Functional approach to a Gelfand–Tsetlin-type basis for $\mathfrak{o}_5$

We consider a realization of representations of the Lie algebra $\mathfrak{o}_5$ in the space of functions on the group $Spin_5\simeq Sp_4$. In the representations, we take a Gelfand–Tsetlin-type basis associated with the restriction $\mathfrak{o}_5\downarrow\mathfrak{o}_3$. Such a basis is useful in problems appearing in quantum mechanics. We explicitly construct functions on the group that correspond to basis vectors. As in the cases of $\mathfrak{gl}_3$ and $\mathfrak{sp}_4$ Lie algebras, these functions can be expressed in terms of $A$-hypergeometric functions (this does not hold for higher-rank algebras of these series). Using this realization, we obtain formulas for the action of generators.