Demazure Resolutions as Varieties of Lattices with Infinitesimal Structure

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