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 invariant moment map $ψ_{G,X}:X\to \g$, that is, the composition of the moment map ${\mu }_{G,X}:X\to g:=Lie(G)$ and the quotient morphism $g\to g\quo G$. We obtain some results on the dimensions of fibers of ${\psi }_{G,X}$ and the corresponding morphism of quotients $X\quo G\to g\quo G$. 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 $g\quo G$-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.

v1 46 pages, v2 37 pages, major corrections are made, Theorem 1.5 and its proof are removed, v3 38 pages, final version to appear in Math. Z

References in corpus (3)

• Invariant functions on symplectic representations
• Equivariant symplectic geometry of cotangent bundles, II
• On fibers of algebraic invariant moment maps