Graded and Geometric Parabolic Induction for Category $\mathcal{O}$

TL;DR

This paper demonstrates that parabolic induction in category O can be lifted to a graded setting, connecting algebraic, geometric, and motivic perspectives, and describing its effects on modules.

Contribution

It proves the gradability of parabolic induction functors in category O and constructs a geometric version using stratified mixed Tate motives.

Findings

Parabolic induction functor is gradable and lifts to graded category O.

Constructs a geometric parabolic induction functor on stratified mixed Tate motives.

Describes the impact of parabolic induction on Soergel modules.

Abstract

We prove that the parabolic induction functor on BGG-category $\mathcal{O}$ associated to a complex reductive Lie algebra is gradable, that is, lifts to graded category $\mathcal{O}$ as constructed by Beilinson-Ginzburg-Soergel. Graded category $\mathcal{O}$ is equivalent to a category of stratified mixed Tate motives on a corresponding flag variety as recently defined by Soergel-Wendt. The graded version of parabolic induction is induced by a geometric parabolic induction functor we construct on the level of stratified mixed Tate motives. We also describe the effect of parabolic induction on the level of Soergel modules.