Smooth components of Springer fibers
Annales de l'Institut Fourier, Volume 61 (2011) no. 5, p. 2139-2182

This article studies components of Springer fibers for 𝔤𝔩(n) that are associated to closed orbits of GL(p)×GL(q) on the flag variety of GL(n),n=p+q. These components occur in any Springer fiber. In contrast to the case of arbitrary components, these components are smooth varieties. Using results of Barchini and Zierau we show these components are iterated bundles and are stable under the action of a maximal torus of GL(n). We prove that if is a line bundle on the flag variety associated to a dominant weight, then the higher cohomology groups of the restriction of to these components vanish. We derive some consequences of localization theorems in equivariant cohomology and K-theory, applied to these components. In the appendix we identify the tableaux corresponding to these components, under the bijective correspondence between components of Springer fibers for GL(n) and standard tableaux.

Cet article étudie les composantes des fibres de Springer pour 𝔤𝔩(n) qui sont associées à des orbites fermées de GL(p)×GL(q) dans la variété de drapeaux de GL(n),n=p+q. Ces composantes apparaîssent dans toute fibre de Springer. En contraste avec le cas de composantes arbitraires, ces composantes sont des variétés lisses. En utilisant des résultats de Barchini et Zierau, nous montrons que ces composantes sont des fibrés itérés et sont stables sous l’action d’un tore maximal de GL(n). Nous démontrons que si est un fibré en droites sur la variété de drapeaux associée à un poids dominant, alors les groupes de cohomologie de degré supérieur de la restriction de à ces composantes s’annulent. Nous déduisons quelques conséquences des théorèmes de localisation en cohomologie équivariante et K-théorie, appliqués à ces composantes. Dans l’appendice, nous indentifions les tableaux correspondants à ces composantes, via la correspondance bijective entre les composantes des fibres de Springer pour GL(n) et les tableaux standard.

DOI : https://doi.org/10.5802/aif.2669
Classification:  14L35,  14M15,  20G20,  22E46
Keywords: Springer fibers, iterated bundles, flag varieties, nilpotent orbits
@article{AIF_2011__61_5_2139_0,
     author = {Graham, William and Zierau, R.},
     title = {Smooth components of Springer fibers},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {61},
     number = {5},
     year = {2011},
     pages = {2139-2182},
     doi = {10.5802/aif.2669},
     mrnumber = {2961851},
     zbl = {1248.14056},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2011__61_5_2139_0}
}
Graham, William; Zierau, R. Smooth components of Springer fibers. Annales de l'Institut Fourier, Volume 61 (2011) no. 5, pp. 2139-2182. doi : 10.5802/aif.2669. http://www.numdam.org/item/AIF_2011__61_5_2139_0/

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