Invariant Hilbert schemes and desingularizations of symplectic reductions for classical groups

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 $$\mu ^{-1}(0)/\!/G$$ .