A Gelfand–Tsetlin-type basis for the algebra $\mathfrak{sp}_4$ and hypergeometric functions

We consider a realization of a representation of the $\mathfrak{sp}_4$ Lie algebra in the space of functions on a Lie group $Sp_4$. We find a function corresponding to a Gelfand–Tsetlin-type basis vector for $\mathfrak{sp}_4$ constructed by Zhelobenko. This function is expressed in terms of an $A$-hypergeometric function. Developing a new technique for working with such functions, we analytically find formulas for the action of the algebra generators in this basis (previously unknown formulas). These formulas turn out to be more complicated than the formulas for the action of generators in the Gelfand–Tsetlin-type basis constructed by Molev.