Exercise 5, § 5

a) Let $a\in$ Aut($\mathfrak{g}$), and $\mathfrak{n}$ the nilspace of $a-1$. Show that the following conditions are equivalent:

(i) Ker($a-1$) is a Cartan subalgebra of $\mathfrak{g}$.

(ii) $\mathfrak{n}$ is a Cartan subalgebra of $\mathfrak{g}$, and $a\in$ Aut$_0(\mathfrak{g})$.

(iii) dim $\mathfrak{n}=$ rk($\mathfrak{g}$) and $a\in$ Aut$_0(\mathfrak{g})$.

b) Assume from now on that $k$ is algebraically closed. Let V be a vector space, R a root system in $V, T_Q$ the group Hom(Q(R)$, k^*$)$,n$ an integer $\geq 1$ and $T_n$ the subgroup of $T_Q$ consisting of the elements whose order divides $n$. Let $\zeta$ be a primitive $n$th root of unity in $k$. For all $H\in P(R^{\vee})$, let $\psi (H)$ be the element $\gamma \rightarrow \zeta^{\gamma(H)}$ of $T_Q$. The map $\psi$ is a homomorphism from $P(R^{\vee})$ to $T_n$, with kernel $nP(R^{\vee})$. Let $t\in T_n$, and let C be an alcove in $P(R^{\vee})\otimes \mathbf{R}$. There exists $w\in W(R)$ and $H\in P(R^{\vee})$ such that $\frac{1}{n}H\in C$ and $\psi (wH) =t$. (Use Chap. VI, §2, no. 1.)

c) Let $\mathfrak{h}$ be a Cartan subalgebra of $\mathfrak{g}$, and R the root system of $(\mathfrak{g},\mathfrak{h})$. Let $n, \zeta , \psi$ be as in b), and $f$ as in no. 2. Let $H\in P(R^{\vee})$. The set of elements of $\mathfrak{g}$ invariant under $f(\psi (H))$ is $\mathfrak{h}\oplus \sum_{\alpha\in R'}\mathfrak{g}^{\alpha}$, where $R'$ is the set of $\alpha \in R$ such that $\alpha (H)\in n\mathbf{Z}$, and $f(\psi (H))$ satisfies the conditions of a) if and only if $\frac{1}{n}H$ belongs to an alcove.

d) Assume from now on that $\mathfrak{g}$ is simple. Let $\mathfrak{h}$ and R be as in $c), (\alpha_1, . . . , \alpha_l)$ a basis of $R,h$ the Coxeter number of $R,\zeta$ a primitive $h$th root of unity in $k$, and $H$ the element of $\mathfrak{h}$ such that $\alpha_i(H) = 1$ for $i= 1, . . . , l$. Prove the following properties:

(i) The homomorphism $\gamma \rightarrow \zeta^{\gamma(H)}$ from Q(R) to $k^*$ defines an element of Aut($\mathfrak{g},\mathfrak{h}$) which satisfies the conditions in a), and has order $h$. (Use $c$), Chap. VI, §2, Prop. 5 and Chap. VI, §1, Prop. 31.)

(ii) Every automorphism of $\mathfrak{g}$ of finite order satisfies the conditions in a) and is of order $\geq h$.

(iii) The automorphisms of $\mathfrak{g}$ of order $h$ satisfying the conditions of a) form a conjugacy class in Aut$_e(\mathfrak{g})$. (Use Prop. 5 of no. 3.)

e) Let $\mathfrak{h}$ and R be as in c), and let $w$ be a Coxeter transformation in W(R). Let $s\in$ Aut($\mathfrak{g},\mathfrak{h}$) be such that $\varepsilon (s) =w$. Show that $s$ satisfies the conditions in a), and is of order $h$. (Use Chap. VI, §1, Prop. 33, Chap. V, §6, no. 2 and Chap. VII, §4, Prop. 9.)

f) If $s\in$ Aut($\mathfrak{g}$), the following conditions are equivalent:

(i) $s$ satisfies the conditions in a), and is of order $h$;

(ii) there exists a Cartan subalgebra $\mathfrak{h}$ of $\mathfrak{g}$ stable under $s$ such that $s|\mathfrak{h}$ is a Coxeter transformation in the Weyl group of $(\mathfrak{g},\mathfrak{h})$. (Use $d$) and $e).)$

g) The characteristic polynomial of the automorphism in d) (i) is

That of the automorphism $s$ in e) is

$B(T) = (T^h-1)^l\prod_{i=1}^l(T-\zeta^{m_i})(m_i$ being the exponents of R).

(Use Prop. 33 (iv) of Chap. VI, §1, no. 11.) Deduce from the relation A(T) = B(T) that, for all $j\geq 1$, the number of $i$ such that $m_i\geq j$ is equal to the number of $\alpha \in R_+$ such that $\alpha (H) =j$; hence recover the result of Exerc. 6 c) of Chap. VI, §$4.^{12}$