Localization of \(\hat{\mathfrak{g}}\)-modules on the Affine Grassmannian

Abstract

We consider the category of modules over the affine Kac-Moody algebra (\hat{\mathfrak{g}}) of critical level with regular central character. In our previous paper we conjectured that this category is equivalent to the category of Hecke eigen-D-modules on the affine Grassmannian (G((t))/G[[t]]). This conjecture was motivated by our proposal for a local geometric Langlands correspondence. In this paper we prove this conjecture for the corresponding (I^0) equivariant categories, where (I^0) is the radical of the Iwahori subgroup of (G((t))). Our result may be viewed as an affine analogue of the equivalence of categories of (\mathfrak{g})-modules and D-modules on the flag variety (G/B), due to Beilinson-Bernstein and Brylinski-Kashiwara.