In this paper we study tilting modules for reductive algebraic groups in pr
ime 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 sometim
es makes it possible to find summands in tilting modules. Our theory also a
pplies in the case of quantum groups at roots of unity. (C) 2000 Elsevier S
cience B.V. All rights reserved.