Abstract
In this paper we introduce Baxter integral \({\mathcal{Q}}\) -operators for finite-dimensional Lie algebras \({\mathfrak{gl}_{\ell+1}}\) and \({\mathfrak{so}_{2\ell+1}}\) . Whittaker functions corresponding to these algebras are eigenfunctions of the \({\mathcal{Q}}\)-operators with the eigenvalues expressed in terms of Gamma-functions. The appearance of the Gamma-functions is one of the manifestations of an interesting connection between Mellin-Barnes and Givental integral representations of Whittaker functions, which are in a sense dual to each other. We define a dual Baxter operator and derive a family of mixed Mellin-Barnes-Givental integral representations. Givental and Mellin-Barnes integral representations are used to provide a short proof of the Friedberg-Bump and Bump conjectures for G = GL(ℓ + 1) proved earlier by Stade. We also identify eigenvalues of the Baxter \({\mathcal{Q}}\)-operator acting on Whittaker functions with local Archimedean L-factors. The Baxter \({\mathcal{Q}}\)-operator introduced in this paper is then described as a particular realization of the explicitly defined universal Baxter operator in the spherical Hecke algebra \({\mathcal {H}(G(\mathbb{R}), K)}\) , K being a maximal compact subgroup of G. Finally we stress an analogy between \({\mathcal{Q}}\)-operators and certain elements of the non-Archimedean Hecke algebra \({\mathcal {H}(G(\mathbb{Q}_p),G(\mathbb{Z}_p))}\) .
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Gerasimov, A., Lebedev, D. & Oblezin, S. Baxter Operator and Archimedean Hecke Algebra. Commun. Math. Phys. 284, 867–896 (2008). https://doi.org/10.1007/s00220-008-0547-9
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DOI: https://doi.org/10.1007/s00220-008-0547-9