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

TL;DR

This paper establishes a direct link between Mellin-Barnes and Givental integral representations of gl(3)-Whittaker functions, using integral identities to interpret their representation theory and potential applications in mirror symmetry.

Contribution

It constructs an integral transformation connecting different realizations of gl(3) representations, clarifying their relation and providing a basis for further geometric and mirror symmetry studies.

Findings

Identified an explicit integral transformation between Gelfand-Tsetlin and Givental representations.

Provided a representation theory interpretation of Barnes and Gustafson integral identities.

Suggested applications to mirror symmetry for the flag manifold GL(3)/B.

Abstract

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.