Exercise 7, § 9

(a) Let G be a semi-simple connected real Lie group, Z its centre and $\rho$ a continuous linear representation of G on a finite-dimensional complex vector space. There exists an integer $p$ such that, for all $z \in Z$, $\rho(z)$ is diagonalizable and all the eigenvalues of $\rho(z)$ are $p$-th roots of unity. (It can be assumed that $\rho$ is irreducible. Then $\rho(z)$ is scalar by Schur's Lemma. On the other hand, $\det \rho(g) = 1$ for all $g \in G$ because $G = \mathscr{D}G$.) (b) Deduce that, if G admits an injective finite-dimensional continuous linear representation, Z is finite.