Lemma 4

Let $H$ be a finite-dimensional simply connected Lie group. (i) For all $u \in \mathrm{Aut}\, L(H)$, let $\theta(u)$ be the unique automorphism of $H$ such that $L(\theta(u)) = u$. Then the mapping $(u, g) \mapsto \theta(u)g$ of $(\mathrm{Aut}\, L(H)) \times H$ into $H$ is analytic. (ii) Let $N$ be a Lie subgroup of $H$ and $\mathrm{Aut}(H, N)$ the set of $v \in \mathrm{Aut}\, H$ such that $v(N) = N$. Then $0^{-1}(\mathrm{Aut}(H, N))$ is a Lie subgroup of $\mathrm{Aut}\, L(H)$. (iii) Suppose that $N$ is discrete and normal, so that the Lie algebra of $G = H/N$ is identified with $L(H)$. For all $w \in \mathrm{Aut}\, G$, let $\eta(w)$ be the unique automorphism of $H$ such that $L(\eta(w)) = L(w)$. Then the mapping $\eta$ is an isomorphism of the group $\mathrm{Aut}\, G$ onto the group $\mathrm{Aut}(H, N)$. To prove (i), it suffices, by Lemma 3 of no. 1, to verify that the mapping $(u, g) \mapsto \theta(u)g$ is analytic in a neighbourhood of $(\mathrm{Id}_{L(H)}, e)$. There exists an open neighbourhood B of 0 in $L(H)$ such that $\psi = \exp_H|B$ is an analytic isomorphism of B onto an open neighbourhood of e in H. There exist an open neighbourhood U of $\mathrm{Id}_{L(H)}$ in Aut $L(H)$ and an open neighbourhood $B'$ of 0 in $L(H)$ such that $U(B') \subset B$. Then the mapping $(u, g) \mapsto \theta(u)g$ of $U \times \psi(B')$ into H is composed of the following mappings:

the mapping $(u, g) \mapsto (u, \psi^{-1}(g))$ of $U \times \psi(B')$ into $U \times B'$; the mapping $(u, x) \mapsto u(x)$ of $U \times B'$ into $B$; the mapping $y \mapsto \psi(y)$ of $B$ into $G$.

Hence this mapping is analytic.

Let $p$ be the canonical mapping of H into the homogeneous space $H/N$. Then $\theta^{-1}(\mathrm{Aut}(H, N))$ is the set of $u \in \mathrm{Aut}\ L(H)$ such that

for all $g \in N$. By § 8, no. 2, Theorem 2 and Corollary 2 to Theorem 2, this proves (ii).

Suppose that N is discrete and normal. Let $w \in \mathrm{Aut}\ G$. Then

hence $p \circ \eta(w) = w \circ p$ and therefore $\eta(w) \in \mathrm{Aut}(H, N)$. Clearly the mapping $\eta$ of $\mathrm{Aut}\ G$ into $\mathrm{Aut}(H, N)$ is an injective homomorphism. This homomorphism is surjective because $p : H \to G$ is a submersion.

Let G be a locally compact group and $\Gamma$ the automorphism group of G. Recall that a topology $\mathscr{T}_\beta$ has been defined on $\Gamma$ (General Topology, Chapter X, § 3, no. 5). It is the coarsest topology for which the mappings $v \mapsto v$ and $v \mapsto v^{-1}$ of $\Gamma$ into $C_c(G; G)$ (space of continuous mappings of G into G with the compact convergence topology) are continuous. The topology $\mathscr{T}_\beta$ is compatible with the group structure on $\Gamma$ (loc. cit.). For every compact subset L of G and every neighbourhood U of $e_G$ in G, let $N(L, U)$ be the set of $\phi \in \Gamma$ such that $\phi(g) \in gU$ and $\phi^{-1}(g) \in gU$ for all $g \in L$; then the $N(L, U)$ form a fundamental system of neighbourhoods of $e_\Gamma$. If G is generated by a compact subset C, the topology $\mathscr{T}_\beta$ is also the coarsest topology for which the mappings $v \mapsto v|C$ and $v \mapsto v^{-1}|C$ of $\Gamma$ into $C_u(C; G)$ are continuous (for every compact subset of G is contained in $(C \cup C^{-1})^n$ for sufficiently large n). If K is locally compact and V is a finite-dimensional vector space over K, the topology $\mathscr{T}_\beta$ on $\mathbf{GL}(V)$ is just the usual topology.