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 $W=(V\oplus V^{\ast }{)}^{\oplus m}$, and let $\mu :W\to Lie(G{)}^{\ast }$ 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 ${\mu }^{-1}(0)\mathrm{/}\text{\hspace{0.17em}⁣}\mathrm{/}G$ for classes of examples where $G=GL(V)$, $O(V)$, or $Sp(V)$. For these classes of examples, ${\mu }^{-1}(0)\mathrm{/}\text{\hspace{0.17em}⁣}\mathrm{/}G$ 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 ${\mu }^{-1}(0)\mathrm{/}\text{\hspace{0.17em}⁣}\mathrm{/}G$.