Springer fiber components in the two columns case for types A and D are normal

Nicolas Perrin; Evgeny Smirnov

Bulletin de la Société Mathématique de France (2012)

  • Volume: 140, Issue: 3, page 309-333
  • ISSN: 0037-9484

Abstract

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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.

How to cite

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Perrin, Nicolas, and Smirnov, Evgeny. "Springer fiber components in the two columns case for types $A$ and $D$ are normal." Bulletin de la Société Mathématique de France 140.3 (2012): 309-333. <http://eudml.org/doc/272550>.

@article{Perrin2012,
abstract = {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.},
author = {Perrin, Nicolas, Smirnov, Evgeny},
journal = {Bulletin de la Société Mathématique de France},
keywords = {Springer fiber; Frobenius splitting; normality; rational resolution; rational singularities},
language = {eng},
number = {3},
pages = {309-333},
publisher = {Société mathématique de France},
title = {Springer fiber components in the two columns case for types $A$ and $D$ are normal},
url = {http://eudml.org/doc/272550},
volume = {140},
year = {2012},
}

TY - JOUR
AU - Perrin, Nicolas
AU - Smirnov, Evgeny
TI - Springer fiber components in the two columns case for types $A$ and $D$ are normal
JO - Bulletin de la Société Mathématique de France
PY - 2012
PB - Société mathématique de France
VL - 140
IS - 3
SP - 309
EP - 333
AB - 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.
LA - eng
KW - Springer fiber; Frobenius splitting; normality; rational resolution; rational singularities
UR - http://eudml.org/doc/272550
ER -

References

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  9. [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. Zbl0797.22001MR1349140
  10. [10] H. Grauert & O. Riemenschneider – « Verschwindungssätze für analytische Kohomologiegruppen auf komplexen Räumen », Invent. Math.11 (1970), p. 263–292. Zbl0202.07602MR302938
  11. [11] X. He & J. F. Thomsen – « Frobenius splitting and geometry of G -Schubert varieties », Adv. Math.219 (2008), p. 1469–1512. Zbl1160.14035MR2458144
  12. [12] —, « On Frobenius splitting of orbit closures of spherical subgroups in flag varieties », preprint arXiv:1006.5175. Zbl1264.14066
  13. [13] S. Kumar – Kac-Moody groups, their flag varieties and representation theory, Progress in Math., vol. 204, Birkhäuser, 2002. Zbl1026.17030MR1923198
  14. [14] M. A. A. van Leeuwen – « A Robinson-Schensted algorithm in the geometry of flags for classical groups », Thèse, Rijksuniversiteit Utrecht, 1989. 
  15. [15] N. Spaltenstein – Classes unipotentes et sous-groupes de Borel, Lecture Notes in Math., vol. 946, Springer, 1982. Zbl0486.20025MR672610
  16. [16] C. Stroppel & B. Webster – « 2-block Springer fibers: convolution algebras, coherent sheaves and embedded TQFT », preprint arXiv:0802.1943. Zbl1241.14009MR2914857

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