D-modules on the affine Grassmannian and representations of affine Kac-Moody algebras (original) (raw)
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For any Kac-Moody group G with Borel B, we give a monoidal equivalence between the derived category of B-equivariant mixed complexes on the flag variety G/B and (a certain completion of) the derived category of G ∨monodromic mixed complexes on the enhanced flag variety G ∨ /U ∨ , here G ∨ is the Langlands dual of G. We also prove variants of this equivalence, one of which is the equivalence between the derived category of U -equivariant mixed complexes on the partial flag variety G/P and a certain "Whittaker model" category of mixed complexes on G ∨ /B ∨ . In all these equivalences, intersection cohomology sheaves correspond to (free-monodromic) tilting sheaves. Our results generalize the Koszul duality patterns for reductive groups in [BGS96]. 1 2 ROMAN BEZRUKAVNIKOV AND ZHIWEI YUN 4.2. The Whittaker category 25 4.3. Convolution 27 4.4. Averaging functors 30 4.5. The functor V 35 4.6. The pro-sheaf P 37 4.7. Proof of Proposition 4.5.7 40 5. Equivalences 44 5.1. Langlands duality for Kac-Moody groups 44 5.2. Equivariant-monodromic duality 45 5.3. Koszul "self-duality" 50 5.4. Parabolic-Whittaker duality 54 5.5. "Paradromic-Whittavariant" duality 58 Appendix A. Completions of monodromic categories 60 A.1. Unipotently monodromic complexes 60 A.2. Pro-objects in a filtered triangulated category 62 A.3. The completion 69 A.4. The case of a trivial A-torsor 73 A.5. The mixed case 78 A.6. The stratified case 81 A.7. Free-monodromic tilting sheaves 83 Appendix B. Construction of DG models 84 B.1. A simple subcategory 85 B.2. The DG model 88 B.3. Functoriality of the DG model 91 B.4. Application to equivariant categories 92 B.5. Application to monodromic categories 93 Appendix C. Calculations for SL(2) 94 List of symbols 96 Acknowledgement 96 References 97 c instead of D b m ). A local system on G m with unipotent monodromy is given by a representation of the pro-quotient of π 1 (G m ⊗ kk ). Taking the logarithm of the unipotent monodromy, such a sheaf corresponds to a finite dimensional Q [[t]]module on which t acts nilpotently. Denote the category of such Q [[t]]-modules by Mod nil (Q [[t]]), then