Let $G$ be a simply-connected simple compact Lie group. Its center $Z(G)$ is a finite abelian group, say $Z(G) = \mathbb Z/k\mathbb Z$ for $G=SU(k)$.

I find the following interpretation of $Z(G)$ in terms of McKay correspondence:

Let $\Gamma$ be a finite subgroup of $SU(2)$. Let $\{ \rho_i\}_{i\in I}$ be the set of (isomorphism classes of) irreducible representations of $\Gamma$. Let $Q$ be the $2$-dimensional representation of $\Gamma$ given by the inclusion $\Gamma\subset SU(2)$. Consider the tensor product decomposition $\rho_i\otimes Q = \bigoplus \rho_j^{\oplus a_{ij}}$. McKay correspondence says $2\delta_{ij} - a_{ij}$ is an affine Cartan matrix of type ADE.

Let $J$ be the subset of $I$ consisting of $1$-dimensional irreducible representations. It consists of $i$ such that the coefficient of $\alpha_i$ in the imaginary root $\delta$ is $1$. (Sometimes they are called special vertices.) We put $J$ an abelian group structure by the tensor product as representations of $\Gamma$. Then $J$ is isomorphic to $Z(G)$.

This observation can be checked by case-by-case analysis. It is easy for type $A_{k-1}$. The group $\Gamma$ is $\mathbb Z/k\mathbb Z$, $J$ is the whole $I$, and is isomorphic to $\Gamma$ itself as a group. For type $D_n$, $J$ consists of $4$ extremal vertices. One needs to know the dual representation of $\rho_j$ for $j\in J$ in order to determine the group $J$. It is given in Gozalez-Sprinberg Verdier paper, and the answer is: $J$ is either $\mathbb Z/2\mathbb Z\oplus \mathbb Z/2\mathbb Z$ or $\mathbb Z/4\mathbb Z$ according to $n$ is even or odd. This coincides with the center of $Spin(2n)$. Exceptional cases can be determined in the same way.

My questions are

Did anybody find this observation before ?

Is there a conceptual explanation of this observation ? In particular, is there a proof without case-by-case check ?