Abstract. Building on the theory of parity sheaves due to Juteau-MautnerWilliamson, we develop a formalism of "mixed modular perverse sheaves" for varieties equipped with a stratification by affine spaces. We then give two applications: (1) a "Koszul-type" derived equivalence relating a given flag variety to the Langlands dual flag variety, and (2) a formality theorem for the modular derived category of a flag variety (extending the main result of [RSW]).

We study the orbits of $G=\mathrm{GL}(V)$ in the enhanced nilpotent cone $V\times\mathcal{N}$, where $\mathcal{N}$ is the variety of nilpotent endomorphisms of $V$. These orbits are parametrized by bipartitions of $n=\dim V$, and we prove that the closure ordering corresponds to a natural partial order on bipartitions. Moreover, we prove that the local intersection cohomology of the orbit closures is given by certain bipartition analogues of Kostka polynomials, defined by Shoji. Finally, we make a connection with Kato's exotic nilpotent cone in type C, proving that the closure ordering is the same, and conjecturing that the intersection cohomology is the same but with degrees doubled.Comment: 32 pages. Update (August 2010): There is an error in the proof of Theorem 4.7, in this version and the almost-identical published version. See the corrigendum arXiv:1008.1117 for independent proofs of later results that depend on that statemen

A fundamental result of Beilinson-Ginzburg-Soergel states that on flag varieties and related spaces, a certain modified version of the category of ℓ-adic perverse sheaves exhibits a phenomenon known as Koszul duality.The modification essentially consists of discarding objects whose stalks carry a nonsemisimple action of Frobenius. In this paper, we prove that a number of common sheaf functors (various pull-backs and push-forwards) induce corresponding functors on the modified category or its triangulated analogue. In particular, we show that these functors preserve semisimplicity of the Frobenius action.

In this paper we propose a construction of a monoidal category of "free-monodromic" tilting perverse sheaves on (Kac-Moody) flag varieties in the setting of the "mixed modular derived category" introduced by the first and third authors. This category shares most of the properties of their counterpart in characteristic 0, defined by Bezrukavnikov-Yun using certain pro-objects in triangulated categories. This construction is the main new ingredient in the forthcoming construction of a "modular Koszul duality" equivalence for constructible sheaves on flag varieties. Contents Chapter 1. Introduction 1.1. Koszul duality 1.2. Koszul duality for constructible sheaves on flag varieties 1.3. The mixed derived category 1.4. The case of Kac-Moody flag varieties 1.5. Monodromy in the mixed modular derived category 1.6. The free-monodromic derived category 1.7. Convolution 1.8. Application to Koszul duality 1.9. The diagrammatic language 1.10. Contents 1.11. Acknowledgements Chapter 2. Coxeter groups and Elias-Williamson calculus 2.1. Realizations of Coxeter groups 2.2. The Elias-Williamson diagrammatic category 2.3. Additive hull of D BS (h, W ) 2.4. More on 2-colored quantum numbers Chapter 3. Bigraded modules and dgg algebras 3.1. Gradings and differentials 3.2. Symmetric algebras associated to realizations 3.3. Derivations of symmetric and exterior algebras 3.4. Graded modules over polynomial rings Chapter 4. Complexes of Elias-Williamson diagrams 4.1. D ⊕ BS -sequences 4.2. Biequivariant complexes 4.3. Right-equivariant complexes 4.4. Left-monodromic complexes 4.5. Triangulated structure 4.6. Right-equivariant versus left-monodromic complexes 4.7. The left monodromy action 4.8. Another triangulated structure on LM(h, W ) 4.9. Karoubian envelopes Chapter 5. Free-monodromic complexes 5.1. Definitions 5.2. The left and right monodromy actions 5.3. Examples of free-monodromic complexes iii iv CONTENTS Chapter 6. Free-monodromic convolution 6.1. Convolutive complexes 6.2. Convolution on dgg Hom spaces 6.3. Convolution of objects 6.4. Convolution of morphisms 6.5. The categories Conv ⊲ FM (h, W ) and Conv ⊳ FM (h, W ) 6.6. Action of Conv FM (h, W ) on Conv LM (h, W ) 6.7. Karoubian envelopes Chapter 7. Hints of functoriality 7.1. Unitor isomorphisms 7.2. Associator isomorphism 7.3. Coherence conditions and n-fold convolution product 7.4. Free-monodromic standard and costandard objects 7.5. Convolution of morphisms revisited 7.6. Convolution as a functor in one variable 7.7. Tilting complexes and the functoriality conjecture Chapter 8. Finite dihedral groups 8.1. Roots for dihedral groups 8.2. Jones-Wenzl projectors 8.3. Further properties of Jones-Wenzl projectors 8.4. "Inductive" description of indecomposable objects 8.5. Morphisms between indecomposables 8.6. Breaking and two-dot morphisms Chapter 9. Rouquier complexes 9.1. Biequivariant minimal Rouquier complexes 9.2. Lifting minimal Rouquier complexes 9.3. Convolution of free-monodromic minimal Rouquier complexes Chapter 10. Flag varieties for Kac-Moody groups 10.1. Cartan...

We establish a character formula for indecomposable tilting modules for connected reductive groups in characteristic ℓ in terms of ℓ-Kazhdan-Lusztig polynomials, for ℓ > h the Coxeter number. Using results of Andersen, one may deduce a character formula for simple modules if ℓ ≥ 2h − 2. Our results are a consequence of an extension to modular coefficients of a monoidal Koszul duality equivalence established by Bezrukavnikov and Yun.1.3. The Kac-Moody case and quantum groups. These ideas were later generalized by Bezrukavnikov-Yun [BY] to the case where G is replaced by a general Kac-Moody group G . Let B ⊂ G be a Borel subgroup, and let U ⊂ B be its unipotent radical. An important new idea in [BY] (also suggested in [BG]) is that a richer version of Koszul duality can be obtained if one "deforms" the categories of semisimple complexes on G /B and tilting perverse sheaves on G ∨ /B ∨ along a polynomial ring. The B-constructible semisimple complexes are thus replaced by the B-equivariant semisimple complexes, and the tilting perverse sheaves are replaced by the so-called "free-monodromic" objects constructed (via a very technical procedure) by Yun using certain pro-objects in the derived category of G ∨ /U ∨ , see [BY, Appendix A]. These deformed categories each have a monoidal structure, given by an appropriate kind of convolution product. The main result of [BY] is an equivalence of monoidal categoriesrelating B-equivariant semisimple complexes on G /B and free-monodromic tilting perverse sheaves attached to G ∨ . From this, Bezrukavnikov-Yun then deduce a Kac-Moody analogue of (1.1). As in §1.2, this result has a combinatorial motivation in terms of Kazhdan-Lusztig polynomials [Yu], and a representation-theoretic motivation in terms of analogues of the category O for Kac-Moody Lie algebras.But a third motivation for the work in [BY], specifically in the case of affine Kac-Moody groups, came from the hope of uniting two geometric approaches to the study of representations of Lusztig's quantum groups at a root of unity (see e.g. [Be, §1.2]), which we review below. Let Rep 0 (U ζ ) denote the principal block of the category of finite-dimensional representations of Lusztig's quantum group U ζ associated with an adjoint semisimple complex algebraic group G, specialized at a root of unity ζ.The first approach comes from [ABG]. The main result of [ABG, Part I] relates 3 Rep 0 (U ζ ) to the derived category of equivariant coherent sheaves on the Springer resolution N of G, denoted by D b Coh G×Gm ( N ). Then the main result of [ABG, Part II] states that D b Coh G×Gm ( N ) is equivalent to the derived category of Iwahoriconstructible perverse sheaves on the affine Grassmannian Gr of the Langlands dual semisimple group G ∨ . Together, these results give a new proof of Lusztig's character formula for simple modules in Rep 0 (U ζ ). (This character formula was already known when [ABG] appeared, by combining work of Kazhdan-Lusztig [KL2], Lusztig [Lu2] and Kashiwara-Tanisaki [KT].)For each s ∈ J, choose, once and for all, a...