Theorem 1

Let G be a finite-dimensional Lie group and $G_0$ its identity component. Suppose that G is generated by $G_0$ and a finite number of elements.

(i) There exists on $\mathrm{Aut}\ G$ one and only one analytic manifold structure satisfying the following condition: (AUT) for every analytic manifold M and every mapping f of M into $\mathrm{Aut}\ G$, f is analytic if and only if the mapping $(m, g) \mapsto f(m)g$ of $M \times G$ into G is analytic.

Suppose in the rest of the statement that Aut G has this structure. (ii) Aut G is a finite-dimensional Lie group. (iii) The morphism $\phi : u \mapsto L(u)$ of Aut G into Aut L(G) is analytic. (iv) If G is connected, $\phi$ is an isomorphism of the Lie group Aut G onto a Lie subgroup of Aut L(G); this Lie subgroup is equal to Aut L(G) if G is simply connected. (v) Let $a$ be the set of infinitesimal automorphisms of G. Then $a$ is a Lie algebra of vector fields and the law of infinitesimal operation associated with the mapping $(u, g) \mapsto u(g)$ of (Aut G) $\times$ G into G is an isomorphism of L(Aut G) onto $a$. (vi) The topology of the Lie group Aut G is the topology $\mathscr{T}_\beta$. (a) The uniqueness of the analytic structure considered in (i) is obvious. (b) Suppose that G is connected. Let H be the universal covering space of G, $p$ the canonical morphism of H onto G and N = Ker $p$. We introduce the notation $\theta$, $\eta$ and Aut(H, N) of Lemma 4. We transport the Lie group structure of Aut L(G) to Aut H by means of $\theta$. Then Aut H becomes a finite-dimensional Lie group and Aut(H, N) a Lie subgroup of Aut H (Lemma 4 (ii)). We transport the Lie group structure of Aut(H, N) to Aut G by means of $\eta^{-1}$. Then Aut G becomes a finite-dimensional Lie group. Properties (ii), (iii) and (iv) of the theorem are satisfied and the mapping $(u, g) \mapsto u(g)$ of (Aut G) $\times$ G into G is analytic (Lemma 4 (i)). Let M be an analytic manifold, $f$ a mapping of M into Aut G and $\phi$ the mapping $(m, g) \mapsto f(m)g$ of M $\times$ G into G. Clearly, if $f$ is analytic, $\phi$ is analytic. Suppose that $\phi$ is analytic. Then the T$\phi$: TM $\times$ TG $\to$ TG is analytic; its restriction to M $\times$ L(G), that is the mapping $(m, x) \mapsto L(f(m))x$ of M $\times$ L(G) into L(G) is therefore analytic; as L(G) is finite-dimensional, it follows that the mapping $m \mapsto L(f(m))$ of M into Aut L(G) is analytic and hence that $f$ is analytic. Thus (i) holds.

Let L(G) be given a norm. For all $\lambda > 0$, let B_\lambda be the open ball of centre 0 and radius $\lambda$ in L(G). We choose $\lambda > 0$ sufficiently small for $\psi = \exp_G|_{B_\lambda}$ to be an isomorphism of the analytic manifold B_\lambda onto the open submanifold $\psi(B_\lambda)$ of G. Let $\Phi$ be a filter on Aut G. For $\Phi$ to converge to $\mathrm{Id}_G$ in Aut G, it is necessary and sufficient that $L(\Phi)$ converge to $\mathrm{Id}_{L(G)}$ in Aut L(G) and hence that $L(\Phi)|_{B_{\lambda/2}}$ and $L(\Phi)^{-1}|_{B_{\lambda/2}}$ converge uniformly to $\mathrm{Id}_{B_{\lambda/2}}$. This condition implies that $\Phi|\psi(B_{\lambda/2})$ and $\Phi^{-1}|\psi(B_{\lambda/2})$ converge uniformly to $\mathrm{Id}_{\psi(B_{\lambda/2})}$. Conversely, suppose that $\Phi|\psi(B_{\lambda/2})$ converges uniformly to $\mathrm{Id}_{\psi(B_{\lambda/2})}$. There exists M $\in \Phi$ such that, if $u \in M$, then $u(\psi(B_{\lambda/2})) \subset \psi(B_{2\lambda/3})$; then L(u)(B_{\lambda/2}) is a connected subset of L(G) whose image under $\exp_G$ is contained in $\psi(B_{2\lambda/3})$, hence L(u)(B_{\lambda/2}) does not meet $B_\lambda - B_{2\lambda/3}$ and therefore L(u)(B_{\lambda/2}) $\subset$ B_\lambda; then the hypothesis that $\Phi|\psi(B_{\lambda/2})$ converges uniformly to $\mathrm{Id}_{\psi(B_{\lambda/2})}$ implies that $L(\Phi)|_{B_{\lambda/2}}$ converges uniformly to $\mathrm{Id}_{B_{\lambda/2}}$. It then follows that:

This proves (vi).

Let D be the law of infinitesimal operation associated with the law of left operation of Aut(G) on G. By Propositions 1 and 2 of no. 1, $D(L(\text{Aut } G)) = a$. Hence a is a Lie algebra of vector fields and D is a morphism of $L(\text{Aut } G)$ onto a. Let $x_1$ and $x_2$ be elements of $L(\text{Aut } G)$ such that $D(x_1) = D(x_2)$. Then the laws of operation $(\lambda, g) \mapsto (\exp \lambda x_1)g$ and $(\lambda, g) \mapsto (\exp \lambda x_2)g$ of K on G have the same associated law of infinitesimal operation; hence, for $|\lambda|$ sufficiently small, $\exp \lambda x_1$ and $\exp \lambda x_2$ coincide on a neighbourhood of e ($§ 4$, no. 7, Theorem 6), whence $\exp \lambda x_1 = \exp \lambda x_2$. It follows that $x_1 = x_2$ and hence D is an isomorphism of $L(\text{Aut } G)$ onto a.

The theorem has thus been completely proved for G connected.

(c) We pass to the general case. By hypothesis, G is generated by $G_0$ and a finite number of elements $x_1, x_2, \ldots, x_n$. Every $u \in \text{Aut } G$ leaves $G_0$ stable. Let $\text{Aut}_1 G$ be the set of $u \in \text{Aut } G$ which, on passing to the quotient, give the identity automorphism of $G/G_0$. This is a normal subgroup of $\text{Aut } G$. By part (b) of the proof, $\text{Aut } G_0$ has a canonical Lie group structure and the mapping $(g_1, g_2, \ldots, g_n, u) \mapsto (ug_1, ug_2, \ldots, ug_n)$ of $G_0^n \times \text{Aut } G_0$ into $G_0^n$ is analytic. Let P be the corresponding semidirect product of $\text{Aut } G_0$ by $G_0^n$; it is a finite-dimensional Lie group ($§ 1$, no. 4, Proposition 7).

If $w \in \text{Aut}_1 G$, we write

For all $w, w'$ in $\text{Aut}_1 G$, and hence $\zeta$ is a homomorphism of $\text{Aut}_1 G$ into P. This homomorphism is obviously injective.

We show that $\zeta(\text{Aut}_1 G)$ is closed in P. Let $\Phi$ be a filter on $\text{Aut}_1 G$ such that $\zeta(\Phi)$ converges to a point $((w_1, \ldots, w_n), w_0)$ of P. Then $\Phi$ converges pointwise to a mapping v of G into G. Clearly v is an endomorphism of the group G. Moreover, v leaves each coset modulo $G_0$ stable and $v|G_0 = w_0$. It follows that $v \in \text{Aut}_1 G$. As $\zeta(v) = ((w_1, \ldots, w_n), w_0)$, we have shown that $\zeta(\text{Aut}_1 G)$ is closed in P.

(d) In part (d) of the proof we assume that $K = \mathbf{R}$. By $§ 8$, no. 2, Theorem 2, $\zeta(\text{Aut}_1 G)$ is a Lie subgroup of P. We transport the real Lie group structure on $\zeta(\text{Aut}_1 G)$ to $\text{Aut}_1 G$ by means of $\zeta^{-1}$. Thus $\text{Aut}_1 G$ becomes a finite-dimensional Lie group.

Let M be an analytic manifold, f a mapping of M into Aut_1G and φ the mapping (m, g) ↦ f(m)g of M × G into G. We have the following equivalences:

f analytic ⇔ the mappings m: ↦ (f(m))_i, where 0 ≤ i ≤ n, are analytic ⇔ {the mappings m ↦ f(m)x_i of M into G, for 1 ≤ i ≤ n, are analytic and the mapping (m, g) ↦ f(m)g of M × G_0 into G is analytic ⇔ φ is analytic.

For w ∈ Aut_1G, L(w) = L(w_0) and hence the morphism w ↦ L(w) of Aut_1G into Aut L(G) is analytic. We see as in (b) that the law of infinitesimal operation associated with the law of operation of Aut_1G on G is an isomorphism of L(Aut_1G) onto a.

Let C be a compact subset of G_0 generating G_0. For a filter Φ to converge to $\mathrm{Id}_G$ on Aut_1G, it is necessary and sufficient that Φ|(C ∪ {x_1} ∪ ... ∪ {x_n}) and Φ^{-1}|(C ∪ {x_1} ∪ ... ∪ {x_n}) converge uniformly to

Id_a|(C ∪ {x_1} ∪ ... ∪ {x_n}).

The topology of Aut_1G is therefore the topology $\mathscr{T}_\beta$.

Clearly Aut_1G is open in Aut G with the topology $\mathscr{T}_\beta$. There exists on Aut G a Lie group structure compatible with this topology and inducing on Aut_1G the structure constructed above (§ 8, no. 1, Corollary 2 to Theorem 1). The fact that the Lie group Aut G has the properties of the theorem follows from the corresponding properties for Aut_1G.

(e) In part (e) of the proof we assume that K = C. By (c) and Theorem 2 of § 8, no. 2, there exists on Aut_1G a real Lie group structure such that ζ is an isomorphism of Aut_1G onto a real Lie subgroup of P.

The law of operation (w, g) ↦ wg of (Aut_1G) × G on G is real analytic. Let D be the associated law of infinitesimal operation. By Propositions 1 and 2 of no. 1, D(L(Aut_1G)) = a.

For all α ∈ a, let α_0 denote the restriction of α to G_0; it is an infinitesimal automorphism of G_0 which we identify, because of part (b) of the proof, with an element of L(Aut G_0). For 1 ≤ i ≤ n, we write

Finally, we write $f(\alpha) = ((\alpha_1, ..., \alpha_n), \alpha_0) \in L(P)$. Then f is a C-linear mapping of a into L(P).

On the other hand, clearly $L(\zeta) = f \circ D$. Hence $L(\zeta)(L(Aut_1G)) = f(a)$ is a complex vector subspace of L(P). By Proposition 2 of § 4, no. 2, $\zeta(Aut_1G)$ is a complex Lie subgroup of P and we can proceed exactly as in (d): we transport the complex Lie group structure on $\zeta(Aut_1G)$ to Aut_1G by means of $\zeta^{-1}$ and we see as in (d) that Aut_1G has the properties analogous to properties (i), (ii), (iii), (v) and (vi) of the theorem.

Clearly Aut₁G is open in Aut G with the topology $\mathscr{T}_\beta$. Let $w \in \mathrm{Aut}\ G$. Let $\sigma$ be the automorphism $v \mapsto wvw^{-1}$ of Aut₁G. It is real analytic ($§ 8$, no. 1, Theorem 1), $L(\sigma)$ is an $\mathbf{R}$-automorphism of $L(\mathrm{Aut}_1 G)$ and

is an $\mathbf{R}$-automorphism of $a$. This automorphism is also the automorphism of $a$ derived from $w$ by transport of structure; as $w$ is K-analytic, we see that $L(\sigma)$ is K-linear. Hence $\sigma$ is K-analytic ($§ 3$, no. 8, Proposition 32). By $§ 1$, no. 9, Proposition 18, there exists on Aut G one and only one Lie K-group structure such that $\mathrm{Aut}_1 G$ is an open Lie subgroup of Aut G. The fact that this structure has the properties of the theorem follows from the corresponding properties for $\mathrm{Aut}_1 G$.