### 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.