A Bruhat atlas for the Mehta-van der Kallen stratification of $T^{\ast }G{L}_n/B$

Abstract

Mehta and van der Kallen put a Frobenius splitting on the type A cotangent bundle $T^{\ast }G{L}_n/B$, thereby defining a stratification by compatibly split subvarieties, and they determined a few of the elements of this stratification. We embed $T^{\ast }G{L}_n/B$ as a stratum in a larger stratified (and Frobenius split) space $G{L}_n/B\times Ma{t}_n$ whose stratification we determine, thereby giving a full description of the one of Mehta-van der Kallen. The main technique is to endow $G{L}_n/B\times Ma{t}_n$ with a Bruhat atlas, covering it with open sets that are stratified-isomorphic to Bruhat cells (in $G{L}_{2n}/B_{2n}$). Among the consequences are that each stratum closure is normal and Cohen-Macaulay.

arXiv comment10 pages