Mathematische Zeitschrift

, Volume 272, Issue 3–4, pp 697–727 | Cite as

Modular intersection cohomology complexes on flag varieties

  • Geordie Williamson
  • Tom Braden


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.


Weyl Group Direct Summand Coxeter Group Decomposition Theorem Borel Subgroup 
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© Springer-Verlag 2011

Authors and Affiliations

  1. 1.Mathematical InstituteUniversity of OxfordOxfordUK
  2. 2.Department of Mathematics and StatisticsUniversity of MassachusettsAmherstUSA

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