Semialgebraic groups and generalized affine buildings

Abstract

We develop the theory of algebraic groups over real closed fields and apply the results to construct a geometric object $\mathcal{B}$ and to prove that $\mathcal{B}$ is an affine $Λ$-building. We use a model theoretic transfer principle to prove generalizations of statements about semisimple Lie groups. In this direction we give proofs for the Iwasawa-decomposition $KAU$, the Cartan-decomposition $KAK$ and the Bruhat-decomposition $BWB$. For unipotent subgroups we prove the Baker-Campbell-Hausdorff formula and use it to analyse root groups. We give a proof of the Jacobson-Morozov Lemma about subgroups whose Lie algebra is isomorphic to $\mathfrak{sl}_2$ and we describe other rank 1 subgroups which are the semisimple parts of Levi-subgroups. We prove a semialgebraic version of Kostant's convexity. Over the reals, semisimple Lie groups are closely related to the symmetry groups of symmetric spaces of non-compact type. These symmetric spaces can be described semialgebraically, which allows us to consider their semialgebraic extension over any real closed field. Starting from these non-standard symmetric spaces we use a valuation (with image some non-discrete ordered abelian group $Λ$) on the fields to define a $Λ$-pseudometric. Identifying points of distance zero results in a $Λ$-metric space $\mathcal{B}$. Assuming that the root system of the associated Lie group is reduced, we prove that $\mathcal{B}$ is an affine $Λ$-building. The proof relies on a thorough analysis of stabilizers.