Tensor products of complementary series of rank one Lie groups

TL;DR

This paper investigates the tensor products of complementary series representations of rank one Lie groups, establishing conditions for the existence of discrete components and describing their finite multiplicity structure.

Contribution

It proves the existence of discrete components in tensor products of complementary series and characterizes their finite multiplicity structure for classical rank one groups.

Findings

Existence of a discrete component bla_{ ext{alpha}+eta} for small parameters bla_{ ext{alpha}, eta}

Finiteness of complementary series components bla_{ ext{alpha}+eta + 2j} in tensor products for SO_0(n,1)

Dependence of the number of components on parameters bla_{ ext{alpha}, eta, n}

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 $\alpha, \beta, n$.