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., inside ) need not be maximal under surjection.

Let be a connected reductive algebraic group over a local non-archimedian field . A subgroup of is called hyperspecial maximal compact subgroup if

- is a maximal compact subgroup of ,
- There is a group scheme such that and is a connected reductive group over , where is a uniformizer in and is the cardinality of the residue field.

Denote by the canonical surjection

(over some local field , or surjection of group schemes, to be pedantic). Choose a non-unit element of . Then

,

but this matrix is in because

.

Hence the image 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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