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

We consider spherical principal series representations of the semisimple Lie group of rank one G=SO(n,1;K)G=SO(n,1;K), K=R,C,HK=R,C,H. There is a family of unitarizable representations πνπν of G for ν   in an interval on RR, the so-called complementary series, and subquotients or subrepresentations of G for ν   being negative integers. We consider the restriction of (πν,G)(πν,G) under the subgroup H=SO(n−1,1;K)H=SO(n−1,1;K). We prove the appearing of discrete components. The corresponding results for the exceptional Lie group F4(−20)F4(−20) and its subgroup Spin0(8,1)Spin0(8,1) are also obtained.