Exercise 6, § 4

Let H be a closed subgroup of G containing $N_G(T)$.

a) Show that H is equal to its own normalizer in G, and that $H/H_0$ is isomorphic to $W/W_{H_0}(T)$.

b) Show that $R(H_0,T)$ is stable under W. Conversely, if K is a connected closed subgroup containing T such that $R(K,T)$ is stable under W, the normalizer of K contains that of T.

c) Assume that G is almost simple. Prove that H is equal to $N_G$(T), or to G, or that we are (up to isomorphism) in one of the following situations:

$\alpha )$ (resp. $\alpha '$)$) G =$ Spin$(2l+ 1,\mathbf{R})$ (resp. $G =\mathbf{S}\mathbf{O}(2l+ 1,\mathbf{R})$) and $H_0$ is the fixer of a non-zero vector of $\mathbf{R}^{2l+1}$, isomorphic to Spin$(2l,\mathbf{R})$ (resp. $\mathbf{S}\mathbf{O}(2l,\mathbf{R})$);

$\beta ) G = U(l,\mathbf{H})$ and $H_0$ is the subgroup D consisting of the diagonal matrices; $\beta ') G = U(l,\mathbf{H})/\{\pm 1\}$ and $H_0= D/\{\pm 1\}$;

$\gamma ) G$ is of type $F_4$ and $H_0$ is isomorphic to Spin$(8,\mathbf{R})$;

$\delta ) G$ is of type $G_2$ and $H_0$ is the subgroup (isomorphic to $\mathbf{S}\mathbf{U}(3,\mathbf{C})$) defined in Exerc. 7 of §3.