Duality in elliptic Ruijsenaars system and elliptic symmetric functions

Abstract

Abstract We demonstrate that the symmetric elliptic polynomials $$E_\lambda (x)$$ E λ ( 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 $$y_i$$ y i (substitute of the Young-diagram variable $$\lambda $$ λ ). This means they are eigenfunctions of the dual eRS system. At the same time, their orthogonal complements in the Schur scalar product, $$P_\lambda (x)$$ P λ ( 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 $$x_i$$ 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.