Z-graded oscillator representations of symplectic Lie algebras

In this paper, we study two-parameter Z-graded oscillator representations of symplectic Lie algebras obtained from the natural oscillator representation by partially swapping differential operators and multiplication operators. We prove that the homogeneous subspaces with nonzero degree or unequal parameters are infinite-dimensional irreducible weight submodules with finite-dimensional weight subspaces. When the two parameters are equal, the homogeneous subspace of degree zero is exactly a direct sum of two infinite-dimensional irreducible weight submodules with finite-dimensional weight subspaces. In generic case, these irreducible modules are non-unitary modules without highest-weight vectors. Our results are extensions of Howe's oscillator construction of infinite-dimensional multiplicity-free irreducible representations for special linear Lie algebras.