Strong multiplicity one theorems for locally homogeneous spaces of compact type

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 $G$.