The Heat Kernel and Theta Inversion on SL2(C)

I. Gaussians, Spherical Inversion, and the Heat Kernel

1. Introduction

2. Spherical Inversion on SL2(C)

3. The Heat Gaussian and Heat Kernel

4. QED, LEG, Transpose, and Casimir

II. Enter ?: The General Trace Formula

5. Convergence and Divergence of the Selberg Trace

6. The Cuspidal and Noncuspidal Traces

III. The Heat Kernel on ?G/K

7. The Fundamental Domain

8. ?-Periodization of the Heat Kernel

9. Heat Kernel Convolution on $$L_{{rm{cusp}}}^2 $$ (?G/K)

IV. Fourier-Eisenstein Eigenfunction Expansions

10. The Tube Domain for ??

11. The ?/U-Fourier Expansion of Eisenstein Series

12. Adjointness Formula and the ?G
-Eigenfunction Expansion

V. The Eisenstein-Cuspidal Affair

13. The EisensteinY-Asymptotics

14. The Cuspidal TraceY-Asymptotics

15. Analytic Evaluations

Keywords: Mathematics, Number Theory, Group Theory and Generalizations, Algebra, Analysis