Sometimes, results on automorphic representations are available only under local assumptions. Typically, one could require the representation to be a $\xi$-*cohomological* cuspidal representation, and I would like to understand the meaning of it.

Let $G$ be a connected reductive group over $\mathbb{Q}$. Let $\xi$ be an irreducible (finite dimensional) algebraic representation of $G$ over $\mathbb{C}$. An (isomorphism classes of) irreducible admissible representations $\pi_\infty$ of $G(\mathbb R)$ is $\xi$-cohomological if it has *the same central character and infinitesimal character as $\xi$*.

I would like an automorphic/classical interpretation of this condition: is it related to being infinitesimally equivalent of a certain discrete series? or to being in a certain $L$-packet?

The reason why it is termed cohomological condition also puzzles me (and this may be part of my lack of understanding).