Discrete components in restriction of unitary representations of rank one semisimple Lie groups

Details Discrete components in restriction of unitary representations of rank one semisimple Lie groups Discrete components in restriction of unitary representations of rank one semisimple Lie groups

2011-11-28

arxiv.org/abs/1111.6406v2

Mathematics

Representation Theory

Revised version

Scientific paper

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.

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