Orthogonality relations for Poincar\'e series

Abstract

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) $.