Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 19 (2023), 063, 22 pages      arXiv:2211.01586      https://doi.org/10.3842/SIGMA.2023.063

Spectral Theory of the Nazarov-Sklyanin Lax Operator

Ryan Mickler a and Alexander Moll b
a) Singulariti Research, Melbourne, Victoria, Australia
b) Department of Mathematics and Statistics, Reed College, Portland, Oregon, USA

Received March 19, 2023, in final form August 27, 2023; Published online September 10, 2023

Abstract
In their study of Jack polynomials, Nazarov-Sklyanin introduced a remarkable new graded linear operator ${\mathcal L} \colon F[w] \rightarrow F[w]$ where $F$ is the ring of symmetric functions and $w$ is a variable. In this paper, we (1) establish a cyclic decomposition $F[w] \cong \bigoplus_{\lambda} Z(j_{\lambda}, {\mathcal L})$ 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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