Non-discrete affine buildings and convexity

TL;DR

This paper extends Kostant's convexity theorem to thick non-discrete affine buildings, using metric geometry methods, and explores implications for groups acting on such buildings.

Contribution

It proves a convexity theorem for affine buildings analogous to symmetric spaces, without relying on automorphism groups, and establishes geometric properties of segments and retractions.

Findings

Proves a convexity theorem for affine buildings similar to symmetric spaces.

Shows segments are contained in apartments in affine buildings.

Demonstrates certain retractions onto apartments are distance diminishing.

Abstract

Affine buildings are in a certain sense analogs of symmetric spaces. It is therefore natural to try to find analogs of results for symmetric spaces in the theory of buildings. In this paper we prove a version of Kostant's convexity theorem for thick non-discrete affine buildings. Kostant proves that the image of a certain orbit of a point $x$ in a symmetric space under a projection onto a maximal flat is the convex hull of the Weyl group orbit of $x$. We obtain the same result for a projection of a certain orbit of a point in an affine building to an apartment. The methods we use are mostly borrowed from metric geometry. Our proof makes no appeal to the automorphism group of the building. However the final result has an interesting application for groups acting nicely on non-discrete buildings, such as groups admitting a root datum with non-discrete valuation. Along the proofs we obtain that segments are contained in apartments and that certain retractions onto apartments are distance diminishing.