Orthogonality relations for Poincar\'e series

TL;DR

This paper derives a formula for the inner product of cuspidal automorphic forms expressed as Poincaré series on semisimple Lie groups, extending known results and providing new proofs for classical cases like Siegel cusp forms.

Contribution

It introduces a new formula for inner products of Poincaré series of automorphic forms on semisimple Lie groups, generalizing previous work on SL(2,R).

Findings

Derived a formula for inner products of Poincaré series on semisimple Lie groups.

Provided a new proof of Petersson inner product for vector-valued Siegel cusp forms.

Extended results previously obtained for SL(2,R) to more general groups.

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) $.