On the non-vanishing of Poincaré series on irreducible bounded symmetric domains

Abstract

Let $\mathcal{D}\equiv G/K$ be an irreducible bounded symmetric domain. Using a vector-valued version of Muić's integral non-vanishing criterion for Poincaré series on locally compact Hausdorff groups, we study the non-vanishing of holomorphic automorphic forms on $\mathcal{D}$ that are given by Poincaré series of polynomial type and correspond via the classical lift to the Poincaré series of certain $K$-finite matrix coefficients of integrable discrete series representations of $G$. We provide an example application of our results in the case when $G=\mathrm{SU}(p,q)$ and $K=\mathrm{S}(\mathrm{U}(p)\times \mathrm{U}(q))$ with $p\ge q\ge 1$.

arXiv comment24 pages, 2 tables