I am trying to make sense of how the following two statements fit together. Here, $X$ is a compact Riemann surface and $E$ is a holomorphic vector bundle of rank $r$ over $X$.
Theorem 1 (Weil): Let $E$ be indecomposable. Then, deg($E$) = 0 if and only if $E$ arises from a representation $\rho: \pi_1(X) \rightarrow GL(r, \mathbb{C})$.
Theorem 2 (Narasimhan-Seshadri): A degree 0 vector bundle is stable if and only if $E$ arises from an irreducible representation $\rho: \pi_1(X) \rightarrow U(r)$.
Can I view Theorem 2 as a ``restriction" of Theorem 1? I have a feeling that it would be naive to do so because Weil's theorem does not say that a representation $\rho: \pi_1(X) \rightarrow GL(r, \mathbb{C})$ defines an indecomposable, degree 0 vector bundles.
Edit: Furthermore, perhaps one can't view Theorem 2 as a ``restriction" of Theorem 1 because Weil hasn't proved that two degree 0 vector bundles are biholomorphic if and only if their corresponding representations are equivalent. I think the picture we get from these two theorems is but I could be wrong.
