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Proceedings of the American Mathematical Society

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Strong multiplicity one theorems for locally homogeneous spaces of compact-type


Authors: Emilio A. Lauret and Roberto J. Miatello
Journal: Proc. Amer. Math. Soc. 148 (2020), 3163-3173
MSC (2010): Primary 22C05, 22E46
DOI: https://doi.org/10.1090/proc/14980
Published electronically: March 2, 2020
MathSciNet review: 4099801
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Abstract:

Let $G$ be a compact connected semisimple Lie group, let $K$ be a closed subgroup of $G$, let $\Gamma$ be a finite subgroup of $G$, and let $\tau$ be a finite-dimensional representation of $K$. For $\pi$ in the unitary dual $\widehat G$ of $G$, denote by $n_\Gamma (\pi )$ its multiplicity in $L^2(\Gamma \backslash G)$.

We prove a strong multiplicity one theorem in the spirit of Bhagwat and Rajan, for the $n_\Gamma (\pi )$ for $\pi$ in the set $\widehat G_\tau$ of irreducible $\tau$-spherical representations of $G$. More precisely, for $\Gamma$ and $\Gamma ’$ finite subgroups of $G$, we prove that if $n_{\Gamma }(\pi )= n_{\Gamma ’}(\pi )$ for all but finitely many $\pi \in \widehat G_\tau$, then $\Gamma$ and $\Gamma ’$ are $\tau$-representation equivalent, that is, $n_{\Gamma }(\pi )=n_{\Gamma ’}(\pi )$ for all $\pi \in \widehat G_\tau$.

Moreover, when $\widehat G_\tau$ can be written as a finite union of strings of representations, we prove a finite version of the above result. For any finite subset $\widehat {F}_{\tau }$ of $\widehat G_{\tau }$ verifying some mild conditions, the values of the $n_\Gamma (\pi )$ for $\pi \in \widehat F_{\tau }$ determine the $n_\Gamma (\pi )$’s for all $\pi \in \widehat G_\tau$. In particular, for two finite subgroups $\Gamma$ and $\Gamma ’$ of $G$, if $n_\Gamma (\pi ) = n_{\Gamma ’}(\pi )$ for all $\pi \in \widehat F_{\tau }$, then the equality holds for every $\pi \in \widehat G_\tau$. We use algebraic methods involving generating functions and some facts from the representation theory of $G$.


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Additional Information

Emilio A. Lauret
Affiliation: Instituto de Matemática (INMABB), Departamento de Matemática, Universidad Nacional del Sur (UNS)-CONICET, Bahía Blanca B8000CPB, Argentina
MR Author ID: 1016885
ORCID: 0000-0003-3729-5300
Email: emilio.lauret@uns.edu.ar

Roberto J. Miatello
Affiliation: CIEM–FaMAF (CONICET), Universidad Nacional de Córdoba, Medina Allende s/n, Ciudad Universitaria, 5000 Córdoba, Argentina
MR Author ID: 124160
Email: miatello@famaf.unc.edu.ar

Keywords: Strong multiplicity one theorem, right regular representation, representation equivalent
Received by editor(s): May 19, 2019
Received by editor(s) in revised form: November 7, 2019, and November 11, 2019
Published electronically: March 2, 2020
Additional Notes: This research was supported by grants from CONICET, FONCyT, and SeCyT
The first-named author was supported by the Alexander von Humboldt Foundation
Communicated by: Martin Liebeck
Article copyright: © Copyright 2020 American Mathematical Society