The $\widehat{W}$-orbit of $\rho$, Kostant's formula for powers of the Euler product and affine Weyl groups as permutations of Z

Let an affine Weyl group $\widehat{W}$ act as a group of affine transformations on a real vector space V. We analyze the $\widehat{W}$-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 $\widehat{W}$ as permutations of the integers.