  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.
