Stationary characters on lattices of semisimple Lie groups

TL;DR

This paper proves that stationary characters on higher-rank semisimple Lie group lattices are conjugation invariant, leading to new insights in representation theory, ergodic theory, and operator algebras, including a structure theorem for stationary actions.

Contribution

It establishes conjugation invariance of stationary characters on irreducible lattices in higher-rank semisimple Lie groups and introduces a novel structure theorem for stationary actions on von Neumann algebras.

Findings

Stationary characters are conjugation invariant on these lattices.

The left regular representation is weakly contained in any weakly mixing representation.

Any uniformly recurrent subgroup of such a lattice is finite.

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.