Comptes Rendus
no. 19-20
Group Theory/Lie Algebras
Exterior powers of the reflection representation in the cohomology of Springer fibres
[Les puissances extérieures de la représentation géométrique dans la cohomologie des fibres de Springer]

Présenté par : Gérard Laumon

Anthony Henderson1
1 School of Mathematics and Statistics, University of Sydney, NSW 2006, Australia
Comptes Rendus. Mathématique, Volume 348 (2010) no. 19-20, pp. 1055-1058.

Soit H⁎(Be) la cohomologie de la fibre de Springer pour l'élément nilpotent e de l'algèbre de Lie simple g. Soit ΛiV la i-ème puissance extérieure de la représentation géométrique de W. Nous trouvons les degrés des contributions de ΛiV à la représentation graduée H⁎(Be), si e est régulier dans une sous-algèbre de Levi et satisfait à une autre condition qui est vraie si g est de type A, B, ou C. Ce résultat démontre partiellement une conjecture de Lehrer et Shoji.

Let H⁎(Be) be the cohomology of the Springer fibre for the nilpotent element e in a simple Lie algebra g. Let ΛiV denote the ith exterior power of the reflection representation of W. We determine the degrees in which ΛiV occurs in the graded representation H⁎(Be), under the assumption that e is regular in a Levi subalgebra and satisfies a certain extra condition which holds automatically if g is of type A, B, or C. This partially verifies a conjecture of Lehrer and Shoji.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2010.09.015

Anthony Henderson 1

1 School of Mathematics and Statistics, University of Sydney, NSW 2006, Australia 
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Anthony Henderson. Exterior powers of the reflection representation in the cohomology of Springer fibres. Comptes Rendus. Mathématique, Volume 348 (2010) no. 19-20, pp. 1055-1058. doi : 10.1016/j.crma.2010.09.015. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2010.09.015/

[1] R. Hotta; T.A. Springer A specialization theorem for certain Weyl group representations and an application to the Green polynomials of unitary groups, Invent. Math., Volume 41 (1977), pp. 113-127

[2] J.E. Humphreys Conjugacy Classes in Semisimple Algebraic Groups, Mathematical Surveys and Monographs, vol. 43, American Mathematical Society, Providence, Rhode Island, 1995

[3] G.I. Lehrer; T. Shoji On flag varieties, hyperplane complements and Springer representations of Weyl groups, J. Austral. Math. Soc. Ser. A, Volume 49 (1990) no. 3, pp. 449-485

[4] G. Lusztig An induction theorem for Springer's representations, Representation Theory of Algebraic Groups and Quantum Groups, Adv. Stud. Pure Math., vol. 40, Math. Soc. Japan, Kinokuniya, 2004, pp. 253-259

[5] T. Shoji Geometry of orbits and Springer correspondence, Orbites unipotentes et représentations, I, Astérisque, vol. 168, Soc. Math. de France, Paris, 1988, pp. 61-140

[6] L. Solomon Invariants of finite reflection groups, Nagoya Math. J., Volume 22 (1963), pp. 57-64

[7] E. Sommers Exterior powers of the reflection representation in Springer theory | arXiv

[8] E. Sommers; P. Trapa The adjoint representation in rings of functions, Represent. Theory, Volume 1 (1997), pp. 182-189

[9] N. Spaltenstein On the reflection representation in Springer's theory, Comment. Math. Helv., Volume 66 (1991) no. 4, pp. 618-636

[10] R. Steinberg Invariants of finite reflection groups, Canad. J. Math., Volume 12 (1960), pp. 616-618

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