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In this paper, we used the free fields of Wakimoto to construct a class ofirreducible representations for the general linear Lie superalgebra$\mathfrak{gl}_{m|n}(\mathbb{C})$. The structures of the representations overthe general linear Lie superalgebra and the special linear Lie superalgebra arestudied in this paper. Then we extend the construction to the affine Kac-MoodyLie superalgebra $\widehat{\mathfrak{gl}_{m|n}}(\mathbb{C})$ on the tensorproduct of a polynomial algebra and an exterior algebra with infinitely manyvariables involving one parameter $\mu$, and we also obtain the necessary andsufficient condition for the representations to be irreducible. In fact, therepresentation is irreducible if and only if the parameter $\mu$ is nonzero.

A family of representations of the affine Lie superalgebra $\widehat{\mathfrak{gl}_{m|n}}(\mathbb{C})$