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

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- by Sébastien Jansou and Nicolas Ressayre PDF
- Represent. Theory
**13**(2009), 50-62 Request permission

## 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.

## References

- Valery Alexeev and Michel Brion,
*Moduli of affine schemes with reductive group action*, J. Algebraic Geom.**14**(2005), no. 1, 83–117. MR**2092127**, DOI 10.1090/S1056-3911-04-00377-7 - Klaus Bongartz,
*Schichten von Matrizen sind rationale Varietäten*, Math. Ann.**283**(1989), no. 1, 53–64 (German). MR**973803**, DOI 10.1007/BF01457501 - 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**, DOI 10.1016/j.aim.2007.11.009 - Joe Harris,
*Algebraic geometry*, Graduate Texts in Mathematics, vol. 133, Springer-Verlag, New York, 1992. A first course. MR**1182558**, DOI 10.1007/978-1-4757-2189-8 - Sébastien Jansou,
*Déformations des cônes de vecteurs primitifs*, Math. Ann.**338**(2007), no. 3, 627–667 (French, with French summary). MR**2317933**, DOI 10.1007/s00208-007-0090-6 - P. I. Katsylo,
*Sections of sheets in a reductive algebraic Lie algebra*, Izv. Akad. Nauk SSSR Ser. Mat.**46**(1982), no. 3, 477–486, 670 (Russian). MR**661143** - Bertram Kostant,
*Lie group representations on polynomial rings*, Amer. J. Math.**85**(1963), 327–404. MR**158024**, DOI 10.2307/2373130 - Hanspeter Kraft,
*Parametrisierung von Konjugationsklassen in ${\mathfrak {s}}{\mathfrak {l}}_{n}$*, Math. Ann.**234**(1978), no. 3, 209–220 (German). MR**491855**, DOI 10.1007/BF01420644 - Hanspeter Kraft and Claudio Procesi,
*Closures of conjugacy classes of matrices are normal*, Invent. Math.**53**(1979), no. 3, 227–247. MR**549399**, DOI 10.1007/BF01389764 - Serge Lang,
*Algebra*, 3rd ed., Graduate Texts in Mathematics, vol. 211, Springer-Verlag, New York, 2002. MR**1878556**, DOI 10.1007/978-1-4613-0041-0 - V. Popov and E. Vinberg,
*Invariant Theory*, Encyclopaedia of Mathematical Sciences, vol. 55, pp. 123-278, Springer-Verlag, 1994. - Gerald W. Schwarz,
*Representations of simple Lie groups with a free module of covariants*, Invent. Math.**50**(1978/79), no. 1, 1–12. MR**516601**, DOI 10.1007/BF01406465 - Igor R. Shafarevich,
*Basic algebraic geometry. 1*, 2nd ed., Springer-Verlag, Berlin, 1994. Varieties in projective space; Translated from the 1988 Russian edition and with notes by Miles Reid. MR**1328833** - J. Weyman,
*The equations of conjugacy classes of nilpotent matrices*, Invent. Math.**98**(1989), no. 2, 229–245. MR**1016262**, DOI 10.1007/BF01388851

## 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
- Received by editor(s): July 23, 2007
- Received by editor(s) in revised form: March 12, 2008
- Published electronically: March 5, 2009
- © Copyright 2009 American Mathematical Society
- Journal: Represent. Theory
**13**(2009), 50-62 - MSC (2000): Primary 14D22, 14L24
- DOI: https://doi.org/10.1090/S1088-4165-09-00331-8
- MathSciNet review: 2485792