A filtration on the ring of Laurent polynomials and representations of the general linear Lie algebra

We first present a filtration on the ring $L_n$ of Laurent polynomials such that the direct sum decomposition of its associated graded ring $\operatorname{gr} L_n$ agrees with the direct sum decomposition of $\operatorname{gr} L_n$, as a module over the complex general linear Lie algebra $\mathfrak{gl}(n)$, into its simple submodules. Next, generalizing the simple modules occurring in the associated graded ring $\operatorname{gr} L_n$, we give some explicit constructions of weight multiplicity-free irreducible representations of $\mathfrak{gl}(n)$.