  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.
