Baxter Operator and Archimedean Hecke Algebra

By A. Gerasimov, D. Lebedev, S. Oblezin. 2007

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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 $${\mathcal {H}(G(\mathbb{Q}_p),G(\mathbb{Z}_p))}$$ .

Subfields: Geometry and Topology, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/1ccb771476ebb053658783382a8d00b2f0da5e99

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Citations in the corpus

Macdonald processes (2011, background): “Let us note the orthogonality and completeness relations which can be found in [90, 102] or [45] Theorem 2.”
Tropical Combinatorics and Whittaker functions (2011, background): “We remark that, while this gives a new interpretation of the Bump-Stade Whittaker integral identity, we do not give a new proof of this identity in the present paper; in fact, we use it to obtain the simplification (1.3).”