Sekiguchi-Debiard operators at infinity

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.

final version

References in corpus (3)

• Gromov-Witten invariants of target curves via Symplectic Field Theory
• Applications of Laplace-Beltrami operator for Jack polynomials
• Irreducible representations of Yangians

Cited by in corpus (12)

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• Instanton R-matrix and W-symmetry
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• On Spin Calogero-Moser system at infinity
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• Exact Bohr-Sommerfeld Conditions for the Quantum Periodic Benjamin-Ono Equation
• Free field theory and observables of periodic Macdonald processes
• Fermionic limit of the Calogero-Sutherland system
• Jack-Laurent symmetric functions
• On Cherednik and Nazarov-Sklyanin large N limit construction for integrable many-body systems with elliptic dependence on momenta
• Free field approach to the Macdonald process
• Matrix elements of vertex operators and fermionic limit of spin Calogero-Sutherland system