Twisted Cherednik systems and non-symmetric Macdonald polynomials

TL;DR

This paper introduces a new class of eigenfunctions related to twisted Cherednik operators, extending the theory of Macdonald polynomials and integrable systems by incorporating a twist parameter, and providing explicit construction formulas.

Contribution

It develops a framework for eigenfunctions of twisted Cherednik systems, connecting non-symmetric Macdonald polynomials with twisted Baker-Akhiezer functions, expanding integrable systems theory.

Findings

Eigenfunctions are constructed from non-symmetric Macdonald polynomials.

Dependence on the twist parameter is contained in ground state eigenfunctions.

A general formula for constructing these eigenfunctions is proposed.

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.