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.

  • $\begingroup$ Category O for any group carries an action of the corresponding Artin braid group by autoequivalences. This comes from an action on the full category of representations with given generalized infinitesimal character commuting with the action of $G$. This gives an action on the Hochschild cohomology (or center) (though maybe more naturally on Hochschild homology, or cocenter), which I expect gives the Springer action you're looking for. $\endgroup$ – David Ben-Zvi Apr 13 '13 at 3:40
  • $\begingroup$ Great, thanks for your comment. I'll have a think about this and see where I get. $\endgroup$ – George Melvin Apr 13 '13 at 4:44

David's comment is correct. This is proven in Section 3 of this paper of Stroppel: http://arxiv.org/abs/math.RT/0608234

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  • $\begingroup$ Thanks for the reference, Ben. It's very much appreciated. $\endgroup$ – George Melvin Apr 13 '13 at 19:38

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