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

Abstract

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 ℋ {\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 σ, then the vector is close to a sub-representation isomorphic to σ: this makes quantitative a result of P. S. Wang. We use that to give a new proof of a result by D. Kerr, H. Li and M. Pichot, that a group G with Property (T) and such that C * ⁢ ( G ) {C^{*}(G)} is residually finite-dimensional, admits a unitary representation which is generic (i.e. the orbit of this representation in R ⁢ e ⁢ p ⁢ ( G , ℋ ) {Rep(G,\mathcal{H})} under the unitary group U ⁢ ( ℋ ) {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 Rep ⁢ ( G , ℋ ) {\mathrm{Rep}(G,\mathcal{H})} .