Tensor products of complementary series of rank one Lie groups

We consider the tensor product ${\pi }_{\alpha }\otimes {\pi }_{\beta }$ of complementary series representations ${\pi }_{\alpha }$ and ${\pi }_{\beta }$ of classical rank one groups $S{O}_0(n,1)$, $SU(n,1)$ 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 $G=SO_0(n, 1)$therearefinitelymanycomplementaryseriesoftheform$\pi_{\alpha+\beta

• 2j}$,$j=0, 1, \cdots, k$,appearinginthetensorproduct$\pi_{\alpha} \otimes \pi_{\beta} $oftwocomplementaryseries$\pi_{\alpha}$and$\pi_{\beta}$,where$k=k(\alpha, \beta, n)$dependson$\alpha, \beta, n$.