Singular components of Springer fibers in the two-column case
[Composantes singulières des fibres de Springer dans le cas deux-colonnes]
Annales de l'Institut Fourier, Tome 59 (2009) no. 6, pp. 2429-2444.

On considère la fibre de Springer u pour un endomorphisme nilpotent u tel que u 2 =0. On décrit les éléments des composantes de u qui sont fixés par l’action du tore standard relatif à une base de Jordan de u. À l’aide de ce résultat, on établit une condition nécessaire et suffisante de singularité pour les composantes de u .

We consider the Springer fiber u corresponding to a nilpotent endomorphism u of nilpotent order 2. As a first result, we give a description of the elements of a given component of u which are fixed by the action of the standard torus relative to some Jordan basis of u. By using this result, we establish a necessary and sufficient condition of singularity for the components of u .

DOI : 10.5802/aif.2495
Classification : 14M15, 14B05
Keywords: Flag varieties, Springer fibers, singularity criteria, Young tableaux
Mot clés : variétés de drapeaux, fibres de Springer, critères de singularité, tableaux de Young
Fresse, Lucas 1

1 Weizmann Institute of Science Department of Mathematics Rehovot 76100 (Israel)
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Fresse, Lucas. Singular components of Springer fibers in the two-column case. Annales de l'Institut Fourier, Tome 59 (2009) no. 6, pp. 2429-2444. doi : 10.5802/aif.2495. http://www.numdam.org/articles/10.5802/aif.2495/

[1] Fung, F. Y. C On the topology of components of some Springer fibers and their relation to Kazhdan-Lusztig theory, Adv. Math. (2003) no. 178, pp. 244-276 | DOI | MR | Zbl

[2] Melnikov, A. Description of B-orbit closures of order 2 in upper triangular matrices, Transf. Groups (2006) no. 11, pp. 217-247 | DOI | MR

[3] Melnikov, A.; Pagnon, N. G. J. Reducibility of the intersections of components of a Springer fiber (to appear in Indag. Math.)

[4] Slodowy, P. Four lectures on simple groups and singularities, Communications of the Mathematical Institute, Volume 11, Rijksuniversiteit Utrecht, 1980 | MR | Zbl

[5] Spaltenstein, N. Classes unipotentes et sous-groupes de Borel, Lecture Notes in Mathematics, Volume 946, Springer-Verlag, Berlin-New York, 1982 | MR | Zbl

[6] Vargas, J. A. Fixed points under the action of unipotent elements of SL(n) in the flag variety, Bol. Soc. Mat. Mexicana (1979) no. 24, pp. 1-14 | MR | Zbl

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