Property (T), finite-dimensional representations, and generic representations

Abstract

Let $G$ be a discrete group with property (T). It is a standard fact that, in a unitary representation of $G$ on a Hilbert space $\mathcal{H}$, almost invariant vectors are close to invariant vectors, in a quantitative way. We begin by showing that, if a unitary representation has some vector whose coefficient function is close to a coefficient function of some finite-dimensional unitary representation $\sigma$, then the vector is close to a sub-representation isomorphic to $\sigma$: this makes quantitative a result of P.S. Wang [Wa]. We use that to give a new proof of a result by D. Kerr, H. Li and M. Pichot [KLP], that a group $G$ with property (T) and such that $C^*(G)$ is residually finite-dimensional, admits a unitary representation which is generic (i.e. the orbit of this representation in $\mathrm{Rep}(G,\mathcal{H})$ under the unitary group $U(\mathcal{H})$ is comeager). We also show that, under the same assumptions, the set of representations equivalent to a Koopman representation is comeager in $\mathrm{Rep}(G,\mathcal{H})$.