Corollary 1

*Let $\rho, \rho_1, \rho_2$ be finite-dimensional semi-simple analytic linear repre-

LIE GROUPS

sentations of $G$ and $n$ an integer $\geq 0$. Then $\rho_1 \otimes \rho_2, T^n \rho, S^n \rho, \wedge^n \rho$ (Appendix) are semi-simple.

The semi-simplicity of $\rho_1 \otimes \rho_2$ follows from Theorem 1 and Chapter I, § 6, Corollary 1 to Theorem 4. The semi-simplicity of $T^n \rho, S^n \rho, \wedge^n \rho$ follows from the semi-simplicity of $\rho_1 \otimes \rho_2$.

We shall see later that, if $k$ is a commutative field of characteristic 0, $\Gamma$ a group and $\rho_1$ and $\rho_2$ finite-dimensional semi-simple linear $k$-representations of $\Gamma$, then $\rho_1 \otimes \rho_2$ is semi-simple.