Reconstructing vector bundles on curves from their direct image on symmetric powers

Abstract: Let $C$ be an irreducible smooth complex projective curve, and let $E$ be an algebraic vector bundle of rank $r$ on $C$. Associated to $E$, there are vector bundles ${\mathcal F}_n(E)$ of rank $nr$ on $S^n(C)$, where $S^n(C)$ is $ $n$-th symmetric power of $C$. We prove the following: Let $E_1$ and $E_2$ be two semistable vector bundles on $C$, with ${\rm genus}(C)\, \geq\, 2$. If ${\mathcal F}_n(E_1)\,= \, {\mathcal F}_n(E_2)$ for a fixed $n$, then $E_1 \,=\, E_2$.