A strong multiplicity one theorem for dimension data

Abstract

We call the dimension data ${\mathcal{D}}_{H_1}$ and ${\mathcal{D}}_{H_2}$ of two closed subgroups $H_1$ and $H_2$ of a given compact Lie group $G$ {\it almost equal} if ${\mathcal{D}}_{H_1}(\rho )={\mathcal{D}}_{H_2}(\rho )$ for all but finitely many irreducible complex linear representations $\rho$ of $G$ up to equivalence. When $G$ is connected, we show that: if ${\mathcal{D}}_{H_1}$ and ${\mathcal{D}}_{H_2}$ are almost equal, then they are equal. When $G$ is non-connected, $G^0\subset H_1\cap H_2$ is a trivial sufficient condition for ${\mathcal{D}}_{H_1}$ and ${\mathcal{D}}_{H_2}$ to be almost equal. In this case assume that ${\mathcal{D}}_{H_1}$ and ${\mathcal{D}}_{H_2}$ are almost equal but non-equal. We show strong relations between $H_1$ and $H_2$ and we construct an example which indicates that $G^0\subset H_1\cap H_2$ is not a necessary condition.

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