Cherednik operators and Ruijsenaars-Schneider model at infinity

Abstract

Heckman introduced $N$ operators on the space of polynomials in $N$ variables, such that these operators form a covariant set relative to permutations of the operators and variables, and such that Jack symmetric polynomials are eigenfunctions of the power sums of these operators. We introduce the analogues of these $N$ operators for Macdonald symmetric polynomials, by using Cherednik operators. The latter operators pairwise commute, and Macdonald polynomials are eigenfunctions of their power sums. We compute the limits of our operators at $N\to \infty$. These limits yield the same Lax operator for Macdonald symmetric functions as constructed in our previous work.

21 pages

References in corpus (2)

• On Spin Calogero-Moser system at infinity
• Lax operator for Macdonald symmetric functions

Cited by in corpus (7)

• Commutative families in DIM algebra, integrable many-body systems and $q,t$ matrix models
• Spin-Ruijsenaars, q-deformed Haldane-Shastry and Macdonald polynomials
• Dunkl and Cherednik operators
• Generating twisted Cherednik eigenfunctions
• Symmetric polynomials: DIM integrable systems versus twisted Cherednik systems
• On Cherednik and Nazarov-Sklyanin large N limit construction for integrable many-body systems with elliptic dependence on momenta
• Twisted Cherednik spectrum as a $q,t$-deformation