Tensor products of complementary series of rank one Lie groups

Abstract

We consider the tensor product $π_α\otimes π_β$ 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 $G=SO_0(n, 1)$ there are finitely many complementary series of the form $π_{α+β+ 2j}$, $j=0, 1, \cdots, k$, appearing in the tensor product $π_α \otimes π_β $ of two complementary series $π_α$ and $π_β$, where $k=k(α, β, n)$ depends on $α, β, n$.

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