Canonical Gelfand-Zeitlin modules over orthogonal Gelfand-Zeitlin algebras

TL;DR

This paper constructs explicit simple modules over orthogonal Gelfand-Zeitlin algebras that realize all Gelfand-Zeitlin characters, using divided difference operators to describe module actions.

Contribution

It provides a new explicit construction of simple modules for orthogonal Gelfand-Zeitlin algebras that realize all Gelfand-Zeitlin characters, extending the understanding of their representation theory.

Findings

Every Gelfand-Zeitlin character is realizable in a U-module.

Gelfand-Zeitlin formulae can be rewritten with divided difference operators.

Explicit bases for certain Gelfand-Zeitlin submodules are constructed.

Abstract

We prove that every orthogonal Gelfand-Zeitlin algebra $U$ acts on its Gelfand-Zeitlin subalgebra $\Gamma$. Considering the dual module, we show that every Gelfand-Zeitlin character of $\Gamma$ is realizable in a $U$-module. We observe that the Gelfand-Zeitlin formulae can be rewritten using divided difference operators. It turns out that the action of the latter operators on $\Gamma$ gives rise to an explicit basis in a certain Gelfand-Zeitlin submodule of the dual module mentioned above. This gives, generically, both in the case of regular and singular Gelfand-Zeitlin characters, an explicit construction of simple modules which realize given Gelfand-Zeitlin characters.