Bergman kernels and Poincaré series

Abstract

We show that the Bergman kernel of a finite-volume quotient of a Hermitian manifold $\tilde{X}$ with bounded geometry by a discrete group $\hat{I}“$ of its isometries is the same as the averaging over $\hat{I}“$ of the Bergman kernel on $\tilde{X}$. We then use these results when $\tilde{X}$ is a Hermitian symmetric space to show that a large class of relative Poincaré series does not vanish. This extends the results of Borthwick-Paul-Uribe and Barron (formerly Foth) to the case of general locally symmetric spaces of finite volume.

22 pages