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 $\widehat G$ 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 $G$.