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

TL;DR

This paper studies the diagonalizability and spectral properties of Bethe algebras acting on tensor products of unitarizable modules over classical Lie (super)algebras, establishing dualities and conditions for simple spectra.

Contribution

It introduces a duality of Bethe algebras for general linear Lie superalgebras and proves diagonalizability and simplicity of spectra under certain conditions.

Findings

Bethe algebra is diagonalizable on finite-dimensional submodules when certain conditions hold.

A duality between Bethe algebras for $rak{gl}_d$ and $rak{gl}_{p+m|q+n}$ is established.

Under generic conditions, Bethe algebras have simple spectra on tensor products of unitarizable modules.

Abstract

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 $q=n=0$.