The Ŵ-orbit of ρρ, Kostant’s formula for powers of the Euler product and affine Weyl groups as permutations of ZZ

Let an affine Weyl group Ŵ act as a group of affine transformations on a real vector space VV. We analyze the Ŵ-orbit of a regular element in VV and deduce applications to Kostant’s formula for powers of the Euler product and to the representations of Ŵ as permutations of the integers.