On monoidal Koszul duality for the Hecke category

TL;DR

This paper introduces and explains monoidal Koszul duality in the Hecke category, illustrating key concepts through the concrete example of SL2 using Soergel bimodules.

Contribution

It provides a gentle, accessible introduction to Koszul duality phenomena in the Hecke category, focusing on a specific duality studied in recent joint work.

Findings

Illustrates key phenomena of Koszul duality in the Hecke category.

Uses Soergel bimodules to model the duality concretely.

Focuses on the simplest nontrivial case of SL2.

Abstract

We attempt to give a gentle (though ahistorical) introduction to Koszul duality phenomena for the Hecke category, focusing on the form of this duality studied in joint work of Achar, Riche, Williamson, and the author. We illustrate some key phenomena and constructions for the simplest nontrivial case of (finite) $\mathrm{SL}_2$ using Soergel bimodules, a concrete algebraic model of the Hecke category.