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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
Published electronically: March 2, 2020
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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

Roberto J. Miatello
Affiliation: CIEM–FaMAF (CONICET), Universidad Nacional de Córdoba, Medina Allende s/n, Ciudad Universitaria, 5000 Córdoba, Argentina

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