A family of representations of the Lie superalgebra glm|nˆ(Cq)

We construct an action of the Lie superalgebras glm|nˆ(Cq) on the tensor product of the polynomial algebra and the exterior algebra with infinitely many variables. This construction is involving one parameter μ and we show that it is irreducible if and only if μ   is nonzero. We also discuss the module structure and irreducibility over a Lie subsuperalgebra graded by the root system A(m−1,n−1)A(m−1,n−1).