viii CONTENT S
8. The Poisson summation formula 108
9. Tate zeta functions 119
9.1. Definition and meromorphic continuation 119
9.2. Functional equation and behavior at s = 1 124
10. Intertwining operators and matrix coefficients 128
10.1. Linear algebra 129
10.2. Representation theory 134
10.3. Orthogonality of matrix coefficients 140
11. The discrete series of GL2(R) 151
11.1. if-type decompositions 151
11.2. Representations of 0(2) 155
11.3. K-type decomposition of an induced representation 156
11.4. The (s, If)-modules (VX)K 162
11.5. Classification of irreducible (g, K)-modules for GL2(R) 165
11.6. A detailed look at the Lie algebra action 181
11.7. Characterization of the weight k discrete series of GL2(R) 186
Groundwork
12. Cusp forms as elements of
LQ(UJ)
195
12.1. From Dirichlet characters to Hecke characters 195
12.2. From cusp forms to functions on G(A) 197
12.3. Comparison of classical and adelic Fourier coefficients 198
12.4. Characterizing the image of Sk(iV,u/) in LQ(U) 201
13. Construction of the test function / 206
13.1. The non-archimedean component of / 206
13.2. Spectral properties of R(f) 213
14. Explicit computations for /
k
and /* 221
The Trace Formula
15. Introduction to the trace formula for R(f) 227
16. Terms that contribute to K(x, y) 230
17. Truncation of the kernel 231
18. Bounds for £
7
l/fcrSfe)! 240
19. Integrability of kj(g) 248
20. The hyperbolic terms as weighted orbital integrals 259
21. Simplifying the unipotent term 270
22. The trace formula 276
Computation of the Trace
23. The identity term 279
24. The hyperbolic terms 280
24.1. A lemma about orbital integrals 280
24.2. The archimedean orbital integral and weighted orbital integral 281
24.3. Simplification of the hyperbolic term 283
24.4. Calculation of the local orbital integrals 283
24.5. The global hyperbolic result 286
25. The unipotent term 288
25.1. Explicit evaluation of the zeta integral at oo 288
25.2. Computation of the non-archimedean local zeta functions 291
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