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

TL;DR

This paper constructs a family of Gelfand-Tsetlin modules over rak{gl}(n,c) that realize arbitrary characters, providing explicit examples and analyzing their support and multiplicities, advancing understanding of these modules.

Contribution

The authors explicitly construct Gelfand-Tsetlin modules for arbitrary characters, extending known examples beyond special cases and detailing their support and multiplicities.

Findings

Constructed a family of modules for each character

Determined the support and multiplicities of these modules

Extended the class of known Gelfand-Tsetlin modules

Abstract

A Gelfand-Tsetlin tableau $T(v)$ induces a character $\chi_v$ of the Gelfand-Tsetlin subalgebra $\Gamma$ of $U = U(\mathfrak{gl}(n,\mathbb C))$. 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.