Sheaves on a loop group and langlands duality

  title={Sheaves on a loop group and langlands duality},
  author={Victor Ginzburg},
  journal={Functional Analysis and Its Applications},
  • V. Ginzburg
  • Published 13 November 1995
  • Mathematics
  • Functional Analysis and Its Applications
An intrinsic construction of the tensor category of finite dimensional representations of the Langlands dual group of G in terms of a tensor category of perverse sheaves on the loop group, LG, is given. The construction is applied to the study of the topology of the affine Grassmannian of G and to establishing a Langlands type correspondence for "automorphic" sheaves on the moduli space of G-bundles. 
Perverse sheaves on affine flags and langlands dual group
This is the first in a series of papers devoted to describing the category of sheaves on the affine flag manifold of a simple algebraic group in terms of the Langlands dual group. In the present
Twisted Whittaker model and factorizable sheaves
Abstract.Let G be a reductive group. The geometric Satake equivalence realized the category of representations of the Langlands dual group Ğ in terms of spherical perverse sheaves (or D-modules) on
Dynamical Weyl groups and equivariant cohomology of transversal slices on affine Grassmannians
Let G be a reductive group; in this note we give an interpretation of the dynamical Weyl group of of the Langlands dual group $\check{G}$ defined by Etingof and Varchenko in terms of the geometry of
Satake equivalence for Hodge modules on affine Grassmannians
For a reductive group G we equip the category of GO-equivariant polarizable pure Hodge modules on the affine Grassmannian GrG with a structure of neutral Tannakian category. We show that it is
Integral homology of loop groups via Langlands dual groups
Let K be a connected compact Lie group, and G its complexification. The homology of the based loop group ΩK with integer coefficients is naturally a Z-Hopf algebra. After possibly inverting 2 or 3,
On tensor categories attached to cells in affine Weyl groups, III
We prove a weak version of Lusztig’s conjecture on explicit description of the asymptotic Hecke algebras (both finite and affine) related to monodromic sheaves on the base affine space (both finite
Modular affine Hecke category and regular unipotent centralizer, I
In this paper we provide, under some mild explicit assumptions, a geometric description of the category of representations of the centralizer of a regular unipotent element in a reductive algebraic
An example of the derived geometrical Satake correspondence over integers
Let G^v be a complex simple algebraic group. We describe certain morphisms of G^v (O)-equivariant complexes of sheaves on the affine Grassmannian Gr of G^v in terms of certain morphisms of
The global nilpotent variety is Lagrangian
The purpose of this note is to present a short elementary proof of a theorem due to Faltings and Laumon, saying that the global nilpotent cone is a Lagrangian substack in the cotangent bundle of the
We describe the equivariant cohomology of cofibers of spherical perverse sheaves on the affine Grassmannian of a reductive algebraic group in terms of the geometry of the Langlands dual group. In


The Principal Three-Dimensional Subgroup and the Betti Numbers of a Complex Simple Lie Group
Let g be a complex simple Lie algebra and let G be the adjoint group of g. It is by now classical that the Poincare polynomial p G (t) of G factors into the form
Lie Group Representations on Polynomial Rings
Let G be a group of linear transformations on a finite dimensional real or complex vector space X. Assume X is completely reducible as a G-module. Let S be the ring of all complex-valued polynomials
Limits of weight spaces, Lusztig’s $q$-analogs, and fiberings of adjoint orbits
Let G be a connected complex semisimple algebraic group, and T a maximal torus inside a Borel subgroup B, with g, t, and b their Lie algebras. Let V be a representation in the category a for g. The
Convexity and Loop Groups
The purpose of this paper is to extend certain convexity results associated with compact Lie groups to an infinite-dimensional setting, in which the Lie group is replaced by the corresponding loop
Hodge Cycles, Motives, and Shimura Varieties
General Introduction.- Notations and Conventions.- Hodge Cycles on Abelian Varieties.- Tannakian Categories.- Langlands's Construction of the Taniyama Group.- Motifs et Groupes de Taniyama.-
Perverse sheaves and *-actions
where t * x denotes the action of t on x. The set Xw is known to be a locally-closed C* -stable algebraic subvariety of X isomorphic to an affine space. The pieces Xw form a cell decomposition X =
Intersection cohomology methods in representation theory
In recent years, the theory of group representations has greatly benefited from a new approach provided by the topology of singular spaces, namely intersection cohomology (IC ) theory. Let G be a
On the cohomology of algebraic and related finite groups
In the above theorem, H*(GL,,M~ ~ denotes the rational cohomology of the algebraic group GL, with coefficients in M(, ~), the rational GL.-module obtained from the adjoint module M . = M ~ ~ through
Two-Dimensional l-Adic Representations of the Fundamental Group of a Curve Over a Finite Field and Automorphic Forms on GL(2)
def Here 7rV denotes a prime element of Ov, Mv {h e Mat(2, Ov)ldet h E 7r 01* }, dg is the Haar measure on GL(2, kv) such that mes GL(2, 0v) = 1. The geometric Frobenius element of xr1(X)