  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.
