Tensor products of complementary series of rank one Lie groups

Abstract

ABSTRACT. We consider the tensor product π α ⊗ π β of complementary series representations π α and π β of classical rank one groups SO 0 (n, 1), SU (n, 1) and Sp(n, 1). We prove that there is a discrete component π α+β for small parameters α, β (in our parametrization). We prove further that for SO o (n, 1) there are finitely many complementary series of the form π α+β+2j , j = 0, 1, · · · , k, appearing in the tensor product π α ⊗ π β of two complementary series π α and π β , where k = k(α, β, n) depends on α, β, n.