Localization of $\mathfrak g$-modules on the affine Grassmannian


We consider the category of modules over the affine Kac-Moody algebra $\widehat{\mathfrak g}$ 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 $G(\!(t)\!)/G[\mskip-2mu[t]\mskip-2mu]$. 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.