Cotangent bundles of partial flag varieties and conormal varieties of their schubert divisors

Let P be a parabolic subgroup in G = SL n ( k ), for 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.