The universal monodromic Arkhipov-Bezrukavnikov equivalence identifies universal monodromic Iwahori-Whittaker sheaves with quasicoherent sheaves on the Grothendieck alteration, plus a monoidal bi-Whittaker equivalence.
Gaitsgory's central functor and the Arkhipov-Bezrukavnikov equivalence in mixed characteristic
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abstract
We show that the nearby cycles functor for the $p$-adic Hecke stack at parahoric level is perverse t-exact, by developing a theory of Wakimoto filtrations at Iwahori level, and that it lifts to the $\mathbb{E}_1$-center. We apply these tools to construct the Arkhipov-Bezrukavnikov functor for $p$-adic affine flag varieties at Iwahori level, and prove that it is an equivalence for all classical groups and also exceptional groups of type $E_6$ and $E_7$.
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2025 1verdicts
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The universal monodromic Arkhipov--Bezrukavnikov equivalence
The universal monodromic Arkhipov-Bezrukavnikov equivalence identifies universal monodromic Iwahori-Whittaker sheaves with quasicoherent sheaves on the Grothendieck alteration, plus a monoidal bi-Whittaker equivalence.