  The main result of this paper is a quasi-hamiltonian analogue of a special
case of the O'Shea-Sjamaar convexity theorem for usual momentum maps. We denote
by U a simply connected compact connected Lie group and we fix an involutive
automorphism of maximal rank on this Lie group (such an automorphism always
exists). We then denote by M a quasi-hamiltonian U-space and we prove that the
image under the momentum map of the fixed-point set of a form-reversing
compatible involution of M is a convex polytope, which is in fact equal to the
full momentum polytope. This theorem was announced in arXiv:math/0609517v1. As
an application, we obtain an example of lagrangian subspace in representation
spaces of surface groups.
