References for the Langlands classification of irreducible admissible representations

Question

can someone provide a few references for the Langlands classification result? I am interested in the details of the result, explained rather carefully. I am also interested in knowing where it stands with respect to the Langlands program, meaning, how it is a special case of the general Langlands program. My first impression is that the literature seems to be vast for the general Langlands program, and all its incarnations, yet much less so for the specific Langlands classification result (having to do with admissible representations); or at least, I could use some guidance from an expert, to know what to read first.

Answer 1

It seems that you've already found plenty of references, but here are some comments that you might find helpful.

The Langlands classification is a result about representations of reductive groups over local fields, so there is a dichotomy between the archimedean and non-archimedean cases. The result ends up being true in either situation (at least with the correct interpretations), but the proofs are going to be a bit different. If I had to guess, I'd say you're referring to the archimedean case (e.g for $SL_2(R)$)? I've never really looked into that; but a good reference for the non-archimedean case is the final chapter of David Renard's book Representations des groupes reductifs p-adique.

The more serious of my comments is that you seem a bit confused about the naming of Langlands classification versus the Langlands programme. It's certainly true that these are related, but this is in a very technical nature that you shouldn't worry about to begin with. The Langlands classification is a technical result about the representations of some group $G$ defined over a local field, which essentially says that the classification of its irreducible representations can be reduced to classifying the discrete series. The (local) Langlands programme is a load of very general (and, in many cases, still very speculative) conjectures that relate the representation theory of groups such as $G$ to Galois theory. A very nice, standard reference for an introduction to the Langlands programme is Stephan Gelbart's article An elementary introduction to the Langlands programme; if I recall correctly it was in Bulletin of the AMS and should be easy to find online (for free).

I don't think it's really worth commenting very much on the relationship between Langlands classification and the local Langlands correspondence (or conjectures). There is one, and it is technically important, but it essentially comes down to understanding how the image of a Galois representation affects the structure of the corresponding representations(s) of $G$. But the problem is that, for groups other than $GL(N)$, this is very, very difficult to understand (and, indeed, isn't entirely known). Probably this is better understood for groups defined over archimedean fields, but I don't know a great deal about that case.

Answer 2

Upon emailing Prof. Knapp, he suggested to me several references, that I thought I would reproduce here, in case other people may find them useful:

• "Representation Theory of Semisimple Groups: an Overview Based on Examples”, by Knapp (1986)
• An expository paper by Peter Trapa and Anthony Knapp published in 2000 in the AMS book series called “IAS/Park City Mathematics Series.”
• 1980 book by Borel and Wallach in the series Annals Studies
• of course, Langlands's original paper.

I have just started reading the expository paper by Knapp and Trapa, and find it extremely well written. Some proofs are omitted there, but the amount of Math covered there is quite big, and the paper contains several examples. A very nice paper.