# Action of $N(H)/H$ on the colors of a spherical homogeneous space $G/H$

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?

• What is a color of a spherical homogeneous space? – LSpice Jun 9 '17 at 12:06
• @LSpice: A homogeneous space $X=G/H$ is called spherical if $B$ has an open orbis in $X$. A color is a $B$-orbit of codimension 1 in $X$. The set of colors $\mathcal D$ is finite. – Mikhail Borovoi Jun 9 '17 at 12:11

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.