Weyl groups of Hamiltonian manifolds, I

Abstract

We consider a connected compact Lie group K acting on a symplectic manifold M such that a moment map m exists. A pull-back function via m Poisson commutes with all K-invariants. Guillemin-Sternberg raised the problem to find a converse. In this paper, we solve this problem by determining the Poisson commutant of the algebra of K-invariants. It is completely controlled by the image of m and a certain subquotient W_M of the Weyl group of K. The group W_M is also a reflection group and forms a symplectic analogue of the little Weyl group of a symmetric space. The proof rests ultimately on techniques from algebraic geometry. In fact, a major part of the paper is of independent interest: it establishes connectivity and reducedness properties of the fibers of the (complex algebraic) moment map of a complex cotangent bundle.

33 pages, TeX

Cited by in corpus (8)

• Convexity of Hamiltonian manifolds
• Quantized symplectic actions and W-algebras
• Computation of Weyl groups of G-varieties
• Equivariant symplectic geometry of cotangent bundles, II
• On fibers of algebraic invariant moment maps
• Algebraic Hamiltonian actions
• Computation of weight lattices of G-varieties
• Lifting central invariants of quantized Hamiltonian actions