Filtrations and tilting modules

In this paper we consider the modular analogue of a recent theorem by Soergel on tilting modules for quantum groups at roots of 1. The modular case is the case of a semisimple algebraic group over a field of characteristic p > 0. A natural conjecture is that the tilting modules in this situation should have the same characters as in the quantum case as long as the highest weights belong to the lowest p(2)-alcove. The character of a tilting module Q (modular or quantized) is determined by the spaces of homomorphisms from the Weyl modulus into Q. We introduce a ''Jantzen type'' filtration on each such Hem-space and we prove that if these filtrations behave in the expected way with respect to translations through walls then Soergel's theorem and its modular analogue follow. Our filtrations also exist outside the lowest p(2)-alcove but it is still a wide open problem to find a conjecture for the characters of tilting modules here.