The universal monodromic Arkhipov--Bezrukavnikov equivalence

TL;DR

This paper establishes a deep equivalence between equivariant sheaves on a Grothendieck alteration and monodromic sheaves on the affine flag variety, advancing the geometric Langlands program.

Contribution

It extends Arkhipov--Bezrukavnikov's equivalence to the universal monodromic setting and introduces a new proof technique via semi-simple localization.

Findings

Identifies equivariant quasicoherent sheaves with monodromic Iwahori--Whittaker sheaves.

Provides a monoidal equivalence involving the loop group of G.

Supports the proof of the tame local Betti geometric Langlands conjecture.

Abstract

We identify equivariant quasicoherent sheaves on the Grothendieck alteration of a reductive group $\mathsf{G}$ with universal monodromic Iwahori--Whittaker sheaves on the enhanced affine flag variety of the Langlands dual group $G$. This extends a similar result for equivariant quasicoherent sheaves on the Springer resolution due to Arkhipov--Bezrukavnikov. We further give a monoidal identification between adjoint equivariant coherent sheaves on the group $\mathsf{G}$ itself and bi-Iwahori--Whittaker sheaves on the loop group of $G$. These results are used in the sequel to this paper to prove the tame local Betti geometric Langlands conjecture of Ben-Zvi--Nadler. Our proof of fully faithfulness provides an alternative to the argument of Arkhipov--Bezrukavnikov. Namely, while they localize in unipotent directions, we localize in semi-simple directions, thereby reducing fully faithfulness to an order of vanishing calculation in semi-simple rank one.