Oscillator Variations of the Classical Theorem on Harmonic Polynomials

TL;DR

This paper explores oscillator variations of harmonic polynomial theorems, establishing dualities and module structures for noncanonical representations of classical Lie algebras using PDE methods.

Contribution

It introduces new oscillator representations of sl(n), o(n), and sp(2n), and characterizes their irreducibility and module structures, revealing novel dualities and explicit modules.

Findings

Established local Howe dualities for oscillator variations.

Identified conditions for irreducibility of solution spaces.

Constructed explicit irreducible modules for o(n) and sp(2n).

Abstract

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.