Kostant Convexity for affine buildings

TL;DR

This paper extends Kostant's convexity theorem to thick affine buildings, demonstrating a convex hull property for certain retractions and applying it to groups with affine BN-pairs, using Coxeter complexes and representation theory.

Contribution

It introduces an affine building analogue of Kostant's convexity theorem, linking geometric retractions to convex hulls and representation theory.

Findings

Proves a convexity property for affine buildings analogous to Kostant's theorem.

Establishes a convex hull description for retracted sets in affine buildings.

Connects the geometric structure of affine buildings with algebraic group representations.

Abstract

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.