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.