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...



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> First some definitions:
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• 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, $\omega^\vee\_0:=0$.



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> Then my question:
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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 $\Lambda\_{coweight}/\Lambda\_{coroot}$.



on the center of a Lie group
I have found a reference that contains the exact statement which I wanted:
It is Theorem 3, on page 14 of [this paper](http://www.sciencedirect.com/science/article/pii/S0001870899919102) of Stephen Sawin.


The vertices of the Weyl alcove which are in bijection with $Z(G)$ are called *sharp corners* (for self-explanatory reasons).
The other corners are called *dull corners*.


