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LIMIT MULTIPLICITIES FOR PRINCIPAL CONGRUENCE SUBGROUPS OF $\text{GL}(n)$ AND $\text{SL}(n)$

Part of: Lie groups Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  08 May 2014

Tobias Finis ,
Erez Lapid  and
Werner Müller
Tobias Finis
Affiliation:
Freie Universität Berlin, Institut für Mathematik, Arnimallee 3, D-14195 Berlin, Germany ([email protected])
Erez Lapid
Affiliation:
Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Jerusalem, 91904, Israel Department of Mathematics, The Weizmann Institute of Science, Rehovot 76100, Israel ([email protected])
Werner Müller
Affiliation:
Mathematisches Institut, Rheinische Friedrich-Wilhelms-Universität Bonn, Endenicher Allee 60, D-53115 Bonn, Germany ([email protected])
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Abstract

We study the limiting behavior of the discrete spectra associated to the principal congruence subgroups of a reductive group over a number field. While this problem is well understood in the cocompact case (i.e., when the group is anisotropic modulo the center), we treat groups of unbounded rank. For the groups $\text{GL}(n)$ and $\text{SL}(n)$ we show that the suitably normalized spectra converge to the Plancherel measure (the limit multiplicity property). For general reductive groups we obtain a substantial reduction of the problem. Our main tool is the recent refinement of the spectral side of Arthur’s trace formula obtained in [Finis, Lapid, and Müller, Ann. of Math. (2) 174(1) (2011), 173–195; Finis and Lapid, Ann. of Math. (2) 174(1) (2011), 197–223], which allows us to show that for $\text{GL}(n)$ and $\text{SL}(n)$ the contribution of the continuous spectrum is negligible in the limit.

Keywords

limit multiplicitieslattices in Lie groupstrace formula

MSC classification

Primary: 11F72: Spectral theory; Selberg trace formula
Secondary: 22E40: Discrete subgroups of Lie groups 22E55: Representations of Lie and linear algebraic groups over global fields and adèle rings 11F70: Representation-theoretic methods; automorphic representations over local and global fields
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 14 , Issue 3 , July 2015 , pp. 589 - 638
Copyright
© Cambridge University Press 2014 

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