Smooth components of Springer fibers

Graham, William; Zierau, R.

Annales de l'Institut Fourier, Tome 61 (2011) no. 5, pp. 2139-2182.

Cet article étudie les composantes des fibres de Springer pour $\mathrm{𝔤𝔩}\left(n\right)$ qui sont associées à des orbites fermées de $GL\left(p\right)\times GL\left(q\right)$ dans la variété de drapeaux de $GL\left(n\right),n=p+q$. 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 $GL\left(n\right)$. Nous démontrons que si $ℒ$ 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 $ℒ$ 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 $GL\left(n\right)$ et les tableaux standard.

This article studies components of Springer fibers for $\mathrm{𝔤𝔩}\left(n\right)$ that are associated to closed orbits of $GL\left(p\right)\times GL\left(q\right)$ on the flag variety of $GL\left(n\right),n=p+q$. 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 $GL\left(n\right)$. We prove that if $ℒ$ is a line bundle on the flag variety associated to a dominant weight, then the higher cohomology groups of the restriction of $ℒ$ 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 $GL\left(n\right)$ and standard tableaux.

Classification : 14L35, 14M15, 20G20, 22E46
Keywords: Springer fibers, iterated bundles, flag varieties, nilpotent orbits
Mot clés : fibres de Springer, fibrés itérés, variété de drapeaux, orbites nilpotents

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