Multiplicity one theorems: the Archimedean case

Abstract

Let $G$ be one of the classical Lie groups $\mathrm{GL}_{n+1}(\mathbb{R})$, $\mathrm{GL}_{n+1}(\mathbb{C})$, $\mathrm{U}(p,q+1)$, $\mathrm{O}(p,q+1)$, $\mathrm{O}_{n+1}(\mathbb{C})$, $\mathrm{SO}(p,q+1)$, $\mathrm{SO}_{n+1}(\mathbb{C})$, and let $G’$ be respectively the subgroup $\mathrm{GL}_{n}(\mathbb{R})$, $\mathrm{GL}_{n}(\mathbb{C})$, $\mathrm{U}(p,q)$, $\mathrm{O}(p,q)$, $\mathrm{O}_n(\mathbb{C})$, $\mathrm{SO}(p,q)$, $\mathrm{SO}_n(\mathbb{C})$, 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 $\mathrm{GL}_{n}(\mathbb{R})\ltimes \mathrm{H}_{2n+1}(\mathbb{R})$, $\mathrm{GL}_{n}(\mathbb{C})\ltimes \mathrm{H}_{2n+1}(\mathbb{C})$, $\mathrm{U}(p,q)\ltimes \mathrm{H}_{2p+2q+1}(\mathbb{R})$, $\mathrm{Sp}_{2n}(\mathbb{R})\ltimes \mathrm{H}_{2n+1}(\mathbb{R})$, $\mathrm{Sp}_{2n}(\mathbb{C})\ltimes \mathrm{H}_{2n+1}(\mathbb{C})$, with their respective subgroups $\mathrm{GL}_{n}(\mathbb{R})$, $\mathrm{GL}_{n}(\mathbb{C})$, $\mathrm{U}(p,q)$, $\mathrm{Sp}_{2n}(\mathbb{R})$, and $\mathrm{Sp}_{2n}(\mathbb{C})$.

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Authors

Binyong Sun

Academy of Mathematics and Systems Science
Chinese Academy of Sciences
No. 55 Zhongguancun East Road
Beijing 100190
P. R. China

Chen-Bo Zhu

Department of Mathematics
National University of Singapore
Block S17, 10 Lower Kent Ridge Road
Singapore 119076
Republic of Singapore