Strong multiplicity one theorems for locally homogeneous spaces of compact type

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 $\hat{G}$ 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 ${\hat{G}}_{\tau }$ of irreducible $\tau$-spherical representations of $G$. More precisely, for $\Gamma$ and ${\Gamma }^{\prime }$ finite subgroups of $G$, we prove that if $n_{\Gamma }(\pi )=n_{{\Gamma }^{\prime }}(\pi )$ for all but finitely many $\pi \in {\hat{G}}_{\tau }$, then $\Gamma$ and ${\Gamma }^{\prime }$ are $\tau$-representation equivalent, that is, $n_{\Gamma }(\pi )=n_{{\Gamma }^{\prime }}(\pi )$ for all $\pi \in {\hat{G}}_{\tau }$. Moreover, when ${\hat{G}}_{\tau }$ can be written as a finite union of strings of representations, we prove a finite version of the above result. For any finite subset ${\hat{F}}_{\tau }$ of ${\hat{G}}_{\tau }$ verifying some mild conditions, the values of the $n_{\Gamma }(\pi )$ for $\pi \in {\hat{F}}_{\tau }$ determine the $n_{\Gamma }(\pi )$'s for all $\pi \in {\hat{G}}_{\tau }$. In particular, for two finite subgroups $\Gamma$ and ${\Gamma }^{\prime }$ of $G$, if $n_{\Gamma }(\pi )=n_{{\Gamma }^{\prime }}(\pi )$ for all $\pi \in {\hat{F}}_{\tau }$ then the equality holds for every $\pi \in {\hat{G}}_{\tau }$. We use algebraic methods involving generating functions and some facts from the representation theory of $G$.

Final version, to appear in Proc. AMS