**Mathematical Surveys and Monographs**

Volume: 67;
2000;
260 pp;
Softcover

MSC: Primary 22;
Secondary 57

**Print ISBN: 978-1-4704-1225-8
Product Code: SURV/67.S**

List Price: $77.00

AMS Member Price: $61.60

MAA Member Price: $69.30

**Electronic ISBN: 978-1-4704-1294-4
Product Code: SURV/67.E**

List Price: $72.00

AMS Member Price: $57.60

MAA Member Price: $64.80

# Continuous Cohomology, Discrete Subgroups, and Representations of Reductive Groups: Second Edition

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*A. Borel; N. Wallach*

It has been nearly twenty years since the first edition of this work. In the intervening years, there has been immense progress in the use of homological algebra to construct admissible representations and in the study of arithmetic groups. This second edition is a corrected and expanded version of the original, which was an important catalyst in the expansion of the field. Besides the fundamental material on cohomology and discrete subgroups present in the first edition, this edition also contains expositions of some of the most important developments of the last two decades.

#### Readership

Graduate students and research mathematicians working in continuous cohomology.

#### Table of Contents

# Table of Contents

## Continuous Cohomology, Discrete Subgroups, and Representations of Reductive Groups: Second Edition

- Contents vii8 free
- Introduction to the First Edition xi12 free
- Introduction to the Second Edition xvii18 free
- Chapter 0. Notation and Preliminaries 120 free
- Chapter I. Relative Lie Algebra Cohomology 726
- 1. Lie algebra cohomology 726
- 2. The Ext functors for (g,t)-modules 928
- 3. Long exact sequences and Ext 1332
- 4. A vanishing theorem 1534
- 5. Extension to (g,K)-modules 1635
- 6. (g,t,L)-modules. A Hochschild-Serre spectral sequence in the relative case 1938
- 7. Poincaré duality 2241
- 8. The Zuckerman functors 2544

- Chapter II. Scalar Product, Laplacian and Casimir Element 3150
- 1. Notation and general remarks 3150
- 2. Scalar product 3352
- 3. Special cases 3655
- 4. The bigrading in the bounded symmetric domain case 3756
- 5. Cohomology with respect to square integrable representations 4059
- 6. Spinors and the spin Laplacian 4362
- 7. Vanishing theorems using spinors 4766
- 8. Matsushima's vanishing theorem 5069
- 9. Direct products 5473
- 10. Sharp vanishing theorems 5574

- Chapter III. Cohomology with Respect to an Induced Representation 5978
- 1. Notation and conventions 5978
- 2. Induced representations and their K-finite vectors 6180
- 3. Cohomology with respect to principal series representations 6483
- 4. Fundamental parabolic subgroups 6685
- 5. Tempered representations 6988
- 6. Representations induced from tempered ones 7089
- 7. Appendix: C[sup(∞)]vectors in certain induced representations 7089

- Chapter IV. The Langlands Classification and Uniformly Bounded Representations 7594
- 1. Some results of Harish-Chandra 7594
- 2. Some ideas of Casselman 7897
- 3. The Langlands classification (first step) 81100
- 4. The Langlands classification (second step) 84103
- 5. A necessary condition for uniform boundedness 87106
- 6. Appendix: Langlands' geometric lemmas 91110
- 7. Appendix: A lemma on exponential polynomial series 94113

- Chapter V. Cohomology with Coefficients in Π[sub(∞)](G) 97116
- Chapter VI. The Computation of Certain Cohomology Groups 115134
- 0. Translation functors 115134
- 1. Cohomology with respect to minimal non-tempered representations. I 117136
- 2. Cohomology with respect to minimal non-tempered representations. II 120139
- 3. Semi-simple Lie groups with R-rank 1 122141
- 4. The groups SO(n,1) and SU(n,1) 127146
- 5. The Vogan-Zuckerman theorem 134153

- Chapter VII. Cohomology of Discrete Subgroups and Lie Algebra Cohomology 137156
- Chapter VIII. The Construction of Certain Unitary Representations and the Computation of the Corresponding Cohomology Groups 151170
- Chapter IX. Continuous Cohomology and Differentiable Cohomology 169188
- Chapter X. Continuous and Differentiable Cohomology for Locally Compact Totally Disconnected Groups 191210
- Chapter XI. Cohomology with Coefficients in Π[sub(∞)](G): The p-adic Case 211230
- Chapter XII. Differentiable Cohomology for Products of Real Lie Groups and T.D. Groups 225244
- Chapter XIII. Cohomology of Discrete Cocompact Subgroups 233252
- Chapter XIV. Non-cocompact S-arithmetic Subgroups 247266
- Bibliography 253272
- Index 259278