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Sekiguchi-Debiard operators at infinity

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JournalCommunications in Mathematical Physics
DateE-pub ahead of print - 18 Oct 2013
DatePublished (current) - Oct 2013
Issue numbern/a
VolumeEarly Online
Number of pages20
Pages (from-to)n/a
Early online date18/10/13
Original languageEnglish

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.

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