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

We realise Stroppel’s extended arc algebra \[13, 51] in the Fukaya–Seidel category of a natural Lefschetz fibration on generic fibre adjoint quotient map type $A$ nilpotent slice with two Jordan blocks, and hence obtain symplectic interpretation certain parabolic two-block versions Bernstein–Gel’fand–Gel’fand $\\mathcal{O}$. As an application, we give new geometric construction spectral sequence from annular to ordinary Khovanov homology. The heart paper is development cylindrical model compute Fukaya categories (affine open subsets of) Hilbert schemes quasi-projective surfaces, which may be independent interest.