element w of the Weyl group W such that where is the affine action of the Weyl group. The Verma module is called singular, if there is no dominant weight on the affine orbit of λ. In this case, there exists a weight so that is on the wall of the fundamental Weyl chamber (δ is the sum of all fundamental weights). Homomorphisms of GVMs By a homomorphism of GVMs we mean -homomorphism. For any two weights a homomorphism may exist only if and are linked with an affine action of the Weyl group of the Lie algebra . This follows easily from the Harish-Chandra theorem on infinitesimal central characters. Unlike in the case of (true) Verma modules, the homomorphisms of GVM's are in general not injective and the dimension may be larger than one in some specific cases. If is a homomorphism of (true) Verma modules, resp. is the kernels of the projection , resp. , then there exists a homomorphism and f factors to a homomorphism of generalized Verma modules . Such a homomorphism (that is a factor of a homomorphism of Verma modules) is called standard. However, the standard homomorphism may be zero in some cases. Standard Let us suppose that there exists a nontrivial homomorphism of true Verma modules . Let be the set of those simple roots α such that the negative root spaces of root are in (like in section Properties). The following theorem is proved by Lepowsky: The standard homomorphism is zero if and only if there exists such that is isomorphic to a submodule of ( is the corresponding root reflection and is the affine action). The structure of GVMs on the affine orbit of a -dominant and -integral weight can be described explicitly. If W is the Weyl group of , there exists a subset of such elements, so that is -dominant. It can be shown that where is the Weyl group of (in particular, does not depend on the choice of ). The map is a bijection between and the set of GVM's with highest weights on the affine orbit of . Let as suppose that , and in the Bruhat ordering (otherwise, there is no homomorphism of (true) Verma modules and the standard homomorphism does not make sense, see Homomorphisms of Verma modules). The following statements follow from the above theorem and the structure of : Theorem. If for