Generating twisted Cherednik eigenfunctions

TL;DR

This paper constructs explicit twisted Cherednik eigenfunctions related to integrable systems from the DIM algebra, generalizing known operators and linking to DIM Hamiltonian eigenstates.

Contribution

It provides a recursive construction of twisted Cherednik eigenfunctions, extending the Kirillov-Noumi operators framework in the context of DIM algebra.

Findings

Explicit recursive construction of eigenfunctions

Connection between twisted Cherednik and DIM Hamiltonians

Generalization of Kirillov-Noumi operators

Abstract

Hamiltonians ${\cal H}^{a}_k$ of new integrable systems associated with the integer rays $(1,a)$ (commutative subalgebras) of Ding-Iohara-Miki (DIM) algebra in the $N$-body representation are closely related to commuting twisted Cherednik Hamiltonians $\mathfrak{C}_i^{(a)}$, ${\cal H}^{a}_k = \sum_{i=1}^N (\mathfrak{C}_i^{(a)})^k$. Moreover, symmetric combinations of eigenfunctions in the twisted Cherednik system were recently shown to produce the DIM Hamiltonian eigenstates. We explicitly construct these twisted Cherednik eigenfunctions recurrently by action of some (creation and permutation) operations. It resembles of a far-going generalization of Kirillov-Noumi operators, but exact relation remains to be specified.