Exposé XX. Reductive groups of semisimple rank 1

by M. Demazure

1. Elementary systems. The groups $U_{\alpha }$ and $U_{-\alpha }$

Recollection 1.1. Let $S=\mathrm{Spec}(k)$, where $k$ is an algebraically closed field, and let $G$ be a reductive $S$-group of semisimple rank 1, $T$ a maximal torus (necessarily split) of $G$. One then has

g = t ⊕ g_α ⊕ g_{−α},

where $\alpha$ and $-\alpha$ are the roots of $G$ with respect to $T$. Moreover, there exist two group monomorphisms

p_α : G_{a, S} → G    and    p_{−α} : G_{a, S} → G

such that

t p_α(x) t⁻¹ = p_α(α(t) x)    and    t p_{−α}(x) t⁻¹ = p_{−α}(α(t)⁻¹ x),

for every $S^{\prime }\to S$ and all $t\in T(S^{\prime })$, $x\in G_a(S^{\prime })$, and such that the morphism

G_{a, S} ×_S T ×_S G_{a, S} → G,

defined by $(y,t,x)\mapsto p_{-\alpha }(y)t{p}_{\alpha }(x)$, is radicial and dominant (Bible, § 13.4, cor. 2 to th. 3).

Since the tangent map at the identity is bijective, this morphism is also separable, hence birational; by Zariski's "Main Theorem" (EGA III₁, 4.4.9), it is therefore an open immersion.

Lemma 1.2. Let $S$ be a scheme, $G$ an $S$-group scheme, $T$ a torus of $G$, $Q$ a subtorus of $T$, $\alpha$ a character of $T$ inducing on $Q$ a non-trivial character on each fiber. Let $p_{\alpha }:G_{a,S}\to G$ (resp. $p_{-\alpha }:G_{a,S}\to G$) be a group morphism normalized by $T$ with multiplier $\alpha$ (resp. $-\alpha$). Suppose that the morphism

u : G_{a, S} ×_S T ×_S G_{a, S} → G,

defined set-theoretically by $u(y,t,x)=p_{-\alpha }(y)t{p}_{\alpha }(x)$, is an open immersion. Let finally $q$ be an integer $⩾0$ and

p : G_{a, S} → G

a group morphism such that for every $S^{\prime }\to S$ and all $t\in Q(S^{\prime })$, $x\in G_a(S^{\prime })$ one has

$$int(t)(p(x))=p(\alpha (t{)}^{q}x).$$

Then there exists a unique $\nu \in G_a(S)$ such that $p(x)=p_{\alpha }(\nu x^q)$.

Indeed, let $\Omega$ be the image of $u$ and $U=p^{-1}(\Omega )$. This is an open subset of $G_{a,S}$ containing the zero section. For every section $t$ of $Q$, the automorphism of $G_{a,S}$ defined by multiplication by $\alpha (t)$ leaves $U$ globally fixed. We have $U=G_{a,S}$; indeed, it suffices to check this when $S$ is the spectrum of an algebraically closed field $k$; then $\alpha :Q(k)\to k\ast$ is surjective, which proves at once that $U(k)\supset k\ast$, hence $U=G_{a,k}$. There therefore exist three morphisms

a : G_{a, S} → G_{a, S},   b : G_{a, S} → T,   c : G_{a, S} → G_{a, S},

such that

p(x) = p_{−α}(a(x)) b(x) p_α(c(x)).

The condition on $p$ translates to

a(α(t) x) = α(t)^{−q} a(x),
b(α(t) x) = b(x),
c(α(t) x) = α(t)^q c(x).

For the same reason as before, one therefore has for every $S^{\prime }\to S$ and every $z\in G_m(S^{\prime })$,

a(z x) = z^{−q} a(x),   b(z x) = b(x),   c(z x) = z^q c(x),

hence

z^q a(z) = a(1),   b(z) = b(1),   c(z) = z^q c(1).

Since $G_{m,S}$ is schematically dense in $G_{a,S}$, one has at once for every $x\in G_a(S^{\prime })$, $S^{\prime }\to S$:

x^q a(x) = a(1) = a(0) = 0,    hence a = 0,
c(x) = x^q c(1) = ν x^q,       for some ν ∈ G_a(S),
b(x) = b(1) = b(0) = e,        hence b = e,

which completes the proof.

Definition 1.3. Let $S$ be a scheme. By an $S$-elementary system one means a triple $(G,T,\alpha )$ where

(i) $G$ is a reductive $S$-group of semisimple rank 1 (Exp. XIX 2.7),

(ii) $T$ is a maximal torus of $G$,

(iii) $\alpha$ is a root of $G$ with respect to $T$ (Exp. XIX 3.2).

One thus has a direct sum decomposition (Exp. XIX 3.5)¹

g = t ⊕ g_α ⊕ g_{−α},

$g_{\alpha }$ and $g_{-\alpha }$ being locally free of rank one.

1.4. If $(G,T,\alpha )$ is an $S$-elementary system, then $(G_{S^{\prime }},T_{S^{\prime }},{\alpha }_{S^{\prime }})$ is an $S^{\prime }$-elementary system for every $S^{\prime }\to S$. If $(G,T,\alpha )$ is an $S$-elementary system, then $(G,T,-\alpha )$ is also one.

If $S$ is a scheme, $G$ a reductive $S$-group, $T$ a maximal torus of $G$, $\alpha$ a root of $G$ with respect to $T$, then (Exp. XIX 3.9), $(Z_{\alpha },T,\alpha )$ is an $S$-elementary system.

Let $(G,T,\alpha )$ be an $S$-elementary system. The invertible module $g_{\alpha }$ is canonically endowed with a $T$-module structure. One therefore has also a $T$-module structure on the vector bundle $W(g_{\alpha })$. On the other hand, the inner automorphisms of $T$ define on $G$ a structure of group with group of operators $T$.

Theorem 1.5. Let $(G,T,\alpha )$ be an $S$-elementary system.

(i) There exists a unique morphism of groups with group of operators $T$

$$\exp :W(g_{\alpha })\to G$$

which induces on the Lie algebras the canonical morphism $g_{\alpha }\to g$.²

In other words, exp is the unique morphism satisfying the following conditions: for every $S^{\prime }\to S$ and every $t\in T(S^{\prime })$, $X,X^{\prime }\in W(g_{\alpha })(S^{\prime })$, one has

exp(X + X′) = exp(X) exp(X′),
int(t)(exp(X)) = exp(α(t) X),
Lie(exp)(X) = X.

(ii) If one defines analogously (in the $S$-elementary system $(G,T,-\alpha )$)

$$\exp :W(g_{-\alpha })\to G,$$

then the morphism

W(g_{−α}) ×_S T ×_S W(g_α) → G

defined set-theoretically by $(Y,t,X)\mapsto \exp (Y)\cdot t\cdot \exp (X)$ is an open immersion.

Suppose we have proved the existence of the desired exp morphisms, and let us prove the other assertions of the theorem. We first prove (ii). Since both sides are of finite presentation and flat over $S$, it suffices to do so when $S$ is the spectrum of an algebraically closed field (SGA 1, I 5.7 and VIII 5.5). Let then $S=\mathrm{Spec}k$. Let $X\in \Gamma (S,g_{\alpha }{)}^{\times }$, $Y\in \Gamma (S,g_{-\alpha }{)}^{\times }$. It suffices to prove that the morphism

G_{a, k} ×_k T ×_k G_{a, k} → G,   (y, t, x) ↦ exp(yY) t exp(xX)

is an open immersion. Now by 1.1 and 1.2, there exist $a,b\in k$ such that

exp(yY) = p_{−α}(a y)    and    exp(xX) = p_α(b x).

Since $\exp :W(g_{-\alpha })\to G$ induces a monomorphism on Lie algebras, one has $a\ne 0$; likewise $b\ne 0$, and one is reduced to 1.1.

The uniqueness of the morphism exp may be proved locally on $S$; one then reduces to the case where $g_{\alpha }$ and $g_{-\alpha }$ are free, and one has only to apply 1.2 (with $Q=T$ and $q=1$).

It remains then to prove the existence of the desired morphism exp. Let us first remark that, by virtue of the theory of faithfully flat descent and the uniqueness assertion just proved, it suffices to demonstrate this existence locally on $S$ for the (fpqc) topology. By the usual arguments using finite presentation, one reduces to the case where $S$ is noetherian, then to the case where it is noetherian local. By virtue of the preceding remark, one can therefore content oneself with proving the existence of the desired morphism exp when $S=\mathrm{Spec}(A)$, with $A$ a complete noetherian local ring with algebraically closed residue field $k$. Let then $p_0:G_{a,k}\to G_k$ be a monomorphism of $k$-groups normalized by $T_k$ with multiplier ${\alpha }_0=\alpha {\otimes }_{A}k$ (one exists by 1.1). One knows (1.1 and 1.2) that the corresponding morphism $T_k{\cdot }_{{\alpha }_0}{G}_{a,k}\to G_k$ is an immersion, hence in particular a monomorphism. Let us provisionally admit the following two lemmas:

Lemma 1.6. Let $S$ be a scheme, $G$ an $S$-group of finite presentation, $T$ an $S$-torus, $\alpha$ a character non-trivial on each fiber of $T$, $s_0$ a point of $S$. Let

f : T ·_α G_{a, S} → G

be an $S$-group morphism such that $f_{s_0}$ is a monomorphism and the restriction of $f$ to $T$ is a monomorphism. There exists an open neighborhood $U$ of $s_0$ such that $f|U$ is a monomorphism.

Lemma 1.7. Let $A$ be a complete noetherian local ring with algebraically closed residue field $k$, $(G,T,\alpha )$ an $A$-elementary system, $p_0:G_{a,k}\to G_k$ a morphism of $k$-groups normalized by $T_k$ with multiplier $\alpha {\otimes }_{A}k$. There exists a group morphism $p:G_{a,A}\to G$ normalized by $T$ with multiplier $\alpha$.

Let $p$ be the morphism whose existence is asserted by 1.7. Let $f:T{\cdot }_{\alpha }{G}_{a,S}\to G$ be the corresponding morphism. It satisfies the hypotheses of 1.6, hence is a monomorphism; in particular $p$ is a monomorphism. One then concludes by Exp. XIX 4.9.

Proof of 1.6. Denote by $ϵ:S\to T{\cdot }_{\alpha }{G}_{a,S}$ the unit section. Since $f$ is unramified at $ϵ(s_0)$, it is so at $ϵ(s)$ for all $s$ in an open neighborhood $U$ of $s_0$; $f|U$ is therefore unramified (Exp. X 3.5)³, hence its kernel $Ker(f{)}_U$ is unramified over $U$. To prove that $f|U$ is a monomorphism, it suffices therefore⁴ to prove that $Ker(f{)}_U$ is radicial over $U$, which is a set-theoretic question. One is thus reduced to proving:

Lemma 1.8. Let $k$ be an algebraically closed field; let $N$ be an invariant subgroup of $T{\cdot }_{\alpha }{G}_{a,k}$ ($\alpha$ a non-trivial character of the torus $T$), étale over $k$ and such that $N\cap T=e$. Then $N=e$.

One has $int(t^{\prime })(t,x)=(t,\alpha (t^{\prime })x)$. If $(t,x)$ is a point of $N$, with $x\ne 0$, then (t, z x) is also a point of $N$ for $z\in k\ast$, and $(t,x)$ is not isolated; hence $N$ is not quasi-finite. One therefore has set-theoretically $N\subset T$, and we are done.

Proof of 1.7. Let $m$ be the radical of $A$, and $S_n=\mathrm{Spec}(A/m^{n+1})$, $n⩾0$. We first show, by induction on $n$, that $p_0$ can be extended for each $n$ to a morphism of $S_n$-groups

p_n : G_{a, S_n} → G_{S_n}

normalized by $T_{S_n}$ with multiplier ${\alpha }_{S_n}$, the $p_n$ further satisfying the condition $p_{n+1}{\times }_{S_{n+1}}{S}_n=p_n$.

Let $H=T{\cdot }_{\alpha }{G}_{a,S}$. The morphism $H_{S_n}\to G_{S_n}$ defined by $p_n$ is denoted $f_n$. Let us admit the following lemma:

Lemma 1.9. If $(G,T,\alpha )$ is a $k$-elementary system, $k$ algebraically closed, and if $p:G_{a,k}\to G$ is a monomorphism normalized by $T$ with multiplier $\alpha$, one has

H²(T ·_α G_{a, k}, g) = 0.

(One lets $T{\cdot }_{\alpha }{G}_{a,k}$ act on $g$ through the morphism $T{\cdot }_{\alpha }{G}_{a,k}\to G$ defined by $p$, and the adjoint representation of $G$).

Then, by virtue of Exp. III 2.8, $f_n$ extends to a morphism of $S_{n+1}$-groups

$$f_{n+1}^{\prime }:H_{S_{n+1}}\to G_{S_{n+1}}.$$

Now $f_{n+1}^{\prime }$ and the canonical immersion of $T_{S_{n+1}}$ into $G_{S_{n+1}}$ have the same restriction to $T_{S_n}$. By Exp. III 2.5, there exists an element $g\in G(S_{n+1})$ such that $g{\times }_{S_{n+1}}{S}_n=e$ and such that $f_{n+1}=int(g)\circ f_{n+1}^{\prime }$ restricts to $T_{n+1}$ along the canonical immersion of $T_{n+1}$. Let $p_{n+1}$ be the restriction of $f_{n+1}$ to $G_{a,S_{n+1}}$. It is a morphism normalized by $T_{S_{n+1}}$ with multiplier ${\alpha }_{S_{n+1}}$, which extends $p_n$.

One has thus constructed a coherent system $(f_n)$ and one must now algebraize it. Now one has:

Lemma 1.10. Let $A$ be a complete noetherian local ring, $m$ its maximal ideal, $S=\mathrm{Spec}(A)$, $S_n=\mathrm{Spec}(A/m^{n+1})$, $T$ an $S$-torus, $\alpha$ a non-zero character of $T$, $X$ an affine $S$-scheme on which $T$ acts. Let $T$ act on $G_{a,S}$ via $\alpha$. Let $q$ be an integer $⩾0$, and let $(f_n{)}_{n⩾0}$ be a coherent system of morphisms

f_n : G_{a, S_n}^q → X_{S_n}

of objects with operators $T_{S_n}$. There exists a unique morphism of objects with operators $T$

f : G_{a, S}^q → X

which induces the $f_n$ (compare with Exp. IX 7.1).

Corollary 1.11. If $X$ is a group with group of operators $T$ and if the $f_n$ are group morphisms, then $f$ is one too.

It suffices to apply the uniqueness assertion of the lemma to the two morphisms $G_{a,S}^{2q}\to X$ deduced from $f$ in the usual way.

Proof of 1.10. Suppose $T$ split, which is moreover the case in the application of 1.10 to the proof of 1.5. One knows (Exp. I 4.7.3.1) that $X\mapsto A(X)$ realizes an equivalence between the category of affine $S$-schemes equipped with a $T$-operation and the opposite category of $S$-graded algebras of type $M={\mathrm{Hom}}_{S-gr.}(T,G_{m,S})$.

One therefore has gradings

B = A(X) = ⨁_{m ∈ M} B_m    and    C = A(G_{a, S}^q) = ⨁_{m ∈ M} C_m.

One sees at once that each $C_m$ is free of finite type over $A$. (Indeed, one has $C_m=0$ if $m$ is not a multiple of $\alpha$, and if $m=d\alpha$, $C_m$ is isomorphic to the $A$-module of homogeneous polynomials of degree $d$, in $q$ variables.) Set

B̂_m = lim_n B_m ⊗_A (A/m^{n+1}),
Ĉ_m = lim_n C_m ⊗_A (A/m^{n+1}),
B̂ = ⨁_{m ∈ M} B̂_m,    Ĉ = ⨁_{m ∈ M} Ĉ_m.

One then has canonical morphisms of $M$-graded algebras

g_B : B → B̂    and    g_C : C → Ĉ.

It follows from the remark made above that $g_C$ is an isomorphism. To give a coherent system $(f_n)$ as in the statement is equivalent to giving a morphism of graded $A$-algebras

$$\hat{F}:\hat{B}\to \hat{C}.$$

To find a morphism $f$ as in the statement is equivalent to finding a morphism of graded $A$-algebras $F:B\to C$ rendering commutative the diagram

B  ──F──→  C
│           │
g_B         g_C
↓           ↓
B̂  ──F̂──→  Ĉ.

Since $g_C$ is an isomorphism, the existence and uniqueness of $F$ are immediate. This proves 1.10.

To complete the proof of 1.5, it remains only to prove 1.9.

1.12. Proof of 1.9. One has $g=t\oplus g_{\alpha }\oplus g_{-\alpha }$. As explained in 1.9, consider $g$ as a $(T{\cdot }_{\alpha }{G}_{a,k})$-module. It is clear that $t\oplus g_{\alpha }$ is a submodule of $g$, the quotient being isomorphic to $g_{-\alpha }$ as a $k$-vector space and even as a $T$-module. It is clear that $G_{a,k}$ acts trivially on this quotient, which is of dimension 1 (for every group morphism from $G_{a,k}$ to $G_{m,k}$ is trivial). Similarly $g_{\alpha }$ is a submodule of $t\oplus g_{\alpha }$, the quotient being isomorphic to $t$ as a $T$-module, $G_{a,k}$ acting trivially on it. To summarize:

Lemma 1.13. Under the conditions of 1.9, $g$ admits a composition series as $(T{\cdot }_{\alpha }{G}_{a,k})$-module whose successive quotients are

$$g_{-\alpha },t,g_{\alpha },$$

viewed as $(T{\cdot }_{\alpha }{G}_{a,k})$-modules via the projection $T{\cdot }_{\alpha }{G}_{a,k}\to T$.

One is therefore reduced to computing the cohomology of $T{\cdot }_{\alpha }{G}_{a,k}$ acting via the projection $T{\cdot }_{\alpha }{G}_{a,k}\to T$ and the character $\beta$ of $T$ (here $\beta =0$, $\alpha$, or $-\alpha$) on $W(k)$.⁵ Let $k[x_1,\cdots ,x_n]$ denote the algebra of polynomials over $k$ in $n$ variables and $k_q[x_1,\cdots ,x_n]$ the subspace of homogeneous polynomials of degree $q$.

Lemma 1.14. With the preceding notations, one has $H^n(T{\cdot }_{\alpha }{G}_{a,k},k)=H^n(C_{\alpha ,\beta }^{\ast })$, where the complex $C_{\alpha ,\beta }^{\ast }$ is defined by

C^n_{α, β} = { k_q[x_1, …, x_n]   if β = q α, with q ∈ ℕ*;
             { 0                  otherwise,

and

δf(x_1, x_2, …, x_{n+1}) = f(x_2, …, x_{n+1})
                         + Σ_{i=1}^n (−1)^i f(x_1, …, x_i x_{i+1}, …, x_{n+1})
                         + (−1)^{n+1} f(x_1, …, x_n).

Indeed, the functor $M\mapsto H^0(T,M)$ is exact on the category of $T$-modules (and the $H^q(T,-)$ vanish), by Exp. I 5.3.2. It follows, as in the usual case of group cohomology, that $H^n(T{\cdot }_{\alpha }{G}_{a,k},k)$ can be computed as the $n$-th cohomology group of the complex of cochains of $G_{a,k}$ in $k$, invariant under $T$, i.e. satisfying

f(α(t) x_1, …, α(t) x_n) = β(t) f(x_1, …, x_n).

This indeed gives the announced complex.

To prove 1.9, it therefore suffices to prove that $H^2(C_{\alpha ,\beta }^{\ast })=0$ for $\beta =0,\alpha ,-\alpha$, which is done at once.

Remark 1.15. One can explicitly compute the groups $H^n(C_{\alpha ,\beta }^{\ast })$ for $\beta =q\alpha$ (see M. Lazard, Lois de groupes et analyseurs*, Annales E.N.S., 1955). In particular, one finds $H^n(C_{\alpha ,q\alpha }^{\ast })=0$ for $n>q$.*

Notations 1.16. The image of the canonical immersion

W(g_{−α}) ×_S T ×_S W(g_α) → G

will be denoted $\Omega$. It is an open subset of $G$ containing the unit section. The image of

W(g_{−α}),   resp. W(g_α),   resp. W(g_{−α}) ×_S T,   resp. T ×_S W(g_α)

will be denoted⁶

U_{−α},   resp. U_α,   resp. U_{−α} · T,   resp. T · U_α.

Then $U_{\alpha }$ (resp. $U_{-\alpha }$) is a subgroup of $G$ canonically endowed with a vector bundle structure, and one has

int(t)(x) = x^{α(t)}    (resp. x^{−α(t)}),

for every $S^{\prime }\to S$, $t\in T(S^{\prime })$, $x\in U_{\alpha }(S^{\prime })$ (resp. $x\in U_{-\alpha }(S^{\prime })$).

One has canonical isomorphisms

T · U_α ≅ T ·_α U_α    and    T · U_{−α} ≅ T ·_{−α} U_{−α}.

The open set $\Omega$ is stable under $int(T)$: one has

int(t′)(y · t · x) = y^{−α(t′)} · t · x^{α(t′)}.

Corollary 1.17. One has $Lie(U_{\alpha }/S)=g_{\alpha }$ and $Lie(U_{-\alpha }/S)=g_{-\alpha }$. The isomorphisms

W(g_α)  ──exp──→  U_α    and    W(g_{−α})  ──exp──→  U_{−α}

are those of Exp. XIX 4.2.

Corollary 1.18. The open set $\Omega$ is relatively schematically dense in $G$ (cf. XVIII, § 1).

Clear by Exp. XVIII, 1.3.⁷

Corollary 1.19. The center of $G$ is $Centr(G)=Ker(\alpha )$. It is therefore a closed subgroup of $G$, of multiplicative type and of finite type.

The second assertion follows from the first by Exp. IX 2.7. Let us therefore prove the first. The inner automorphism defined by a section of $Ker(\alpha )$ acts trivially on $\Omega$ (last formula of 1.16), hence on $G$ by 1.18. Conversely, if $g\in G(S)$ centralizes $G$, then it centralizes $T$ and $U_{\alpha }$, hence is a section of $T$ (Exp. XIX 2.8) annihilating $\alpha$; since this also holds after any base change, one indeed has $Centr(G)=Ker(\alpha )$.

Corollary 1.20. For there to exist a monomorphism $p_{\alpha }:G_{a,S}\to G$ normalized by $T$ with multiplier $\alpha$, it is necessary and sufficient that the O_S-module $g_{\alpha }$ be free. More precisely, one has a bijection given by

X_α ↦ (x ↦ exp(x X_α))    and    p_α ↦ Lie(p_α)

between $\Gamma (S,g_{\alpha }{)}^{\times }$ and the set of monomorphisms $p_{\alpha }$ as above (which is also the set of isomorphisms of vector group schemes $G_{a,S}\stackrel{\sim }{\to }{U}_{\alpha }$).⁸

Corollary 1.21. The subgroups $U_{\alpha }$ and $U_{-\alpha }$ of $G$ commute on no fiber.

Indeed, if $(U_{\alpha }{)}_s$ and $(U_{-\alpha }{)}_s$ commute, then ${\Omega }_s$ is a subgroup of $G_s$, hence ${\Omega }_s=G_s$⁹ and $G_s$ is solvable, which contradicts the hypothesis that $G_s$ is reductive of semisimple rank 1.

2. Structure of elementary systems

Theorem 2.1. Let $S$ be a scheme, $(G,T,\alpha )$ an $S$-elementary system. There exists a morphism of O_S-modules

g_α ⊗_{O_S} g_{−α} → O_S,   (X, Y) ↦ ⟨X, Y⟩,

and a morphism of $S$-groups

α* : G_{m, S} → T

such that for every $S^{\prime }\to S$ and all $X\in \Gamma (S^{\prime },g_{\alpha }\otimes O_{S^{\prime }})$, $Y\in \Gamma (S^{\prime },g_{-\alpha }\otimes O_{S^{\prime }})$ one has the equivalence:

exp(X) · exp(Y) ∈ Ω(S′) ⇐⇒ 1 + ⟨X, Y⟩ ∈ G_m(S′),

and under these conditions one has the formula:

(F)    exp(X) · exp(Y) = exp( Y / (1 + ⟨X, Y⟩) ) · α*(1 + ⟨X, Y⟩) · exp( X / (1 + ⟨X, Y⟩) ).

Moreover, the morphisms $(X,Y)\mapsto ⟨X,Y⟩$ and $\alpha \ast$ are uniquely determined, the former is an isomorphism, hence puts the modules $g_{\alpha }$ and $g_{-\alpha }$ in duality, and one has $\alpha \circ \alpha \ast =2$ (squaring in $G_{m,S}$).

In view of the uniqueness assertions of the theorem, it suffices to do the proof locally on $S$. One can therefore assume $g_{\alpha }$ and $g_{-\alpha }$ free on $S$. Take then $X\in \Gamma (S,g_{\alpha }{)}^{\times }$, $Y\in \Gamma (S,g_{-\alpha }{)}^{\times }$ and set $p_{\alpha }(x)=\exp (xX)$, $p_{-\alpha }(y)=\exp (yY)$, for $x,y\in G_a(S^{\prime })$, $S^{\prime }\to S$. By 1.5 and 1.21, it suffices to prove:

Proposition 2.2. Let $S$ be a scheme, $G$ an $S$-group, $T$ a torus of $G$, $\alpha$ a character of $T$ non-trivial on each fiber, $p_{\alpha }:G_{a,S}\to G$ (resp. $p_{-\alpha }:G_{a,S}\to G$) a group monomorphism normalized by $T$ with multiplier $\alpha$ (resp. $-\alpha$). Suppose that:

(i) The morphism $G_{a,S}{\times }_{S}T{\times }_S{G}_{a,S}\to G$ defined by $(y,t,x)\mapsto p_{-\alpha }(y)t{p}_{\alpha }(x)$ is an open immersion. Denote its image by $\Omega$.

(ii) For every $s\in S$, $(p_{\alpha }{)}_s(G_{a,\kappa (s)})$ and $(p_{-\alpha }{)}_s(G_{a,\kappa (s)})$ do not commute.

Then there exist $a\in G_a(S)$ and $\alpha \ast \in {\mathrm{Hom}}_{S-gr.}(G_{m,S},T)$, uniquely determined, having the following properties: for every $S^{\prime }\to S$ and all $x,y\in G_a(S^{\prime })$, one has

p_α(x) p_{−α}(y) ∈ Ω(S′) ⇐⇒ 1 + a x y ∈ G_m(S′),

and, under this condition, one has the formula

p_α(x) p_{−α}(y) = p_{−α}( y / (1 + a x y) ) · α*(1 + a x y) · p_α( x / (1 + a x y) ).

Moreover, $a$ is invertible (i.e. $a\in G_m(S)$) and $\alpha \circ \alpha \ast =2$.

Proof:

A) Consider the morphism

$$G_{a,S}^2\to G$$

defined by $(x,y)\mapsto p_{\alpha }(x)p_{-\alpha }(y)$. Let $U$ be the inverse image of $\Omega$ under this morphism. It is an open subset of $G_{a,S}^2$ containing $0{\times }_S{G}_{a,S}$ and $G_{a,S}{\times }_{S}0$. There therefore exist uniquely determined $S$-scheme morphisms

A : U → G_{a, S},   C : U → G_{a, S},   B : U → T

satisfying the set-theoretic relation:

p_α(u) p_{−α}(v) = p_{−α}(A(u, v)) B(u, v) p_α(C(u, v)).

One has immediately the relations

A(0, v) = v,   A(u, 0) = 0,   C(u, 0) = u,   C(0, v) = 0,
B(u, 0) = B(0, v) = e.

Let $S^{\prime }$ be a separated $S$-scheme and let $t\in T(S^{\prime })$ be a point of $T$. Since ${\Omega }_{S^{\prime }}$ is stable under $int(t)$, then, by the last formula of 1.16, $U_{S^{\prime }}$ is stable under the automorphism $(x,y)\mapsto (\alpha (t)x,\alpha (t{)}^{-1}y)$ of $G_{a,S^{\prime }}^2$, and one has the relations:

A(α(t) u, α(t)⁻¹ v) = α(t)⁻¹ A(u, v),
C(α(t) u, α(t)⁻¹ v) = α(t) C(u, v),
B(α(t) u, α(t)⁻¹ v) = B(u, v).

Since $\alpha$ is faithfully flat, one deduces that for every $S^{\prime }\to S$ and every $z\in G_m(S^{\prime })$, $U_{S^{\prime }}$ is stable under the transformation $(x,y)\mapsto (zx,z^{-1}y)$ and one has

A(z u, z⁻¹ v) = z⁻¹ A(u, v),
C(z u, z⁻¹ v) = z C(u, v),
B(z u, z⁻¹ v) = B(u, v).

Suppose first that $v$ is invertible; setting $z=v$, one deduces that if $(u,v)$ is a section of $U$, then (u v, 1) is also one, and one has

A(u v, 1) = v⁻¹ A(u, v),    B(u v, 1) = B(u, v).

Let then $V$ be the open set of $G_{a,S}^2$ defined by¹⁰

(u, v) ∈ V(S′) ⇐⇒ (u, v), (u v, 1) and (1, u v) belong to U(S′).

Since $U$ is an open set of $G_{a,S}^2$ containing $0{\times }_S{G}_{a,S}$ and $G_{a,S}{\times }_{S}0$, then $V$ is a neighborhood of the zero section of $G_{a,S}^2$ and we have just seen that the morphisms

(u, v) ↦ A(u, v) and (u, v) ↦ v A(u v, 1)
resp. (u, v) ↦ B(u, v) and (u, v) ↦ B(u v, 1)

coincide on $V\cap (G_{a,S}{\times }_S{G}_{m,S})$. Since $G_{a,S}{\times }_S{G}_{m,S}$ is schematically dense in $G_{a,S}^2$, these morphisms therefore coincide on $V$.

One knows that $A(0,1)=1$; it follows that there exists an open set W_1 of $G_{a,S}$ containing the zero section such that for every section $x$ of W_1, $A(x,1)$ is invertible; setting $A(x,1{)}^{-1}=F(x)$, one obtains that if $(u,v)\in V(S^{\prime })$¹¹ and $uv\in W_1(S^{\prime })$, $S^{\prime }\to S$, then $A(u,v)=vA(uv,1)=vF(uv{)}^{-1}$. Arguing similarly for $C$, one obtains that there exists an open set W_2 of $G_{a,S}$ containing the zero section, and an element $E$¹² of $O(W_2{)}^{\times }$, such that $C(u,v)=uC(1,uv)=uE(uv{)}^{-1}$, if $(u,v)\in V(S^{\prime })$ and $uv\in W_2(S^{\prime })$. Consequently, setting $W=W_1\cap W_2$, one obtains:

There exists an open set $W$ of $G_{a,S}$ containing the zero section, and $S$-morphisms

F : W → G_{m, S},   F(0) = 1,
H : W → T,         H(0) = e,
E : W → G_{m, S},   E(0) = 1,

such that if $(u,v)\in V(S^{\prime })$ and $uv\in W(S^{\prime })$, $S^{\prime }\to S$, one has

(+)    p_α(u) p_{−α}(v) = p_{−α}(v F(u v)⁻¹) H(u v) p_α(u E(u v)⁻¹).

B) Let us now use the associativity of $G$ to write

p_α(u) p_{−α}(v) p_{−α}(w) = p_α(u) p_{−α}(v + w).

There exists an open set $L$ of $G_{a,S}^3$, containing the unit section, such that $(u,v,w)\in L(S^{\prime })$ is equivalent to

(u, v) ∈ V(S′),   (u E(u v)⁻¹, w) ∈ V(S′),   (u, v + w) ∈ V(S′),
u v ∈ W(S′),     u w E(u v)⁻¹ ∈ W(S′),       u(v + w) ∈ W(S′).

Using then the formula (+), one writes at once for $(u,v,w)\in L(S^{\prime })$ the relations:

(1)   E(u v + u w) = E(u w E(u v)⁻¹) E(u v),
(2)   H(u v + u w) = H(u w E(u v)⁻¹) H(u v),
(3)   (v + w) F(u v + u w)⁻¹ = α(H(u v)⁻¹) w F(u w E(u v)⁻¹)⁻¹ + v F(u v)⁻¹.

It is immediate from the definition of $L$ that $(1,0,0)\in L(S)$. Consider therefore

L ∩_T (1 ×_S G_{a, S} ×_S G_{a, S}) = 1 ×_S M;

$M$ is an open set of $G_{a,S}^2$, containing the section (0, 0), and for $(v,w)\in M(S^{\prime })$, one has $v,wE(v{)}^{-1},v+w\in W(S^{\prime })$ and

(1′)   E(v + w) = E(w E(v)⁻¹) E(v),
(2′)   H(v + w) = H(w E(v)⁻¹) H(v),
(3′)   (v + w) F(v + w)⁻¹ = α(H(v))⁻¹ w F(w E(v)⁻¹)⁻¹ + v F(v)⁻¹.

Consider finally the morphism from $M$ to $G_{a,S}^2$ defined set-theoretically by $(v,w)\mapsto (v,wE(v{)}^{-1})$.¹³ It preserves the section (0, 0) and induces an isomorphism of $M$ onto an open set $N$ of $G_{a,S}^2$ containing the zero section (the inverse isomorphism being given by $(x,y)\mapsto (x,yE(x))$).¹⁴ One has thus proved the following assertion:

There exists an open set $N$ of $G_{a,S}^2$, containing the zero section, such that if $(x,y)\in N(S^{\prime })$, then x, y and $x+yE(x)$¹⁵ belong to $W(S^{\prime })$ and:

(1″)   E(x + y E(x)) = E(x) E(y),
(2″)   H(x + y E(x)) = H(x) H(y),
(3″)   (x + y E(x)) F(x + y E(x))⁻¹ = x F(x)⁻¹ + α(H(x))⁻¹ y E(x) F(y)⁻¹.

C) Arguing similarly with left associativity, one demonstrates the following assertion:¹⁶

There exists an open set $N^{\prime }$ of $G_{a,S}^2$, containing the zero section, such that if $(x,y)\in N^{\prime }(S^{\prime })$, then x, y and $x+yF(x)$¹⁷ belong to $W(S^{\prime })$, and

(4″)   F(x + y F(x)) = F(x) F(y),
(5″)   H(x + y F(x)) = H(x) H(y),
(6″)   (x + y F(x)) E(x + y F(x))⁻¹ = x E(x)⁻¹ + α(H(x))⁻¹ y F(x) E(y)⁻¹.

We are therefore led to solve the "functional equation" (1″).

Lemma 2.3. Let $S$ be a scheme, $W$ an open set of $G_{a,S}$ containing the unit section, $F:W\to G_{m,S}$ an $S$-morphism. Suppose that $F(0)=1$ and that there exists an open set $N$ of $G_{a,S}^2$ containing the zero section such that for $(x,y)\in N(S^{\prime })$, x, y, and $x+yF(x)$¹⁷ belong to $W(S^{\prime })$ and that one has:

(†)    F(x + y F(x)) = F(x) F(y).

(i) If $S$ is the spectrum of a field $k$, there exists $a\in k$ such that $F(x)=1+ax$.

(ii) If $a=F^{\prime }(0)\in \Gamma (S,O_S)$ is invertible, then $F(x)=1+ax$.

By the hypotheses, we can differentiate the given equation at $x=0$ (resp. at $y=0$) and we find that

(∗)    F′(y) (1 + y F′(0)) = F′(0) F(y)    for (0, y) ∈ N(S′),

resp.

F′(x) F(x) = F(x) F′(0)    for (x, 0) ∈ N(S′).

Since $F$ takes its values in $G_m$, the second relation gives us

(∗′)   F′(x) = F′(0)    for (x, 0) ∈ N(S′);

whence, by the first,

F′(0)(1 + y F′(0)) = F′(0) F(y)    for (y, 0), (0, y) ∈ N(S′).

If $a=F^{\prime }(0)$ is invertible, this gives us

F(y) = 1 + a y,

for $y$ a section of an open set of $W$ containing the unit section, hence schematically dense in $W$, which proves (ii). This also proves (i) when $F^{\prime }(0)\ne 0$.

If $F^{\prime }(0)=0$, then, by (∗′), $F^{\prime }(x)=0$ when $x$ is "near 0", hence $F^{\prime }=0$ by schematic density. If $k$ is of characteristic 0, $F$ is a rational fraction with zero derivative, hence constant and equal to $F(0)=1$.

If $k$ is of characteristic $p$, and if $F$ is not constant,¹⁸ there exists an integer $n>0$ and a rational fraction $F_1\in k(X)$ such that $F_1^{\prime }(X)\ne 0$ and

$$F(X)=F_1(X^{p^n})=F_1(X{)}^{p^n}.$$

Substituting in the functional equation, one finds

(†_1)   F_1(x + y F_1(x)^{p^n}) = F_1(x) F_1(y).

Differentiating at $x=0$, one finds

$$({\ast }_1)F_1^{\prime }(y)=F_1^{\prime }(0)F_1(y),$$

and differentiating (†_1) at $y=0$, one obtains

(∗′_1)  F′_1(x) F_1(x)^{p^n} = F_1(x) F′_1(0).

Since, by hypothesis, $F_1^{\prime }(X)$ is an invertible element of $k(X)$, one deduces from these two equalities that

$$F_1(X{)}^{p^n}=1,$$

hence F_1 is a constant, contradicting the initial hypothesis. This shows that $F$ is constant, and equal to $1=F(0)$.

D) Suppose $S$ is the spectrum of a field. If $F^{\prime }(0)=0$, then $F=1$. Formula (5″) then gives us $H(x+y)=H(x)H(y)$, which shows that $H$ extends to a group morphism $G_{a,S}\to T$ (Exp. XVIII 2.3), which is necessarily constant of value $e$. On the other hand, by Lemma 2.3, one will also have $E(x)=1+bx$ for some $b\in k$. But then (6″) gives, for $(x,y)\in N(S^{\prime })$,

(x + y) E(x + y)⁻¹ = x E(x)⁻¹ + y E(y)⁻¹,

hence,¹⁹ by Exp. XVIII 2.3 again, $x\mapsto xE(x{)}^{-1}$ extends to a morphism of $k$-groups $G_{a,k}\to G_{a,k}$, hence $x/(1+bx)=cx$ for some $c\in k$, whence $b=0$ (and $c=1$).

This shows that F, H, E are constant of value (1, e, 1) in a neighborhood of the unit section, hence everywhere, which by (+) shows that $U_{\alpha }$ and $U_{-\alpha }$ commute, contrary to hypothesis (ii).

If $S$ is now arbitrary, one has therefore proved that $F^{\prime }(0)$ is non-zero on no fiber, hence is invertible. The same evidently holds for $E^{\prime }(0)$, which by Lemma 2.3 shows that there exist $a,b\in G_m(S)$ such that

(♦_1)   F(x) = 1 + a x,    E(x) = 1 + b x,    for x ∈ W(S′).

E) The rest is now easy. Substituting the preceding results into (3″), one finds

y α(H(x)) (1 + a y) = y (1 + a x + a y(1 + b x)) (1 + a x).

This formula is valid for every section $(x,y)$ of $N$. But since $G_{a,S}{\times }_S{G}_{m,S}$ is schematically dense in $G_{a,S}^2$, one deduces

(1 + a y) α(H(x)) = (1 + a x + a y(1 + b x)) (1 + a x).

Setting $y=0$, this gives $\alpha (H(x))=(1+ax{)}^2$. Substituting this into the preceding equality, one finds²⁰

a² x y = a b x y.

Since $G_{m,S}$ is schematically dense in $G_{a,S}$, one deduces $a^2=ab$, whence, since $a$ is invertible,

(♦_2)   a = b    and    α(H(x)) = (1 + a x)².

Since $a$ is invertible, $x\mapsto 1+ax$ is an automorphism of $G_{a,S}$; one can therefore find an open set $W^{\prime }$ of $G_{a,S}$ containing the section 1 and a morphism

$$P:W^{\prime }\to T$$

such that $P(1+ax)=H(x)$.²¹

Substituting in the relation (2′), one finds at once for $(x,y)\in N(S^{\prime })$,

P(1 + a x + a y) = P((1 + a x + a y) / (1 + a x)) P(1 + a x),

which proves that there exists an open neighborhood of 1 in $G_{m,S}$ such that one has for $x$ and $y$ in this neighborhood $P(x)P(y)=P(xy)$. By Exp. XVIII 2.3, there exists a group morphism

(♦_3)   α* : G_{m, S} → T

extending $P$. Since $\alpha (H(x))=(1+ax{)}^2$ near the section 0, one has $\alpha (\alpha \ast (z))=z^2$ near the section 1, hence

(♦_4)   α ∘ α* = 2.

F)²² Assembling the results (+) and (♦_1 — ♦_4), one sees that there exist $a\in G_m(S)$ and $\alpha \ast \in {\mathrm{Hom}}_{S-gr.}(G_{m,S},T)$ such that $\alpha \circ \alpha \ast =2$ and that, if $(u,v)\in V(S^{\prime })$ and $uv\in W(S^{\prime })$, then $1+auv$ is invertible and

p_α(u) p_{−α}(v) = p_{−α}( v / (1 + a u v) ) · α*(1 + a u v) · p_α( u / (1 + a u v) ).

Consider the open set $V^{\prime }$ of $G_{a,S}^2$ defined by "$1+auv$ invertible", i.e. $V^{\prime }=(G_{a,S}^2{)}_f$ where $f(u,v)=1+auv$. The two sides of the preceding formula define morphisms from $V^{\prime }$ to $G$ which coincide in a neighborhood of the section 0, hence coincide on $V^{\prime }$. The preceding formula is therefore valid for every section $(u,v)$ of $V^{\prime }$. In particular, it follows that $V^{\prime }\subset U$, where $U$ is the open set introduced at the beginning of A).

Let us prove that $U=V^{\prime }$. Returning to the notations of A), one has a morphism

A : U → G_{a, S}

which, on $V^{\prime }$, is defined by $A(u,v)=v(1+auv{)}^{-1}$. To show that $U=V^{\prime }$, which is a set-theoretic question, one is reduced to the case where $S$ is the spectrum of a field $k$, hence to the obvious assertion: the domain of definition of the rational map $G_{a,k}^2\to G_{a,k}$ defined by the rational fraction $Y/(1+aXY)$ is the open set defined by the function $1+aXY$.

G) One has thus proved the existence of $a$ and $\alpha \ast$, as well as the two additional properties announced. It remains to prove uniqueness. Let then $a^{\prime }$ and $\alpha {\ast }^{\prime }$ also satisfy the required conditions. If $u,v\in G_a(S^{\prime }{)}^2$, one has at once:

1 + a u v invertible ⇒ 1 + a′ u v invertible and v / (1 + a u v) = v / (1 + a′ u v);

one therefore has for every section $u$ of $G_a(S^{\prime })$

1 + a u invertible ⇒ 1 + a u = 1 + a′ u,

which proves at once $a=a^{\prime }$.

With the same notations, one then has

1 + a u invertible ⇒ α*(1 + a u) = α*′(1 + a u),

hence also $\alpha \ast =\alpha {\ast }^{\prime }$.

Corollary 2.4. Let $\exp (Y)t\exp (X)$ and $\exp (Y^{\prime })t^{\prime }\exp (X^{\prime })$ be two elements of $\Omega (S^{\prime })$. Then their product is in $\Omega (S^{\prime })$ if and only if $u=1+⟨X,Y^{\prime }⟩$ is invertible, and one has then

(F′)    exp(Y) t exp(X) · exp(Y′) t′ exp(X′) =
             exp(Y + u⁻¹ α(t)⁻¹ Y′) · t t′ α*(u) · exp(u⁻¹ α(t′)⁻¹ X + X′).

Remark 2.5. One can also write formula (F) of Theorem 2.1 without invoking the morphisms exp. Indeed, transporting through these morphisms the duality $g_{\alpha }\otimes g_{-\alpha }\to O_S$, one obtains a canonical pairing of vector bundles:

U_α ×_S U_{−α} → G_{a, S},

which we shall still denote $(x,y)\mapsto ⟨x,y⟩$. One therefore has

⟨exp X, exp Y⟩ = ⟨X, Y⟩.

If $x\in U_{\alpha }(S^{\prime })$, $y\in U_{-\alpha }(S^{\prime })$ and if $1+⟨x,y⟩\in G_m(S^{\prime })$, one has

(F)    x · y = y^{(1 + ⟨x, y⟩)⁻¹} · α*(1 + ⟨x, y⟩) · x^{(1 + ⟨x, y⟩)⁻¹}.

Corollary 2.6. The pairing

W(g_α) ×_S W(g_{−α}) → G_{a, S}

defines a pairing of principal $G_{m,S}$-bundles

W(g_α)^× ×_S W(g_{−α})^× → G_{m, S}.

This pairing will be denoted $(X,Y)\mapsto ⟨X,Y⟩$, or more simply $(X,Y)\mapsto XY$.

For every section $X\in \Gamma (S,g_{\alpha }{)}^{\times }$, there therefore exists a unique section $X^{-1}$ of $\Gamma (S,g_{-\alpha }{)}^{\times }$ such that $X{X}^{-1}=1$. One has $(zX{)}^{-1}=z^{-1}{X}^{-1}$. The morphism

$$s:W(g_{\alpha }{)}^{\times }\to W(g_{-\alpha }{)}^{\times }$$

thus defined is therefore an isomorphism of schemes, compatible with the isomorphism $s:z\mapsto z^{-1}$ on the operator groups.

Definition 2.6.1. One says that $X$ and $s(X)=X^{-1}$ are paired.

Apply Corollary 2.4 to $Y=0=X^{\prime }$ and $Y^{\prime }=a{X}^{-1}$, $a\in O_S(S)$. Then $u=1+a$ and $u^{-1}{Y}^{\prime }=u^{-1}(u-1)X^{-1}=(1-u^{-1})X^{-1}$, whence:

Corollary 2.7. Let $X\in \Gamma (S,g_{\alpha }{)}^{\times }$ and $u\in \Gamma (S,O_S{)}^{\times }$. One has

α*(u) = exp((u⁻¹ − 1) X⁻¹) exp(X) exp((u − 1) X⁻¹) exp(−u⁻¹ X).

Definition 2.8. The morphism $\alpha \ast$ is called the coroot associated with the root $\alpha$.

Remark 2.9. If $(G,T,\alpha )$ is an $S$-elementary system, $(G,T,-\alpha )$ is also one. By Theorem 2.1 one therefore has a duality between $g_{-\alpha }$ and $g_{\alpha }$, and a coroot $(-\alpha )\ast$. Taking the inverse of formula (F), one proves at once

⟨X, Y⟩ = ⟨Y, X⟩,    (−α)* = −α*.

Let us now pass to the Lie algebra of $G$. The root $\alpha$ and the coroot $\alpha \ast$ define the linear forms

O_S  ──α*──→  t  ──α──→  O_S.

One will denote $H_{\alpha }=\alpha \ast (1)$. One calls $\alpha$ the infinitesimal root associated with $\alpha$, and $H_{\alpha }$ the corresponding infinitesimal coroot.

Lemma 2.10. Let $S^{\prime }\to S$ and $X,X^{\prime }\in W(g_{\alpha })(S^{\prime })$, $H\in W(t)(S^{\prime })$, $Y,Y^{\prime }\in W(g_{-\alpha })(S^{\prime })$, $t\in T(S^{\prime })$. One has

(1)   Ad(t) H = H,    Ad(t) X = α(t) X,    Ad(t) Y = α(t)⁻¹ Y.

(2)   { Ad(exp(X)) H = H − α(H) X,    Ad(exp(X)) X′ = X′,
      { Ad(exp(X)) Y = Y + ⟨X, Y⟩ H_α − ⟨X, Y⟩ X.

(2′)  { Ad(exp(Y)) H = H + α(H) Y,    Ad(exp(Y)) Y′ = Y′,
      { Ad(exp(Y)) X = X + ⟨X, Y⟩ H_{−α} − ⟨X, Y⟩ Y.

(3)   [H, X] = α(H) X,    [H, Y] = −α(H) Y,    [X, Y] = ⟨X, Y⟩ H_α.

$$(4)H_{-\alpha }=-H_{\alpha }.$$

$$(5)\alpha (H_{\alpha })=2.$$

The proof of these various formulas is either trivial or an immediate consequence of formula (F) of 2.1.

Corollary 2.11. Suppose $H_{\alpha }$ is non-zero on every fiber (which is in particular the case if 2 is invertible on $S$, by (5)). Then $X_{\alpha }\in \Gamma (S,g_{\alpha }{)}^{\times }$ and $X_{-\alpha }\in \Gamma (S,g_{-\alpha }{)}^{\times }$ are paired if and only if $[X_{\alpha },X_{-\alpha }]=H_{\alpha }$.

2.12. Let $(G,T,\alpha )$ be an $S$-elementary system. We know (1.19) that the center of $G$ is $Centr(G)=Ker(\alpha )$, a group of multiplicative type and of finite type. If $Q$ is a subgroup of multiplicative type of $Centr(G)$, the quotient $G/Q$ is affine over $S$ (Exp. IX 2.5), smooth over $S$ (Exp. VI_B 9.2) with connected reductive fibers of semisimple rank 1 (Exp. XIX 1.8).

Set $G^{\prime }=G/Q$; this is a reductive $S$-group of semisimple rank 1; $T^{\prime }=T/Q$ is a maximal torus of it. The open set $U_{-\alpha }\cdot T\cdot U_{\alpha }$ of $G$ is stable under $Q$, and one sees at once that the quotient is isomorphic to $U_{-\alpha }{\times }_S(T/Q){\times }_S{U}_{\alpha }$. If one denotes by ${\alpha }^{\prime }$ the character of $T^{\prime }$ induced by $\alpha$, it follows that the morphism derived from the canonical morphism $G\to G^{\prime }$ induces isomorphisms

g_α  ──∼──→  g′_{α′}    and    g_{−α}  ──∼──→  g′_{−α′}.

In particular, ${\alpha }^{\prime }$ is a root of $G^{\prime }$ with respect to $T^{\prime }$. Hence, denoting by $\alpha /Q$ the character $T/Q\to G_{m,S}$ induced by $\alpha$, one has:

Lemma 2.13. If $Q$ is a subgroup of multiplicative type of $Ker(\alpha )$, then

$$(G/Q,T/Q,\alpha /Q)$$

is an elementary system.

Lemma 2.14. Under the preceding conditions, the following diagrams are commutative:

W(g_α)  ──exp──→  G  ←──exp──  W(g_{−α})
   │                │                │
   can ≀            can              can ≀
   ↓                ↓                ↓
W(g′_{α′})  ──exp──→  G′  ←──exp──  W(g′_{−α′})

g_α ⊗ g_{−α}  ──∼──→  O_S
     │                  │
     can ≀              id
     ↓                  ↓
g′_{α′} ⊗ g′_{−α′}  ──∼──→  O_S

              T
        α*  ↗  ↘ α
G_{m, S}        G_{m, S}
        ↓ can   ↑
G_{m, S}  ⤴  ⤵  G_{m, S}
       α′*  ↘  ↗ α′
              T′

3. The Weyl group

Notations 3.0.²³ If $(G,T,\alpha )$ is an $S$-elementary system, one will denote

N = Norm_G(T),    W = Norm_G(T) / T,

(cf. Exp. XIX 6.3); $N$ is a closed subgroup of $G$, smooth over $S$. One will denote by $N^{\times }=N-T$ the open subscheme of $N$ induced on the complement of $T$.²⁴ Let $R$ be the (unique) maximal torus of $Ker(\alpha )$, and $T^{\prime }$ the image of $\alpha \ast :G_{m,S}\to T$, which is a subtorus of dimension 1 of $T$.

The morphism

T′ ×_S R → T

induced by the product in $T$ is surjective (hence faithfully flat); indeed, one is reduced to checking this on the geometric fibers, and this follows at once from the formula $\alpha \circ \alpha \ast =2$.

Theorem 3.1. With the preceding notations:

(i) $W$ is isomorphic to the constant group $(\mathbb{Z}/2\mathbb{Z}{)}_S$.

(ii) $N^{\times }$ is a principal homogeneous bundle locally trivial under $T$, on the left by the law $(t,q)\mapsto tq$ (resp. on the right by the law $(q,t)\mapsto qt$).

(iii) One has the formula

int(w) t = t · α*(α(t)⁻¹)

for $w\in N^{\times }(S^{\prime })$, $t\in T(S^{\prime })$, $S^{\prime }\to S$. In the decomposition $T_{S^{\prime }}=T_{S^{\prime }}^{\prime }\cdot R_{S^{\prime }}$, $int(w)$ induces the identity on $R_{S^{\prime }}$ and the symmetry on $T_{S^{\prime }}^{\prime }$. One has the relations

α ∘ int(w) = α⁻¹,    int(w) ∘ α* = (α*)⁻¹.

(iv) For $X\in W(g_{\alpha }{)}^{\times }(S^{\prime })$, $S^{\prime }\to S$, set

w_α(X) = exp(X) exp(−X⁻¹) exp(X).

Then $w_{\alpha }(X)\in N^{\times }(S^{\prime })$ and the morphism $w_{\alpha }:W(g_{\alpha }{)}^{\times }\to N^{\times }$ thus defined satisfies

w_α(z X) = α*(z) w_α(X) = w_α(X) α*(z)⁻¹,

for $z\in G_m(S^{\prime })$, $X\in W(g_{\alpha }{)}^{\times }(S^{\prime })$, $S^{\prime }\to S$.

(v)²⁵ For $X,Y\in W(g_{\alpha }{)}^{\times }(S^{\prime })$, $S^{\prime }\to S$, one has, with the notations of 2.6,

w_α(X) w_α(Y) = α∨(−X Y⁻¹).

In particular,

w_α(X)² = α*(−1) ∈ ₂T(S) ∩ Centr(G)(S),
w_α(X)⁻¹ = w_α(−X) = α*(−1) w_α(X).

(vi) If one defines analogously, for $Y\in W(g_{-\alpha }{)}^{\times }(S^{\prime })$,

w_{−α}(Y) = exp(Y) exp(−Y⁻¹) exp(Y),

one has (in addition to formulas analogous to the preceding)

w_{−α}(X⁻¹) = w_α(X)⁻¹ = w_α(−X),
w_α(X) w_{−α}(Y) = α*(X Y).

Proof. (i) has already been seen in Exp. XIX 2.4; it follows at once that $N^{\times }$ is indeed a principal homogeneous bundle under $T$ for the laws defined in (ii); the fact that it is locally trivial (for the Zariski topology) follows notably from (iv).

Let us prove (iii); if $w\in N^{\times }(S)$, it is clear that $\alpha \circ int(w)$ is a root of $G$ with respect to $T$, hence is locally equal to $\alpha$ or $-\alpha$; since on each fiber it is $-\alpha$ (Bible, 12-05, proof of the corollary to prop. 1), one has $\alpha \circ int(w)=-\alpha$. By transport of structure, one deduces

−α* = int(w)⁻¹ ∘ α* = int(w) ∘ α*,

since $int(w{)}^2=int(w^2)$ and $w^2$ is a section of $T$. Therefore $int(w)$ induces the symmetry on $T^{\prime }$; since $R$ is central, $int(w)$ induces the identity on $R$. The formula of (iii) defines a morphism $T\to T$ which satisfies the same properties, hence coincides with $int(w)$.

Let us prove (iv). One has successively

w_α(X) t w_α(X)⁻¹ = exp(X) exp(−X⁻¹) exp(X) t exp(−X) exp(X⁻¹) exp(−X)
                  = exp(X) exp(−X⁻¹) exp(X − α(t) X) exp(α(t)⁻¹ X⁻¹) exp(−α(t) X) t.

By application of formula (F), one has

exp(−X⁻¹) exp((1 − α(t)) X) = exp((α(t)⁻¹ − 1) X) α*(α(t)⁻¹) exp(−α(t)⁻¹ X⁻¹).

Substituting in the preceding relation, one finds

int(w_α(X)) t = exp(α(t)⁻¹ X) α*(α(t)⁻¹) exp(−α(t) X) t
              = exp(a X) α*(α(t)⁻¹) t,

where

a = α(t)⁻¹ − (α ∘ α*)(α(t)⁻¹) α(t),

but $\alpha \circ \alpha \ast =2$, which gives at once $a=0$ and $w_{\alpha }(X)\in N^{\times }(S^{\prime })$.

Let us prove the second assertion of (iv). One has²⁶

α*(z) w_α(X) = exp(z² X) exp(−z⁻² X⁻¹) exp(z² X) α*(z)
             = exp(z X) exp((z² − z) X) exp(−z⁻² X⁻¹) exp(z² X) α*(z)
             = exp(z X) exp(−z⁻¹ X⁻¹) α*(z)⁻¹ exp((z³ − z²) X) exp(z² X) α*(z)
             = exp(z X) exp(−z⁻¹ X⁻¹) exp(z X) = w_α(z X).

Let us prove (v). By virtue of the preceding result, the first formula of (v) follows at once from the second; let us prove the second:

w_α(X)² = exp(X) exp(−X⁻¹) exp(2 X) exp(−X⁻¹) exp(X)
        = exp(X) exp(−X⁻¹) exp(X⁻¹) α*(−1) exp(−2 X) exp(X)
        = exp(X) α*(−1) exp(−X) = α*(−1),

since $\alpha (\alpha \ast (-1))=(-1{)}^2=1$, which proves that $\alpha \ast (-1)\in Centr(G)(S)$.

Let us prove (vi). The first assertion is a particular case of the second; let us prove the second. Both sides of this formula define morphisms $W(g_{\alpha }{)}^{\times }{\times }_{S}W(g_{-\alpha }{)}^{\times }\to G$. To prove that they coincide, it suffices to do so on a non-empty open set on each fiber (Exp. XVIII 1.4); it therefore suffices to verify the relation when $1+XY$ is invertible. One then has successively:

w_α(X) w_{−α}(Y) = exp(X) exp(−X⁻¹) exp(X) exp(Y) exp(−Y⁻¹) exp(Y)
                 = exp(X) exp(−X⁻¹) exp(Y / (1 + X Y)) α*(1 + X Y) exp(X / (1 + X Y)) exp(−Y⁻¹) exp(Y)
                 = exp(X) exp(−X⁻¹ / (1 + X Y)) α*(1 + X Y) exp(−Y⁻¹ / (1 + X Y)) exp(Y)
                 = exp(−X⁻² Y⁻¹) α*(X Y / (1 + X Y)) exp(X + Y⁻¹) α*(1 + X Y) exp(−Y⁻¹ / (1 + X Y)) exp(Y)
                 = exp(−X⁻² Y⁻¹) α*(X Y) exp((Y⁻¹ + X) / (1 + X Y)²) exp(−Y⁻¹ / (1 + X Y)) exp(Y)
                 = α*(X Y) exp(−Y) exp(Y) = α*(X Y).

Corollary 3.2. Let $n\in \mathbb{Z}$, $n\ne 0$. For every $w\in G(S)$, the following conditions are equivalent:

(i) $w\in N^{\times }(S)$,

(ii) one has $int(w)\circ n\alpha \ast =-n\alpha \ast$ (recall that $(n\alpha \ast )(z)=\alpha \ast (z{)}^n$).

One has (i) ⇒ (ii) (assertion (iii) of Theorem 3.1); conversely, one may suppose that $N^{\times }$ admits a section, and one is reduced to proving:

Lemma 3.3. One has $Cent{r}_G(n\alpha \ast )=T$ for $n\ne 0$.

Indeed, the image $T^{\prime }$ of $n\alpha \ast$ is a subtorus of $G$. It follows (Exp. XIX 2.8) that $Cent{r}_G(n\alpha \ast )$ is a reductive subgroup of $G$ containing $T$. Since on each fiber one has $Cent{r}_{G_s}(n\alpha {\ast }_s)\ne G_s$, then $Cent{r}_{G_s}(n\alpha {\ast }_s)=T_s$ (Exp. XIX 1.6.3)²⁷, hence $Cent{r}_G(n\alpha \ast )=T$, since these are smooth subgroups of $G$.

Remark 3.4. The construction of $w_{\alpha }$ and the fact that $w_{\alpha }(X)$ normalizes $T$ rely only on formula (F). In particular, if $G$ is an $S$-group satisfying the conditions of 2.2, $Nor{m}_G(T)$ differs from $T$ on each fiber. It follows that if $G$ is an affine $S$-group with connected fibers satisfying the conditions of 2.2, it is reductive of semisimple rank 1. Indeed, it is smooth in a neighborhood of the unit section, hence smooth, and one can apply the criterion of Exp. XIX 1.11.

3.5. Before stating the following theorem, let us make a few remarks. We identify as usual $g_{-\alpha }$ with $(g_{\alpha }{)}^{\otimes -1}$. Similarly, we shall identify ${\mathrm{Hom}}_{O_S}(g_{-\alpha },g_{\alpha })$ with $(g_{\alpha }{)}^{\otimes 2}$ and hence

$$Iso{m}_{O_S-mod.}(W(g_{-\alpha }),W(g_{\alpha }))\cong W((g_{\alpha }{)}^{\otimes 2}{)}^{\times }.$$

If $w\in N^{\times }(S)$, then $Ad(w)$ permutes $g_{\alpha }$ and $g_{-\alpha }$ (3.1, (iii)), hence defines an isomorphism:

a_α(w) : g_{−α} ──∼──→ g_α,

which we shall therefore identify with a section $a_{\alpha }(w)\in \Gamma (S,(g_{\alpha }{)}^{\otimes 2}{)}^{\times }$. This construction is compatible with base change and therefore defines a morphism

$$a_{\alpha }:N^{\times }\to W((g_{\alpha }{)}^{\otimes 2}{)}^{\times },$$

such that $a_{\alpha }(w)Y=Ad(w)Y$ for all $w\in N^{\times }(S^{\prime })$, $Y\in \Gamma (S^{\prime },g_{-\alpha }{)}^{\times }$, $S^{\prime }\to S$.

Theorem 3.6. (i) One has

int(w) exp(Y) = exp(a_α(w) Y)

for every $S^{\prime }\to S$ and all $w\in N^{\times }(S^{\prime })$, $Y\in W(g_{-\alpha })(S^{\prime })$.

(ii) One has

a_α(t w) = α(t) a_α(w),    a_α(w t) = α(t)⁻¹ a_α(w).

(iii) If one defines analogously $a_{-\alpha }:N^{\times }\to W((g_{-\alpha }{)}^{\otimes 2}{)}^{\times }$, one has

$$a_{-\alpha }(w)=a_{\alpha }(w{)}^{-1}.$$

²⁸