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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?
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It's established convention from what I understand that the title of a post should ask a question, so do change your title to something like "References for Harish-Chandra pairs and modules, category “O”?" –  Thanos D. Papaïoannou Nov 27 '09 at 4:31
Thanks, I've changed the title as you suggested. –  Leonid Positselski Nov 27 '09 at 9:56
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6 Answers

up vote 4 down vote accepted

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").
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Thank you very much, these are helpful references. –  Leonid Positselski Nov 28 '09 at 11:25
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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).

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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.

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Thanks, that's definitely better than nothing. But perhaps there are some -- I don't know -- monographs? expository articles? long papers with various definitions being discussed? original articles? –  Leonid Positselski Nov 27 '09 at 14:36
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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.

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Unfortunately, I cannot read German. It's a pity that there is no English translation. –  Leonid Positselski Mar 26 '10 at 16:28
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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.

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Yes, I just wanted to mention the book Enveloping algebras by Dixmer. Rosenberg's old book. "Noncommutative algebraic geometry and representation theory of quantized algebra" also contains the Harish-Chandra modules for one chapter. –  Shizhuo Zhang Dec 24 '09 at 11:52
Could you kindly say a couple of words about what Rosenberg does with Harish-Chandra modules in his book? The book itself unfortunately is not available for online view. –  Leonid Positselski Dec 24 '09 at 14:46
Harish-Chandra modules have by now been studied extensively in a fairly algebraic context by Joseph (in many papers), Jantzen (especially in his enveloping algebra book, written in German), and many others. Even if your motivation comes from unitary representations of real Lie groups, as in work of Vogan and others, you are led to complexify the Lie algebras. The BGG category O theory illuminates mainly the complex Lie group part of the theory, but my book just tells the algebraic story leading up to the use of geometric/homological methods. This is already a long story. –  Jim Humphreys Mar 22 '10 at 13:26
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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.

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It is a little late in the precise sense that the book for which I was collecting these references has been published already in September 2010. But thank you, nevertheless; these may turn out to be helpful references. –  Leonid Positselski Mar 23 '12 at 13:42
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