  We consider a family of examples falling into the following context (first
considered by Vinberg): Let G be a connected reductive algebraic group over the
complex numbers. A subgroup, K, of fixed points of a finite-order automorphism
acts on the Lie algebra of G. Each eigenspace of the automorphism is a
representation of K. Let g1 be one of the eigenspaces. We consider the harmonic
polynomials on g1 as a representation of K, which is graded by homogeneous
degree. Given any irreducible representation of K, we will see how its
multiplicity in the harmonic polynomials is distributed among the various
graded components. The results are described geometrically by counting integral
points on faces of a polyhedron. The multiplicity in each graded component is
given by intersecting these faces with an expanding sequence of shells.
