CONTENTS
Preface
Introduction
Part I
1. Measures on homogeneous spaces = 1
Appendix 1. Surjectivity of Cc (G) → Cc (H＼G) = 15
Appendix 2. Surjectivity of Cc (A＼G) → Cc (A＼G／B) = 17
2. The spectral theorem for a family of compact operators = 19
3. Applications of Theorem 2.7 to certain unitary representations = 27
4. The decomposition of L2 ( Γ ＼G) = 32
5. The Selberg trace formula for compact quotients = 38
6. Spinor spaces and spin modules = 45
Appendix 1. More on the spin group = 66
Part II
7. Discrete series multiplicities = 74
Appendix 1. Harish-Chandra's theorem on the discrete series = 99
Appendix 2. The holomorphic discrete series of SL(2,R) = 100
Appendix 3. Dirac operators on homogeneous vector bundles over G／K = 103
Appendix 4. Langlands' technical assumption on Γ = 108
8. Cohomo1ogica1 interpretation of derived functor module multiplicities = 110
Appendix 1. Action of the center of U(g) on Rq j W = 138
Appendix 2. The discrete series as a derived functor module = 143
9. Class 1 representations and spherical functions = 148
Appendix 1. Commutativity of C4 (G) = 167
Appendix 2. Iwasawa decomposition and Harish-Chandra c-function for SL(2,R) = 168
Appendix 3. Spherical functions on SL(2,R) = 170
10. The Matsushima-Murakami formula revisited = 176
11. The Selberg trace formula in the rank 1 case = 179
Appendix 1. Ix ( φ ) for x elliptic = 200
Appendix 2. The trace formula for SL(2,R) = 208
12. The Class 1 spectrum of Γ ＼G／K = 214
Appendix 1. Solution of z'／z＝ ψ = 232
13. The theta transform and its adjoint = 235
Appendix 1. Computation of the measure dn ∞ for SL(2,R) = 259
Appendix 2. Inversion formula for the Mellin transform = 261
14. The Poisson summation formula = 265
Appendix 1. A lemma involving the divisor function σν = 274
15. Eisenstein series for SL(2,R), SL(2,Z) = 278
16. The spectrum of L2 ( Γ ＼G／K) = 290
Appendix 1. Two trivial observations = 315
17. H u ·· ber's formula = 318
18. Cohomological interpretation of limits of discrete series multiplicities = 331
References = 338
