Modular intersection cohomology complexes on flag varieties

Details Modular intersection cohomology complexes on flag varieties Modular intersection cohomology complexes on flag varieties

2007-09-03

arxiv.org/abs/0709.0207v5

Mathematics

Representation Theory

38 pages. v2: added an appendix by Tom Braden, fixed minor errors and typos. v3: changed title, many minor changes, added mate

Scientific paper

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 \le 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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