In mathematics, generalized Verma modules are a generalization of a (true) Verma module, and are objects in the representation theory of Lie algebras. They were studied originally by James Lepowsky in the 1970s. The motivation for their study is that their homomorphisms correspond to invariant differential operators over generalized flag manifolds. The study of these operators is an important part of the theory of parabolic geometries. Definition Let be a semisimple Lie algebra and a parabolic subalgebra of . For any irreducible finite-dimensional representation of we define the generalized Verma module to be the relative tensor product . The action of is left multiplication in . If λ is the highest weight of V, we sometimes denote the Verma module by . Note that makes sense only for -dominant and -integral weights (see weight) . It is well known that a parabolic subalgebra of determines a unique grading so that . Let . It follows from the Poincaré–Birkhoff–Witt theorem that, as a vector space (and even as a -module and as a -module), . In further text, we will denote a generalized Verma module simply by GVM. Properties of GVMs GVM's are highest weight modules and their highest weight λ is the highest weight of the representation V. If is the highest weight vector in V, then is the highest weight vector in . GVM's are weight modules, i.e. they are direct sum of its weight spaces and these weight spaces are finite-dimensional. As all highest weight modules, GVM's are quotients of Verma modules. The kernel of the projection is where is the set of those simple roots α such that the negative root spaces of root are in (the set S determines uniquely the subalgebra ), is the root reflection with respect to the root α and is the affine action of on λ. It follows from the theory of (true) Verma modules that is isomorphic to a unique submodule of . In (1), we identified . The sum in (1) is not direct. In the special case when , the parabolic subalgebra is the Borel subalgebra and the GVM coincides with (true) Verma module. In the other extremal case when , and the GVM is isomorphic to the inducing representation V. The GVM is called regular, if its highest weight λ is on the affine Weyl orbit of a dominant weight . In other word, there exist an