Oscillator Variations of the Classical Theorem on Harmonic Polynomials
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.
Cite
@article{arxiv.1012.2931,
title = {Oscillator Variations of the Classical Theorem on Harmonic Polynomials},
author = {Cuiling Luo and Xiaoping Xu},
journal= {arXiv preprint arXiv:1012.2931},
year = {2010}
}
Comments
49pages; This paper is a two-parameter extension of the first author's work "arXiv:0804.0305v2[math.RT]," which is equivalent to the one-paratemter special case of this work. The approaches have changed. The mistakes there have been corrected