  Let G be a finite, complex reflection group and f its discriminant
polynomial. The fibers of f admit commuting actions of G and a cyclic group.
The virtual $G\times C_m$ character given by the Euler characteristic of the
fiber is a refinement of the zeta function of the geometric monodromy,
calculated in a paper of Denef and Loeser. We compute the virtual character
explicitly, in terms of the poset of normalizers of centralizers of regular
elements of G, and of the subspace arrangement given by proper eigenspaces of
elements of G. As a consequence, we compute orbifold Euler characteristics and
find some new "case-free" information about the discriminant.
