Research Papers

On equivalence of the Mellin–Barnes and the Givental integral realizations of the Whittaker functions as matrix elements

A. A. Gerasimov^a, D. R. Lebedev^ba, S. V. Oblezin<sup>c

a</sup> Laboratory for Quantum Field Theory and Information, Institute for Information Transmission Problems, RAS, 127994, Moscow, Russia
^b Moscow Center for Continuous Mathematical Education, Bol. Vlasyevsky per. 11, 119002 Moscow, Russia
^c Beijing Institute of Mathematical Sciences and Applications, Huairou District, Beijing 101408, China

Abstract: An integral transformation is constructed that intertwines the Gelfand–Tsetlin and the (modified) Gauss–Givental realizations of principal series representations of $\mathfrak{gl}(3,\mathbb{R})$. This provides a direct identification of the corresponding integral representations for the $\mathfrak{gl}(3,\mathbb{R})$-Whittaker function. The construction makes an essential use of 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 an explicit analytic realization of the mirror symmetry map in the case of the flag manifold $GL(3,\mathbb{C})/B$.

Keywords: Principal series representations, Gelfand–Tsetlin realization, Gauss–Givental realization, Whittaker function of a real Lie group, integral identities and transformations.

Received: 06.04.2025

Language: English