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arXiv · 2501.17832

Orthogonality relations for Poincar\'e series

Sonja Žunar
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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) $.

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Sonja Žunar. 2025-01-29. Orthogonality relations for Poincar\'e series. https://arxiv.org/abs/2501.17832

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