Let $F$ be a totally real field. Let $G$ be a (split) connected reductive group over $F$. Let $\pi$ be an irreducible automorphic representation of $G$.
Recall in the work of Buzzard and Gee "The conjectural connections between automorphic representations and Galois representations". .
- (Definition 2.3.1) $\pi$ is $L$-algebraic, if the infinitesimal character $\lambda_{\pi_v} \in X^*(T)$ for any real place $v$ of $F$.
- (Definition 2.3.3) $\pi$ is $C$-algebraic (e.g. cohomological), if $\lambda_{\pi_v} - \rho_B \in X^*(T)$ for any real place $v$ of $F$.
- (Conjecture) If $\pi$ is $L$-algebraic, then conjecturally we could attach $\ell$-adic Langlands parameter $\rho_\pi: Gal(F^{alg}/F) \to {}^LG(\overline{\mathbb Q}_\ell)$ to $\pi$, which is also compatible with Satake isomorphism at any finite place $v$ of $F$ where $\pi_v$ is unramified.
Assuming $G=GL_n$ or a classical group, in good cases (e.g. regular), we know how to attach $\rho_\pi$ for cohomological cuspidal $\pi$. Assume the conjecture holds for cuspidal $\pi$.
The question: Do we currently know how to attach $\rho_\pi$ for automorphic representation $\pi$ in the discrete spectrum? Is the local-global compatibility known at unramified places?
In the case $G=GL_n$, we have the classification of the discrete spectrum (via residue of Eisenstein series) in the book of Moeglin-Waldspurger. See e.g. Definition of discrete spectrum and continuous and basic properties. If we know $\pi$ comes from a cuspidal auto rep $GL_m$ ($m|n$). Do we just construct $\rho_\pi$ via an inclusion map of $GL_m$ into ${}^LG$? It seems this is true only up to semi-simplification of the Langlands parameter.
