  Let $S$ be a Riemann surface obtained by deleting a finite number of points,
called cusps, from a compact Riemann surface. Let $\rho: \pi_1(S)\to Sl(n,
\mathbb{C})$ be a semisimple linear representation of $\pi_1(S)$ which is
unipotent near the cusps. We investigate various cohomologies associated to
$\rho$ of $\bar S$ with degenerating coefficients $L_{\rho}$ (considered as a
local system -- a flat vector bundle, a Higgs bundle, or a
$\mathcal{D}$-module, depending on the context): the \v{C}ech cohomology of
$j_*L_{\rho}$, the $L^2$-cohomology, the $L^2$-Dolbeault cohomology, and the
$L^2$-Higgs cohomology, and the relationships between them. This paper is meant
to be a part of the general program of studying cohomologies with degenerating
coefficients on quasiprojective varieties and their K\"ahlerian
generalizations. The general aim here is not restricted to the case of curves
nor to the one of representations that are unipotent near the divisor. The
purpose of this note therefore is to illuminate at this particular case where
many of the (analytic and geometric) difficulties of the general case are not
present what differences will appear when we consider unipotent harmonic
bundles instead of Variations of Hodge Structures where the results are known.
