@YCor Yeah, it was terribly general. I changed it. Feel free to modify it further if you so desire.

Right, I mistakenly equated it with Bruhat decomposition in my mind.

Now I am confused. I thought parahorics are always compact. If B denotes the Iwahori and any parahoric is a finite union on BwB, and B is compact, then so is any parahoric. What am I missing?

I see, so stabilizers of vertices being parahoric are always compact but they may not be maximal compact.

So my statement that "maximal compacts are exactly the same as maximal parahorics" holds true only in simply connected case? So if I understand things correctly, in non simply connected case, compacts are not stabilizers of vertices...

@FriedrichKnop Thanks, for some reason I thought we needed simply connected or some simplicity assumption. But if I understand things correctly, isn't the notion of valued root data only defined for \emph{semisimple} groups as opposed to reductive?

I thought what you described as special is hyperspecial. Am I missing something?