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

0498908 - MÚ 2019 RIV DE eng J - Článek v odborném periodiku
Doucha, Michal - Malicki, M. - Valette, A.
Property (T), finite-dimensional representations, and generic representations.
Journal of Group Theory. Roč. 22, č. 1 (2019), s. 1-13. ISSN 1433-5883. E-ISSN 1435-4446
Grant CEP: GA ČR GF16-34860L
Institucionální podpora: RVO:67985840
Klíčová slova: generic representations
Obor OECD: Applied mathematics
Impakt faktor: 0.466, rok: 2019
https://www.degruyter.com/view/j/jgth.2019.22.issue-1/jgth-2018-0030/jgth-2018-0030.xml

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 Rep(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})}.
Trvalý link: http://hdl.handle.net/11104/0291195