Homomorphisms of vector bundles on curves and parabolic vector bundles on a symmetric product

Let $S^n(X)$ be the symmetric product of an irreducible smooth complex projective curve $X$. Given a vector bundle $E$ on $X$, there is a corresponding parabolic vector bundle ${\mathcal V}_{E*}$ on $S^n(X)$. If $E$ is nontrivial, it is known that ${\mathcal V}_{E*}$ is stable if and only if $E$ is stable. We prove that

$$H^0(S^n(X),\, {\mathcal H}om_{\rm par}({\mathcal V}_{E*}\, , {\ma... ...\!=\! H^0(X,\, F\otimes E^\vee )\oplus (H^0(X,\, F)\otimes H^0(X,\, E^\vee )).$$

As a consequence, the map from a moduli space of vector bundles on $X$ to the corresponding moduli space of parabolic vector bundles on $S^n(X)$ is injective.