%0 Journal Article %T Kazhdan-Lusztig combinatorics via sheaves on Bruhat graphs %A Peter Fiebig %J Mathematics %D 2005 %I arXiv %X We study the projective objects in an exact category naturally associated to a Coxeter system. We discuss an analog of the Kazhdan-Lusztig conjecture and show how it follows from a "genericity" conjecture and how the latter follows from a "Hard Lefschetz" conjecture. %U http://arxiv.org/abs/math/0512311v2