Satake diagram

Definition

Let $\mathfrak{g}_0$ be a finite-dimensional real semisimple Lie algebra with a Cartan decomposition $\mathfrak{g}_0 = \mathfrak{k} \oplus \mathfrak{p}$. Let $\mathfrak{a}$ be a maximal abelian subspace of $\mathfrak{p}$, and let $\mathfrak{t}$ be a maximal abelian subspace of the centralizer of $\mathfrak{a}$ in $\mathfrak{k}$. Then $\mathfrak{h} = (\mathfrak{t} \oplus \mathfrak{a}) \otimes_\R \C$ is a Cartan subalgebra of the complexification $\mathfrak{g} = \mathfrak{g}_0 \otimes_\R \C$, of which $\mathfrak{g}_0$ is a real form. Let $\Phi$ be the roots of $(\mathfrak{g}, \mathfrak{h})$; they are real on $i\mathfrak{t} \oplus \mathfrak{a}$. Order the dual of $i\mathfrak{t} \oplus \mathfrak{a}$ lexicographically with respect to a basis whose first vectors span the dual of $\mathfrak{a}$. Let $\Delta$ be the base of simple roots for the positive roots of this order. The Satake diagram of $\mathfrak{g}_0$ is the Dynkin diagram of $\Phi$ with vertex set $\Delta$, decorated as follows:

1. a vertex $\alpha$ is painted black if $\alpha$ vanishes on $\mathfrak{a}$, and white otherwise;
2. two distinct white vertices $\alpha, \beta$ are joined by a double-headed arrow if $\alpha$ and $\beta$ have the same restriction to $\mathfrak{a}$.