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Journal of Lie Theory 23 (2013), No. 4, 979--1003
Copyright Heldermann Verlag 2013

Graded Oscillator Generalizations of the Classical Theorem on Harmonic Polynomials

Cuiling Luo
College of Science, Hebei United University, Tangshan, Hebei 063009, P. R. China

Xiaoping Xu
Hua Loo-Keng Key Mathematical Laboratory, Institute of Mathematics, Academy of Mathematics and System Sciences, Chinese Academy of Sciences, Beijing 100190, P. R. China


\def\C{{\Bbb C}} \def\R{{\Bbb R}} 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 $(sl(2,\R),o(n,\R))$ Howe duality. In this paper, we study two-parameter oscillator variations of the above theorem associated with noncanonical oscillator representations of $o(n,\C)$. We find the condition when the homogeneous solution spaces of the variated Laplace equation are irreducible modules of $o(n,\C)$ and the homogeneous subspaces are direct sums of the images of these solution subspaces under the powers of the dual differential operator. This establishes an $(sl(2,\C),o(n,\C))$ Howe duality on some homogeneous subspaces. In generic case, the obtained irreducible $o(n,\C)$-modules are infinite-dimensional non-unitary modules without highest-weight vectors. When both parameters are equal to the maximal allowed value, we obtain explicit irreducible $({\cal G}, {\cal K})$-modules for $o(n,\C)$.

Keywords: Orthogonal Lie algebra, harmonic polynomial, oscillator representation, irreducible module, invariant operator.

MSC: 17B10, 17B20; 42B37

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