Kostant Convexity for affine buildings

We prove an analogue of Kostants convexity theorem for thick affine buildings and give an application for groups with affine BN-pair. Recall that there are two natural retractions of the affine building onto a fixed apartment A: The retraction r centered at an alcove in A and the retraction $\rho$ centered at a chamber in the spherical building at infinity. We prove that for each special vertex x in A the set $\rho(r^{-1}(W.x))$ is a certain convex hull of W.x. The proof can be reduced to a statement about Coxeter complexes and heavily relies on a character formula for highest weight representations of algebraic groups.