Heat Kernel and Theta Inversion on SL2(C) (eBook)

Preface 5
Contents 6
Introduction 10
Spherical Inversion 11
Fourier and Eigenfunction Expansions 11
Gaussians and the Trace Formula 12
The General Path to Theta Inversion 12
Zetas 14
Ladders 15
Connection with Analytic Number Theory 15
Connections with Geometry 16
Towers of Ladders 18
Part I: Gaussians, Spherical Inversion, and the Heat Kernel 19
Chapter 2 20
Spherical Inversion on SL2(C) 20
1.1 The Iwasawa Decomposition, Polar Decomposition,and Characters 22
1.1.1 Characters 23
1.1.2 K-bi-invariant Functions 24
1.2 Haar Measures 26
1.3 The Harish Transform and the Orbital Integral 30
1.4 The Mellin and Spherical Transforms 32
1.5 Computation of the Orbital Integral 35
1.6 Gaussians on G and Their Spherical Transform 39
1.6.1 The Polar Height 42
1.7 The Polar Haar Measure and Inversion 44
1.8 Point-Pair Invariants, the Polar Height,and the Polar Distance 48
Chapter 3 51
The Heat Gaussian and Heat Kernel 51
2.1 Dirac Families of Gaussians 51
2.1.1 Scaling 52
2.1.2 Decay Property 54
2.2 Convolution, Semigroup, and Approximations Properties 55
2.2.1 Approximation Properties 57
2.3 Complexifying t and the Null Space of Heat Convolution 60
2.4 The Casimir Operator 61
2.4.1 Scaling 68
2.5 The Heat Equation 69
2.5.1 Scaling 71
Chapter 4 73
QED, LEG, Transpose, and Casimir 73
3.1 Growth and Decay, QEDinfinity and LEGinfinity 73
3.2 Casimir, Transpose, and Harmonicity 76
3.3 DUTIS 82
3.4 Heat and Casimir Eigenfunctions 84
Onward 87
Part II: Enter cap gamma: The General Trace Formula 88
Chapter 5 89
Convergence and Divergence of the Selberg Trace 89
4.1 The Hermitian Norm 90
4.2 Divergence for Standard Cuspidal Elements 93
4.2.1 Cuspidal and Parabolic Subgroups 93
4.3 Convergence for the Other Elements of Cap Gamma 96
What Next? 99
Chapter 6 100
The Cuspidal and Noncuspidal Traces 100
5.1 Some Group Theory 101
5.1.1 Conjugacy Classes 104
5.2 The Double Trace and its Decomposition 105
5.3 Explicit Determination of the Noncuspidal Terms 109
5.3.1 The Volume Computation 110
5.3.2 The Orbital Integral 111
5.4 Cuspidal Conjugacy Classes 113
Part III: The Heat Kernal on cap gamma/G/K 118
Chapter 7 119
The Fundamental Domain 119
6.1 SL2(C) and the Upper Half-Space H3 120
6.2 Fundamental Domain and cap gamma infinity 123
6.3 Finiteness Properties 126
6.4 Uniformities in Lemma 6.2.3 132
6.5 Integration on cap gamma/G/K 133
6.6 Other Fundamental Domains 135
Chapter 8 137
Cap Gamma-Periodization of the Heat Kernel 137
7.1 The Basic Estimate 137
7.1.1 Convolution 138
7.2 Heat Convolution and Eigenfunctions on Cap Gamma/G/K 142
7.3 Casimir on Cap Gamma/G/K 147
7.4 Measure-Theoretic Estimate for Convolution on Cap Gamma/G 149
7.5 Asymptotic Behavior of KCap Gammat for t.Infinity 151
Chapter 9 153
Heat Kernel Convolution on L2cusp (cap gamma/G/K) 153
8.1 General Criteria for Compactness 154
8.2 Estimates for the (cap gamma cap gamma infinity)-Periodization 157
8.3 Fourier Series for the cap gamma"U , cap gamma infinity-Periodizations of Gaussians 159
8.3.1 Preliminaries: The cap gamma"U and cap gamma infinity-Periodizations 159
8.4 The Convolution Cuspidal Estimate 162
8.5 Application to the Heat Kernel 163
Part IV: Fourier - Eisenstein Eigenfunction Expansions 166
Chapter 10 167
The Tube Domain for cap gamma infinity 167
9.1 Differential-Geometric Aspects 167
9.2 The Tube of FR and its Boundary Relation with deltaFR 169
9.3 The F-Normalizer of cap gamma 171
9.4 Totally Geodesic Surface in H3 172
9.4.1 The Half-Plane H2j 173
9.5 Some Boundary Behavior of F in H3 Under cap gamma 175
9.5.1 The Faces Bi of F and their Boundaries 175
9.5.2 H-triangle 176
9.5.3 Isometries of F 178
9.6 The Group ˜ and a Basic Boundary Inclusion 180
9.7 The Set I, its Boundary Behavior, and the Tube T 181
9.8 Tilings 182
9.8.1 Coset Representatives 184
9.9 Truncations 185
Chapter 11 190
The Cap GammaU/U-Fourier Expansion of Eisenstein Series 190
10.1 Our Goal: The Eigenfunction Expansion 190
10.2 Epstein and Eisenstein Series 192
10.3 The K-Bessel Function 196
10.3.1 Gamma Function Identities 198
10.3.2 Differential and Difference Relations 200
10.4 Functional Equation of the Dedekind Zeta Function 201
10.5 The Bessel–Fourier Cap GammaU/U-Expansion of Eisenstein Series 205
10.5.1 The Constant Term 210
10.6 Estimates in Vertical Strips 212
10.7 The Volume–Residue Formula 215
10.8 The Integral over F and Orthogonalities 217
Chapter 12 222
Adjointness Formulaand the cap gamma/G-Eigenfunction Expansion 222
11.1 Haar Measure and the Mellin Transform 223
11.1.1 Appendix on Fourier Inversion 225
11.2 Adjointness Formula and the Constant Term 228
11.2.1 Adjointness Formula 229
11.3 The Eisenstein Coefficient E * f and the Expansion for f member of C infinity c (cap gamma/G/K) 231
11.4 The Heat Kernel Eigenfunction Expansion 236
Part V: The Eisenstein - Cuspidal Affair 240
Chapter 13 241
The Eisenstein Y-Asymptotics 241
12.1 The Improper Integral of Eigenfunction Expansion over Cap Gamma/G 241
12.1.1 L2-Cuspidal Trace 242
12.2 Green’s Theorem on FLessthen EqualY 245
12.3 Application to Eisenstein Functions 249
12.4 The Constant-Term Integral Asymptotics 253
12.4.1 Appendix 255
12.5 The Nonconstant-Term Error Estimate 256
Chapter 14 258
The Cuspidal Trace Y-Asymptotics 258
13.1 The Nonregular Cuspidal Integral over F less than equal to Y 259
13.2 Asymptotic Expansion of the Nonregular Cuspidal Trace 264
13.3 The Regular Cuspidal Integral over F lass than equal to Y 269
13.4 Nonspecial Regular Cuspidal Asymptotics 272
13.5 Action of the Special Subset 274
13.6 Special Regular Cuspidal Asymptotics 277
Chapter 15 284
Analytic Evaluations 284
14.1 Partial Sums Asymptotics for ZetaQ and the Euler Constant 284
14.2 Estimates Using Lattice-Point Counting 287
14.3 Partial-Sums Asymptotics for ZetaQ(i) and the Euler Constant 289
14.4 The Hurwitz Constant 293
14.4.1 The Complex Case, with Z[i] 294
14.4.2 Average of the Hurwitz Constant 295
14.5 f 80 f phi(r)rh(r)dr when phi = gt 298
14.6 Evaluation of C PrimeYo and C1 300
14.7 The Theta Inversion Formula 305
References 307
Index 312