Nearby cycles on Drinfeld-Gaitsgory-Vinberg Interpolation Grassmannian and long intertwining functor

TL;DR

This paper establishes a duality between certain equivariant D-modules on the affine Grassmannian via nearby cycles on the Drinfeld-Gaitsgory-Vinberg interpolation Grassmannian, extending to Beilinson-Drinfeld Grassmannian and affine flag varieties.

Contribution

It proves a canonical duality between $U( ext{t})$- and $U^-( ext{t})$-equivariant D-modules using nearby cycles, introducing new insights into their properties and generalizations.

Findings

Duality between equivariant D-modules established

Nearby cycles characterized and compared with previous studies

Results extended to Beilinson-Drinfeld Grassmannian and affine flag variety

Abstract

Let $G$ be a reductive group and $U,U^-$ 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 $Gr_G$ 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 arXiv:1805.07721. We study various properties of the mentioned nearby cycles, in particular compare them with the nearby cycles studied in arXiv:1411.4206 and arXiv:1607.00586. We also generalize our results to the Beilinson-Drinfeld Grassmannian $Gr_{G,X^I}$ and to the affine flag variety $Fl_G$.