$F$ is a non archimedean field here. To be more precise, I would actually prefer a set of representative in $B(F)$ for the discrete space $B(F) / B(o)Z(F)$?


This can be phrased also as question about lattices in $F^n$, but I would prefer to stay on the group level.



What is a canonical set of representatives in $GL(n,F)$ for the vertices in the Bruhat Tits building?
You somehow want to parametrise the vertices of the building of $G={\rm GL}(n,F)$ :
$$
G/F^\times K = BK/F^\times K= B(F)/B({\mathfrak o})Z(F)
$$
(by Iwasawa decomposition).


For $n=2$ ou can easily find representatives, but for $n>2$, it's going to be tricky!


I just give some hints. Write $N$ for the unipotent radical of $B$ and $T$ for the diagonal torus so that $B=T\ltimes N$.


-- If $n,n'\in N$ and $t, t'\in T$, then if $nt\sim nt'$ mod $B(O)Z(F)$, one has $t\sim t'$ mod $Z(F)T(O)$. So you may assume that $t$ is of the form


$$
t= {\rm diag}(\varpi^{k\_1}, ...,\varpi^{k\_n})
$$
where $(k\_1 ,...,k\_n )$ is well defined modulo the diagonal action of $\mathbb Z$ on ${\mathbb Z}^n$. 


-- You have $nt \sim n't$ mod $Z(F)B(O)$ iff $n\sim n'$ mod $tN(0)t^{-1}$.


So for each $t$ as above, you need to find a system of representatives of
$$
N(F)/tN(O)t^{-1}
$$
For $n=2$, this is easy. For $n>2$, this seems tricky. I've never tried ...


