Proposition 11

Let E be a finite dimensional $\mathfrak{g}$-module, and consider the $\mathfrak{g}$-module $E^*=$ Hom$_k(E, k)$.

(i) An element $\lambda \in P$ is a weight of $E^*$ if and only if $-\lambda$ is a weight of E, and the multiplicity of $\lambda$ in $E^*$ is equal to that of $-\lambda$ in E.

(ii) If E is simple and has highest weight $\omega ,E^*$ is simple and has highest weight $-w_0(\omega )$ (cf. no. 2, Remark 2).

Consider $k$ as a trivial $\mathfrak{g}$-module whose elements are of weight 0. By what was said above, the elements of $E^*$ of weight $\lambda$ are the homomorphisms from E to $k$ which vanish on $E^\mu$ if $\mu\not=-\lambda$. This proves (i). If E is simple, $E^*$ is simple (Chap. I, §3, no. 3), and the last assertion follows from Remark 2 of no. 2.