Here $F$ is a locally compact non-archimedean non-discrete field.

Let $X$ be the reduced (affine) Bruhat-Tits building of ${\rm GL}(n,F)$. Fix a maximal split torus $T$. Let $B$ be a Borel subgroup containing $T$ and write $U$ for the unipotent radical of $B$. Let $A$ be the unique apartment of $X$ stabilized by the normalizer $N_G (T)$ of $T$ in $G$. Finally fix a vertex $x$ of $X$.

It is not difficult to see that there exists a unique point $x_U$ of $A$ such that $x_U =u.x\in A$ for some $u\in U$ (use Iwasawa decomposition). Making $B$ vary among the Borel subgroups containing $T$, one gets $n!$ points $x_U$ in $A$.

My first question is:

What is the link between the points $x_U$ and the projection $x_A$ of $x$ onto the apartment $A$ ($A$ is a closed convex subset of the CAT$(0)$ space $X$ and this projection is well defined) ? (For $n=2$, $x_A$ is the isobarycenter.)

Now fix $B$ and consider $x$ and $u$ as above. Assume that $x\not\in A$. Then $u$ fixes a sector $C$ of $A$. Let $c$ be a point of $C$. Consider the geodesic segments $[c,x]$ and $[c, x_U ]$ (so that $[c,x_U ]=u.[c,x]$). Let $c_0$ be the unique point of $[c,x]$ such that $[c,c_0 ]\subset A$ and $(c_0 ,x]\cap A =\emptyset$.

My second question is:

Is the point $c_0$ close to $x_A$ ? Can one choose $c$ such that $c_0 =x_A$ ?

My third question is :

Do we have $d(x,x_A )=\frac{1}{2}d(x,x_U )$ ?

7more comments