I'm trying to understand how to identify groups that come out of manipulating a local Dynkin diagram in a simple way. Ultimately, I want to understand the following, which I add mostly as some form of context, although I'd be very happy to hear of any references that discuss this:
Let $G$ be a linear algebraic group defined over a number field $K$, and let $\upsilon < \infty$ be a finite place of $K$. One can consider the group $G(K_\upsilon)$ of $K_\upsilon$-rational points of $G$ and the parahoric subgroups of $G(K_\upsilon)$. (These are the groups $P_\upsilon \subseteq G(K_\upsilon)$ which contain an Iwahori subgroup, which is essentially the inverse image $\pi^{-1} (B)$ in $G(O_\upsilon)$ of a Borel subgroup $B \subseteq G(k_\upsilon)$, where $k_\upsilon = O_\upsilon / \pi_\upsilon O_\upsilon$ is the residue field of $K$ at $\upsilon$ and $O_\upsilon$ the ring of integers of $K_\upsilon$.) Each such parahoric subgroup $P_\upsilon$ is associated to a (smooth, affine) Bruhat-Tits $O_\upsilon$-group scheme ${\bf G_\upsilon}$ with the property that ${\bf G_\upsilon}(O_\upsilon) \simeq P_\upsilon$. Hence it makes sense to talk about the group $P_\upsilon$ over the finite residue field $k_\upsilon$ in the form of the projection $\pi ( P_\upsilon) = \overline{G}_\upsilon (k_\upsilon)$ onto the base change $\overline{G}_\upsilon = {\bf G_\upsilon} \times_{O_\upsilon} k_\upsilon$ (which is just the "same group as $G$, but with its defining equations reduced from $O_\upsilon$ to $k_\upsilon$"). The group $\overline{G}_\upsilon (k_\upsilon)$ has a Levi decomposition $\overline{G}_\upsilon (k_\upsilon) \simeq M_\upsilon \ltimes U_\upsilon$ where $M_\upsilon$ is reductive (sometimes called the maximal reductive quotient, I believe) and $U_\upsilon$ is the unipotent radical. Now, if $G$ is an inner form of some classical group and you can narrow down the type of $G$ over $K_\upsilon$ (in the form of a (couple of) local Dynkin diagram(s) or a Tits index), how can you write down all the possible candidates for $M_\upsilon$ by looking at these diagrams/indices?
Now, the question: In the paper On Volumes of Arithmetic Quotients of $\mathrm{SO}(1,n)$ by Belolipetsky, the author considers the local Dynkin diagram $\Delta_1$ of type $B_r$
and identifies groups ($M_\upsilon$ in my notation above) based on the Dynkin diagrams that result when certain vertices are removed. For example:
- The diagram $\Delta_1 \setminus \lbrace \alpha_0 \rbrace$ is just the usual Dynkin diagram of type $B_r$, and the associated classical group is $\mathrm{SO}_{2r+1}$. This is also the group which the author identifies. So far so good.
- The diagram $\Delta_1 \setminus \lbrace \alpha_2 \rbrace$ is the disjoint union of two Dynkin diagrams of type $A_1$ and one diagram of type $B_{r-2}$. The group identified from this diagram is $\mathrm{O}_4 \times \mathrm{SO}_{2r-3}$. This seems reasonable since $\mathrm{O}_4$ has type $D_2 \simeq A_1 \times A_1$, and $\mathrm{SO}_{2r-3}$ has type $B_{r-2}$.
- However, the diagram $\Delta_1 \setminus \lbrace \alpha_0, \alpha_1 \rbrace$ is the Dynkin diagram $B_{r-1}$ associated to $\mathrm{SO}_{2r-1}$, but the group that "comes out of this manipulation" is not $\mathrm{SO}_{2r-1}$, but $\mathrm{GL}_1 \times \mathrm{SO}_{2r-1}$.
Where does the $\mathrm{GL}_1$-factor come from? And what is actually going on when you remove vertices (and edges) and identify groups based on the resulting diagrams? How do the "constituent parts" of all of these diagrams and indices relate to properties of the groups they describe?
Part of my problem is probably that I don't see the full picture of how local diagrams or Tits indices encode the group structure. It seems that the canonical reference for all of this is the paper Reductive groups over local fields by Tits (1979) since every paper I've seen that deals with these types of questions and procedures simply references that paper and doesn't offer many other insights into what is going on. Unfortunately, my understanding of these things has not improved at all by reading Tits' paper (even though it is intended for an audience wishing to apply the theory and not necessarily understand it in detail). Is there a good alternative reference (a book, lecture notes, anything) that explains these things in a less dense way? I heard good things about the manuscript Bruhat-Tits theory and buildings by Jiu-Kang Yu, but this seems unavailable online except in an abridged form where a number of things appear to be missing.