Multiplicity one theorems: the Archimedean case

Binyong Sun, Chen-Bo Zhu

Published 2009-03-08, updated 2011-06-17Version 2

Let $G$ be one of the classical Lie groups $\GL_{n+1}(\R)$, $\GL_{n+1}(\C)$, $\oU(p,q+1)$, $\oO(p,q+1)$, $\oO_{n+1}(\C)$, $\SO(p,q+1)$, $\SO_{n+1}(\C)$, and let $G'$ be respectively the subgroup $\GL_{n}(\R)$, $\GL_{n}(\C)$, $\oU(p,q)$, $\oO(p,q)$, $\oO_n(\C)$, $\SO(p,q)$, $\SO_n(\C)$, embedded in $G$ in the standard way. We show that every irreducible Casselman-Wallach representation of $G'$ occurs with multiplicity at most one in every irreducible Casselman-Wallach representation of $G$. Similar results are proved for the Jacobi groups $\GL_{n}(\R)\ltimes \oH_{2n+1}(\R)$, $\GL_{n}(\C)\ltimes \oH_{2n+1}(\C)$, $\oU(p,q)\ltimes \oH_{2p+2q+1}(\R)$, $\Sp_{2n}(\R)\ltimes \oH_{2n+1}(\R)$, $\Sp_{2n}(\C)\ltimes \oH_{2n+1}(\C)$, with their respective subgroups $\GL_{n}(\R)$, $\GL_{n}(\C)$, $\oU(p,q)$, $\Sp_{2n}(\R)$, $\Sp_{2n}(\C)$.