2010/07/21 by Bertrand Lemaire, Lemaire, Bertrand · 1 citation
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Advanced Topics in Algebra
paper · pdf · doi:10.48550/arxiv.1007.3576
Let F be a non--Archimedean locally compact field (\rm car(F)≥ 0), \bf G be a connected reductive group defined over F, θ be an F--automorphism of \bf G, and ω be a character of \bf G(F). We fix a Haar measure dg on \bf G(F). For a smooth irreducible (θ,ω)--stable complex representation π of \bf G(F), that is such that π∘ θ≃ π⊗ ω, the choice of an isomorphism A from π⊗ ω to π∘ θ defines a distribution ΘπA, called the \og (A--)twisted character of π\fg: for a compactly supported locally constant function f on \bf G(F), we put ΘπA(f)=\rm trace(π(fdg)∘ A). In this paper, we study these distributions ΘπA, without any restrictive hypothesis on F, \bf G or θ. We prove in particular that the restriction of ΘπA on the open dense subset of \bf G(F) formed of those elements which are θ--quasi--regular is given by a locally constant function, and we describe how this function behaves with respect to parabolic induction and Jacquet restriction. This leads us to take up again the Steinberg theory of automorphisms of an algebraic group, from a rationnal point of view.