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

Abstract

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.

Latex, 27 pages, minor corrections, to appear in Journal of Pure and Applied Algebra