A topological approach to Soergel theory

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 $\mathbb{k}$. 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 $\mathbb{k}$-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 $T_{\mathbb{k}}^{\vee }{\times }_{(T_{\mathbb{k}}^{\vee }{)}^W}{T}_{\mathbb{k}}^{\vee }$, where $T_{\mathbb{k}}^{\vee }$ is the $\mathbb{k}$-torus dual to $T$.

v2: minor corrections