no. 13-14
Group Theory/Algebraic Geometry
On geometric properties of orbital varieties in type A
[Sur des propriétés géométriques des variétés orbitales pour le type A]

Présenté par : the Editorial Board

Fresse, Lucas1 ; Melnikov, Anna2
1 Einstein Institute of Mathematics, Hebrew University, Jerusalem 91904, Israel
2 Department of Mathematics, University of Haifa, Haifa 31905, Israel
Comptes Rendus. Mathématique, Tome 349 (2011) no. 13-14, pp. 735-739.

Lʼintersection entre une orbite nilpotente O⊂gln(C) et lʼalgèbre de Lie Lie(B)⊂gln(C) dʼun sous-groupe de Borel B⊂GLn(C) est une variété algébrique quasi-affine équidimensionnelle. Ses composantes irréductibles sont appelées variétés orbitales. Dans cette Note, on propose des critères pour quʼune variété orbitale soit lisse ou bien possède une orbite dense pour lʼaction adjointe de B. De plus, on souligne un lien possible entre ces deux propriétés.

The intersection between a nilpotent orbit O⊂gln(C) and the Lie algebra Lie(B)⊂gln(C) of a Borel subgroup B⊂GLn(C) is an equidimensional, quasi-affine algebraic variety. Its irreducible components are called orbital varieties. In this Note, we provide criteria to guarantee that an orbital variety is smooth or has a dense orbit for the adjoint action of B. In addition, we point out a possible relation between these two properties.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2011.05.016
Fresse, Lucas 1 ; Melnikov, Anna 2

1 Einstein Institute of Mathematics, Hebrew University, Jerusalem 91904, Israel
2 Department of Mathematics, University of Haifa, Haifa 31905, Israel 
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Fresse, Lucas; Melnikov, Anna. On geometric properties of orbital varieties in type A. Comptes Rendus. Mathématique, Tome 349 (2011) no. 13-14, pp. 735-739. doi : 10.1016/j.crma.2011.05.016. http://www.numdam.org/articles/10.1016/j.crma.2011.05.016/

[1] Fresse, L.; Melnikov, A. On the singularity of the irreducible components of a Springer fiber in sl(n), Selecta Math. (N.S.), Volume 16 (2010), pp. 393-418

[2] L. Fresse, A. Melnikov, Some characterizations of singular components of Springer fibers in the two-column case, Algebr. Rep. Theory, , in press. | DOI

[3] Fresse, L. On the singularity of some special components of Springer fibers, J. Lie Theory, Volume 21 (2011), pp. 205-242

[4] Joseph, A. On the variety of a highest weight module, J. Algebra, Volume 88 (1984), pp. 238-278

[5] Spaltenstein, N. Classes unipotentes et sous-groupes de Borel, Lecture Notes in Math., vol. 946, Springer-Verlag, 1982

[6] Steinberg, R. On the desingularization of the unipotent variety, Invent. Math., Volume 36 (1976), pp. 209-224

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