Demazure resolutions as varieties of lattices with infinitesimal structure

Let \(k\) be a field of positive characteristic. We construct, for each dominant coweight \(\lambda\) of the standard maximal torus in the special linear group, a closed subvariety \(D(\lambda)\) of the multigraded Hilbert scheme of an affine space over \(k\), such that the \(k\)-valued points of \(D(\lambda)\) can be interpreted as lattices in \(k((z))^n\) endowed with infinitesimal structure. Moreover, for any \(\lambda\) we construct a universal homeomorphism from \(D(\lambda)\) to a Demazure resolution of the Schubert variety associated with \(\lambda\) in the affine Grassmannian. Lattices in \(D(\lambda)\) have non-trivial infinitesimal structure if and only if they lie over the boundary of the big cell.