Demazure resolutions as varieties of lattices with infinitesimal structure

Abstract

Let k be a field of positive characteristic. We construct, for each dominant cocharacter λ of the standard maximal torus in Sl n , 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. The variety D ( λ ) carries a natural Sl n ( k 〚 z 〛 ) -action. Moreover, for any λ we construct an Sl n ( 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 ) .