Bethe algebras for unitarizable modules over classical Lie (super)algebras and a duality

Let g denote the classical Lie algebra gld, sp2d, or so2d with a fixed ∗-structure σ. Let M1, …, Mℓ be unitarizable g-modules (with respect to σ), and let z=(z1,…,zℓ)∈Cℓ. We investigate the action of the Bethe algebra Bgμ for g with respect to μ∈g* on the tensor product M̲(z)≔M1(z1)⊗⋯⊗Mℓ(zℓ) of evaluation g[t]-modules. We show that if μ◦σ equals the complex conjugation of μ, then Bgμ is diagonalizable on any finite-dimensional Bgμ-submodule of M̲(z) for z∈Rℓ. This, together with the result derived from the duality of Bethe algebras (see below), suggests that a simple spectrum conjecture for Bgμ should hold. We establish a duality of Bethe algebras for the general linear Lie (super)algebras gld and glp+m|q+n. As an application, we show that under a generic condition, the Bethe algebra for glp+m|q+n with respect to z∈Cp+q+m+n is diagonalizable with a simple spectrum on any weight space of L1(w1) ⊗⋯⊗ Ld(wd), where the Li are (infinite-dimensional) unitarizable highest weight glp+m|q+n-modules corresponding to generalized partitions of depth 1, and w1,…,wd∈C. We also obtain the corresponding result for glp+m by setting q = n = 0.