On a family of Siegel Poincaré series

Let [Formula: see text] be a congruence subgroup of text]. Using Poincaré series text]-finite matrix coefficients integrable discrete representations text], we construct spanning set for the space Siegel cusp forms weight We prove non-vanishing certain elements this using Muić’s integral criterion on locally compact Hausdorff groups. Moreover, representation theory study Petersson inner products corresponding cuspidal automorphic forms, thereby recovering representation-theoretic proof some well-known results reproducing kernel function Our are obtained by generalizing methods developed Muić in his work holomorphic upper half-plane to setting higher degree.