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The Heat Kernel and Theta Inversion on SL2(C)

The Heat Kernel and Theta Inversion on SL2(C)
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Field name Details
Dewey Class 512.7
Title The Heat Kernel and Theta Inversion on SL2(C) (EB) / by Jay Jorgenson, Serge Lang.
Author Jorgenson, Jay
Added Personal Name Lang, Serge
Other name(s) SpringerLink (Online service)
Publication New York, NY : Springer , 2008.
Physical Details X, 319 pages : online resource.
Series Springer monographs in mathematics 1439-7382
ISBN 9780387380322
Summary Note 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.:
Contents note Introduction -- Spherical Inversion on SL2(C) -- The Heat Gaussian and Kernel -- QED, LEG, Transpose, and Casimir -- Convergence and Divergence of the Selberg Trace -- The Cuspidal and Non-Cuspidal Traces -- The Heat Kernel -- The Fundamental Domain -- Gamma Periodization of the Heat Kernel -- Heat Kernel Convolution -- The Tube Domain -- The Fourier Expansion of Eisenstein Series -- Adjointness Formula and the Eigenfunction Expansion -- The Eisenstein Y-Asymptotics -- The Cuspidal Trace Y-Asymptotics -- Analytic Evaluations -- Index -- References.
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