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On Baxter \({\mathcal Q}\)-operators and their arithmetic implications. (English) Zbl 1171.81009

Summary: We consider Baxter \({\mathcal Q}\)-operators for various versions of quantum affine Toda chain. The interpretation of eigenvalues of the finite Toda chain Baxter operators as local Archimedean \(L\)-functions proposed recently is generalized to the case of affine Lie algebras. We also introduce a simple generalization of Baxter operators and local \(L\)-functions compatible with this identification. This gives a connection of the Toda chain Baxter \({\mathcal Q}\)-operators with an Archimedean version of the Polya-Hilbert operator proposed by Berry-Keating. We also elucidate the Dorey-Tateo spectral interpretation of eigenvalues of \({\mathcal Q}\)-operators. Using explicit expressions for eigenfunctions of affine/relativistic Toda chain we obtain an Archimedean analog of Casselman-Shalika-Shintani formula for Whittaker function in terms of characters.

MSC:

81R12 Groups and algebras in quantum theory and relations with integrable systems
81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
39A13 Difference equations, scaling (\(q\)-differences)
11M06 \(\zeta (s)\) and \(L(s, \chi)\)
11M36 Selberg zeta functions and regularized determinants; applications to spectral theory, Dirichlet series, Eisenstein series, etc. (explicit formulas)
17B37 Quantum groups (quantized enveloping algebras) and related deformations
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
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