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

Let $P$ be a parabolic subgroup in $G=S{L}_n(\mathbf{k})$, for $\mathbf{k}$ an algebraically closed field. We show that there is a $G$-stable closed subvariety of an affine Schubert variety in an affine partial flag variety which is a natural compactification of the cotangent bundle $T^{\ast }G\mathrm{/}P$. Restricting this identification to the conormal variety $N^{\ast }X(w)$ of a Schubert divisor $X(w)$ in $G\mathrm{/}P$, we show that there is a compactification of $N^{\ast }X(w)$ as an affine Schubert variety. It follows that $N^{\ast }X(w)$ is normal, Cohen-Macaulay, and Frobenius split.