Let $k$ be a number field and $S$ be a finite set of places of $k$. Let $G$ be a connected semisimple algebraic group over $k$. Let $k_S=\prod_{v\in S}k_v$ where $k_v$ is the completion of $k$ at $v$.

Question: Is maximal compact subgroup of $G(k_S)$ unique up to conjugation? If it is not unique, are there finitely many of them up to conjugation?

  • 3
    $\begingroup$ Uniqueness certainly doesn't hold: take $k=\Bbb{Q}$, $S=\{p\}$, $G=SL_2$. Then $G(\Bbb{Q}_p)$ has two conjugacy classes of maximal compacts; you can take as representatives $SL_2(\Bbb{Z}_p)$ and its conjugate under $\begin{pmatrix}0 & 1\\ p & 0\end{pmatrix}$. $\endgroup$ – user19918273 Sep 6 '14 at 19:11

Since the maximal compact subgroup question has a complicated history, and is treated at very different levels of generality in the literature (Bruhat-Tits papers in particular), it may be helpful to fill in Paul's answer a bit. There was early work in special cases by Bruhat over half a century ago, in the aftermath of Chevalley's uniform 1955 construction of split groups over an arbitrary field coming from simple Lie algebras over $\mathbb{C}$. But the clearest picture began to emerge from the important 1965 paper by Iwahori and Matsumoto (see in particular their Prop. 2.32): here.

In this approach and the further work of Bruhat-Tits one considers in particular a simple, simply connected algebraic group $G$ such as $\mathrm{SL}_n$ defined over a complete non-archimedian field $K$, obtaining a $(B,N)$-pair structure and Bruhat decomposition which leads eventually to a determination of the conjugacy classes of maximal compact subgroups of $G(K)$ (which are maximal "parahoric" subgroups): the number of these is $\ell +1$, where $\ell$ is the $K$-rank of $G$. The minimal "parahoric" subgroups are themselves all conjugate. Here the usual Weyl group is expanded to an affine (or extended affine) Weyl group.

The later papers by Bruhat and Tits develop such ideas in vast generality, but as early as 1966 their announcements of results show clearly the direction in which they were going. To state the technical results for $G(k_S)$ in the question here takes some care, but the basic example cited by Paul shows how the $\ell+1$ arises (the rank in his split example being $n-1$).

As Paul indicates, a direct computation can be done in the smallest case $\mathrm{SL}_2$. See for example the end of $\S15$ in my old Springer Lecture Notes 789 on arithmetic groups, where the Bruhat-Tits building appears as simply a tree and Serre's ideas about groups acting on trees can be used.

  • $\begingroup$ P.S. "Paul" here refers to Paul Garrett, who gave a concise but helpful partial answer to the question. $\endgroup$ – Jim Humphreys Sep 11 '14 at 15:20

As @user19918273 noted, uniqueness fails immediately: somewhat more generally, for $SL_n(k_v)$ for non-archimedean $k_v$, there are $n$ conjugacy classes of maximal compacts. However, there is a unique conjugacy class of Iwahori subgroup, and all the choices of maximal compacts are describable in terms of a refined Cartan decomposition relative to an Iwahori: choice of generator to omit from the collection of reflections generating the affine Weyl group corresponds to the choice of maximal compact containing the given Iwahori. These things are not trivial to prove, though it's possible to do so for $SL_2$. Generally, things are kept most orderly by using some parts of the theory of affine buildings.

  • $\begingroup$ Thanks! How about general semisimple group $G$? $\endgroup$ – ronggang Sep 6 '14 at 19:27
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    $\begingroup$ The same thing is true, for almost the same reasons, for unramified, quasi-split classical semi-simple groups. If any of those adjectives fail to apply, then there are complications either in the statement or in the proofs. One difference in general is that, unlike $SL_n$, some maximal compacts are "special/nice/whatever", but not all. In $SL_n$, all are "special", meaning that the Coxeter group obtained from the elements that give the compact from the Iwahori surjects to the spherical Weyl group. In $Sp_4$ (4-by-4) this already fails, so there is a non-special maximal compact. $\endgroup$ – paul garrett Sep 6 '14 at 19:42
  • $\begingroup$ does uniqueness hold for $GL_n(\mathbb{Q}_p)$? $\endgroup$ – Abdelmalek Abdesselam Sep 9 '14 at 14:09
  • $\begingroup$ @AbdelmalekAbdesselam, yes for $GL_n$ instead of $SL_n$, but for most other (all?) reductive groups there is no such adjustment to have uniqueness. $\endgroup$ – paul garrett Sep 9 '14 at 14:49
  • $\begingroup$ @Paul: thanks! I just wanted to make sure. I suppose a proof could go as follows: construct a G invariant ultrametric norm by taking $\max_{g\in G} ||g\cdot||$ where $||\cdot||$ is say the max of the p-adic absolute values of components. Then by maximality G has to be the group which conserves this norm. By an analogue of diagonalization of symmetric real matrices the latter norm has to be $||M\cdot||$ for a suitable invertible matrix $M$. Correct? $\endgroup$ – Abdelmalek Abdesselam Sep 9 '14 at 16:19

If G is a simply connected semisimple group (e.g. ${\rm SL}(N)$, ${\rm Sp}_N$, ${\rm Spin_N}$, ...), then it is a theorem of Bruhat and Tits that there are exactly $l+1$ conjugacy classes of maximal compact subgroups in $G (k_v )$, where $l$ is the rank of $G$ (the dimension of a maximal split torus). If $G$ is not simply connected, this number may be strictly less that $l+1$.

If you are a number theorist a nice summary of the Bruhat-Tits theory may be find in :

Platonov, Vladimir; Rapinchuk, Andrei Algebraic groups and number theory. Translated from the 1991 Russian original by Rachel Rowen. Pure and Applied Mathematics, 139. Academic Press, Inc., Boston, MA, 1994.

  • $\begingroup$ The "connected center" requirement usually doesn't apply to the simply connected semisimple groups you've mentioned, so your formulation should be tightened. (And it isn't clear what "simply connected" should mean for a reductive algebraic group in general.) $\endgroup$ – Jim Humphreys Sep 10 '14 at 12:55
  • $\begingroup$ @Jim Humpreys I modified my answer according to your remarks. $\endgroup$ – Paul Broussous Sep 10 '14 at 13:05
  • $\begingroup$ Your parenthetic statement about the split rank is still out of focus. It's helpful in any case to add a reference to the work of Bruhat-Tits (IHES papers freely available online through numdam). $\endgroup$ – Jim Humphreys Sep 10 '14 at 15:52

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