A transversal for the $\operatorname{Ad}(K)$ action on a sphere in $\mathfrak{p}$

This exercise level question has been unanswered on MSE for a few years. I hope you can answer it either there or here.

$$G$$ is a semisimple Lie group with a choice of Cartan decomposition on its Lie algebra $$\mathfrak{g} = \mathfrak{k} \oplus \mathfrak{p}$$. The Cartan involution also induces an inner product on $$\mathfrak{g}$$ and we denote the sphere in $$\mathfrak{p}$$ by $$\mathfrak{p}_1$$. Let $$K$$ be the connected subgroup associated to $$\mathfrak{k}$$ and let $$\mathfrak{a}\subset \mathfrak{p}$$ be a maximal abelian subalgebra.

The action of $$\mathfrak{a}$$ on $$\mathfrak{g}$$ gives rise to a root system $$\Sigma$$ on $$\mathfrak{a}^*$$. Fix a notion of positivity here and let $$\mathfrak{a}^+ := \lbrace H \in \mathfrak{a}: \lambda(H)>0 \text{ for all } \lambda \in \Sigma^+\rbrace$$

Question: Why is it that the unit vectors in $${\mathfrak{a}^+}$$ is a transversal for the adjoint action of $$K$$ on $$\mathfrak{p}_1$$? More precisely, for every $$X \in \mathfrak{p}_1$$, there is a unique unit vector $$H$$ in $$\overline{\mathfrak{a}^+}$$ which is in the $$K$$-orbit of $$X$$.

$$\mathfrak{p} = \bigcup \operatorname{Ad}(k)\mathfrak{a}$$, coincidence of the analytic Weyl group and algebraic Weyl group, and the fact that $$\overline{\mathfrak{a}^+}$$ is a fundamental domain for the algebraic Weyl group acting on $$\mathfrak{a}$$, together imply that each $$K$$ orbit in $$\mathfrak{p}$$ passes through at least once. But it's not yet clear to me why two unit vectors in $$\overline{\mathfrak{a}^+}$$ cannot be related by some $$k \in K\smallsetminus N_K(\mathfrak{a})$$.

• As an intersection of 2 open subsets
– YCor
May 12, 2020 at 22:34
• Open in $\mathfrak{a}$? $g\mathfrak{a}^+$ might not be contained in $\mathfrak{a}$, right? May 12, 2020 at 22:37
• Oops sorry, it's not open. I'm not used to "fundamental domain" in such context, would you way what it means exactly for $D=\overline{\mathfrak{a}^+}$ to be a fundamental domain? It ought to mean that for every $g\in K-\{1\}$ we have $gD\cap D$ is small, but in which sense exactly? (it's not empty, since it contains zero and I'd guess it may contain more than zero) does it mean it's contained in the relative boundary of $D$ (boundary of $D$ in $\mathfrak{a}$)?
– YCor
May 12, 2020 at 22:41
• I made an edit. Does the question make sense? May 12, 2020 at 22:44
• @LSpice both. OP passed to the 1-sphere to avoid intersections, but in the original setting, the question can currently be stated as: is it true that every $K$-orbit in $\mathfrak{p}$ meets the closure of $\mathfrak{a}^+$ in a singleton.
– YCor
May 13, 2020 at 4:14