Lax-operator proof of the two-parameter Macdonald identity

Develop a Lax-operator proof of the two-parameter identity for Macdonald symmetric functions, namely, prove directly via the Nazarov–Sklyanin Macdonald Lax operator that for all partitions \(\mu\) and \(\nu\), \(\kappa_{u,v}(J_\mu J_\nu)=\kappa_{u,v}(J_\mu)\kappa_{u,v}(J_\nu)T_{\mu\star\nu}(uv)\), where \(\kappa_{u,v}(J_\lambda)=\epsilon_u(J_\lambda)\epsilon_v(J_\lambda)/\epsilon_{uv}(J_\lambda)\) and \(T_{\mu\star\nu}\) is the box-star factor.

Background

The paper constructs a full Nazarov–Sklyanin Lax operator for Macdonald symmetric functions and uses its spectral theory to prove a product-resolvent formula and a Macdonald box-sum identity for Littlewood–Richardson coefficients. It then introduces a two-parameter functional $\kappa_{u,v}$ and proves the corresponding multiplicativity identity, strengthening the box-sum formula.

The two-parameter identity is established instead by reducing it to a finite-rank product formula of Warnaar and extending the resulting equality by rational-function interpolation. The unresolved problem is to obtain a proof of this same identity directly from the Lax-operator framework, analogous to the Lax-based proof given earlier in the paper for the one-parameter box-sum identity.