Proposition 20

Let G be a finite-dimensional real Lie group.

(i) There exist a complex Lie group $\tilde{G}$ and an $\mathbf{R}$-analytic morphism $\gamma$ of G into $\tilde{G}$ with the following properties: for every complex Lie group H and every $\mathbf{R}$-analytic morphism $\phi$ of G into H, there exists one and only one $\mathbf{C}$-analytic morphism $\psi$ of $\tilde{G}$ into H such that $\phi = \psi \circ \gamma$.

(ii) If $(\tilde{G}', \gamma')$ has the same properties as $(\tilde{G}, \gamma)$, there exists one and only one isomorphism $\theta$ of $\tilde{G}$ onto $\tilde{G}'$ such that $\theta \circ \gamma = \gamma'$.

(iii) The $\mathbf{C}$-linear mapping of $L(G) \otimes \mathbf{C}$ into $L(\tilde{G})$ which extends $L(\gamma)$ is surjective; in particular $\dim_{\mathbf{C}}(\tilde{G}) \leq \dim_{\mathbf{R}}(G)$.

Assertion (ii) is obvious. We prove the existence of an ordered pair $(\tilde{G}, \gamma)$ with properties (i) and (iii).

(a) Suppose first that G is connected. Let $g = L(G)$, $g_C = g \otimes_{\mathbf{R}} \mathbf{C}$ be the complexification of g, S (resp. S') the simply connected real (resp. complex) Lie group with Lie algebra g (resp. $g_C$) and $\sigma$ the unique $\mathbf{R}$-analytic morphism of S into S' such that $L(\sigma)$ is the canonical injection of g into $g_C$. Let $\pi$ be the unique $\mathbf{R}$-analytic morphism of S onto G such that $L(\pi) = \mathrm{Id}_{L(G)}$ and F = Ker $\pi$.

For every complex Lie group H and every $\mathbf{R}$-analytic morphism $\phi$ of G into H, $L(\phi): g \to L(H)$ has a unique $\mathbf{C}$-linear extension to $g_C$ and this extension is of the form $L(\phi^*)$, where $\phi^*$ is a $\mathbf{C}$-analytic morphism of S' into H. Then

and hence $\phi \circ \pi = \phi^* \circ \sigma$. Therefore $\phi^*(\sigma(F)) = \phi(\pi(F)) = \{e\}$, whence Let P be the intersection of the $\mathrm{Ker}\, \phi^*$ for variable $\phi$. This is a normal Lie subgroup of S' (no. 2, Corollary 3 to Proposition 1). Let $\tilde{G} = S'/P$ and λ: S' → Ġ be the canonical morphism. Then σ(F) ⊂ P and hence there exists one and only one R-analytic morphism γ of G into Ġ such that γ ∘ π = λ ∘ σ. If ψ: Ġ → H denotes the morphism derived from φ* when passing to the quotient, then

whence ψ ∘ γ = φ. Clearly L(ψ), and hence ψ, are determined uniquely by the equality ψ ∘ γ = φ. Thus we have proved that the ordered pair (\tilde{G}, γ) has properties (i) and (iii).

(b) We pass now to the general case. Let F be the identity component of G, M = G/F and i: F → G and p: G → M be the canonical morphisms. We apply part (a) of the proof to F. We obtain an ordered pair (\tilde{F}, δ). For all g ∈ G, Int g|F = ω'(g) is an automorphism of F. By the universal property of \tilde{F}, there exists one and only one automorphism ω(g) of the complex Lie group \tilde{F} such that δ ∘ ω'(g) = ω(g) ∘ δ. Clearly ω is a morphism of G into Aut(\tilde{H}). If g ∈ G and f ∈ F, then

If f ∈ F, then δ ∘ (Int_F f) = (Int_\tilde{F} δ(f)) ∘ δ and Int_\tilde{F} δ(f) is an automorphism of the complex Lie group \tilde{F}; hence Int_\tilde{F} δ(f) = ω(f).

Hence Lemma 7 can be applied, which gives a diagram

Let \tilde{F} be identified with a normal subgroup of \tilde{G} by means of ī. The group \tilde{G} is generated by \tilde{F} and γ(G); hence the automorphisms of \tilde{F} defined by the elements of \tilde{G} are automorphisms of the complex Lie group structure. By § 1, no. 9, Proposition 18, there exists one and only one complex Lie group structure on \tilde{G} such that \tilde{F} is an open Lie subgroup of \tilde{G}. Henceforth \tilde{G} will have this structure. As δ is R-analytic, γ is R-analytic.

The ordered pair (\tilde{G}, γ) has property (iii) of the proposition. We show that it has property (i). Let H be a complex Lie group and ψ an R-analytic morphism of G into H. There exists a C-analytic morphism η of \tilde{F} into H such that η ∘ δ = φ|F. Let g ∈ G. The mappings

of \tilde{F} into H are C-analytic morphisms; they coincide on δ(F), for, if f' ∈ F, then

therefore $\eta(\omega(g)f) = \phi(g)\eta(f)\phi(g)^{-1}$ for all $g \in G$ and all $f \in \tilde{F}$. If $G'$ denotes the semi-direct product of $G$ by $\tilde{F}$ relative to $\omega$, there then exists a morphism $\zeta$ of the group $G'$ into $H$ which coincides with $\phi$ on $G$ and with $\eta$ on $\tilde{F}$. For $f \in F$,

Hence $\zeta$ defines on passing to the quotient a morphism $\psi$ of $\tilde{G}$ into $H$. Then $\psi \circ \gamma = \phi$ and $\psi \circ \tilde{i} = \eta$; the latter inequality implies that $\psi$ is $\mathbf{C}$-analytic.

Finally, let $\psi'$ be a $\mathbf{C}$-analytic morphism of $\tilde{G}$ into $H$ such that $\phi = \phi' \circ \gamma$. Then

and hence $\psi' \circ \tilde{i} = \psi \circ \tilde{i}$. As $\tilde{G}$ is generated by $\tilde{i}(\tilde{F})$ and $\gamma(G)$, $\psi' = \psi$.