  In this paper we give a combinatorial characterization of projections of
geodesics in Euclidean buildings to Weyl chambers. We apply these results to
the representation theory of complex semisimple Lie groups and to spherical
Hecke rings associated with nonarchimedean reductive Lie groups. Our main
application is a generalization of the saturation theorem of Knutson and Tao
for SL(n) to other complex semisimple Lie groups.
