Person:

Gaitsgory, Dennis

We extend the theory of chiral and factorization algebras, developed for curves by Beilinson and Drinfeld (American Mathematical Society Colloquium Publications, 51. American Mathematical Society, Providence, RI, 2004), to higher-dimensional varieties. This extension entails the development of the homotopy theory of chiral and factorization structures, in a sense analogous to Quillen’s homotopy theory of differential graded Lie algebras. We prove the equivalence of higher-dimensional chiral and factorization algebras by embedding factorization algebras into a larger category of chiral commutative coalgebras, then realizing this interrelation as a chiral form of Koszul duality. We apply these techniques to rederive some fundamental results of Beilinson and Drinfeld (American Mathematical Society Colloquium Publications, 51. American Mathematical Society, Providence, RI, 2004) on chiral enveloping algebras of (\star)-Lie algebras.

The present paper studies the connection between the category of modules over the affine Kac-Moody Lie algebra at the critical level, and the category of D-modules on the affine flag scheme (G((t))/I), where (I) is the Iwahori subgroup. We prove a localization-type result, which establishes an equivalence between certain subcategories on both sides. We also establish an equivalence between a certain subcategory of Kac-Moody modules, and the category of quasi-coherent sheaves on the scheme of Miura opers for the Langlands dual group, thereby proving a conjecture of the authors in 2006.

Let (X) be an algebraic variety, (f) a regular function, (j:U \hookrightarrow X) the complement to the locus of vanishing of (f), and (M) a holonomic D-module on (U). Consider the (D_U[s])-module (M\otimes "f^s"). The goal of this note is to describe all (D_X[s])-submodules (N \hookrightarrow j_(M\otimes "f^s")) such that (j^(N)\simeq M\otimes "f^s").

Let (G) be a reductive group. The geometric Satake equivalence realized the category of representations of the Langlands dual group (\check G) in terms of spherical perverse sheaves (or D-modules) on the affine Grassmannian (Gr_G=G((t))/G[[t]]) of the original group G. In the present paper we perform a first step in realizing the category of representations of the quantum group corresponding to (\check G) in terms of the geometry of (Gr_G). The idea of the construction belongs to Jacob Lurie.

We consider the category of modules over the affine Kac-Moody algebra (\hat{\mathfrak{g}}) of critical level with regular central character. In our previous paper we conjectured that this category is equivalent to the category of Hecke eigen-D-modules on the affine Grassmannian (G((t))/G[[t]]). This conjecture was motivated by our proposal for a local geometric Langlands correspondence. In this paper we prove this conjecture for the corresponding (I^0) equivariant categories, where (I^0) is the radical of the Iwahori subgroup of (G((t))). Our result may be viewed as an affine analogue of the equivalence of categories of (\mathfrak{g})-modules and D-modules on the flag variety (G/B), due to Beilinson-Bernstein and Brylinski-Kashiwara.

We develop the theory of ind-coherent sheaves on schemes and stacks. The category of ind-coherent sheaves is closely related, but inequivalent, to the category of quasi-coherent sheaves, and the difference becomes crucial for the formulation of the categorical Geometric Langlands Correspondence.

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 quasi-coherent sheaves is continuous.

We prove that the algebra of endomorphisms of a Weyl module of critical level is isomorphic to the algebra of functions on the space of monodromy-free opers on the disc with regular singularity and residue determined by the highest weight of the Weyl module. This result may be used to test the local geometric Langlands correspondence proposed in our earlier work.

The goal of the paper is to show that the (derived) category of D-modules on the stack (Bun_G(X)) is compactly generated. Here X is a smooth complete curve, and G is a reductive group. The problem is that (Bun_G(X)) is not quasi-compact, so the above compact generation is not automatic. The proof is based on the following observation: (Bun_G(X)) can be written as a union of quasi-compact open substacks, which are "co-truncative", i.e., the (j_!) extension functor is defined on the entire category of D-modules.

We define an algebro-geometric model for the space of rational maps from a smooth curve X to an algebraic group G, and show that this space is homologically contractible. As a consequence, we deduce that the moduli space BunG of G-bundles on X is uniformized by the appropriate rational version of the affine Grassmannian, where the uniformizing map has contractible fibers.