From Kajihara's transformation formula to deformed Macdonald-Ruijsenaars and Noumi-Sano operators

TL;DR

This paper derives kernel identities for deformed Macdonald-Ruijsenaars and Noumi-Sano operators from Kajihara's transformation formula, proving their commutativity and diagonalization by super-Macdonald polynomials, and describing their algebraic structure.

Contribution

It introduces new kernel identities linking hypergeometric series to deformed integrable operators, establishing their algebraic properties and connections to inverse limits.

Findings

Operators pairwise commute

Operators are diagonalized by super-Macdonald polynomials

Explicit algebraic description via Harish-Chandra isomorphism

Abstract

Kajihara obtained in 2004 a remarkable transformation formula connecting multiple basic hypergeometric series associated with $A$-type root systems of different ranks. By multiple principle specialisations of his formula, we deduce kernel identities for deformed Macdonald-Ruijsenaars (MR) and Noumi-Sano (NS) operators. The deformed MR operators were introduced by Sergeev and Veselov in the first order case and by Feigin and Silantyev in the higher order cases. As applications of our kernel identities, we prove that all of these operators pairwise commute and are simultaneously diagonalised by the super-Macdonald polynomials. We also provide an explicit description of the algebra generated by the deformed MR and/or NS operators by a Harish-Chandra type isomorphism and show that the deformed MR (NS) operators can be viewed as restrictions of inverse limits of ordinary MR (NS) operators.