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

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.

MR Zbl

DOI : 10.24033/bsmf.2629

Classification : 14B05, 14N20
Keywords: Springer fiber, Frobenius splitting, normality, rational resolution, rational singularities

@article{BSMF_2012__140_3_309_0,
     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://geodesic.mathdoc.fr/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://geodesic.mathdoc.fr/articles/10.24033/bsmf.2629/

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