Narrow Results

Actions of Lie groups on presymplectic manifolds are analyzed, introducing the suitable comomentum and momentum maps. The subsequent theory of reduction of presymplectic dynamical systems with symmetry is studied. In this way, we give a method of reduction which enables us to remove gauge symmetries as well as non-gauge "rigid" symmetries at once. This method is compared with other step-by-step reduction procedures. As particular examples in this framework, we discuss the reduction of time-dependent dynamical systems with symmetry, the reduction of a mechanical model of field theories with gauge and non-gauge symmetries, and the gauge reduction of the system made of a conformal particle.

Suppose an irreducible unitary representation π of a Lie group G is obtained as a geometric quantization of a coadjoint orbit in the Kirillov–Kostant–Duflo orbit philosophy. Let H be a closed subgroup of G, and we compare the following two conditions.

(1) The restriction π|_H is discretely decomposable in the sense of Kobayashi.

(2) The momentum map is proper.

In this article, we prove that (1) is equivalent to (2) when π is any holomorphic discrete series representation of scalar type of a semisimple Lie group G and (G, H) is any symmetric pair.

The main result of this paper is Theorem 2.12 which says that the quotient μ^-1({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 M_K 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 quasi-hamiltonian space (M, ω, μ: M → U), the associated quotient M//U := μ^-1({1})/U is a disjoint union of symplectic manifolds (Theorem 2.12):