In this post, I will give a counter-example, that the image of a hyperspecial maximal compact subgroup under a surjective map need not (even) be maximal. In particular, this will tell us that a maximal compact subgroup (e.g., $SO(n)$ inside $SL_n(\mathbb R)$) need not be maximal under surjection.

Let $G$ be a connected reductive algebraic group over a local non-archimedian field $F$. A subgroup $K$ of $G(F)$ is called hyperspecial maximal compact subgroup if

• $K$ is a maximal compact subgroup of $G(F)$,
• There is a group scheme $\mathcal G$ such that $\mathcal G(\mathcal O_F)=K$ and $\mathcal G(\mathcal O_F / \varpi \mathcal O_F)$ is a connected reductive group over $\mathbb F_q$, where $\varpi$ is a uniformizer in $\mathcal O_F$ and $q$ is the cardinality of the residue field.

Denote by $\phi$ the canonical surjection

$\phi : SL_2 \to PGL_2 = GL_2 / G_m$

(over some local field $F$, or surjection of group schemes, to be pedantic). Choose a non-unit element $x$ of $\mathcal O_F$. Then

$\begin{pmatrix}x & \\ & x^{-1} \end{pmatrix} \not\in SL_2(\mathcal O_F)$,

but this matrix is in $PGL_2(\mathcal O_F)$ because

$\begin{pmatrix} x & \\ &x^{-1} \end{pmatrix} = \begin{pmatrix} x & \\ & x^{-1} \end{pmatrix} . \begin{pmatrix} x & \\ & x \end{pmatrix} = \begin{pmatrix} x^2 & \\ & 1 \end{pmatrix}$.

Hence the image $\phi(SL_2(O_F) \subsetneq PGL_2(O_F)$ is compact but not maximal compact (so in particular, not hyperspecial).

Remarks:

• A good reference to read about hyperspecial maximal compact subgroups is Tits’ article in Corvallis (Volume 1). Nick and I are planning to start a reading seminar next semester in Spring 14 on the 40-odd page article.
• I found this counter-example somewhere on MathOverflow, while I was preparing the Borel-Tate articles (Corvallis, Volume 2) to speak in our “Arthur seminar”.