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

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 $G_{n-1}$.