Contents

Introduction to the First Edition xi

Introduction to the Second Edition xvii

Chapter 0. Notation and Preliminaries 1

1. Notation 1

2. Representations of Lie groups 2

3. Linear algebraic and reductive groups 4

Chapter I. Relative Lie Algebra Cohomology 7

1. Lie algebra cohomology 7

2. The Ext functors for (g,£)-modules 9

3. Long exact sequences and Ext 13

4. A vanishing theorem 15

5. Extension to (g, K)-modules 16

6. (g,£,L)-modules.

A Hochschild-Serre spectral sequence in the relative case 19

7. Poincare duality 22

8. The Zuckerman functors 25

Chapter II. Scalar Product, Laplacian and Casimir Element 31

1. Notation and general remarks 31

2. Scalar product 33

3. Special cases 36

4. The bigrading in the bounded symmetric domain case 37

5. Cohomology with respect to square integrable representations 40

6. Spinors and the spin Laplacian 43

7. Vanishing theorems using spinors 47

8. Matsushima's vanishing theorem 50

9. Direct products 54

10. Sharp vanishing theorems 55

Chapter III. Cohomology with Respect to an Induced Representation 59

1. Notation and conventions 59

2. Induced representations and their if-finite vectors 61

3. Cohomology with respect to principal series representations 64

4. Fundamental parabolic subgroups 66

5. Tempered representations 69

6. Representations induced from tempered ones 70

7. Appendix: C°° vectors in certain induced representations 70