The classical theory of Eichler-Shimura realizes the space of cusp forms of certain weights and levels in the parabolic cohomology of modular curves, which is the image of the cohomology with compact support in the ordinary cohomology. In Langlands’ Antwerp paper, he also uses parabolic cohomology (of the “Shimura variety” version of modular curve $G(\mathbb{Q})\backslash \mathcal{H}^\pm\times G(\mathbb{A}_f)/K$) and obtain a Matsushima type decomposition of it. But the Matsushima’s formula (also in Borel-Wallach’s book and J.Arthur’s 1981 Invent paper) uses $L^2$-cohomology of the locally symmetric spaces. In Blasius & Rogawski’s paper in Motives, they use intersection cohomology of (the Baily-Borel-Satake compactification of) certain Shimura varieties and use the Matsushima type decomposition because in this case the $L^2$ cohomology is the same as the intersection cohomology. More recently in Sophie Morel’s work, she also uses intersection cohomology.

I am not familiar with algebraic topology and the various cohomology theories mentioned above. The only intuition I have is, some Shimura varieties are non-compact, and their compactifications are probably singular, so one has to put some effort on the cohomology theory used. So my questions are:

1: What is the original motivation for using parabolic cohomology in the early works of Eichler-Shimura, since this seems strange at first. Is there any conceptual reason that it works?

2: Are the $L^2$ and parabolic cohomologies of the locally symmetric space isomorphic?

3: It seems that intersection cohomology is the most powerful one, so can one override the theory of Eichler-Shimura by using intersection cohomology? If so, is there any reference on this (or it is too trivial to be wrote down in any reference? )

4: In the early works of Eichler-Shimura and Langlands, they used different decompositions of the parabolic cohomology, say of Hodge type and Matsushima type respectively. Are they essentially different approaches (to the construction of $\ell$-adic Galois representations) or they are related in some way?