Let $G$ be a semisimple group over $\mathbb C$, and $X=G/H$ be a spherical homogeneous space of $G$. Let $T\subset B\subset G$ be a maximal torus and a Borel subgroup. Let $S=S(G,T,B)$ denote the corresponding set of simple roots.

Let ${\mathcal{P}}(S)$ denote the set of subsets of $S$. Let $M$ denote the weight lattice of $X$, and set $N:={\rm Hom}(M,\mathbb Z)$. Let $\mathcal D$ denote the set of colors of $X$.

We have maps $\rho\colon \mathcal D\to N$ and ${\varsigma}\colon\mathcal D\to{\mathcal{P}}(S)$.
Here ${\varsigma}(D)$ for $D\in\mathcal D$ is the set of simple roots $\alpha\in S$ such that the corresponding minimal parabolic subgroup $P_\alpha\supset B$ *moves* the color $D$.
Thus we obtain a map
$$ {\varsigma}\times\rho\colon\ \mathcal D\ \longrightarrow\ {\mathcal{P}}(S)\times N.$$
This map need not be injective, but by Proposition 3.2.3 of
Losev's paper "Uniqueness property for spherical homogeneous spaces" each of its fibers has $\le 2$ elements.

Now consider the group ${{\rm Aut}}_G(X)=\mathcal N_G(H)/H$, this group acts on $\mathcal D$. One can easily see that ${{\rm Aut}}_G(X)$ acts on the fibers of ${\varsigma}\times\rho$.

Question 1.Is it true that ${{\rm Aut}}_G(X)$ acts transitively on each fiber of ${\varsigma}\times\rho$ ?

Question 2.In particular, is it true that if $\mathcal N_G(H)=H$, then then the map ${\varsigma}\times\rho$ is injective?

sphericalif $B$ has an open orbis in $X$. Acoloris a $B$-orbit of codimension 1 in $X$. The set of colors $\mathcal D$ is finite. $\endgroup$ – Mikhail Borovoi Jun 9 '17 at 12:11