Connectivity properties of moment maps on based loop groups

Let $G$ be a connected compact Lie group, and let $M$ be a connected Hamiltonian $G$-manifold with equivariant moment map $\phi$. We prove that if there is a simply connected orbit $G\cdot x$, then $\pi_1(M)\cong\pi_1(M/G)$; if additionally $\phi$ is proper, then $\pi_1(M)\cong\pi_1(\phi^{-1}(G\cdot a))$, where $a=\phi(x)$. We also prove that if a maximal torus of $G$ has a fixed point $x$, then $\pi_1(M)\cong\pi_1(M/K)$, where $K$ is any connected subgroup of $G$; if add...

The convexity theorem of Atiyah and Guillemin-Sternberg says that any connected compact manifold with Hamiltonian torus action has a moment map whose image is the convex hull of the image of the fixed point set. Sjamaar-Lerman proved that the Marsden-Weinstein reduction of a connected Hamitonian $G$-manifold is a stratified symplectic space. Suppose $1\ra A\ra G\ra T\ra 1$ is an exact sequence of compact Lie groups and $T$ is a torus. Then the reduction of a Hamiltonian $G$-m...

We study generalized moment maps for a Hamiltonian action on a connected compact $H$-twisted generalized complex manifold introduced by Lin and Tolman and prove the convexity and connectedness properties of the generalized moment maps for a Hamiltonian torus action.

Let $(G,K)$ be a Riemannian symmetric pair of maximal rank, where $G$ is a compact simply connected Lie group and $K$ the fixed point set of an involutive automorphism $\sigma$. This induces an involutive automorphism $\tau$ of the based loop space $\Omega(G)$. There exists a maximal torus $T\subset G$ such that the canonical action of $T\times S^1$ on $\Omega(G)$ is compatible with $\tau$ (in the sense of Duistermaat). This allows us to formulate and prove a version of Duist...

This paper studies Hamiltonian circle actions, i.e. circle subgroups of the group Ham(M,\om) of Hamiltonian symplectomorphisms of a closed symplectic manifold (M,\om). Our main tool is the Seidel representation of \pi_1(\Ham(M,\om)) in the units of the quantum homology ring. We show that if the weights of the action at the points at which the moment map is a maximum are sufficiently small then the circle represents a nonzero element of \pi_1(\Ham(M,\om)). Further, if the isot...

We consider presymplectic manifolds equipped with Hamiltonian G-actions, G being a connected compact Lie group. A presymplectic manifold is foliated by the integral submanifolds of the kernel of the presymplectic form. For a presymplectic Hamiltonian G-manifold, Lin and Sjamaar propose a condition under which they show that the moment map image has the same ``convex and polyhedral" property as the moment map image of a symplectic Hamiltonian G-manifold, a result proved indepe...

Consider a Hamiltonian action of a compact Lie group on a compact symplectic manifold. A theorem of Kirwan's says that the image of the momentum mapping intersects the positive Weyl chamber in a convex polytope. I present a new proof of Kirwan's theorem, which gives explicit information on how the vertices of the polytope come about and on how the shape of the polytope near any point can be read off from infinitesimal data on the manifold. It also applies to some interesting ...

We extend the famous convexity theorem of Atiyah, Guillemin and Sternberg to the case of non-Hamiltonian actions. We show that the image of a generalized momentum map is a bounded polytope times a vector space. We prove that this picture is stable for small perturbations of the symplectic form. We also observe that an n-dimensional torus symplectic action on a 2n-dimensional symplectic manifold, with fixed point, is Hamiltonian. We finally prove that an extension of Kirwan's ...

For a smooth projective unitary representation of a locally convex Lie group G, the projective space of smooth vectors is a locally convex Kaehler manifold. We show that the action of G on this space is weakly Hamiltonian, and lifts to a Hamiltonian action of the central U(1)-extension of G obtained from the projective representation. We identify the non-equivariance cocycles obtained from the weakly Hamiltonian action with those obtained from the projective representation, a...

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...