Let $G$ be a simply connected simple algebraic group over $\mathbb C$, $B\subset G$ a Borel subgroup, and $T\subset B$ a maximal torus. Let $\mathcal{S}=\mathcal{S}(G,T,B)$ denote the set of simple roots. For $\alpha\in \mathcal{S}$, let $P_\alpha\supset B$ denote the corresponding minimal parabolic subgroup.

Let $Y=G/H$ be a spherical homogeneous space of $G$. The word "spherical" means that the Borel subgroup $B$ has an open orbit in $Y$.

Let ${\mathcal{D}}$ denote the (finite) set of colors of $Y$, that is, of $B$-orbits of codimension one in $Y$. Let $\mathcal{X}\subset X^*(B)$ denote the weight lattice of $Y$. There is a canonical map $$\rho\colon {\mathcal{D}}\to V:={\rm Hom}_{\mathbb Z} (\mathcal{X}, \mathbb{Q}).$$

Let $\mathcal{P}(\mathcal{S})$ denote the set of subsets of $\mathcal{S}$. For a color $D\in{\mathcal{D}}$, let $\varsigma(D)$ denote the set of $\alpha\in\mathcal{S}$ such that $P_\alpha\cdot D\neq D$. We obtain a canonical map $$\varsigma\colon {\mathcal{D}}\to\mathcal{P}(\mathcal{S}).$$

We say that two colors $D,D'\in{\mathcal{D}}$ are a pair of colors if $$\rho(D)=\rho(D')\in V\quad \text{and}\quad\varsigma(D)=\varsigma(D')\in\mathcal{P}(\mathcal{S}).$$ There cannot be three different colors $D,D',D''$ with the same images in $V$ and $\mathcal{P}(\mathcal{S})$.

Question. What is an example of a spherical homogeneous space $Y=G/H$ having a pair of colors and such that the center $Z(G)$ is not contained in $H$? Where can I find a number of such examples?

Any comments or references are welcome!


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