# Cherednik operators and Ruijsenaars-Schneider model at infinity

## TL;DR

This paper extends Heckman's operators to Macdonald symmetric polynomials using Cherednik operators, analyzes their behavior at infinity, and connects to the Lax operator for Macdonald functions.

## Contribution

It introduces Cherednik operator analogues for Macdonald polynomials and computes their limits as the number of variables approaches infinity.

## Key findings

- Cherednik operators commute pairwise.
- Macdonald polynomials are eigenfunctions of these operators.
- Limits yield the same Lax operator as in previous work.

## 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.