Exercise 12, § 11

Assume that $\mathfrak{g}$ is semi-simple. Let $x\in \mathfrak{g}$ and let $s$ (resp. $n$) be the semi-simple (resp. nilpotent) component of $x$. Let $\mathfrak{a}_x$ (resp. $\mathfrak{a}_s$) be the commutant of $x$ (resp. $s$) in $\mathfrak{g}$.

a) Show that $n$ is a nilpotent element of the semi-simple algebra $\mathscr{D}(\mathfrak{a}_s)$, and that the commutant of $n$ in $\mathfrak{a}_s$ is equal to $\mathfrak{a}_x$. Deduce that dim$\mathfrak{a}_x<$ dim $\mathfrak{a}_s$ if $n\not= 0$, i.e. if $x$ is not semi-simple.

b) Show that dim$\mathfrak{a}_x=$ rk($\mathfrak{g}$) if and only if $n$ is a principal nilpotent element of $\mathscr{D}(\mathfrak{a}_s)$.

c) Put G = Aut$_e(\mathfrak{g})$. Show that, for all $\lambda \in k$, there exists $\sigma_{\lambda}\in G$ such that $\sigma_{\lambda}x=s+\lambda^2n$. (Show that, if $n\not= 0$, there exists an $\mathfrak{s}\mathfrak{l}_2$-triplet in $\mathfrak{a}_s$ of which the first component is $n$, and deduce a homomorphism $\varphi :\mathbf{S}\mathbf{L}(2, k)\rightarrow G$; take $\sigma_{\lambda}$ to be the image under $\varphi$ of a suitable diagonal element of $\mathbf{S}\mathbf{L}(2, k).$) Deduce that $s$ belongs to the closure of $G.x$ in the Zariski topology.

d) Show that, if $x$ is not semi-simple, $x$ does not belong to the closure of $G.s$ in the Zariski topology (use the inequality dim$\mathfrak{a}_x<$ dim $\mathfrak{a}_s$, cf. $a$)$)$.

$^{20}$For more details on Exercises 6 to 11, see: B. KOSTANT, The principal threedimensional subgroup and the Betti numbers of a complex simple Lie group,

Amer. J. Math., Vol. LXXXI (1959), pp. 973-1032. e) Assume that $k$ is algebraically closed. Prove the equivalence of the following properties:

(i) $x$ is semi-simple;

(ii) $G.x$ is closed in $\mathfrak{g}$ in the Zariski topology.

(The implication (ii) $=\Rightarrow$ (i) follows from $c$). If (i) is satisfied, and if $x'$ is in the closure of $G.x$, Exerc. 15 of §8 shows that the semi-simple component $s'$ of $x'$ belongs to $G.x$, so $x'$ belongs to the closure of $G.s'$; conclude by applying d) to $x'$ and $s'.)$

f) Assume that $k$ is algebraically closed. Let $F_x$ be the set of elements $y\in \mathfrak{g}$ such that $f(x) =f(y)$ for every invariant polynomial function $f$ on $\mathfrak{g}$; we have $y\in F_x$ if and only if the semi-simple component of $y$ is G-conjugate to $s($§8, Exerc. 17). Show that $F_x$ is the union of a finite number of orbits of G, and that this number is $\leq 3^{l(x)}$, where $l(x)$ is the rank of $\mathscr{D}(\mathfrak{a}_s)$. Only one of these orbits is closed: that of $s$; only one is open in $F_x:$ that consisting of the elements $y\in F_x$ such that dim $\mathfrak{a}_y=$ rk($\mathfrak{g}$).