Strong multiplicity one theorems for locally homogeneous spaces of compact type

LAURET, EMILIO A.; MIATELLO, ROBERTO J.

PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY

AMER MATHEMATICAL SOC

Año: 2020 vol. 148 p. 3163 - 3173

0002-9939

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 $au$ 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 $L^2(Gammaackslash G)$.We prove a strong multiplicity one theorem in the spirit of Bhagwat and Rajan, for the $n_Gamma(pi)$ for $pi$ in the set $widehat G_au$ of irreducible $au$-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 $piin widehat G_au$, then $Gamma$ and $Gamma´$ are $au$-representation equivalent, that is, $n_{Gamma}(pi)=n_{Gamma´}(pi)$ for all $piin widehat G_au$. Moreover, when $widehat G_au$ can be written as a finite union of strings of representations, we prove a finite version of the above result. For any finite subset $widehat {F}_{au}$ of $widehat G_{au}$ verifying some mild conditions, the values of the $n_Gamma(pi)$ for $piinwidehat F_{au}$ determine the $n_Gamma(pi)$´s for all $pi in widehat G_au$.In particular, for two finite subgroups $Gamma$ and $Gamma´$ of $G$, if $n_Gamma(pi) = n_{Gamma´}(pi)$ for all $piin widehat F_{au}$ then the equality holds for every $pi in widehat G_au$. We use algebraic methods involving generating functions and some facts from the representation theory of $G$.