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The Heat Kernel and Theta Inversion on SL2(C)

By: Jay Jorgenson, Serge Lang

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Hardcover | 30 October 2008

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332 Pages

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23.39 x 15.6 x 1.91

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The worthy purpose of this text is to provide a complete, self-contained development of the trace formula and theta inversion formula for SL(2,Z[i])SL(2,C). Unlike other treatments of the theory, the approach taken here is to begin with the heat kernel on SL(2,C) associated to the invariant Laplacian, which is derived using spherical inversion. The heat kernel on the quotient space SL(2,Z[i])SL(2,C) is arrived at through periodization, and further expanded in an eigenfunction expansion. A theta inversion formula is obtained by studying the trace of the heat kernel. Following the author's previous work, the inversion formula then leads to zeta functions through the Gauss transform.</

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From the reviews:

"The book under review ... provides an introduction to the general theory of semisimple or reductive groups G, with symmetric space G/K (K maximal compact). ... It is ... meant for experienced insiders, even as the presentation of the material is excellent and accessible. A well-prepared graduate student would do well with this book. More experienced analytic number theorists will find it enjoyable and spellbinding. ... I heartily recommend to other analytic number theorists of a similar disposition." (Michael Berg, MAA Online, December, 2008)

"This book is part of a program of the authors to develop a systematic theory of theta and zeta functions on homogeneous spaces, using techniques of harmonic analysis and, in particular, heat kernels. ... the book includes many details that would likely have been left for the reader to work out by her/himself in a more streamlined monograph. ... all in all, an enjoyable book to read." (Fredrik Stroemberg, Zentralblatt MATH, Vol. 1192, 2010)

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Non-Fiction Mathematics Calculus & Mathematical Analysis Functional Analysis & Transforms Number Theory Algebra Groups & Group Theory Differential Calculus & Equations Topology Algebraic Topology

Preface	p. v
Introduction	p. 1
Gaussians, Spherical Inversion, and the Heat Kernel
Spherical Inversion on SL₂(C)	p. 13
The Iwasawa Decomposition, Polar Decomposition, and Characters	p. 15
Characters	p. 16
K-bi-invariant Functions	p. 17
Haar Measures	p. 19
The Harish Transform and the Orbital Integral	p. 23
The Mellin and Spherical Transforms	p. 25
Computation of the Orbital Integral	p. 28
Gaussians on G and Their Spherical Transform	p. 32
The Polar Height	p. 35
The Polar Haar Measure and Inversion	p. 37
Point-Pair Invariants, the Polar Height, and the Polar Distance	p. 41
The Heat Gaussian and Heat Kernel	p. 45
Dirac Families of Gaussians	p. 45
Scaling	p. 46
Decay Property	p. 48
Convolution, Semigroup, and Approximations Properties	p. 49
Approximation Properties	p. 51
Complexifying t and the Null Space of Heat Convolution	p. 54
The Casimir Operator	p. 55
Scaling	p. 62
The Heat Equation	p. 63
Scaling	p. 65
QED, LEG, Transpose, and Casimir	p. 67
Growth and Decay, QED and LEG	p. 67
Casimir, Transpose, and Harmonicity	p. 70
DUTIS	p. 76
Heat and Casimir Eigenfunctions	p. 78
Enter ¿: The General Trace Formula
Convergence and Divergence of the Selberg Trace	p. 85
The Hermitian Norm	p. 86
Divergence for Standard Cuspidal Elements	p. 89
Cuspidal and Parabolic Subgroups	p. 89
Convergence for the Other Elements of	p. 92
The Cuspidal and Noncuspidal Traces	p. 97
Some Group Theory	p. 98
Conjugacy Classes	p. 101
The Double Trace and its Decomposition	p. 102
Explicit Determination of the Noncuspidal Terms	p. 106
The Volume Computation	p. 107
The Orbital Integral	p. 108
Cuspidal Conjugacy Classes	p. 110
The Heat Kernel on ¿G/K
The Fundamental Domain	p. 117
SL₂(C) and the Upper Half-Space H3	p. 118
Fundamental Domain and ¿	p. 121
Finiteness Properties	p. 124
Uniformities in Lemma 6.2.3	p. 130
Integration on ¿G/K	p. 131
Other Fundamental Domains	p. 133
¿-Periodization of the Heat Kernel	p. 135
The Basic Estimate	p. 135
Convolution	p. 136
Heat Convolution and Eigenfunctions on ¿G/K	p. 140
Casimir on ¿G/K	p. 145
Measure-Theoretic Estimate for Convolution on ¿G	p. 147
Asymptotic Behavior of K^¿t for t	p. 149
Heat Kernel Convolution on I&sp (¿G/K)	p. 151
General Criteria for Compactness	p. 152
Estimates for the - Periodization	p. 155
Fourier Series for the Periodizations of Gaussians	p. 157
Preliminaries: The r and Periodizations	p. 157
The Fourier Series	p. 158
The Convolution Cuspidal Estimate	p. 160
Application to the Heat Kernel	p. 161
Fourier-Eisenstein Eigenfunction Expansions
The Tube Domain for ¿	p. 167
Differential-Geometric Aspects	p. 167
The Tube of F_Rand its Boundary Relation with 3R	p. 169
The F-Normalizer of ¿	p. 171
Totally Geodesic Surface in H₃	p. 172
The Half-Plane H_2j	p. 173
Some Boundary Behavior of F in H₃ Under ¿	p. 175
The Faces B_i of & and their Boundaries	p. 175
H-triangle	p. 176
Isometrics of F	p. 178
The Group ¿ and a Basic Boundary Inclusion	p. 180
The Set y, its Boundary Behavior, and the Tube T	p. 181
Tilings	p. 182
Coset Representatives	p. 184
Truncations	p. 185
The ¿uU-Fourier Expansion of Eisenstein Series	p. 191
Our Goal: The Eigenfunction Expansion	p. 191
Epstein and Eisenstein Series	p. 193
The K-Bessel Function	p. 197
Gamma Function Identities	p. 199
Differential and Difference Relations	p. 201
Functional Equation of the Dedekind Zeta Function	p. 202
The Bessel-Fourier ¿_UU-Expansion of Eisenstein Series	p. 206
The Constant Term	p. 211
Estimates in Vertical Strips	p. 21313
The Integral over F and Orthogonalities	p. 218
Adjointness Formula and the ¿G-Eigenfunction Expansion	p. 223
Haar Measure and the Mellin Transform	p. 224
Appendix on Fourier Inversion	p. 226
Adjointness Formula and the Constant Term	p. 229
Adjointness Formula	p. 230
The Eisenstein Coefficient E_*f and the Expansion for/ e C(¿G/K)	p. 232
The Heat Kernel Eigenfunction Expansion	p. 237
The Eisenstein-Cuspidal Affair
The Eisenstein Y-Asymptotics	p. 243
The Improper Integral of Eigenfunction Expansion over ¿G	p. 243
¿2-Cuspidal Trace	p. 244
Green's Theorem on F<Y	p. 247
Application to Eisenstein Functions	p. 251
The Constant-Term Integral Asymptotics	p. 255
Appendix	p. 257
The Nonconstant-Term Error Estimate	p. 258
The Cuspidal Trace Y-Asymptotics	p. 261
The Nonregular Cuspidal Integral over &<Y	p. 262
Asymptotic Expansion of the Nonregular Cuspidal Trace	p. 267
The Regular Cuspidal Integral overF<Y	p. 272
Nonspecial Regular Cuspidal Asymptotics	p. 275
Action of the Special Subset	p. 277
Special Regular Cuspidal Asymptotics	p. 280
Analytic Evaluations	p. 287
Partial Sums Asymptotics for ú_Qand the Euler Constant	p. 287
Estimates Using Lattice-Point Counting	p. 290
Partial-Sums Asymptotics for úQ and the Euler Constant	p. 292
The Hurwitz Constant	p. 296
The Complex Case, with Z[i]	p. 297
Average of the Hurwitz Constant	p. 298
Jq f9(r)rh(r)dr when (p = gr	p. 301
Evaluation of C'_yo and C₁	p. 303
The Theta Inversion Formula	p. 308
References	p. 311
Index	p. 317
Table of Contents provided by Ingram. All Rights Reserved.

The Heat Kernel and Theta Inversion on SL2(C)

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