Exercise 4, § 9

Assume that $\mathfrak{g}$ is simple, and denote by $\gamma$ the element of $R_+$ such that $H_{\gamma}$ is the highest root of $R^{\vee}$; write $H_{\gamma}=\sum_{\alpha\in B}n_{\alpha}H_{\alpha}$. We have $\langle \rho , H_{\gamma}\rangle =$ $\sum n_{\alpha}=h-1$, where $h$ is the Coxeter number of R (Chap. VI, §1, no. 11, Prop. 31).

a) Let $\alpha \in B$. Show that, for all $\beta \in R_+$,

and that equality holds if $\beta =\gamma$. Deduce that every prime factor of dim $E(\varpi_{\alpha})$ is $\leq h+n_{\alpha}-1$.

b) Assume that $\varpi_{\alpha}$ is not minuscule, i.e. $n_{\alpha}\geq 2$. Let $m\in 2, n_{\alpha}$ and $p=h+m-1$. Verify (cf. Chap. VI, Plates) that there exists $\beta \in R_+$ such that $\langle \varpi_{\alpha}, H_{\beta}\rangle =n_{\alpha}$ and $\langle \rho , H_{\beta}\rangle =h-1-(n_{\alpha}-m)$, hence $\langle \varpi_{\alpha}+\rho , H_{\beta}\rangle =p$. Deduce that, if $p$ is prime, $p$ divides dim $E(\varpi_{\alpha})$. (Remark that $p$ does not divide any of the $\langle \rho , H_{\beta}\rangle$ for $\beta \in R_+$, cf. Exerc. 3.)

c) When $\mathfrak{g}$ is of type $G_2$ (resp. $F_4,E_8$), we have $h= 6$ (resp. $12,30$), and dim $E(\varpi_{\alpha})$ is divisible by 7 (resp. $13,31$), When $\mathfrak{g}$ is of type $E_6$ (resp. $E_7$), and $\varpi_{\alpha}$ is not minuscule, dim $E(\varpi_{\alpha})$ is divisible by 13 (resp. 19).