Now showing items 1-20 of 24

    • Another Realization of the Category of Modules Over the Small Quantum Group 

      Arkhipov, Sergey; Gaitsgory, Dennis (Elsevier BV, 2003-01)
      Let g be a semi-simple simply-connected Lie algebra and let U-l be the corresponding quantum group with divided powers, where is an even order root of unity. Let in addition u(l) subset of U-l be the corresponding "small" ...
    • The Category of Singularities as a Crystal and Global Springer Fibers 

      Arinkin, Dmitro; Gaitsgory, Dennis (American Mathematical Society (AMS), 2017-05-08)
      We prove the "gluing conjecture" on the spectral side of the categorical geometric Langlands conjecture. The key tool is the structure of crystal on the category of singularities, which allows one to reduce the conjecture ...
    • 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 (2014-06-06)
      This thesis begins a study of principal series categories in geometric representation theory using the Beilinson-Drinfeld theory of chiral algebras. We study Whittaker objects in the unramified principal series category. ...
    • Compact Generation of the Category of D-Modules on the Stack of G-Bundles on a Curve 

      Drinfeld, Vladimir; Gaitsgory, Dennis (2013)
      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 \(Bun_G(X)\) ...
    • Contractibility of the space of rational maps 

      Gaitsgory, Dennis (Springer Science + Business Media, 2012)
      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 ...
    • A Corollary of the B-function 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 D-module on \(U\). Consider the \(D_U[s]\)-module \(M\otimes ...
    • D-Modules on Spaces of Rational Maps and on Other Generic Data 

      Barlev, Jonathan (2012-12-13)
      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 ...
    • D-Modules on the Affine Flag Variety and Representations of Affine Kac-Moody Algebras 

      Frenkel, Edward; Gaitsgory, Dennis (American Mathematical Society, 2009)
      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 ...
    • 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 quasi-coherent sheaves and ind-coherent 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 G-variety X. The space Z may be thought ...
    • Ind-Coherent Sheaves 

      Gaitsgory, Dennis (Independent University of Moscow, 2013)
      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 ...
    • 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 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 ...
    • 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 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 ...
    • 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 ...
    • Parameters and Duality for the Metaplectic Geometric Langlands Theory 

      Gaitsgory, Dennis; Lysenko, S (Springer Nature, 2017-10-11)
      We introduce the space of parameters for the metaplectic Langlands theory as factorization gerbes on the affine Grassmannian, and develop metaplectic Langlands duality in the incarnation of the metaplectic geometric Satake ...
    • Picard-Lefschetz Oscillators for the Drinfeld-Lafforgue-Vinberg Compactification 

      Schieder, Simon Fabian (2015-05-01)
      We study the singularities of the Drinfeld-Lafforgue-Vinberg compactification of the moduli stack of G-bundles 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 1-affineness 

      Gaitsgory, Dennis (American Mathematical Society (AMS), 2015)
      We define the notion of 1-affineness for a prestack, and prove an array of results that establish 1-affineness of certain types of prestacks.