Weyl groups of Hamiltonian manifolds, I

TL;DR

This paper characterizes the Poisson commutant of K-invariant functions on symplectic manifolds with moment maps, revealing its structure is governed by a Weyl group analogue, using algebraic geometry techniques.

Contribution

It determines the Poisson commutant in terms of a new Weyl group analogue, solving a problem posed by Guillemin and Sternberg.

Findings

The Poisson commutant is controlled by the image of the moment map and a subquotient Weyl group.

W_M is a reflection group and a symplectic analogue of the little Weyl group.

Connectivity and reducedness of fibers of complex algebraic moment maps are established.

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.