Invariant Hilbert schemes and desingularizations of symplectic reductions for classical groups

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