A geometric approach to $1$-singular Gelfand-Tsetlin ${\mathfrak{g}\mathfrak{l}}_n$-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 $\mathcal S=\mathcal S(\mathcal D_o)$withbasis$\mathcal B_o = \mathcal B(\mathcal D_o)$ 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 $\mathcal U(\mathfrak{gl}_n(\mathbb C)) \to \mathcal D_o$\mathrm{.}Usingthisobservationweobtainaconstructionoftheuniversal$1$-singularGelfand-Tsetlin$\mathfrak{gl}_n(\mathbb C)$-module from [FRG].