Remark

Assume that $\mathfrak{g}$ is simple. Let $\Sigma^{'\vee}$ be the Coxeter graph of the affine Weyl group $W_a(R^{\vee})$. Recall that the vertices of $\Sigma^{'\vee}$ are the vertices of the Coxeter graph $\Sigma^{\vee}$ of $W(R^{\vee})$, together with a supplementary vertex 0. The group $A(R^{\vee})$ operates on $\Sigma^{'\vee}$ leaving 0 fixed. The group Aut($\Sigma^{'\vee}$) is canonically isomorphic to the semi-direct product of $A(R^{\vee})/W(R^{\vee})$ with a group $\Gamma_C$ (cf. Chap. VI, §2, no. 3, and Chap. VI, §4, no. 3); clearly (Aut $\Sigma^{'\vee}$)$(0) =\Gamma_C(0)$; and $\Gamma_C(0)$ consists of 0 and the vertices of $\Sigma^{\vee}$ corresponding to the $\varpi_i$ for $i\in J$ (cf. Chap. VI, §2, Prop. 5 and Remark 1 of no. 3). In summary, the minuscule weights are the fundamental weights corresponding to the vertices of $\Sigma^{\vee}$ which can be obtained from 0 by the operation of an element of Aut($\Sigma^{'\vee}$).

With the notations of Chap. VI, Plates I to IX, we deduce from the preceding that the minuscule weights are the following:

For type $A_l(l\geq 1):\varpi_1, . . . , \varpi_l$.

For type $B_l(l\geq 2):\varpi_l$.

For type $C_l(l\geq 2):\varpi_1$.

For type $D_l(l\geq 3):\varpi_1, \varpi_{l-1}, \varpi_l$.

For type $E_6:\varpi_1, \varpi_6$.

For type $E_7:\varpi_7$.

For types $E_8,F_4,G_2$ there are no minuscule weights.