A new way to construct $1$-singular Gelfand-Tsetlin modules

We present a simplified way to construct the Gelfand-Tsetlin modules over $\mathfrak{gl}(n,\mathbb{C})$ related to a $1$-singular GT-tableau defined in [6]. We begin by reframing the classical construction of generic Gelfand-Tsetlin modules found in [3], showing that they form a flat family over generic points of $\mathbb{C}^{\binom{n}{2}}$. We then show that this family can be extended to a flat family over a variety including generic points and $1$-singular points for a fixed singular pair of entries. The $1$-singular modules are precisely the fibers over these points.