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

Let \(P\) be a parabolic subgroup in \(G=SL_n(\mathbf k)\), 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 \(T^*G/P\). Restricting this identification to the conormal variety \(N^*X(w)\) of a Schubert divisor \(X(w)\) 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.