We define a moment map for a torus action on a smooth manifold without a two-form. Cobordisms of such structures are meaningful even if the manifolds are non compact, as long as the moment maps are proper. We prove that a compact manifold with a torus action and a moment map is cobordant the normal bundle of its fixed point set. Two formulas follow easily: Guillemin’s topological version of the...

Let M be a compact manifold with a Hamiltonian T action and moment map Φ. The restriction map in equivariant cohomology from M to a level set Φ(p) is a surjection, and we denote the kernel by Ip. When T has isolated fixed points, we show that Ip distinguishes the chambers of the moment polytope for M . In particular, counting the number of distinct ideals Ip as p varies over different chambers ...

In this paper we give a classification of parabolic subalgebras of simple Lie algebras over C that satisfy two properties. The first property is Lynch’s sufficient condition for the vanishing of certain Lie algebra cohomology spaces for generalized Whittaker modules associated with the parabolic subalgebra and the second is that the moment map of the cotangent bundle of the corresponding genera...

The integration of the exponential of the square of the moment map of the circle action is studied by a direct stationary phase computation and by applying the DuistermaatHeckman formula. Both methods yield two distinct formulas expressing the integral in terms of contributions from the critical set of the square of the moment map. The cohomological pairings on the symplectic quotient, includin...

We consider a connected symplectic manifold M acted on by a connected Lie group G in a Hamiltonian fashion. If G is compact we study the smooth function f =‖ μ ‖. We prove that if a point x ∈ M realizes a local maximum of the squared moment map ‖ μ ‖ then the orbit Gx is symplectic and Gμ(μ(x)) is G-equivariantly symplectomorphic to a product of a flag manifold and a symplectic manifold which i...

Let (M,ω,Φ) be a Hamiltonian T -space and let H ⊆ T be a closed Lie subtorus. Under some technical hypotheses on the moment map Φ, we prove that there is no additive torsion in the integral full orbifoldK-theory of the orbifold symplectic quotient [M//H ]. Our main technical tool is an extension to the case of moment map level sets the well-known result that components of the moment map of a Ha...