On a family of Siegel Poincar\'e series

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.