Sur les champs de vecteurs invariants sur l'espace tangent d'un espace symétrique

2013/08/23 by Abderrazak Bouaziz, Bouaziz, Abderrazak, Nouri Kamoun +1
Mathematics · #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic Geometry and Number Theory #math.GR #math.RT

paper · pdf · doi:10.48550/arxiv.1308.5108

18 pages, in French

arxiv created 2013/08/23 · arxiv updated 2013/08/26

Abstract

Let G be a real reductive connected Lie group and σ an involution of G. Let H denote the identity component of the group of fixed points of σ, \mathfrak g the Lie algebra of G and \mathfrak q the -1 eigenspace of σ in \mathfrak g. The group H acts naturally on \mathfrak q via the adjoint representation. Let C^∞(\mathfrak q)^H denote the algebra of H-invariant smooth functions on \mathfrak q, and \mathfrak X(\mathfrak q)^H the space of H-invariant smooth vetor fields on \mathfrak q. Any vetor field X∈ \mathfrak X(\mathfrak q)^H defines naturally a derivation D_X of the algebra C^∞(\mathfrak q)^H. We prove that the image of the map X↦ D_X is the set of derivations of the algebra C^∞(\mathfrak q)^H preserving the ideal \itΦC^∞(\mathfrak q)^H of C^∞(\mathfrak q)^H, where \itΦ is a discriminant function on \mathfrak q.

Related

• Involutions of reductive Lie algebras in positive characteristic
• Invariant hypersurfaces for derivations in positive characteristic
• Vecteurs distributions H-invariants de representations induites, pour un espace symetrique reductif p-adique G/H
• Harmonic analysis on real reductive symmetric spaces
• The algebra of K-invariant vector fields on a symmetric space G/K