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Symmetric functions, parabolic category $$\mathcal O$$, and the Springer fiber. (English) Zbl 1145.20003
In this very interesting paper the author proves that the center of a regular block of the parabolic category $$\mathcal O$$ for the general linear Lie algebra is isomorphic to the cohomology algebra of the corresponding Springer fiber, which was conjectured by M. Khovanov [in Commun. Contemp. Math. 6, No. 4, 561-577 (2004; Zbl 1079.57009)]. The same result was also proved by C. Stroppel [in “Perverse sheaves on Grassmannians, Springer fibres and Khovanov cohomology”, to appear in Compos. Math.] using completely different methods, however, which are still based on an earlier result of the author [J. Brundan, “Centers of degenerate cyclotomic Hecke algebras and parabolic category $$\mathcal O$$”, preprint arXiv:math/0607717].
Additionally to the above mentioned main result the author also finds presentations for the centers of singular blocks, which are cohomology algebras of Spaltenstein varieties. The key idea in the proofs is the construction of an action of the general linear Lie algebra $$\mathfrak{gl}_\infty(\mathbb{C})$$ on the direct sum of the centers of all integral blocks of $$\mathcal O$$.

##### MSC:
 20C08 Hecke algebras and their representations 17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) 17B35 Universal enveloping (super)algebras
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