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Invariant deformations of orbit closures in $ \mathfrak{sl}(n)$


Authors: Sébastien Jansou and Nicolas Ressayre
Journal: Represent. Theory 13 (2009), 50-62
MSC (2000): Primary 14D22, 14L24
DOI: https://doi.org/10.1090/S1088-4165-09-00331-8
Published electronically: March 5, 2009
MathSciNet review: 2485792
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Abstract: We study deformations of orbit closures for the action of a connected semisimple group $ G$ on its Lie algebra $ \mathfrak{g}$, especially when $ G$ is the special linear group.

The tools we use are the invariant Hilbert scheme and the sheets of $ \mathfrak{g}$. We show that when $ G$ is the special linear group, the connected components of the invariant Hilbert schemes we get are the geometric quotients of the sheets of $ \mathfrak{g}$. These quotients were constructed by Katsylo for a general semisimple Lie algebra $ \mathfrak{g}$; in our case, they happen to be affine spaces.


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  • [AB] V. Alexeev and M. Brion, Moduli of affine schemes with reductive group action, J. Algebraic Geom. 14, 83-117, 2005. MR 2092127 (2006a:14017)
  • [Bo] K. Bongartz, Schichten von Matrizen sind rationale Varietdten, Math. Ann. 283, no. 1, 53-64, 1989. MR 973803 (90g:14030)
  • [BC] P. Bravi and S. Cupit-Foutou, Equivariant deformations of the affine multicone over a flag variety, Adv. Math. 217 (2008), no. 6, 2800-2821, MR 2397467 (2009a:14061)
  • [Hr] J. Harris, Algebraic geometry: A first course, GTM 133, Springer-Verlag, 1992. MR 1182558 (93j:14001)
  • [J] S. Jansou, Déformations des cônes de vecteurs primitifs, Math. Ann. 338, no 3, 627-667, 2007. MR 2317933 (2008d:14069)
  • [Ka] P.I. Katsylo, Sections of sheets in a reductive algebraic Lie algebra, Izv. Akad. Nauk SSSR Ser. Mat. 46 no. 3, 477-486, 1982. MR 661143 (84k:17005)
  • [Ko] B. Kostant, Lie group representations on polynomial rings, Amer. J. Math. 85, 327-404, 1963. MR 0158024 (28:1252)
  • [Kr] H. Kraft, Parametrisierung von Konjugationsklassen in $ \sln$, Math. Ann. 234 no. 3, 209-220, 1978. MR 0491855 (58:11047)
  • [KP] H. Kraft and C. Procesi, Closures of conjugacy classes of matrices are normal, Invent. Math. 53 no. 3, 227-247, 1979. MR 549399 (80m:14037)
  • [La] S. Lang, Algebra, GTM 211 Springer-Verlag, pp. xvi+914, 2002. MR 1878556 (2003e:00003)
  • [PV] V. Popov and E. Vinberg, Invariant Theory, Encyclopaedia of Mathematical Sciences, vol. 55, pp. 123-278, Springer-Verlag, 1994.
  • [Sch] G.W. Schwarz, Representations of simple Lie groups with a free module of covariants, Invent. Math. 50 no. 1, 1-12, 1978/79. MR 516601 (80c:14008)
  • [Sha] I.R. Shafarevich, Basic algebraic geometry. 1. Varieties in projective space, Springer-Verlag, 1994. MR 1328833 (95m:14001)
  • [W] J. Weyman, The equations of conjugacy classes of nilpotent matrices, Invent. Math. 98 no. 2, 229-245, 1989. MR 1016262 (91g:20070)

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Additional Information

Sébastien Jansou
Affiliation: Le Mas des Landes, 87170 Isle, France

Nicolas Ressayre
Affiliation: Department of Mathematics, University of Montpellier II, Place Eugène Bataillon, Montpellier, France

DOI: https://doi.org/10.1090/S1088-4165-09-00331-8
Received by editor(s): July 23, 2007
Received by editor(s) in revised form: March 12, 2008
Published electronically: March 5, 2009
Article copyright: © Copyright 2009 American Mathematical Society

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