Exercise 12, § 4

Assume that G is simply-connected; for all $\alpha \in R(G,T)$, put $t_{\alpha}=$ exp($\frac{1}{2}K_{\alpha}$). Put $N = N_G$(T), and denote by $\varphi : N\rightarrow W$ the canonical map. Let B be a basis of $R(G,T)$; choose an element $n_{\alpha}$ of $(N\cap S_{\alpha})$ -- $(T\cap S_{\alpha})$ for all $\alpha \in B$.

a) Show that $\varphi (n_{\alpha}) =s_{\alpha}$ and $n^2_{\alpha}=t_{\alpha}$, so that $n^4_{\alpha}= 1$.

b) Let $\alpha$ and $\beta$ be two distinct elements of B, and let $m_{\alpha \beta}$ be the order of $s_{\alpha}s_{\beta}$ in W. Prove that

$n_{\alpha}n_{\beta}=n_{\beta}n_{\alpha}$ if $m_{\alpha \beta}= 2$

$n_{\alpha}n_{\beta}n_{\alpha}=n_{\beta}n_{\alpha}n_{\beta}$ if $m_{\alpha \beta}= 3$

$(n_{\alpha}n_{\beta})^2= (n_{\beta}n_{\alpha})^2$ if $m_{\alpha \beta}= 4$

$(n_{\alpha}n_{\beta})^3= (n_{\beta}n_{\alpha})^3$ if $m_{\alpha \beta}= 6$

(if for example $m_{\alpha \beta}= 3$, then $(s_{\alpha}s_{\beta})s_{\alpha}(s_{\alpha}s_{\beta})^{-1}=s_{\beta}$ and $s_{\alpha}s_{\beta}(\alpha ) =\beta$; show that $n_{\alpha}n_{\beta}n_{\alpha}n^{-1}_{\beta}n^{-1}_{\alpha}n^{-1}_{\beta}$ belongs to $S_{\beta}$, and conclude by remarking that $S_{\alpha}\cap S_{\beta}\cap T =\{e\}$).

c) Deduce from b) that there exists a unique section $\nu : W\rightarrow N$ of $\varphi$ such that $\nu (s_{\alpha}) =n_{\alpha}$ and $\nu (ww') =\nu (w)\nu (w')$ if $l(ww') =l(w) +l(w')$ (where $l(w)$ denotes the length of $w$ with respect to the generating system $(s_{\alpha})_{\alpha\in B}$; cf. Chap. VI, §1, no. 5, Prop. 5). Put $n_w=\nu (w)$.

d) Let $W^*$ be the subgroup of N generated by the $n_{\alpha}$; show that $W^*\cap T$ is the subgroup $T_2$ of T consisting of the elements of order $\leq 2$ and that W can be identified with $W^*/T_2$.

e) Let $w\in W$ be such that $w^2= 1$; show that $n^2_w=\prod_{\alpha\in R_w}t_{\alpha}$, where $R_w$ is

the set of positive roots $\alpha$ such that $w(\alpha )<0$ (write $w=s_{\alpha_1}. . . s_{\alpha_r}$, with $r=l(w)$ and $\alpha_i\in B$; apply $c$) and Chap. VI, §1, no. 6, Cor. 2 of Prop. 17). f) Assume that G is almost simple. Let $c$ be a Coxeter transformation in W and $h$ the Coxeter number of W (Chap. VI, §1, no. 11). Show that $n^h_c=$ $\prod_{\alpha\in R_+}t_{\alpha}$ (use $c$)$,e)$ and Exerc. 2 of Chap. V, §6).