Un th\'eor\`eme de convexit\'e r\'eel pour les applications moment \`a valeurs dans un groupe de Lie

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.