I'm trying to set straight my various pieces of knowledge about the center of a compact Lie group, and I'm running in circles...

First some definitions:

• Let $G$ be compact, simple, and simply connected Lie group, with Dynkin diagram $\Gamma$.

• Let $\{\alpha_i\}_{i=1...n}$ be the simple roots of $G$; they form by definition the vertex set of $\Gamma$

(sometimes, I'll write just $i$ instead of $\alpha_i$ for a vertex of $\Gamma$).

• Let $\Gamma^e=\Gamma\cup\{\alpha_0\}$ be the extended Dynkin diagram, where one adds the lowest root to $\Gamma$.

• Let $Z(\Gamma^e)$ be the subset of $\Gamma^e$ defined as the orbit of $\alpha_0$ under the automorphism group of $\Gamma^e$.

• Let $Z(G)$ be the center of $G$, which is also the quotient $\Lambda_{coweight}/\Lambda_{coroot}$.

• Let $\omega^\vee_i\in\Lambda_{coweight}$ be the fundamental coweights, defined by $\alpha_i(\omega^\vee_j)=\delta_{ij}$.

Let us also agree that, by convention, $\omega^\vee_0:=0$.

Then my question:

Is it true (and if so, where can I find a proof of this fact) that $$\{\omega^\vee_i \,|\, i\in Z(\Gamma^e)\}$$ forms a complete set of representatives of $\Lambda_{coweight}/\Lambda_{coroot}$.