On a family of Siegel Poincaré series

Abstract

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