Quasi-hamiltonian Quotients as Disjoint Unions of Symplectic Manifolds

The main result of this paper is Theorem 2.13 which says that the quotient μ({1})/U associated to a quasi-hamiltonian space (M,ω, μ : M → U) has a symplectic structure even when 1 is not a regular value of the momentum map μ. Namely, it is a disjoint union of symplectic manifolds of possibly different dimensions, which generalizes the result of Alekseev, Malkin and Meinrenken in [AMM98]. We illustrate this theorem with the example of representation spaces of surface groups. As an intermediary step, we give a new class of examples of quasi-hamiltonian spaces: the isotropy submanifold MK whose points are the points of M with isotropy group K ⊂ U . The notion of quasi-hamiltonian space was introduced by Alekseev, Malkin and Meinrenken in their paper [AMM98]. The main motivation for it was the existence, under some regularity assumptions, of a symplectic structure on the associated quasi-hamiltonian quotient. Throughout their paper, the analogy with usual hamiltonian spaces is often used as a guiding principle, replacing Lie-algebra-valued momentum maps with Lie-group-valued momentum maps. In the hamiltonian setting, when the usual regularity assumptions on the group action or the momentum map are dropped, Lerman and Sjamaar showed in [LS91] that the quotient associated to a hamiltonian space carries a stratified symplectic structure. In particular, this quotient space is a disjoint union of symplectic manifolds.. In this paper, we prove an analogous result for quasi-hamiltonian quotients. More precisely, we show that for any quasihamiltonian space (M,ω, μ : M → U), the associated quotient M//U := μ({1})/U is a disjoint union of symplectic manifolds (Theorem 2.13):