A functional realization of the Gelfand–Tsetlin base

A realization of a finite dimensional irreducible representation of the Lie algebra $\mathfrak{gl}_n$ in the space of functions on the group $\mathrm{GL}_n$ is considered. It is proved that functions corresponding to Gelfand–Tsetlin diagrams are linear combinations of some new functions of hypergeometric type which are closely related to $A$-hypergeometric functions. These new functions are solution of a system of partial differential equations which follows from the Gelfand–Kapranov–Zelevinsky by an “antisymmetrization”. The coefficients in the constructed linear combination are hypergeometric constants, that is, they are values of some hypergeometric functions when instead of all arguments ones are substituted.