Smooth components of Springer fibers

Abstract

This article studies components of Springer fibers for gl(n) that are associated to closed orbits of GL(p) X 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 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(n) and standard tableaux.