Spectral Theory of the Nazarov–Sklyanin Lax Operator

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.