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

TL;DR

The paper probes O. Chalykh's program linking Macdonald polynomials to Baker–Akhiezer (BA) functions and their twisted variants, with a focus on eigenfunctions of commuting DIM Hamiltonians along DIM rays $(-1,a)$. It develops explicit BA- and twisted BA-function constructions for $N=2$ and generalizes to arbitrary $N$, analyzes the generic $N=3$ structure, and derives integral (CMM-type) identities that relate BA functions to Macdonald polynomials. It then argues and provides evidence that twisted BA functions serve as eigenfunctions of the DIM Hamiltonians in the $N$-body representation, highlighting subtle distinctions between DIM-generated operators and Cherednik realizations. The work points to broader generalizations, including elliptic Macdonald polynomials and connections to knot invariants, while acknowledging that a complete general theory remains to be established.

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.