Harmonic Analysis on Reductive, p-adic Groups

294 Pages·2011·2.44 MB·English

by  Robert S. Doran, Paul J., Jr. Sally, Loren Spice (ed.)

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
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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
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Dedicated to the memory of
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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
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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