Cotangent Bundle to the Flag Variety - II

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