Exercise 15, § 11

Assume that $\mathfrak{g}$ is semi-simple.

a) Let $x$ be a nilpotent element of $\mathfrak{g}$. Show that $x$ is contained in Im(ad $x$)$^2$ (use Prop. 2). Deduce that (ad $x$)$^2= 0$ implies $x= 0$.

b) Let $(x, h, y)$ be an $\mathfrak{s}\mathfrak{l}_2$-triplet in $\mathfrak{g}$, and let $\mathfrak{s}=kx\oplus kh\oplus ky$. Prove the equivalence of the following conditions:

(i) Im(ad $x$)$^2=k.x$;

(ii) the $\mathfrak{s}$-module $\mathfrak{g}/\mathfrak{s}$ is a sum of simple modules of dimension 1 or 2;

(iii) the only eigenvalues of ad$_{\mathfrak{g}}h$ distinct from 0, 1 and $-1$ are 2 and $-2$, and their multiplicity is equal to 1.

c) Assume that $\mathfrak{g}$ is splittable simple. Let $\mathfrak{h}$ be a splitting Cartan subalgebra of $\mathfrak{g}, B$ a basis of $R(\mathfrak{g},\mathfrak{h})$, and $\gamma$ the highest root of $R(\mathfrak{g},\mathfrak{h})$ relative to B. Let $(x, h, y)$ be an $\mathfrak{s}\mathfrak{l}_2$-triplet such that $h\in \mathfrak{h}_{\mathbf{Q}}$ and $\alpha (h)\geq 0$ for all $\alpha \in B$ (cf. Prop. 5). Show that conditions (i), (ii), (iii) of b) are satisfied if and only if $h=H_{\gamma}$, in which case $x\in \mathfrak{g}^{\gamma}$ and $y\in \mathfrak{g}^{-\gamma}$.

d) Retain the hypotheses of c), and put G = Aut$_0(\mathfrak{g})$. Show that the $\mathfrak{s}\mathfrak{l}_2$-triplets satisfying (i), (ii) and (iii) are G-conjugate (use Exerc. 10). Show that, if $x$ is a non-zero nilpotent element of $\mathfrak{g}$ satisfying (i), the closure of $G.x$ in the Zariski topology is equal to $\{0\} \cup G.x$.