Invariant Hilbert schemes and desingularizations of symplectic reductions for classical groups

TL;DR

This paper constructs canonical desingularizations of symplectic reductions for classical groups using invariant Hilbert schemes, linking geometric invariant theory with Lie algebra orbit closures.

Contribution

It introduces a method to resolve singularities of symplectic reductions for classical groups via invariant Hilbert schemes, connecting to nilpotent orbit closures.

Findings

Constructed canonical desingularizations for specific symplectic reductions.

Compared Hilbert-Chow morphism with known symplectic desingularizations.

Established links between invariant Hilbert schemes and nilpotent orbit closures.

Abstract

Let $G \subset GL(V)$ be a reductive algebraic subgroup acting on the symplectic vector space $W=(V \oplus V^*)^{\oplus m}$, and let $\mu:\ W \rightarrow Lie(G)^*$ be the corresponding moment map. In this article, we use the theory of invariant Hilbert schemes to construct a canonical desingularization of the symplectic reduction $\mu^{-1}(0)/\!/G$ for classes of examples where $G=GL(V)$, $O(V)$, or $Sp(V)$. For these classes of examples, $\mu^{-1}(0)/\!/G$ is isomorphic to the closure of a nilpotent orbit in a simple Lie algebra, and we compare the Hilbert-Chow morphism with the (well-known) symplectic desingularizations of $\mu^{-1}(0)/\!/G$.