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We develop basic constructions of the Baxter operator formalism for theMacdonald polynomials associated with root systems of type A. Precisely weconstruct a dual pair of mutually commuting Baxter operators such that theMacdonald polynomials are their common eigenfunctions. The dual pair of Baxteroperators is closely related to the dual pair of recursive operators forMacdonald polynomials leading to various families of their integralrepresentations. We also construct the Baxter operator formalism for theq-deformed gl(l+1)-Whittaker functions and the Jack polynomials obtained bydegenerations of the Macdonald polynomials associated with the type A_l rootsystem. This note provides a generalization of our previous results on theBaxter operator formalism for the Whittaker functions. It was demonstratedpreviously that Baxter operator formalism for the Whittaker functions has deepconnections with representation theory. In particular the Baxter operatorsshould be considered as elements of appropriate spherical Hecke algebras andtheir eigenvalues are identified with local Archimedean L-factors associatedwith admissible representations of reductive groups over R. We expect that theBaxter operator formalism for the Macdonald polynomials has an interpretationin representation theory of higher-dimensional arithmetic fields.

Baxter operator formalism for Macdonald polynomials