Exercise 17, § 4

We retain the notations of the preceding exercise; assume further that the subgroup H is clean (Exerc. 15).

a) Show that $R(G,T)$ is contained in the subgroup of X(T) generated by the union of the $R(\lambda )$ for $\lambda \in B_H$.

b) Let $\Delta$ be the identity component of the subgroup of S consisting of the $s\in S$ such that $\lambda (s) =\mu(s)$ for all $\lambda , \mu$ in $B_H$. Show that there exists a basis $\{\alpha_1, . . . , \alpha_l\}$ of $R(G,T)$ and an integer $k$, with $0\leq k\leq l-1$, such that $\Delta$ is the identity component of the set of $t\in T$ satisfying

(Let $x$ be the element of L(S) such that $\delta (\lambda )(x) = 2\pi i$ for all $\lambda \in B_H$; deduce from $a$) that exp $x\in C(G)$. Choose $\{\alpha_1, . . . , \alpha_l\}$ so that $i\delta (\alpha_j)(x)$ is zero for $1\leq j\leq k$ and $<0$ for $k+ 1\leq j\leq l$; then show that for all $\lambda \in B_H$ every root in $R(\lambda )$ can be written

with $j\geq k+ 1$ and $n_i\in \mathbf{N}$. Conclude by using $a).)$

c) Show that the integer $k$ does not depend on the choice of the tori $S,T$ or of the bases $B_H,\{\alpha_1, . . . , \alpha_l\}$; it is equal to the rank of the derived group of $Z_G(\Delta )$.