Singular components of Springer fibers in the two-column case.
(Composantes singulières des fibres de Springer dans le cas deux-colonnes.)

*(French. Abridged English version)*Zbl 1167.14035The Springer fibers of type \(A\), i.e. the fibers of the Springer resolution of the nilcone in \(gl_n\), first defined by T. A. Springer [Algebr. Geom., Bombay Colloq. 1968, 373–391 (1969; Zbl 0195.50803)] are of great interest in representation theory. The Springer fiber corresponding to a nilpotent element \(u\) is defined as the variety of all Borel subalgebras in \(gl_n\) containing \(u\), or, equivalently, the variety of all full flags that are stabilized by \(u\). These varieties are connected, equidimensional, but not irreducible (unless \(u\) is regular or zero) and hence singular. Their irreducible components admit a nice combinatorial description: they can be indexed by the standard Young tableaux whose shape is the partition of \(n\) given by the sizes of the Jordan blocks of \(u\) [cf. N. Spaltenstein, Classes unipotentes et sous-groupes de Borel. Lecture Notes in Mathematics. 946. Berlin etc.: Springer-Verlag (1982; Zbl 0486.20025)]. The geometry of these irreducible components is, however, far from being well understood.

It is known that all the components of the Springer fiber over \(u\) are smooth if the Jordan normal form of \(u\) consists of two blocks or contains only one Jordan block of size greater than one (these are the so-called two-line and hook cases). Recently, L. Fresse and A. Melnikov [“On the singularity of the irreducible components of a Springer fiber in \(sl(n)\)”, arXiv:math/0905.1617 (2009)] described all \(u\) such that all the components of the Springer fiber are smooth.

In this paper, the author deals with the simplest case when the singularities actually arise, that is, the case of \(u\) satisfying the equation \(u^2=0\) (the two-column case). He provides a smoothness criterion for the components of such Springer fibers. This criterion is formulated in combinatorial terms: a component is singular if and only if the number of Young tableaux produced from its standard Young tableau by means of a certain combinatorial rule (the so-called line-standard tableaux) is greater than \(\frac{\mu(\mu-1)}{2}\), where \(\mu=\dim \ker u\).

It is known that all the components of the Springer fiber over \(u\) are smooth if the Jordan normal form of \(u\) consists of two blocks or contains only one Jordan block of size greater than one (these are the so-called two-line and hook cases). Recently, L. Fresse and A. Melnikov [“On the singularity of the irreducible components of a Springer fiber in \(sl(n)\)”, arXiv:math/0905.1617 (2009)] described all \(u\) such that all the components of the Springer fiber are smooth.

In this paper, the author deals with the simplest case when the singularities actually arise, that is, the case of \(u\) satisfying the equation \(u^2=0\) (the two-column case). He provides a smoothness criterion for the components of such Springer fibers. This criterion is formulated in combinatorial terms: a component is singular if and only if the number of Young tableaux produced from its standard Young tableau by means of a certain combinatorial rule (the so-called line-standard tableaux) is greater than \(\frac{\mu(\mu-1)}{2}\), where \(\mu=\dim \ker u\).

Reviewer: Evgeny Smirnov (Moscow)

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\textit{L. Fresse}, C. R., Math., Acad. Sci. Paris 347, No. 11--12, 631--636 (2009; Zbl 1167.14035)

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##### References:

[1] | Fung, F.Y.C., On the topology of components of some Springer fibers and their relation to kazhdan – lusztig theory, Adv. math., 178, 244-276, (2003) · Zbl 1035.20004 |

[2] | Spaltenstein, N., Classes unipotentes et sous-groupes de Borel, Lecture notes in math., vol. 946, (1982), Springer-Verlag Berlin, New York · Zbl 0486.20025 |

[3] | Springer, T.A., The unipotent variety of a semisimple group, (), 373-391 · Zbl 0195.50803 |

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