Let $C$ be a smooth projective curve over $\mathbb{C}$ and $G$ be a finite group acting on $C$. Consider the quotient $f:C\rightarrow C'=C/G$. Suppose that $C'$ is smooth (I think this will be true, since the quotient by a finite group is normal, and hence non-singular for curves).
Consider a line bundle $L$ on $C$ with a $G$-action. It is possible that $L$ descends to $C'$, that is, there is a line bundle $L'$ in $C'$ such that $f^*L'=L$.
Suppose it doesn't consider the maximal subsheaf of $L$ which descends, say $L^G\subset L$. This has to either zero or a rank one sheaf.
1) Is $L^G$ locally free. Can it be a line bundle not equal to $L$. For example, this will be ruled out if $L/L^G$ is locally free.
2)We can similarly talk about $V^G$ for a vector bundle $G$. Will $V^G$ be a direct summand of $V$?
I would be grateful for some direction.
