  We find a simple product formula for the characteristic polynomial of the
permutations with a fixed descent set under the weak order. As a corollary we
obtain a simple product formula for the characteristic polynomial of
alternating permutations. We generalize these results to Coxeter groups. We
also find a formula for the generating function for the characteristic
polynomials of classical Coxeter groups, which is then related to affine
Coxeter groups.
