Generalized Verma module

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.

Definition

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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 (see weight)  .

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 (and even as a  -module and as a  -module),

 .

In further text, we will denote a generalized Verma module simply by GVM.

Properties of GVMs

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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   (the set S determines uniquely the subalgebra  ),   is the root reflection with respect to the root α and   is the affine action of   on λ. It follows from the theory of (true) Verma modules 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 all fundamental weights).

Homomorphisms of GVMs

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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 (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   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.

Standard

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Let us suppose that there exists a nontrivial homomorphism of true Verma modules  . Let   be the set of those simple roots α such that the negative root spaces of root   are in   (like in section Properties). The following theorem is proved by Lepowsky:[2]

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 the affine action).

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   (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 the affine orbit of  . Let as suppose that  ,   and   in the Bruhat 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 (see Bruhat 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 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

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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

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Examples

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See also

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References

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  1. ^ Named after Daya-Nand Verma.
  2. ^ Lepowsky J., A generalization of the Bernstein-Gelfand-Gelfand resolution, J. Algebra, 49 (1977), 496-511.
  3. ^ Penedones, João; Trevisani, Emilio; Yamazaki, Masahito (2016). "Recursion relations for conformal blocks". Journal of High Energy Physics. 2016 (9). doi:10.1007/JHEP09(2016)070. hdl:11449/173478. ISSN 1029-8479.