Stationary characters on lattices of semisimple Lie groups

Abstract

We show that stationary characters on irreducible lattices $\Gamma < G$ Γ < G of higher-rank connected semisimple Lie groups are conjugation invariant, that is, they are genuine characters. This result has several applications in representation theory, operator algebras, ergodic theory and topological dynamics. In particular, we show that for any such irreducible lattice $\Gamma < G$ Γ < G , the left regular representation $\lambda _{\Gamma }$ λ Γ is weakly contained in any weakly mixing representation $\pi $ π . We prove that for any such irreducible lattice $\Gamma < G$ Γ < G , any Uniformly Recurrent Subgroup (URS) of $\Gamma $ Γ is finite, answering a question of Glasner–Weiss. We also obtain a new proof of Peterson’s character rigidity result for irreducible lattices $\Gamma < G$ Γ < G . The main novelty of our paper is a structure theorem for stationary actions of lattices on von Neumann algebras.

Charmenability of higher rank arithmetic groups

“…We will prove that this is incompatible with claim 2 by adapting the argument of Ge-Kadison (which applies in factorial situations). Strictly speaking, we could derive the contradiction from Ge-Kadison's result and a direct integral argument as was done in [BH19]. But we provide a different argument here, not involving direct integrals.…”

“…Both the statement and the proof of Theorem C are similar to [BH19,Theorem 5.1], but extra difficulties appear in the proof to handle simple algebraic groups in positive characteristic. Assume that E : M → L ∞ (G/P ) is not invariant.…”

“…The first class of charfinite groups were obtained in [BH19]. More precisely, the main results of [BH19] show that lattices in higher rank connected simple Lie groups with finite center are charfinite. New classes of charmenable and charfinite groups were subsequently obtained in [BBHP20].…”

“…In the commutative setting, this projection is not relevant. This consideration already appeared in [BH19].…”

Noncommutative ergodic theory of higher rank lattices

“…We now present in a unified way the main results we obtained in [13] (joint work with R. Boutonnet) and [2] (joint work with U. Bader, R. Boutonnet and J. Peterson). Our first main result deals with the existence of characters.…”