Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On $b$-Whittaker functions   The $b$-Whittaker functions are eigenfunctions of the modular $q$-deformed
$\mathfrak{gl}_n$ 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 Teichm\"uller 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-Sepp\"al\"ainen-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 $U_q(\mathfrak{sl}_n)$, 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.

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The categories of this paper are: astro-ph
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The categories of this paper are: q-alg
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The categories of this paper are: hep-lat
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The categories of this paper are: nucl-th
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The categories of this paper are: nlin
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The categories of this paper are: q-bio
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The categories of this paper are: eess
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The categories of this paper are: patt-sol
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The categories of this paper are: acc-phys
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The categories of this paper are: econ
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The categories of this paper are: mtrl-th
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The categories of this paper are: bayes-an
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The categories of this paper are: quant-ph
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The categories of this paper are: cmp-lg
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The categories of this paper are: atom-ph
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The categories of this paper are: plasm-ph