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Invariant Hilbert schemes and desingularizations of symplectic reductions for classical groups

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

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Acknowledgments

I am deeply thankful to Michel Brion for proposing this subject to me, for a lot of helpful discussions, and for his patience. I thank Tanja Becker for exchange of knowledge on invariant Hilbert schemes by e-mail and during her stay in Grenoble in October 2010. I also thank Bart Van Steirteghem for helpful discussions during his stay in Grenoble in Summer 2011.

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  1. Institut Fourier, UMR 5582 CNRS-UJF, Université Grenoble I, BP 74, 38402 , St. Martin d’Hères Cédex, France

    Ronan Terpereau

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  1. Ronan Terpereau
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Correspondence to Ronan Terpereau.

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Terpereau, R. Invariant Hilbert schemes and desingularizations of symplectic reductions for classical groups. Math. Z. 277, 339–359 (2014). https://doi.org/10.1007/s00209-013-1259-1

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  • DOI: https://doi.org/10.1007/s00209-013-1259-1

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