The companion section for classical groups

We use the companion matrix construction for ${\mathrm{G}\mathrm{L}}_n$ to build canonical sections of the Chevalley map $[\mathfrak{g}\mathrm{/}G]\to \mathfrak{g}\mathrm{/}\text{\hspace{0.17em}⁣}\mathrm{/}G$ for classical groups $G$ as well as the group $G_2$. To do so, we construct canonical tensors on the associated spectral covers. As an application, we make explicit lattice descriptions of affine Springer fibers and Hitchin fibers for classical groups and $G_2$.