**Contemporary Mathematics**

Volume: 543;
2011;
277 pp;
Softcover

MSC: Primary 22; 11; 20;

Print ISBN: 978-0-8218-4985-9

Product Code: CONM/543

List Price: $104.00

Individual Member Price: $83.20

**Electronic ISBN: 978-0-8218-8222-1
Product Code: CONM/543.E**

List Price: $104.00

Individual Member Price: $83.20

# Harmonic Analysis on Reductive, $p$-adic Groups

Share this page *Edited by *
*Robert S. Doran; Paul J. Sally, Jr.; Loren Spice*

This volume contains the proceedings of the AMS Special Session on Harmonic Analysis and Representations of Reductive, \(p\)-adic Groups, which was held on January 16, 2010, in San Francisco, California.

One of the original guiding philosophies of harmonic analysis on \(p\)-adic groups was Harish-Chandra's Lefschetz principle, which suggested a strong analogy with real groups. From this beginning, the subject has developed a surprising variety of tools and applications. To mention just a few, Moy-Prasad's development of Bruhat-Tits theory relates analysis to group actions on locally finite polysimplicial complexes; the Aubert-Baum-Plymen conjecture relates the local Langlands conjecture to the Baum-Connes conjecture via a geometric description of the Bernstein spectrum; the \(p\)-adic analogues of classical symmetric spaces play an essential role in classifying representations; and character sheaves, originally developed by Lusztig in the context of finite groups of Lie type, also have connections to characters of \(p\)-adic groups.

The papers in this volume present both expository and research articles on these and related topics, presenting a broad picture of the current state of the art in \(p\)-adic harmonic analysis. The concepts are liberally illustrated with examples, usually appropriate for an upper-level graduate student in representation theory or number theory. The concrete case of the two-by-two special linear group is a constant touchstone.

#### Table of Contents

# Table of Contents

## Harmonic Analysis on Reductive, $p$-adic Groups

- Contents vii8 free
- Preface ix10 free
- List of Participants xi12 free
- Toward a Mackey formula for compact restriction of character sheaves 114 free
- Supercuspidal characters of SL2 over a p-adic field 1932
- 1. Introduction 2033
- 2. Field extensions 2639
- 3. Tori 2740
- 4. A principal-value integral 3043
- 5. The building and filtrations 3245
- 6. Haar measure 3447
- 7. Duality, Fourier transforms, and orbital integrals 3548
- 8. Unrefined minimal K-types 3750
- 9. Representations of depth zero 3851
- 10. Representations of positive depth 3952
- 11. Parametrization of supercuspidal representations 4356
- 12. Inducing representations 4457
- 13. Murnaghan–Kirillov theory 4861
- 14. ‘Ordinary’ supercuspidal characters 5265
- 15. ‘Exceptional’ supercuspidal characters 6477
- References 6780

- Geometric structure in the representation theory of reductive p-adic groups II 7184
- The construction of Hecke algebras associated to a Coxeter group 91104
- Distinguished supercuspidal representations of SL2 103116
- 1. Introduction 103116
- 2. Elliptic tori 105118
- 3. Involutions of SL2 106119
- 4. Multiplicity constants 111124
- 5. Supercuspidal representations 112125
- 6. Distinguished toral supercuspidal representations 115128
- 7. Distinguished depth-zero supercuspidal representations 117130
- Appendix A. The building and the Moy-Prasad groups 121134
- References 133146

- Twisted Levi sequences and explicit types on Sp4 135148
- Regularity and distinction of supercuspidal representations 155168
- Patterns in branching rules for irreducible representations of SL2(k), for k a p-adic field 185198
- Parametrizing nilpotent orbits in p-adic symmetric spaces 201214
- An integration formula of Shahidi 215228
- Managing metaplectiphobia: Covering p-adic groups 237250

#### Readership

Graduate students and research mathematicians interested in representations of \(p\)-adic groups.