  In this paper we study the derived category of sheaves on the affine
Grassmannian of a complex reductive group G, contructible with respect to the
stratification by G(C[[x]])-orbits. Following ideas of Ginzburg and
Arkhipov-Bezrukavnikov-Ginzburg, we describe this category (and a mixed
version) in terms of coherent sheaves on the nilpotent cone of its Langlands
dual reductive group. We also show, in the mixed case, that restriction to the
nilpotent cone of a Levi subgroup corresponds to hyperbolic localization on
affine Grassmannians.
