Character rigidity of simple algebraic groups

Bachir Bekka (1)

1 IRMAR - Institut de Recherche Mathématique de Rennes

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 P G(k) + the quotient of G(k) + by its center. Then every normalized function of positive type on P G(k) + which is constant on conjugacy classes is a convex combination of 1 P G(k) + and δ e. As corollary, we obtain that, when k is countable, the only ergodic IRS's (invariant random subgroups) of P G(k) + are δ P G(k) + and δ {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.

Domaines: Théorie des groupes [math.GR]; Algèbres d'opérateurs [math.OA]

Bachir Bekka. Character rigidity of simple algebraic groups. Mathematische Annalen, 2020, 378 (3-4), pp.1223-1243. ⟨10.1007/s00208-020-02061-x⟩. ⟨hal-02267868v2⟩