Andersen-Filtrierung und harter Lefschetz

We consider the principal block of category O and its graded version. On the space of homomorphisms from a Verma module to an indecomposable tilting module we may define natural filtrations following Andersen. The arguments given in this article prove that these filtrations are compatible with the graded structure, although explicitely we only show that the dimensions of the sucessive subquotients of the filtration are compatible with this intuition. This statement is very similar to the semisimplicity of the subquotients of the Jantzen filtration proved by Beilinson and Bernstein, but the method of proof is quite different. I would like to know how to directly relate both results, as this would give an alternative proof of said semisimplicity.