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 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 \(\mathbf {1}_{PG(k)^+}\) 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 \(G(k)_{\mathrm{ab}}\) or is essentially free.
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Acknowledgements
We are grateful to Jesse Peterson for a useful suggestion concerning the proof of Corollary D.ii.
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Communicated by Andreas Thom.
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The author acknowledges the support by the ANR (French Agence Nationale de la Recherche) through the projects Labex Lebesgue (ANR-11-LABX-0020-01) and GAMME (ANR-14-CE25-0004)
