Appears in collection : Tropical Geometry, Berkovich Spaces, Arithmetic D-Modules and p-adic Local Systems

Derived localization theorem for modules over g = Lie(G) where G is a reductive algebraic group over a filed of positive characteristic relates g-modules to crystalline D-modules on the flag variety G/B. It can be composed with Cartier transform to relate representations of g to coherent sheaves on the cotangent bundle T *G/B. A related description of modules over the Frobenius kernel Gr involves coherent sheaves on a certain derived scheme S mapping to T *G/B. That derived scheme S plays a role in (local) geometric Langlands duality which leads to old and possibly new connections between modular representations and geometric Langlands duality.