Sekiguchi-Debiard operators at infinity

TL;DR

This paper constructs a family of commuting differential operators for infinitely many variables, with Jack symmetric functions as eigenfunctions, linking to the quantum Calogero-Sutherland model and providing explicit shift operators.

Contribution

It introduces a novel hierarchy of commuting operators at infinity, extending the Sekiguchi-Debiard operators and connecting them to integrable models and symmetric functions.

Findings

Operators are limits of Sekiguchi-Debiard operators as N→∞

Jack symmetric functions are eigenfunctions of these operators

Provides explicit shift operators for Jack functions

Abstract

We construct a family of pairwise commuting operators such that the Jack symmetric functions of infinitely many variables $x_1,x_2,...$ are their eigenfunctions. These operators are defined as limits at $N\to\infty$ of renormalised Sekiguchi-Debiard 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 Jack reproducing kernel. Our result yields a hierarchy of commuting Hamiltonians for the quantum Calogero-Sutherland model with infinite number of bosonic particles in terms of the collective variables of the model. Our result also yields explicit shift operators for the Jack symmetric functions.