Generalizations 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. Algebraically, this gives an $(sl(2,\mathbb R),o(n,\mathbb R))$ Howe duality. In this talk, we will represent various generalizations of the above theorem. In our noncanonical generalizations, the constant-coefficient Laplace equation changes to variable-coefficient partial differential equations.