Alex Heaton: Graded multiplicity in harmonic polynomials

Abstract

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 the intersection of two unbounded polyhedra.