Tensor products of complementary series of rank one Lie groups

Abstract

We consider the tensor product $\pi_{\alpha}\otimes \pi_{\beta}$ 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 $\alpha, \beta$ (in our parametrization). We prove further that for $G=SO_0(n, 1)$ there are finitely many complementary series of the form $\pi_{\alpha+\beta + 2j}$, $j=0, 1, \cdots, k$, appearing in the tensor product $\pi_{\alpha} \otimes \pi_{\beta} $ of two complementary series $\pi_{\alpha}$ and $\pi_{\beta}$, where $k=k(\alpha, \beta, n)$ depends on $\alpha, \beta, n$.