Construction and non-vanishing of a family of vector-valued Siegel Poincaré series

Using Poincaré series of $K$-finite matrix coefficients of integrable antiholomorphic discrete series representations of ${\mathrm{S}\mathrm{p}}_{2n}(\mathbb{R})$, we construct a spanning set for the space $S_{\rho }(\mathrm{\Gamma })$ of Siegel cusp forms of weight $\rho$ for $\mathrm{\Gamma }$, where $\rho$ is an irreducible polynomial representation of ${\mathrm{G}\mathrm{L}}_n(\mathbb{C})$ of highest weight $\omega \in {\mathbb{Z}}^n$ with ${\omega }_1\ge \dots \ge {\omega }_n>2n$, and $\mathrm{\Gamma }$ is a discrete subgroup of ${\mathrm{S}\mathrm{p}}_{2n}(\mathbb{R})$ commensurable with ${\mathrm{S}\mathrm{p}}_{2n}(\mathbb{Z})$. Moreover, using a variant of Muić’s integral non-vanishing criterion for Poincaré series on unimodular locally compact Hausdorff groups, we prove a result on the non-vanishing of constructed Siegel Poincaré series.