Relation between $\xi$-cohomological and discrete series

Question

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).

Answer 1

This condition comes up because of $(\mathfrak{g}, K)$-cohomology, which is an extremely important invariant of automorphic representations.

If $\xi$ is an algebraic rep, then $\xi$ defines a locally-constant sheaf on the locally-symmetric space $Y_G(U)$ for any open compact $U \subset G(\mathbf{A}_f)$. The Betti cohomology $H^\star_B(Y_G(U), L_\xi)$ can be computed in terms of automorphic forms. If $G$ is anisotropic over $\mathbf{Q}$ (so $Y_G(U)$ is compact, and all automorphic reps of $G$ are cuspidal) then Matsushima's formula says that this Betti cohomology is given by $$\bigoplus_\pi m(\pi) \cdot \pi_f^U \otimes H^*(\mathfrak{g}, K; \pi_\infty \otimes \xi^\vee),$$ where the sum is over cuspidal automorphic reps of $G$. There is a generalisation of this to arbitrary $G$ by Franke (but you have to be careful to isolate the "cuspidal part" of the cohomology).

So the correct definition of "cohomological" should really be "$H^*(\mathfrak{g}, K; \pi_\infty \otimes \xi^\vee)$ is non-zero for some $\xi$"; these are the representations which contribute to the Betti cohomology of symmetric spaces. However, an obvious necessary condition for this is that $\pi_\infty$ have the same central and infinitesimal character as $\xi$, since otherwise the cohomology vanishes for trivial reasons.

Discrete series representations are always cohomological. However, there are more cohomological representations which are not discrete series (fortunately, since lots of groups, e.g. GL(n) for n > 2, don't have any discrete series representations).