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Let $P$ be a parabolic subgroup in $G=SL_n(\mathbf k)$, for $\mathbf k$ analgebraically closed field. We show that there is a $G$-stable closedsubvariety of an affine Schubert variety in an affine partial flag varietywhich is a natural compactification of the cotangent bundle $T^*G/P$.Restricting this identification to the conormal variety $N^*X(w)$ of a Schubertdivisor $X(w)$ in $G/P$, we show that there is a compactification of $N^*X(w)$as an affine Schubert variety. It follows that $N^*X(w)$ is normal,Cohen-Macaulay, and Frobenius split.

Cotangent Bundles of Partial Flag Varieties and Conormal Varieties of their Schubert Divisors