Corollary 2

Assume that $k$ is $\mathbf{R}$ or $\mathbf{C}$ or a non-discrete complete ultrametric field. Let G be a Lie group with Lie algebra $\mathfrak{g}$. Assume that, for any fundamental representation $\rho$ of $\mathfrak{g}$, there exists an analytic linear representation $\rho '$ of G such that $\rho = L(\rho ')$. Then, for any finite dimensional linear representation $\pi$ of $\mathfrak{g}$, there exists an analytic linear representation $\pi '$ of G such that $\pi = L(\pi ')$.

We use the notations of Cor. 1. There exists a representation $\sigma$ of G on $X =\bigotimes_{\alpha\in B}(\bigotimes^{m_{\alpha}}E_{\alpha})$ such that $L(\sigma )$ corresponds to the $\mathfrak{g}$-module structure of X (Chap. III, §3, no. 11, Cor. 3 of Prop. 41). Let C be the isotypical component of X of highest weight $\lambda$. In view of Chap. III, §3, no. 11, Prop. 40, it suffices to prove that C is stable under $\sigma (G)$. Let $g\in G$ and $\varphi =$ Ad($g$). Then $\sigma (g)a_X\sigma (g)^{-1}= (\varphi (a))_X$ for all $a\in \mathfrak{g}$. On the other hand, $\varphi$ is an automorphism of $\mathfrak{g}$ that takes $\mathfrak{h}$ to $\mathfrak{h}', R$ to $R'= R(\mathfrak{g},\mathfrak{h}'), B$ to a basis $B'$ of $R'$, and $\varpi_{\alpha}$ to the highest weight $\varpi '_{\alpha}$ of $E_{\alpha}$ relative to $\mathfrak{h}'$ and $B'$ (since $\varphi$ transforms $E_{\alpha}$ into a $\mathfrak{g}$-module isomorphic to $E_{\alpha}$). Hence $\varphi$ takes $\lambda$ to $\sum m_{\alpha}\varpi '_{\alpha}$. By the Remark above, $\sigma (g)(C) = C$.