The companion section for classical groups

TL;DR

This paper constructs canonical sections for classical groups and G_2 using companion matrices, enabling explicit descriptions of affine Springer and Hitchin fibers through spectral cover tensors.

Contribution

It introduces a novel method for building canonical sections for classical groups and G_2 via companion matrices and spectral tensors, advancing geometric representation theory.

Findings

Explicit lattice descriptions of affine Springer fibers

Explicit lattice descriptions of Hitchin fibers

Construction of canonical sections for classical groups and G_2

Abstract

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