some positive root and the length (see Bruhat ordering) l(w')=l(w)+1, then there exists a nonzero standard homomorphism . Theorem. The standard homomorphism is zero if and only if there exists such that and . However, if is only dominant but not integral, there may still exist -dominant and -integral weights on its affine orbit. The situation is even more complicated if the GVM's have singular character, i.e. there and are on the affine orbit of some such that is on the wall of the fundamental Weyl chamber. Nonstandard A homomorphism is called nonstandard, if it is not standard. It may happen that the standard homomorphism of GVMs is zero but there still exists a nonstandard homomorphism. Bernstein–Gelfand–Gelfand resolution Examples The fields of conformal field theory belong to generalized Verma modules of the conformal algebra. See also Verma module Parabolic geometry External links Code for constructing the BGG resolution of Lie algebra modules and computing its cohomology References Representation theory of Lie algebras