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.