= Solution
Let $\rho:\pi_1(X)\to U(V)$ be the <unitary representation>. The associated <unitary flat holomorphic vector bundle> is
$$
E_V=(\widetilde X\times V)/\pi_1(X),
$$
where deck transformations act on the first factor and $\rho$ acts on the second. Local lifts to the <universal cover> give holomorphic orthonormal frames with locally constant unitary transition matrices. The induced <Hermitian metric> is therefore well defined, and the induced flat <unitary connection> is its <Chern connection>.
Let $s$ be a global <holomorphic section>. In one of these parallel orthonormal frames write $s=(f_1,\ldots,f_r)$, where every $f_j$ is holomorphic. Its squared norm $u=|s|^2=\sum_j|f_j|^2$ is a globally defined smooth function because the transition matrices are unitary. In a local complex coordinate,
$$
\frac{\partial^2u}{\partial z\,\partial\bar z}=\sum_{j=1}^r|f'_j(z)|^2\ge0.
$$
Choose any smooth conformal metric on the curve, locally $\kappa(z)|dz|^2$. Its Laplacian satisfies
$$
\Delta u=\frac4\kappa\sum_j|f'_j|^2\ge0.
$$
On the compact curve without boundary, <Stokes theorem> gives $\int_X\Delta u\,d\mathrm{vol}=0$. A continuous nonnegative function with zero integral vanishes everywhere, so each $f'_j=0$. Thus the section is locally constant in every parallel frame and is globally parallel. This proves that <holomorphic sections of a unitary flat bundle are parallel>.
A parallel section is determined by its value $v$ at a chosen base point. Following it around a loop returns the value $\rho(\gamma)v$, so a global parallel section exists exactly when $\rho(\gamma)v=v$ for every $\gamma\in\pi_1(X)$. Conversely, such a fixed vector gives the constant equivariant section on $\widetilde X$, which descends holomorphically. Evaluation and this construction are inverse linear maps, giving the canonical identification
$$
\boxed{H^0(X,E_V)\cong V^{\pi_1(X)}.}
$$
Unitarity is what makes the local squared norms glue and the integrated nonnegative identity available; no assumption that the universal cover is compact is made.
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