On the non-vanishing of Poincar\'e series on irreducible bounded symmetric domains

TL;DR

This paper investigates conditions under which Poincaré series on irreducible bounded symmetric domains do not vanish, using vector-valued integral criteria, with applications to specific groups like SU(p,q).

Contribution

It extends Muić's non-vanishing criterion to vector-valued Poincaré series on symmetric domains, providing new non-vanishing results for automorphic forms.

Findings

Established non-vanishing criteria for vector-valued Poincaré series.

Applied results to the case of G=SU(p,q) and K=U(p)×U(q).

Demonstrated the existence of non-zero automorphic forms from Poincaré series.

Abstract

Let $ \mathcal D\equiv G/K $ be an irreducible bounded symmetric domain. Using a vector-valued version of Mui\'c's integral non-vanishing criterion for Poincar\'e series on locally compact Hausdorff groups, we study the non-vanishing of holomorphic automorphic forms on $ \mathcal D $ that are given by Poincar\'e series of polynomial type and correspond via the classical lift to the Poincar\'e 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\geq q\geq1 $.