Restriction to symmetric subgroups of unitary representations of rank one semisimple Lie groups

TL;DR

This paper investigates how certain unitary representations of rank one semisimple Lie groups decompose when restricted to symmetric subgroups, revealing finitely many discrete components through holomorphic series embeddings.

Contribution

It establishes the finiteness of discrete components in restrictions of spherical complementary series of rank one Lie groups by embedding into holomorphic discrete series.

Findings

Finitely many discrete components in restrictions of complementary series.

Embedding into holomorphic discrete series facilitates analysis.

Branching rules derived for specific subgroup restrictions.

Abstract

We consider the spherical complementary series of rank one Lie groups $H_n=\SO_0(n, 1; \mathbb F)$ for $\mathbb F=\mathbb R, \mathbb C, \mathbb H$. We prove that there exist finitely many discrete components in its restriction under the subgroup $H_{n-1}=\SO_0(n-1, 1; \mathbb F)$. This is proved by imbedding the complementary series into analytic continuation of holomorphic discrete series of $G_n=SU(n, 1)$, $SU(n, 1)\times SU(n, 1)$ and $SU(2n, 2)$ and by the branching of holomorphic representations under the corresponding subgroup $G_{n-1}$.