On a family of Siegel Poincar\'e series

Abstract

Let $ \Gamma $ be a congruence subgroup of $ \mathrm{Sp}_{2n}(\mathbb Z) $. Using Poincar\'e series of $ K $-finite matrix coefficients of integrable discrete series representations of $ \mathrm{Sp}_{2n}(\mathbb R) $, we construct a spanning set for the space $ S_m(\Gamma) $ of Siegel cusp forms of weight $ m\in\mathbb Z_{>2n} $. We prove the non-vanishing of certain elements of this spanning set using Mui\'c's integral non-vanishing criterion for Poincar\'e series on locally compact Hausdorff groups. Moreover, using the representation theory of $ \mathrm{Sp}_{2n}(\mathbb R) $, we study the Petersson inner products of corresponding cuspidal automorphic forms, thereby recovering a representation-theoretic proof of some well-known results on the reproducing kernel function of $ S_m(\Gamma) $. Our results are obtained by generalizing representation-theoretic methods developed by Mui\'c in his work on holomorphic cusp forms on the upper half-plane to the setting of Siegel cusp forms of a higher degree.