Character rigidity of simple algebraic groups

TL;DR

This paper extends Tits' simplicity theorem to algebraic groups over infinite fields, showing rigidity properties of their subgroups and implications for invariant random subgroups and ergodic actions.

Contribution

It proves a new extension of Tits' simplicity theorem for algebraic groups over infinite fields, revealing rigidity of positive type functions and classifying invariant random subgroups.

Findings

Normalized positive type functions are convex combinations of trivial and delta functions.

The only ergodic invariant random subgroups are trivial or singleton.

Ergodic actions either factor through abelianization or are essentially free.

Abstract

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 $\delta_e.$ As corollary, we obtain that the only ergodic invariant random subgroups (IRS) of $PG(k)^+$ are $\delta_{PG(k)^+}$ and $\delta_{\{e\}},$ 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.