A structure theorem for euclidean buildings

Abstract

We prove an affine analog of Scharlau’s reduction theorem for spherical buildings. To be a bit more precise let X be a euclidean building with spherical building $$\partial X$$ at infinity. Then there exists a euclidean building $$\bar{X}$$ such that X splits as a product of $$\bar{X}$$ with some euclidean k-space such that $$\partial \bar{X}$$ is the thick reduction of $$\partial X$$ in the sense of Scharlau. In addition we prove a converse statement saying that an embedding of a thick spherical building at infinity extends to an embedding of the euclidean building having the extended spherical building as its boundary.