Basic Properties
Verma modules, considered as -modules, are highest weight modules, i.e. they are generated by a highest weight vector. This highest weight vector is (the first is the unit in and the second is the unit in the field, considered as the -module ) and it has weight .
Verma modules are weight modules, i.e. is a direct sum of all its weight spaces. Each weight space in is finite dimensional and the dimension of the -weight space is the number of possibilities how to obtain as a sum of positive roots (this is closely related to the so-called Kostant partition function).
Verma modules have a very important property: If is any representation generated by a highest weight vector of weight, there is a surjective -homomorphism That is, all representations with highest weight that are generated by the highest weight vector (so called highest weight modules) are quotients of
contains a unique maximal submodule, and its quotient is the unique (up to isomorphism) irreducible representation with highest weight
The Verma module itself is irreducible if and only if none of the coordinates of in the basis of fundamental weights is from the set .
The Verma module 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).
Read more about this topic: Verma Module
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