A Cubic E6-Generalization of the Classical Theorem on Harmonic Polynomials

Classical harmonic analysis says that the spaces of homogeneous harmonic polynomials (solutions of Laplace equation) are irreducible modules of the corresponding orthogonal Lie group (algebra) and the whole polynomial algebra is a free module over the invariant polynomials generated by harmonic polynomials.Dickson invariant trilinear form is the unique fundamental invariant in the polynomial algebra over the basic irreducible module of E 6 .In this paper, we prove that the space of homogeneous polynomial solutions with degree m for the dual cubic Dickson invariant differential operator is exactly a direct sum of m/2 + 1 explicitly determined irreducible E 6 -submodules and the whole polynomial algebra is a free module over the polynomial algebra in the Dickson invariant generated by these solutions.Thus we obtain a cubic E 6 -generalization of the above classical theorem on harmonic polynomials.