Normalizer with respect to a compact subgroup

Question

Let $G$ be a semisimple Lie group with $K$ to be the compact subgroup in the Iwasawa decomposition $G=KAN.$ Then, if $\mathfrak a$ is the Lie algebra of $A$ what is $N_K(\mathfrak a)$ which is referred as normalizer in Knapp's book? I can understand the meaning of $N_{\mathfrak g}(\mathfrak a)$ which is the normalizer of $\mathfrak a$ in $\mathfrak g,$ i.e. the smallest subalegbra in which $\mathfrak a$ becomes an ideal.

Answer 1

Knapp does not define it explicitly but this notation usually means $$N_K(\mathfrak{a})=\{ k\in K: Ad(k)(\mathfrak{a})=\mathfrak{a}\},$$ which is consistent with how he uses it (see e.g. Prop. 6.52). This is how Duistermaat (Section 2.8) defines it.