# Fukaya-Seidel categories of Hilbert schemes and parabolic category $\mathcal{O}$

## TL;DR

This paper connects Fukaya-Seidel categories with parabolic category O and Hilbert schemes, providing new geometric insights and a spectral sequence relating Khovanov homologies.

## Contribution

It realises Stroppel's extended arc algebra within Fukaya-Seidel categories and develops a cylindrical model for computing Fukaya categories of Hilbert schemes.

## Key findings

- Symplectic interpretation of parabolic two-block category O.
- New geometric construction of the spectral sequence from annular to ordinary Khovanov homology.
- Development of a cylindrical model for Fukaya categories of Hilbert schemes.

## Abstract

We realise Stroppel's extended arc algebra in the Fukaya-Seidel category of a natural Lefschetz fibration on the generic fiber of the adjoint quotient map on a type $A$ nilpotent slice with two Jordan blocks, and hence obtain a symplectic interpretation of certain parabolic two-block versions of Bernstein-Gelfan'd-Gelfan'd category $\mathcal{O}$. As an application, we give a new geometric construction of the spectral sequence from annular to ordinary Khovanov homology. The heart of the paper is the development of a cylindrical model to compute Fukaya categories of (affine open subsets of) Hilbert schemes of quasi-projective surfaces, which may be of independent interest.