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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 \(\mathcal O\).

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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[1] J. Bernstein [I. N. BernšTeĭN], “Trace in categories” in Operator Algebras, Unitary Representations, Enveloping Algebras, and Invariant Theory (Paris, 1989) , Progr. Math. 92 , Birkhäuser, Boston, 1990, 417–423. · Zbl 0747.17007
[2] J. Bernstein [I. N. BernšTeĭN], I. Frenkel, and M. Khovanov, A categorification of the Temperley-Lieb algebra and Schur quotients of \(U(\mathfraksl_2)\) via projective and Zuckerman functors , Selecta Math. (N.S.) 5 (1999), 199–241. · Zbl 0981.17001
[3] I. N. BernšTeĭN [J. Bernstein], I. M. Gel’Fand [Gelfand], and S. I. Gel’Fand [Gelfand], A certain category of \(\mathfrak g\)-modules (in Russian), Funkcaional. Anal. i Priložen. 10 , no. 2 (1976), 1–8.; English translation in Funct. Anal. Appl. 10 (1976), 87–92.
[4] W. Borho and R. Macpherson, “Partial resolutions of nilpotent varieties” in Analyse et topologie sur les espaces singuliers, II, III (Luminy, France, 1981) , Astérisque 101 –102., Soc. Math. France, Montrouge, 1983, 23–74. · Zbl 0576.14046
[5] A. Braverman and D. Gaitsgory, On Ginzburg’s Lagrangian construction of representations of, \(\mathrm GL(n)\) , Math. Res. Lett. 6 (1999), 195–201. · Zbl 0971.22010
[6] J. Brundan, Centers of degenerate cyclotomic Hecke algebras and parabolic category \(\mathcal O\) , preprint,\arxivmath/0607717v3[math.RT] · Zbl 1202.20008
[7] J. Brundan and A. Kleshchev, Representations of shifted Yangians and finite \(W\)-algebras , preprint,\arxivmath/0508003v3[math.RT] · Zbl 1169.17009
[8] -, Schur-Weyl duality for higher levels , preprint,\arxivmath/0605217v2[math.RT]
[9] C. Chevalley, Invariants of finite groups generated by reflections , Amer. J. Math. 77 (1955), 778–782. JSTOR: · Zbl 0065.26103
[10] J. Chuang and R. Rouquier, Derived equivalences for symmetric groups and \(\mathfraksl_2\)-categorification , Ann. of Math. (2) 167 (2008), 245–298. · Zbl 1144.20001
[11] C. De Concini and C. Procesi, Symmetric functions, conjugacy classes and the flag variety , Invent. Math. 64 (1981), 203–219. · Zbl 0475.14041
[12] I. Frenkel, M. Khovanov, and C. Stroppel, A categorification of finite-dimensional irreducible representations of quantum \(\mathfraksl_ 2\) and their tensor products , Selecta Math. (N.S.) 12 (2006), 379–431. · Zbl 1188.17011
[13] W. Fulton, Young Tableaux , London Math. Soc. Stud. Texts 35 , Cambridge Univ. Press, Cambridge, 1997. · Zbl 0878.14034
[14] P. Gabriel, Des catégories abéliennes , Bull. Soc. Math. France 90 (1962), 323–448. · Zbl 0201.35602
[15] A. M. Garsia and C. Procesi, On certain graded \(S_n\)-modules and the \(q\)-Kostka polynomials , Adv. Math. 94 (1992), 82–138. · Zbl 0797.20012
[16] V. A. Ginzburg, Lagrangian construction of the enveloping algebra \(U(\mathrm sl_n)\) (in Russian), Funktsional. Anal. i Prilozhen 26 , no. 1 (1992), 64–66.; English translation in Funct. Anal. Appl. 26 (1992), 51–52.
[17] -, “Geometric methods in the representation theory of Hecke algebras and quantum groups” in Representation Theories and Algebraic Geometry (Montréal, 1997) , NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 514 , Kluwer, Dordrecht, Netherlands, 1998, 127–183. · Zbl 1009.17011
[18] I. Gordon, Baby Verma modules for rational Cherednik algebras , Bull. London Math. Soc. 35 (2003), 321–336. · Zbl 1042.16017
[19] R. Hotta and N. Shimomura, The fixed-point subvarieties of unipotent transformations on generalized flag varieties and the Green functions , Math. Ann. 241 (1979), 193–208. · Zbl 0463.20032
[20] R. Hotta and T. A. Springer, A specialization theorem for certain Weyl group representations and an application to the Green polynomials of unitary groups , Invent. Math. 41 (1977), 113–127. · Zbl 0389.20037
[21] M. Khovanov, Crossingless matchings and the cohomology of \((n,n)\) Springer varieties , Commun. Contemp. Math. 6 (2004), 561–577. · Zbl 1079.57009
[22] I. G. Macdonald, Symmetric functions and Hall polynomials , 2nd ed., Oxford Math. Monogr., Oxford Sci. Publ., Oxford Univ. Press, New York, 1995. · Zbl 0824.05059
[23] V. Mazorchuk and C. Stroppel, Projective-injective modules, Serre functors and symmetric algebras , preprint,\arxivmath/0508119v2[math.RT] · Zbl 1235.16013
[24] W. Soergel, Kategorie \(\mathcal O\), perverse Garben und Moduln über den Koinvarianten zur Weylgruppe , J. Amer. Math. Soc. 3 (1990), 421–445. JSTOR: · Zbl 0747.17008
[25] N. Spaltenstein, The fixed point set of a unipotent transformation on the flag manifold , Nederl. Akad. Wetensch. Proc. Ser. A 79 (1976), 452–456. · Zbl 0343.20029
[26] C. Stroppel, Category \(\mathcal O\): Quivers and endomorphism rings of projectives , Represent. Theory 7 (2003), 322–345. · Zbl 1050.17005
[27] -, Parabolic category \(\mathcal O\), perverse sheaves on Grassmannians, Springer fibres and Khovanov homology , preprint,\arxivmath/0608234v2[math.RT]
[28] T. Tanisaki, Defining ideals of the closures of the conjugacy classes and representations of the Weyl groups , Tôhoku Math. J. (2) 34 (1982), 575–585. · Zbl 0544.14030
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