Harmonic Analysis on Reductive, p-adic Groups PDF
Preview Harmonic Analysis on Reductive, p-adic Groups
C ONTEMPORARY M ATHEMATICS 543 Harmonic Analysis on Reductive, p-adic Groups AMS Special Session on Harmonic Analysis and Representations of Reductive, p-adic Groups January 16, 2010 San Francisco, CA Robert S. Doran Paul J. Sally, Jr. Loren Spice Editors American Mathematical Society conm-543-doran6-cov.indd 1 3/7/11 4:26 PM Harmonic Analysis on Reductive, p-adic Groups This page intentionally left blank C ONTEMPORARY M ATHEMATICS 543 Harmonic Analysis on Reductive, p-adic Groups AMS Special Session on Harmonic Analysis and Representations of Reductive, p-adic Groups January 16, 2010 San Francisco, CA Robert S. Doran Paul J. Sally, Jr. Loren Spice Editors American Mathematical Society Providence, Rhode Island Editorial Board Dennis DeTurck, managing editor George Andrews Abel Klein Martin J. Strauss 2010 Mathematics Subject Classification. Primary 22E50, 11F70, 22E35, 20G25,20C33, 20G40, 20G05. Library of Congress Cataloging-in-Publication Data Harmonic analysis on reductive, p-adic groups : AMS special session, harmonic analysis and representationsofreductive,p-adicgroups,January16,2010,SanFrancisco,California/RobertS. Doran,PaulJ.Sally,Jr.,LorenSpice,editors. p.cm. —(Contemporarymathematics;v.543) Includesbibliographicalreferences. ISBN978-0-8218-4985-9(alk.paper) 1. Representations of Lie groups—Congresses. 2. p-adic groups—Congresses. 3. Harmonic analysis—Congresses. I.Doran,RobertS.,1937– II.Sally,Paul. III.Spice,Loren,1981– QA387.H37 2011 512(cid:2).482—dc22 2011000976 Copying and reprinting. Materialinthisbookmaybereproducedbyanymeansfor edu- cationaland scientific purposes without fee or permissionwith the exception ofreproduction by servicesthatcollectfeesfordeliveryofdocumentsandprovidedthatthecustomaryacknowledg- 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 commercialuseofmaterialshouldbeaddressedtotheAcquisitionsDepartment,AmericanMath- ematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. Requests can [email protected]. Excludedfromtheseprovisionsismaterialinarticlesforwhichtheauthorholdscopyright. In suchcases,requestsforpermissiontouseorreprintshouldbeaddresseddirectlytotheauthor(s). (Copyrightownershipisindicatedinthenoticeinthelowerright-handcornerofthefirstpageof eacharticle.) (cid:2)c 2011bytheAmericanMathematicalSociety. Allrightsreserved. TheAmericanMathematicalSocietyretainsallrights exceptthosegrantedtotheUnitedStatesGovernment. Copyrightofindividualarticlesmayreverttothepublicdomain28years afterpublication. ContacttheAMSforcopyrightstatusofindividualarticles. PrintedintheUnitedStatesofAmerica. (cid:2)∞ Thepaperusedinthisbookisacid-freeandfallswithintheguidelines establishedtoensurepermanenceanddurability. VisittheAMShomepageathttp://www.ams.org/ 10987654321 161514131211 Dedicated to the memory of Joseph Shalika This page intentionally left blank Contents Preface ix List of Participants xi Toward a Mackey formula for compact restriction of character sheaves Pramod N. Achar and Clifton L. R. Cunningham 1 Supercuspidal characters of SL over a p-adic field 2 Jeffrey D. Adler, Stephen DeBacker, Paul J. Sally, Jr., and Loren Spice 19 Geometric structure in the representation theory of reductive p-adic groups II Anne-Marie Aubert, Paul Baum, and Roger Plymen 71 The construction of Hecke algebras associated to a Coxeter group Bill Casselman 91 Distinguished supercuspidal representations of SL 2 Jeffrey Hakim and Joshua M. Lansky 103 Twisted Levi sequences and explicit types on Sp 4 Ju-Lee Kim and Jiu-Kang Yu 135 Regularity and distinction of supercuspidal representations Fiona Murnaghan 155 Patterns in branching rules for irreducible representations of SL (k), for k a 2 p-adic field Monica Nevins 185 Parametrizing nilpotent orbits in p-adic symmetric spaces Ricardo Portilla 201 An integration formula of Shahidi Steven Spallone 215 Managing metaplectiphobia: Covering p-adic groups Martin H. Weissman 237 vii This page intentionally left blank Preface This volume contains the proceedings of the AMS Special Session Harmonic analysis and representations of reductive, p-adic groups, which was held in San Francisco,CA,onJanuary16,2010,duringthatyear’sJointMathematicsMeetings. The purpose of the session, as amplified in this volume, was to bring together the mostup-to-dateperspectivesonthemanyexotictoolsthathavebeendevelopedfor the study of p-adic harmonic analysis. (To be explicit, we note here that “p-adic harmonic analysis” will always refer to the analysis of complex-valued functions and distributions on p-adic groups; we do not consider p-adic-valued functions in this volume.) Our subject was born from the idea that one should be able to develop a theory of harmonic analysis for functions and distributions on p-adic groups to parallel the corresponding theory for real groups. Harish-Chandra articulated this in his p-adic Lefschetz principle, which was exemplified throughout his work; and Langlands’s suite of conjectures makes precise many of the ways in which the two sorts of analysis can be regarded as manifestations of a common theme. With that said, though the ends are, in some cases, the same, the techniques in use have often been extremely different, and it can be hard to keep pace with the state of the art in p-adic harmonic analysis. It is therefore our pleasure to present here both expository and research articles by an assortment of expert researchers, both introducing and reporting on the latest developments along a number of fronts. Twoparticularlyexcitingthemesaretheapplicationsofgeometrytorepresentation theory (cf. Achar–Cunningham and Aubert–Baum–Plymen) and generalisations of harmonic analysis onlinear groups tothesettings ofsymmetric spaces (cf.Hakim– Lansky, Murnaghan, and Portilla) and covering groups (cf. Weissman). Most papers are based closely on their authors’ talks. In many cases authors have elected to focus on a specific example rather than a general theory; constant reference is made throughout the volume to the most familiar case of the 2×2 special linear group, so that the reader can glean deep general information about thesubjectwithoutfearofgettinglostinstructuretheory. Wearealsofortunateto havetwofurthercontributedarticles, bothwrittenspecificallyforthisvolume: one by Bill Casselman, on “The construction of Hecke algebras associated to Coxeter groups”; and another by Marty Weissman, on “Managing metaplectiphobia”. Theeffortputinbothbytheauthors, toassemble theexcellentpapersthatwe present, and the referees, who picked through each one with a fine-toothed comb, has been considerable; the editors are pleased to take this opportunity to express ourgratitudetothem,andtoalltheparticipantsinthisspecialsession, formaking the present volume possible. We also thank Sergei Gel’fand, Christine Thivierge, ix
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