La convexité de l'application moment d'un groupe de Lie

Abstract

AbstractLet π be a unitary representation of a Lie group G. The moment mapping Ψπ of π assigns to every C∞ vector ξ in the Hilbert space H of π the linear functional Ψπ(ξ) of the Lie algebra g of G by the rule ψπ(ξ)(X)=1i〈dπ(X)ξ, ξ〉H, X ϵ g In this paper, we study the moment set Iπ of π, i.e., the closure of the image of Ψπ. It is shown that for solvable G, Iπ is always convex and that if furthermore π is irreducible, then Iπ is the closure (in g∗) of the convex hull of the Kirillov-Pukanszky orbit of π. If G is compact and if π is irreducible, then we show that Iπ is the convex hull of the orbit of the highest weight Λ of π, if and only if the number Πi = 1n 〈2Λ − αi, αi〉 is different from 0. Here α1, …, αn denote the simple roots of g