Stationary characters on lattices of semisimple Lie groups

Abstract We show that stationary characters on irreducible lattices $\Gamma < G$ Γ < G of higher-rank connected semisimple Lie groups are conjugation invariant, is, they genuine characters. This result has several applications in representation theory, operator algebras, ergodic theory and topological dynamics. In particular, we for any such lattice , the left regular $\lambda _{\Gamma }$ λ is weakly contained mixing $\pi $ π . prove Uniformly Recurrent Subgroup (URS) finite, answering a question Glasner–Weiss. also obtain new proof Peterson’s character rigidity The main novelty our paper structure theorem actions von Neumann algebras.