Un Théorème De Convexité Réel Pour Les Applications Moment À Valeurs Dans Un Groupe De Lie

Dans cette note, on présente l’énoncé et les principales idées de la démonstration d’un théorème de convexité réel pour les applications moment à valeurs dans un groupe de Lie. Ce résultat est un analogue quasi-hamiltonien du théorème de O’Shea et Sjamaar dans le cadre hamiltonien usuel. On démontre que l’image par l’application moment du lieu des points fixes d’une involution renversant la 2-forme de structure d’un espace quasi-hamiltonien est un polytope convexe, et l’on décrit ce polytope comme sous-polytope du polytope moment. In this note, we state and give the main ideas of the proof of a real convexity theorem for group-valued momentum maps. This result is a quasi-Hamiltonian analogue of the O’Shea-Sjamaar theorem in the usual Hamiltonian setting. We prove here that the image under the momentum map of the fixed-point set of a form-reversing involution defined on a quasi-Hamiltonian space is a convex polytope, that we describe as a subpolytope of the momentum polytope. Abridged English version Let (G, (. | .)) be a compact connected Lie group whose Lie algebra g := Lie(G) is endowed with an Ad-invariant (positive definite) scalar product (. | .). We denote by θ = g.dg and by θ = dg.g the Maurer-Cartan 1-forms on G and by χ the Cartan 3-form, that is to say the bi-invariant 3-form defined for X,Y, Z ∈ g = T1G by χ1(X,Y, Z) := ([X,Y ] | Z). When G acts on a manifold M , we denote by X x := d dt |t=0(exp(tX).x) the fundamental vector field associated to X ∈ g by the action of G on M . A quasi-Hamiltonian G-space (M,ω, μ : M → U) is a manifold M , acted upon by the group G, endowed with an invariant 2-form ω such that there exists an equivariant map μ : M → G (for the conjugacy action of G on itself) satisfying : (i) dω = −μχ (ii) for all x ∈ M , kerωx = {X x : X ∈ g |Adμ(x).X = −X} (iii) for all X ∈ g, the interior product ιX♯ω satisfies ιX♯ω = 1 2 μ(θ + θ | X) This notion was introduced by Alekseev, Malkin and Meinrenken in [1] and basic examples include the conjugacy classes of the Lie group G. The map μ is called the momentum map, underlining the important analogies with usual Hamiltonian spaces that one encounters in the study of quasiHamiltonian spaces. In the rest of this paper, the compact connected Lie group G will be assumed to be simply connected as well. In [1], Alekseev, Malkin and Meinrenken showed that in this case the intersection of the image μ(M) of the momentum map μ with a fundamental domain exp(W) for the conjugacy action (W ⊂ g is a Weyl alcove in the Lie algebra of G) is a convex polytope (a result which they derived from Meinrenken and Woodward in [12]). Here, we study the image μ(M) under the momentum map of the fixed-point set M of an involution β defined on M satisfying βω = −ω (we shall say that β reverses the structural 2-form ω) and additional compatibility relations with the action and the momentum map (see theorem 2.1). To formulate these compatibility conditions, the group G is assumed to be endowed with an involutive automorphism τ . Additionally, we assume that the involution τ(g) := τ(g) leaves a maximal torus T of G pointwise fixed. Such an involution always exists on a compact connected simply connected Lie group (see [11]). Under this assumption, we show (theorem 2.1) that the set μ(M) ∩ exp(W) is a convex set, and that it is in fact equal to the whole momentum polytope μ(M) ∩ exp(W). The proof is sketched in section 3 and details are Supported by the Japanese Society for Promotion of Science (JSPS).