Let $G$ be the $k$-points of a connected, reductive group $\mathbf G$ over a local field $k$. I have heard several statements about compact subgroups and Iwasawa decomposition, mostly in the context of semisimple real lie groups. I wanted to know what was true for reductive groups. So my question is, which of the following six statements are true? (one is a definition)

1 . There exist finitely many conjugacy classes of maximal compact subgroups of $G$. If $k = \mathbf R$, then all maximal compact subgroups are conjugate.

2 . If $k$ is nonarchimedean, then maximal compact subgroups are always open.

Let $K$ be a compact subgroup of $G$, and $P$ a parabolic subgroup of $G$. Suppose that the product mapping $P \times K \rightarrow G$ is a homeomorphism.

3 . If $P, K$ are as above, $G = PK$ is called an Iwasawa decomposition.

If my definition of Iwasawa decomposition is incorrect, please keep in mind my meaning of the term in the following questions.

4 . If $P$ is a parabolic subgroup of $G$, then there exists a compact subgroup $K$ such that $PK$ is an Iwasawa decomposition.

5 . If $P_0$ is a minimal parabolic, there exists a maximal compact subgroup $K$ such that $P_0K$ is an Iwasawa decomposition.

Let $P_0$ be a minimal parabolic of $G$, let $\mathbf A_0$ be a maximal $k$-split torus of $\mathbf G$ with $\mathbf A_0(k) \subseteq P_0$, and let $\Phi = \Phi(\mathbf A_0,\mathbf G)$ be the roots of $\mathbf A_0$ in $G$. Let $\Delta$ be the base of $\Phi$ corresponding to $P_0$, so that the subsets of $\Delta$ correspond to $P_0$-standard parabolics of $G$. Let $\theta \subseteq \Delta$, and let $w \in G(k)$ be an element in the Weyl group $N_G(A_0)/Z_G(A_0)$ such that $w(\theta) \subseteq \Delta$. Let $P, P'$ be the standard parabolics of $G$ corresponding to $\theta, w(\theta)$.

6 . If $K$ is a compact subgroup such that $PK$ is an Iwasawa decomposition, then $P'K$ is also one.

  • 1
    $\begingroup$ Your notion of "Iwasawa decomposition" is a bit too restrictive if $k$ is non-archimedean. If $K$ is open then $P \times K \to G$ can never be a homeomorphism; the notion of "Iwasawa decomposition" one usually encounters is just to require that the map be surjective. $\endgroup$ – David Loeffler Apr 20 '17 at 8:48
  • $\begingroup$ Then, under what circumstances can the measure $dg$ be written as $dp \, dk$? $\endgroup$ – D_S Apr 20 '17 at 12:36
  • 1
    $\begingroup$ I don't have precise answers to your questions, but the Iwasawa decomposition can be found in the article by Cartier in Corvallis. Any two maximal compact subgroups of any real connected Lie group are conjugate. $\endgroup$ – Grad student May 1 '17 at 5:10
  • $\begingroup$ In general,, I would advise you to read Bruhat-Tits papers "Groupes reductifs sur un corps local, I, II". If $G$ is semisimple, then 1 is proved in part I (3.3.3), and 2 is immediate from the definition of "borne subgroup". 3 is no longer true in general - for "special" (or "good") maximal compact subgroups, one has an Iwasawa decomposition $ G = PK $ but the above map is no longer a homeomorphism (see part I, 4.4.3). 5 follows from (4.4.6.) together with the table (1.3.12) estblishing the existence of a special vertex in each of the affine Dynkin diagrams. 4 simply follows from 5. $\endgroup$ – assaferan May 14 '18 at 11:34

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