Discrete components in restriction 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 subgroups, revealing discrete components and extending results to exceptional groups.

Contribution

It demonstrates the appearance of discrete components in the restriction of principal series representations of rank one semisimple Lie groups, including exceptional cases.

Findings

Discrete components appear in restrictions of principal series representations.

Results extend to exceptional Lie groups like F4(-20) and Spin(8,1).

Identification of unitarizable and subquotient representations in restrictions.

Abstract

We consider spherical principal series representations of the semisimple Lie group of rank one $G=SO(n, 1; \mathbb K)$, $\mathbb K=\br, \bc, \bh$. There is a family of unitarizable representations $\pi_{\nu}$ of $G$ for $\nu$ in an interval on $\mathbb R^+$, the so-called complementary series, and subquotient or subrepresentations of $G$ for $\nu$ being negative integers. We consider the restriction of $(\pi_{\nu}, G)$ under the subgroup $H=SO(n-1, 1; \mathbb K)$. We prove the appearing of discrete components. The corresponding results for the exceptional Lie group $F_{4(-20)}$ and its subgroup $Spin(8,1)$ are also obtained.