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"](https://arxiv.org/abs/math/0703543) 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 ${\varsigma}\times\rho$ ?
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> 
> **Question 2.** In particular, is it true that if $\mathcal N\_G(H)=H$, then then the map ${\varsigma}\times\rho$ is injective?
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> 



Action of $N(H)/H$ on the colors of a spherical homogeneous space $G/H$
The answer to both questions is "yes" as was communicated to me by Losev.


The main reason is that already $\varsigma$ is almost injective. More precisely, the sets $\varsigma(D)$ are always non-empty and they are pairwise disjoint except for a very well controlled situation.


Assume $I:=\varsigma(D\_1)\cap\varsigma(D\_2)\ne\emptyset$. Then $D\_1$ and $D\_2$ are moved by the same simple root $\alpha$. Luna has shown that this can only happen if $\alpha$ is also a spherical root. He showed further that then all elements of $\rho(D\_i)$ are simple roots which are spherical. The Luna axioms easily imply $I=\{\alpha\}$. 


So the only way for $\varsigma(D\_1)=\varsigma(D\_2)$ is when $\varsigma(D\_1)=\varsigma(D\_2)=\{\alpha\}$. At this point one has to look at $\rho$. If $\rho(D\_1)=\rho(D\_2)$ then both are equal to $\frac12\alpha^\vee$ (since their sum is $\alpha^\vee$). This implies that it is possible to replace $\alpha$ by $2\alpha$ in the homogeneous spherical datum for $G/H$. This modified datum corresponds to a subgroup $\tilde H$ in which $H$ has index two. So $G/H\to G/\tilde H$ is obtained an automorphism $\phi$ of order $2$. Since $D\_1$ and $D\_2$ are mapped to the same color in $G/\tilde H$ they are swapped by $\phi$.


As an extra bonus one gets that $\phi$ interchanges *only* $D\_1$ and $D\_2$ and leaves all other colors fixed.


