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

In the paper a realization of representation of a Lie algebra $\mathfrak{sp}_4$ in the space of function on the Lie group $Sp_4$ is considered. We find a function corresponding to a Gelfand-Tsetlin type vector for $\mathfrak{sp}_4$ constructed by D.P. Zhelobenko. This function is expressed though a $A$-hypergeometric function. After developing some new technique we derive analytically formulas for the action of generators of algebra in this base (the were not known before). These formula turn out to be much more difficult than the formulas for the action of generators in the Gelfand-Tsetlin type base constructed by Molev.