Local systems of shapovalov elements

Shapovalov introduced elements in the enveloping algebra of a semi-simple Lie algebra that when applied to the highest weight vector of a Verma module generated a Verma submodule. A Shapovalov element induces a containment of Verma modules uniformly across all highest weights in a particular root hyperplane. As a consequence of the Jantzen conjecture, the generator of a Verma submodule is known to lie in a definite level of the Jantzen filtration of the Verma module. Is it possible to choose a Shapovalov element that will place itself naturally into the correct filtration level uniformly across the root hyperplane? This paper considers this problem in the case of sl(n). There are weights, called singular weights, where such Shapovalov elements cannot exist even in a local sense. What then can be said for the non-singular weights? A system of elements is introduced using Carter’s theory of lowering operators ([C]). In the examples, given any non-singular weight, there is an element of this system that provides a local solution to the problem. It is conjectured that this description extends to the general case and characterizes the singular weights.