Macdonald operators at infinity

TL;DR

This paper constructs a family of commuting operators for Macdonald symmetric functions in infinitely many variables, derived as limits of finite-variable operators, with explicit symbols and applications to step operators.

Contribution

It introduces a new family of commuting differential operators at infinity for Macdonald functions, expanding the algebraic tools available for their analysis.

Findings

Operators are limits of finite-variable Macdonald operators as N→∞

Explicit symbols of these operators expressed via Hall-Littlewood functions

Derived elementary step operators for Macdonald symmetric functions

Abstract

We construct a family of pairwise commuting operators such that the Macdonald symmetric functions of infinitely many variables $x_1,x_2,...$ and of two parameters $q,t$ are their eigenfunctions. These operators are defined as limits at $N\to\infty$ of renormalised Macdonald operators acting on symmetric polynomials in the variables $x_1,...,x_N$. They are differential operators in terms of the power sum variables $p_n=x_1^n+x_2^n+...$ and we compute their symbols by using the Macdonald reproducing kernel. We express these symbols in terms of the Hall-Littlewood symmetric functions of the variables $x_1,x_2,...$. Our result also yields elementary step operators for the Macdonald symmetric functions.