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 $$h(\hat{w})=\hat{w}\hat{I}»,\hat{I}»\in \mathbb{O}.$$ After fixing a complex slice $\mathbb{R}\oplus \mathbb{R}\hat{u}\subset \mathbb{O}$, the problem becomes a real spectral problem for $H_{u,Q}=h-Q{R}_{\hat{u}}$, 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 $\hat{u}$ to vary over the unit sphere in $\mathrm{Im}\mathbb{O}$.

56 pages, 1 figure