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