Difference in Bruhat-Tits Building of $PGL_n$ and $SL_n$

Question

I have a basic question/ clarification required about Bruhat-Tits buildings of the groups of $GL_n$, $PGL_n$ and $SL_n$ over say a finite extension of $\mathbb{Q}_p$. I have seen in different documents the groups $PGL_n$ and $SL_n$ being used interchangeably to define the same Bruhat-Tits building, using lattice chains for example (Can't think of good references off the top of my head, but I can try to have exact references if required). But I have also heard people say that there are subtle differences between these. Can someone explain or point out a reference where the examples of these groups and their differences (if any) are discussed?

Answer 1

The groups $G={\rm PGL}_n (F)$ and $H={\rm SL}_n (F)$ ($F$ a locally compact non archimedean field) share the same Bruhat-Tits building $X$. The building $X$ is a $(n-1)$ dimensional simplicial complex on which $G$ and $H$ act via simplicial automorphisms.

As any building, $X$ is labelable. This means that there exists a map $l$ : $X^0 \longrightarrow \{ 0,1,2,...,n-1\}$ (where $X^0$ is the set of vertices in $X$) such that the restriction of $l$ to any simplex of $X$ is injective.

The main difference between $G$ and $H$ is that if $H$ acts on $X$ by preserving the labelling, $G$ does not. In fact, if $v_F$ denotes the normalized valuation of $F$, an element $g\in G$ preserves the labelling of $X$ iff $n$ divides $v_F ({\rm det}(g))$.