A geometric approach to $1$-singular Gelfand-Tsetlin $\mathfrak {gl}_n$-modules

This paper is devoted to an elementary new construction of $1$-singular Gelfand-Tsetlin modules using complex geometry. We introduce a universal ring $\mathcal D_o$ together with the vector space $\mathcal S=\mathcal S(\mathcal D_o)$ with basis $\mathcal B_o = \mathcal B(\mathcal D_o)$ formed from some local distributions such that $\mathcal S$ is a natural $\mathcal D_o$-module. For any homomorphism of rings $\mathcal U(\mathfrak{h}) \to \mathcal D_o$, where $\mathfrak{h}$ is a Lie algebra, it follows that $\mathcal S$ is also an $\mathfrak{h}$-module. We observe that the homomorphism of rings constructed in [FO] is a homomorphism of type $\mathcal U(\mathfrak{gl}_n(\mathbb C)) \to \mathcal D_o$. Using this observation we obtain a construction of the universal $1$-singular Gelfand-Tsetlin $\mathfrak{gl}_n(\mathbb C)$-module from [FRG].