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.



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. 


