Affine category O, Koszul duality and Zuckerman functors

TL;DR

This paper explores the relationship between affine category O, Koszul duality, and Zuckerman functors, revealing how certain functors are dual and how categories at different levels relate through decomposition.

Contribution

It establishes that functors in affine category O are Koszul dual to Zuckerman functors and demonstrates a decomposition approach across different levels using categorical actions.

Findings

Functors E and F are Koszul dual to Zuckerman functors.

Decomposition of F functor across levels using subcategory structures.

General criteria for functors to be Koszul dual to Zuckerman functors.

Abstract

The parabolic category $\mathcal{O}$ for affine ${\mathfrak{gl}}_N$ at level $-N-e$ admits a structure of a categorical representation of $\widetilde{\mathfrak{sl}}_e$ with respect to some endofunctors $E$ and $F$. This category contains a smaller category $\mathbf{A}$ that categorifies the higher level Fock space. We prove that the functors $E$ and $F$ in the category $\mathbf{A}$ are Koszul dual to Zuckerman functors. The key point of the proof is to show that the functor $F$ for the category $\mathbf{A}$ at level $-N-e$ can be decomposed in terms of the components of the functor $F$ for the category $\mathbf{A}$ at level $-N-e-1$. To prove this, we use the following fact: a category with an action of $\widetilde{\mathfrak sl}_{e+1}$ contains a (canonically defined) subcategory with an action of $\widetilde{\mathfrak sl}_{e}$. We also prove a general statement that says that in some general situation a functor that satisfies a list of axioms is automatically Koszul dual to some sort of Zuckerman functor.