Modular intersection cohomology complexes on flag varieties

TL;DR

This paper introduces a combinatorial method based on W-graphs to determine the characters of intersection cohomology complexes on low rank flag varieties, confirming parts of Lusztig's conjecture for SL(n) with n ≤ 7.

Contribution

It develops a new combinatorial procedure utilizing parity sheaves to identify characters of intersection cohomology complexes on flag varieties, extending known results to certain low rank cases.

Findings

Characters of intersection cohomology complexes match Kazhdan-Lusztig basis elements for n < 7

The method confirms Lusztig's conjecture for SL(n) with n ≤ 7

Examples show limitations of the technique and instances of torsion in stalks and costalks

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 \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.