References for Harish-Chandra pairs and modules, category "O"?

Question

I am looking for references related to the terms "Harish-Chandra pair" and "Harish-Chandra modules", and also to the term "category O". I know what these are, or I think I do (a Harish-Chandra pair is a pair (Lie algebra; subgroup) with the subgroup acting in the Lie algebra, satisfying some natural conditions). The question is about any standard or classical sources I could refer to.

1. What are the standard or classical references for the terms "Harish-Chandra pair" and "Harish-Chandra module"?
2. The same question for algebraic Harish-Chandra pairs and modules (with an algebraic or proalgebraic subgroup).
3. One example of a category of Harish-Chandra modules is the category "O" of representations of, e.g., a simple Lie algebra, integrable to the Borel subgroup. Another version is the category of representations integrable to the maximal unipotent subgroup. What are the standard or classical references for either or both of the above definitions of the category "O"?
4. My understanding is that what was called "Harish-Chandra modules" in the classical representation theory was not the above example 3. at all, but rather the modules over a real Lie algebra integrable to the maximal compact subgroup. What are the standard or classical references for this notion of Harish-Chandra modules?

Answer 1

Harish-Chandra pairs and their use in localization is discussed in Beilinson-Bernstein A proof of Jantzen Conjectures, available on Joseph Bernstein's web page. See in particular sections 1.8 and 3.3.

Other sources:

1. Beilinson-Drinfeld, Quantization of Hitchin's Integrable System and Hecke Eigensheaves, sections 7.7-7.9. Available on line.
2. Beilinson-Feigin-Mazur Notes on Conformal Field Theory (Incomplete). The discussion is centered around groupoids and the Virasoro algebra. This is on Barry Mazur's web page (link on the side to "older material").

Answer 2

I'm commenting on part 4) of your question really I think: Harish-Chandra was interested in studying the unitary dual, but in the process realized that the whole unitary structure was rather more than one needed to carry around, and moreover, that it was necessary also to consider representations which were not necessarily unitary, and in this wider context the analytic issues of when to consider two representations "equivalent" intrude more prominently (because, for example, there are lots of different kinds of function spaces on a given G-space).

The notion of "admissible module" -- a module where each irreducible representation of the maximal compact subgroup occurs with finite multiplicity gave a natural class of representations which includes all unitary representations, and once in this class it is pretty natural to restrict attention to the action to the action of the Lie algebra g, and the maximal compact K (as the Lie algebra acts on K-finite vectors). Thus certainly all the motiviation for the notion of a (g,K)-module must be due to Harish-Chandra, but I'm not sure if he actually abstracts the idea in his papers.

I can't tell from BGG's original paper if they are thinking of category O in the context of (g,K)-modules: their stated motivation is much more from modular representation theory. One nice reference for the representations of real groups is the book "Representation Theory of Lie groups" from the Park City summer school -- for example there's an article by Knapp and Trapa which discusses the work of Harish-Chandra.

Also, I'd just like to mention that the idea of considering the pair (g,K) where g is the Lie algebra and K is the maximal compact of a real Lie group G is quite natural if one considers that the g action contains the infinitesimal information while the maximal K is a retract of G, and so one could hope that its action captures enough of the "global" information in the representation. (I think it was Graeme Segal who pointed this out to me -- perhaps an obvious comment for him, but I found it insightful and at least psychologically useful).

Answer 3

If you are able to read german then there might be a standard reference to harish chandra modules (the only own I know which is actually more then just the definition stuff from papers): jantzen - Einhüllende Algebren halbeinfacher Lie-Algebren - especially Chapter 6&7.

Answer 4

The first sections of Gaitsgory's notes on geometric representation theory contain basic facts on the category O; but I don't know whether that reference is either classical or standard.

Answer 5

The book "Enveloping algebras" by Dixmier appears to contain some material on Harish-Chandra modules in the classical sense (i.e., with respect to the maximal compact subgroup) and also on Verma modules, though not on Harish-Chandra pairs or category "O".

Besides, there is a new book "Representations of semisimple Lie algebras in the BGG category O", by Humphreys, AMS 2008. It discusses category O at great length and contains also some words about the classical Harish-Chandra modules. Abstract Harish-Chandra pairs aren't mentioned.

A discussion of algebraic Harish-Chandra pairs in the infinite-dimensional proalgebraic setting (over the complex numbers) can be found in the unpublished paper "Notes on Conformal Field Theory" by Beilinson-Feigin-Mazur.

Answer 6

This answer may be a little late, considering the date it was asked, but anyway...

Concerning question 4), a "standard" reference is perhaps the two volume treatise by Wallach on "Reductive Groups". Some aspects of Harish-Chandra modules (with a view towards globalisation questions) are also developed in a recent preprint by Bernstein and Kroetz (see B. Kroetz's web page). Jantzen's Habilitationsschrift has already been mentioned. If you can read German, this also an excellent source.

Over the last few years, Penkov, Serganova and Zuckermann have been developing the theory of (g,k)-modules in the general setting where g is some Lie algebra and k is some subalgebra. The modules one considers are g-modules which are semi-simple over k (plus possibly additional conditions). Check the list of recent preprints and publications of these authors.