Proposition 32

*Let $G$ be a finite-dimensional connected real Lie group. Suppose that $L(G)$ is reductive. The following conditions are equivalent:

(i) $G/\overline{D^1}G$ is compact; (ii) (resp. (ii')) every finite-dimensional analytic linear representation of $G$ on a complex (resp. real) vector space is semi-simple.

(i) $\Rightarrow$ (ii'): Suppose that $G/\overline{D^1}G$ is compact. Then every continuous linear representation of $G/\overline{D^1}G$ on a finite-dimensional real vector space is semi-simple (Integration, Chapter VII, § 3, Proposition 1). Let $\rho$ be a finite-dimensional analytic linear representation of $G$ on a real vector space. Then $\rho|D^1G$ is analytic, $D^1G$ is semi-simple (Chapter I, § 6, Proposition 5) and hence $\rho|D^1G$ is semi-simple (Corollary 3 to Theorem 1). Hence $\rho$ is semi-simple (Lemma 2).

It can be seen similarly that (i) $\Rightarrow$ (ii').

(i) $\Rightarrow$ (ii): Suppose that $G/\overline{D^1}G$ is not compact and hence isomorphic to a group of the form $\mathbf{R}^p \times \mathbf{T}^q$ with $p > 0$ ($§ 6$, no. 4, Proposition 11 (ii)). Then there exists a surjective morphism of $G/\overline{D^1}G$ onto $\mathbf{R}$ and hence a surjective morphism of $G$ onto $\mathbf{R}$. The mapping

is an analytic linear representation of $G$ on $\mathbf{R}^2$ which is not semi-simple, for the only 1-dimensional vector subspace of $\mathbf{R}^2$ which is stable under $\sigma(G)$ is $\mathbf{R}(0, 1)$.

It can be seen similarly that (ii) $\Rightarrow$ (i).