BGG algebras and the BGG reciprocity principle

Abstract

The BGG reciprocity principle is a famous result in the representation theary of complex, semisimple Lie algebras proved by Bernstein, Gelfand, and Gelfand in [BGG 21. It may be viewed as an analogue for the category 6 of the-Brauer reciprocity principle in the modular representation theory of finite groups and of Humphreys’ reciprocity principle in the representation theory of restricted Lie algebras. Indeed, the result of Humphreys served as motivation for the BGG result. Similar results have since been proved in a number of other settings, by various approaches. For all these results, one works in a suitable category of modules in which each simple module L has a projective cover P. In addition there is a family of intermediate modules {Mi} such that each P has a filtration with M,“s as quotients. Then the appropriate reciprocity principle states that M, occurs as a filtration quotient in P as often as L occurs as a composition factor in Mi. In the case of the category 0, the simple modules are the simple highest weight modules, the category is introduced as a setting for projective covers to exist, and the intermediate modules are the Verma modules. The proof of BGG reciprocity in [BGG 21 is fairly straightforward, once the basic facts about the category 0 are developed. The crucial point is the construction of enough projective modules in 0. Once these are constructed, the existence of projective covers, the existence of suitable filtrations on these, and the reciprocity principle are ail proved, with reference to the construction, However, another proof of reciprocity exists which is more natural in the sense that it makes no further reference to the construction of projective modules. This proof can be’used in some of the other settings as well. Although in any given setting no drastic simplifications result, I believe it is still worthwhile to draw attention to this more conceptual approach. The properties of the category 0 which are needed for the proof may be