Noncanonical Polynomial Representations of Classical Lie Algebras

Using the skew-symmetry of the differential operators and multiplication operators in the canonical representations of finite-dimensional classical Lie algebras, we obtain some noncanonical polynomial representations of the classical Lie algebras. The representation spaces of all polynomials are decomposed into irreducible submodules, which are infinite-dimensional. Bases for the irreducible submodules are constructed. In particular, we obtain some new infinite-dimensional irreducible modules of symplectic Lie algebras that are not of highest weight type.