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 $S{O}_0(n,1)$, $SU(n,1)$ 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 $G=S{O}_0(n,1)$ 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 $\alpha ,\beta ,n$.

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• Invariant differential operators on H-type groups and discrete components in restrictions of complementary series of rank one semisimple groups