On equivalence of the Mellin-Barnes and the Givental integral representations of the Whittaker functions

We construct an integral transformation intertwining the Gelfand-Tsetlin and the (modified) Gauss-Givental realizations of principle series representations of gl(3). This provides a direct identification of the corresponding integral representations for the gl(3)-Whittaker function. The construction essentially uses integral identities due to Barnes and Gustafson thus providing a basis for their representation theory interpretation. The result of this paper might be useful for constructing the explicit analytic realization of the mirror symmetry map in the case of the flag manifold GL(3)/B.