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Perverse sheaves, Koszul IC-modules, and the quiver for the category \(\mathcal{O}\)

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Abstract

For a stratified topological space we introduce the category of IC-modules, which are linear algebra devices with the relations described by the equation d 2=0. We prove that the category of (mixed) IC-modules is equivalent to the category of (mixed) perverse sheaves for flag varieties. As an application, we describe an algorithm calculating the quiver underlying the BGG category \(\mathcal{O}\) for arbitrary simple Lie algebra, thus answering a question which goes back to I. M. Gelfand.

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Authors and Affiliations

  1. Department of Mathematics, MIT, 77 Mass Ave, Cambridge, 02139, MA, USA

    Maxim Vybornov

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  1. Maxim Vybornov
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Correspondence to Maxim Vybornov.

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Dedicated to George Lusztig on the occasion of his 60-th birthday

Mathematics Subject Classification (1991)

14F43, 17B10, 32S60

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Vybornov, M. Perverse sheaves, Koszul IC-modules, and the quiver for the category \(\mathcal{O}\) . Invent. math. 167, 19–46 (2007). https://doi.org/10.1007/s00222-006-0005-2

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