Constructing homomorphisms between Verma modules

We describe a practical method for constructing a nontrivial homomorphism between two Verma modules of an arbitrary semisimple Lie algebra. With some additions the method generalises to the affine case. A theorem of Verma, Bernstein-Gel’fand-Gel’fand gives a straightforward criterion for the existence of a nontrivial homomorphism between Verma modules. Moreover, the theorem states that such homomorphisms are always injective. In this paper we consider the problem of explicitly constructing such a homomorphism if it exists. This boils down to constructing a certain element in the universal enveloping algebra of the negative part of the semismiple Lie algebra. There are several methods known to solve this problem. Firstly, one can try and find explicit formulas. In this approach one fixes the type (but not the rank). This has been carried out for type An in [18], Section 5, and for the similar problem in the quantum group case in [4], [5], [6]. In [4] root systems of all types are considered, and the solution is given relative to so-called straight roots, using a special basis of the universal enveloping algebra (not of Poincaré-Birkhoff-Witt type). In [5], [6] the solution is given for types An and Dn for all roots, in a Poincaré-Birkhoff-Witt basis. Our approach compares to this in the sense that we have an algorithm that, given any root of a fixed root system, computes a general formula relative to any given Poincaré-Birkhoff-Witt basis (see Section 3). A second approach is described in [18], which gives a general construction of homomorphisms between Verma modules. However, it is not easy to see how to carry out this construction in practice. The method described here is a variant of the construction in [18], the difference being that we are able to obtain the homomorphism explicitly. In Section 1 of this paper we review the theoretical concepts and notation that we use, and describe the problem we deal with. In Section 2 we derive a few commutation formulas in the field of fractions of U(n). Then in Section 3 the construction of a homomorphism between Verma modules is described. In Section 4 we briefly comment on the problem of finding compositions of inclusions. In Section 5 we comment on the analogous problem for