Relative Serre duality for Coxeter groups

TL;DR

This paper proves a conjecture relating adjoint functors in the homotopy categories of Soergel bimodules for all finite Coxeter groups, using advanced bimodule theory and mixed derived categories.

Contribution

It establishes the conjecture for all finite Coxeter groups, extending previous partial results with new theoretical tools.

Findings

Confirmed the conjecture for all finite Coxeter groups.

Connected the adjoint functors via the relative full twist.

Utilized Abe-Bott-Samelson bimodules and mixed derived categories.

Abstract

It was conjectured by Gorsky, Hogancamp, Mellit, and Nakagane that the left and right adjoints of the parabolic induction functor between homotopy categories of Soergel bimodules associated to a finite Coxeter group are related by the relative full twist. Several cases of this conjecture are known including for symmetric groups, crystallographic Coxeter groups, and dihedral groups. We prove this conjecture in complete generality using the theory of Abe-Bott-Samelson bimodules and the Achar-Riche-Vay mixed derived category.