On two geometric realizations of the anti-spherical module

The anti-spherical modules over the affine Hecke algebras admit two different realizations: one realization is in terms of the space of Whittaker functions on the affine flag manifolds and the other realization, due to Kazhdan-Lusztig, is in terms of the equivariant K-theory of the Springer resolution of the nilpotent cone for the dual group. I will explain the work of Arkhipov-Bezrukavnikov on the equivalence between the Iwahoric-Whittaker category and the equivariant derived category for the Springer resolution, which provides a geometric lift (or categorification) of the above two realizations of the anti-spherical modules.