A topological approach to Soergel theory

TL;DR

This paper develops a topological framework for Soergel theory on flag varieties, establishing algebraic and categorical equivalences that deepen understanding of perverse sheaves and their monodromic counterparts over arbitrary fields.

Contribution

It introduces a topological approach to Soergel theory, describing endomorphisms via coinvariant algebras and establishing equivalences with Soergel modules, extending the theory to arbitrary fields.

Findings

Endomorphisms described by coinvariant algebra

Equivalence between projective/tilting objects and Soergel modules

Derived category of T-monodromic sheaves characterized as a monoidal category

Abstract

We develop a "Soergel theory" for Bruhat-constructible perverse sheaves on the flag variety $G/B$ of a complex reductive group $G$, with coefficients in an arbitrary field $\Bbbk$. Namely, we describe the endomorphisms of the projective cover of the skyscraper sheaf in terms of a "multiplicative" coinvariant algebra, and then establish an equivalence of categories between projective (or tilting) objects in this category and a certain category of "Soergel modules" over this algebra. We also obtain a description of the derived category of $T$-monodromic $\Bbbk$-sheaves on $G/U$ (where $U$, $T\subset B$ are the unipotent radical and the maximal torus), as a monoidal category, in terms of coherent sheaves on the formal neighborhood of the base point in $T^\vee_\Bbbk \times_{(T^\vee_\Bbbk)^W} T^\vee_\Bbbk$, where $T^\vee_\Bbbk$ is the $\Bbbk$-torus dual to $T$.