Gelfand-Tsetlin modules over $\mathfrak{g}\mathfrak{l}(n,\mathbb{C})$ with arbitrary characters

A Gelfand-Tsetlin tableau $T(v)$ induces a character ${\chi }_v$ of the Gelfand-Tsetlin subalgebra $\mathrm{\Gamma }$ of $U=U(\mathfrak{g}\mathfrak{l}(n,\mathbb{C}))$. By a theorem due to Ovsienko, for each tableau $T(v)$ there exists a finite number of nonisomorphic irreducible Gelfand-Tsetlin modules with ${\chi }_v$ in its support, though explicit examples of such modules are only known for special families of characters. In this article we build a family of Gelfand-Tsetlin modules parametrized by characters, such that each character appears in its corresponding module. We also find the support of these modules, with multiplicities.