# Baxter Operator and Archimedean Hecke Algebra

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Communications in Mathematical Physics

, 284:867

First Online: 19 August 2008Received: 20 October 2007Accepted: 13 February 2008

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 = GLℓ + 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}\ .

Communicated by L. Takhtajan

Autor: A. Gerasimov - D. Lebedev - S. Oblezin