Nearby cycles on Drinfeld–Gaitsgory–Vinberg interpolation Grassmannian and long intertwining functor

Abstract

Let G be a reductive group, and let U, $U^{-}$ be the unipotent radicals of a pair of opposite parabolic subgroups P, $P^{-}$. We prove that the DG categories of $U((t))$-equivariant and $U^{-}((t))$-equivariant D-modules on the affine Grassmannian ${\mathrm{Gr}}_G$ are canonically dual to each other. We show that the unit object witnessing this duality is given by nearby cycles on the Drinfeld–Gaitsgory–Vinberg interpolation Grassmannian defined in a recent work by Finkelberg, Krylov, and Mirković. We study various properties of the mentioned nearby cycles, and in particular compare them with the nearby cycles studied in works by Schieder. We also generalize our results to the Beilinson–Drinfeld Grassmannian ${\mathrm{Gr}}_{G,X^I}$ and to the affine flag variety ${\mathrm{Fl}}_G$.

This version of the paper contains fewer appendices than the version submitted to arXiv.