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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Strong multiplicity one theorems for locally homogeneous spaces of compact-type
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by Emilio A. Lauret and Roberto J. Miatello PDF
Proc. Amer. Math. Soc. 148 (2020), 3163-3173 Request permission

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
  • 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
  • © Copyright 2020 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 148 (2020), 3163-3173
  • MSC (2010): Primary 22C05, 22E46
  • DOI: https://doi.org/10.1090/proc/14980
  • MathSciNet review: 4099801