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Representation Theory 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 This page intentionally left blank 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 acknowledg- ment 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 This page intentionally left blank 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 Pr¨ufer modules Claus Michael Ringel 163 v 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 partic- ipants. 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 Brou´e 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
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