Multiplicity one theorems: the Archimedean case

TL;DR

This paper proves multiplicity one theorems for irreducible representations of classical Lie groups and their subgroups, extending known results to a broader class of groups and representations.

Contribution

It establishes new multiplicity one results for Casselman-Wallach representations of classical Lie groups and Jacobi groups, including their subgroups, in the Archimedean setting.

Findings

Every irreducible Casselman-Wallach representation of a subgroup occurs with multiplicity at most one in the larger group.

Results extend multiplicity one theorems to classical Lie groups and Jacobi groups.

Theorems hold for a wide range of groups including real, complex, and quaternionic cases.

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