Admissibility Conjecture and Kazhdan's Property (T) for quantum groups

TL;DR

This paper advances the understanding of quantum groups by proving the Admissibility Conjecture for a broad class, analyzing Property (T), and establishing new characterizations and implications for finite-dimensional representations and unimodularity.

Contribution

It provides partial solutions to the Admissibility Conjecture, extends Property (T) characterizations, and generalizes key theorems for quantum groups with non-trivial scaling groups.

Findings

Finite-dimensional representations factor through matrix quantum groups.

Characterization of Property (T) via spectral isolation of representations.

Quantum groups with Property (T) are unimodular.

Abstract

We give a partial solution to a long-standing open problem in the theory of quantum groups, namely we prove that all finite-dimensional representations of a wide class of locally compact quantum groups factor through matrix quantum groups (Admissibility Conjecture for quantum group representations). We use this to study Kazhdan's Property (T) for quantum groups with non-trivial scaling group, strengthening and generalising some of the earlier results obtained by Fima, Kyed and So{\l}tan, Chen and Ng, Daws, Skalski and Viselter, and Brannan and Kerr. Our main results are: (i) All finite-dimensional unitary representations of locally compact quantum groups which are either unimodular or arise through a special bicrossed product construction are admissible. (ii) A generalisation of a theorem of Wang which characterises Property (T) in terms of isolation of finite-dimensional irreducible representations in the spectrum. (iii) A very short proof of the fact that quantum groups with Property (T) are unimodular. (iv) A generalisation of a quantum version of a theorem of Bekka--Valette proven earlier for quantum groups with trivial scaling group, which characterises Property (T) in terms of non-existence of almost invariant vectors for weakly mixing representations. (v) A generalisation of a quantum version of Kerr-Pichot theorem, proven earlier for quantum groups with trivial scaling group, which characterises Property (T) in terms of denseness properties of weakly mixing representations.