We establish equivalences of the following three triangulated categories: \[ D quantum ( g ) ⟷ D coherent G ( N ~ ) ⟷ D perverse ( G r ) . D_\text {quantum}(\mathfrak {g})\enspace \longleftrightarrow \enspace D^G_\text {coherent}(\widetilde {{\mathcal N}})\enspace \longleftrightarrow \enspace D_\text {perverse}(\mathsf {Gr}). \] Here, D quantum ( g ) D_\text {quantum}(\mathfrak {g}) is the derived category of the principal block of finite-dimensional representations of the quantized enveloping algebra (at an odd root of unity) of a complex semisimple Lie algebra g \mathfrak {g} ; the category D coherent G ( N ~ ) D^G_\text {coherent}(\widetilde {{\mathcal N}}) is defined in terms of coherent sheaves on the cotangent bundle on the (finite-dimensional) flag manifold for G G ( = = semisimple group with Lie algebra g \mathfrak {g} ), and the category D perverse ( G r ) D_\text {perverse}({\mathsf {Gr}}) is the derived category of perverse sheaves on the Grassmannian G r {\mathsf {Gr}} associated with the loop group L G ∨ LG^\vee , where G ∨ G^\vee is the Langlands dual group, smooth along the Schubert stratification. The equivalence between D quantum ( g ) D_\text {quantum}(\mathfrak {g}) and D coherent G ( N ~ ) D^G_\text {coherent}(\widetilde {{\mathcal N}}) is an “enhancement” of the known expression (due to Ginzburg and Kumar) for quantum group cohomology in terms of nilpotent variety. The equivalence between D perverse ( G r ) D_\text {perverse}(\mathsf {Gr}) and D coherent G ( N ~ ) D^G_\text {coherent}(\widetilde {{\mathcal N}}) can be viewed as a “categorification” of the isomorphism between two completely different geometric realizations of the (fundamental polynomial representation of the) affine Hecke algebra that has played a key role in the proof of the Deligne-Langlands-Lusztig conjecture. One realization is in terms of locally constant functions on the flag manifold of a p p -adic reductive group, while the other is in terms of equivariant K K -theory of a complex (Steinberg) variety for the dual group. The composite of the two equivalences above yields an equivalence between abelian categories of quantum group representations and perverse sheaves. A similar equivalence at an even root of unity can be deduced, following the Lusztig program, from earlier deep results of Kazhdan-Lusztig and Kashiwara-Tanisaki. Our approach is independent of these results and is totally different (it does not rely on the representation theory of Kac-Moody algebras). It also gives way to proving Humphreys’ conjectures on tilting U q ( g ) U_q(\mathfrak {g}) -modules, as will be explained in a separate paper.

We show that, on the level of derived categories, representations of the Lie algebra of a semisimple algebraic group over a field of finite characteristic with a given (generalized) regular central character are the same as coherent sheaves on the formal neighborhood of the corresponding (generalized) Springer fiber.The first step is to observe that the derived functor of global sections provides an equivalence between the derived category of D-modules (with no divided powers) on the flag variety and the appropriate derived category of modules over the corresponding Lie algebra. Thus the "derived" version of the Beilinson-Bernstein localization theorem holds in sufficiently large positive characteristic. Next, one finds that for any smooth variety this algebra of differential operators is an Azumaya algebra on the cotangent bundle. In the case of the flag variety it splits on Springer fibers, and this allows us to pass from D-modules to coherent sheaves. The argument also generalizes to twisted D-modules. As an application we prove Lusztig's conjecture on the number of irreducible modules with a fixed central character. We also give a formula for behavior of dimension of a module under translation functors and reprove the Kac-Weisfeiler conjecture.The sequel to this paper [BMR2] treats singular infinitesimal characters. To Boris Weisfeiler, missing since 1985 Contents

Abstract. In [BMR] we observed that, on the level of derived categories, representations of the Lie algebra of a semisimple algebraic group over a field of finite characteristic with a given (generalized) regular central character can be identified with coherent sheaves on the formal neighborhood of the corresponding (generalized) Springer fiber. In the present paper we treat singular central characters.The basic step is the Beilinson-Bernstein localization of modules with a fixed (generalized) central character λ as sheaves on the partial flag variety corresponding to the singularity of λ. These sheaves are modules over a sheaf of algebras which is a version of twisted crystalline differential operators. We discuss translation functors and intertwining functors. The latter generate an action of the affine braid group on the derived category of modules with a regular (generalized) central character, which intertwines different localization functors. We also describe the standard duality on Lie algebra modules in terms of D-modules and coherent sheaves. §0. Introduction This is a sequel to [BMR]. In the first chapter we extend the localization construction for modular representations from [BMR] to arbitrary infinitesimal characters. This is used in the second chapter to study the translation functors. We use translation functors to construct intertwining functors, which generate an action of the affine braid group on the derived

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Abstract. In this paper we construct and study an action of the affine braid group associated to a semi-simple algebraic group on derived categories of coherent sheaves on various varieties related to the Springer resolution of the nilpotent cone. In particular, we describe explicitly the action of the Artin braid group. This action is a "categorical version" of Kazhdan-Lusztig-Ginzburg's construction of the affine Hecke algebra, and is used in particular in [BM] in the course of the proof of Lusztig's conjectures on equivariant K-theory of Springer fibers.

We consider the problem of quantization of smooth symplectic varieties in the algebro-geometric setting. We show that, under appropriate cohomological assumptions, the Fedosov quantization procedure goes through with minimal changes. The assumptions are satisfied, for example, for affine and for projective varieties. We also give a classification of all possible quantizations.

To Joseph Bernstein with admiration and gratitude ROMAN BEZRUKAVNIKOV AND IVAN MIRKOVI Ć 4.2. Lifting to characteristic zero 4.3. Shifting the alcoves 5. Applications to Representation Theory 5.1. Generic independence of p 5.2. Equivariant versions and Slodowy slices 5.3. Gradings and bases in K-theory 5.4. Proofs for subsection 5.3 5.5. Koszul property 6. Grading that satisfies property (⋆)