  In this paper we introduce Baxter integral Q-operators for finite-dimensional
Lie algebras gl(n+1) and so(2n+1). Whittaker functions corresponding to these
algebras are eigenfunctions of the 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(n+1) proved earlier by Stade. We
also identify eigenvalues of the Baxter Q-operator acting on Whittaker
functions with local Archimedean L-factors. The Baxter 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 H(G(R),K), K
being a maximal compact subgroup of G. Finally we stress an analogy between
Q-operators and certain elements of the non-Archimedean Hecke algebra
H(G(Q_p),G(Z_p)).
