On Chalykh's approach to eigenfunctions of DIM-induced integrable Hamiltonians

TL;DR

This paper explores Oleg Chalykh's theory connecting Macdonald polynomials, Baker-Akhiezer functions, and their potential role as eigenfunctions of integrable Hamiltonians related to the Ding-Iohara-Miki algebra, providing supporting evidence.

Contribution

It investigates the conjecture that twisted Baker-Akhiezer functions serve as eigenfunctions for certain DIM algebra Hamiltonians, extending Chalykh's earlier work.

Findings

Evidence supporting the role of twisted Baker-Akhiezer functions as eigenfunctions

Connections established between Macdonald polynomials and integrable Hamiltonians

Potential new methods for constructing eigenfunctions in algebraic integrable systems

Abstract

Quite some years ago, Oleg Chalykh has built a nice theory from the observation that the Macdonald polynomial reduces at $t=q^{-m}$ to a sum over permutations of simpler polynomials called Baker-Akhiezer functions, which can be unambiguously constructed from a system of linear difference equations. Moreover, he also proposed a generalization of these polynomials to the twisted Baker-Akhiezer functions. Recently, in a private communication Oleg suggested that these twisted Baker-Akhiezer functions could provide eigenfunctions of the commuting Hamiltonians associated with the $(-1,a)$ rays of the Ding-Iohara-Miki algebra. In the paper, we discuss this suggestion and some evidence in its support.