Lie Groups and Automorphic Forms This page intentionally left blank https://doi.org/10.1090/amsip/037 AMS/IP Studies in Advanced Mathematics Volume 37 Lie Groups and Automorphic Forms Proceedings of the 2003 Summer Program Zhejiang University Center of Mathematical Sciences Hangzhou, China Lizhen Ji, Managing Editor Jian-Shu Li, H. W. Xu, and Shing-Tung Yau, Editors American Mathematical Society • International Press Shing-Tung Yau, General Editor 2000 Mathematics Subject Classification. Primary 20Gxx, 22Exx, 11-XX, 55Nxx. The paper, "On the cohomology of locally symmetric spaces and of their compactifi- cations," by Leslie Saper, starting on page 169, is used by permission. © 2002, Current Developments in Mathematics, International Press, Somerville, MA. The photograph of Armand Borel on page v is courtesy of Armand Borel's daughter, Dominique Borel. Library of Congress Cataloging-in-Publication Data Lie groups and automorphic forms / [Lizhen Ji, editor]. p. cm. — (Studies in advanced mathematics, ISSN 1089-3288 ; v. 37) Includes bibliographical references. ISBN-13: 978-0-8218-4198-3 (alk. paper) ISBN-10: 0-8218-4198-X (alk. paper) 1. Lie groups. 2. Automorphic forms. I. Ji, Lizhen, 1964- QA387.L52 2006 512'.482—dc22 2006048421 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.) © 2006 by the American Mathematical Society and International Press. All rights reserved. The American Mathematical Society and International Press retain all rights except those granted to the United States Government. 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/ Visit the International Press home page at URL: http://www.intlpress.com/ 10 9 8 7 6 5 4 3 2 1 11 10 09 08 07 06 Dedicated to the memory of Professor Armand Borel May 21, 1923 - August 11, 2003 Professor Armand Borel was an internationally recognized and respected mathematician. As a giant of 20th Century mathematics, his work was fundamental to the development and formation of modern mathematics. In 1991, Professor Borel received the Steele Prize from the American Mathematical Society for his lifelong contributions to modern mathematics. In recent years, Armand Borel devoted himself to organizing the Summer Program at the Center of Mathematical Science at Zhejiang University, China. Thousands of young mathematicians and students in China were fortunate enough to learn from his lectures and were greatly inspired by him. Professor Borel passed away on August 11, 2003 in Princeton, New Jersey. His wife, Gaby Borel, lives in Princeton and his daughters, Dominique and Ann Borel, reside in New York City. This page intentionally left blank CONTENTS Preface L.Ji, J.S.Li, H. W.Xu and S.T.Yau ix Lie Groups and Linear Algebraic Groups I. Complex and Real Groups Armand Borel 1 Introduction to the Cohomology of Arithmetic Groups Armand Borel 51 Lectures on locally symmetric spaces and arithmetic groups Lizhen Ji 87 Petersson and Kuznetsov Trace Formulas Jianya Liu and Yangbo Ye 147 On the Cohomology of Locally Symmetric Spaces and of their Compact- ifications Leslie Saper 169 vii This page intentionally left blank Preface Upon the invitation of Professor S.T. Yau, the director of the Center of Math• ematical Sciences at Zhejiang University, Professor Armand Borel, together with several of us in 2002, started to organize a multi-year summer school program on Lie groups, automorphic forms, and related topics at the Center. The plan was to carry out a self-contained, annual program that would start with the basics of Lie groups, Lie algebras and representation theories, and would lead to modern theories on automorphic forms, in particular, the Langlands program and Shimura varieties. The first year of the program was to commence in 2003. Professor Borel worked out a detailed plan for the summer program to be held in June and July, and everything was arranged and settled in April 2003. Unfortunately, the program was interrupted and eventually canceled due to the unexpected outbreak of the SARS epidemic in China in the spring and early summer of 2003. With the strong support of Professor S.T. Yau, two of Professor Borel's col• leagues, L. Ji and J.S. Li, started in the middle of May 2003 to organize a shorter summer program, which did take place at the Center that August. The short summer program was a success with over eighty participants attending. Though Professor Borel could not participate in and give lectures at this summer program, his lecture notes were distributed to the participants. The summer program, ac• cording to Professor Borel's original plan, was held successfully in July 2004 and again in 2005 at the Center. This book consists of excerpted notes from the lecture series of the 2003 Summer Program, together with two sets of Professor Borel's lecture notes, which were circulated during the program, and a set of L. Saper's lecture notes, which were presented in the 2004 program. Because the first year of the program prepared participants for more advanced topics to come in future programs, many courses in this book are foundational and elementary. The first set of lecture notes by Professor Borel gives a comprehensive intro• duction to structures of complex and real groups. The lecture is less formal and hence more user-friendly than most existing treatments of such topics and is an excellent introduction to them. The second set of notes by Professor Borel motivates the study of cohomology groups of arithmetic groups and discusses many methods used to understand them and related compactifications. Cohomology groups of arithmetic groups have been studied intensively by many people, and this set of notes serves as a very useful introduction to this active field. The lecture notes by L. Ji study arithmetic groups and locally symmetric spaces by emphasizing important examples and applications in many different fields. By presenting the various versions of the reduction theories, the differences and simi• larities of these theories become clear. The lecture notes by J. Liu and Y. Ye introduce the Petersson trace formula and Kuznetsov trace formula, which are closely related to the Selberg trace formula. These notes explain how the formulas can be used to prove subconvexity bounds for certain Rankin-Selberg L-functions, an important topic in modern analytic number theory. ix x Preface to Lie Groups and Automorphic Forms The chapter by L. Saper gives a comprehensive introduction to L2-cohomology and intersection cohomology groups of locally symmetric spaces and related com- pactifications. This work is actually a previously published paper by L. Saper and was distributed at the 2004 Summer Program. This is a useful supplement to the 2003 lectures found in this book and are reprinted here at the recommendation of S.T. Yau and J. Li. Besides the speakers whose works are included in this book, many notable people also gave lectures at the summer school that year, namely, D.L. Li and H.W. Lu on Number theory, H.L. Li on Lie algebras and groups of Lie type, J.S. Li on Elliptic curves, J.P. Wang on Modular forms, Y.B. Shen on Differential and complex geometry, and N.H. Xi on Linear algebraic groups. We would like to take this opportunity to thank all the speakers for presenting exceptional lectures on such short notice and all the participants for braving the extremely hot weather in Hangzhou to attend the 2003 Summer Program. In particular, we would like to thank the local organizers, L. Fang and S.L. Wang, for arranging and overseeing all the logistical aspects of the program. Without their help, the 2003 Summer Program could not have taken place. We would also like to thank the Center, especially its director, Professor S.T. Yau, its executive director, K.F. Liu, and its staff members for their support and devoted assistance during the program. We would also like to thank L. Saper for his thorough proofreading and type• setting work on the two sets of Professor Borel's lecture notes found in this book. Finally, we would like to thank J. Dai, H. Rutledge, R. Rogers, and L. Lin for their efforts and help in typesetting and publishing this book. L. Ji, Managing Editor J.S. Li H.W. Xu S.T. Yau May 2006 This page intentionally left blank Titles in This Series 37 Lizhen Ji, Jian-Shu Li, H. W. Xu, and Shing-Tung Yau, Editors, Lie Groups and Automorphic Forms, 2006 36 Chuu-Lian Terng, Editor, Integrable Systems, Geometry, and Topology, 2006 35 Felix Finster, The Principle of the Fermionic Projector, 2006 34 Ren-Hong Wong, Editor, Computational Geometry, 2003 33 Eric D'Hoker, Duong Phong, and Shing-Tung Yau, Mirror Symmetry IV, 2002 32 Xi-Ping Zhu, Lectures on Mean Curvature Flows, 2002 31 Kiyoshi Igusa, Higher Franz-Reidemeister Torsion, 2002 30 Weiman Han and Mircea Sofonea, Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity, 2002 29 S.