We consider spherical principal series representations of the semisimple Lie group of rank one G = SO(n, 1; K), K = R, C, H. There is a family of unitarizable representations π ν of G for ν in an interval on R + , the so-called complementary series, and subquotients or subrepresentations of G for ν being negative integers. We consider the restriction of (π ν , G) under the subgroup H = SO(n − 1, 1; 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.

ABSTRACT. We consider the spherical complementary series of rank one Lie groups H n = SO 0 (n, 1; F) for F = R, C, H. We prove that there exist finitely many discrete components in its restriction under the subgroup H n−1 = SO 0 (n − 1, 1; 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) × SU (n, 1) and SU (2n, 2) and by the branching of holomorphic representations under the corresponding subgroup G n−1 .

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