The wave kernel for the Laplacian on the classical locally symmetric spaces of rank one, theta functions, trace formulas and the Selberg zeta function

Abstract

We calculate the wave kernels for the classical rank-one symmetric spaces. The result is employed in order to provide a meromorphic extension of the theta function of an even-dimensional compact locally symmetric space of non-compact type. Moreover we give a short derivation of the Selberg trace formula. We discuss the relation between the right hand side of the functional equation of the Selberg zeta function, the Plancherel measure, Weyl's dimension formula and the wave kernel on the non-compact symmetric space and on its compact dual in an explicit manner.