Lemma 1

Let $\mathfrak{g}$ be a semi-simple Lie algebra, V a finite dimensional vector space, $\rho$ a linear representation of $\mathfrak{g}$ on V, D a vector subspace of V$,\mathfrak{h}$ a Cartan subalgebra of $\mathfrak{g},\mathfrak{s}$ (resp. $\mathfrak{s}'$) the set of $x\in \mathfrak{h}$ such that $\rho (x)D\subset D$ (resp. $\rho (x)D = 0$), and $\Phi$ the bilinear form on $\mathfrak{g}$ associated $^5$ to $\rho$.

(i) If $\mathfrak{h}$ is splitting, the vector subspaces $\mathfrak{s}$ and $\mathfrak{s}'$ of $\mathfrak{h}$ are rational over $\mathbf{Q}$.

(ii) If $\rho$ is injective, the restriction of $\Phi$ to $\mathfrak{s}$ (resp. $\mathfrak{s}'$) is non-degenerate.

Assume that the Cartan subalgebra $\mathfrak{h}$ is splitting. Let $d$ be the dimension of D; put $W =\bigwedge^d(V)$ and $\sigma =\bigwedge^d(\rho )$; denote also by $(e_1, . . . , e_d)$ a basis of D and $e=e_1\wedge \cdots \wedge e_d$ a decomposable $d$-vector associated to D. Let P be the set of weights of $\sigma$ with respect to $\mathfrak{h}$; denote by $W^\mu$ the subspace of

$^5$ In other words, $\Phi (x, y) =$ Tr($\rho (x)\rho (y)$) for $x, y\in \mathfrak{g}$. W associated to the weight $\mu$, and put $e=\sum_{\mu\in P}e^\mu$ (with $e^\mu\in W^\mu$ for all

$\mu\in P)$; finally, let $P'$ be the set of weights $\mu$ such that $e^\mu\not= 0$ and let $P''$ be the set of differences of elements of $P'$. Let $x$ be in $\mathfrak{h}$; then $x$ belongs to $\mathfrak{s}$ if and only if there exists $c$ in $k$ such that $\rho (x).e=c.e$ (Chap. VII, §5, no. 4, Lemma 2 (i)). Since $\rho (x).e^\mu=\mu(x).e^\mu$, we see that $x\in \mathfrak{s}$ is equivalent to the relation “$\mu(x) = 0$ for all $\mu\in P''$ ”. Now, the $\mathbf{Q}$-structure of $\mathfrak{h}$ is the $\mathbf{Q}$-vector subspace $\mathfrak{h}_{\mathbf{Q}}$ of $\mathfrak{h}$ generated by the coroots $H_{\alpha}$ and all $\mu$ in $P''$ take rational values on $\mathfrak{h}_{\mathbf{Q}}$; it follows (Algebra, Chap. II, §8, no. 4, Prop. 5) that $\mathfrak{s}$ is a subspace of $\mathfrak{h}$ rational over $\mathbf{Q}$.

For any weight $\mu\in P$, let $p_\mu$ be the projection onto $V^\mu$ associated to the decomposition $V =\bigoplus_{\mu\in P}V^\mu$; denote by $P_1$ the set of $\mu\in P$ such that

$p_\mu(D)\not= 0$. It is immediate that $\mathfrak{s}'$ is the intersection of the kernels (in $\mathfrak{h}$) of the elements of $P_1$; it follows, in the same way as for $\mathfrak{s}$, that $\mathfrak{s}'$ is a subspace of $\mathfrak{h}$ rational over $\mathbf{Q}$. This proves (i).

By extension of scalars, it suffices to prove (ii) when $k$ is algebraically closed, hence when $\mathfrak{h}$ is splitting. Let $\mathfrak{m}$ be a vector subspace of $\mathfrak{h}$ rational over $\mathbf{Q}$; for all non-zero $x$ in $\mathfrak{m}_{\mathbf{Q}}=\mathfrak{m}\cap \mathfrak{h}_{\mathbf{Q}}$, we have $\Phi (x, x)>0$ by the Cor. of Prop. 1 of §7, no. 1. The restriction of $\Phi$ to $\mathfrak{m}_{\mathbf{Q}}$ is non-degenerate, and hence so is the restriction of $\Phi$ to $\mathfrak{m}$ since $\mathfrak{m}$ is canonically isomorphic to $k\otimes_{\mathbf{Q}}\mathfrak{m}_{\mathbf{Q}}$.