# Multiplicity one theorems: the Archimedean case

Research paper by **Binyong Sun, Chen-Bo Zhu**

Indexed on: **17 Jun '11**Published on: **17 Jun '11**Published in: **Mathematics - Representation Theory**

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#### Abstract

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 $\GL_{n}(\R)$, $\GL_{n}(\C)$, $\oU(p,q)$, $\Sp_{2n}(\R)$,
$\Sp_{2n}(\C)$.