Kazhdan-Lusztig combinatorics via sheaves on Bruhat graphs

Abstract

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.

10 pages; references added; to appear in the proceedings for the conference "Algebraic and Geometric Combinatorics" in Anogia, Crete, in August 2005