Local Geometric Langlands Correspondence: The Spherical Case

Abstract

A module over an affine Kac–Moody algebra $\hat{g}$ is called spherical if the action of the Lie subalgebra g[[t]] on it integrates to an algebraic action of the corresponding group G[[t]]. Consider the category of spherical $\hat{g}$-modules of critical level. In this paper we prove that this category is equivalent to the category of quasi-coherent sheaves on the ind-scheme of opers on the punctured disc which are unramified as local systems. This result is a categorical version of the well-known description of spherical vectors in representations of groups over local non-archimedian fields. It may be viewed as a special case of the local geometric Langlands correspondence proposed in [FG2].