| 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 FRand its Boundary Relation with 3R | p. 169 |
| The F-Normalizer of ¿ | p. 171 |
| Totally Geodesic Surface in H3 | p. 172 |
| The Half-Plane H2j | p. 173 |
| Some Boundary Behavior of F in H3 Under ¿ | p. 175 |
| The Faces Bi 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 úQand 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 |
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