**University Lecture Series**

Volume: 16;
1999;
97 pp;
Softcover

MSC: Primary 22;

Print ISBN: 978-0-8218-2025-4

Product Code: ULECT/16

List Price: $25.00

Individual Member Price: $20.00

**Electronic ISBN: 978-1-4704-2165-6
Product Code: ULECT/16.E**

List Price: $25.00

Individual Member Price: $20.00

# Admissible Invariant Distributions on Reductive $p$-adic Groups

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*Harish-Chandra; Stephen DeBacker; Paul J. Sally, Jr.*

Harish-Chandra presented these lectures on admissible invariant distributions for \(p\)-adic groups at the Institute for Advanced Study in the early 1970s. He published a short sketch of this material as his famous “Queen's Notes”. This book, which was prepared and edited by DeBacker and Sally, presents a faithful rendering of Harish-Chandra's original lecture notes.

The main purpose of Harish-Chandra's lectures was to show that the character of an irreducible admissible representation of a connected reductive \(p\)-adic group \(G\) is represented by a locally summable function on \(G\). A key ingredient in this proof is the study of the Fourier transforms of distributions on \(\mathfrak g\), the Lie algebra of \(G\). In particular, Harish-Chandra shows that if the support of a \(G\)-invariant distribution on \(\mathfrak g\) is compactly generated, then its Fourier transform has an asymptotic expansion about any semisimple point of \(\mathfrak g\).

Harish-Chandra's remarkable theorem on the local summability of characters for \(p\)-adic groups was a major result in representation theory that spawned many other significant results. This book presents, for the first time in print, a complete account of Harish-Chandra's original lectures on this subject, including his extension and proof of Howe's Theorem.

In addition to the original Harish-Chandra notes, DeBacker and Sally provide a nice summary of developments in this area of mathematics since the lectures were originally delivered. In particular, they discuss quantitative results related to the local character expansion.

#### Table of Contents

# Table of Contents

## Admissible Invariant Distributions on Reductive $p$-adic Groups

- Cover Cover11 free
- Title v6 free
- Copyright vi7 free
- Contents vii8 free
- Preface ix10 free
- Introduction 116 free
- Part I. Fourier transforms on the Lie algebra 520 free
- 1. The mapping f [omitted] ø[(sub)f] 520
- 2. Some results about neighborhoods of semisimple elements 1328
- 3. Proof of Theorem 3.1 1732
- 4. Some consequences of Theorem 3.1 3045
- 5. Proof of Theorem 5.11 3247
- 6. Application of the induction hypothesis 3853
- 7. Reformulation of the problem and completion of the proof 4257
- 8. Some results on Shalika's germs 4863
- 9. Proof of Theorem 9.6 5166

- Part II. An extension and proof of Howe's Theorem 5570
- Part III. Theory on the group 7186
- 14. Representations of compact groups 7186
- 15. Admissible distributions 7489
- 16. Statement of the main results 7489
- 17. Recapitulation of Howe's theory 7590
- 18. Application to admissible invariant distributions 7691
- 19. First step of the reduction from G to M 7994
- 20. Second step 8297
- 21. Completion of the proof 8499
- 22. Formal degree of a supercuspidal representation 86101

- Bibliography 91106
- List of Symbols 95110
- Index 97112 free
- Back Cover Back Cover1113

#### Readership

Graduate students and research mathematicians interested in representations of Lie groups.

#### Reviews

This branch of representation theory is particularly hard going. In addition, Harish-Chandra's notes were extremely terse, and were tucked away in an obscure source … the authors have done us all a favour by writing a complete modern treatment which should prove more accessible (in both senses) to modern PhD students.

-- Bulletin of the London Mathematical Society

DeBacker and Sally are to be commended for their excellent work.

-- Mathematical Reviews