From Baxter operator to Archimedean Hecke algebra

In this note we introduce Baxter integral Q-operators for finite-dimensional Lie algebras gll+1 and so2l+1. Corresponding Whittaker functions are eigenfunctions of the Q-operators with the eigenvalues expressed in terms of Gamma-functions. This property is one of the manifestations of an interesting connection between MellinBarnes and Givental integral representations of Whittaker functions. The two types of integral representations appear to be dual to each other and allow a family of mixed Mellin-Barnes-Givental integral representations introduced in this note. We use this duality to provide a simple proof of the Bump conjecture for G = GL(l+1) and discuss a relation with the proof given previously by Stade. We also identify eigenvalues of the Baxter Q-operator acting on Whittaker functions with local Archimedean Lfactors and stress an analogy between Q-operators and certain elements of the nonArchimedean Hecke algebra H(G(Qp), G(Zp)). Thus Baxter Q-operator should be naturally considered as an element of the Archimedean Hecke algebra H(G(R),K), K being a maximal compact subgroup of G. E-mail: [email protected] E-mail: [email protected]