Geometric Langlands duality and representations of algebraic groups over commutative rings

Abstract

In this paper we give a geometric version of the Satake isomorphism. Given a connected complex reductive algebraic group, we show that the category of representations of its Langlands dual is naturally equivalent to a certain category of perverse sheaves on the complex affine Grassmannian. We can work with perverse sheaves with coefficients in an arbitrary commutative ring and then we recover the representation theory of the split form of the dual group over the commutative ring.

This version contains a new Appendix B. In the original version there appears to be a gap in section 12. To fix the gap we explain in Appendix B how to read through section 12 without having to pose condition (12.11c)

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