A real convexity theorem for quasi-hamiltonian actions

TL;DR

This paper proves a convexity theorem for quasi-hamiltonian spaces, showing that the image of fixed-point sets under the momentum map forms a convex polytope, extending classical convexity results to a new setting.

Contribution

It establishes a convexity theorem for quasi-hamiltonian actions, generalizing the O'Shea-Sjamaar convexity theorem to this framework.

Findings

The momentum map image of fixed-point sets is a convex polytope.

The convex polytope coincides with the full momentum polytope.

Application to Lagrangian subspaces in surface group representations.

Abstract

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.