For now I always worked in the setting of the Bruhat-Tits tree in the $SL(2)$ setting (like in the book of Serre), without any further background about Bruhat-Tits buildings. I would like to adapt some proofs in the unitary group case, let us say the quasi-split unitary group $U(1,1)$ in two variables. So I wonder if there is a more general and handable definition of the Bruhat-Tits tree. I sketch here what I would to to adapt the definition, and will be gald to read any remark or corrections of my misunderstandings:
Let $F$ be a non-archimedean local field. The Bruhat-Tits tree of $U(1,1)(F)$ is a graph made of:
- vertices: the homothetic classes of rank two lattices in $F^2$
- edges: two classes are connected if they have representants such that $\varpi \Lambda \subsetneq \Lambda' \subsetneq \Lambda$ where $\varpi$ is a uniformizer
It is a tree for the same reasons as the $SL(2)$ case, and I believe the vertices are also all of degree $q+1$ where $q$ is the cardinality of the residue field (all that does not depend on the group, and that is what bother me).
Now we can define the standard apartment as the line consisting of the lattices $\Lambda_i = (e_1, \varpi^i e_2)$ where $(e_1, e_2)$ is a basis of $F^2$. I would like to relate every lattice to it, computing its distance. In the case of $GL(2)$, the Iwasawa decomposition implies that every lattive is of the form $\Lambda_{a,s} = (e_1, ae_1 + \varpi^s e_2)$ with $a \in F$ and $s \in \mathbf{Z}$, and the distance to the standard apartment is $-v_F(a)$. I would like to do the same for $U(1,1)(F)$, but do we now such a "standard form" for lattices in this case ?
