Let $K$ be a (locally compact) local field and $G$ be a linear algebraic $K$-group.


Does the topological group $G(K)$ have a cocompact solvable closed subgroup?


If $\mathrm{char}(K)=0$, it is true that $G(K)$ has a cocompact solvable closed subgroup, see Proposition 9.3 in "A. Borel and J. Tits: Groupes reductifs" (Publ. IHES, 1965: [link](https://eudml.org/doc/103858)). So the only case left is when the characteristic is positive.


It is also well-known that for any local field $K$, $\mathrm{GL}(n,K)$ has a cocompact solvable closed subgroup.


What about $G=H\_{2n+1}\rtimes \mathrm{Sp}\_{2n}$, which is the semidirect product of the Heisenberg group $H\_{2n+1}$ with the symplectic group $\mathrm{Sp}\_{2n}$?



Does the group G(K) have a cocompact solvable closed subgroup?
Yes. By replacing $G$ with its maximal smooth closed subgroup (whose formation commutes with any separable extension on $K$; see Lemma C.4.1 and Remark C.4.2 in the book *Pseudo-reductive Groups*), we may and do assume $G$ is $K$-smooth. Alternatively, if you prefer, replace $G$ with the Zariski-closure of $G(K)$ to achieve the same end. In such cases we shall describe the solvable group in rather concrete terms.


Let's first treat the case when $G$ is reductive (but possibly not connected), as inspiration for the general case. With $G$ reductive, let $H$ be a maximal $K$-split smooth connected solvable $K$-subgroup. Explicitly, there exists a minimal parabolic $K$-subgroup $P$ of $G^0$ such that $H = S \ltimes U$ where $U = \mathscr{R}\_{u,K}(P)$ is the $K$-unipotent radical of $P$ and $S$ is a maximal split $K$-torus in $P$. We claim that $G(K)/H(K)$ is compact. 


Since $H$ is smooth, the natural map $G(K)/H(K) \rightarrow (G/H)(K)$ is a homeomorphism onto an open image (via the $K$-analytic implicit function theorem and the Zariski-local structure of smooth morphisms, applied to $G \rightarrow G/H$). But this open image is full since the smooth connected $H$ is $K$-split solvable, so it is the same to show that $(G/H)(K)$ is compact. It suffices to show that $(G^0/H)(K)=G^0(K)/H(K)$ is compact, since $G^0(K)$ is open of finite index in $G(K)$. Thus, we may and do rename $G^0$ as $G$ so that $G$ is connected.


By minimality, $P = Z\_{G}(S) \ltimes U$. Note that $H$ is normal in $P$. Since $G/P$ is projective, clearly $(G/P)(K)$ is compact. The map $G/H \rightarrow G/P$ is a right torsor for $P/H = Z\_{G}(S)/S$, and $Z\_{G}(S)/S$ is a $K$-anisotropic connected reductive group. Thus, $(Z\_{G}(S)/S)(K)$ is compact, or in other words $(P/H)(K)$ is compact. It follows that the map $(G/H)(K) \rightarrow (G/P)(K)$ has open image over which its source is a topological torsor for the compact group $(P/H)(K)$. It is therefore enough to show that $(G/H)(K) \rightarrow (G/P)(K)$ is surjective. Even better, the Borel-Tits structure theorem for connected reductive groups over general fields gives that $G(K) \rightarrow (G/Q)(K)$ is surjective for any parabolic $K$-subgroup $Q$ of $G$ (even when ${\rm{H}}^1(K,Q)$ is nontrivial, as can happen). 




---


How to go beyond the reductive case? This is rather trivial when ${\rm{char}}(K)=0$, as then $G$ has a $K$-split unipotent normal smooth connected $K$-subgroup modulo which it is reductive, so let's assume ${\rm{char}}(K)=p>0$. The imperfectness of such $K$ creates many difficulties. I tried some elementary arguments involving Frobenius isogenies to try to justify direct passage to the reductive case over such $K$. Unfortunately, this involves the intervention of various non-smooth group schemes which messed up all of my attempts to try to show various subsets of topological spaces are closed. 


So in the end I failed to find an elementary reduction to the reductive case when $K$ is a local function field. All I can find is a proof that involves the full force of the structure theory of pseudo-reductive groups, which you may find somewhat heavy for the task at hand. If someone else can find a way to rigorously justify passage to the reductive case in a more direct manner in positive characteristic then that would be swell. In what follows, [CGP] refers to the book *Pseudo-reductive Groups* (2nd ed.) by Conrad, Gabber, and Prasad.




---


Let $H$ be a maximal $K$-split solvable smooth connected $K$-subgroup of $G$. We claim that $G(K)/H(K) = (G/H)(K)$ is compact. Clearly $H$ contains $V := \mathscr{R}\_{us,K}(G)$, the maximal $K$-split unipotent smooth connected normal $K$-subgroup of $G$, and $H/V$ is a maximal
$K$-split solvable smooth connected $K$-subgroup of $G/V$, with $G/H = (G/V)/(H/V)$. By [CGP, Cor. B.3.5], $G/V$ has $K$-unipotent radical that is $K$-wound (i.e., does not contain $\mathbf{G}\_{\rm{a}}$ as a $K$-subgroup); in other words, $G/V$ is "quasi-reductive" in the sense of [CGP, Def. C.2.11]. Thus, we can replace $G$ with $G/V$ to reduce to the case that $G$ is quasi-reductive. 


Let $P$ be a minimal pseudo-parabolic $K$-subgroup of $G$ and $S \subset P$ a maximal $K$-split torus. By [CGP, Prop. C.2.4] we have $P = Z\_G(S) \cdot \mathscr{R}\_{us,K}(P)$; this is a semi-direct product by the first paragraph in [CGP, Rem. C.2.33]. Let $H = S \ltimes \mathscr{R}\_{us,K}(P)$, so $H$ is normal in $P$ with $H(K)$ solvable and $G(K)/H(K)=(G/H)(K)$ as in the reductive case. We claim that $G(K)/H(K)$ is compact. As in the reductive case, $G/H \rightarrow G/P$ is a $P/H$-torsor and so $(G/H)(K) \rightarrow (G/P)(K)$ is a topological $(P/H)(K)$-torsor over its open image. This image is full (as in the reductive case) because $G(K) \rightarrow (G/P)(K)$ is surjective [CGP, Lemma C.2.1]. Thus, as in the reductive case it suffices to prove the compactness of $(P/H)(K)$ and $(G/P)(K)$. 


We first analyze $G/P$. Recall that $W := \mathscr{R}\_{u,K}(G)$ is $K$-wound. By definition of pseudo-parabolicity we have $W \subset P$, and $G/P = (G/W)/(P/W)$ with $P/W$ a minimal pseudo-parabolic $K$-subgroup of the pseudo-reductive $G/W$ (see [CGP, Prop. 2.2.10]). Thus, to prove compactness of $(G/P)(K)$ we may and do assume $G$ is pseudo-reductive. This case is rather subtle since $G/P$ is essentially *never* proper when $G$ is pseudo-reductive but non-reductive. (The ur-example is $G={\rm{R}}\_{k'/k}(G')$ for a non-separable finite extension field $k'/k$ and a nontrivial connected semisimple $k'$-group $G'$, in which case $P = {\rm{R}}\_{k'/k}(P')$ for a parabolic $k'$-subgroup $P' \subset G'$. In such cases $G/P={\rm{R}}\_{k'/k}(G'/P')$, and this is *never* proper when $P' \ne G'$ due to [CGP, Example A.5.6].) 


In view of the good behavior of pseudo-parabolic $K$-subgroups with respect to quotients of pseudo-reductive groups by central smooth connected $K$-subgroups, the general structure theorem for $G$ (in terms of the standard construction away from characteristics 2 and 3, the "generalized standard" construction in characteristics 2 and 3, and the "totally non-reduced" construction in characteristic 2: see [CGP, Prop. 10.1.4, Theorem 10.2.1(1)]) reduces the compactness of $(G/P)(K)$ to the case when $G = {\rm{R}}\_{K'/K}(G')$ for a nonzero finite reduced $K$-algebra $K'$ and a smooth affine $K'$-group $G'$ whose fibers over the factor fields of $G'$ are one of the following: (i) connected semisimple, absolutely simple, and simply connected, (ii) basic exotic in characteristics 2 or 3, (iii) basic non-reduced pseudo-simple in characteristic 2. 


By [CGP, Prop. 2.2.13], the pseudo-parabolic $K$-subgroups of such a Weil restriction $G$ are precisely ${\rm{R}}\_{K'/K}(Q')$ for fiberwise-pseudo-parabolic $K'$-subgroups $Q' \subset G'$. Thus, $P={\rm{R}}\_{K'/K}(P')$ for a $K'$-subgroupp $P' \subset G'$ that is fiberwise minimal pseudo-parabolic. Since $G/P = {\rm{R}}\_{K'/K}(G'/P')$, so 
$(G/P)(K)=(G'/P')(K')$ as topological spaces, we can pass to the fibers over the factor fields of $K'$ separately (renaming such a factor field as $K$) so that $G$ falls into one of the cases (i), (ii), or (iii) above. 


Case (i) is easy, as then $G/P$ is projective. For case (ii), [CGP, Prop. 11.4.6(1)] identifies $G/P$ as a closed subscheme of ${\rm{R}}\_{K^{1/p}/K}(G'/P')$ for a canonically determined connected semisimple (even absolutely simple and simply connected) $K^{1/p}$-group $G'$ and (minimal) parabolic $K^{1/p}$-subgroup $P'$. Thus, we get the desired compactness in these cases due to the compactness of ${\rm{R}}\_{K^{1/p}/K}(G'/P')(K)=(G'/P')(K^{1/p})$ that follows from the projectivity of $G'/P'$ over $K^{1/p}$. 


How about case (iii)? In these cases we run into a temporary snag because the natural map $\xi:G/P \rightarrow {\rm{R}}\_{K^{1/2}/K}(G'/P')$ analogous to the basic exotic case turns out *not* to be a closed immersion, and in fact has positive-dimensional fiber over the identity point of the target; see [CGP, Rem. 11.4.5]. But fortunately in these cases a separate miracle is available: the map $\xi$ is bijective on $K$-points. To be precise, topologically this map on $K$-points is identified with $G(K)/P(K)\rightarrow G'(K^{1/2})/P'(K^{1/2})$, and this is bijective by the end of [CGP, Prop. 11.4.4]. But $G(K) \rightarrow G'(K^{1/2})$ is a homeomorphism by [CGP, Prop. 9.9.4(2)], so we win!


It remains to analyze $P/H$. Recall as in the reductive case that $P/H = Z\_G(S)/S$. We claim that this group has no $K$-subgroup isomorphic to either of $\mathbf{G}\_{\rm{a}}$ or $\mathbf{G}\_{\rm{m}}$. The $K$-wound $W$ is a normal $K$-subgroup modulo which its quotient is $Z\_{G/W}(S)/S$, so it suffices to show that $Z\_{G/W}(S)/S$ has neither $\mathbf{G}\_{\rm{a}}$ nor $\mathbf{G}\_{\rm{m}}$ as a $K$-subgroup. But $Z\_{G/W}(S)$ inherits pseudo-reductivity of $G/W$, so $Z\_{G/W}(S)/S$ is a pseudo-reductive group that is $K$-anisotropic in the sense that it does not contain $\mathbf{G}\_{\rm{m}}$ as a $K$-subgroup. Hence, $Z\_{G/W}(S)/S$ certainly has no proper pseudo-parabolic $K$-subgroup, so it does not contain $\mathbf{G}\_{\rm{a}}$ as a $K$-subgroup either, due to [CGP, Thm. C.3.8]!


To summarize, the proof of compactness of $(P/H)(K)$ is reduced to proving the compactness of $\mathcal{G}(K)$ for any smooth connected affine $K$-group $\mathcal{G}$ that contains neither $\mathbf{G}\_{\rm{a}}$ nor $\mathbf{G}\_{\rm{m}}$ as a $K$-subgroup. This is Proposition A.5.7 in the paper "Finiteness theorems for algebraic groups over function fields" in Compositio 148.


