Algebraic Hamiltonian actions

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 $${\mu_{G,X}//G : X//G \rightarrow \mathfrak{g} //G}$$ of the moment map $${\mu_{G,X} : X\rightarrow \mathfrak{g}}$$ . We prove that for a wide class of Hamiltonian actions (including, for example, actions on generically symplectic varieties) all fibers of the morphism μ G,X //G have the same dimension. We also study the “Stein factorization” of μ G,X //G. Namely, let C G,X denote the spectrum of the integral closure of $${\mu_{G,X}^{*}(\mathbb{K}[\mathfrak{g}]^G)}$$ in $${\mathbb{K}(X)^G}$$ . We investigate the structure of the $${\mathfrak{g} //G}$$ -scheme C G,X . Our results partially generalize those obtained by F. Knop for the actions on cotangent bundles and symplectic vector spaces.