I recently learned of a result of Brundan describing the centre of the regular block of parabolic category $\mathcal{O}$ for $\mathfrak{gl}_{n}$ as the cohomology of a corresponding Springer fibre (available here); this is an isomorphism at the level of $\mathbb{C}$-algebras. However, we also have an action of the symmetric group on these cohomology algebras.

Question: Is there a way to define an $S_{n}$-action on the centre of (parabolic) category $\mathcal{O}$ for $\mathfrak{gl}_{n}$, and in such a way that Brundan's isomorphism becomes a morphism of graded $S_{n}$-modules?

Of course, I can simply transport the $S_{n}$-action in the obvious way but I was looking for a more intrinsic action on the centre that arises from (parabolic) category $\mathcal{O}$ itself. Also, I've had a look at Brundan's paper but it doesn't appear to be discussed there (if I'm being blind then please let me know!)

Thanks in advance.