Generalized Verma module


In mathematics, generalized Verma modules are a generalization of a 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 .
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,
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 , is the root reflection with respect to the root α and
is the affine action of on λ. It follows from the theory of Verma modules that is isomorphic to a unique submodule of. In, we identified. The sum in is not direct.
In the special case when, the parabolic subalgebra is the Borel subalgebra and the GVM coincides with 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 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.

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 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 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 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 moduls.
Let be the set of those simple roots α such that the negative root spaces of root are in .
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 .

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 . 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.
The following statements follow from the above theorem and the structure of :

Theorem. If for some positive root and the length l=l+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