TITLE: Why the BGG category O?
QUESTION [27 upvotes]: Given a finite-dimensional semisimple complex Lie algebra $\mathfrak{g}$, the Bernstein-Gelfand-Gelfand category $\mathcal O$ is the full subcategory of $\mathfrak g$-modules satisfying some finiteness conditions. It contains all finite-dimensional modules as well as all highest-weight modules, it's Noetherian and Artinian, and it's Abelian. It's clear to me why you would want to work in some full subcategory of $\mathfrak g$-modules which has the above properties, but why $\mathcal O$? Is it minimal in some sense with respect to these properties and/or some other important properties?

REPLY [17 votes]: It might be worth pointing out a different motivation for Category O, namely the theory of Harish Chandra (g,K) modules. These are algebraic models for continuous representations of real reductive groups (the real forms of g). Harish Chandra's amazing theory reduces many questions in the representation theory of real groups to this algebraic theory, which can then be studied geometrically, for example by Beilinson-Bernstein localization.
In any case Category O is essentially the category of Harish Chandra modules associated to the complex reductive group G, when considered as a real Lie group. There are slight subtleties in what kind of semisimplicity/local finiteness etc we require for the center of the enveloping algebra or the maximal torus, but in broad strokes the two coincide, and category O is thus a nice combinatorially accessible model of a very basic object in representation theory of Lie $groups$.