Exercise 10, § 5

Let $\mathfrak{h}$ be a splitting Cartan subalgebra of $\mathfrak{g}$, and $(X_{\alpha})_{\alpha\in R}$ a Chevalley system in $(\mathfrak{g},\mathfrak{h})$, cf. §2, no. 4. If $\alpha \in R$, put

and denote by $\overline{W}$ the subgroup of Aut$_e(\mathfrak{g},\mathfrak{h})$ generated by the $\theta_{\alpha}$.

a) Show that $\varepsilon (W) = W(R)$.

b) Let $s\in W$, and let $w=\varepsilon (s)$. Show that

$s(X_{\alpha}) =\pm X_{w(\alpha)}$ for all $\alpha \in R$.

(Use Exerc. 5 of §2.)

c) Let M be the kernel of $\varepsilon :\overline{W}\rightarrow W(R)$. Show that M is contained in the subgroup of $f(T_Q)$ consisting of the elements $f(\varphi )$ such that $\varphi^2= 1$. Show that M contains the elements $f(\varphi_{\alpha})$ defined by $\varphi_{\alpha}(\beta ) = (-1)^{\langle\beta ,\alpha^{\vee}\rangle}$ (remark that $\theta^2_{\alpha}=f(\varphi_{\alpha})$).

$d)^*$Let $\varphi \in$ Hom(Q$,\{\pm 1\}$). Show that $f(\varphi )$ belongs to M if and only if $\varphi$ extends to a homomorphism from P to $\{\pm 1\}$. (Sufficiency follows from the fact that M contains the $f(\varphi_{\alpha})$. To prove necessity, reduce to the case in which $k=\mathbf{Q}$, and use the fact that M is contained in $f(T_Q)\cap$ Aut$_e(\mathfrak{g}) =$ Im(T$_P$), cf. §7, Exerc. $26d$)$.)$ Deduce that M is isomorphic to the dual of the group $Q/(Q\cap 2P).^{13}$\*