Let $G$ be a complex, reductive, algebraic group and let $G=KB$ be the complexified Iwasawa decomposition of $G$, see also [[SW02](http://www.emis.de/journals/JLT/12-2/sinla2e.pdf)]. Let $T$ be a maximal torus of $B$, therefore a maximal torus of $G$. Let $N := \operatorname N\_G(T)$. My question is: Can I always achieve $N\subseteq KT$? Or differently put, can I find a representative of each $w\in W=N/T$ inside $K$?


My intuition comes from the $G=\operatorname{GL}\_n(\mathbb C)$ case where $K$ is the orthogonal group which contains all the permutation matrices which in turn are a set of representatives for the Weyl group with respect to the torus consisting of diagonal matrices.


If this is always possible, I would be happy to have a reference which I can quote for this statement.



Weyl group action on complexified Iwasawa decomposition
You need more data before you can say "the" complexified Iwasawa decomposition, namely the choice of real group of which $G$ is the complexification (as in your reference). For example, if your $GL\_2(\mathbb C)$ was the complexification of $U(1,1)$, its maximal compact is $U(1)\times U(1)$, and you lose.


If $G\_{\mathbb R}$ is the *split* real form, though (as in your example), then it's the realification of the Chevalley $\mathbb Z$-form, which has enough $\mathbb Z$-points to contain a maximal torus, $(\mathbb Z^\times)^{rank(G)} \cong (Z\_2)^{rank(G)}$. Its normalizer is again a finite group, so contained in a maximal compact of $G\_{\mathbb R}$. ADDED: Hence with this choice of real form, you can indeed find such representatives, inside the $\mathbb Z$-form.


