The draft of the present book was ?rst completed in August 2005, one month prior to the passing of Serge Lang. As such, this book is the last completed in its entirety by Lang. Beginning the early 1990’s, Lang became fascinated with the prospect of - ing heat kernels and heat kernel analysis in analytic number theory. Speci?cally, we developed a program of study where one would de?ne Selberg-type zeta fu- tions associated to ?nite volume quotients of symmetric spaces, and we speculated that each such zeta function would admit a functional equation where lower level Selberg-type zeta functions would appear. In the case of the Riemann surface asso- ated to PSL (Z), the Selberg zeta function has a functional equation which involves 2 the Riemann zeta function. Lang and I began the work necessary to carry out our proposed analysis for quotients of SL (C)bySL (Z[i]). As with other mathema- n n cal works undertaken by Lang, he wanted to develop the foundations himself. The present book is the result of establishing, as only Lang could, the case n= 2. During the reviewing process, I have refrained from making any changes in order to preserve Lang’s style of exposition. New York Jay Jorgenson May 2008 v Contents Preface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Part I Gaussians, Spherical Inversion, and the Heat Kernel 1 Spherical Inversion on SL (C). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2 1. 1 The Iwasawa Decomposition, Polar Decomposition, andCharacters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1. 1. 1 Characters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1. 1. 2 K-bi-invariant Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .