Stationary characters on lattices of semisimple Lie groups
Rémi Boutonnet, Cyril Houdayer
Abstract
We show that stationary characters on irreducible lattices <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Γ</mml:mi> <mml:mo><</mml:mo> <mml:mi>G</mml:mi> </mml:mrow> </mml:math> 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Γ</mml:mi> <mml:mo><</mml:mo> <mml:mi>G</mml:mi> </mml:mrow> </mml:math> , the left regular representation <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>λ</mml:mi> <mml:mi>Γ</mml:mi> </mml:msub> </mml:math> is weakly contained in any weakly mixing representation <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>π</mml:mi> </mml:math> . We prove that for any such irreducible lattice <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Γ</mml:mi> <mml:mo><</mml:mo> <mml:mi>G</mml:mi> </mml:mrow> </mml:math> , any Uniformly Recurrent Subgroup (URS) of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Γ</mml:mi> </mml:math> is finite, answering a question of Glasner–Weiss. We also obtain a new proof of Peterson’s character rigidity result for irreducible lattices <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Γ</mml:mi> <mml:mo><</mml:mo> <mml:mi>G</mml:mi> </mml:mrow> </mml:math> . The main novelty of our paper is a structure theorem for stationary actions of lattices on von Neumann algebras.