Let g be a semisimple Lie algebra and p a parabolic subalgebra of g . For any irreducible finite-dimensional representation V of p we define the generalized Verma module to be the relative tensor product
M p ( V ) := U ( g ) ⊗ U ( p ) V .
The action of g is left multiplication in U ( g ) .
If λ is the highest weight of V, we sometimes denote the Verma module by M p ( λ ) .
Note that M p ( λ ) makes sense only for p -dominant and p -integral weights (see weight) λ .
It is well known that a parabolic subalgebra p of g determines a unique grading g = ⊕ j = − k k g j so that p = ⊕ j ≥ 0 g j . Let g − := ⊕ j < 0 g j . It follows from the Poincaré–Birkhoff–Witt theorem that, as a vector space (and even as a g − -module and as a g 0 -module),
M p ( V ) ≃ U ( g − ) ⊗ V .
In further text, we will denote a generalized Verma module simply by GVM.
GVM's are highest weight modules and their highest weight λ is the highest weight of the representation V. If v λ is the highest weight vector in V, then 1 ⊗ v λ is the highest weight vector in M p ( λ ) .
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 M λ → M p ( λ ) is
( 1 ) K λ := ∑ α ∈ S M s α ⋅ λ ⊂ M λ where S ⊂ Δ is the set of those simple roots α such that the negative root spaces of root − α are in p (the set S determines uniquely the subalgebra p ), s α is the root reflection with respect to the root α and s α ⋅ λ is the affine action of s α on λ. It follows from the theory of (true) Verma modules that M s α ⋅ λ is isomorphic to a unique submodule of M λ . In (1), we identified M s α ⋅ λ ⊂ M λ . The sum in (1) is not direct.
In the special case when S = ∅ , the parabolic subalgebra p is the Borel subalgebra and the GVM coincides with (true) Verma module. In the other extremal case when S = Δ , p = g and the GVM is isomorphic to the inducing representation V.
The GVM M p ( λ ) is called regular, if its highest weight λ is on the affine Weyl orbit of a dominant weight λ ~ . In other word, there exist an element w of the Weyl group W such that
λ = w ⋅ λ ~ where ⋅ is the affine action of the Weyl group.
The Verma module M λ is called singular, if there is no dominant weight on the affine orbit of λ. In this case, there exists a weight λ ~ so that λ ~ + δ is on the wall of the fundamental Weyl chamber (δ is the sum of all fundamental weights).
By a homomorphism of GVMs we mean g -homomorphism.
For any two weights λ , μ a homomorphism
M p ( μ ) → M p ( λ ) may exist only if μ and λ are linked with an affine action of the Weyl group W of the Lie algebra g . 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
d i m ( H o m ( M p ( μ ) , M p ( λ ) ) ) may be larger than one in some specific cases.
If f : M μ → M λ is a homomorphism of (true) Verma modules, K μ resp. K λ is the kernels of the projection M μ → M p ( μ ) , resp. M λ → M p ( λ ) , then there exists a homomorphism K μ → K λ and f factors to a homomorphism of generalized Verma modules M p ( μ ) → M p ( λ ) . 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 M μ → M λ . Let S ⊂ Δ be the set of those simple roots α such that the negative root spaces of root − α are in p (like in section Properties). The following theorem is proved by Lepowsky:
The standard homomorphism M p ( μ ) → M p ( λ ) is zero if and only if there exists α ∈ S such that M μ is isomorphic to a submodule of M s α ⋅ λ ( s α is the corresponding root reflection and ⋅ is the affine action).
The structure of GVMs on the affine orbit of a g -dominant and g -integral weight λ ~ can be described explicitly. If W is the Weyl group of g , there exists a subset W p ⊂ W of such elements, so that w ∈ W p ⇔ w ( λ ~ ) is p -dominant. It can be shown that W p ≃ W p ∖ W where W p is the Weyl group of p (in particular, W p does not depend on the choice of λ ~ ). The map w ∈ W p ↦ M p ( w ⋅ λ ~ ) is a bijection between W p and the set of GVM's with highest weights on the affine orbit of λ ~ . Let as suppose that μ = w ′ ⋅ λ ~ , λ = w ⋅ λ ~ and w ≤ w ′ in the Bruhat ordering (otherwise, there is no homomorphism of (true) Verma modules M μ → M λ 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 W p :
Theorem. If w ′ = s γ w for some positive root γ and the length (see Bruhat ordering) l(w')=l(w)+1, then there exists a nonzero standard homomorphism M p ( μ ) → M p ( λ ) .
Theorem. The standard homomorphism M p ( μ ) → M p ( λ ) is zero if and only if there exists w ″ ∈ W such that w ≤ w ″ ≤ w ′ and w ″ ∉ W p .
However, if λ ~ is only dominant but not integral, there may still exist p -dominant and p -integral weights on its affine orbit.
The situation is even more complicated if the GVM's have singular character, i.e. there μ and λ are on the affine orbit of some λ ~ such that λ ~ + δ is on the wall of the fundamental Weyl chamber.
Nonstandard
A homomorphism M p ( μ ) → M p ( λ ) 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.