Related papers: The Arinkin-Gaitsgory temperedness conjecture

In arXiv:1807.09038 we formulated a conjecture describing the derived category D-mod(Gr$_{GL(n)}$) of (all) D-modules on the affine Grassmannian of the group $GL(n)$ as the category of quasi-coherent sheaves on a certain stack (it is…

In this paper we prove two results pertaining to the (unramified and global) geometric Langlands program. The first result is an analogue of the Ramanujan conjecture: any cuspidal D-module on Bun_G is tempered. We actually prove a more…

Let $Bun_G(X)$ be the moduli stack of $G$-torsors on a smooth projective curve $X$ for a reductive group $G$. We prove a conjecture made by Drinfeld-Wang and Gaitsgory on the Deligne-Lusztig duality for D-modules on $Bun_G(X)$. This…

In this paper, we introduce the category of quasi-tempered automorphic D-modules, which is a rather natural class of D-modules from the point of view of geometric Langlands. We provide a characterization of this category in terms of…

Given an algebraic stack $X$, one may compare the derived category of quasi-coherent sheaves on $X$ with the category of dg-modules over the dg-ring of functions on $X$. We study the analogous question in stable homotopy theory, for derived…

The so called theory of derived D-modules is an extension of classical D-modules to derived algebraic geometry, which uses the derived information of the base scheme. We prove that the three different definitions of derived D-modules, given…

The theory of $\Theta$-stratifications generalizes a classical stratification of the moduli of vector bundles on a smooth curve, the Harder-Narasimhan-Shatz stratification, to any moduli problem that can be represented by an algebraic…

We show that the category of ind-coherent sheaves on a quasi-smooth scheme is naturally tensored over the category of sheared D-modules on its shifted cotangent bundle, commuting with its natural action of categorified Hoschschild cochains.…

We associate a t-structure to a family of objects in D(A), the derived category of a Grothendieck category A. Using general results on t-structures, we give a new proof of Rickard's theorem on equivalence of bounded derived categories of…

Let $G$ and $\check{G}$ be Langlands dual connected reductive groups. We establish a monoidal equivalence of $\infty$-categories between equivariant quasicoherent sheaves on the formal neighborhood of the nilpotent cone in $G$ and…

We prove that under a certain mild hypothesis, the DG category of D-modules on a quasi-compact algebraic stack is compactly generated. We also show that under the same hypothesis, the functor of global sections on the DG category of…

We define the derived category of quasi--coherent modules for certain Artin stacks as the homotopy category of two Quillen monoidal model structures on the corresponding category of unbounded complexes of quasi--coherent modules.

Let $G$ be a unimodular locally compact group. We define a property of irreducible unitary $G$-representations $V$ which we call c-temperedness, and which for the trivial $V$ boils down to F{\o}lner's condition (equivalent to the trivial…

We give a moduli-theoretic proof of the classical theorem of Gabriel, stating that a scheme can be reconstructed from the abelian category of quasi-coherent sheaves over it. The methods employed are elementary and allow us to extend the…

In $\rm{GL}_N$, a series of subgroups indexed by $0\leq M\leq N-1$ were noticed by H. Jacquet-I. Piatetski Shapiro-J. Shalika, J. Cogdell, and D. Gaiotto. It was conjectured by D. Gaiotto that the categories of twisted D-modules on the…

We introduce quasi-BPS categories for twisted Higgs bundles, which are building blocks of the derived category of coherent sheaves on the moduli stack of semistable twisted Higgs bundles over a smooth projective curve. Under some condition…

We characterize all fields of definition for a given coherent sheaf over a projective scheme in terms of projective modules over a finite-dimensional endomorphism algebra. This yields general results on the essential dimension of such…

In this article we prove a conjecture of Braverman-Kazhdan in \cite{BK} on the acyclicity of gamma sheaves in the de Rham setting. The proof relies on the techniques developed in \cite{BFO} on Drinfeld center of Harish-Chandra bimodules and…

This elementary survey article was prepared for a talk at the 2016 Superschool on Derived Categories and D-branes. The goal is to outline an identification of the bounded derived category of coherent sheaves on a Calabi-Yau threefold with…

In this article we give a proof of Serre's conjecture for the case of odd level and arbitrary weight. Our proof does not depend on any generalization of Kisin's modularity lifting results to characteristic 2 (moreover, we will not consider…