# Is every compact simply-connected reductive p-adic group perfect?

Let $$k$$ be a nonarchimedean local field and $$G$$ a reductive $$k$$-group, which we assume to be semisimple and simply-connected. Recall that an abstract group $$H$$ is perfect if it is generated by commutators, that is, equals its derived subgroup.

Question: Is $$G(k)$$ perfect?

When $$G$$ is isotropic, $$G(k)$$ is known to be perfect. This is a consequence of the solution to the Kneser–Tits problem for $$k$$: the group $$G(k)^+$$ generated by (the rational points of) additive subgroups of $$G$$ is the same as the group generated by root subgroups of $$G$$ for a fixed maximal split torus, and the latter group is generated by commutators. It is known that $$G(k)=G(k)^+$$ when $$G$$ is simply-connected.

So the main thrust of my question is when $$G$$ is anistropic, or equivalently (hence the title), when $$G(k)$$ is compact in the analytic topology. In type $$A$$, Platonov and Jančevskiĭ have proved (On a conjecture of Harder) by computations with division rings that $$G(k)$$ is perfect. (Clarification: This result is for $$D^\times$$, not $$\text{SL}_1(D)$$.)

Related question: Are there any anisotropic simple $$k$$-groups not of type $$A$$?

I realize that if the answer to the related question is negative then this work of Platonov and Jančevskiĭ answers my first question. I once looked through the tables in Tits's Boulder notes Classification of semisimple algebraic groups and it seemed like there were compact simple reductive groups only in type $$A$$. But this result was so surprising, and the tables so hard for me to follow, that I was not confident in the correctness of my understanding.

Bonus points for questions that work with a more general $$k$$, say, a complete discretely valued field with residue field of dimension $$\leq1$$.

Clarification: I was too blithe about what happens in type $$A$$. As krl pointed out in the comments, $$\operatorname{SL}_1(D)$$ is not perfect, and in fact, its commutator subgroup is its pro-unipotent radical. Riehm's paper "The norm 1 group of a $$\mathfrak P$$-adic division algebra" carefully analyzes the situation. So in the end, the answer to my question is "no" overall, but "yes" in many cases.

• The answer to the Related question is "no", there are no other such groups, but I know of no better reference than Tits's Boulder notes. Jul 5 at 14:40
• That's really surprising: there are so many compact reductive groups over the reals. It would be nice to have a simpler explanation of this fact than a case-by-case analysis. Anyway, your comment completely resolves my question, so if you write it as an answer then I'll select it. Jul 5 at 16:12
• The surprise works the other way for me: I'm always amazed by the abundance of compact groups in the real case! Jul 5 at 16:54
• Maybe I'm missing something: isn't the answer to the question "no" in general? Can't one calculate that $[\textrm{SL}_1(D),\textrm{SL}_1(D)] \neq \textrm{SL}_1(D)$ (cf. Platonov-Rapinchuk, Thm. 1.9) ?
– krl
Jul 6 at 2:05
• Yes, that makes sense -- I appreciate your generosity. @krl, I will accept your comment as an answer if you post it. Jul 7 at 15:51

It seems that the answer to the Question is "no" in general (in the anisotropic case): for example, if $$D$$ is a finite-dimensional central division algebra over $$k$$, then Theorem 1.9 in Platonov-Rapinchuk (based on the article of Riehm quoted above) implies that $$[\textrm{SL}_1(D), \textrm{SL}_1(D)] \neq \textrm{SL}_1(D).$$
At your request, I post my comment as an answer: the answer to the Related question is "no", i.e., all simple, anisotropic groups over a non-Archimedean local field are of type $$\mathsf A$$; but I know of no better reference than Tits's Boulder notes Classification of algebraic semisimple groups.