Stationary characters on lattices of semisimple Lie groups

Rémi Boutonnet; Cyril Houdayer

doi:10.1007/s10240-021-00122-8

Article

Stationary characters on lattices of semisimple Lie groups

Rémi Boutonnet ¹; Cyril Houdayer ¹

¹

Publications Mathématiques de l'IHÉS, Volume 133 (2021), pp. 1-46

Abstract

We show that stationary characters on irreducible lattices $Γ < G$ 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 $Γ < G$ , the left regular representation $λ_{Γ}$ is weakly contained in any weakly mixing representation $π$ . We prove that for any such irreducible lattice $Γ < G$ , any Uniformly Recurrent Subgroup (URS) of $Γ$ is finite, answering a question of Glasner–Weiss. We also obtain a new proof of Peterson’s character rigidity result for irreducible lattices $Γ < G$ . The main novelty of our paper is a structure theorem for stationary actions of lattices on von Neumann algebras.

Received: 2019-09-18
Accepted: 2021-02-15
Online First: 2021-03-02
Published online: 2021-06-07

Zbl

DOI: 10.1007/s10240-021-00122-8

Author's affiliations:

Rémi Boutonnet ¹; Cyril Houdayer ¹

¹

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     title = {Stationary characters on lattices of semisimple {Lie} groups},
     journal = {Publications Math\'ematiques de l'IH\'ES},
     pages = {1--46},
     year = {2021},
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Rémi Boutonnet; Cyril Houdayer. Stationary characters on lattices of semisimple Lie groups. Publications Mathématiques de l'IHÉS, Volume 133 (2021), pp. 1-46. doi: 10.1007/s10240-021-00122-8

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