Item

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.