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Invariant Hilbert schemes and desingularizations of quotients by classical groups

Let $W$ be a finite-dimensional representation of a reductive algebraic group $G$. The invariant Hilbert scheme $\mathcal{H}$ is a moduli space that classifies the $G$-stable closed subschemes $Z$ of $W$ such that the affine algebra $k[Z]$ is the direct sum of simple $G$-modules with prescribed multiplicities. In this article, we consider the case where $G$ is a classical group acting on a classical representation $W$ and $k[Z]$ is isomorphic to the regular representation of $G$ as a $G$-module. We obtain families of examples where $\mathcal{H}$ is a smooth variety, and thus for which the Hilbert-Chow morphism $γ: \mathcal{H} \rightarrow W//G$ is a canonical desingularization of the categorical quotient.