Exercise 17, § 6

(a) Let $G, \overline{G}, G_0$ be as in Exercise 16 (b). Let $V$ be a finite-dimensional complex vector space and $\rho$ an irreducible linear representation of $G_0$ on $V$. Show that there exist finite-dimensional complex vector spaces $X, Y$, an analytic irreducible linear representation $\sigma$ (resp. $\tau$) of $G$ (resp. $\overline{G}$) on $X$ (resp. $Y$) and an isomorphism of $V$ onto $X \otimes Y$ which transforms $\rho$ into $\sigma \otimes \tau$. (Use Exercise 16 (a), Proposition 2 of Chapter I, § 2, and the Corollary to Proposition 8 of Algebra, Chapter VIII, § 7.)

(b) Show that the conclusion of (a) is not necessarily true if we omit the hypothesis that $G$ is simply connected. (Consider the complex Lie group $\mathbf{C}^*$.)