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”.
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