Exercise 16, § 9

a) Take $\mathfrak{g}$ to be of type $F_4$ and use the notations of Chap. VI, Plate VIII. If $i= 1,2,3,4$, put

The set of weights of $E(\varpi_i)$ is the disjoint union of the $W\omega$, where $\omega$ belongs to $\mathscr{X}_i$ (cf. §7, Prop. 5 (iv)). We have:

b) Show, by means of Th. 2, that:

dim $E(\varpi_1) = 52$, dim $E(\varpi_2) = 1274$, dim $E(\varpi_3) = 273$, dim $E(\varpi_4) = 26$. c) By using Chap. V, §3, Prop. 1, and the plates of Chap. VI, show that

Card(W$\varpi_1$) $= 2^73^22^{-3}$(3!)$^{-1}= 24$.

Calculate Card(W$\varpi_2$)$, . . .$, Card(W$.2\varpi_4$) similarly. Deduce that the number of weights of $E(\varpi_2)$ is 553; since this number is strictly less than dim $E(\varpi_2)$, one of these weights is of multiplicity $\geq 2$. d) Make analogous calculations for $\varpi_1, \varpi_3, \varpi_4$. By using Exerc. 21 of §7, deduce that, if $\rho$ is a non-zero simple representation of $\mathfrak{g},\rho$ admits a weight of multiplicity $\geq 2$.

e) Prove the same result for a simple algebra of type $E_8$.

f) Let $\mathfrak{a}$ be a splittable simple Lie algebra. Deduce from $d),e)$ and Prop. 7 and 8 of §7 the equivalence of the following properties:

(i) $\mathfrak{a}$ admits a non-zero simple representation all of whose weights are of multiplicity 1;

(ii) $\mathfrak{a}$ is neither of type $F_4$ nor of type $E_8$.