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 $PG(k)^+$ 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 $δ_e.$ As corollary, we obtain that the only ergodic invariant random subgroups (IRS) of $PG(k)^+$ are $δ_{PG(k)^+}$ and $δ_{\{e\}},$ when $k$ is countable. 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 of $G(k)$ or is essentially free.

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