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!


Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.