Verma Module - Homomorphisms of Verma Modules

Homomorphisms of Verma Modules

For any two weights a non-trivial 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.

Each homomorphism of Verma modules is injective and the dimension

for any . So, there exists a nonzero if and only if is isomorphic to a (unique) submodule of .

The full classification of Verma module homomorphisms was done by Bernstein-Gelfand-Gelfand and Verma and can be summed up in the following statement:

There exists a nonzero homomorphism if and only if there exists a sequence of weights

such that for some positive roots (and is the corresponding root reflection and is the sum of all fundamental weights) and for each is a natural number ( is the coroot associated to the root ).

If the Verma modules and are regular, then there exists a unique dominant weight and unique elements w, w′ of the Weyl group W such that

P

and

where is the affine action of the Weyl group. If the weights are further integral, then there exists a nonzero homomorphism

if and only if

in the Bruhat ordering of the Weyl group.