Eulerian polynomials of spherical type

The Eulerian polynomial of a finite Coxeter system (W, S) of rank n records, for each k ¿ {1, . . . , n}, the number of elements w ¿ W with an ascent set {s ¿ S | l(ws) > l(w)} of size k, where l(w) denotes the length of w with respect to S. The classical Eulerian polynomial occurs when the Coxeter group has type An, so W is the symmetric group on n + 1 letters. Victor Reiner gave a formula for arbitrary Eulerian polynomials and showed how to compute them in the classical cases. In this note, we compute the Eulerian polynomial for any spherical type.