Algebraic Hamiltonian actions

Abstract

Abstract. In this paper we deal with a Hamiltonian action of a reductive algebraic group G on an irreducible normal affine Poisson variety X. We study the quotient morphism µ G

On fibers of algebraic invariant moment maps

“…Subsection 2.2 is devoted to conical Hamiltonian varieties introduced in [Lo2]. In Subsection 2.3 we study a local structure of Hamiltonian actions.…”

“…Let us, at first, define two important invariants of a Hamiltonian variety: its Cartan space and Weyl group. The proofs of the facts below concerning these invariants can be found in [Lo2], Subsection 5.2. Let L be the principal centralizer and X L an L-cross-section of a Hamiltonian G-variety…”

“…By Remark 3.1.2 from [Lo2], the action G 0 : X 0 is Hamiltonian with the moment map µ G 0 ,X 0 := p • µ e G 0 ,X 0 , where p denotes the natural projection g 0 → g 0 .…”

“…On the other hand, Theorems 1.2.5,1.2.7 from [Lo2] describe the image of ψ G,X . This description is particularly easy when X satisfies some additional conditions that can be described as a presence of a grading on C[X] compatible with the structure of a Hamiltonian variety.…”

“…If X is conical, then C G,X is a quotient of a vector space by a finite group and ψ G,X is surjective, see [Lo2], Theorem 1.2.7. More precisely, there is a subspace a ⊂ g (called the Cartan space of X) and a subgroup W ⊂ N G (a)/Z G (a) (the Weyl group) such that C G,X ∼ = a/W and the finite morphism τ G,X : C G,X → g//G is induced by the embedding a ֒→ g. So the subspace a ⊂ g and the group W encode the difference between ψ G,X and ψ G,X .…”