Generalized affine buildings for semisimple algebraic groups over real closed fields

TL;DR

This paper constructs affine $ ext{Lambda}$-buildings for semisimple algebraic groups over real closed fields using real algebraic geometry, characterizing their structure and group actions.

Contribution

It introduces a novel construction of affine $ ext{Lambda}$-buildings for these groups and analyzes their geometric and group-theoretic properties.

Findings

Characterization of the spherical building at infinity

Description of the local building at a base point

Computation of stabilizers and group decompositions

Abstract

We use real algebraic geometry to construct an affine $\Lambda$-building $B$ associated to the $\mathbb{F}$-points of a semisimple algebraic group, where $\mathbb{F}$ is a valued real closed field. We characterize the spherical building at infinity and the local building at a base point. We compute stabilizers of various subsets of $B$ and obtain group decompositions.