Symmetric polynomials: DIM integrable systems versus twisted Cherednik systems

TL;DR

The paper investigates the link between eigenfunctions of the DIM algebra’s commutative subalgebras and those of the twisted Cherednik system, focusing on the twisted setting with $t=q^{-m}$. It develops two parallel constructions of symmetric eigenfunctions: (i) by symmetrizing twisted Baker-Akhiezer functions from the DIM side, and (ii) by Weyl-summing twisted non-symmetric Macdonald polynomials from the Cherednik side, showing that both yield the same symmetric basis that are eigenfunctions of the DIM Hamiltonians and the twisted Cherednik Hamiltonians. A central result is the equivalence, for each partition $\lambda$, of the symmetric Baker-Akhiezer sum $\Psi^{(a)}_m(\vec{x},\vec{y}^{(\lambda)})$ with the twisted Macdonald symmetric polynomial $\mathcal{M}^{(a,m)}_{\lambda}(\vec{x})$, highlighting a concrete DIM–DAHA correspondence in the twisted regime. The work provides explicit formulas and examples (notably for $N=2$) to illustrate the construction and discusses open questions, including a more explicit description of the twisting $SL(2,\mathbb{Z})$ transform and extensions to general $t$.

Abstract

We discuss interrelations between eigenfunctions of the Hamiltonians associated with the commutative (integer ray) subalgebras of the Ding-Iohara-Miki algebra and those of the twisted Cherednik system. In the case of $t=q^{-m}$ with natural $m$, eigenfunctions of the first system of Hamiltonians are the twisted Baker-Akhiezer functions (BAFs) introduced by O. Chalykh, while eigenfunctions of the twisted Cherednik Hamiltonians are twisted non-symmetric Macdonald polynomials. Actually, the twisted Cherednik ground state is symmetric and coincides with a peculiar symmetric BAF. We lift this correspondence to excited states, and claim that both Cherednik eigenfunctions and BAF's can be combined to produce symmetric functions, which coincide with each other and are eigenfunctions of the both DIM Hamiltonians and power sums of the twisted Cherednik Hamiltonians at once. This reflects the correspondence between the DIM algebra and the spherical DAHA explicitly.