Invariant vector bundles and Hitchin systems

Let $X\to Y$ be a Galois cover with Galois group $\mathrm{\Gamma }$, where $X$ and $Y$ are smooth complex projective curve of genus $⩾2$. In this paper, we study the moduli spaces of semistable $\mathrm{\Gamma }-$invariant vector bundles on $X$ and classify their connected components. We also study the Hitchin systems on these moduli spaces and determine their fibers in the smooth case.