Algebraic Hamiltonian actions

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 invariant moment map ${\psi }_{G,X}:X\to \text{g}$, that is, the composition of the moment map ${\mu }_{G,X}:X\to g:=Lie(G)$ and the quotient morphism $g\to g\text{quo}G$. We obtain some results on the dimensions of fibers of ${\psi }_{G,X}$ and the corresponding morphism of quotients $X\text{quo}G\to g\text{quo}G$. We also study the “Stein factorisation” of ${\psi }_{G,X}$. Namely, let $C_{G,X}$ denote the spectrum of the integral closure of ${\psi }_{G,X}^{\ast }(K[g{]}^G)$ in $K(X{)}^G$. We investigate the structure of the $g\text{quo}G$-scheme $C_{G,X}$. Our results partially generalize those obtained by F. Knop in the case of the actions on cotangent bundles and symplectic vector spaces.