The way the Narasimhan-Seshadri Theorem is stated classically is that the semi-stable, degree-0 holomorphic vector bundles with structure group $G=U(n)$ on a Riemann surface $X$, are in one-to-one correspondence with representations of the fundamental group $\pi_{1}(X)$ in $G=U(n)$, modulo conjugation. Moreover, the stable locus coincides precisely with the irreducible representations, and one can extend to arbitrary degree with mild adjustments. This gives a nice three-way correspondence between (i) semi-stable holomorphic bundles with structure group $G=U(n)$, (ii) representations of $\pi_{1}(X)$ in $G=U(n)$, and (iii) flat connections on $U(n)$-bundles on $X$.
My main question is: certainly the gauge theory and topology easily generalize to $G$ an arbitrary compact Lie Group, but does this level of generality hold in the holomorphic vector bundles? In other words, can we write the same "trinity" above where we instead speak of semi-stable, degree-0 holomorphic vector bundles with structure group $G^{\mathbb{C}}$?
I suppose I'm confused about something even more elementary. Certainly the most general holomorphic vector bundles have structure group $\rm{GL}(n,\mathbb{C})$. And I think you can still have semi-stable bundles with this structure group. So why isn't the Narasimhan-Seshadri Theorem stated for $G=\rm{GL}(n, \mathbb{C})$?
In addition, I've heard vaguely that for a certain choice of $G$, one can arrive at moduli spaces of Higgs bundles? If this is indeed the case is there a popular reference which explains this correspondence?