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Using the skew-symmetry of the differential operators and multiplicationoperators in the canonical representations of finite-dimensional classical Liealgebras, we obtain some noncanonical polynomial representations of theclassical Lie algebras. The representation spaces of all polynomials aredecomposed into irreducible submodules, which are infinite-dimensional. Basesfor the irreducible submodules are constructed. In particular, we obtain somenew infinite-dimensional irreducible modules of symplectic Lie algebras thatare not of highest weight type.

Noncanonical Polynomial Representations of Classical Lie Algebras