Spectral Theory of the Nazarov-Sklyanin Lax Operator

Abstract

In their study of Jack polynomials, Nazarov-Sklyanin introduced a remarkable new graded linear operator $\mathcal{L}:F[w]\to F[w]$ where $F$ is the ring of symmetric functions and $w$ is a variable. In this paper, we (1) establish a cyclic decomposition $F[w]\cong {⨁}_{\lambda }Z(j_{\lambda },\mathcal{L})$ into finite-dimensional $\mathcal{L}$-cyclic subspaces in which Jack polynomials $j_{\lambda }$ may be taken as cyclic vectors and (2) prove that the restriction of $\mathcal{L}$ to each $Z(j_{\lambda },\mathcal{L})$ has simple spectrum given by the anisotropic contents $[s]$ of the addable corners $s$ of the Young diagram of $\lambda$. Our proofs of (1) and (2) rely on the commutativity and spectral theorem for the integrable hierarchy associated to $\mathcal{L}$, both established by Nazarov-Sklyanin. Finally, we conjecture that the $\mathcal{L}$-eigenfunctions ${\psi }_{\lambda }^s\in F[w]$ {with eigenvalue $[s]$ and constant term} ${\psi }_{\lambda }^s{|}_{w=0}=j_{\lambda }$ are polynomials in the rescaled power sum basis $V_{\mu }{w}^l$ of $F[w]$ with integer coefficients.

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