 as well. This implies that Γ1 × Γ2 is also a duality group. (Note that EΓ1 × EΓ2 is a manifold with corners. But the corners can be smoothed out as pointed out in Remark 4.6.) In fact, a stronger result is true. Specifically, by [Bi, Theorem 9.10], the class of duality groups is closed under extension. Specifically, assume that Γ1, Γ2 are duality groups of dimensions n1,n2, and that Γ fits into an exact sequence:

1 → Γ1 → Γ → Γ2 → 1, then Γ is a duality group of dimension n1 + n2.

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An important procedure to construct a new group from a given group Γ is to consider the outer automorphism group Out(Γ) = Aut(Γ)/Inn(Γ). Since Inn(Γ) is isomorphic to Γ/Z(Γ), where Z(Γ) is the center of Γ, it is reasonable to divide out Inn(Γ) in order to get a new group. Based on the examples discussed in this paper, it is reasonable to formulate the following general conjecture. Conjecture 11.1. If Γ is a duality group, then Out(Γ) is also a duality group. Consequently, if Z(Γ) is trivial, then Aut(Γ) is also a duality group. In fact, if Z(Γ) is trivial, then Aut(Γ) fits into an exact sequence 1 → Γ → Aut(Γ) → Out(Γ) → 1, and the duality of Out(Γ) implies the duality of Aut(Γ). Remark 11.2. It might be worthwhile to point out that when Γ is a duality group, subgroups and quotient subgroups of Γ are usually not virtual duality groups. For example, the free group Fn contains subgroups and are also mapped onto some quotient groups which are not finitely generated. Since duality groups are of type FP (Proposition 4.1) and hence finitely generated, the above subgroups and quotient groups are not duality groups. Now we examine cases where the above conjecture holds. It is known that the free group Fn is a duality group [Br1, p. 223, Example 5]. (Another geometric way to understand this is to note that if Γ ⊂ SL(2, R) is a torsion-free non-uniform lattice subgroup, then Γ is a free group on finitely many generators and the surface Γ\SL(2, R)/SO(2) admits a Borel-Serre compactification, which implies that Γ is a duality group of dimension 1 as in §6.) By the results in §10, Out(Fn)isavirtual duality group. Suppose that Γ = Zn, the free abelian group, then Out(Zn)=GL(n, Z)andis a virtual duality group by §6. Suppose that S is a compact surface of genus g ≥ 1andΓ=π1(S), a surface group. Then it is known that Out(π1(S)) = Mod(S)=Modg [I1, Theorem 2.9.A]. By the results of §8, Out(π1(S)) is a virtual duality group. (As pointed out before, if S is non-compact, then π1(S) is a free group.) If Γ is a finitely generated nilpotent group, it is a virtual duality group since it admits a torsion-free subgroup of finite index and then by applying either the discussions in §4 or the above result that the class of duality groups is closed under extensions, together with the fact that Z is a duality group. By [Se], Out(Γ) is a finite extension of an arithmetic group. Therefore, Out(Γ) is a virtual duality group. (By an arithmetic group, we mean a group that is isomorphic to an arithmetic subgroup of a linear algebraic group defined over Q.) If Γ is a polycyclic-by-finite group, then Γ is a virtual duality group by the same arguments as in the previous paragraph, i.e., the class of duality groups is closed under extension. By [BaG], Out(Γ) is an arithmetic group, and hence by §6, Out(Γ) is also a virtual duality group. If Γ is an arithmetic subgroup of a linear semisimple algebraic G over Q,then Γ is a virtual duality group by the results in §6. Assume that the associated symmetric space X = G/K is not the Poincaré upper half-plane, then the Mostow strong rigidity implies that Out(Γ) = Out(G) is a finite group and hence is a virtual duality group. If Γ is an arithmetic subgroup of G = SL(2) in the exceptional case,

STEINBERG REPRESENTATIONS AND DUALITY PROPERTIES OF GROUPS97 27 then Out(Γ) is essentially a mapping class group of the surface Γ\SL(2, R)/SO(2) and is a virtual duality group as well. The same result holds for any lattice subgroup of a semisimple (or more generally a reductive) Lie group G. Given the above discussions, it is conceivable that if G is any (virtually) connected Lie group, which is not necessarily reductive, and Γ is a lattice subgroup of G,thenOut(Γ) is a virtual duality group.

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[Ha1] J.Harer, The virtual cohomological dimension of the mapping class group of an orientable surface, Invent. Math. 84 (1986) 157–176. [Ha2] J.Harer, The cohomology of the moduli space of curves,inTheory of moduli (Montecatini Terme, 1985), pp. 138–221, Lecture Notes in Math., 1337, 1988. [Hav] W.Harvey, Boundary structure of the modular group,inRiemann surfaces and related topics, pp. 245–251, Ann. of Math. Stud., 97, Princeton Univ. Press, 1981. [HV] A.Hatcher, K.Vogtmann, The complex of free factors of a free group, Quart. J. Math. Oxford Ser. 49 (1998) 459–468. [Hu] J.Humphreys, The Steinberg representation, Bull. Amer. Math. Soc. 16 (1987) 247–263. [I1] N.Ivanov, Mapping class groups,inHandbook of geometric topology, pp. 523–633, North- Holland, Amsterdam, 2002. [I2] N.Ivanov, Complexes of curves and Teichmüller modular groups, Russian Math Surveys, V. 42, No. 3 (1987) 55-107. [I3] N.Ivanov, Attaching corners to Teichmüller space, Leningrad Math. J. 1 (1990) 1177–1205. [I4] N.Ivanov, Complexes of curves and Teichmüller spaces, Math. Notes 49 (1991), no. 5-6, 479–484. [IJ] N.Ivanov, L.Ji, Infinite topology of curve complex and non-Poincaré duality of mapping class group, to appear in L’Enseignement Mathématique. [Ive] B.Iversen, Cohomology of sheaves, Universitext. Springer-Verlag, Berlin, 1986. xii+464 pp. [Ji] L.Ji, Buildings and their applications in geometry and topology, Asian J. Math. 10 (2006) 11–80. [M] A.Malcev, On a class of homogeneous spaces, Amer. Math. Soc. Translation 1951, no. 39, 33 pp. [Ma] W.Massey, Homology and cohomology theory. An approach based on Alexander-Spanier cochains, Monographs and Textbooks in Pure and Applied Mathematics, Vol. 46, Marcel Dekker, Inc., 1978. xiv+412 pp. [Mu] J.Munkres, Elements of algebraic topology, Addison-Wesley Publishing Company, 1984. ix+454 pp. [Ra1] M.S.Raghunathan, Discrete subgroups of Lie groups, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 68. Springer-Verlag, 1972. ix+227 pp. [Ra2] M.S.Raghunathan, A note on quotients of real algebraic groups by arithmetic subgroups, Invent. Math. 4 (1967/1968) 318–335. [Sa] L.Saper, Tilings and finite energy retractions of locally symmetric spaces, Comment. Math. Helv. 72 (1997) 167–202. [Se] D.Segal, On the outer automorphism group of a polycyclic group,inProceedings of the Second International Group Theory Conference, (Bressanone, 1989), Rend. Circ. Mat. Palermo (2) Suppl. No. 23 (1990), 265–278. [So] L.Solomon, The Steinberg character of a finite group with BN-pair,inTheory of Finite Groups, pp. 213–221, Benjamin, New York, 1969. [S] N.Steenrod, Homology with local coefficients, Ann. of Math. 44 (1943) 610–627. [St] R.Steinberg, Robert Steinberg collected papers, American Mathematical Society, 1997. xx+599 pp. [Str] R.Strebel, A remark on subgroups of infinite index in Poincaréduality groups, Comment. Math. Helvetici. 52 (1977) 317–324. [To] A.Toth, On the Steinberg module of Chevalley groups, Manuscripta Math. 116 (2005) 277–295. [V1] K.Vogtmann, Automorphisms of free groups and outer space, Geom. Dedicata 94 (2002) 1–31. [V2] K.Vogtmann, The cohomology of automorphism groups of free groups, Proc. of Interna- tional Congress of Mathematicians. Vol. II, pp. 1101–1117, Eur. Math. Soc., 2006.

Department of Mathematics, University of Michigan, Ann Arbor, MI 48109 E-mail address: [email protected]

Contemporary Mathematics Contemporary Mathematics Volume 478, 2009 Volume 00, XXXX

Characters of simplylaced nonconnected groups versus characters of nonsimplylaced connected groups

Shrawan Kumar, George Lusztig and Dipendra Prasad

Abstract. Let G be a connected, simply-connected, almost simple semisimple group over C of simplylaced type and let σ be a nontrivial diagram automorphism of G.LetGσ be the (disconnected) group generated by G and σ. As a consequence of a theorem of Jantzen the character of an irreducible representation of Gσ (also irreducible on G)onGσ can be expressed in terms of a character of an irreducible representation of a certain connected simply connected semisimple group Gσ of nonsimplylaced type. We show how Jantzen’s theorem can be deduced from properties of the canonical bases.

Let G be a connected, simply-connected, almost simple algebraic group of simplylaced type over C.LetT be a maximal torus of G.Letxi : C −→ G, yi : C −→ G (i ∈ I) be homomorphisms which together with T form a pinning (épinglage) of G. We fix a nontrivial automorphism σ of G such that σ(T )=T ,andsuchthat → ˜ for some permutation i i of I we have σ(xi(a)) = x˜i(a), σ(yi(a)) = y˜i(a)for all a ∈ C.Fori ∈ I we write σ(i)=˜i.Letσ be the finite subgroup of the automorphism group of G generated by σ and let Gσ be the semidirect product of G with σ. Let X be the group of characters T −→ C∗;letY be the group of one parameter subgroups C∗ −→ T and let , : Y × X −→ Z be the standard pairing. For i ∈ I we −1 −1 −1 define αi ∈ X by xi(αi(t)) = txi(1)t , yi(αi(t) )=tyi(1)t for all t ∈ T .This is a root of G.Letˇαi ∈ Y be the corresponding coroot. Note that (a) (Y,X,, , αˇi,αi(i ∈ I)) is the root datum of G.Nowσ induces automorphisms of X, Y denoted again by σ; these are compatible with , and we have σ(αi)=ασ(i),σ(ˇαi)=ˇασ(i) for i ∈ I. + Let X = {λ ∈ X; αˇi,λ∈N∀i ∈ I}. σ We set Yσ = Y/(σ −1)Y , X = {λ ∈ X; σ(λ)=λ}.Notethat, : Y ×X −→ Z σ induces a perfect pairing Yσ × X −→ Z denoted again by , .LetIσ be the set of σ-orbits on I. For any O∈Iσ letα ˇO ∈ Yσ be the image ofα ˇi under Y −→ Yσ where O { ∈ } { O∈ } i is any element of .Since αˇi; i I is a Z-basis of Y we see that αˇO; Iσ is O∈ h ∈ σ a Z-basis of Yσ. For any Iσ let αO =2 i∈O αi X where h is the number of unordered pairs (i, j) such that i, j ∈O,andαi + αj is a root. Note that h =0

The ﬁrst author was supported in part by NSF Grant

ccXXXX2009 American American Mathematical Mathematical Society 991

2100 KUMAR, LUSZTIG, AND PRASAD

except when G is of type A2n when h =0forallO but one and h =1foroneO. Note that σ (b) (Yσ, X, , , αˇO,αO(O∈Iσ)) σ + σ is a root datum, see [Ja, p.29]. Let X = {λ ∈ X; αˇO,λ∈N∀O ∈ Iσ} = σ + X ∩ X .LetGσ be the connected semisimple group over C with root datum (b). By definition, Gσ is provided with an épinglage (Tσ,xO,yO (O∈Iσ)where ∗ −1 Tσ := C ⊗ Yσ = T/{σ(t)t ; t ∈ T } is a maximal torus of Gσ and xO : C −→ Gσ, −→ −1 −1 −1 yO : C Gσ satisfy xO(αO(t1)) = t1xO(1)t1 , yO(αO(t1) )=t1yO(1)t1 for all σ ∗ t1 ∈ Tσ.(Wehave X =Hom(Tσ, C ) canonically.) ∼ L L σ 0 L Note that Gσ is simply connected and that Gσ = ((( G) ) )where () denotes the Langlands dual group and (LG)σ denotes the fixed point set of the automor- L L σ σ phism of G induced by σ.NowGσ is only isogenous to (G )whereG is the fixed point set of σ : G −→ G. Let λ ∈ σX+.Wecanviewλ both as a character of T and as a character of Tσ.LetV (resp. V ) be a finite dimensional complex irreducible representation of G (resp. Gσ) with a non-zero vector η (resp. η ) such that xi(a)η =0forall i ∈ I,a ∈ C (resp. xO(a)η =0forallO∈Iσ,a∈ C)andtη = λ(t)η for all t ∈ T (resp. t η = λ(t )η for all t ∈ Tσ). Now V can be regarded as a representation of Gσ whose restriction to G is as above and on which the action of σ satisfies σ(η)=η. Let µ ∈ X.LetVµ = {x ∈ V ; tx = µ(t)x ∀t ∈ T }.Notethatσ : V −→ V permutes the weight spaces Vµ among themselves. A weight space Vµ is σ-stable if σ and only if µ ∈ X;inthiscaseµ can be viewed as a character of Tσ and we set { ∈ ∀ ∈ } Vµ = x V ; t x = µ(t )x t Tσ . σ Theorem (Jantzen [Ja, Satz 9]). For µ ∈ X we have tr(σ : Vµ −→ Vµ)= dim Vµ.

Corollary. Let : T −→ Tσ be the canonical homomorphism. For any t ∈ T we have tr(tσ : V −→ V )=tr((t),V). The corollary describes completely the character of V on Gσ in terms of the character of V since any semisimple element in Gσ is G-conjugate to an element of the form tσ with t ∈ T . Note also that there is a well defined bijection between the set of semisimple G-conjugacy classes in Gσ and the set of semisimple Gσ- conjugacy classes in Gσ which for any t ∈ T maps the G-conjugacy class of tσ to the Gσ-conjugacy class of (t); see [L2, 6.26], [Mo]. We now show (assuming that G is not of type A2n) how Jantzen’s theorem can be deduced from properties of canonical bases in [L1]. According to [L1], V has a canonical basis Bλ and V has a canonical basis Bλ.Also,Bλ (resp. Bλ) can be naturally viewed as a subset of B (resp. B), the canonical basis of the + part of the universal enveloping algebra attached to the root datum (a) (resp. (b)). Now σ acts naturally on B (preserving the subset Bλ)and[L1, Theorem 14.4.9] provides a canonical bijection between B and the fixed point set of σ on B. (This theorem is applicable since the Cartan datum of (b) is obtained from the Cartan datum of (a) by the general ”folding” procedure [L1, 14.1] which appplies to any simplylaced Cartan datum of not necessarily finite type together with an admissible automorphism; here we use that G is not of type A2n.) This restricts to σ a bijection between Bλ and the fixed point set Bλ of σ on Bλ.Nextwenotethat Bλ (resp. Bλ) is compatible with the decomposition of V (resp. V )intoweight ∩ spaces and from the definitions we see that the bijection above carries Bλ Vµ

CHARACTERS OF SIMPLYLACED NONCONNECTED GROUPS 1013

σ bijectively onto Bλ ∩ Vµ.SinceBλ ∩ Vµ is a basis of Vµ which is σ-stable we have −→ σ ∩ ∩ tr(σ : Vµ Vµ)=( Bλ Vµ). Using the bijection above this equals (Bλ Vµ) ∩ and this is equal to dim Vµ since Bλ Vµ is a basis of Vµ. This gives the desired result. We refer the reader to [FSS, FRS, NS, N1, N2, We] for other approaches to Jantzen’s theorem. We thank S. Naito and the referee for pointing out these references to us. The ﬁrst two authors were supported in part by the National Science Founda- tion. The third author thanks the Institute for Advanced Study where this work was done, and gratefully acknowledges receiving support through grants to the Institute by the Friends of the Institute, and the von Neumann Fund.

References [FSS] J. Fuchs, B. Schellekens and G. Schweigert, From Dynkin diagrams symmetries to fixed point structures, Comm. Math. Phys. 180 (1996), 39-97. [FRS] J. Fuchs, U. Ray and G. Schweigert, Some automorphisms of generalized Kac-Moody algebras,J.Algebra191 (1997), 518-590. [Ja]J.C.Jantzen,Darstellungen Halbeinfacher Algebraischer Groupen,, Bonner Math. Schriften 67 (1973). [L1] G. Lusztig, Introduction to quantum groups, Progress in Math., vol. 110, Birkhäuser, 1993. [L2] G. Lusztig, Classification of unipotent representations in simple p-adic groups, II, Repre- sent. Theory (2002), 243-289. [Mo] S. Mohrdieck, Conjugacy classes of non-connected semisimple algebraic groups, Transfor. Groups 8 (2003), 377-395. [N1] S.Naito, Twining character formulas of Borel-Weil-Bott type, J. Math. Sci. Univ. Tokyo 9 (2002), 637-658. [N2] S. Naito, Twining characters, Kostant’s homology formula and the Bernstein-Gelfand- Gelfand resolution, J. Math. Kyoto Univ. 42 (2002), 83-103. [NS] S. Naito and D. Sagaki, Lakshmibai-Seshadri paths fixed by a diagram automorphism,J. Algebra 245 (2001), 395-412. [We] R. Wendt, Weyl’s character formula for non connected Lie groups and orbital theory for twisted affine Lie algebras, J. Funct. Anal. 180 (2001), 31-65.

S.K.: Department of Mathematics, University of North Carolina, Chapel Hill, NC 27599-3250, USA

G.L.: Department of Mathematics, M.I.T., Cambridge, MA 02139, USA E-mail address: [email protected]

D.P.: School of Mathematics, Tata Institute of Fundamental Research, Colaba, Mumbai 400005, India, and The Institute for Advanced Study, Princeton, NJ 08540, USA

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Contemporary Mathematics Volume 478, 2009

Classiﬁcation of Finite-dimensional Basic Hopf Algebras According to Their Representation Type

Gongxiang Liu

Abstract. The main aim of this paper is to give the classification of finite- dimensional basic Hopf algebras according to their representation type. We attach every finite-dimensional basic Hopf algebra H a natural number nH , which will help us to determine the representation type of H.Theclassof finite-dimensional basic Hopf algebras of finite representation type is determined completely. A complete list of local Frobenius algebras of tame type is given. By using this list, we get all possible algebraic structures of tame basic Hopf algebras.

Contents

1. Introduction 2. Representation type of basic Hopf algebras 3. Classification of basic Hopf algebras of finite representation type 4. Classification of basic Hopf algebras of tame type Acknowledgement References

1. Introduction 1.1. Throughout this paper k denotes an algebraically closed ﬁeld. All spaces are k-spaces. By an algebra we mean an associative algebra with identity element. For an algebra A, JA denotes its Jacobson radical. We freely use the results, notation, and conventions of [49].

1.2. The classiﬁcation of Hopf algebras is one of central problems of Hopf algebra theory. The ﬁrst celebrated result on this problem is now known as the following Cartier-Kostant-Milnor-Moore theorem.

1991 Mathematics Subject Classiﬁcation. Primary 16W30, 16G30; Secondary 16G20. Key words and phrases. Basic Hopf algebra, representation type, Nakayama algebra, Frobe- nius algebra.

c XXXXc 2009 American Mathematical Society 1031

2104 GONGXIANG LIU

Theorem 1.1. A cocommutative Hopf algebra over an algebraically closed field k of characteristic 0 is a semidirect product of a group algebra and the enveloping algebra of a Lie algebra. In particular, a finite-dimensional cocommutative Hopf algebra over k is a group algebra. 1.3. In recent years, some substantial classification results in the infinite- dimensional case are given. All possible cotriangular Hopf algebras were determined [24], and a class of pointed Hopf algebras with finite Gelfand-Kirillov dimension is classified [2][5]. Lu, Wu and Zhang introduced the concept of homological integral, which generalizes the usual integral defined for finite-dimensional Hopf algebras to a large class of infinite-dimensional Hopf algebras, and use it to research particularly noetherian affine Hopf algebras of Gelfand-Kirillov dimension 1 [43]. Although these results shed some light on the structure of infinite-dimensional Hopf algebras, it is still very hard to handle infinite-dimensional Hopf algebras in general, and the classification of finite-dimensional Hopf algebras is of more interest for us. We can use the following diagram (see [1]) to explain the general procedure to classify finite-dimensional Hopf algebras.

semisimple (⇔ H = H/JH ) 

Classification of fin.-dim. H/J Hopf algebra H H is a Hopf algebra @ @R non-semisimple @ @R others Over the last decade, under various assumption, considerable progress has been made in classifying finite-dimensional Hopf algebras. To the author’s knowledge, the classification of finite-dimensional Hopf algebras mainly consists of the following four aspects: (1) Classification of semisimple cosemisimple Hopf algebras; (2) Classification of non-semisimple Hopf algebras; (3) Classification of all Hopf algebras of a prescribed dimension; (4) Classification of triangular Hopf algebras.

1.4. For (1), if the characteristic of k is 0, then we know that a Hopf algebra is semisimple is equivalent to that it is cosemisimple by [39]. By a beautiful result of Etingof and Gelaki [20], problems in positive characteristic can be reduced to similar problems in characteristic 0. Therefore, we often consider the classiﬁcation of semisimple Hopf algebras over an algebraically closed ﬁeld of characteristic 0. Some semisimple Hopf algebras of special dimensions, particularly for dimensions p, p2,p3,pq,pq2 [22][30][45][46][52][53][65] and dimensions ≤ 60 [54], were inten- sively studied. For example, semisimple Hopf algebras of dimension pq are shown to be trivial. That is, they are isomorphic to group algebras or dual group algebras. The class of semisimple Hopf algebras that are simple as Hopf algebras is researched recently [27][28]. Under a systematic study of of fusion categories, Etingof, Dik- shych and Ostrik asked an interesting question (Question 8.45 in [25]) about the

CLASSIFICATION OF BASIC HOPF ALGEBRAS 1053 semisimple Hopf algebras: dose there exist a finite-dimensional semisimple Hopf algebra whose representation category is not group-theoretical? This question was answered affirmatively by Nikshych [59]. But, the classification of semisimple Hopf algebras is still a widely open question. We refer to the survey papers [1][50]fora more detailed exposition.

1.5. For (2), substantial results in this case are known for the class of pointed Hopf algebras over an algebraically closed field of characteristic 0. There are two different methods which were used to classify pointed Hopf algebras. One method was formulated by N.Andruskiewitsch and H.-J.Schneider. They reduced the study of pointed Hopf algebras to the study of Nichols algebras via bosonization given by Radford [61]andMajid[44]. This method gets great success. One of most remarkable properties of this method is that it allows Lie theory to enter into the picture through quantum groups [7]. Many new examples about pointed Hopf algebras were found through this way. It can also help us to give counterexamples for Kaplansky’s Conjecture 10. For details see [6][7][9][31][33][34][35][36]. Recently, Andruskiewitsch and Schneider have classified all finite-dimensional pointed Hopf algebras whose group of group-like elements G(H) is abelian such that all prime divisors of the order of G(H)are> 7. See [10]. Another method, mainly due to Pu Zhang and his co-workers, is to use quivers and their representation theory. This method depends heavily on one of Cibils- Rosso’s conclusions [14]. One of merits of this method is that it introduces the combinatorial methods to enter into the field of the classification of pointed Hopf algebras. By using this method, the classification of so called Monomial Hopf algebras was gotten. Locally finite simple-pointed Hopf algebras can also be classified. For details see [13][60].

1.6. For (3), the starting point of this direction should be the following Y.Zhu’s result [65]. Theorem 1.2. Let p be a prime number number and k an algebraically closed field of characteristic 0. Then a Hopf algebra of dimension p over k is necessarily semisimple and isomorphic to the group algebra of Zp. By using [8]and[48], S-H. Ng [55] classified all Hopf algebras of dimension p2 and showed that they are the group algebras and the Taft algebras. For dimension pq with p = q, a folklore conjecture says that such Hopf algebras are semisimple. If it is true, the results given in Subsection 1.4 imply that Hopf algebras of dimension pq,wherep and q are distinct, are trivial. This conjecture was verified for some particular values of p and q [4][12][23][56][57][58]. There are other classification results in low dimension. All Hopf algebras of dimension ≤ 15 were classified and the most recent result in dimension 16 is [29]. See [29] and references therein.

1.7. For (4), the works of N. Andruskiewitsch, P. Etingof and S. Gelaki should be considered the most important. P. Etingof and S. Gelaki indeed show that semisimple triangular Hopf algebras are very closed to group algebras [21]. The structure of minimal triangular Hopf algebras is also given [3]. There are some nice surveys about classiﬁcation of ﬁnite-dimensional Hopf algebras, see for instance [1].

4106 GONGXIANG LIU

1.8. According to the fundamental theorem of Drozd [17], the category of finite-dimensional algebras over k can be divided into disjoint classes of finite representation, tame and wild algebras. This fact stimulates us to classify finite- dimensional Hopf algebras through their representation type. In order to realize this idea, we need add some conditions on the Hopf algebra H: (1): We assume that H/JH is a quotient Hopf algebra. By the general procedure to classify finite-dimensional Hopf algebras, this requirement is not strange. Note that H satisfies this condition if and only if the coradical of the dual Hopf algebra H∗ is a Hopf subalgebra of H∗. (2): By 1.4, the classification of semisimple Hopf algebras is still a widely open question. This suggests us that we should consider semisimple Hopf algebra H/JH which can be described easily. A good candidate satisfying these conditions is the class of basic Hopf algebras. That is, as an algebra, it is basic. This implies that H/JH is a Hopf algebra automatically (see Lemma 1.1 in [32]). Since the field k is algebraically closed and ∼ ∗ H/JH is a Hopf algebra, the condition “basic” implies that H/JH = (kG) for some finite group G. Before giving the classification of finite-dimensional basic Hopf algebras, we should give an effective way to determine their representation type at first. This is indeed what we will do in the next section. Explicitly, we can attach to every finite-dimensional basic Hopf algebra H a natural number nH and prove that (i) H is of finite representation type if and only if nH =0ornH = 1; (ii) if H is tame, then nH = 2 and (iii) if nH ≥ 3, then H is wild. The dual of a basic Hopf algebra is a pointed Hopf algebra, and vice versa. So, all classification results on pointed Hopf algebras (some of them mentioned in subsection 1.5) can be applied by duality to basic Hopf algebras. But, I think, there is no possibility to give structures of all basic Hopf algebras. Inspired by the case of path algebras, it is quite natural to give the classification of finite-dimensional basic Hopf algebras of finite representation type and tame type, and Section 3 and Section 4 are devoted to classifying finite-dimensional basic Hopf algebras of finite representation type and tame type respectively. Indeed, in Section 3, the class of finite-dimensional basic Hopf algebras of finite representation type is classified completely (see Theorem 3.1). In the case of tame type, we give a list of algebras which contains all possible tame basic Hopf algebras in Section 4. Notice that it is still a problem to give the actual determination of tame basic Hopf algebras in this list (see Problem 4.1 in Section 4).

1.9. A finite-dimensional algebra A is said to be of finite representation type provided there are finitely many non-isomorphic indecomposable A-modules. A is of tame type or A is a tame algebra if A is not of finite representation type, whereas for any dimension d>0, there are finite number of A-k[T ]-bimodules Mi which are free as right k[T ]-modules such that all but a finite number of indecomposable A-modules of dimension d are isomorphic to Mi ⊗k[T ] k[T ]/(T − λ)forλ ∈ k.We say that A is of wild type or A is a wild algebra if there is a finitely generated A- k-bimodule B whichisfreeasarightk-module such that the functor B ⊗k − from mod-k, the category of finitely generated k-modules, to mod-A, the category of finitely generated A-modules, preserves indecomposability and reflects isomorphisms. See [18] for more details.

CLASSIFICATION OF BASIC HOPF ALGEBRAS 1075

For other unexplained notations about representation theory of ﬁnite-dimensional algebras in this paper, see [11][18].

2. Representation type of basic Hopf algebras In the rest of this paper, all algebras are assumed to be finite-dimensional. In this section, the definition of the covering quiver ΓG(W ), introduced by Green and Solberg [32], is given at first. Then we observe that we can associate to this covering quiver ΓG(W ) a natural number n , which can help us to determine ΓG(W ) the representation type of the finite-dimensional algebra A whose Ext-quiver is ΓG(W ). For a finite-dimensional basic Hopf algebra H,itisknownthatitsExt- quiver is a covering quiver. So the above results can be applied to the case of finite-dimensional basic Hopf algebras directly.

Definition 2.1. Let G be a ﬁnite group and let W =(w1,w2,...,wn) be a sequence of elements of G.WesayW is a weight sequence if, for each g ∈ G,the −1 −1 −1 sequences W and (gw1g ,gw2g ,...,gwng ) are the same up to a permutation. Deﬁne a quiver, denoted by ΓG(W ), as follows. The vertices of ΓG(W ) is the set {vg}g∈G and the arrows are given by

{ −1 → −1 | ∈ } (ai,g): vg vwig i =1, 2,...,n,g G . We call this quiver the covering quiver (with respect to G and W ). Example 2.1. (1): Let G =,gn =1andW =(g). The corresponding covering quiver is

•v1H (a1,g) 3 H (a1, 1) H HHj • vgn−1 • vg 6 −1 2 (a1,g ) (a1,g ) ? • v n−2 • HYg vg2 H · · (a ,g3) H · 1 • v·gn−3

We call such quiver a basic cycle of length n. (2): Let G = K4 = {1,a,b,ab}, the Klein four group, and W = (1). Then the corresponding covering quiver is •1 , •a , •b , •ab 

For a covering quiver ΓG(W ), define nΓG(W ) to be the length of W . For an algebra A, it is Morita equivalent to a unique basic algebra B(A) and for this basic algebra B(A), the Gabriel’s theorem says that there exists a unique quiver Q and an admissible ideal I (i.e. J N ⊆ I ⊆ J 2 where J is the ideal generated by all arrows ∼ of Q) such that B(A) = kQ/I.See[11]. This quiver is called the Ext-quiver of A. It is known that for a finite quiver Q, the path algebra kQ is of finite representation type if and only if the underlying graph Q of Q is one of Dynkin diagrams: An,Dn,E6,E7,E8, and is of tame type if and only if the underlying graph Q is one of Euclidean diagrams: An, Dn, E6, E7, E8. For details, see [11][63]. These facts will be used freely in the proof of the following conclusion.

6108 GONGXIANG LIU

Theorem . 2.1 Let ΓG(W ) be a covering quiver, nΓG(W ) deﬁned as the above and assume A is an algebra with Ext-quiver ΓG(W ).Then

(i) A is of ﬁnite representation type if and only if nΓG(W ) =0or nΓG(W ) =1;

(ii) A is tame only if nΓG(W ) =2; ≥ (iii) If nΓG(W ) 3,thenA is wild. Proof. (i): “If part: ” When nΓG(W ) = 0, there is no any arrow in nΓG(W ). This implies A is a semisimple algebra and so is of finite representation type. When nΓG(W ) =1,ΓG(W ) is a finite union of basic cycles. It is well known that a basic algebra is Nakayama if and only if its Ext-quiver is An or a basic cycle. Thus the basic algebra of A is a Nakayama algebra. Since every Nakayama algebra must be of finite representation type ( [11], p. 197) and A is Morita equivalent to its basic algebra, A is of finite representation type. “Only if part: ” It is sufficient to prove that A is not of finite representation ≥ type if nΓG(W ) 2. In order to prove this, it is enough to consider the case ≥ nΓG(W ) = 2 (in fact, we will show later that if nΓG(W ) 3, then A is wild). We denote the basic algebra of A by B(A). We need only to prove that B(A)isof ∼ infinite representation type. By the Gabriel’s theorem, kΓG(W )/I = B(A) for an admissible ideal I. Denote the ideal generating all arrows in kΓG(W )byJ.Bythe definition of admissible ideal, we have an algebra epimorphism 2 B(A) kΓG(W )/J . 2 Thus it is enough to prove that kΓG(W )/J is not of finite representation type. 2 2 Since the Jacobson radical of kΓG(W )/J is clearly 2-nilpotent, kΓG(W )/J is stably equivalent to the following hereditary algebra (see Theorem 2.4 in Chapter Xin[11]): kΓG(W )/J 0 Λ= 2 J/J kΓG(W )/J

The Ext-quiver of Λ is indeed the separated quiver of ΓG(W ) (see the proof of Theorem 2.6 in Chapter X of [11]). Assume W =(w1,w2). If w1 = w2, we can ﬁnd that the separated quiver of ΓG(W ) is a disjoint union of quivers of following form:

- i • - • j

This means Λ is not of ﬁnite representation type since clearly above quiver is a Kronecher quiver which is not a Dynkin diagram (see also Theorem 2.6 in Chapter Xof[11]). If w1 = w2,ΓG(W )s must contain the following sub-quiver:

i2 i1 • •

? 1 ? • • - • j2 j1

CLASSIFICATION OF BASIC HOPF ALGEBRAS 1097

Here 1 is the identity element of G.Ifi1 = i2,ΓG(W )s is not a Dynkin diagram and thus Λ is of inﬁnite representation type. If it is not, ΓG(W )s contains the following sub-quiver:

i2 i1 • • • - • j3 j4

? 1 ? • • - • j2 j1

If j4 = jl for l =1, 2, 3, ΓG(W )s is not a Dynkin diagram and thus Λ is of infinite representation type. If it is not, repeats above process and by the definition of covering quiver, there exit it,is or jt,js satisfying it = is or jt = js.Inaword, ΓG(W )s is not a Dynkin diagram and thus Λ is of infinite representation type. A celebrated result of H.Krause [38] states that two stably equivalent algebras have 2 the same representation type. Thus kΓG(W )/J is not of finite representation type since Λ is so. Therefore B(A) is not of finite representation type. ⇔ ≥ Clearly, (ii) (iii). So it is enough to prove (iii). Since nΓG(W ) 3, we assume W =(w1,w2,w3,...). Just like analysis of the “Only if part” of (i), we consider the 2 separated quiver of kΓG(W )/J .Ifw1 = w2, we have the following form sub-quiver of ΓG(W )s:

j •2 6

• - • 1 j1

2 It is clearly not a Euclidean diagram and thus kΓG(W )/J is a wild algebra. If w1 = w2, not loss generality, we can assume wi = wj for 1 ≤ i = j ≤ 3. This implies ΓG(W )s contains the following sub-quiver

 i21 j i11 • •3 • 6

- ? - ? i22 • • • • • i12 j2 1 j1 2 which is clearly not Euclidean diagram and thus kΓG(W )/J is a wild algebra. Therefore B(A)andthusA is a wild algebra. The following conclusion (see Theorem 2.3 in [32]) states the importance of covering quivers. Lemma 2.2. Let H be a ﬁnite-dimensional basic Hopf algebra over k.Then there exists a ﬁnite group G and a weight sequence W =(w1,w2,...,wn) of G, ∼ such that H = kΓG(W )/I for an admissible ideal I.

8110 GONGXIANG LIU

This result indeed tells us that the Ext-quiver of a basic Hopf algebra H must be a covering quiver ΓG(W ). By this, deﬁne nH := nΓG(W ).

Corollary 2.3. Let H be a finite-dimensional basic Hopf algebra and nH defined as above. Then (i) H is of finite representation type if and only if nH =0or nH =1; (ii) If H is tame, then nH =2; (iii) If nH ≥ 3,thenH is of wild type. We want to take this opportunity to give two applications of Theorem 2.1. The first one is to give a new proof of Theorem 3.1 in [42]: Corollary 2.4. [Theorem 3.1 in [42]] Let H be a finite-dimensional basic Hopf algebra. Then H is of finite representation type if and only if it is a Nakayama algebra. Proof. It is enough to prove the necessity since every Nakayama algebra must be of finite representation type. By the Theorem 2.1, we know that H is of finite representation type if and only if nH =0ornH =1.WhennH = 0, there is no arrow in ΓG(W ). This means H is semisimple and of course Nakayama. When nH =1,ΓG(W ) is a disjoint union of basic cycles and H is Nakayama too (see the first paragraph of the proof of Theorem 2.1).

The second one is to give an easy way to determine the representation type of a kind of Drinfeld doubles. Consider the basic cycle of length n (Example 2.1 (1)) m and we denote this quiver by Zn and by γi the path of length m starting at the vertex ei (i =1,...,n). d We consider the quotient algebra Γn,d := kZn/J with d|n. It is a Hopf algebra with comultiplication ∆, counit ε and antipode deﬁned as follows. We ﬁx a primitive d-th root of unity q. ⊗ 1 ⊗ 1 l 1 ⊗ ∆(et)= ej el, ∆(γt )= ej γl + q γj el, j+l=t j+l=t

1 1 − t+1 1 ε(et)=δt0,ε(γt )=0,S(et)=e−t,S(γt )= q γ−t−1. ∗cop As a Hopf algebra, (Γn,d) is isomorphic to the generalized Taft algebra Tnd(q)[37] which as an associative algebra is generated by two elements g and x with relations gn =1,xd =0,xg= qgx, with comultiplication ∆, counit ε, and antipode S given by ∆(g)=g ⊗ g, ∆(x)=1⊗ x + x ⊗ g,

ε(g)=1,ε(x)=0,

S(g)=g−1,S(x)=−xg−1. For details, see [19]. In [19], the authors studied the representation theory of the Drinfeld Double D(Γn,d) and proved the following conclusion.

CLASSIFICATION OF BASIC HOPF ALGEBRAS 1119

Lemma . D n2 2.5 [Theorem 2.25 in [19]] The Ext-quiver of (Γn,d) has d isolated n(d−1) vertices which correspond to the simple projective modules, and 2 copies of the quiver

b¨ • Hb ¨ ¨* HY Hj • ¨ H • · · b b . · ..

. . . .

. · .. · · • Hb ¨ • HY Hj ¨b H • ¨¨* b b

2n 4n with d vertices and d arrows. The relations on this quiver are bb, bb and bb − bb.

From this lemma, the authors of [19] ﬁnd that D(Γn,d) is a special biserial algebra and thus it is of ﬁnite representation type or tame type (see [19]). After listing all indecomposable modules of D(Γn,d), they get that D(Γn,d) is a tame algebra. Indeed, even without the complete list of indecomposable D(Γn,d)-modules, we also can prove that it is tame now.

Corollary 2.6. D(Γn,d) is a tame algebra.

Proof. We have known that D(Γn,d) is a special biserial algebra and thus it is tame or of finite representation type. Thus in order to prove that it is tame, it is enough to show that it is not of finite representation type. Note that the above 2n −1 quiver is a covering quiver ΓG(W ) by setting G = and W =(g, g ), and then nΓG(W ) = 2. Therefore, Theorem 2.1 gives us the desire conclusion. 3. Classification of basic Hopf algebras of finite representation type The classification of basic Hopf algebras of finite representation type indeed has been given by the author (with F. Li) in [42]. In [42], the conclusion is given in the language of pointed Hopf algebras. Note that the dual of pointed Hopf algebras are basic ones. For our purpose, we rewrite the result out in the language of basic Hopf algebras without proof (see Theorem 4.6 in [42]). Theorem 3.1. Let H be a finite-dimensional basic Hopf algebra. Then ∼ ∗ (i) H is semisimple if and only if H = (kG) for some finite group G; (ii) Assume the characteristic of k is zero and H is not semisimple, then H ∗ ∼ is of finite representation type if and only if H = A(α) for some group datum α =(G, g, χ, µ); (iii) Assume the characteristic of k is p and H is not semisimple, then H is of finite representation type if and only if there exist two natural numbers n>0,r≥

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r 0,ad0-th primitive root of unity q ∈ k with d0|n,andd = p d0 ≥ 2 such that ∗ ∼ H = Cd(n) ⊕···⊕Cd(n) as coalgebras and ∗ ∼ H = Cd(n)#σk(G/N) as Hopf algebras, where G = G(H) and N = G(Cd(n)). Remark 3.2. (i) Here a group datum (for details, see [13]) over k is defined to be a sequence α =(G, g, χ, µ) consisting of (1) a finite group G, with an element g in its center, (2) a one-dimensional k-representation χ of G, (3) an element µ ∈ k such that µ =0ifo(g)=o(χ(g)), and if µ =0then χo(χ(g)) =1. For a group datum α =(G, g, χ, µ)overk, the corresponding Hopf algebra A(α) was defined in [13], which is generated as an algebra by x and all h ∈ G with relations xd = µ(1 − gd),xh= χ(h)hx, ∀ h ∈ G where d = o(χ(g)). Its comultiplication ∆, counit ε, and antipode S are defined by ∆(x)=g ⊗ x + x ⊗ 1,ε(x)=0,

∆(h)=h ⊗ h, ε(h)=1 ∀ h ∈ G, S(x)=−g−1x, S(h)=h−1, ∀ h ∈ G. When d is a prime, the corresponding Hopf algebra A(α) appeared before [13]in [16]. ⊕d−1 ≥ (ii) For any quiver Γ, we deﬁne Cd(Γ) := i=1 kΓ(i)ford 2, where Γ(i)is the set of all paths of length i in Γ. We denote the basic cycle of length n (Example 2.1 (1)) by Zn and denote Cd(Zn)byCd(n). For more details about this theorem, see [42].

4. Classification of basic Hopf algebras of tame type For the radically graded tame basic Hopf algebras, all possible structure are determined in the author’s paper [41]. In this section, we determine the structure of tame basic Hopf algebras (without the assumption of radical grading) completely. We now give a short description of our method which is a kind of generalization of the method used in [41]. Let H be a finite-dimensional basic Hopf algebra over k. Then we have a Hopf epimorphism H H/JH where JH is the Jacobson ∼ radical of H. By a work of H.-J. Schneider (see [64]), we have H = RH #σH/JH , where RH = {a ∈ H|(id ⊗ π)∆(a)=a ⊗ 1} and π : H → H/JH the canonical epimorphism. We will show that RH is a local Frobenius algebra. By [40], we know that H and RH have the same representation type. These results help us to reduce the study of tame basic Hopf algebras to that of tame local Frobenius algebras. Fortunately, we classify all tame local Frobenius algebras and show that there are only ten classes of local algebras which are tame Frobenius (see Theorem 4.1). By this, we find one possible structure given in [41] will not happen and the detail will be given at the end of this section.

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4.1. A complete list of tame local Frobenius algebras. Denote the characteristic of k by chark. The main result of this subsection is the following. Theorem 4.1. Let Λ be a tame local Frobenius algebra. If chark =2 ,then ∼ Λ = k/Iwhere I is one of forms: (1): I =(xm − yn,yx− axm,xy) for a ∈ k and m, n ≥ 2; (2): I =(x2,y2, (xy)m − a(yx)m) for 0 = a ∈ k and m ≥ 1; (3): I =(x2 − (yx)m,y2, (xy)m +(yx)m) for m ≥ 1; (4): I =(x2 − (yx)m,y2 − (xy)m, (xy)m +(yx)m, (xy)mx) for m ≥ 1; (5): I =(x2,y2, (xy)mx − (yx)my) for m ≥ 1; (6): I =(x2 − (yx)m−1y − b(xy)m,y2, (xy)m − a(yx)m) for a, b ∈ k with a =0 and m ≥ 2; (7): I =(x2 − (yx)m−1y − b(xy)m,y2 − (xy)m, (xy)m +(yx)m, (xy)mx) for a, b ∈ k with a =0 and m ≥ 2; (8): I =(x2−(yx)m−1y−f(xy)m,y2−(xy)m−1x−e(xy)m, (xy)m−a(yx)m, (xy)mx) for a, e, f ∈ k with a =0 and m ≥ 2; (9): I =(x2 − (yx)m,y2, (xy)mx − a(yx)my) for 0 = a ∈ k and m ≥ 1; (10): I =(x2 − (yx)m,y2 − (xy)m, (xy)mx − a(yx)my, (xy)m+1) for 0 = a ∈ k and m ≥ 1. We want to prove Theorem 4.1 now. Some preliminaries must be given at ﬁrst. It is easy to see that a local algebra is Frobenius if and only if the dimension of its socle equals to one. In this section, Λ always denotes a local Frobenius algebra and JΛ its Jacobson radical. Recall that for any self-injective algebra Λ, we always have soc ΛΛ=soc ΛΛ (see [51]). This fact will be used frequently. Any tame local algebra A must have a quiver of the form '$'$

• ? 6 &%&% x y ∼ We denote this quiver by Q. By the Gabriel’s Theorem, we know A = k< x, y > /I for some ideal J 2 ⊆ I ⊆ J N where J is the ideal of kgenerated by x, y and N ≥ 2. Therefore, if A is Frobenius then dimkA ≥ 4. For convenience, we always denote the image of x, y in A by x, y too. Proposition 4.2. All local algebras listed in Theorem 4.1 are tame local Frobe- nius algebras. Proof. By checking the dimension of the socle, it is easy to see that they are Frobenius algebras. It is known that Λ and Λ/socΛ have the same representation type. Now we can ﬁnd all Λ/socΛ are images of maximal tame local algebras which given by C. Ringel [62]. Thus they are tame or of ﬁnite representation type. But it is known that k/(x, y)2 is tame and clearly there is a natural algebra epimorphism Λ k/(x, y)2 for any Λ in Theorem 4.1. Therefore, they are all tame. Lemma . 2 4.3 Let Λ=kQ/I be a local Frobenius algebra such that JΛ is generated by x2 and y2.Thenxy =0if and only if I =(xm − yn,yx− axm,xy) for 0 = a ∈ k with m, n ≥ 2 or xy = yx =0. Moreover, if xy = yx =0,then I =(xm − yn,xy,yx) for m, n ≥ 2.

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Proof. It is enough to prove the necessity. By assumption, we have that Λ is spanned by 1,x,x2, ··· ,y,y2, ···.Wemaywriteyx = xcw + ydz where w, z are units in k[[x]] and k[[y]] respectively and c, d ≥ 2. Then 0 = xyx = xc+1w and then xc+1 =0.Sincealsoxcy = 0, it follows that xc ∈ socΛ, the socle of Λ. Moreover, 0=yxy = yd+1z and we deduce that yd+1 =0.Sincealsoxyd = 0, it follows that yd ∈ socΛ. This shows that yx ∈ socΛsincesocΛ is an ideal of Λ. Assume yx =0now.Let m, n be the maximal integers such that xm =0 ,yn = m+1 n+1 ≥ m n ∈ 0andx =0,y = 0. Clearly, m, n 2andx ,y socΛ. By dimksoc√Λ= 1, there are a, b ∈ k with ab =0suchthat xm = ayn and yx = bxm.Lety = n ay, then xm = yn. The last statement is clear and the lemma is proved.

Lemma 4.4. Assume that chark =2 and Λ is a 4-dimensional local Frobenius algebra. Then Λ is isomorphic to one of the following algebras: (1): kQ/(x2 − y2,yx− ax2,xy) for 0 = a ∈ k; (2): kQ/(x2,y2,xy− ayx) for 0 = a ∈ k.

Proof. Let x, y be generators of JΛ.SincedimkΛ = 4 and Λ is Frobenius, xy and yx belong to the socle of Λ. 2 2 (I): Assume xy =0.Ifyx =0,then y =0and x =0since dimksocΛ=1. Therefore, by above lemma, we can ﬁnd m =2,n= 2 since otherwise the dimension ∼ of Λ will bigger than 4. Thus, Λ = kQ/(x2 − y2,yx− ax2,xy)for0= a ∈ k. If yx = 0. In this case, we know that x2 = ay2 for 0 = a ∈ k.Letu ∈ k with u2 = −a, then, by chark =2, X = x + uy, Y = x − uy are generators. And, X2 = Y 2 = x2 + u2y2 = x2 − ay2 =0,XY = YX = x2 − u2y2 = x2 + ay2. ∼ Therefore, Λ = k/(X2,Y2,XY− YX) which is a special case of (2). (II): Assume xy =0 = yx.Thenxy = cyx for 0 = c ∈ k.BydimksocΛ=1, ∼ we have x2 = axy and y2 = bxy.Ifa = b =0,thenΛ= kQ/(x2,y2,xy− cyx). Otherwise, no loss generality, assume a =0.Let Y = x − ay,thenxY =0. Therefore we are in case (I) again.

Lemma 4.5. Let Λ be a local Frobenius algebra. Then 2 3 ≤ (i): If Λ is tame then dimkJΛ/JΛ 2. 2 3 ≤ (ii): If chark =2and dimkJΛ/JΛ 1 then dimkΛ=4or Λ is an algebra as in Theorem 4.1 (1). Proof. 2 3 ≥ (i) If dimkJΛ/JΛ 3, then there is a homomorphic image which is wild (see (2.1) of [62] ). This implies Λ is wild which contradict the assumption that Λ is tame. 2 3 ≤ (ii) Suppose now that dimkJΛ/JΛ 1. Then the dimension must be 1, since otherwise x, y would lie in socΛandsocΛ would not be simple. By this, we know 3 that dimkΛ/JΛ =4. 3 Case (1): If Λ/JΛ is Frobenius, then by Lemma 4.4 we have 3 ∼ 2 − 2 − 2 3 ∼ 2 2 − Λ/JΛ = kQ/(x y ,yx ax ,xy)orΛ/JΛ = kQ/(x ,y,xy ayx) for a =0. 3 ∼ 2 − 2 − 2 − 2 2 − 2 ∈ 3 If Λ/JΛ = kQ/(x y ,yx ax ,xy), then xy, yx ax ,x y JΛ.By ∈ 3 2 2 ∈ 4 2 − 2 ∈ 3 3 − 2 ∈ 4 3 ∈ 4 xy JΛ, x y, yxy, xy ,xyx JΛ.Byx y JΛ, x xy JΛ and thus x JΛ. 2 − 2 ∈ 3 3 − 2 ∈ 4 2 ∈ 4 Using x y JΛ again, we can ﬁnd x y x JΛ and thus y x JΛ. Similarly, 2 − 3 ∈ 4 2 − 3 ∈ 4 2 3 ∈ 4 3 ⊆ 4 by yx ax JΛ and x y y JΛ,wehaveyx ,y JΛ. Therefore, JΛ JΛ 3 and thus JΛ = 0. This implies dimkΛ=4.

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3 ∼ 2 2 − 2 2 − ∈ 3 If Λ/JΛ = kQ/(x ,y,xy ayx), we have x ,y,xy ayx JΛ.Bythis,it 2 2 2 2 3 3 ∈ 4 3 ⊆ 4 is easy to show that xy ,xy, yx ,yx, x ,y,xyx,yxy JΛ.ThusJΛ JΛ 3 and so JΛ = 0. This also means that dimkΛ=4. 3 3 ≥ Case (2): Assume now Λ/JΛ is not Frobenius. Therefore, dimksoc(Λ/JΛ) 2. ∈ 3 ∈ 3 ∈ This implies x soc(Λ/JΛ)ory soc(Λ/JΛ). Not loss generality, assume x 3 2 ∈ 3 2 2 soc(Λ/JΛ). Thus we have x ,xy,yx JΛ. This means JΛ is generated by y and thus xy = uyl where u is a unit of k[[y]] and l ≥ 3. Let x = x − uyl−1 and we have xy = 0. By Lemma 4.3, we know that yx =0orI =(xm − yn,yx − axm,xy). If I =(xm − yn,yx − axm,xy), the proof is done. In the case of yx =0, we write x2 = vys for u a unit of k[[y]] and s ≥ 3. Thus x3 = vysx =0andso x2 ∈ socΛ. Take m to be the maximal integer such that ym =0and ym+1 =0. Therefore, ym ∈ socΛandthusx2 = aym.Lety = λy. Take a suitable λ,wehave ∼ x 2 = y m and Λ = kQ/I for I =(x 2 − y m,xy ,yx ). This is a special case of Theorem 4.1 (1).

The following lemma is given in [18] (page 84). Lemma 4.6. Let A be a tame local algebra with the quiver Q, of dimension 5, ∼ with J 3 =0.ThenΛ = kQ/L where L is one of the following ideals: (1): (xy, yx); (2): (yx − x2,xy); (3): (yx − x2,xy− ay2) where a ∈ k and 0 = a =1 ; (4): (x2,y2); (5): (yx − x2,y2). Lemma . 3 4.7 Let Λ be a local Frobenius algebra such that xy and yx lie in JΛ. Assume chark =2 ,thendimkΛ=4or Λ is an algebra as in Theorem 4.1 (1).

Proof. By Lemma 4.4, we may assume dimkΛ > 4. The algebra has a basis of the form {1, x,...,xs, y,...,yt}. If xy = 0, then by Lemma 4.3, Λ is of the form given in Theorem 4.1 (1). We consider now the case when xy =0 = yx.Letp be as large as possible ∈ p ≥ p+1 such that xy and yx JΛ. Clearly, p 3. Then one of them does not lie in JΛ . ≡ p p ≡ p p p+1 ∈ Write xy x u + y v and yx x w + y z (modulo JΛ )whereu, v, w, z k. We may replace x, y by x,y where x = x − yp−1z and y = y − xp−1u.Thenwe have new relations xy ≡ cyp and yx ≡ dxp for c, d ∈ k. Moreover, one of them, p+1 p+1 p+1 p+1 ≡ ≡ c say, is non-zero. Now, JΛ is generated by x and y ,andcy yxy p ≡ p−1 p ∈ p+2 p+1 p+1 dx y dcx y JΛ and already JΛ =(x ). p+1 ∈ p+2 p+1 p ⊆ If d = 0, then similarly x JΛ and thus JΛ =0.SoJΛ socΛ. We ∈ p p p p assumed that 0 = xy JΛ and it follows that JΛ = socΛ. We have x = ay for 0 = a ∈ k and also xy = cyp and yx = dxp with cd = 0. Replace now x, y by x = x − cyp−1 and y.Thenxy = 0 and thus we can use Lemma 4.3 again. ∈ p+1 p+1 Suppose now that d =0.Thenyx JΛ =(x ). Note that yx =0and consequently yx = xmu for m ≥ p +1,where u is a unit of k[[x]]. Replace y by y = y − xm−1u,thenyx = 0. We also can use Lemma 4.3 again.

Lemma 4.8. Let Λ be a local Frobenius algebra such that yx − x2 and xy lie in 3 ∼ m − n − m ∈ JΛ.ThenΛ = kQ/(x y ,yx ax ,xy) for 0 = a k.

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Proof. Claim: JΛ does not have generators x ,y with (x y ) and (y x ) lying 3 in JΛ. This claim was proved in [18] (see Lemma III.7 of [18]). 3 ≡ 4 ∈ 4 3 ∈ 4 3 We have that x xyx (modulo JΛ). But xyx JΛ and thus x JΛ.SoJΛ is generated by y3.Sowemayhavexy = ysu where u is a unit in k[[y]] and s ≥ 3. Let x = x − ys−1u,thenwehavexy =0.Bytheclaim,yx = 0 and thus Lemma 4.3 is applied.

Lemma 4.9. Let Λ be a local Frobenius algebra such that yx − x2 and xy − ay2 3 lie in JΛ where 0 = a =1.ThendimkΛ=4. Proof. 3 ≡ ≡ 2 ≡ 2 ≡ 3 4 By assumption, we have x xyx ay x ayx ax (modulo JΛ) 2 ≡ 2 ≡ 2 3 ≡ ≡ 2 4 3 ∈ 4 and x y axy a y ayxy ax y (modulo JΛ). Since a =1,x JΛ and 2 ∈ 4 4 x y JΛ.Sincea = 0, it follows that all the other monomials occurring lie in JΛ. 3 ⊆ 4 3 2 ⊆ This means that JΛ JΛ and thus JΛ =0.So,JΛ socΛ. By Λ is Frobenius, dimkΛ=4. Lemma . 2 2 3 4.10 Let Λ be a local Frobenius algebra such that x and y lie in JΛ. Assume chark =2 ,thenΛ is isomorphic to one of algebras in Theorem 4.1.

Proof. By Lemma 4.4 and Lemma 4.5, we can assume that dimkΛ > 4and 2 3 2 dimkJΛ/JΛ =2.Thusxy and yx are generators of JΛ which are independent. Case (1): Assume x2 and y2 lie in socΛ. Let m ≥ 1 be the integer such that (xy)m =0and( xy)m+1 = 0. We claim that socΛ=((xy)m)orsocΛ=((xy)mx). Indeed, since y2 ∈ socΛ, we always have (xy)my =0.Thusif(xy)mx =0then m m m 0 =( xy) ∈ socΛ. By dimksocΛ=1,socΛ=((xy) ). Otherwise, (xy) x =0. By (xy)m+1 =0and(xy)mx2 =0,socΛ=((xy)mx). Thus the claim is proved. If socΛ=((xy)m), then there exists 0 = a ∈ k such that (yx)m = a(xy)m. By x2 and y2 lie in socΛ, we have x2 = b(xy)m and y2 = c(xy)m for b, c ∈ k. − b c − m−1 − m−1 If a = 1, let d = 1+a , e = 1+a and x = x dy(xy) , y = y e(xy) y, then we have x2 =0andy2 = 0. Thus it is isomorphic to the one of algebras in Theorem 4.1 (2). If a = −1, then consider b, c.Ifb = c = 0, it is isomorphic to the one of algebras in Theorem 4.1 (2). If one of b, c is zero while the other is not zero, say b =0and c =0.Letx = λx and y = µy for λ, µ ∈ k. By suitable choice of λ, µ, we can assume x2 =(xy)m and y2 = 0. Thus the algebra has the form as in Theorem 4.1 (3). Similarly, if bc = 0 then we can show the algebra has the form as in Theorem 4.1 (4). If socΛ=((xy)mx), then there exists 0 = a ∈ k such that (yx)my = a(xy)mx. By x2 and y2 lie in socΛ, we have x2 = b(xy)mx and y2 = cy(xy)m for b, c ∈ k. Let x = b(xy)m − x and y = c(yx)m − y. Then we can ﬁnd x2 =0andy2 =0. Moreover, clearly we can take a to be 1. Thus the algebra has the form as in Theorem 4.1 (5). 2 ∈ n − n+1 2 ∈ l Case (2): Otherwise, choose n, l such that x JΛ JΛ and y JΛ with l ≥ n.Takealson as large as possible with respect to these conditions. n+1 ⊆ Claim: JΛ socΛ. This claim was proved in [18] (see Lemma III.10 of [18]). Now we consider two possibilities: socΛisanevenpowerofJΛ or socΛisanodd power of JΛ. 2m (i) If socΛisanevenpowerofJΛ,thensocΛ=JΛ .Bythehypothesisat n ⊆ n+1 n+1 the beginning, JΛ socΛ. So JΛ =0andthusJΛ = socΛ. Therefore, n +1 = 2m. Clearly, socΛ=((xy)m)and(yx)m = a(xy)m for 0 = a ∈ k. Then x2 = c(xy)m−1x + d(yx)m−1y + f(xy)m and (c, d) =(0 , 0). Without loss

CLASSIFICATION OF BASIC HOPF ALGEBRAS 11715 of generality, we can assume c = 0 since otherwise we can replace x by x = x − c(xy)m−1. 2 2 Now consider y . By the hypothesis, the element lies in soc2Λ. If y ∈ socΛ, then we show similarly as in Case (1) that Λ is an algebra in Theorem 4.1 (6), (7). Otherwise, y2 = a(xy)m−1x + b(yx)m−1y + e(xy)m and (a, b) =(0 , 0). Similarly, we can assume b =0.Thusa =0. We have now x2 = d(yx)m−1y + f(xy)m and y2 = a(xy)m−1x + e(xy)m.Let x = λx and y = µy. By a suitable choice of λ, µ, we can assume a =1=d.This is an algebra in Theorem 4.1 (8). n+1 (ii) Otherwise, socΛisanoddpowerofJΛ. Similarly, we have socΛ=JΛ = ((xy)mx)and(xy)mx = a(yx)my for a = 0. By assumption, x2 = b(xy)m + c(yx)m + f(xy)mx. As before, we can assume b = 0. Also, we consider y2.If y2 ∈ socΛ, then by the discussion of Case (1), the algebra is isomorphic to one of algebras in Theorem 4.1 (9). If not, we have y2 = d(xy)m + e(yx)m + g(xy)mx for d, e, g ∈ k. Similarly, we can assume that e =0andd = 0. As in Case (1), we also can assume f = g =0.Sonowwehavex2 = c(yx)m and y2 = d(xy)m. Similarly, let x = λx and y = µy and choose suitable λ, µ, we may assume c = d =1.Thus it is an algebra in Theorem 4.1 (10). Lemma . 3 4.11 There is no local Frobenius algebra Λ such that Λ/JΛ satisﬁes Lemma 4.6 (5). Proof. Suppose such algebra exists. 3 ⊆ 3 Claim: JΛ =(xyx) socΛ. We have that JΛ is generated by xyx and yxy by 4 ≡ 3 ≡ ≡ 2 ≡ the given relations. Moreover, modulo JΛ we have that xyx x yxx y x 0 3 4 2 ⊆ 4 ⊆ 5 4 and therefore JΛ =(yxy). This implies JΛ =((yx) ) yJΛ JΛ.ThusJΛ =0as required. 3 We claim yxy must be zero now. Otherwise, assume yxy =0andthusJΛ = 4 2 ∈ (yxy)=socΛ. Since JΛ =0,weknowxyx =0andxy = 0. This means xy socΛ. Clearly, xy = 0 since otherwise yxy =0.SincedimksocΛ = 1, there exists nonzero c ∈ k such that xy = cyxy.Sowehavexy = cyxy = c2y2xy =0.It’sa 3 2 ⊆ contradiction. This means yxy =0andthusJΛ =0andJΛ socΛ. Therefor socΛ is not simple, which is absurd. 2 3 ≤ Proof of Theorem 4.1: Since Λ is tame, dimkJΛ/JΛ 2 by Lemma 4.5. 2 3 If dimkJΛ/JΛ = 1, Lemma 4.5 shows that Λ is one of algebras of this list. 2 3 3 3 If dimkJΛ/JΛ =2,thendimkΛ/JΛ = 5. This means that Λ/JΛ satisﬁes the conditions of Lemma 4.6. Therefore, Lemma 4.7-4.11 give our desired conclusion.

4.2. Tame basic Hopf algebras. The main aim of this subsection is to describe the structure of tame basic Hopf algebras (see Theorem 4.15). ∼ Let H be a basic Hopf algebra and JH be its Jacobson radical. Recall H/JH = (kG)∗ for some ﬁnite group G. In this section, we always assume chark =2and chark |G|.ThuskG is always semisimple. Denote H/JH by H. Now we have a Hopf algebra epimorphism H H.

By a result which given by H.-J. Schneider [64], there is an algebra RH such that ∼ H = RH #σH.

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Lemma 4.12. RH is a local algebra. Proof. ⊕ 2 ⊕ For any ﬁnite-dimensional algebra A,wewritegrA = A/JA JA/JA ···.By[61]and[44], there is an algebra RgrH, which is a graded braided Hopf H YD grH ∼ R H R algebra in H , such that = grH# . Hence grH is Frobenius by [26]and is local since the degree 0 part is k.ThusRgrH is a local Frobenius algebra. By Blattner-Montgomery Duality Theorem (see Section 9.4 in [49]), we have ∗ ∼ (RH #σH)#(H) = Mn(RH ), ∗ ∼ (RgrH#H)#(H) = Mn(RgrH) ∗ where n = dim H.Notethat(H) is a group algebra now, thus we have J ∗ = k (RH #σ H)#(H) (J )#(H)∗. This means we have the following isomorphism RH #σ H ∼ ∗ grMn(RH ) = (gr(RH #σH))#(H) . Thus, ∼ ∼ ∗ ∼ ∗ ∼ Mn(grRH ) = grMn(RH ) = (grH)#(H) = (RgrH#H)#(H) = Mn(RgrH). ∼ ∼ So we have Mn(grRH ) = Mn(RgrH) and thus grRH = RgrH.ByRgrH is local, grRH and thus RH is local.

Lemma 4.13. RH is a Frobenius algebra.

Proof. By the Lemma 4.12, it is enough to show that RH is self-injective since any basic self-injective algebra must be Frobenius. By Blattner-Montgomery Duality Theorem, we need only show H#(H)∗ is self-injective. Let P be a projective H#(H)∗-module, we need to show that P is also injective. For H#(H)∗-modules M, N,leti : M→ N and h : M → P be two H#(H)∗- module morphisms such that i is injective. In order to prove that P is injective ∗ ∈ as an H#(H) -module, it is enough to find an f HomH#(H)∗ (N,P) satisfying h = fi .ItisknownthatH#(H)∗ is a free H-module. Thus P is also a projective H-module. By H is Frobenius, P is injective as an H-module. Thus there exists an H-morphism f such that h = fi. Define f(n)= S(t1) · f(t2 · n)forn ∈ N, where t is a non-zero right integral with ε(t)=1.Thenf is H#(H)∗-linear by [15] and satisfies h = fi . The following lemma is proved in [40] (see Theorem 2.6 in [40]). Lemma 4.14. Let A be a finite-dimensional algebra and H a finite-dimensional ∗ Hopf algebra. If H and H are semisimple, then A#σH and A have the same representation type. The next conclusion will give us all possible structures of tame basic Hopf algebras. Theorem 4.15. Let H be a basic Hopf algebra. Assume chark =2 and ∼ dimkH/JH is invertible in k,thenH is tame if and only if H = k ∗ /I#σ(kG) for some finite group G and some ideal I which is one of the following forms: (1): I =(xm − yn,yx− axm,xy) for a ∈ k and m, n ≥ 2; (2): I =(x2,y2, (xy)m − a(yx)m) for 0 = a ∈ k and m ≥ 1; (3): I =(x2 − (yx)m,y2, (xy)m +(yx)m, ) for m ≥ 1;

CLASSIFICATION OF BASIC HOPF ALGEBRAS 11917

(4): I =(x2 − (yx)m,y2 − (xy)m, (xy)m +(yx)m, (xy)mx) for m ≥ 1; (5): I =(x2,y2, (xy)mx − a(yx)my) for 0 = a ∈ k and m ≥ 1; (6): I =(x2 − (yx)m−1y − b(xy)m,y2, (xy)m − a(yx)m) for a, b ∈ k with a =0 and m ≥ 2; (7): I =(x2 − (yx)m−1y − b(xy)m,y2 − (xy)m, (xy)m +(yx)m, (xy)mx) for a, b ∈ k with a =0 and m ≥ 2; (8): I =(x2−(yx)m−1y−f(xy)m,y2−(xy)m−1x−e(xy)m, (xy)m−a(yx)m, (xy)mx) for a, e f ∈ k with a =0 and m ≥ 2; (9): I =(x2 − (yx)m,y2, (xy)mx − a(yx)my) for 0 = a ∈ k and m ≥ 1; (10): I =(x2 − (yx)m,y2 − (xy)m, (xy)mx − a(yx)my, (xy)m+1) for 0 = a ∈ k and m ≥ 1. Proof. “Only if part: ” On one hand, by Lemma 4.12, Lemma 4.13 and Lemma 4.14, RH is a tame local Frobenius algebra. On the other hand, H/JH is ∼ ∗ a commutative semisimple Hopf algebra and thus H/JH = (kG) for some finite ∼ group. Therefore, by Theorem 4.1 and H = RH #σH/JH , we get the desired conclusion. “If part: ” By Proposition 4.2, we know that k/Iis a tame algebra. So the sufficiency is gotten from Lemma 4.14. Remark 4.16. (1) In order to apply Lemma 4.14, we need kG to be semisimple and thus the hypothesis chark |G|, posed at the beginning of this subsection, is needed. Note that at the end of proof of Lemma 4.13, this hypothesis was also used to guarantee the existence of the right integral t satisfying ε(t)=1. (2) By a conclusion of Radford or Majid (see [61][44]), if Λ is a braided Hopf ∗ (kG) YD algebra in (kG)∗ for some finite group G, then we can form the bosonization Λ × (kG)∗ which is a Hopf algebra. For a tame local Frobenius algebra A,above theorem dose not imply the existence of finite group G satisfying A is a braided ∗ (kG) YD Hopf algebra in (kG)∗ . Problem 4.1. For a tame local Frobenius algebra A, give an effective method to determine that whether there is a finite group G satisfying A isabraidedHopf ∗ (kG) YD algebra in (kG)∗ .IfsuchaG exists, then find all of them. A similar problem was given in [41]([41], Problem 5.1). For a tame local radically graded Frobenius algebra, this problem has been solved by the author with his co-workers. The details will appear elsewhere. Example 4.1. (Tensor products of Taft algebras) Let Tn2 (q),Tm2 (q ) be two Taft algebras. Direct computation shows that ∼ Tn2 (q) ⊗k Tm2 (q ) = k/I#k(Zn × Zm) n m where I =(x ,y ,xy− yx). Thus by Theorem 4.15, Tn2 (q) ⊗k Tm2 (q )istameif and only if m = n =2. Example 4.2. (Book Algebras) Let q be a n-th primitive root of unity and m a positive integer satisfying (m, n)=1.LetH = h(q, m)=k /(xn,yn,gn − 1,gx− qxg, gy − qmyg, xy − yx) and with comultiplication, antipode and counit given by ∆(x)=x ⊗ g +1⊗ x, ∆(y)=y ⊗ 1+gm ⊗ y, ∆(g)=g ⊗ g S(x)=−xg−1,S(y)=−g−my, S(g)=g−1,ε(x)=ε(y)=0,ε(g)=1.

18120 GONGXIANG LIU

It is a Hopf algebra and called book algebra. As in [41], we have ∼ h(q, m) = k/I#kZn where I =(xn,yn,xy− qmyx). Thus by Theorem 4.15, h(q, m)istameifand only if n =2.Inthiscase,q must equal to −1andm =1.Thusonlyh(−1, 1) is tame and the others are all wild. Example 4.3. (The dual of Frobenius-Lusztig kernel) Let p be an odd number and q a p-th primitive root of unity. By definition, the Frobenius-Lusztig kernel uq(sl2) is an associative algebra generated by E,F,K with relations K − Kp−1 Kp =1,Ep =0,Fp =0,KE= q2EK, KF = q−2FK, EF−FE = . q − q−1 Its comultiplication, counit and antipode are defined by ∆(E)=1⊗ E + E ⊗ K, ∆(F )=K−1 ⊗ F + F ⊗ 1, ∆(K)=K ⊗ K; ε(E)=ε(F )=0,ε(K)=1; S(E)=−q2K−1E, S(F )=−KF, S(K)=K−1. ∗ It is a pointed Hopf algebra and thus uq(sl2) is a basic Hopf algebra. We now give ∗ the Hopf structure of uq(sl2) explicitly. l i j It is known that uq(sl2) has a basis {K E F |0 ≤ l, i, j ≤ p − 1} and thus 3 l i j ∗ ∗ dimkuq(sl2)=p .Wedenoteby(K E F ) the element of uq(sl2) which sent KlEiF j to 1 and the other element in the above basis to 0. Let p−1 p−1 p−1 a = qi(Ki)∗ + qi(KiEF)∗,b= qi(KiE)∗, i=0 i=0 i=0 p−1 p−1 c = q−i(KiF )∗,d= q−i(Ki)∗. i=0 i=0 By direct computations, the following relations hold. ba = qab, db = qbd, ca = qac. dc = qcd, bc = cb, ad − da =(q−1 − q)bc, da − qbc =1,dp =1,cp = bp =0. For example, let us check the relationbc = cb and the other relations can be i−j i ∗ j ∗ checked similarly. By definition, bc = i,j q (K E) (K F ) .Inordertomake (KiE)∗(KjF )∗(KlEmF n) =0,wemusthave m = n =1.But ∆(KlEF)=Kl−1 ⊗ KlEF + q2Kl−1E ⊗ Kl+1F + KlF ⊗ KlE + KlEF ⊗ Kl. This implies if (KiE)∗(KjF )∗ =0then j = i +2.Thus p−1 p−1 bc = qi−j(KiE)∗(KjF )∗ = q−2q2(KlEF)∗ = (KlEF)∗. i,j l=0 l=0 p−1 l ∗ Similarly, we can show cb = l=0 (K EF) also. By da − qbc =1anddp =1,wehavea = d−1(1 + qbc). It is straightforward to show that the algebra, which is generated by a, b, c, d with above relations, has 3 ∗ dimension p . Thus algebra is just uq(sl2) . The comultiplication, counit and the antipode are given as follows. ∆(a)=a ⊗ a + b ⊗ c, ∆(b)=a ⊗ b + b ⊗ d;

CLASSIFICATION OF BASIC HOPF ALGEBRAS 12119

∆(c)=c ⊗ a + d ⊗ c, ∆(d)=c ⊗ b + d ⊗ d; ε(a)=ε(d)=1,ε(b)=ε(c)=0; S(a)=d, S(b)=−qb, S(c)=−q−1c, S(d)=a. ∗ ∼ ∗ Z Clearly, Juq (sl2) =(b, c)anduq(sl2) /(b, c) = k p.Thus ∗ ∼ ∗ Z uq(sl2) = Ruq (sl2) #σk p ∗ ∗ { ∈ | ⊗ ⊗ } where by deﬁnition Ruq (sl2) = x uq(sl2) (id π)∆(x)=x 1 .Here ∗ ∗ π : uq(sl2) → uq(sl2) /(b, c) is the canonical map. Thus it is easy to see that − 1 ∈ ∗ ∗ dc, d b Ruq (sl2) which generate Ruq (sl2) and satisfy the following relations (dc)p =0, (d−1b)p =0,dc· d−1b = q2(d−1b) · dc. Denote dc by x and d−1b by y,wehave

∗ ∼ Ruq (sl2) = k/I where I =(xp,yp,xy− q2yx) which is not an algebra in Theorem 4.1. Thus, by ∗ Theorem 4.15, uq(sl2) is wild. At last, I want to take this chance to give an addendum to [41]. In Section 3 of [41], we give the following conclusion (see Theorem 4.1 in [41]). ∼ “Let Λ be a tame local graded Frobenius algebra. If chark =2,thenΛ = k< x, y > /I where I is one of forms: (1): I =(x2 − y2,yx− ax2,xy)for0= a ∈ k; (2): I =(x2,y2, (xy)m − a(yx)m)for0= a ∈ k and m ≥ 1; (3): I =(xn − yn,xy,yx)forn ≥ 2; (4): I =(x2,y2, (xy)mx − (yx)my)form ≥ 1; (5): I =(yx − x2,y2).”

By the Lemma 4.11, we know that the case (5) of above conclusion will not appear. Thus the better form of Theorem 4.1 of [41] is the following. Theorem Let Λ be a tame local graded Frobenius algebra. If chark =2 ,then ∼ Λ = k/Iwhere I is one of forms: (1): I =(x2 − y2,yx− ax2,xy) for 0 = a ∈ k; (2): I =(x2,y2, (xy)m − a(yx)m) for 0 = a ∈ k and m ≥ 1; (3): I =(xn − yn,xy,yx) for n ≥ 2; (4): I =(x2,y2, (xy)mx − (yx)my) for m ≥ 1. Of course, Theorem 5.4 of [41] should be changed accordingly. That is, delete the case (5) of Theorem 5.4 in [41].

Acknowledgement I would like to express my hearty thanks to Professor Nanhua Xi for his encour- agements. The work in Section 4 was finished when the author visit the Oxford University under the financial support from the Leverhulme Trust through the Academic Interchange Network Algebras, Representations and Applications. I am grateful to Professors K. Erdmann and A. Henke for stimulating conversations. I thank their hospitality from the university and the financial support from the Leverhulme Trust. I also want to thank my supervisor, Professor Fang Li, for his

20122 GONGXIANG LIU helpful comments. I am very grateful to the organizers for their excellent work. I also would like to thank the referee for very valuable comments.

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Department of Mathematics, Nanjing University, Nanjing 210093, China E-mail address: [email protected]

Contemporary Mathematics Contemporary Mathematics Volume 478, 2009 Volume 00, 1997

Twelve bridges from a reductive group to its Langlands dual

G. Lusztig

Introduction. These notes are based on a series of lectures given by the author at the Central China Normal University in Wuhan (July 2007). The aim of the lectures was to provide an introduction to the Langlands philosophy. According to a theorem of Chevalley, connected reductive split algebraic groups over a fixed field A are in bijection with certain combinatorial objects called root data. Now there is a natural involution on the collection of root data in which roots and coroots are interchanged. This corresponds by Chevalley’s theorem to an involution on the collection of connected reductive split algebraic groups over A. The image of a group under this involution is called the Langlands dual of that group. In the 1960’s Langlands made the remarkable discovery that some features about the representations of a reductive group (such as classification) should be recorded in terms of data in the Langlands dual group. He thus formulated two conjectures: one involving groups over a local field and one involving automorphic representations with respect to a group over a global field. In these notes we try to give several examples of “bridges” which connect some aspect of the collection (GA) of Chevalley groups attached to a root datum R and ∗ to various commutative rings A and some aspect of the analogous collection (GA)of Chevalley groups attached to the dual root datum R∗ and to various commutative rings A. By “aspect” we mean something about the structure of one of the groups GA or of its representations or of an associated object such as the affine Hecke algebra H. In each case the existence of the bridge is surprising due to the fact ∗ that (GA)and(GA) are related only through a very weak connection (via their root data); in particular there is no direct, elementary construction which produces the Langlands dual group from a given group. In fact we describe twelve such bridges (some conjectural) the first three of which are very famous and were found by Langlands himself. (I) A (conjectural) bridge from irreducible admissible representations of GK (where K is a finite extension of Qp) to certain conjugacy classes of homomorphisms ∗ of the Weil group WK to GC.(See§10.) This bridge contains almost as a special

Supported in part by the National Science Foundation

cc20091997 American American Mathematical Mathematical Society 1251

2G126 .LUSZTIG case a bridge from irreducible representations of H specialized at a non-root of 1 ∗ and conjugacy classes of certain pairs of elements in GC.(See§9.) (II) A bridge from irreducible admissible representations of GR to certain con- ∗ jugacy classes of homomorphisms of the Weil group WR to GC.(See§11.) (III) A (conjectural) bridge connecting certain automorphic representations attached to Gk (k a function ﬁeld over Fp) and certain homomorphisms of Gal(k/k¯ ) ∗ into GC.(See§12.) (IV) A bridge from cells in the aﬃne Weyl group constructed from H to unipo- ∗ tent classes in GC.(See§13.) (V) A bridge from “special unipotent pieces” in G to “special unipotent Fp pieces” in G∗ .(See§14.) Fp

(VI) A bridge from irreducible representations of GFq to certain “special” con- ∗ jugacy classes in GC.(See§16.) (VII) A bridge from character sheaves on GC to “special” conjugacy classes in ∗ GC.(See§17.) (VIII) A bridge constructed by Vogan connecting certain intersection cohomol- ∗ ogy spaces associated to symmetric spaces of GC and similar objects for GC.(See §18.) (IX) A (partly conjectural) bridge connecting multiplicities in standard modules of GK (as in (I)) or GR with intersection cohomology spaces arising from the ∗ geometry of GC.(See§19.) (X) A bridge connecting the tensor product of two irreducible finite dimensional representations of GC with the convolution of certain perverse sheaves on the affine ∗ § Grassmannian attached to GC(()).(See 20.) (XI) A bridge connecting the canonical basis of the plus part of the enveloping algebra attached to GQ with certain subsets of the totally positive part of the upper ∗ § triangular subgroup of GA where A = R[[]]. (See 21.) (XII) A (partly conjectural) bridge connecting the characters of irreducible modular representations of G with certain intersection cohomology spaces asso- Fp ∗ § ciated with the geometry of GC(()).(See 22.) Note that in some cases (such as the very important Case III) our treatment is only a very brief sketch. Moreover to simplify the exposition we restrict ourselves to the case of split groups. We now describe the contents of these notes. In §1weintroducerootdata.In §2 we use an idea of McKay (extended by Slodowy) to construct the irreducible simply connected root data. In §3 we introduce the affine Weyl group. In §4we introduce the affine Hecke algebra and its asymptotic version [L9]. In §5 we define the Z-form of the coordinate ring of a Chevalley group. We do not follow the original approach of [C] but rather the approach of Kostant [Ko]. In §6 we define the Chevalley groups. In §7 we define the Weyl modules. In §8 we define the Langlands dual group. In §9-§22 we discuss the various bridges mentioned above. I wish to thank David Vogan for some useful comments on a first version of these notes.

1. Root data. A root datum is a collection R =(Y,X,, , αˇi,αi(i ∈ I)) where Y,X are ﬁnitely generated free abelian groups, , : Y × X −→ Z is a perfect bilinear pairing, I is a ﬁnite set,α ˇi(i ∈ I) are elements of Y and αi(i ∈ I)are

TWELVE BRIDGES FROM A REDUCTIVE GROUP TO ITS LANGLANDS DUAL127 3

elements of X such that αˇi,αi =2foralli, αˇi,αj∈−N for all i = j;itis assumed that the following (equivalent) conditions are satisfied: (i) there exist ci ∈ Z>0(i ∈ I) such that the matrix (ciαˇi,αj)i,j∈I is symmetric, positive definite; ∈ ∈ (ii) there exist ci Z>0(i I) such that the matrix ( αˇi,αj cj)i,j∈I is symmetric, positive definite. By the equivalence of (i),(ii), ∗ R =(X, Y, , ,αi, αˇi(i ∈ I)) where x, y = y, x for x ∈ X, y ∈ Y , is again a root datum, said to be the dual of R.NotethatR∗∗ = R in an obvious way. For R as above let W be the (finite) subgroup of Aut(Y ) generated by the automorphisms si : y → y −y, αiαˇi (i ∈ I) or equivalently the subgroup of Aut(X) generated by the automorphisms si : x → x −αˇi,xαi (i ∈ I); these two subgroups may be identified by taking contragredients. We say that W is the Weyl group of R; it is also the Weyl group of R∗.Let + X = {λ ∈ X; αˇi,λ∈N for all i ∈ I}, + { ∈ ∈ ∈ } Y = y Y ; y, αi N for all iI . ∈ ≥ − ∈ ≥ For λ, λ X write λ λ if λ λ i Nαi and λ >λif λ λ, λ = λ. R R We say that is simply connected if Y = i Zαˇi.Wesaythat is adjoint R if X= i Zαi.Wesaythat is semisimple if X/ i Zαi is finite or equivalently R ∅ Y/ i Zαˇi is finite. We say that is irreducible if I = and there is no partition I = I ∪ I of I such that I ,I are = ∅ and αˇi,αj =0foralli ∈ I ,j ∈ I .

2. Subgroups of SL2(C) and root data. In [MK] McKay discovered a remarkable direct connection between finite subgroups of SL2(C) and “simply laced affine Dynkin diagrams”. Slodowy [SL] extended this to a connection between certain pairs of subgroups of SL2(C) (one contained in the other) and “affine Dynkin diagrams”. Let Γ, Γ be two finite subgroups of SL2(C) such that Γ is a normal subgroup of Γ with Γ/Γ cyclic. We show how to attach to (Γ, Γ) a root datum (we use an argument generalizing one in [L12, 1.2]). Let X be the category of finite dimensional complex representations of Γ which can be extended to representations of Γ .Letρi(i ∈ I˜) be the indecomposable objects of X up to isomorphism, that is, the representations of Γ which are restrictions of irreducible representations of ∈ ˜ Γ .Leti0 I be such that ρi0 is the trivial representation of Γ on C.Letσ be the 2 obvious representation of Γ on C ;wehaveσ ∈X.LetV be theR-vector space { ∈ ˜} X ∈ with basis i; i I . Any object ρ of gives rise to a vector ρ = i nii V where ⊕ ∼ ⊕ ni ∈ ∈ ˜ ⊗ ∈X ⊗ ρ = iρi ;hereni N.Fori I we have ρi σ and ρi σ = j∈I˜ cijj with cij ∈ N. For i ∈ I˜, ρi is the direct sum of mi irreducible representations of Γ (each with multiplicity 1). Let [, ] be the bilinear form on V with values in R given by − ∈ ˜ ∈ ∈ [i, j]=(2δij cij)mj for i, j I.Letx = i xii V , x = i xii V where ∈ ∈ 2 xi,xi R.Forγ Γletλg be an eigenvalue of γ on C .Wehave | |−1 − − [x, x ]= Γ xixjtr(γ,ρi)tr(γ,ρj)(2 λg λg) ∈ i,j;γ Γ | |−1 | − | | − | = Γ ( xitr(γ,ρi) 1 λγ )( xjtr(γ,ρj) 1 λγ ). γ∈Γ i j

4G128 .LUSZTIG

In particular, −1 2 [x, x]=|Γ| | xitr(γ,ρi)|1 − λγ || ≥ 0. γ∈Γ i ∈ | − | If [x, x] = 0 then for any γ Γwehave i xitr(γ,ρi) 1 λγ =0,thatis,forany ∈ −{ } ∈ γ Γ 1 we have i xitr(γ,ρi) = 0, that is, there exists c R such that x = cr ∈X where r is the regular representation of Γ; if in addition we have xi0 =0then we see that c = 0 hence x =0. Let I = I˜ −{i0}.LetY be the subgroup of V generated by {i; i ∈ I}.For i ∈ I letα ˇi = i ∈ Y .LetX =Hom(Y,Z). Let , : Y × X −→ Z be the obvious pairing. For j ∈ I define αj ∈ X by αˇi,αj =(i, j). We have αˇi,αi =2forall i ∈ I and αˇi,αj = −cij ∈−N for i = j in I. By the argument above the matrix (αˇi,αjmj)i,j∈I is symmetric and positive definite. Hence R =(Y,X,, , αˇi,αi(i ∈ I)) is a (simply connected) root datum. Note that all simply connected irreducible root data are obtained by this construction exactly once (up to isomorphism) from pairs (Γ, Γ)asabove(uptoconju- gacy) with Γ = {1} and with the property that any element of Γ which commutes with any element of Γ is contained in Γ. Such pairs are classified as follows: (a) Γ = Γ is a cyclic group Zn of order n ≥ 2; (b) Γ = Γ is a binary dihedral group D4n of order 4n ≥ 8; (c) Γ = Γ is a binary tetrahedral group G24 of order 24; (d) Γ = Γ is a binary octahedral group G48 of order 48; (e) Γ = Γ is a binary icosahedral group G120 of order 120; (f) Γ = D4n, Γ=Z2n with n ≥ 2; (g) Γ = D8n,Γ=D4n with n ≥ 2; (h) Γ = G48,Γ=G24; (i) Γ = G24, Γ=D8.

3. Affine Weyl group. Let R =(Y,X,, , αˇi,αi(i ∈ I)) be a root datum. Let W , si(i ∈ I)beasin§1. Let W be the semidirect product W · Y .Wehave W = {way; w ∈ W, y ∈ Y } where a is a symbol; the multiplication is given by −1 (way)(way )=wwaw (y)+y for w, w ∈ W , y, y ∈ Y .WeidentifyY with its image under the homomorphism y → 1ay (a normal subgroup of W)andW with its image under the homomorphism w → wa0. Let R be the set of elements of X of the form w(αi)forsomew ∈ W, i ∈ I.Let Rˇ be the set of elements of Y of the form w(ˇαi)forsomew ∈ W, i ∈ I.Thereisa unique W -equivariant bijection α ↔ αˇ between R and Rˇ such that αi ↔ αˇi for any −1 i ∈ I.Forα ∈ R we set sα = wsiw ∈ W where α = w(αi),w ∈ W, i ∈ I.Note that sα is well defined. Let Rmin be the set of all α ∈ R such that the following holds: if α ∈ R, α ≤α then α = α. + ∩ − − + + ∪ − Let R = R ( i Nαi), R = R .WehaveR = R R . Following Iwahori and Matsumoto [IM], we define a function l : W −→ N by l(way)= |y, α +1| + |y, α|. α∈R+;w(α)∈R− α∈R+;w(α)∈R+

Let S be the subset of W consisting of the involutions si(i ∈ I) and the involutions αˇ ∈ | sαa with α Rmin.Notethatl S=1. ∈ + y ∈ + If y Y we have l(a )= α∈R+ y, α . Hence for y, y Y we have l(ay · ay )=l(ay+y )=l(ay)+l(ay ).

TWELVE BRIDGES FROM A REDUCTIVE GROUP TO ITS LANGLANDS DUAL129 5

4. Affine Hecke algebra. We preserve the notation of §3. Let A = Z[v, v−1] where v is an indeterminate. Let H be the associative A-algebra with 1 with generators Tw(w ∈ W )andrelations −1 (Ts − v)(Ts + v )=0fors ∈ S, TwTw = Tww for w, w ∈ W such that l(w)+l(w )=l(ww ). † We have T1 =1and{Tw; w ∈ W } is an A-basis of H.Leth → h be the A- H † − l(w) −1 → ¯ algebra involution of such that Tw =( 1) Tw .Leth h be the ring H −1 ∈ n −n ∈ involution of such that Tw = Tw−1 for w W and v = v for n Z.Let ∈ ∈H z W . According to [KL1] there is a unique element cz such that cz = cz ∈ −1 −1 and cz = w∈Wf pw,zTw where pw,z v Z[v ] for all w = z, pz,z =1and { ∈ } A H pw,z = 0 for all but finitely manyw.Notethat cw; w W is an -basis of . ∈ ∈A For x, y W we write cxcy = z∈Wf hx,y,z cz where hx,y,z is 0 for all but finitely many z.Letz ∈ W. Accordingto[L9] there is a unique a(z) ∈ N such a(z) −1 a(z)−1 −1 that hx,y,z ∈ v Z[v ] for any x, y and hx,y,z ∈/ v Z[v ]forsomex, y.We a(z) a(z)−1 −1 have hx,y,z = γx,y,z−1 v mod v Z[v ]whereγx,y,z−1 ∈ Z.LetJ be the { ∈ } free abelian group with basis tw; w W . Consider the Z-algebra structure on − J such that txty = z∈Wf γx,y,z 1 tz for all x, y inW . This structure is associative D and the ring J has a unit element 1 of the form d∈D td where is a finite subset of W consisting of involutions [L9]. For x, y in W we write x ∼ y when txtuty =0 for some u ∈ W. This is an equivalence relation on W; the equivalence classes are called two-sided cells. For any two-sided cell c letJc be the subgroup of J spanned { ∈ } ⊕ by tz; z c . This is a subring of J with unit d∈D∩c td.WehaveJ = cJc as rings. Consider the A-linear map φ : H−→A⊗J given by † φ(cx)= hx,d,ztz z∈W,df ∈D;a(d)=a(z) for any x ∈ W.ThisisanA-algebra homomorphism [L9].

5. Coordinate ring. Let R =(Y,X,, , αˇi,αi(i ∈ I)) be a root datum. Let f be the associative Q-algebra with 1 deﬁned by the generators θi(i ∈ I)andthe Serre relations − a a b ( 1) (θi /a!)θj(θi /b!)

a,b∈N;a+b=1− αˇi,αj for i = j in I.Let0U be the symmetric algebra of Q ⊗ Y .LetU be the Q-algebra with 1 defined by the generators x+,x−(x ∈ f), a ∈ 0U and the relations: x → x+ is an algebra homomorphism f −→ U respecting 1; x → x− is an algebra homomorphism f −→ U respecting 1; a → a is an algebra homomorphism 0U −→ U respecting 1; + − + + ∈ ∈ yθi θi y = y, αi θi for y Y,i I; − − − − − ∈ ∈ yθi θi y = y, αi θi for y Y,i I; + − − − + θi θj θj θi = δijαˇi for i, j in I. −→ ⊗ + + ⊗ ⊗ + Define a Q-algebra homomorphism ∆ : U U U by ∆(θi )=θi 1+1 θi , − − ⊗ ⊗ − ∈ ⊗ ⊗ ∈ ∆(θi )=θi 1+1 θi for i I,∆(y)=y 1+1 y for y Y . Define a −→ opp + − + − − − ∈ Q-algebra isomorphism Σ : U U by Σ(θi )= θi ,Σ(θi )= θi for i I, Σ(y)=−y for y ∈ Y .

6G130 .LUSZTIG

(n) n ∈ Let f Z be the subring of f generated by the elements θi := θi /n!, (i ∈ I,n N). Note that f Z is a lattice in the Q-vector space f. Following [Ko]we define U to be the subring of U generated by the elements x+,x− (x ∈ f )and Z Z y y(y−1)...(y−n+1) ∈ ∈ n := n! ,(y Y,n N). Note that U Z is a lattice in the Q-vector space U. Hence U Z ⊗Z U Z is a lattice in U ⊗Q U. Note that ∆ restricts to a ring homomorphism U Z −→ U Z ⊗Z U Z denoted again by ∆. Also Σ restricts to a ring opp isomorphism U Z −→ U Z denoted again by Σ. For any U-module M and any x ∈ X we set M x = {m ∈ M; ym = y, xm for any y ∈ Y }. Let C be the category whose objects are U-modules M with dimQ M<∞ such x † that M = ⊕x∈X M . For any Q-vector space V we set V =HomQ(V,Q). For † † M ∈Cwe define cM : M ⊗ M −→ U by m ⊗ ξ → [u → ξ(um)]. Let † O = cM (M ⊗ M ), M∈C † a Q-subspace of U . (This agrees with the definition in [Ko]whenR is semisim- ∈O −→ → ple.) Forf,f we define ff : U Q by u s f(us)f (us)where ⊗ ∈ ∈O ∆(u)= s us us, us,us U.Wehaveff . This defines a structure of associative, commutative algebra on O. This algebra has a unit element: the −→ + → − → ∈ → algebra homomorphism U Q such that θi 0, θi 0fori I, y 0for y ∈ Y and 1 → 1. For f ∈Owe define a linear function δ(f):U ⊗ U −→ Q by † u1 ⊗ u2 → f(u1u2). Note that O⊗Ois naturally a subspace of (U ⊗ U) and that the image of δ : O−→ (U ⊗ U)† is contained in the subspace O⊗O so that δ defines a linear map O−→O⊗Odenoted again by δ. This is an algebra homomorphism. For f ∈Owe define a linear function σ(f):U −→ Q by u → f(Σ(u)). We have σ(f) ∈O;thusσ defines a linear map O−→Odenoted again by σ.Thisisanalge- bra homomorphism. Define : O−→ Q by f → f(1). Note that the commutative algebra O with the comultiplication δ, the antipode σ and the counit is a Hopf algebra over Q. x Let f ∈O. There is a unique collection (f )x∈X of numbers in Q such that f x = 0 for all but finitely many x ∈ X and n1 n2 nr n1 n2 nr x f(y1 y2 ...yr )= y1,x y2,x ... yr,x f x∈X for any y1,y2,...,yr in Y and n1,n2,...,nr in N. For example if f = cM (m ⊗ ξ) ∈C ∈ ∈ † x ∈ x where M ,m M,ξ M ,wehavef = ξ(mx)wheremx M are defined ∈ → x O−→ by m = x∈X mx. Note that for any x X, f f is a linear function Q. Let OZ = {f ∈O; f(U Z) ⊂ Z}. (This agrees with the definition in [Ko]when R is semisimple.) Note that OZ is a subring of O. One can show that OZ is a lattice in the Q-vector space O. Hence OZ ⊗Z OZ is a lattice in the Q-vector space O⊗O. Note that δ : O−→O⊗Orestricts to a ring homomorphism δZ : OZ −→OZ ⊗Z OZ; σ : O−→Orestricts to a ring isomorphism σZ : OZ −→OZ; : O−→ Q restricts to a ring homomorphism Z : OZ −→ Z. The commutative ring OZ together with the comultiplication δZ, the antipode σZ and the counit Z is a Hopf ring. For any x x ∈ X and f ∈OZ we have f ∈ Z. For any commutative ring A with 1 we set OA = A ⊗OZ. By extension of scalars, from δZ,σZ,Z we get A-algebra homomorphisms δA : OA −→OA ⊗A OA, σA : OA −→OA, A : OA −→ A. The commutative A-algebra OA together with the

TWELVE BRIDGES FROM A REDUCTIVE GROUP TO ITS LANGLANDS DUAL131 7

comultiplication δA, the antipode σA and the counit A is a Hopf algebra over A. For any x ∈ X the homomorphism OZ −→ Z gives rise by extension of scalars to an x A-linear map OA −→ A denoted again by f . The following two properties are proved in [L19]: (i) the A-algebra OA is ﬁnitely generated; (ii) if A is an integral domain then OA is an integral domain.

6. Chevalley groups. We preserve the notation of §5. Let W, si be as in §1. Let A be a commutative ring with 1. As in [Ko] we define GA to be the set O −→ ∈ of A-algebra homomorphisms A A respecting 1. For g, g GA we define O −→ → ⊗ O gg : A A by f s g(fs)g (fs)whereδA(f)= s fs fs with fs,fs in A. Then gg ∈ GA and (g, g ) → gg is a group structure on GA with unit element A. We say that GA is the Chevalley group attached to the root datum R and to the R commutative ring A.WewritealsoGA instead of GA when we want to emphasize the dependence on R. If κ : A −→ A is a homomorphism of commutative rings with 1 and g : OA −→ A is in GA then applying to g the functor A ⊗A?(whereA is regarded as an A-algebra via κ)weobtainanA -algebra homomorphism OA −→ A respecting 1 which is denoted byg ˜.Nowg → g˜ is a group homomorphism GA −→ GA said to be induced by κ. For any i ∈ I,b ∈ A we define an A-linear map xi(b):OA −→ A by ⊗ → n (n) + as fs asb fs((θi ) ). s s n∈N

∈ ∈O ∈O (n) + Here as A, fs Z.Sincefs we have fs((θi ) ) = 0 for large enough n so that the last sum is finite. From the definitions we see that xi(b) ∈ GA and that b → xi(b) is an (injective) group homomorphism A −→ GA. Similarly, for any i ∈ I,b ∈ A we define an A-linear map yi(b):OA −→ A by ⊗ → n (n) − as fs asb fs((θi ) ). s s n∈N

Here as ∈ A, fs ∈OZ. Again the last sum is finite. From the definitions we see that ∈ → −→ yi(b) GA and that b yi(b) is an (injective) group homomorphism A GA. ∗ ⊗ ∈ ∗ ⊗ Let A be the group of units of A.Lett = r ar yr A Y with ∗ ar ∈ A ,yr ∈ Y . We define t : OA −→ A by

→ yr ,x x f ar f . r x∈X ∈ ∗ n ∈ (Since ar A , ar is defined for any n Z.) From the definitions we see that ∗ t ∈ GA and that t → t is an injective group homomorphism A ⊗ Y −→ GA with ∗ image denoted by TA (a commutative subgroup of GA). We identify A ⊗ Y with R TA via this homomorphism. We write also TA instead of TA when we want to emphasize the dependence on R. For i ∈ I we define an elements ˙i ∈ GA bys ˙i = xi(1)yi(−1)xi(1). We have 2 − ⊗ ∈ ∗ ⊗ ··· s˙i =( 1) αˇi A Y = TA.Moreover,fori = j we haves ˙is˙js˙i =˙sjs˙is˙j ... (both sides have n factors where n is the order of sisj in W ). It follows that for any ∈ ∈ w W there is a well defined elementw ˙ GA such thatw ˙ =˙si1 s˙i2 ...s˙ir whenever −1 w = si1 si2 ...sir with r = l(w). Note thatwT ˙ Aw˙ = TA. More precisely, for

8G132 .LUSZTIG

−1 t ∈ TA we havewt ˙ w˙ = w(t)wherew : t → w(t)istheW -action on TA given by a ⊗ y → a ⊗ w(y)fora ∈ A∗, y ∈ Y . | +| For any sequence i1,i2,...,ir in I such that l(si1 si2 ...sir )=r = R ,the r map A −→ GA given by

(a1,a2,...,ar) → x (a )˙s x (a )˙s−1 ...s˙ s˙ ...s˙ x (a )˙s−1 ...s˙−1s˙−1 i1 1 i1 i2 2 i1 i1 i2 ir−1 ir r ir−1 i2 i1

+ is injective and its image is a subgroup UA of GA independent of the choice of i1,i2,...,ir.(See[L19].)

Similarly, for any sequence i1,i2,...,ir in I such that l(si1 si2 ...sir )=r = + r |R |,themapA −→ GA given by

(a1,a2,...,ar) → y (a )˙s y (a )˙s−1 ...s˙ s˙ ...s˙ y (a )˙s−1 ...s˙−1s˙−1 i1 1 i1 i2 2 i1 i1 i2 ir−1 ir r ir−1 i2 i1 − is injective and its image is a subgroup UA of GA independent of the choice of i1,i2,...,ir. + − + + + The subgroups UA ,UA are normalized by TA.WesetBA = UA TA = TAUA , − − − BA = UA TA = TAUA . ∪ + + If A is a field, we have a partition GA = w∈W BA wB˙ A . 7. Weyl modules. We preserve the notation of §6. For any λ ∈ X+ let T αˇi,λ +1 T λ = i fθi , a left ideal of f;letΛλ = f/ λ, a finite dimensional Q-vector − space. Let η be the image of 1 ∈ f in Λλ. We regard Λλ as a U-module in which x ∈ + ∈ ∈ acts as left multiplication by x(x f); θi η =0fori I; yη = y, λ η for y Y . We say that Λλ is a Weyl module.WehaveΛλ ∈C. For λ ∈ X+ let T (n) T ∩ λ,Z = fθi = λ f Z, i,n;n≥ αˇi,λ +1 T a left ideal of f Z.LetΛλ,Z = f Z/ λ,Z.ThenΛλ,Z is a lattice in the Q-vector space Λλ and a U Z-submodule of Λλ. For a commutative ring A with 1 we set Λλ,A =Λλ,Z ⊗ A. Oopp Oopp O O We write , A for , A with the opposite comultiplication. Define Ξ : −→O⊗ → ⊗ ⊗ Λλ Λλ by e j cΛl (e ξj) ej where (ej)isaQ-basis of Λλ and (ξj) † Oopp O ⊗ is the dual basis of Λλ.ThismakesΛλ into a -comodule. Now Z Z Λλ,Z is a lattice in O⊗Λλ and Ξ restricts to ΞZ :Λλ,Z −→OZ ⊗Z Λλ,Z. By extension of scalars we obtain an A-linear map ΞA :Λλ,A −→OA ⊗A Λλ,A making Λλ,A into a Oopp O −→ −→ A -comodule. For any g : A A which is in GA, we define ρg :Λλ,A Λλ,A → ⊗ ⊗ ∈O ∈ by e h g(fh) eh where ΞA(e)= h fh eh, fh A,eh Λλ,A.Notethat g → ρg is a group action. Thus Λλ,A is a GA-module. 8. The Langlands dual group. If A is an algebraically closed field then R GA = GA is a reductive connected algebraic group over A with coordinate ring O R → R A. Moreover, according to Chevalley, GA is a bijection ∼ {root data up to isom.} −→ {reductive connected algebraic groups over A up to isom.}.

TWELVE BRIDGES FROM A REDUCTIVE GROUP TO ITS LANGLANDS DUAL133 9

An element of GA is said to be semisimple if it is conjugate to an element in TA. + An element of GA is said to be unipotent if it is conjugate to an element in UA . (a) In the remainder of these notes (except in §15) we fix a root datum R = (Y,X,, , αˇi,αi(i ∈ I)). Define W, W,H in terms of R as in §1,§3,§4. Define W∗, H∗ like W, W∗ but in terms of R∗ instead of R. R R For a commutative ring A with 1 we write GA,TA instead of GA ,TA and ∗ ∗ R∗ R∗ ∗ GA,TA instead of GA ,TA .WesaythatGA is the Langlands dual group to GA. B + Let A be the set of subgroups of GA that are conjugate to BA (or equivalently to − B∗ R∗ R ∈ ∗ BA ). Let A be the analogous set defined in terms of instead of .Forg GA B∗ { ∈B∗ ∈ } we set A,g = B A; g B . 9. Representations of affine Hecke algebras. Assume that X/ i Zαi ∗ has no torsion. We fix v ∈ C .LetHv = C ⊗A H where C is regarded as an ∗ A-algebra via v → v.LetΨv be the set of all pairs (s, u)uptoGC-conjugation ∗ ∗ −1 v2 where s ∈ GC is semisimple, u ∈ GC is unipotent and sus = u .(Thereisa ∗ z unique morphism of algebraic groups C −→ GC, z → u such that for any n ∈ N, un is the n-th power of u.) Assume that v is not a root of 1. The Deligne-Langlands conjecture states that there is a canonical finite to one surjective map

(a) {irred. Hv-modules up to isom.}−→ Ψv. A refinement of this conjecture was stated in [L5] namely that the fibre of (a) at (s, u) should be in natural bijection with the set of irreducible representations (up to isomorphism) of the group of connected components of the centralizer of (s, u) ∗ in GC which appear in the natural representation of this group on the cohomology ∗ of {B ∈BC; s ∈ B,u ∈ B}. This came from a study of examples connected with ∗ “subregular” unipotent elements in GC.In[L7]itwasshownthatHv acts naturally ∗ on the equivariant K-theory of BC where the parameter of the Hecke algebra comes from equivariance with respect to a C∗-action. In [L7] it was also suggested that one should construct representations of Hv using the equivariant K-theory of the B∗ ∈ ∗ varieties C,u for u GC unipotent. This was established in [KL2]whichgavea proof of (a) (in the refined form). In [Xi] it is shown (by a reduction to [KL2]) that a statement analogous to (a) (in the refined form) holds also when v is allowed to be a root of 1 in the complement of a specific finite set of roots of 1 depending on R. 10. p-adic groups. Let K be a finite extension of the field of p-adic numbers (p aprimenumber).Leta be the integral closure in K of the ring of p-adic integers and let m be the unique maximal ideal of a so that a/m is a finite field with q elements. Let K¯ be an algebraic closure of K.Leta¯ be the integral closure of a in K¯ and let m¯ be the unique maximal ideal of a¯ so that a¯/m¯ is an algebraic closure of a/m.LetWK be the Weil group of K, that is, the inverse image under the natural homomorphism π :Gal(K/K¯ ) −→ Gal(a¯/m¯ , a/m) of the subgroup Z of Gal(a¯/m¯ , a/m) consisting of the integer powers of the automorphism x → xq.Let ω I be the kernel of π. We have an exact sequence 1 −→I−→ WK −→ Z −→ 1where ∗ ω is the restriction of π. A homomorphism ρ : WK −→ GC is said to be admissible ∗ −1 if ρ(I) is finite and ρ(γ) is semisimple in GC for some/any γ ∈ ω (1). Let K ∗ ∗ ∗ Φ (GC) be the set of all pairs (ρ, u)(uptoGC-conjugacy) where ρ : WK −→ GC ∗ is an admissible homomorphism and u ∈ GC is a unipotent element such that

10134 G. LUSZTIG

−1 qω(w) ρ(w)uρ(w) = u for any w ∈ WK . We regard GK as a topological group with the p-adic topology. An irreducible representation GK −→ GL(E)(whereE is a C-vector space) is said to be admissible if the stabilizer of any vector of E is open in GK and if for any open subgroup H of GK the space of H-invariant vectors in E has ﬁnite dimension. According to the local Langlands conjecture there is a canonical ﬁnite to one surjective map

K ∗ (a) {irred. admissible representations of GK up to isom.}−→ Φ (GC).

This is known to be true in the case where GK = GLn(K), see [HT]. In the general case but assuming that R is adjoint, a class of irreducible admissible representations (called “unipotent”) has been described in [L15] where a canonical finite to one surjective map { }−→ K ∗ (b) unipotent representations of GK up to isom. Φ1 (GC) K ∗ { ∈ ∗ I { }} was constructed; here Φ1 (GC)= (ρ, u) Φ(GC); ρ( )= 1 .Notethat K ∗ √ § Φ1 (GC) may be identified with the set Ψ q in 9 and that (b) constitutes a verification of (a) in a special case. Note that some of the unipotent representations can be understood by the method described in §9; to understand the remaining ones one needs the theory of character sheaves and a geometric construction of certain affine Hecke algebras with unequal parameters in terms of equivariant homology. K ∗ For any (ρ, u) ∈ Φ (GC)wedenotebyΞρ,u the set of irreducible representations (up to isomorphism) of the group of connected components of the simultaneous ∗ ∗ centralizer of (ρ, u)inGC on which the action of the centre of GC is trivial. ∈ K ∗ According to [L15], for any (ρ, u) Φ1 (GC) the fibre of the map (b) at (ρ, u)is K ∗ in bijection with Ξρ,u. This suggests that more generally for any (ρ, u) ∈ Φ (GC), the fibre of the (conjectural) map (a) at (ρ, u) is in bijection with Ξρ,u. Note that in general neither side of (a) is well understood. But recent results of J.-L.Kim [Ki] give a classification of “supercuspidal representations” of GK (assuming that p is sufficiently large) which gives some hope that the left hand side of (a) can be understood for such p. ∗ 11. Real groups. The Weil group of R is by definition WR = C ×Gal(C/R) (τ )(τ ) with the group structure (z1,τ1)(z2,τ2)=((−1) 1 2 z1τ1(z2),τ1τ2)where (τ)=0ifτ =1and(τ)=1ifτ =1.Wecanidentify WR with the subgroup of the group of nonzero quaternions a+bi+cj +dk generated by {a+bi;(a, b) ∈ R2 −{0}} and by j. We regard WR as a Lie group with two connected components. Let R ∗ ∗ Φ (GC) be the set of all continuous homomorphisms WR −→ GC whose image ∗ consists of semisimple elements, up to conjugation by GC.LetK be a maximal compact subgroup of the Lie group GR. An “irreducible admissible” representation of GR is by definition a C-vector space E with an action of K and one of Lie (GR) such that any vector in E is contained in a finite dimensional K-stable subspace of E; the two actions induce the same action on Lie (K); the action of Lie (GR)is compatible with the K-actiononLie(GR)andtheK-action on E.Moreover,this should be irreducible in the obvious sense. According to Langlands [La2] there is a canonical finite to one surjective map

R ∗ (a) {irred. admissible representations of GR up to isom.}−→ Φ (GC).

The ﬁbres of (a) have been described by Knapp and Zuckerman.

TWELVE BRIDGES FROM A REDUCTIVE GROUP TO ITS LANGLANDS DUAL135 11

R ∗ ∗ We now give some examples of elements of Φ (GC) in the case where GC = GL(V )withV a finite dimensional C-vector space. Assume that we are given a p,q direct sum decomposition V = ⊕p,qV with (p, q) ∈ C × C,p− q ∈ Z; assume also that we are given a C-linear isomorphism : V −→ V such that 2 =1and p,q q,p (V )=V for all p, q. We define an action of WR on V by specifying the action of (z,1) with z ∈ C∗ and the action of (1,τ)withτ ∈ Gal(C/R) −{1}.If (p, q) runs only in Z × Z we have (z,1) · x = zpz¯qx for z ∈ C∗,x∈ V p,q.Thesame formula holds in the general case: we interpret zpz¯q as (zz¯)(p+q)/2( √z )p−q.(The zz¯ strictly positive real number zz¯ can be raised to any complex power.) We have p−q p,q R ∗ (1,τ) · x = (−1) (x)forx ∈ V . This is an object of Φ (GC). 12. Global fields. Let k be a field which is a finite algebraic extension of the field of rational functions in one variable over the finite field Fp.LetA be the ring of adeles of k and let k −→ A be the canonical imbedding. Let k¯ be an algebraic closure of k.Letl be a prime number = p. The global Langlands conjecture [La1] predicts a connection between the set consisting of irreducible “cuspidal” representations of GA with nonzero vectors fixed ¯ ∗ by Gk on the one hand and a certain set of homomorphisms of Gal(k/k)intoG ¯ Ql which are irreducible in a suitable sense, on the other hand. This conjecture has been proved in the case where Gk = GLn(k). (See [Dr]for n =2and[Lf] for any n.) There is an analogous conjecture in which k is replaced by a finite extension of Q and also a geometric analogue of the conjecture in which the curve over a finite field represented by k is replaced by a smooth projective curve over C.(See [KW].) 13. Cells in affine Weyl groups and unipotent classes. Define φ : H−→ A⊗J in terms of R as in §4. Let K be an algebraic closure of the field C(v)of rational functions with coefficients in C in an indeterminate v.Letc be a two-sided cell of W .LetJc be the corresponding direct summand of the ring J. We can find some simple module E of the C-algebra C ⊗ Jc. It is necessarily of finite dimension over C. We can regard K⊗CE as a K⊗QJ-module in which the summands K⊗QJc + act as zero for c = c.Fory, y in Y we have Tay Tay = Tay Tay = Tay+y (see + §3). Hence the operators φ(Tay ):K ⊗C E −→ K ⊗C E (with y ∈ Y )commute. We can find e ∈ K ⊗C E −{0} which is a simultaneous eigenvector for these + ∗ operators. Thus we have φ(Tay )e = b(y)e for all y ∈ Y where b(y) ∈ K satisfy b(y)b(y)=b(y + y) for any y, y in Y +. There is a unique element t ∈ K∗ ⊗ X ⊗ ∈ ∗ ∈ y,x such that, if t = s ks xs with ks K ,xs X then b(y)= s ks for any y ∈ Y +. One can show [L9]thatt is a very special element of K∗ ⊗ T :wecan ∈ ∗ ⊗ ∈{ n ∈ }⊗ ⊂ ∗ ⊂ ∗ write uniquely t = t t where t C X and t v ; n Z X TK GK is v 0 ∗ equal to ϑ − for some homomorphism of algebraic groups ϑ : SL (K) −→ G . 0 v 1 2 K C G∗ ϑ 11 G∗ Let be the conjugacy class in C such that 01 is conjugate in K to some element of C. One can show [L9]thatC is well defined by c (it is independent of the choice of E,e,ϑ)andthatc → C is a bijection ∼ ∗ (a) {two-sided cells of W } −→{unipotent conjugacy classes in GC}. 14. Special unipotent classes. We preserve the setup in §13. The intersection of W with a two-sided cell of W is said to be a two-sided cell of W if it is

12136 G. LUSZTIG

nonempty. Note that the two-sided cells of W form a partition of W . A unipotent ∗ conjugacy class in GC is said to be special if it corresponds under the bijection §13(a) to a two-sided cell of W which has a nonempty intersection with W . ∗ The special unipotent classes of GC were introduced in a different (but equivalent) way in [L1] as the unipotent classes such that the corresponding irreducible representation of W (under the Springer correspondence) is in the class SW defined in [L1]. This definition makes sense when C∗ is replaced by any algebraically closed field A.Forρ ∈SW we denote by Cρ,A the corresponding special unipotent element ∗ ∗ of GA and by Cρ,A the corresponding special unipotent element of GA.(Thesets ∗ SW for R, R coincide.) Let Cˆρ,A = C¯ρ,A −∪C C¯ where C¯ρ,A is the closure of Cρ,A and C runs over the special unipotent classes contained in C¯ρ,A −Cρ,A.Itisknown that the subsets Cˆρ,A form a partition of the unipotent variety of GA into locally closed subvarieties which are rational homology manifolds. We define similarly the ˆ∗ ∗ subvarieties Cρ,A of the unipotent variety of GA. We have the following result (see [L16],[L17]): For any ρ ∈SW there exists a polynomial Pρ with integer coefficients such that for any q (a power of a prime number) we have

ˆ ˆ∗ ∗ |C F ∩ GF | = |C ∩ GF | = Pρ(q). ρ, q q ρ,Fq q

15. Preparatory results. Let Y,X be two free abelian groups of finite rank and let , : Y × X −→ Z be a perfect pairing. Let A : Y −→ Y be a homomorphism such that det(A) = 0, that is, such that |Y/AY | < ∞.Wethenhave|Y/AY | = ± det(A). Define a homomorphism A : X −→ X by y, A(x) = A(y),x for all y ∈ Y,x ∈ X.Thendet(A)=det(A) hence |X/A(X)| < ∞.NowA (resp. A) induces endomorphisms of Q ⊗ Y and of Q/Z ⊗ Y (resp. Q ⊗ X and Q/Z ⊗ X) denoted again by A (resp. A). Also, , induces a Q-linear pairing (Q⊗Y )×(Q⊗X) −→ Q denoted again by , . We define a pairing (, ):Y/A(Y ) × X/A(X) −→ Q/Z by

(y, x)=A−1(y),x mod Z = y, A−1(x) mod Z where y ∈ Y,A−1(y) ∈ Q ⊗ Y , x ∈ X, A−1(x) ∈ Q ⊗ X.Nowx → [y → (y, x)] is an isomorphism ∼ (a) X/A(X) −→ Hom(Y/A(Y ), Q/Z). ∼ We deﬁne an isomorphism Y/A(Y ) −→ (Q/Z ⊗ Y )A+1 (ﬁxed point set of A +1)by

y → image of A−1(y) under Q ⊗ Y −→ Q/Z ⊗ Y.

∼ Similarly we have an isomorphism X/A(X) −→ (Q/Z ⊗ X)A +1.Viathelasttwo isomorphisms, (a) becomes an isomorphism

∼ (Q/Z ⊗ X)A +1 −→ Hom((Q/Z ⊗ Y )A+1, Q/Z).

This is induced by ξ → [η →A(η),ξ mod Z where ξ ∈ Q ⊗ X, A(ξ) ∈ X, η ∈ Q ⊗ Y,A(η) ∈ Y .Letp be a prime number and let (Q/Z) be the subgroup of Q/Z consisting of elements of order not divisible by p. Assume now that p does not divide det(A). Then (Q/Z⊗Y )A+1 =((Q/Z) ⊗Y )A+1 and we get an isomorphism

∼ (b) (Q/Z ⊗ X)A +1 −→ Hom(((Q/Z) ⊗ Y )A+1, (Q/Z)).

TWELVE BRIDGES FROM A REDUCTIVE GROUP TO ITS LANGLANDS DUAL137 13

∗ A+1 Now let k be an algebraic closure of the finite field Fp.Let(k ⊗ Y ) be the fixed point set of the endomorphism z ⊗ y → z ⊗ (A +1)y of k∗ ⊗ Y . We define a canonical isomorphism ∼ (c) Hom(((Q/Z) ⊗ Y )A+1, (Q/Z)) −→ Hom((k∗ ⊗ Y )A+1,k∗) ∼ as follows. We choose an isomorphism ζ :(Q/Z) −→ k∗.Thenζ ⊗ 1:(Q/Z) ⊗ ∼ ∗ A+1 ∼ ∗ A+1 Y ) −→ k ⊗ Y restricts to an isomorphism ζ1 :(Q/Z) ⊗ Y ) −→ (k ⊗ Y ) and (c) carries a homomorphism φ :((Q/Z) ⊗ Y )A+1 −→ (Q/Z) to ζφζ−1.We 1 ∼ must show that the map (c) is independent of the choice of ζ.Letκ :(Q/Z) −→ ∼ (Q/Z) be an isomorphism. Then κ ⊗ 1:(Q/Z) ⊗ Y ) −→ (Q/Z) ⊗ Y restricts A+1 ∼ A+1 to an isomorphism κ1 :(Q/Z) ⊗ Y ) −→ ((Q/Z) ⊗ Y ) and it is enough to show that for any homomorphism φ :((Q/Z) ⊗ Y )A+1 −→ (Q/Z) we have −1 κφκ1 = φ.Since(Q/Z) is an injective Z-module, there exists a homomorphism φ˜ :(Q/Z) ⊗ Y −→ (Q/Z) whose restriction to ((Q/Z) ⊗ Y )A+1 is φ. It is enough to show that κφ˜(κ ⊗ 1)−1 = φ˜. By choosing a basis of Y we see that it is enough to show that for any homomorphism ψ :(Q/Z) −→ (Q/Z) we have κψκ−1 = ψ. This follows from the fact that the ring of endomorphisms of the group (Q/Z) is commutative (it is a product of rings of l-adic integers for various primes l = p). Let µ be the group of roots of 1 in C. We note that the isomorphism ζ : Q/Z −→ µ given by r → exp(2πir) induces an isomorphism ζ ⊗ 1:Q/Z ⊗ X −→ µ ⊗ X ∼ ⊗ A +1 −→ ⊗ A +1 and this restricts to an isomorphism ζ1 :(Q/Z X) (µ X) where (µ ⊗ X)A +1 is the fixed point set of the endomorphism z ⊗ x → z ⊗ (A +1)x of ⊗ µ X.Viaζ1 and (c), the isomorphism (b) becomes a canonical isomorphism ∼ (d) (µ ⊗ X)A +1 −→ Hom((k∗ ⊗ Y )A+1,k∗).

16. Groups over Fq. We return to the setup in §8(a). Let k be an algebraic closure of the finite field Fp.LetK be an algebraically closed field of characteristic 0 with a fixed imbedding of groups ι : k∗ −→ K∗. ∗ ∗ ∗ We have Tk = k ⊗ Y , TC = C ⊗ X.LetFq be the subfield of k such that q |Fq| = q. The ring homomorphism k −→ k, c → c induces (as in §6) a group homomorphism F : Gk −→ Gk (Frobenius map) whose fixed point set is the finite ∈ ˙ { ∈ group GFq . Following [DL] we consider for any w W the set Xw = g −1 ∈ +} w { ∈ q −1 } Gk; g F (g) wU˙ k , an algebraic variety over k.LetTk = t Tk; t = w (t) , × w ˙ → a finite subgroup of Tk. The finite group GFq Tk acts on Xw by (g1,t):g −1 × w −→ ∈ g1gt .Letχw : GFq Tk Z be the class function which to any (g1,t) × w ∗ GFq Tk associates the alternating sum of traces of (g1,t) on the l-adic cohomology with compact support of X˙ w. (Here l is any prime number = p but the resulting

class function is known to be independent of l;see[DL].) For any irreducible GFq - E ∈ ∈ w ∗ module ρ over K let ρ be the set of all pairs (w, θ)wherew W , θ Hom(Tk ,K ) and ∈ × w θ(t)tr(g1,ρ)χw(g1,t) =0.Accordingto[ DL]wehaveEρ = ∅ (g1,t) GFq Tk for any ρ. ∗w−1 ∗ q To any (w, θ) ∈Eρ we associate an element θˆ ∈ TC := {t ∈ TC; t = w(t)} as follows. Define A : Y −→ Y by y → qw(y)−y and A : X −→ X by x → qw−1(x)−x. w ∗ ⊗ A+1 w −→ ∗ Then Tk =(k Y ) is a finite group of order prime to p. Hence θ : Tk K has values in the group of roots of 1 of order prime to p in K∗ which can be identified with k∗ via ι.Thusθ can be viewed as an element of Hom((k∗ ⊗ Y )A+1,k∗) so that it corresponds under §15(d) to an element θˆ of (µ ⊗ X)A +1.Thislast

14138 G. LUSZTIG

group is a subgroup of (C∗ ⊗ X)A +1 (the ﬁxed point set of the endomorphism ∗ ∗w−1 z ⊗ x → z ⊗ (A +1)x of C ⊗ X)whichisthesameasTC . From the results in ∗ [DL] we see that the W -orbit of θˆ in TC depends only on ρ and not on the choice of (w, θ)inEρ. We thus have a well deﬁned map { }−→ irred. GFq -modules over K up to isom. ∗ q (a) {semisimple conjugacy classes in GC stable under g → g };

∗ it is given by ρ → GC-conjugacy class of θˆ (as above). This map appears in [DL] ∗ ∗ in a somewhat different form. In [DL] GC is replaced by Gk. But the method of [DL] is less canonical: it is based on two choices (see [DL, (5.0.1), (5,0.2)]) while the present method is based on only one choice, that of ι; the choice of ι can be also eliminated as we will see below). ∗ An element g ∈ GC is said to be special if the unipotent part gu of g is a special unipotent element (see §14) of the connected centralizer of the semisimple part gs ∗ of g (a reductive connected group). A conjugacy class in GC is said to be special if some/any element of it is special. The map (a) can be refined to a canonical map { }−→ irred. GFq -modules over K up to isom. ∗ q (b) {special conjugacy classes in GC stable under g → g }. (See [L6].) Note that (a) is the composition of (b) with the map which to the ∗ ∗ GC-conjugacy class of a special element g associates the GC-conjugacy class of gs. The map (b) is surjective and its fibres are described explicitly in [L6], [L10]. Note that the maps (a),(b) depend on the choice of the imbedding ι : k∗ −→ K∗. However if we take K to be an algebraic closure of the quotient field of the ring of Witt vectors of k then there is a canonical choice of ι and the maps (a),(b) become completely canonical. + + 17. Character sheaves. Define BC,UC , w˙ in terms of C, R as in §6. Let E be a C-local system of rank 1 on TC with finite monodromy. The monodromy of ∗ E is a homomorphism f : Y −→ C with finite image which can be viewed as an ∈ ∗ ⊗ ⊗ element of finite order χE C X given by χE = j f(yj) xj where (yj)isa basis of Y and (xj) is the dual basis of X.MoreoverE → χE is a bijection ∼ {C − local systems of rank 1 on TC with finite monodromy up to isom.} −→ (a) ∗ {elements of finite order of TC}. + Let c : GC −→ GC/UC be the obvious map. An irreducible intersection cohomology complex K on GC is said to be a character sheaf on GC if it is GC-equivariant and if for any w ∈ W and any j ∈ Z the j-th cohomology sheaf of c!K restricted to + + + BCwB˙ C/UC is a local system LK,w,j with finite monodromy. We can find w, j as above and a local system E of rank 1 on TC with finite monodromy such that E is a + + + direct summand of the inverse image of LK,w,j under the map TC −→ BCwB˙ C/UC , + ∗ t → wtU˙ C . One can show that the corresponding element χE ∈ TC is well defined (up to the action of W ) that is, it does not depend on the choice of w, j, E.Thus we have a well defined map

{character sheaves on GC up to isom.}−→ ∗ (a) {conjugacy classes of elements of ﬁnite order in GC};

TWELVE BRIDGES FROM A REDUCTIVE GROUP TO ITS LANGLANDS DUAL139 15

∗ it is given by K → GC-conjugacy class of χE (as above). The map (a) can be reﬁned to a canonical map

{character sheaves on GC up to isom.}−→ (b) ∗ {special conjugacy classes in GC of elements g with gs of ﬁnite order}. (See [L8].) Note that (a) is the composition of (b) with the map which to the ∗ ∗ GC-conjugacy class of a special element g associates the GC-conjugacy class of gs. The map (b) is surjective and its ﬁbres are described explicitly in [L8].

18. Vogan duality. To GC we associate a finite collection of polynomials: those recording the restrictions of the cohomology sheaves of the simple perverse sheaves on BC, equivariant under the conjugation action of the centralizers of the various involutions of GC, to the various orbits of these centralizers. (These polynomials were studied in [LV].) We consider also the analogous collection of poly- ∗ nomials associated to GC. Vogan duality [V1] states that these two collections of polynomials are related to each other by a simple algebraic rule: essentially the inversion of a matrix. This is a generalization of the inversion formula in [KL1]. 19. Multiplicities in standard modules. Let K be either as in §10 or K = R. The Grothendieck group whose basis consists of admissible irreducible representations of GK has another natural basis consisting of “standard representations” in natural bijection with the first basis. The representations in the second basis are easier to describe and understand. Hence the (upper triangular) matrix M expressing the second basis in terms of the first basis (“multiplicity matrix”) is of interest. In every known case the entries of M canbeexpressedinterms ∗ of intersection cohomology coming from the geometry of GC. For the case where K = R we refer the reader to [ABV]; in this case some of the polynomials in §18 ∗ (attached to GC) evaluated at 1 appear as entries of M. In the remainder of this subsection we assume that K is as in §10. For simplicity we assume that R is of adjoint type. We use the notation in §10. 0 0 K We fix an element w ∈ WK such that ω(w )=1.LetΦ˜ be the set of ∗ ∗ all triples (ρ, u, E)(uptoGC-conjugacy) where ρ : WK −→ GC is an admissible ∗ −1 qω(w) homomorphism, u is a unipotent element of GC such that ρ(w)uρ(w) = u for all w ∈ WK and E is an irreducible representation of the group of connected ∗ { ∈ ∗ −1 ∈ } components of Gρ,u := g GC; ρ(w)gρ(w) = g for all w WK ,gu = ug on ∗ which the image of the centre of GC acts trivially. By §10 it is expected that Φ˜ K is an index set for both the rows and the columns of M. We shall describe a matrix M indexed by Φ˜ K whichisdefinedintermsof ∗ geometry of GC. ∗ Let Ψ be the set of homomorphisms ψ : I−→ GC such that ψ(I) is finite and ∗ −1 0 0−1 such that Γψ := {g ∈ GC; gψ(w)g = ψ(w ww ) for all w ∈I}is non-empty. ∗ Let Ψbethesetof¯ GC-orbits (by conjugacy) on Ψ. ˜ K ¯ Define κ : Φ −→ Ψby(ρ, u, E) → ρ|I .Theentriesm(ρ,u,E),(ρ,u,E) of M can be described as follows. If (ρ, u, E), (ρ,u,E) are not in the same fibre of κ then m(ρ,u,E),(ρ,u,E) =0. ∈ ∗ { ∈ ∗ −1 ∈I} We now fix ψ Ψ. Let Gψ = g GC; gψ(w)g = ψ(w) for all w . ∗ This is the centralizer of a finite subgroup of GC hence is a (possibly disconnected) ∗ ∗ I ∗ reductive subgroup of GC.LetGψ be the normalizer of ψ( )inGC.Notethat

16140 G. LUSZTIG

∗ ∗ ∗ Gψ is a normal subgroup of finite index of Gψ and that Γψ is a single Gψ-coset ∗ in Gψ . The fibre of κ at ψ can be identified with the set of all triples (s, N, E) ∗ ∈ ∗ ∈ (up to Gψ-conjugacy) where s Gψ is a semisimple element such that s Γψ, { ∈ ∗ } N is an element of Xs := N1 Lie (Gψ); Ad(s)N1 = qN (necessarily nilpotent) and E is an irreducible representation of the group of connected components of ∗ { ∈ ∗ } Gψ,s,N := g Gψ; gs = sg, Ad(g)N = N on which the image of the centre ∗ 0 of GC acts trivially. (The identification is given by (ρ, u, E) → (ρ(w ), log(u),E) −1 where ρ|I = ψ.) We now consider two elements (s, N, E), (s ,N ,E )inκ (ψ). If ∗ s, s are not in the same Gψ-orbit then m(s,N,E),(s,N ,E) = 0. Now assume that ∗ (s, N, E), (s ,N ,E )aresuchthats, s are in the same Gψ-orbit. We can assume ∗ { ∈ ∗ } that s = s .LetGψ,s = g Gψ; gs = sg . This is an algebraic group which ∗ acts on Xs by conjugation with finitely many orbits. Let C be the Gψ,s-orbit of ∗ N and let C be the Gψ,s-orbit of N .NotethatE (resp. E ) determines a local ∗ system E (resp. E )onC (resp. C )whichisGψ,s-equivariant and is irreducible ∗ as a Gψ,s-equivariant local system. If C is not contained in the closure of C then m(s,N,E),(s,N ,E) = 0. Now assume that C is contained in the closure of C .Let E be the intersection cohomology complex on the closure of C determined by E. For every integer j we consider the j-th cohomology sheaf of E restricted to C; ∗ this is a Gψ,s-equivariant local system on C in which E appears say nj times. We − j set m(s,N,E),(s ,N ,E ) = j( 1) nj. We see that the intersection cohomology complexes on Xs considered above are essentially of the type considered in [L18]. We also see that the objects in κ−1(ψ) behave like the parameters for the unipotent representations for a collection of not necessarily split and not necessarily connected p-adic groups smaller than GK . It is known that M, M coincide as far as the entries with both indices contained in κ−1(1) are concerned; these correspond to unipotent representations. (This was conjectured by the author and independently, in a special case connected with GLn,in[Ze]; the proof was given by Ginzburg in a special case connected with the affine Hecke algebra H and by the author in the general case.) We expect that M = M.(Seealso[V2].) 20. Multiplicities in tensor products. Assume that R is simply connected. + For λ, λ ,λ in X let mλ,λ,λ be the multiplicity of Λλ in the tensor product ∗ ∗ Λλ ⊗ Λλ (an object of C,see§5). On the other hand let l : W −→ N,Tw,cw,pw,z ∗ be defined like l : W −→ N,Tw,cw,pw,z in §3,§4 but with respect to R instead of R.WehaveW∗ = {wax; w ∈ W, x ∈ X}. + λ For any λ ∈ X there is a unique element Mλ in the double coset Wa W on which l∗ : WaλW −→ N achieves its maximum value. For any λ, λ in X+ we have ∗ in H : − − − 1 1 1 (P cMλ )(P cMλ )= m˜ λ,λ ,λ (P cMλ ) λ∈X+

∈A ∈A where P is given by cM0 cM0 = PcM0 andm ˜ λ,λ ,λ .In[L4]itisshown that

(a) mλ,λ,λ =˜mλ,λ,λ ; in particular,

(b) m˜ λ,λ,λ is a constant.

TWELVE BRIDGES FROM A REDUCTIVE GROUP TO ITS LANGLANDS DUAL141 17

(In the special case where GC is a general linear group, this was proved earlier in [L3] using the theory of Hall-Littlewood functions.) In [L4]itisalsoshownthat for λ, λ in X+, λ (c) dim Λλ (that is, the multiplicity of the weight λ in the U-module Λλ)is equal to pMl ,Mλ (1). Note that at the time when [L4] was written it was known that the product of −1 ∗ elements of the form P cMλ corresponds to the convolution of GC[[]]-equivariant ∗ ∗ simple perverse sheaves on the “affine Grassmannian” GC(())/GC[[]] so that (b) is equivalent to the statement that such a convolution is a direct sum of simple perverse sheaves of the same type (without shift). Thus it was clear that the ∗ category whose objects are finite direct sums of GC[[]]-equivariant simple perverse sheaves on the “affine Grassmannian” has a natural monoidal structure given by convolution; moreover (b) showed that this monoidal category was very similar to that of representations of GC (identical at the level of Grothendieck groups). But it was not clear how to construct the commutativity isomorphism for the convolution product. This was accomplished around 1989 by V.Ginzburg [Gi]andlaterina more elegant form by V.Drinfeld. As a result, GC can be reconstructed from the ∗ ∗ ∗ tensor category of GC[[]]-equivariant perverse sheaves on GC(())/GC[[]],see[Gi]. 21. Canonical bases. Define f as in §5intermsofR.LetA = R[[]] where ∗+ R∗ + is an indeterminate. Define UA in terms of in the same way as UA was defined in §6intermsofR.By[L14,§10] there is a canonical bijection between the canonical basis of f (defined as in [L11], [L13]) and a certain collection of ∗+ ∗+ subsets of UA which form a partition of the totally positive part of UA .The bijection is not defined directly; instead it is shown that both sets are parametrized by the same combinatorial objects.

22. Modular representations. Let k be an algebraic closure of the ﬁnite + ﬁeld Fp. Assume that R is simply connected. For λ ∈ X the Gk-module Λλ,k (see § 7) is not necessarily irreducible but has a unique irreducible quotient Λλ,k.For + λ, λ in X let mλ,λ be the number of times that Λλ,k appears in a composition series of the Gk-module Λλ,k. Note that the knowledge of the multiplicities mλ,λ

implies the knowledge of the character of the Gk-modules Λλ,k since the character of Λλ,k is known by Weyl’s character formula. Conjecturally (see [L2]) if p is suﬃciently large with respect to R, the multiplicities mλ,λ can be expressed in ∗ terms of polynomials pw,z (as in §4) where w, z are elements in W which have maximal length in their left W -coset; they can be also expressed in terms of certain ∗ intersection cohomology spaces associated with the geometry of GC(()),where is an indeterminate. A proof of the conjecture (without an explicit bound for p)is provided by combining [AJS], [KT], [KL3] or alternatively by combining [AJS], [ABG].

References [ABV] J. Adams, D. Barbasch and D. A. Vogan, Jr., The Langlands classiﬁcation of irreducible characters of real reductive groups, Progress in Math (1992), Birkhauser. [AJS]H.H.Andersen,W.SoergelandJ.C.Jantzen,Representations of quantum groups at a p-th root of unity and of semisimple groups in characteristic p: independence of p, Ast´erisque 220 (1994).

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[ABG] S. Arkhipov, R. Bezrukavnikov and V. Ginzburg, Quantum groups, the loop grassmannian and the Springer resolution, Jour.Amer.Math.Soc. 17 (2004), 595-678. [C] C. Chevalley, Certains schémas de groupes semi-simples,Sém. Bourbaki 1960/61, Soc. Math. France, 1995. [DL] P. Deligne and G. Lusztig, Representations of reductive groups over finite fields, Ann. Math. 103 (1976), 103-161. [Dr] V. Drinfeld, Two dimensional l-adic representations of the fundamental group of a curve over a finite field and automorphic forms on GL(2), Amer.Jour.Math. 105 (1983), 85-114. [Gi] V. Ginzburg, Perverse sheaves on a loop group and Langlands duality, math.AG/9511007. [HT] M. Harris and R. Taylor, On the geometry and cohomology of some simple Shimura varieties, Ann.Math.Studies, vol. 151, Princeton Univ.Press, 2001. [IM] N. Iwahori and H. Matsumoto, On some Bruhat decomposition and the structure of the Hecke ring of p-adic Chevalley groups, Publ.Math.IHES 25 (1965), 5-48. [KW] A. Kapustin and E. Witten, Electric-magnetic duality and the geometric Langlands program, arXiv:hep-th/0604151. [KT] M. Kashiwara and T. Tanisaki, The Kazhdan-Lusztig conjecture for affine Lie algebras with negative level, Duke Math.J. 77 (1995), 21-62. [KL1] D. Kazhdan and G. Lusztig, Representations of Coxeter groups and Hecke algebras,In- vent.Math. 53 (1979), 165-184. [KL2] D. Kazhdan and G. Lusztig, Proof of the Deligne-Langlands conjecture for Hecke algebras, Invent.Math. 87 (1987), 153-215. [KL3] D. Kazhdan and G. Lusztig, Tensor structures arising from affine Lie algebras, IV,Jour. Amer. Math. Soc. 7 (1994), 383-453. [Ki] J.-L. Kim, Supercuspidal representations: an exhaustion theorem, Jour.Amer.Math.Soc. 20 (2007), 273-320. [Ko] B. Kostant, Groups over Z, Algebraic Groups and Their Discontinuous Subgroups, Proc. Symp. Pure Math., vol. 8 publ. Amer. Math. Soc., 1966, pp. 90-98. [Lf] L. Lafforgue, Chtoukas de Drinfeld et correspondance de Langlands,Invent.Math.147 (2002), 1-242. [La1] R. P. Langlands, Problems in the theory of automorphic forms, Lectures in Modern Anal- ysis and Applications, Lecture Notes in Math, vol. 170, Springer Verlag, 1970, pp. 18-61. [La2] R. P. Langlands, On the classification of irreducible representations of real algebraic groups, Representation theory and harmonic analysis on semisimple Lie groups, Math. Surveys Monogr., vol. 31 Amer. Math. Soc., Providence, RI, 1989, pp. 101-170. [L1] G. Lusztig, A class of irreducible representations of a Weyl group, Proc. Kon. Nederl. Akad. (A) 82 (1979), 323-335. [L2] G. Lusztig, Some problems in the representation theory of finite Chevalley groups,Proc. Symp. Pure Math. Amer. Math. Soc. 37 (1980), 313-317. [L3] G. Lusztig, Green polynomials and singularities of unipotent classes, Adv. Math. 42 (1981), 169-178. [L4] G. Lusztig, Singularities, character formulas and a q-analog of weight multiplicities, Astérisque 101-102 (1983), 208-229. [L5] G. Lusztig, Some examples of square integrable representations of semisimple p-adic groups, Trans.Amer.Math.Soc. 227 (1983), 623-653. [L6] G. Lusztig, Characters of reductive groups over a finite field, Ann.Math.Studies, vol. 107, Princeton Univ.Press, 1984. [L7] G. Lusztig, Equivariant K-theory and representations of Hecke algebras, Proc. Amer. Math. Soc. 94 (1985), 337-342. [L8] G. Lusztig, Character sheaves, V, Adv.Math. 61 (1986), 103-155. [L9] G. Lusztig, Cells in affine Weyl groups, II, J.Alg. 109 (1987), 536-548; IV, J. Fac. Sci. Tokyo U.(IA) 36 (1989), 297-328. [L10] G. Lusztig, On representations of reductive groups with disconnected center,Astérisque 168 (1988), 157-166. [L11] G. Lusztig, Canonical bases arising from quantized enveloping algebras, Jour. Amer. Math. Soc. 3 (1990), 447-498. [L12] G. Lusztig, Affine quivers and canonical bases, Publ.Math.IHES 76 (1992), 111-163. [L13] G. Lusztig, Introduction to quantum groups, Progress in Math., vol. 110, Birkhauser, 1993.

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[L14] G. Lusztig, Total positivity in reductive groups, Lie theory and geometry, Progr.in Math., vol. 123, Birkhäuser Boston, 1994, pp. 531-568. [L15] G. Lusztig, Classification of unipotent representations of simple p-adic groups,Int.Math. Res. Notices (1995), 517-589; II, Represent.Th. 6 (2002), 243-289. [L16] G. Lusztig, Notes on unipotent classes,AsianJ.Math.1 (1997), 194-207. [L17] G. Lusztig, Unipotent elements in small characteristic, Transform. Groups 10 (2005), 449-487; II,arXiv:RT/0612320. [L18] G. Lusztig, Graded Lie algebras and intersection cohomology, arXiv:RT/0604535. [L19] G. Lusztig, Study of a Z-form of the coordinate ring of a reductive group, arxiv:0709.1286. [LV] G. Lusztig and D.A.Vogan, Jr., Singularities of closures of K-orbits on a flag manifold, Invent.Math. 71 (1983), 365-379. [MK] J. McKay, Graphs, singularities and finite groups, Proc. Symp. Pure Math. Amer. Math. Soc. 37 (1980), 183-186. [SL] P. Slodowy, Simple singularities and simple algebraic groups, Lecture Notes in Math., vol. 815, Springer Verlag, 1980. [V1] D. A. Vogan, Jr., Irreducible characters of semisimple Lie groups, IV: character multiplicity duality, Duke Math.J. 4 (1982), 943-1073. [V2] D. A. Vogan, Jr., The local Langlands conjecture, Representation theory of groups and algebras, Contemp. Math., vol. 145 Amer. Math. Soc., Providence, RI, 1993, pp. 305-379. [Xi] N. Xi, Representations of affine Hecke algebras and based rings of affine Weyl groups, Jour.Amer.Math.Soc. 20 (2007), 211-217. [Ze] A. Zelevinsky, A p-adic analogue of the Kazhdan-Lusztig conjecture, Funkt.Anal.Pril. 15 (1981), 9-21.

Department of Mathematics, M.I.T., Cambridge, MA 02139 E-mail address: [email protected]

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Contemporary Mathematics Volume 478, 2009

Some New Highest Weight Categories

Brian J. Parshall and Leonard L. Scott

Abstract. Let G be a semisimple, simply connected algebraic group deﬁned over an algebraically closed ﬁeld k of positive characteristic p. Assume that p is larger than the Coxeter number of G and that the Lusztig character formula holds for regular restricted dominant weights. In this paper, we introduce two Creg Creg new highest weight categories even and odd, both full subcategories of the category C of rational G-modules. The standard and costandard modules for these categories arise from “reduction mod p” from the quantum enveloping algebra associated to G.

1. Introduction

Let Uζ be a quantum enveloping algebra (Lusztig form) associated to a finite root system Φ and a pth root of unity ζ. Here, p is a positive prime integer. For each dominant weight λ,letLζ (λ) be the finite dimensional, irreducible (type 1 and integrable) representation of Uζ of high weight λ.Letk be an algebraically closed field of characteristic p,andletG be the semisimple, simply connected algebraic group over k having root system Φ. By a process of “reduction mod p,” the module Lζ (λ) defines rational modules for G. The rational module obtained from a minimal (resp., maximal) lattice (with respect to an appropriate integral form of Uζ over a red PID O) is denoted ∆ (λ)(resp.,∇red(λ)). These modules were (with a different notation) first defined by Lusztig [10]. The homological properties of the modules red ∆ (λ)and∇red(λ) were investigated in [6](seealso[12]). In that work, the modules played an important role in the authors’ efforts (with E. Cline) to attack growth issues on 1-cohomology arising from Guralnick’s universal bound conjecture [7] for finite groups. Let C be the category of rational G-modules; it is a highest weight category in a precise sense [3]. Assume that p>h(the Coxeter number of G). For each regular dominant weight λ,writeλ = w · λ−,whereλ− belongs to the anti-dominant p- alcove (the alcove containing −2ρ, ρ being the half-sum of the positive roots) and where w ∈ Wp, the affine p-Weyl group. Define the length (λ)ofλ to be the Creg Creg length (w)ofw in the Coxeter group Wp.Thenlet even (resp., odd)bethe full subcategory of C generated by the irreducible rational G-modules L(λ)ofhigh weight λ for λ regular and (λ) an even integer (resp., odd integer). The main result, proved in Theorem 3.2, shows that, if the Lusztig character formula holds

Research supported in part by the National Science Foundation.

1451

2146 BRIAN J. PARSHALL AND LEONARD L. SCOTT

Creg Creg for all restricted, regular dominant weights, then even and odd are highest weight red categories with standard objects ∆ (λ) and costandard objects ∇red(λ). Given a dominant weight λ,writeλ = λ0 +pλ1,whereλ0 is restricted dominant red and λ1 is dominant. Then ∆ (λ) has an elegant alternative description, due to Z. red ∼ red (1) (1) Lin [9]. Explicitly, ∆ (λ) = ∆ (λ0)⊗∆(λ1) ,where∆(λ1) denotes the twist red (1) of ∆(λ1) through the Frobenius map on G. Also, define ∆ (λ) = L(λ0)⊗∆(λ1) . When p is sufficiently large so that the Lusztig character formula holds for the red ∼ red irreducible modules L(λ0), we have ∆ (λ) = ∆ (λ) . Generally, one naturally asks if the standard modules ∆(λ)have∆red-filtrations, i. e., filtrations by G- submodules with sections of the form ∆red(µ). The same question, but using the modules ∆red(λ), goes back at least to J. Jantzen [8, §§3.8–12, 5.9], who answers it positively subject to restrictions on λ;see[1] for some history of this problem.1 This question remains open, but it is conjectured in [6] to have a positive answer for all dominant weights λ,solongasp>h,and[6, Thm. 6.9] does present a partial result in this direction. We revisit this result in Theorem 4.3 below, providing another proof based on the methods of this paper; see Remark 4.4. Finally, Theorem 4.5 uses the main result again, giving a new description of the modules ∆red(λ)and ∇red(λ)forλ regular. 2. Review of some preliminary notation and results Let G be a simply connected, semisimple algebraic group over k, defined and split over Fp. Fix a maximal split torus T and a Borel subgroup B ⊃ T .Let X = X(T )(resp.,X∨ = X∨(T )) be the character (resp., cocharacter) group on T . There is a natural pairing X × X∨ → Z, γ,σ → (γ,σ) ∈ Z. We let Φ ⊂ X be the set of roots of T in the Lie algebra g of G, ordered so that the set Φ+ of positive roots coincides with the Borel subgroup B+ ⊃ T opposite − + to B = B .LetΠ={α1,...,αn} be the simple roots in Φ , listed as in [2, Appendix]. For convenience, we will assume that Φ is irreducible (so that G is a simple algebraic group). Let X+ = X+(T ) be the set of dominant weights on T ;thus,X is a free Z- module with basis the fundamental dominant weights 1,...,n which are defined ∨ ∈ ∨ ∈ ∨ by (i,αj )=δi,j . Here, for a root α Φ, α X is the corresponding coroot. Let ρ = 1 + ···+ n be the Weyl weight, and let α0 be the maximal short root + ∨ in Φ .Leth =(ρ, α0 ) + 1 be the Coxeter number of Φ. + For a positive integer r, Xr is the set of dominant weights λ which satisfy ∨ r ∈ + (λ, α )

1In particular, [1, Cor. 3.7] would imply that if p ≥ 2h − 2, then every ∆(λ)hasa∆red - ﬁltration, for each λ ∈ X+. However, as pointed out to us by Andersen (in a private communication), the proof of [1, Lemma 3.3] does not hold, and so [1, Cor. 3.7] remains unproved.

SOME NEW HIGHEST WEIGHT CATEGORIES 1473 the complex simple Lie algebra associated to G. Thus, at the character level, ch ∆(λ)=ch∇(λ), and these characters are given by Weyl’s character formula. The category C is a highest weight category in the sense of [3]. This means the following conditions hold: (HWC1) For λ ∈ X+, ∇(λ)hassocleL(λ), and if L(µ) is a composition factor of ∇(λ)withµ = λ,thenµ<λ. (HWC2) For λ ∈ X+,letI(λ) be the injective envelope in C of L(λ). Then I(λ) has an increasing filtration F • = F •(λ): 0=F 0 ⊂ F 1 ⊂ F 2 ··· , i 1 0 ∼ i i−1 ∼ such that F = I(λ), F /F = ∇(λ), and, for i>1, F /F = ∇(µi)forsome + µi ∈ X satisfying µi >λ. The category C admits a strong duality D (in the sense of [4]) and we have D∇(λ)=∆(λ). However, C does not have enough projective objects. Let F : G → G be the Frobenius morphism defined by the Fp-structure on G. If V ∈Cand r is a positive integer, V (r) denotes the rational G-module obtained from V by making g ∈ G act on V by F r(g). In this paper, we only consider the case of r =1. Let E := R ⊗Z X be endowed with a positive definite, symmetric bilinear form ( , ), invariant under the Weyl group W of Φ. We identify X∨ as a subgroup of E,so ∨ 2 × ∨ → Z that α = (α,α) α and the pairing X X is compatible with the inner product. The affine Weyl group Wp = pZΦ W is the group of affine transformations on E generated by W and the normal subgroup consisting of translations by elements in ∨ pZΦ. If α ∈ Φandr ∈ Z, define sα,r : E → E by sα,r(x)=x−((x, α )−rp)α.Then ∈ { ··· } sα,r Wp.Also,(Wp,Sp) is a Coxeter system, putting Sp = sα1 , ,sαn ,sα0,−1 . In this paper, we use the “dot” action of Wp on E, given by setting w · x = w(x + ρ) − ρ.LetC+ ⊂ E be the dominant fundamental alcove; it consists of all ∈ E ∨ ··· ∨ x satisfying the inequalities 0 < (x + ρ, αi ), i =1, ,n,and(x + ρ, α0 ) 3. We fix a p-modular system (K, O,k). Thus, O is a discrete valuation ring with maximal ideal m =(π), fraction field K of characteristic 0, and residue field k = O/m. We can assume that O has a primitive pth root of unity ζ.(See[6,Rem. 1.4] and the discussion there.) Choose an O-form Uζ of Uζ .PutU ζ = Uζ /πUζ ,and let I be the ideal in U ζ generated by the images of the elements Ki − 1, 1 ≤ i ≤ n,

4148 BRIAN J. PARSHALL AND LEONARD L. SCOTT in the usual notation. By [10, (8.15)], ∼ (2.1) U ζ /I = hy(G), the distribution algebra of G over k. The category of finite dimensional integrable, type 1 Uζ -modules will be denoted by Cζ . It is a highest weight category with irreducible (resp. standard, costandard) + + modules Lζ (λ)(resp.,∆ζ (λ), ∇ζ (λ)), λ ∈ X .Forµ ∈ X ,ch∆ζ (µ)=ch∇ζ (µ)= ∈ + ch ∆(µ). We will always assume that given λ Xreg,chLζ (λ)=χKL(λ). This assumption is always valid if p>h, though in many cases this restriction is too severe; see [13] for the precise result. + As discussed in [6, §1.5], for λ ∈ X , the module Lζ (λ) has a minimal min max (resp., maximal) admissible lattice L (λ)(resp.,L (λ)forUζ which, upon red reduction to k, defines a rational G-module ∆ (λ)(resp.,∇red(λ)). We recall that these modules have, according to Lin [9], an alternative description as red ∼ red (1) ∼ (1) ∆ (λ) = ∆ (λ0)⊗∆(λ1) and ∇red(λ) = ∇red(λ0)⊗∇red(λ1) if λ = λ0+pλ1, ∈ + ∈ + 2 with λ0 X1 , λ1 X ;seealso[6, Prop. 1.6]. By [6, Thm. 5.4, Thm.6.7] if + p>hand if the LCF holds for all regular weights in X1 ,then n red λ , ∇ µ n L λ ,L µ , (2.2) dim ExtG(∆ ( ) red( )) = dim ExtCζ ( ζ( ) ζ ( )) λ, µ ∈ X+ n L λ ,L µ for all reg. The groups ExtCζ ( ζ ( ) ζ( )) can be explicitly calculated; see, e. g., [6, (1.4.2)]. In particular, we will often use without mention the fact that, ∈ + given λ, µ Xreg, n L λ ,L µ ⇒ n ≡ λ − µ . ExtCζ ( ζ ( ) ζ( )) =0 = ( ) ( )mod2 We record the following result, which we have largely already discussed. See [9]and[6, §1.5]. Lemma 2.1. Let λ ∈ X+. red (1) ∆ (λ) (resp., ∇red(λ)) has irreducible head (resp., socle) L(λ). All other red composition factors L(µ) of ∆ (λ) (resp., ∇red(λ)) satisfy µ<λ. + (2) Assume that p>hand that the LCF holds for all regular weights in X1 . red ∼ (1) ∼ (1) Then ∆ (λ) = L(λ0) ⊗ ∆(λ1) and ∇red(λ) = L(λ0) ⊗∇red(λ1) .Hereλ = λ0 + pλ1 as above. − − − For a regular dominant weight λ,writeλ = w · λ ,whereλ ∈ CZ .Asinthe introduction, define (λ)=(w). We say that L(λ) has even (resp., odd) parity ≡ ≡ + + provided that (λ) 0(resp.,(λ) 0) mod 2. Let Xreg, even (resp., Xreg,odd)be set of regular dominant weights λ such that (λ) ≡ 0(resp.,(λ) ≡ 0) mod 2.

3. The highest weight categories Creg Creg C Let even (resp., odd) be the full subcategory of generated by the irreducible modules L(λ) having even (resp., odd) parity. For example, a rational G-module Creg belongs to even if the composition factors of any ﬁnite dimensional submodule have even parity.

2More generally, [9] describes, in the same spirit, the rational G-modules obtained from the irreducible modules for quantum enveloping algebras at a prth root of unity by a reduction mod p process. The result we have quoted is merely the r = 1 case of this result (which is suﬃcient for the applications in [6]). We do not investigate here how the results of this paper might generalize to the r>1case.

SOME NEW HIGHEST WEIGHT CATEGORIES 1495

Lemma 3.1. Assume that p>hand that the LCF holds for all regular weights + ∈ + ∈ + in X1 .Letλ, µ Xreg, even,andletτ X . Then: red ∇ ∈ + (1) If [∆ (λ):L(τ)] =0or [ red(λ):L(τ)] =0,thenτ Xreg, even.In red ∇ Creg particular, ∆ (λ) and red(λ) both belong to even. 1 red 1 ∇ (2) If ExtG(∆ (λ),L(µ)) =0or ExtG(L(µ), red(λ)),thenµ>λ. 1 red red 1 ∇ ∇ (3) If ExtG(∆ (λ), ∆ (µ)) =0or ExtG( red(µ), red(λ)),thenµ>λ. + + Similar statements holds with Xreg, even replaced throughout by Xreg,odd. Proof. red + We only prove the ∆ -statements for the poset Xreg, even. ∈ + ∈ + red ∼ Writing λ = λ0 + pλ1 with λ0 X1 and λ1 X ,wehavethat∆ (λ) = (1) red L(λ0)⊗∆(λ1) .Thus,ifL(τ) is a composition factor of ∆ (λ), then τ = λ0 +pσ, where L(σ) is a composition factor of ∆(λ1). Hence, σ = λ1 − δ,whereδ ∈ ZΦ. Therefore, τ = λ − pδ,soτ and λ have the same parity. This proves (1). 1 red It is easy to see that (3) follows from (2). So suppose ExtG(∆ (λ),L(µ)) =0. Since λ and µ have the same parity, (2.2) implies that 1 red λ , ∇ λ 1 L λ ,L µ . dim ExtG(∆ ( ) red( )) = dim ExtUζ ( ζ ( ) ζ( )) = 0

Therefore, forming the exact sequence 0 → L(µ) →∇red(µ) → Q(µ) → 0, we see 1 red red that ExtG(∆ (λ),L(µ)) is a homomorphic image of HomG(∆ (λ),Q(µ)). Since ∆red(λ) has head L(λ), it follows that if this Hom-space is non-zero, then λ<µ, as required.

Now we can prove the main result of this section. Theorem 3.2. Assume that p>hand that the LCF holds for all regular Creg restricted dominant weights. Then even is a highest weight category with weight + ∈ + poset Xreg, even.Forλ Xreg, even, the corresponding standard (resp., costandard) red object is ∆ (λ) (resp., ∇red(λ)). Creg + ∈ Similarly, odd is a highest weight category with weight poset Xreg,odd.Forλ + red Xreg,odd, the corresponding standard (resp., costandard) object is ∆ (λ) (resp., ∇red(λ). Proof. Creg We will prove only the assertion for even. We follow the deﬁnition of highest weight categories as given in [3]. By Lemma 3.1(1), each ∇red(λ), ∈ + Creg ∇ λ Xreg, even, belongs to even. In addition, by Lemma 2.1(2), red(λ)hasso- cle isomorphic to L(λ), while the other composition factors L(µ)satisfyµ<λ ∈ + (and, of course, µ Xreg, even). ∈ + Therefore, it remains to show that, given λ Xreg, even, L(λ) has injective Creg ∇ ∇ envelope in even which has an red-ﬁltration with bottom section red(λ)and higher sections ∇red(µ)forµ>λ. Consider the set of dominant weights µ ∈ + ≥ Xreg, even which satisfy µ λ. Enumerate this set as λ0 = λ, λ1,λ2,... so that λ ≤ µ<τ,thenµ = λi and τ = λj for i

(i) Ij(λ) has a ﬁltration with sections ∇red(λi), i ≤ j; (ii) soc Ij(λ)=L(λ); 1 (iii) ExtG(L(µ),Ij(λ)) = 0 implies that µ = λi for some i>j. ∼ (iv) Ij(λ) = Ij(λ)/πIj(λ)forsomeUζ -lattice Ij(λ).

6150 BRIAN J. PARSHALL AND LEONARD L. SCOTT

Then I(λ):= Ij(λ) j Creg is the required injective envelope of L(λ)in even. To begin, let I0(λ)=∇red(λ). Then Lemma 3.1(1),(2) implies that (i)–(iii) min hold, while we can take I0(λ)=L (λ) to satisfy (iv). Suppose then that Ij(λ) has been constructed satisfying properties (i)–(iv). Choose a basis ξ1,...,ξn of 1 ∇ ∞ ExtG( red(λj+1),Ij(λ)). Of course, n< . Deﬁne Ij+1(λ) by means of an extension

⊕n (3.1) ξ :0→ Ij(λ) → Ij+1(λ) →∇red(λj+1) → 0,

⊕n with the following property: if ιi : ∇red(λj+1) →∇red(λj+1) maps ∇red(λj+1) ∇ ⊕n ∗ isomorphically onto the ith coordinate of red(λj+1) , then the pull-back ιi ξ deﬁnes ξi. The map 1 ∇ → 1 (3.2) ExtG( red(λj+1),Ij(λ)) ExtG(L(λj+1),Ij(λ)) is injective, because ∇red(λj+1)/L(λj+1) has only composition factors with high π weights smaller than λj+1. The exact sequence 0 → Ij(λ) → Ij(λ) → Ij(λ) → 0 gives an exact sequence

0 → HomG(∇red(λj+1),Ij(λ))

α 1 max → Ext e (L (λj+1), Ij(λ)) Uζ

π 1 max β 1 → Ext e (L (λj+1), Ij(λ)) → Ext (∇red(λj+1),Ij(λ)). Uζ G

1 max (See [6, (1.4.5)]. Notice that Ext e (L (λj+1), Ij(λ)) is a torsion module by [6, Uζ (1.4.4)] and the fact that λj+1 and the high weights of the composition factors of Ij(λ)K have even parity. The image of α is the submodule of all elements killed by π, which has the same dimension as Im β.So,wehave

dim HomG(∇red(λj+1),Ij(λ)) = dim Im β ≤ 1 ∇ dim ExtG( red(λj+1),Ij(λ)) ≤ 1 dim ExtG(L(λj+1),Ij(λ)) by the injectivity of (3.2). Using (2.2), we obtain by the long exact sequence of coho- 1 ∼ red mology that Ext (L(λj+1),Ij(λ)) = HomG(rad ∆ (λj+1),Ij(λ)), which has the G ∼ same dimension as HomG(∇red(λj+1)/L(λj+1),Ij(λ)) = HomG(∇red(λj+1),Ij(λ)). red (To see the equality of dimension, ﬁrst observe that rad ∆ (λj+1)and ∇red(λj+1)/L(λj+1) have the same composition factors. Then use condition (iii), which implies that the functor HomG(−,Ij(λ)) is exact on the subcategory of G- modules having composition factors L(λi), i =0,...,j.) Thus, the inequalities in (3.3) are all equalities. In particular, we see that the map (3.2) is an isomorphism. Now diagram chasing easily implies that Ij+1(λ) satisﬁes conditions (i)–(iii). For example, the long exact sequence of cohomology

SOME NEW HIGHEST WEIGHT CATEGORIES 1517 applied to the exact sequence (3.1), together with Lemma 3.1, gives an exact sequence → ∇ ⊕n → 1 0 HomG(L(λj+1), red(λj+1) ) ExtG(L(λj+1),Ij) → 1 → ExtG(L(λj+1),Ij+1)) 0. But ⊕n dim HomG(L(λj), ∇red(λj+1) )=n, which also equals 1 ∇ dim ExtG( red(λj+1),Ij(λ)). Now the isomorphism (3.2) implies that is an isomorphism. Hence, we conclude 1 that ExtG(L(λj+1),Ij+1(λ)) = 0. Finally, we have also shown that β is surjective, so that Ij+1(λ) lifts to a lattice Ij+1(λ)forUζ . Thus, condition (iv) also holds for Ij+1(λ).

Remark 3.3. It is interesting to compare the above result to the discussion in § + [11, 5]. Let Γ be a finite ideal in Xreg, even. (A similar discussion would work for + D b C C Xreg,odd.) Form the derived category = D ( ), regarding as fully embedding in D as complexes concentrated in degree 0. Using Lemma 2.1, we see that the modules ∆red(λ), λ ∈ Γ, satisfy the conditions of [11, Th. 5.9]. Therefore, there exists a strict full subcategory CΓ of D and a highest weight category CΓ (having weight poset Γ) with the following properties: CΓ is the exact full subcategory of CΓ red consisting of objects with a ∆-filtration. Each ∆ (λ) ∈CΓ identifies in CΓ with the standard module indexed by λ ∈ Γ. For λ ∈ Γ, the projective indecomposable cover PΓ(λ)ofL(λ)inCΓ is obtained by a recursive construction exactly dual to the construction of the Ij(λ) in the proof of the above theorem. Letting T = op ⊕λ∈ΓPΓ(λ), we can take CΓ = (EndC(T )-mod) . However, in the abstract setting of [11], there is no guarantee that the constructed highest weight category CΓ will fully embed into the original category D (or, in the present case, C). The above Creg theorem shows that this embedding does happen in the case of even.

4. Applications + Creg →C Maintain the notation of the previous section. Let i∗ : even (resp., − Creg →C + i∗ : odd ) denote the canonical full embedding of categories. The functor i∗ is ! C→Creg exact, and admits left (resp., right) exact right (resp., left) adjoint i+ : even ∗ C→Creg ∈C ! ∗ (resp., i+ : even). Explicitly, for M , i+M (resp., i+M)isthelargest submodule (resp., quotient module) of M having composition factors L(λ)with ∈ + − ! ∗ λ Xreg, even. Similarly, i∗ admits a right (resp., left) adjoint i− (resp., i−). ∈Creg ∈Creg Suppose that M,N even (resp., M,N odd), then there is an induced ho- n + n n n − momorphism R i (M,N) : Ext reg (M,N) → Ext (M,N)(resp.,R i (M,N): ∗ Ceven G ∗ n n ExtCreg (M,N) → Ext (M,N)). Clearly, odd G Lemma . ∈Creg 4.1 For M,N even, n + n n R i (M,N) : Ext reg (M,N) → Ext (M,N) ∗ Ceven G Creg is an isomorphism for n =0, 1. The analogous statement holds for odd.

8152 BRIAN J. PARSHALL AND LEONARD L. SCOTT

n + + For n>1, R i∗ (M,N) is generally not an isomorphism, i. e., i∗ does not induce a full embedding at the level of derived categories. (And, of course, a n − ∈ + similar statement holds for R i∗ (M,N).) In fact, take λ, µ Xreg, even so that 2 L λ ,L µ 2 red λ , ∇ µ ExtCζ ( ζ ( ) ζ( )) =0.ThenExtG(∆ ( ) red( )) = 0 by (2.2). On the other 2 red hand, Ext reg (∆ (λ), ∇ (λ)) = 0 by Theorem 3.2 and standard properties of Ceven red highest weight categories (e. g., use Lemma 4.2 below). We will make use of the following well-known criterion, first proved independently in the context of rational G-modules by S. Donkin and L. Scott. Lemma 4.2. Let C be a highest weight category with finite poset Λ.LetM ∈C. 1 Then M has a ∆-filtration (resp., ∇-filtration) if and only if ExtC (M,∇(ω)) = 0 1 (resp., ExtC (∆(ω),M)=0)forallω ∈ Λ. ∈ + red For λ Xreg, define rad (λ) by means of the following exact sequence

(4.1) 0 → radred(λ) → ∆(λ) → ∆red(λ) → 0.

Also, deﬁne socred(λ) by the following exact sequence

(4.2) 0 →∇red(λ) →∇(λ) → socred(λ) → 0. Creg Creg We state the following theorem for even. A similar result holds for odd. Theorem 4.3. Assume that p>hand that the LCF holds for all regu- ∈ + ∗ red lar restricted dominant weights. Let λ Xreg, even.Theni− rad (λ) (resp., ! red ∇ Creg i− socred(λ)) has a ∆ -ﬁltration (resp., red-ﬁltration) in odd. ∗ red Proof. We prove the assertion for i− rad (λ), leaving the dual case for ! ∗ red red i− socred ∇(λ) to the reader. First, write i− rad (λ)asaquotientrad (λ)/M (λ), and let E(λ):=∆(λ)/M (λ). Thus, we have an exact sequence ∗ red red 0 → i− rad (λ) → E(λ) → ∆ (λ) → 0.

∈ + 1 Claim 1: Let ω Xreg,odd and assume that ω<λ.ThenExtG(E(λ),L(ω)) = 0. In fact, suppose that 0 → L(ω) → F →ι E(λ) → 0isanextension.The module E(λ) is a cyclic G-module generated by a high weight vector vλ of weight λ. Thus, the universal mapping property of ∆(λ) implies there is a morphism → ◦ ∈ + σ :∆(λ) F such that ι σ is surjective. Because ω Xreg,odd, σ(M(λ)) = 0, so that (up to a nonzero scalar), the morphism ι is split by the induced morphism σ¯ : E(λ)=∆(λ)/M (λ) → F . This proves Claim 1. ∈ + 1 ∇ Claim 2: Let ω Xreg,odd and assume that ω<λ.ThenExtG(E(λ), red(ω)) = 0. This claim follows immediately from Claim 1, since all the composition factors ∇ ∈ + L(τ)of red(ω)satisfyτ Xreg,odd and τ<λ. ∈ + 1 ∗ red ∇ Claim 3: Assume that ω Xreg,odd and ω<λ.ThenExtG(i− rad (λ), red(ω)) = 0. ∗ red • For convenience, denote i− rad (λ)byR(λ). The long exact sequence of ExtG applied to the short exact sequence above Claim 1 give the following exact sequence → 1 ∇ → 2 red ∇ 0 ExtG(R(λ), red(ω)) ExtG(∆ (λ), red(ω)) in view of Claim 2. Since ω and λ have opposite parity, Claim 3 follows from (2.2).

SOME NEW HIGHEST WEIGHT CATEGORIES 1539

Creg −∞ Now consider the highest weight category odd[Γ], where Γ = ( ,λ), the ideal + Creg in Xreg,odd consisting of dominant weights <λ. It is the full subcategory of odd ∈ ∗ red ∈Creg whose objects have composition factors L(γ), γ Γ. Then i− rad (λ) odd[Γ]. Since 1 ∗ red ∇ ExtG(i− rad (λ), red(ω)) = 0 ∗ red red for all ω ∈ Γ, Lemma 4.1 and Lemma 4.2 imply that i− rad (λ)hasa∆ - filtration, as required. Remark 4.4. It is interesting to compare the above result with that given in [6, Thm. 6.9], which proves a similar filtration result. We will show in this remark that the results are the same, though we require the original argument for [6,Thm. 6.9] to see the equivalence. We still assume that p>hand that the LCF holds for + ∈ + all regular weights in X1 .Forλ Xreg, put 2 Eζ (λ)=∆ζ (λ)/rad ∆ζ (λ). Let E(λ) be the image of ∆ζ (λ)inEζ (λ), and set E (λ) be the rational G-module obtained by reducing E(λ) to the field k.LetD(λ) be the kernel of the natural surjection E(λ) Lmin(λ). Then [6, Thm. 6.9] shows that E(λ)hasa∆red- filtration with top section ∆red(λ). ∈ + An essential step in the proof of this result was to show that if ω Xreg satisfies 1 ω<λand if ω has parity opposite to that of λ,thenExtG(E (λ),L(ω)) = 0. (See, for example, Claim 4 in the proof given in [6].) It follows easily from this ∼ fact that E (λ) = E(λ) (as defined in the proof of Theorem 4.3. In particular, ∼ ∗ red this isomorphism means that D(λ) = D(λ)/πD(λ) is isomorphic to i+ rad (λ)if ∈ + λ Xreg, even, and so the conclusion of Theorem 4.3 and [6, Thm. 6.9] are really the same, at least over k. (It is then easy to extend Theorem 4.3 to the integral case.) Finally, we conclude this section with the following alternative characterization red of the modules ∆ (λ)and∇red(λ). Theorem 4.5. Assume that p>hand that the LCF holds for all regular ∈ + restricted dominant weights. Let λ Xreg, even.Then ∗ ∼ red ! ∇ ∼ ∇ i+∆(λ) = ∆ (λ)and i+ (λ) = red(λ). ∈ + ∗ ! A similar result holds if λ Xreg,odd, using i− and i−. Proof. ∈ + ∗ ∼ We only consider the case λ Xreg, even, and prove that i+∆(λ) = red ∈ + ∗ ∆ (λ)forλ Xreg, even. The dual statement is left to the reader. Apply i+ to (4.1) to obtain using Lemma 3.1 the exact sequence ∗ red → ∗ → red → i+ rad (λ) i+∆(λ) ∆ (λ) 0. ∗ red ∈ + Thus, it suffices to prove that i+ rad (λ) = 0. If not, there exists ν Xreg, even ∗ red such that ν<λand L(ν) lies in the head of i+ rad (λ). Since ∆(λ) has irreducible head L(λ), it follows from Lemma 4.1 that 1 red 1 red Ext (∆ (λ),L(ν)) = Ext reg (∆ (λ),L(ν)) =0 . G Ceven But this situation is impossible by standard highest weight theory, cf. [3, Lemma 3.2(b)] (use the dual version for standard modules).

10154 BRIAN J. PARSHALL AND LEONARD L. SCOTT

References [1] H. Andersen, p-Filtrations and the Steinberg module, J. Algebra 244 (2001), 664–683. [2] N. Bourbaki, Groupes et algèbres de Lie, IV, V, VI, Hermann (1968). [3] E. Cline, B. Parshall, Finite dimensional algebras and highest weight categories, J. reine angew. Math. 391 (1988), 85-99. [4] E. Cline, B. Parshall, and L. Scott, Duality in highest weight categories, Comtemp. Math. 82 (1989), 7-22. [5] E. Cline, B. Parshall, and L. Scott, Abstract Kazhdan-Lusztig theories, Tôhoku Math. J. 45 (1993), 511-534. [6] E. Cline, B. Parshall, and L. Scott, Reduced standard modules and cohomology, Transactions of Amer. Math. Soc.,toappear. [7] R. Guralnick, The dimension of the first cohomology group, in: Representation Theory, II, Ottawa (1984), in: Lectures Notes in Math. 1178, Springer (1986), 94–97. [8]J.C.Jantzen,Darstellungen halbeinfacher Gruppen und ihrer Frobenius-Kerne, J. Reine und Angew. Math. 317 (1980), 157-199. . [9] Z. Lin, Highest weight modules for algebraic groups arising from quantum groups, J. Algebra 208 (1998), 276–303. [10] G. Lusztig, Quantum groups at roots of 1, Geom. Dedicata 35 (1990), 89–114. [11] B. Parshall and L. Scott, Derived categories, quasi-hereditary algebras, and algebraic groups, Carlton Univ. Math. Notes 3 (1989), 1–111. [12] B. Parshall and L. Scott, Beyond the Jantzen region, Oberwolfach Report No. 15/2006 (2006), 26–30. [13] T. Tanisaki, Character formulas of Kazhdan-Lusztig type, Representations of finite dimensional algebras and related topics in Lie theory and geometry, Fields Inst. Commun. 40, Amer. Math. Soc., Providence, RI (2004) 261–276.

Department of Mathematics, University of Virginia, Charlottesville, VA 22903 E-mail address: [email protected] Department of Mathematics, University of Virginia, Charlottesville, VA 22903 E-mail address: [email protected]

Contemporary Mathematics Volume 478, 2009

Classiﬁcation of quasi-trigonometric solutions of the classical Yang–Baxter equation

Iulia Pop and Alexander Stolin

Abstract. It was proved by Montaner and Zelmanov that up to classical twisting Lie bialgebra structures on g[u] fall into four classes. Here g is a simple complex ﬁnite-dimensional Lie algebra. It turns out that classical twists within one of these four classes are in a one-to-one correspondence with the so-called quasi-trigonometric solutions of the classical Yang-Baxter equation. In this paper we give a complete list of the quasi-trigonometric solutions in terms of sub-diagrams of the certain Dynkin diagrams related to g.Wealso explain how to quantize the corresponding Lie bialgebra structures.

1. Introduction The present paper constitutes a step towards the classification of quantum groups. We describe an algorithm for the quantization of all Lie bialgebra structures on the polynomial Lie algebra P = g[u], where g is a simple complex finite- dimensional Lie algebra. Lie bialgebra structures on P , up to so-called classical twisting, have been classified by F. Montaner and E. Zelmanov in [6]. We recall that given a Lie co-bracket δ on P ,aclassical twist is an element s ∈ P ∧ P such that (1.1) CYB(s)+Alt(δ ⊗ id)(s)=0, where CYB is the l.h.s. of the classical Yang-Baxter equation. We also note that a classical twist does not change the classical double Dδ(P ) associated to a given Lie bialgebra structure δ.Ifδs is the twisting co-bracket via s, then the Lie bialgebras (P, δ)and(P, δs) are in the same class, i.e. there exists a Lie algebra isomorphism between Dδ(P )andDδs (P ), preserving the canonical forms and compatible with the canonical embeddings of P into the doubles. According to the results of Montaner and Zelmanov, there are four Lie bialgebra structures on P up to classical twisting. Let us present them:

2000 Mathematics Subject Classiﬁcation. Primary 17B37, 17B62; Secondary 17B81. Key words and phrases. Classical Yang–Baxter equation, r-matrix, Manin triple, parabolic subalgebra, generalized Belavin–Drinfeld data. Acknowledgment. The ﬁrst author was supported by European Community Research Train- ing Network LIEGRITS, grant no. MRTN-CT-2003-505078.

c 2009 Americanc 0000 Mathematical (copyright Societyholder) 1551

2156 IULIA POP AND ALEXANDER STOLIN

∗ 2 Case 1. Consider δ1 =0.Consequently,D1(P )=P + εP ,whereε =0. The symmetric nondegenerate invariant form Q is given by the canonical pairing between P and εP ∗. Lie bialgebra structures which fall in this class are the elements s ∈ P ∧ P satisfying CYB(s) = 0. Such elements are in a one-to-one correspondence with ﬁnite-dimensional quasi-Frobenius Lie subalgebras of P .

Case 2. Let us consider the co-bracket δ2 given by

(1.2) δ2(p(u)) = [r2(u, v),p(u) ⊗ 1+1⊗ p(v)], where r2(u, v)=Ω/(u − v). Here Ω denotes the quadratic Casimir element on g. −1 It was proved in [7] that the associated classical double is D2(P )=g((u )), together with the canonical invariant form

(1.3) Q(f(u),g(u)) = Resu=0K(f,g), where K denotes the Killing form of the Lie algebra g((u−1)) over C((u−1)). Moreover, the Lie bialgebra structures which are obtained by twisting δ2 are in a one-to-one correspondence with so-called rational solutions of the CYBE, according to [7].

Case 3. In this case, let us consider the Lie bialgebra structure given by

(1.4) δ3(p(u)) = [r3(u, v),p(u) ⊗ 1+1⊗ p(v)], − ⊗ 1 with r3(u, v)=vΩ/(u v)+Σαeα fα + 2 Ω0,whereeα,fα are root vectors of g and Ω0 is the Cartan part of Ω. −1 It was proved in [4] that the associated classical double is D3(P )=g((u ))×g, together with the invariant nondegenerate form Q deﬁned by

(1.5) Q((f(u),a), (g(u),b)) = K(f(u),g(u))0 − K(a, b), where the index zero means that one takes the free term in the series expansion. According to [4], there is a one-to-one correspondence between Lie bialgebra structures which are obtained by twisting δ3 and so-called quasi-trigonometric solutions of the CYBE.

Case 4. We consider the co-bracket on P given by

(1.6) δ4(p(u)) = [r4(u, v),p(u) ⊗ 1+1⊗ p(v)], with r4(u, v)=uvΩ/(v − u). It was shown in [10] that the classical double associated to the Lie bialgebra −1 × ⊗C 2 structure δ4 is D4(P )= g((u )) (g [ε]), where ε =0.TheformQ is described N k N k as follows: if f(u)= −∞ aku and g(u)= −∞ bku ,then

−2 Q(f(u)+A0 +A1ε, g(u)+B0 +B1ε)=Resu=0u K(f,g)−K(A0,B1)−K(A1,B0).

Lie bialgebra structures which are in the same class as δ4 are in a one-to-one correspondence with quasi-rational r-matrices,asitwasprovedin[10]. Regarding the quantization of these Lie bialgebra structures on P , the following conjecture stated in [4] and proved by G. Halbout in [3] plays a crucial role.

CLASSIFICATION OF QUASI-TRIGONOMETRIC SOLUTIONS 1573

Theorem 1.1. Any classical twist can be extended to a quantum twist, i.e., if (L, δ) is any Lie bialgebra, s is a classical twist, and (A, ∆,ε) is a quantization of (L, δ), there exists F ∈ A ⊗ A such that (1) F =1+O() and F − F 21 = s + O(2), (2) (∆ ⊗ id)(F )F 12 − (id ⊗ ∆)(F )F 23 =0, (3) (ε ⊗ id)(F )=(id⊗ ε)(F )=1. Moreover gauge equivalence classes of quantum twists for A are in bijection 2 with gauge equivalence classes of -dependent classical twists s = s1 + O( ) for L. Let us suppose that we have a Lie bialgebra structure δ on P .Thenδ is obtained by twisting one of the four structures δi from Cases 1–4. This above theorem implies that in order to find a quantization for (P, δ), it is sufficient to determine the quantization of δi and then find the quantum twist whose classical limit is s.Let us note that the quantization of (P, δ3) is well-known. The corresponding quantum algebra was introduced by V. Tolstoy in [11] and it is denoted by Uq(g[u]). The quasi-trigonometric solutions of the CYBE were studied in [5], where it was proved that they fall into classes, which are in a one-to-one correspondence with vertices of the extended Dynkin diagram of g. Let us consider corresponding roots, namely simple roots α1,α2, ···αr and α0 = −αmax.In[5] quasi-trigonometric solutions corresponding to the simple roots which have coefficient one in the decomposition of the maximal root were classified. It was also proved there that quasi-trigonometric solutions corresponding to α0 are in a one-to-one correspondence with constant solutions of the modified CYBE classified in [1]andthepoly- nomial part of these solutions is constant. The aim of our paper is to obtain a complete classification of quasi-trigonometric solutions of the CYBE. In particular, we describe all the quasi-trigonometric solutions with non-trivial polynomial part for g = o(5).

2. Lie bialgebra structures associated with quasi-trigonometric solutions Definition 2.1. AsolutionX of the CYBE is called quasi-trigonometric if it is of the form X(u, v)=vΩ/(u−v)+p(u, v), where p is a polynomial with coefficients in g ⊗ g. The class of quasi-trigonometric solutions is closed under gauge transformations. We first need to introduce the following notation: Let R be a commutative ring and let L be a Lie algebra over R. Let us denote by AutR(L) the group of automorphisms of L over R. In other words we consider such automorphisms of L, which satisfy the condition f(rl)=rf(l), where r ∈ R, l ∈ L. At this point we note that there exists a natural embedding −1 AutC[u](g[u]) → AutC((u−1))(g((u ))), defined by the formula σ(u−kx)=u−kσ(x) , for any σ ∈ AutC[u](g[u]) and x ∈ g[u]. Now if X is a quasi-trigonometric solution and σ(u) ∈ AutC[u](g[u]), one can check that the function Y (u, v):=(σ(u) ⊗ σ(v))(X(u, v)) is again a quasi- trigonometric solution. X and Y are said to be gauge equivalent.

4158 IULIA POP AND ALEXANDER STOLIN

Theorem 2.2. There exists a natural one-to-one correspondence between quasi– trigonometric solutions of CYBE for g and linear subspaces W of g((u−1))×g which satisfy the following properties: (1) W is a Lie subalgebra of g((u−1)) × g and W ⊇ u−N g[[u−1]] for some positive integer N. (2) W ⊕ g[u]=g((u−1)) × g. (3) W is a Lagrangian subspace of g((u−1)) × g with respect to the invariant bilinear form Q given by (1.5).

Let σ(u) ∈ AutC[u](g[u]). Let σ(u)=σ(u)⊕σ(0) be the induced automorphism of g((u−1)) × g.

Definition 2.3. We will say that W1 and W2 are gauge equivalent if there exists σ(u) ∈ AutC[u](g[u]) such that W1 = σ(u)W2. It was checked in [4] that two quasi-trigonometric solutions are gauge equivalent if and only if the corresponding subalgebras are gauge equivalent. Let h be a Cartan subalgebra of g with the corresponding set of roots R and a choice of simple roots Γ. Denote by gα the root space corresponding to a root α.Leth(R)bethesetofallh ∈ h such that α(h) ∈ R for all α ∈ R.Consider C −1 −k the valuation on ((u )) deﬁned by v( k≥n aku )=n. For any root α and any −1 h ∈ h(R), set Mα(h):={f ∈ C((u )) : v(f) ≥ α(h)}.Consider −1 (2.1) Oh := h[[u ]] ⊕ (⊕α∈RMα(h) ⊗ gα). As it was shown in [5], any maximal order W which corresponds to a quasi- trigonometric solution of the CYBE, can be embedded (up to some gauge equivalence) into Oh × g.Moreoverh may be taken as a vertex of the standard simplex ∆st = {h ∈ h(R):α(h) ≥ 0 for all α ∈ Γandαmax ≤ 1}. Vertices of the above simplex correspond to vertices of the extended Dynkin diagram of g, the correspondence being given by the following rule:

0 ↔ αmax h ↔ α , i i where αi(hj)=δij/kj and kj are given by the relation kjαj = αmax. We will write Oα instead of Oh if α is the root which corresponds to the vertex h. By straightforward computations, one can check the following two results: Lemma 2.4. Let R be the set of all roots and α an arbitrary simple root. Let k be the coefficient of α in the decomposition of αmax. − ≤ ≤ For each r, k r k,letRr denote the set of all roots which contain α with coefficient r.Letg = h ⊕ g and g = g .Then 0 β∈R0 β r β∈Rr β k 0 −1 (2.2) Oα = u Ogr + Ogr + uOg−k, r=1 r=1−k where O := C[[u−1]]. Lemma 2.5. Let α be a simple root and k its coefficient in the decomposition of αmax.Let∆α denote the set of all pairs (a, b), a ∈ g0 +g−k, b ∈ g0 +g−1 +...+g−k, a = a0 + a−k, b = b0 + b−1 + ... + b−k and a0 = b0.Then

CLASSIFICATION OF QUASI-TRIGONOMETRIC SOLUTIONS 1595

(i) The orthogonal complement of Oα × g with respect to Q is given by −1 k−1 ⊥ −1 −2 (2.3) (Oα × g) = Ogr + u Ogr + u Ogk. r=−k r=0 (ii) There exists an isomorphism σ O × α g ∼ ⊕ ⊕ × (2.4) ⊥ = (gk g0 g−k) g (Oα × g) given by ⊥ σ((f,a)+(Oα × g) )=(a0 + b0 + c0,a), where the element f ∈ Oα is decomposed according to Lemma 2.4: −1 −1 −1 −1 f = u (a0 + a1u + ...)+(b0 + b1u + ...)+u(c0 + c1u + ...)+..., ai ∈ gk, bi ∈ g0, ci ∈ g−k and a ∈ g. (iii) (Oα × g) ∩ g[u] is sent via the isomorphism σ to ∆α.

Let us make an important remark. The Lie subalgebra gk + g0 + g−k of g coincides with the semisimple Lie algebra whose Dynkin diagram is obtained from the extended Dynkin diagram of g by crossing out α. Let us denote this subalgebra by Lα.TheLiealgebraLα × g is endowed with the following invariant bilinear form: (2.5) Q((a, b), (c, d)) = K(a, c) − K(b, d), for any a, c ∈ Lα and b, d ∈ g. + − P L On the other hand, g0 + g k is the parabolic subalgebra −αmax of α which corresponds to −αmax. The Lie subalgebra g0 + g−1 + ... + g−k is the parabolic − subalgebra Pα of g which corresponds to the root α and contains the negative Borel − subalgebra. Let us also note that g0 is precisely the reductive part of Pα and of P + a, b ∈ P + × P − −αmax . We can conclude that the set ∆α consists of all pairs ( ) −αmax α whose reductive parts are equal. Theorem 2.6. Let α be a simple root. There is a one-to-one correspondence −1 between Lagrangian subalgebras W of g((u )) × g which are contained in Oα × g and transversal to g[u], and Lagrangian subalgebras l of Lα × g transversal to ∆α (with respect to the bilinear form Q).

Proof. Since W is a subspace of Oα×g,letl be its image in Lα×g. Because W is transversal to g[u], one can check that l is transversal to the image of (Oα×g)∩g[u] in Lα × g,whichisexactly∆α.ThefactthatW is Lagrangian implies that l is also Lagrangian. Conversely, if l is a Lagrangian subalgebra of Lα × g transversal to ∆α,then its preimage W in Oα × g is transversal to g[u] and Lagrangian as well.

The Lagrangian subalgebras l of Lα × g which are transversal to ∆α,canbe determined using results of P. Delorme [2] on the classiﬁcation of Manin triples. We are interested in determining Manin triples of the form (Q , ∆α, l). Let us recall Delorme’s construction of so-called generalized Belavin-Drinfeld data.Letr be a ﬁnite-dimensional complex, reductive, Lie algebra and B asym- metric, invariant, nondegenerate bilinear form on r. The goal in [2] is to classify all

6160 IULIA POP AND ALEXANDER STOLIN

Manin triples of r up to conjugacy under the action on r of the simply connected Lie group R whose Lie algebra is r. One denotes by r+ and r− respectively the sum of the simple ideals of r for which the restriction of B is equal to a positive (negative) multiple of the Killing form. Then the derived ideal of r is the sum of r+ and r−. Let j0 be a Cartan subalgebra of r, b0 a Borel subalgebra containing j0 and b0 ∩ ∩ be its opposite. Choose b0 r+ as Borel subalgebra of r+ and b0 r− as Borel subalgebra of r−.DenotebyΣ+ (resp., Σ−)thesetofsimplerootsofr+ (resp., r−) with respect to the above Borel subalgebras. Let Σ = Σ+ ∪ Σ− and denote by W =(Hα,Xα,Yα)α∈Σ+ a Weyl system of generators of [r, r]. Definition 2.7 (Delorme, [2]). One calls (A, A , ia, ia ) generalized Belavin- Drinfeld data with respect to B when the following ﬁve conditions are satisﬁed: (1) A is a bijection from a subset Γ+ of Σ+ on a subset Γ− of Σ− such that

B(HAα,HAβ)=−B(Hα,Hβ),α,β ∈ Γ+. (2) A is a bijection from a subset Γ+ of Σ+ on a subset Γ− of Σ− such that − ∈ B(HA α,HA β)= B(Hα,Hβ),α,β Γ+. −1 { ∈ ∈ } (3) If C = A A is the map defined on dom(C) = α Γ+ :Aα Γ− by Cα = A−1Aα,thenC satisfies: For all α ∈ dom(C), there exists a positive integer n such that α,..., Cn−1α ∈ dom(C) and Cnα/∈ dom(C). (4) ia (resp., ia ) is a complex vector subspace of j0, included and Lagrangian in the orthogonal a (resp., a ) to the subspace generated by Hα, α ∈ Γ+ ∪ Γ− (resp., ∪ Γ+ Γ−). (5) If f is the subspace of j0 generated by the family Hα + HAα, α ∈ Γ+,and f is defined similarly, then (f ⊕ ia) ∩ (f ⊕ ia )=0.

Let R+ be the set of roots of j0 in r which are linear combinations of elements of Γ+. One defines similarly R−, R+ and R−. The bijections A and A can then be extended by linearity to bijections from R+ to R− (resp., R+ to R−). If A satisfies condition (1), then there exists a unique isomorphism τ between the subalgebra m+ of r spanned by Xα, Hα and Yα, α ∈ Γ+, and the subalgebra m− spanned by Xα, Hα and Yα, α ∈ Γ−, such that τ(Hα)=HAα, τ(Xα)=XAα, τ(Yα)=YAα for all α ∈ Γ+.IfA satisfies (2), then one defines similarly an isomorphism τ between m + and m −. Theorem 2.8 (Delorme, [2]). (i) Let BD =(A, A , ia, ia ) be generalized Belavin- Drinfeld data, with respect to B.Letn be the sum of the root spaces relative to roots α of j0 in b0, which are not in R+ ∪ R−.Leti := k ⊕ ia ⊕ n,where k := {X + τ(X):X ∈ m+}. Let n be the sum of the root spaces relative to roots α of j0 in b0, which are ∪ ⊕ ⊕ { ∈ } not in R+ R−.Leti := k ia n ,wherek := X + τ (X):X m + . Then (B,i, i) is a Manin triple. (ii) Every Manin triple is conjugate by an element of R to a unique Manin triple of this type. ext Let us consider the particular case r = Lα ×g.WesetΣ+ := (Γ \{α})×{0}, Σ− := {0}×ΓandΣ:=Σ+ ∪ Σ−.

CLASSIFICATION OF QUASI-TRIGONOMETRIC SOLUTIONS 1617

Denote by (Xγ ,Yγ ,Hγ )γ∈Γ a Weyl system of generators for g with respect to − the root system Γ. Denote αmax by α0.LetHα0 be the coroot of α0.Wechoose ∈ ∈ Xα0 gα0 , Yα0 g−α0 such that [Xα0 ,Yα0 ]=Hα0 . A Weyl system of generators in Lα × g (with respect to the root system Σ) is the following: X(β,0) =(Xβ, 0), H(β,0) =(Hβ, 0), Y(β,0) =(Yβ, 0), for any ext β ∈ Γ \{α},andX(0,γ) =(0,Xγ), H(0,γ) =(0,Hγ ), Y(0,γ) =(0,Yγ ), for any γ ∈ Γ. By applying the general result of Delorme, one can deduce the description of the Manin triples of the form (Q , ∆α, l).

Corollary 2.9. Let S := Γ \{α} and ζS := {h ∈ h : β(h)=0, ∀β ∈ S}. For any Manin triple (Q , ∆α, l), there exists a unique generalized Belavin-Drinfeld data BD =(A, A , ia, ia ) where A : S ×{0}−→{0}×S, A(γ,0) = (0,γ) and ia =diag(ζS), such that (Q , ∆α, l) is conjugate to the Manin triple TBD =(Q , i, i ). Moreover, up to a conjugation which preserves ∆α, one has l = i . Proof. Let us suppose that (Q , ∆α, l) is a Manin triple. Then there exists a unique generalized Belavin-Drinfeld data BD =(A, A , ia, ia ) such that the corresponding TBD =(Q , i, i ) is conjugate to (Q , ∆α, l). Since i and ∆α are conjugate P + × P − and ∆α is “under” the parabolic subalgebra α0 α , it follows that i is also “under” this parabolic and thus a = ζS × ζS. According to [2], p. 136, the map A should be an isometry between S ×{0} and {0}×S, so it is given by an isometry A˜ of S. Let m be spanned by Xβ, Hβ, Yβ for all β ∈ S.Theni contains k := {(X, τ(X)) : ∈ } X m ,whereτ has the property that τ(Xβ)=XA˜β, τ(Hβ)=HA˜β, τ(Yβ)=YA˜β, for all β ∈ S. Since i and ∆α are conjugate, τ has to be an inner automorphism of m and ia =diag(ζS). It follows that A˜ = id. This ends the proof. ˜ ⊆ ext \{ } ⊆ We will consider triples of the form (Γ1, Γ2, A ), where Γ1 Γ α ,Γ2 Γ ˜ and A is an isometry between Γ1 and Γ2. ∈ ˜ We say that such a triple is of type I if α/Γ2 and (Γ1, Γ2, A ) is an admissible ∈ ˜ triple in the sense of [1]. A triple is of type II if α Γ2 and A (β)=α,forsome ∈ \{ } \{ } ˜ β Γ1 and (Γ1 β , Γ2 α ), A ) is an admissible triple in the sense of [1]. By using the deﬁnition of generalized Belavin-Drinfeld data, one can easily check the following:

Lemma 2.10. Let A : S ×{0}−→{0}×S, A(γ,0) = (0,γ) and ia =diag(ζS). A quadruple (A, A , ia, ia ) is generalized Belavin-Drinfeld data if and only if the pair (A , ia ) satisfies the following conditions: ×{ }−→{ }× ˜ ˜ (1) A :Γ1 0 0 Γ2 is given by A (γ,0) = (0, A (γ)) and (Γ1, Γ2, A ) is of type I or II from above. × ∈ (2) Let f be the subspace of h h spanned by pairs (Hβ,HA˜(β)) for all β Γ1. Let ia be Lagrangian subspace of a := {(h1,h2) ∈ h×h : β(h1)=0,γ(h2)=0, ∀β ∈ ∀ ∈ } Γ1, γ Γ2 .Then (2.6) (f ⊕ ia ) ∩ diag(h)=0. Remark 2.11. One can always find ia which is a Lagrangian subspace of a and satisfies condition (2.6). This is a consequence of [2] Remark 2, p. 142. A proof of this elementary fact can also be found in [9], Lemma 5.2.

8162 IULIA POP AND ALEXANDER STOLIN

Summing up the previous results we conclude the following: Theorem 2.12. Let α be a simple root. Suppose that l is a Lagrangian subalgebra of Lα × g transversal to ∆α. Then, up to a conjugation which preserves ∆α, ˜ one has l = i ,wherei is constructed from a pair formed by a triple (Γ1, Γ2, A ) of type I or II and a Lagrangian subspace ia of a such that (2.6) is satisﬁed. Let us apply this result to classify solutions in g = o(5).

Corollary 2.13. Let α1, α2 be the simple roots in o(5) and α0 = −2α1 − α2. Up to gauge equivalence, there exist two quasi-trigonometric solutions with nontrivial polynomial part.

Proof. The root α2 has coeﬃcient k = 1 in the decomposition of the max- ˜ ⊆{ } imal root. The only possible choice for a triple (Γ1, Γ2, A )withΓ1 α0,α1 , ⊆{ } { } { } ˜ Γ2 α1,α2 tobeoftypeIorIIisΓ1 = α0, ,Γ2 = α2, , A (α0)=α2. One can check that ia is a√ 1-dimensional√ space spanned by the following pair − − − (diag( 2, 1, 0, 1, 2), diag(0, 5, 0, 5, 0)). The Lagrangian subalgebra i2 con- × structed from this triple is transversal to ∆α2 in g g. The root α1 has coeﬃcient k = 2 in the decomposition of the maximal root. ˜ ⊆{ } ⊆{ } The only possible choice for a triple (Γ1, Γ2, A )withΓ1 α0,α2 ,Γ2 α1,α2 { } { } ˜ is again Γ1 = α0, ,Γ2 = α2, , A (α0)=α2 and ia is as in the previous case. The Lagrangian subalgebra i1 constructed from this triple is transversal to ∆α1 in × Lα1 g.

References [1] A. Belavin, V. Drinfeld, Triangle equations and simple Lie algebras. Math. Phys. Reviews, Vol. 4 (1984), Harwood Academic, 93–165. [2] P. Delorme, Classification des triples de Manin pour les algèbres de Lie réductives complexes. J. Algebra, 246 (2001), 97–174. [3] G. Halbout, Formality theorem for Lie bialgebras and quantization of twists and coboundary r-matrices. Adv. Math. 207 (2006), 617–633. [4] S. Khoroshkin, I. Pop, A. Stolin, V. Tolstoy, On some Lie bialgebra structures on polynomial algebras and their quantization. Preprint no. 21, 2003/2004, Mittag-Leffler Institute, Sweden. [5] S. Khoroshkin, I. Pop, M. Samsonov, A. Stolin, V. Tolstoy, On some Lie bialgebra structures on polynomial algebras and their quantization. ArXiv math. QA/0706.1651v1. To appear in Comm. Math. Phys. [6] F. Montaner, E. Zelmanov, Bialgebra structures on current Lie algebras. Preprint, University of Wisconsin, Madison, 1993. [7] A. Stolin, On rational solutions of Yang-Baxter equations. Maximal orders in loop algebra. Comm. Math. Phys. 141 (1991), 533–548. [8] A. Stolin, A geometrical approach to rational solutions of the classical Yang-Baxter equation. Part I. Symposia Gaussiana, Conf.A, Walter de Gruyter, Berlin, New York, 1995, 347–357. [9] A. Stolin, Some remarks on Lie bialgebra structures for simple complex Lie algebras. Comm. Alg. 27 (9) (1999), 4289–4302. [10] A. Stolin, J. Yermolova–Magnusson, The 4th structure. Czech. J. Phys, Vol. 56, No. 10/11 (2006), 1293–1927. [11] V. Tolstoy, From quantum affine Kac–Moody algebras to Drinfeldians and Yangians. in Kac– Moody Lie algebras and related topics, Contemp. Math. 343, Amer. Math. Soc. 2004, 349–370.

Department of Mathematical Sciences, University of Gothenburg, Sweden E-mail address: [email protected], [email protected]

Contemporary Mathematics Contemporary Mathematics Volume 478, 2009 Volume 00, XXXX

The Relevance and the Ubiquity of Pr¨ufer Modules

Claus Michael Ringel

Abstract. Let R be a ring. An R-module M is called a Prüfer module provided there exists a locally nilpotent, surjective endomorphism of M with kernel of finite length. We want to outline the relevance, but also the ubiquity of Prüfer modules. The main assertion will be that any Prüfer module which is not of finite type gives rise to a generic module, thus to infinite families of indecomposable modules with fixed endo-length (here we are in the setting of the second Brauer-Thrall conjecture). In addition, we will report on a construction procedure which yields a wealth of Prüfermodules. Unfortunately, we do not know which modules obtained in this way are of finite type.

Introduction. This is the written account of a lecture given at 4th International Conference on Representation Theory (ICRT-IV), Lhasa, July 16–20, 2007. Section 1 recalls the definition of a Prüfer module as introduced in [R6] and provides some examples, in Section 2 we show that the degeneration theory of modules concerns certain Prüfer modules of finite type, whereas Section 4 provides a construction of Prüfer modules using pairs of monomorphisms U0 → U1; these two sections 2 and 4 are reports on some of the results of [R6]. In the last Section 6 we discuss the question how to search for pairs of monomorphisms U0 → U1. This is an announcement of results of the forthcoming paper [R9], where “take-off categories” are introduced. The central part is section 3. There, we show that the existence of a Prüfer module which is not of finite type implies the existence of a generic module, thus of infinite families of indecomposable modules with fixed (and arbitrarily large) endolength. This has not yet appeared in print (but see [R8]) and has been announced under the title: Prüfer modules which are not of finite type. Also section 5 is new, here we show that given a tame hereditary algebra, a finite length module N can generate a non-finite-type module M only in case N has a preprojective direct summand which is sincere. This gives an indication why it seems to be reasonable to look for module embeddings in take-off categories.

1991 Mathematics Subject Classiﬁcation. Primary 16D10, 16G60. Secondary: 16D70, 16D90 16S50, 16G20, 16P99.

ccXXXX2009 American American Mathematical Mathematical Society 1631

2164 CLAUS MICHAEL RINGEL

1. Pr¨ufer modules.

(1.1). Let R be any ring. We deal with (left) R-modules. An R-module M is called a Prüfer module provided there exists an endomorphism φ of M with the following properties: φ is locally nilpotent, surjective, and the kernel W of φ is non-zero and of finite length. The module W is called the basis of M.LetW [n]be n ∞ the kernel of φ ,thenM = n W [n], thus we also may write M = W [ ]. (1.2) The classical example. Let R = Z and p a prime number. Let S = Z[p−1] the subring of Q generated by p−1.Then S/R = lim Z/Zpn → is the Prüfergroup for the prime p. These Prüfer groups are all the indecomposable Z-modules which are Prüfer modules. (1.3). More generally, let R be a Dedekind ring. Then any indecomposable R-module of finite length is of the from W [n], where W is the basis of an indecomposable Prüfer module and n ∈ N. In particular, this applies to R = Z, but also say to the polynomial ring R = k[T ] in one variable with coefficients in the field k. Note that a k[T ]-module is just a pair (V,f), where V is a k-space and f is a linear d operator on V .Ifchark = 0, then the pair (k[T ], d t )isaPrüfer module. As Atiyah [A] has shown, a corresponding assertion holds for the coherent sheaves over an elliptic curve: any indecomposable coherent sheaf of finite length is of the from W [n], where W is the basis of an indecomposable Prüfermodule and n ∈ N. (1.4) Example. Consider the Kronecker algebra Λ = kQ, this is the path algebra of the quiver Q ...... ◦◦...... The embedding functor 1 ...... → .. mod k[T ] mod kΛ (V,f) VV...... f preserves Prüfer modules. Using this functor, we obtain all indecomposable Prüfer modules for the Kronecker algebra with one exception, the remaining one is of the form ..f ...... VV...... 1 where (V,f) is the indecomposable Prüfer k[T ]-module such that f has 0 as eigenvalue. (1.5). We also should mention a famous theorem of Crawley-Boevey [CB1]: Let Λ be a finite-dimensional k-algebra and k an algebraically closed field. If Λ is tame, and d ∈ N, then almost all indecomposable Λ-modules of length d are of the from W [n], where W [∞] is an indecomposable Prüfer module. (1.6) Warning. The Prüfer modules as defined above do not have to be indecomposable: for example the countable direct sum W (N) of copies of W with the shift endomorphism (w1,w2,...) → (w2,w3,...)isaPrüfer module: the trivial Prüfer module with basis W . Less trivial examples will be seen in the next section.

THE RELEVANCE AND THE UBIQUITY OF PRUFER¨ MODULES 1653

(1.7) Lemma. Let M be a Pr¨ufer module with basis W . If the endomorphism ring of W is a division ring, then either M = W (N) or else M is indecomposable. Proof. This is an immediate consequence of the process of simpliﬁcation [R1].

A module M is of finite type provided it is the direct sum of finitely generated modules and such that there are only finitely many isomorphism classes of indecomposable direct summands. Our main concern will be Prüfer modules which are not of finite type. But first we consider some Prüfer modules of finite type.

2. Degenerations of modules In this and the next section we consider artin algebras (these are rings which are module finite over the center, the center being artinian). (2.1). In order to motivate the notion of a degeneration, let us consider first the case of Λ being a finite-dimensional k-algebra where k is an algebraically closed field. Assume that Λ is generated as a k-algebra by a1,a2,...,at subject to relations ρi.Ford ∈ N, we consider the variety t M(d)={(A1,...,At) ∈ M(d×d, k) | ρi(A1,...,At) = 0 for all i}, of d-dimensional Λ-modules: its elements are the d-dimensional Λ-modules with underlying vector space kd (thus, up to isomorphism, all d-dimensional Λ-modules). The group GL(d, k)operatesonM(d) by simultaneous conjugation. Elements of M(d) belong to the same orbit if and only if they are isomorphic. Theorem (Zwara). Let X, Y be d-dimensional Λ-modules. Then Y is in the orbit closure of X if and only if there exists a finitely generated Λ-module U and an exact sequence 0 → U → X ⊕ U → Y → 0. Such a sequence should be called a Riedtmann-Zwara sequence. (2.2). Now let Λ be an arbitrary artin algebra and X, Y Λ-modules of finite length. We call Y a degeneration of X provided there exists a finitely generated Λ-module U and an exact sequence 0 → U → X ⊕ U → Y → 0

Proposition (Zwara). Y is a degeneration of X if and only if there is a Pr¨ufer module M with basis Y such that Y [t +1] Y [t] ⊕ X for some t,or, equivalently, for almost all t.

Proof. See [Z2] and [R6].

Note that an isomorphism Y [t +1] Y [t] ⊕ X yields directly a Riedtmann- Zwara sequence as well as a co-Riedtmann-Zwara sequence, using the canonical exact sequences 0 → Y [t] → Y [t +1]→ Y → 0 , 0 → Y → Y [t +1]→ Y [t] → 0

4166 CLAUS MICHAEL RINGEL

and replacing the middle term by Y [t] ⊕ X. (2.3). Note that the Pr¨ufer modules Y [∞] obtained in (2.2) satisfy Y [∞] Y [t] ⊕ X(N) for some t. In particular, these are modules of ﬁnite type.

3. Pr¨ufer modules and the second Brauer-Thrall conjecture

(3.1). Let Λ be an artin algebra. The Krull-Remak-Schmidt Theorem asserts that any finitely generated Λ-module can be written as a direct sum of indecomposable modules, and such a decomposition is unique up to isomorphism. The artin algebra Λ is called representation-infinite provided there are infinitely many isomorphism classes of indecomposable Λ-modules, otherwise representation- finite. (3.2). The first Brauer-Thrall conjecture was solved by Roiter in 1968: Theorem (Roiter [Ro]). If Λ is a representation-infinite artin algebra, then there are indecomposable modules of arbitrarily large finite length.

(3.3). The second Brauer-Thrall conjecture has been solved only for finite dimensional k-algebras, where k is an algebraically closed (or at least perfect) field: Theorem (Bautista [Ba], Bongartz [Bo]). If Λ is a representation-infinite k-algebra, where k is an infinite perfect field, then there are infinitely many natural numbers d such that there are infinitely many indecomposable Λ-modules of length d.

It has been conjectured by Brauer-Thrall that the assertion holds for any infinite field. For finite fields, or, more generally, for an arbitrary artin algebra Λ, one may conjecture the following: if Λ is representation-infinite, then there are infinitely many natural numbers d such that there are infinitely many indecomposable Λ- modules of endo-length d.(Theendo-length of a module M is the length of M when M is considered as a module over its endomorphism ring). (3.3). Let Λ be an artin algebra. A Λ-module M is said to be generic provided M is indecomposable, of infinite length, but of finite endo-length. Theorem (Crawley-Boevey [CB2]). Let Λ be a finite-dimensional k-algebra (k a field). Let M be a generic Λ-module. Then there are infinitely many natural numbers d such that there are infinitely many indecomposable Λ-modules of endo- length d.

(3.4) Prüfer modules yield generic modules. Theorem. Let M be a Prüfer module. The following conditions are equivalent: (i) M is not of finite type. (ii) There is an infinite index I set such that the product module M I has a generic direct summand. (iii) For every infinite index I set, the product module M I has a generic direct summand.

THE RELEVANCE AND THE UBIQUITY OF PRUFER¨ MODULES 1675

Proof. The implication (iii) =⇒ (ii) is trivial. Also (ii) =⇒ (i) is obvious: If M is of finite type, then all product modules M I are of finite type. We only have to show (i) =⇒ (iii). (It is sufficient to consider I = N in (iii), since any infinite index set I can be written as the disjoint union of N and some other index set I, and then M I = M N ⊕ M I , however, there is no problem to work in general.) Now assume that I is an infinite index set and that M I has no indecomposable direct summand which is endo-finite and of infinite length. Since M is a Prüfer module, there is a surjective, locally nilpotent endomorphism φ with kernel W = W [1] non-zero and of finite length. Let W [n]bethekernelofφn.Thus W [1] ⊂ W [2] ⊂···⊂ W [n]=M n is a filtration of M with finite length modules W [n]. We obtain a corresponding chain of inclusions W [1]I ⊂ W [2]I ⊂···⊂ W [n]I = M . n It has been shown in [R3] (see also [K]) that M is isomorphic to a direct sum of copies of M and itself a direct summand of M I ; there is an endo-finite submodule E of M I such that M I = M ⊕ E.

Any endo-finite module E can be written as a direct sum of copies of finitely many indecomposable endo-finite modules, say E1,...,Et. By assumption, all these modules Ei are of finite length. A well-known lemma of Auslander asserts that any indecomposable direct summand of M I of finite length is a direct summand of M itself, thus the modules E1,...,Et occur as direct summands of M. Since M is artinian as a module over its endomorphism ring, M is Σ-algebraic compact, thus it is a direct sum of indecomposable modules with local endomorphism ring. Write M = A ⊕ B,whereA is a direct sum of copies of the various Ei and B has no direct summand of the form Ei, for any i. We want to show that B is of finite length. This then shows that M is of finite type. ∩ ∩ The modules A, B are also filtered, with An = A W [n],Bn = B W [n] (it is obvious that A = n An,B= n Bn). For any n there is some n with W [n] ⊆ An ⊕ Bn . (Namely, let x ∈ W [n], write x = a + b with a ∈ A, b ∈ B. ∈ ∈ ∈ Then there is somen with a, b Mn ,thusa An ,b Bn .) I I We write A = i Ai and B = i Bi . Then M = A ⊕ B

⊇ I ⊆ ⊕ I I ⊕ (the inclusion is obvious, the other follows from W [n] (An Bn ) = An I ⊆ ⊕ Bn A B .). We see that (AI /A) ⊕ (BI /B)=M I /M = E,

I I thus A /A = EA and B /B = EB with E = EA ⊕ EB.Inparticular,EA and EB are direct sums of copies of E1,...,Et. Since the direct sum of the inclusion maps A → AI and B → BI

6168 CLAUS MICHAEL RINGEL

is a split monomorphism, the maps themselves are split monomorphisms, thus

I I A A ⊕ EA and B B ⊕ EB.

Consider the last isomorphism. If Ei is a direct summand of EB, then it is a direct summand of B (Auslander Lemma), impossible. This shows that EB =0. But then I B = B implies that B = Bn for some n,thusB ⊆ W [n]. This shows that B is of ﬁnite length.

May-be one should record: Assume that M I /M is the direct sum of copies of indecomposable modules E1,...,Et of finite length, then M is the direct sum of a finite length module B and of copies of the modules Ei. (3.5). As we have mentioned, the second Brauer-Thrall conjecture claims the following: if Λ is a representation-infinite algebra, then there are infinitely many natural numbers d such that there are infinitely many indecomposable Λ-modules of endo-length d. Note that this is an assertion which concerns only modules of finite length. But it seems that it may be reasonable to look for a solution using modules of infinite length. As we have seen, it will be sufficient to show that a representation-infinite algebra has a Prüfer module which is not of finite type, since this implies the existence of a generic module and thus the existence of infinitely many indecomposable Λ-modules of endo-length d.

4. The ladder construction of Pr¨ufer modules

We return to rings and modules in general.

(4.1). This construction was exhibited in [R6], let us recall here the essential steps: We start with a proper inclusion U0 ⊂ U1 (say with cokernel W )andamap v0 : U0 → U1, and we form the pushout of w0 and v0

..... w0 ...... 0 ...... U0 ...... U1 ...... W ...... 0 ...... v0 .....v1 ...... w1 ...... 0 ...... U1 ...... U2 ...... W ...... 0

we obtain a module U2, as well as a monomorphism w1 : U1 ⊂ U2 (again with cokernel W )andamapv1 : U1 → U2. Using induction, we obtain in this way modules Ui, monomorphisms wi : Ui ⊂ Ui+1 (all with cokernel W )aswellasmapsvi : Ui → Ui+1 such that vi+1wi = wi+1vi for all i ≥ 0. This means that we obtain the following ladder of commutative squares: U −−−−w0→ U −−−−w1→ U −−−−w2→ U −−−−w3→ ··· ⏐0 ⏐1 ⏐2 ⏐3 ⏐ ⏐ ⏐ ⏐ v0! v1! v2! v3!

w1 w2 w3 w4 U1 −−−−→ U2 −−−−→ U3 −−−−→ U4 −−−−→ ···

THE RELEVANCE AND THE UBIQUITY OF PRUFER¨ MODULES 1697 We form the inductive limit U∞ = i Ui (along the maps wi). Since all the squares commute, the maps vi induce a map U∞ → U∞ whichwedenotebyv∞:

w0 w1 w2 w3 U −−−−→ U −−−−→ U −−−−→ U −−−−→ ··· U = U∞ ⏐0 ⏐1 ⏐2 ⏐3 i ⏐i ⏐ ⏐ ⏐ ⏐ ⏐ v0! v1! v2! v3! !v∞ −−−−w1→ −−−−w2→ −−−−w3→ −−−−w4→ ··· U1 U2 U3 U4 i Ui = U∞

We also may consider the factor modules U∞/U0 and U∞/U1.Themap v∞ : U∞ → U∞ maps U0 into U1, thus it induces a map

v : U∞/U0 −→ U∞/U1.

and this map v is an isomorphism. Namely, there are the commutative diagrams with exact rows:

wi−1 0 −−−−→ U − −−−−→ U −−−−→ W −−−−→ 0 i⏐ 1 ⏐i " ⏐ ⏐ " !vi−1 !vi "

wi 0 −−−−→ Ui −−−−→ Ui+1 −−−−→ W −−−−→ 0

which means that the cokernel Ui/Ui−1 = W of wi−1 is mapped under the restriction vi of v isomorphically onto the cokernel Ui+1/Ui = W of wi. Thus, we see that the map v is a map from a ﬁltered module with factors Ui/Ui−1 (where i ≥ 1) to a ﬁltered module with factors Ui+1/Ui (again with i ≥ 1), and the maps vi are just those induced on the factors. Since all the maps vi are isomorphisms, also v itself is an isomorphism. It follows: The composition of maps

p v−1 U∞/U0 −−−−→ U∞/U1 −−−−→ U∞/U0

(p the projection map) is an epimorphism φ with kernel U1/U0. It is easy to see that φ is locally nilpotent.

Proposition. The module U∞/U0 is a Pr¨ufer module with respect to the en- −1 domorphism φ = v ◦ p, its basis is W = U1/U0.

This shows that starting with a proper inclusion U0 ⊂ U1 and a map v0 : U0 → U1, the ladder construction yields a Pr¨ufer module U∞/U0 with bases U1/U0.

8170 CLAUS MICHAEL RINGEL

(4.2). In case also v0 is injective, we obtain a second Pr¨ufer module. Namely, there is the following chessboard: U −−−−w0→ U −−−−w1→ U −−−−w2 → U −−−−w3→ ··· ⏐0 ⏐1 ⏐2 ⏐3 ⏐ ⏐ ⏐ ⏐ v0! v1! v2! v3!

U −−−−w1→ U −−−−w2→ U −−−−w3 → ··· ⏐1 ⏐2 ⏐3 ⏐ ⏐ ⏐ v1! v2! v3!

U −−−−w2→ U −−−−w3→ ··· ⏐2 ⏐3 ⏐ ⏐ v2! v3!

U −−−−w3→ ··· ⏐4 ⏐ v3! ··· We see both horizontally as well as vertically ladders: the horizontal ladders yield U∞ and its endomorphism v∞; the vertical ladders yield U∞ with an endomorphism w∞. (4.3) Examples. First, let us show that the ordinary Prüfer groups (as considered in abelian group theory) are obtained in this way. Let R = Z be the ring of integers. Module homomorphisms Z → Z are given by the multiplication with some integer n,thuswedenotesuchamapjustbyn.LetU0 = U1 = Z and w0 =2, v0 = n. If n is odd, then the Prüfer module U∞/U0 is just the Prüfer group for the Z 1 Q 1 prime 2(andU∞(2,n)= [ 2 ] is the subring of generated by 2 ). Note that if n is even, then the Prüfer module U∞/U0 is an elementary abelian 2-group.

Second, let R = K(2) be the Kronecker algebra over some field k.LetU0 be simple projective, U1 indecomposable projective of length 3 and w0 : U0 → U1 a non-zero map with cokernel H (one of the indecomposable modules of length 2). For any map v0 : U0 → U1, we obtain a Prüfer module M = U∞/U0.Incase v0 ∈/ kw0, this module M is indecomposable (and it is the Prüfermodule H[∞]as considered in [R2]), otherwise M it is a direct sum of copies of H.

(4.4) Lemma. The modules U∞ as well as U∞/U0 are generated by U1. Proof. We only have to consider U∞ = i Ui. The pushout construction shows that for i ≥ 2, the module Ui is a factor module of Ui−1 ⊕ Ui−1,thusby induction Ui is generated by U1.

(4.5). A self-extension 0 → W → W [2] → W → 0 is called a ladder extension provided there is a commutative diagram with exact rows w q 0 −−−−→ U −−−−0→ U −−−−→ W −−−−→ 0 ⏐0 ⏐1 " ⏐ ⏐ " α! ! "

0 −−−−→ W −−−−→ W [2] −−−−→ W −−−−→ 0

THE RELEVANCE AND THE UBIQUITY OF PRUFER¨ MODULES 1719

such that α = qv0 for some v0 : U0 → U1. In this case, the given self-extension is the W [2] part of the Prüfer module W [∞] which is obtained from the maps w0,v0 using the ladder construction. One should note that not every self-extension of a module is a ladder extension (for example, if S is a simple R-module, where R is artinian, then no non-trivial self-extension of S is a ladder extension). On the other hand, for R hereditary, every self-extension is a ladder extension [R6]. (4.5). Assume that Λ is a finite-dimensional hereditary k-algebra. (For example, Λ may be the path algebra kQ of a finite quiver Q without oriented cycles.) Recall that the Euler characteristic (−1)i dim Exti(M,M) i≥0 yields a quadratic form q on the Grothendieck group K0(Λ) and q is positive definite if and only if Λ is representation-finite. (In the quiver case, this means that Q is the disjoint union of quivers of Dynkin type An,Dn,E6,E7,E8.) We see: Any Λ-module M with End(M) a division ring and q([M]) ≤ 0 is the basis of an indecomposable Prüfermodule. The Prüfer module is unique if and only if q([M]) = 0.

5. The search for pairs of embeddings

Let Λ be a representation-infinite artin algebra. The aim is to find pairs of embeddings w0,v0 : U0 → U1 such that the corresponding Prüfer module U∞/U0 is not of finite type. (5.1). For dealing with Prüfer modules obtained using the ladder construction, it seems to be of interest to relate the finite type properties of U∞ and U∞/U0.

Lemma. U∞/U0 is of finite type iff U∞ is of finite type. Proof. First, assume that U∞ is of finite type, say U∞ = i∈I Mi with all Mi indecomposable of finite length, and with only finitely many isomorphism classes ⊆ of modules involved. Now U0 i∈I Mi = M with I a finite subset of I.Then U∞/U0 = M /U0 ⊕ Mi, i∈I\I is a direct sum of indecomposable modules of finite length (one has to decompose M /U0) and only finitely many isomorphism classes are involved. The converse follows from Roiter’s extension argument, see for example [R5].

(5.2) Proposition. Assume that Λ is tame hereditary (or tame concealed). Let M,N be Λ-modules such that N is of ﬁnite length and generates M.IfM is not of ﬁnite type, then N has a direct summand which is sincere and preprojective.

Note that we cannot claim that N has an indecomposable direct summand which is sincere and preprojective. A typical example will be N = ΛΛ, this module generates all the Λ-modules, but usually has no indecomposable sincere direct summand.

10172 CLAUS MICHAEL RINGEL

Proof of Proposition. Let k be the center of Λ, this is a field and Λ is a finite-dimensional k-algebra. We will assume that Λ is hereditary (the case when Λ is concealed requires only few modifications). Let N = P ⊕ R ⊕ Q with P preprojective, R regular and Q preinjective. Assume P is not sincere, thus there is a simple Λ-module S which does not occur as a composition factor of P . We can order the indecomposable preprojective modules P1,P2,... and the indecomposable preinjective modules ...,Q2,Q1 such that Hom(Pi,Pj)=0for i>jand Hom(Qi,Qj)=0forinare sincere. Note that there is a bound b such that dimk Hom(R, Qi) ≤ b for all i.Namely, Q(i)=τ tI(y)forsomet ≥ 0andsomevertexy in the quiver of Λ. Since R is regular, it is τ-periodic with period at most 6, thus dimk Hom(R, Qi) is bounded by t the maximum of the numbers dimk Hom(R, τ I(y)) with 0 ≤ t ≤ 5andy avertex of the quiver. This implies the following: If Qi is generated by R, then there is a surjective b map R → Qi, and therefore the multiplicity [Qi : S]ofS as a composition factor of Qi is bounded by b[R : S]. There are only finitely many Qi with [Qi : S] ≤ b[R : S], thus there is some m such that [Qi : S] >b[R : S] for all i>m. But this implies that a module Qi with i>mcannot be generated by P ⊕ R (the trace of R in Qi is a submodule with at most b[R : S] composition factors S and the trace of P in Qi does not provide any such composition factor). We can assume in addition that m is chosen in such a way that all the indecomposable direct summands of Q are of the form Qi with i ≤ m.Thenwesee that the modules Qi with i>mcannot be generated by N = P ⊕ R ⊕ Q. Now let M be a (not necessarily finitely generated) module which is generated by N. We want to show that M is of finite type. According to [R2], we can write M = M1 ⊕ M2 ⊕ M3 where M1 is a direct sum of modules of the form Pi with 1 ≤ i ≤ n,whereM3 is a direct sum of modules of the form Qi with 1 ≤ i ≤ m, and where M2 has no direct summand of the form Pi with 1 ≤ i ≤ n,orQi with 1 ≤ i ≤ m.WithM also M2 is generated by N, and we want to see that M2 is of finite type (then also M is of finite type). Thus we see that we can assume that M = M2, this means that we consider a module generated by N which has no direct summand of the form Pi with 1 ≤ i ≤ n, or Qi with 1 ≤ i ≤ m. First of all, M cannot have any indecomposable preprojective direct summand M .Namely,Hom(R⊕Q, M )=0,thusM would be generated by P , but P is not sincere, whereas M is sincere. Second, we note that M cannot have any indecomposable preinjective direct summand. Namely, it would be generated by N, but an indecomposable preinjective module Qi which is generated by N satisfies i ≤ m. This means that M is regular (as defined in [R2]). Also, we see that Hom(Q, M)= 0, thus M is generated by P ⊕ R.LetM be the trace of R in M.SinceR is a regular module of finite length, it follows that M is regular and of finite type (it is a direct sum of copies of the regular factor modules of R). Now M/M is generated by P , thus it is not sincere and therefore of finite type. Write M/M as a direct sum of indecomposable modules, and collect these modules according to the property of being preprojective, regular or preinjective. Thus M/M = P ⊕ R ⊕ Q,whereP is a direct sum of modules of finite length modules which are preprojective, R is a direct sum of modules of finite length which are regular, and Q is a direct sum of modules of finite length which are preinjective.

THE RELEVANCE AND THE UBIQUITY OF PRUFER¨ MODULES 17311

Now P = 0, since otherwise M would have a proper factor module which is preprojective (and therefore a direct summand). Also, Q = 0, since otherwise M cannot be regular. This shows that M is an extension of M by M/M = R. As we have seen, M is a direct sum of copies of the regular factor modules of R,whereasR is a direct sum of ﬁnite length modules which are regular and do not contain the composition factor S. But this implies that M is regular and is of ﬁnite type.

6. Take-oﬀ subcategories

Let Λ be a representation-infinite artin algebra and mod Λ the category of finitely generated Λ-modules. A full subcategory C of mod Λ is said to be a take-off subcategory, provided the following conditions are satisfied: (1) C is closed under direct sums and under submodules. (2) C contains infinitely many isomorphism classes of indecomposable modules. (3) No proper subcategory of C satisfies (1) and (2). (6.1) Theorem. Any subcategory satisfying (1) and (2) contains a take-off subcategory.

In particular, this means that the module category of any representation-infinite artin algebra has at least one take-off subcategory: take-off subcategories always do exist! (6.2) Examples. If Λ is a connected hereditary algebra which is representation- infinite, then the preprojective modules form a take-off subcategory. In general, there may be several take-off subcategories: For example, if Λ has several minimal representation-infinite factor algebras, then any such factor algebra yields a take-off subcategory of mod Λ. Remark. Observe that the existence of take-off subcategories is in sharp contrast to the usual characterization of “infinity” (a set is infinite iff it contains proper subsets of the same cardinality)! (6.3) Properties of a take-off subcategory C. Let C be a take-off subcategory of mod Λ. (1) For any d, there are only finitely many isomorphism classes of modules of length d which belong to C. Thus: C contains indecomposable modules of arbitrarily large finite length.

Let C be the class of all Λ-modules M such that any finitely generated submodule of M belongs to C. (2) There are indecomposable modules M in C of infinite length. (3) If M is an indecomposable module M in C of infinite length, then any indecomposable module N in C embeds into M — even a countable direct sum N (N) embeds into M.

12174 CLAUS MICHAEL RINGEL

(6.4). We have seen that mod Λ contains take-off subcategories C, such a subcategory C contains indecomposable modules M of arbitrarily large finite length, and thus indecomposable modules with arbitrarily large socle. Conjecture. Let U be an indecomposable Λ-module belonging to a take-off subcategory. If there is a simple module S such that S7 embeds into U, then there are two embeddings w0,v0 : S → U such that the corresponding Prüfer module is not of finite type.

Remark. The bound 7 cannot be lowered, as the path algebra kQ of the E8-quiver Q with subspace orientation shows. Note that kQ is a tame hereditary algebra, and for a tame hereditary algebra, the preprojective modules form the unique take-off subcategory. Now the indecomposable representation U with dimension vector 54320 642 3 is preprojective and S6 embeds into U,whereS is the simple projective kQ- module. Note that U is not faithful, thus the Prüfer modules constructed by pairs w0,v0 : S → U are also not faithful. But all non-faithful kQ-modules are of finite type.

References [A] Atiyah, M., Vector bundles over an elliptic curve, Proceedings of the London Mathematical Society 7 (1957), 414–452. [Ba] Bautista, R., On algebras of strongly unbounded representation type, Comment. Math. Helv. 60 (1985), 392–399. [Bo] Bongartz, K., Indecomposables are standard, Comment. Math. Helv. 60 (1985), 400–410. [CB1] Crawley-Boevey, W.W., On tame algebras and bocses, Proc. London Math. Soc.(3) 56 (1988), 451–483. [CB2] Crawley-Boevey, W.W., Modules of finite length over their endomorphism ring,Represen- tations of algebras and related topics (ed: Brenner, Tachikawa) London Math. Soc. Lec. Note Series, vol. 168, 1992, pp. 127–184. [K] Krause, H., Generic modules over artin algebras, Proc. London Math. Soc. 76 (1998), 276–306. [R1] Ringel, C.M., Representations of k-species and bimodules, J.Algebra 41 (1976), 269–302. [R2] Ringel, C.M., Infinite dimensional representations of finite dimensional hereditary algebras, Symposia Math. 23, 321–412. [R3] Ringel, C.M., A construction of endofinite modules, Advances in Algebra and Model The- ory, (ed. M. Droste, R. Göbel), Gordon-Breach, London, 1997, pp. 387–399. [R4] Ringel, C.M., The Gabriel-Roiter measure, Bull. Sci. math. 129 (2005), 726–748. [R5] Ringel, C.M., Foundation of the Representation Theory of Artin Algebras, Using the Gabriel-Roiter Measure., Proceedings ICRA 11. Queretaro 2004. Contemporary Math., vol. 406., Amer.Math.Soc., 2006, pp. 105–135. [R6] Ringel, C.M., The ladder construction of Prüfer modules, Revista de la Union Matematica Argentina 48-2 (2007), 47–65. [R7] Ringel, C.M., The first Brauer-Thrall conjecture, Models, Modules and Abelian Groups, In Memory of A.L.S. Corner. Walter de Gruyter, Berlin 2008 (ed: R. Göbel, B. Goldsmith) (To appear). [R8] Ringel, C.M., Prüfer modules of finite type., Selected Topics in Representation Theory. Lecture Notes. Bielefeld, 2006. [R9] Ringel, C.M., Take-off subcategories., (In preparation). A preliminary version is in: Selected Topics in Representation Theory. Lecture Notes. Bielefeld, 2006. [Ro] Roiter, A.V., Unboundedness of the dimension of the indecomposable representations of an algebra which has infinitely many indecomposable representations, Izv. Akad. Nauk SSSR. Ser. Mat. 32 (1968), 1275–1282.

THE RELEVANCE AND THE UBIQUITY OF PRUFER¨ MODULES 17513

[Z1] Zwara, G., A degeneration-like order for modules, Arch. Math. 71 (1998), 437–444. [Z2] Zwara, G., Degenerations of ﬁnite-dimensional modules are given by extensions,Compo- sitio Mathematica 121 (2001), 205–218.

Fakultat¨ Mathematik, Universitat¨ Bielefeld, POBox 100131, D33501, Bielefeld, Germany E-mail address: [email protected]

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Contemporary Mathematics Volume 478, 2009

Quivers and the Euclidean group

Alistair Savage

Abstract. We show that the category of representations of the Euclidean group of orientation-preserving isometries of two-dimensional Euclidean space is equivalent to the category of representations of the preprojective algebra of type A∞. We also consider the moduli space of representations of the Euclidean group along with a set of generators. We show that these moduli spaces are quiver varieties of the type considered by Nakajima. Using these identiﬁcations, we prove various results about the representation theory of the Euclidean group. In particular, we prove it is of wild representation type but that if we impose certain restrictions on weight decompositions, we obtain only a ﬁnite number of indecomposable representations.

1. Introduction The Euclidean group E(n)=Rn SO(n) is the group of orientation-preserving isometries of n-dimensional Euclidean space. The study of these objects, at least for n =2, 3, predates even the concept of a group. In this paper we will focus on the Euclidean group E(2). Even in this case, much is still unknown about the representation theory. Since E(2) is solvable, all its finite-dimensional irreducible representations are one-dimensional. The finite-dimensional unitary representations, which are of interest in quantum mechanics, are completely reducible and thus isomorphic to direct sums of such one-dimensional representations. The infinite-dimensional unitary irreducible representations have received considerable attention (see [1, 3, 4]). There also exist finite-dimensional nonunitary indecomposable representations (which are not irreducible) and much less is known about these. However, they play an important role in mathematical physics and the representation theory of the Poincaré group. The Poincarégroup is the group of isometries of Minkowski spacetime. It is the semidirect product of the translations of R3 and the Lorentz transformations. In 1939, Wigner [25] studied the subgroups of the Lorentz group leaving invariant the four-momentum of a given free particle. The maximal such subgroup is called the little group. The little group governs the internal space-time symmetries of the relativistic particle in question. The little groups of massive particles are locally

2000 Mathematics Subject Classiﬁcation. Primary: 17B10, 22E47; Secondary: 22E43. This research was supported in part by the Natural Sciences and Engineering Research Coun- cil (NSERC) of Canada.

c 0000c 2009 (copyright Alistair holder) Savage 1771

2178 ALISTAIR SAVAGE isomorphic to the group O(3) while the little groups of massless particles are locally isomorphic to E(2). That is, their Lie algebras are isomorphic to those of O(3) and E(2) respectively. We refer the reader to [2, 5, 16, 20] for further details. The group E(2) also appears in the Chern-Simons formulation of Einstein gravity in 2 + 1 dimensions. In the case when the space-time has Euclidean signature and the cosmological constant vanishes, the phase space of gravity is the moduli space of flat E(2)-connections. In the current paper, we relate the representation theory of the Euclidean group E(2) to the representation theory of preprojective algebras of quivers of type A∞. In fact, we show that the categories of representations of the two are equivalent. To prove this, we introduce a modified enveloping algebra of the Lie algebra of E(2) and show that it is isomorphic to the preprojective algebra of type A∞.Furthermore,we consider the moduli space of representations of E(2) along with a set of generators. We show that these moduli spaces are quiver varieties of the type considered by Nakajima in [21, 22]. These identifications allow us to draw on known results about preprojective algebras and quiver varieties to prove various statements about representations of E(2). In particular, we show that E(2) is of wild representation type but that if we impose certain restrictions on the weight decomposition of a representation, we obtain only a finite number of indecomposable representations. We conclude with some potential directions for future investigation.

2. The Euclidean algebra Let E(2) = R2 SO(2) be the Euclidean group of motions in the plane and let e(2) be the complexiﬁcation of its Lie algebra. We call e(2) the (three-dimensional) Euclidean algebra. It has basis {p+,p−,l} and commutation relations

(2.1) [p+,p−]=0, [l, p±]=±p±. Since SO(2) is compact, the category of ﬁnite-dimensional E(2)-modules is equivalent to the category of ﬁnite-dimensional e(2)-modules in which l acts semisimply with integer eigenvalues. Will will use the term e(2)-module to refer only to such modules. For k ∈ Z, we shall write Vk to indicate the eigenspace of l with eigenvalue k (the k-weight space). Thus, for an e(2)-module V ,wehavetheweightspace decomposition V = Vk,Vk = {v ∈ V | l · v = kv},k∈ Z, k and

p+Vk ⊆ Vk+1,p−Vk ⊆ Vk−1. We may form the tensor product of any representation V with the character χn for n ∈ Z.Hereχn is the one-dimensional module C on which p± act by zero and l acts by multiplication by n.ThenaweightspaceVk of weight k becomes a weight space Vk ⊗ χn of weight k + n. In this way, we may “shift weights” as we please. k Z For k ∈ Z,lete be the element of (Z≥0) with kth component equal to one and all others equal to zero. For an e(2)-module V we deﬁne k dim V = (dim Vk)e . k∈Z

QUIVERS AND THE EUCLIDEAN GROUP 1793

Let U be the universal enveloping algebra of e(2) and let U +, U − and U 0 be the subalgebras generated by p+, p− and l respectively. Then we have the triangular decomposition ∼ + 0 − U = U ⊗ U ⊗ U (as vector spaces). Note that the category of representations of U is equivalent to the category of representations of e(2). In [18, Chapter 23], Lusztig introduced the modiﬁed quantized enveloping algebra of a Kac-Moody algebra. Following this idea, we introduce the modiﬁed enveloping algebra U˜ by replacing U 0 with a sum of 1-dimensional algebras + − U˜ = U ⊗ Cak ⊗ U . k∈Z Multiplication is given by

akal = δklak,

p+ak = ak+1p+,p−ak = ak−1p−,

p+p−ak = p−p+ak.

One can think of ak as projection onto the kth weight space. Note that U˜ is an algebra without unit. We say a U˜-module V is unital if ∈ ∈ (1) for any v V ,wehaveakv = 0 for all but ﬁnitely many k Z,and ∈ (2) for any v V ,wehave k∈Z akv = v. A unital U˜-module can be thought of as a U-module with weight decomposition. Thus we have the following proposition.

Proposition 2.1. The category of unital U˜-modules is equivalent to the category of U-modules and hence the category of e(2)-modules.

3. Preprojective algebras In this section, we review some basic results about preprojective algebras. The reader is referred to [12] for further details. A quiver is a 4-tuple (I,H,out, in) where I and H are disjoint sets and out and in are functions from H to I.ThesetsI and H are called the vertex set and arrow set respectively. We think of an element h ∈ H as an arrow from the vertex out(h) to the vertex in(h).

out(• h)in(h •/h)

An arrow h ∈ H is called a loop if out(h)=in(h). A quiver is said to be finite if both its vertex and arrow sets are finite. We shall be especially concerned with the following quivers. For a, b ∈ Z with a ≤ b,letQa,b be the quiver with vertex set I = {k ∈ Z | a ≤ k ≤ b} and arrows H = {hi | a ≤ i ≤ b − 1} with out(hi)=i and in(hi)=i +1.WesaythatQa,b is aquiveroftypeAb−a+1 since this is the type of its underlying graph. The quiver Q∞ has vertex set I = Z and arrows H = {hi | i ∈ Z} with out(hi)=i and in(hi)=i + 1. We say that the quiver Q∞ is of type A∞. Note that the quivers Qa,b are finite while the quiver Q∞ is not.

4180 ALISTAIR SAVAGE

Let Q =(I,H,out, in) be a quiver without loops and let Q∗ =(I,H∗, out∗, in∗) be the double quiver of Q. By definition, H∗ = {h | h ∈ H}∪{h¯ | h ∈ H}, out∗(h)=out(h), in∗(h)=in(h), out∗(h¯)=in(h), in∗(h¯)=out(h). ∗ ∗ ∗ From now on, we will write in and out for in and out respectively. Since in |H =in ∗ and out |H = out, this should cause no confusion. A path in a quiver Q is a sequence p = hnhn−1 ···h1 of arrows such that in(hi)=out(hi+1)for1≤ i ≤ n − 1. We call the integer n the length of p and define out(p)=out(h1)andin(p)=in(hn). The path algebra CQ is the algebra spanned by the paths in Q with multiplication given by pp if in(p)=out(p) p · p = 0otherwise and extended by linearity. We note that there is a trivial path i starting and ∈ C ending at i for each i I. The path algebra Q has a unit (namely i∈I i)ifand only if the quiver Q is finite. k ∈ C A relation in a quiver Q is a sum of the form j=1 ajpj, aj , pj a path for 1 ≤ j ≤ k.Fori ∈ I let ri = hh¯ − hh¯ h∈H, out(h)=i h∈H, in(h)=i be the Gelfand-Ponomarev relation in Q∗ associated to i.Thepreprojective algebra P (Q) corresponding to Q is defined to be P (Q)=CQ∗/J where J is the two-sided ideal generated by the relations ri for i ∈ I. Let V(I) denote the category of finite-dimensional I-graded vector spaces with morphisms being linear maps respecting the grading. For V ∈V(I), we let dim V = ∗ (dim Vi)i∈I be the I-graded dimension of V .Arepresentation of the quiver Q is ∗ an element V ∈V(I) along with a linear map xh : Vout(h) → Vin(h) for each h ∈ H . We let ∗ rep(Q ,V)= HomC(Vout(h),Vin(h)) h∈H∗ be the affine variety consisting of representations of Q∗ with underlying graded vector space V . A representation of a quiver can be naturally interpreted as a ∗ ∗ CQ -module structure on V .Forapathp = hnhn−1 ...h1 in Q ,welet ··· xp = xhn xhn−1 xh1 . ∈ ∗ k We say a representation x rep(Q ,V) satisfies the relation j=1 ajpj,if k

ajxpj =0. j=1 If R is a set of relations, we denote by rep(Q∗,R,V)thesetofallrepresentationsin rep(Q∗,V) satisfying all relations in R. This is a closed subvariety of rep(Q∗,V). Every element of rep(Q∗,J,V) can be naturally interpreted as a P (Q)-module structure on V andsowealsowrite mod(P (Q),V)=rep(Q∗,J,V)

QUIVERS AND THE EUCLIDEAN GROUP 1815 for the aﬃne variety of P (Q)-modules with underlying vector space V . The algebraic group GV = i∈I GL(Vi)actsonmod(P (Q),V)by −1 · · ∗ ∗ g x =(gi)i∈I (xh)h∈H =(gin(h)xhgout(h))h∈H . Two P (Q)-modules are isomorphic if and only if they lie in the same orbit. For a I dimension vector v ∈ (Z≥ ) ,let 0 v vi v V = C , mod(P (Q), v)=mod(P (Q),V ),Gv = GV v . i∈I ∼ Then we have that mod(P (Q),V) = mod(P (Q), dim V ) for all V ∈V(I). There- fore, we will blur the distinction between mod(P (Q),V)andmod(P (Q), dim V ). We say an element x ∈ mod(P (Q),V)isnilpotent if there exists an N ∈ Z>0 such that for any path p of length greater than N,wehavexp =0.Denotethe closed subset of nilpotent elements of mod(P (Q),V)byΛV,Q and let Λv,Q =ΛV v,Q. The varieties ΛV,Q are called nilpotent varieties or Lusztig quiver varieties.Lusztig [17, Theorem 12.3] has shown that the ΛV,Q have pure dimesion dim(rep(Q, V )). Proposition 3.1. For a quiver Q, the following are equivalent: (1) P (Q) is ﬁnite-dimensional, (2) ΛV,Q =mod(P (Q),V) for all V ∈V(I), (3) Q is a Dynkin quiver (i.e. its underlying graph is of ADE type). Proof. The equivalence of (1) and (3) is well-known (see for example [23]). That (2) implies (3) was proven by Crawley-Boevey [6] and the converse was proven by Lusztig [17, Proposition 14.2].

Thus, for a Dynkin quiver Q, nilpotency holds automatically and ΛV,Q is just the variety of representations of the preprojective algebra P (Q) with underlying vector space V . The representation type of the preprojective algebras is known. Proposition 3.2 ([7, 13]). Let Q be a ﬁnite quiver. Then the following hold: (1) P (Q) is of ﬁnite representation type if and only if Q is of Dynkin type An, n ≤ 4,and (2) P (Q) is of tame representation type if and only if Q is of Dynkin type A5 or D4.

Thus P (Q) is of wild representation type if Q is not of Dynkin type An, n ≤ 5,or D4.

In the sequel, we will refer to the preprojective algebra P (Q∞). While Q∞ is not a finite quiver, any finite-dimensional representation is supported on finitely many vertices and thus is a representation of a quiver of type An for sufficiently large n. Thus we deduce the following.

Corollary 3.3. All ﬁnite-dimensional representations of P (Q∞) are nilpotent and P (Q∞) is of wild representation type.

For a ﬁnite quiver Q,letgQ denote the Kac-Moody algebra whose Dynkin graph − is the underlying graph of Q and let U(gQ) denote the lower half of its universal enveloping algebra. It turns out that Lusztig quiver varieties are intimately related − to U(gQ) .Namely,Lusztig[17] has shown that there is a space of constructible I functions on the varieties Λv,Q, v ∈ (Z≥0) , and a natural convolution product such

6182 ALISTAIR SAVAGE

− that this space of functions is isomorphic as an algebra to U(gQ). The functions − on an individual Λv,Q correspond to the weight space of weight i∈I viαi,where the αi are the simple roots of gQ. Furthermore, the irreducible components of Λv,Q are in one-to-one correspondence with a basis of this weight space. Under this correspondence, each irreducible component is associated to the unique function equal to one on an open dense subset of that component and equal to zero on an open dense subset of all other components. The set of these functions yields − a basis of U(gQ) , called the semicanonical basis, with very nice integrality and positivity properties (see [19]). If instead of constructible functions one works with the Grothendieck group of a certain class of perverse sheaves, a similar construction − yields a realization of (the lower half of) the quantum group Uq(gQ) and the canonical basis (see [17]).

4. Representations of the Euclidean algebra and preprojective algebras In this section we examine the close relationship between representations of the Euclidean algebra e(2) and the preprojective algebras of types An and A∞. Theorem 4.1. The modiﬁed universal enveloping algebra U˜ is isomorphic to the preprojective algebra P (Q∞). ∗ Proof. Deﬁne a map ψ : CQ∞ → U˜ by

ψ(i)=ai,ψ(hi)=p+ai = ai+1p+,ψ(h¯i)=aip− = p−ai+1,i∈ I. It is easily veriﬁed that this extends to a surjective map of algebras with kernel J and thus the result follows.

Let Mod e(2) be the category of e(2)-modules. For a ≤ b,letModa,b e(2) be the full subcategory consisting of representations V such that Vk =0forkb.Forv ( ≥0) , we also define Moda,b e(2) and Mod e(2) to be the full subcategories of Moda,b e(2) and Mod e(2) consisting of representations V such that dim V = v. Let Mod P (Q) be the category of finite-dimensional P (Q)-modules and for I v v ∈ (Z≥0) ,letMod P (Q) be the full subcategory consisting of modules of graded dimension v. Corollary 4.2. We have the following equivalences of categories. v ∼ v ∼ (1) Mod e(2) = Mod P (Q∞), Mod e(2) = Mod P (Q∞), v ∼ v ∼ (2) Moda,b e(2) = Mod P (Qa,b), Moda,b e(2) = Mod P (Qa,b). Proof. Statement (1) follows from Theorem 4.1 and Proposition 2.1. State- ment (2) is obtained by restricting weights to lie between a and b. Theorem 4.3. The following statements hold. (1) The Euclidean algebra e(2), and hence the Euclidean group E(2), have wild representation type, and (2) for a, b ∈ Z with 0 ≤ b − a ≤ 3, there are a finite number of isomorphism classes of indecomposable e(2)-modules V whose weights lie between a and b; that is, such that Vk =0for kb. Proof. These statements follow immediately from Corollary 4.2, Proposi- tion 3.2 and Corollary 3.3.

QUIVERS AND THE EUCLIDEAN GROUP 1837

Corollary 4.4. Let A be a finite subset of Z with the property that A does not contain any five consecutive integers. Then there are a finite number of isomorphism classes of indecomposable e(2)-modules V with the property that Vk =0 if k ∈ A.

Proof. Partition A into subsets A1,...,An such that Aj = {aj,aj +1,...,aj + mj} and |a − b| > 1fora ∈ Ai, b ∈ Aj with i = j. By hypothesis, we have mj ≤ 3 ≤ ≤ ∈ for 1 j n.LetV be an e(2)-module such that Vk =0ifk A.ThenV n j j decomposes as a direct sum of modules V = j=1 V where Vk =0ifkaj + mk.Thus,ifV is indecomposable, we must have V = V for some j.But there are a ﬁnite number of such V j, up to isomorphism, by Theorem 4.3. The result follows.

For a ∈ Z,wesayane(2)-module V has lowest weight a if Va =0and Vk =0 for k

Proof. By tensoring with the character χ−a we may assume that a =0. In order for an e(2)-module to be indecomposable, its set of weights must be a set of consecutive integers. By Corollary 4.4, it suffices to consider the modules of dimension 5. Again, by Corollary 4.4, we need only consider the case when dim Vk =1for0≤ k ≤ 4. We consider the equivalent problem of classifying the ∈ C ≤ ≤ GV -orbits of indecomposable elements x ΛV,Q0,4 where Vk = for 0 k 4. ∗ Fixing the standard basis in each Vk, we can view the maps xh, h ∈ H ,ascomplex r x¯ x numbers. Considering the Gelfand-Ponomarev relation 0,weseethat h0 h0 =0. r x x Then the relation 1 implies h¯1 h1 = 0. Continuing in this manner, we see that x x ≤ i ≤ x x ≤ i ≤ x h¯i hi =0for0 3. Thus hi =0or h¯i =0for0 3. Since x x i is indecomposable, we cannot have both hi =0and h¯i =0forany .Thus, 4 there are precisely 2 =16GV -orbits in ΛV,Q0,4 . Representatives for these orbits x x correspond to setting one of hi or h¯i equal to one and the other to zero for each 0 ≤ i ≤ 3. We note that Douglas [9] has shown that there are finitely many indecomposable e(2)-modules (up to isomorphism) of dimensions five and six. The proof of Corollary 4.5 shows how Corollary 4.4 can simply such proofs. We also point out that the graphs appearing in [9] roughly correspond, under the equivalence of categories in Corollary 4.2, to the diagrams appearing in the enumeration of irreducible components of quiver varieties given in [11]. Remark 4.6. As noted at the end of Section 3, the Lusztig quiver varieties Λv,Q are closely related to the Kac-Moody algebra gQ. Thus, the results of this section show that there is a relationship between the representation theory of the Euclidean group E(2) and the Lie algebra sl∞ (or the Lie groups SL(n)).

5. Naka jima quiver varieties In this section we brieﬂy review the quiver varieties introduced by Nakajima [21, 22]. We restrict our attention to the case when the quiver involved is of type A.

8184 ALISTAIR SAVAGE

Let Q be the quiver Q∞ or Qa,b for some a ≤ b.ForV,W ∈V(I) deﬁne LQ(V,W)=ΛV,Q ⊕ HomC(Wi,Vi). i∈I

∗ ∈ We denote points of LQ(V,W)by(x, s)wherex =(xh)h∈H ΛV,Q and s = ∈ (si)i∈I i∈I HomC(Wi,Vi). We say an I-graded subspace S of V is x-invariant ∗ if xh(Sout(h)) ⊆ Sin(h) for all h ∈ H .Wesayapoint(x, s) ∈ LQ(V,W)isstable if the following property holds: If S is an I-graded x-invariant subspace of V st containing im s,thenS = V .WedenotebyLQ(V,W) the set of stable points. The group GV acts on LQ(V,W)by −1 · · ∗ ∗ g (x, s)=(gi)i∈I ((xh)h∈H , (si)i∈I )=((gin(h)xhgout(h))h∈H , (gisi)i∈I ).

The action of GV preserves the stability condition and the stabilizer in GV of a stable point is trivial. We form the quotient st LQ(V,W)=LQ(V,W) /GV . I The LQ(V,W) are called Nakajima quiver varieties.Forv, w ∈ (Z≥0) ,weset v w st v w st v w LQ(v, w)=LQ(V ,V ),LQ(v, w) = LQ(V ,V ) , LQ(v, w)=LQ(V ,V ). We then have ∼ st ∼ st LQ(V,W) = LQ(dim V,dim W ),LQ(V,W) = LQ(dim V,dim W ) , ∼ LQ(V,W) = LQ(dim V,dim W ), and so we often blur the distinction between these pairs of isomorphic varieties. Let Irr ΛV,Q (resp. Irr LQ(V,W)) denote the set of irreducible components of Λ (resp. L (V,W)). Then Irr L (V,W) can be identiﬁed with V,Q Q⎧ & Q ⎫ & ⎨ & st ⎬ & Y ∈ Irr ΛV,Q & Y ⊕ HomC(Wi,Vi) = ∅ . ⎩ & ⎭ i∈I

Speciﬁcally, the irreducible components of LQ(V,W) are precisely those ⎛ ⎞ st ⎝ ⎠ Y ⊕ HomC(Wi,Vi) /GV i∈I which are nonempty. Proposition 5.1 ([22, Corollary 3.12]). The dimension of the Nakajima quiver varieties associated to the quiver Q∞ are given by L − 2 dimC Q∞ (v, w)= (viwi vi + vivi+1). i∈Z In a manner analogous to the way in which Lusztig quiver varieties are re- − lated to U(gQ) (see Section 3), Nakajima quiver varieties are closely related to the representation theory of gQ.Inparticular,Nakajima[22]hasshownthat L v Htop( Q(v, w)) is isomorphic to the irreducible integrable highest-weight representation of gQ of highest weight i∈I wiωi where the ωi are the fundamental weights of gQ.HereHtop is top-dimensional Borel-Moore homology. The action of the Chevalley generators of gQ are given by certain convolution operations. The vector space Htop(LQ(v, w)) corresponds to the weight space of weight

QUIVERS AND THE EUCLIDEAN GROUP 1859 − i∈I (wiωi viαi). In [21], Nakajima gave a similar realization of these representations using a space of constructible functions on the quiver varieties rather than their homology. The irreducible components of Nakajima quiver varieties enumerate a natural basis in the representations of gQ. These bases are given by the fundamental classes of the irreducible components in the Borel-Moore homology construction and by functions equal to one on an open dense subset of an irreducible component (and equal to zero on an open dense subset of all other irreducible components) in the constructible function realization.

6. Moduli spaces of representations of the Euclidean algebra Given that e(2) has wild representation type, it is prudent to restrict one’s attention to certain subclasses of modules and to attempt a classification of the modules belonging to these classes. One possible approach is to impose a restriction on the number of generators of a representation (see [8, 9] for some results in this direction and [10] for other classes). In this section we will examine the relationship between moduli spaces of representations of the Euclidean algebra along with a set of generating vectors and Nakajima quiver varieties. Let V be a finite-dimensional e(2)-module. For u1,u2,...,un ∈ V ,wedenote by u1,...,un the submodule of V generated by {u1,...,un}. It is defined to be the smallest submodule of V containing all the ui. A element u ∈ V is called a weight vector if it lies in some weight space Vk of V . For a weight vector u,welet ∈ { } wt u = k where u Vk.Wesaythat u1,...,un is a set of generators ofV if each ∈ Z Z | | ui is a weight vector and u1,...,un = V .Forv ( ≥0) ,welet v = k∈Z vk. Z Definition 6.1. For v, w ∈ (Z≥0) ,letE(v, w) be the set of all j (V,(uk)k∈Z, 1≤j≤wk ) { j } where V is a finite-dimensional e(2)-module with dim V = v and uk k∈Z, 1≤j≤wk j j is a set of generators of V such that wt uk = k. We say that two elements (V,(uk)) ˜ j and (V,(˜uk)) of E(v, w) are equivalent if there exists a e(2)-module isomorphism → ˜ j j φ : V V such that φ(uk)=˜uk. We denote the set of equivalence classes by E(v, w). Theorem 6.2. There is a natural one-to-one correspondence between E(v, w) L and Q∞ (v, w). Proof. j ∈ Let (V,(uk)k∈Z, 1≤j≤wk ) E(v, w)andletV = Vk be the weight v space decomposition of V .ThusVk is isomorphic to C k and we identity the two j ∈ via this isomorphism. We then define a point ϕ(V,(uk)) = (x, s) LQ∞ (v, w)by setting

x p | ,x p−| ,i∈ Z, hi = + Vi h¯i = Vi+1 j j ∈ Z ≤ ≤ s(wk)=uk,k , 1 j wk, j w { } C k where wk 1≤j≤wk is the standard basis of and the map s is extended by linearity. It follows from the results of Section 4 that x ∈ ΛV v,Q and so (x, s) ∈ j LQ∞ (v, w). Furthermore, it follows from the fact that (uk) is a set of genera- → st tors, that (x, s) is a stable point. Thus ϕ : E(v, w) LQ∞ (v, w) .Itiseas- j ˜ j ily veriﬁed that two elements (V,(uk)) and (V,(˜uk)) are equivalent if and only j ˜ j if ϕ(V,(uk)) and ϕ(V,(˜uk)) lie in the same Gv-orbit. Thus ϕ induces a map

10186 ALISTAIR SAVAGE

v E →L ∼ C k ϕ : (v, w) Q∞ (v, w) which is independent of the isomorphism V = chosen in our construction. It is easily seen that ϕ is a bijection. As noted in Section 5, the irreducible components of Nakajima quiver varieties can be identified with the irreducible components of Lusztig quiver varieties that are not killed by the stability condition. In the language of e(2)-modules, passing from Lusztig quiver varieties to Nakajima quiver varieties amounts to imposing the condition that the module be generated by a set of |w| weight vectors with weights prescribed by w. A partition is a sequence of non-increasing natural numbers λ =(λ1,λ2,...,λl). The corresponding Young diagram is a collection of rows of square boxes which are left justified, with λi boxes in the ith row, 1 ≤ i ≤ l. We will identify a partition and its Young diagram and we denote by Y the set of all partitions (or Young diagrams). If b is a box in a Young diagram λ,wewritex ∈ λ and we denote the box in the ith column and jth row of λ by xi,j (if such a box exists). The residue of λ,a Z xi,j ∈ λ is defined to be res xi,j = i − j.Forλ ∈Yand a ∈ Z, define v ∈ (Z≥0) λ,a by setting vi+a to be the number of boxes in λ of residue i. Proposition 6.3. For λ ∈Y, there exists a unique e(2)-module V (up to isomorphism) with a single generator of weight a ∈ Z and dim V = vλ,a.Itis given by

V = SpanC{x | x ∈ λ}

l(xi,j)=(a +resxi,j )xi,j =(a + i − j)xi,j

p+(xi,j )=xi+1,j

p−(xi,j )=xi,j+1, where we set xi,j =0if there is no box of λ in the ith column and jth row. Z λ,a For v ∈ (Z≥0) such that v = v for all λ ∈Yand a ∈ Z, there are no e(2)-modules V with a single generator and dim V = v

Proof. By tensoring with an appropriate χn, we may assume that the generator of our module has weight zero. It is shown in [11, §5.1] that λ,0,λ∈Y, L 0 1ifv = v dimC Q∞ (v, w )= , 0otherwise 0 0 where w0 =1andwi =0fori = 0 (the ﬁrst case can be deduced from the dimension formula in Proposition 5.1). It then follows from Theorem 6.2 that if V is an e(2)-module with a single generator v of weight zero, we must have dim V = vλ,0. Furthermore, up to isomorphism, there is only one such pair (V,v) and thus only one such module V . Thus e(2)-modules with a single generator of a ﬁxed weight are determined completely by the dimensions of their weight spaces. This was proven directly by Gruber and Henneberger in [14]. As in the proof of Corollary 4.5, we see that our knowledge of the precise relationship between quivers and the Euclidean algebra allows us to use known results about quivers and quiver varieties to simplify such proofs. Remark 6.4. As explained at the end of Section 5, the Nakajima quiver varieties LQ(v, w) are closely connected to the representation theory of gQ. Therefore,

QUIVERS AND THE EUCLIDEAN GROUP 18711 the relationship noted in Remark 4.6 between the representation theory of the Eu- clidean group and the Lie algebra sl∞ (or the Lie groups SL(n)) is emphasized further by the above results. Namely, the moduli space of representations of the Eu- clidean group along with a set of generators is closely related to the representation theory of sl∞ and the Lie groups SL(n). Remark 6.5. Although Theorem 4.3 tells us that the Euclidean group has wild representation type, the results of this section produce a method of approaching the unwieldy problem of classifying its representations. Namely, if we ﬁx the cardinality and weights of a generating set, the resulting moduli space of representations (along with a set of generators) is enumerated by a countable number of ﬁnite-dimensional varieties, one variety for the representations of each graded dimension.

7. Further directions The ideas presented in this paper open up some possible avenues of further investigation. We present here two of these. Consider the Euclidean algebra over a field k of characteristic p instead of over the complex numbers. This algebra is still spanned by {p+,p−,l} with commutation relations (2.1) but the weights of representations are elements of Z/pZ (if we restrict our attention to “integral” weights as usual) instead of Z. One can then show that this category of representations is equivalent to the category of representations of the preprojective algebra of the quiver of affine type Aˆp−1.Inthiscase, the representations with one generator are, in general, more complicated than in the complex case. We refer the reader to [11] for an analysis of the corresponding quiver varieties. There a graphical depiction of the irreducible components of these varieties is developed. These quiver varieties are related to moduli spaces of solutions of anti-self-dual Yang-Mills equations and Hilbert schemes of points in C2 and it would be interesting to further examine the relationship between these spaces and the Euclidean algebra. In [15]and[24], Kashiwara and Saito defined a crystal structure on the sets of irreducible components of Lusztig and Nakajima quiver varieties. Using this structure, each irreducible component can be identified with a sequence of crystal operators acting on the highest weight element of the crystal. Under the identification of quiver varieties with (moduli spaces of) e(2)-modules, these sequences correspond to the Jordan-Hölder decomposition of e(2)-modules. It could be fruitful to further examine the implications of this correspondence.

References [1] H. Ahmedov and I. H. Duru. Unitary representations of the two-dimensional Euclidean group in the Heisenberg algebra. J. Phys. A, 33(23):4277–4281, 2000. [2] A. O. Barut and R. Raczka. Theory of group representations and applications. World Scien- tific Publishing Co., Singapore, second edition, 1986. [3] K. Baumann. Vector and ray representations of the Euclidean group E(2). Rep. Math. Phys., 34(2):171–180, 1994. [4] A. M. Boyarski˘ı and T. V. Skrypnik. Singular orbits of a coadjoint representation of Euclidean groups. Uspekhi Mat. Nauk, 55(3(333)):169–170, 2000. [5] G. Cassinelli, G. Olivieri, P. Truini, and V. S. Varadarajan. On some nonunitary representations of the Poincaré group and their use for the construction of free quantum fields. J. Math. Phys., 30(11):2692–2707, 1989. [6] W. Crawley-Boevey. Geometry of the moment map for representations of quivers. Compositio Math., 126(3):257–293, 2001.

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[7] V. Dlab and C. M. Ringel. The module theoretical approach to quasi-hereditary algebras. In Representations of algebras and related topics (Kyoto, 1990), volume 168 of London Math. Soc. Lecture Note Ser., pages 200–224. Cambridge Univ. Press, Cambridge, 1992. [8] A. Douglas. A classification of the finite dimensional, indecomposable representations of the Euclidean algebra e(2) having two generators. PhD thesis, University of Toronto, 2006. [9] A. Douglas. Finite dimensional representations of the Euclidean algebra e(2) having two generators. J. Math. Phys., 47(5):053506, 14, 2006. [10] A. Douglas and A. Premat. A class of nonunitary, finite dimensional representations of the Euclidean algebra e(2). Comm. Algebra, 35(5):1433–1448, 2007. [11] I. B. Frenkel and A. Savage. Bases of representations of type A affine Lie algebras via quiver varieties and statistical mechanics. Int. Math. Res. Not., (28):1521–1547, 2003. [12] C. Geiss, B. Leclerc, and J. Schröer. Semicanonical bases and preprojective algebras. Ann. Sci. Ecole´ Norm. Sup. (4), 38(2):193–253, 2005. [13] C. Geiss and J. Schröer. Varieties of modules over tubular algebras. Colloq. Math., 95(2):163– 183, 2003. [14] B. Gruber and W. C. Henneberger. Representations of the Euclidean group in the plane. Nuovo Cimento B (11), 77(2):203–233, 1983. [15] M. Kashiwara and Y. Saito. Geometric construction of crystal bases. Duke Math. J., 89(1):9– 36, 1997. [16] Y. S. Kim and M. E. Noz. Theory and applications of the Poincarégroup. Fundamental Theories of Physics. D. Reidel Publishing Co., Dordrecht, 1986. [17] G. Lusztig. Quivers, perverse sheaves, and quantized enveloping algebras. J. Amer. Math. Soc., 4(2):365–421, 1991. [18] G. Lusztig. Introduction to quantum groups, volume 110 of Progress in Mathematics. Birkhäuser Boston Inc., Boston, MA, 1993. [19] G. Lusztig. Semicanonical bases arising from enveloping algebras. Adv. Math., 151(2):129– 139, 2000. [20] R. Mirman. Poincaré zero-mass representations. Internat. J. Modern Phys. A, 9(1):127–156, 1994. [21] H. Nakajima. Instantons on ALE spaces, quiver varieties, and Kac-Moody algebras. Duke Math. J., 76(2):365–416, 1994. [22] H. Nakajima. Quiver varieties and Kac-Moody algebras. Duke Math. J., 91(3):515–560, 1998. [23] I. Reiten. Dynkin diagrams and the representation theory of algebras. Notices Amer. Math. Soc., 44(5):546–556, 1997. [24] Y. Saito. Crystal bases and quiver varieties. Math. Ann., 324(4):675–688, 2002. [25] E. Wigner. On unitary representations of the inhomogeneous Lorentz group. Ann. of Math. (2), 40(1):149–204, 1939.

University of Ottawa, Ottawa, Ontario, Canada E-mail address: [email protected]

Contemporary Mathematics Volume 478, 2009

eu2-Lie admissible algebras and Steinberg unitary Lie algebras

Shikui Shang and Yun Gao

Abstract. In this paper, we will give a necessary and suﬃcient condition such that a k-algebra R becomes eu2-Lie admissible. Then, we will work out the second homology group of the Lie algebra eu2(R, −, γ).

Introduction

In this paper, we will study the eu2-Lie admissible algebras and Steinberg unitary Lie algebras stu2(R, −, γ). Steinberg unitary Lie algebras were introduced by Allison and Faulkner [AF] as a generalization of Steinberg Lie algebras (see [KL] and [F]). They were further studied in [G1] for the case R is associative and in [AG] for the case R is structurable. More precisely, let k be a field and R be an associative k-algebras with identity, equipped with an involution −(an anti-automorphism which squares to the identity). The elementary unitary Lie algebra eun(R, −, γ) is the subalgebra − −1 of the matrice Lie algebra gln(R) generated by ξij(a)=eij(a) γiγj eji(¯a), for 1 ≤ i = j ≤ n, a ∈ R,whereeij is the standard matrix unit and γ is an n-tuple of × nonzero scalars (γ1, ··· ,γn), γi ∈ k ,1≤ i ≤ n. The elements ξij(a) satisfy certain canonical relations. When n ≥ 3, the Steinberg unitary Lie algebra stun(R, −, γ)is defined by generators uij(a) corresponding to ξij(a) and those same canonical relations. So one has a Lie algebra homomorphism φ from stun(R, −, γ)toeun(R, −, γ). It is known that the above homomorphism φ yields a central extension(see [AF]). It is easy to see that both stun(R, −, γ)andeun(R, −, γ) are perfect since the generators are contained in their derived algebras. With some assumptions on R, ∼ φ becomes the universal covering of eun(R, −, γ)withkerφ = −1HD1(R), where −1HD1(R) is the first skew-dihedral homology group of R (see [G1]). For n ≥ 3 and an arbitrary nonassociative algebra (R, −) with involution, the canonical relations might cause a nontrivial relation uij(a)=0witha =0.Denote the kernel of uij by In = In(R, −, γ), which is independent of the choice of γ and

Key words and phrases. Steinberg unitary Lie algebra, eu2-Lie admissible algebra, Lie triple system. 2000 Mathematics Subject Classiﬁcation: 17B60, 17B55, 17D25, 17A30. Research of the second author (the corresponding author) was partially supported by NSERC of Canada and Chinese Academy of Science.

c 2009 Americanc 0000 Mathematical (copyright Societyholder) 1891

2190 SHIKUI SHANG AND YUN GAO

∼ i = j. It is easy to show that In is an ideal of R and stun(R,¯ −, γ) = stun(R, −, γ) where R¯ = R/In.IfIn(R, −, γ)=0,wesaythat(R, −)isn-faithful. In fact, if n ≥ 4, (R, −)isn-faithful if and only if R is associative. Allison and Faulkner proved that (R, −) is a 3-faithful nonassociative algebra if and only if (R, −)is structurable in [AF]. On the other hand, when n =2,sl2(R)andst2(R) were studied in [G2], which demonstrated some intrigue phenomena. In this paper, we will deal with eu2(R, −, γ) and introduce the Steinberg unitary Lie algebra stu2(R, −, γ). We will further study their relations. Our approach is to use the Lie triple system and some technics developed in [H] and [S]. This not only simplified a lot of verifications as was done in [G2] but also enabled us to obtain a necessary and sufficient condition for I2 = 0 which was not able to do in [G2]. This revised approach (using the Lie triple system) is due to the generous referee who provided many critical and instructive suggestions with great details. The organization of this paper is as follows. In Section 1, we study eu2(R, −, γ) and give a necessary and sufficient condition for (R, −) such that eu2(R, −, γ) becomes a Lie algebra. Then in Section 2, we define the Steinberg Lie algebra stu2(R, −, γ) and give a necessary and sufficient condition for I2(R, −, γ)=0using some knowledge of Lie triple systems. Moreover, we prove that stu2(R, −, γ)isthe universal covering of eu2(R, −, γ)whenR is an eu2-Lie admissible algebra satisfying some assumptions given in Section 1. We investigate the structure of ker φ in Section 3. In the last section, we apply the results of Section 2 and 3 to study sl2(S)andst2(S), and recover the main theorems in [G2]. We would like to thank the referee for his lengthy comments by showing us the adoption of Lie triple system which significantly improved this new version of our paper.

1. Some basics for the algebra eu2(R, −, γ) Let k be a ﬁeld and R be a nonassociative k-algebra with identity, equipped with an involution −. We always assume that char k = 2. Then, we have R = R+ ⊕ R− where R+ = {a ∈ R| a¯ = a} and R− = {a ∈ R| a¯ = −a}.Anya ∈ R has a 1 1 − 1 ∈ 1 − ∈ decomposition a = 2 (a +¯a)+ 2 (a a¯) such that 2 (a +¯a) R+ and 2 (a a¯) R−. For each positive integer n ≥ 2, the algebra Mn(R)ofn × n matrices with − coeﬃcients in R forms a nonassociative algebra gln(R)=Mn(R) over k under the commutator product. Assume that n ≥ 2, Let γ be an n-tuple of nonzero scalars ··· ∈ × ≤ ≤ − −1 (γ1, ,γn), γi k ,1 i n.Wedenoteξij(a)=eij(a) γiγj eji(¯a)for 1 ≤ i = j ≤ n,a ∈ R,whereeij is the standard matrix unit, and eun(R, −, γ)is the subalgebra of gln(R) generated by all ξij(a). An algebra g with product [ , ]is called perfect if [g, g]=g.Whenn ≥ 3, eun(R, −, γ) is perfect. But we will see the case n = 2 is a bit more complicated. In this paper, we use the following notations. Notation . For a, b, c ∈ R,wedenote 1 [a, b]=ab − ba; a ∗ b = (ab + ba); (a, b, c)=(ab)c − a(bc); 2 a, b = ab − ab; a, b, c = a, bc + cb, a J(a, b, c)=[[a, b],c]+[[b, c],a]+[[c, a],b]

eu2-LIE ADMISSIBLE ALGEBRAS AND STEINBERG UNITARY LIE ALGEBRAS191 3 and [A, B],A∗ B, (A, B, C), A, B, A, B, C are the linear spans of [a, b],a∗ b, (a, b, c), a, b, a, b, c,fora ∈ A, b ∈ B and c ∈ C. People often call [a, b](or (a, b, c)) a commutator (or associator) of R. Lemma 1.1 . Let R be a nonassociative algebra with involution − and decomposition R = R+ ⊕ R−. For any a, b, c ∈ R,

(i) a, b = −a, b, i.e. R, R⊂R−, (ii) a, b = ab, 1 = 1,ab, a, 1 = 1,a = a − a,¯

In particular, a, 1 = 1,a =2a, if a ∈ R−, (iii) [a, b]=[¯b, a¯]=−[¯a,¯b], we have [R, R]=[R, R],

(iv) R, R = R−, { − | ∈ } ∩ (v)spank a, b b, a a, b R = R− [R, R], (vi)[R−,R−] ⊂ R−, i.e. R− is closed under the Lie product, (vii) a, b, c = −b, a, c¯, we have R, R, R = R, R, R. Proof. (i),(ii) and (iii) are obvious. (iv) follows from (i) and (ii). For (v), since (iii), (iv) and a, b−b, a = ab − ab − ba + ba =[a, b] − [a, b], we have { − | ∈ }⊂ ∩ spank a, b b, a a, b R R− [R, R]. Conversely, if we take t =Σi[ai,bi] ∈ R− ∩ [R, R], then 1 1 1 t = (Σ [a ,b ] − Σ [a ,b ]) = Σ ([a ,b ] − [a ,b ]) = Σ (a ,b −b ,a ) 2 i i i i i i 2 i i i i i 2 i i i i i { − | ∈ }⊃ ∩ which shows spank a, b b, a a, b R R− [R, R]. (vi) and (vii) are easily veriﬁed. − −1 − γ Since ξ12(a)= γ1γ2 ξ21(¯a), we have ξ12(R)=ξ21(R)andeu2(R, , )is generated by ξ12(R). Set H(a, b)=[ξ12(a),ξ21(b)] for a, b ∈ R,wehave

H(a, b)=[ξ12(a),ξ21(b)] − −1 − −1 ¯ =[e12(a) γ1γ2 e21(¯a),e21(b) γ2γ1 e12(b)]

=e11(ab − ab) − e22(ba − ba)

=e11(a, b) − e22(b, a)

Let H(R, R)=[ξ12(R),ξ21(R)] be the linear span of H(a, b),a,b∈ R,then Lemma 1.2 .

H(R, R)={A = e11(x)+e22(y)| x, y ∈ R−and Tr(A) ∈ R− ∩ [R, R]}. Proof. “⊂” can be obtained from Lemma 1.1 (i) and (v). For “⊃”, Let A be a diagonal matrix with Tr(A) ∈ R− ∩ [R, R]}.Since Tr(H(a, b)) = a, b−b, a, Lemma 1.1(v) shows Tr(H(R, R)) = R− ∩ [R, R]. Thus, we can adjust A by an element of H(R, R) to assume that Tr(A) = 0; i.e. 1 A = e (y) − e (y)= (e (y, 1) − e (1,y)) 11 22 2 11 22 1 = H(y, 1) ∈ H(R, R), 2 for some y ∈ R.

4192 SHIKUI SHANG AND YUN GAO

Lemma 1.3 . The nonassociative algebra eu2(R, −, γ) has a vector space decomposition eu2(R, −, γ)=ξ12(R) ⊕ H(R, R). Proof. By Lemma 1.1 (vi), for any a, b, c ∈ R, − − −1 [H(a, b),ξ12(c)] = [e11( a, b ) e22( b, a ),e12(c) γ1γ2 e21(¯c)] − −1 − =e12( a, b c + c b, a ) γ1γ2 e21( ( b, a c¯+¯c a, b )) − −1 =e12( a, b, c ) γ1γ2 e21( a, b, c )

=ξ12(a, b, c) We only need to show that H(R, R) is closed under [ , ]. Indeed, we have

[H(a, b),H(c, d)] = [e11(a, b) − e22(b, a),e11(c, d) − e22(d, c)]

=e11([a, b, c, d]) + e22([b, a, d, c]).

By Lemma 1.1 (i) and (vi), [a, b, c, d]+[b, a, d, c] ∈ R− ∩ [R, R], for any a, b, c, d ∈ R. It is contained in H(R, R) because of Lemma 1.2.

Now we give the necessary and suﬃcient condition for eu2(R, −, γ) is perfect.

Theorem 1.1 . eu2(R, −, γ) is perfect if and only if R = R, R, R.

Proof. By Lemma 1.3, eu2(R, −, γ)=ξ12(R) ⊕ H(R, R), then

[eu2(R, −, γ), eu2(R, −, γ)] = [H(R, R),ξ12(R)] ⊕ H(R, R).

We also have [H(a, b),ξ12(c)] = ξ12(a, b, c). Thus, the theorem is proved.

Recall the deﬁnition of the center Z(R) of a nonassociative algebra R is the set of all elements in R which commute and associate with all elements. Particularly, Z(R) is contained in the nucleus N(R)ofR,where N(R)={n ∈ R|(n, R, R)=(R, n, R)=(R, R, n)=0}. We show some assumptions on R, from which imply R = R, R, R. Assumption 1.1 . Suppose that there exists an element e ∈ R such that (i) e ∈ R−,namely,¯e = −e, (ii) e lies in the center Z(R)ofR, (iii) e is a unit.

Assumption 1.2 . Suppose that R− ∗ R = R,where∗ is the Jordan product ∗ 1 ∗ { ∗ | ∈ ∈ } a b = 2 (ab + ba)andR− R = i ai bi ai R−,bi R . Obviously, the Assumption 1.1 implies 1.2. These assumptions are not so restrictive. For example, if −1 has no square root in k,andS is an associa- − tive k-algebra√ with identity,√ equipped an involution , then√ consider the algebra√ − R = S ⊗k k( −1) = S ⊕ −1S, extending on R by a + −1b =¯a − −1 ¯b. One can check that R is an√ associative k-algebra equipped an involution. In this case, we may choose e = −1 and Assumption 1.1 holds. Also, one may check that Assumption 1.2 holds for any associative composition algebra with the natural involution −, but Assumption 1.1 may fail.

eu2-LIE ADMISSIBLE ALGEBRAS AND STEINBERG UNITARY LIE ALGEBRAS193 5

For a ∈ R−,b∈ R,sincea, 1 = 1,a = a − a¯ =2a,wesee 1 1 1 a ∗ b = (ab + ba)= ((a − a¯)b + b(a − a¯)) = a, 1,b 2 4 4

Therefore, R− ∗ R ⊂R, R, R. So, assumption 1.2 implies eu2(R, −, γ) is perfect. Next, we shall investigate when eu2(R, −, γ) is a Lie algebra. A nonassociative algebra R with an involution − over k is called eu2-Lie admissible if eu2(R, −, γ)isa Lie algebra over k. (We will see that it is independent of γ.) Certainly, associative algebras are eu2-Lie admissible. Let T be a nonassociative k-algebra. For x1,x2,x3 ∈ T ,we deﬁne

J(x1,x2,x3)=[[x1,x2],x3]+[[x2,x3],x1]+[[x3,x1],x2].

Then, T is a Lie algebra over k if and only if it is anticommutative and J(x1,x2,x3)= 0 for any x1,x2,x3 ∈ T . Lemma 1.4 . Let T be an anticommutative k-algebra, σ (i) J(x1,x2,x3)=(−1) J(xσ(1),xσ(2),xσ(3))forxi ∈ T and σ ∈ S3,theper- mutation group of {1, 2, 3}. (ii) If S ⊂ T generates T and J(S, T, T )=0,thenT is a Lie algebra over k.

Proof. Clearly, (i) holds for σ = (12) and σ = (123), and hence for all σ ∈ S3. For x ∈ T ,wenotethatJ(x, T, T ) = 0 if and only if ad(x):y → [x, y]isa derivation of T .IfJ(x, T, T )=J(y, T, T)=0,ad(x)andad(y) are derivations. Then, J(x, y, T )=0showsthatad([x, y]) = [ad(x),ad(y)], which is also a derivation of T .Thus,J([x, y],T,T) = 0. This shows (ii).

Now, we prove the main theorem of this section.

Theorem 1.2 . (R, −)iseu2-Lie admissible if and only if the following identities hold

(i)(a,¯b, c)+(b, c,¯ a)+(c, a,¯ b)=(a, c,¯ b)+(c,¯b, a)+(b, a,¯ c) (ii)(a, b, c, d)+(c, d,a,b)+(a, d, c,b)=(¯b, a,¯ c, d)+(c, d, ¯b, a¯)+(¯b, d, c, a¯) (iii)(a, b, c, d,e)+(e, b, a, d, c)+(a, b,e,d, c) =(c, d, a, b,e)+(e, d, c, b, a)+(c, d,e,b, a) for any a, b, c, d, e ∈ R.

Proof. Obviously, eu2(R, −, γ) is anticommutative. Thus, it is a Lie algebra if and only if J(x, y, z) = 0 for any x, y, z ∈ eu2(R, −, γ). Since eu2(R, −, γ) is generated by ξ12(a),a ∈ R, by Lemma 1.3 and Lemma 1.4, we need to consider the following three cases. Case 1. x, y, z ∈ ξ12(R). We can assume x = ξ12(a),y = ξ12(b)andz = ξ12(c). − −1 ¯ [[x, y],z]=[[ξ12(a),ξ12(b)],ξ12(c)] = γ1γ2 [[ξ12(a),ξ21(b)],ξ12(c)] − −1 ¯ − −1 ¯ = γ1γ2 [H(a, b),ξ12(c)] = γ1γ2 ξ12( a, b, c ) − −1 ¯ Thus, J(x, y, z)= γ1γ2 ξ12( a, b, c + b, c,¯ a + c, a,¯ b ).

6194 SHIKUI SHANG AND YUN GAO

From direct calculation, a,¯b, c + b, c,¯ a + c, a,¯ b =(a¯b)c − (ba¯)c + c(¯ba) − c(¯ab) +(bc¯)a − (c¯b)a + a(¯cb) − a(¯bc) +(ca¯)b − (ac¯)b + b(¯ac) − b(¯ca) =((a,¯b, c)+(b, c,¯ a)+(c, a,¯ b))b − ((a, c,¯ )+(c,¯b, a)+(b, a,¯ c)), which yields the identity (i). Case 2. x, y ∈ ξ12(R)andz ∈ H(R, R). We assume x = ξ12(a),y = ξ21(b)andz = H(c, d).

[[x, y],z]=[[ξ12(a),ξ21(b)],H(c, d)] = [H(a, b),H(c, d)]

= e11([a, b, c, d]) + e22([b, a, d, c])

[[y, z],x]=[[ξ21(b),H(c, d)],ξ12(a)] = [ξ21(bc, d + d, cb),ξ12(a)]

=[ξ21(d, c, b),ξ12(a)] = −H(a, d, c, b)

[[z,x],y]=[[H(c, d),ξ12(a)],ξ21(b)] = [ξ12(c, d, a),ξ21(b)] = H(c, d, a,b) Then,

J(x, y, x)=e11([a, b, c, d] −a, d, c, b + c, d, a,b)

+ e22([b, a, d, c]+d, c, b,a−b, c, d, a) Furthermore, we have, [a, b, c, d] −a, d, c, b + c, d, a,b = a, bc, d−c, da, b−a, d, cb + bc, d + c, da + ad, c,b =(ab)c, d−(¯ba¯)c, d−c, d(ab)+c, d(¯ba¯) −c a(d, b) − a(bc, d) − (¯bd, c)¯a − (c, d¯b)¯a +(c, da)b +(ad, c)b + ¯b(¯ac, d)+¯b(d, ca¯) =(a, b, c, d) − (¯b, a,¯ c, d)+(c, d,a,b) − (c, d, ¯b, a¯)+(a, d, c,b) − (¯b, d, c, a¯) Exchanging a, c with b, d, we can obtain that J(x, y, z)=0 ⇐⇒ (ii) holds. Case 3. x, y ∈ H(R, R)andz ∈ ξ12(R). Assume that x = H(a, b),y = H(c, d)andz = ξ12(e),

[[x, y],z]=[[H(a, b),H(c, d)],ξ12(e)]

=[e11([a, b, c, d]) + e22([b, a, d, c]),ξ12(e)]

= ξ12([a, b, c, d]e − e[b, a, d, c])

[[y, z],x]=[[H(c, d),ξ12(e)],H(a, b)] = [ξ12(c, d, e),H(a, b)]

= −ξ12(a, b, c, d, e)

[[z,x],y]=[[ξ12(e),H(a, b)],H(c, d)] = [H(c, d),ξ12(a, b, e)]

= ξ12(c, d, a, b, e)

eu2-LIE ADMISSIBLE ALGEBRAS AND STEINBERG UNITARY LIE ALGEBRAS195 7

So in this case, J(x, y, x)=

ξ12([a, b, c, d]e − e[b, a, d, c] −a, b, c, d, e + c, d, a, b, e) and [a, b, c, d]e − e[b, a, d, c] −a, b, c, d, e + c, d, a, b, e =(a, bc, d−c, da, b)e − e(b, ad, c−d, cb, a) − (a, bc, d, e + c, d, ea b, )+(c, da, b, e + a, b, ed, c) =(a, bc, d)e − (c, da, b)e − e(b, ad, c)+e(d, cb, a) −a, b(c, de) −a, b(ed, c) − (c, de)b, a−(ed, c)b, a + c, d(a, be)+c, d(eab, )+(a, be)d, c +(eb, a)d, c =(a, b, c, d,e) − (c, d, a, b,e)+(e, b, a, d, c) − (e, d, c, b, a)+(a, b,e,d, c) − (c, d,e,b, a) which shows that J(x, y, x)=0⇐⇒ (iii). So, (R, −)iseu2-Lie admissible ⇐⇒ (i)-(iii) hold.

Corollary 1.1 . Let R be a nonassociative k-algebra with an involution −. If R is eu2-Lie admissible, then R− is Lie admissible, i.e R− is a Lie algebra with [ , ]overk.

Proof. Let x = H(a, b),y = H(c, d),z = H(e, f) ∈ eu2(R, −, γ), we can easily see

J(x, y, z)=e11(J(a, b, c, d, e, f)) − e22(J(b, a, d, c, f,e)).

By Lemma 1.1 (iv) and (vi), J(x, y, z) = 0 implies J(R−,R−,R−)=0.We have this corollary.

An associative k-algebra is eu2-Lie admissible. Finally, we give a class of eu2-Lie admissible algebras which are not associative. Example 1.1. Let V = X ⊕ Y be a k-vector space with a symplectic form f : V × V → k satisfying f(X, X)=f(Y,Y )=0.LetW = k1 ⊕ V be the k-algebra with identity 1 and uv = f(u, v)1 for u, v ∈ V and involution given by 1=1and¯ x + y = x − y for x ∈ X, y ∈ Y . With this involution, we have W+ = k1 ⊕ X and W− = Y .Ifu, v, w ∈ V ,we have (u, v, w)=f(u, v)w − f(v, w)u = f(u, v)w + f(w, v)u. By the non-degeneracy of f,iff(u, v) =0,then( v, v, u)=f(u, v)v =0.Thus, W is not associative. But we will show that W is eu2-Lie admissible, i.e. it satisﬁes the identities (i) − (iii) in Theorem 1.2. For (i), it suﬃces to check a, b, c ∈ V . LHS =(a,¯b, c)+(b, c,¯ a)+(c, a,¯ b) =f(a,¯b)c + f(c,¯b)a + f(b, c¯)a + f(a, c¯)b + f(c, a¯)b + f(b, a¯)c =(f(c,¯b)+f(b, c¯))a +(f(a, c¯)+f(c, a¯))b +(f(a,¯b)+f(b, a¯))c,

8196 SHIKUI SHANG AND YUN GAO and RHS =(a, c,¯ b)+(c,¯b, a)+(b, a,¯ c) =f(a, c¯)b + f(b, c¯)a + f(c,¯b)a + f(a,¯b)c + f(b, a¯)c + f(c, a¯)b =(f(c,¯b)+f(b, c¯))a +(f(a, c¯)+f(c, a¯))b +(f(a,¯b)+f(b, a¯))c. So, (i) holds. Furthermore, since k1,k1 = V,V =0and1,v = v, 1 = v − v¯ for v ∈ V . we see that a, b = b, a∈Y for all a, b ∈ W .Also,wehave(u, v, w)=f(u, v)w + f(w, v)u =(w, v, u)and (u, v, w)+(v, w, u)+(w, u, v) =f(u, v)w + f(w, v)u + f(v, w)u + f(u, w)v + f(w, u)uv + f(v, )w =0 for u, v, w ∈ V . Therefore, both sides of (ii) and (iii) are 0. We have W is eu2-Lie admissible. Particularly, if dimkX =dimkY = n<∞,thendimkW =2n +1. In Section 4, we will give another class of eu2-Lie admissible algebras which may be not associative (See Example 4.1).

2. Steinberg unitary Lie algebras stu2(R, −, γ) For an nonassociative k-algebra R with identity, equipped with an involution −, the Steinberg unitary Lie algebra stu2(R, −, γ) is deﬁned by the generators u12(a),u21(a),h(a, b), a, b ∈ R, subject to the relations:

(2.1) a → u12(a)isak-linear mapping, − −1 (2.2) u21(a)=u12( γ2γ1 a¯)

(2.3) h(a, b)=[u12(a),u21(b)]

(2.4) [h(a, b),u12(c)] = u12(a, b, c) for all a, b, c ∈ R. By the result of Allison and Faulkner in [AF], if n ≥ 3, In(R)={a ∈ R|uij (a)= 0} are ideals of R.Whenn = 2, the subspace I2(R)={a ∈ R | u12(a)=0} of R may not be an ideal. We will give a necessary and sufficient condition for I2(R)=0 using Lie triple systems. Recall that a Lie triple system T over k is a k-vector space with a triple product [ ,,]:T × T × T → T (a trilinear map) satisfying (LTS1) [x, x, y]=0, (LTS2) [x, y, z]+[y, z, x]+[z,x,y]=0, (LTS3) [x, y, [u, v, w]] = [[x, y, u],v,w]+[u, [x, y, v],w]+[u, v, [x, y, w]], for any x, y, z, u, v, w ∈ T . (More details can be found in [H].) For any Lie algebra L, we define a triple product on L by [x, y, z]=[[x, y],z] for x, y, z ∈ L.ThenL becomes a Lie triple system. On the other hand, for a Lie triple system T , we define L(x, y)(z)=[x, y, z]forx, y, z ∈ T . The standard

eu2-LIE ADMISSIBLE ALGEBRAS AND STEINBERG UNITARY LIE ALGEBRAS197 9 embedding of T is the k-vector space S(T )=L(T,T) ⊕ T with the bracket product [A + x, b + y]=(AB − BA + L(x, y)) + (A(y) − B(x)) for A, B ∈ L(T,T),x,y ∈ T . Then, S(T ) is closed under the bracket product [ , ] and becomes a Lie algebra. Let T be a vector space with any triple product [ ,,]. The universal embedding U(T )ofT is the Lie algebra over k generated by u(x),x∈ T , subject to the relations: (2.5) x → u(x)isak-linear mapping, (2.6) u([x, y, z]) = [[u(x),u(y)],u(z)] for all x, y, z ∈ T . From (2.6), we see that u(T ) is a Lie triple system, which is a subsystem of the Lie triple system U(T ). Lemma 2.1 . T is a Lie triple system over k if and only if ker u =0. Proof. Clearly, u : T → U(T ) preserves the triple product. If ker u =0,then ∼ T = u(T ) is a Lie triple system. Conversely, if T is a Lie triple system, by the universal property of U(T ), there exists a Lie homomorphism U(T ) → S(T ) such that u(x) → x.So,keru =0.

Take T = T (R, −,γ)tobeR with triple bracket product − −1 ¯ [a, b, c]= γ1γ2 a, b, c , for a, b, c ∈ R. By comparing the relations (2.5)-(2.6) to (2.1)-(2.4), we have U(T (R, −,γ)) = stu2(R, −, γ)andkeru = I2(R). Thus, Theorem 2.1 . If R is a nonassociative k-algebra with identity, equipped with an involution −,thenI2(R)=0⇐⇒ the following equations hold (i)(a,¯b, c)+(b, c,¯ a)+(c, a,¯ b)=(a, c,¯ b)+(c,¯b, a)+(b, a,¯ c), (ii) a, b, c, d, e = a, b, c,d,e−c, b, a, d,e + c, d, a, b, e, for any a, b, c, d, e ∈ R.

Proof. We have seen that I2(R)=0⇔ T (R, −,γ) is a Lie triple system. First, if we deﬁne [x, y, z] = λ[x, y, z]forsomeλ ∈ k×, whether (LTS1)-(LTS3) hold is independent of λ. So, we can assume that [a, b, c]=a,¯b, c. Since a, a¯ = aa¯ − aa¯ =0, [a, a, c]=a, a,¯ c = a, a¯c + ca,¯ a =0. (LTS1) is automatic in T (R, −,γ). (LTS2) becomes a,¯b, c + b, c,¯ a + c, a,¯ b =0, which is just the equation (i) in Theorem 1.2. Using Lemma 1.1(vii), we see that (LTS3) with x = a, y = ¯b, u = c,v = d,¯ w = e becomes a, b, c, d, e = a, b, c,d,e−c, b, a, d,e + c, d, a, b, e.

Thus, I2(R)=0⇐⇒ the equations (i) and (ii) hold.

10198 SHIKUI SHANG AND YUN GAO

If (R, −)iseu2-Lie admissible, we have that eu2(R, −, γ) is a Lie algebra over k.Sinceξ12(a),ξ21(a)andH(a, b) satisfy certain canonical relations (2.1)-(2.4) for a, b ∈ R, by the universal property of stu2(R, −, γ), there exits a Lie homomorphism

φ : stu2(R, −, γ) → eu2(R, −, γ) such that

φ(ξij(a)) = eij(a),φ(h(a, b)) = H(a, b), for a, b ∈ R, 1 ≤ i = j ≤ 2. Since ξ12(R) generates eu2(R, −, γ), φ is a Lie | → epimorphism. The restriction of φ u12(R) : u12(R) ξ12(R) is a linear isomorphism. Thus, I2(R)=0. Furthermore, we have

Lemma 2.2 . If (R, −)iseu2-Lie admissible,

φ 0 → ker φ → stu2(R, −, γ) → eu2(R, −, γ) → 0 is a central extension of the Lie algebra eu2(R, −, γ). Proof. By Jacobi identity, we have

[h(a, b),h(c, d)] = [h(a, b), [u12(c),u21(d)]]

=[u12(a, b, c),u21(d)] + [u12(c),u21(a, b, d)] =h(a, b, c,d) − h(c, b, a, d). Thus, [h(R, R),h(R, R)] ⊂ h(R, R).

We obtain stu2(R, −, γ)=h(R, R)+u12(R). Moreover, since

φ(h(R, R) ∩ u12(R)) ⊂ H(R, R) ∩ ξ12(R)=0 | ∩ and φ u12(R) is injective, we have h(R, R) u12(R) = 0. This shows that

stu2(R, −, γ)=h(R, R) ⊕ u12(R) is a vector space decomposition of stu2(R, −, γ). | ⊂ Because φ u12(R) is a linear isomorphism, ker φ h(R, R)and

[ker φ, u12(R)] ⊂ u12(R).

On the other hand, ker φ is an ideal of stu2(R, −, γ), which yields

[ker φ, u12(R)] ⊂ ker φ ⊂ h(R, R).

So, [ker φ, u12(R)] = 0. But u12(R) generates the whole Lie algebra stu2(R, −, γ), so ker φ is contained in center of stu2(R, −, γ).

If we assume that R has the element e satisfying the condition in Assumption 1.1, then R, R, R = R, stu2(R, −, γ)andeu2(R, −, γ) are perfect. For this case, we can prove that stu2(R, −, γ) is the universal covering of eu2(R, −, γ).

Theorem 2.2 . If R is eu2-Lie admissible and satisﬁes Assumption 1.1, then stu2(R, −, γ) is the universal covering of eu2(R, −, γ), i.e. it is centrally closed.

eu2-LIE ADMISSIBLE ALGEBRAS AND STEINBERG UNITARY LIE ALGEBRAS199 11

Proof. With the assumption of the existence of the element e,wehave

1,e,e−1a =2e · e−1a + e−1a · 2e =4a for any a ∈ R. Suppose that π 0 −→ V −→ L −→ eu2(R, −, γ) −→ 0 is a central extension of eu2(R, −, γ). We will show that there exists a Lie homomorphism τ : stu2(R, −, γ) → L such that φ = π ◦ τ. Let {rλ}λ∈Λ be a basis of R containing 1(Λ is an index set). Letu ˜12(rλ)bea preimage in L of ξ12(rλ) under π. For general a ∈ R, deﬁne the preimageu ˜12(a)of − −1 ˜ ξ12(a) by linearity and deﬁneu ˜21(a)=˜u12( γ2γ1 a¯), h(a, b)=[˜u12(a), u˜21(b)]. We know that [H(a, b),ξ12(c)] = ξ12(a, b, c), so −1 −1 [h˜(1,e), u˜12(e a)] =u ˜12(1,e,e a)+v(a)=4˜u12(a)+v(a), → 1 where v : R V is a k-linear map. Replaceu ˜12(a)by˜u12(a)+ 4 v(a), and set − −1 ˜ u˜21(a)=˜u12( γ2γ1 a¯). Note that h(a, b) is unchanged since it dose not depend on the choice of preimages ofu ˜12(a)and˜u21(b). So we have −1 (2.7) [h˜(1,e), u˜12(e a)] = 4˜u12(a).

Since e is contained in the center of R,ineu2(R, −, γ) [H(a, b),H(1,e)]

=e11([a, b, 1,e]) + e22([b, a, e, 1])

=2e11([a, b,e]) + 2e22([b, a,e]) = 0. Therefore, [h˜(a, b), h˜(1,e)] ∈ V . By (2.7), we have 1 [h˜(a, b), u˜ (c)] = [h˜(a, b), [h˜(1,e), u˜ (e−1c)]] 12 4 12 1 = ([[h˜(a, b), h˜(1,e)], u˜ (e−1c)] + [h˜(1,e), [h˜(a, b), u˜ (e−1c)]]) 4 12 12 1 = (0 + [h˜(1,e), u˜ (a, b, e−1c)]) 4 12 1 = [h˜(1,e), u˜ (e−1a, b, c)] =u ˜ (a, b, c). 4 12 12

Then,u ˜12(a), u˜21(a), h˜(a, b) satisfy the relations (2.1)-(2.4). By the universal property of stu2(R, −, γ), there exists a unique homomorphism τ from stu2(R, −, γ) to L so that τ(u12(a)) =u ˜12(a), τ(u21(a)) =u ˜12(a), τ(h(a, b)) = h˜(a, b). Evidently, π◦τ = φ.Sostu2(R, −, γ) is the universal covering of eu2(R, −, γ) and it is centrally closed.

Remark 2.1 Note that ker φ is also independent of the choice of γ =(γ1,γ2).

3. The structure of ker φ

Throughout this section we assume that (R, −)isaneu2-Lie admissible k- algebra, where − is an involution of R. By Lemma 2.2, φ : stu2(R, −, γ) → eu2(R, −, γ) is a central extension of eu2(R, −, γ). We will give the exact and explicit structure of ker φ. In Section 2, we have seen that stu2(R, −, γ) U(T ), − − −1 ¯ where T = T (R, ,γ)isR with triple product [a, b, c]= γ1γ2 a, b, c and U(T )

12200 SHIKUI SHANG AND YUN GAO is the universal embedding of it. When R is eu2-Lie admissible, we have I2(R)=0 and T (R, −,γ) is a Lie triple system. We will use the following theorem for a Lie triple system T (See [S, Theorem 3.5]). Theorem 3.1 . Let T be a Lie triple system over k,letI be the subspace of T ⊗2 spanned by all x ⊗ y + y ⊗ x, [x, y, z] ⊗ w + z ⊗ [x, y, w]+[z,w,x] ⊗ y + x ⊗ [z,w,y] for x, y, z, w ∈ T ,andsetl(x, y)=x⊗y+I ∈ T ⊗2/I.ThenT ⊗2/I ⊕T with bracket product given by [x, y]=l(x, y) [l(x, y),z]=−[z,l(x, y)] = [x, y, z] [l(x, y),l(z,w)] = l([x, y, z],w)+l(z,[x, y, w]) is a Lie algebra isomorphic to the universal embedding U(T ). Using Theorem 3.1 for the Lie triple system T ((R, −γ)), we get

Corollary 3.1 . Suppose that (R, −)iseu2-Lie admissible. Let ∆ be the subspace of R⊗2 spanned by all (3.1) a ⊗ b + ¯b ⊗ a¯ (3.2) a, b, c⊗d − c ⊗b, a, d + c, d, a⊗b − a ⊗d, c, b and set h(a, b)=a ⊗ b +∆∈ R⊗2/∆. − −1 ⊗2 ⊕ Let u12(R)beacopyofR and set u21(a)=u12( γ2γ1 a¯). Then, R /∆ u12(R) with product given by [u12(a),u21(b)] = h (a, b), − [h (a, b),u12(c)] = [u12(c),h(a, b)] = u12( a, b, c ), [h(a, b),h(c, d)] = h(a, b, c,d) − h(c, b, a, d) is a Lie algebra isomorphic to the Steinberg unitary Lie algebra stu2(R, −, γ). Proof. − − −1 ¯ We know that R = T (R, ,γ)with[a, b, c]= γ1γ2 a, b, c is a Lie ⊗2 ⊕ triple system. It suﬃces to show that R /∆ u12(R) is isomorphic to the universal embedding U(T (R, −γ)) of T (R, −,γ). Let θ˜ : R⊗2 → R⊗2 be given by ˜ ⊗ − −1 ⊗ ¯ θ(a b)= γ1γ2 a b. Applying θ˜ to the generators of I,weget ˜ ⊗ ⊗ − −1 ⊗ ¯ ⊗ θ(a b + b a)= γ1γ2 (a b + b a¯), θ˜([a, b, c] ⊗ d + c ⊗ [a, b, d]+[c, d, a] ⊗ b + a ⊗ [c, d, b]) − −1 ¯ ⊗ ¯− ⊗¯ ¯ ¯ ⊗¯ − ⊗¯ ¯ = γ1γ2 ( a, b, c d c b, a, d + c, d, a b a d, c, b ), so θ˜(I)=∆,andθ˜ induces a linear isomorphism θ : R⊗2/I → R⊗2/∆with − −1 ¯ ⊗2 ⊕ → ⊗2 ⊕ θ(l(a, b)) = γ1γ2 h (a, b). Extending θ on R /I R R /∆ u12(R)by → u + a θ(u)+u12(a)

eu2-LIE ADMISSIBLE ALGEBRAS AND STEINBERG UNITARY LIE ALGEBRAS201 13 for u ∈ R⊗2/I and a ∈ R, it is easy to check that θ is an isomorphism of Lie algebra ⊗2 ⊕ over k. By Theorem 3.1, we have R /∆ u12(R) is isomorphic to the Steinberg unitary Lie algebra stu2(R, −, γ). We get the main theorem of this section

Theorem 3.2 . Suppose that (R, −)isaneu2-Lie admissible algebra and ⊗2 φ : stu2(R, −, γ) → eu2(R, −, γ) is the central extension. Let ∂ : R → R− with ∂(a ⊗ b)=a, b−b, a. Then, ker φ ker ∂/∆˜ where ∆=∆˜ ⊕ (1 ⊗ R−). ⊗2 Proof. Deﬁne ∂i : R → R−, i =1, 2with∂1(a ⊗ b)=a, b and ∂2(a ⊗ b)= b, a.Thus,∂ = ∂1 − ∂2 and ∂i(1 ⊗ s)=2s for s ∈ R−. By Corollary 3.1, We can − γ ⊗2 ⊕ take stu2(R, , )=R /∆ u12(R)withφ(u12(a)) = ξ12(a)and

φ(a ⊗ b +∆)=H(a, b)=e11(∂1(a ⊗ b)) − e22(∂2(a ⊗ b)).

Thus, ker φ = K/∆whereK =ker∂1 ∩ ker ∂2.WeseethatK, 1 ⊗ R− ⊂ ker ∂ and

K ∩ (1 ⊗ R−)=0. ∈ − ⊗ 1 Also, if u ker ∂ and v = u 1 2 ∂2(u), then 1 ∂ (v)=∂ (u) − ∂ (1 ⊗ ∂ (u)) = ∂ (u) − ∂ (u)=∂(u)=0 1 1 2 1 2 1 2 and 1 ∂ (v)=∂ (u) − ∂ (1 ⊗ ∂ (u)) = ∂ (u) − ∂ (u)=0. 2 2 2 2 2 2 2 ∈ ⊗ 1 ⊕ ⊗ So, v K and u = v +1 2 ∂2(u). Thus, ker ∂ = K (1 R−), and the canonical projection π :ker∂ → K/∆, (u, v) → u +∆ for u ∈ K, v ∈ 1 ⊗ R−,givesus K/∆ ker ∂/ker π,wherekerπ =∆+(1⊗ R−) and the sum is direct. Finally, we have ker φ K/∆ ker ∂/∆˜ for ∆=∆˜ ⊕ (1 ⊗ R−).

By Theorem 2.2, if we assume that (R, −)iseu2-Lie admissible with Assumption 1.1, then (stu2(R, −, γ),φ) is the universal covering of eu2(R, −, γ). Associating with Theorem 3.2, we have ∼ Corollary 3.2 . H2(eu2(R, −, γ)) = ker ∂/∆.˜

4. Applications on sl2(S) and st2(S)

In the last section, we use the above theorems for dealing with sl2(S)andst2(S) for the k-algebra S, and obtain the corresponding results in [G2]. Suppose that S is a nonassociative k-algebra with identity 1. Let (R, −)=(S ⊕ op S , ex) be the k-algebra with identity (1, 1), satisfying (s1,s2)(t1,t2)=(s1t1,t2s2) and (s1,s2)=(s2,s1)fors1,s2,t1,t2 ∈ S. Then, we have R+ = {(s, s) ∈ S ⊕ op op S | for s ∈ S} and R− = {(s, −s) ∈ S ⊕ S | for s ∈ S}. It is clear that R is associative if and only if so is S. As usual, we consider the k-algebra sl2(S) as the subalgebra of gl2(S)gen- × 2 erated by e12(s)ande21(s)fors ∈ S.Takingγ =(1, 1) ∈ (k ) ,thereisan

14202 SHIKUI SHANG AND YUN GAO isomorphism ϕ : eu2(R, −, γ) → sl2(S)givenbyϕ(ξ12((s1,s2))) = e12(s1) − e21(s2) and ϕ(H(((s1,s2), (t1,t2)))) = e11(s1t1 − t2s2)+e22(s2t2 − t1s1) (see Lemma 1.3). Especially, ϕ(H(((s, 0), (t, 0)))) = e11(st) − e22(ts). ∼ In [G2], S is called sl2-Lie admissible if sl2(S) is a Lie algebra. Since sl2(S) = eu2(R, −, γ), we have that S is sl2-Lie admissible if and only if R is eu2-Lie admissible. From Theorem 1.2, we have ([G2, Theorem 1])

Theorem 4.1 . Let S is a nonassociative k-algebra, then S is sl2-Lie admissible if and only if (4.1) (p, q, rs)+(p, sr, q)+(rs, p, q)=0 for all p, q, r, s ∈ S. Proof. For any (R, −), we have (a, b, c)=−(¯c,¯b, a¯). Letting z =(a, b, c, d)+(c, d,a,b)+(a, d, c,b), we see that Theorem 1.2 (ii) is equivalent to z =¯z.For(R, −)=(S ⊕ Sop,ex), if we take a ∈ S ∪ Sop,thenz ∈ S ∪ Sop.Soz =¯z is equivalent to z =0.Thus,in this case, we can replace (ii) with (ii’) (a, b, c, d)+(c, d,a,b)+(a, d, c,b)=0. ⇔ We denote the S-component of (i) by (i)S, etc., and claim (i)S, (ii’)S, (iii)S (4.1) for S. Again we can take a, b, c, etc. in S ∪ Sop.IftheS-component of some term of (i) is nonzero, we can permute the variables to assume it is (a,¯b, c), so ¯ ∈ ¯ op a, b, c S, Except for (c, b, a), the other terms are in S ,so(i)S is equivalent to (4.2) (p, q, r)+(r, p, q)=0 ∈ ∈ op for p, q, r S. The terms in (ii’)S are zero unless a, b S.SinceS, S =0 − ¯ ∈ and c, d = d, c¯ ,wecantakec, d S. Thus, (ii’)S is equivalent to (4.1) for S. Similarly, (iii)S is equivalent to (4.3) (rq, ts, p)+(p, qr, st)+(rq, p, st)=(ts, rq, p)+(p, st, qr)+(ts, p, qr) for S. [G2, Section 2] shows (4.1) implies (4.2) and (4.3), so the claim holds. Since op op op (S ) = S, we can reserve the roles of S and S to see that (i)Sop , (ii’)Sop , (iii)Sop ⇔ (4.1) for Sop. However, since (a, b, c)op = −(c, b, a), we see that (4.1)for S ⇔(4.1) for Sop, and the result follows.

Example 4.1 By the above Theorem, let S be a sl2-Lie admissible k-algebra which op is not associative, then we have (S ⊕ S , ex) is a eu2-Lie admissible k-algebra with involution which is also not associative. In [G2, Section 4], a class of sl2-Lie admissible algebras R(n) which are not associative is given, so (R(n) ⊕ R(n)op, ex) are examples of the eu2-Lie admissible algebras which are not associative. Since for ﬁnite n, the dimension of R(n) ⊕ R(n)op is even, the algebra W given in Example 1.1 can not be isomorphic to one of these algebras.

Suppose that S is an sl2-Lie admissible k-algebra. If we take e =(1, −1) ∈ R ∼ which satisﬁes the Assumption 1.1, then we have eu2(R, −, γ) = sl2(S) is perfect. Straightforwardly, we can also obtain it since the generators 1 e (s)= [[e (1),e (1)],e (s)] 12 2 12 21 12 1 and e21(s)= 2 [[e12(1),e21(1)],e21(s)] are contained in [sl2(S),sl2(S)].

eu2-LIE ADMISSIBLE ALGEBRAS AND STEINBERG UNITARY LIE ALGEBRAS203 15

Theorem 4.2 . Suppose that S is an sl2-Lie admissible k-algebra. Let ∆ be the subspace of S⊗2 spanned by all (4.4) a ⊗ b + b ⊗ a (a(bc)+(cb)a) ⊗ d − c ⊗ ((ba)d + d(ab)) (4.5) +((cd)a + a(dc)) ⊗ b − a ⊗ ((dc)b +(bc)d). Then, H2(sl2(S)) ker ∂ /∆ where ∂ : S ⊗ S → S is deﬁned by ∂(a ⊗ b)=[a, b]. op Proof. First, since S is an sl2-Lie admissible, we have (R, −, )=(S ⊕S , ex) is an eu2-Lie admissible k-algebra with Assumption 1.1 for e =(1, −1). So, eu2(R, −, γ) sl2(S)andH2(sl2(S)) ker φ by Theorem 3.2. We claim that ker φ ker ∂/∆. Recall the deﬁnition of ∆ and ∂ in Theorem 3.2. We take a =(1, 1),b=(1, −1),c=(r, 0),d=(0,s) in (3.2), then (1, 1), (1, −1), (r, 0)⊗(0,s) − (r, 0) ⊗(1, −1), (1, 1), (0,s) + (r, 0), (0,s), (1, 1)⊗(1, −1) − (1, 1) ⊗(0,s), (r, 0), (1, −1) =4(1, −1)(r, 0) ⊗ (0,s) − (r, 0) ⊗ 4(1, −1)(0,s) =8(r, 0) ⊗ (0,s) ∈ ∆. Thus, S ⊗ Sop ⊂ ∆. Similarly, taking a =(1, 1),b =(1, −1),c =(0,s),d =(r, 0) in (3.2), we have Sop ⊗ S ⊂ ∆. Also, if T is the span of all the a ⊗ b + ¯b ⊗ a¯ for a, b ∈ S,thenT ⊂ ∆. It is easy to see that R⊗2 = S⊗2 ⊕ (S ⊗ Sop) ⊕ (Sop ⊗ S) ⊕ T. Denote Ω = (S ⊗ Sop) ⊕ (Sop ⊗ S) ⊕ T and let π be the projection from R⊗2 onto S⊗2.SinceΩ⊂ ∆ ⊂ ∆˜ , using Corollary 3.1, we have ker φ ker ∂/∆˜ π(ker ∂)/π(∆)˜ . Clearly, π applied on (3.1) is 0. We can take a, b, c, d in (3.2) to be in S ∪ Sop. If a ∈ S, b ∈ Sop or vice versa, the π applied to (3.2) is 0. Since a, b, c = −¯b, a,¯ c and π(c, d, a⊗b)=−π(¯b ⊗ c, d, a)=−π(¯b ⊗c, d, a¯), we can take a, b ∈ S. Similarly, we can take c, d ∈ S.Inthiscase,π applied to (3.2) gives us (4.5). Thus, π(∆) is spanned by (4.5). Letting b = d = 1 in (4.5) gives (4.6) (ac + ca) ⊗ 1 − c ⊗ a − a ⊗ c ∈ π(∆).

Since π((r1,r2) ⊗ (s1,s2)) = r1 ⊗ s1 − s2 ⊗ r2,weseethat (4.7) π((1, 1) ⊗ (a, −a)) = 1 ⊗ a + a ⊗ 1.

Since ∆=∆˜ ⊕(1⊗R−), π(∆)˜ is spanned by (4.5) and all 1⊗a+a⊗1, for a ∈ S.We have π(∆)˜ ⊂ ∆. On the other hand, taking c = 1 in (4.6) shows a⊗1−1⊗a ∈ π(∆). Thus, a ⊗ 1 ∈ π(∆)˜ for a ∈ S. Then, we have c ⊗ a + a ⊗ c ∈ π(∆)˜ for any a, c ∈ S and π(∆)˜ ⊃ ∆.So,π(∆)˜ = ∆. Since Ω ⊂ ∆ ⊂ ker ∂,weseethatπ(ker ∂)=π(ker ∂ ∩ S⊗2). Since ∂((a, 0) ⊗ (b, 0)) = ([a, b], −[b, a]), we have π(ker ∂)=ker∂. Finally, ker φ π(ker ∂)/π(∆)˜ ker ∂/∆.

16204 SHIKUI SHANG AND YUN GAO

For any nonassociative k-algebra S, the Steinberg Lie algebra st2(S) is deﬁned by generators X12(r),X21(r),T(r, s),r,s∈ S, subject to the relations( See [G2])

r → Xij(r)isak-linear mapping,

T (r, s)=[X12(r),X21(s)],

[T (r, s),X12(t)] = X12((rs)t + t(sr)),

[T (r, s),X21(t)] = −X21((sr)t + t(rs)),

[Xij(r),Xij(s)] = 0, where r, s, t ∈ S and (i, j)=(1, 2) or (2, 1). For (R, −, γ)=(S ⊕ Sop, ex, 1), we also have an isomorphism ψ : stu2(R, −, γ) → st2(S)givenbyψ(u12((r, s))) = X12(r) − X21(s)andψ(h((r1,r2), (s1,s2))) = T (r1,s1) − T (r2,s2). If S is sl2- Lie admissible, there also exists a Lie algebra epimorphism φ : st2(S) → sl2(S) satisfying φ (Xij(s)) = eij(s),φ(T (s, t)) = e11(st) − e22(ts), for s, t ∈ S, 1 ≤ i = j ≤ 2. The following diagram commutes, φ 0 → ker φ → stu2(R, −, γ) → eu2(R, −, γ) → 0 |ψ |ϕ φ 0 → ker φ → st2(S) −→ sl2(S) → 0 .

Thus, the above commutative diagram induces the isomorphism ψ|ker φ :kerφ → ker φ by the Five Lemma. By Theorem 4.2, we have the following result, Corollary 4.1 . ker φ ker ∂/∆. Remark 4.1 If S is associative, the structure of ker φ is given by as hC1(S)in [G2, Section 3]. Comparing it with ker ∂/∆, we deﬁne ∆ with less generators in (4.4)-(4.5).

Finally, we give the remark which is similar with [G1, Remark 2.64]. Remark 4.2 Let est2(R, −, γ) be the subalgebra of st2(R) generated by the ele- − −1 ∈ ≤ ≤ − ments Xij(a) γiγj Xji(¯a), a R,1 i = j 2. It is easy to see that Xij(a) −1 γiγj Xji(¯a) satisfies the defining relations (2.1)-(2.4), thus by the universal property of stu2(R, −, γ), there exists a Lie algebra epimorphism ϕ : stu2(R, −, γ) → − γ − −1 est2(R, , ) such that ϕ(uij(a)) = Xij(a) γiγj Xji(¯a). Moreover, ϕ is an isomorphism from stu2(R, −, γ)toest2(R, −, γ)whenR is associative. References [AF] B. N. Allison and J. R. Faulkner, Nonassociative coefficient algebras for Steinberg unitary Lie algebras, J. Algebra 161 (1993) 1–19. [AG] B. N. Allison and Y. Gao, Central quotients and coverings of Steinberg unitary Lie algebras, Canad. J. Math, 48 (1996) 449–482. [BK] S. Berman and Y. S. Krylyuk, Universal central extensions of twisted and untwisted Lie algebras extended over commutative rings, J. Algebra 173(1995), 302–347. [Bl] S. Bloch, The dilogarithm and extensions of Lie algebras, Alg. K-theory, Evanston 1980, Springer Lecture Notes in Math 854 (1981) 1–23. [F] J. R. Faulkner, Barbilian planes, Geom. Dedicata 30 (1989) 125–181. [G1] Y. Gao, Steinberg Unitary Lie algebras and Skew-Dihedral Homology, J.Algebra,17 (1996),261-304. [G2] Y. Gao, On the Steinberg Lie algebras st2(R), Comm. in Alg. 21 (1993) 3691–3706. [Ga] H. Garland, The arithmetic theory of loop groups, Publ. Math. IHES 52 (1980) 5–136.

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[H] T. Hodge, Lie triple systems, restricted Lie triple systems, and algebra groups, J. Algebra, 244 (2001),533–380. [Ka] C. Kassel, Kähler differentials and coverings of complex simple Lie algebras extended over a commutative ring, J. Pure and Appl. Alg. 34 (1984) 265–275. [KL] C. Kassel and J-L. Loday, Extensions centrales d’algèbres de Lie, Ann. Inst. Fourier 32 (4) (1982) 119–142. [L1] J-L. Loday, Homologies diédrale et quaternionique, Adv. in Math 66 (1987) 119–148. [L2] J-L. Loday, Cyclic homology, Grundlehren der mathematischen Wissenschaften 301, Springer 1992. [S] O. N. Smirnov , Imbedding of Lie systems in Lie algebras, to appear. [St] R. Steinberg, Lectures on Chevalley groups,(notes by J. Faulkner & R. Wilson), Yale Univ. Lect. Notes 1967.

Department of Mathematics, University of Science and Technology of China, Hefei, Anhui P.R.China 230026 E-mail address: [email protected] Department of Mathematics and Statistics, York University, Toronto, Ontario, Canada M3J 1P3 E-mail address: [email protected]

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Contemporary Mathematics Volume 478, 2009

Lusztig’s conjecture for ﬁnite classical groups with even characteristic

Toshiaki Shoji

Abstract. The determination of scalars involved in Lusztig’s conjecture concerning the characters of finite reductive groups was achieved by Waldspurger in the case of finite classical groups Sp2n(Fq)orOn(Fq)whenp, q are large enough. Here p is the characteristic of the finite field Fq.Inthispaper,we determine the scalars in the case of Sp2n(Fq)withp = 2, by applying the theory of symmetric spaces over a finite field due to Kawanaka and Lusztig. We also obtain a weaker result for SO2n(Fq)withp = 2, of split type.

0. Introduction

Let G be a connected reductive group defined over a finite field Fq of characteristic p with Frobenius map F . Lusztig’s conjecture asserts that, under a suitable parametrization, almost characters of the finite reductive group GF coincide with the characteristic functions of character sheaves of G up to scalar. Once Lusztig’s conjecture is settled, and the scalars involved there are determined, one obtains a uniform algorithm of computing irreducible characters of GF . Lusztig’s conjecture was solved in [S1] in the case where the center of G is connected. In [S2], the scalars in question were determined in the case where G is a classical group with connected center, when p is odd, and the scalars are related to the unipotent characters of GF . By extending the method there, Waldspurger [W] proved Lusztig’s conjecture (or its appropriate generalization) for Sp2n and On assuming that p, q are large enough. He also determined the scalars involved in the conjecture. But these methods cannot be applied to the case of classical groups with even characteristic. In this paper we take up the problem of determining the scalars in the case of classical groups with p = 2. We show that the scalars are determined explicitly in the case where G = Sp2n with p = 2. We also obtain a somewhat weaker result for the case SO2n of split type, when p = 2, containing the case related to the unipotent characters. The main ingredient for the proof is the theory of symmetric spaces over finite fields due to Kawanaka [K] and Lusztig [L4]. They determined the 2 multiplicity of irreducible representations of GF occurring in the induced module

1991 Mathematics Subject Classiﬁcation. Primary 20G40; Secondary 20G05. Key words and phrases. ﬁnite classical groups, representation theory.

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2 GF IndGF 1 in the case where the center of G is connected (for arbitrary characteristic). 2 Using this, one can determine the scalars for GF in many cases for a connected classical group with connected center, with arbitrary characteristic. On the other hand, it was shown in [S3] that there exists a good representatives F of C for a unipotent class C in Sp2n or SON for arbitrary characteristic. This m implies that the generalized Green functions of GF turn out to be polynomials in q (more precisely, rational functions in q if p = 2) for various extension ﬁeld Fqm , and so certain values of almost characters are also rational functions in q.This makes it possible to apply some sort of specialization argument for the character m values of GF for any m ≥ 1, and one can determine the scalars of GF which are 2 related to the unipotent characters, from the result for GF . Thus we rediscover the results in [S2]. But this method works also for p =2,andfromthiswecan deduce the result for GF . The main result of this paper was announced in the 4th International Confer- ence on Representation Theory, Lhasa, 2007.

1. Lusztig’s conjecture

1.1. Let k be an algebraic closure of a finite field Fq of characteristic p.LetG be a connected reductive algebraic group defined over k. We fix a Borel subgroup B of G, and a maximal torus T contained in B, and a Weyl group W = NG(T )/T of G with respect to T .LetDG be the bounded derived category of constructible Q¯ l-sheaves on G,andletMG be the full subcategory of DG consisting of perverse sheaves. Let S(T ) be the set of isomorphism classes of tame local systems on T , i.e., ⊗n the local systems L of rank 1 such that L Q¯ l for some integer n ≥ 1, invertible in k.TakealocalsystemL∈S(T ) such that w∗L Lfor some w ∈ W .Then L ∈D L∈S one can construct a complex Kw G as in [L2, III, 12.1]. For each (T )we denote by GL the set of isomorphism classes of irreducible perverse sheaves A on G p i L L such that A is a constituent of the i-th perverse cohomology sheaf H (Kw )ofKw for any i, w.ThesetG of character sheaves on G is defined as G = L∈S(T ) GL.

1.2. We consider the Fq-structure of G, and assume that G is defined over Fq with Frobenius map F . We assume that B and T are both F -stable. We assume further that the center of G is connected. Let G∗ be the dual group of G and T ∗ a maximal torus of G∗ dual to T . By fixing an isomorphism ι : k∗ Q/Z (Q is the subring of Q consisting of elements whose numerator is invertible in k), we have an ∗ ∗ ∗ ∗ isomorphism f : T S(T ) (see e.g., [S1, II, 1.4, 3.1]). Let W = NG(T )/T be ∗ ∗ the Weyl group of G .ThenW may be identified with W = NG(T )/T , compatible with f. F acts naturally on S(T ), via F −1 : L → F ∗L,andtheactionofF on S(T ) corresponds to the action of F −1 on T ∗ via f. For each s ∈ T ∗ such that the conjugacy class {s} in G∗ is F -stable, we put ∗ Ws = {w ∈ W | w(s)=s}, ∗ Zs = {w ∈ W | F (s)=w(s)}.

Then Zs is non-empty, and one can write Zs = z1Ws for some element z1 ∈ Zs. Since the center of G is connected, ZG∗ (s) is connected reductive, and Ws is a Weyl −1 ∗ → group of ZG (s). We choose z1 so that γ = γs = z1 F : Ws Ws leaves invariant the set of simple roots of ZG∗ (s) determined naturally from B and T .

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Similarly, for any L∈S(T ) such that F ∗L L, we deﬁne ∗ WL = {w ∈ W | w L L}, ∗ −1 ∗ ZL = {w ∈ W | F L (w ) L}.

Then WL (resp. ZL) is naturally identified with Ws (resp. Zs). 1.3. Let Irr GF be the set of irreducible characters of GF . Then Irr GF is partitioned into a disjoint union of subsets E(GF , {s}), where s ∈ T ∗ and {s} runs over all the F -stable semisimple classes in G∗. According to [L1], two parameter F sets X(Ws,γ)andX(Ws,γ) are attached to E(G , {s}), and a non-degenerate pairing { , } : X(Ws,γ) × X(Ws,γ) → Q¯ l is defined. Here X(Ws,γ) is a finite set, and X(Ws,γ) is an infinite set with a free action of the group M of all roots of ¯ ∗ unity in Ql . More precisely, there exists a set X(Ws)withγ-action, and a natural γ map X(Ws,γ) → X(Ws) whose image coincides with X(Ws) ,thesetofγ-fixed points in X(Ws). In the case where γ acts trivially on Ws, X(Ws,γ)coincideswith γ X(Ws,γ) × M. In general, the orbits set X(Ws,γ)/M is in bijection with X(Ws) . F Now the set E(G , {s}) is parametrized by X(Ws,γ). We denote by ρy the F irreducible character in E(G , {s}) corresponding to y ∈ X(Ws,γ). In turn, for each x ∈ X(W ,γ), an almost character R is defined as s x l(z1) (1.3.1) Rx =(−1) {y, x}∆(y)ρy,

y∈X(Ws,γ) where ∆(y)=±1 is a certain adjustment in the case of exceptional groups E7,E8. If c is the order of γ on Ws, the almost characters Rx are determined, up to a c-th root unity multiple, by the M-orbit of x in X(Ws,γ). 1.4. It is known by [L2, V], that the set GL is parametrized by X(WL)= X(Ws) under the identification WL Ws.Foreachy ∈ X(Ws), we denote by Ay F the corresponding character sheaf in GL.LetG be the set of F -stable character ∗ F F sheaves, i.e., the set of A ∈ G such that F A A.ThenG = L GL ,where L runs over the elements in S(T ) such that (Fw)∗L Lfor some w ∈ W .The F γ F set GL is parametrized by X(Ws) .ForeachA ∈ GL , we fix an isomorphism ∗ ∼ φA : F A −→ A as in [L2, V, 25.1]. Then φA is unique up to a root of unity multiple. F → ¯ We define a class function χA = χA,φA as the characteristic function G Ql of A. In the case of classical groups, we have the following theorem, which is a (partial) solution to the Lusztig’ conjecture. Theorem 1.5 ([S1,II, Theorem 3.2]). Assume that G is a (connected) classical group with connected center. Then for each x ∈ X(Ws,γ), there exists an algebraic number ζx of absolute value 1 such that

Rx = ζxχAx¯ , γ where x¯ is the image of x under the map X(Ws,γ) → X(Ws) .

1.6. Assume that G is a connected classical group with connected center. Let P be an F -stable parabolic subgroup of G containing B,andL be an F -stable Levi subgroup of P containing T , UP the unipotent radical of P .ThenWL = NL(T )/T is a Weyl subgroup of W ,andBL = B ∩ L is a Borel subgroup of L containing T . Let L be the set of character sheaves on L. We assume that LL contains a cuspidal character sheaf A0 for L∈S(T ), where L is Fw-stable for some w ∈ W .ThenA0

4210 TOSHIAKI SHOJI may be expressed by the intersection cohomology complex as A0 =IC(Σ,E)[dim Σ], where Σ is the inverse image of a conjugacy class in G = G/Z0(G) under the natural map π : G → G,andE is a cuspidal local system on Σ. The pair (Σ,E), or its restriction on the conjugacy class, is called a cuspidal pair on G. Then either L is the maximal torus or L has the same type as G,andA0 is a unique cuspidal G character sheaf contained in LL. Consider the induced complex K =indP A0 on G.ThenK is a semisimple perverse sheaf on G whose components are contained in G. By Lemma 5.9 in [S1,I], the endomorphism algebra EndMG K is isomorphic to the group algebra Q¯ l[WE ]ofWE ,where −1 ∗ WE = {n ∈ NG(L) | nΣn = Σ,ad(n) E E}/L, −1 ∗ ZE = {n ∈ NG(L) | F (nΣn )=Σ,(Fn) E E}/L. On the other hand, if we choose a positive integer r large enough, the set r r E(LF , {s}) contains a unique cuspidal character δ of LF ,wheres ∈ T ∗ corresponds to L under f. We deﬁne −1 w Wδ = {w ∈ NW (WL) | wBLw = BL, δ δ}, −1 Fw Zδ = {w ∈ NW (WL) | wBLw = BL, δ δ}. Since LL contains a unique cuspidal character sheaf, we have Wδ WE ,Zδ ZE by [S1, I, (5.16.1)]. Moreover there exists w1 ∈ Zδ such that Zδ = w1Wδ and that γ1 = Fw1 : Wδ → Wδ gives rise to an automorphism of the Coxeter group Wδ. ∧ ∧ We denote by (Wδ) the set of irreducible characters of Wδ,and(Wδ)ex the subset ∧ of (Wδ) consisting of γ1-stable characters. Then A0 is Fw1-stable, and for each ∈ ∧ ∈ ∈ γ E (Wδ)ex,thereexistsxE X(Ws,γ)and¯xE X(Ws) such that AE = Ax¯E is an F -stable character sheaf in GL which is a simple component of K corresponding ∈ ∈E F r { } to E EndM K.Moreover,ρxE (G , s )isanF -stable irreducible character r GF which is a constituent of the Harish-Chandra induction IndP F r δ corresponding to ∧ ∈ r E (Wδ)ex, and the image of the Shintani descent ShF /F of ρxE determines the F almost character RxE of G . (For the Shintani descent, see [S1]).

1.7. Let Lw1 be an F -stable Levi subgroup twisted by F (w1), i.e., Lw1 = −1 −1 αLα for α ∈ G such that α F (α)=F (˙w1) for a representativew ˙ 1 ∈ NG(L) ∈Z −1 ∼ −1 ∗ of w1 E . Then by ad(α ):Lw1 −→ L,ad(α ) A0 gives rise to an F -stable cuspidal character sheaf on Lw1 which we denote by A0. If we ﬁx an isomorphism ∗E ∼ E w1 ∗ ∼ ϕ0 :(F w˙ 1) −→ , ϕ0 induces an isomorphism ϕ0 : F A0 −→ A0 on Lw1 .Wechoose ∗ ∼ w1 φ : F A A as φ = ϕ . Then by Theorem 1.5, we have A0 0 −→ 0 A0 0

Lw (1.7.1) R 1 = ζ χ 0 0 A0 L ζ ∈ ¯ ∗ R w1 LF for some 0 Ql of absolute value 1, where 0 is the almost character of w1 ∈ corresponding to A0 Lw1 . The following result was proved in [S1] in the course of the proof of the main theorem. (Note that in [S1], the constants ε0ξA0 and ε0ξAE are used. But the proof shows that these constants are indeed given by ζ0 = ε0ξA0 .)

Lemma 1.8 ([S1, II, Lemma 3.7]). Let ζ0 be as in (1.7.1). Then we have − dim Σ RxE =( 1) ζ0χAE ∈ ∧ for any E (Wδ)ex.

LUSZTIG’S CONJECTURELUSZTIG’S FOR CONJECTURE FINITE CLASSICAL FOR FINITE GROUPSWITH CLASSICAL GROUPSEVEN CHARACTERISTIC2115

1.9. Lemma 1.8 shows that the determination of the scalars ζx appeared in Theorem 1.5 is reduced to the case of cuspidal character sheaves. We note that it is further reduced to the case of adjoint groups. In fact, let A0 be an F -stable cuspidal character sheaf contained in GL.Letπ : G → G be as before. Then ∗ A0 can be written as A0 E0 ⊗ π A¯0[d], where d =dimZ(G), and E0 is a local E ∈S system on G which is the inverse image of 0 (G/Gder) under the natural map G → Gder (Gder is the derived subgroup of G), and A¯0 is a cuspidal character sheaf on G¯.SinceA¯0 is a unique cuspidal character sheaf in G¯L, A¯0 is F -stable. ∗ ¯ E ∗ ∼ Then π A0 is F -stable and so 0 is also F -stable. Then φA0 : F A0 −→ A0 is given ⊗ ∗ ∗ ¯ ∼ ¯ ¯ by φA0 = ϕ0 π φA¯0 ,whereφA¯0 : F A0 −→ A0 is the map chosen for A0,andϕ0 ∗E ∼ E is the pull-back of the canonical isomorphism F 0 −→ 0. Hence χA0 is written ⊗ ∗ ∗ as χA0 = θ0 π χA¯0 ,whereπ χA¯0 is the pull-back of χA¯0 under the induced F F F map π : G → G¯ ,andθ0 is a linear character of G corresponding to E0.A similar description works also for almost characters. Let R0 (resp. R¯0)bethe F F almost character of G (resp. G¯ ) corresponding to A0 (resp. A¯0). Then we have ∗ R0 = θ0 ⊗ π R¯0. (This follows from the fact that if δ is a cuspidal irreducible F r character of G corresponding to A0 for suﬃciently large r,thenδ can be written F r as δ = θ⊗δ¯,whereδ¯ is a cuspidal irreducible character of G¯ corresponding to A¯0, r and θ is an F -stable linear character of GF , and by applying the Shintani descent on δ.) Thus ζ0 for A0 coincides with ζ0 for A¯0.

2. Generalized Green functions

2.1. Under the setting in 1.6, we further assume that GF is of split type. Let L be as before. Assume that A0 is a cuspidal character sheaf on L of the form 0 A0 = IC(Σ,E)[dim Σ], where Σ = Z (L) × C with a unipotent class C in L and E = Q¯ l E for a cuspidal local system E on C.ThenWE = W = NG(L)/L. For each w ∈W,letLw be an F -stable Levi subgroup of G obtained from L by −1 −1 twisting w as in 1.7, i.e., Lw = αLα with α ∈ G such that α F (α)=F (˙w) −1 −1 ∗ for a representativew ˙ ∈ NG(L)ofw.PutΣw = αΣα , Ew =ad(α ) E,alocal system on Σw. We assume that the pair (C, E )isF -stable, and ﬁx an isomorphism ∗ ∼ ∗ ∼ ϕ0 : F E −→ E . Then one can construct an isomorphism (ϕ0)w : F Ew −→ Ew as in ∗ ∼ [L2, II, 10.6], and this induces an isomorphism ϕw : F Kw −→ Kw,whereKw is a E G complex induced from the pair (Σw, w). Note that Kw is isomorphic to indP A0, ∈W with a speciﬁc mixed structure twisted by w . WedenotebyχKw,ϕw the characteristic function of Kw with respect to ϕw. W→W Since L is of the same type as G,andF is of split type, γ1 : is identity. G ⊗ Let K =indP A0 = E∈W∧ VE AE be the decomposition of K into simple ∧ components, where AE is a character sheaf corresponding to E ∈W ,andVE is the multiplicity space of AE which has a natural structure of irreducible W-module ∗ ∼ corresponding to E. Then there exists a unique isomorphism φAE : F AE −→ AE for each E ∈W∧ such that

(2.1.1) χKw,ϕw = Tr (w, VE)χAE . E∈W∧ F Let Guni be the unipotent variety of G,andGuni be the set of F -ﬁxed points in Guni. F G The restriction of χKw,ϕw on G is the generalized Green function Q E uni Lw ,Cw, w ,(ϕ0)w −1 E −1 ∗E ([L2, II, 8.3]), where Cw = αCα , w =ad(α ) ,localsystemonCw,and

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E (ϕ0)w is the restriction of (ϕ0)w on w. On the other hand, by the generalized ∧ Springer correspondence, for each E ∈W ,thereexistsapair(C1, E1), where C1 is a unipotent class in G and E1 is a G-equivariant simple local system on C1,such that | E 0 (2.1.2) AE Guni IC(C1, 1)[dim C1 +dimZ (L)]. E ∗ ∼ Now the pair (C1, 1)isF -stable, and φAE : F AE −→ AE determines an isomor- ∗E ∼ E phism ψE1 : F 1 −→ 1 via (2.1.2) (cf. [L2, V, 24.2]). In other words, the choice ∗E ∼ E F → ¯ of ϕ0 : F −→ determines ψE1 .Letχ(C1,E1) : C1 Ql be the characteristic E function of 1 with respect to ψE1 . Lusztig ([L2, V, 24]) gave an algorithm of com- F puting χ = χ on G .Hereχ | F is expressed in terms of a linear AE AE ,φAE uni AE G uni E combination of various χ(C ,E ),where(C1, 1) is a pair as above corresponding to ∧ 1 1 some E ∈W .Letp be the coefficient of χ E in the expansion of χ . E,E (C1, 1) AE m Then pE,E satisfies the following property; if we replace F by F for any integer F m m>0, we obtain a similar coefficient pE,E (with respect to G ) starting from (m) m ∗E ∼ E (m) ϕ0 :(F ) −→ induced naturally from ϕ0, which we denote by pE,E .Then (m) m there exists a rational function PE,E (x) such that pE,E = PE,E (q ). (Note : It is shown in [L2, V] that PE,E turns out to be a polynomial if p is good.) ∈ F 0 Now χ(C1,E1) is described as follows; take u C1 and put AG(u)=ZG(u)/ZG(u). Then F acts naturally on AG(u), and in our setting F acts trivially on it. The set F F of G -conjugacy classes in C1 is in 1:1 correspondence with the set AG(u)(note: F F AG(u) is abelian). We denote by au arepresentativeofaG -class in C1 corresponding to a ∈ AG(u). On the other hand, the set of G-equivariant simple local ∧ systems on C1 is in 1:1 correspondence with the set AG(u) of irreducible characters ∧ F of AG(u). For each ρ ∈ AG(u) , we define a function fρ on G by uni ρ(a)ifv = u ∈ CF , (2.1.3) f (v)= a 1 ρ ∈ F 0ifv/C1 ∈ F ∈ ∧ E for v Guni.Letρ AG(u) be the character corresponding to 1. Then there ∈ ¯ ∗ exists ηE Ql of absolute value 1 such that

(2.1.4) χ(C1,E1) = ηE fρ. ∗E ∼ E ∈ F Note that ηE depends on the choice of ϕ0 : F −→ and on the choice of u C1 . We have the following theorem. Theorem 2.2 ([S3]). Let G be a classical group, simple modulo center. Assume that the derived subgroup of G does not contain the Spin group. Further assume ∈ F that G is of split type. Then for each unipotent class C1 in G, there exists u C1 (called a split unipotent element) satisfying the following; Let (C, E ) be the pair in F ∗ ∼ L as in 2.1 and u0 ∈ C be a split element. Choose ϕ0 : F E −→ E so that the ϕ E →E E E u isomorphism ( 0)u0 : u0 u0 induced on the stalk u0 of at 0 is identity. ∈ F Choose a split element u C1 for deﬁning fρ in (2.1.3). Then ηE =1for any E ∈W∧.

∈ F 2.3. Returning to the setting in 2.1, we choose a split element u C1 for ∈ F m each unipotent class C1 of G.Thenu C1 for any integer m>0, (in fact, it is a F m (m) ∈ F m split element with respect to G ), and we choose ua C1 , a representative of F m F m ∈ G -class in C1 for each a AG(u). For later discussion, we prepare a notation.

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(m) (m) Assume given a family of functions h = {h }m>0,whereh is a class function F m on Guni .Thenwesaythath is a rational function in q if there exists a rational (m) (m) m function HC1,a(x)foreachpair(C1,a) such that h (ua )=HC1,a(q ). For each E ∈W∧, we have an isomorphism φ(m) :(F m)∗A ∼ A ,andone AE E −→ E (m) F m can deﬁne a function χ = χ (m) on G . Then in view of Theorem 2.2, AE AE ,φ AE (m) Lusztig’s algorithm implies that {χ | F m }m>0 is a rational function in q. AE Guni Thus, by (2.1.1) we have the following corollary. Corollary 2.4. Assume that G is as in Theorem 2.2, and that GF is of split G type. The generalized Green function Q E can be expressed as a rational Lw ,Cw , w,(ϕ0)w function in q.

(m) 2.5. More generally, if there exists a family of values h = {h ∈ Q¯ l}m>0 such that h(m) = H(qm) for some rational function H(x), we say that h is a rational function in q.

3. Cuspidal character sheaves

3.1. Let G be an adjoint simple group of classical type. We assume that 0 G is of split type over Fq.LetG be the set of cuspidal character sheaves on G. A ∈ G0 isgivenintheformA = IC(C,E)[dim C], where C is a conjugacy class in G,andE is a simple G-equivariant local system on C. We shall describe the cuspidal character sheaves on G (cf. [L2, V, 22.2, 23.2], see also [S2, 6.6]) and their mixed structures. 0 (a) G = PSp2n (n ≥ 1) with p : odd. G is empty if n is even. Assume that 2 n is odd. Then for each pair (N1,N2) such that Ni = di + di for some integers di ≥ 0andthatn = N1 + N2, one can associate cuspidal character sheaves on G as follows. Let C be a conjugacy class of g = su = us,wheres is a semisimple 0 × {± } element of G such that ZG(s) is isomorphic to H =(Sp2N1 Sp2N2 )/ 1 ,andu 0 is a unipotent element of ZG(s) H such that the unipotent class C0 containing u gives a unique cuspidal pair (C0, E0) with unipotent support of H.Here(C0, E0) E is described as follows. There exists a cuspidal pair (Ci, i)forSp2Ni such that C0 = C1 × C2 and E0 = E1 E2.Chooseu =(u1,u2) ∈ C0 such that ui ∈ Ci. ∈ ∧ E ∈ Let ρi AHi (ui) corresponding to i,whereHi = Sp2Ni .Thenρ1 ρ2 × ∧ (AH1 (u1) AH2 (u2)) factors through AH (u) and deﬁnes an irreducible character ρ0 of AH (u) corresponding to E0. Now assume that N1 = N2.ThenZG(s) is connected, and so AG(g)=AH (u), ∧ and ρ0 gives an irreducible character ρ ∈ AG(g) which determines a local system E E on C, and we denote by AN1,N2 the character sheaf corresponding to (C, ). Next assume that N1 = N2.ThenAG(s) Z/2Z and AH (u) is a subgroup of AG(g) AG(g) of index 2. We have Ind ρ0 = ρ + ρ ,whereρ, ρ are linear characters of AH (u) AG(u). If we write E, E the simple local system on C corresponding to ρ, ρ ,then the pairs (C, E), (C, E ) are both cuspidal pairs of G.WedenotebyA ,A N1,N2 N1,N2 the cuspidal character sheaves on G corresponding to (C, E), (C, E ), respectively. The set G0 consists of these elements. We shall ﬁx a mixed structure on (C, E). Since s ∈ GF , H is F -stable, and E ∈ F so (C0, 0)isalsoF -stable. Choose u =(u1,u2) C0 such that ui are split

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∈ F ∗E ∼ E elements in Sp2Ni ,andﬁxs T appropriately. We choose ϕ0 : F −→ so that the induced isomorphism (ϕ0)g : Eg →Eg on the stalk Eg at g is identity. Then ϕ induces an isomorphism ϕ : F ∗A ∼ A . We deﬁne φ = φ by 0 N1,N2 −→ N1,N2 A AN1,N2 (dim G−dim C)/2 φA = q ϕ. A similar construction is applied also for (C, E ).

0 (b) G = PSOm (m ≥ 3) with p : odd. G is empty unless m is either odd or divisible by 8. Note that PSOm = SOm if m is odd. To each pair (N1,N2)such 2 ≥ that Ni = di for some di 1andthatm = N1 + N2, one can associate cuspidal character sheaves A associated to (C, E) as follows. Let C be the conjugacy class of 0 G containing g = su = us,wheres is a semisimple element such that H = ZG(s) × × {± } is isomorphic to SON1 SON2 if m is odd, and to (SON1 SON2 )/ 1 if m is 0 even, and u is a unipotent element in ZG(s) H such that the unipotent class C0 containing u gives a unique cuspidal pair (C0, E0) with unipotent support on H. × E E E E Here C0 = C1 C2, 0 1 2 with the cuspidal pair (Ci, i)onSONi .Choose ∈ ∈ ∈ ∧ E u =(u1,u2) C0 such that ui Ci.Letρi AHi (ui) corresponding to i,where ∈ × ∧ Hi = SONi .Thenρ1 ρ2 (AH1 (u1) AH2 (u2)) factors through AH (u)and ∧ gives an irreducible character ρ0 ∈ AH (u) corresponding to E0. Depending on the structure of AG(s), the three cases occur. (i) The case where N1 =0orN2 =0.Inthiscase,ZG(s) is connected and so ∧ AG(u) AH (u). ρ0 gives ρ ∈ AG(g) , which determines a local system E on C, E and (C, ) corresponds to the cuspidal character sheaf AN1,N2 . (ii) The case where N1 > 0,N2 > 0,N1 = N2.ThenAG(s) Z/2Z,and AG(g) AH (u) is regarded as an index 2 subgroup of AG(g). We have Ind ρ0 = ρ + ρ AH (u) ∧ for ρ, ρ ∈ AG(g) .IfwewriteE, E the simple local system corresponding to ρ, ρ , (C, E), (C, E ) are both cuspidal pairs for G. WedenotebyA ,A the N1,N2 N1,N2 cuspidal character sheaves corresponding to them. (iii) The case where N1 = N2.InthiscaseAG(s) Z/2Z × Z/2Z and so AG(g) AG(g)/AH (u) Z/2Z × Z/2Z.Ind ρ0 decomposes into 4 irreducible (lin- AH (u) ear) characters, ρ, ρ ,ρ ,ρ of AG(g). Correspondingly, we have simple local systems E, E , E , E on C, and all of them give cuspidal pairs on G. Wedenoteby A ,A , A , A the cuspidal character sheaves corresponding to N1,N2 N1,N2 N1,N2 N1,N2 them. All of the above three cases give the set G0. We shall fix a mixed structure F on cuspidal character sheaves. Since s ∈ G , H is F -stable, and so (C0, E0)is ∈ F F -stable. Take u =(u1,u2) C0 such that ui are split elements in SONi ,andfix F ∗ ∼ s ∈ T .Wechooseϕ0 : F E −→ E so that the induced isomorphism (ϕ0)g : Eg →Eg ∗ ∼ on the stalk Eg at g is identity. ϕ0 induces an isomorphism ϕ : F A −→ A for (dim G−dim C)/2 A = AN1,N2 . We define φA by φA = q ϕ. We define similarly for A ,A ,A . N1,N2 N1,N2 N1,N2

0 2 (c) G = Sp2n (n ≥ 1) with p =2. G is empty unless n = d + d for some d ≥ 1. Assume that n = d2 +d.ThenG contains a unique cuspidal pair (C, E). The set G0 consists of a single character sheaf A associated to (C, E). We fix a mixed structure of A. C is an F -stable unipotent class of G, and we take a split element F ∗ ∼ u ∈ C . We fix an isomorphism ϕ0 : F E −→ E so that the induced isomorphism ∗ ∼ (ϕ0)u : Eu →Eu on the stalk Eu of E at u is identity. ϕ0 induces ϕ : F A −→ A.We (dim G−dim C)/2 define φA by φA = q ϕ.

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0 2 (d) G = SO2n with p =2(n ≥ 1). G is empty unless n =4d for some d ≥ 1. Assume that n =4d2.ThenG contains a unique cuspidal pair (C, E). The set G0 consists of a single character sheaf A associated to (C, E). C is an F -stable F unipotent class of G, and we take a split element u ∈ C . We deﬁne φA in a similar (dim G−dim C)/2 way as in the case (c), by φA = q ϕ. We show the following lemma.

Lemma 3.2. Let ρ be the irreducible character of AG(g) corresponding to the local system E on C,asin3.1.Thenρ is a linear character such that ρ2 =1.A similar fact holds also for ρ,ρ,ρ if there exists any. Proof. Let ρ be one of the characters ρ, ρ,ρ,ρ if there exists any. It is 2 enough to show that ρ(a ) = 1 for any a ∈ AG(g). By investigating the structure of AG(g), we see that AG(g) is an elementary abelian 2-group if N1 = N2.Thus 2 in this case, ρ(a ) = 1. We assume that N1 = N2.Thenp is odd, and G is PSp2n or PSO2n. Assume that G = PSp2n.WehaveAG(g) σ AH (u), where σ is an element of order 2 permuting two factors of AH (u). In this case a ∈ AG(g)isof order 2 or 4. If a has order 2, there is nothing to prove. Assume that a has order 2 AG(g) 4. Then we have a ∈ AH (u). Put θ =Ind ρ0.SinceAH (u)isanelementary AH (u) 2 abelian 2-group, and ρ0 is σ-stable, we see that θ(a )=|AG(g)|/|AH (u)|.This 2 shows that ρ(a ) = 1 for any irreducible factor ρ of θ. Next assume that G = SO2n. In this case, AG(g) σ AH (u), where AH (u) is an elementary abelian 2-group containing AH (u) as an index 2 subgroup, and σ is an element of order 2 acting on AH (u). σ stabilizes AH (u) permuting their two factors, and ρ0 is σ-stable. Thus a similar argument shows that ρ(a2) = 1. The lemma is proved.

4. Symbols and unipotent characters

4.1. Irreducible characters contained in E(GF , {1}) are called unipotent characters. In the case of classical groups, unipotent characters are parametrized by a combinatorial object called symbols. In this section, we review unipotent characters of classical groups. Let G be a classical group over Fq of type Bn,Cn or Dn. We assume that F F G is of split type if G is of type Dn. The set of unipotent characters of G is parameterized by symbols. A symbol is an (unordered) pair S of finite subsets T { } S ∼ S { }∪ of 0, 1, 2,... modulo the shift operation T T with S = 0 (S +1), { }∪ S T = 0 (T + 1). The rank of a symbol Λ = T is defined by . / |S| + |T |−1 2 r(Λ)= λ + µ − , 2 λ∈S µ∈T where [z] denotes the largest integer which does not exceed z. The defect d(Λ) of Λ is defined by the absolute value of |S|−|T |. The rank and the defect are independent of the shift operation. ≥ d For each integer d 0, we denote by Φn the set of symbols of rank n and defect S d. In the case where Λ = T is defect 0, Λ is said to be degenerate if S = T ,andis 0 said to be non-degenerate otherwise. We denote by Φn the set of symbols of rank

10216 TOSHIAKI SHOJI n and defect 0, where the degenerate symbols are counted twice. We put d + 0 d Φn = Φn, Φn = Φn Φn . d:odd d≡0(mod4) F Then the unipotent characters of G of type Bn or Cn (resp. Dn of split type) + are parametrized by Φn (resp. Φn ). In the notation of 1.3, Ws = W and γ =1 + since F is of split type, and Φn or Φn is nothing but X(Ws,γ)=X(W, 1). We F ∈ + denote by ρΛ the unipotent character of G corresponding to Λ Φn or Φn .The unipotent cuspidal character exists if and only if n = d2 +d (resp. n =4d2)forsome integer d ≥ 1ifG is of type Bn or Cn (resp. Dn). In these cases, the symbol Λc (the cuspidal symbol) corresponding to the (unique) cuspidal unipotent character is given as follows. 0, 1, 2,...,2d Λ ∈ 2d+1 G B C ,n d2 d , c = − Φn ( :type n or n = + ) 0, 1, 2,...,4d − 1 Λ ∈ 4d G D ,n d2 . c = − Φn ( :type n =4 )

+ 4.2. We introduce a notion of families in Φn or Φn.TwosymbolsΛ,Λ belong S S ∪ ∪ to the same family if Λ,Λ are represented by T , T such that S T = S T ∩ ∩ + ∈ and that S T = S T . Families give a partition of Φn or Φn .AsymbolΛ Φn S { } of defect 1 is called a special symbol if Λ = T with S = a0,a1,...,am and { } ≤ ≤ ≤ ··· ≤ ≤ T = b1,...,bm such that a0 b1 a1 bm am. Similarly, a symbol ∈ + S { } Λ Φn of defect 0 is called a special symbol if Λ = T with S = a1,...,am , T = {b1,...,bm} such that a1 ≤ b1 ≤ ··· ≤ am ≤ bm. Each family contains a unique special symbol. Let F be a (non-degenerate) family. Then any symbol Λ ∈F can be expressed as 0 Z2 (Z1 − M) Λ = ΛM = 0 , Z2 M for some M,whereZ1,Z2 are determined by F; Z2 is the set of elements which appearinbothrowsofΛ, Z1 is the set of singles in Λ,andM is a subset of Z1. The map M → ΛM gives a bijective correspondence between the set of subsets M of Z1 such that |M|≡d1 (mod 2) and F,where|Z1| =2d1 +1(resp. |Z1| =2d1) F⊂ F⊂ + ≥ F⊂ + if Φn (resp. Φn ) for some integer d1 1. (In the case of Φn ,we further assume that the smallest element in M is bigger that that of Z1 − M.) In F ⊂ particular, the special symbol in can be written as ΛM0 for some M0 Z1 such that |M0| = d1.ForM ∈ Z1, put M = M0 ∪ M − M0 ∩ M. We deﬁne a pairing { , } : F×F→Q by

1 |M ∩M | (4.2.1) {Λ ,Λ } = (−1) , M M 2f − F⊂ F⊂ + where f = d1 (resp. f = d1 1) if Φn (resp. Φn ). We extend this pairing + F F F F to the pairing on Φn or Φn by requiring that and are orthogonal if = , { } + which we denote by the same symbol. Note that the pairing , on Φn or Φn coincides with the pairing { , } on X(W, 1) given in 1.3. Hence, for each Λ ∈F, the almost character R is given as Λ (4.2.2) RΛ = {Λ,Λ}ρΛ , Λ ∈Xn

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+ where Xn =Φn or Φn according to the cases G is of type Bn or Cn,orG is of type Dn. By the property of the pairing { , }, one can also write, for each Λ ∈F, (4.2.3) ρΛ = {Λ,Λ}RΛ . Λ ∈Xn

4.3. Assume that GF contains a cuspidal unipotent character, and denote by Fc the family containing the cuspidal symbol Λc. Then the special symbol Λ0 contained in F is given as follows. c 0, 2, 4,...,2d Λ = (G :typeB or C ,n= d2 + d), 0 , ,..., d − n n 1 3 2 1 0, 2,...,4d − 2 Λ = (G :typeD ,n=4d2). 0 1, 3,...,4d − 1 n

In the case where G is of type Bn or Cn,wehaveZ1 = {0, 1,...,2d} and M0 = {1, 3,...,2d−1}. In the case where G is of type Dn,wehaveZ1 = {0, 1,...,4d−1} and M0 = {1, 3,...,4d − 1}. In both cases, Λc is given by Λc = ΛM with M = ∅.

We denote by R0 = RΛc the cuspidal almost character.

4.4. Let A0 be the cuspidal character sheaf of G corresponding to R0 = RΛc . Let (C, E) be the cuspidal pair corresponding to A0. Then it is contained in the list in 3.1. If p = 2, it is uniquely determined since G0 consits of a single element. In the case where p = 2, the explicit correspondence is known by [S2, Prop. 6.7], see also [L3]. The conjugacy class is given as follows (though we don’t need it in later discussions). We use the notation in 3.1. Let g = su = us ∈ C with 0 2 H = ZG(s). Assume that G = PSp2n with n = d + d.ThenH is isogeneous 2 to Spd2+d × Spd2+d. Assume that G = PSO2n+1 with n = d + d.ThenH is 2 isogenous to SO(d+1)2 × SOd2 . Assume that G = PSO2n with n =4d .ThenH is isogeneous to SO4d2 × SO4d2 .

5. Symmetric space over finite fields In this section, we apply the theory of symmetric space over finite fields to the problem of determining the scalars ζx occurring in Lusztig’s conjecture (Theorem 2 1.5)] for GF .

5.1. Let G be a connected reductive group over a finite field Fq with Frobenius 2 2 map F . We consider the symmetric space GF /GF . For a class function f on GF , we define m2(f)by

2 GF 1 m (f)=Ind F 1,f 2 = f(x). 2 G GF |GF | x∈GF

In the case where G has a connected center, m2(ρ) is determined by Kawanaka [K], 2 Lusztig [L4] for any irreducible character ρ of GF . F 5.2. Let C be an F -stable conjugacy class in G.Takex ∈ C and let AG(x) be the component group of ZG(x) as before. F acts naturally on AG(x). We assume F F that F acts trivially on AG(x). Then the set of G -conjugacy classes in C is in bijection with the set AG(x)/∼ of conjugacy classes in AG(x). The correspondence is given as follows; for each a ∈ AG(x), take a representativea ˙ ∈ ZG(x). There ∈ −1 −1 F exists ha G such that ha F (ha)=˙a.Thenxa = haxha in contained in C ,

12218 TOSHIAKI SHOJI and the set {xa | a ∈ AG(x)/ ∼} gives a complete set of representatives of the GF -conjugacy classes in CF . F 2 The above description works also for the case of C .Wedenoteby{ya | a ∈ F 2 F 2 AG(x)/∼} the set of G -conjugacy classes in C . We deﬁne a class function fτ 2 on GF for each τ ∈ A (x)∧ as follows. G F 2 τ(a)ifg is G -conjugate to ya, (5.2.1) fτ (g)= 2 0ifg/∈ CF . We have the following lemma.

∧ 2 Lemma 5.3. Let τ ∈ AG(x) be a linear character such that τ =1.Thenwe have F F m2(fτ )=|C |/|G |. F F F Proof. Take g ∈ C .Theng is G -conjugate to an xa ∈ C for some −1 −1 a ∈ AG(x). Then there exists h ∈ G such that g = hxh and that h F (h)=˙a ∈ F ZG(x). Since F acts trivially on AG(x), we may choosea ˙ ∈ ZG(x) .Wehave h−1F 2(h)=h−1F (h) · F (h−1F (h)) =aF ˙ (˙a)=a ˙2.

F F 2 It follows that any element g ∈ C is G -conjugate to ga2 for some a ∈ AG(x). 2 Hence we have fτ (g)=τ(a ) = 1 by our assumption. We have 1 m (f )= f (g)=|CF |/|GF | 2 τ |GF | τ g∈CF as asserted. The lemma is proved.

5.4. Assume that G is as in 3.1, and we use the notation there. Let A = IC(C,E)[dim C] be the cuspidal character sheaf on G. (Here E represents one of the simple local systems E, E ,... on C if there exist more than one). Then A is ∧ F -stable. Let ρ ∈ AG(g) be the irreducible character corresponding to E.Note F 2 that F acts trivially on AG(g). We consider the class function fρ on Guni deﬁned as in (5.2.1) for τ = ρ. We show that −(dim G−dim C) (5.4.1) m2(fρ)=q .

In fact thanks to Lemma 3.2, one can apply Lemma 5.3 and we have m2(fρ)= F F F F |C |/|G |.SinceF acts trivially on AG(g), C splits into several G -conjugacy classes, which are parametrized by AG(g)/∼.Leta1,...,ar be the representatives F F of the conjugacy classes in AG(g), and let Ci the G -conjugacy classes in C cor- ∈C | F | | || 0 F | responding to ai.Wechoosegi i.SinceZG(gi) = AA(ai) ZG(gi) ,where A = AG(g), we have r r | F | | F | |C | | F | | |−1| 0 F |−1 C / G = i / G = ZA(ai) ZG(gi) . i=1 i=1 Here Z0 (g )F Z0 (u )F ,whereg = s u = u s and H = Z0 (s ). Note that G i Hi i i i i i i i G i Z0 (u ) Z0 (u), and since u ∈ C ,where(C , E ) is a cuspial pair with unipotent Hi i H 0 0 0 0 support, it is known by [L2,I, Prop.3.12] that ZH (u) is a unipotent group. It follows that − − |Z0 g F | |Z0 u F | qdim H dim C0 qdim G dim C . G( i) = Hi ( i) = =

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Hence we have r F F −(dim G−dim C) −1 −(dim G−dim C) |C |/|G | = q |ZA(ai)| = q . i=1 This proves (5.4.1). 2 ∗ ∼ 2 ∗ ∼ 2 ∗ ∼ Let ϕ0 :(F ) E −→ E, ϕ :(F ) A −→ A and φA :(F ) A −→ A be as in 3.1, but 2 F 2 replacing F by F . Then the characteristic function χE,ϕ0 on C coincides with fρ| F 2 .SinceA is clean, the function χA,ϕ coincides with fρ| F 2 . It follows that C C dim G−dim C χA coincides with q fρ. This implies that Lemma 5.5. Let A be a cuspidal character sheaf of G. Then we have

m2(χA)=1.

5.6. Let s ∈ T ∗ be such that the class {s} is F 2-stable. Then there exists ∈{ } 2 0 2 s0 s such that F (s0)=s0.LetH = ZG(s0). Then H is an F -stable reductive subgroup of G. It is known, since the center of G is connected, that there exists a F 2 F 2 natural bijection E(G , {s}) ↔E(H , {1}), ρ ↔ ρuni. Concerning the values of m2(ρ)andm2(ρuni), the following result is known. (For unipotent characters, we follow the notation in Section 4.) Theorem 5.7 ([K], [L4]). Let G be a connected classical group with connected center. Then ∈{ }F 2 −1 (i) If there does not exist s0 s such that F (s0)=s0 ,thenm2(ρ)=0 F 2 for any ρ ∈E(G , {s}). If there exists such s0, then under the notation F 2 of 5.6, m2(ρ)=m2(ρuni) for any ρ ∈E(G , {s}). (ii) Assume that G is of type Bn or Cn.LetF be a family in Φn such that |Z | =2d +1 (cf. 4.2). Then we have 1 1 2d1 if Λ is special, m2(ρΛ)= 0 otherwise. F + (iii) Assume that G is of type Dn.Let be a non-degenerate family in Φn such that |Z | =2d (cf. 4.2). Then we have 1 1 − 2d1 1 if Λ is special, m2(ρΛ)= 0 otherwise. If F = {Λ,Λ} is a degenerate family, then we have 1 if F is of split type, m2(ρΛ)=m2(ρΛ )= 0 otherwise. In view of (4.2.2), we have the following corollary. Corollary 5.8 ([L4]). Let G be a classical group of split type. Assume that F is a non-degenerate family. Then for any Λ ∈F, we have m2(RΛ)=1.

∗ 5.9. Let A be as in 5.4 and assume that A ∈ GL.Lets ∈ T be such that 2 the class {s} corresponds to L via f in 1.2. Then s =1.LetR0 be the almost character of GF corresponding to A as given in Theorem 1.5. Then by Lemma 5.5, we have m2(R0) =0.Notethat R0 is a linear combination of irreducible characters

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F 2 contained in E(G , {s}). Thus by Theorem 5.7, (i), there exists s0 ∈{s} such that ∈ F 2 −1 0 s0 G and that F (s0)=s0 = s0.ThenH = ZG(s0)isanF -stable reductive subgroup of G,andsoH is split over Fq2 . Recall that the almost character R0 is given as in (1.3.1). Then it is known that (−1)l(z1) = σ(H)σ(G), where σ(G)is a split rank of G with respect to Fq2 ,andviceversaforH.SinceG, H are split, H F 2 we have l(z1)=σ(H)σ(G)=1.LetR0 be the almost character of H obtained from R0 under the correspondence ρ ↔ ρuni.Thenweseethat H (5.9.1) m2(R0)=m2(R0 )=1. In fact, the ﬁrst equality follows from Theorem 5.7, (i), together with the fact that l(z1) = 1. The second equality follows from Theorem 5.7, (ii), (iii). By Theorem 1.5, we know that R0 = ζχA with some scalar ζ,ifR0 is the almost character corresponding to the character sheaf A.Sincem2(R0)=ζm2(χA), we have the following theorem by combining Lemma 5.5 with (5.9.1). Theorem 5.10. Let G be an adjoint simple group of classical type. Let A be a cuspidal character sheaf, and χA = χA,φA be the characteristic function of A on F 2 F 2 G (deﬁned as in 3.1). Let R0 be the almost character of G corresponding to A. Then we have R0 = χA. As a corollary we have the following result, which holds without any restriction on p nor q. Corollary 5.11. Let G be a connected classical group with connected center. Then the constants ζx appearing in Lusztig’s, conjecture (Theorem 1.5) can be de- 2 termined for GF in the following cases; under the notation of 1.6, assume that Wδ = Zδ. Then we have − dim Σ RxE =( 1) χAE ∈ ∧ − dim Σ for any E Wδ . In other words, we have ζxE =( 1) . Proof. Lemma 1.8 together with the argument in 1.9 shows that the determination of ζx is reduced to the case of ζ0 (the one corresponding to the cuspidal character sheaf) in the case of adjoint simple groups. We know that ζ0 =1by Theorem 5.10. Then the corollary follows from Lemma 1.8, together with 1.9 since 2 2 LF LF w1 = .

Remark 5.12. In the case where G = PSp2n or SO2n+1, Corollary 5.11 gives F 2 a complete answer for the determination of constants ζx for G since we have always Wδ = Zδ in that case. In the case where G = PSO2n, the corollary holds R A ∈ G ¯ if xE is a linear combination of unipotent characters, i.e., if xE Ql .Butit happens that Wδ = Zδ for some δ.

2 6. From GF to GF

6.1. The results Theorem 5.10 and Corollary 5.11 in the previous section 2 are only valid for the group GF . In this section, by using a certain specialization F argument, we extend those results to the group G as far as χA are concerned with unipotent characters. In the case of p = 2, this implies the extension of Theorem

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5.10 and Corollary 5.11 for the group GF of split type. We have the following theorem.

Theorem 6.2. Let G be a classical group of split type over Fq.LetA be a G ¯ χ χ cuspidal character sheaf contained in Ql ,and A = A,φA be the characteristic F F function of A over G .LetR0 be the almost character of G corresponding to A. Then we have R0 = χA. As a corollary, we have F Corollary 6.3. Let G = Sp2n or SO2n with p =2. Assume that G is of split type. Then the constants ζx appearing in Lusztig’s conjecture (Theorem 1.5) can be determined completely for Sp2n,andpartlyforSO2n. More precisely, under the notation of 1.6, assume that Wδ = Zδ. Then we have − dim Σ (6.3.1) RxE =( 1) χAE ∧ E ∈ W A ∈ G ¯ for any δ . In particular, (6.3.1) holds if E Ql in the case where G = SO2n.

Proof. As in the proof of Corollary 5.11, the determination of ζx is reduced to that of ζ0. Assume that G = Sp2n or SO2n with p = 2. In this case, it is known by 3.1, (c), (d), that a unique cuspidal character sheaf (if it exists) is always contained G ¯ in Ql . Hence Theorem 6.2 can be applied, and the corollary follows from Lemma 1.8 (see also Remark 5.12). Remark 6.4. A similar argument as in Corollary 6.3 works also for the case p A ∈ G ¯ where = 2. In particular the formula (6.3.1) holds for the case where E Ql . Thus we rediscover the results in Theorem 6.2 in [S2]. Although the argument given in [S2] can not be applied to the case where p = 2, the proof here works simultaneously for arbitrary p.

6.5. The remainder of this paper is devoted to the proof of Theorem 6.2. In what follows, we assume that G is a classical group containing a cuspidal unipotent 2 character. Hence G is of type Bn or Cn with n = d + d,oroftypeDn with n =4d2. We assume further that GF is of split type. We follow the notation in Section 4. F GF F F Let Iq be the G -module IndBF 1 induced from B to G . Then the irreducible ∧ component of Iq is in bijective correspondence with W .WedenotebyρE the F ∧ irreducible G -module occurring in Vq corresponding to E ∈ W . ρE gives a ∈ H unipotent character, which we denote by ρΛE with ΛE Xn.Let be the Iwahori- Hecke algebra over Q[u1/2,u−1/2] associated to the Coxeter system (W, S)with generators {Ts | s ∈ S}. H has a basis {Tw | w ∈ W },whereTw is deﬁned as T = T ...T for a reduced expression w = s ···s . H is characterized by w si1 sik i1 ik the following properties; (Ts − u)(Ts +1)=0,

TsTw = Tsw if l(sw)=l(w)+1, where l : W → Z>0 is the length function of W . In the case of type Bn or Dn, the generator set S of W is described as follows. Assume that W is the Weyl group of type Bn.ThenW is realized as a group of

16222 TOSHIAKI SHOJI signed permutations of I = {1, 1¯, 2, 2¯,...,n,n¯}.ThesetS of generators is given as S = {s0,s1,...,sn−1} with s0 =(1, 1)¯ ,s1 =(1, 2),...,sn−1 =(n − 1,n), and we denote by Ti the generator of H corresponding to si. Note that the subalgebra of H generated by T1,...,Tn−1 is isomorphic to the Iwahori-Hecke algebra of type An−1. Next assume that W is of tyep Dn.ThenW is a subgroup of the Weyl group of type Bn, generated by s0 =(1, 1)(2, 2),s1,...,sn−1.WedenotebyT0,T1,...,Tn−1 the corresponding generators of H. The endomorphism algebra EndGF Iq is isomorphic to the specialized algebra 1/2 −1/2 Q¯ l ⊗H via the algebra homomorphism Q[u ,u ] → Q¯ l by u → q,whichwe denote by Hq.WedenotebyEq the irreducible representation of Hq corresponding ∧ F to E ∈ W .NowIq has a structure of G ×Hq-module, and the trace for g ∈ F G ,Tw ∈Hq is written as (6.5.1) Tr ((g, Tw),Iq)= Tr (g, ρE)Tr(Tw,Eq). E∈W ∧ By replacing ρ = ρ by R by using (4.2.3), we have E ΛE Λ (6.5.2) Tr ((g, Tw),Iq)= fΛ(w)RΛ(g),

Λ∈Xn where (6.5.3) fΛ(w)= {Λ,ΛE } Tr (Tw,Eq). E∈W ∧

It is known that Tr (Tw,Eq) is a polynomial in q in the sense of 2.3. Hence fΛ(w) is also a polynomial in q. We are interested in fΛ(w) in the case where Λ is the cuspidal symbol Λc, and we want to ﬁnd some special w ∈ W such that fΛc (w) =0.LetW be the Weyl group of type Bn or Dn. Then any element of W can be expressed as a product of positive cycles and negative cycles, where the number of negative cycles is even if W is of type Dn. We have the following proposition. Proposition . ∈ 6.6 There exists an element w W such that fΛc (w) =0,where either w is a Coxeter element in W ,orw contains a positive cycle of length ≥ 2.

6.7. The proof of the proposition will be given in Section 7. Here assuming the proposition, we continue the proof of the theorem. We prove the theorem by induction on the semisimple rank of G, and so we assume that the theorem holds for the classical groups of the smaller semisimple rank. Let A be the cuspidal character sheaf on G as in the theorem, and let C be the conjugacy class which is the support of A.Wechooseg = su = us ∈ CF as in 3.1. We choose w ∈ W as in Proposition m 6.6. We consider the equation (6.5.2) simultaneously for the groups GF for any m integer m ≥ 1. Note that g ∈ CF and it has a uniform description for any m ≥ 1 m since the split unipotent element for GF is split for any extended group GF .Then we can write (6.5.2) as m (m) m (m) (6.7.1) Tr ((g, T ),I m )= f (w)(q )R (g)+f (w)(q )R (g), w q Λ Λ Λc Λc Λ= Λc H (m) F m where Tw is an element of qm ,andRΛ denotes the almost character of G . (m) By induction hypothesis and by Remark 6.4, the formula (6.3.1) holds for RΛ if Λ = Λc. We show the following lemma.

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Lemma 6.8. Assume that Λ = Λc. Then under the induction hypothesis, (m) RΛ (g) is a rational function in q.

Proof. Let A the character sheaf corresponding to R , and denote by χ(m) Λ Λ AΛ F m the characteristic function on G associated to AΛ. In view of (6.3.1), it is enough (m) to show that χ (g) ia a rational function in q.SinceA ∈ G ¯ , by applying AΛ Ql Corollary 4.10 in [S2] (a0 = 1 in the notation there since F is of split type), the m computation of χ(m)(g) is reduced to that of χ(m)(u) for a subgroup HF ,where AΛ AΛ H is a connected centralizer of some semisimple element in G.Soweconsider (m) F m G χA on Guni . AΛ is a direct summand of a certain complex K =indP A0,where Λ A ∈ L ¯ L 0 Ql is a cuspidal character sheaf of a Levi subgroup . Then the computation (m) of χ | F m is reduced to the computation of the generalized Green functions of AΛ Guni m m GF associated to LF . Hence by Corollary 2.4, we obtain the lemma.

F 6.9. Next we consider the left hand side of (6.7.1). The G -module Iq is a permutation representation of GF on GF /BF . Since the action of GF on GF /BF is independent of the isogeny, we may assume that G = SpN or SON and that F g ∈ G .LetV be the vector space over Fq of dim V = N, equipped with a non- degenerated alternating form (resp. symmetric bilinear form) f on V if G = SpN (resp. SON ). In the case where G = SON with p =2andN is even, we also consider the quadratic form Q on V .NowthesetGF /BF may be identified with the set of flags Fq as follows; A flag F =(V0 ⊂ V1 ⊂ ··· ⊂ Vn) is a sequence of subspaces of V such that dim Vi = i and that Vi are isotropic with respect to f, where N =2n, 2n+1 or 2n according to the cases where G = Sp2n,SO2n+1 or SO2n. In the case where p =2andG = SO2n, we assume further that the restriction of Q on Vn is zero. Now in the case where G = Sp2n or SO2n+1, Fq consists of all F flags on V . G acts naturally on Fq via x :(V0 ⊂ ··· ⊂ Vn) → (xV0 ⊂ ··· ⊂ xVn) F F F F F for x ∈ G ,andtheG -set Fq is identified with the G -set G /B .Inthecase F where G = SO2n, we define Fq as the set of all flags on V as above, then G F F F F acts on q with two G -orbits, q and q. Either of them can be identified with F F F F G /B , and we have a natural bijection between q and q, which is given in the ⊂ ⊂ ··· ⊂ → ⊂ ⊂ ··· ⊂ form (V0 V1 Vn) (V0 V1 Vn), (only the term Vn is changed to Vn). We consider the vector space Jq over Q¯ l with basis Fq, which is identified with Iq. By the identification Iq Jq, Hq acts on Jq, whose action is given as follows; let F =(V0 ⊂···⊂Vn). For i =1,...,n− 1, we have FTn−i = (V0 ⊂···⊂Vi−1 ⊂ W ⊂ Vi+1 ⊂···⊂Vn),

W = Vi where the sum is taken over all the isotopic subspaces W such that Vi−1 ⊂ W ⊂ Vi+1 and that dim W = i, W = Vi.InthecaseofBn,wehave FT0 = (V0 ⊂···⊂Vn−1 ⊂ W ),

W ⊃Vn−1

18224 TOSHIAKI SHOJI where the sum is taken over all the isotropic subspaces W such that dim W = n and W = V .InthecaseoftypeD ,wehave n n ⊂···⊂ ⊂ ⊂ FT0 = (V0 Vn−2 W W ), W ⊃W ⊃Vn−2 where the sum is taken over the isotropic subspaces W ⊃ W such that dim W = n, dim W = n − 1andW contains Vn−2 and some more conditions. It follows from the description of the action of H on Iq,weseethat (6.9.1) Assume that w ∈ W contains a positive cycle of length ≥ 2. Then there exists k ≥ 1 such that for any F =(V0 ⊂···⊂Vn), FTw is a linear combination of ⊂···⊂ F =(V0 Vn) such that Vk = Vk. We show the following lemma.

Lemma 6.10. Under the induction hypothesis, Tr ((g, Tw),Iqm ) is a rational function in q. Proof. The following argument was inspired by [HR2], where the combinatorial properties of Tr ((u, Tw),Iq) is discussed in the case of GLn(Fq) with a unipotent element u. First assume that w contains a positive cycle of length ≥ 2. Then by (6.9.1), for any flag F =(V0 ⊂ ··· ⊂ Vn), FTw is a linear combination of ⊂ ··· ⊂ ∈J F =(V0 Vn) such that Vk = Vk. We now prepare a notation. If v q, and F ∈Fq,wedenotebyv|F the coefficient of F in the expression of v as a linear combination of base vectors. Let F =(V0 ⊂ ··· ⊂ Vn) ∈Fq, and assume that − | | | − 1 ⊂ ··· ⊂ gFTw F =0.SincegFTw F = FTw g 1F, g F =(V0 Vn)isoftheform −1 that Vk = V = g Vk. It follows that Vk is stabilized by g.Thuswehave k Tr ((g, Tw),Iq)= gFTw|F F∈F q = gFTw|F W F=(V ⊂···⊂W ⊂···⊂V ) 0 n = gF Tw|F gF Tw|F W F =(V0⊂···⊂W ) F =(W ⊂··· ) where W runs over all the isotropic subspaces in V such that dim W = k and that gW = W .LetH = GL(W ), and H be the group of isometries Sp(W)orSO(W) ⊥ W JW F for W = W /W .LetIq q be the corresponding induced modules for H , W JW F and similarly define Iq q for H . g acts naturally on W (resp. on W ), and ∈ F ∈ F we denote by gW H (resp. gW H ) the corresponding elements. Also the I J W J W action of Tw on q induces an action on q (resp. on q )whichisgivenbyTw (resp. Tw ) with an element w (resp. w ) in the Weyl group of H (resp. H ). Then the last sum can be written as W W (6.10.1) Tr ((g, Tw),Iq)= Tr ((gW ,Tw ),Iq )Tr((gW ,Tw ),Iq ). W 0 ⊂ ··· ⊂ 0 F F Let F0 =(V0 Vn ) be the standard flag whose stabilizer in G is B , 0 and put Wk = Vk . Then there exists an F -stable maximal parabolic subgroup F F P of G containing B such that P is the stabilizer of Wk in G .LetL be an F -stable Levi subgroup of P containing T .ThenL is isomorphic to L × L,where − L = GLk and L is a similar group as G of rank n k.Letg1,...,gr (resp.

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F F g1 ,...,gs ) be representatives of the conjugacy classes in L (resp. L ) such that ∈ F F ij gij =(gi,gj ) L is conjugate to g under G .WedenotebyXq the set of −1 F W such that W = xWk and that x gx is conjugate to gij in L . Then (6.10.1) implies that r s ij L L | | (6.10.2) Tr ((g, Tw),Iq)= Xq Tr ((gi,Tw ),Iq )Tr((gj ,Tw ),Iq ), i=1 j=1

L L F F ∈ F where Iq ,Iq are corresponding induced modules for L ,L .Choosexij G −1 ij F F F such that g = xijgijxij for i, j.ThenXq is in bijection with the set ZG(g) xijL /L , | ij| | F | | F | hence Xq = ZG(g) / ZL(gij) . m We now consider (6.10.2) for any GF . Then the choice of representatives ∈ F m | ij | gij G does not depend on m, and we see that Xqm is a rational function in q. On the other hand, one can write as in (6.5.1) L Tr ((gj ,Tw ),Iq )= Tr (gj ,ρE)Tr(Tw ,Eq), ∈ ∧ E WL

(m) where WL is the Weyl group of L . By induction hypothesis, RΛ (gj ) is a rational F (m) function in q for any almost character RΛ of L . Hence Tr (gj ,ρE ) is a rational L function in q. It follows that Tr ((gj ,Tw ),Iqm ) is a rational function in q. Similarly, L andasitisknownsinceL = GLk,Tr((gi,Tw ),Iqm ) is a rational function in q. Thus we conclude that Tr ((g, Tw),Iqm ) is a rational function in q as asserted. Next assume that w is a Coxeter elment of W .Wenotethat r q if xu is regular unipotent, (6.10.3) Tr ((x, Tw),Iq)= 0otherwise,

F for x ∈ G ,wherexu is the unipotent part of x and r is the semisimple rank of G. In fact (6.10.3) is discussed in [HR2, Prop. 3.2] in the case where G = GLn. The argument there works in general if we notice that ZG(v)=ZU (v) for a regular ∈ F | F | r unipotent element v U and that ZU (v) = q ,whereU is the unipotent radical of B. (6.10.3) implies that Tr ((g, Tw),Iqm ) is a polynomial in q. Hence the lemma holds.

6.11 We now prove the theorem. In the formula (6.7.1) the left hand side (m) is a rational function in q by Lemma 6.10. For for Λ = Λc, RΛ (g) is a rational function in q by Lemma 6.8. Since fΛ(w) is a polynomial in q,andfΛc (w) =0 by Proposition 6.6, we see that R(m)(g) is a rational function in q.ByTheorem Λc 1.5, one can write R(m)(g)=ζ(m)χ(m)(g)withsomeζ(m) ∈ Q¯ ∗ of absolute value Λc A0 l 1, where A = A is the cuspidal character sheaf. We know that χ(m)(g)isa 0 Λc A0 non-zero polynomial in q. We also know by Theorem 5.10 that ζ(m) =1foreven m. It follows that R(m)(q)/χ(m)(g) is a rational function in q, and takes the value Λc A0 1 for any power of q2. Hence R(m)(g)=χ(m)(g) for any m,andwehaveζ(m) =1. Λc A0

This shows that RΛc = χA0 , and the theorem is proved (modulo Proposition 6.6).

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7. Proof of Proposition 6.6

1 7.1. Recall that Φn is the set of symbols of rank n and defect 1 as in 4.1. In the ∧ 1 case where W is the Weyl group of type Bn,thesetW is in bijection with Φn.The correspondence is given as follows; let Pn be the set of of double partitions (λ, µ) ∧ such that |λ| + |µ| = n.ThenW is parametrized by Pn. For a double partition (λ, µ) ∈Pn,wewriteλ =(λ0 ≤ λ1 ≤ ··· ≤ λm)andµ =(µ1 ≤ µ2 ≤ ··· ≤ µm) ≥ − with λi,µi 0 for some integer m. We put ai = λi+i, bi = µi+(i 1), and deﬁne the { } { } S ∈ 1 sets S, T by S = a0,a1,...,am and T = b1,...,bm .ThenΛ = T Φn,and 1 P this gives a bijective correspondence between Φn and n, and so gives a bijection 1 ∧ 1 between Φn and W .Asin6.5,wedenotebyΛE the symbol in Φn corresponding ∧ 2 to E ∈ W . Assume that n = d + d,andletΛ0 be the special symbol in Fc.Then Λ0 = ΛE ,whereE corresponds to (λ, µ) ∈Pn such that λ =(0≤ 1 ≤ 2 ≤···≤d), µ =(1≤ 2 ≤···≤d). 0 Recall that Φn is the set of symbols of rank n and defect 0 as in 4.1. In the case ∧ 0 where W is the Weyl group of type Dn, W is in bijection with Φn,whichisgiven as follows; let Pn be the set of unordered partitions (λ, µ) such that |λ| + |µ| = n, ∧ where (λ, λ) is counted twice. Then W is parametrized by Pn.For(λ, µ) ∈ Pn, we write λ =(λ1 ≤ ··· ≤ λm), µ =(µ1 ≤ ··· ≤ µm)withλi,µj ≥ 0forsome ≥ − − integer m 1. We put ai = λi +(i 1), bi = µi +(i 1), and deﬁne the sets { } { } S ∈ 0 S, T by S = a1,...,am , T = b1,...,bm .ThenΛ = T Φn and this gives 0 ∧ a bijective correspondence between Φn and W . Asin6.5,wedenotebyΛE the 0 ∈ ∧ 2 symbol in Φn corresponding to E W . Assume that n =4d and let Λ0 be the special symbol in Fc.ThenΛ0 = ΛE ,whereE corresponds to (λ, µ) ∈ Pn such that λ =(0≤ 1 ≤···≤2d − 1), µ =(1≤ 2 ≤···≤2d) First we show the following lemma.

∧ Lemma 7.2. Assume that E ∈ W is such that ΛE ∈Fc.IfE corresponds to (λ, µ) ∈Pn (resp. (λ, µ) ∈ Pn), in the case where W is of type Bn (resp. Dn), then we have ⎧ ⎪ 1 | | ⎪ (−1) λ +d(d+1)/2 if G is of type B , ⎨2d n { } Λc,ΛE = ⎪ ⎪ 1 ⎩ (−1)|λ|+d(2d−1) if G is of type D . 22d−1 n

Proof. In the case where F = Fc is the cuspidal family, Z1 = {0, 1,...,2d} (resp. Z1 = {0, 1,...,4d−1})andM0 = {1, 3,...,2d−1} (resp. M0 = {1, 3,...,4d− 1})ifG is of type Bn (resp. Dn)by4.3.MoreoverΛc = ΛM with M = ∅.Then ⊂ | | | | M = M0,andforanyM Z1 such that M = d (resp. M0 =2d), we ∩ − have M M = M0 M .NowM is written as M = Modd Mev,where Modd (resp. Mev) is the subset of M consisting of odd numbers (resp. even num- ∩ − − bers). We have M M0 =0Modd and so M0 M = M0 Modd.Moreoverwehave − − − Z1 M =((Z1)ev Mev) (M0 Modd), where (Z1)ev is deﬁned similarly. Assume ∈P ∈ P d that M corresponds to (λ, µ) n (resp. (λ, µ) n). Let γ = i=1 i = d(d+1)/2

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− (resp. γ = 2d 1 i = d(2d − 1)). Then i=1 |λ| =( a) − γ ∈ − a Z1M ≡ ( a) − γ (mod 2) ∈ − a M0 Modd ≡| − |− M0 Modd γ (mod 2). This proves the lemma.

∈ Z 1/2 −1/2 ∈ 7.3. We consider fΛc (w) [u ,u ]forsomew W , and compute it by making use of the Murnaghan-Nakayama formula for H due to Halverson and Ram [HR1]. We use the notation in 6.5. First assume that W is of type Bn and H is the corresponding Hecke algebra. For each 1 ≤ k ≤ n, we deﬁne Lk ∈Hinductively as L1 = T0 and Lk = Tk−1Lk−1Tk−1 for k =2,...,n.For1≤ k

For an element i or ¯i in I, we put |i| = |¯i| = i. For a sequence r =(r1,...,rk)of elements of I such that |r1| < |r2| < ···< |rk|, we deﬁne Tr ∈Hby ··· Tr = R1,r1 R|r1|+1,r2 R|rk−1|+1,rk . ∈ Then Tr coincides with Twr ,wherewr W is given by a cyclic notation of the signed permutation,

wr =(1, 2,...,|r1|−1,r1)(|r1| +1, |r1| +2,...,|r2|−1,r2) (7.3.1) ···(|rk−1| +1,...,|rk|−1,rk).

We also use the following cycle type expression of wr

(7.3.2) wr =[l1,...,lr], where li ∈ I is such that |li| = |ri|−|ri−1| and li is barred if ri is barred. For example, if r =(1¯, 4¯, 7, 12), then wr =[1¯, 3¯, 3, 5]. We now prepare some notation related to the skew diagram. Let λ be a double partition of size n. Apart from the notation in 7.1, we express it as λ =(λα,λβ), α β α α β β where λ ,λ are partitions. For µ ∈Pn,wewriteµ ⊆ λ if µ ⊆ λ and µ ⊆ λ . The Young diagram of λ is defined as a pair of Young diagrams of λα and λβ. We often identify the double partition and the corresponding Young diagram. For double partitions µ ⊆ λ, the skew diagram λ/µ =((λ/µ)α, (λ/µ)β) is defined naturally. For each node x in the skew diagram λ/µ, the content ct(x) is defined as follows; uj−i+1 if x is in position (i, j)in(λ/µ)α, (7.3.3) ct(x)= −uj−i if x is in position (i, j)in(λ/µ)β. The skew diagram X is called a border strip if it is connected and does not contain any 2×2 block of nodes (“connected” means that two nodes are connected horizontally or vertically). The skew diagram X is called a broken border strip if its connected components are border strip. Note that a double partition (α, β)with both α, β non-empty consists of two connected components. For a border strip X, a sharp corner is a node with no node above it and no node to its left. A dull corner

22228 TOSHIAKI SHOJI in a border strip is a node which has a node to its left and a node above it (and so has no node directly northwest of it). For a skew diagram X,letC be the set of connected components of X,and put m = |C|, the number of connected components of X. We deﬁne ∆(X), ∆(X) ∈ Z[u1/2,u−1/2] as follows; ⎧ ⎪ u1/2 − u−/2 m−1 u1/2 c(Y )−1 −u−1/2 r(Y )−1 ⎨⎪( ) ( ) ( ) Y ∈C ∆(X)= ⎪ if X is a broken border strip, ⎩⎪ 0otherwise.

⎧ ⎪(u1/2)c(X)−1(−u−1/2)r(X)−1 ct(y)−1 ct(z) ⎨⎪ y∈DC z∈SC ∆(X)= ⎪ if X is a (connected) border strip, ⎩⎪ 0otherwise, where SC and DC denote the set of sharp corners and dull corners in a border strip, and r(X)(resp. c(X)) is the number of rows (resp. columns) in the border strip X. The Murnaghan-Nakayam formula for H by Halverson-Ram is given as follows. Note that in the formula below, l (w) denotes the number of s1,...,sn−1 (excluding s0) occurring in the reduced expression of w ∈ W Theorem . λ 7.4 ([HR1, Theorem 2.20]) Assume that W is of type Bn.LetEu be the irreducible representation of H associated to λ ∈Pn.Then λ l (wr)/2 (1) (2) (1) ··· (k) (k−1) Tr (Twr ,Eu )=u ∆(µ )∆(µ /µ ) ∆(µ /µ ), ∅=µ(0)⊆µ(1)⊆···⊆µ(k)=λ where the sum is taken over all the sequences ∅ = µ(0) ⊆ µ(1) ⊆···⊆µ(k) = λ such (k) (k−1) (k) (k−1) that |µ /µ | = |rk|−|rk−1| and the factor ∆(µ /µ ) is barred if rk in r is barred.

7.5. Next assume that W is the Weyl group of type Dn and H is the corresponding Hecke algebra. Then under the notation of 6.5, we deﬁne Lk by ≤ ≤ L1 =1,L2 = T0T1,andLk = Tk−1Lk−1Tk−1 for k =3,...,n.For1 k

For a sequence r =(r1,...,rk)ofelementsI such that |r1| < |r2| < ···< |rk| and that the even numbers of ri are barred, we deﬁne Tr ∈Hby ··· Tr = R1,r1 R|r1|+1,r2 R|rk−1|+1,rk .

Note that Tr does not always correspond to Tw for some w, but it corresponds to ∈ Twr for wr W as given in (7.3.1) in the following two cases, (i) ri > 0fori =1,...,k, (ii) r1 = −1,r2 < 0andri > 0fori =3,...,k.

In what follows, we only consider r as above, and so assume that Tr = Twr . Note that wr is regarded as an element of the Weyl group of type Bn in the notation of 7.3.

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As in 7.3, we consider the skew diagram λ/µ, and define ct(x)bymodifying (7.3.2), uj−i if x is in position (i, j)in(λ/µ)α, ct(x)= −uj−i if x is in position (i, j)in(λ/µ)β. For a skew diagram X, we define ∆(X), ∆ (X) ∈ Z[u1/2,u−1/2] by the formula in 7.3, but for ∆ (X), we modify the definition of ∆(X) by replacing ct(x)byct(x). Concerning the irreducible characters of H, we have the following result. Theorem . λ 7.6 ([HR1, Theorem 4.21]) Assume that W is of type Dn.LetEu α β be the irreducible representation of H associated to λ ∈ Pn such that λ = λ . λ Then Tr (Twr ,Eu ) can be computed by the formula in Theorem 7.4 for type Bn,by replacing ∆(X) by ∆ (X).

F F + 7.7. Let c is the cuspidal family as in 4.3, where c is a subset of Φn or Φn . F 1 F 0 Let c (resp. c ) be the set of symbols of defect 1 (resp. defect 0) contained in F F ⊂ Pd c. First assume that c Φn.Let n be the set of double partitions λ such that ∈F ∈Pd ≥ ≥ ··· ≥ ΛEλ c.Thenλ n can be written as λ =(α, β)withα : α1 α2 ≥ ≥ ≥ ≥ ··· ≥ ≥ ∗ ∗ ≥ ∗ ≥ ··· αd αd+1 0, and β : β1 β2 βd 0. Let β : β1 β2 be the dual partition of β. The following fact is easily checked. F 1 Pd { ∈P | ∗ ≤ ≤ } (7.7.1) c n = λ =(α, β) n αi + βd−i+2 = d for 1 i d +1 . F ⊂ + Pd ∈ P Next assume that c Φn .Let n be the set of double partitions λ n such that ΛEλ ∈Fc,whereλ =(α, β)withα : α1 ≥···≥αd ≥ 0, β : β1 ≥···≥βd ≥ 0 (in this case always α = β). Let β∗ be the dual partition of β.Thenwehave F 0 Pd { ∈ P | ∗ ≤ ≤ } (7.7.2) c n = λ =(α, β) n αi + β2d−i+1 =2d for 1 i d . ∈Pd d+1 (7.7.1) shows that λ =(α, β) n,2 is obtained from a diagram γ =(d )of rectangular shape as follows; take any partition α ⊆ γ,andletβ be the dual of the partition obtained by rearranging the skew diagram γ/α. Similarly, (7.7.2) shows ∈ Pd 2d that λ =(α, β) n is obtained from the diagram γ = ((2d) ) of rectangular shape by a similar process as above. For example, in the case of type Bn with d =2,wehaven = d(d +1)=6and P2 { 2 2 2 3 − 2 6 = (21; 21), (2 ;1 ), (1 ; 31), (2 ; ), (2 1; 1), (212;2), (13;3), (2; 22), (1; 32), (−;32)}.

2 In the case of type Dn with d =1,wehaven =4d =4and P1 { 2 − 2 2 } 4 = (2 ; ), (21 ;1), (2, 1 ) .

7.8. Let wr be an element in W associated to some r =(r1,...,rk)asin (7.3.1). By Lemma 7.2 and by (6.5.3), we have − |α| (α,β) (7.8.1) fΛc (wr)=δ ( 1) Tr (Twr ,Eu ), (α,β) Pd Pd where the sum is taken over all λ =(α, β)in n (resp. in n), and the constant δ d(d+1)/2 −d d(2d−1) −2d+1 is given as δ =(−1) 2 (resp. (−1) 2 )ifG is of type Bn (resp. Dn).

24230 TOSHIAKI SHOJI

We shall compute this sum for some speciﬁc choice of r by applying the Murnaghan-Nakayama formula (Theorem 7.4 or Theorem 7.6). In order to discuss the case Bn and Dn simultaneously, we consider the following setting. Let Pa,b ∈P ≥ ··· ≥ ≥ n be the set of double partitions λ =(α, β) n,whereα : α1 αa 0,β : β1 ≥···≥βb ≥ 0andn = ab, such that Pa,b { ∈P | ∗ ≤ ≤ } (7.8.2) n = λ =(α, β) n αi + βa−i+1 = b for 1 i a . Pa,b a Hence n is the set of λ contained in the Young diagram γ =(b ) of rectangular Pd,d+1 Pd P2d,2d shape in the above sense. In particular, n coincides with n,and n (under Pd the identiﬁcation (α, β)=(β,α)) coincides with n. Put − |α| (α,β) (7.8.3) fa,b(wr)= ( 1) Tr (Twr ,Eu ). a,b (α,β)∈Pn

We take rk = n, rk−1 = n − (2a +2b − 6) so that in applying Theorem 7.4 or Theorem 7.6, µ(k)/µ(k−1) is a broken border strip of length 2a +2b − 6. Let X be − ∈Pa,b a broken border0 strip of length 2a +2b 6 contained in λ =(α, β) n .Wecan write X = Y Z with Y ⊂ α, Z ⊂ β, broken border strips. Since the maximum length of a border strip is a + b − 1, we have only to consider the following 5 cases. Case I. |Y | = a + b − 1, |Z| = a + b − 5, Case II. |Y | = a + b − 2, |Z| = a + b − 4, Case III. |Y | = a + b − 3, |Z| = a + b − 3, Case IV. |Y | = a + b − 4, |Z| = a + b − 2, Case V. |Y | = a + b − 5, |Z| = a + b − 1. We consider the diagram γ =(ba) of rectangular shape so that α ∪ β∗ = γ.In the following discussion, we regard Y and Z as paths in γ, instead of considering α and β separately. For example, the following ﬁgure explains an example of the case I, where λ =(α, β)=(42212, 4221) with n = 20, Y is a border strip of length 8andZ is a border strip of length 4. In the ﬁgure, • (resp. ×) denotes the starting point and the ending point of Y (resp. Z).

• • × × - × × • •

γ =(45) α =(42212) β =(422)

In computing fa,b(wr), we use the following cancellation property. Lemma 7.9. Let x be the top rightmost node of γ,andy the west of x, z the south of x. Assume that x, y ∈ Y and that z ∈ Z.Thenα : a1 ≥ α2 ≥··· ,β : β1 ≥ ≥··· − ∈P − β2 with α1 = a, β1 = b 1.Letλ =(α ,β ) n be deﬁned by a1 = α1 1, ∈Pa,b −{ } β1 = β1 +1 and αj = αj,βj = βj for j =1.Thenλ n .PutY = Y x and Z = Z 0+ {x}.ThenY (resp. Z ) is a broken border strip of α (resp. β ). Let X = Y Z. Then we have ∆(X)=∆(X).

LUSZTIG’S CONJECTURELUSZTIG’S FOR CONJECTURE FINITE CLASSICAL FOR FINITE GROUPSWITH CLASSICAL GROUPSEVEN CHARACTERISTIC23125

Moreover, the double partition (α − Y,β − Z) coincides with (α − Y ,β − Z). Hence in the computation of fa,b(wr), the broken border strip X of this type may be ignored. Similar situations occur also for the cases, such as x is the bottom leftmost node of γ and y is the north of x, z is the east of x,andx, y ∈ Y , z ∈ Z.

Proof. Since y ∈ Y and z ∈ Z, the number of border strips in X is the same as the numbers in X.Sincec(Y )=c(Y ) − 1,c(Z)=c(Z) + 1, and r(Y )= r(Y ),r(Z)=r(Z), we see that ∆(X)=∆(X). Clearly, we have (α − Y,β − Z)= (α − Y ,β − Z). It follows that in the sequence ∅⊂µ(1) ⊂ ··· ⊂ µ(k) = λ in Theorem 7.4 or Theorem 7.6, ∅⊂···⊂µ(k−1) is common for λ if µ(k)/µ(k−1) = X. Since |α| = |α|−1, two terms starting from X and from X are canceled in the computation of fa,b(wr) by using these theorems.

0 7.10. We shall classify the broken border strip X = Y Z which is needed for the computation of fa,b(wr). Let xt be the top rightmost node and xb be the bottom leftmost node of the Young diagram α.Letyt be the top rightmost node of β and yb be the bottom leftmost node of β.Notethatweembedβ in γ by using ∗ β , and regard yt,yb as a box in γ. First we consider the Case I. We have α1 = b, αa =0,and xt is the top rightmost node and xb is the bottom leftmost node of γ. Y is a unique border strip connecting xt and xb. Assume that α2

26232 TOSHIAKI SHOJI

Case I. |Y | = a + b − 1, |Z| = a + b − 5.

• • • • × × × ×

× × • • • × • ×

(1) −U (2) u−1U (3) uU (4) −U

Case II. |Y | = a + b − 2, |Z| = a + b − 4.

• • • • × • × × ×

• × • × • × • × • × ×

(1) u1/2U 2 (2) −u1/2U 2 (3) u1/2U 2 (4) −u−1/2U 2

• × • × × ×

× • • ×

(5) −u−1/2U 2 (6) u1/2U 2

Case III. |Y | = a + b − 3, |Z| = a + b − 3.

• × • × × • × • × •

• × • • × • • × • ×

(1) U 3 (2) −U (3) U 3 (4) uU

LUSZTIG’S CONJECTURELUSZTIG’S FOR CONJECTURE FINITE CLASSICAL FOR FINITE GROUPSWITH CLASSICAL GROUPSEVEN CHARACTERISTIC23327

• × • × • × • × • ×

• × × • × × • • × ×

(5) U 3 (6) u−1U (7) −U (8) U 3

• • • • × × × × • • • • × × × ×

(9) −U (10) u−1U (11) uU (12) −U

∈P 7.11. In each case listed in 7.10, one obtains a unique λ 0 n (n = n − (2a +2b − 6)) from λ by removing the broken border strip X = Y Z,where λ =(α,β)withα = α − Y,β = β − Z.LetJ be the set of classes in the list in 7.10, and let Qj be the set of λ ∈Pn satisfying the condition in each case j.For example we consider the case j =I-(1).ThenQ = QI−(1) is the set of λ =(α ,β ) satisfying the following conditions. ⎧ ⎪ ⎨α =(α1,α2,...,αa−3), ∗ ∗ ∗ ∗ ∗ ∗ β =(β ,β ,...,β − )withβ = b − 1,β − =1, ⎩⎪ 1 2 a 1 1 a 1 ∗ − − αi + β a−1−i = b 2fori =1,...,a 3.

For each λ =(α ,β ) ∈Q, we consider a broken0 border strip X of length 2a+2b− 10. Then X is unique, and is given as X = Y Z,whereY is a unique border strip in α of length a+b−7, and Z is a unique border strip in β of length a+b−3. By removing X from λ ,oneobtainsλ ∈Pn ,wheren = n − (2a +2b − 10) = (a − 4)(b − 4). Then it is easy to check that

a−4,b−4 { ∈P | ∈Q} P (7.11.1) λ n λ = n , Pa−4,b−4 − − and one recovers the original set n by replacing a, b by a 4,b 4. We can compute that ∆(X)=−(u1/2 − u−1/2), which is independent from λ ∈Q.For example, let a =7,b =6withn = 42. Take λ = (6432212, 62542) ∈P7,6.Then 42 ∗ λ =(α,β) = (3212, 64312), and under an appropriate rearrangement, γ = α∪β can be drawn as in the following ﬁgure. Here Y (resp. Z) is a unique border strip of length 6, (resp. length 10), and • (resp. × ) denotes the starting point and the ending point of Y (resp. Z). From this, we obtain γ =(23), and ∈P3,2 λ =(α ,β )=(1, 32) 6 .

28234 TOSHIAKI SHOJI

× •

- - • ×

∗ ∗ γ = α ∪ β γ = α ∪ β α =(1) β = (32)

In fact, similar arguments work for all other cases, and the set Qj is described in ∗ a similar way. In particular, γ = α ∪ β is of the shape obtained from a rectangle by attaching two nodes, one on the above or right of the northeast corner, and the other on the below or left of the0 southwest corner of the rectangle. In all the cases, the broken border strip X = Y Z of length 2a+2b−10 is determined uniquely, Pa−4,b−4 and we always ﬁnd the set n after removing the border strips X .Moreover, ∆(X) takes the common value −(u1/2 − u−1/2) for all the cases through Case I ∼ Case V. Recall that wr =(r1,...,rk)withrk = n. Assume that

rk − rk−1 =2a +2b − 6,rk−1 − rk−2 =2a +2b − 10. Thus rk−2 = n with n =(a − 4)(b − 4). We put r =(r1,...,rk−2) and consider wr ∈ Wn . By investigating the above list, we have the following lemma. Lemma 7.12. Under the notation above, there exists a non-zero (Laurent) polynomial h(u) such that

fa,b(wr)=h(u)fa−4,b−4(wr ).

Proof. For each j ∈ J,wedenoteby∆j(u) the Laurent polynomial ∆(X) |α| attached to the broken border strip X for Qj as in the list. We put εj =(−1) for λ =(α, β) ∈ Qj . (Note that εj is independent of the choice of λ ∈Qj). By Pa−4,b−4 (7.11.1), the cardinality of Qj coincides with the cardinality of n , hence is independent of j ∈ J. Then the investigation in 7.11 shows that 1/2 −1/2 a−4,b−4 − |P | (7.12.1) fa,b(wr)= (u u ) n εj∆j(u) fa−4,b−4(wr ). j∈J Thus in order to show the lemma, it is enough to see that j εj∆j(u) =0.Forthis 3/2 we compare the highest degree term u in ∆j . It follows from the list in 7.10, ∆j contains the term u2/3 in the following cases, where the coeﬃcients are always 1. Case I. (3), Case II. (1), (3), (6), Case III. (1), (3), (4), (5), (8), (11), Case IV. (2), (4), (5), Case V. (2), where the numbering in Case IV and V is given by the bijective correspondence with Case II and Case I through the rotation of γ.Notethatεj takes the constant value for each case, I ∼ V. They have the common value for Case I, III, or V, and

LUSZTIG’S CONJECTURELUSZTIG’S FOR CONJECTURE FINITE CLASSICAL FOR FINITE GROUPSWITH CLASSICAL GROUPSEVEN CHARACTERISTIC23529 have a different common value for Case II or IV. This shows that the coefficient of 2/3 u in j εj∆j = 0. Hence h(u) = 0 as asserted. Returning to the original setting, we show the following two propositions, which give the proof of Proposition 6.6. 2 Proposition 7.13. Assume that W is of type Bn with n = d + d. We define an element wr ∈ W by the cycle type expression given in (7.3.2) as follows. [2, 12, 16, 12 + 16, 16 + 16,...,12 + 16k, 16 + 16k], if d ≡ 1(mod4), [6, 16, 20, 16 + 16, 20 + 16,...,16 + 16k, 20 + 16k], if d ≡ 2(mod4), [4, 8, 4+16, 8+16,...,4+16k, 8+16k], if d ≡ 3(mod4), [8, 12, 8+16, 12 + 16,...,8+16k, 12 + 16k], if d ≡ 0(mod4), for some k ≥ 1, where the last term is equal to 4d − 4, and the next term is equal − to 4d 8, and so on. Then we have fΛc (wr) =0. Proof. We apply Lemma 7.12 with a = d +1,b = d.ThenfΛc (wr)= h(u)fΛc (wr ) for some non-zero h(u), where r =(r1,...,rk)andr =(r1,...,rk−2) with rk = n = d(d+1), rk −rk−1 =4d−4, rk−1−rk−2 =4d−8, rk−2 =(d−4)(d−3).

Thus the computation of fΛc (wr) is reduced to the case where d =1, 2, 3. Assume that d =1,thenn = 2. One can check by using the formula for ∆(X)that fΛc (w) =0forw =(2). (Note that fΛc (w)=0forw = (2)). Next assume that d =2,thenn = 6. The direct computation shows that fΛc(w) =0forw =(6). Finally assume that d =3,thenn = 12. By using a similar method as in the proof of Lemma 7.12, one can show that fΛc (w) =0forw =(4, 8) (we obtain a similar list, but some classes in the list in 7.10 don’t appear for this case). This proves the proposition. 2 Proposition 7.14. Assume that W is of type Dn with n =4d . We deﬁne an element wr ∈ W as follows. [1, 3, 14, 18, 14 + 16, 18 + 16,...,14 + 16k, 18 + 16k], if d ≡ 1(mod2), [6, 10, 6+16, 10 + 16,...,6+16k, 10 + 16k], if d ≡ 0(mod2), for some k ≥ 1, where the last term is equal to 4d − 6 and the next term is equal to − 4d 10, and so on. Then we have fΛc (wr) =0. Proof. We apply Lemma 7.12 with a = b =2d.Notethatsinceα = β for ∈P2d,2d any λ =(α, β) n , the argument for type Bn can be applied without change. Then fΛc (wr)=h(u)fΛc (wr ) for some non-zero h(u), where r =(r1,...,rk)and 2 r =(r1,...,rk−2)withrk = n =4d , rk − rk−1 =4d − 6, rk−1 − rk−2 =4d − 10, − 2 rk−2 =4(d 2) . Thus the computation of fΛc (wr) is reduced to the case where d =1, 2. Assume that d =2.Thenn = 16. Since a = b = 4, Lemma 7.12 can be applied, and we see that fΛc (w) =0forw =(6, 10). Next assume that d =1. Then n = 4. One can check by using ∆ (X)thatfΛc (w) =0forw =(1, 3). This proves the proposition. Remark . 7.15 As the formula (7.12.1) shows, our element fΛc (wr) turns out to be 0 if u → 1. So our computation cannot be performed in the level of Weyl groups. On the other hand, Lusztig showed in [L5] that there exists an element ∈ | w W such that fΛc (w) u=1 =0.ThusfΛc (w) = 0 as a polynomial. This w has a simpler form than ours, but since it is a product of negative cycles of the full length

30236 TOSHIAKI SHOJI

(i.e., the sum of the lengths of negative cycles is equal to n), it is not appropriate for the computation of the bitrace on the ﬂags (cf. Lemma 6.10).

References [HR1] T. Halverson and A. Ram; Murnaghan-Nakayama rues for characters of Iwahori-Hecke algebras of classical type, Trans. Amer. Math. Soc. 348 (1996), 3976 - 3995.

[HR2] T. Halverson and A. Ram; Bitraces for GLn(Fq) and the Iwahori-Hecke algebra of type An−1, Indag. Math. New Series 10 (1999), 247 - 268. [K] N. Kawanaka; On subfield symmetric spaces over a finite field, Osaka J. Math. 28 (1991) 759 - 791. [L1] G. Lusztig; “Characters of Reductive groups over a finite field”, Ann. of Math. Studies, Vol. 107, Princeton Univ. Press, Princeton, 1984. [L2] G. Lusztig; Character sheaves, I Adv. in Math. 56 (1985), 193–237, II Adv. in Math. 57 (1985), 226–265, III, Adv. in Math. 57 (1985), 266–315, IV, Adv. in Math. 59 (1986), 1–63, V, Adv. in Math. 61 (1986), 103–155. [L3] G. Lusztig; Remarks on computing irreducible characters, J. of Amer. Math. Soc., 5 (1992), 971–986.

[L4] G. Lusztig; G(Fq)-invariants in irreducible G(Fq2 )-modules, Represent. Theory, 4 (2000), 446 - 465. [L5] G. Lusztig; Rationality properties of unipotent representations, J. Algebra 258 (2002), 1 -22. [S1] T. Shoji; Character sheaves and almost characters of reductive groups, Adv. in Math. 111 (1995), 244 - 313, II, Adv. in Math. 111 (1995), 314 - 354. [S2] T. Shoji; Unipotent characters of finite classical groups, in “Finite reductive groups; related structures and representations,” Progress in Math., Vol.141 (1997), 373 - 413. [S3] T. Shoji; Generalized Green functions and unipotent classes for finite reductive groups, II. To appear in Nagoya Math. Journal. [W] J.-L. Waldspurger; “Une conjecture de Lusztig pour les groupes classiques”, Mémoires de la Soc. Math. France, No. 96, Soc. Math. Fracne, 2004.

Graduate School of Mathematics, Nagoya University, Chikusa-ku, Nagoya 464-8602, Japan

Contemporary Mathematics Volume 478, 2009

A survey on quasiﬁnite representations of Weyl type Lie algebras

Yucai Su

Dedicated to my teacher Professor Zhexian Wan for his 80th birthday

Abstract. The purpose of this survey article, mainly based on materials from the papers by the author and B. Xin [26, 30, 31], and also by other authors [5, 14, 16, 37], is to present some results on classifications of quasifinite representations and unitary representations of Lie algebras related to W -infinity dN algebras, especially, W1+∞, W∞ and D . In particular, one main result states that an irreducible quasifinite module over a Lie algebra of this kind is either a highest/lowest weight module or a module of the intermediate series.

Introduction The W -infinity algebras naturally arise in various physical theories, such as conformal field theory, the theory of the quantum Hall effect and integrable systems, etc. and receive intensive studies in the literature (e.g., [1, 2, 3, 4, 5, 8, 9, 15, 16, 17, 20, 34]). The W1+∞ algebra, also known as the unique nontrivial central extension D of the Lie algebra D of differential operators on the circle [15], is the most fundamental one among these algebras. A systematic study of representation theory of the Lie algebra W1+∞ was initiated in [16]. In that paper, irreducible quasifinite highest weight representations of W1+∞ were classified and it was shown that they can be realized in terms of irreducible highest weight representations of the central extension gl∞ of the Lie algebra gl∞ of infinite matrices with only finitely many nonzero diagonals. Furthermore, unitary representations were described. This study was continued in [9, 17] in the framework of vertex algebra theory. In particular, character formulas for primitive representations of W1+∞ with central charge N were established, and the vertex operator algebra structures on the vacuum modules of W1+∞ and the W -algebra W(glN ) were studied. Later on, Ref. [5] gave a complete description of the irreducible quasifinite highest weight

1991 Mathematics Subject Classification. Primary 17B68; Secondary 17B10, 17B65, 17B66. Key words and phrases. W -infinity algebras, Weyl type Lie algebras, quasifinite representations. The author was supported Supported by a NSF grant 10471091 of China and “One Hundred Talents Program” from USTC, China.

2238 YUCAI SU modules over the Lie algebra DN (the unique nontrivial central extension of the Lie algebra DN of N × N-matrix differential operators on the circle), and classified the unitary ones, and obtained them in terms of representation theory of the cen- m m tral extension gl∞(R )oftheLiealgebragl∞(R ) of infinite matrices with only finitely many nonzero diagonals over the algebras Rm = C[t]/(tm+1) for various m ∈ Z+. In [14], all irreducible quasifinite highest weight modules of the subalgebras W(k) W W(k) 1+∞ of 1+∞ were classified, where 1+∞ is the unique nontrivial central ex- D k tension of the Lie algebra ∂t of differential operators on the circle that annihilate all polynomials of degree less than k (the most important of these subalgebras is W W(1) another W -infinity algebra ∞ = 1+∞). Furthermore, unitary modules over W∞ are also considered. In particular, it is obtained surprisingly that the list of unitary modules over W∞ is much richer than that over W1+∞. In [26, 30, 31], irreducible quasifinite modules over Weyl type Lie algebras W(Γ,n), W(Γ, 1), W(Γ,n)(1), W(Γ, 1)(1), DN were classified. Here W(Γ,n)isthe Lie algebra of generalized differential operators, which can be defined as the tensor product space of the group algebra C[Γ] with basis {tα | α ∈ Γ} (for any non- n degenerate additive subgroup Γ of C ) and the polynomial algebra C[D1, ..., Dn] (1) (where Di’s are degree operators of C[Γ]), while W(Γ,n) is the Lie subalgebra of W(Γ,n) of generalized differential operators of degree at least 1, and W(Γ,n), W(Γ, 1) are respectively the universal central extensions of W(Γ, 1), W(Γ, 1)(1). (1) Thus in particular, W1+∞ = W(Z, 1), W∞ = W(Z, 1) , and so, Lie algebras W(Γ,n), W(Γ, 1), W(Γ,n)(1), W(Γ, 1)(1) can be regarded as multi-variable generalizations of W -infinity algebras. A particular result of [26, 30, 31] states that an irreducible quasifinite module over a Lie algebra of this kind is either a highest/lowest weight module or a module of the intermediate series. As is pointed in [14, 16], when studying the representation theory of a Lie algebra of this kind, we encounter the difficulty that though it admits a Z-gradation, each of the graded subspaces is still infinite dimensional, and thus the study of modules which satisfy the quasifiniteness condition that its graded subspaces have finite dimensions, becomes a nontrivial problem. The aim of this survey article is to give an overview of the above mentioned papers, and to present the main techniques and results of these papers.

1. Weyl type Lie algebras and some subalgebras 1.1. Weyl type Lie algebras. A (classical) Weyl algebra of rank n is the associative algebra ∂ ∂ ∂ ∂ + C C ±1 ±1 An = [t1, ..., tn, , ..., ]orAn = [t1 , ..., tn , , ..., ] ∂t1 ∂tn ∂t1 ∂tn of diﬀerential operators over C. Under the commutator, An is a Lie algebra of Weyl n type, denoted by W(n). Then W(n)=⊕α∈Zn W(n)α is a Z -graded Lie algebra n such that the graded subspace W(n)α for α =(α1, ..., αn) ∈ Z has basis elements α µ α1 ··· αn µ1 ··· µn ∈ Zn (1.1) t D = t1 tn D1 Dn ,µ=(µ1, ..., µn) +, ∂ where Di = ti .Itisknown[8, 22]thatW(n) has a nontrivial universal central ∂ti extension if and only if n =1.TheW -inﬁnity algebra W1+∞ is the universal central extension of W(1) (cf. (1.4)).

REPRESENTATIONS OF WEYL TYPE ALGEBRAS 2393

Based on a classification (in [32]) of pairs (A, D) of a commutative associative algebra A and a finite-dimensional locally finite Abelian derivation subalgebra D such that A is D-simple, a large class of Lie algebras of Weyl type was constructed in [33], whose structure theory was studied in [34, 36]. This type of Lie algebras of Weyl type is essentially a generalization of the Lie algebra W(n). Now we introduce a special subclass of such Lie algebras, which can be regarded as a multi-variable generalization of W -infinity algebras. Consider the n-dimensional vector space Cn. Denote an element in Cn by

α =(α1, ..., αn). Let Γ be an additive subgroup of Cn such that Γ is nondegenerate, i.e., it contains a C-basis of Cn.LetC[Γ] be the group algebra of the group Γ with basis element α written as t ,α∈ Γ. We define the degree operators Di to be the derivations α α Di : t → αit for α ∈ Γ, where i ∈ [1,n]. Then the generalized Weyl algebra A(Γ,n) of rank n is the associative subalgebra of EndC(C[Γ]) generated by Di’s and left multiplications by tα,α ∈ Γ. Thus A(Γ,n) is a tensor product space of C[Γ] and the polynomial C { α µ | ∈ ∈ Zn } µ algebra [D1, ..., Dn], with basis t D α Γ,µ + , where D stands for n µi i=1 Di , and product: µ − (1.2) tαDµ · tβDν = βλtα+βDµ+ν λ, ∈Zn λ λ + where βλ = n βλi ,(µ)= n ( µi ), and for i, j ∈ C, the binomial coefficient i=1 i λ i=1 λi i j−1 − ∈ Z (j )= k=0(i k)/j!ifj +,or0otherwise. Under the commutator, A(Γ,n)isaLie algebra of generalized Weyl type of rank n (also called a Lie algebra of generalized differential operators)[26], denoted by W(Γ,n), whose Lie bracket is thus given by (1.3) [tαDµ,tβDν ]=tαDµ · tβDν − tβDν · tαDµ. The classical Lie algebra W(n) of Weyl type is simply the Lie algebra W(Zn,n). The Lie algebra W(Γ,n) has a nontrivial universal central extension if and only if n =1[22]. The universal central extension W(Γ, 1) of W(Γ, 1) is defined by: the bracket (1.3) is replaced as [tα[D] ,tβ[D] ]=tα[D] · tβ[D] − tβ[D] · tα[D] µ ν µ ν ν µ − µ α + µ (1.4) +δα,−β( 1) µ!ν! µ+ν+1 κ, W d µ−1 − where κ is a central element of (Γ, 1), D = t dt and [D]µ = i=0 (D i)(herewe have to use this new notation in order to have a simple expression of the coefficient of κ). In case Γ = Z, (1.4) can be rewritten into the better form (1.5) [tif(D),tjg(D)] = ti+j f(D−j)g(D)−f(D)g(D+i) +ϕ tif(D),tjg(D) κ, where, the 2-cocycle ϕ, which seems to appear first in [15], is defined by f(k)g(k + j)ifi = −j>0, ϕ tif(D),tjg(D) = −i≤k≤−1 0ifi + j =0or i = j =0, for i, j ∈ Z and all polynomials f(D),g(D).

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Let k>0. Denote the Lie subalgebra of W(Γ,n) of (generalized) differential operators of degree at least k by W(Γ,n)(k),namely,W(Γ,n)(k) is the Lie subalgebra with basis elements (1.6) tαDµ for α ∈ Γ, |µ|≥k, | | n W (k) where µ = i=1 µi is the level of µ. Similarly, one can define (Γ, 1) to be W (k) W(k) the universal central extension of (Γ, 1) . In particular, we denote 1+∞ = W Z (k) W W(1) ( , 1) . Then another W -infinity algebra ∞ = 1+∞ is the universal central extension of W(Z, 1)(1). A Lie algebra L is called

∗ Γ-graded if L = ⊕α∈ΓLα is decomposed as a direct sum of graded subspaces such that the Lie bracket respects the gradation, namely, [Lα, Lβ] ⊂ Lα+β for α, β ∈ Γ; ∗ finitely Γ-graded if all graded subspaces Lα are finite dimensional; ∗ not-finitely Γ-graded if dim Lα = ∞ for some α ∈ Γ. Thus, W(Γ,n), W(Γ,n)(k), and their central extensions are not-finitely Γ-graded Lie algebras such that the basis element tαDµ has degree α ∈ Γ (cf. (1.1)), and κ has degree 0. For the case W(Γ,n), we can write W(Γ,n)=⊕α∈ΓW(Γ,n)α such that each graded subspace W(Γ,n)α has basis elements α µ ∈ Zn (1.7) t D for µ +.

Thus W(Γ,n)0 is commutative which is also a Cartan subalgebra (i.e., a maximal self-normalized nilpotent subalgebra), and it contains the (unique) maximal torus (i.e., a maximal commutative subalgebra with each element being ad-semi-simple) 0 0 H with basis elements 1 := t D ,Di for i ∈ [1,n] (the uniqueness can be see from [34, 36]). N DN Next, we introduce the W -infinity algebra D (e.g., [5, 30]). Denote by as the associative algebra which is the tensor product of two associative algebras W(Z, 1) DN and glN .Thus as has basis elements i j (1.8) t D A for i ∈ Z,j∈ Z+,A∈ glN , with product j i j · k l i+k j l j s j+k j+l−s (1.9) t D A t D B = t (D + k) D AB = s k t D AB. s=0 DN × Under the commutator, as is the Lie algebra of N N-matrix differential operators on the circle, denoted by DN , whose Lie bracket is thus given by (1.10) [tiDjA, tkDlB]=tiDjA · tkDlB − tkDlB · tiDjA. The Lie algebra DN has a unique nontrivial central extension [8, 22]. The W -infinity algebra DN is the universal central extension of DN , defined by [5, 9]: the bracket (1.10) is replaced as ti[D] A, tk[D] B = ti[D] A · tk[D] B − tk[D] B · ti[D] A j l j l l j j i + j (1.11) + δ − (−1) j!l! tr(AB)κ. i, k j+l+1

1 Thus D is simply the Lie algebra W1+∞.

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Let {Ep,q | p, q ∈ [1,N]} be the standard basis of glN ,whereEp,q is the matrix with entry 1 at (p, q) and 0 otherwise. Then DN has the principal Z-gradation DN = ⊕i∈ZDN i with the graded subspace [5]

k j (1.12) DN i =span{t D Ep,q | k ∈Z,j∈Z+,p,q∈[1,N],kN+p−q =i}⊕δi,0Cκ.

j Note that t has degree N in this case. In particular, DN 0 with basis {D Ep,p, κ | j ∈ Z+,p ∈ [1,N]} is commutative (a Cartan subalgebra), and H with basis {Epp,DEpp,κ| p ∈ [1,N]} is a maximal torus.

1.2. Some subalgebras. It might be worth mentioning that Weyl type Lie algebras contain some interesting Lie subalgebras. It is well known [19] that the Virasoro algebra Vi r is a finitely Z-graded Lie algebra which can be defined as the universal central extension of the Lie algebra of linear differential operators on the circle. Thus Vi r has a basis {Li,κ| i ∈ Z} such that κ is central and

i3 − i (1.13) [L ,L ]=(j − i)L + κ. i j i+j 12 Ref. [21, 35] generalized the notion of the Virasoro algebra to generalized Virasoro algebras Vi r (Γ), for any nonzero additive subgroup Γ of C, which is the ﬁnitely Γ-graded Lie algebra with basis {Lα,κ| α ∈ Γ} and relation

α3 − α (1.14) [L ,L ]=(α − β)L + κ. α β α+β 12 When Γ is ﬁnitely generated additive subgroup of C, Vi r (Γ) is also called a higher hank Virasoro algebra [21]. One immediately sees that W(Γ, 1)(k) contains a generalized Virasoro subalgebra if and only if k ≤ 1, and when k ≤ 1, the subspace a of W(Γ, 1) with basis {Lα := t D, κ | α ∈ Γ} is the generalized Virasoro algebra Vi r (Γ). One also sees that W(Γ,n)(k),k≤ 1 contains the ﬁnitely Γ-graded Lie subalgebra, denoted by Wi t t (Γ,n) and called a generalized Witt algebra (e.g., [32, 38]), α which has basis {t Di | α ∈ Γ,i∈ [1,n]} and relation

α β α+β (1.15) [t Di,t Dj ]=t (βiDj − αjDi).

Since any finitely Γ-graded Cartan type Lie algebra can be realized as a subalgebra of Wi t t (Γ,n)forsomen (e.g., [38]), we see that W(Γ,n)(k),k≤ 1 for various n also contain all finitely Γ-graded Cartan type Lie algebras. Moreover, when k ≥ 2, W(Γ,n)(k) does not contain a generalized Witt algebra. Next, we see that DN not only contains a Virasoro subalgebra with basis {Li := tiDI, κ | i ∈ Z},whereI is the N × N identity matrix, but also contains the finitely i Z-graded Lie subalgebra linearly spanned by {t A, κ, D | i ∈ Z,A∈ glN },whichis in fact the affine glN Lie algebra, whose nontrivial Lie brackets are given by

i j i+j i i (1.16) [t A, t B]=t [A, B]+δi,−j i tr(AB)κ, [D, t A]=it A.

Since every affine Kac-Moody algebra can be embedded into the affine glN Lie algebra for some N, we obtained that DN for various N contain all affine Kac- Moody algebras.

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1.3. Quasiﬁnite representations. Let L = ⊕α∈ΓLα be a Γ-graded Lie algebra such that L0 is commutative, and in general we assume dim Lα = ∞ for α ∈ Γ. We always assume that L0 is a Cartan subalgebra containing a ﬁnite-dimensional maximal torus H such that L = ⊕α∈ΓLα is also the root space decomposition with ∗ respect to H, namely, there exists ϕα ∈H (the dual space of H) such that

(1.17) Lα = {x ∈L|[h, x]=ϕα(h)x for all h ∈H} for α ∈ Γ.

In this case, ϕα is called a root of H. We combine several concepts related to the representation theory into the following deﬁnition.

Definition 1.1. An L-module V is called a λ (1) generalized weight module if V = ⊕λ∈H∗ V is decomposed as a direct sum of generalized weight spaces, where λ n V = {v ∈ V |∀h ∈H, ∃ n ∈ Z+ with h − λ(h) v =0} is the generalized weight space of weight λ; 0,λ (2) weight module if V = ⊕λ∈H∗ V is decomposed as a direct sum of weight spaces, where V 0,λ = {v ∈ V | hv = λ(h)v for h ∈H}is the weight space of weight λ; (3) Γ-graded module if V = ⊕α∈ΓVα is decomposed as a direct sum of graded subspaces such that the action of L respects the gradation, namely, LαVβ⊂ Vα+β for α, β ∈ Γ; (4) quasifinite module [16]ifV is Γ-graded such that all graded subspaces are finite dimensional, i.e., dim Vα < ∞ for α ∈ Γ; (5) uniformly bounded module if V is quasifinite and the dimensions of graded subspaces are uniformly bounded, i.e., there exists an integer K>0such that dim Vα ≤ K for α ∈ Γ; (6) module of the intermediate series if it is quasifinite and dim Vα ≤ 1for α ∈ Γ; (7) highest/lowest weight module if there exists a total ordering ) on Γ compatible with its group structure (i.e., if α ) β, λ * µ for α, β, λ, µ ∈ Γ ) ∈L∗ L then α + λ β + µ), and there exist λ 0 (the dual space of 0)and nonzero vector vλ ∈ V (in this case λ is called the highest/lowest weight of V and vλ a highest/lowest weight vector of V ) such that V is generated by vλ satisfying

L±vλ =0, and xvλ = λ(x)vλ for x ∈L0,

where L± = ⊕±α0Lα (so that L = L− ⊕L0 ⊕L+ has a triangular decomposition); (8) Verma module if V = V (λ), where V (λ) is deﬁned as follows: there exist ) ∈L∗ a total ordering on Γ compatible with its group structure, and λ 0 and 0 = vλ ∈ V (λ) such that L+vλ =0,xvλ = λ(x)vλ for x ∈L0, and L ∼ V λ Cv U L ⊗ L ⊕L Cv U L− ⊗C Cv , (1.18) ( )=IndL0⊕L+ λ = ( ) U( 0 +) λ = ( ) λ where, U(L) denotes the universal enveloping algebra of L .

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Remark 1.2. (1) A quasifinite module V may not be a weight module. But V is necessarily a generalized weight module since each finite- dimensional homogenous subspace Vα is an H-module, thus has a generalized weight space decomposition. However Vα may contain more than one weights. (2) By (1), the category of quasifinite modules is equivalent to the category of generalized weight modules with finite-dimensional generalized weight spaces. (3) A highest/lowest weight module V is Γ-graded: V = ⊕α∈ΓVα if we define vλ to have degree zero, but it may not be quasifinite. Also, V is necessarily a weight module, whose highest weight vector vλ is a common eigenvector for both H and L0 (a weight vector may not be a common eigenvector for α L0; for instance, in the adjoint module W(Γ,n), the vector v = t with α ∈ Γ\{0} is a weight vector but not a common eigenvector for W(Γ,n)0). (4) By (1.17), one can prove that any submodule of a generalized weight module is again a generalized weight module. (5) Let V (λ) be the Verma module defined in (1.18), and M a proper submodule of V (λ). Then vλ ∈/ M. By (4), the sum of all proper submodules is still proper, thus it is the maximal proper submodule, denoted by M(λ). Then (1.19) L(λ)=V (λ)/M (λ), is the (unique) irreducible highest weight module with highest weight λ. Note that Verma modules for finitely Z-graded Lie algebras (in particular, for all finite dimensional Lie algebras [10], affine Kac-Moody Lie algebras [12], the Virasoro algebra [19], etc) are always quasifinite in the sense of Definition 1.1(4). ∼ However, when Γ = Z, Verma modules for finitely Γ-graded Lie algebras might not be quasifinite. In fact, it is proved in [18, 27] that every nontrivial highest ∼ weight module for Vi r (Γ) or Wi t t (Γ,n) is not quasifinite when Γ = Z.Furthermore, one sees immediately from (1.18) that every Verma module V (λ)foranot-finitely Γ-graded Lie algebra is not quasifinite (in fact all graded subspaces of V (λ)are infinite dimensional except the one with degree 0 which is one dimensional spanned by the highest weight vector vλ). Thus, as stated in the introduction, when we study the representation theory of a not-finitely Γ-graded Lie algebra L,weencounterthe difficulty that although it admits a Γ-gradation, each of the graded subspaces is still infinite dimensional, and therefore the study of quasifinite modules becomes a nontrivial and interesting problem. Some natural problems arise (where a singular vector in a highest weight module is a nonzero weight vector v other than the highest weight vector such that L+v =0). ∼ Problem 1.3. Let Γ be a torsion-free Abelian group such that Γ = Z,andlet L = ⊕α∈ΓLα be a finitely Γ-graded Lie algebra such that Lα =0for α ∈ Γ. Is every Verma module V (λ)overL not quasifinite? (We conjecture that the answer is “yes”.) Problem 1.4. Let W be one of Weyl type Lie algebras W(Γ,n)(k), W(Γ, 1)(k), DN . (1) Determine when the Verma W-module V (λ) is irreducible.

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(2) If the Verma W-module V (λ) is not irreducible, determine all singular vectors (or equivalently, all composition factors). (3) Classify irreducible quasifinite highest weight W-modules. (4) Classify irreducible quasifinite W-modules. (5) Classify indecomposable uniformly bounded (not necessarily weight) W- modules. (6) Classify indecomposable quasifinite (not necessarily weight) W-modules.

(k) Problem 1.5. Consider similar problems for W(Γ,n) ⊗ glN or its central extension.

In the following sections, we shall present partial results for the problems listed in Problem 1.4. The proofs of some obtained results heavily depend on the representation theories of the Virasoro algebra and Witt algebra (cf. (1.13)–(1.15)) which are contained as subalgebras in the Weyl type Lie algebras. Since when k ≥ 2, the Lie algebras W(Γ,n)(k) and W(Γ, 1)(k) do not contain a Virasoro subalgebra or Witt subalgebra, the techniques used for the Virasoro algebra and Witt algebra cannot be applied, most problems concerning them remain unsolved. For convenience, we list below what has been done for the problems listed in Problem 1.4: (1) is done for W = W(Γ, 1) [37], thus also for W = W(Γ, 1). (2) is not done in any case. ∼ (3) is done for W = W(Γ,n)(k), W(Γ, 1)(k) with k ≤ 1orΓ= Z,andDN [5, 9, 14, 16, 17, 26, 30, 31]. (4) is done for W = W(Γ,n)(k), W(Γ, 1)(k) with k ≤ 1, and DN [26, 30, 31]. (5) is done for W = W(Γ,n), W(Γ, 1) [26, 30]. ∼ (6) is done for W = W(Γ,n), W(Γ, 1) with Γ = Z [26, 30].

2. Irreducible and unitary quasifinite highest weight representations Highest weight representations are the most important objects in the study of representation theory of Lie algebras; for instance, finite dimensional simple Lie algebras [10], the Virasoro algebra [15, 19, 23], affine Kac-Moody Lie algebras [12]. We shall present some results of [5, 14, 16, 37] which give characterizations of irreducible (including unitary) quasifinite highest weight W-modules L(λ)for W = W1+∞, W∞ or DN . Note that L(λ) is uniquely determined by its highest weight λ.Thusthe following definition of generating series is natural. Definition . W W ∈ W ∗ 2.1 (1) If = 1+∞,anelementλ ( 1+∞)0 is charac- n terized by the central charge cλ = λ(κ)anditslabels ∆n = −λ(D ),n≥ 0. We define the generating series to be ∞ xn (2.1) ∆λ(x)= ∆n. n=0 n!

W W ∈ W ∗ − n+1 ≥ (2) If = ∞, we deﬁne the labels of λ ( ∞)0 to be ∆n = λ(D ),n 0, and deﬁne the generating series as in (2.1).

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W N ∈ N ∗ − n (3) If = D , we define the labels of λ (D )0 to be ∆p,n = λ(D Epp), p ∈ [1,N],n≥ 0 and define the generating series ∞ xn ∆p,λ(x)= ∆p,n. n=0 n! A function is called a quasipolynomial if it is a linear combination of functions of the form p(x)eax,wherep(x)isapolynomial,a ∈ C. Then a well-known fact [9, 14, 16] states that a formal power series is a quasipolynomial if and only if it satisfies a nontrivial linear differential equation with constant coefficients.

Theorem 2.2. (1) Suppose W = W1+∞.Then φ (x) L(λ) is quasiﬁnite ⇐⇒ ∆ (x)= λ (see [16]) λ ex − 1 ⇐⇒ L(λ) is a proper quotient of V (λ)(see [37]),

for some quasipolynomial φλ(x) with φλ(0) = 0. (2) Suppose W = W∞.Then d L(λ) is quasiﬁnite ⇐⇒ ∆ (x)+ (ex − 1)∆ (x) is a quasipolynomial λ dx λ d φ (x) ⇐⇒ ∆ (x)= λ λ dx ex − 1

for some quasipolynomial φλ(x) with φλ(0) = 0 (see [14]). (3) Suppose W = DN .Then φ (x) L(λ) is quasiﬁnite ⇐⇒ ∆ (x)= N,λ ,and N,λ ex − 1 ∆p,λ(x)=φp,λ(x)+∆N,λ(x),p=1, 2, ..., N − 1,

for some quasipolynomials φp,λ(x) with φN,λ(0) = 0 (see [5]). Thus the last equivalence in Theorem 2.2(1) in particular shows that the Verma x module V (λ)overW1+∞ is irreducible if and only if (e − 1)∆λ(x) is a quasipoly- ∼ nomial. For the case W(Γ, 1) with Γ = Z,Ref.[37] obtained the following. Theorem . L W ∈L∗ 2.3 Let = (Γ, 1),andletλ 0. (1) Suppose the order ) on Γ is dense in the sense that for each α ∈ Γ+ = {β ∈ Γ | β ) 0}, the subset B(α)={β ∈ Γ | α ) β ) 0} of Γ is inﬁnite. Then the Verma L-module V (λ) is irreducible if and only if λ =0 . (2) Suppose the order ) on Γ is discrete in the sense that there exists a ∈ Γ+, called the minimal element of Γ+ such that B(a)=∅. Then the Verma L-module V (λ) is irreducible if and only if the Verma module with highest ∼ weight λ over the Lie algebra W(aZ) = W1+∞ is irreducible. The main techniques in obtaining Theorem 2.2 are the descriptions of parabolic subalgebras and generalized Verma modules. We brieﬂy overview below. For convenience, we assume W = W1+∞ or W∞ (we refer to [5]ifW = DN ). A parabolic subalgebra of W is a subalgebra P such that P ⊃ W ⊕W W ⊕ W = 0 +,where + = i. i>0

Then P = ⊕i∈ZPi such that Pi = Wi for i ≥ 0andPi = {0} for some i<0. Let ∈W∗ | W ⊕W C λ 0 be such that λ W0∩[P,P] = 0. The 1-dimensional ( 0 +)-module vλ

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(cf. Deﬁnition 1.1(8)) extends to a P-module by letting Pi act as zero for i<0. The generalized Verma module is the highest weight module

V (P,λ)=U(W) ⊗U(P) Cλ. ∈W Pa ⊕ Pa For a nonzero element a −1,denote = i∈Z i to be the minimal parabolic Pa W ≥ subalgebra containing a.Then i = i for i 0, and Pa { W W } Pa Pa Pa −1 =span [...[[a, 0], 0], ..., ] , −i−1 =[ −1, −i], Wa Pa Pa ∩W W P furthermore, 0 := [ , ] 0 =[a, 1]. A parabolic subalgebra is nondegenerate if P−i has finite codimension in W−i for all i>0. A nonzero element a a ∈W−1 is nondegenerate if P is nondegenerate. Then the following results can be proved without too much difficulty [5, 14, 16]. Lemma 2.4. The Lie algebra W satisfies the following properties.

(1) If a ∈W−β for some β>0 and [a, W1]=0,thena =0. (2) If P is a nondegenerate parabolic subalgebra of W, then there exists a nondegenerate element a such that Pa ⊂P. (3) Any nonzero element a ∈W−1 is nondegenerate. (4) Any parabolic subalgebra of W is nondegenerate. Lemma . ∈W∗ 2.5 The following conditions on λ 0 are equivalent: (1) V (λ) contains a singular vector a·vλ in V (λ)−1,wherea is nondegenerate; ∈W Wa (2) there exists a nondegenerate element a −1 such that λ( 0 )=0; (3) L(λ) is quasiﬁnite; (4) there exists a nondegenerate element a ∈W−1 such that L(λ) is the irreducible quotient of the generalized Verma module V (Pa,λ). Now suppose L(λ) is quasiﬁnite. By Lemma 2.5(2) and (4), there exists some monic polynomial fλ(x) of minimal degree, uniquely determined by the highest weight λ, called the characteristic polynomial of L(λ), such that −1 (2.2) t fλ(D) vλ =0.

Denote x (1 − e )∆λ(x)ifW = W1+∞, (2.3) φλ(x)= d x − W W ∆λ(x)+ dx (e 1)∆λ(x) if = ∞. Then Theorem 2.2 is obtained by proving that (2.2) is equivalent to the following diﬀerential equation [14, 16] d (2.4) f φ (x)=0, λ dx λ which in particular shows that φλ(x) is a quasipolynomial. For example, let us W W m j ∈ C prove (2.4) for the case = 1+∞.Writefλ(x)= j=0 fλ,jx for some fλ,j k with fλ,m =1.Fork ≥ 0, applying the element Sk := t(D +1) of W1 (the graded subspace of W of degree 1) to (2.2), using (1.5), we obtain k k λ D fλ(D) − (D +1) fλ(D +1)+fλ(0)δk,0κ =0, which is equivalent to m (2.5) fλ,jFj+k =0 forallk =0, 1, ..., j=0

REPRESENTATIONS OF WEYL TYPE ALGEBRAS 24711 where j−1 j (2.6) Fj = δj,0c + s ∆s. s=0 ∞ xs Introducing the generating series F (x)= s=0 s! Fs, we can rewrite (2.5) into the form m d j (2.7) fλ,j F (x)=0. j=0 dx Thus L(λ) is quasifinite if and only if F (x) is a quasipolynomial. But (2.6) in fact x means that cλ − F (x)=(e − 1)∆λ(x)=φλ(x), and (2.4) follows from (2.7) (note that in case W = W∞, since the element Sk is not in W∞, we need to replace Sk k by t(D +1) D ∈W∞ and in this case we need to define φλ(x) as the second case of (2.3) in order to have the differential equation (2.4), see [14] for detail). Remark 2.6. For all not-finitely Z-graded Lie algebras L which have properties of Lemmas 2.4 and 2.5, one can consider quasifinite highest weight L-modules as above (e.g., [28, 29]). Now let us describe unitary irreducible quasifinite highest weight modules. First we introduce the semi-linear anti-involution ω on DN (for W1+∞, it is simply the special case D1), i.e., ω is R-linear satisfying ω2 =1,ω(ax)=aω(x),ω([x, y]) = [ω(y),ω(x)], for a ∈ C,x,y∈ DN , where the overbar denotes the conjugate number. Thus ω is −1 uniquely determined by ω(t)=t ,ω(D)=D and ω(Eij)=Eji. Precisely, ω is defined by i −i −i (2.8) ω(t f(D)Ekl)=f(D)t Ekl = t f(D − i)Ekl,ω(κ)=κ, j j ∈ C W where f(D)= j≥0 f jD for f(D)= j≥0 fj D ,fj . For the case ∞,since −i the element t f(D − i) is in general not in W∞, we need to define ω to satisfy ω(tif(D)D)=f(D)t−iD = t−if(D − i)D, ω(κ)=κ. A module V over W = W1+∞, W∞, DN is unitary (with respect to the anti- involution ω) if there exists a positive definite Hermitian form H(·, ·)onV such that it is contravariant, i.e., ω(a)anda are adjoint operators on V with respect to the Hermitian form H:

(2.9) H(av1,v2)=H(v1,ω(a)v2)forv1,v2 ∈ V and a ∈W.

For the irreducible highest weight module L(λ) such that the central charge cλ = λ(κ) and all labels are real numbers, there exists a unique contravariant Hermitian form H(·, ·)onL(λ) such that H(vλ,vλ) = 1, which is deﬁned by

H(avλ,bvλ)=ω(a)bvλ for a, b ∈ U(W), where v is the coeﬃcient of vλ in the decomposition of v with respect to the gradation of L(λ)forv ∈ L(λ). We have the following (note that Theorem 2.7(1) is the special case of (3) for N =1).

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Theorem 2.7. (1) (see [16]) The irreducible quasifinite highest weight W rix − 1+∞-module L(λ) is unitary if and only if φλ(x)= i ni(e 1) for some positive integers ni and distinct real numbers ri, such that cλ = i ni,andφλ(x) is as in Theorem 2.2. (2) (see [14]) The nontrivial W∞-module L(λ) is unitary if and only if the following conditions are satisfied: rix (a) φλ(x)+c = i nie for some nonzero integers ni and distinct real numbers ri. (b) For each i, ni > 0 whenever ri ∈ Z. (c) For 0 <α<1,letEα = {i | ri ∈ α + Z}. Then for each α with ∅ { | ∈ } Eα = ,theset ni i Eα contains exactly one negative number, and m = − n > 0 and r − r ≤ m for all i, j ∈ E . α i∈Eα i i j α α (3) (see [5]) A nontrivial DN -module L(λ) is unitary if and only if c is a positive integer and there exist r1, ..., rc ∈ R and a partition {1, ..., c} = I0 ∪···∪IN−1, such that x r x (r +1)x (e − 1)∆p,λ(x)= e i + e i − c. i∈I0∪···∪Ip−1 i∈Ip∪···IN−1 The main technique to obtain the results is the description of relations of m m W1+∞, W∞ and DN to the central extension gl∞(R )oftheLiealgebragl∞(R ) of infinite matrices with only finitely many nonzero diagonals over the algebras m m+1 R = C[t]/(t )form ∈ Z+. A key in obtaining Theorem 2.7(1) is the following (similar results also hold for W∞, DN , and even for other not-finitely Z-graded Lie algebras, e.g., Lie algebras considered in [28, 29]).

Lemma 2.8. If W1+∞-module L(λ) is unitary, then the characteristic polynomial fλ(x) has only simple real roots. − 1 2 − − ∈W W This can be proved as follows. Let T = 2 (D ∆2 1) = 1+∞, j −1 −1 j −1 one can check T (t vλ)=(t D )vλ for j ≥ 0. Since t fλ(D) vλ =0,we −1 j see {(t D )vλ | 0 ≤ j

3. Classification of quasifinite modules The classification of irreducible modules in some category is definitely one of the most important problems in the representation theory of Lie algebras. In this section, we shall consider the problem of classification of quasifinite modules over Weyl type Lie algebras. Before we state the main results, let us present some related

REPRESENTATIONS OF WEYL TYPE ALGEBRAS 24913 results for some well-known Lie algebras (some results may be used to prove the results for Weyl type Lie algebras).

3.1. Some results for the Virasoro algebra and aﬃne Kac-Moody algebras. It is well known [11, 19, 23, 25, 27] that a module of the intermediate series over Vi r (cf. (1.13)) is a subquotient of modules Aa,b,A(α),B(α)fora, b, α ∈ C,whereAa,b,A(α),B(α) all have basis {vi | i ∈ Z} such that the central element κ acts trivially and

(3.1) Aa,b : Livj =(a + j + bi)vi+j,

(3.2) A(α): Livj =(i + j)vi+j (j =0) ,Liv0 = i(i + α)vi,

(3.3) B(α): Livj = jvi+j (j = −i),Liv−i = −i(i + α)v0. The following famous theorem was originally conjectured by Kac [11] in 1982, and proved by Mathieu [19] in 1992 (with partial results proved in [23]). Theorem 3.1. A Harish-Chandra Vi r -module (i.e., an irreducible quasiﬁnite Vi r -module) is either a module of the intermediate series or a highest/lowest weight module. The same theorem also holds for the super-Virasoro algebras [24]. It is also obtained in [25] that a uniformly bounded Harish-Chandra module over a higher rank Virasoro algebra is a module of the intermediate series. Furthermore, Theorem 3.1 was generalized to the higher hank Virasoro algebras [18, 27]: Theorem 3.2. Any Harish-Chandra module over the higher rank Virasoro algebra Vi r (Γ) is (1) a module of the intermediate series or a highest/lowest weight module if ∼ Γ = Z; ∼ (2) a module of the intermediate series if rank Γ = 1 and Γ = Z; (3) a module of the intermediate series or isomorphic to L(a, b, Γ,α) for some a, b ∈ C,α∈ Γ and a subgroup Γ of Γ with Γ=Γ ⊕ Zα if rank Γ > 1. Here the Vir(Γ)-module V (a, b, Γ,α) is deﬁned as follows: Suppose Γ = Γ⊕Zα for some subgroup Γ of Γ and some α ∈ Γ. Let a, b ∈ C,andletMa,b be the (unique) nontrivial irreducible quotient of the Vi r (Γ)-module of the intermediate series of the form Aa,b (cf. (3.1)). Write Vi r (Γ) = ⊕i∈ZVi r (Γ)i such that Vi r (Γ)i has basis {Lβ+iα,δi,0κ | β ∈ Γ } (thus Vi r (Γ )=Vi r (Γ)0). Then

Vi r (Γ) = Vi r (Γ)− ⊕ Vi r (Γ)0 ⊕ Vi r (Γ)+ is a triangular decomposition of Vi r (Γ), where Vi r (Γ)± = ⊕±i>0Vi r (Γ)i.Put Vi r (Γ)+Ma,b = 0 and deﬁne the induced module (sometimes also called a generalized Verma module)

Vi r (Γ) V (Ma,b)=Ind Ma,b Vi r (Γ)+⊕Vi r (Γ)0 ⊗ ∼ ⊗ = U(Vi r (Γ)) U(Vi r (Γ)+⊕Vi r (Γ)0) Ma,b = U(Vi r (Γ)−) C Ma,b. Then L(a, b, Γ ,α)=V (Ma,b)/J is the (unique) irreducible quotient of V (Ma,b), where J is the maximal proper submodule of V (Ma,b). The following is also obtained in [27].

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Theorem 3.3. Suppose there is a group injection Z × Z → Γ. Then a Harish- Chandra module over Wi t t (Γ,n) is either a uniformly bounded module with all nonzero weights having the same multiplicity, or a finitely-dense module (see [27] for definition). Earlier than the time when Theorem 3.1 was proved, an analogous result of Theorem 3.1 was obtained in [6, 7] in 1986 and 1988 for integrable irreducible modules over (nontwisted and twisted) affine Kac-Moody algebras. Here a module over a Kac-Moody algebra g is called integrable [12] if it is diagonalizable with respect to a Cartan subalgebra h of g and if, as a representation of each of the sl2-subalgebras of g corresponding to the Chevalley generators of g,itbreaksup as a direct sum of finite-dimensional representations (such representations are interesting because they automatically lift to representations of the corresponding Kac-Moody group). The main results of [6, 7] can be stated as the following (for convenience we only state the result for the nontwisted case). Theorem 3.4. Let V be an integrable irreducible module over the nontwisted affine Kac-Moody algebra g. Then the center κ (here κ is a proper scalar multiple of κ such that in (1.16), κ is replaced by κ and tr(AB) is replaced by minus the Killing form of the corresponding finite dimensional simple Lie algebra g0 ) acts on V by an integer l(V )(called the level of V ), and there are three cases: (1) if l(V ) > 0 then V is a highest weight module (usually called a standard module); (2) if l(V ) < 0 then V is a lowest weight module; (3) if l(V )=0then V is a loop module (see [6] for definition). In the following subsections, we shall see that Theorem 3.1 can be generalized to Weyl type Lie algebras. We would like to mention that similar results might also hold for some other Γ-graded Lie algebras which contain the Virasoro algebra (or Witt algebra) as a subalgebra; for example, Block type Lie algebras [28, 29].

3.2. Classification of quasifinite modules over W(Γ,n) and W(Γ, 1). Denote the set of n-tuples of commuting p × p matrices by n { | ∈ } glp = G =(G1, ..., Gn) Gi glp,GiGj = GjGi for i, j =1, ..., n . ∈ n C ⊗ Cp Let G =(G1, ..., Gn) glp . The tensor space [Γ] can be defined as a module over the associative algebra A(Γ,n) (cf.(1.2)), thus also a W(Γ,n)-module (denoted by Ap,G), such that t acts as the multiplication and Di acts as the derivation on p C[Γ] and as Gi on C .Namely, α µ · β α+β · µ ∈ ∈ Zn ∈ Cp (3.4) Ap,G : t D t v = t [β 11 + G] v for α, β Γ,µ +,v , ∈ n × · µ where 11 =(1p, ..., 1p) glp with 1p being the p p identity matrix, [β 11 + G] = n · µi · i=1(βi 1p + Gi) and βi 1p denotes the scalar multiplication of the identity matrix, and where the action of a matrix on v is defined by the matrix-vector multiplication (by regarding v as a column vector). Obviously, Ap,G is a uniformly bounded W(Γ,n)-module. ∼ By [34], there exists a Lie algebra isomorphism (involution) σ : W(Γ,n) = W(Γ,n) such that

α µ |µ|+1 µ α (3.5) σ(t D )=(−1) D · t for α ∈ Γ,µ∈ Z+,

REPRESENTATIONS OF WEYL TYPE ALGEBRAS 25115 | | n · where µ = i=1 µp is as in (1.6), and the “ ” in the right-hand side means the product deﬁned by (1.2). Using this isomorphism, we have another W(Γ,n)-module Ap,G, called the twisted module of Ap,G, for the pair (p, G), deﬁned by

α µ β |µ|+1 α+β µ (3.6) Ap,G : t D · t v =(−1) t [(α + β) · 11 + G] v, ∈ ∈ Zn ∈ Cp for α, β Γ,µ +,v . Clearly, Ap,G or Ap,G is decomposable if and only if there exist some invertible matrix P and some integer p1 with 1 ≤ p1

(3.7) Ap,G or Ap,G is irreducible ⇐⇒ p =1,

n (in this case G ∈ C ,andA1,G, A1,G are modules of the intermediate series). Now we can state the result obtained in [26].

Theorem 3.5. (1) Let W be the Lie algebra W(Z, 1) or W1+∞. (i) A uniformly bounded W-module is a direct sum of a trivial module, a module Ap,G andamoduleAp,G for some p, p ∈ Z+, G ∈ glp,G ∈ glp (in the central extension case, the central element κ acts trivially). (ii) An irreducible quasifinite W-module is a highest/lowest weight module or a module of the intermediate series (i.e., A1,G or A1,G for some G ∈ C). ∼ (2) Suppose W = W(Γ,n), W(Γ, 1) with Γ = Z. (i) A quasifinite W-module is a direct sum of a trivial module and a uniformly bounded module. (ii) A uniformly bounded W-module is a direct sum of a trivial module, ∈ Z ∈ n amoduleAp,G and a module Ap ,G for some p, p +, G glp , ∈ n G glp . (iii) A nontrivial irreducible quasifinite W-module is a module of the in- n termediate series (i.e., A1,G or A1,G for some G ∈ C ).

Remark 3.6. (1) (cf. Remark 1.2(1)) If V is a weight module, each Gi in (3.4) is diagonalizable, and all uniformly bounded modules are completely reducible. (2) In Theorem 3.5(2), if a module have infinite number of the trivial composition factors, then it may not be uniformly bounded since any Γ-graded vector space can be defined as a trivial module. The main techniques in obtaining Theorem 3.5 are developed from the techniques of Mathieu [19] used in his proof of Theorem 3.1. These techniques seem to be useful in obtaining analogous results for some Z-graded Lie algebras (e.g., [28, 29]). The main idea is to make use of results of representations of the Vira- soro subalgebra (cf. (1.13)). Let us first consider W = W1+∞, and denote (cf. (1.4)): d L = ti[D] = ti+j( )j for i ∈ Z,j∈ Z . i,j j dt + Using Theorems 3.1 and 3.2, one can obtain the following 2 lemmas, from which, Theorem 3.5(1) follows.

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Lemma 3.7. (1) Suppose V is a uniformly bounded Vi r -module generated by a vector v such that there exists some k0 ∈ Z satisfying Liv =0for all i ∈ Zk0. Then V = Cv is trivial. (2) Suppose V is an irreducible quasiﬁnite W-module without a highest/lowest weight. Then

L1,0 : Vi → Vi+1 and L−1,0 : Vi → Vi−1 are injective and thus bijective for all i ∈ Z.Inparticular,V is uniformly bounded.

Lemma 3.8. Suppose V = ⊕i∈ZVi is an indecomposable uniformly bounded W-module without the trivial composition factor. Then

(1) L0,0 acts as a constant, and κ acts as zero. (2) Li,0 acts nondegenerately on V for all i =0 . (3) V is a module of the form Ap,G or Ap,G. The proof of Lemma 3.7(1) is straightforward, while Lemma 3.7(2) can be obtained by following the arguments in the proof of Lemma 1.6 in [19]. To prove Lemma 3.8, by Lemma 3.7(2), there exists p ∈ Z+ such that dim Vi = p for ∈ Z (1) (p) i , and hence we can choose a basis Y0 =(y0 , ..., y0 )ofV0, and deﬁne (1) (p) abasisYi =(yi , ..., yi )ofVi satisfying Yi = L1,0Yi−1. Then by supposing Li,j Yk = Yi+kPi+j,j,k for some Pi+j,j,k ∈ glp, and using relation (1.4), one can easily deduce

Pi,0,n = Pi,Pi,1,n =¯nPi + Qi,

Pi,2,n =[¯n]2Pi +2¯nQi + Ri,Pi,3,n =[¯n]3Pi +3[¯n]2Qi +3¯nRi + Si, for some Pi,Qi,Ri,Si ∈ glp,where¯n+G for some fixed G ∈ Mp×p (here we identify a scalar with the corresponding p × p scalar matrix), [¯n]j is a similar notation to [D]j ,andQ1 = 0 (we use notationn ¯ + G in order to be able to take Q1 =0;note from [L0,1,Li,j ]=iLi,j that G commutes with all other matrices involved in the discussion). Then by further use of relation (1.4), we can deduce Pi = ±1. Thus by (3.5), we can suppose Pi = 1, and then deduce Qi = Ri =0.¿Fromthis,the lemma follows. ∼ For the case W = W(Γ,n)withΓ= Z,letWi t t (Γ,n) be the generalized Witt algebra defined in (1.15). Using Theorem 3.3, one can obtain the following result, from which, Theorem 3.5(2) can be then deduced. Lemma 3.9. (1) Any quasifinite W-module V with a finite number of the trivial composition factor is a uniformly bounded module. (2) Let V be a uniformly bounded W-module without the trivial composition factor. Then tα · v =0 for all α ∈ Γ\{0},v∈ V \{0}. 3.3. Classification of quasifinite representations of W(Γ,n)(1) and W(Γ, 1)(1). Although W(Γ,n)(1) is a subalgebra of W(Γ,n), the representations of W(Γ,n)(1) seem to be more complicated than those of W(Γ,n), due to the crucial fact that the elements tβ = tβD0 for β ∈ Γ do not appear in W(Γ,n)(1) (thus we do not have some nice results such as Lemma 3.8(1) and (2)). Because of this, we are unable to work out indecomposable uniformly bounded W(Γ,n)(1)-module. But we can still determine its irreducible modules, and the proof of the results seems to be more technical than that of W(Γ,n)[31].

REPRESENTATIONS OF WEYL TYPE ALGEBRAS 25317

n For α ∈ C , we have two families of modules Aα, Aα of the intermediate series over W(Γ,n)(1) or W(Γ, 1)(1) as follows (which correspond to (3.4), (3.6) with p = 1): They have basis {yβ | β ∈ Γ} such that the central element κ acts trivially and β µ µ Aα :(t D )yγ =(α + γ) yβ+γ , β µ |µ|+1 µ Aα :(t D )yγ =(−1) (α + β + γ) yβ+γ , ∈ ∈ Zn \{ } µ λ for β,γ Γ,µ + 0 (where (α + γ) is a notation as β in (1.2)). Obviously, Aα or Aα is irreducible if and only if α/∈ Γ.

Remark 3.10. It is straightforward to prove that Aα, Aα are the only W(Γ,n)(1)-modules of the intermediate series. In particular, unlike the Virasoro (1) algebra case (cf. (3.2) and (3.3)), W(Γ,n) -modules Aα, Aα do not have any deformation. The main result of this subsection is the following result obtained in [31]. (1) Theorem 3.11. (1) Suppose W = W(Z, 1) or W∞. (i) An irreducible quasifinite W-module is a highest/lowest weight module, or a module of the intermediate series. (ii) A nontrivial uniformly bounded indecomposable weight W-module is a module of the intermediate series. ∼ (2) Suppose W = W(Γ,n)(1) or W(Γ,n)(1) such that Γ = Z. (i) A nontrivial irreducible quasifinite W-module is a module of the intermediate series. (ii) A nontrivial quasifinite indecomposable weight W-module is a module of the intermediate series.

This theorem mainly follows from the following 2 lemmas, where W = W∞.

Lemma 3.12. Let S be a subspace of W0 with finite codimension. Given i0 > 0, W i0 i0+1 i0 2 let Mi0,S denote the subalgebra of generated by t D, t D, t D and S.Then W ⊕ W ⊂ there exists some integer K>0 such that [K,∞) := i≥K i Mi0,S . Lemma 3.13. Assume that V is an irreducible quasifinite W-module without a highest/lowest weight. For any i, j ∈ Z,i=0 , −1, the linear map i | ⊕ i+1 | ⊕ i 2| → ⊕ ⊕ t D Vj t D Vj t D Vj : Vj Vi+j Vi+j+1 Vi+j is injective. In particular, dim Vj ≤ 2dimV0 +dimV1 for j ∈ Z,thusV is uniformly bounded. 3.4. Classification of quasifinite representations of DN . We define 2 families of DN -modules V (α), V (α), α ∈ C, of the intermediate series below. For afixedα ∈ C, the obvious representation of DN (with trivial action of the central element κ) on the vector space V (α)=tαCN [t, t−1] defines an irreducible module T N V (α). Let {εp =(δp1, ..., δpN ) | p ∈ [1,N]} be the standard basis of C ,where the superscript “T” means the transpose of vectors or matrices (thus the elements N i j of C are column vectors). Then an element t D Ep,q ∈ DN acts on a vector k+α t εr ∈ V (α)by i j k+α j i+k+α (3.8) (t D Ep,q)(t εr)=δq,r(k + α) t εp, k+α for i, k ∈ Z,j ∈ Z+,p,q,r ∈ [1,N]. For j ∈ Z,letV (α)j = Ct εr, where k ∈ Z,r∈ [1,N] are unique such that j +1=kN + r, then V (α)=⊕j∈ZV (α)j is

18254 YUCAI SU a Z-graded space such that dim V (α)j =1forj ∈ Z.ThusV (α) is a module of the intermediate series. N k+α For v ∈ C ,k∈ Z,denotevk = t v ∈ V (α) (note that vk is in general not a homogeneous vector). For A ∈ glN , define Avk =(Av)k,whereAv is the natural action of A on v defined linearly by Ep,qεr = δq,rεp (i.e., the action is defined by the matrix-vector multiplication). Then (3.8) can be rewritten as i j j (3.9) (t D A)vk =(k + α) Avi+k, N N for i, k ∈ Z,j∈ Z+,A∈ glN ,v∈ C . Clearly V (α)isalsoaD -module (cf. (1.8) as ∼ and (1.9)). Note that (cf. (3.5)) there exists a Lie algebra isomorphism σ : DN = DN such that (3.10) σ(tiDjA)=(−1)j+1ti(D + i)jAT, for i ∈ Z,j∈ Z+,A∈ glN . Using this isomorphism, we have another DN -module V (α) (again with trivial action of κ), called the twisted module of V (α), defined by i j j+1 j T (3.11) (t D A)vk =(−1) (i + k + α) A vi+k, N for i, k ∈ Z,j∈ Z+,A∈ glN ,v∈ C .Infact,V (α)isthedual module of V (−α): i+α j−α If we define a nondegenerate bilinear form on V (α) × V (−α)byt εp,t εq = δi+j,0δp,q,then xv, v = −v, xv for x ∈ DN , v ∈ V (α),v∈ V (−α). DN Obviously, V (α)isnota as-module. Now we can generalize the above modules V (α)andV (α) as follows: Let α be an indecomposable linear transformation on Cm (thus up to equivalences, α is N m uniquely determined by its unique eigenvalue λ). Let glN and α act on C ⊗ C defined by N m A(u ⊗ v)=Au ⊗ v, α(u ⊗ v)=u ⊗ αv for A ∈ glN ,u∈ C ,u∈ C . Then in (3.9) and (3.11), by allowing v to be in CN ⊗ Cm, we obtain 2 families of indecomposable uniformly bounded modules V (m, α), V (m, α). The main result of this subsection is the following theorem obtained in [30].

Theorem 3.14. (1) An irreducible quasifinite module over DN is a highest/lowest weight module or else a module of the intermediate series. (2) A nontrivial module of the intermediate series over DN is a module V (α) or V (α) for some α ∈ C. (3) A nontrivial indecomposable uniformly bounded module over DN is a module V (m, α) or V (m, α) for some m ∈ Z+\{0} and some indecomposable linear transformation α of Cm. Thus in particular, a nontrivial indecomposable uniformly bounded module over DN DN is simply a as -module or its twist, and there is an equivalence between the DN category of uniformly bounded as-modules without the trivial composition factor and the category of linear transformations on finite-dimensional vector spaces. Since irreducible quasifinite highest weight modules have been classified in [5](Theorem 2.2(3)) and irreducible lowest weight modules are simply dual modules of irreducible highest weight modules, Theorem 3.14 in fact classifies all irreducible quasifinite modules over DN and over DN .

REPRESENTATIONS OF WEYL TYPE ALGEBRAS 25519

Note that there is a one to one correspondence between Lie conformal algebras and maximal formal distribution Lie algebras, and the Lie algebra DN is simply the formal distribution Lie algebra associated to the general Lie conformal algebra gcN . Thus in the language of conformal algebras, this theorem in particular also gives proofs of Theorems 6.1 and 6.2 of [13] on the classification of finite indecomposable modules over the conformal algebras CendN and gcN . The proof of Theorem 3.14 mainly follows from Theorem 3.5 and the following result. Lemma 3.15. Suppose V is an irreducible quasifinite DN -module without highest and lowest weight vectors. Then | → −1| → t Vi : Vi Vi+N and t Vi : Vi Vi−N are injective and thus bijective for all i ∈ Z (recall from (1.12) that t has degree N). In particular, by letting K =max{dim Vp | p ∈ [1,N]}, we have dim Vi ≤ K for i ∈ Z;thusV is uniformly bounded. Acknowledgement. The author would like to thank Professor Zongzhu Lin and the referee for useful comments and suggestions.

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Department of Mathematics, University of Science and Technology of China, Hefei 230026, China

Contemporary Mathematics Volume 478, 2009

Maximal and Primitive Elements in Baby Verma Modules for Type B2

Nanhua Xi

The purpose of this paper is to find maximal and primitive elements of baby Verma modules for a quantum group of type B2. As a consequence the composition factors of the baby Verma modules are determined. A similar approach can be used to find maximal and primitive elements of Weyl modules for type B2. In principle the results can be used to determine the module structure of a baby Verma module, but the calculations are rather involved, much more complicated than the case of type A2. For type A2, submodule structure of a Weyl module has been determined in [DS1, I, K] and by Cline (unpublished). For type B2, the socle series of Weyl modules was determined in [DS2]. In [X2] we determine the maximal and primitive elements in Weyl modules for type A2, so that the Weyl modules are understood more explicitly. This paper is a sequel to [X2], but less complete, since submodule structure of a baby Verma module is not determined. In this paper we only work with quantized enveloping algebras at roots of 1 (Lusztig version). For hyperalge- bras the approach is completely similar, actually simpler. The contents of the paper are as follows. In section 1 we recall some definitions and results about maximal and primitive elements. In section 2 we recall some facts about a quantized enveloping algebra of type B2. In section 3 we determine the maximal and primitive elements in a Verma module of the (slightly enlarged) Frobenius kernel of type B2. In section 4 we indicate that the maximal and primitive elements in a Weyl module for type B2 can be worked out similarly, but we omit the results. To avoid complicated expressions and for simplicity we assume that the order of the involved root of 1 is odd and greater than 3 and we only work with some special weights. The approach for general cases is completely similar.

1. Maximal and Primitive Elements In this section we ﬁx notation and recall the deﬁnition and some results for maximal and primitive elements. We refer to [L1-4, X1-2] for additional information.

2000 Mathematics Subject Classiﬁcation: Primary 17B37; Secondary 20G05. Key words: maximal element, primitive element, baby Verma module . The author was supported in part by the National Natural Science Foundation of China (No. 10671193).

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1.1. Let Uξ be a quantized enveloping algebra (over Q(ξ)) at a root ξ of 1 (Lusztig version). We assume that the rank of the associated Cartan matrix is n and ≥ (a) (a) −1 the order of ξ 3. As usual, the generators of Uξ are denoted by ei ,fi ,ki,ki , etc. Let u be the Frobenius kernel and u˜ the subalgebra of Uξ generated by all n elements in u and in the zero part of Uξ.Forλ ∈ Z and a Uξ-module (or u˜- module M)wedenotebyMλ the λ-weight space of M. A nonzero element in Mλ will be called a vector of weight λ or a weight vector. Let m be a weight vector of (a) a Uξ-module (resp. u˜-module) M.Wecallm maximal if ei m =0foralli and a ≥ 1(resp. eαm = 0 for all root vectors eα in the positive part of u˜). We call m a primitive element if there exist two submodules M2 ⊂ M1 of M such that m ∈ M1 and the image in M1/M2 of m is maximal. Obviously, maximal elements areprimitive.Wehave(see[X2]): (a) Let m ∈ M be a weight vector and let P1 be the submodule of M generated by m.Thenm is primitive if and only if the image in P1/P2 of m is maximal for some proper submodule P2 of P1. We shall write L(λ)(resp. L˜(λ)) for an irreducible Uξ-module (resp. u˜-module) of highest weight λ. (b) If m is a primitive element of weight λ,thenL(λ)(orL˜(λ)) is a composition factor of M (depending whether M is a Uξ-module or a u˜-module). (c) Let M and N be modules and φ : M → N a homomorphism. Let m be a weight vector in M.Ifφ(m) is a primitive element of N,thenm is a primitive element of M. (d) Let M,N,φ,m be as in (c) and assume φ(m) =0. Ifm is a primitive element of M, then either φ(m) is a primitive element of N or φ(P1)=φ(P2), where P1 is the submodule of M generated by m and P2 is any submodule of P1 such that the image in P1/P2 of m is maximal. (e) Let M,N,φ,m be as in (c) and assume φ(m) =0. Ifm is a maximal element of M,thenφ(m) is a maximal element of N. We shall denote by Z˜(λ) the (baby) Verma module of u˜ with highest weight λ and denote by 1˜λ a nonzero element in Z˜(λ)λ. Recall that to deﬁne Uξ we need to choose di ∈{1, 2, 3} such that (diaij) is symmetric, where (aij) is the concerned 2di n n × n Cartan matrix. Let li be the order of ξ .Forλ =(λ1, ..., λn) ∈ Z we set lλ =(l1λ1, ..., lnλn). We call λ l-restricted if 0 ≤ λi ≤ li − 1 for all i.Denoteby n n Nl the set of all l-restricted elements in Z . The following fact is well known and (a) (b) follows easily from the commutation formula for ej fi (see [L1, 4.1(a)]). ∈ n ∈ n ∈ n (λi+1)˜ (f) Let λ Nl ,λ Z .Setµ = λ + lλ Z .Thenfi 1µ is maximal in Z˜(µ)ifλi = li − 1.

2. Some basic facts

2.1. From now on we assume that Uξ is of type B2. In this section we recall some basic facts about Uξ and the Verma modules Z˜(λ). For completeness and in order to ﬁx notations, we give the deﬁnition of Uξ and Z˜(λ). Let aii =2,a12 = −2,a21 = −1. Let U be the associative algebra over Q(v) −1 (v an indeterminate) generated by ei,fi,ki,ki (i =1, 2) with relations −1 −1 k1k2 = k2k1,kiki = ki ki =1,

iaij −iaij kiej = v ejki,kifj = v fjki,

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k − k−1 e f − f e = δ i i , i j j i ij − −1 vi vi 2 − 2 −2 2 e1e2 (v + v )e2e1e2 + e2e1 =0, 3 − 2 −2 2 2 −2 2 − 3 e1e2 (v +1+v )e1e2e1 +(v +1+v )e1e2e1 e2e1 =0, 2 − 2 −2 2 f1f2 (v + v )f2f1f2 + f2 f1 =0, 3 − 2 −2 2 2 −2 2 − 3 f1 f2 (v +1+v )f1 f2f1 +(v +1+v )f1f2f1 f2f1 =0, 2 −1 where v1 = v and v2 = v .LetU be the A = Z[v, v ]-subalgebra of U generated e(a) ea/ a ,f(a) f a/ a ,k ,k−1,a∈ ,i , a by all i = i [ ]i! i = i [ ]i! i i N =12, where [ ]i!= a vih−v−ih ≥ h=1 vi−v−i if a 1and[0]i! = 1. Note that the element . / a c−h+1 −c+h−1 −1 k ,c v k − v k i = i i i i a h − −h vi vi h=1 . / k ,c is in U for all c ∈ Z,a∈ N. We understand that i =1ifa =0.Note a (a) − 2 a (a) − 2 a that f12 =(f1f2 v f2f1) /[a]! and f12 =(f2f1 v f1f2) /[a]! are in U for all a ∈ N.Also − a − a (a) (f1f12 f12f1) (a) (f12f1 f1f12) f112 = −1 a and f112 = −1 a (v + v ) [a]2! (v + v ) [a]2! are in U for all a ∈ N. Regard Q(ξ)asanA-algebra by specializing v to ξ.Then Uξ = U ⊗A Q(ξ). See [L3]. (a) (a) (a) (a) (a) (a) −1 . For/ convenience, the images in Uξ of ei ,fi ,f12 ,f12 ,f112 ,f112,ki,ki , k ,c i etc. will be denoted by the same notation respectively. Let l be the order a 2i li li of ξ and li be the order of ξ .InUξ we have ei = fi =0. For simplicity in this paper we assume that l is odd. Then l1 = l2 = l. The Frobenius kernel u of Uξ −1 − is the subalgebra of Uξ generated by all ei,fi,ki,ki ,i=1, 2. Its negative part u (a) (a) (a) (a) − ≤ ≤ − is generated by all fi.Notethatf12 ,f12 ,f112 ,f112, are. in u /if 0 a l 1. k ,c The subalgebra u˜ of U is generated by all e ,f ,k ,k−1, i ,i=1, 2; c ∈ ξ i i i i a Z,a∈ N. ∈ 2 ˜ For λ =(λ1,λ2) . Z ,wedenoteby/ . /Iλ the left ideal of .Uξ generated/ by all (a) ki,c λi + c b e (a>0),k − ξiλi , − . (We denote by the value at i i a a a ξi ξi − − i a vb h+1−vb+h 1 ∈ ∈ ξ of h=1 vh−v−h for any b Z, a N and i =1, 2.) The Verma module Z(λ)ofUξ is deﬁned to be Uξ/I˜λ.Let1˜λ be the image in Z(λ)of1.TheVerma module Z˜(λ)ofu˜ is deﬁned to be the u˜-submodule of Z(λ) generated by 1˜λ .Given non-negative integers a and b,weset (a) (a+b) (a+2b) (b) (b) (a+2b) (a+b) (a) xa,b = f1 f2 f1 f2 = f2 f1 f2 f1 . Recall that l is the order of ξ. The following result is a special case of [X1, 4.2 (ii)]. (a) Assume 0 ≤ a, b ≤ l − 1, c, d ∈ Z,andletµ =(lc − 1+a, ld − 1+b). − Then the element xa,b is in u and xa,b1˜µ is maximal in Z˜(µ) and generates the unique irreducible submodule of Z˜(µ). The irreducible submodule is isomorphic to L˜(lc − 1 − a, ld − 1 − b).

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The argument for [X1, 4.4(iv)] also gives the following result. (b) Keep the assumption and notations in (a). Let p, q, s, t ∈ N such that x = (a+2b−pl) (a+b−ql) (a) (a+b−sl) (a+2b−tl) (b) f1 f2 f1 and y = f2 f1 f2 are nonzero elements, then − eix1˜µ = eiy1˜µ =0fori =1, 2. If x and y are further in u ,thenx1˜µ and y1˜µ are maximal in Z˜(µ). We shall need a few formulas, which are due to Lusztig (see [L3, L4]). In Uξ we have . / a + b (c) f (a)f (b) = f (a+b), i i a i ξi (i) (j) 2ij (j) (i) (d) f12 f2 = ξ f2 f12 , (i) (j) 2ij (j) (i) (e) f112f12 = ξ f12 f112, (i) (j) 2ij (j) (i) (f) f1 f112 = ξ f112f1 , s (i) (j) −2rs−2st −4h+2 − (r) (2s) (t) (g) f2 f112 = ξ (ξ 1)f112f12 f2 , r,s,t∈N h=1 r+s=j s+t=i s (i) (j) −rs−st+s −2h (r) (s) (t) (h) f12 f1 = ξ (ξ +1)f1 f112f12 , r,s,t∈N h=1 r+s=j s+t=i (i) (j) 2ru+2su+rt (r) (s) (t) (u) (i) f2 f1 = ξ f1 f112f12 f2 . r,s,t,u∈N s+t+u=i r+2s+t=j s (i) (j) −2rs−2st −4h+2 − (r) (2s) (t) (j) f 112f2 = ξ (ξ 1)f2 f 12 f 112, r,s,t∈N h=1 r+s=j s+t=i s (i) (j) −rs−st+s −2h (r) (s) (t) (k) f1 f 12 = ξ ξ +1)f 12 f 112f1 , r,s,t∈N h=1 r+s=j s+t=i (i) (j) 2ru+2rt+us (r) (s) (t) (u) (l) f1 f2 = ξ f2 f 12 f 112f1 . r,s,t,u∈N r+s+t=j s+2t+u=i (m) Assume 0 ≤ a0,b0 ≤ l − 1anda1,b1 ∈ Z.Wehave . / . / a + a l a a 0 1 = 0 1 , b b l b b 0 + 1 ξi 0 ξi 1 a where 1 is the ordinary binomial coeﬃcient. b1 Using (i), (l) and (m) we get ≤ ≤ − (l) (a) − (a) (l) (l) (a) − (a) (l) − (n) If 0 a l 1, then f1 f2 f2 f1 and f2 f1 f1 f2 are in u . ≤ ≤ − (a) (b) (c) (a) (l+b) (c) − (o) Let 0 a, b, c l 1. Then f1 f2 f1 =0andf1 f2 f1 is in u ,if − ≥ (a) (b) (c) (a) (l+b) (c) − − ≥ a + c 2b l. Similarly f2 f1 f2 =0andf2 f1 f2 is in u ,ifa + c b l The assertions (n) and (o) will be frequently used in computations. 2 + Let α1 =(2, −1),α2 =(−2, 2) ∈ Z . The set of positive roots is R = {α1,α2,α1 + α2, 2α1 + α2}.LetW be the Weyl group generated by the simple

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∨ reﬂections si corresponding to αi. Assume that l ≥ 5. Then ρ, β 

(3) Let µ = λ +(l − 1)α2. The following elements are maximal in Z˜(µ): ˜ (3)˜ (l−1)˜ (2) (3)˜ (l−1)˜ 1µ,f1 1µ,f2 1µ,f2 f1 1µ,f1f2 1µ, (2) (3)˜ (2) (l+1) (l−1)˜ (3) (l+2) (2l+1) (l−1)˜ f1f2 f1 1µ,f2 f1 f2 1µ,f1 f2 f1 f2 1µ.

(4) Let µ = λ +(l − 1)α1 +(l − 2)α2. The following elements are maximal in Z˜(µ): ˜ (3)˜ (l−2)˜ (3)˜ 1µ,f1 1µ,f2 1µ,f2f1 1µ, (l−1) (l−2)˜ (l−1) (l−2)˜ f1 f2 1µ,f2f1 f2 1µ. (l−1) (l+1) (3)˜ (3) (l+1) (2l−1) (l−2)˜ f1 f2 f1 1µ,f1 f2 f1 f2 1µ.

(5) Let µ = λ +(l − 3)α1 +(l − 1)α2. The following elements are maximal in Z˜(µ): ˜ (l−3)˜ (2)˜ (l−1) (l−3)˜ 1µ,f1 1µ,f2 1µ,f2 f1 1µ, (2)˜ (l−1) (l−3)˜ f1f2 1µ,f1f2 f1 1µ, (l−1) (l+1) (2)˜ (2) (l+1) (l−1) (l−3)˜ f2 f1 f2 1µ,f2 f1 f2 f1 1µ.

(6) Let µ = λ +(l − 4)α1 +(l − 2)α2. The following elements are maximal in Z˜(µ): ˜ (l−3)˜ ˜ (l−2) (l−3)˜ 1µ,f1 1µ,f21µ,f2 f1 1µ, (l−1) ˜ (l−1) (l−2) (l−3)˜ f1 f21µ,f1 f2 f1 1µ. (l−2) (l−1) ˜ (l−1) (l−2) (l−3)˜ f2 f1 f21µ,f2f1 f2 f1 1µ.

(7) Let µ = λ +(l − 3)α1 +(l − 3)α2. The following elements are maximal in Z˜(µ): ˜ ˜ (l−2)˜ (l−1) ˜ 1µ,f11µ,f2 1µ,f2 f11µ, (l−3) (l−2)˜ (l−3) (l−1) ˜ f1 f2 1µ,f1 f2 f11µ. (l−1) (2l−3) (l−2)˜ (l−2) (2l−3) (l−1) ˜ f2 f1 f2 1µ,f2 f1 f2 f11µ.

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(8) Let µ = λ +(l − 4)α1 +(l − 3)α2. The following elements are maximal in Z˜(µ): ˜ (l−1)˜ (l−1)˜ (l−2) (l−1)˜ 1µ,f1 1µ,f2 1µ,f2 f1 1µ, (l−3) (l−1)˜ (l−3) (l−2) (l−1)˜ f1 f2 1µ,f1 f2 f1 1µ. (l−2) (2l−3) (l−1)˜ (l−1) (3l−3) (2l−2) (l−1)˜ f2 f1 f2 1µ,f2 f1 f2 f1 1µ.

3. maximal and primitive elements of Z˜(λ) for type B2 In this section we determine the maximal and primitive elements in Z˜(λ)(or equivalently in any highest weight module of u˜). To avoid complicated expressions we only work with some weights in the W -orbit of (0, 0). For general cases the approach is completely similar. From now on we assume that the odd integer l is at least 5. Theorem 3.1. Let a, b be integers and λ =(la, lb).Then (i) The following 8 elements are maximal in Z˜(λ): ˜ ˜ ˜ (2) ˜ (3) ˜ 1λ,f11λ,f21λ,f2 f11λ,f1 f21λ, (2) (3) ˜ (3) (2) ˜ (2) (3) ˜ f2 f1 f21λ,f1 f2 f11λ,f1f2 f1 f21λ. (ii) The following 12 elements are primitive elements in Z˜(λ) but not maximal: x − f (3),f(l) f ˜ , l 1,2 ˜ , f (2),f(l) f (3)f ˜ [ 1 2 ] 21λ (l−1) 1λ [ 2 1 ] 1 21λ f1 x − x − f (2),f(l) f ˜ , 3,l 1 ˜ , 3,l 1 ˜ , f (3),f(l) f (2)f ˜ , [ 2 1 ] 11λ (l−1) 1λ (l) (l−1) 1λ [ 1 2 ] 2 11λ f2 f1 f2 x − − − 3,l 2 ˜ ,f(l 1)f (l)f ˜ ,ff (l 1)f (l)f ˜ , (l−1) (l−2) 1λ 2 1 21λ 1 2 1 21λ f1 f2 x − − x − − l 1,l 1 f ˜ , l 1,l 1 f ˜ . (l−3) (l−2) (l−1) 21λ (l−2) (2l−3) (l−1) 11λ f1 f2 f1 f2 f1 f2 − x (See 2.1 for the deﬁnition of xa,b. Convention: [x, y]=xy yx and y stands for an arbitrary homogeneous element z in u− such that zy = x.) Moreover no maximal element in Z˜(λ) has the same weight as any of the above 12 elements. (iii) The maximal and primitive elements in (i-ii) provide 20 composition factors of Z˜(λ), which are

L˜(λ), L˜(λ − α1), L˜(λ − α2),

L˜(λ − α1 − 2α2), L˜(λ − 3α1 − α2)

L˜(λ − 3α1 − 3α2), L˜(λ − 4α1 − 2α2), L˜(λ − 4α1 − 3α2),

L˜(λ − 3α1 − (l +1)α2), L˜(λ − (l +3)α1 − (l +3)a2), L˜(λ − (l +3)α1 − 3α2),

L˜(λ−(l+1)α1−2α2), L˜(λ−(2l+4)α1−(l+2)α2), L˜(λ−(l+4)α1−(l+2)α2),

L˜(λ − 4α1 − (l +2)α2), L˜(λ − (l +3)α1 − (l +1)α2), L˜(λ − lα1 − lα2),

L˜(λ − (l +1)α1 − lα2), L˜(λ − 2lα1 − 2lα2), L˜(λ − 2lα1 − lα2). Moreover, Z˜(λ) has only the 20 composition factors.

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Proof. (i) According to 1.1 (e-f), we see that (i) is true. (ii) Now we argue for (ii). (1) Consider the homomorphism: ˜ → ˜ − ˜ → (l−1)˜ ϕ1 : Z(λ) Z(λ +(l 1)α1)), 1λ t1 = f1 1λ+(l−1)α1 . Let (3) (l) − (l) (3) ∈ − x1 =(f1 f2 f2 f1 )f2 u . Note that (3) (l+1) (3) (l+1) (l−1)˜ f1 f2 t1 = f1 f2 f1 1λ+(l−1)α1 = x1t1.

Using 1.1 (c) we see that x11˜λ is a primitive element of weight γ38 = λ − 3α1 − (l + 1)α2. Let (2) (l) − (l) (2) ∈ − y1 =(f2 f1 f1 f2 )x1 u . (2) (3) (l+1) (l−1) Note that f2 f1 f2 f1 = 0. We then can check that (2) (l+3) (l+1) (2) (l+3) (l+1) (l−1)˜ f2 f1 f2 t1 = f2 f1 f2 f1 1λ+(l−1)α1 = y1t1.

Using 1.1 (c) we see that y11˜λ is a primitive element of weight γ48 = λ − (l +3)α1 − xl−1,2 (l +3)α2.Notethatwehavey1 = (l−1) . f1 (l+1) (l−1) (l) (l−1) (l+1) (l−1) Note that f2 f1 = f2f2 f1 ,sowehavexf2 f1 =0ifxf2 =0 ∈ − ˜ and x u . Thus we have a homomorphism (recall that 1λ+(l−1)α1+lα2 is also an element of the Verma module Z(λ +(l − 1)α1 + lα2)ofUξ): ˜ → (l+1) (l−1)˜ ψ1 : u˜f21λ u˜f2 f1 1λ+(l−1)α1+lα2 , ˜ → (l+1) (l−1)˜ f21λ f2 f1 1λ+(l−1)α1+lα2 . (3) ˜ (3) (l+1) (l−1)˜ (3) (l+1) (l−1) Note that ψ1(f1 f21λ)=f1 f2 f1 1λ+(l−1)α1+lα2 and f1 f2 f1 is in u−. (2) (l) − (l) (2) ∈ − (3) ˜ Let z1 = f2 f1 f1 f2 u . Using 1.1 (c) we see that z1f1 f21λ is a − − (3) xl−1,2 primitive element of weight γ47 = λ (l+3)α1 3α2.Notethatz1f1 f2 = (l) (l−1) . f2 f1 (2) Now we consider the homomorphism: ˜ → ˜ − ˜ → (l−1)˜ ϕ2 : Z(λ) Z(λ +(l 1)α2)), 1λ t2 = f2 1λ+(l−1)α2 . Let (2) (l) − (l) (2) ∈ − x2 =(f2 f1 f1 f2 )f1 u . Note that (2) (l+1) (2) (l+1) (l−1)˜ f2 f1 t2 = f2 f1 f2 1λ+(l−1)α2 = x2t2.

Using 1.1 (c) we see that x21˜λ is a primitive element of weight γ31 = λ−(l +1)α1 − 2α2. − (3) (l+2) (2l+1) (l−1)˜ Let y2 be homogeneous in u such that y2t2 = f1 f2 f1 f2 1µ,here µ = λ +(l − 1)α2. According to 1.1 (c) we know that y21˜λ is a primitive element − − x3,l−1 of weight γ41 = λ (2l +4)α1 (l +2)α2.Notethaty2 = (l−1) . f2 As the reason for ψ1, we have a homomorphism: ˜ → (l+1) (l−1)˜ ψ2 : u˜f11λ u˜f1 f2 1λ+lα1+(l−1)α2 , ˜ → (l+1) (l−1)˜ f11λ f1 f2 1λ+lα1+(l−1)α2 .

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(2) ˜ (2) (l+1) (l−1)˜ Note that ψ2(f2 f11λ)=t2 = f2 f1 f2 1λ+lα1+(l−1)α2 and the element (2) (l+1) (l−1) − − f2 f1 f2 is in u .Letz2 be homogeneous in u such that (3) (l+2) (2l+1) (l−1)˜ z2t2 = f1 f2 f1 f2 1λ+lα1+(l−1)α2 . (2) ˜ − Using 1.1 (c) we see that z2f2 f11λ is a primitive element of weight γ42 = λ (l + − (2) x3,l−1 4)α1 (l +2)α2.Notethatz2f2 f1 = (l) (l−1) . f1 f2 ˜ We also have a homomorphism (recall that 1λ+2lα1+(l−1)α2 is also an element of the Verma module Z(λ +2lα1 +(l − 1)α2)ofUξ): ˜ → (2l+1) (l−1)˜ θ2 : u˜f11λ u˜f1 f2 1λ+2lα1+(l−1)α2 , ˜ → (2l+1) (l−1)˜ f11λ t2 = f1 f2 1λ+2lα1+(l−1)α2 . (3) (l) ∈ − Let w2 =[f1 ,f2 ] u .Then (2) (3) (l+2) (2l+1) (l−1)˜ w2f2 t2 = f1 f2 f1 f2 1λ+2lα1+(l−1)α2 . (2) ˜ − Using 1.1 (c) we see that w2f2 f11λ is a primitive element of weight γ44 = λ − (2) x3,l−1 4α1 (l +2)α2.Notethatw2f2 f1 = (2l) (l−1) . f1 f2 (3) Now we consider the homomorphism:

Z˜(λ) → Z˜(λ +(l − 1)α1 +(l − 2)α2), ˜ → (l−1) (l−2)˜ 1λ t3 = f1 f2 1λ+(l−1)α1+(l−2)α2 . − Let x3 be homogeneous in u such that (3) (l+1) (2l−1) (l−2)˜ x3t3 = f1 f2 f1 f2 1λ+(l−1)α1+(l−2)α2 .

Using 1.1 (c) we know that x31˜λ is a primitive element of weight γ37 = λ − (l + − x3,l−2 3)α1 (l +1)α2.Notethatx3 = (l−1) (l−2) . f1 f2 (l−3) (l−2) ∈ − (4) Since f1 f2 f2u (see 2.1), we have a surjective homomorphism ˜ → (l−3) (l−2)˜ ϕ4 : u˜f21λ u˜f1 f2 1λ+(l−3)α1+(l−3)α2 ˜ → (l−3) (l−2)˜ f21λ f1 f2 1λ+(l−3)α1+(l−3)α2 . (l−1) (l) − (l) (l−1) ∈ − Let x4 = f2 f1 f1 f2 u . Then ˜ (l−1) (l) ˜ x4f21λ = f2 f1 f21λ is a primitive element of weight γ34 = λ − lα1 − lα2. Let y4 = f1x4f2.Theny41˜λ is a primitive element of weight γ45 = λ − (l + 1)α1 − lα2. (5) Consider the homomorphism ˜ → (l−3) (l−2) (l−1)˜ u˜f21λ u˜f1 f2 f1 1λ+(2l−4)α1+(l−3)α2 ˜ → (l−3) (l−2) (l−1)˜ f21λ f1 f2 f1 1λ+(2l−4)α1+(l−3)α2 . − Let x5 be homogeneous in u such that (l−3) (l−2) (l−1) (l−1) (3l−3) (2l−2) (l−1) x5f1 f2 f1 = f2 f1 f2 f1 .

By 1.1 (c), x5f21˜λ is primitive and is of weight γ35 = λ − 2lα1 − 2lα2.Notethat xl−1,l−1 x5 = (l−3) (l−2) (l−1) . f1 f2 f1

MAXIMAL AND PRIMITIVE ELEMENTS 2659

Consider the surjective homomorphism ˜ → (l−2) (2l−3) (l−1)˜ u˜f11λ u˜f2 f1 f2 1λ+(2l−4)α1+(2l−3)α2 ˜ → (l−2) (2l−3) (l−1)˜ f11λ f2 f1 f2 1λ+(2l−4)α1+(2l−3)α2 . − Let y5 be homogeneous in u such that (l−2) (2l−3) (l−1) (l−1) (3l−3) (2l−2) (l−1) y5f2 f1 f2 = f2 f1 f2 f1 .

By 1.1 (c), y5f11˜λ is primitive and is of weight γ32 = λ − 2lα1 − lα2.Notethat xl−1,l−1 y5 = (l−2) (2l−3) (l−1) . f2 f1 f2 We may also consider the homomorphism: ˜ → (l−3) (2l−2) (l−1)˜ u˜f21λ u˜f1 f2 f1 1λ+(2l−4)α1+(2l−2)α2 , ˜ → (l−3) (2l−2) (l−1)˜ u˜f11λ f1 f2 f1 1λ+(2l−4)α1+(2l−2)α2 . − (l 1) (l) (l) ˜ xl−1,l−1 ˜ Using 1.1 (c) we know that [[f2 ,f1 ],f1 ]f21λ = (l−3) (2l−2) (l−1) f21λ is also a f1 f2 f1 primitive element of weight λ − 2lα1 − lα2. (2) (3) ˜ ˜ The element f1f2 f1 f21λ generates the unique irreducible submodule of Z(λ). Clearly, the weight of any element in (ii) is not greater than λ−4α1 −3α2, therefore no maximal element in Z˜(λ) has the same weight as any of the elements in (ii). (iii) Using (i), (ii) and 1.1 (b), we see that Z˜(λ) has the 20 composition factors. The dimensions of irreducible u˜-modules are known (see [APW]). By a comparison of dimensions we know that Z˜(λ) has only the 20 composition factors. The theorem is proved. Theorem 3.2. Let a, b be integers and λ =(la, lb − 3).Then (i) The following elements are maximal in Z˜(λ): ˜ ˜ (l−2)˜ (l−1) ˜ 1λ,f11λ,f2 1λ,f2 f11λ, (l−3) (l−2)˜ (l−3) (l−1) ˜ f1 f2 1λ,f1 f2 f11λ. (l−1) (2l−3) (l−2)˜ (l−2) (2l−3) (l−1) ˜ f2 f1 f2 1λ,f2 f1 f2 f11λ, (l−1) (l−3) (l−1) (l−3) f f f f f − − 2 1 ˜ , 1 2 1 ˜ ,f(l 3) f (l 1),f(l) f ˜ . (2) 1λ (2) 1λ 1 [ 2 1 ] 11λ f2 f2 (ii) The following elements are primitive elements in Z˜(λ) but not maximal: x − − x − − − l 1,l 1 ˜ , l 1,l 1 ˜ , f (l 1),f(l) f ˜ , (l−1) 1λ (l) (l−1) 1λ [ 2 1 ] 11λ f1 f2 f1 x − x − − 3,l 1 ˜ , 3,l 1 ˜ ,f(l 1)f (l)f ˜ , (2) (3) 1λ (l+2) (3) 1λ 1 2 11λ f2 f1 f2 f1 x − − x − 3,l 2 f ˜ , f (l 2),f(l) ˜ , l 1,2 f ˜ . (3) 11λ [ 2 1 ]1λ (3) (2) 11λ f2f1 f2f1 f2 Moreover no maximal element in Z˜(λ) has the same weight as any of the above 9 elements. (iii) The maximal and primitive elements in (i-ii) provide 20 composition factors of Z˜(λ), which are

L˜(λ), L˜(λ − α1), L˜(λ − (l − 2)α2),

L˜(λ − α1 − (l − 1)α2), L˜(λ − (l − 3)α1 − (l − 2)α2)

10266 NANHUA XI

L˜(λ−(l−2)α1−(l−1)α2), L˜(λ−(2l−3)α1−(2l−3)α2), L˜(λ−(2l−2)α1−(2l−3)α2),

L˜(λ−(l−3)α1−(l−3)α2), L˜(λ−(l−2)α1−(l−3)a2), L˜(λ−(3l−3)α1−(3l−3)α2),

L˜(λ−(3l−3)α1 −(2l−3)α2), L˜(λ−(l+1)α1 −(l−1)α2), L˜(λ−(2l−2)α1 −(l−1)α2),

L˜(λ − (2l +1)α1 − (2l − 1)α2), L˜(λ − (2l +1)α1 − (l − 1)α2), L˜(λ − lα1 − lα2),

L˜(λ − lα1 − (2l − 2)α2), L˜(λ − lα1 − (l − 2)α2), L˜(λ − 2lα1 − lα2). Moreover, Z˜(λ) has only the 20 composition factors. Proof. (i) According to 1.1 (e-f), we see that the ﬁrst 8 elements in (i) are maximal. Consider the homomorphism: ˜ → ˜ ˜ → (2)˜ Z(λ) Z(λ +2α2), 1λ t1 = f2 1λ+2α2 . − − (l−1) (l−3) (2) f (l 1)f (l 3) − 2 1 ˜ Since f2 f1 is in u f2 , using 1.1 (c) we see that (2) 1λ and f2 − − f f (l 1)f (l 3) 1 2 1 ˜ − − − − (2) 1λ are primitive elements of weights λ (l 3)α1 (l 3)α2 and f2 λ − (l − 2)α1 − (l − 3)α2 respectively. One can check directly that the two elements are maximal. We will show that the last element in (i) is maximal in part (2) of the argument for (ii). (ii) Now we argue for (ii). (1) Consider the homomorphism: ˜ → ˜ − ˜ → (l−1)˜ Z(λ) Z(λ +(l 1)α1)), 1λ t1 = f1 1λ+(l−1)α1 . xl−1,l−1 ∈ − ˜ Let x1 = (l−1) u . Using 1.1 (c) we see that x11λ is a primitive element of f1 weight λ − (3l − 3)α1 − (3l − 3)α2. Consider the homomorphism: (l−2)˜ → (2l−2) (l−1)˜ u˜f2 1λ u˜f2 f1 1λ+(l−1)α1+lα2 , (l−2)˜ → (2l−2) (l−1)˜ f2 1λ f2 f1 1λ+(l−1)α1+lα2 . Let 1 − − y = [[f (l 1),f(l)],f(l)]f (l 3). 1 2 2 1 1 1 (2l−2) (l−1) (l−2)˜ Then y1f2 f1 = xl−1,l−1. Using 1.1 (c) we see that y1f2 1λ is a primitive − − − − (l−2) (l) (l−1) element of weight λ (3l 3)α1 (2l 3)α2.Notethaty1f2 = xl−1,l−1/f2 f1 is in u−. (2) Now we consider the homomorphism: ˜ → ˜ ˜ → (2)˜ Z(λ) Z(λ +2α2), 1λ t2 = f2 1λ+2α2 . Let (l−1) (l) − (l) (l−1) ∈ − x2 =(f2 f1 f1 f2 )f1 u . Using 1.1 (c) we see that x21˜λ is a primitive element of weight λ−(l+1)α1−(l−1)α2. (l−3) ∈ − ˜ Let y2 = f1 x2 u . According to 1.1 (c) we know that y21λ is a primitive element of weight λ − (2l − 2)α1 − (l − 1)α2. It is easy to see that y21˜λ is maximal. (3) Now we consider the homomorphism:

Z˜(λ) → Z˜(λ +3α1 +2α2), ˜ → (2) (3)˜ 1λ t3 = f2 f1 1λ+3α1+2α2 .

MAXIMAL AND PRIMITIVE ELEMENTS 26711

x3,l−1 ∈ − ˜ Let x3 = (2) (3) u . Using 1.1 (c) we know that x31λ is a primitive element of f2 f1 weight λ − (2l +1)α1 − (2l − 1)α2. Consider the homomorphism: ˜ → (l+2) (3)˜ Z(λ) u˜f2 f1 1λ+3α1+(l+2)α2 , ˜ → (l+2) (3)˜ 1λ t3 = f2 f1 1λ+3α1+(l+2)α2 . − (l 1) (l) (l) x3,l−1 ∈ − ˜ Let y3 =[[f2 ,f1 ],f1 ]f1 = (l+2) (3) u . Using 1.1 (c) we know that y31λ is f2 f1 a primitive element of weight λ − (2l +1)α1 − (l − 1)α2. (4) We consider the homomorphism: ˜ → (3)˜ u˜f11λ u˜f2f1 1λ+2α1+α2 , ˜ → (3)˜ f11λ t4 = f2f1 1λ+2α1+α2 . (l−1) (l) ∈ − ˜ Let x4 = f1 f2 f1 u . Using 1.1 (c) we know that x41λ is a primitive element of weight λ − lα1 − lα2. x3,l−2 ∈ − ˜ Let y4 = (3) u . Using 1.1 (c) we know that y4f11λ is a primitive element f2f1 of weight λ − 2lα1 − (2l − 2)α2. (5) Now we consider the homomorphism: ˜ → ˜ ˜ → (l−1) (l+1) (3)˜ Z(λ) Z(λ+(l+2)α1 +(l+1)α2), 1λ t3 = f1 f2 f1 1λ+(l+2)α1+(l+1)α2 . Let (l−2) (l) − (l) (l−2) ∈ − x5 =(f2 f1 f1 f2 ) u .

Using 1.1 (c) we see that x51˜λ is a primitive element of weight λ − lα1 − (l − 2)α2. (6) Consider the homomorphism ˜ → (3) (2)˜ u˜f11λ u˜f2f1 f2 1λ+2α1+3α2 ˜ → (3) (2)˜ f11λ f2f1 f2 1λ+2α1+3α2 . xl−1,2 ∈ − ˜ − − Let x6 = (3) (2) u By 1.1 (c), x6f11λ is primitive and is of weight λ 2lα1 f2f1 f2 lα2. (l−2) (2l−3) (l−1) ˜ Note that the element m = f2 f1 f2 f11λ generates the unique irreducible submodule of Z˜(λ). By comparing the weights of the following 6 elements with the weight of m, x − − x − − x − l 1,l 1 ˜ , l 1,l 1 ˜ , 3,l 1 ˜ , (l−1) 1λ (l) (l−1) 1λ (2) (3) 1λ f1 f2 f1 f2 f1 x − − − x − 3,l 1 ˜ ,f(l 2)f (l 1)f (l)f ˜ , l 1,2 f ˜ , (l+2) (3) 1λ 2 1 2 11λ (3) (2) 11λ f2 f1 f2f1 f2 we see that there are no maximal elements in Z˜(λ) that have the same weight with any of above 6 elements. Now we show that there are no maximal elements in Z˜(λ) that have the same weight with any of other 3 elements in (ii) by assuming (iii). By (iii), Z˜(λ) has only one composition factor isomorphic to L˜(λ − (l +1)α1 − (l − 1)α2). Suppose that there is a maximal element m in Z˜(λ)ofweightλ − (l + − − (l−1) (l) ˜ 1)α1 (l 1)α2.Thenm is in u˜y,herey =[f2 ,f1 ]f11λ. It is clear that ⊂ − − (l−3) (l−1) ˜ (3) (l−3) (l−1) ˜ u˜y u˜ y +u˜ f1 f2 f11λ.Thusm = ay +bf1 f1 f2 f11λ for some a, b

12268 NANHUA XI in Q(ξ). Thus m = ay.Buty is not maximal. So there are no maximal elements ˜ (l−1) (l) ˜ in Z(λ) that have the same weight with [f2 ,f1 ]f11λ. Similarly, we see that there are no maximal elements in Z˜(λ) that have the (l−2) (l) ˜ (l−1) (l) ˜ same weight with any of [f2 ,f1 ]1λ,f1 f2 f11λ. (iii) Using (i), (ii) and 1.1 (b), we see that Z˜(λ) has the 20 composition factors. By a comparison of dimensions we know that Z˜(λ) has only the 20 composition factors. The theorem is proved.

Theorem 3.3. Let a, b be integers and λ =(la + l − 2,lb+1).Then (i) The following elements are maximal in Z˜(λ): ˜ (l−1)˜ (2)˜ (l−1)˜ 1λ,f1 1λ,f2 1λ,f2f1 1λ, (3) (2)˜ (3) (l+1) (l−1)˜ f1 f2 1λ,f1 f2 f1 1λ. (3) (2)˜ (2) (l+3) (l+1) (l−1)˜ f2f1 f2 1λ,f2 f1 f2 f1 1λ, (3) (2) (3) f f f f f x − 1 2 ˜ , 2 1 2 ˜ , 3,l 2 ˜ . 1λ 1λ (l−2) 1λ f1 f1 f2 (ii) The following elements are primitive elements in Z˜(λ) but not maximal:

(l−1) (l) (l−1) (l) x − − f f f f f f f f (2),f(l) ˜ , l 1,l 1 ˜ , 2 1 2 ˜ , 1 2 1 2 ˜ , [ 2 1 ]1λ (l−2) (2l−3) (l−1) 1λ 1λ 1λ f1 f1 f2 f1 f2

(2) (l) (3) [f ,f ]f f x − x − x − 2 1 1 2 ˜ , 3,l 1 ˜ , 3,l 1 ˜ , 3,l 1 ˜ , 1λ (l−1) 1λ (l+1) (l−1) 1λ (2l+1) (l−1) 1λ f1 f1f2 f1 f2 f1 f2 x − − l 1,l 1 f (2)˜ . (l−3) (l−1) 2 1λ f1 f2 Moreover there are no maximal elements in Z˜(λ) which have the same weight with any of above 9 elements. (iii) The maximal and primitive elements in (i-ii) provide 20 composition factors of Z˜(λ), which are

L˜(λ), L˜(λ − (l − 1)α1), L˜(λ − 2α2),

L˜(λ − (l − 1)α1 − α2), L˜(λ − 3α1 − 2α2)

L˜(λ − (l +2)α1 − (l +1)α2), L˜(λ − 3α1 − 3α2),

L˜(λ − (2l +2)α1 − (l +3)α2) L˜(λ − 2α1 − α2),

L˜(λ − 2α1 − 3α2), L˜(λ − lα1 − 2α2), L˜(λ − (2l − 1)α1 − lα2),

L˜(λ − (l − 1)α1 − lα2), L˜(λ − lα1 − lα2), L˜(λ − (l +2)α1 − 3α2),

L˜(λ−(2l+3)α1−(l+2)α2), L˜(λ−(l+3)α1−(l+2)α2), L˜(λ−3α1−(l+2)α2),

L˜(λ − (2l +2)α1 − (l +1)α2), L˜(λ − (3l − 1)α1 − 2lα2). Moreover, Z˜(λ) has only the 20 composition factors.

MAXIMAL AND PRIMITIVE ELEMENTS 26913

Proof. (i) According to 1.1 (e-f), we see that the ﬁrst 8 elements in (i) are maximal. Consider the homomorphism: ˜ → ˜ ˜ → ˜ Z(λ) Z(λ + α1), 1λ t1 = f11λ+α1 . (3) (2) (3) (3) − f1 f2 f2 f1 f2 Since f f2 is in u f1, using 1.1 (c) we see that 1˜λ and 1˜λ are 1 f1 f1 primitive elements of weights λ − 2α1 − α2 and λ − 2α1 − 3α2 respectively. Now we consider the homomorphism: ˜ → ˜ − ˜ → (l−2)˜ Z(λ) Z(λ +(l 2)α2)), 1λ t2 = f2 1λ+(l−2)α2 . − (3) (l) (l) (l 1)˜ x3,l−2 ˜ Using 1.1 (c) we see that f1 f2[f2 ,f1 ]f1 1λ = (l−2) 1λ is a primitive element f2 of weight λ − (2l +2)α1 − (l +1)α2. One can check directly that the three elements are maximal. (ii) Now we argue for (ii). (1) Consider the homomorphism: ˜ → ˜ ˜ → ˜ Z(λ) Z(λ + α1), 1λ t1 = f11λ+α1 . Using Theorem 3.1 and 1.1 (c) we see that the ﬁrst 8 elements are primitive. (2) Now we consider the homomorphism: (2)˜ → (l−3) (l−1)˜ u˜f2 1λ u˜f1 f2 1λ+(l−3)α1+(l−3)α2 , (2)˜ → (l−3) (l−1)˜ f2 1λ f1 f2 1λ+(l−3)α1+(l−3)α2 . xl−1,l−1 (2)˜ − Using 1.1 (c) we know that (l−3) (l−1) f2 1λ is a primitive element of weight λ f1 f2 (3l − 1)α1 − 2lα2. Consider the homomorphism: ˜ → ˜ ˜ → ˜ Z(λ) Z(λ + α1)), 1λ t1 = f11λ+α1 . It is easy to see that all the 9 primitive elements have non-zero image. By Theorem 3.1 (ii) and 1.1 (e) we know that there are no maximal elements in Z˜(λ) which have the same weight with any of the 9 primitive elements. (iii) Using (i), (ii) and 1.1 (b), we see that Z˜(λ) has the 20 composition factors. By a comparison of dimensions we know that Z˜(λ) has only the 20 composition factors. The theorem is proved. Theorem 3.4. Let a, b be integers and λ =(la +2,lb+ l − 2).Then (i) The following elements are maximal in Z˜(λ): ˜ (3)˜ (l−1)˜ (2) (3)˜ 1λ,f1 1λ,f2 1λ,f2 f1 1λ, (l−1)˜ (2) (3)˜ f1f2 1λ,f1f2 f1 1λ. (2) (l+1) (l−1)˜ (l−1) (2l+1) (l+2) (3)˜ f2 f1 f2 1λ,f2 f1 f2 f1 1λ, (2) (3) (2) f2 f1 f1 f2 f1 1˜λ, 1˜λ. f2 f2 (ii) The following elements are primitive elements in Z˜(λ) but not maximal: − x − − x − f (l 1),f(l) ˜ , l 1,l 1 ˜ , 3,l 2 ˜ , f (3),f(l) ˜ , [ 2 1 ]1λ (l−3) (l−2) (l−1) 1λ (l−1) (l−2) 1λ [ 1 2 ]1λ f1 f2 f1 f2f1 f2

14270 NANHUA XI

x − x − − x − 3,l 1 ˜ , 3,l 1 ˜ ,ff (l 1),f(l) ˜ , l 1,2 ˜ , (2l) (l−1) 1λ (l) (l−1) 1λ 1[ 2 1 ]1λ (l−1) 1λ f2f1 f2 f2f1 f2 f2f1 x − − f (2),f(l) f (3)˜ , l 1,l 1 ˜ . [ 2 1 ] 1 1λ (l−3) (2l−2) (l−1) 1λ f1 f2 f1 Moreover no maximal element in Z˜(λ) has the same weight as any of above 10 elements. (iii) The maximal and primitive elements in (i-ii) provide 20 composition factors of Z˜(λ), which are

L˜(λ), L˜(λ − 3α1), L˜(λ − (l − 1)α2),

L˜(λ − 3α1 − 2α2), L˜(λ − α1 − (l − 1)α2)

L˜(λ−4α1 −2α2), L˜(λ−(l+1)α1 −(l+1)α2), L˜(λ−(2l+4)α1 −(2l+1)α2),

L˜(λ − α1 − α2), L˜(λ − 4α1 − α2), L˜(λ − lα1 − (l − 1)α2),

L˜(λ − 2lα1 − (2l − 1)α2), L˜(λ − (l +3)α1 − lα2), L˜(λ − 3α1 − lα2),

L˜(λ−4α1 −(l+1)α2), L˜(λ−(l+4)α1−(l+1)α2), L˜(λ−(l+1)α1−(l−1)α2),

L˜(λ − (l +3)α1 − (l +2)α2), L˜(λ − (l +3)α1 − 2α2), L˜(λ − 2lα1 − (l − 1)α2). Moreover, Z˜(λ) has only the 20 composition factors. Proof. (i) According to 1.1 (e-f), we see the ﬁrst 8 elements in (i) are maximal. Consider the homomorphism: ˜ → ˜ ˜ → ˜ Z(λ) Z(λ + α2), 1λ t1 = f21λ+α2 . (2) (3) (2) (2) − f2 f1 f1 f2 f1 Since f f1 is in u f2, using 1.1 (c) we see that 1˜λ and 1˜λ are 2 f2 f2 primitive elements of weights λ − α1 − α2 and λ − 4α1 − α2 respectively. One can check directly that the two elements are maximal. (ii) Now we argue for (ii). (1) Consider the homomorphism: ˜ → ˜ ˜ → ˜ Z(λ) Z(λ + α2)), 1λ t1 = f21λ+α2 . Using Theorem 3.1 and 1.1 (c) we see that the ﬁrst 9 elements are primitive. (2) Now we consider the homomorphism: ˜ → (l−3) (2l−2) (l−1)˜ Z(λ) u˜f1 f2 f1 1λ+(2l−4)α1+(2l−2)α2 , ˜ → (l−3) (2l−2) (l−1)˜ 1λ f1 f2 f1 1λ+(2l−4)α1+(2l−2)α2 . − (l 1) (l) (l) ˜ xl−1,l−1 ˜ Using 1.1 (c) we know that [[f2 ,f1 ],f1 ]1λ = (l−3) (2l−2) (l−1) 1λ is a primitive f1 f2 f1 element of weight λ − 2lα1 − (l − 1)α2. Consider the homomorphism: ˜ → ˜ ˜ → ˜ Z(λ) Z(λ + α2)), 1λ t1 = f21λ+α2 . It is easy to see that all the 10 primitive elements have non-zero image. By Theorem 3.1 (ii) and 1.1 (e) we know that there are no maximal elements in Z˜(λ) which have the same weight with any of the 10 primitive elements. (iii) Using (i), (ii) and 1.1 (b), we see that Z˜(λ) has the 20 composition factors. By a comparison of dimensions we know that Z˜(λ) has only the 20 composition factors. The theorem is proved.

MAXIMAL AND PRIMITIVE ELEMENTS 27115

4. Weyl Modules for Type B2 ∈ 2 For λ =(λ1,λ2) Z+ we denote by. Iλ the/ left. ideal of/ Uξ generated by all (a) (a ) ki,c λi + c e (a ≥ 1), f i (a ≥ λ +1), k −ξλi , − . The Weyl module i i i i i a a ξi V (λ)ofUξ is deﬁned to be Uξ/Iλ, its dimension is (λ1 +1)(λ2 +1)(λ1 +λ2 +2)(λ1 + 2λ2 +3)/6. Let vλ be a nonzero element in V (λ)λ. We can work out the maximal and primitive elements in V (λ) as in section 3 (cf. [X2]). We omit the results here. Acknowledgement: The work was completed during my visit to the University of Sydney. It is a great pleasure to thank Professor G. Lehrer for the invitation. Part of the work was done during my visit to Bielefeld University. I am grateful to the SFB 343 in Bielefeld University for ﬁnancial support. Finally I would like to thank the referee for helpful comments.

References [APW] H.H. Andersen, P. Polo, K. Wen, Representations of quantum algebras. Invent. Math. 104 (1991), no. 1, 1–59. [DS1] S.R. Doty and J.B. Sullivan, The submodule structure of Weyl module for SL3, J. Alg. 96 (1985), 78-93. [DS2] S.R. Doty and J.B. Sullivan, On the structure of the higher cohomology modules of line bundles on G/B, J. Alg. 114(1988), 286-332. [I] R.S. Irving, The structure of certain highest weight modules for SL3, J. Alg. 99 (1986), 438-457. [K] Kühne-Hausmann, K, Zur Untermodulstruktur der Weylmoduln für Sl3, Bonner Mathe- matische Schriften 162, Universität Bonn, Mathematisches Institut, Bonn, 1985. vi+190 pp. [L1] G. Lusztig, Modular representations and quantum groups, Contemp. Math. 82 (1989), 59-77. [L2] G. Lusztig, Finite dimensional Hopf algebras arising from quantized universal enveloping algebras, Jour. Amer. Math. Soc. 3 (1990), 257-296. [L3] G. Lusztig, Quantum groups at roots of 1, Geom. Ded. 35 (1990), 89-114. [L4] G. Lusztig, Introduction to quantum groups, Progress in Mathematics 110, Birkháuser, Boston · Basel · Berlin, 1993. [X1] N. Xi, Irreducible modules of quantized enveloping algebras at roots of 1, Publ. RIMS. Kyoto Univ. 32 (1996), 235-276. [X2] N. Xi, Maximal and primitive elements in Weyl modules for type A2, J. Alg., 215(1999), 735-756.

Institute of Mathematics, Chinese Academy of Sciences, Beijing 100080, China E-mail address: [email protected]

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Contemporary Mathematics Volume 478, 2009

Irreducible representations of the special algebras in prime characteristic

Yu-Feng Yao and Bin Shu

Abstract. Let L = S(m; n) be a graded Lie algebra of special type in the Cartan type series over an algebraically closed ﬁeld of characteristic p>0. In this paper we study irreducible modules of L. For this we adopt the notion of the category C which was introduced in [11] by Skryabin for the study of representations of generalized Witt algebras. In the generalized restricted Lie algebra setting, a class of induced modules of L of (generalized) p-character χ are proved to belong to C. All irreducible modules of L of p-character χ are n n − determined when the height of χ is not bigger than min{p i − p i 1 | 1 ≤ i ≤ m}−2. These irreducible modules turn out to be just induced modules in the so-called non-exceptional cases. As to the exceptional cases, irreducible modules, as well as the dimensions, are precisely obtained through a complex of induced modules.

1. Introduction This paper is a continuation of the work in [10]. In the previous paper, we studied irreducible modules of the generalized Jacobson-Witt algebras W (m; n) which is a so-called generalized restricted Lie algebra as introduced in [8]. There we determined simple modules of W (m; n) of generalized p-character χ when the n n −1 height of χ

2000 Mathematics Subject Classiﬁcation. 17B50, 17B70.

c XXXXc 2009 American Mathematical Society 2731

2274 YU-FENG YAO AND BIN SHU

2. Preliminaries In this paper, we always assume that the ground ﬁeld F is algebraically closed with prime characteristic, m ∈ N, m ≥ 3.

2.1. Graded simple Lie algebras of series S of Cartan type. Let a = m (a1,a2, ··· ,am),b=(b1,b2, ··· ,bm) ∈ Z ,wewritea ≤ b (resp. a ≥ b)ifai ≤ bi ≥ ≤ ≤ ≤ (resp. ai bi) for all 1 i m and we write ab)ifa b(resp. a ≥ b), but a = b.Ifa, b ≥ 0, define a = ai ,where ai is the usual binomial b bi bi i ai coefficient with the convention = 0 unless ai ≥ bi.SetA(m; n)={a = bi n n n (a1,a2, ··· ,am) | 0 ≤ ai ≤ p i − 1, ∀ i =1, 2, ···m}.Denoteτ =(p 1 − 1,p 2 − n m 1, ··· ,p m − 1), then A(m; n)={a =(a1,a2, ··· ,am) ∈ Z | 0 ≤ a ≤ τ}.The (a) divided power algebra A(m; n)isanF -algebra having F -basis {x | a ∈ A(m; n)} (a) (b) a+b (a+b) (c) ∈ with the multiplication rule x x = a x ,wherex =0ifc/A(m; n). For each i, 1 ≤ i ≤ m,denoteεi =(δi1,δi2, ··· ,δim). The following formula assures that A(m; n) is an associative and commutative algebra which can be easily verified by straightforward computation: for any α, β, γ ∈ A(m; n), γ γ − α γ γ − β (2.1) = . α β β α (a) Let Di denote the derivation of A(m; n) uniquely determined by the rule: Di(x )= − x(a εi). The generalized Jacobson-Witt algebra W (m; n) is a Lie algebra consisting of all special derivations of A(m; n). According to [12, 4.2], W (m; n)=F - (a) span{x Di | a ∈ A(m; n), 1 ≤ i ≤ m}⊆Der(A(m; n)). For a ∈ A(m; n), set m (a) |a| := ai and W (m; n)[i] := F -span{x Dj ||a| = i +1,j =1, 2, ··· ,m}, i=1 s then W (m; n)= W (m; n)[i] is a natural Z-gradation of W (m; n), where s = i=−1 m (pni −1)−1. Then W (m; n) has a natural filtration associated with the gradation i=1 above: W (m; n)=W (m; n)− ⊃ W (m; n) ⊃ W (m; n) ⊃··· 1 0 1 where W (m; n)i = W (m; n)[j]. j≥i (a) (a) (a−εi) Define a linear map div : W (m; n) → A(m; n), x Di → Di(x )=x ∈ m m A(m; n). Let S(m; n):=Ker (div)={ fiDi ∈ W (m; n) | Di(fi)=0}. i=1 i=1 The special algebra S(m; n) is defined by the derived algebra of S(m; n), i.e. (1) S(m; n)=S(m; n) =[S(m; n), S(m; n)].Accordingto[12, 4.3], S(m; n)=F - (a) span{Dij (x ) | a ∈ A(m; n), 1 ≤ i

2.2. Generalized restricted Lie algebras and generalized χ-reduced representations. Although S(m; n) is not restricted whenever n = 1,itisalways a generalized restricted Lie algebra in the sense as below (cf. [8]).

REPRESENTATIONS OF S(m; n)3275

Definition 2.1. A generalized restricted Lie algebra L over F is a Lie algebra associated with an ordered basis E =(ei)|i∈I and a mapping ϕs : E → L sending → ϕs | ∈ Z ϕs psi ∈ ei ei with s =(si) i∈I ,wheresi + such that ad ei =(adei) for all i I.

By a straightforward verification, L0 = S(m; n)0 is restricted under the mapping D → D[p],whereD[p] is the usual pth power of the derivation D.So [p] p for a basis E1 of S(m; n)0,wehaveadx =(adx) for any x ∈ E1.Take E = E1 ∪{D1,D2, ··· ,Dm},thenE is a basis of S(m; n). We can assume that E =(ei)|i∈I such that ei = Di, i =1, 2, ··· ,m and ej ∈ E1 for j>m.Thenlet s =(n1,n2, ··· ,nm, 1, 1, ··· , 1), define ϕs : E → L sending ei → 0for1≤ i ≤ m → [p] ϕs psi ∀ ∈ and ej ej for j>m,thenadei =(adei) , i I.SoS(m; n)isageneral- ized restricted Lie algebra in the sense of Definition 2.1 . For a generalized restricted Lie algebra over F , by Schur lemma, we have the following fact.

Lemma 2.2. If (L, ϕs) is a generalized restricted Lie algebra over F associated with a basis E =(ei)|i∈I and ϕs, s =(si)|i∈I . Suppose (V, ρ) is an irreducible representation of L, then there exists a unique χ ∈ L∗ such that : psi − ϕs psi ∈ (2.2) ρ(ei) ρ(ei )=χ(ei) idV ,ei E. Here the function χ is called a (generalized) p-character of V . A representation (module) of L satisfying ( 2.2) is called a generalized χ-reduced representation (module). Especially when χ =0, it is called a generalized restricted representation (module) of L.

Assume that (L, ϕs) is a generalized restricted Lie algebra associated with a ∗ basis E =(ei)|i∈I and ϕs, s =(si)|i∈I .Foraχ ∈ L , we deﬁne Ups (L, χ):= s s p i − ϕs − psi | ∈ p i − ϕs − psi | ∈ U(L)/(ei ei χ(ei) ei E), where (ei ei χ(ei) ei E)denote s p i − ϕs − psi the ideal in U(L) generated by these central elements ei ei χ(ei) for all ei ∈ E. Ups (L, χ) is called the generalized χ-reduced enveloping algebra of L (cf. [8]). When χ =0,Ups (L, 0) is often called the generalized restricted enveloping algebra of L, simply denoted by Ups (L). We have category equivalence between the generalized χ-reduced (resp. generalized restricted) module category of L and Ups (L, χ)(resp.Ups (L))-module category. Remark 2.3. A restricted Lie algebra (g, [p]) is naturally a generalized restricted Lie algebra associated with an arbitrary given basis E and s = 1 := (1, 1, ··· , 1). Furthermore, in this case , a generalized χ-reduced module (algebra) coincides with the χ-reduced module (algebra). 2.3. The primitive p-envelope of S(m; n). Recall that associated with a generalized restricted Lie algebra L, there is a special p-envelope consisting of primitive elements of the generalized restricted enveloping algebra of L, called a primitive p-envelope (cf. [9]). We turn to L = S(m; n). Then, there is a natural realization in Der(R) of the primitive p-envelope of L which is as follows:

m ni−1 pdi L = S(m; n) FDi . i=1 di=1 ∗ ∗ ∗ For χ ∈ L ,wedenoteby¯χ ∈ L the trivial extension of χ to L , i.e.χ ¯|S(m;n) = χ, m ni−1 ∈ pdi whileχ ¯(x) = 0, for any x FDi . The following lemma will be useful. i=1 di=1

4276 YU-FENG YAO AND BIN SHU

Lemma 2.4. ([9] and [10]) Keep notations as above. There is an algebra isomorphism: ∼ (2.3) Ups (L, χ) = U(L, χ¯) and thus ∼ Φ (2.4) Ups (L, χ) = Ups (L, χ ), where Φ ∈ Aut(L) and χΦ(D):=χ(Φ−1(D)), ∀ D ∈ L.

3. Skryabin’s C-category and independence of diﬀerential operators 3.1. In the sequel, set L = S(m; n). As stated above, L admits a natural s m ni α gradation: L = L[i],wheres = (p − 1) − 2andL[t] = F -span{Dij (x ) | i=−1 i=1 m |α| = αk = t +2, 1 ≤ i

Definition 3.1. (M,σ) is called a discrete L0-module if for any x ∈ M,there exists some nonnegative integer l such that σ(Ll)x =0. Definition 3.2. Set R = A(m; n)andL = S(m; n). Denote by C the category whose objects are additive groups M endowed with an R-module structure (M, ρR), an L-module structure (M, ρL)andanL0-module structure (M, σ)sothatM is discrete as σ(L0) module and the following properties are satisﬁed :

(R1) [ρL(D),ρR(f)] = ρR(Df); (R2) [σ(D ),ρR(f)] = 0; (R3) [σ(D ),ρL(Di)] = 0; (R4) ρ (D (f)) = ρ (D (f)) ◦ ρ (D ) − ρ (D (f)) ◦ ρ (D )+ L ij R j L i R i L j α α ρR(D f) ◦ σ(Dij(x )), |α|≥2 where f ∈ R, D ∈ L, D ∈ L0,i,j =1, 2, ··· ,m. The morphisms in C-category are the mappings admissible with the three module structures. A module M in C-category is called a C-module. Definition 3.3. Let R, L be as above. Denote by (R, L)-mod the category whose objects are additive groups M endowed with an R-module structure (M, ρR) and an L-module structure (M, ρL) which satisﬁes (R1) above. The morphisms in (R, L)-mod are the mappings admissible with the two module structures. A module M in (R, L)-mod is called an (R, L)-module.

REPRESENTATIONS OF S(m; n)5277

3.3. In this subsection, we will recall some facts about independent systems of diﬀerential operators introduced in [11] which we will use to study submodules and homomorphisms later. Assume R is a commutative algebra with unit over F . Endow the endomorphism algebra EndF R with an R-module structure by putting (f · ϕ)(g)=fϕ(g), for f,g ∈ R, ϕ ∈ EndF R.

Definition 3.4. ([11]) A system of endomorphisms Φ ⊆ EndF R is called independent if for any ﬁnite subset Φ = {ϕ1,ϕ2, ··· ,ϕn}⊆Φ, the submodule n ValΦ of the free F -module R generated by all n-tuples (ϕ1(g),ϕ2(g), ··· ,ϕn(g)) with g ∈ R coincides with Rn. Proposition . { pri | ≤ ≤ ≤ } 3.5 ([11])Suppose ∂i 1 i m, 0 ri

4. Submodules and homomorphisms in the category C In the following sequel, we always assume R = A(m; n). According to Remark { pri | ≤ ≤ ≤ } 3.6(3), Di 1 i m, 0 ri

Definition 4.1. For an irreducible L0-module (M,σ), deﬁne the height of M be the smallest nonnegative integer l such that M is annihilated by σ(Ll). ∼ A restricted simple L[0] = sl(m)-module can be regarded a restricted simple L0-module with L1-trivial action. Recall that the set of isomorphism classes of

6278 YU-FENG YAO AND BIN SHU restricted simple sl(m)-modules can be parameterized by restricted weights of sl(m) (cf. [3]and[4, Part II, Ch2-3]), which play roles as “highest weights”. The simple restricted L0-module corresponding to fundamental weights of sl(m) will be called exceptional (weighted-) module. Lemma 4.2. (i) Let M ∈ C, assume that

α (4.1) σ(Dij(x )) = 0 for α ∈ A(m; n)\A (m; n) and 1 ≤ i

Theorem 4.3. (i) Let M ∈ C. Assume that :

(4.2) M is completely reducible as σ(L0)-module and none of its irreducible

REPRESENTATIONS OF S(m; n)7279 summand is exceptional.

α (4.3) σ(Dij(x )) = 0 for α ∈ A(m; n)\A (m; n),i,j =1, 2, ··· ,m,i= j. Then any L-submodule M of M is a C-submodule. (ii) Let M,N be two objects of C satisfying ( 4.2), ( 4.3). Then any L-module homomorphism ϕ : M → N is a morphism in C. Proof. As in Lemma 4.2, we know that (ii) is a sequence of (i). The same arguments as in the proof of Lemma 4.2 reduce the proof of (i) to the case when M is annihilated by σ(Ll)forsomel ≥ 0. From Lemma 4.2, we only need to prove that M is an R-submodule of M. We use a similar method developed by Skryabin in [11]. Let P = {m ∈ M | Rm ⊆ M } be the largest R-submodule contained in M and Q = RM be the smallest R-submodule containing M .By(R1),P, Q are L-submodules. Hence by Lemma 4.2, P, Q are C-submodules. Then we can now pass to an object Q/P ∈ C and its L-submodule M /P . Hence, at the beginning we can impose the additional assumption that M contains no nonzero R-submodule of M and RM = M. Then it’s sufficient for us to prove that M =0. We use a similar strategy developed by Skryabin in [11]. The main ideal is as follows. We want to seek endomorphisms ϕ of M lying in the associative algebra generated by the endomorphisms σ(D ), D ∈ L0 with the property that for any f ∈ R, the endomorphism fϕ belongs to the associative subalgebra generated by the endomorphisms ρL(D), D ∈ L.TheL-submodule M is stable under fϕ for any f ∈ R. Hence, it contains the R-submodule Rϕ(M ). By the hypothesis, then ϕ(M ) = 0. From (R2), ϕ is an R-module endomorphism , so ϕ(M)=ϕ(RM )= Rϕ(M ) = 0, i.e. ϕ = 0. This gives a certain relations between the endomorphisms σ(D ), D ∈ L0. It appears that there are sufficiently many relations of a similar kind so that they can’t satisfy simultaneously unless M =0. Assume M =0andlet l be the least positive integer such that M is annihilated by σ(Ll). First of all, consider l ≤ 1. Then M can be viewed as the factor ∼ algebra L0/L1 = sl(D)-module, where D =F -span{D1,D2, ··· ,Dm}. Given in- ∈{ ··· } ∈ dices s1,s2,s1,s2 1, 2, ,m .Letfν ,gν R be a finite number of elements { ∈ || |≤ } satisfying (3.1) for A = α A(m; n) α 4 and γ = εs1 + εs2 + εs + εs , i.e. 1 2 u , α ε + ε + ε + ε , α 1 if = s1 s2 s1 s2 (4.4) fν D gν = , ν=1 0 otherwise. Hence for any f ∈ R, i, j, s, t ∈{1, 2, ··· ,m},wehave: ρL(Dij(ffν ))ρL(Dst(gν )) ν = ρL(Dj(ffν )Di − Di(ffν )Dj)ρL(Dt(gν )Ds − Ds(gν )Dt) ν = ρR(Dj(ffν ))ρL(Di) − ρR(Di(ffν ))ρL(Dj)+ ν α α ρR(D (ffν ))σ(Dij(x )) ρR(Dt(gν ))ρL(Ds) − α β β ρR(Ds(gν ))ρL(Dt)+ ρR(D (gν ))σ(Dst(x )) β

8280 YU-FENG YAO AND BIN SHU = ρR(Dj(ffν ))ρL(Di) − ρR(Di(ffν ))ρL(Dj)+ ν ρR(DiDj (ffν ))σ(Eii − Ejj)+ ρR(DjDl(ffν ))σ(Eli) − l= i ρR(DiDq(ffν ))σ(Eqj) ρR(Dt(gν ))ρL(Ds) − q= j ρ (D (g ))ρ (D )+ρ (D D (g ))σ(E − E )+ R s ν L t R st ν ss tt ρR(DtDr(gν ))σ(Ers) − ρR(DsDu(gν ))σ(Eut) r= s u= t = ρR(f) ρR(fν DiDjDsDt(gν ))σ(Eii − Ejj)σ(Ess − Ett)+ ν ρR(fν DiDj DtDr(gν ))σ(Eii − Ejj)σ(Ers) − ν r=s ρR(fν DiDj DsDu(gν ))σ(Eii − Ejj)σ(Eut)+ ν u=t ρR(fν Dj DlDsDt(gν ))σ(Eli)σ(Ess − Ett)+ ν l=i ρR(fν DjDlDtDr(gν ))σ(Eli)σ(Ers) − ν l=i r=s ρR(fν DjDlDsDu(gν ))σ(Eli)σ(Eut) − ν l=i u=t ρR(fν DiDqDsDt(gν ))σ(Eqj)σ(Ess − Ett) − ν q=j ρR(fν DiDqDtDr(gν ))σ(Eqj)σ(Ers)+ ν q= j r= s ρR(fν DiDqDsDu(gν ))σ(Eqj)σ(Eut) ν q= j u= t By the remark in the ﬁrst paragraph and (4.4), we obtain the following relation (4.5) δ{i,j,s,t},{s ,s ,s ,s }σ(Eii − Ejj)σ(Ess − Ett)+ 1 2 1 2 δ{ } { }σ(E − E )σ(E )− i,j,t,r , s1,s2,s1,s2 ii jj rs r=s δ{ } { }σ(E − E )σ(E )+ i,j,s,u , s1,s2,s1,s2 ii jj ut u=t δ{ } { }σ(E )σ(E − E )+ j,l,s,t , s1,s2,s1,s2 li ss tt l=i δ{ } { }σ(E )σ(E )− j,l,t,r , s1,s2,s1,s2 li rs l=i r=s δ{ } { }σ(E )σ(E )− j,l,s,u , s1,s2,s1,s2 li ut l= i u= t

REPRESENTATIONS OF S(m; n)9281 δ{ } { }σ(E )σ(E − E )− i,q,s,t , s1,s2,s1,s2 qj ss tt q=j δ{ } { }σ(E )σ(E )+ i,q,t,r , s1,s2,s1,s2 qj rs q=j r=s δ{ } { }σ(E )σ(E ) i,q,s,u , s1,s2,s1,s2 qj ut q= j u= t =0

Lemma 4.4. If an irreducible representation σ of the Lie algebra sl(m) in a vector space V satisﬁes ( 4.5), then V is exceptional.

+ Proof. Set n := F -span{Eij | ij}. For any a, b ∈{1, 2, ··· ,m},a = b, take s1 = s2 = s1 = s2 = b, i = s = a, j = t = b in (4.5). From (4.5), we get 2 + − σ(Eba) =0.Son and n act on V nilpotently. Then there exists a maximal vector v in V with respect to the Borel subalgebra h + n+. Assume λ is the corresponding maximal weight. Set λk = λ(Ek,k − Ek+1,k+1),k =1, 2, ··· ,m− 1. Take s1 = k +1, s1 = k +1,s2 = k, s2 = k . i = k, j = k +1,s = k , t = k +1in (4.5). Then from (4.5), we have:

(4.6) σ(Ekk − Ek+1,k+1)σ(Ekk − Ek+1,k+1)+(1− δk,k )σ(Ekk)σ(Ekk )

− (1 − δk+1,k)σ(Ek+1,k)σ(Ek,k+1) − (1 − δk+1,k )σ(Ek,k+1)σ(Ek+1,k )

+(1− δk,k )σ(Ek+1,k+1)σ(Ek+1,k+1) =0.

2 − Taking k = k in (4.6), and then letting both sides of (4.6) act on v,wegetλk λk = 0. So λk = 0 or 1 for k =1, 2, ··· ,m− 1. If all λk =0,k =1, 2, ··· ,m− 1, then v is an exceptional-weight vector. Otherwise there exists some k ∈{1, 2, ··· ,m− 1} such that λk =1,andallλi =0fori

If V is an irreducible L0-submodule of M,thenV is exceptional, due to Lemma 4.4. By the assumption of (4.2), there occurs a contradiction M = 0. Therefore, l>1, so σ(Ll)=0.Andσ(Ll−1) is a nonzero abelian ideal of σ(L0). Deﬁne a total order “≺”onthesetofn-tuples A(m; n) by putting α ≺ β (or β ) α) for α, β ∈ A(m; n)ifeither|α| < |β| or |α| = |β| and β is greater than α in lexicographical order. For f,g ∈ R,wehave:

(4.7) ρL(Dij(f))ρL(Dij(g)) = ρL(Dj(f)Di − Di(f)Dj)ρL(Dj(g)Di − Di(g)Dj) =(ρ (D (f))ρ (D ) − ρ (D (f))ρ (D )+ R j L i R i L j α α ρR(D (f))σ(Dij(x )))(ρR(Dj(g))ρL(Di) − α β β ρR(Di(g))ρL(Dj)+ ρR(D (g))σ(Dij(x ))) β

10282 YU-FENG YAO AND BIN SHU

2 = ρR(Dj(f)Dj(g))ρL(Di) + ρR(Dj(f)DiDj(g))ρL(Di) −

ρR(Dj(f)Di(g))ρL(Di)ρL(Dj) − ρR(Dj(f)DiDi(g))ρL(Dj) −

ρR(Di(f)Dj(g))ρL(Dj)ρL(Di) − ρR(Di(f)DjDj (g))ρL(Di)+ ρ (D (f)D (g))ρ (D )2 + ρ (D (f)D D (g))ρ (D )+ R i i L j R i j i L j β+εi β ρR(Dj(f)D (g))σ(Dij(x )) + β β β ρR(Dj(f)D (g))ρL(Di)σ(Dij(x )) − β β β ρR(Di(f)D (g))ρL(Dj)σ(Dij(x )) − β β+εj β ρR(Di(f)D (g))σ(Dij(x )) + β α α ρR(D (f)Dj(g))σ(Dij(x ))ρL(Di) − α α α ρR(D (f)Di(g))σ(Dij(x ))ρL(Dj)+ α α β α β ρR(D (f)D (g))σ(Dij(x ))σ(Dij(x )) α β 2 2 = ρR(Dj(fDj(g)))ρL(Di) − ρR(fDjDj(g))ρL(Di) +

ρR(Dj(fDiDj (g)))ρL(Di) − ρR(fDiDj Dj(g))ρL(Di) −

ρR(Dj(fDi(g)))ρL(Di)ρL(Dj)+ρR(fDjDi(g))ρL(Di)ρL(Dj) −

ρR(Dj(fDiDi(g)))ρL(Dj)+ρR(fDjDiDi(g))ρL(Dj) −

ρR(Di(fDj(g)))ρL(Dj)ρL(Di)+ρR(fDiDj(g))ρL(Dj)ρL(Di) −

ρR(Di(fDjDj (g)))ρL(Di)+ρR(fDiDj Dj(g))ρL(Di)+ 2 2 ρR(Di(fDi(g)))ρL(Dj) − ρR(fDiDi(g))ρL(Dj) + ρ (D (fD D (g)))ρ (D ) − ρ (fD D D (g))ρ (D )+ R i j i L j R i j i L j β β ρR(Dj(fD (g)))ρL(Di)σ(Dij(x )) − β β+εj β ρR(fD (g))ρL(Di)σ(Dij(x )) + β β+εi β ρR(Dj(fD (g)))σ(Dij(x )) − β β β ρR(Di(fD (g)))ρL(Dj)σ(Dij(x )) + β β+εi β ρR(fD (g))σ(Dij(x )) − β β+εj β ρR(Di(fD (g)))σ(Dij(x )) + β | | α (−1) α ρ (Dα (fDα +εj (g)))σ(D (xα))ρ (D )− α R ij L i α α+α=α

REPRESENTATIONS OF S(m; n)15287

[ρ (D (xα)),ρ (xα )](Eβ ⊗ v) L ij R β α + α − ε − ε =(−1)|α|+|α |−1 i j − α + α − ε − ε α − ε i j i α + α − εi − εj − − Eβ+εi+εj α α ⊗ v. α − εj On the other hand:

α α β ρR(Dij(x )(x ))(E ⊗ v) α−εj α−εi α β =ρR((x Di − x Dj )(x ))(E ⊗ v) − − − − =ρ (xα εj xα εi − xα εi xα εj )(Eβ ⊗ v) R α + α − εi − εj α + α − εi − εj − − = − ρ (xα+α εi εj )(Eβ ⊗ v) α − ε α − ε R j i α + α − ε − ε =(−1)|α|+|α |−2 i j − α − ε j α + α − εi − εj β − − Eβ+εi+εj α α ⊗ v α − ε α + α − ε − ε i i j β α + α − ε − ε =(−1)|α|+|α |−1 i j − α + α − ε − ε α − ε i j i α + α − εi − εj − − Eβ+εi+εj α α ⊗ v. α − εj α α α α Therefore [ρL(Dij(x )),ρR(x )] = ρR(Dij(x )(x )). Hence (R1) holds.

α α β (2) ∀ Dij(x ) ∈ L0,x ∈ R, E ⊗ v ∈V, then: α α β [σ(Dij(x )),ρR(x )](E ⊗ v) =σ(D (xα)) ◦ ρ (xα )(Eβ ⊗ v) − ρ (xα ) ◦ σ(D (xα))(Eβ ⊗ v) ij R R ij β =σ(D (xα)) (−1)|α | Eβ−α ⊗ v − ρ (xα ) Eβ ⊗ ρ (D (xα))v ij α R 0 ij β β =(−1)|α | Eβ−α ⊗ ρ (D (xα))v − (−1)|α | Eβ−α ⊗ ρ (D (xα))v α 0 ij α 0 ij =0.

α α Therefore [σ(Dij(x )),ρR(x )] = 0. Hence (R2) holds.

α β (3) ∀ Dk ∈ L[−1],k =1, 2, ··· ,m, Dij(x ) ∈ L0, E ⊗ v ∈V, then: α β [ρL(Dk),σ(Dij(x ))](E ⊗ v) α β α β =ρL(Dk) ◦ σ(Dij(x ))(E ⊗ v) − σ(Dij(x )) ◦ ρL(Dk)(E ⊗ v)

β α α β+εk =ρL(Dk)(E ⊗ ρ0(Dij(x ))v) − σ(Dij(x ))(E ⊗ v)

β+εk α β+εk α =E ⊗ ρ0(Dij(x ))v − E ⊗ ρ0(Dij(x ))v =0. α Therefore [ρL(Dk),σ(Dij(x ))] = 0. Hence (R3) holds.

REPRESENTATIONS OF S(m; n)11283 | | α (−1) α ρ (Dα (fDα +εi (g)))σ(D (xα))ρ (D )+ α R ij L j α α+α=α α (−1)|α | ρ (Dα (fDα +β(g)))σ(D (xα))σ(D (xβ)). α R ij ij α α+α=α β The third and the fourth equations in (4.7) hold because of the equations − α − |α| α α α Di(f)g = Di(fg) fDi(g)andD (f)g = ( 1) α D (fD (g)). α+α=α For a family of R-linear endomorphisms of M:Φ=(ϕα)α∈A(m;n), we deﬁne its support SuppΦ = {α ∈ A(m; n) | ϕα =0 }. Deﬁne the degree degΦ of Φ to be the largest integer t such that there exists α ∈SuppΦ with |α| = t. The next lemma is due to Skryabin ([11]), which is needed to complete the proof of Theorem 4.3.

Lemma 4.5. ([11])Lett ≥ 0 be an integer and Φ=(ϕα)α∈A(m;n) be a family of R-linear endomorphisms of M satisfying : (1) degΦ ≤ t. | | (2) The endomorphisms ϕα with α = t are pairwise commuting. α (3) M is stable under all endomorphisms Φ(f)= ρR(D (f))ϕα, ∀ f ∈ R. α∈A(m;n) Then the endomorphisms ϕα are nilpotent for all α with |α| = t.

For a ﬁxed pair i, j ∈{1, 2, ··· ,m},i < j, put ϕα =0for|α|≤1, ϕα = α σ(Dij(x )) for |α|≥2andt = l +1. The properties (1) and (2) in Lemma 4.5 hold, but (3) is lacking, so we can’t directly apply Lemma 4.5. However, fortunately M is stable under the endomorphism ρL(Dij(f)) = ρL(Dj(f)Di − Di(f)Dj)= ρR(Dj(f))ρL(Di) − ρR(Di(f))ρL(Dj)+Φ(f). If γ,γ ∈ A(m; n)andfν ,gν ∈ R are chosen as in the proof of Lemma 4.5 in [11], i.e. 1, when α = γ + γ, (4.8) f Dαg = ν ν ∈ | |≤ ν 0, when α A(m; n), α 2t, α = γ + γ . From (4.7) and Lemma 4.5, we see that ρL(Dij(ffν ))ρL(Dij(gν )) = Φ(ffν )Φ(gν )=Ψ(f) ν ν α whereΨ=(ψα )α∈A(m;n) and Ψ(f)= ρR(D (f))ψα ,degΨ≤ t − t ,t = |γ |. α Furthermore, t α ψ = (−1) ϕ ϕ α α α β α β whenever |α| = t − t, and the sum is taken over all α, β ∈ A(m; n) such that |α| = |β| = t, α * α,α−α +β = γ +γ.SoM is stable under the endomorphisms Ψ(f). The arguments used in the proof of Lemma 4.5 in [11]nowworkwithout α any change. We deduce as Lemma 4.5 in [11]thatσ(Dij(x )) are nilpotent for all α ∈ A(m; n)with|α| = l +1 and1 ≤ i

12284 YU-FENG YAO AND BIN SHU that σ(Ll−1) = 0. It contradicts the deﬁnition of l. The proof of Theorem 4.3 is completed.

5. Simple modules for S(m; n) in non-exceptional cases ∗ 5.1. For χ ∈ L , deﬁne the height of χ htχ := min{i ≥−1 | χ(Li)=0}.Let (V,ρ )beaχ| -reduced representation of the restricted Lie algebra L .Thenwe 0 L0 0 Ups (L, χ) have an induced module V:=Ind V = Ups (L, χ) V ,wheres is the U(L0,χ) U(L0,χ) same one as in §2.2, and U(L0,χ) is the so-called χ-reduced enveloping algebra p − [p] − p of L0 which is a quotient of U(L0) by the ideal generated by x x χ(x) , ∀ ∈ V β ⊗ α α1 α2 ··· αm ≤ ≤ x L0. As a vector space, = FE V ,whereE = e−1e−2 e−m, 0 αi m β p i − 1,e−i = Di, 1 ≤ i ≤ m.

5.2. Below we will introduce some module operators on V, ﬁnally present a desired module category after three steps as follows.

Step 1: The R-module structure ρR deﬁned via β (5.1) ρ (xα)Eβ ⊗ v =(−1)|α| Eβ−α ⊗ v. R α

By (2.1), one can easily see that V becomes an R-module concerning the R-module structure deﬁned by (5.1). Step 2: The L-module structure ρL deﬁned via :

(5.2) ρ (D (xα))Eβ ⊗ v L ij | |− β + εi β + εj − =(−1) α 1 − Eβ+εi+εj α ⊗ v+ α − ε α − ε j i β (−1)|α|−|γ| Eβ+γ−α ⊗ ρ (D (xγ))v. α − γ 0 ij 0<γ≤α, |γ|≥2

Denote by ind the induced representation of L associated with the induced module

Ups (L, χ) structure on V=Ind V from the L0-module (V,ρ0). We have the following U(L0,χ) proposition.

Proposition 5.1. The action of L on V deﬁned by ( 5.2) coincides with ind.So V becomes an L-module with the L-module structure deﬁned by ( 5.2). Furthermore it is a generalized χ-reduced L-module.

Proof. We need to prove that the following equation holds for any α, β ∈ A(m; n), 1 ≤ i

α β α β ρL(Dij(x ))E ⊗ v = ind(Dij(x ))E ⊗ v.

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This can be veriﬁed by a direct computation: | |− β + εi β + εj − (LHS) =(−1) α 1 − Eβ+εi+εj α ⊗ v+ α − εj α − εi β (−1)|α|−|γ| Eβ+γ−α ⊗ ρ (D (xγ))v α − γ 0 ij 0<γ≤α, |γ|≥2 β + ε β =(−1)|α|−1 i − − α − εj α − εj β + εj β − − Eβ+εi+εj α ⊗ v+ α − εi α − εi | |− β − (−1) α 1 Eβ+εi+εj α ⊗ v− α − εj | |− β − (−1) α 1 Eβ+εi+εj α ⊗ v+ α − ε i β (−1)|γ| Eβ−γ ⊗ ρ (D (xα−γ ))v γ 0 ij 0≤γ<α |α−γ|≥2 | |− β β − =(−1) α 1 − Eβ+εi+εj α ⊗ v α − εi − εj α − εi − εj α β + ind(Dij(x ))E ⊗ v α β =ind(Dij(x ))E ⊗ v =RHS.

The third equation above is because in the generalized χ-reduced algebra Ups (L, χ), we have the following formula which will be used frequently: β D (xα)Eβ = (−1)|γ| Eβ−γ D (xα−γ ) ij γ ij 0≤γ≤β

Step 3: The L0-module structure defined via : β β (5.3) σ(D )E ⊗ v = E ⊗ ρ0(D )vD∈ L0,β ∈ A(m; n),v ∈ V. V | It is clear that becomes a χ L0 -reduced L0-module with the module structure | defined by (5.3) from the assumption that (V,ρ0)isaχ L0 -reduced representation of L0. Next we will prove that V belongs to the category C for the three module structures defined by (5.1), (5.2) and (5.3). We need the following combinatorial formulae as (2.1) which can be directly verified. Lemma 5.2. For any α, β, α,γ ∈ A(m; n), 1 ≤ i

14286 YU-FENG YAO AND BIN SHU

β β − α ε β − α ε + j − + i (5.5) + α α − εi α − εj − β + εi + εj α β + εi − β + εj α α − εj α − εi β α α − ε − ε α α − ε − ε + i j − + i j = α + α − εi − εj α − εj α − εi Theorem 5.3. Notations are as above, and the R-module structure, the L- module structure and the L0-module structure on V are deﬁned by ( 5.1), ( 5.2)and ( 5.3) respectively. Then the module V must belong to the category C. Proof. We need to prove that (R1)–(R4) in Deﬁnition 3.2 hold. α α β (1) ∀∈Dij(x ) ∈ L, x ∈ R, E ⊗ v ∈V,then: α α β [ρL(Dij(x )),ρR(x )](E ⊗ v) =ρ (D (xα)) ◦ ρ (xα )(Eβ ⊗ v) − ρ (xα ) ◦ ρ (D (xα))(Eβ ⊗ v) L ij R R L ij β =ρ (D (xα)) (−1)|α | Eβ−α ⊗ v − L ij α β ε β ε α |α|−1 + i + j β+εi+εj −α ρR(x ) (−1) − E ⊗ v α − εj α − εi β − ρ (xα )( (−1)|α|−|γ| Eβ+γ−α ⊗ ρ (D (xγ))v ) R α − γ 0 ij 0<γ≤α | |≥ γ 2 | | β | |− β − α + εi β − α + εj − − =(−1) α ((−1) α 1 − Eβ α +εi+εj α ⊗ v)− α α − ε α − ε j i | |− β + εi β + εj | | β + εi + εj − α − − (−1) α 1( − )(−1) α Eβ+εi+εj α α ⊗ v+ α − ε α − ε α j i β β − α (−1)|α | (−1)|α|−|γ| Eβ−α +γ−α ⊗ ρ (D (xγ ))v− α α − γ 0 ij 0<γ≤α |γ|≥2 β β + γ − α (−1)|α |(−1)|α|−|γ| Eβ+γ−α−α ⊗ ρ (D (xγ ))v α − γ α 0 ij 0<γ≤α | |≥ γ 2 β β − α + ε β − α + ε =(−1)|α|+|α |−1 i − j − α α − ε α − ε j i β + εi + εj − α β + εi β + εj − − − Eβ+εi+εj α α ⊗ v+ α α − ε α − ε j i β β − α (−1)|α|+|α |−|γ | − α α − γ 0<γ≤α | |≥ γ 2 β β + γ − α Eβ+γ−α−α ⊗ ρ (D (xγ))v. α − γ α 0 ij By Lemma 5.2, we have:

16288 YU-FENG YAO AND BIN SHU

α β (4) ∀ Dij(x ) ∈ L, E ⊗ v ∈V, then:

ρ (D (xα))(Eβ ⊗ v) L ij | |− β + εi β + εj − =(−1) α 1 − Eβ+εi+εj α ⊗ v+ α − ε α − ε j i β (−1)|α|−|γ| Eβ+γ−α ⊗ ρ (D (xγ ))v. α − γ 0 ij 0<γ≤α |γ|≥2 On the other hand: α α ρR(Dj(x )) ◦ ρL(Di) − ρR(Di(x )) ◦ ρL(Dj)+ γ α γ β ρR(D (x )) ◦ σ(Dij(x )) (E ⊗ v) 0<γ≤α |γ|≥2

α−εj β+εi α−εi β+εj =ρR(x )(E ⊗ v) − ρR(x )(E ⊗ v)+ α−γ β γ ρR(x ) E ⊗ ρ0(Dij(x ))v 0<γ≤α | |≥ γ 2 | |− β + εi − | |− β + εj − =(−1) α 1 Eβ+εi+εj α ⊗ v − (−1) α 1 Eβ+εi+εj α ⊗ v+ α − ε α − ε j i β (−1)|α|−|γ| Eβ+γ−α ⊗ ρ (D (xγ ))v α − γ 0 ij 0<γ≤α | |≥ γ 2 | |− β + εi β + εj − =(−1) α 1 − Eβ+εi+εj α ⊗ v+ α − ε α − ε j i β (−1)|α|−|γ| Eβ+γ−α ⊗ ρ (D (xγ ))v. α − γ 0 ij 0<γ≤α |γ|≥2

α α α So ρL(Dij(x )) = ρR(Dj(x )) ◦ ρL(Di) − ρR(Di(x )) ◦ ρL(Dj)+ γ α γ ρR(D (x )) ◦ σ(Dij(x )). Therefore (R4) holds. 0<γ≤α |γ|≥2 Summing up, we know that V belongs to the category C. The proof is completed.

∼ εi+εi+1 5.3. According to [12, 4.3], L[0] = sl(m, F ). Denote hi := Di,i+1(x )= ε ε x i Di −x i+1 Di+1,i=1, 2, ···m−1, then H :=F -span{hi | i =1, 2, ··· ,m−1} is a canonical torus of L[0].LetV be an irreducible L0-module, λ =(λ1,λ2, ··· ,λm−1) ∈ m−1 F ,0= v ∈ V is called a weight vector of weight λ if ρ(hi)v = λiv, i = 1, 2, ···m − 1, where ρ is the representation of L0 corresponding to the L0-module N N { εl+εj εl | ≤ V . If in addition ρ( )v =0,where :=F -span Dij (x )=x Di 1 l

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For χ ∈ L∗, then each simple generalized χ-reduced module of L is some ho- V | momorphic image of an induced module from a χ L0 -reduced simple L0-module V .Whenhtχ ≤ 0, then L1 act on V trivially and any simple L[0]-module can be regarded as L0-module via trivial L1 action. In this case, all simple L[0]-module ∗ are parameterized by the restricted weight set Λ : = {λ ∈H | λ(hi) ∈ Fp ,i = ··· − } Fm−1 ∈ 1, 2, ,m 1 = p , as “highest weight” (cf. [3]). Conversely each λ Λ, as “highest weight”, corresponds to a unique simple L[0]-module denoted by L0(λ). In this case, we may say “exceptional” or “nonexceptional” for a simple L0-module, depending on what the “highest weight” is. Thanks to Theorems 4.3 and 5.3, we have the following result concerning the representations of special algebra L = S(m; n). ∗ − Theorem 5.4. Let χ ∈ L with htχ ≤ min{pni − pni 1 | 1 ≤ i ≤ m}−2.If V is a nonexceptional irreducible L0-module with character χ,then(V,ρL) is an irreducible L-module. Proof. Set R = A(m; n),L = S(m; n). By Theorem 5.3, V belongs to the V { θ ⊗ | ∈ } ∈ V category C.Set θ = F -span E v v V for θ A(m; n). Then = V V ∼ V θ∈A(m;n) θ and θ = V as σ(L0)-modules. Therefore θ is an irreducible submodule of V as σ(L0)-module. Furthermore Vθ is not exceptional, so the ﬁrst condi- − tion in Theorem 4.3 holds. The assumption htχ ≤ min{pni −pni 1 | 1 ≤ i ≤ m}−2 assures the second condition in Theorem 4.3. So any L-submodule V of V is also an R-submodule of V by Theorem 4.3. Assume V is a nonzero L-submodule of V. Next we will prove that V = V. n θ Assume 0 = v = E i ⊗ vi ∈V,whereθi ∈ A(m; n), 0 = vi ∈ V . We can deﬁne i=1 a total order in A(m; n) by lexicographical order, i.e. α =(α1,α2, ··· ,αm) ≺ β =(β1,β2, ··· ,βm) if and only if |α| < |β| or |α| = |β| and there exists some i ∈{1, 2, ··· ,m} such that αj = βj for j

θ1 |θ1| ρR(x )v =(−1) 1 ⊗ v1 ∈V. Therefore V = V by the simplicity of V as L0-module. We are done.

Remark 5.5. For the case of the restricted Lie algebras, the results in this Theorem have been obtained by Zhang Chao-Wen in [14].

6. Simple modules for S(m; n) with exceptional weights 6.1. Let χ ∈ L∗ and let M be a simple generalized χ-reduced L-module. As § V | noted in 5.3, M is a homomorphic image of a module induced from some χ L0 - reduced simple L0-module V . By the arguments in § 5, we know that V is simple (and hence isomorphic to M)ifV is not exceptional and the height of χ is no more − than min{pni − pni 1 | i =1, 2, ··· ,m}−2. In this section we will take up the case where V is exceptional and construct M as a quotient of V. The dimension of M will be determined as well. When htχ = −1 i.e. χ = 0, the determination of the simple exceptional-weight modules were completed respectively by Shen and Nakano (cf. [7]and[5]). So in this section we always assume χ ∈ L∗, htχ = 0. Maintain the notations before. In particular R = A(m; n),L= S(m; n).

18290 YU-FENG YAO AND BIN SHU

6.2. We need the following lemma to simplify the sequent argument. Lemma 6.1. For χ ∈ L∗, htχ =0,thereexistsΦ ∈ Aut(L) and l ∈{1, 2, ··· ,m} Φ −1 such that χ (Di):=χ(Φ (Di)) = 0 for i = l. Furthermore one can choose a suit- Φ able Φ ∈ Aut(L) such that χ (Di)=δil. ∼ Proof. Since S(m; n) = S(m; n )forn =(n1,n2, ··· ,nm), n =(nσ(1),nσ(2), ··· ,nσ(m)),σ ∈ Sm. We might suppose n1 ≤ n2 ≤ ··· ≤ nm.Ashtχ =0,then there exists i ∈{1, 2, ··· ,m} such that χ(Di) =0.Set l =max{i | χ(Di) =0 }.De- l−1 −1 ﬁne ϕ ∈ Aut(A(m; n)) via ϕ(xi)=xi if i = l and ϕ(xl)=xl − χ(Dl) χ(Di)xi. i=1 Set Φ : = ϕ : W (m; n) → W (m; n) sending X ∈ W (m; n)toϕ−1Xϕ ∈ W (m; n). Accordingto[13], Φ is an automorphism of W (m; n). By direct computation, one α −1 α −1 −1 α can easily see that Φ(x Di)=ϕ (x )Di − δi

β For any Dij(x ) ∈ S(m; n), then β Φ(Dij(x ))

−1 β−εj −1 −1 β−εj =ϕ (x )Di − δi

−1 β−εi −1 −1 β−εi (ϕ (x )Dj − δj

−1 β−εj −1 β−εi −1 −1 β−εi Di(ϕ (x )) − Dj (ϕ (x )) + δj

−1 −1 β−εj − δi

Therefore Φ|S(m;n) is an automorphism of S(m; n), we also denote it by Φ. Φ−1 By straightforward computation, one can obtain that χ (Di)=χ(Φ(Di)) = Φ−1 0fori = l and χ (Dl)=χ(Φ(Dl)) = χ(Dl) =0.Set ψ := ϕl ◦ Φwhere −1 ψ−1 ϕl(xi)=xi for i = l and ϕl(xl)=χ(Dl) ϕ(xl). Then χ (Di)=δil, 1 ≤ i ≤ m, as required.

6.3. Combining Lemma 2.4 and Lemma 6.1, we might as well assume χ ∈ L∗ with htχ = 0 satisfying χ(Di)=δil for a certain positive integer l, without loss of generality, up to module equivalence. One can realize exceptional module L0(ωk)byk-fold exterior product of the ∼ natural module V of L[0] = sl(m). i.e. V is an m-dimensional vector space with an ordered basis {vi | i =1, 2, ··· ,m} and the action xiDj := Eij(i = j): → → εi+εi+l − → → vj vi,vs 0fors = j; Di, i+1(x ):=Eii Ei+1,i+1 : vi vi,vi+1 ∼ k −vi+1,vs → 0,s = i, i + 1; the action of L1 is trivial. Then L0(ωk) = Vk := V with exceptional weight vector v1 ∧ v2 ∧···∧vk of exceptional weight ωk. Set I :=IndV = U s (L, χ) V ,whereu(L ) is the restricted envelop- k k p u(L0) k 0 ing algebra of L0. By Theorem 5.3, Ik is a module in the C-category. By PBW

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m { β ⊗∧k | ∈ β βi ∧k Theorem, Ik has an F -basis E vΓ β A(m; n),E = Di , vΓ = i=1 ∧ ∧···∧ { ··· }⊆{ ··· }} vγ1 vγ2 vγk , Γ= γ1 <γ2 < <γk 1, 2, ,m . β k β Define a linear mapping : dk : Ik → Ik+1 by dk(E ⊗∧ vΓ)= ρL(Di)E ⊗ i/∈Γ k β+εi k ∧ vΓ ∧ vi = E ⊗∧ vΓ ∧ vi. In the following sequel, for simplicity, we often i/∈Γ k omit the subscript k of ∧ vΓ if the context is clear. We will take use of some notations in the sequel, following Holmes in [1]: assume Γ is a subset of {1, 2, ··· ,m},thendenote|iΓj| =#{k ∈ Γ | ik| =#{i ∈ Γ | i>k} and denote Γ\i∪j a tuple obtained by replacing i by j and reordering which contribute to the sign change of the corresponding exterior product by (−1)|iΓj|. For a statement P ,Set 1, if P is true, δP = 0, if P is false. We have the following proposition, the proof of which can be achieved with the χ aid of Theorem 3.9 in [1] if one notes that Ωk in [1] can be similarly defined for the generalized restricted Cartan-Type series W and S,andthereisanL-module χ isomorphism between Ik and Ωk . Proposition 6.2. Maintain the notations as above. The following statements are true. (1) dk is an L-module homomorphism. (2) dk+1 ◦ dk =0. Proof. As argument in the previous paragraph, one can similarly construct χ Ωk as in [1]withtheL-module structure similar to the one in Proposition 3.7 in [1]. Then define −→ χ Ψ: Ik Ωk β k (τ−β) E ⊗∧ vΓ −→ y ⊗ dxΓ where τ =(pn1 −1,pn2 −1, ··· ,pnm −1). One can see that Ψ is actually an L-module χ isomorphism between Ik and Ωk . Furthermore, by the arguments parallel to the proof of Theorem 3.9 in [1], the same statements also hold for the nonrestricted case in the generalized restricted Lie algebra setting. Having this in mind, one immediately get the results of this proposition, parallel to Theorem 3.9 in [1]. By Proposition 6.2, we obtain a complex of L-modules:

d0 d1 d2 dm−1 (6.1) 0 → I0 → I1 → I2 →··· → Im → 0.

Set I¯k = Ik/Ker dk. By Proposition 6.2, dk is an L-module homomorphism. So Ker dk is a submodule of Ik, therefore I¯k is an L-module. We need the following lemma which is similar to Proposition 3.10 in [1] to obtain our main theorems. Lemma 6.3. The following statements hold β (1) For any k ∈{0, 1, ··· ,m−1}, I¯k has a basis {E ⊗∧vΓ | β ∈ A(m; n), Γ ⊆ {1, 2, ··· ,m}, |Γ| = k, l∈ / Γ},wherel is the same as in Lemma 6.1, i.e. χ(Di)=δil. (2) The sequence ( 6.1) is exact. We ﬁnally obtain the main theorem of this section.

20292 YU-FENG YAO AND BIN SHU

Theorem 6.4. Assume htχ =0. The following statements hold. (1) If 0 β(l)orα(l)=β(l)andα is greater than β in the lexicographical order. In the sense of this order, we choose β0 =max{β | β ∈ A}. n n We ﬁrst claim that β0(l)

nl − 0=ρ (x(p 1)εl D )w L i − n − β0+εi (p l 1)εl ⊗∧ ∗ β ⊗∧ = aβ0,ΓE vΓ + E vΓ i/∈Γ β n where all β appearing in ∗E ⊗∧vΓ are different from β0 + εi − (p l − 1)εl.The choice of i makes the above equation a contradiction. n So, we finally obtain that β0(l) lsuch that β0(j) ≥ 2, then 0=ρ (xβ0(j)εj D )w L l − − β0(j) β0+εl β0(j)εj ⊗ ∗ β ⊗ =( 1) E uβ0 + E uβ β where all β appearing in ∗E ⊗ uβ are different from β0 + εl − β0(j)εj.So − − β0(j) β0+εl β0(j)εj ⊗ ( 1) E uβ0 = 0. It is still a contradiction. If there exists j, k > l such that β0(j)=β0(k)=1,then 0=ρ (xεj +εk D )w L l β0+εl−εj −εk ⊗ ∗ β ⊗ =E uβ0 + E uβ

REPRESENTATIONS OF S(m; n)21293 β where all β appearing in ∗E ⊗ uβ are diﬀerent from β0 + εl − εj − εk.So − − β0+εl εj εk ⊗ E uβ0 = 0, a contradiction. If there exists k>lsuch that β0(k)=1 ≥ ∈ and β0(l) 1. Assume i Γ such that aβ0,Γ = 0(this is possible, since we have assumed k>0), then

0=ρ (x(β0(l)+1)εl D )w L i | | − β0(l) − iΓl ⊗∧ ∗ β ⊗ =( 1) δi∈Γaβ0,Γ( 1) Dk vΓ\i∪l + E uβ β where all β appearing in ∗E ⊗ uβ are diﬀerent from εk.So | | − β0(l) − iΓl ⊗∧ ( 1) δi∈Γaβ0,Γ( 1) Dk vΓ\i∪l =0, | | | | ⊗ ⊗ a contradiction. Therefore β0 < 2. If β0 =1,thenw = Di uεi + 1 u0. If w = Di ⊗ v + 1 ⊗ u0,i

εi 0=ρL(x Dm)w − ⊗ ⊗ ⊗ = Dm v + Di v + 1 u0 which is a contradiction. So w = Dm ⊗ v + 1 ⊗ u0.Choose1≤ i

εi 0=ρL(x Dj)w ⊗ · ⊗ =Dm εiDj v + 1 u0 ε which implies that x i Dj · v =0∀ 1 ≤ i

2εm+ε1 =ρL(Dm1(x ))w

2εm εm+ε1 =ρL(x Dm − x D1)w

=1 ⊗ av1 which is a contradiction. k So β0 = 0. i.e. w = 1 ⊗ u0.SoM = I¯k by the simplicity of ∧ V as L0- module. This is to say that I¯k is simple for 0

Applying Theorem 5.3, Theorem 5.4 and Theorem 6.4, we obtain the following results.

22294 YU-FENG YAO AND BIN SHU

Theorem 6.5. Assume htχ =0, then the following statements hold. m−1 (1) There are p − 1 distinct (up to isomorphism) simple Ups (L, χ) modules. { χ | ∈ Fm−1} They are represented by L (λ) λ p . χ ∼ χ (2) L (λ) Z (λ):=U s (L, χ) L (λ) if and only if λ/∈{ω ,ω , ··· , = p u(L0) 0 1 2 χ ∼ χ ωm−1} and L (ωm) = L (ωm−1). χ n (3) If λ is not exceptional, then dimF L (λ)=p i dimF L0(λ) and m−1 p ni , ≤ k

χ ∼ χ Proof. We will prove that L (λ) = L (µ) unless λ = µ or λ, µ ∈{ωm−1,ωm}, case by case. Case 1: Both λ and µ are nonexceptional. χ ∼ According to Theorem 5.3 and Theorem 5.4, in this case L (λ) = Ind(L0(λ)), χ ∼ χ χ L (µ) = Ind(L0(µ)). Both L (λ)andL (µ) are modules in the C-category. As in the proof of Theorem 5.4, any L-module homomorphism between Lχ(λ)andLχ(µ) is a homomorphism in the C-category, therefore it is an R-module homomorphism. Assume ϕ : Lχ(λ) → Lχ(µ)isanL-module isomorphism, we will prove λ = µ. Assume vλ is a maximal vector of L0(λ) with maximal weight λ and suppose θ ϕ(1 ⊗ vλ)= E ⊗ uθ where 0 = uθ ∈ L0(µ). Recall that A(m; n) has a natural lexicographical order. If uθ =0forsome θ =0,set θ0 =max{θ | uθ =0 }=0.As ϕ is also an R-module homomorphism, then θ0 ⊗ θ0 ⊗ θ0 θ ⊗ ⊗ 0=ϕ(ρR(x )(1 vλ)) = ρR(x )ϕ(1 vλ)=ρR(x )( E uθ)=1 uθ0 a contradiction. Hence ϕ(1 ⊗ vλ)=1⊗ u0.Then1⊗ u0 is a maximal vector with χ maximal weight λ in L (µ). Therefore u0 is a maximal vector with maximal weight λ in L0(µ), then λ = µ. Case 2: Both λ and µ are exceptional. We assume λ, µ ∈{ω1, ··· ,ωm−1} and λ = µ. i.e. λ = ωi,µ = ωj,i = j. From the proof of Theorem 6.4, we know that the maximal vector of I¯k is of weight ∼ ωk for k =1, ···m − 2, while for k = m − 1, it is ωm−1 or ω0.SoI¯i = I¯j. i.e. χ ∼ χ L (ωi) = L (ωj). Case 3: λ is nonexceptional and µ is exceptional. χ ∼ χ We can assume µ = ωi,i∈{1, ··· ,m−1}. We will prove that L (λ) = L (ωi). χ χ Suppose there is an L-module isomorphism ϕ : L (λ) → L (ωi). We can ﬁx a maximal weight vector of L0(λ) with maximal weight λ,sayvλ.Then1⊗ vλ is a χ maximal weight vector of L (λ) with maximal weight λ.Soϕ(1 ⊗ vλ) is a maximal χ weight vector of L (ωi) with maximal weight λ. By the proof of Theorem 6.4, the χ weight of any maximal vector in L (ωi) is exceptional, Concretely speaking, it is ωi when 1 ≤ i

Remark 6.6. When p =2,m =3andn = 1, the results in this theorem coincide with those ones in [6].

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References [1] Holmes R. R., Simple modules with character height at most one for the restricted Witt algebras, J.Algebra 237 No.2(2001), 446-469. [2] Holmes R. R. and Zhang Chao-Wen, Some simple modules for the restricted Cartan type Lie algebras, Journal of Pure and Applied Algebra 173(2002), 135-165. [3] Humphreys J. E., Modular representations of classical Lie algebras and semi-simple algebraic groups, 19(1971), 51-79. [4] Jantzen J. C., Representations of Algebraic Groups (second edition), American Mathematical Society, 2003. [5] Nakano D., Projective modules over Lie algebras of Cartan type, Memoirs of AMS No.470 (1992). [6] Shan Chui-Ping and Jiang Zhi-Hong, Irreducible representations of S(3,1)(in chinese), Preprint(2007). [7] Shen Guang-Yu, Graded modules of graded Lie algebras of Cartan type, III , Chinese Ann. Math. Ser.B 9(1988), 404-417. [8] Shu Bin, The generalized restricted representations of graded Lie algebras of cartan type, J.Algebra 194(1997), 157-177. [9] Shu Bin, Quasi p-mappings and representations of modular Lie algebras, in “Proceedings of the international conference on representation theorey”, CHEP & Springer-Verlag, Beijing 2000, 375-401. [10] Shu Bin and Yao Yu-Feng, Irreducible modules of the generalized Jacobson-Witt algebras, preprint [11] Skryabin S., Independent systems of derivations and Lie algebra representations,in“Algebra and Analysis, Eds: Archipov / Parshin / Shafarvich.” Walter de Gruyter& Co, Berlin-New York, 1994, 115-150. [12] Strade H. and Farnsteiner R., Modular Lie algebras and their representations ,Marcel Dekker, New York, 1988. [13] Wilson R. L., Classiﬁcation of generalized witt algebras over algebraically closed ﬁlds, Trans. Amer. Math. Soc. Vol.153(1971), 191-210. [14] Zhang Chao-Wen, Representations of the restricted Lie algebras of Cartan-type, J.Algebra 290(2005), 408-432.

Department of Mathematics, East China Normal University, Shanghai 200241, China. E-mail address: [email protected] Department of Mathematics, East China Normal University, Shanghai 200241, China. E-mail address: [email protected]

Articles in this volume cover topics related to representation theory of various algebraic objects such as algebraic groups, quantum groups, Lie algebras, (finite- and infinite-dimensional) finite groups, and quivers. Collected in one book, these articles show deep relations between all these aspects of Representation Theory, as well as the diversity of algebraic, geometric, topological, and categorical techniques used in studying representations.

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