Proposition 12

Let E be a finite dimensional simple $\mathfrak{g}$-module, $\omega$ its highest weight, and $\mathscr{B}$ the vector space of $\mathfrak{g}$-invariant bilinear forms on E. Let $m$ be the integer $\sum_{\alpha\in R_+}\omega (H_{\alpha})$, so that $m/2$ is the sum of the coordinates of $\omega$ with respect to B (Chap. VI, §1, no. 10, Cor. of Prop. 29). Let $w_0$ be the element of W such that $w_0(B) =-B$.

(i) If $w_0(\omega )\not=-\omega$, then $\mathscr{B}= 0$.

(ii) Assume that $w_0(\omega ) =-\omega$. Then $\mathscr{B}$ is of dimension 1, and every non-zero element of $\mathscr{B}$ is non-degenerate. If $m$ is even (resp. odd ), every element of $\mathscr{B}$ is symmetric (resp. alternating).

a) Let $\Phi \in \mathscr{B}$. The map $\varphi$ from E to $E^*$ defined, for $x, y\in E$, by $\varphi (x)(y) =$ $\Phi (x, y)$ is a homomorphism of $\mathfrak{g}$-modules. If $\Phi \not= 0$, then $\varphi \not= 0$, so $\varphi$ is an isomorphism by Schur’s lemma, and hence $\Phi$ is non-degenerate. Consequently, the $\mathfrak{g}$-module E is isomorphic to the $\mathfrak{g}$-module $E^*$, so that $w_0(\omega ) =-\omega$. We have thus proved (i).

b) Assume from now on that $w_0(\omega)=-\omega$. Then E is isomorphic to $E^*$. The vector space $\mathscr B$ is isomorphic to $\operatorname{Hom}_{\mathfrak g}(E,E^*)$, and hence to $\operatorname{Hom}_{\mathfrak g}(E,E)$ which is of dimension 1 (§6, no. 1, Prop. 1 (iii)). Hence $\dim \mathscr B=1$. Every non-zero element $\Phi$ of $\mathscr B$ is non-degenerate by a). Put $\Phi_1(x,y)=\Phi(y,x)$ for $x,y\in E$. By the preceding, there exists $\lambda\in k$ such that $\Phi_1(x,y)=\lambda\Phi(x,y)$ for all $x,y\in E$. Then $\Phi(y,x)=\lambda\Phi(x,y)=\lambda^2\Phi(y,x)$, so $\lambda^2=1$ and $\lambda=\pm1$. Thus, $\Phi$ is either symmetric or alternating.

c) By Chap. VII, §1, no. 3, Prop. 9 (v), $E^\lambda$ and $E^\mu$ are orthogonal with respect to $\Phi$ if $\lambda+\mu\ne0$. Since $\Phi$ is non-degenerate, it follows that $E^\omega,E^{-\omega}$ are not orthogonal with respect to $\Phi$.

d) There exists a homomorphism $\varphi$ from $\mathfrak{sl}(2,k)$ onto a subalgebra of $\mathfrak g$ that takes $H$ to

(Lemma 2). Consider E as an $\mathfrak{sl}(2,k)$-module via this homomorphism. Then the elements of $E^\lambda$ are of weight $\lambda\left(\sum_{\alpha\in R_+}H_\alpha\right)$. If $\lambda\in P$ is such that $E^\lambda\ne0$ and $\lambda\ne\omega,\lambda\ne-\omega$, then $-\omega<\lambda<\omega$, so Let G be the isotypical component of type V$(m)$ of the $\mathfrak{sl}(2,k)$-module E. By the preceding, G is of length 1 and contains $E^\omega,E^{-\omega}$. By c), the restriction of $\Phi$ to G is non-zero. By §1, no. 3, Remark 3, $\pi$ is even or odd according as this restriction is symmetric or alternating. In view of b), this completes the proof.