Smooth components of Springer fibers

William Graham[1]; R. Zierau[2]

  • [1] University of Georgia Mathematics Department Athens, Georgia 30602 (USA)
  • [2] Oklahoma State University Mathematics Department Stillwater, Oklahoma 74078 (USA)

Annales de l’institut Fourier (2011)

  • Volume: 61, Issue: 5, page 2139-2182
  • ISSN: 0373-0956

Abstract

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This article studies components of Springer fibers for 𝔤𝔩 ( n ) that are associated to closed orbits of G L ( p ) × G L ( q ) on the flag variety of G L ( 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 G L ( n ) . 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 G L ( n ) and standard tableaux.

How to cite

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Graham, William, and Zierau, R.. "Smooth components of Springer fibers." Annales de l’institut Fourier 61.5 (2011): 2139-2182. <http://eudml.org/doc/219859>.

@article{Graham2011,
abstract = {This article studies components of Springer fibers for $\mathfrak\{gl\}(n)$ that are associated to closed orbits of $GL(p)\times GL(q)$ 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 $\mathcal\{L\}$ is a line bundle on the flag variety associated to a dominant weight, then the higher cohomology groups of the restriction of $\mathcal\{L\}$ 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 $GL(n)$ and standard tableaux.},
affiliation = {University of Georgia Mathematics Department Athens, Georgia 30602 (USA); Oklahoma State University Mathematics Department Stillwater, Oklahoma 74078 (USA)},
author = {Graham, William, Zierau, R.},
journal = {Annales de l’institut Fourier},
keywords = {Springer fibers; iterated bundles; flag varieties; nilpotent orbits},
language = {eng},
number = {5},
pages = {2139-2182},
publisher = {Association des Annales de l’institut Fourier},
title = {Smooth components of Springer fibers},
url = {http://eudml.org/doc/219859},
volume = {61},
year = {2011},
}

TY - JOUR
AU - Graham, William
AU - Zierau, R.
TI - Smooth components of Springer fibers
JO - Annales de l’institut Fourier
PY - 2011
PB - Association des Annales de l’institut Fourier
VL - 61
IS - 5
SP - 2139
EP - 2182
AB - This article studies components of Springer fibers for $\mathfrak{gl}(n)$ that are associated to closed orbits of $GL(p)\times GL(q)$ 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 $\mathcal{L}$ is a line bundle on the flag variety associated to a dominant weight, then the higher cohomology groups of the restriction of $\mathcal{L}$ 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 $GL(n)$ and standard tableaux.
LA - eng
KW - Springer fibers; iterated bundles; flag varieties; nilpotent orbits
UR - http://eudml.org/doc/219859
ER -

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