Narrow Results

We use the companion matrix construction for ${\mathrm{\text{GL}}}_n$ to build canonical sections of the Chevalley map $[𝔤/G]\to 𝔤//G$ 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 $G_2$.