arXiv subjects
Publications and source records attributed to Dennis Gaitsgory.
The excursion algebra associated to a scheme X over a finite field and a reductive group G is the algebra of global functions on the stack of arithmetic G-local systems on X. When X is a curve, this algebra acts on the space of automorphic functions. In this paper we establish some basic properties of this algebra.
We introduce and study the filtration on the space of automorphic functions (in the everywhere unramified situation for the function field case) obtained by transferring the filtration on the spectral side of the classical Langlands conjecture, induced by coherent singular support. We propose a number of conjectures that tie this filtration (which, by design, arises from the notion of cohomological support) to a filtration on the space of C-valued automorphic functions that arises by considering the analytic spectrum of Hecke operators.
This is the final paper in the series of five, in which we prove the geometric Langlands conjecture (GLC). We conclude the proof of GLC by showing that there exists a unique (up to tensoring up by a vector space) Hecke eigensheaf corresponding to an irreducible local system (hence, the title of the paper). We achieve this by analyzing the geometry of the stack of local systems.
In this paper we record the formalism of algebro-geometric DG categories (in short AGCat) following a suggestion of V. Drinfeld. This formalism will be applied to ``real-world" problems in papers sequel to this one, [GRV2] and [GRV3].
We show that the (2-)category of categorical representations of the loop group embeds fully faithfully into the (2-)category of factorization module categories with respect to the affine Grassmannian.
We construct the geometric Langlands functor in one direction (from the automorphic to the spectral side) in characteristic zero settings (i.e., de Rham and Betti). We prove that various forms of the conjecture (de Rham vs Betti, restricted vs. non-restricted, tempered vs. non-tempered) are equivalent. We also discuss structural properties of Hecke eigensheaves.
This is a write-up for the plenary ICM talk, 2026. The goal of this paper is to propose a set of conjectures whose aim is to answer the basic question of the Langlands program (over function fields): how to describe the space of automorphic functions in terms of the spectral side (i.e., Langlands parameters)?
We establish part of the statement of the geometric Langlands conjecture for l-adic sheaves over a field of positive characteristic. Namely, we show that the category of automorphic sheaves with nilpotent singular support is equivalent to the appropriately defined category of ind-coherent sheaves on the union of some of the connected components of the stack of Langlands parameters.
We establish the compatibility of the Langlands functor with the operations of Eisenstein series constant term, and deduce that the Langlands functor induces an equivalence on Eisenstein-generated subcategories.
In this paper we introduce the categorical "true local terms" maps for Artin stacks and show that they are additive and commute with proper pushforwards, smooth pullbacks and specializations. In particular, we generalizing results of [Va2] to this setting. As an application, we supply proofs of two theorems stated in [AGKRRV]. Namely, we show that the "true local terms" of the Frobenius endomorphism coincide with the "naive local terms" and that the "naive local terms" commute with !-pushforwards. The latter result is a categorical version of the classical Grothendieck--Lefschetz trace formula.
We develop the notion of indscheme in the context of derived algebraic geometry, and study the categories of quasi-coherent sheaves and ind-coherent sheaves on indschemes. The main results concern the relation between classical and derived indschemes and the notion of formal smoothness.
This paper is a sequel to [Ga1]. We study the semi-infinite category on the Ran version of the affine Grassmannian, and study a particular object in it that we call the semi-infinite intersection cohomology sheaf. Unlike the situation of [Ga1], this version of the semi-infinite intersection IC sheaf is defined as the middle of extension of the constant (more precisely, dualizing) sheaf on the basic stratum, in a certain t-structure. We give several explicit description and characterizations of our semi-infinite intersection IC sheaf: we describe its !- and *- stalks; we present it explicitly as a colimit; we relate it to the IC sheaf of Drinfeld's relative compactification; we describe it via the Drinfeld-Plucker formalism.
We introduce the semi-infinite category of sheaves on the affine Grassmannian, and construct a particular object in it, which we call the the semi-infinite intersection cohomology sheaf. We relate it to several other entities naturally appearing in the geometric Langlands theory.
Given a category C acted on by the loop group G((t)), we define its Whittaker model Whit(C) as C^{N((t)),χ}, where χis a non-degenerate character. We study the properties of this construction. When C is the category of sheaves on the quotient of G((t)) by a congruence subgroup, we find a "finite-dimensional" model for Whit(C); the corresponding geometric object is Drinfeld's compactification, denoted \overline{Bun}_N (with poles and level structure).
We study factorization algebras on configuration spaces of points on the curved, colored by elements of the root lattice. We show that the factorization algebra attached to Lusztig's quantum group can be obtained as a direct image of a twisted Whittaker sheaf on the Zastava space.
A theorem of Kazhdan and Lusztig establishes an equivalence between the category of G(CO)-integrable representations of the Kac-Moody algebra \hat{g}_{-κ} at a negative level -κand the category \Rep_q(G) of (algebraic) representations of the "big" (a.k.a. Lusztig's) quantum group. In this paper we propose a conjecture that describes the category of Iwahori-integrable Kac-Moody modules. The corresponding object on the quantum group side, denoted Rep^{mxd}_q(G), involves Lusztig's version of the quantum group for the Borel and the De Concini-Kac version for the negative Borel.
This note corrects a certain construction in (the old version of) the paper [GL]. The construction in question starts from a Brylinski-Deligne datum, which is an extension of a group G by K_2, and produces a factorization line bundle on the affine Grassmannian of the group G.
In this paper we prove, for G a connected reductive algebraic group satisfying a technical assumption, that the Satake category of G (with coefficients in a finite field, a finite extension of Q_l, or the ring of integers of such a field) can be described via Iwahori-Whittaker perverse sheaves on the affine Grassmannian. As an application, we confirm a conjecture of Juteau-Mautner-Williamson describing the tilting objects in the Satake category.