The Hitchin Fibration for Symmetric Pairs

This thesis consists of two chapters: The first and primary component is dedicated to the Hitchin morphism for symmetric spaces, which is joint work with B. Morrissey. We introduce and describe the ``regular quotient'' and explain some basic consequences for Higgs bundles. We include an invariant theoretic approach to spectral covers in this setting for the space $\GL_{2n}/\GL_n\times \GL_n$. We also include some work towards an enhanced Grothendieck-Springer style correspondence in for quasi-split pairs, which classifies certain parabolics of $H$ rather than Borels of $G$. We study the component groups of such covers, including their importance in describing regular centralizers. In the second chapter, we recount some joint work with B.C. Ng\^o on companion matrices for clssical groups and $G_2$. We use the companion matrix construction for $\GL_n$ to build canonical sections of the Chevalley map $[\fg/G]\to \fg\git G$ 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 $G_2$.