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).

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).