The lowest two-sided cell of a Coxeter group with complete graph

In this paper we give a description of the structure of the based ring of the lowest two-sided cell for a weighted Coxeter group with complete graph. It is proved that this ring is generated, in a simple way, by some subrings which are isomorphic to the based rings of the lowest two-sided cells for affine Weyl groups of type A˜1 or A˜2.