Free-monodromic mixed tilting sheaves on flag varieties

Abstract

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...

Dualité de Koszul formelle et théorie des représentations des groupes algébriques réductifs en caractéristique positive

“…) pour tout w ∈ W. Cette égalité peut également se démontrer directement (c'est-à-dire sans utiliser le Théorème 7.4) en utilisant les "objets libre-monodromiques" considérés dans [AMRW1]. Remarque 8.6.…”

How to glue parity sheaves

“…The notion of a graded parity sheaf is equivalent to the notion of a "parity sequence" from [AMRW1]. The category of graded parity sheaves is denoted by Parity Z Gm (X , k).…”

“…This object is a "tilting perverse sheaf" in the sense of [BBM]. See [BBM,Remark 1.1(ii)], and see [AMRW1,Example 4.6.4] for a related object in the context of flag varieties.…”

“…To solve this problem, we introduce a variant of D mix (X , k) called the monodromic derived category, and denoted by D mix mon (X , k). This category, whose definition is closely inspired by that of "free-monodromic objects" in [AMRW1], does allow both nonconstructible objects and nonsemisimple local systems. It fits into a diagram with the G m -equivariant and ordinary derived categories as shown below:…”

Nearby cycles for parity sheaves on a divisor with simple normal crossings

“…In this language, the shifts [1] along the arrows are absorbed into the objects, and the whole picture looks more like a "classical" chain complex of parity sheaves. (Objects in mixed modular derived categories in [AR1,AR2,AMRW] were drawn in this way.) For n = 2, our object Z looks like Finally, on Gr ̟1 , we can redraw these complexes using the Elias-Williamson calculus to indicate the maps in the differentials.…”