Construction and non-vanishing of a family of vector-valued Siegel Poincar\'e series

Using Poincar\'e series of K -finite matrix coefficients of integrable antiholomorphic discrete series representations of Sp₂₍ (R), we construct a spanning set for the space S_ () of Siegel cusp forms of weight for, where is an irreducible polynomial representation of GLₙ (C) of highest weight Zⁿ with ₁ₙ>2n, and is a discrete subgroup of Sp₂₍ (R) commensurable with Sp₂₍ (Z). Moreover, using a variant of Mui\'c's integral non-vanishing criterion for Poincar\'e series on unimodular locally compact Hausdorff groups, we prove a result on the non-vanishing of constructed Siegel Poincar\'e series.