Character rigidity of simple algebraic groups
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Mathematische Annalen
Character rigidity of simple algebraic groups Bachir Bekka1 Received: 10 November 2019 / Revised: 5 May 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
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 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 δe . 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. Mathematics Subject Classification 20G05 · 22D10 · 22D25 · 22D40
1 Introduction Given a locally compact group G, recall that a continuous function ϕ : G → C is of positive type (or positive definite) if, for all g1 , . . . , gn ∈ G, the matrix (ϕ(g −1 j gi ))1≤i, j≤n is positive semi-definite. As is well-known, such functions are exactly the diagonal matrix coefficients of unitary representations of G in Hilbert spaces (see Sect. 2). The set P(G) of functions ϕ of positive type on G, normalized by the condition ϕ(e) = 1, is convex and the extreme points in P(G) are the diagonal matrix coefficients, given by unit vectors, of irreducible unitary representations of G.
Communicated by Andreas Thom. 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).
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Bachir Bekka [email protected] Univ Rennes, CNRS, IRMAR-UMR 6625, Campus Beaulieu, 35042 Rennes Cedex, France
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B. Bekka
Assume now that G is a discrete group. By results of Glimm [15] and Thoma that is, the set of irreducible unitary [29], the classification of the unitary dual G, representations of G up to unitary equivalence, is hopeless, unless G is virtually abelian. However, the set of characters or traces of G, that we are going to define, seems to be more tractable. Definition 1.1 A function ϕ ∈ P(G) is a trace on G if ϕ is central (that is, constant on conjugacy classes of G). The set Tr(G) of traces on G is a convex compact subset of the unit ball of ∞ (G) for the topology of pointwise convergence. An extreme point of Tr(G) is a character of G. We denote by Char(G) the set of characters of G. Several authors studied the problem of the description of Char(G) for various discrete groups (see e.g. [4,11,17,20–22,25,28]). Traces on groups arise in various settings. An immediate example of a trace on G is the usual normalized trace associated t
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