Vectorial metric compactification of symmetric spaces and affine buildings
1. The vectorial metric
Let be a symmetric space or a building. Flats (apartments) are modelled on a Euclidean space equipped with the action of a finite Euclidean Coxeter group , the Weyl group. Weyl chambers are sectors forming fundamental domains for in . Pick one of them, .
Let be the folding projection .
Given two points , , there is a flat (isometric copy of ) containing and . Put at the origin. Then does not depend on choices, let us denote it by . This refines the canonical metric, given by , where is a -invariant Euclidean norm on .
satisfies a triangle inequality with respect to the following partial order:
This appears only in a hidden manner in Bruhat-Tits or Kostant, and in the special case of , in a paper by Ky Fan (1950).
The inequality is an equality iff belongs to the diamond of and , i.e. the intersection . In other words, iff belongs to a geodesic between and for any Finsler metric whose restriction to is linear.
Theorem 1 The restriction of to pairs of geodesics is convex
This appears hidden in a paper by Kostant.
2. Vectorial horo-compactification
The idea is to modify the classical horo-bordification of metric spaces. Use to embed into the space of functions from to , equipped with the topology of uniform convergence on bounded subsets. Divide by the -action by translation.
Theorem 2 If is locally compact, this gives an embedding whose closure is compact. It is equivariant under isometries of . This compactification dominates all linear Finsler horocompactifications (Kapovich-Leeb-Porti).
The boundary consists of functions called vectorial horofunctions.
Beware that does not dominate the compactification.
3. Facets and transverse spaces at infinity
Facets in are images of walls of in flats. Say two facets are asymptotic if they stay at bounded distance from each other. They are strictly asymptotic if their distance tends to 0 far from walls. Facets at infinity are strict asymptoticity equivalence classes of facets, viewed as subset of the visual compactification. They form the simplices of a simplicial complex structure on the visual compactification (Tits boundary).
Let be a facet at infinity. The tube at infinity in direction is the set of facets converging to . It is the product of a Euclidean space (the linear span of a facet) and a subsymmetric space.
Theorem 3 There is a 1-1 correspondence between strong asymptoticity classes of facets and points of the boundary . Thus is stratified by subsymmetric spaces. In other words, coincides with the polyhedral or Satake compactification of .
For instance, the closure of a flat in has one point for each Weyl chamber, and a whole vectorsubspace for each wall.
Vector horofunctions can be described explicitly.