Journal of Lie Theory 26 (2016), No. 2, 497--533
Copyright Heldermann Verlag 2016
Nested Punctual Hilbert Schemes and Commuting Varieties of Parabolic Subalgebras
Michaël Bulois
Institut Camille Jordan, Université Jean Monnet, Maison de l'Université, 10 rue Tréfilerie, 42023 Saint-Etienne Cedex 2, France
michael.bulois@univ-st-etienne.fr
Laurent Evain
Dép. de Maths, Faculté des Sciences, Université d'Angers, 2 Boulevard Lavoisier, 49045 Angers Cedex 01, France
laurent.evain@univ-angers.fr
It is known that the variety parametrizing pairs of commuting nilpotent matrices is irreducible and that this provides a proof of the irreducibility of the punctual Hilbert scheme in the plane. We extend this link to the nilpotent commuting variety of some parabolic subalgebras of Mn(K) and to the punctual nested Hilbert scheme. By this method, we obtain a lower bound on the dimension of these moduli spaces. We characterize the cases where they are irreducible. In some reducible cases, we describe the irreducible components and their dimensions.
Keywords: Hilbert scheme, Commuting variety, GIT, parabolic algebra, nilpotent orbit.
MSC: 14C05, 14L30, 14L24, 17B08, 15A27
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