Strong multiplicity one theorems for locally homogeneous spaces of compact type

TL;DR

This paper establishes strong multiplicity one theorems for the spectra of locally homogeneous spaces of compact type, showing that spectral data for finitely many representations determines the entire spectrum and the subgroup structure.

Contribution

It proves new strong multiplicity one results for $ au$-spherical representations, extending previous work to a broader class of spaces and finite subsets of the spectrum.

Findings

Spectral data for all but finitely many representations determines the entire spectrum.

Finite spectral data on a carefully chosen subset suffices to determine the full spectrum.

Results apply to finite subgroups and $ au$-representation equivalence in compact semisimple Lie groups.

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 $G$.