Orthogonality relations for Poincaré series
Sonja Žunar
TL;DR
The paper addresses the problem of computing the Petersson inner product of cuspidal automorphic forms on a general connected semisimple Lie group $G$ with finite center, when these forms are given by Poincaré series of $K$-finite matrix coefficients from an integrable discrete series $\pi$. Building on Muić's methods for ${\rm SL}_2({\mathbb{R}})$, the authors develop a representation-theoretic framework that yields an explicit inner-product formula: for $h,h',v,v'\in H_K$, $\left<P_\Gamma c_{h,h'},P_\Gamma c_{v,v'}\right>_{L^2(\Gamma\backslash G)}=\frac{1}{d(\pi)}\left<h,v\right>_H\left(P_\Gamma c_{v',h'}\right)(1_G)$, with orthogonality when the involved discrete series are not equivalent. The work further analyzes the structure of $P_\Gamma c_{H_K,h'}$, proving a unitary $G$-equivalence to the corresponding isotypic component and decomposing $L^2(\Gamma\backslash G)_{[\pi]}$ into a Hilbert-space direct sum of closed $G$-invariant subspaces; multiplicities $m_\Gamma(\pi)$ arise as dimensions of the relevant spaces. As an application, the paper provides a representation-theoretic proof of a classical Petersson inner product formula for vector-valued Siegel cusp forms, connecting Poincaré lifts to reproducing kernels and extending Muić's results to the Siegel setting. Overall, the results unify and extend orthogonality phenomena for automorphic forms across general semisimple groups and illuminate the link between Poincaré series, discrete-series representations, and reproducing-kernel structures.
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 Poincaré 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. Muić for cusp forms on the upper half-plane, i.e., in the case when $ G=\mathrm{SL}_2(\mathbb R) $.