Baxter Operator Formalism for Macdonald Polynomials

Gerasimov, Anton; Lebedev, Dimitri; Oblezin, Sergey

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 bispectral pair of mutually commuting Baxter operators such that the Macdonald polynomials are their common eigenfunctions. The bispectral pair of Baxter operators is closely related to the bispectral 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 glℓ+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 over higher-dimensional local/global fields. © 2013 Springer Science+Business Media Dordrecht.

Citation

Gerasimov, A., Lebedev, D., & Oblezin, S. (2014). Baxter Operator Formalism for Macdonald Polynomials. Letters in Mathematical Physics, 104(2), 115-139. https://doi.org/10.1007/s11005-013-0659-9

• Journal Article Type: Article
• Acceptance Date: Oct 8, 2013
• Online Publication Date: Nov 24, 2013
• Publication Date: 2014-02
• Deposit Date: Nov 25, 2017
• Publicly Available Date: Jan 15, 2020
• Journal: Letters in Mathematical Physics
• Print ISSN: 0377-9017
• Electronic ISSN: 1573-0530
• Publisher: Springer Verlag
• Peer Reviewed: Peer Reviewed
• Volume: 104
• Issue: 2
• Pages: 115-139
• DOI: https://doi.org/10.1007/s11005-013-0659-9
• Public URL: https://nottingham-repository.worktribe.com/output/1096393
• Publisher URL: https://link.springer.com/article/10.1007%2Fs11005-013-0659-9
• PMID: 00033012