Bergman kernels and Poincar\'e series

TL;DR

This paper demonstrates that the Bergman kernel on a finite-volume quotient of a Hermitian manifold can be obtained by averaging the kernel on the universal cover, and applies this to show non-vanishing of certain Poincaré series in symmetric spaces.

Contribution

It extends previous results on Bergman kernels and Poincaré series to general locally symmetric spaces of finite volume.

Findings

Bergman kernel equals the average over the group of the universal cover's kernel.

Large class of relative Poincaré series are non-vanishing.

Results generalize prior work to broader class of symmetric spaces.

Abstract

We show that the Bergman kernel of a finite-volume quotient of a Hermitian manifold $\widetilde{X}$ with bounded geometry by a discrete group $\Gamma$ of its isometries is the same as the averaging over $\Gamma$ of the Bergman kernel on $\widetilde{X}$. We then use these results when $\widetilde{X}$ is a Hermitian symmetric space to show that a large class of relative Poincar\'e 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.