Consider a reductive group over a local field. What is the normalizer of a maximal compact subgroup? 


If this is to general, what is the normalizer of $GL(n, \mathbb{Z}\_p)$ in $GL(n, \mathbb{Q}\_p)$, $U(n)$ in $GL(n, \mathbb{C})$, and $O(n)$ in $GL(n, \mathbb{R})$?



Normalizers of maximal compact groups?
Hints :


-- For $K= {\rm GL}(n,{\mathbb Z}\_p )$. Make $G={\rm GL}(n,{\mathbb Q}\_p )$ acts on ${\mathbb Z}\_p$-lattices of ${\mathbb Q}\_p^n$. Prove that the lattices stabilized by $K$ are the $p^k {\mathbb Z}\_p^n$, $k\in {\mathbb Z}$. Observe that the normalizer $\tilde K$ of $K$ permutes these lattices and conclude that ${\tilde K}={\mathbb Q}\_p^\times K$.


-- For $K=O(n)$ (or $U(n)$). Do someting similar by making $G$ act on the set of positive definite symmetric (hermitian) matrices via $(X,A)\mapsto XA{\bar X}^{t}$.


In general, for a non-archimedean base field, you can make $G$ act on the extended Bruhat-Tits building, but the answer is going to be technical according to whether $G$ has a center or not, is simply connected or not. Over archimedean fields, I guess you have to use symmetric spaces.


