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# On Limit Multiplicities for Spaces of Automorphic Forms

Published:1999-10-01
Printed: Oct 1999
• Anton Deitmar
• Werner Hoffmann
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## Abstract

Let $\Gamma$ be a rank-one arithmetic subgroup of a semisimple Lie group~$G$. For fixed $K$-Type, the spectral side of the Selberg trace formula defines a distribution on the space of infinitesimal characters of~$G$, whose discrete part encodes the dimensions of the spaces of square-integrable $\Gamma$-automorphic forms. It is shown that this distribution converges to the Plancherel measure of $G$ when $\Ga$ shrinks to the trivial group in a certain restricted way. The analogous assertion for cocompact lattices $\Gamma$ follows from results of DeGeorge-Wallach and Delorme.
 Keywords: limit multiplicities, automorphic forms, noncompact quotients, Selberg trace formula, functional calculus
 MSC Classifications: 11F72 - Spectral theory; Selberg trace formula 22E30 - Analysis on real and complex Lie groups [See also 33C80, 43-XX] 22E40 - Discrete subgroups of Lie groups [See also 20Hxx, 32Nxx] 43A85 - Analysis on homogeneous spaces 58G25 - unknown classification 58G25

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