# Example of a spherical homogeneous space $G/H$ with a pairs of colors and with the center of $G$ not contained in $H$?

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!