In his work on the classification of automorphic representations of a group $G$, Arthur has conjectured that the parameterization of such representations involves a homomorphism $\rho : SL_2 \times Gal(\bar{F}/F) \rightarrow G^\vee$. This is a refinement of earlier conjectures that involved homomorphisms $\tilde{\rho} : Gal(\bar{F}/F) \rightarrow G^\vee$ and thus the $SL_2$ appearing in $\rho$ is sometimes referred to as "Arthur's $SL_2$". My questions are the following :

For arbitrary $G$, is there a bijection between certain kinds of autormorphic representations and the instances where the $SL_2$ plays a non-trivial role ? Ex : $G=GL_n$, my understanding is that the $SL_2$ does not play a non-trivial role and that this is related to the absence of non-tempered cuspidal unitary representations (I learnt the statement from Frenkel's review). But, it is not clear to me if there is a bijection between non-tempered cuspidal unitary representations and cases where $SL_2$ plays a non-trivial role on the Galois side.

I would like to ask a question similar in spirit to the above one for the

*local*case. Is there one ? (Arthur's article has a few comments about what the existence of such parameters means for the local case, but I could not use it to come up with a question for the local case).

**References** : Arthur's paper "Unipotent automorphic representations : conjectures (1989)" is available here and the Frenkel review where I came across Arthur's $SL_2$ is here.