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

Let $\mathfrak{g}$ denote the classical Lie algebra $\mathfrak{gl}_d$, $\mathfrak{sp}_{2d}$, or $\mathfrak{so}_{2d}$ with a fixed $*$-structure $\sigma$. Let $M_1, \ldots, M_\ell$ be unitarizable $\mathfrak{g}$-modules (with respect to $\sigma$), and let ${\bf z}=(z_1, \ldots, z_\ell) \in \mathbb{C}^\ell$. We investigate the action of the Bethe algebra $\mathcal{B}_{\mathfrak{g}}^\mu$ for $\mathfrak{g}$ with respect to $\mu \in \mathfrak{g}^*$ on the tensor product $\underline{M}({\bf z}):=M_1(z_1) \otimes \cdots \otimes M_\ell(z_\ell)$ of evaluation $\mathfrak{g}[t]$-modules. We show that if $\mu \circ \sigma$ equals the complex conjugation of $\mu$, then $\mathcal{B}_{\mathfrak{g}}^\mu$ is diagonalizable on any finite-dimensional $\mathcal{B}_{\mathfrak{g}}^\mu$-submodule of $\underline{M}({\bf z})$ for ${\bf z} \in \mathbb{R}^\ell$. This, together with the result derived from the duality of Bethe algebras (see below), suggests that a simple spectrum conjecture for $\mathcal{B}_{\mathfrak{g}}^\mu$ should hold. We establish a duality of Bethe algebras for the general linear Lie (super)algebras $\mathfrak{gl}_d$ and $\mathfrak{gl}_{p+m|q+n}$. As an application, we show that under a generic condition, the Bethe algebra for $\mathfrak{gl}_{p+m|q+n}$ with respect to ${\bf z} \in \mathbb{C}^{p+q+m+n}$ is diagonalizable with a simple spectrum on any weight space of $L_1(w_1) \otimes \cdots \otimes L_d(w_d)$, where the $L_i$ are (infinite-dimensional) unitarizable highest weight $\mathfrak{gl}_{p+m|q+n}$-modules corresponding to generalized partitions of depth 1, and $w_1, \ldots, w_d \in \mathbb{C}$. We also obtain the corresponding result for $\mathfrak{gl}_{p+m}$ by setting $q=n=0$.