Baxter operator formalism for Macdonald polynomials

TL;DR

This paper develops a Baxter operator formalism for Macdonald polynomials, establishing dual pairs of commuting operators with these polynomials as eigenfunctions, and extends the framework to related functions like q-deformed Whittaker and Jack polynomials.

Contribution

It introduces a new Baxter operator formalism for Macdonald polynomials and related functions, generalizing previous results and linking to representation theory and integral representations.

Findings

Constructed dual pairs of Baxter operators for Macdonald polynomials.

Established integral representations via recursive operators.

Connected Baxter operators to representation theory and L-factors.

Abstract

We develop basic constructions of the Baxter operator formalism for the Macdonald polynomials associated with root systems of type A. Precisely we construct a dual pair of mutually commuting Baxter operators such that the Macdonald polynomials are their common eigenfunctions. The dual pair of Baxter operators is closely related to the dual pair of recursive operators for Macdonald polynomials leading to various families of their integral representations. We also construct the Baxter operator formalism for the q-deformed gl(l+1)-Whittaker functions and the Jack polynomials obtained by degenerations of the Macdonald polynomials associated with the type A_l root system. This note provides a generalization of our previous results on the Baxter operator formalism for the Whittaker functions. It was demonstrated previously that Baxter operator formalism for the Whittaker functions has deep connections with representation theory. In particular the Baxter operators should be considered as elements of appropriate spherical Hecke algebras and their eigenvalues are identified with local Archimedean L-factors associated with admissible representations of reductive groups over R. We expect that the Baxter operator formalism for the Macdonald polynomials has an interpretation in representation theory of higher-dimensional arithmetic fields.