A duality of Bethe algebras for general linear Lie (super)algebras

We study the joint action of the Bethe algebra Bdʷ for the general linear Lie algebra gld with respect to w ∈ Cᵈ and the Bethe algebra Bp+m|q+nᶻ for the general linear Lie superalgebra glp+m|q+n with respect to z ∈ Cp+q+m+n on the Fock space of $d(p+m)$ bosonic and $d(q+n)$ fermionic oscillators. We establish a duality, called the Bethe duality of (gld, glp+m|q+n), which is an equivalence between the actions of the algebras Bdʷ and Bp+m|q+nᶻ on the Fock space. As an application, we show that the action of Bp+m|q+nᶻ on each weight space of the evaluation module M( w), where M is a d-fold tensor product of certain classes of infinite-dimensional unitarizable highest weight glp+m|q+n-modules, is a cyclic Bp+m|q+nᶻ-module, and that Bp+m|q+nᶻ is diagonalizable with a simple spectrum on the weight space for generic w and z.