TITLE: Philosophy behind cohomological representations
QUESTION [13 upvotes]: For a given real reductive Lie group $G$, we have the notion of a representation being cohomological using the Lie algebra cohomology. In particular we know that the discrete series representations of $G$(whenever it exists) is cohomological. I am trying to understand what information do we exactly obtain when we know that these representations are cohomological. Also, why are cohomological representations important and what role do they play in the global setting, i.e in the setting of automorphic representations?
It would be great even if you could direct me to an appropriate reference.
Thanks in advance.

REPLY [5 votes]: 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.