Demazure resolutions as varieties of lattices with infinitesimal structure

Let k be a field of positive characteristic. We construct, for each dominant cocharacter 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(n(z)) endowed with infinitesimal structure. The variety D(λ) carries a natural Sln(k〚z〛)-action. Moreover, for any λ we construct an Sln(k〚z〛)-equivariant 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).