On b-Whittaker functions

Gus Schrader, Alexander Shapiro

Research output: Working paper

Abstract

The b-Whittaker functions are eigenfunctions of the modular q-deformed gln open Toda system introduced by Kharchev, Lebedev, and Semenov-Tian-Shansky. Using the quantum inverse scattering method, the named authors obtained a Mellin-Barnes integral representation for these eigenfunctions. In the present paper, we develop the analytic theory of the b-Whittaker functions from the perspective of quantum cluster algebras. We obtain a formula for the modular open Toda system's Baxter operator as a sequence of quantum cluster transformations, and thereby derive a new modular b-analog of Givental's integral formula for the undeformed Whittaker function. We also show that the b-Whittaker functions are eigenvectors of the Dehn twist operator from quantum higher Teichmüller theory, and obtain b-analogs of various integral identities satisfied by the undeformed Whittaker functions, including the continuous Cauchy-Littlewood identity of Stade and Corwin-O'Connell-Seppäläinen-Zygouras. Using these results, we prove the unitarity of the b-Whittaker transform, thereby completing the analytic part of the proof of the conjecture of Frenkel and Ip on tensor products of positive representations of Uq(sln), as well as the main step in the modular functor conjecture of Fock and Goncharov. We conclude by explaining how the theory of b-Whittaker functions can be used to derive certain hyperbolic hypergeometric integral evaluations found by Rains.
Original languageEnglish
PublisherArXiv
Number of pages36
Publication statusPublished - 3 Jun 2018

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