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 Sp₂ₙ( R), we construct a spanning set for the space S_ρ(Γ) of Siegel cusp forms of weight ρ for Γ, where ρ is an irreducible polynomial representation of GLₙ( C) of highest weight ω∈ Zⁿ with ω₁≥…≥ωₙ>2n, and Γ is a discrete subgroup of Sp₂ₙ( R) commensurable with Sp₂ₙ( 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.