Images of eigenvalue distributions under power maps

In [earlier work by the author], it was shown that if U is a random n x n unitary matrix, then for any p>=n, the eigenvalues of U^p are i.i.d. uniform; similar results were also shown for general compact Lie groups. We study what happens when p<n instead. For the classical groups, we find that we can describe the eigenvalue distribution of U^p in terms of the eigenvalue distributions of smaller classical groups; the earlier result is then a special case. The proofs rely on the fact that a certain subgroup of the Weyl group is itself a Weyl group. We generalize this fact, and use it to study the power-map problem for general compact Lie groups.