Weyl groups of Hamiltonian manifolds, I

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 (1997)

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Remark: The paper is very incoherent in nature. It combines parts with purely topological, differentiable, and algebraic geometric content. Therefore, I abandoned the idea to publish it in a journal. Instead, I split it in four different parts. So far the topological part (covering §2) and the first algebraic part (covering §6) are finished.

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Last updated: August 10, 2001