The Heat Kernel and Theta Inversion on SL2(C)

Jay Jorgenson, Serge Lang (auth.)0387380310, 9780387380322, 9780387380315

The present monograph develops the fundamental ideas and results surrounding heat kernels, spectral theory, and regularized traces associated to the full modular group acting on SL2(C). The authors begin with the realization of the heat kernel on SL2(C) through spherical transform, from which one manifestation of the heat kernel on quotient spaces is obtained through group periodization. From a different point of view, one constructs the heat kernel on the group space using an eigenfunction, or spectral, expansion, which then leads to a theta function and a theta inversion 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 Selberg trace formula and uncover the second step in their envisioned “ladder” of geometrically defined zeta functions, where each conjectured step would include lower level zeta functions as factors in functional equations.

Table of contents :
Front Matter….Pages i-x
Introduction….Pages 1-9
Spherical Inversion on SL 2 (C)….Pages 13-43
The Heat Gaussian and Heat Kernel….Pages 45-66
QED, LEG, Transpose, and Casimir….Pages 67-81
Convergence and Divergence of the Selberg Trace….Pages 85-95
The Cuspidal and Noncuspidal Traces….Pages 97-114
The Fundamental Domain….Pages 117-134
Γ-Periodization of the Heat Kernel….Pages 135-150
Heat Kernel Convolution on $$L_{{rm{cusp}}}^2 $$ (ΓG/K)….Pages 151-163
The Tube Domain for Γ∞….Pages 167-189
The Γ/ U -Fourier Expansion of Eisenstein Series….Pages 191-222
Adjointness Formula and the Γ G -Eigenfunction Expansion….Pages 223-240
The Eisenstein Y -Asymptotics….Pages 243-259
The Cuspidal Trace Y -Asymptotics….Pages 261-286
Analytic Evaluations….Pages 287-309
Back Matter….Pages 311-319