**CBMS Regional Conference Series in Mathematics**

Volume: 59;
1985;
76 pp;
Softcover

MSC: Primary 22;
Secondary 20

**Print ISBN: 978-0-8218-0709-5
Product Code: CBMS/59**

List Price: $27.00

Individual Price: $21.60

**Electronic ISBN: 978-1-4704-2420-6
Product Code: CBMS/59.E**

List Price: $25.00

Individual Price: $20.00

# Harish-Chandra Homomorphisms for \(\mathfrak p\)-Adic Groups

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*Roger Howe*

A co-publication of the AMS and CBMS

This book introduces a systematic new approach to the construction and
analysis of semisimple \(p\)-adic groups. The basic construction
presented here provides an analogue in certain cases of the
Harish-Chandra homomorphism, which has played an essential role in the
theory of semisimple Lie groups. The book begins with an overview of
the representation theory of GL\(_n\) over finite groups. The author
then explicitly establishes isomorphisms between certain convolution
algebras of functions on two different groups. Because of the form of
the isomorphisms, basic properties of representations are preserved,
thus giving a concrete example to the correspondences predicted by the
general philosphy of Langlands.

The first chapter, suitable as an introduction for graduate students,
requires only a basic knowledge of representation theory of finite
groups and some familiarity with the general linear group and the
symmetric group. The later chapters introduce researchers in the
field to a new method for the explicit construction and analysis of
representations of \(p\)-adic groups, a powerful method clearly capable of
extensive further development.

#### Readership

#### Table of Contents

# Table of Contents

## Harish-Chandra Homomorphisms for $\mathfrak p$-Adic Groups

- Cover Cover11 free
- Title iii4 free
- Copyright iv5 free
- Contents vii8 free
- Preface ix10 free
- Chapter 1. A Hecke Algebra Approach to the Representations of GL[sub(n)](F[sub(q)]) 114 free
- Chapter 2. Hecke Algebras for GL[sub(n)] over Local Fields: Introduction 1326
- Chapter 3. The Harish-Chandra Homomorphism in the Unramif iedAnisotropic Case 2942
- Appendix 1. Plancherel Measure and Hecke Algebras 5972
- Appendix 2. The Representation Ind[sup(G)][sub(B)]1 6578
- Appendix 3. Cuspidal Representations of GL[sub(n)](F[sub(q)]) 6982
- References 7588
- Back Cover Back Cover190