The distribution of descents and length in a Coxeter group

We give a method for computing the q-Eulerian distribution W (t, q) = w∈W t des(w) q l(w) as a rational function in t and q, where (W, S) is an arbitrary Coxeter system, l(w) is the length function in W , and des(w) is the number of simple reflections s ∈ S for which l(ws) < l(w). Using this we compute generating functions encompassing the q-Eulerian distributions of the classical infinite families of finite and affine Weyl groups.