We present a new version of the path model for representations of a semisimple group G. In the new setting the LS-paths are replaced by certain galleries in the affine Coxeter complex. Using the corresponding affine Tits building, we associate in a canonical way to the galleries some finite dimensional projective varieties in the affine grassmannian associated to G. It turns out that these are precisely the intersection homology cycles investigated by Mirkovic and Vilonen.