A sum formula for tilting filtrations

In this paper we study tilting modules for reductive algebraic groups in prime characteristics. These modules are characterized by having filtrations both by Weyl modules and by dual Weyl modules. For a given dominant weight there is a Jantzen type filtration for the space of homomorphisms from the Weyl module with that highest weight into a tilting module. We prove a sum formula for these filtrations. A few examples show how this formula sometimes makes it possible to find summands in tilting modules. Our theory also applies in the case of quantum groups at roots of unity.