# References for Langlands classification

I kindly ask about some references concerning the representation theory of the Langlands dual of a compact Lie group, and how it relates to things related to the original compact Lie group.

My background: I know some basic facts about Lie groups/algebras, such as their root systems, Weyl groups etc. I am not familiar yet with the Langlands program, and related things. I am mostly interested for now in working over $\mathbb{R}$ and $\mathbb{C}$.

Edit 1: after some research, I realize that what I want is a reference on the Langlands classification, done by Langlands himself. So I will start by reading that article by Langlands ("On the classification of irreducible representations of real algebraic groups").

Edit 2: I found some introductory notes on endoscopy by J-P Labesse, which look very promising to me! http://www.math.utah.edu/~ptrapa/src2006/labesse.pdf

Edit 3: Knapp's article, suggested by Desiderius Severus, is indeed a really good introduction to the Langlands classification, which is part of the local Langlands program in the Archimedean case. In some of my comments, one can see that I was confused between the Langlands classification, and the geometric Satake isomorphism, which plays a role in the Geometric Langlands program. I apologize for this confusion. It took me some time to get used to some of the jargon of the Langlands program (and even now, I cannot claim to have mastered the jargon, but I have improved a little).

• You might be interested in the geometric Satake theorem, which expresses the representation theory of $G^\vee$ in terms of sheaves on the affine Grassmannian for $G$. See Zhu's notes for the full story. – Arun Debray Nov 2 '17 at 15:21
• @ArunDebray, thank you. The subject is tough but interesting. – Malkoun Nov 2 '17 at 15:26
• @Malkoun: Probably you should add one or two tags, such as 'lie-groups' and 'rt.representation-theory'. The Langlands program is especially concerned with connecting such subjects to number theory. You might take a look at somewhat older contributions (the two Bulletin articles being freely available online): mathscinet.ams.org/mathscinet-getitem?mr=1990371 and ams.org/journals/bull/1984-10-02/S0273-0979-1984-15237-6, /mathscinet.ams.org/mathscinet-getitem?mr=2823020 – Jim Humphreys Nov 2 '17 at 17:26
• @Jim Humphreys, thank you so much for these references. I have plenty to read now. – Malkoun Nov 2 '17 at 17:38
• @JimHumphreys, Prof. Humphreys, I love the topic, and particularly the links to Number Theory. In my opinion, the reciprocity theorems are extremely beautiful. But, I would like to get faster to what I want specifically, meaning while avoiding "adèles" and "idèles", for the time being (only). Do you happen to know of a specific reference on the Langlands classification theorem? I started reading a paper by Langlands, but I am missing some definitions. – Malkoun Nov 2 '17 at 19:17

The first source in which I really discovered quite explicitly the archimedean local Langlands classification is in this beautiful article of Knapp, reviewing it in some pages. Moreover, it has the appeal to give a short historical motivation, to deal with the $SL_2$ case, and then to turn to the general one in an explicit manner.