Exercise 20, § 7

Assume that $k=\mathbf{R}$ or $\mathbf{C}$. Let $\Gamma$ be a simply-connected Lie group with Lie algebra $\mathfrak{g}$. Let $\lambda , \mu,E,F,G$ be as in Prop. 10. Let $(e_1, . . . , e_n)$ (resp. $(f_1, . . . , f_p)$) be a basis of E (resp. F) consisting of eigenvectors of $\mathfrak{h}$, with $e_1\in E^{\lambda}$ and $f_1\in F^\mu$. We can consider $E,F,G$ as $\Gamma$ -modules. If $\gamma \in \Gamma$, denote by $a_i(\gamma )$ the coordinate of $\gamma .e_1$ with index $i$, and $b_j(\gamma )$ the coordinate of $\gamma .f_1$ with index $j$. Show that the function $a_ib_j$ on $\Gamma$ is not identically zero. Deduce that, for all $(i, j)$, there exists an element of $G\subset E\otimes F$ whose coordinate of index $(i, j)$ is $\not= 0$; for $k=\mathbf{R}$ or $\mathbf{C}$, this gives a new proof of Prop. 10. Pass from this to the case $k=\mathbf{Q}$, and then to the case of an arbitrary field, cf. Exerc. 4.