On a family of Siegel Poincaré series

Abstract

Let $\Gamma$ be a congruence subgroup of ${\mathrm{Sp}}_{2n}(\mathbb{Z})$. Using Poincaré series of $K$-finite matrix coefficients of integrable discrete series representations of ${\mathrm{Sp}}_{2n}(\mathbb{R})$, we construct a spanning set for the space $S_m(\Gamma )$ of Siegel cusp forms of weight $m\in {\mathbb{Z}}_{>2n}$. 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 ${\mathrm{Sp}}_{2n}(\mathbb{R})$, 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 $S_m(\Gamma )$. 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.

21 pages