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 $G_{n-1}$.