Quasi-flat representations of uniform groups and quantum groups

TL;DR

This paper studies special unitary representations of discrete and quantum groups where eigenvalues are evenly spread among roots of unity, analyzing their universal models and stationarity properties.

Contribution

It computes the universal model space for quasi-flat representations of various groups, including metabelian groups, and explores stationarity in classical and quantum group contexts.

Findings

Universal model space computed for certain groups.

Stationarity results for classical and quantum groups.

Negative results on the structure of these models.

Abstract

Given a discrete group $\Gamma=<g_1,\ldots,g_M>$ and a number $K\in\mathbb N$, a unitary representation $\rho:\Gamma\to U_K$ is called quasi-flat when the eigenvalues of each $\rho(g_i)\in U_K$ are uniformly distributed among the $K$-th roots of unity. The quasi-flat representations of $\Gamma$ form altogether a parametric matrix model $\pi:\Gamma\to C(X,U_K)$. We compute here the universal model space $X$ for various classes of discrete groups, notably with results in the case where $\Gamma$ is metabelian. We are particularly interested in the case where $X$ is a union of compact homogeneous spaces, and where the induced representation $\tilde{\pi}:C^*(\Gamma)\to C(X,U_K)$ is stationary in the sense that it commutes with the Haar functionals. We present several positive and negative results on this subject. We also discuss similar questions for the discrete quantum groups, proving a stationarity result for the discrete dual of the twisted orthogonal group $O_2^{-1}$.