Intersection Cohomology Complexes on Low Rank Flag Varieties

We study intersection cohomology complexes on flag varieties with coefficients in a field of positive characteristic and present a combinatorial procedure (based on the W -graph of the Coxeter group) which determines their characters in many cases on low rank flag varieties. Our procedure works uniformly in almost all characteristics (p > 5 is always sufficient) and, if successful, verifies a version of the decomposition theorem. In particular we are able to show that the characters of all intersection cohomology complexes with coefficients in a field of characteristic 6= 2 on the flag variety G/B of type An for n < 7 are given by Kazhdan-Lusztig basis elements and that the decomposition theorem is true. By results of Soergel, this implies a part of Lusztig’s conjecture for representations of SL(n, Fp) with n ≤ 7 and p > n. We also give examples where our techniques fail.