Spherical varieties and Langlands duality

Let G be a connected reductive complex algebraic group. This paper is devoted to the space Z of meromorphic quasimaps from a curve into an affine spherical G-variety X. The space Z may be thought of as an algebraic model for the loop space of X. In this paper, we associate to X a connected reductive complex algebraic subgroup $\check H$ of the dual group $\check G$. The construction of $\check H$ is via Tannakian formalism: we identify a certain tensor category Q(Z) of perverse sheaves on Z with the category of finite-dimensional representations of $\check H$. Combinatorial shadows of the group $\check H$ govern many aspects of the geometry of X such as its compactifications and invariant differential operators. When X is a symmetric variety, the group $\check H$ coincides with that associated to the corresponding real form of G via the (real) geometric Satake correspondence

On a vanishing conjecture appearing in the geometric Langlands correspondence

Let $X$ be a smooth complete curve, and let $Bun_n$ be the moduli stack of rank $n$ vector bundles on $X$. Let $E$ be a local system on $X$. In a recent paper of E.Frenkel, K.Vilonen and the author, it was shown that the vanishing of a certian functor $Av_E^d$ acting from the category $D(Bun_n)$ to itself, implies the geometric Langlands conjecture. In this paper we establish the required vanishing result. Our proof works for sheaves with char=0 coefficients, or with torsion coefficients when the parameter $d$ is less than the characteristic.Comment: Revised versio

D-modules on the affine Grassmannian and representations of affine Kac-Moody algebras

Let ${\mathfrak g}$ be a simple Lie algebra. For a level $\kappa$ (thought of as a symmetric ${\mathfrak g}$-invariant form of ${\mathfrak g}$), let $\hat{\mathfrak g}_\kappa$ be the corresponding affine Kac-Moody algebra. Let $Gr_G$ be the affine Grassmannian of ${\mathfrak g}$, and let $D_\kappa(Gr_G)-mod$ be the category of $\kappa$-twisted right D-modules on $Gr_G$. By taking global sections of a D-module, we obtain a functor $\Gamma:D_\kappa(Gr_G)-mod\to {\mathfrak g}_\kappa-mod$. It is known that this functor is exact and faithful when $\kappa$ is negative or irrational. In this paper, we show that the functor $\Gamma$ is exact and faithful also when $\kappa$ is the critical level.Comment: Revised version; to appear in Duke Math. Journa

Differential operators and the loop group via chiral algebras

Let $G$ be an algebraic group and let $\widetilde{\mathfrak g}$ be the corresponding affine algebra on some level. Consider the induced module V:=Ind^{\widetilde{\mathfrak g}}_{{\mathfrak g}[[t]](O_{G[[t]]}), where $O_{G[[t]]}$ is the ring of regular functions on the group $G[[t]]$. In this paper we show that $V$ is naturally a vertex operator algebra, which is "responsible" for D-modules on the loop group $G((t))$. Using the techiques of VOA we show that $V$ is in fact a bimodule over the affine algebra. In addition, we show that $V$ possesses a remarkable property related to its BRST reduction with respect to $\widetilde{\mathfrak g}$. This paper has a considerable intersection with a recent preprint of Gorbunov, Malikov and Schechtman.Comment: Revised version, Section 6 adde

The gerbe of Higgs bundles

The purpose of this work is to describe the (category of) Higgs bundles on a complex scheme X having a given cameral cover X~. We show that this category is a T_{X~}-gerbe, where T_{X~} is a certain sheaf of abelian groups on X, and we describe the class of this gerbe precisely. In particular, it follows that the set of isomorphism classes of Higgs bundles with a fixed cameral cover X~ is a torsor over the group H^1(X, T_{X~}), which itself parametrizes T_{X~}-torsors on X. This underlying group can be described as a generalized Prym variety, whose connected component is either an abelian variety or a degeneration thereof.Comment: 47 pages; two added sections contain applications to Higgs bundles with values and to bundles on elliptic fibration

Operads, Grothendieck topologies and Deformation theory

The idea of the work is to find an invariant way to pass from deformation theory to cohomology, which does not use any explicit cocycles. The appropriate cohomology theory is based on considering sheaves on a certain site. An advantage of the approach is that it enables one to give a simultanious treatement to all types of algebras. As an application, we prove the PBW theorem in cases where it is not yet known.Comment: 13 pages, AmsTe

A "strange" functional equation for Eisenstein series and miraculous duality on the moduli stack of bundles

We show that the failure of the usual Verdier duality on Bun(G) leads to a new duality functor on the category of D-modules, and we study its relation to the operation of Eisenstein series

Geometric realizations of Wakimoto modules at the critical level

We study the Wakimoto modules over the affine Kac-Moody algebras at the critical level from the point of view of the equivalences of categories proposed in our previous works, relating categories of representations and certain categories of sheaves. In particular, we describe explicitly geometric realizations of the Wakimoto modules as Hecke eigen-D-modules on the affine Grassmannian and as quasi-coherent sheaves on the flag variety of the Langlands dual group

Deformations of local systems and Eisenstein series

Let $X$ be a (smooth and complete) curve and $G$ a reductive group. In [BG] we introduced the object that we called "geometric Eisenstein series". This is a perverse sheaf $\bar{Eis}_E$ (or rather a complex of such) on the moduli stack $Bun_G(X)$ of principal $G$-bundles on $X$, which is attached to a local system $E$ on $X$ with respect to the torus $\check{T}$, Langlands dual to the Cartan subgroup $T\subset G$. In loc. cit. we showed that$\bar{Eis}_E$ corresponds to the $\check{G}$-local system induced from $E$, in the sense of the geometric Langlands correspondence. In the present paper we address the following question, suggested by V. Drinfeld: what is the perverse sheaf on $Bun_G(X)$ that corresponds to the universal deformation of $E$ as a local system with respect to the Borel subgroup $\check{B}\subset \check{G}$? We prove, following a conjecture of Drinfeld, that the resulting perverse sheaf if the classical, i.e., non-compactified Eisenstein series

The category of singularities as a crystal and global Springer fibers

We prove the "Gluing Conjecture" on the spectral side of the categorical geometric Langlands correspondence. The key tool is the structure of crystal on the category of singularities, which allows to reduce the conjecture to the question of homological triviality of certain homotopy types. These homotopy types are obtained by gluing from a global version of Springer fibers