TITLE: Simple vector bundle isomorphic to one of its twistings
QUESTION [5 upvotes]: Let $V$ be a vector bundle over an algebraic curve $C$, ad assume that that $V \cong V \otimes L$ for some line bundle $L$. 
If $V$ is decomposable this is clearly possible, for example take $V \cong \mathcal{O}_C \oplus L$ with $L^2 \cong \mathcal{O}_C$. 

Q. Does $V \cong V \otimes L$ imply $L \cong \mathcal{O}_C$ if  $V$ is simple?

REPLY [5 votes]: The answer is no, as shown by the following classical example due to Atiyah.
Take an elliptic curve $E$, choose a point $p \in E$ and consider the unique non-split extension $$0 \longrightarrow \mathcal{O}_E \longrightarrow V_p \longrightarrow \mathcal{O}_E(p) \longrightarrow 0.$$
Then $V_p \simeq V_p \otimes L$ for every $2$-torsion line bundle $L$ on $E$, see Remark p. 35 of [F].
Note that $V_p$ is stable and thus simple, see [F], Theorem 9 p. 89.
References.
[F] R. Friedman: Algebraic surfaces and holomorphic vector bundles, Springer 1998.