It is known that if $Q$ is an indefinite non-degenerate quadratic form on $ \mathbb{R}^n$ with $n \ge 3$, then any maximal compact subgroup $K$ of the orthogonal group $SO(Q)$ acts transitively on the projectiviziation of the light cone. In other words, if $Q(x)=Q(y)=0$ for non-zero vectors $x,y \in \mathbb{R}^n$, then there exists $g \in K$, with $gx= \lambda y$ for some $ \lambda \neq 0$. My question concerned the $p$-adic situation: suppose $Q$ is non-degenerate quadratic form on $ \mathbb{Q}\_p^n$ which is isotropic, i.e. the equation $Q(x)=0$ has a non-zero solution $x \in \mathbb{Q}\_p^n$ and $n \ge 3$. It is well-know that there may be more than one conjugacy classes of the maximal compact subgroups. My question is: is there at least one such maximal compact subgroup that acts transitively on the projectiviziation of the light cone? 



Orbits of the maximal compact subgroup on the light cone for $p$-adic groups
Let $(V,(x,y)\mapsto x.y)$ be a bilinear space. Let $L$ be a lattice on $V$, and let $G$ be its stabilizer in $O(V)$. 


To an isotropic line $\ell$ in $V$ one can associate the ideal $I(\ell):=(\ell\cap L).L$, and of course, two lines $\ell\_0$ and $\ell\_1$ belong to a same orbit under $G$ only if $I(\ell\_0)=I(\ell\_1)$. 


Now, when $p$ is odd, the maximal compact groups are the stabilizers of the lattices $L$ that have a decomposition :
$$L\simeq H^k\perp <p>\otimes H^{r-k}\perp A$$ where 


* $r$ is the Witt index of $V$,
* $k$ is an integer in $[0..r]$,
* $H$ is the hyperbolic plane over $\mathbf Z\_p$ ($(x,y)\mapsto x\_1y\_2+x\_2y\_1$),
* $<p>\otimes H$ is the hyperbolic plane scaled by $p$ ($(x,y)\mapsto px\_1y\_2+px\_2y\_1$) and
* $A$ is a maximal integral anisotropic lattice.


Thus we see that there are only two favourable cases : $k=0$ and $k=r$. In both cases, a little more work will show that the action of $G$ is indeed transitive on the isotropic lines of $V$.


When $p=2$ one has to be a little more carefull in the description of the maximal compact groups, but the result is the same.


Reference : P. Garrett, Buildings and Classical Groups.


