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In this paper we give a combinatorial characterization of projections ofgeodesics in Euclidean buildings to Weyl chambers. We apply these results tothe representation theory of complex semisimple Lie groups and to sphericalHecke rings associated with nonarchimedean reductive Lie groups. Our mainapplication is a generalization of the saturation theorem of Knutson and Taofor SL(n) to other complex semisimple Lie groups.

A path model for geodesics in Euclidean buildings and its applications to representation theory