Towards a Tamely Ramified Geometric Langlands Correspondence

Abstract

Motivated by the recent proof of the unramified global geometric Langlands correspondence, we make progress towards constructing a tamely ramified version of the global geometric Langlands correspondence. We show that the DG category of D-modules on the moduli stack of principal G-bundles on a smooth, projective, connected curve with Iwahori level structure at a point is compactly generated, which allows the Langlands functor to be constructed in this setting. We then study the local behaviour of this correspondence by constructing factorization module categories at a point and constructing equivalences between them. These equivalences are factorization module categorical versions of the Arkhipov-Bezrukavnikov equivalence, Bezrukavnikov’s equivalence, and an Iwahori invariant Fundamental Local Equivalence.