- Generalized Verma module
Generalized Verma modules are object in the
representation theory ofLie algebras , a field inmathematics . They were studied originally byJames Lepowsky in seventies. The motivation for their study is that their homomorphisms correspond toinvariant differential operator s overgeneralized flag manifold s. The study of these operators is an important part of the theory of parabolic geometries.Definition
Let be a
semisimple Lie algebra and aparabolic subalgebra of . For any irreducible finite dimensional representation of we define the generalized Verma module to be therelative 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 thePoincaré-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 GVM's
GVM's are
highest weight module s and theirhighest 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 module s, i.e. they are direct sum of itsweight spaces and these weight spaces are finite dimensional.As all
highest weight module s, GVM's are quotients of Verma modules. The kernel of the projection is :where is the set of thosesimple root s α such that the negative root spaces of root are in (the set S determines uniquely the subalgebra ), is theroot reflection with respect to the root α and is theaffine action of on λ. It follows from the theory of (true)Verma module s 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 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 (δ is the sum of allfundamental weight s).Homomorphisms of GVM's
By a homomorphism of GVM's we mean -homomorphism.
For any two weights a
homomorphism :
may exist only if and are linked with an
affine action of theWeyl group of the Lie algebra . This follows easily from theHarish-Chandra theorem oninfinitesimal central character s.Unlike in the case of (true)
Verma module s, the homomorphisms of GVM's are in general not injective and thedimension :
may be larger than one in some specific cases.
If is a homomorphism of (true) 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 (that is a factor of a homomorphism of Verma modules) is called standard. However, the standard homomorphism may be zero in some cases.
tandard
Let us suppose that there exists a nontrivial homomorphism of true Verma moduls .Let be the set of those
simple root s α such that the negative root spaces of root are in (like in section Properties).The following theorem is proved by Lepowsky [Lepowski J., A generalization of the Bernstein-Gelfand-Gelfand resolution, J. Algebra, 49 (1977), 496-511.] :The standard homomorphism is zero if and only if there exists such that is isomorphic to a submodule of ( is the corresponding
root reflection and is theaffine action ).The structure of GVM's on the affine orbit of a -dominant and -integral weight can be described explicitely. 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 theWeyl group of (in particular, does not depend on the choice of ). The map is a bijection between and the set of GVM's with highest weights on theaffine orbit of . Let as suppose that , and in theBruhat ordering (otherwise, there is no homomorphism of (true) Verma modules 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 :
"Theorem." If for some
positive root and the length (seeBruhat ordering ) l(w')=l(w)+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 stil 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 GVM's is zero but there stil exists a nonstandard homomorphism.
Bernstein-Gelfand-Gelfand resolution
Examples
References
ee also
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Verma module
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