As is well known, maximal compact subgroup of real Lie group is just the fixed points of Cartan involution.
Now the question is what's the possible padic analog?
As is well known, maximal compact subgroup of real Lie group is just the fixed points of Cartan involution. Now the question is what's the possible padic analog? 


There are some complications in the padic case. For GL_n, every maximal compact subgroup is conjugate to GL_n of the ring of integers, but for more general groups (eg PGL_2, SU_3) there can be more than one conjugacy class of maximal compact subgroups. One thing that you can say is that every maximal compact subgroup is the stabaliser of a point on the BruhatTits building. 


Here is how the real and padic 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 padic 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 padic cases. Think of the fields themselves. In the padic 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 padic case. Thus, even in the rare cases where $G^\theta$ is compact, it is not maximal. 

