I will present recent joint work with Cyril Houdayer about a stationary version of characters on discrete groups. Our main result states that such stationary characters on lattices of semisimple Lie groups are actually genuine characters. I will present applications of this result to representation theory and operator algebras, as well as topological dynamics. I will also mention a result in non-commutative ergodic theory, which is the key to our approach and is of independent interest.