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.
- What are the standard or classical references for the terms "Harish-Chandra pair" and "Harish-Chandra module"?
- The same question for algebraic Harish-Chandra pairs and modules (with an algebraic or proalgebraic subgroup).
- 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"?
- 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?