Cotangent Bundle to the Flag Variety - I

We show that there is a $S{L}_n$-stable closed subset of an affine Schubert variety in the infinite dimensional Flag variety (associated to the Kac-Moody group $\widehat{S{L}_n}$) which is a natural compactification of the cotangent bundle to the finite-dimensional Flag variety $S{L}_n\mathrm{/}B$.