In mathematics, generalized Verma modules are a generalization of a (true) Verma module,[1] 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.
we define the generalized Verma module to be the relative tensor product The action of
If λ is the highest weight of V, we sometimes denote the Verma module by
determines a unique grading
It follows from the Poincaré–Birkhoff–Witt theorem that, as a vector space (and even as a
-module), In further text, we will denote a generalized Verma module simply by GVM.
(the set S determines uniquely the subalgebra
It follows from the theory of (true) Verma modules that
is the Borel subalgebra and the GVM coincides with (true) Verma module.
is called regular, if its highest weight λ is on the affine Weyl orbit of a dominant weight
In other word, there exist an element w of the Weyl group W such that where
is the affine action of the Weyl group.
is called singular, if there is no dominant weight on the affine orbit of λ.
is on the wall of the fundamental Weyl chamber (δ is the sum of all fundamental weights).
are linked with an affine action of the Weyl group
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.
is a homomorphism of (true) Verma modules,
and f factors to a homomorphism of generalized Verma modules
Let us suppose that there exists a nontrivial homomorphism of true Verma modules
The following theorem is proved by Lepowsky:[2] The standard homomorphism
The structure of GVMs on the affine orbit of a
and the set of GVM's with highest weights on the affine orbit of
in the Bruhat ordering (otherwise, there is no homomorphism of (true) Verma modules
and the length (see Bruhat ordering) l(w')=l(w)+1, then there exists a nonzero standard homomorphism
The situation is even more complicated if the GVM's have singular character, i.e. there
is on the wall of the fundamental Weyl chamber.