[Paper Review] Modular affine Hecke category and regular centralizer

Roman Bezrukavnikov, Simon Riche|arXiv (Cornell University)|Jun 8, 2022

Algebraic structures and combinatorial models44 references4 citations

TL;DR

This paper provides a combinatorial description of tilting perverse sheaves on the affine flag variety of a reductive group over a field of positive characteristic, using a novel realization of Soergel bimodules via the universal centralizer of the Langlands dual group. It establishes an equivalence between a category of free-monodromic perverse sheaves and Soergel bimodules, enabling character formulas for indecomposable tilting sheaves in terms of $\ell$-Kazhdan–Lusztig polynomials—extending classical Soergel theory to positive characteristic.

ABSTRACT

In this paper we provide a "combinatorial" description of the category of tilting perverse sheaves on the affine flag variety of a reductive algebraic group, and its free-monodromic variant, with coefficients in a field of positive characteristic. This provides a replacement for the familiar "Soergel theory" for characteristic-0 coefficients, and the second step in our project towards the construction of an equivalence of categories relating the two natural geometric realizations of the associated affine Hecke algebra in the case of positive-characteristic coefficients.

Motivation & Objective

• To develop a positive-characteristic analogue of Soergel theory for affine flag varieties, replacing the classical $\mathbb{C}$-coefficients with coefficients in an algebraically closed field $\Bbbk$ of characteristic $\ell > 0$.
• To construct a category of free-monodromic perverse sheaves on the universal cover $\widetilde{\mathrm{Fl}}_G$ that models the constructible side of a modular tamely ramified local Langlands correspondence.
• To establish an equivalence between this category of perverse sheaves and a category of Soergel bimodules realized via representations of the universal centralizer of the Langlands dual group $G^\vee_\Bbbk$.
• To derive character formulas for indecomposable tilting perverse sheaves using $\ell$-Kazhdan–Lusztig polynomials, extending known results from characteristic zero and finite flag varieties.

Proposed method

• Realize Soergel bimodules not via graded bimodules over a polynomial ring, but via representations of the universal centralizer of the Langlands dual group $G^\vee_\Bbbk$, which captures the geometric Satake correspondence and affine Weyl group structure.
• Construct the category of free-monodromic perverse sheaves on $\widetilde{\mathrm{Fl}}_G$ as a completion of the bounded derived category of constructible sheaves with respect to a pro-system of nilpotent ideals.
• Use a projective limit construction involving quotients $R^\wedge_A / (\mathfrak{m}_A)^n \cdot R^\wedge_A$ to model the completed category $\widehat{D}(R^\wedge_A)$, ensuring compatibility with t-structures.
• Apply Verdier and Serre quotient formalisms to relate the derived categories of sheaves on the Steinberg variety and the completed flag variety, preserving t-structures.
• Leverage the t-exactness of pushforward functors $j_!$ and $j_*$ along strata to lift t-structure conditions from finite-type strata to the completed setting.
• Establish an equivalence between the category of tilting perverse sheaves on $\widetilde{\mathrm{Fl}}_G$ and the category of Soergel bimodules via a degrading functor, linking mixed and ordinary perverse sheaves.

Experimental results

Research questions

• RQ1 How can Soergel theory be extended to positive characteristic coefficients for affine flag varieties, where classical constructions fail due to non-semisimplicity?
• RQ2 What is the correct categorical realization of Soergel bimodules in the affine Weyl group setting over a field of positive characteristic, that incorporates the geometry of the Langlands dual group?
• RQ3 Can character formulas for indecomposable tilting perverse sheaves on the affine flag variety be expressed combinatorially in terms of $\ell$-Kazhdan–Lusztig polynomials?
• RQ4 How can the completed derived category of sheaves on the Steinberg variety be described using pro-objects and projective limits, and how does this relate to t-structures?
• RQ5 What is the role of the universal centralizer in realizing Soergel bimodules in the positive characteristic setting, and how does it relate to geometric Satake?

Key findings

• The paper constructs an equivalence between the category of free-monodromic tilting perverse sheaves on $\widetilde{\mathrm{Fl}}_G$ and a category of Soergel bimodules realized via representations of the universal centralizer of $G^\vee_\Bbbk$, providing a positive-characteristic analog of Soergel theory.
• It proves that the characters of indecomposable tilting perverse sheaves on $\mathrm{Fl}_G$ are given by the $\ell$-Kazhdan–Lusztig polynomials, extending the classical result from characteristic zero.
• The completed derived category $\widehat{D}(R^\wedge_A)$ is identified with the projective limit of bounded derived categories of finitely generated modules over $R^\wedge_A / (\mathfrak{m}_A)^n$, preserving t-structures.
• The t-structure on the completed category is characterized via projective systems of objects in the bounded derived category, with $\mathscr{F} \in {}^p\widehat{D}_{\mathcal{S}}(X\setminus A,\Bbbk)^{\leq 0}$ if and only if it is isomorphic to a projective limit of objects in ${}^pD^b_{\mathcal{S}}(X,\Bbbk)^{\leq 0}$.
• The degrading functor relating mixed and ordinary perverse sheaves is constructed via the completion process, showing that the two categories are related through the same underlying Soergel-theoretic framework.
• The results provide a foundational step toward constructing modular tamely ramified local Langlands equivalences, linking Iwahori-equivariant perverse sheaves on $\mathrm{Fl}_G$ with coherent sheaves on the Steinberg variety of $G^\vee_\Bbbk$.