Ruslan Maksimau

"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.