A Bruhat atlas for the Mehta-van der Kallen stratification of $T^* GL_n/B$

TL;DR

This paper provides a comprehensive Bruhat atlas for the Mehta-van der Kallen stratification of the cotangent bundle of the flag variety, revealing detailed geometric properties of the stratification.

Contribution

It constructs a Bruhat atlas for the stratified space, enabling full description and analysis of the stratification's geometric structure.

Findings

Stratification is fully described within a larger Frobenius split space.

Each stratum closure is proven to be normal and Cohen-Macaulay.

The approach uses stratified-isomorphic Bruhat cells in an extended space.

Abstract

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