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 modul ar case is the case of a semisimple algebraic group over a field of ch aracteristic p > 0. A natural conjecture is that the tilting modules i n this situation should have the same characters as in the quantum cas e 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 introd uce a ''Jantzen type'' filtration on each such Hem-space and we prove that if these filtrations behave in the expected way with respect to t ranslations through walls then Soergel's theorem and its modular analo gue 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 chara cters of tilting modules here.