On fibers of algebraic invariant moment maps

Abstract

Abstract. In this paper we study some properties of fibers of the invariant moment map for a Hamiltonian action of a reductive group on an affine symplectic variety. We prove that all fibers have equal dimension. Further, under some additional restrictions, we show that the quotients of fibers are irreducible normal schemes.

Computation of Weyl groups of 𝐺-varieties

“…For a subgroup Γ ⊂ W (g) we denote by ∆ Γ the subset of ∆(g) consisting of all α with s α ∈ Γ. Proposition 3.3.17 ([Lo3,Corollaries 4.16,4.19]). …”

“…At first, we consider the case when X is affine. Here we present some results on W G,X obtained in [Lo3]. Those results can be applied here because W G,X coincides up to conjugacy with the Weyl group of the Hamiltonian G-variety T * X (see [Kn4]).…”

“…Restrictions of the first type are valid for smooth affine varieties. They are derived from results of [Lo3]. The computation in Section 5 is based on these restrictions.…”

Computation of weight lattices of G-varieties

“…Let V be a symplectic vector space and G be a reductive group acting on V by linear symplectomorphisms. Then the action [Lo1] or [Lo7], Example 2.5. By the model variety M G (H, η, V ) we mean the set of all points of X, where ω is nondegenerate.…”

“…As in the computation of Weyl groups in [Lo5], the main ingredient in the computation of the lattices Λ G,G/H is the theory of Hamiltonian actions of reductive groups developed in [K1], [Lo1], [Lo2], [Lo6], [Lo7].…”