Philosophy behind cohomological representations

The real place of an automorphic representation is an irreducible unitary representation $\pi$ of $G(\mathbb R)$. A fundamental basic case is when $\pi$ has the same infinitesimal character as a finite dimensional representation (also known as regular and integral infinitesimal character). By a result of Susana Salamanca-Riba$^*$ every such representation is cohomological. This includes all discrete series representations, and the trivial representation.

* MR1671213, On the unitary dual of real reductive Lie groups and the $A_{\mathfrak q}(λ)$ modules: the strongly regular case., Duke Math. J. 96 (1999), no. 3, 521–546.

This is just a long comment, and I'm not sure if it is the kind of thing you're looking for. Hopefully someone can give a proper answer to your question.

In an old question, Paul Garrett made the following enlightening remark, regarding cohomological representations and automorphic forms:

"We probably must reconcile ourselves to "cohomological" being the "right" generalization of "holomorphic"."

I highly recommend you to read the complete answer.

Since I read it I tend to think of cohomological as a "the right generalization" of holomorphic, particularly in light of the "discrete implies cohomological" result that Garret mentions. Then again, I don't have to think about cohomological all that much.

I'm far from an expert, but here is a comment. In the case of a Shimura variety, the Matsushima-Murakami formula and the (proof of the) Zucker conjecture shows that cohomological representations are precisely the ones that contribute to the intersection cohomology of the Shimura variety with coefficients in some local system. This "piece" of the intersection cohomology is a candidate for a compatible system of $\ell$-adic Galois representations attached to the representation (or a "motive"). Thus cohomological representations are precisely the ones for which Deligne's original construction of Galois representations attached to Hecke eigenforms of weight $\geq 2$ could be optimistically hoped to generalize. For instance, Weissauer has constructed 4-dimensional Galois representations attached to Siegel cusp forms of genus 2 in the cohomological case, using this strategy. But e.g. for the interesting case of Siegel cusp forms of weight $(2,2)$, which are expected to have Galois representations given by $H^1$ of an abelian surface, such a construction is wide open.