  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.
