  We study two-parameter oscillator variations of the classical theorem on
harmonic polynomials, associated with noncanonical oscillator representations
of sl(n) and o(n). We find the condition when the homogeneous solution spaces
of the variated Laplace equation are irreducible modules of the concerned
algebras and the homogeneous subspaces are direct sums of the images of these
solution subspaces under the powers of the dual differential operator. This
establishes a local (sl(2),sl(n)) and (sl(2),o(n)) Howe duality, respectively.
In generic case, the obtained irreducible o(n)-modules are infinite-dimensional
non-unitary modules without highest-weight vectors. As an application, we
determine the structure of noncanonical oscillator representations of sp(2n).
When both parameters are equal to the maximal allowed value, we obtain an
infinite family of explicit irreducible (G,K)-modules for o(n) and sp(2n).
Methodologically we have extensively used partial differential equations to
solve representation problems.
