# Ext-enhanced monoidal Koszul duality for $\mathrm{GL}_2$

## TL;DR

This paper proposes an enhancement to the monoidal Koszul duality for the Hecke category related to GL2 by adding an extra grading, supported by evidence in the context of HOMFLYPT link homology.

## Contribution

It introduces an enhanced grading structure to the monoidal Koszul duality for the Hecke category, specifically for GL2, connecting to HOMFLYPT link homology conjectures.

## Key findings

- Evidence supporting the enhanced duality for GL2.
- Connections established between the duality and HOMFLYPT link homology.
- Framework developed for future generalizations.

## Abstract

The Hecke category participates in an equivalence called monoidal Koszul duality, which exchanges it with the category of (Langlands-dual) "free-monodromic tilting sheaves." Motivated by a recent conjecture of Gorsky and the first-named author on HOMFLYPT link homology, we propose to enhance this duality with an additional grading. We provide evidence for this enhancement in the case of $\mathrm{GL}_2$, working in the language of the second-named author's joint work with Achar, Riche, and Williamson.