Tensor products of complementary series of rank one Lie groups

Journal article, 2017

We consider the tensor product pi(alpha) aSu pi(beta) of complementary series representations pi(alpha) and pi(beta) of classical rank one groups SO (0)(n; 1), SU(n; 1) and Sp(n; 1). We prove that there is a discrete component pi(alpha+beta) for small parameters alpha and beta (in our parametrization). We prove further that for SO0(n; 1) there are finitely many complementary series of the form pi(alpha+beta+2j) , j = 0, 1,..., k, appearing in the tensor product pi(alpha) aSu pi(beta) of two complementary series pi(alpha) and pi(beta) where k = k(alpha, beta n) depends on alpha, beta and n.