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Mehta and van der Kallen put a Frobenius splitting on the type A cotangentbundle $T^* GL_n/B$, thereby defining a stratification by compatibly splitsubvarieties, 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 afull description of the one of Mehta-van der Kallen. The main technique is toendow $GL_n/B \times Mat_n$ with a Bruhat atlas, covering it with open setsthat are stratified-isomorphic to Bruhat cells (in $GL_{2n}/B_{2n}$). Among theconsequences are that each stratum closure is normal and Cohen-Macaulay.

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