Orthogonality relations for Poincaré series

Let Let ???? be a connected semisimple Lie group with finite center. We prove a formula for the inner product of two cuspidal automorphic forms on ???? that are given by Poincaré series of ????-finite matrix coefficients of an integrable discrete series representation of ????. 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 Muić for cusp forms on the upper half-plane, i.e., in the case when ???? = SL2 (R).