Is this the CAT(0) metric on an affine building?

Question

Let $R$ be a discrete valuation ring qith quotient field $Q$ and let $t\in R$ be a generator of the unique maximal ideal in $R$. Let $V$ be a finite-dimensional $Q$-vector space. Then one can consider the set of all homothety classes of $R$-lattices in $Q$ (i.e. finitely generated $R$-submodules of the same rank).

The affine building is the simplicial complex whose vertices are homothety classes of lattices (meaning $L\sim \lambda L$ for $\lambda\in Q\setminus \{0\}$) and a set $\{[L_0],\ldots ,[L_m]\}$ of homothety classes spans a simplex if there are represenatatives $L_0,\ldots,L_n$ with $L_0\subset L_1\subset \ldots \subset L_m\subset t^{-1}L_0$.\

I read several times that there is a Euclidean structure on this building that turns it into a CAT(0)-space but I never read explicitly what the metric on the simplex is.

In general for any maximal simplex (aka chamber) one can find a basis $e_1,\ldots,e_n$ of $V$ such that $e_1,\ldots,e_i, te_{i+1},\ldots te_n$ is an $R$-basis for $L_i$ (for $i=0,\ldots, n-1$).

So the question is: what is the metric on this simplex. My guess is that the lattice $L_i$ corresponds to the point $(0,\ldots,0,1,\ldots,1)\in \mathbb{R}^n$ (with $i$ zeros) and that its homothety class corresponds to the orthogonal projection of this point onto the plane $\{x\in \mathbb{R}^n| \sum_i x_i=0\}$.

Then we can take the convex hull of those $n$- points in that plane to get an Euclidean simplex which gives the building the structure of an Euclidean simplicial complex.

However I could not verify that it is a CAT(0) space. One would have to show that the links are CAT(1) (with the induced spherical metric) and I do not have a clue how this works. However I had the impression that this is rather well known and I just don't see it.

Answer 1

The important thing seems to be that one needs to understand the connection between the CAT(0)-realization of the Coxeter complex that corresponds to the Weyl group and the description of an apartment given in terms of lattices. Since a geometric realization of this Coxeter complex will basically describe a geometric realization of one of the apartments in the building. By theorem 11.16 in the book by Abramenko and Brown (see Stefan Witzel's comment) the euclidean metric on one apartment determines the CAT(0)-metric on the entire building. So it is enough to understand the metric on one apartment.

Apartments correspond to ordered bases and, as you've said, the fundamental domain of the Weyl group action is the chamber (i.e. maximal simplex) with vertices $[[e_1, ..., e_i, te_{i+1}, ..., t e_n]]$, let's call it $c_0$.

One can realize the Weyl group as a euclidean reflection group. See Abramenko-Brown chapter 2.5.1. for a construction of the canonical linear representation. In addition, the corollary to proposition 5 in Bourbaki (Lie groups and Lie-algebras chapters 4-6) VI §2 chapter 2 gives explicit coordinates of the vertices of the fundamental chamber (called 'alcove' in Bourbaki, a 'chamber' is something else there) in terms of root systems. I.e. the chamber is described as a certain subset/simplex in some $R^n$.

In order to obtain the "standard-CAT(0)-metric" on the apartment defined by ${e_1, ... e_n}$ equip the simplex $c_0$ with the metric the fundamental chamber given in Bourbaki has. It should then be possible to check whether this is the same metric you've suggested above.

The following might be helpful, too: Davis, "Buildings are CAT(0)" http://www.math.osu.edu/~mdavis/buildings.pdf

Hope this makes sense and helps.

Answer 2

You might also want to look at chapters 10 and 11 in the book by Abramenko and Brown. In particular Theorem 11.16.

Answer 3

See this paper. In particular, the second paragraph on page 3.