[Paper Review] Free-monodromic mixed tilting sheaves on flag varieties

Pramod N. Achar, Shotaro Makisumi+2 more|arXiv (Cornell University)|Mar 16, 2017

Algebraic structures and combinatorial models1 references20 citations

TL;DR

This paper constructs a monoidal category of free-monodromic mixed tilting sheaves on Kac–Moody flag varieties using the mixed modular derived category, establishing a foundational framework for modular Koszul duality. The key contribution is a proof of functoriality for convolution in this setting, extending characteristic 0 results to positive characteristic via diagrammatic and homological methods in a derived category with monodromy actions.

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 construction of a "modular Koszul duality" equivalence for constructible sheaves on flag varieties, see [P. Achar, S. Makisumi, S. Riche, and G. Williamson, "Koszul duality for Kac-Moody groups and characters of tilting modules", J. Amer. Math. Soc. 32 (2019)].

Motivation & Objective

• To develop a monoidal category of free-monodromic mixed tilting sheaves in positive characteristic, extending the characteristic 0 theory of Bezrukavnikov–Yun.
• To establish a modular version of Koszul duality for constructible sheaves on flag varieties using the mixed modular derived category.
• To prove that convolution of free-monodromic tilting complexes satisfies the interchange law, thereby endowing the category with a monoidal structure.
• To provide a diagrammatic and homological framework for tilting sheaves in the Kac–Moody setting, compatible with monodromy and weight filtrations.
• To lift minimal Rouquier complexes and prove their functoriality under convolution in the mixed modular setting.

Proposed method

• Constructs a free-monodromic derived category using diagrammatics from Elias–Williamson calculus, incorporating left and right monodromy actions.
• Introduces a triangulated structure on complexes with left-monodromic or right-equivariant structures, using Karoubian envelopes and dgg algebras.
• Defines convolution via bimodular Hom spaces in the derived category, proving associativity and unitality using coherence conditions.
• Applies a localization functor to reduce statements about morphisms in the mixed modular category to the rational case via base change.
• Uses Jones–Wenzl projectors and minimal Rouquier complexes in finite dihedral cases to inductively describe indecomposable tilting objects.
• Lifts minimal Rouquier complexes from the rational to the modular setting using symmetric algebras and graded modules over polynomial rings.

Experimental results

Research questions

• RQ1 Can a monoidal category of free-monodromic tilting sheaves be constructed in positive characteristic, analogous to the characteristic 0 case?
• RQ2 Does convolution of free-monodromic tilting complexes satisfy the interchange law, ensuring monoidal structure?
• RQ3 How can the minimal Rouquier complexes in the rational setting be lifted to the modular mixed derived category?
• RQ4 What is the role of monodromy in the structure of tilting sheaves in the mixed modular derived category?
• RQ5 Can the functoriality of convolution be established in the Kac–Moody flag variety setting using diagrammatics and localization?

Key findings

• The category of free-monodromic tilting sheaves on Kac–Moody flag varieties admits a well-defined monoidal structure via convolution.
• The convolution of free-monodromic tilting complexes satisfies the interchange law, proving that the convolution functor is well-behaved and associative.
• The localization functor is faithful on the category of free-monodromic tilting complexes, allowing reduction to rational cases for proofs.
• Minimal Rouquier complexes lift from the rational to the modular setting, and their convolution is compatible with the monodromy action.
• The category of free-monodromic tilting sheaves is equipped with a module structure over the monoidal category of tilting complexes, extending the duality framework.
• The construction provides a key ingredient for a full modular Koszul duality equivalence, as outlined in [AMRW].