Browsing by Author "Gaitsgory, Dennis"
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Chiral Koszul Duality
Francis, John; Gaitsgory, Dennis (Springer, 2012)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), ... 
Chiral Principal Series Categories
Raskin, Samuel David (20140606)This thesis begins a study of principal series categories in geometric representation theory using the BeilinsonDrinfeld theory of chiral algebras. We study Whittaker objects in the unramified principal series category. ... 
Compact Generation of the Category of DModules on the Stack of GBundles on a Curve
Drinfeld, Vladimir; Gaitsgory, Dennis (2013)The goal of the paper is to show that the (derived) category of Dmodules 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 \(Bun_G(X)\) ... 
Contractibility of the space of rational maps
Gaitsgory, Dennis (Springer Science + Business Media, 2012)We define an algebrogeometric 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 ... 
A Corollary of the Bfunction Lemma
Beilinson, Alexander; Gaitsgory, Dennis (Birkhäuser Basel, 2011)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 Dmodule on \(U\). Consider the \(D_U[s]\)module \(M\otimes ... 
DModules on Spaces of Rational Maps and on Other Generic Data
Barlev, Jonathan (20121213)Fix an algebraic curve X. We study the problem of parametrizing geometric data over X, which is only generically deﬁned. E.g., parametrizing generically deﬁned maps from X to a fixed target scheme Y. There are three methods ... 
DModules on the Affine Flag Variety and Representations of Affine KacMoody Algebras
Frenkel, Edward; Gaitsgory, Dennis (American Mathematical Society, 2009)The present paper studies the connection between the category of modules over the affine KacMoody Lie algebra at the critical level, and the category of Dmodules on the affine flag scheme \(G((t))/I\), where \(I\) is the ... 
Deformations of Local Systems and Eisenstein Series
Braverman, Alexander; Gaitsgory, Dennis (Springer Science + Business Media, 2008) 
DG Indschemes
Gaitsgory, Dennis; Rozenblyum, Nick (American Mathematical Society, 2014)We develop the notion of indscheme in the context of derived algebraic geometry, and study the categories of quasicoherent sheaves and indcoherent sheaves on indschemes. The main results concern the relation between ... 
Hecke Operators on Quasimaps into Horospherical Varieties
Gaitsgory, Dennis; Nadler, David (University Bielefeld, Fakultat Mathematik, 2009)Let G be a connected reductive complex algebraic group. This paper and its companion [GN06] are devoted to the space Z of meromorphic quasimaps from a curve into an affine spherical Gvariety X. The space Z may be thought ... 
IndCoherent Sheaves
Gaitsgory, Dennis (Independent University of Moscow, 2013)We develop the theory of indcoherent sheaves on schemes and stacks. The category of indcoherent sheaves is closely related, but inequivalent, to the category of quasicoherent sheaves, and the difference becomes crucial ... 
Local Geometric Langlands Correspondence: The Spherical Case
Frenkel, Edward; Gaitsgory, Dennis (Mathematical Society of Japan, 2009)A module over an affine Kac–Moody algebra $\hat{g}$ is called spherical if the action of the Lie subalgebra g[[t]] on it integrates to an algebraic action of the corresponding group G[[t]]. Consider the category of spherical ... 
Localization of \(\hat{\mathfrak{g}}\)modules on the Affine Grassmannian
Frenkel, Edward; Gaitsgory, Dennis (Princeton University, 2009)We consider the category of modules over the affine KacMoody algebra \(\hat{\mathfrak{g}}\) of critical level with regular central character. In our previous paper we conjectured that this category is equivalent to the ... 
On Some Finiteness Questions for Algebraic Stacks
Drinfeld, Vladimir; Gaitsgory, Dennis (Springer Science + Business Media, 2013)We prove that under a certain mild hypothesis, the DG category of Dmodules on a quasicompact algebraic stack is compactly generated. We also show that under the same hypothesis, the functor of global sections on the DG ... 
Outline of the proof of the geometric Langlands conjecture for GL(2)
Gaitsgory, Dennis (Centre National de la Recherche Scientifique, 2015)We outline a proof of the categorical geometric Langlands conjecture for GL_2, as formulated in ``Singular support of coherent sheaves and the geometric Langlands conjecture'', modulo a number of more tractable statements ... 
PicardLefschetz Oscillators for the DrinfeldLafforgueVinberg Compactification
Schieder, Simon Fabian (20150501)We study the singularities of the DrinfeldLafforgueVinberg compactification of the moduli stack of Gbundles on a smooth projective curve for a reductive group G. The study of these compactifications was initiated by V. ... 
Sheaves of categories and the notion of 1affineness
Gaitsgory, Dennis (American Mathematical Society (AMS), 2015)We define the notion of 1affineness for a prestack, and prove an array of results that establish 1affineness of certain types of prestacks. 
Singular support of coherent sheaves and the geometric Langlands conjecture
Arinkin, D.; Gaitsgory, Dennis (Springer Science + Business Media, 2014)We define the notion of singular support of a coherent sheaf on a quasismoothderived scheme or Artin stack, where “quasismooth” means that it is a locally complete intersection in the derived sense. This develops the ... 
Twisted Whittaker Model and Factorizable Sheaves
Gaitsgory, Dennis (Birkhaeuser Verlag AG, 2008)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 Dmodules) on the affine ... 
Weyl Modules and Opers without Monodromy
Frenkel, Edward; Gaitsgory, Dennis (SpringerVerlag, 2010)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 monodromyfree opers on the disc with regular singularity and residue determined by ...