Unipotent and Nilpotent Classes in Simple Algebraic Groups and Lie Algebras

Home , Nilpotent, Unipotent

Mathematical Surveys and Monographs Volume 180

Martin W. Liebeck Gary M. Seitz

American Mathematical Society http://dx.doi.org/10.1090/surv/180

Unipotent and Nilpotent Classes in Simple Algebraic Groups and Lie Algebras

Mathematical Surveys and Monographs Volume 180

Unipotent and Nilpotent Classes in Simple Algebraic Groups and Lie Algebras

Martin W. Liebeck Gary M. Seitz

American Mathematical Society Providence, Rhode Island EDITORIAL COMMITTEE Ralph L. Cohen, Chair MichaelA.Singer Jordan S. Ellenberg Benjamin Sudakov MichaelI.Weinstein

2010 Mathematics Subject Classiﬁcation. Primary 20G15, 20G40, 20G41, 20E45, 20E32, 17B45, 17B08.

For additional information and updates on this book, visit www.ams.org/bookpages/surv-180

Library of Congress Cataloging-in-Publication Data Liebeck, M. W. (Martin W.), 1954– Unipotent and nilpotent classes in simple algebraic groups and lie algebras / Martin W. Liebeck, Gary M. Seitz. p. cm. — (Mathematical surveys and monographs ; v. 180) Includes bibliographical references and index. ISBN 978-0-8218-6920-8 (alk. paper) 1. Linear algebraic groups. 2. Lie algebras. I. Seitz, Gary M., 1943– II. Title.

QA179.L54 2012 512.482—dc23 2011043518

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Contents

Preface xi

Chapter 1. Introduction 1

Chapter 2. Preliminaries 9 2.1. Notation 9 2.2. Subgroups 13 2.3. Automorphisms and Lie algebras 15 2.4. Frobenius morphisms and the Lang-Steinberg theorem 19 2.5. Nilpotent and unipotent elements 20 2.6. Distinguished parabolic subgroups 24 2.7. Centralizers of nilpotent elements 28 2.8. Distinguished elements in characteristic zero 34

Chapter 3. Classical groups in good characteristic 39 3.1. Preliminary lemmas 40 3.2. Proof of Theorem 3.1 43 3.3. Additional information 49

Chapter 4. Classical groups in bad characteristic: statement of results 59

Chapter 5. Nilpotent elements: the symplectic and orthogonal cases, p = 2 65 5.1. Indecomposables 65 5.2. Distinguished nilpotent elements 67 5.3. Distinguished normal form 69 5.4. Centralizers of nilpotent elements I: connected centralizers 71 5.5. Centralizers of nilpotent elements II: component groups 78 5.6. Orthogonal groups of odd dimension 86 5.7. Splitting 89 5.8. Labellings of some nilpotent classes 90

Chapter 6. Unipotent elements in symplectic and orthogonal groups, p = 2 91 6.1. Indecomposables 91 6.2. Distinguished unipotent elements 92 6.3. A map from unipotents to nilpotents in SO(V ) 93 6.4. A lemma on representations 94 6.5. Centralizers of unipotents I: connected centralizers 96 6.6. Centralizers of unipotents II: component groups 101 6.7. Splitting 109 6.8. Orthogonal groups of odd dimension 110 6.9. Labellings of some unipotent classes 110

vii viii CONTENTS

Chapter 7. Finite classical groups 113 7.1. Good characteristic 113 7.2. Bad characteristic 116

Chapter 8. Tables of examples in low dimensions 119

Chapter 9. Exceptional groups: statement of results for nilpotent elements 129

Chapter 10. Parabolic subgroups and labellings 133 10.1. T -labellings and associated parabolic subgroups 133 10.2. Labellings of some distinguished classes in classical groups 136

Chapter 11. Reductive subgroups 139

Chapter 12. Annihilator spaces of nilpotent elements 153 12.1. Lemmas on representation theory 153 12.2. Annihilator spaces of distinguished nilpotent elements 156 12.3. Further results 165

Chapter 13. Standard distinguished nilpotent elements 169 13.1. Distinguished nilpotent elements corresponding to distinguished parabolic subgroups 169 13.2. Component groups 181 13.3. Subgroups J, R 201

Chapter 14. Exceptional distinguished nilpotent elements 203

Chapter 15. Nilpotent classes and centralizers in E8 219 15.1. Preliminary lemmas 220 15.2. Proof of Theorem 15.1, I: strategy 225 15.3. Proof of Theorem 15.1, II: calculation of the centralizers 228 15.4. Proof of Theorem 15.1, III: completeness of the list 258

Chapter 16. Nilpotent elements in the other exceptional types 263 16.1. The cases where (G0, p) 6= (F4, 2) or (G2, 3) 263 16.2. The case (G0, p) = (G2, 3) 267 16.3. The case (G0, p) = (F4, 2) 268 16.4. Dual pairs 276

Chapter 17. Exceptional groups: statement of results for unipotent elements 281

Chapter 18. Corresponding unipotent and nilpotent elements 287

Chapter 19. Distinguished unipotent elements 299 19.1. The proof of Theorem 19.1 301 19.2. The proof of Theorem 19.2 310

Chapter 20. Non-distinguished unipotent classes 317 20.1. The case G = E6,E7,E8 317 20.2. The case G = F4, p 6= 2 329 20.3. The case G = F4, p = 2 330 20.4. The case G = G2 332 20.5. Proofs of the results 17.1 - 17.10 332 CONTENTS ix

Chapter 21. Proofs of Theorems 1, 2 and Corollaries 3 - 8 341 Chapter 22. Tables of nilpotent and unipotent classes in the exceptional groups 351 22.1. Classes and centralizers in exceptional algebraic groups 351 22.2. Unipotent classes and centralizers in ﬁnite exceptional groups 351 22.3. The dual pairs J, R 352 Bibliography 373 Glossary of symbols 377 Index 379

Preface

This book concerns the theory of unipotent elements in simple algebraic groups over algebraically closed or finite fields, and nilpotent elements in the corresponding simple Lie algebras. These topics have been an important area of study for decades, with applications to representation theory, character theory, the subgroup structure of algebraic groups and finite groups, and the classification of the finite simple groups. Even detailed information on centralizers is important. For example, information regarding the component groups of centralizers is useful in studying representations of Weyl groups. There is a great deal of literature on unipotent and nilpotent elements, and many beautiful general results have been proved. In addition to the general theory, there are many situations where precise information on conjugacy classes is of great importance, such as class representatives and precise centralizers. Here the literature is less satisfactory. More than anything else, our reason for writing this book is that we believe that the information on centralizers is of sufficient importance that it deserves a single source, where results are presented completely in all characteristics, and with consistent notation. In particular the detailed tables of results for exceptional algebraic and finite groups in Chapter 22 should be easily understandable and usable by readers, and likewise tables for some low-dimensional classical groups in Chapter 8. This is our aim and our approach to this, while using ideas from the literature, is in many parts new. Our results go beyond what is currently known in several ways. For example, the literature on centralizers of unipotent and nilpotent elements in classical groups and Lie algebras in characteristic 2 is not complete, and we obtain complete information. We establish a number of new structural results on centralizers, their embeddings in certain parabolic subgroups, and how the reductive part of the centralizer is embedded in the ambient group. The book is divided into 22 chapters. The first is an introduction to the topic and overview of the results in the book, and the second contains a number of basic results on algebraic groups that will be used throughout; some of these are standard, others less so, but proofs are provided in most cases. Our results for classical groups are proved in Chapters 3–6. Chapter 3 concerns the case where the characteristic of the underlying field is “good” (meaning that it is not 2 for symplectic and orthogonal groups), and the analysis is fairly short and elementary. This is not the case for characteristic 2, covered in Chapters 4,5 and 6. Here our approach is for the most part new, as are many of the results, and takes substantial effort. In Chapter 7, these results are applied to give corresponding results on

xi xii PREFACE classes and centralizers in finite classical groups, and some tables illustrating our results for various classical groups of dimension up to 10 are given in Chapter 8. The remainder of the book, Chapters 9–22, is devoted to the exceptional groups G2,F4,E6,E7 and E8. A key feature of our approach is that we first focus on the classes and centralizers of nilpotent elements, and then use these results to deal with the unipotent elements. This approach has the advantage that our theory for nilpotent elements e has a number of structural features that are not present for unipotent elements, such as the existence of a naturally defined 1-dimensional torus acting on the 1-space spanned by e, and an associated parabolic subgroup, which turns out to contain the centralizer of e. The main results for nilpotent elements are stated in Chapter 9, and proved in the following seven chapters. Unipotent elements are then handled in Chapters 17–20. Finally, Chapter 21 contains proofs of some of our general results on the structure and embedding of centralizers, together with various corollaries of our main results; and Chapter 22 has detailed tables of classes and centralizers in the exceptional algebraic groups, and also in the associated finite groups of Lie type. It will be apparent even from this brief discussion that in this book we are focussing almost exclusively on the classification and centralizer structure of unipotent and nilpotent classes. There are many other issues concerning these classes which are of great interest in algebraic group theory, algebraic geometry and representation theory. We shall not touch upon these subjects directly, although a number of proofs do require a certain amount of representation theory. This book does not contain an introduction to the theory of algebraic groups; neither does it contain definitions and basic properties of the simple groups. Never- theless, we have written it with the intention of being comprehensible to graduate students and researchers who have a basic knowledge of these topics. We would like to thank Tim Burness for reading the manuscript and suggesting many corrections, Ross Lawther for double-checking some of the calculations in Chapter 13, and Donna Testerman and Bob Guralnick for helpful comments.

Martin Liebeck and Gary Seitz

Author addresses: Department of Mathematics, Imperial College, London SW7 2AZ, England email: [email protected]

Department of Mathematics, University of Oregon, Eugene, Oregon 97403, USA email: [email protected] Bibliography

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Glossary of symbols

x A˜i, subsystem Ai of short roots, 11 s , image of s under x, 13 An(q), Dn(q), E6(q), 19 sα, reflection in the root α, 11 Altn, alternating group of degree n, 12 SLn(K), 9 Aut(G), automorphism group of G, 17 SOn(K), On(K), 9 2 BG, Borel subgroup, 11 S V , symmetric square of V , 42 C3(a1), 129, 269 Sx, fixed points of x in S, 13 CG(T, e) = CG(T ) ∩ CG(e), 5 Sp2n(K), 9 CV (T, e), 158 Symn, symmetric group of degree n, 12 CV (e), annihilator space of e, 153 T , 1-dimensional torus, 4 dτ, differential of τ, 41 T -labelling, 133 D(m), indecomposable module, 86 T (G)1, tangent space at the identity, 35 T , maximal torus of G, 9 Dn(ai), distinguished class in Dn, 58, 90, G 110 TX (λ), tilting module of high weight λ, 11 u, unipotent element, 4 Dih2n, dihedral group of order 2n, 12 e, nilpotent element, 3 Uα, root subgroup, 11 e, u corresponding nilpotent and unipotent Ui, connected unipotent group of elements, 287 dimension i, 9 Uc ...c , notation for U , 11 eα, root vector in L(G), 11 1 r c1α1+...+cr αr Uij..., notation for Uα +α +..., 11 ec ...c , notation for ec α +...+c α , 11 i j 1 r 1 1 r r V ↓ Y , restriction of V to Y , 13 eij..., notation for eα +α +..., 11 i j V (m),W (m), indecomposables for u, 59, 91 fα = e−α, 11 G(q), finite group of Lie type, 1 V (m),W (m),Wl(m), indecomposables for e, 59, 65, 66 Gσ, fixed point group of σ in G, 114 VX (λ) (or just λ), irreducible KX-module Gτ , fixed point group of τ, 9 of high weight λ, 11 h (c), element of maximal torus T , 11 α G W (G), Weyl group of G, 11 Inndiag(G(q)), 352 W (λ), Weyl module of high weight λ, 11 J , Jordan block, 39 X i X.Y , extension of X by Y , 12 K, algebraically closed field, 1 Z , cyclic group of order p, 5 L(G), Lie algebra of G, 1 p ∆-module, 142 L(G)(q), Lie algebra over , 22 Fq ∆(λ; µ), 141 L(G) , L(Q) , 136 i i Π(G), system of fundamental roots, 9 L(G) , L(Q) , 136 ≥i ≥i Σ(G), root system, 9 L(Q)(i), 12 αij..., notation for αi + αj + ..., 11 L(Q) , 31 k χV , χ-function, 59 L(Q)≥k, 31 κ, map from unipotents to nilpotents, 94 (≥i) L(Q ), 12 λi, fundamental dominant weight, 11 [m; l], a χ-function, 59 ω, semilinear map on L(G), 258 M1/M2/ ··· , notation for a module, 12 σ, Frobenius morphism, 114, 258 P , parabolic subgroup, 4 σq, q-field morphism, 19 P −, opposite parabolic, 11 ∧2V , alternating square of V , 42 Pij..., parabolic subgroup, 25 Q(≥i)/Q(≥i+1), ith level of Q, 12 Q≥2, 4 Q≥k, 31, 136 Ru(X), unipotent radical of X, 9

377

Index

annihilator space, 153 dual pair J,R, 220, 276 exceptional distinguished nilpotents, 203 bad primes, 2 Levi subgroups, 139 Bala-Carter theory, 3, 35, 39, 40 maximal rank subgroups, 139, 142, 143 nilpotent classes, 130, 263 classical groups, 1, 9, 39 nilpotent elements, 129 canonical form for nilpotents, 50, 60 non-distinguished unipotent classes, 317 canonical form for unipotents, 50, 61, 93 reductive subgroups, 139 centralizers of nilpotents, 39, 71, 78 regular nilpotent classes, 169 centralizers of unipotents, 39, 96, 101 standard distinguished nilpotents, 169 distinguished nilpotent classes, 67, 136 tables of dual pairs, 352 distinguished nilpotent element, 60 tables of nilpotent classes, 351 distinguished unipotent classes, 43, 61, tables of unipotent classes, 351 92 unipotent classes, 281 dual pair J,R, 46 exceptional unipotent classes, 299 finite, 113 in bad characteristic, 59 field morphism, 19 in good characteristic, 39 finite classical groups, 113 Lie algebras of, 16 splitting of classes, 115 natural module, 39 finite exceptional groups, 284 splitting of centralizers, 77 unipotent classes, 351 splitting of classes, 49, 109 finite group of Lie type, 1, 19 tables of examples, 119 twisted groups, 19 unipotent classes, 43, 91 Frobenius morphism, 19, 21, 114, 258 component group, 9 q-field morphism, 19 correspondence e → u, 287 fundamental dominant weight, 11 dense double coset, 7 good characteristic, 2 distinguished nilpotent element, 20 good primes, 2 annihilator space of, 156 graph automorphism exceptional, 203 exceptional, 19 standard, 169 standard, 17 distinguished normal form, 39, 49, 60, 61, 69 height of a root, 12 distinguished parabolic subgroup, 24 high weight, 11 in classical groups, 54 distinguished unipotent element, 20, 92, ideal, 311 299 exceptional, 299 labelled diagram, 133 standard, 299 labelling, 4, 32, 54, 133 dual pair J,R, 6, 46, 48, 51, 201, 276, 352 Lang-Steinberg theorem, 3, 19, 114 dual root system, 18 level of Q, 12 Dynkin diagram, 1, 10 level of a root, 12 Levi subgroup, 3, 11, 20 exceptional graph morphism, 19 exceptional groups, 129 maximal rank subgroup, 11 distinguished unipotent classes, 299 minimal module, 12, 153

379 380 INDEX natural module, 39 unipotent element, 1 nilpotent classes centralizer, 39, 62, 281 centralizers in E8, 219 distinguished, 3, 20, 92, 299 distinguished, 20 regular, 57, 61, 110, 288 in E8, 219 standard distinguished, 299 in classical groups, 39, 65 unipotent radical, 5 in exceptional groups, 219, 263 notation Ru(X), 9 regular, 60, 90, 169 untwisted diagonal subgroup, 226 simply connected groups, 130 splitting of, 89 Weyl group, 11 nilpotent element, 1 Weyl module, 11 annihilator space of, 153 centralizer of, 39, 71, 78, 130 centralizers in E8, 219 distinguished, 20, 60, 169, 203 regular, 60, 90, 169 orders of unipotent elements, 45, 92, 284 parabolic subgroup, 4 distinguished, 24 labelling, 133 opposite, 11, 33 standard, 11 unipotent radical, 11 real element, 6 Ree groups, 19 regular nilpotent class, 60, 90, 169 regular unipotent class, 57, 61, 110, 288 shape of a root, 12 simple algebraic group, 1, 9 simply connected group, 40, 57, 130, 285 spin group, 57 spin module, 148 splitting of centralizers, 77, 283 splitting of classes, 49, 109, 115, 285 Springer correspondence, 21 variations of, 281 Springer map, 2, 21 standard basis, 16, 113 subsystem subgroup, 11 surj-inj property, 154 Suzuki groups, 19 tilting module, 11 unipotent classes distinguished, 43, 92, 299 exceptional, 299 in classical groups, 43, 91 in exceptional groups, 281 non-distinguished, 317 numbers of, 284 regular, 57, 61, 110, 288 simply connected groups, 40, 57, 285 spin groups, 57 splitting of, 49, 109, 115, 285 standard distinguished, 299 This book concerns the theory of unipotent elements in simple algebraic groups over algebraically closed or finite fields, and nilpotent elements in the corresponding simple Lie algebras. These topics have been an important area of study for decades, with applications to representation theory, character theory, the subgroup structure of algebraic groups and finite groups, and the classification of the finite simple groups. The main focus is on obtaining full information on class representatives and centralizers of unipotent and nilpotent elements. Although there is a substantial literature on this topic, this book is the first single source where such information is presented completely in all characteristics. In addition, many of the results are new—for example, those concerning centralizers of nilpotent elements in small characteristics. Indeed, the whole approach, while using some ideas from the literature, is novel, and yields many new general and specific facts concerning the structure and embeddings of centralizers.

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