Generalized Verma module

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 formula_1 be a semisimple Lie algebra and formula_2 a parabolic subalgebra of formula_1. For any irreducible finite-dimensional representation formula_4 of formula_2 we define the generalized Verma module to be the relative tensor product
The action of formula_1 is left multiplication in formula_8.
If λ is the highest weight of V, we sometimes denote the Verma module by formula_9.
Note that formula_9 makes sense only for formula_2-dominant and formula_2-integral weights (see weight) formula_13.
It is well known that a parabolic subalgebra formula_2 of formula_1 determines a unique grading formula_16 so that formula_17.
Let formula_18.
It follows from the Poincaré–Birkhoff–Witt theorem that, as a vector space (and even as a formula_19-module and as a formula_20-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 formula_22 is the highest weight vector in V, then formula_23 is the highest weight vector in formula_9.
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 formula_25 is
where formula_27 is the set of those simple roots α such that the negative root spaces of root formula_28 are in formula_2 (the set S determines uniquely the subalgebra formula_2), formula_31 is the root reflection with respect to the root α and
formula_32 is the affine action of formula_31 on λ. It follows from the theory of (true) Verma modules that formula_34 is isomorphic to a unique submodule of formula_35. In (1), we identified formula_36. The sum in (1) is not direct.
In the special case when formula_37, the parabolic subalgebra formula_2 is the Borel subalgebra and the GVM coincides with (true) Verma module. In the other extremal case when formula_39, formula_40 and the GVM is isomorphic to the inducing representation V.
The GVM formula_9 is called "regular", if its highest weight λ is on the affine Weyl orbit of a dominant weight formula_42. In other word, there exist an element w of the Weyl group W such that
where formula_44 is the affine action of the Weyl group.
The Verma module formula_35 is called "singular", if there is no dominant weight on the affine orbit of λ. In this case, there exists a weight formula_42 so that formula_47 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 formula_1-homomorphism.
For any two weights formula_49 a homomorphism
may exist only if formula_51 and formula_13 are linked with an affine action of the Weyl group formula_53 of the Lie algebra formula_1. 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 formula_56 is a homomorphism of (true) Verma modules, formula_57 resp. formula_58 is the kernels of the projection formula_59, resp. formula_25, then there exists a homomorphism formula_61 and f factors to a homomorphism of generalized Verma modules formula_62. 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 moduls formula_63.
Let formula_27 be the set of those simple roots α such that the negative root spaces of root formula_28 are in formula_2 (like in section Properties).
The following theorem is proved by Lepowsky:
The standard homomorphism formula_62 is zero if and only if there exists formula_68 such that formula_69 is isomorphic to a submodule of formula_70 (formula_31 is the corresponding root reflection and formula_44 is the affine action).
The structure of GVMs on the affine orbit of a formula_1-dominant and formula_1-integral weight formula_42 can be described explicitly. If W is the Weyl group of formula_1, there exists a subset formula_77 of such elements, so that formula_78 is formula_2-dominant. It can be shown that formula_80 where formula_81 is the Weyl group of formula_2 (in particular, formula_83 does not depend on the choice of formula_42). The map formula_85 is a bijection between formula_83 and the set of GVM's with highest weights on the affine orbit of formula_42. Let as suppose that formula_88, formula_43 and formula_90 in the Bruhat ordering (otherwise, there is no homomorphism of (true) Verma modules formula_91 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 formula_83:
"Theorem." If formula_93 for some positive root formula_94 and the length (see Bruhat ordering) l(w')=l(w)+1, then there exists a nonzero standard homomorphism formula_62.
"Theorem". The standard homomorphism formula_62 is zero if and only if there exists formula_97 such that formula_98 and formula_99.
However, if formula_42 is only dominant but not integral, there may still exist formula_2-dominant and formula_2-integral weights on its affine orbit.
The situation is even more complicated if the GVM's have singular character, i.e. there formula_51 and formula_13 are on the affine orbit of some formula_42 such that formula_47 is on the wall of the fundamental Weyl chamber.
Nonstandard.
A homomorphism formula_62 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.