maximal compact subgroup as fixed points of some involution on p-adic group?

Here is how the real and p-adic situations are the same.

Let $G$ be a connected reductive algebraic group defined over a field $F$ not of characteristic two. Let $theta$ be an involution of $G$ defined over $F$. Then the group $G^theta$ of fixed points is a reductive algebraic subgroup of $G$.

Here are two ways in which they are different.

In the real case, one can always choose $theta$ so that the group of rational points of $G^theta$ is compact. In the p-adic case, compact reductive groups are quite rare, and so in most cases there is no analogous way to choose $theta$.

Second, compact subgroups do not play the same roles in the real and p-adic cases. Think of the fields themselves. In the p-adic case, the maximal compact subring is the ring of integers. In the real case, there are no nontrivial compact subrings. There is a ring of integers, but it is not compact. Moreover, since $G^theta$ has smaller dimension than $G$, it cannot be an open subgroup, and maximal compact subgroups are always open in the p-adic case. Thus, even in the rare cases where $G^theta$ is compact, it is not maximal.

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