  We complete the study of characters on higher rank semisimple lattices
initiated in [BH19,BBHP20], 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 [BH19] 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.
