Smooth components of Springer fibers
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 are identified the tableaux corresponding to the considered components under the bijective correspondence between components of Springer fibers for \(\mathrm{GL}(n)\) and standard tableaux.
scientific article; zbMATH DE number 6032137
scientific article