Exercise 2, § 7

Let $(R_i)_{i\in I}$ be the family of irreducible components of R, and $\mathfrak{g}=\prod_{i\in I}\mathfrak{g}_i$ the

corresponding decomposition of $\mathfrak{g}$ into a product of simple algebras. Identify P with the product of the $P(R_i)$, and give each $P(R_i)$ the order relation defined by the basis $B_i= B\cap R_i$.

a) Let $\omega = (\omega_i)_{i\in I}$ be an element of $P_{++}=\prod_{i\in I}P_{++}(R_i)$. Show that the simple

$\mathfrak{g}$-module $E(\omega )$ is isomorphic to the tensor product of the simple $\mathfrak{g}_i$-modules $E(\omega_i)$.

b) Let $\mathscr{M}$ (resp. $\mathscr{M}_i$) be the set of elements of $P_{++}$ (resp. of $P_{++}(R_i)$) having the equivalent properties (i), (ii), (iii), (iv) of Prop. 6 and 7. Show that $\mathscr{M}=\prod_{i\in I}\mathscr{M}_i$, in other words that $\omega \in \mathscr{M}$ if and only if, for all $i\in I,\omega_i$

is either zero or a minuscule weight of $R_i$. Deduce that $\mathscr{M}$ is a system of representatives in P of the elements of $P/Q$.

c) Let E be a simple $\mathfrak{g}$-module, and $\mathscr{X}$ its set of weights. Show that $\mathscr{X}$ contains a unique element of $\mathscr{M}$, and that the multiplicity of this element is equal to the upper bound of the multiplicities of the elements of $\mathscr{X}$.