Preface ......................................................... v
Introduction .................................................... 1
Part I Gaussians, Spherical Inversion, and the Heat Kernel
1. Spherical Inversion on SL2(C) ................................ 13
1.1. The Iwasawa Decomposition, Polar Decomposition,
and Characters ............................................ 15
1.1.1. Characters ......................................... 16
1.1.2. K-bi-invariant Functions ........................... 17
1.2. Haar Measures ............................................. 19
1.3. The Harish Transform and the Orbital Integral ............. 23
1.4. The Mellin and Spherical Transforms ....................... 25
1.5. Computation of the Orbital Integral ....................... 28
1.6. Gaussians on G and Their Spherical Transform .............. 32
1.6.1. The Polar Height ................................... 35
1.7. The Polar Haar Measure and Inversion ...................... 37
1.8. Point-Pair Invariants, the Polar Height,
and the Polar Distance .................................... 41
2. The Heat Gaussian and Heat Kernel ........................... 45
2.1. Dirac Families of Gaussians ............................... 45
2.1.1. Scaling ............................................ 46
2.1.2. Decay Property ..................................... 48
2.2. Convolution, Semigroup, and Approximations Properties ..... 49
2.2.1. Approximation Properties ........................... 51
2.3. Complexifying t and the Null Space of Heat Convolution .... 54
2.4. The Casimir Operator ...................................... 55
2.4.1. Scaling ............................................ 62
2.5. The Heat Equation ......................................... 63
2.5.1. Scaling ............................................ 65
3. QED, LEG, Transpose, and Casimir ............................ 67
3.1. Growth and Decay, QED∞ and LEG∞ ........................... 67
3.2. Casimir, Transpose, and Harmonicity ....................... 70
3.3. DUTIS ..................................................... 76
3.4. Heat and Casimir Eigenfunctions ........................... 78
Part II Enter Г: The General Trace Formula
4. Convergence and Divergence of the Selberg Trace ............. 85
4.1. The Hermitian Norm ........................................ 86
4.2. Divergence for Standard Cuspidal Elements ................. 89
4.2.1.Cuspidal and Parabolic Subgroups .................... 89
4.3. Convergence for the Other Elements of Г ................... 92
5. The Cuspidal and Noncuspidal Traces ......................... 97
5.1. Some Group Theory ......................................... 98
5.1.1. Conjugacy Classes ................................. 101
5.2. The Double Trace and its Decomposition ................... 102
5.3. Explicit Determination of the Noncuspidal Terms .......... 106
5.3.1. The Volume Computation ............................ 107
5.3.2. The Orbital Integral .............................. 108
5.4. Cuspidal Conjugacy Classes ............................... 110
Part III The Heat Kernel on Г\G/K
6. The Fundamental Domain ..................................... 117
6.1. SL2(C) and the Upper Half-Space H3 ....................... 118
6.2. Fundamental Domain and Г∞ ................................ 121
6.3. Finiteness Properties .................................... 124
6.4. Uniformities in Lemma 6.2.3 .............................. 130
6.5. Integration on Г\G/K ..................................... 131
6.6. Other Fundamental Domains ................................ 133
7. Г-Periodization of the Heat Kernel ......................... 135
7.1. The Basic Estimate ....................................... 135
7.1.1. Convolution ....................................... 136
7.2. Heat Convolution and Eigenfunctions on Г\G/K ............. 140
7.3. Casimir on Г\G/K ......................................... 145
7.4. Measure-Theoretic Estimate for Convolution on T\G ........ 147
7.5. Asymptotic Behavior of KГt for t → ∞ ...................... 149
8. Heat Kernel Convolution on L2cusp (Г\G/K) ................... 151
8.1. General Criteria for Compactness ......................... 152
8.2. Estimates for the (Г - Г∞)-Periodization ................. 155
8.3. Fourier Series for the Г"U, Г∞-Periodizations of
Gaussians ................................................ 157
8.3.1. Preliminaries: The Г"U and Г∞-Periodizations ....... 157
8.3.2. The Fourier Series ................................ 158
8.4. The Convolution Cuspidal Estimate ........................ 160
8.5. Application to the Heat Kernel ........................... 161
Part IV Fourier-Eisenstein Eigenfunction Expansions
9. The Tube Domain for ....................................... 167
9.1. Differential-Geometric Aspects ........................... 167
9.2. The Tube of  R and its Boundary Relation with ∂R ...... 169
9.3. The -Normalizer of Г ................................... 171
9.4. Totally Geodesic Surface in H3 ........................... 172
9.4.1. The Half-Plane H2j ................................ 173
9.5. Some Boundary Behavior of R in H3 Under Г .............. 175
9.5.1. The Faces  i of and their Boundaries ......... 175
9.5.2. H-triangle ........................................ 176
9.5.3. Isometries of ................................. 178
9.6. The Group Г and a Basic Boundary Inclusion ............... 180
9.7. The Set  , its Boundary Behavior, and the Tube  ....... 181
9.8. Tilings .................................................. 182
9.8.1. Coset Representatives ............................. 184
9.9. Truncations .............................................. 185
10. The u\U-Fourier Expansion of Eisenstein Series ............ 191
10.1.Our Goal: The Eigenfunction Expansion .................... 191
10.2.Epstein and Eisenstein Series ............................ 193
10.3.The K-Bessel Function .................................... 197
10.3.1. Gamma Function Identities ........................ 199
10.3.2. Differential and Difference Relations ............ 201
10.4.Functional Equation of the Dedekind Zeta Function ........ 202
10.5.The Bessel-Fourier Гu\U-Expansion of Eisenstein
Series ................................................... 206
10.5.The Constant Term ........................................ 211
10.6.Estimates in Vertical Strips ............................. 213
10.7.The Volume-Residue Formula ............................... 216
10.8.The Integral over and Orthogonalities ................. 218
11. Adjointness Formula and the Г\G-Eigenfunction Expansion ... 223
11.1.Haar Measure and the Mellin Transform .................... 224
11.1.1.Appendix on Fourier Inversion ..................... 226
11.2.Adjointness Formula and the Constant Term ................ 229
11.2.1 Adjointness Formula ............................... 230
11.3.The Eisenstein Coefficient E * f and the Expansion
for f ∈ Cc∞ (Г\G/K) ....................................... 232
11.4.The Heat Kernel Eigenfunction Expansion .................. 237
Part V The Eisenstein-Cuspidal Affair
12.The Eisenstein Y-Asymptotics ............................... 243
12.1.The Improper Integral of Eigenfunction Expansion
over Г\G ................................................. 243
12.1.1.L2-Cuspidal Trace ................................. 244
12.2.Green's Theorem on ≤ Υ ................................ 247
12.3.Application to Eisenstein Functions ...................... 251
12.4.The Constant-Term Integral Asymptotics ................... 255
12.4.1. Appendix ......................................... 257
12.5.The Nonconstant-Term Error Estimate ...................... 258
13.The Cuspidal Trace Υ-Asymptotics ........................... 261
13.1.The Nonregular Cuspidal Integral over ≤ Υ ............. 262
13.2.Asymptotic Expansion of the Nonregular Cuspidal Trace .... 267
13.3.The Regular Cuspidal Integral over ≤ Υ ................ 272
13.4.Nonspecial Regular Cuspidal Asymptotics .................. 275
13.5.Action of the Special Subset ............................. 277
13.6.Special Regular Cuspidal Asymptotics ..................... 280
14.Analytic Evaluations ....................................... 287
14.1.Partial Sums Asymptotics for  and the Euler Constant ... 287
14.2.Estimates Using Lattice-Point Counting ................... 290
14.3.Partial-Sums Asymptotics for (i) and the Euler
Constant ................................................. 292
14.4.The Hurwitz Constant ..................................... 296
14.4.1.The Complex Case, with Z[i] ....................... 297
14.4.2.Average of the Hurwitz Constant ................... 298
14.5.∫0∞ fφ(r) rh (h)dr when φ = gt ............................ 301
14.6.Evaluation of C`Y0 and C1 ................................. 303
14.7.The Theta Inversion Formula .............................. 308
References .................................................... 311
Index ......................................................... 317
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