A tropical view on Bruhat-Tits buildings and their compactifications

TL;DR

This paper explores the connection between Bruhat-Tits buildings, their compactifications, and tropical geometry, revealing how algebraic representations influence the structure and stabilization properties within these geometric objects.

Contribution

It introduces a novel link between tropical linear algebra and the structure of Bruhat-Tits buildings and their compactifications derived from algebraic representations.

Findings

The compactification fan corresponds to the weight polytope of the representation.

Stabilizers of points are described by tropical linear algebra.

The compactification relates to the tropicalization of character hypersurfaces.

Abstract

We relate some features of Bruhat-Tits buildings and their compactifications to tropical geometry. If G is a semisimple group over a suitable non-Archimedean field, the stabilizers of points in the Bruhat-Tits building of G and in some of its compactifications are described by tropical linear algebra. The compactifications we consider arise from algebraic representations of G. We show that the fan which is used to compactify an apartment in this theory is given by the weight polytope of the representation and that it is related to the tropicalization of the hypersurface given by the character of the representation.