Exercise 19, § 4

We retain the notations of Exerc. 16 to 18; assume that the subgroup H is principal. Give $X(T)\otimes \mathbf{R}$ (resp. $X(S)\otimes \mathbf{R}$) a scalar product invariant under $W_G(T)$ (resp. under $W_H(S)$). Let $\lambda , \mu$ be two roots in $B_H$.

a) Show that, if $\lambda$ and $\mu$ are orthogonal, the sets $R(\lambda )$ and $R(\mu)$ are orthogonal. Deduce that if G is almost simple then so is H.

b) Assume from now on that $n(\lambda , \mu) =-1$. Show that there exists a surjective map $u: R(\lambda )\rightarrow R(\mu)$ such that, for $\alpha \in R(\lambda ),u(\alpha )$ is the unique root of $R(\mu)$ linked (that is, not orthogonal) to $\alpha$; we have $K_\mu=\sum_{\beta\in R(\mu)}K_{\beta}$, and the roots

in $R(\mu)$ are pairwise orthogonal (write $K_\mu$ and $K_{\lambda}$ as linear combinations of the $K_{\alpha}$, then identify the coefficients).

c) If $n(\mu, \lambda ) =-1$, the map $u$ is bijective, and the only pairs of linked roots in $R(\lambda )\cup R(\mu)$ are the pairs $(\alpha , u(\alpha ))$ for $\alpha \in R(\lambda )$.

d) Assume that $n(\mu, \lambda ) =-2$; let $\beta \in R(\mu)$. Show that $u^{-1}(\beta )$ contains either one or two elements; if $u^{-1}(\beta ) =\{\alpha_1, \alpha_2\}$, the root $\alpha_i(i= 1,2)$ is orthogonal to the other roots in $R(\lambda )$, and has the same length as $\beta$; if $u^{-1}(\beta ) =\{\alpha \}$ and $\|\alpha \|=\|\beta \|,\alpha$ is linked to a unique root $\alpha '\in R(\lambda )$, of the same length as $\alpha$, and $\{\alpha , \alpha '\}$ is orthogonal to the remainder of $R(\lambda )$; if $u^{-1}(\beta ) =\{\alpha \}$ and $\|\alpha \| \not=\|\beta \|,\alpha$ is orthogonal to the other roots in $R(\lambda )$.

e) Study similarly the case $n(\mu, \lambda ) =-3$.