Smooth components of Springer fibers
type: Item
instance of: scholarly article
title: Smooth components of Springer fibers
zbMATH Open document ID: 1248.14056
publication date: 2012-05-09T00:00:00Z
full work available at URL: https://eudml.org/doc/219859
full work available at URL: https://hdl.handle.net/11244/301407
reviewed by: Jacopo Gandini
Mathematics Subject Classification ID: 14M15
Mathematics Subject Classification ID: 14L35
Mathematics Subject Classification ID: 20G20
Mathematics Subject Classification ID: 22E46
zbMATH DE Number: 6032137
zbMATH Keywords: Springer fibers
zbMATH Keywords: iterated bundles
zbMATH Keywords: flag varieties
zbMATH Keywords: nilpotent orbits
author: William Graham
author: R. Zierau
MaRDI profile type: Publication
OpenAlex ID: W2315647508
cites work: The moment map and equivariant cohomology
cites work: Certain components of Springer fibers and associated cycles for discrete series representations of 𝑆𝑈(𝑝,𝑞)
cites work: Q3999894
cites work: Q4215528
cites work: Characteristic Cycles of Discrete Series for ℝ-Rank One Groups
cites work: Representation Theory and Complex Geometry
cites work: Homology of the Zero-Set of a Nilpotent Vector Field on a Flag Manifold
cites work: Localization in equivariant intersection theory and the Bott residue formula
cites work: Betti numbers of Springer fibers in type \(A\)
cites work: Singular components of Springer fibers in the two-column case
cites work: A unified approach on Springer fibers in the hook, two-row and two-column cases
cites work: On the topology of components of some Springer fibers and their relation to Kazhdan-Lusztig theory.
cites work: The Annihilators of Irreducible Harish-Chandra Modules for SU(p,q) and Other Type A n-1 Groups
cites work: Excited Young diagrams, equivariant $K$-theory, and Schubert varieties
cites work: Positivity in equivariant Schubert calculus.
cites work: On Positivity in T-Equivariant K-Theory of Flag Varieties
cites work: Q5602063
cites work: Q3483467
cites work: Q3312347
cites work: A fixed point formula for action of tori on algebraic varieties
cites work: Schubert patches degenerate to subword complexes
cites work: Kac-Moody groups, their flag varieties and representation theory
cites work: Green polynomials and singularities of unipotent classes
cites work: Geometric Invariant Theory
cites work: Adjacency of Young tableaux and the Springer fibers
cites work: Classes unipotentes et sous-groupes de Borel
cites work: Trigonometric sums, Green functions of finite groups and representations of Weyl groups
cites work: A construction of representations of Weyl groups
cites work: An occurrence of the Robinson-Schensted correspondence
cites work: Q4720030
cites work: Q3907682
cites work: Cohomologie et K-théorie équivariantes des variétés de Bott-Samelson et des variétés de drapeaux
DOI: 10.5802/AIF.2669
published in: Annales de l’institut Fourier
review text: The present paper studies some components of the Springer fibers for \(\mathrm{GL}(n)\) which are associated to closed orbits of \(\mathrm{GL}(p) \times \mathrm{GL}(q)\) on the flag variety of \(\mathrm{GL}(n)\), where \(n =p+q\). Such components occur in any Springer fiber and, in contrast to the case of arbitrary components, they are always smooth. Building on the work of \textit{L.~Barchini} and \textit{R.~Zierau} [Represent. Theory 12, 403--434 (2008; Zbl 1186.22017)], the authors show that these components are iterated bundles and are stable for the action of a maximal torus \(H\) of \(\mathrm{GL}(n)\). Using this description of the components, the authors calculate the Betti numbers and they determine the \(H\)-fixed points and the weights of \(H\) acting on the tangent spaces at these points.NEWLINENEWLINEIf \(\mathcal L\) is a line bundle on the flag variety of \(\mathrm{GL}(n)\) and if \(Z\) is one of the considered components of a Springer fiber regarded as a subvariety of the flag variety, using localization theorems in equivariant \(K\)-theory and Borel-Moore homology the authors obtain a character formula for the cohomology of \(\mathcal L_Z\) as a \(H\)-module, where \(\mathcal L_Z\) denotes the restriction of \(\mathcal L\) to \(Z\). If the weight corresponding to \(\mathcal L\) is sufficiently dominant, then it is shown that the higher cohomology groups of \(\mathcal L_Z\) vanish, therefore the character formula gives in this case the character of \(H^0(Z,\mathcal L_Z)\). This is of interest because of a result of \textit{J.~T.~Chang} [Trans. Am. Math. Soc. 341, No. 2, 603--622 (1994; Zbl 0817.22009)], which relates the dimension of \(H^0(Z,\mathcal L_Z)\) with the associated cycle of a discrete series representation of \(U(p,q)\). Applying localization theorems, they are also obtained formulas which express the classes defined by the considered components in equivariant cohomology and in \(K\)-theory in terms of Schubert bases. In the appendix, they…
description: scientific article; zbMATH DE number 6032137