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Stationary characters on lattices of semisimple Lie groups

Publications mathématiques de l'IHÉS volume 133, pages 1–46 (2021)Cite this article

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.

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Authors and Affiliations

  1. Institut de Mathématiques de Bordeaux, CNRS, Université Bordeaux I, 33405, Talence, France

    Rémi Boutonnet

  2. Université Paris-Saclay, Institut Universitaire de France, Laboratoire de Mathématiques d’Orsay, CNRS, 91405, Orsay, France

    Cyril Houdayer

Authors
  1. Rémi Boutonnet
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  2. Cyril Houdayer
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Correspondence to Cyril Houdayer.

Additional information

RB is supported by a PEPS grant from CNRS and ANR grant AODynG 19-CE40-0008. CH is supported by ERC Starting Grant GAN 637601 and Institut Universitaire de France.

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Boutonnet, R., Houdayer, C. Stationary characters on lattices of semisimple Lie groups. Publ.math.IHES 133, 1–46 (2021). https://doi.org/10.1007/s10240-021-00122-8

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