Quasi-flat representations of uniform groups and quantum groups

Given a discrete group $\mathrm{\Gamma }=<g_1,\dots ,g_M>$ and a number $K\in\mathbb N$,aunitaryrepresentation$\rho:\Gamma\to U_K$ 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 $\mathrm{\Gamma }$ form altogether a parametric matrix model $\pi :\mathrm{\Gamma }\to C(X,U_K)$. We compute here the universal model space $X$ for various classes of discrete groups, notably with results in the case where $\mathrm{\Gamma }$ 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 $O_2^{-1}$.