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 ${\mathrm{Sp}}_{2n}\left(R\right)$, we construct a spanning set for the space $S_{\rho }\left(\Gamma \right)$ of Siegel cusp forms of weight ρ for Γ, where ρ is an irreducible polynomial representation of ${\mathrm{GL}}_n\left(C\right)$ of highest weight $\omega \in Z^n$ with ${\omega }_1\ge \dots \ge {\omega }_n>2n$, and Γ is a discrete subgroup of ${\mathrm{Sp}}_{2n}\left(R\right)$ commensurable with ${\mathrm{Sp}}_{2n}\left(Z\right)$. 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.