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.

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About metric2011

metric2011 is a program of Centre Emile Borel, an activity of Institut Henri Poincaré, 11 rue Pierre et Marie Curie, 75005 Paris, France. See http://www.math.ens.fr/metric2011/
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