Generalized Jones traces and Kazhdan--Lusztig bases
  We develop some applications of certain algebraic and combinatorial conditions on the elements of Coxeter groups, such as elementary proofs of the positivity of certain structure constants for the associated Kazhdan--Lusztig basis. We also explore some consequences of the existence of a Jones-type trace on the Hecke algebra of a Coxeter group, such as simple procedures for computing leading terms of certain Kazhdan--Lusztig polynomials.   The sequel to this paper is math.QA/0509363. 
R.M. Green