Vogan diagram

Definition

Let $\mathfrak{g}_0$ be a real semisimple Lie algebra with complexification $\mathfrak{g}$. Let $\theta$ be a Cartan involution of $\mathfrak{g}_0$, with Cartan decomposition $\mathfrak{g}_0 = \mathfrak{k}_0 \oplus \mathfrak{p}_0$. Let $\mathfrak{h}_0 = \mathfrak{t}_0 \oplus \mathfrak{a}_0$ be a $\theta$-stable Cartan subalgebra with $\mathfrak{t}_0 \subseteq \mathfrak{k}_0$ and $\mathfrak{a}_0 \subseteq \mathfrak{p}_0$, chosen so that $\dim \mathfrak{t}_0$ is as large as possible. Let $\Delta$ be the set of roots of $\mathfrak{g}$ with respect to the complexification $\mathfrak{h}$ of $\mathfrak{h}_0$. Choose positive roots $\Delta^+$ from an ordering in which $i\mathfrak{t}_0$ comes before $\mathfrak{a}_0$. Then $\theta \Delta^+ = \Delta^+$. So $\theta$ permutes the simple roots of $\Delta^+$. Each simple root fixed by $\theta$ is imaginary, that is, it vanishes on $\mathfrak{a}_0$. Let $\mathfrak{k}$ and $\mathfrak{p}$ be the complexifications of $\mathfrak{k}_0$ and $\mathfrak{p}_0$. An imaginary root $\alpha$ is compact if its root space lies in $\mathfrak{k}$, and noncompact if it lies in $\mathfrak{p}$.

The Vogan diagram of $(\mathfrak{g}_0, \mathfrak{h}_0, \Delta^+)$ is the Dynkin diagram of $\Delta^+$ with these marks:

1. Each pair of simple roots exchanged by $\theta$ is joined by a two-headed arrow.
2. Each simple root fixed by $\theta$ is painted if it is noncompact, and left unpainted if it is compact.