Orthogonality relations for Poincar\'e series

Let $G$ be a connected semisimple Lie group with finite center. We prove a formula for the inner product of two cuspidal automorphic forms on $G$ that are given by Poincar\'e series of $K$-finite matrix coefficients of an integrable discrete series representation of $G$. As an application, we give a new proof of a well-known result on the Petersson inner product of certain vector-valued Siegel cusp forms. In this way, we extend results previously obtained by G. Mui\'c for cusp forms on the upper half-plane, i.e., in the case when $G=\mathrm{SL}_2(\mathbb R)$.