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Modular intersection cohomology complexes on flag varieties

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Abstract

We present a combinatorial procedure (based on the W-graph of the Coxeter group) which shows that the characters of many intersection cohomology complexes on low rank complex flag varieties with coefficients in an arbitrary field are given by Kazhdan–Lusztig basis elements. Our procedure exploits the existence and uniqueness of parity sheaves. In particular we are able to show that the characters of all intersection cohomology complexes with coefficients in a field on the flag variety of type A n for n < 7 are given by Kazhdan–Lusztig basis elements. By results of Soergel, this implies a part of Lusztig’s conjecture for SL(n) with n ≤ 7. We also give examples where our techniques fail. In the appendix by Tom Braden examples are given of intersection cohomology complexes on the flag varities for SL(8) and SO(8) which have torsion in their stalks or costalks.

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Authors and Affiliations

  1. Mathematical Institute, University of Oxford, 24-29 St Giles’, Oxford, OX1 3LB, UK

    Geordie Williamson

  2. Department of Mathematics and Statistics, University of Massachusetts, Amherst, USA

    Tom Braden

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  1. Geordie Williamson
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  2. Tom Braden
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Correspondence to Geordie Williamson.

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Williamson, G., Braden, T. Modular intersection cohomology complexes on flag varieties. Math. Z. 272, 697–727 (2012). https://doi.org/10.1007/s00209-011-0955-y

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