Continuous cohomology, discrete subgroups, and representations of reductive groups

Contents......Page 3
Introduction to the First Edition......Page 7
Introduction to the Second Edition......Page 13
1. Notation......Page 15
2. Representations of Lie groups......Page 16
3. Linear algebraic and reductive groups......Page 18
1. Lie algebra cohomology......Page 21
2. The Ext functors for (g, t)-modules......Page 23
3. Long exact sequences and Ext......Page 27
4. A vanishing theorem......Page 29
5. Extension to (g, K)-modules......Page 30
6. (g,t, L)-modules. A Hochschild-Serre spectral sequence in the relative case......Page 33
7. Poincaré duality......Page 36
8. The Zuckerman functors......Page 39
1. Notation and general remarks......Page 45
2. Scalar product......Page 47
3. Special cases......Page 50
4. The bigrading in the bounded symmetric domain case......Page 51
5. Cohomology with respect to square integrable representations......Page 54
6. Spinors and the spin Laplacian......Page 57
7. Vanishing theorems using spinors......Page 61
8. Matsushima's vanishing theorem......Page 64
9. Direct products......Page 68
10. Sharp vanishing theorems......Page 69
1. Notation and conventions......Page 73
2. Induced representations and their K-finite vectors......Page 75
3. Cohomology with respect to principal series representations......Page 78
4. Fundamental parabolic subgroups......Page 80
5. Tempered representations......Page 83
7. Appendix: C^\infty vectors in certain induced representations......Page 84
1. Some results of Harish-Chandra......Page 89
2. Some ideas of Casselman......Page 92
3. The Langlands classification (first step)......Page 95
4. The Langlands classification (second step)......Page 98
5. A necessary condition for uniform boundedness......Page 101
6. Appendix: Langlands' geometric lemmas......Page 105
7. Appendix: A lemma on exponential polynomial series......Page 108
1. Preliminaries......Page 111
3. A vanishing theorem for the class \Pi_\infty(G)......Page 114
4. Cohomology with coefficients in the Steinberg representation......Page 117
5. H^1 and the topology of E(G)......Page 121
6. A more detailed examination of first cohomology......Page 124
0. Translation functors......Page 129
1. Cohomology with respect to minimal non-tempered representations. I......Page 131
2. Cohomology with respect to minimal non-tempered representations. II......Page 134
3. Semi-simple Lie groups with R-rank 1......Page 136
4. The groups SO(n, 1) and SU(n, 1)......Page 141
5. The Vogan-Zuckerman theorem......Page 148
1. Manifolds......Page 151
2. Discrete subgroups......Page 153
3. \Gamma cocompact, E a unitary \Gamma-module......Page 156
4. G semi-simple, \Gamma cocompact, E a unitary \Gamma-module......Page 159
5. \Gamma cocompact, E a G-module......Page 161
6. G semi-simple, \Gamma cocompact, E a G-module......Page 163
1. The oscillator representation......Page 165
2. The decomposition of the restriction of the oscillator representation to certain subgroups......Page 169
3. The theta distributions......Page 175
4. The reciprocity formula......Page 178
5. The imbedding of V_l into L^2(\Gamma \ G)......Page 179
Introduction......Page 183
1. Continuous cohomology for locally compact groups......Page 184
2. Shapiro's lemma......Page 189
3. Hausdorff cohomology......Page 191
4. Spectral sequences......Page 192
5. Differentiate cohomology and continuous cohomology for Lie groups......Page 194
6. Further results on differentiable cohomology......Page 198
1. Continuous and smooth cohomology......Page 205
2. Cohomology of reductive groups and buildings......Page 210
3. Representations of reductive groups......Page 213
4. Cohomology with respect to irreducible admissible representations......Page 214
5. Forgetting the topology......Page 219
6. Cohomology of products......Page 221
1. Some results of Harish-Chandra......Page 225
2. The Langlands classification (p-adic case)......Page 229
3. Uniformly bounded representations and \Pi_\infty(G)......Page 232
4. Another proof of the non-unitarizability of the V_J's......Page 235
0. Homological algebra over idempotented algebras......Page 239
1. Differentiable cohomology......Page 240
2. Modules of K-finite vectors......Page 242
3. Cohomology of products......Page 244
1. Subgroups of products of Lie groups and t.d. groups......Page 247
2. Products of reductive groups......Page 250
3. Irreducible subgroups of semi-simple groups......Page 253
4. The \Gamma-module E is the restriction of a rational G-module......Page 257
2. Stable cohomology......Page 261
3. The use of L^2 cohomology......Page 263
4. S-arithmetic subgroups......Page 265
Bibliography......Page 267
Index......Page 273