The $\hat W$-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 $\hat W$ act as a group of affine transformations on a real vector space V. We analyze the $\hat W$-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 $\hat W$ as permutations of the integers. 
Paola Cellini, Pierluigi Moseneder Frajria, Paolo Papi