Cotangent Bundle to the Flag Variety - II

Let $P$ be a parabolic subgroup in $S{L}_n(\mathbb{C})$. We show that there is a $S{L}_n(\mathbb{C})$-stable closed subvariety of an affine Schubert variety in an infinite dimensional partial Flag variety (associated to the Kac-Moody group $\widehat{S{L}_n}(\mathbb{C})$) which is a natural compactification of the cotangent bundle to $S{L}_n(\mathbb{C})\mathrm{/}P$. As a consequence, we recover the Springer resolution for any orbit closure inside the variety of nilpotent matrices.