Exercise 9, § 5

Let R be a reduced root system, B a basis of R, and $(\mathfrak{g}_0,\mathfrak{h}_0,B,(X_{\alpha})_{\alpha\in B})$ a corresponding framed semi-simple Lie algebra (§4). Let $\overline{k}$ be an algebraic closure of $k$, and $\rho :$ Gal($\overline{k}/k$)$\rightarrow$ Aut(R$,B$) a continuous homomorphism (cf. Exerc. 8); if $\sigma \in$ Gal($\overline{k}/k$), denote by $\rho_{\sigma}$ the $k$-linear automorphism of $\overline{\mathfrak{g}}_0= \overline{k}\otimes_k\mathfrak{g}_0$ such that $\rho_{\sigma}(X_{\alpha}) =X_{\rho(\sigma)\alpha}$. On the other hand, the natural operation of Gal($\overline{k}/k$) on $\overline{k}$ can be extended to an operation on $\overline{\mathfrak{g}}_0$. Let $\mathfrak{g}$ be the subset of $\overline{\mathfrak{g}}_0$ consisting of the elements $x$ such that $\rho_{\sigma}(x) =\sigma^{-1}.x$ for all $\sigma \in$ Gal($\overline{k}/k$).

a) Show that $\mathfrak{g}$ is a $k$-Lie subalgebra of $\overline{\mathfrak{g}}_0$, and that the injection of $\mathfrak{g}$ into $\overline{\mathfrak{g}}_0$ extends to an isomorphism from $\overline{k}\otimes_k\mathfrak{g}$ to $\overline{\mathfrak{g}}_0$. In particular, $\mathfrak{g}$ is semi-simple.

b) Let $\mathfrak{b}_0$ be the subalgebra of $\mathfrak{g}_0$ generated by $\mathfrak{h}_0$ and the $X_{\alpha}$. Put

Show that $\overline{k}\otimes_k\mathfrak{b}= \overline{\mathfrak{b}}_0$ and that $\overline{k}\otimes_k\mathfrak{h}= \overline{\mathfrak{h}}_0$, so that $\mathfrak{b}$ is a Borel subalgebra of $\mathfrak{g}$ and $\mathfrak{h}$ is a Cartan subalgebra contained in it.

$^{12}$For more details on this exercise, 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. c) Show that the homomorphism $\pi$ associated to the algebra $\mathfrak{g}$ (cf. Exerc. 8) is equal to $\rho$.