Duality in elliptic Ruijsenaars system and elliptic symmetric functions

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 yi (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 lambda (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 xi 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.

Automated summary

This study demonstrates that the symmetric elliptic polynomials are eigenfunctions of the elliptic Ruijsenaars-Schneider Hamiltonians, while their orthogonal complements are eigenfunctions of the elliptic reduction of the Koroteev-Shakirov Hamiltonians. These two sets of eigenfunctions are related by a mysterious orthogonality transformation and the coordinates xi only appear after the Miwa transform, which explains the difficulties in obtaining self-dual Hamiltonians from the double elliptic version of the KS Hamiltonians.