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Journal of Lie Theory 22 (2012), No. 4, 1091--1107
Copyright Heldermann Verlag 2012

On the Inner Product of Certain Automorphic Forms and Applications

Goran Muic
Dept. of Mathematics, University of Zagreb, Bijenicka cesta 30, 10000 Zagreb, Croatia


\def\R{{\Bbb R}} Let $\Gamma\subset {\rm SL}_2(\R)$ be a discrete subgroup such that the quotient $\Gamma\backslash{\rm SL}_2(\R)$ has a finite volume. In this paper we compute the Petersson inner product of automorphic cuspidal forms with Poincar\' e series constructed out of matrix coefficients of a holomorphic discrete series of lowest weight $m\ge 3$. We apply the result to give new and representation-theoretic proofs of previous results, some of which were known to Petersson, and are anyway not surprising to experts.

Keywords: Fuchsian groups, automorphic forms, modular forms, Poincare series.

MSC: 11F70, 11F20

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