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

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