A sort-of explanation, at least to connect to some little examples: a subgroup of a reductive $p$-adic group $G$ is "special" or "good" if it fixes a "special/good" *vertex* in the affine building. A vertex is special/good when it is special/good in every *apartment* in which it lies, in the building (in the unique maximal apartment system). Each apartment is an affine Coxeter complex attached to Coxeter group $(G,S)$ with generators $S$. A vertex in an affine Coxeter complex is special when its stabilizer *surjects* to the "linear parts" quotient of the affine Coxeter group by its *translations*. That is, the fixer of that point surjects to the spherical Weyl group of $G$.

Examples: all maximal compacts in $SL_n(\mathbb Q_p)$ are special/good, because all vertices look the same. E.g., $SL_3(\mathbb Q_p)$'s apartments are simplicial complexes that look like the equilateral-triangle tesselation of $\mathbb R^2$. But for $Sp_4(\mathbb Q_p)$ (four-by-four matrices) there are two different types of vertices: the apartments look like planes tesselated by squares with *diagonals* added touching vertices $(m,n)$ with $m,n$ of the same parity (is one way to describe it in words...). Thus, half the vertices will have the vertical and horizontal $1$-simplices *and* four diagonals connected to them, while the other half of the vertices have only the four vertical and horizontal simplices connected to them. The former are special, the latter are *not* special.