  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.
