Invariant Hilbert schemes and desingularizations of symplectic reductions for classical groups
Résumé
Let $G \subset GL(V)$ be a reductive algebraic subgroup acting on the symplectic vector space $W=V^{\oplus n'} \oplus V*^{ \oplus n'}$, and $\mu: W \rightarrow Lie(G)*$ the corresponding moment map. In this article, we use the theory of invariant Hilbert schemes to construct canonical desingularizations of the irreducible components of the symplectic reduction $\mu^{-1}(0)//G$ for classes of examples where $G=GL(V)$, $O(V)$, or $Sp(V)$. Such components are isomorphic to closures of nilpotent orbits in simple Lie algebras and, for these classes of examples, we determine all the cases where the Hilbert-Chow morphism is a symplectic desingularization.
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