The Laurent polynomials My,ws in the Hecke algebra with unequal parameters

Let (W,S) be a Coxeter system and H the associated Hecke algebra with unequal parameters. The Laurent polynomials My,ws and py,w for y,w∈W and s∈S play an important role in the representations of H. We study the properties of My,ws and py,w, the relations among them, as well as with the left, right and two-sided cells of W.