Fake degrees for reflection actions on roots

Abstract

A finite irreducible real reflection group of rank l and Coxeter number h has root system of cardinality h*l. It is shown that the fake degree for the permutation action on its roots is divisible by [h]_q = 1+q+q^2+...+q^{h-1}, and that in simply-laced types, it equals [h]_q times the summation of q^{e_i - 1} where e_i runs through the exponents, so that e_i - 1 are the codegrees.