Notes of Anne Parreau’s Cambridge lecture 10-05-2017

Vectorial metric compactification of symmetric spaces and affine buildings

1. The vectorial metric

Let ${X}$ be a symmetric space or a building. Flats (apartments) are modelled on a Euclidean space ${A}$ equipped with the action of a finite Euclidean Coxeter group ${W}$, the Weyl group. Weyl chambers are sectors forming fundamental domains for ${W}$ in ${A}$. Pick one of them, ${C}$.

Let ${\Theta:A\rightarrow C}$ be the folding projection ${A\rightarrow A/W\sim C}$.

Given two points ${x}$, ${y\in X}$, there is a flat (isometric copy of ${A}$) containing ${x}$ and ${y}$. Put ${x}$ at the origin. Then ${\Theta(y)\in C}$ does not depend on choices, let us denote it by ${\vec d(x,y)}$. This refines the canonical ${CAT(0)}$ metric, given by ${d(x,y)=|\vec d(x,y)|}$, where ${|\cdot|}$ is a ${W}$-invariant Euclidean norm on ${A}$.

${\vec d}$ satisfies a triangle inequality with respect to the following partial order:

$\displaystyle \begin{array}{rcl} u\geq_A 0 \Leftrightarrow \forall\in C,~\langle u|v\rangle \geq 0. \end{array}$

This appears only in a hidden manner in Bruhat-Tits or Kostant, and in the special case of ${Sl(n,{\mathbb R})}$, in a paper by Ky Fan (1950).

The inequality is an equality ${\vec d(x,z)=\vec d(x,y)+\vec d(y,z)}$ iff ${y}$ belongs to the diamond of ${x}$ and ${z}$, i.e. the intersection ${(x+C)\cap (z-C)}$. In other words, iff ${y}$ belongs to a geodesic between ${x}$ and ${z}$ for any Finsler metric whose restriction to ${C}$ is linear.

Theorem 1 The restriction of ${\vec d}$ to pairs of ${CAT(0)}$ 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 ${\vec d(x,\cdot)}$ to embed ${X}$ into the space of functions from ${X}$ to ${A}$, equipped with the topology of uniform convergence on bounded subsets. Divide by the ${A}$-action by translation.

Theorem 2 If ${X}$ is locally compact, this gives an embedding whose closure ${\vec X}$ is compact. It is equivariant under isometries of ${X}$. This compactification dominates all linear Finsler horocompactifications (Kapovich-Leeb-Porti).

The boundary ${\partial \vec X=\vec X \setminus X}$ consists of functions called vectorial horofunctions.

Beware that ${\vec X}$ does not dominate the ${CAT(0)}$ compactification.

3. Facets and transverse spaces at infinity

Facets in ${X}$ are images of walls of ${C}$ 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 ${\sigma}$ be a facet at infinity. The tube at infinity in direction ${\sigma}$ is the set of facets converging to ${\sigma}$. 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 ${\partial \vec X}$. Thus ${\partial \vec X}$ is stratified by subsymmetric spaces. In other words, ${\partial \vec X}$ coincides with the polyhedral or Satake compactification of ${X}$.

For instance, the closure of a flat in ${\partial\vec X}$ has one point for each Weyl chamber, and a whole vectorsubspace for each wall.

Vector horofunctions can be described explicitly.