Macdonald operators at infinity

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.

💡 Research Summary

The paper addresses the long‑standing problem of extending the family of commuting Macdonald operators, originally defined on symmetric polynomials in a finite set of variables, to the setting of infinitely many variables. Starting from the well‑known Macdonald difference operators $D_N$ acting on the algebra $\Lambda_N$ of symmetric polynomials in $x_1,\dots,x_N$, the authors first introduce a renormalisation factor—essentially $t^{\binom N2}$—that renders the operators stable under the limit $N\to\infty$. By taking this limit in the topology of formal power series in the power‑sum generators $p_n=\sum_{i\ge1}x_i^n$, they obtain a new operator $\mathcal D$ acting on the full symmetric function space $\Lambda$. Unlike $D_N$, which is a finite‑difference operator, $\mathcal D$ is expressed as an infinite‑order differential operator in the $p_n$ variables.

A central technical achievement is the computation of the symbol of $\mathcal D$ using the Macdonald reproducing kernel
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