Local geometric Langlands correspondence: the spherical case

Abstract

A module over an affine Kac–Moody algebra $\widehat{\mathfrak{g}}$ is called spherical if the action of the Lie subalgebra $\mathfrak{g} [[t]]$ on it integrates to an algebraic action of the corresponding group $G [[t]]$. Consider the category of spherical $\widehat{\mathfrak{g}}$-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].