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Localization of g^-modules on the affine Grassmannian

Edward Frenkel·Dennis Gaitsgory
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Abstract

We consider the category of modules over the affine Kac-Moody algebra g^ of critical level with regular central character. In our previous paper math.RT/0508382 we conjectured that this category is equivalent to the category of Hecke eigen-D-modules on the affine Grassmannian G((t))/G[[t]]. This conjecture was motivated by our proposal for a local geometric Langlands correspondence. In this paper we prove this conjecture for the corresponding I^0 equivariant categories, where I^0 is the radical of the Iwahori subgroup of G((t)). Our result may be viewed as an affine analogue of the equivalence of categories of g-modules and D-modules on the flag variety G/B, due to Beilinson-Bernstein and Brylinski-Kashiwara.

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Edward Frenkel, Dennis Gaitsgory. 2005-12-26. Localization of g^-modules on the affine Grassmannian. https://arxiv.org/abs/math/0512562

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