Demazure resolutions as varieties of lattices with infinitesimal structure

Let k be a field of positive characteristic. We construct, for each dominant coweight λof the standard maximal torus in the special linear group, 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 λwe construct a universal homeomorphism from D(λ) to a Demazure resolution of the Schubert variety associated with λin 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.

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