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In this paper we introduce Baxter integral Q-operators for finite-dimensionalLie algebras gl(n+1) and so(2n+1). Whittaker functions corresponding to thesealgebras are eigenfunctions of the Q-operators with the eigenvalues expressedin terms of Gamma-functions. The appearance of the Gamma-functions is one ofthe manifestations of an interesting connection between Mellin-Barnes andGivental integral representations of Whittaker functions, which are in a sensedual to each other. We define a dual Baxter operator and derive a family ofmixed Mellin-Barnes-Givental integral representations. Givental andMellin-Barnes integral representations are used to provide a short proof of theFriedberg-Bump and Bump conjectures for G=GL(n+1) proved earlier by Stade. Wealso identify eigenvalues of the Baxter Q-operator acting on Whittakerfunctions with local Archimedean L-factors. The Baxter Q-operator introduced inthis paper is then described as a particular realization of the explicitlydefined universal Baxter operator in the spherical Hecke algebra H(G(R),K), Kbeing a maximal compact subgroup of G. Finally we stress an analogy betweenQ-operators and certain elements of the non-Archimedean Hecke algebraH(G(Q_p),G(Z_p)).

Baxter operator and Archimedean Hecke algebra