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Supersymmetyric Analogues 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. In this paper, we first establish two-parameter $\mbb{Z}^2$-graded supersymmetric oscillator generalizations of the above theorem for the Lie superalgebra $gl(n|m)$. Then we extend the result to two-parameter $\mbb{Z}$-graded supersymmetric oscillator generalizations of the above theorem for the Lie superalgebras $osp(2n|2m)$ and $osp(2n+1|2m)$.