Compactifications polygonales d'un immeuble affine

Abstract

We define a compactification of an affine building $\I$ indexed by a family of partitions of the director space $\vec A$ of one of its appartments $A$. This compactification is similar to Satake's compatification of a symetric space, and it generalizes the quite well known polygonal compactification of an affine building in the sense that it is independant of the action of a group on the building, and that it allows some variations depending on the choice of the partition of $\vec A$. The different choices will mainly lead to different subgroups of the Weyl group acting on the border of $A$. Along the proofs, we get some results to help one find subsets of the building wich are included in an apartment, for exemple we prove that two sector facets can always be reduced so that they fit in one apartment.