The Heat Kernel and Theta Inversion on SL2(C)
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ISBN-13:
9781441922823
Veröffentl:
2010
Einband:
Paperback
Erscheinungsdatum:
19.11.2010
Seiten:
332
Autor:
Serge Lang
Gewicht:
505 g
Format:
235x155x19 mm
Serie:
Springer Monographs in Mathematics
Sprache:
Englisch
Beschreibung:
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.
The 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 gotten 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.
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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