Proposition 40

Let $G$ be a Lie group, $E$ a complete normable space, $\pi$ an analytic linear representation of $G$ on $E$ and $F$ a closed vector subspace of $E$ stable under $\pi(G)$. Suppose that $K$ is of characteristic 0, or that $F$ is a direct factor of $E$.

(i) The subrepresentation $\pi_1$ and the quotient representation $\pi_2$ of $\pi$ defined by $F$ are analytic representations.

(ii) $F$ is stable under $L(\pi)(L(G))$.

(iii) Let $\varrho_1$ and $\varrho_2$ be the subrepresentation and quotient representation of $L(\pi)$ defined by $F$. Then $L(\pi_1) = \varrho_1, L(\pi_2) = \varrho_2$.

Let $A$ be the set of $u \in \mathscr{L}(E)$ such that $u(F) \subset F$. Then $A$ is a closed vector subspace of $\mathscr{L}(E)$ and $\pi$ takes its values in $A$. By virtue of the hypotheses on $K$ and $F$, the mapping $\pi': G \to A$ with the same graph as $\pi$ is analytic (Differentiable and Analytic Manifolds, R, 5.8.5). The canonical mappings $\theta_1 : A \to \mathscr{L}(F)$ and $\theta_2 : A \to \mathscr{L}(E/F)$ are continuous and linear and hence analytic. This proves (i). The mappings $T_e(\pi)$ and $T_e(\pi')$ have the same graph and hence $L(\pi)(L(G)) \subset A$, which proves (ii). We have