A Real Convexity Theorem for Quasi-hamiltonian Actions

When (M,ω, μ : M → U) is a quasi-hamiltonian U -space with U a compact connected and simply connected Lie group, the intersection of μ(M) with the exponential exp(W) of a closed Weyl alcove W ⊂ u = Lie(U) is homeomorphic, via the exponential map, to a convex polytope of u ([AMM98]). In this paper, we fix an involutive automorphism τ of U such that the involution τ− : u 7→ τ(u) leaves a maximal torus T ⊂ U pointwise fixed (such an involutive automorphism always exists on a given compact connected Lie group U). We then show (theorem 3.3) that if β is a form-reversing involution on M with non-empty fixed-point set M , compatible with the action of (U, τ) and with the momentum map μ, then we have the equality μ(M)∩exp(W) = μ(M)∩exp(W). In particular, μ(M)∩ exp(W) is a convex polytope. This theorem is a quasi-hamiltonian analogue of the O’Shea-Sjamaar theorem ([OS00]) when the symmetric pair (U, τ) is of maximal rank. As an application of this result, we obtain an example of lagrangian subspace in representation spaces of surfaces groups (theorem 5.3).