Stationary characters on lattices of semisimple Lie groups

Abstract

We show that stationary characters on irreducible lattices $\Gamma <G$ 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 $\Gamma <G$, the left regular representation ${\lambda }_{\Gamma }$ is weakly contained in any weakly mixing representation $\pi$. We prove that for any such irreducible lattice $\Gamma <G$, any uniformly recurrent subgroup (URS) of $\Gamma$ is finite, answering a question of Glasner-Weiss. We also obtain a new proof of Peterson's character rigidity result for irreducible lattices $\Gamma <G$. The main novelty of our paper is a structure theorem for stationary actions of lattices on von Neumann algebras.

33 pages. v2: minor modifications. To appear in Publ. Math. Inst. Hautes Études Sci

References in corpus (4)

• Charmenability of arithmetic groups of product type
• Boundaries, rigidity of representations, and Lyapunov exponents
• Quasi-regular representations of discrete groups and associated C*-algebras
• Characters of the group ${\mathrm{EL}}_d(R)$ for a commutative Noetherian ring $R$

Cited by in corpus (7)

• Charmenability of arithmetic groups of product type
• Noncommutative ergodic theory of higher rank lattices
• Charmenability of higher rank arithmetic groups
• Subalgebras, subgroups, and singularity
• Characters of the group ${\mathrm{EL}}_d(R)$ for a commutative Noetherian ring $R$
• The noncommutative factor theorem for lattices in product groups
• Crossed product splitting of intermediate operator algebras via 2-cocycles