Charmenability of higher rank arithmetic groups

We complete the study of characters on higher rank semisimple lattices initiated in [BH21, BBHP22], the missing case being the case of lattices in higher rank simple algebraic groups in arbitrary characteristics. More precisely, we investigate dynamical properties of the conjugation action of such lattices on their space of positive definite functions. Our main results deal with the existence and the classification of characters from which we derive applications to topological dynamics, ergodic theory, unitary representations and operator algebras. Our key theorem is an extension of the noncommutative Nevo–Zimmer structure theorem obtained in [BH21] to the case of simple algebraic groups defined over arbitrary local fields. We also deduce a noncommutative analogue of Margulis’ factor theorem for von Neumann subalgebras of the noncommutative Poisson boundary of higher rank arithmetic groups.