Weyl group action on complexified Iwasawa decomposition

Question

Let $G$ be a complex, reductive, algebraic group and let $G=KB$ be the complexified Iwasawa decomposition of $G$, see also [SW02]. 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.

Answer 1

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.