Canonical Gelfand–Zeitlin Modules over Orthogonal Gelfand–Zeitlin Algebras

Abstract We prove that every orthogonal Gelfand–Zeitlin algebra $U$ acts (faithfully) 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 the given Gelfand–Zeitlin characters.