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 $Γ$ be a finite subgroup of $G$, and let $τ$ be a finite-dimensional representation of $K$. For $π$ in the unitary dual $\widehat G$ of $G$, denote by $n_Γ(π)$ its multiplicity in $L^2(Γ\backslash G)$. We prove a strong multiplicity one theorem in the spirit of Bhagwat and Rajan, for the $n_Γ(π)$ for $π$ in the set $\widehat G_τ$ of irreducible $τ$-spherical representations of $G$. More precisely, for $Γ$ and $Γ'$ finite subgroups of $G$, we prove that if $n_Γ(π)= n_{Γ'}(π)$ for all but finitely many $π\in \widehat G_τ$, then $Γ$ and $Γ'$ are $τ$-representation equivalent, that is, $n_Γ(π)=n_{Γ'}(π)$ for all $π\in \widehat G_τ$. Moreover, when $\widehat G_τ$ can be written as a finite union of strings of representations, we prove a finite version of the above result. For any finite subset $\widehat {F}_τ$ of $\widehat G_τ$ verifying some mild conditions, the values of the $n_Γ(π)$ for $π\in\widehat F_τ$ determine the $n_Γ(π)$'s for all $π\in \widehat G_τ$. In particular, for two finite subgroups $Γ$ and $Γ'$ of $G$, if $n_Γ(π) = n_{Γ'}(π)$ for all $π\in \widehat F_τ$ then the equality holds for every $π\in \widehat G_τ$. We use algebraic methods involving generating functions and some facts from the representation theory of $G$.

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