Exercise 6, § 5

Assume that G is simply-connected. Let $p$ be a prime number and $g$ an element of C(G) such that $g^p=e$.

a) Show that there exist elements $u\in T$ and $w\in W$ such that

(i) $w(u)u^{-1}=g$;

(ii) $w^p= 1$;

(iii) $u^p=e$ if $p\not= 2,u^p=e$ or $g$ if $p= 2$.

(Let A be an alcove in $\mathfrak{t}$; for $i= 0,1, . . . , p-1$, let $x_i\in \overline{A}$ be such that exp $x_i=g^i$. Take $u$ to be the element exp $x$, where $x$ is the barycentre of the facet of $\overline{A}$ whose vertices are the $x_i$, and $w$ to be the element of W such that $w(A) = A-x.$)

b) Prove that there exist $u\in T$ and $v\in N_G(T)$ such that

(i) $vuv^{-1}u^{-1}=g$;

(ii) $v^p=e$ if $p\not= 2,v^p=e$ or $g$ if $p= 2$;

(iii) $u^p=e$ if $p\not= 2,u^p=e$ or $g$ if $p= 2$.

(Use the construction in $a$), lifting $w$ to $n_w$ as in Exerc. 12 of §4; take $v=n^{p+1}_w$ if $p\not= 2,v=n_w$ if $p= 2.)$