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

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