Harmonic Analysis on Reductive Groups PDF
Preview Harmonic Analysis on Reductive Groups
Progress in Mathematics Volume 101 SeriesEditors J.Oesterle A.Weinstein Harmonie Analysis on Reduetive Groups r.Bowaoin Co{[ege 1989 William Barker Pau! Sally Editors Springer Science+Business LLC Medi~ William H. Barker Paul J. Sal1y, Jr. Department of Mathematies Department of Mathematies Bowdoin College University of Chieago Brunswiek, ME 04011 Chieago, IL 60637 Library of Congress Cataloging-in-Publieation Data Harmonie analysis on reduetive groups: proceedings of the Bowdoin eonferenee 1989, Bowdoin College, 1989 / William Barker, Pau1 Sally, editors. p. em. -- (Progress in mathematies : vol. 101) Includes bibliographieal referenees. ISBN 978-1-4612-6768-3 ISBN 978-1-4612-0455-8 (eBook) DOI 10.1007/978-1-4612-0455-8 1. P-adie groups--Congresses. 2. Representations of groups -Congresses. 3. Harmonie analysis--Congresses. I. Barker, William, 1946- 11. Sal1y, Paul. III. Series. QA 171.H32 1991 91-33801 512'.74--de20 CIP Printed on acid-free paper © Springer Science+Business Media New York 1991 Originally published by Birkhäuser Boston in 1991 Softcover reprint ofthe hardcover 1s t edition 1991 Copyright is not claimed for works ofU.S. Government employees. All rights reserved. No part ofthis publication may be reproduced, stored in a retrieval system or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without prior permission ofthe copyright owner. Permission to photocopy for internal or personal use, or the internal or personal use ofspecific clients, is granted by Springer Science+Business Media, LLC for libraries and other users registered with the Copyright Clearance Center (CCC), provided that the base fee of$O.OO per copy, plus $.20 per page is paid direcdy to CCC, 21 Congress Street, Salem, MA 01970, U.S.A. Special requests should be addressed direcdy to Springer Science+Business Media, LLC. ISBN 978-1-4612-6768-3 Typeset in AMS-TE X. 987 6 5 4 3 2 1 CONTENTS Lifting ofCharacters Jeffrey Adams 1 Handling the Inverse Spherical Fourier Transform Jean-Philippe Anker . . . . . . . . .. 51 Some Problems in LocalHarmonic Analysis James Arthur . . . . . . . . . . . . . . . . . . . . . . . . . 57 Asymptotic Expansions on Symmetric Spaces Erik van den Ban and Henrik Schlichtkrull . . . . . . . . . . . . . 79 The Admissible Dual ofGLN viaRestriction to Compact Open Subgroups Colin J. Bushnell and PhilipC. Kutzko ......... 89 Invariant Harmonic Analysis on the Schwartz Space ofa Reductive p-Adic Group LaurentClozel . 101 Constructing the SupercuspidalRepresentations ofGLN(F ), F p-Adic Lawrence Corwin . . . . . . . . . . . . 123 A Remark on the Dunkl Differential Difference Operators GerritJ. Heckman . 181 Invariant Differential Operators andWeyl Group Invariants SigurdurHelgason . . . . . . . . . . . . . . . . . 193 The Schwartz Space ofa Semisimple Lie Group Rebecca A. Herb . 201 Intertwining Functors andIrreducibility ofStandard Harish-Chandra Sheaves Dragan MiIi~i(; . . . . 209 vi Contents Fundamental G-Strata Lawrence Morris. . . 223 Construction andClassification ofIrreducible Harish-Chandra Modules Wilfried Schmid . 235 Langlands' Conjecture on Plancherel Measures for p-Adic Groups Freydoon Shahidi . . . . . . . . . . . 277 Transfer and Descent: Some Recent Results Diana Shelstad . 297 On Jacquet Models ofInduced Representations ofp-Adic Symplectic Groups Marko Tadie .. . . . . . . . . . . . . 305 Associated Varieties and Unipotent Representations David A. Vogan, Jr. . 315 PREFACE AconferenceonHarmonic AnalysisonReductiveGroups was heldatBowdoin College in Brunswick, Maine from July 31 to August 11, 1989. Thestatedgoal ofthe conference was to explorerecentadvances in harmonic analysis on both realandp-adicgroups. Itwas thefirstconferencesincetheAMS SummerSym posium on Harmonic Analysis on Homogeneous Spaces, heldatWilliamstown, Massachusetts in 1972, to cover local harmonic analysis on reductive groups in such detail and to such an extent. While the Williamstown conference was longer(three weeks) and somewhat broader (nilpotentgroups, solvable groups, as well as semisimpleand reductivegroups), the structureand timeliness ofthe two meetings was remarkably similar. The program of the Bowdoin Conference consisted of two parts. First, there were six major lecture series, each consisting of several talks addressing those topics in harmonic analysis on real and p-adic groups which were the focus of intensive research during the previous decade. These lectures began at an introductory level and advanced to the current state of research. Sec ond, there was a series of single lectures in which the speakers presented an overviewoftheir latestresearch. The principal speakers and theirtopics for the lecture series were: James Arthur, Some problems in local harmonic analysis; Colin Bushnell, The admissible dual ofGL(n) via restriction to compact, open subgroups; LaurentClozel, lnvariant harmonic analysis on the Schwartz space ofa reductive p-adic group; Lawrence Corwin, Constructing the supercuspi dal representations ofGL(n,F), F p-adic; Wilfried Schmid, Construction and classification ofirreducible Harish-Chandra modules; and David Vogan, As sociated varieties and unipotent representations. The content ofthese lectures reflected accurately most of the major developments in the field of harmonic analysis on reductive groups since the Williamstown Conference. Those devel opments ofimportance which were not covered in the principal lecture series, such as harmonic analysis on semisimple symmetric spaces, were discussed in the individual lectures. By design, the conference did not treat the extensive applications of local representation theory to the theory of automorphic forms. This is, without doubt, oneofthe mostactiveand interestingareas ofmodem mathematics,and conferences on this topic have been held regularly over the past fifteen years. Two ofthe morecomprehensivewere heldatCorvallis in 1977 (AMS Summer SymposiumonAutomorphicForms,Representations,and L-functions,A. Borel andW. Casselman, eds.) and Ann Arborin 1988(Automorphic forms, Shimura varieties,and L-functions,L.ClozelandJ. Milne,eds.). Itshouldbementioned, however, thatmuchoftheeurrentresearchinlocal harmonicanalysis,especially for p-adic groups, is motivated by considerations related to various aspects of the Langlands program. Moreover, it appears that some deep results in local vii viii Preface hannonic analysis are necessary to achieve further progress in some parts of this program. Returning to the Williamstown Conference, we note that a numberofpar ticipants at Williamstown also attended the Bowdoin conference, and, in fact, severalpeoplespokeatbothconferences. On theotherhand, mostofthespeak ers at Bowdoin had not started their research in mathematics in 1972. These facts illustrate both the continuity and the enduring vitality of the research in hannonic analysis on reductive groups. At Williamstown, the principal lec ture series related directly to reductive groups were given by Harish-Chandra (Hannonicanalysisonreductivep-adicgroups); S.Helgason(Functionsonsym metric spaces); and V. S. Varadarajan (The theory ofcharactersand the discrete series for semisimple Lie groups). Aquick glance at the papers in this volume showsthattheselectureseriesarestilldirectlyconnectedtopresentdayresearch. Forexample, Helgason's lectures are related to the papers ofAnkerand Helga son I, Varadarajan's to the papers ofAdams and Herb,and Harish-Chandra's to mostofthe p-adic papers. It is no surprise that the deep and penetrating work ofHarish-Chandraon bothreal andp-adic groups is reflected in the majorityof thepapersappearing here. Furthennore, it was at the Williamstown Conference that Roger Howe stated his famous conjecture about the finite-dimensionality of certain spaces of invariant distributions on p-adic groups. Harish-Chandra regarded this conjecture as the key to the study of invariant distributions on these groups. The Howe conjecture was proved by Clozel in the mid-1980s, and many of the consequences are contained in his paper in this volume. At thesametime, thepaperspresentedat theBowdoinConferencecontainedmuch mathematics that wasdevelopedentirelywithin thepasttwenty years. Instances of this are the work on unipotent representations in the unitary dual for real groups, applications of the local trace formula to hannonic analysis, the intro duction ofD-moduIes into the study ofrepresentation theory, along with many others. This volume is intended to serve as a reference for both graduate students and researchers working in representation theory and harmonic analysis on re ductive groups. While the papers included here represent to a large extent the materialcoveredin thetalksattheConference,theyactuallycontainmuchmore. The principal speakers,and, tosomedegree, the individual speakers have made a serious effort to give a complete exposition of their topics. We expect that these proceedings will providea valuable resoUJCe for many years. Paul Sally, Jr. ACKNOWLEDGMENTS TheOrganizingCommitteefor theconferenceconsistedofWilliam Barker, Rebecca Herb, Paul Sally, and Joseph Wolf. Joseph Bernstein also served on the committeeuntil illnessforced him to withdraw. Theeditorsofthese proceedings wish to thank their fellow committee members for the time and effort they expended in making the conference a success. The editors also wish to thank those colleagues who delivered lec tures at the conference, andespeciallythose whosubsequently contributed manuscripts to the Proceedings. Our thanks further go to the reviewers of the papers-many valuable improvements resulted from their careful reading. We gratefully acknowledge the support provided by the National Sci ence Foundation in grant number DMS-8804695. Additional support specifically designated to aid graduate students- was generously supplied by Bowdoin College. These grants permitted a large and diverse group of mathematicians to participate in the conference. The editors wish to extend a special word of thank to the staff of BirkhauserBostonfor theirsupportandpatienceduringthe preparationof this volume. The delays they had to endure may have made lesser persons change professions. We further thank Ann Kostant, also of Birkhauser Boston, who did an excellent job of converting a number of the longer manuscripts into lEX. During and prior to the conference, the Organizing Committee was fortunate to have the services of Pam Ohlman as Administrative Assis tant. Pam ably coordinated all the daily functions of the conference, and displayed confidence and good humor even under the most trying of cir cumstances. Sue Theberge, the Academic Coordinator for the Bowdoin Mathematics Department, also helped with the administrative work; in addition, she played an important role in polishing and completing the lEXfiles for this volume. Finally, we wish to thank Donald Knuth for inventing the marvelous computertypesettingsystem lEX, andtheAMSforthe developmentofthe lEXmacro package AMS-lEX. All the papers in this volume were typeset by AMS-lEX, version 2.0, preprint style. William Barker Harmonic Analysis on Reductive Groups Participant List Prof.JeffreyAdams University ofMaryland Dr. Jean-Philippe Anker Princeton University Prof.SusumuArild University ofTokyo, Japan Prof.JamesG.Arthur University ofToronto, Canada Prof.MagdyAhmedAssem PurdueUniversity Prof. DanBarbasch Cornel1 University Dr. LeticiaBarchini SUNYatStonyBrook Prof.WilliamH. Barker Bowdoin Col1ege Dr. Robert Baston Oxford University, GreatBritain Prof.RobertBMard UniversireduQuebecIIMontreal,Canada Prof. BirneT. Binegar Oklahoma Slate University Mr. JuergenBlume Mathematisches Institul, Bonn, W. Germany Prof. BrianD.Boe UniversityofGeorgia Dr.MladenBozicevic Mathematical Sciences Research Institute Dr. ColinBushnel1 King's Col1ege, London, GreatBritain Prof. Jen-Tseh Chang OklahomaSlate Prof. LaurentClozel Universirede Paris-Sud,France Prof. David Col1ingwood University of Washington Prof. Mark Copper Florida International University Prof. LawrenceJ. Corwin Rutgers University Prof. JamesE. Daly University ofColorado at Colorado Springs Dr. AntonDeitmar UniversityofMuenster, W. Germany Dr.HideoDoi Hiroshima University, Japan Dr.FokkoduCloux Ecole Polytechnique, Paris, France Dr.EdwardDunne Oklahoma Slate University Prof. MasaakiEguchi Hiroshima University, Japan Prof. Thomas Enright University ofCalifornia, San Diego Prof. SamR. Evens Rutgers University Prof. Mogens Flensted-Jensen Royal Vet. andAgriculturalUn., Denmark Prof. Jeffrey Fox University ofColorado Mr.MasaakiFurusawa Johns Hopkins University Prof. PierreY. Gail1ard SUNYatBuffalo Dr.DevraGarfmkle Rutgers University, Newark Prof. PaulGerardin Penn State University Mr. DavidGoldberg University ofMaryland Mr. DanielGoldstein University ofChicago Prof. RoeGoodman Rutgers University Mr. WilliamGraham Massachusetts Institute ofTechnology Prof. KennethI.Gross UniversityofVermont Mr. JeffHakim Columbia University Prof. Henryk Hecht University ofUtah Prof.GerritHeckman Katholieke Universiteil, The Netherlands Prof.SigurdurHelgason Massachusetts Institute ofTechnology Prof. Aloysius G. Helminck North CarolinaState University Prof. RebeccaHerb University ofMaryland Prof. Michael J. Heumos Columbia University Prof. RogerHowe Yale University Mr. Jing-song Huang Massachusetts Institute ofTechnology
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