Octonionic structure operator and its right spectrum

Abstract

We study a canonical $G_2$-equivariant operator $h:\mathbb{O}\otimes_{\mathbb{R}}V\to \mathbb{O}\otimes_{\mathbb{R}}V$ defined using only octonion multiplication, where $V$ is the standard $7$-dimensional $G_2$-module. We first compute its ordinary real spectrum using the $G_2$-decomposition of $\mathbb{O}\otimes_{\mathbb{R}}V$. We then analyze the octonionic right-eigenvalue problem $$ h(\widehat w)=\widehat w\lambda, \qquad \lambda\in\mathbb{O}. $$ After fixing a complex slice $\mathbb{R}\oplus\mathbb{R}\widehat u\subset\mathbb{O}$, the problem becomes a real spectral problem for $H_{u,Q}=h-Q R_{\widehat u}$, whose residual symmetry is $\mathrm{SU}(3)$. The resulting $\mathrm{SU}(3)$-block decomposition yields two explicit spectral loci in each slice: a quartic curve and a circle. The equations defining these loci are independent of the slice, and the full right spectrum is obtained by allowing $\widehat u$ to vary over the unit sphere in $\operatorname{Im}\mathbb{O}$.