Arthur's refinement of parameters for unitary automorphic representations

Question

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.

Answer 1

For $G=Gl_n$, the $SL_2$ factor of Arthur plays a trivial role in the classification only when you restrict yourself to cuspidal automorphic representations. But Arthur is interested with more general automorphic representations that are in the discrete spectrum, that is (essentially) the irreducible sub-representation (in the naive sense, that is not the one that are in the continuous spectrum) of $L^2(G(\mathbb A)/G(\mathbb Q),\omega)$, where $\omega$ is a central character. For the classification of this, the $SL_2$ factor is absolutely necessary. In fact proving that the classification of the discrete spectrum of $GL_n$ is, modulo the classification of the cuspidal spectrum of $GL_n$, as pre diced by Arthur conjecture with his $SL_2$ factor is a monumental theorem of Langlands (prior to Arthur, who wa motivated by it to introduce its $SL_2$-factor), which has been rewritten with more details by Moeglin and Waldspurger in a large book with evocative sub-title "une paraphrase de l'Écriture"). This answers your question about $GL_n$.

In general, it is still true, according to Arthur's formalism now proved in many cases, that the representations (or rather an $A$-packet of representations) that have an Arthur parameter which is trivial on the $SL_2$ factor are exactly the one which are tempered.

To understand more about Arthur formalism, besides Arthus' articles which are very deep and very hard, and Frankel's survey you quote, you can give a look to the Appendix of my book with Chenevier, Astérisque 328, and also to the new book of Chenevier and Lannes, Chapter 8.