Baxter operator formalism for Macdonald polynomials

Details Baxter operator formalism for Macdonald polynomials Baxter operator formalism for Macdonald polynomials

2012-04-04

arxiv.org/abs/1204.0926v1

Mathematics

Algebraic Geometry

21 pages

Scientific paper

We develop basic constructions of the Baxter operator formalism for the Macdonald polynomials. 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 Whittaker functions and the Jack polynomials obtained by degenerations of the Macdonald polynomials. 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 real numbers. We expect that Baxter operator formalism for the Macdonald polynomials has an interpretation in representation theory of higher-dimensional arithmetic fields.

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