Abstract
We classify all equivalences between the indecomposable abelian categories which appear as blocks in BGG category \({\mathcal {O}}\) for reductive Lie algebras. Our classification implies that a block in category \({\mathcal {O}}\) only depends on the Bruhat order of the relevant parabolic quotient of the Weyl group. As part of the proof, we observe that any finite dimensional algebra with simple preserving duality admits at most one quasi-hereditary structure.
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Beilinson, A., Ginzburg, V., Soergel, W.: Koszul duality patterns in representation theory. J. Am. Math. Soc. 9(2), 473–527 (1996)
Bernstein, I.N., Gel’fand, I.M., Gel’fand, S.I.: A certain category of \({\mathfrak{g}}\)-modules. Funkcional. Anal. i Prilozen. 10(2), 1–8 (1976)
Björner, A., Brenti, F.: Combinatorics of Coxeter groups. Graduate Texts in Mathematics, vol. 231. Springer, New York (2005)
Brundan, J., Stroppel, C.: Semi-infinite highest weight categories. arXiv:1808.08022
Brundan, J., Goodwin, S.M.: Whittaker coinvariants for \(GL(m|n)\). Adv. Math. 347, 273–339 (2019)
Cline, E., Parshall, B., Scott, L.: Finite-dimensional algebras and highest weight categories. J. Reine Angew. Math. 391, 85–99 (1988)
Coulembier, K., Penkov, I.: On an infinite limit of BGG categories O. To appear in Mosc. Math. J. arXiv:1802.06343
Coulembier, K., Serganova, V.: Homological invariants in category \({\cal{O}}\) for the general linear superalgebra. Trans. Am. Math. Soc. 369(11), 7961–7997 (2017)
Deligne, P.: Catégories tannakiennes. The Grothendieck Festschrift, Vol. II, 111–195, Progr. Math., 87, Birkhäuser Boston, Boston, MA (1990)
Dlab, V., Ringel, C.M.: The module theoretical approach to quasi-hereditary algebras. Representations of algebras and related topics (Kyoto, 1990), 200–224, London Math. Soc. Lecture Note Ser., 168, Cambridge Univ. Press, Cambridge (1992)
Ehrig, M., Stroppel, C.: Diagrammatic description for the categories of perverse sheaves on isotropic Grassmannians. Select. Math. 22(3), 1455–1536 (2016)
Graham, J.J., Lehrer, G.I.: Cellular algebras. Invent. Math. 123(1), 1–34 (1996)
Humphreys, J.E.: Representations of semisimple Lie algebras in the BGG category \({\cal{O}}\). Graduate Studies in Mathematics, 94. American Mathematical Society, Providence, RI (2008)
König, S., Xi, C.C.: When is a cellular algebra quasi-hereditary? Math. Ann. 315(2), 281–293 (1999)
Soergel, W.: Kategorie \({\cal{O}}\), perverse Garben und Moduln über den Koinvarianten zur Weylgruppe. J. Am. Math. Soc. 3(2), 421–445 (1990)
Stroppel, C.: Category \({\cal{O}}\): quivers and endomorphism rings of projectives. Represent. Theory 7, 322–345 (2003)
Acknowledgements
The research was supported by the ARC Grant DE170100623. The author thanks Chih-Whi Chen and Geordie Williamson for interesting discussions, and the referee for many useful comments on the exposition.
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Coulembier, K. The classification of blocks in BGG category \({\mathcal {O}}\). Math. Z. 295, 821–837 (2020). https://doi.org/10.1007/s00209-019-02376-9
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DOI: https://doi.org/10.1007/s00209-019-02376-9