Abstract

We prove an analogue of Kostant's convexity theorem for thick affine buildings and give an application for groups with affine BN-pairs. 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 ρ centered at a chamber in the spherical building at infinity. We prove that for each special vertex x ∈ A the set ρ(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.