Twisted Cherednik systems and non-symmetric Macdonald polynomials

TL;DR

This work extends the theory of non-symmetric Macdonald polynomials to twisted Cherednik systems arising from DIM algebra, introducing higher Cherednik operators along integer rays and their ground-state twisting. The central idea is that eigenfunctions of twisted Hamiltonians can be constructed universally as linear combinations of ground-state factors $\Omega^{(a)}$ with $a$-independent coefficients, so that all twist dependence resides in the ground state, i.e., a twisted Baker-Akhiezer function. For $n=2$, explicit polynomial eigenfunctions are obtained at $t=q^{-m}$, including two basic solutions and their symmetric combinations, while for general $n$ a conjectural universal expansion is proposed, with the ground state carrying the twisting via $\Xi^{(a)}_\lambda$. In the $q\to1$ limit, the twisted system reduces to a Dunkl–Jack-type framework with the twisting absorbed into a simple Vandermonde-like factor, clarifying the structural separation between ground-state twisting and the rest of the eigenfunction data. Overall, the paper reveals a new non-symmetric facet of integrable systems tied to the DIM algebra, linking twisted Cherednik theory with Baker-Akhiezer functions and Demazure-like objects, and sets a program for further non-symmetric, twist-controlled generalizations.

Abstract

We consider eigenfunctions of many-body system Hamiltonians associated with generalized (a-twisted) Cherednik operators used in construction of other Hamiltonians: those arising from commutative subalgebras of the Ding-Iohara-Miki (DIM) algebra. The simplest example of these eigenfunctions is provided by non-symmetric Macdonald polynomials, while generally they are constructed basing on the ground state eigenfunction coinciding with the twisted Baker-Akhiezer function being a peculiar (symmetric) eigenfunction of the DIM Hamiltonians. Moreover, the eigenfunctions admit an expansion with universal coefficients so that the dependence on the twist $a$ is hidden only in these ground state eigenfunctions, and we suggest a general formula that allows one to construct these eigenfunctions from non-symmetric Macdonald polynomials. This gives a new twist in theory of integrable systems, which usually puts an accent on symmetric polynomials, and provides a new dimension to the {\it triad} made from the symmetric Macdonald polynomials, untwisted Baker-Akhiezer functions and Noumi-Shiraishi series.