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

## TL;DR

This paper introduces a geometric construction of 1-singular Gelfand-Tsetlin modules for a0n using complex geometry, universal rings, and local distributions, providing a new perspective on these algebraic structures.

## Contribution

It presents a novel geometric approach to constructing 1-singular Gelfand-Tsetlin modules via universal rings and distributions, extending previous algebraic methods.

## Key findings

- Constructed a universal ring a0o and associated module a0s with basis from local distributions.
- Showed that a0s is a natural module over a0o and a Lie algebra a0h.
- Derived a universal 1-singular Gelfand-Tsetlin a0gla0n(a0C) module from previous work.

## Abstract

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].