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 reductive Lie groups and to spherical Hecke rings associated with split nonarchimedean reductive Lie groups. Our main application is a generalization of the saturation theorem of Knutson and Tao for SLn to other complex semisimple Lie groups.
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Michael Kapovich, John J. Millson, A path model for geodesics in Euclidean buildings and its applications to representation theory. Groups Geom. Dyn. 2 (2008), no. 3, pp. 405–480DOI 10.4171/GGD/46