Unitary spherical representations of Drinfeld doubles

TL;DR

This paper classifies irreducible spherical unitary representations of the Drinfeld double of q-deformed compact Lie groups, specifically providing a complete classification for SU_q(3), and demonstrates property (T) for certain quantum groups.

Contribution

It offers a complete classification of spherical unitary representations for the Drinfeld double of SU_q(3) and establishes property (T) for the Drinfeld double of SU_q(2n+1).

Findings

Complete classification of spherical representations for SU_q(3).

Proof that the Drinfeld double of SU_q(2n+1) has property (T).

Implication of central property (T) for the dual of SU_q(2n+1).

Abstract

We study irreducible spherical unitary representations of the Drinfeld double of a $q$-deformation of a connected simply connected compact Lie group, which can be considered as a quantum analogue of the complexification of the Lie group. In the case of $SU_q(3)$, we give a complete classification of such representations. As an application, we show the Drinfeld double of the quantum group $SU_q(2n+1)$ has property (T), which also implies central property (T) of the dual of $SU_q(2n+1)$.