Consequences of quasi-free (Q921669)

An action of an n-dimensional torus T on a complex manifold X is said to be quasi-free if there is a T-invariant subset S of X on which T acts freely, such that the orbit of every point \(p\in X-S\) has dimension \(<n\). One commonly makes this assumption when discussing the reduced spaces of a Hamiltonian action of T on a symplectic manifold, in order for the reduced spaces to be tractable. This paper discusses consequences of this assumption, showing that it is equivalent to a certain integrality condition on the induced action on the tangent spaces to the fixed points. In practice, many interesting T-actions occur as the restrictions of a Hamiltonian action of a compact Lie group G to its maximal torus. (Such G-actions are the quantized version of representations of G.) They find conditions under which these T-actions are quasi-free. This is relevant to the problem of understanding the ``symplectic character'' of the G-action.