Let $\alpha \in R(G,T)$; put $V_{\alpha}= (G/$Z(Ker$\alpha$ ))$\times$ (Ker$\alpha$ ), and let $\varphi_{\alpha}$ be the morphism from $V_{\alpha}$ to G such that, for all $g\in G$ and all $t\in$ Ker $\alpha$, we have $\varphi_{\alpha}(\overline{g}, t) =gtg^{-1}$ (we denote by $\overline{g}$ the coset of $g$ modulo Z(Ker$\alpha$ )). Then $V_{\alpha}$ is a compact real-analytic manifold of dimension

dim $V_{\alpha}=$ dim $G-$ dim Z(Ker $\alpha$ ) $+$ dim Ker$\alpha$

$=n-$ (dim T + 2) + (dim $T-1$) $=n-3$

(§4, no. 5, Th. $1$)$;\varphi_{\alpha}$ is a morphism of real-analytic manifolds, and the image of $\varphi_{\alpha}$ consists of the elements of G conjugate to an element of Ker $\alpha$. It now suffices to take V to be the sum of the manifolds $V_{\alpha}$, and $\varphi$ to be the morphism inducing $\varphi_{\alpha}$ on each $V_{\alpha}$.

b) The group W (resp. $W_a$) operates simply-transitively on the set of chambers (resp. alcoves ).

c) Let C be a chamber and A an alcove. Then $\overline{C}$ (resp. A, resp. A) is a fundamental domain for the operation of W on $\mathfrak{t}$ (resp. of $W_a$ on $\mathfrak{t}$, resp. of $W_a$ on $\mathfrak{t}$-- $\mathfrak{t}_r$). If $x\in \mathfrak{t}_r$ and $w\in W_a$ are such that $w(x) =x$, then $w=$ Id.

d) For every chamber C, there exists a unique alcove A such that $A\subset C$ and$\gamma \in \frac{0}{A}.\in \overline{A}$. For every alcove A, there exists a unique $\gamma \in N(G,T)$ such that

If $w\in W$ and $\gamma \in \Gamma$(T), then $wt_{\gamma}w^{-1}=t_{w(\gamma)}$ and $wt_{\gamma}w^{-1}t^{-1}_{\gamma}=t_{w(\gamma)-\gamma}$, with $w(\gamma )-\gamma \in N(G,T)$; this immediately implies a). The rest of the proposition follows from Chap. VI, §1, no. 5 and §2, nos. 1 and 2.

a) The group $W'_a$ is the semi-direct product of $H_A$ by $W_a$.

b) The exponential map $\overline{A}\rightarrow T$ and the canonical injection $T\rightarrow G$ induce by passage to the quotients and to subsets homeomorphisms

$\overline{A}/H_A\rightarrow T/W\rightarrow G/$Int(G)

$A/H_A\rightarrow T_r/W\rightarrow G_r/$Int(G).

Let $w'\in W'_a$; then $w'(A)$ is an alcove of $\mathfrak{t}$, and there exists (Prop. $2b$)$)$ a unique element $w$ of $W_a$ such that $w(A) =w'$(A), that is $w^{-1}w'\in H_A$. Since $W_a$ is normal in $W'_a$, this proves a).

The canonical injection of $\overline{A}$ into $\mathfrak{t}$ induces a continuous bijection $\theta :\overline{A}\rightarrow$ $\mathfrak{t}/W_a$ (Prop. $2c$)$)$, and this is a homeomorphism since A is compact. Since $W_a$ is normal in $W'_a$, the group $H_A$ operates canonically on $\mathfrak{t}/W_a($Algebra, Chap. I, §5, no. 4) and $\mathfrak{t}/W'_a$ can be identified with the quotient $(\mathfrak{t}/W_a)/H_A$; the map $\theta$ is compatible with the operations of $H_A$, hence induces by passage to the quotient a homeomorphism $A/H_A\rightarrow \mathfrak{t}/W'_a$. Further, exp$_{\Gamma}$ induces a homeomorphism from $\mathfrak{t}/\Gamma (T)$ to T, hence also a homeomorphism from $\mathfrak{t}/W'_a$ to $T/W$. Assertion b) follows from that and Cor. 1 of Prop. 5 of §2, no. 4.

$\overline{C}\rightarrow \mathfrak{t}/W\rightarrow \mathfrak{g}/$Ad(G).

The canonical maps $\overline{C}\rightarrow \mathfrak{t}$ and $\mathfrak{t}\rightarrow \mathfrak{t}/W$ are proper (General Topology, Chap. III, §4, no. 1, Prop. $2c$)$)$. The map $\overline{C}\rightarrow \mathfrak{t}/W$ is continuous, proper and bijective (Prop. $2c$)$)$; thus, it is a homeomorphism, hence the corollary in view of the Cor. of Prop. 6 of §2, no. 5.

We can assume that $g$ belongs to $T_r($§2, no. 2, Th. 2). Let $x$ be an element of $\mathfrak{t}_r$ such that exp $x=g$, and let $A'$ be the alcove of $\mathfrak{t}$ containing $x$. The alcoves A of $\mathfrak{t}$ such that $g\in$ exp(A) are the alcoves $A'-\gamma$ for $\gamma \in \Gamma (T)$; thus, the assertion follows from Prop. $2d)$.

a) H is a closed subgroup of G.

b) If $H_0$ is central in G, then G is commutative (so G = T).

Assertion a) is clear. To prove b), we can replace G by D(G) (§1, Cor. 1 of Prop. 4), and hence can assume that G is semi-simple. Then, if $H_0$ is central in G, we have $L(H) =\{0\}$, so the endomorphism $L(u)-$Id of $\mathfrak{g}$ is bijective. Let $f$ be the endomorphism of the manifold G defined by $f(g) =u(g)^{-1}g$ for $g\in G$; it is étale, for if $g\in G$ and $x\in \mathfrak{g}$, we have $T(f)(xg) =u(g)^{-1}(x-L(u)(x))g$, so the tangent map of $f$ at $g$ is bijective. It follows that the image of $f$ is open and compact, hence coincides with G since G is connected. Now let E be a framing of G (§4, no. 10, Def. 3) and $u(E)$ its image under $u$. By Prop. 19 of loc. cit., there exists an element $h$ of G such that (Int $h$)$(E) =u(E)$. Let $g\in G$ be such that $h=f(g) =u(g)^{-1}g$; then

$u\circ$ Int $g=$ (Int $u(g)$)$\circ u=$ Int $g\circ$ (Int $h$)$^{-1}\circ u$,

so the framing (Int $g$)$(E)$ is stable under $u$. If (Int $g$)$(E) = (T_1,B,(U_{\alpha})_{\alpha\in B})$, then $\sum U_{\alpha}\in L(H)$; since $L(H) =\{0\}$, this implies that $B =\emptyset$, so $G = T_1$, and G is commutative.

For all $\alpha \in R(G,T)$, let $S_{\alpha}$ be the submanifold of S consisting of the elements $s$ of S such that $(xs)^{\alpha}= 1$. If there is no element $s$ of S such that $xs$ is regular, S is the union of the submanifolds $S_{\alpha}$, hence is equal to one of them. Hence there exists $\alpha$ in $R(G,T)$ such that $(xs)^{\alpha}= 1$ for all $s\in S$; but this implies that $x^{\alpha}= 1$ and $\alpha |S = 1$, so $Z(x)\cap Z(S)\supset$ Z(Ker $\alpha$ ), hence the lemma.

Since G is simply-connected, $\Gamma (T)$ is generated by the nodal vectors $K_{\alpha}$ $(\alpha \in R(G,T))$, and hence has a basis consisting of the family of the $K_{\alpha}$ for which $\alpha$ belongs to the basis B(C) defined by C (Chap. VI, §1, no. 10). Thus, it suffices to prove that, if $\varphi$ is an automorphism of the torus T leaving stable a basis of $\Gamma$(T), the set of fixed points of $\varphi$ is connected. Decomposing this basis into the disjoint union of the orbits of the group generated by $\varphi$, we are reduced to the case in which $T =\mathbf{U}^n$ and $\varphi$ is the automorphism $(z_1, . . . , z_n) \rightarrow (z_2, . . . , z_n, z_1)$; in this case the fixed points of $\varphi$ are the points $(z, z, . . . , z)$ for $z\in \mathbf{U}$, which form a connected subgroup of T.

a) There exists an element $a$ of $\mathfrak{g}$, fixed by $L(u)$ and by Ad $x$, such that $x$ exp $a$ is regular.

b) There exists a regular element $g$ of G fixed by $u$ and commuting with $x$.

Let H be the group of fixed points of $u, S$ a maximal torus of $Z(x)\cap H$, and K the identity component of $Z(S)\cap Z(x)$. This is a connected closed subgroup of G; further, by Cor. 5 of Th. 2 of §2, no. 2, there exist maximal tori of G containing S and $x$, so K is of maximal rank and contains S and $x$. On the other hand, K is stable under $u$ since S and $x$ are; denote by V the set of fixed points of $u$ in K. Then

so $V_0$ is contained in the centralizer of S in $(Z(x)\cap H)_0$; but the latter reduces to S (loc. cit., Cor. 6), hence finally $V_0= S$. Lemma 1 now implies that K is commutative, hence is a maximal torus of G (since it is connected and of maximal rank). It contains S and $x$, and is equal to the identity component of $Z(S)\cap Z(x)$; assertion a) now follows from Lemma 2. We deduce b) by taking $g=x$ exp $a$.

Replacing $\mathfrak{s}$ by $\mathscr{D}\mathfrak{s}$, we can assume that $\mathfrak{s}$ is semi-simple. Let S be a compact simply-connected Lie group with Lie algebra $\mathfrak{s}$, and let $u$ be the automorphism of S such that $L(u) =\varphi$. Prop. 3 implies the existence of an element $a$ of $\mathfrak{s}$, fixed by $\varphi$, such that exp $a$ is regular in S; in particular, $a$ is regular in $\mathfrak{s}$ (no. 2, remark 4).

a) The identity component of the group of fixed points of $u$ contains a regular element of G.

b) There exists a maximal torus K of G and a chamber of L(K) that are stable under $u$.

c) If G is simply-connected, the set of fixed points of $u$ is connected.

Assertion a) is the particular case $x=e$ of Prop. 3. We assume now that G is simply-connected and prove b) and c). Let $x$ be an element of G fixed by $u$, and let $g$ be a regular element of G, fixed by $u$ and commuting with $x$ (Prop. 3). The centralizer K of $g$ is a maximal torus of G (no. 2, Remark 2), stable under $u$, and containing $x$ and $g$. By Cor. 3 of Prop. 2 of no. 2, there exists a unique alcove A of L(K) such that $g\in$ exp A and $0\in A$; since $g$ is fixed by $u, L(u)$ leaves A, and hence also the chamber of L(K) containing A, stable. This proves b); further, the set of points of K fixed by $u$ is connected (Lemma 3) and contains $x$ and $e$, hence $c) ($General Topology, Chap. I, §11, no. 1, Prop. 2).

It remains to prove b) in the general case. Now, if $\widetilde{D}(G)$ is the universal covering of D(G), and if $f: \widetilde{D}(G)\rightarrow G$ is the canonical morphism, there exists an automorphism $\widetilde{u}$ of $\widetilde{D}(G)$ such that $f\circ \widetilde{u}=u\circ f$. If $\widetilde{K}$ is a maximal torus of $\widetilde{D}(G)$ and $\widetilde{C}$ a chamber of $L( \widetilde{K})$, stable under $\widetilde{u}$ (this exists by what has already been proved), there exists (§2, no. 3, Remark 2) a unique maximal torus K of G and a unique chamber C of L(K) such that $\widetilde{K} =f^{-1}(K)$ and $\widetilde{C} = L(f)^{-1}$(C), and we see immediately that K and C are stable under $u$, hence assertion b) in the general case.

a) The centralizer of every element of G is connected.

b) Any two commuting elements of G belong to the same maximal torus.

By Cor. 3 of Prop. 11 of §4, no. 6, D(G) is simply-connected. We have $G = C(G)_0.D(G)$; let $x\in G$; write $x=uv$, with $u\in C(G)_0$ and $v\in D(G)$. Then $Z(x) = C(G)_0.Z_{D(G)}(v)$. By Th. $1c), Z_{D(G)}(v)$ is connected, so $Z(x)$ is connected, hence a). Thus, by Cor. 3 of Th. 2 of §2, no. $2, Z(x)$ is the union of the maximal tori of G containing $x$, hence b).

(*) There exist elements $u_1, . . . , u_n$ of $\Gamma$ such that, for all $i$, the closure $\Gamma_i$ of the subgroup of $\Gamma$ generated by $u_1, . . . , u_i$ is a normal subgroup of $\Gamma$, and $\Gamma_n=\Gamma$.

Then, there exists a maximal torus of G stable under the operation of $\Gamma$.

We argue by induction on the dimension of G. Clearly, we can assume that $u_1\not=$ Id; then the subgroup H of fixed points of $u_1$ is distinct from G, and is stable under the operation of $\Gamma$. Moreover, since $\Gamma$ is compact, the image of $\Gamma$ in Aut(H$_0$) is a quotient of $\Gamma$, hence it also satisfies condition (*). By the induction hypothesis, there exists a maximal torus S of H stable under $\Gamma$. The centralizer K of S in G is connected (§2, no. 2, Cor. 5) and stable under $\Gamma$; this is a maximal torus of G, since $H_0$ contains a regular element of G (Th. $1a$)$)$ which is conjugate to an element of S (loc. cit., Cor. 4).

Let H be a compact Lie group, N a nilpotent subgroup of H. Then the closure $\Gamma$ of N is also a nilpotent group (Chap. III, §9, no. 1, Cor. 2 of Prop. 1), and it suffices, in view of Cor. 3, to prove that $\Gamma$ satisfies condition (*). Now $\Gamma_0$ is a connected compact nilpotent Lie group, hence is a torus (§1, no. 4, Cor. 1 of Prop. 4), and there exists an element $u_1$ of $\Gamma$ generating a dense subgroup of $\Gamma_0($General Topology, Chap. VII, §1, no. 3, text preceding Prop. 8). The finite group $\Gamma /\Gamma_0$ is nilpotent and there exist $\widetilde{u}_2, . . . ,\widetilde{u}_n\in \Gamma /\Gamma_0$ generating $\Gamma /\Gamma_0$ and such that the subgroup of $\Gamma /\Gamma_0$ generated by $(\widetilde{u}_2, . . . ,\widetilde{u}_r)$ is normal for $r= 2, . . . , n($Algebra, Chap. I, §6, no. 5, Th. 1 and no. 7, Th. 4). Then, if $u_2, . . . , u_n$ are representatives of $\widetilde{u}_2, . . . ,\widetilde{u}_n$ in $\Gamma$, the sequence $(u_1, . . . , u_n)$ has the required properties.

$f'(\overline{z}, x) =$ (Ad $gt^{-1}$)$(z-$(Ad $t$)$z+x)$.

Let F be the map from $G\times T$ to T such that $F(g, t) =gtg^{-1}$. Since $F\circ (\gamma (g)$, Id$_T) =$ Int $g\circ F$, we have $T_{(g,t)}(F)(gz, tx) = T_t$(Int $g$)$\circ T_{(e,t)}(F)(z, tx)$; by Chap. III, §3, no. 12, Prop. 46,

$T_{(e,t)}(F)(z, tx) =t$((Ad $t^{-1}$)$z-z$) $+tx=t$((Ad $t^{-1}$)$(z-$ (Ad $t$)$z+x$))

and consequently

$T_{(g,t)}(F)(gz, tx) =gtg^{-1}$((Ad $gt^{-1}$)$(z-$ (Ad $t$)$z+x$)).

The lemma follows immediately from this formula by passage to the quotient.

(i) $t\in T_r$ (resp. $x\in \mathfrak{t}_r$).

(i bis) The element $f(\overline{g}, t)$ (resp. $\varphi (\overline{g}, x)$) is regular in G.

(ii) The map $f$ (resp. $\varphi$ ) is a submersion at the point $(\overline{g}, t)$ (resp. $(\overline{g}, x)$).

(ii bis) The map $f$ (resp. $\varphi$ ) is étale at the point $(\overline{g}, t)$ (resp. $(\overline{g}, x)$).

b) The map $f_r$ (resp. $\varphi_r$, resp. $\varphi_A$) makes $(G/T)\times T_r$ (resp. $(G/T)\times \mathfrak{t}_r$, resp. $(G/T)\times A$) a principal covering of $G_r$ with group W (resp. $W'_a$, resp. $H_A$).

a) The equivalence of (i) and (i bis) is clear; that of (ii) and (ii bis) follows from the relations dim((G$/T$)$\times T$) $=$ dim((G$/T$)$\times \mathfrak{t}$) $=$ dim(G). By Lemma 4, $f$ is a submersion at the point $(\overline{g}, t)$ if and only if $\mathfrak{g}=\mathfrak{t}+$ Im(Ad $t-$Id), which means that $t$ is regular. Finally, since $\varphi =f\circ$ (Id$_{G/T}\times$ exp$_T$)$,\varphi$ is étale at the point $(\overline{g}, x)$ if and only if $f$ is étale at the point $(\overline{g}$, exp $x$), which by the preceding means that $x$ belongs to $\mathfrak{t}_r$.

b) The morphisms $f_r, \varphi_r, \varphi_A$ are thus étale. On the other hand, W operates freely on $G/T$, and a fortiori on $(G/T)\times T$. Let $g, g'$ in G and $t, t'$ in $T_r$ be such that $f(\overline{g}, t) =f(\overline{g}', t')$; then Int $g^{-1}g'$ maps $t'$ to $t$, and hence normalizes T, since $T = Z(t)_0= Z(t')_0$, and the class $w$ of $g^{-1}g'$ in W maps $(\overline{g}, t)$ to $(\overline{g}', t')$. It follows that $f_r$ is a principal covering with group W; this immediately implies that $\varphi_r$ is a principal covering with group $W'_a$, and hence by restriction to the connected component $(G/T)\times A$ of $(G/T)\times \mathfrak{t}_r$, that $\varphi_A$ is a principal covering with group $H_A$.