Exercise 18, § 9

Let G be a connected real Lie group, Z its centre, Z_0 the identity component of Z, g the Lie algebra of G and $\mathfrak{z}$ the centre of g. Then Z and Z_0 are Lie quasi-subgroups of G with Lie algebra $\mathfrak{z}$ (Exercise 3). An acceptable norm on g is a norm defining the topology of g and making g into a normed Lie algebra.

(a) For all $r > 0$ and $z \in \mathfrak{z}$, there exists an acceptable norm on g whose value at z is <r. (Let q be an acceptable norm on g. Show that, for suitably chosen $\lambda > 0$, the function $x \mapsto \|x\| = \lambda q(x) + \inf_{y \in \mathfrak{z}} q(x + y)$ has the required properties.)

(b) Show that the following conditions are equivalent: (i) $Z_0$ is simply connected; (ii) for every acceptable norm on g, the restriction of $\exp_G$ to the open ball of centre 0 and radius $\pi$ is injective; (iii) there exists $r > 0$ such that, for every acceptable norm on g, the restriction of $\exp_G$ to the open ball of centre 0 and radius r is injective.

(To prove (iii) $\Rightarrow$ (i), use (a). To prove (i) $\Rightarrow$ (ii), suppose that x, y are in g, $\|x\| < \pi$, $\|y\| < \pi$, $x \neq y$, $\exp_G x = \exp_G y$. Then $\exp \operatorname{ad} x = \exp \operatorname{ad} y$ and hence $\operatorname{ad} x = \operatorname{ad} y$ by Proposition 17 of § 6 applied to a complexification of g. Hence there exists a non-zero z in $\mathfrak{z}$ such that $\exp_G z = e$; it follows that (i) is false.)

(c) By considering the group $G'$ of Exercise 13 (b), show that the conclusion of (b) is no longer true if $\pi$ is replaced by a number $> \pi$. (In the notation of Exercise 13, use the norm $ae_1 + be_2 + ce_3 \mapsto (a^2 + b^2 + c^2)^{\frac{1}{2}}$ on $L(G')$.)