Comparison of unitary duals of Drinfeld doubles and complex semisimple Lie groups

TL;DR

This paper analyzes the unitary representation theory of Drinfeld doubles of q-deformed compact Lie groups, revealing new properties and explicitly determining the unitary duals for certain quantum groups, with implications for property (T).

Contribution

It provides a detailed description of the unitary duals of Drinfeld doubles of q-deformed compact Lie groups and establishes property (T) for their duals, advancing understanding of quantum group representations.

Findings

Unitary duals of Drinfeld doubles are characterized in terms of complexified Lie groups.

All q-deformations of higher rank compact Lie groups have property (T).

The unitary dual of SL_q(n,C) is explicitly determined.

Abstract

We determine a substantial part of the unitary representation theory of the Drinfeld double of a $q$-deformation of a compact Lie group in terms of the complexification of the compact Lie group. Using this, we show that the dual of every $q$-deformation of a higher rank compact Lie group has central property ${\rm (T)}$. We also determine the unitary dual of $SL_q(n,\mathbb{C})$.