# 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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