Geometric Langlands duality and representations of algebraic groups over commutative rings

@article{Mirkovic2004GeometricLD,
  title={Geometric Langlands duality and representations of algebraic groups over commutative rings},
  author={I. Mirkovic and Kari Vilonen},
  journal={Annals of Mathematics},
  year={2004},
  volume={166},
  pages={95-143}
}
As such, it can be viewed as a first step in the geometric Langlands program. The connected complex reductive groups have a combinatorial classification by their root data. In the root datum the roots and the co-roots appear in a symmetric manner and so the connected reductive algebraic groups come in pairs. If G is a reductive group, we write G for its companion and call it the dual group G. The notion of the dual group itself does not appear in Satake's paper, but was introduced by Langlands… 
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References

SHOWING 1-10 OF 50 REFERENCES
Perverse Sheaves on affine Grassmannians and Langlands Duality
This is an expanded version of the text ``Perverse Sheaves on Loop Grassmannians and Langlands Duality'', AG/9703010. The main new result is a topological realization of algebraic representations of
On quasi-reductive group schemes
This paper was motivated by a question of Vilonen, and the main results have been used by Mirkovic and Vilonen to give a geometric interpre- tation of the dual group (as a Chevalley group over Z) of
On De Jong’s conjecture
AbstractLet X be a smooth projective curve over a finite field Fq. Let ρ be a continuous representation π(X) → GLn(F), where F = Fl((t)) with Fl being another finite field of order prime to q.Assume
Perverse sheaves on real loop Grassmannians
The aim of this paper is to identify a certain tensor category of perverse sheaves on the loop Grassmannian Grℝ of a real form Gℝ of a connected reductive complex algebraic group G with the category
Sheaves on a loop group and langlands duality
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
Conformal blocks and generalized theta functions
LetSUXr be the moduli space of rankr vector bundles with trivial determinant on a Riemann surfaceX. This space carries a natural line bundle, the determinant line bundleL. We describe a canonical
Hyperbolic localization of intersection cohomology
AbstractFor a normal variety X defined over an algebraically closed field with an action of the multiplicative group T = Gm, we consider the "hyperbolic localization" functor Db(X) → Db(XT), which
Theory of spherical functions on reductive algebraic groups over p-adic fields
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