# What is “special” maximal compact subgroup of algebraig group over local field?

Learning the theory of Langlands correspondence, I met the notion of "special" maximal compact subgroup of a (reductive) algebraic group over a local field.

Here, I think the word "compact" is used in analytic meaning.

The textbook I used did not explain the definition of "special".

My questions:

(1) What is the definition of "special" maximal compact subgroup?

(2) Is there any concrete example of maximal compact subgroup which is NOT "special"?

Please give me any advice.

## 1 Answer

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.

• I thought what you described as special is hyperspecial. Am I missing something? – Vishal Gupta Feb 29 '16 at 10:19
• @VishalGupta, since the examples given are split, there is no difference between special and hyperspecial. – LSpice Sep 10 '18 at 15:40
• Also, doesn't 'special' usually refer only to the parahoric, not to the potentially larger full point stabiliser? (Or do they always coincide for specials? Certainly they coincide in the split case, but I'm not so sure about unitary groups ....) – LSpice Sep 10 '18 at 15:41