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arXiv · 2501.06876

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

Sonja Žunar
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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 $.

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Sonja Žunar. 2025-01-12. On the non-vanishing of Poincar\'e series on irreducible bounded symmetric domains. https://arxiv.org/abs/2501.06876

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