Seminar in Algebra, Lie Theory, and Geometry

Ruslan Maksimau

(Max Planck Institute for Mathematics, Bonn)

"Affine Category \(\mathcal{O}\) and Categorical Actions"

Date: Thu, April 20, 2017
Time: 11:30
Place: Seminar Room (120), Research I

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 \(F\). Our goal is to prove that the functors \(E\) and \(F\) are Koszul dual to the Zuckerman functors.
To do this, it is enough to show that the functor \(F\) for the category \(\mathcal{O}\) at level \(-N-e\) can be decomposed in terms of the components of the functor \(F\) for the category \(\mathcal{O}\) at level \(-N-e-1\). To get such a decomposition, we prove a general fact about categorical representations: a category with an action of \(\widetilde{\mathfrak{sl}}_{e+1}\) contains a subcategory with an action of \(\widetilde{\mathfrak{sl}}_{e}\). The proof of this claim can be reduced to a general statement about quiver Hecke algebras.