Exercise 13, § 4

a) Choose a basis B of $R(G,T)$, and denote by $R_+$ the set of positive roots of $R(G,T)$. Prove that $z_G=\prod_{\alpha\in R_+}$ exp($\frac{1}{2}K_{\alpha}$) is an element of C(G),

independent of the choice of T and of B; we have $z_G^2=e$.

b) Let H be another connected compact Lie group. Then $z_{G\times H}= (z_G, z_H)$; prove that, if $f: G\rightarrow H$ is a surjective morphism of Lie groups, then $f(z_G) =$ $z_H$.

c) Put $R = R(G,T)$; assume that G is simply-connected, so that X(T) can be identified with P(R). Show that the kernel of the homomorphism $\chi \rightarrow \chi (z_G)$ from X(T) to $\{1,-1\}$ is the subgroup $P'(R)$ defined in Exerc. 8 of Chap. VI, §1.

d) If $G =\mathbf{S}\mathbf{U}(n,\mathbf{C})$, then $z_G= (-1)^{n+1}I_n$;

if $G =\mathbf{S}\mathbf{U}(n,\mathbf{H})$, then $z_G=-I_n$;

if G = Spin$(n,\mathbf{R})$ with $n\equiv 3,4,5$ or 6 (mod 8), then $z_G=-1$;

if G = Spin$(n,\mathbf{R})$ with $n\equiv 0,1,2,7$ (mod 8), then $z_G= 1$;

if G is of type $E_6,E_8,F_4$ or $G_2$, then $z_G=e$;

if G is simply-connected of type $E_7$, then $z_G$ is the unique non-identity element of C(G).

(Use Chap. VI, §4, Exerc. 5.)