Springer fiber components in the two columns case for types A and D are normal
[Les composantes de fibre de Springer, dans le cas de deux colonnes de types A et D, sont normales]
Bulletin de la Société Mathématique de France, Tome 140 (2012) no. 3, pp. 309-333.

We study the singularities of the irreducible components of the Springer fiber over a nilpotent element N with N 2 =0 in a Lie algebra of type A or D (the so-called two columns case). We use Frobenius splitting techniques to prove that these irreducible components are normal, Cohen-Macaulay, and have rational singularities.

DOI : 10.24033/bsmf.2629
Classification : 14B05, 14N20
Mots-clés : Springer fiber, Frobenius splitting, normality, rational resolution, rational singularities 
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     author = {Perrin, Nicolas and Smirnov, Evgeny},
     title = {Springer fiber components in the two columns case for types $A$ and $D$ are normal},
     journal = {Bulletin de la Soci\'et\'e Math\'ematique de France},
     pages = {309--333},
     publisher = {Soci\'et\'e math\'ematique de France},
     volume = {140},
     number = {3},
     year = {2012},
     doi = {10.24033/bsmf.2629},
     mrnumber = {3059118},
     zbl = {1268.14006},
     language = {en},
     url = {http://www.numdam.org/articles/10.24033/bsmf.2629/}
}
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Perrin, Nicolas; Smirnov, Evgeny. Springer fiber components in the two columns case for types $A$ and $D$ are normal. Bulletin de la Société Mathématique de France, Tome 140 (2012) no. 3, pp. 309-333. doi : 10.24033/bsmf.2629. http://www.numdam.org/articles/10.24033/bsmf.2629/

[1] N. Bourbaki - Groupes et algèbres de Lie, Hermann, 1954. | Zbl

[2] M. Brion & S. Kumar - Frobenius splitting methods in geometry and representation theory, Progress in Math., vol. 231, Birkhäuser, 2005. | MR | Zbl

[3] M. Demazure - « Désingularisation des variétés de Schubert généralisées », Ann. Sci. École Norm. Sup. 7 (1974), p. 53-88. | Numdam | MR | Zbl

[4] L. Fresse - « Composantes singulières des fibres de Springer dans le cas deux-colonnes », C. R. Math. Acad. Sci. Paris 347 (2009), p. 631-636. | MR | Zbl

[5] -, « Singular components of Springer fibers in the two-column case », Ann. Inst. Fourier (Grenoble) 59 (2009), p. 2429-2444. | Numdam | MR | Zbl

[6] L. Fresse & A. Melnikov - « On the singularity of the irreducible components of a Springer fiber in 𝔰𝔩 n », Selecta Math. (N.S.) 16 (2010), p. 393-418. | MR | Zbl

[7] W. Fulton - Young tableaux, London Mathematical Society Student Texts, vol. 35, Cambridge Univ. Press, 1997. | MR | Zbl

[8] F. Y. C. Fung - « On the topology of components of some Springer fibers and their relation to Kazhdan-Lusztig theory », Adv. Math. 178 (2003), p. 244-276. | MR | Zbl

[9] V. V. Gorbatsevitch, A. L. Onishchik & È. B. Vinberg (éds.) - Lie groups and Lie algebras, III, Encyclopaedia of Math. Sciences, vol. 41, Springer, 1994. | MR | Zbl

[10] H. Grauert & O. Riemenschneider - « Verschwindungssätze für analytische Kohomologiegruppen auf komplexen Räumen », Invent. Math. 11 (1970), p. 263-292. | MR | Zbl

[11] X. He & J. F. Thomsen - « Frobenius splitting and geometry of G-Schubert varieties », Adv. Math. 219 (2008), p. 1469-1512. | MR | Zbl

[12] -, « On Frobenius splitting of orbit closures of spherical subgroups in flag varieties », preprint arXiv:1006.5175. | Zbl

[13] S. Kumar - Kac-Moody groups, their flag varieties and representation theory, Progress in Math., vol. 204, Birkhäuser, 2002. | MR | Zbl

[14] M. A. A. Van Leeuwen - « A Robinson-Schensted algorithm in the geometry of flags for classical groups », Thèse, Rijksuniversiteit Utrecht, 1989.

[15] N. Spaltenstein - Classes unipotentes et sous-groupes de Borel, Lecture Notes in Math., vol. 946, Springer, 1982. | MR | Zbl

[16] C. Stroppel & B. Webster - « 2-block Springer fibers: convolution algebras, coherent sheaves and embedded TQFT », preprint arXiv:0802.1943. | MR | Zbl

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