Geometric Realizations of Wakimoto Modules at the Critical Level

We study the Wakimoto modules over the affine Kac-Moody algebras at the critical level from the point of view of the equivalences of categories proposed in our previous works, relating categories of representations and certain categories of sheaves. In particular, we describe explicitly geometric realizations of the Wakimoto modules as Hecke eigen-Dmodules on the affine Grassmannian and as quasi-coherent sheaves on the flag variety of the Langlands dual group.