Let W be an irreducible finite or affine Coxeter group and let W c be the set of fully commutative elements in W. We prove that the set W c is closed under the Kazhdan-Lusztig preorder LR if and only if W c is a union of two-sided cells of W .

We study the complex reflection groups G(r, p, n). By considering these groups as subgroups of the wreath products Zr ≀ Sn, and by using Clifford theory, we define combinatorial parameters and descent representations of G(r, p, n), previously known for classical Weyl groups. One of these parameters is the flag major index, which also has an important role in the decomposition of these representations into irreducibles. A Carlitz type identity relating the combinatorial parameters with the degrees of the group, is presented.