Character rigidity of simple algebraic groups

We prove the following extension of Tits' simplicity theorem. Let k be an infinite field, G an algebraic group defined and quasi-simple over k, and G(k) the group of k-rational points of G. Let G(k)(+) be the subgroup of G(k) generated by the unipotent radicals of parabolic subgroups of G defined over k and denote by PG(k)(+) the quotient of G(k)(+) by its center. Then every normalized function of positive type on PG(k)(+) which is constant on conjugacy classes is a convex combination of 1(PG(k)+) and delta(e). As corollary, we obtain that, when k is countable, the only ergodic IRS's (invariant random subgroups) of PG(k)(+) are delta(PG(k)+) and delta({e}). A further consequence is that, when k is a global field and G is k-isotropic and has trivial center, every measure preserving ergodic action of G(k) on a probability space either factorizes through the abelianization G(k)(ab) or is essentially free.