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Gaitsgory, Dennis

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Gaitsgory

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Dennis

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Gaitsgory, Dennis
Author's Publications: search.filters.isAuthorOfPublication.4c5e4143-efe9-4622-8152-b6a5c3f9d8de

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    A “Strange” Functional Equation for Eisenstein Series and Miraculous Duality on the Moduli Stack of Bundles
    (Societe Mathematique de France, 2017-09) Gaitsgory, Dennis
    We show that the failure of the usual Verdier duality on BunG leads to a new duality functor on the category of D-modules, and we study its relation to the operation of Eisenstein series.
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    Deformations of Local Systems and Eisenstein Series
    (Springer Science + Business Media, 2008) Braverman, Alexander; Gaitsgory, Dennis
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    DG Indschemes
    (American Mathematical Society, 2014) Gaitsgory, Dennis; Rozenblyum, Nick
    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.
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    Ind-Coherent Sheaves
    (Independent University of Moscow, 2013) Gaitsgory, Dennis
    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.
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    On Some Finiteness Questions for Algebraic Stacks
    (Springer Science + Business Media, 2013) Drinfeld, Vladimir; Gaitsgory, Dennis
    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.
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    Weyl Modules and Opers without Monodromy
    (Springer-Verlag, 2010) Frenkel, Edward; Gaitsgory, Dennis
    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.
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    Compact Generation of the Category of D-Modules on the Stack of G-Bundles on a Curve
    (2013) Drinfeld, Vladimir; Gaitsgory, Dennis
    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.
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    Contractibility of the space of rational maps
    (Springer Science + Business Media, 2012) Gaitsgory, Dennis
    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.
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    Sheaves of categories and the notion of 1-affineness
    (American Mathematical Society (AMS), 2015) Gaitsgory, Dennis
    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.
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    Singular support of coherent sheaves and the geometric Langlands conjecture
    (Springer Science + Business Media, 2014) Arinkin, D.; Gaitsgory, Dennis
    We define the notion of singular support of a coherent sheaf on a quasi-smooth-derived scheme or Artin stack, where “quasi-smooth” means that it is a locally complete intersection in the derived sense. This develops the idea of “cohomological” support of coherent sheaves on a locally complete intersection scheme introduced by D. Benson, S. B. Iyengar, and H. Krause. We study the behavior of singular support under the direct and inverse image functors for coherent sheaves. We use the theory of singular support of coherent sheaves to formulate the categorical geometric Langlands conjecture. We verify that it passes natural consistency tests: It is compatible with the geometric Satake equivalence and with the Eisenstein series functors. The latter compatibility is particularly important, as it fails in the original “naive” form of the conjecture.