Duality in elliptic Ruijsenaars system and elliptic symmetric functions

TL;DR

The paper reveals a duality between elliptic Ruijsenaars-Schneider Hamiltonians and elliptic symmetric functions, showing their eigenfunction relationships and orthogonality properties, which clarify complexities in double elliptic systems.

Contribution

It establishes the eigenfunction and orthogonality relations of elliptic symmetric functions within dual elliptic Ruijsenaars systems, elucidating their connection to elliptic KS Hamiltonians.

Findings

Elliptic symmetric polynomials are eigenfunctions of dual eRS Hamiltonians.

Orthogonal complements of these polynomials are eigenfunctions of elliptic KS Hamiltonians.

The duality and orthogonality explain challenges in formulating self-dual Hamiltonians in double elliptic systems.

Abstract

We demonstrate that the symmetric elliptic polynomials $E_\lambda(x)$ originally discovered in the study of generalized Noumi-Shiraishi functions are eigenfunctions of the elliptic Ruijsenaars-Schneider (eRS) Hamiltonians that act on the mother function variable $y_i$ (substitute of the Young-diagram variable $\lambda$). This means they are eigenfunctions of the dual eRS system. At the same time, their orthogonal complements in the Schur scalar product, $P_R(x)$ are eigenfunctions of the elliptic reduction of the Koroteev-Shakirov (KS) Hamiltonians. This means that these latter are related to the dual eRS Hamiltonians by a somewhat mysterious orthogonality transformation, which is well defined only on the full space of time variables, while the coordinates $x_i$ appear only after the Miwa transform. This observation explains the difficulties with getting the apparently self-dual Hamiltonians from the double elliptic version of the KS Hamiltonians.