# Is there a comprehensive survey of the discrete series representation of a real reductive group?

Vague form of the question: where can one find a comprehensive and possible modern account of the discrete series representations of a real reductive group?

Motivation: I am a master's student trying to learn about the real local Langlands correspondence, with a background in representation theory given by a solid knowledge of finite dimensional Lie algebras, structure of reductive Lie groups and, more recently, the example of the all the (infitesimal classes of irreducible, admissible) representations of $$\operatorname{GL}(2,\mathbb{R})$$.
While reading I have come to understand that these form somewhat building blocks for the $$L$$-packet of representations associated to an $$L$$-parameter $$\varphi$$.
I have also been surprised by a couple of statements I have encountered and hence I feel that a better grasp on the topic would make me appreciate and perhaps even understand better the subject at large. Let me make an example: Reading the extremely instructive answer by Emerton to Uniqueness of local Langlands correspondence for connected reductive groups over real/complex field, one understands immediately that there are no discrete series for $$\operatorname{GL}(3)$$ but this is a non obvious result, which becomes obvious after knowing that "a (linear connected semisimple Lie) group has discrete series if and only if the rank of the group is equal to the rank of the maximal compact subgroup" as user B R points out in the comments 1 2 of Automorphic forms on GL(3).

1. By modern I mean possibly a graduate/PhD level book which is not one of the original articles. (And hence was not written to publish new results but to teach the existing literature.)
2. By comprehensive I mean that it should address all the major results of the theory for a general reductive $$G$$ (so not just the semisimple case) and possibly give examples. In particular it should contain existance results, (a least sketches of) proofs and (at least sketches of) the constructions. It should also contain a discussion of holomorphic, quaternionic discrete series and other subfamilies I may not be aware of. (It may of course contain further material.)

• Harish-Chandra's papers: "Discrete series for semsimple Lie groups I" and II
• Knapp's "Representation theory of semisimple groups, an overview based on examples"
• Paul Garrett informal note titled "Some facts about discrete series", which actually covers many results and features a long bibliography.
• Daniel Bump's book about automorphic forms and representations.
• The list of references for the construction of discrete series given by Wikipedia.

Clearly, a possible answer could be that there is no single book or article covering this and one has to rely on a list of them (the list being given in the answer).

Bonus points if the answer includes a short summary of what "anybody" should know besides what already written by Garrett and if it highlights what is most relevant to the Langlands programm.

P.s. Why this question might be relevant to others: a beginner with no access in his university to experts on representation theory of real Lie groups might struggle to know what are the relevant objects. I myself did and still do. An answer could help such students and already the question gives several good suggested readings for the curious.

I am not completely sure this is research level, and I apologize if it is not.

In Harish-Chandra’s classification of the discrete series representations for $$G(\mathbf R)$$, there are some unusual aspects of the formulas for their characters. Langlands confronted this in the work that led to his classification, together with the principle of functoriality in his mind, Langlands eventually proposed the theory of endoscopy. And then Shelstad made an endoscopic extension of Langlands’ classification. Here are some great references:
• For classical results of semisimple $$\mathbf R$$-groups due to Harish-Chandra, Knapp‘s book Representation Theory of Semisimple Groups covers almost everything you need to know.
• @LSpice Note that Langlands replaced ‘semisimple Lie groups’ but not ‘semisimple reductive groups’ by real points of a reductive algebraic group over $\mathbf R$. So it’s not a problem of generality, but different working settings. Commented Jul 14 at 13:15