The universal monodromic Arkhipov--Bezrukavnikov equivalence
Gurbir Dhillon, Jeremy Taylor
TL;DR
The paper proves a universal monodromic Arkhipov--Bezrukavnikov equivalence, identifying automorphic category Shv_{(I)}(Fl) with spectral QCoh(\,B/\B) and establishing a monoidal bi-Whittaker correspondence QCoh(G/G) ≃ _χH_χ, all in a universal monodromy deformation of the Langlands setup. It develops a robust toolkit of universal monodromic Wakimoto and central sheaves, Plücker relations, and a fiber functor to control associated gradeds, then leverages semi-simple localization and rank-one order-of-vanishing arguments to prove fully faithfulness and equivalences. The strategy contrasts with Arkhipov--Bezrukavnikov by localizing in semi-simple directions rather than nilpotent directions, enabling a transparent, deformation-based approach that ultimately validates the tame local Betti geometric Langlands conjecture of Ben-Zvi--Nadler. The work also constructs the monoidal correspondence with bi-Iwahori--Whittaker categories and provides a comprehensive framework for extending AB's results to the universal setting, setting the stage for broader applications in geometric Langlands and representation theory.
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.