The Heat Kernel and Theta Inversion on SL2(C) / Edition 1

The Heat Kernel and Theta Inversion on SL2(C) / Edition 1

by Jay Jorgenson, Serge Lang
     
 

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ISBN-10: 0387380310

ISBN-13: 9780387380315

Pub. Date: 10/15/2008

Publisher: Springer New York

The present monograph devlops the fundamental ideas and relults surronding heat kenels, spectral theory, and regularized traces associated the full modular group action on SL2(C) through spherical transform, from which one manifestation of the heat kernel on quotient spaces is obtained through group perodization. Fron a different point of view, one

Overview

The present monograph devlops the fundamental ideas and relults surronding heat kenels, spectral theory, and regularized traces associated the full modular group action on SL2(C) through spherical transform, from which one manifestation of the heat kernel on quotient spaces is obtained through group perodization. Fron a different point of view, one constructs the heat kernel on the group space using an eignfuction, or spectral, expansion, which then leads to a theta function and a theta inbversion formula by equating the two realizations of the heat kernel on the quotient space. The trace of the heat kernel diverges, which naturally leads to a regularization of the trace by studying Eisenstein series on the eigenfunction side and the cuspidal elements on the group periodization side. By focusing on the case of Sl2 (z[i]) acting on SL2(C), the authors are able to emphasize the importance of specific examples of the general theory of the general Selgerg trace formula and uncover the second step in their envisioned "ladder" of geometrically defined zet functions, where each conjectured step would include lower level zeta functions as factors in functional equations.

Product Details

ISBN-13:
9780387380315
Publisher:
Springer New York
Publication date:
10/15/2008
Series:
Springer Monographs in Mathematics Series
Edition description:
2008
Pages:
319
Product dimensions:
6.30(w) x 9.30(h) x 0.80(d)

Table of Contents

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 Γ: 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 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 Computation107

5.3.2 The Orbital Integral 108

5.4 Cuspidal Conjugacy Classes 110

Part III The Heat Kernel on Γ\G/K

6 The Fundamental Domain 117

6.1 SL2(C) and the Upper Half-Space H3 118

6.2 Fundamental Domain and Γ 121

6.3 Finiteness Properties 124

6.4 Uniformities in Lemma 6.2.3 130

6.5 Integration on Γ\G/K 131

6.6 Other Fundamental Domains 133

7 Γ-Periodization of the Heat Kernel 135

7.1 The Basic Estimate 135

7.1.1 Convolution 136

7.2 Heat Convolution and Eigenfunctions on Γ\G/K 140

7.3 Casimir on Γ\G/K 145

7.4 Measure-Theoretic Estimate for Convolution on Γ\G 147

7.5 Asymptotic Behavior of KΓt for t 149

8 Heat Kernel Convolution on I&sp (Γ\G/K) 151

8.1 General Criteria for Compactness 152

8.2 Estimates for the - Periodization 155

8.3 Fourier Series for the Periodizations of Gaussians 157

8.3.1 Preliminaries: The r and 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 Γ 167

9.1 Differential-Geometric Aspects 167

9.2 The Tube of FRand its Boundary Relation with 3R 169

9.3 The F-Normalizer of Γ 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 Γ 175

9.5.1 The Faces Bi of & and their Boundaries 175

9.5.2 H-triangle 176

9.5.3 Isometrics of F 178

9.6 The Group Γ and a Basic Boundary Inclusion 180

9.7 The Set y, its Boundary Behavior, and the Tube T 181

9.8 Tilings 182

9.8.1 Coset Representatives 184

9.9 Truncations 185

10 The Γu\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 ΓU\U-Expansion of Eisenstein Series 206

10.5.1 The Constant Term 211

10.6 Estimates in Vertical Strips 21313 10.7 The Volume-Residue Formula 216

10.8 The Integral over F and Orthogonalities 218

11 Adjointness Formula and the Γ\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/ e C(Γ\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 Γ\G 243

12.1.1 Γ2-Cuspidal Trace 244

12.2 Green's Theorem on F12.3 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 Y-Asymptotics 261

13.1 The Nonregular Cuspidal Integral over &13.2 Asymptotic Expansion of the Nonregular Cuspidal Trace 267

13.3 The Regular Cuspidal Integral overF13.4 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 uQand the Euler Constant 287

14.2 Estimates Using Lattice-Point Counting 290

14.3 Partial-Sums Asymptotics for uQ 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 Jq f9(r)rh(r)dr when (p = gr 301

14.6 Evaluation of C'yo and C1 303

14.7 The Theta Inversion Formula 308

References 311

Index 317

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