The Hitchin Fibration for Symmetric Pairs

TL;DR

This paper extends the Hitchin fibration framework to symmetric spaces, introducing the regular quotient, and explores spectral covers and centralizer group schemes, advancing the understanding of Higgs bundles in this setting.

Contribution

It introduces the regular quotient for the Hitchin fibration in symmetric spaces and provides an invariant theoretic approach to spectral covers for specific symmetric pairs.

Findings

Development of the regular quotient for symmetric space Hitchin fibrations

Invariant theoretic description of spectral covers for GL_{2n}/(GL_n×GL_n)

Analysis of the regular centralizer group scheme for quasisplit pairs

Abstract

We introduce and describe the "regular quotient" for the Hitchin fibration for symmetric spaces and explain some basic consequences for Higgs bundles. We include an invariant theoretic approach to spectral covers in this setting for the particular space $\mathrm{GL}_{2n}/(\mathrm{GL}_n\times \mathrm{GL}_n)$. We also include a study of the regular centralizer group scheme for quasisplit pairs, including a Galois description of a closely related group scheme. We collect some basic consequences for Hitchin systems associated to such pairs.