The Category O of Bernstein, Gelfand, and Gelfand

One of I.M. Gelfand’s many contributions to mathematics was his
construction and study of “A category of g-modules” (joint with
J. Bernstein and S.I. Gelfand), over a complex simple Lie algebra g.
The BGG Category O has since been widely studied in this and other
settings.
We present a setup in which to study the Cateegory O in many of these
settings, with emphasis on central characters, classification of
primitive ideals, and the finest possible block decomposition into
blocks (with BGG Reciprocity).