Hitchin systems for invariant and anti-invariant vector bundles

Abstract

Given a smooth projective complex curve $X$ with an involution $\sigma$, we study the Hitchin systems for the locus of anti-invariant (resp. invariant) stable vector bundles over $X$ under $\sigma$. Using these integrable systems and the theory of the nilpotent cone, we study the irreducibility of these loci. The anti-invariant locus can be thought of as a generalisation of Prym varieties to higher rank.