Exposé XXIII. Reductive groups: uniqueness of pinned groups

by M. Demazure

¹

The aim of this Exposé is the proof of the uniqueness theorem (Theorem 4.1). This was proved by Chevalley in the case of an algebraically closed field; the method of reduction to rank two used here is also due to Chevalley (see Bible, Exp. 23 and 24). Along the way, we obtain an explicit description of reductive groups by generators and relations (3.5).

1. Pinnings

Definition 1.1. Let $S$ be a scheme and $(G,T,M,R)$ an $S$-split group (XXII, 1.13). One calls pinning[^N.D.E-XXIII-butterfly] of this split group the datum of a system $\Delta$ of simple roots of $R$ and, for each $\alpha \in \Delta$, a section $X_{\alpha }\in \Gamma (S,g_{\alpha }{)}^{\times }$.[^N.D.E-XXIII-butterfly-anchor]

In other words, a pinning of the reductive group $G$ over the nonempty scheme $S$ is the datum of:

(i) a maximal torus $T$,

(ii) an abelian group $M$ and an isomorphism $T\simeq D_S(M)$,

(iii) a system of roots $R$ of $G$ with respect to $T$,

(iv) a system of simple roots $\Delta$ of $R$,

(v) an $X_{\alpha }\in \Gamma (S,g_{\alpha }{)}^{\times }$ for every $\alpha \in \Delta$, that is, of a

u_α = exp_α(X_α) ∈ U_α^×(S)    for every α ∈ Δ,

satisfying condition (D 1) of Exp. XXII, 1.13 (indeed condition (D 2) of loc. cit. is automatically satisfied²).

Every split group admits a pinning; in particular, every reductive group is locally pinnable for the étale topology.

1.2.

If $G$ is a pinned $S$-group — that is, an $S$-split group equipped with a pinning — it is canonically endowed with the system of positive roots $R^{+}$ defined by $\Delta$, the corresponding Borel subgroup $B=B_{R^{+}}$, the opposite Borel subgroup $B^{-}=B_{R^{-}}$, the unipotent groups $U=B_u$, $U^{-}=(B^{-}{)}_u$, the open set $U^{-}\cdot T\cdot U$, etc. Likewise, for each $\alpha \in \Delta$, one has a canonical isomorphism of vector groups

p_α : G_{a, S} ⥲ U_α,    x ↦ exp_α(x X_α) = u_α^x,

normalized by $T$ with multiplier $\alpha$, and whose datum is equivalent to that of $X_{\alpha }$ (Exp. XXII, 1.1).

By duality, one deduces an $X_{-\alpha }\in \Gamma (S,g_{-\alpha }{)}^{\times }$ and an isomorphism

p_{-α} : G_{a, S} ⥲ U_{-α}

which is the contragredient of the preceding one (Exp. XXII, 1.3). We shall set (Exp. XX, 3.1)

w_α = w_α(X_α) = p_α(1) p_{-α}(-1) p_α(1) = p_{-α}(-1) p_α(1) p_{-α}(-1).

One then has (loc. cit. 3.1, 3.7)

w_α^2 = α^*(-1),    int(w_α) t = s_α(t) = t · α^*(α(t)^{-1}),

int(w_α) p_α(x) = p_{-α}(-x) = p_{-α}(x)^{-1},
Ad(w_α) X_α = -X_{-α},

int(w_α) p_{-α}(x) = p_α(-x) = p_α(x)^{-1},
Ad(w_α) X_{-α} = -X_α.

We shall use the preceding notation systematically in what follows.

Definition 1.3. Let $S$ be a scheme and $(G,T,M,R,\Delta ,(X_{\alpha }))$ and $(G^{\prime },T^{\prime },M^{\prime },\cdots )$ two pinned $S$-groups. One says that the morphism of split groups (Exp. XXII, 4.2.1)

$$f:G\to G^{\prime }$$

is compatible with the pinnings, or that it defines a morphism of pinned groups, if the bijection $d:R\to R^{\prime }$ associated with it (cf. loc. cit.) satisfies $d(\Delta )={\Delta }^{\prime }$ and if, for every $\alpha \in \Delta$, one has

f(exp_α(X_α)) = exp_{d(α)}(X'_{d(α)}),    i.e.    f(u_α) = u'_{d(α)}.

³

1.4.

If we denote by $q(\alpha )$ the integer of loc. cit., we therefore have

f(p_α(x)) = p'_{d(α)}(x^{q(α)})    for    α ∈ Δ,

and consequently

f(p_{-α}(x)) = p'_{-d(α)}(x^{q(α)}),    f(w_α) = w'_{d(α)}.

Recall (Exp. XXII, 4.2) that, for every $\alpha \in R$ and all $z\in G_m(S^{\prime })$, $t\in T(S^{\prime })$,

f(α^*(z)) = d(α)^*(z)^{q(α)} = d(α)^*(z^{q(α)}),    d(α)(f(t)) = α(t)^{q(α)}.

1.5.

We shall call a pinned root datum a root datum endowed with a system of simple roots, and a $p$-morphism of pinned root data a $p$-morphism of root data (Exp. XXI, 6.8) sending simple roots to simple roots.

If $G$ is a pinned $S$-group, we denote by $R(G)$ the corresponding pinned root datum (this is the root datum of Exp. XXII, 1.14 equipped with $\Delta$). Let $p$ be the integer defined in Exp. XXII, 4.2.2. One then has:

Scholium 1.6. The correspondence $G\mapsto R(G)$ defines a functor from the category of pinned reductive $S$-groups to that of pinned root data (with $p$-morphisms as morphisms).

1.7. The pinned groups $Z_{{\Delta }_1}$

Let ${\Delta }_1$ be a subset of the system of simple roots $\Delta$ of the pinned group $G$. Let $T_{{\Delta }_1}$ be the maximal torus of ${\bigcap }_{\alpha \in {\Delta }_1}Ker(\alpha )$; set

$$Z_{{\Delta }_1}=Cent{r}_G(T_{{\Delta }_1}).$$

Set $R^{\prime }=\mathbb{Z}\cdot {\Delta }_1\cap R$; one knows (Exp. XXII, 5.10.7) that $Z_{{\Delta }_1}$ is a reductive $S$-group with radical $T_{{\Delta }_1}$, that (T, M, R') is a splitting of it, and ${\Delta }_1$ a system of simple roots. It follows that $(Z_{{\Delta }_1},T,M,R^{\prime },{\Delta }_1,(X_{\alpha }{)}_{\alpha \in {\Delta }_1})$ is a pinned $S$-group. We shall always equip $Z_{{\Delta }_1}$ with this pinning. In particular, we shall consider the groups

Z_α = Z_{{α}},    Z_{αβ} = Z_{{α, β}}.

We denote by $B_{{\Delta }_1}=B\cap Z_{{\Delta }_1}$; one knows (loc. cit.) that this is the canonical Borel subgroup⁴ of $Z_{{\Delta }_1}$, and that its unipotent part is $U_{{\Delta }_1}=U\cap Z_{{\Delta }_1}$. In particular,

$$B_{\alpha }=T\cdot U_{\alpha }.$$

We shall denote

U_{αβ} = U_{{α, β}} = U ∩ Z_{αβ} = ∏_{γ ∈ R^+_{αβ}} U_γ,

where $R_{\alpha \beta }^{+}$ denotes the set of positive roots that are linear combinations of $\alpha$ and $\beta$.

Now let $f:G\to G^{\prime }$ be a morphism of pinned $S$-groups. If $d:R\to R^{\prime }$ is the corresponding bijection and if ${\Delta }_1$ is a subset of $\Delta$, then $d({\Delta }_1)={\Delta }_1^{\prime }$ is a subset of ${\Delta }^{\prime }$, and it is clear that $f$ sends $T_{{\Delta }_1}$ into $T_{{\Delta }_1^{\prime }}^{\prime }$, hence $Z_{{\Delta }_1}$ into $Z_{{\Delta }_1^{\prime }}^{\prime }$. The corresponding $S$-morphism

$$f_{{\Delta }_1}:Z_{{\Delta }_1}\to Z_{d({\Delta }_1^{\prime })}^{\prime }$$

is compatible with the canonical pinnings; it defines a morphism of pinned root data

R(f_{Δ_1}) : R(Z_{Δ_1}, T, M, …) → R(Z_{Δ'_1}, T', M', …)

and the underlying morphisms $M^{\prime }\to M$ of $R(f)$ and $R(f_{{\Delta }_1})$ coincide.

1.8. Study of the group $Nor{m}_G(T)$

For each pair $(\alpha ,\beta )$ of simple roots, denote by $n_{\alpha \beta }$ the order of the element $s_{\alpha }{s}_{\beta }$ of the Weyl group $W$. In particular, $n_{\alpha \alpha }=1$. One has therefore $(w_{\alpha }{w}_{\beta }{)}^{n_{\alpha \beta }}\in T(S)$.

Definition 1.8.1. For every $(\alpha ,\beta )\in \Delta \times \Delta$, one sets

t_{αβ} = (w_α w_β)^{n_{αβ}} ∈ T(S).

Moreover, one sets (Exp. XX, 3.1)

t_α = t_{αα} = w_α^2 = α^*(-1) ∈ T(S).

Proposition 1.8.2. Let $H$ be an $S$-functor in groups transforming direct sums of schemes into products (for example a sheaf for the Zariski topology). Let

$$f_T:T\to H$$

be a morphism of groups and $h_{\alpha }(\alpha \in \Delta )$ elements of $H(S)$. In order that there exist a morphism of groups (necessarily unique)

$$f_N:Nor{m}_G(T)\to H$$

inducing $f_T$ on $T$ and such that $f(w_{\alpha })=h_{\alpha }$ for $\alpha \in \Delta$, it is necessary and sufficient that the following two conditions be satisfied:

(i) For every $\alpha \in \Delta$,

f_T(s_α(t)) = h_α f_T(t) h_α^{-1}

for every $t\in T(S^{\prime })$, $S^{\prime }\to S$ (i.e. $f_T\circ s_{\alpha }=int(h_{\alpha })\circ f_T$).

(ii) For every $(\alpha ,\beta )\in \Delta \times \Delta$,

$$f_T(t_{\alpha \beta })=(h_{\alpha }{h}_{\beta }{)}^{n_{\alpha \beta }}.$$

Proof. Indeed, equip (Sch) with the topology $\mathcal{T}$ generated by the pretopology whose covering families are the direct sums; the hypothesis of the statement says that $H$ is a $\mathcal{T}$-sheaf. Let $L$ be the free group with generators $(m_{\alpha }{)}_{\alpha \in R}$ and L_1 the invariant subgroup generated by the elements $(m_{\alpha }{m}_{\beta }{)}^{n_{\alpha \beta }}$, $(\alpha ,\beta )\in \Delta \times \Delta$. Let $g:L\to W$ be the morphism defined by $g(m_{\alpha })=s_{\alpha }$; one knows (Exp. XXI, 5.1) that $g$ induces an isomorphism ḡ of $L/L_1$ onto $W$. Make $L$ act on $T$ through $g$ (or, equivalently, by $m_{\alpha }\cdot t=s_{\alpha }(t)$). Let L_S be the constant group defined by $L$; consider the semidirect product $T\cdot L_S=N$ for the preceding operation. One has a morphism of $S$-groups

h : T · L_S = N → Norm_G(T)

unique such that $h(m_{\alpha })=w_{\alpha }$, $h(t)=t$ for every $t\in T(S^{\prime })$, $S^{\prime }\to S$. Let N_1 be the invariant subsheaf of groups of $N$ generated by the

t_{αβ}^{-1} · (m_α m_β)^{n_{αβ}},    (α, β) ∈ Δ × Δ.

One evidently has $N_1\subset Kerh$; consider the induced morphism

$$h_1:N/N_1\to Nor{m}_G(T).$$

Let us prove that $h_1$ is an isomorphism. Since $h$ induces on $T$ the canonical immersion, which is a monomorphism, the canonical morphism

$$T\to N/N_1$$

is also a monomorphism, hence induces an isomorphism of $T$ onto $T{N}_1/N_1$. For the same reason, $h_1$ induces an isomorphism of $T{N}_1/N_1$ onto $T$; to prove that $h_1$ is an isomorphism, it therefore suffices to verify that the corresponding morphism

$$h_2:N/T{N}_1\to Nor{m}_G(T)/T$$

is an isomorphism. Now TN_1 is the invariant $\mathcal{T}$-subsheaf of groups of $N$ generated by $T$ and the $(m_{\alpha }{m}_{\beta }{)}^{n_{\alpha \beta }}$, that is, the $\mathcal{T}$-subsheaf generated by $T$ and L_1, that is, $T\cdot (L_1{)}_S$. The morphism $h_2$ is therefore identified with the morphism

$$\bar{g}:L_S/(L_1{)}_S\to W_S$$

which is an isomorphism by construction.

The proof of 1.8.2 is now easy; the conditions are evidently necessary. Let us prove that they are sufficient. Condition (i) shows that there exists a morphism

$$u:N\to H$$

such that $u(m_{\alpha })=h_{\alpha }$ for $\alpha \in \Delta$, and $u{|}_T=f_T$. Condition (ii) says that $u$ vanishes on N_1, which yields the result at once.

1.9. Faithfulness of the functor $R$

Proposition 1.9.1. The functor $R$ of 1.6 is faithful: if

f, g : G ⇒ G'

are two morphisms of pinned groups such that $R(f)=R(g)$, then $f=g$.

Indeed, $f$ and $g$ coincide on $T$, $U_{\alpha }(\alpha \in \Delta )$ and $U_{-\alpha }(\alpha \in \Delta )$; it therefore suffices to apply:

Lemma 1.9.2. Let $S$ be a scheme, $G$ a pinned reductive $S$-group, $H$ an $S$-presheaf in groups, separated for (fppf). Let

f, g : G ⇒ H

be two morphisms of $S$-groups. The following conditions are equivalent:

(i) $f=g$.

(ii) $f$ and $g$ coincide on $T$, on each $U_{\alpha }(\alpha \in \Delta )$, on each $U_{-\alpha }(\alpha \in \Delta )$.

(iii) $f$ and $g$ coincide on $T$, on each $U_{\alpha }(\alpha \in \Delta )$, and $f(w_{\alpha })=g(w_{\alpha })$ for each $\alpha \in \Delta$.

Indeed, (i) ⇒ (ii) is trivial, (ii) ⇒ (iii) follows at once from the definition of $w_{\alpha }$ (1.2). It remains to prove (iii) ⇒ (i). If $\alpha \in R$, there exists a sequence ${\alpha }_i\subset \Delta$ with $\alpha =s_{{\alpha }_1}{s}_{{\alpha }_2}\cdots s_{{\alpha }_n}({\alpha }_{n+1})$, hence

U_α = int(w_{α_1} ⋯ w_{α_n}) U_{α_{n+1}},

which shows that $f$ and $g$ coincide on each $U_{\alpha }$. It follows that $f$ and $g$ coincide on $\Omega$, hence coincide (Exp. XXII, 4.1.11).

Remark 1.9.3. If $G$ is semisimple, one may, in (ii) and (iii), drop the hypothesis that $f$ and $g$ coincide on $T$. Indeed, $G$ is generated as an (fppf) sheaf by the $U_{\alpha }$, $\alpha \in R$ (Exp. XXII, 6.2.2 (a)).

2. Generators and relations for a pinned group

In this section, we fix a pinned $S$-group $G$. If $\alpha ,\beta \in \Delta$, we shall use the notation $Z_{\alpha }$, $Z_{\alpha \beta }$, $U_{\alpha \beta }$, $R_{\alpha \beta }^{+}$ of 1.7.

Theorem 2.1. Let $S$ be a scheme, $G$ a pinned $S$-group, $H$ an $S$-sheaf in groups for (fppf). Let

f_N : Norm_G(T) → H,    f_α : U_α → H,    α ∈ R,

be morphisms of groups. In order that there exist a morphism of groups (necessarily unique)

$$f:G\to H$$

extending $f_N$ and the $f_{\alpha }(\alpha \in R)$, it is necessary and sufficient that the following conditions be satisfied:

(i) For every $\alpha \in \Delta$ and every $\beta \in R$,

int(f_N(w_α)) ∘ f_β = f_{s_α(β)} ∘ int(w_α).

(ii) For every $\alpha \in \Delta$, there exists a morphism of groups

$$F_{\alpha }:Z_{\alpha }\to H$$

extending $f_{\alpha }$, $f_{-\alpha }$ and $f_N{|}_{Nor{m}_{Z_{\alpha }}(T)}$.

(iii) For every pair $(\alpha ,\beta )\in \Delta \times \Delta$, there exists a morphism of groups $U_{\alpha \beta }\to H$ inducing $f_{\gamma }$ on $U_{\gamma }$ for every $\gamma \in R_{\alpha \beta }^{+}$ (i.e. $U_{\gamma }\subset U_{\alpha \beta }$).

2.1.1. Proof

The conditions of the statement are evidently necessary. Choose on the other hand a structure of totally ordered group on ${\Gamma }_0(R)$ such that the roots > 0 are the elements of $R^{+}$ (Exp. XXI, 3.5.6); every product indexed by a subset of $R$ will be taken relative to this order.

Denote by $f_T$ the restriction of $f_N$ to $T$, and consider the morphism

$$f:\Omega \to H$$

defined set-theoretically by

f(∏_{α ∈ R^-} y_α · t · ∏_{α ∈ R^+} x_α) = ∏ f_α(y_α) · f_T(t) · ∏ f_α(x_α).

Any morphism satisfying the conditions of the statement must extend $f$; on the other hand any morphism of groups $f:G\to H$ extending $f$ also extends $f_N$: indeed, extending $f_{\alpha }$ and $f_{-\alpha }$, it satisfies $f(w_{\alpha })=F_{\alpha }(w_{\alpha })=f_N(w_{\alpha })$, and it extends $f_T$ by hypothesis. By Exp. XXII, 4.1.11 (ii), one is therefore reduced to proving:

Proposition 2.1.2. The morphism $f:\Omega \to H$ defined above is "generically multiplicative"; more precisely, for every $S^{\prime }\to S$ and all $x,y\in \Omega (S^{\prime })$ such that $xy\in \Omega (S^{\prime })$, one has $f(xy)=f(x)f(y)$.

Lemma 2.1.3. Let $n\in Nor{m}_G(T)(S^{\prime })$, $S^{\prime }\to S$, and let $\alpha ,\beta \in R$ be such that $int(n)U_{\alpha }=U_{\beta }$ (i.e. $n(\alpha )=\beta$); then

int(f_N(n)) ∘ f_α = f_β ∘ int(n).

Indeed, it suffices to verify the formula for a system of generators of the sheaf $Nor{m}_G(T)$; it is true for each $w_{\alpha }$, $\alpha \in \Delta$ (by 2.1 (i)), so it suffices to do it for $n\in T(S^{\prime })$. This is trivial by 2.1 (ii) if $\alpha$ is simple; otherwise, one takes a $w\in W$ such that $w^{-1}(\alpha )\in \Delta$; writing $w$ as a product of simple reflections,⁵ one is reduced to proving that if the formula is true for $\alpha$ and for every $n$, it is also true for $w_{{\alpha }_0}(\alpha )$ and $t\in T(S^{\prime })$, where ${\alpha }_0\in \Delta$. Now, by 2.1 (i),

int(f_N(t)) ∘ f_{w_{α_0}(α)} = int(f_N(t w_{α_0})) ∘ f_α ∘ int(w_{α_0}^{-1})
                             = f_{w_{α_0}(α)} ∘ int(t w_{α_0}) ∘ int(w_{α_0}^{-1}).

Lemma 2.1.4. The restriction of $f$ to $U$ (resp. $U^{-}$) is a morphism of groups. In particular, this restriction is independent of the order chosen on the roots.

It suffices to give the proof for $U$. By virtue of Exp. XXII, 5.5.8, it suffices to verify that for every pair $\alpha <\beta$ of positive roots, one has for all $x_{\alpha }\in U_{\alpha }(S^{\prime })$, $x_{\beta }\in U_{\beta }(S^{\prime })$, $S^{\prime }\to S$,

f_β(x_β) f_α(x_α) f_β(x_β^{-1}) = f(x_β x_α x_β^{-1}).

By Exp. XXII, 5.5.2, there exist $x_{\gamma }\in U_{\gamma }(S^{\prime })$ ($\gamma =i\alpha +j\beta \in R$, $i>0$, $j⩾0$⁶) such that

x_β x_α x_β^{-1} = ∏_γ x_γ,

and we must therefore verify the relation

f_β(x_β) f_α(x_α) f_β(x_β^{-1}) = ∏_{γ = iα + jβ, i > 0, j ⩾ 0} f_γ(x_γ).

By Exp. XXI, 3.5.4, there exists $w\in W$ such that $w(\alpha )={\alpha }_0\in \Delta$, $w(\beta )\in A_{{\alpha }_0{\beta }_0}$ (notation of 1.7), where ${\beta }_0\in \Delta$. Lifting $w$ to an $n\in Nor{m}_G(T)(S)$ (by Exp. XXII, 3.8), it suffices to verify the preceding relation after conjugation by $f_N(n)$. By 2.1.3, one is therefore reduced to the case $\alpha ,\beta \in R_{{\alpha }_0{\beta }_0}^{+}$, a case in which we conclude by condition 2.1 (iii).

Lemma 2.1.5. Let $n\in Nor{m}_G(T)(S)$ be such that $int(n)U=U^{-}$ (i.e. $n$ is the symmetry of the Weyl group⁷ (Exp. XXI, 3.6.14)). For every $u\in U(S^{\prime })$, $S^{\prime }\to S$ (resp. $u\in U^{-}(S^{\prime })$, $S^{\prime }\to S$), one has

f(n u n^{-1}) = f_N(n) f(u) f_N(n^{-1}).

Immediate by 2.1.3 and 2.1.4.

Lemma 2.1.6. Let $u\in B(S^{\prime })$, $v\in B^{-}(S^{\prime })$, $g\in \Omega (S^{\prime })$, $S^{\prime }\to S$. Then

f(v g u) = f(v) f(g) f(u).

Indeed, set $v=v_1{t}_1$, $g=v_2{t}_2{u}_2$, $u=t_3{u}_3$, with $v_i\in U^{-}(S^{\prime })$, $t_i\in T(S^{\prime })$, $u_i\in U(S^{\prime })$. One has

f(v) f(g) f(u) = f(v_1) f_T(t_1) f(v_2) f_T(t_2) f(u_2) f_T(t_3) f(u_3),

on the one hand, and

f(v g u) = f(v_1 t_1 v_2 t_1^{-1} t_1 t_2 t_3 t_3^{-1} u_2 t_3 u_3)
         = f(v_1 · t_1 v_2 t_1^{-1}) f_T(t_1 t_2 t_3) f(t_3^{-1} u_2 t_3 · u_3).

Using 2.1.4 to decompose the two extreme terms of this last expression, one obtains

f(v g u) = f(v_1) f(t_1 v_2 t_1^{-1}) f_T(t_1 t_2 t_3) f(t_3^{-1} u_2 t_3) f(u_3).

One is then reduced to the obvious formulas

f(t_1 v_2 t_1^{-1}) = f_T(t_1) f(v_2) f_T(t_1)^{-1},    f(t_3^{-1} u_2 t_3) = f_T(t_3)^{-1} f(u_2) f_T(t_3),

which follow from the definition of $f$ and from 2.1.3.

Lemma 2.1.7. Let $\alpha \in \Delta$ and $g\in \Omega (S^{\prime })\cap int(w_{\alpha }{)}^{-1}\Omega (S^{\prime })$, $S^{\prime }\to S$. Then

f(w_α g w_α^{-1}) = f_N(w_α) f(g) f_N(w_α)^{-1}.

Indeed, let $g\in \Omega (S^{\prime })$, $S^{\prime }\to S$. Write, by Exp. XXII, 5.6.8,

g = a x_{-α} t x_α b,

with $a\in U_{-\hat{\alpha }}^{-}(S^{\prime })$, $x_{-\alpha }\in U_{-\alpha }(S^{\prime })$, $t\in T(S^{\prime })$, $x_{\alpha }\in U_{\alpha }(S^{\prime })$, $b\in U_{\hat{\alpha }}(S^{\prime })$. By 2.1.3, 2.1.4 and the fact that $s_{\alpha }$ permutes the positive roots $\ne \alpha$ (cf. Exp. XXI, 3.3.2), one has

int(w_α) a ∈ U_{-α̂}^-(S'),    int(w_α) b ∈ U_{α̂}(S')

and the formula to be proved is true for $g=a$ or $g=b$. By 2.1.6, one is therefore reduced to proving the asserted formula when $g=x_{-\alpha }t{x}_{\alpha }\in Z_{\alpha }(S^{\prime })$. But then "everything happens in $Z_{\alpha }$", and one concludes by condition (ii) of 2.1.

Lemma 2.1.8. Let $n\in Nor{m}_G(T)(S)$. For every $S^{\prime }\to S$ and every $g\in \Omega (S^{\prime })$ such that $int(n)g\in \Omega (S^{\prime })$, one has

f(n g n^{-1}) = f_N(n) f(g) f_N(n)^{-1}.

This is trivial if $n\in T(S)$ (by 2.1.3). The two sides of the preceding formula define morphisms of $\Omega \cap int(n{)}^{-1}\Omega$ into $H$; to verify that they coincide, it suffices to verify that there exists an open $V_n$ of $\Omega$ containing the unit section such that $int(n)V_n\subset \Omega$, and that $f\circ int(n)$ and $int(f_N(n))\circ f$ coincide on $V_n$. By virtue of the structure of $Nor{m}_G(T)$, it suffices to prove that if the preceding assertion is true for an $n^{\prime }\in Nor{m}_G(T)(S)$ and if $\alpha \in \Delta$, it is true for $n=n^{\prime }{w}_{\alpha }$. Set

V_n = Ω ∩ int(w_α)^{-1} V_{n'}.

One has $int(n)V_n\subset int(n^{\prime })V_{n^{\prime }}\subset \Omega$. If $g\in V_n(S^{\prime })$, then

int(n) g = int(n') int(w_α) g.

Now $int(w_{\alpha })g\in V_{n^{\prime }}(S^{\prime })$, hence by hypothesis

f(int(n') int(w_α) g) = int(f_N(n')) f(int(w_α) g);

since $int(w_{\alpha })g\in \Omega (S^{\prime })$, one may apply 2.1.7, which gives

f(int(w_α) g) = int(f_N(w_α)) f(g),

and one concludes at once.

Let us now prove 2.1.2. Let $x,x^{\prime }\in \Omega (S^{\prime })$; write as usual

x = v t u,    x' = v' t' u',

whence

x x' = v t (u v') t' u'.

By 2.1.6 and the relation $U^{-}(S^{\prime })\Omega (S^{\prime })U(S^{\prime })=\Omega (S^{\prime })$, one is reduced to proving that if $u{v}^{\prime }\in \Omega (S^{\prime })$, then $f(u{v}^{\prime })=f(u)f(v^{\prime })$. Let $n\in Nor{m}_G(T)(S^{\prime })$ be as in 2.1.5 (ii). One has

f(u) = f_N(n)^{-1} f(n u n^{-1}) f_N(n),    f(v') = f_N(n)^{-1} f(n v' n^{-1}) f_N(n),

by loc. cit., whence

f(u) f(v') = f_N(n)^{-1} f(n u n^{-1}) f(n v' n^{-1}) f_N(n).

But $nu{n}^{-1}\in U^{-}(S^{\prime })$, $n{v}^{\prime }{n}^{-1}\in U(S^{\prime })$, so that

f(n u n^{-1}) f(n v' n^{-1}) = f((n u n^{-1})(n v' n^{-1})) = f(n u v' n^{-1}),

which gives

f(u) f(v') = f_N(n)^{-1} f(n u v' n^{-1}) f_N(n).

If $u{v}^{\prime }\in \Omega (S^{\prime })$, one may apply 2.1.8 and one is done.

Remark 2.2. Instead of giving oneself $f_N$, one may give oneself a morphism of groups $f_T:T\to H$ and sections $h_{\alpha }\in H(S)(\alpha \in \Delta )$ satisfying the conditions of 1.8.2. One must then replace condition (ii) of the theorem by:

(ii') There exists a morphism of groups $F_{\alpha }:Z_{\alpha }\to H$ extending

f_α,    f_{-α},    f_T    and satisfying    F_α(w_α) = h_α.

We shall now give the preceding theorem a more explicit form. Keep the preceding notation.

Theorem 2.3. Let $S$ be a scheme, $G$ a pinned $S$-group. Let $H$ be an $S$-sheaf in groups for (fppf),

f_T : T → H,    f_α : U_α → H,    α ∈ Δ,

morphisms of groups, and

h_α ∈ H(S),    (α ∈ Δ),

sections of $H$. In order that there exist a morphism of groups

$$f:G\to H$$

(necessarily unique) inducing $f_T$ and the $f_{\alpha }(\alpha \in \Delta )$ and satisfying $f(w_{\alpha })=h_{\alpha }$ for every $\alpha \in \Delta$, it is necessary and sufficient that the following conditions be satisfied:

(i) For every $S^{\prime }\to S$, every $t\in T(S^{\prime })$, every $\alpha \in \Delta$ and every $x\in U_{\alpha }(S^{\prime })$,

f_T(t) f_α(x) f_T(t)^{-1} = f_α(int(t) x).            (1)

(ii) For every $\alpha \in \Delta$, every $S^{\prime }\to S$, every $t\in T(S^{\prime })$,

h_α f_T(t) h_α^{-1} = f_T(s_α(t)) = f_T(t · α^* α(t)^{-1}).            (2)

(iii) For every $(\alpha ,\beta )\in \Delta \times \Delta$,

$$(h_{\alpha }{h}_{\beta }{)}^{n_{\alpha \beta }}=f_T(t_{\alpha \beta }).(3)$$

(iv) For every $\alpha \in \Delta$, (recall that we write $u_{\alpha }={\exp }_{\alpha }(X_{\alpha })$)

$$(h_{\alpha }{f}_{\alpha }(u_{\alpha }){)}^3=e.(4)$$

(v) For every $(\alpha ,\beta )\in \Delta \times \Delta$, $\alpha \ne \beta$, there exists a morphism of groups

$$f_{\alpha \beta }:U_{\alpha \beta }\to H$$

satisfying the two following conditions:

a) One has

f_{αβ}|_{U_α} = f_α,    f_{αβ}|_{U_β} = f_β,            (5)

b) For every $\gamma \in R_{\alpha \beta }^{+}$, $\gamma \ne \alpha$ (resp. $\gamma \ne \beta$), and every $x\in U_{\gamma }(S^{\prime })$, $S^{\prime }\to S$,

int(h_α) f_{αβ}(x) = f_{αβ}(int(w_α) x),                                       (6)
(resp. int(h_β) f_{αβ}(x) = f_{αβ}(int(w_α) x)).

Proof. The uniqueness is clear by 1.9.2. Let us prove existence.

Lemma 2.3.1. There exists a morphism of groups

$$f_N:Nor{m}_G(T)\to H$$

extending $f_T$ and satisfying $f_N(w_{\alpha })=h_{\alpha }$.

This is precisely what is asserted by 1.8.2, taking into account conditions (2) and (3).

Lemma 2.3.2. There exists a morphism of groups

F_α : Z_α → H,    α ∈ Δ,

extending $f_T$, $f_{\alpha }$ and satisfying $F_{\alpha }(w_{\alpha })=h_{\alpha }$, hence extending $f_N{|}_{Nor{m}_{Z_{\alpha }}(T)}$.

This is clear from Exp. XX, 6.2 and conditions (1), (3) and (4).

Lemma 2.3.3. For every $(\alpha ,\beta )\in \Delta \times \Delta$, $\alpha \ne \beta$, every $n\in Nor{m}_{Z_{\alpha \beta }}(T)(S)$ such that $n(\alpha )=\alpha$ (resp. $n(\alpha )=\beta$), i.e. $int(n)U_{\alpha }=U_{\alpha }$ (resp. $int(n)U_{\alpha }=U_{\beta }$), every $S^{\prime }\to S$ and every $x\in U_{\alpha }(S^{\prime })$,

int(f_N(n)) f_α(x) = f_α(int(n) x)

(resp.

int(f_N(n)) f_α(x) = f_β(int(n) x)).

Indeed, there exist a $t\in T(S)$ and a sequence ${\alpha }_i\subset \alpha ,\beta$ such that $n=t{w}_{{\alpha }_k}\cdots w_{{\alpha }_1}$. The condition is satisfied for $n=t$ by condition (1). One may therefore suppose $n=w_{{\alpha }_k}\cdots w_{{\alpha }_1}$. We proceed by induction on $k$, i.e. we suppose the assertion proved for every $n$ which can be written as a product of fewer than $k-1$ simple reflections (and satisfies the hypotheses of the lemma). Consider the roots

$${\gamma }_i=s_{{\alpha }_i}\cdots s_{{\alpha }_1}(\alpha ).$$

If all the ${\gamma }_i$ are positive, i.e. belong to $R_{\alpha \beta }^{+}$, one may apply condition (v) of 2.3; using the notation of 2.3 (v), one concludes at once by induction that

int(f_N(w_{α_i} ⋯ w_{α_1})) f_α(x) = f_{αβ}(int(w_{α_i} ⋯ w_{α_1}) x),

which for $i=k$ is the desired property. If not all the ${\gamma }_i$ are positive, there exists an index $j<k$ such that

γ_j ∈ R^+,    γ_{j+1} ∉ R^+.

Since ${\gamma }_{j+1}=s_{{\alpha }_j}({\gamma }_j)$, it follows at once that ${\gamma }_j={\alpha }_j$, by Exp. XXI, 3.3.2, hence that ${\gamma }_j$ is $\alpha$ or $\beta$, and one may decompose $n$ as

$$n=n^{\prime }\cdot n^{\prime \prime },n^{\prime }=w_{{\alpha }_k}\cdots w_{{\alpha }_{j+1}},n^{\prime \prime }=w_{{\alpha }_j}\cdots w_{{\alpha }_1},$$

$n^{\prime }$ and n'' satisfying the hypotheses of the lemma and being a product of fewer than $k-1$ reflections, hence satisfying by the induction hypothesis the conclusion of the lemma.

Lemma 2.3.4. Let $\alpha \in R$. If $n,n^{\prime }\in Nor{m}_G(T)(S)$ and $\beta ,{\beta }^{\prime }\in \Delta$ satisfy $n(\alpha )=\beta$ and $n^{\prime }(\alpha )={\beta }^{\prime }$, one has

int(f_T(n)^{-1}) f_β(n x n^{-1}) = int(f_T(n')^{-1}) f_{β'}(n' x n'^{-1}),

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

It suffices to verify that if $n(\alpha )=\beta$, $\alpha ,\beta \in \Delta$, then $int(f_T(n))\circ f_{\alpha }=f_{\beta }\circ int(n)$. Now, by Tits's lemma (Exp. XXI, 5.6), there exists a sequence of simple roots ${\alpha }_0=\alpha ,{\alpha }_1,\cdots ,{\alpha }_m=\beta$, and a sequence of elements $w_i\in W$, $i=0,1,\cdots ,m-1$, with

n = w_{m-1} ⋯ w_0,
w_i(α_i) = α_{i+1},    i = 0, 1, …, m - 1,

the following condition being moreover satisfied: for every $i=0,1,\cdots ,m-1$, there exists a simple root ${\beta }_i$ such that

w_i ∈ W_{α_i β_i},    α_{i+1} = α_i    or    β_i.

One is then reduced by induction to the case treated in the preceding lemma.

Lemma 2.3.5. There exists a family of morphisms of groups $f_{\gamma }^{\prime }:U_{\gamma }\to H$, $\gamma \in R$, satisfying

(i) For $\alpha \in \Delta$, $f_{\alpha }^{\prime }=f_{\alpha }$ and $f_{-\alpha }^{\prime }=F_{\alpha }{|}_{U_{-\alpha }}$, where $F_{\alpha }$ is defined in 2.3.2.

(ii) For $\alpha ,\beta \in \Delta$ and $\gamma \in R_{\alpha \beta }^{+}$, $f_{\gamma }^{\prime }=f_{\alpha \beta }{|}_{U_{\gamma }}$.

(iii) For every $n\in Nor{m}_G(T)(S)$ and $\alpha ,\beta \in R$ such that $n(\alpha )=\beta$,

int(f_N(n)) f'_α(x) = f'_β(int(n) x)

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

For every root $\alpha \in R$, there exists an $n\in Nor{m}_G(T)(S)$ such that $n(\alpha )\in \Delta$. Then define $f_{\alpha }^{\prime }(x)$ as the expression of 2.3.4. This is independent of the choice of $n$, and $f_{\alpha }^{\prime }$ is indeed a morphism of groups. Property (iii) is satisfied by construction. The first assertion of (i) is clear (take $n=e$), the second too (take $n=w_{\alpha }$ and apply 2.3.2); if $\gamma \in R_{\alpha \beta }^{+}$ ($\alpha ,\beta \in \Delta$), there exists $n\in Nor{m}_{Z_{\alpha \beta }}(T)(S)$ such that $n(t)=\alpha$ or $\beta$; one then applies (iii) and conditions (5) and (6) and has proved (ii).

2.3.6.

Let us now prove the theorem by showing that the conditions of 2.1 are satisfied, for the morphisms $f_N$ and $f_{\alpha }^{\prime }$, $\alpha \in R$.

• 2.1 (i) follows from 2.3.5 (iii),
• 2.1 (ii) follows from 2.3.5 (i) and 2.3.2,
• 2.1 (iii) follows from 2.3.5 (ii) and from the fact that $f_{\alpha \beta }$ is a morphism of groups.

An obvious corollary is:

Corollary 2.4. Let $S$ be a scheme, $G$ a pinned $S$-group of semisimple rank $⩾1$, $H$ an (fppf) $S$-sheaf in groups. For each $(\alpha ,\beta )\in \Delta \times \Delta$, let

$$F_{\alpha \beta }:Z_{\alpha \beta }\to H$$

be a morphism of groups. In order that there exist a morphism of groups

$$f:G\to H$$

inducing the $F_{\alpha \beta }$, it is necessary and sufficient that for every $(\alpha ,\beta )\in \Delta \times \Delta$,

$$F_{\alpha \beta }{|}_{Z_{\alpha }}=F_{\alpha \alpha }.$$

The condition is evidently necessary. Suppose it satisfied. Set $f_T=F_{\alpha \alpha }{|}_T$ (which does not depend on $\alpha$, since $F_{\alpha \alpha }{|}_T=F_{\alpha \beta }{|}_T=F_{\beta \beta }{|}_T$). Set

p_α = F_{αα}|_{U_α},    h_α = F_{αα}(w_α),    f_{αβ} = F_{αβ}|_{U_{αβ}}.

The conditions of 2.3 are evidently satisfied: (1), (2), (4) "are verified" in $Z_{\alpha }$, (3), (5) and (6) in $Z_{\alpha \beta }$. There therefore exists a morphism $f:G\to H$ extending $f_T$, the $f_{\alpha }(\alpha \in \Delta )$ and satisfying $f(w_{\alpha })=h_{\alpha }$; it coincides with $F_{\alpha \beta }$ on generators of $Z_{\alpha \beta }$, hence satisfies $f{|}_{Z_{\alpha \beta }}=F_{\alpha \beta }$.

One also has the following technical corollary.

Corollary 2.5. Let $S$ be a scheme, $G$ and $G^{\prime }$ two pinned $S$-groups of semisimple rank 2, $q$ an integer > 0 such that $x\mapsto x^q$ is an endomorphism of $G_{a,S}$, $h:R(G^{\prime })\to R(G)$ a $q$-morphism of pinned root data. Choose for each $\alpha \in R^{+}$ an $X_{\alpha }\in \Gamma (S,g_{\alpha }{)}^{\times }$ and an $X_{d(\alpha )}^{\prime }\in \Gamma (S,g_{d(\alpha )}^{\prime }{)}^{\times }$ (extending the canonical choices for $\alpha \in \Delta$). Suppose the following conditions are realized:

(i) If $\Delta =\alpha ,\beta$, then $D_S(h)(t_{\alpha \beta })=t_{d(\alpha )d(\beta )}^{\prime }$.

(ii) For every $\alpha \in \Delta$ and every $\beta \in R^{+}$, $\beta \ne \alpha$ (whence $s_{\alpha }(\beta )\in R^{+}$), if $z\in G_m(S)$ is defined by

Ad(w_α) X_β = z X_{s_α(β)},

one has also

Ad(w'_{d(α)}) X'_{d(β)} = z^{q(β)} X'_{d(s_α(β))}.

(iii) There exists a morphism of groups $f:U\to U^{\prime }$ such that for every $\alpha \in R^{+}$,

f(exp(x X_α)) = exp(x^{q(α)} X'_{d(α)})

for every $x\in G_a(S^{\prime })$, `S' → S.

Then there exists a morphism of pinned groups $g:G\to G^{\prime }$ such that $R(g)=h$.

Indeed, one defines $f_{\alpha }:U_{\alpha }\to G^{\prime }$ by

f_α(exp(x X_α)) = exp(x^{q(α)} X'_{d(α)});

one sets $f_T=D_S(h)$, $h_{\alpha }=w_{d(\alpha )}^{\prime }$. The conditions of 2.3 are satisfied (note that $q(s_{\alpha }(\beta ))=q(\beta )$ (Exp. XXI, 6.8.4) and that one always has $D_S(h)(t_{\alpha })=t_{d(\alpha )}^{\prime }$), and one concludes at once.

Remark 2.6. One may make condition (v) of 2.3 more precise as follows. One must first verify:

(a) For every word in $w_{\alpha }$ and $w_{\beta }$ such that the corresponding word transforms $\alpha$ or $\beta$ into $\alpha$ or $\beta$, the relation of type 2.3.3 corresponding is satisfied. In fact the proof of 2.3.3 shows that it suffices to verify it for the words in $w_{\alpha }$ and $w_{\beta }$ that are minimal (in the sense that any nontrivial initial sub-word does not satisfy the imposed conditions).

If condition (a) is satisfied, one may now define for each $\gamma \in R_{\alpha \beta }^{+}$ an $f_{\gamma }:U_{\gamma }\to H$ as in 2.3.5; one must then write:

(b) The morphism defined by the $f_{\gamma }$

U_{αβ} = ∏_{γ ∈ R^+_{αβ}} U_γ → H

is a morphism of groups. By Exp. XXII, 5.5.8, (b) is entailed by:

(b') The preceding morphism respects the commutation relations between $U_{\gamma }$ and $U_{\delta }$ for $\gamma ,\delta \in R_{\alpha \beta }^{+}$, $\gamma >\delta$ (i.e. the relations in $C_{i,j,\gamma ,\delta }$ of Exp. XII, 5.5.2).

It is clear that conversely the set of conditions (a) and (b') is equivalent to (v).

One may even reduce the preceding conditions to conditions bearing only on the elements $h_{\alpha }$, $h_{\beta }$, $f_{\alpha }(u_{\alpha })$, $f_{\beta }(u_{\beta })$ of $H$. A condition of type (a) will be written for example, if $s_{\alpha }{s}_{\beta }{s}_{\alpha }(\beta )=\alpha$:

int(h_α h_β h_α) f_β(x) = f_α(int(w_α w_β w_α) x);            (1)

for every $x\in U_{\alpha }(S^{\prime })$, $S^{\prime }\to S$. In particular, for $x=u_{\beta }$, one has $int(w_{\alpha }{w}_{\beta }{w}_{\alpha })u_{\beta }=u_{\alpha }^z$ for a certain section $z$ of $G_m(S)$, and the preceding relation will give

int(h_α h_β h_α) f_β(u_β) = f_α(u_α)^z.            (1')

Let us show that, conversely, taking into account conditions (i) to (iv) of 2.3, (1') entails (1). If $t\in T(S^{\prime })$, $S^{\prime }\to S$, make $int(t)$ act on (1'); taking into account conditions (i) and (ii), one obtains (1) for $x=int(t)u_{\beta }=u_{\beta }^{\beta (t)}$. It suffices to remark now that $\beta :T\to G_{m,S}$ is faithfully flat, hence that condition (1) is certainly true for $x\in U_{\alpha }(S^{\prime }{)}^{\times }$, $S^{\prime }\to S$. Since it is additive in $x$, and every section of $U_{\alpha }$ can be locally written as a sum of two sections of $U_{\alpha }^{\times }$, one concludes that (1') + (i) + (ii) ⇒ (1).

One reasons similarly with conditions of type (b). It is then necessary to use the fact that if $\gamma$ and ${\gamma }^{\prime }$ are two distinct (hence linearly independent over $\mathbb{Q}$) positive roots, the morphism $T\to G_{m,S}^2$ of components $\gamma$ and ${\gamma }^{\prime }$ is faithfully flat. We leave the details of this transposition to the reader.

3. Groups of semisimple rank 2

3.1. Generalities

Lemma 3.1.1. Let $S$ be a scheme, $G$ a pinned $S$-group, $\alpha$ and $\beta$ two roots of $G$, with $\alpha +\beta \ne 0$.

(i) If $\alpha +\beta \notin R$, then

exp(X_α) exp(X_β) = exp(X_β) exp(X_α)

for all $X_{\alpha }\in W(g_{\alpha })(S^{\prime })$, $X_{\beta }\in W(g_{\beta })(S^{\prime })$, $S^{\prime }\to S$.

(ii) If $\alpha +\beta$ and $\alpha -\beta$ are not roots, then

w_α(z_α) exp(X_β) w_α(z_α)^{-1} = exp(X_β)

for all $X_{\beta }\in W(g_{\beta })(S^{\prime })$, $z_{\alpha }\in W(g_{\alpha }{)}^{\times }(S^{\prime })$, $S^{\prime }\to S$, and

w_α(z_α) w_β(z_β) = w_β(z_β) w_α(z_α)

for all $z_{\alpha }\in W(g_{\alpha }{)}^{\times }(S^{\prime })$ and $z_{\beta }\in W(g_{\beta }{)}^{\times }(S^{\prime })$, $S^{\prime }\to S$.

(iii) Let $X_{\alpha }\in W(g_{\alpha }{)}^{\times }(S^{\prime })$, $X_{\beta }\in W(g_{\beta }{)}^{\times }(S^{\prime })$, and $w\in Nor{m}_G(T)(S^{\prime })$ such that $w(\alpha )=\beta$; define $z\in G_m(S^{\prime })$ by

Ad(w) X_α = z X_β.

Then

int(w) exp(x X_α) = exp(x z X_β),
int(w) exp(y X_α^{-1}) = exp(y z^{-1} X_β^{-1}),
int(w) w_α(X_α) = β^*(z) w_β(X_β).

In particular, if $z=\pm 1$, then

$$int(w)w_{\alpha }(X_{\alpha })=w_{\beta }(X_{\beta }{)}^z.$$

(iv) If one sets $t_{\alpha }={\alpha }^{\ast }(-1)$, $t_{\beta }={\beta }^{\ast }(-1)$, then

s_α(t_β) = t_β t_α^{(β^*, α)},    β(t_α) = (-1)^{(α^*, β)}.

Proof. (i) follows at once from Exp. XXII, 5.5.2; (ii) from Exp. XX, 3.1 and from (i) applied to $(\beta ,\alpha )$, $(\beta ,-\alpha )$, $(-\beta ,\alpha )$, $(-\beta ,-\alpha )$; (iii) is evident from the definitions. For the last assertion of (iii), note that ${\beta }^{\ast }(-1)=w_{\beta }(X_{\beta }{)}^{-2}$. Finally, (iv) is trivial.

Proposition 3.1.2 (Groups of type $A_1\times A_1$). Let $S$ be a scheme, $G$ a pinned $S$-group of type $A_1\times A_1$, and denote $\Delta =R^{+}=\alpha ,\beta$.

(i) One has

t_{αβ} = (w_α w_β)^2 = t_α t_β = (w_β w_α)^2 = t_{βα}.

(ii) One has

Ad(w_α) X_β = X_β,    Ad(w_β) X_α = X_α.

(iii) $U_{\alpha }$ and $U_{\beta }$ commute (i.e. $U$ is commutative).

Indeed, by assertion (ii) of the lemma, $w_{\alpha }{w}_{\beta }=w_{\beta }{w}_{\alpha }$, whence $(w_{\alpha }{w}_{\beta }{)}^2=w_{\alpha }^2{w}_{\beta }^2=t_{\alpha }{t}_{\beta }$, which is (i). By assertion (ii) of the lemma, one has also (ii); finally, (iii) is assertion (i) of the lemma.

3.1.3.

Let us make condition (v) of 2.3 explicit here. Using the method exposed in 2.6, one obtains the two following groups of conditions, setting $v_{\alpha }=f_{\alpha }(u_{\alpha })$ for $\alpha \in \Delta$:

(A)   h_α v_β h_α^{-1} = v_β,           (B)   v_α v_β = v_β v_α.
      h_β v_α h_β^{-1} = v_α

3.2. Groups of type A_2

Proposition 3.2.1. Let $S$ be a scheme, $G$ a pinned $S$-group of type A_2, and denote $\Delta =\alpha ,\beta$, $R^{+}=\alpha ,\beta ,\alpha +\beta$.

(i) One has

t_{αβ} = (w_α w_β)^3 = e = (w_β w_α)^3 = t_{βα}.

(ii) Set $Ad(w_{\beta })X_{\alpha }=X_{\alpha +\beta }$. Then

Ad(w_α) X_β = -X_{α + β},    Ad(w_α) X_{α + β} = X_β,    Ad(w_β) X_{α + β} = -X_α.

(iii) Set $p_{\alpha +\beta }(x)=\exp (x{X}_{\alpha +\beta })=int(w_{\beta })p_{\alpha }(x)$. Then

p_β(y) p_α(x) = p_α(x) p_β(y) p_{α + β}(x y).

3.2.2.

The proof occupies Nos. 3.2.2 to 3.2.7. First, $({\beta }^{\ast },\alpha )=({\alpha }^{\ast },\beta )=-1$, whence $\alpha (t_{\beta })=\beta (t_{\alpha })=-1$.

Set $Ad(w_{\beta })X_{\alpha }=X_{\alpha +\beta }$; one has at once

Ad(w_β) X_{α + β} = α(t_β) X_α = -X_α.

Set

Ad(w_α) X_β = z X_{α + β},    z ∈ G_m(S),

whence

Ad(w_α) X_{α + β} = β(t_α) z^{-1} X_β = -z^{-1} X_β.

We know (Exp. XXII, 5.5.2) that there exists a unique section $A\in G_a(S)$ such that

p_β(y) p_α(x) = p_α(x) p_β(y) p_{α + β}(A x y).            (+)

To prove (ii) and (iii), it therefore remains to show that $A=-z=1$.

3.2.3.

Make $int(w_{\beta })$ act on the preceding formula (+); one obtains:

p_{-β}(-y) p_{α + β}(x) = p_{α + β}(x) p_{-β}(-y) p_α(-A x y).            (++)

3.2.4.

By definition of $p_{\alpha +\beta }$,

w_β p_α(x) w_β^{-1} = p_{α + β}(x),

which is written

p_β(1) p_{-β}(-1) p_β(1) p_α(x) p_β(-1) p_{-β}(1) p_β(-1) = p_{α + β}(x).

Since $p_{\beta }$ and $p_{\alpha +\beta }$ commute, $\alpha +2\beta$ not being a root, this is also written

p_β(1) p_α(x) p_β(-1) = p_{-β}(1) p_{α + β}(x) p_{-β}(-1).

Using now (+) in the left member and (++) in the right, one obtains:

p_α(x) p_β(1) p_{α + β}(A x) p_β(-1) = p_{α + β}(x) p_{-β}(1) p_α(A x) p_{-β}(-1).

Since $\alpha +2\beta$ (resp. $\alpha -\beta$) is not a root, the left (resp. right) member is written

p_α(x) p_{α + β}(A x)    resp. p_{α + β}(x) p_α(A x)

and the right-hand term equals $p_{\alpha }(Ax)p_{\alpha +\beta }(x)$, since $2\alpha +\beta$ is not a root. Therefore

p_α(x) p_{α + β}(A x) = p_α(A x) p_{α + β}(x),

which gives (by XXII 4.1.3) $A=1$.

3.2.5.

Now make $int(w_{\alpha })$ act on formula (+); using the fact that $A=1$, one finds

p_{α + β}(z y) p_{-α}(-x) = p_{-α}(-x) p_{α + β}(z y) p_β(-z^{-1} x y).            (+++)

3.2.6.

Write now, as a moment ago,

w_α p_β(y) w_α^{-1} = p_{α + β}(z y),

whence, since $p_{\alpha }$ and $p_{\alpha +\beta }$ commute,

p_α(1) p_β(y) p_α(-1) = p_{-α}(1) p_{α + β}(z y) p_{-α}(-1).

Using now (+) and (+++), this is also written

p_β(y) p_{α + β}(-y) = p_{α + β}(z y) p_β(-z^{-1} y),

and since $p_{\beta }$ and $p_{\alpha +\beta }$ commute, this gives $z=-1$.

3.2.7.

We have therefore proved (ii) and (iii). Let us prove (i). One has

w_α w_β w_α = w_α w_β w_α^{-1} w_α^2 = w_{α + β}^{-1} t_α,

whence

w_β w_α w_β w_α w_β = w_β w_{α + β}^{-1} t_α w_β = w_β w_{α + β}^{-1} w_β^{-1} · s_β(t_α) · t_β
                    = w_α · t_α t_β · t_β = w_α t_α = w_α^{-1},

which gives

(w_α w_β)^3 = (w_β w_α)^3 = e,

which completes the proof.

Remark 3.2.8. Condition (v) of 2.3 translates here as (setting $v_{\alpha }=f_{\alpha }(u_{\alpha })$):

(A)   int(h_α h_β h_α) v_β = vβ → h_α v_β h_α^{-1} = h_β v_α h_β^{-1}      (B)
      int(h_β h_α h_β) v_α = vα                                                  v_β v_α = v_α v_β · h_β v_α h_β^{-1},
                                                                                 v_β · h_β v_α h_β^{-1} = h_β v_α h_β^{-1} · v_β,
                                                                                 v_α · h_β v_α h_β^{-1} = h_β v_α h_β^{-1} · v_α.

Setting $v_{\alpha +\beta }=int(h_{\beta })v_{\alpha }$, the three last conditions are also written

(B)   v_β v_α = v_α v_β v_{α + β},
      v_α v_{α + β} = v_{α + β} v_α,
      v_β v_{α + β} = v_{α + β} v_β.

3.3. Groups of type B_2

Proposition 3.3.1. Let $S$ be a scheme, $G$ a pinned $S$-group of type B_2, and denote $\Delta =\alpha ,\beta$, $R^{+}=\alpha ,\beta ,\alpha +\beta ,2\alpha +\beta$.

(i) One has

t_{αβ} = (w_α w_β)^4 = t_α = (w_β w_α)^4 = t_{βα}.

(ii) If one sets

Ad(w_β) X_α = X_{α + β},    Ad(w_α) X_β = X_{2α + β},

one has:

Ad(w_α) X_{α + β} = -X_{α + β},
Ad(w_α) X_{2α + β} = X_β,
Ad(w_β) X_{α + β} = -X_α,
Ad(w_β) X_{2α + β} = X_{2α + β}.

(iii) Set

p_{α + β}(x) = exp(x X_{α + β}) = int(w_β) p_α(x),
p_{2α + β}(x) = exp(x X_{2α + β}) = int(w_α) p_β(x).

Then:

p_β(y) p_α(x) = p_α(x) p_β(y) p_{α + β}(x y) p_{2α + β}(x^2 y),
p_{α + β}(y) p_α(x) = p_α(x) p_{α + β}(y) p_{2α + β}(2 x y).

3.3.2.

The proof occupies Nos. 3.3.2 to 3.3.6. One has $({\beta }^{\ast },\alpha )=-1$, $({\alpha }^{\ast },\beta )=-2$, whence $\alpha (t_{\beta })=-1$, $\beta (t_{\alpha })=1$.

Note in passing that $\beta (t_{\alpha })=\alpha (t_{\alpha })=1$, which shows that $t_{\alpha }$ is a section of $Centr(G)$. Set

Ad(w_β) X_α = X_{α + β},    Ad(w_α) X_β = X_{2α + β}.

One has at once

Ad(w_β) X_{α + β} = α(t_β) X_α = -X_α,
Ad(w_α) X_{2α + β} = β(t_α) X_β = X_β.

Since $(2\alpha +\beta )+\beta$ and $(2\alpha +\beta )-\beta$ are not roots,

Ad(w_β) X_{2α + β} = X_{2α + β}.

There exists a scalar $k\in G_m(S)$ such that

Ad(w_α) X_{α + β} = k X_{α + β}.

On the other hand, by Exp. XXII, 5.5.2, there exist sections $A,B,C\in G_a(S)$ such that

p_β(y) p_α(x) = p_α(x) p_β(y) p_{α + β}(A x y) p_{2α + β}(B x^2 y),       (1)
p_{α + β}(y) p_α(x) = p_α(x) p_{α + β}(y) p_{2α + β}(C x y).               (2)

It is therefore necessary, in (ii) and (iii), to prove that $A=B=1$, $C=2$, $k=-1$.

3.3.3.

Make $int(w_{\alpha })$ act on formula (2). One finds

p_{2α + β}(y) p_{-α}(-x) = p_{-α}(-x) p_{2α + β}(y) p_{α + β}(A k x y) p_β(B x^2 y).       (3)

Transforming (2) likewise, one obtains

p_{α + β}(k y) p_{-α}(-x) = p_{-α}(-x) p_{α + β}(k y) p_β(C x y).       (4)

Transforming (1) by $int(w_{\beta })$,

p_{-β}(-y) p_{α + β}(x) = p_{α + β}(x) p_{-β}(-y) p_α(-A x y) p_{2α + β}(B x^2 y).       (5)

3.3.4.

Write

w_β p_α(x) w_β^{-1} = p_{α + β}(x).

Since $\alpha +2\beta$ is not a root, this gives

p_β(1) p_α(x) p_β(-1) = p_{-β}(1) p_{α + β}(x) p_{-β}(-1).

Using (1) in the left member and (5) in the right, one obtains

p_α(x) p_β(1) p_{α + β}(A x) p_{2α + β}(B x^2) p_β(-1) =
    p_{α + β}(x) p_{-β}(1) p_α(A x) p_{2α + β}(-B x^2) p_{-β}(-1).

Since $p_{\beta }$ commutes with $p_{\alpha +\beta }$ and $p_{2\alpha +\beta }$ on the one hand, and $p_{-\beta }$ commutes with $p_{\alpha }$ and $p_{2\alpha +\beta }$ on the other hand, this is written

p_α(x) p_{α + β}(A x) p_{2α + β}(B x^2) = p_{α + β}(x) p_α(A x) p_{2α + β}(-B x^2).

Transforming the right member by (2), one obtains

p_α(x) p_{α + β}(A x) p_{2α + β}(B x^2) = p_α(A x) p_{α + β}(x) p_{2α + β}((A C - B) x^2),

which gives

A = 1,    C = 2 B.

3.3.5.

Write now

w_α p_β(y) w_α^{-1} = p_{2α + β}(y).

Since $3\alpha +\beta$ is not a root, this gives

p_α(1) p_β(y) p_α(-1) = p_{-α}(1) p_{2α + β}(y) p_{-α}(-1).

Using (1) in the left member, (3) in the right, one obtains

p_β(y) p_{α + β}(-A y) p_{2α + β}(B y) = p_{2α + β}(y) p_{α + β}(A k y) p_β(B y).

Since $p_{\alpha +\beta }$, $p_{2\alpha +\beta }$ and $p_{\beta }$ commute, this gives at once

B = 1,    -A = A k,

whence finally

A = 1,    B = 1,    C = 2,    k = -1.

3.3.6.

We have therefore proved (ii) and (iii). Let us prove (i). Taking into account the equality $s_{\beta }(t_{\alpha })=t_{\alpha }{t}_{\beta }^{({\alpha }^{\ast },\beta )}=t_{\alpha }$ (since $({\alpha }^{\ast },\beta )=2$),⁸ one has successively:

w_α w_β w_α = w_α w_β w_α^{-1} t_α = w_{2α + β} t_α,
w_β w_α w_β w_α w_β = w_β w_{2α + β} w_β^{-1} · s_β(t_α) · t_β = w_{2α + β} t_α t_β,
w_α w_β w_α w_β w_α w_β w_α = w_α w_{2α + β} w_α^{-1} · s_α(t_α t_β) · t_α = w_β · t_α · t_β t_α · t_α
                            = w_β^{-1} t_α,

whence

$$(w_{\beta }{w}_{\alpha }{)}^4=t_{\alpha },$$

and

(w_α w_β)^4 = s_β(t_α) = t_α,

which completes the proof.

Remark 3.3.7. Condition (v) of 2.3 translates here as follows, setting $v_{\alpha +\beta }=int(h_{\beta })v_{\alpha }$ and $v_{2\alpha +\beta }=int(h_{\alpha })v_{\beta }$:

(A)   int(h_α h_β h_α) v_β = v_β,        (B)   v_β v_α = v_α v_β v_{α + β} v_{2α + β},
      int(h_β h_α h_β) v_α = v_α,              v_{α + β} v_α = v_α v_{α + β} v_{2α + β}^2,
                                               v_{α + β} v_β = v_β v_{α + β},
                                               v_{2α + β} v_α = v_α v_{2α + β},
                                               v_{2α + β} v_β = v_β v_{2α + β},
                                               v_{2α + β} v_{α + β} = v_{α + β} v_{2α + β}.

3.4. Groups of type G_2

Proposition 3.4.1. Let $S$ be a scheme, $G$ a pinned $S$-group of type G_2, and denote $\Delta =\alpha ,\beta$, $R^{+}=\alpha ,\beta ,\alpha +\beta ,2\alpha +\beta ,3\alpha +\beta ,3\alpha +2\beta$.

(i) One has

t_{αβ} = (w_α w_β)^6 = e = (w_β w_α)^6 = t_{βα}.

(ii) If one sets

Ad(w_β) X_α = X_{α + β},    Ad(w_α) X_{α + β} = X_{2α + β},
Ad(w_α) X_β = -X_{3α + β},    Ad(w_β) X_{3α + β} = X_{3α + 2β},

one has:

Ad(w_α) X_{2α + β} = -X_{α + β},    Ad(w_α) X_{3α + β} = X_β,
Ad(w_α) X_{3α + 2β} = X_{3α + 2β},    Ad(w_β) X_{α + β} = -X_α,
Ad(w_β) X_{2α + β} = X_{2α + β},    Ad(w_β) X_{3α + 2β} = -X_{3α + β}.

(iii) If one sets

p_{α + β}(x) = exp(x X_{α + β}) = int(w_β) p_α(x),
p_{2α + β}(x) = exp(x X_{2α + β}) = int(w_α w_β) p_α(x),
p_{3α + β}(x) = exp(x X_{3α + β}) = int(w_α) p_β(-x),
p_{3α + 2β}(x) = exp(x X_{3α + 2β}) = int(w_β w_α) p_β(-x),

one has:

p_β(y) p_α(x) = p_α(x) p_β(y) p_{α + β}(x y) p_{2α + β}(x^2 y) p_{3α + β}(x^3 y) p_{3α + 2β}(x^3 y^2),
p_{α + β}(y) p_α(x) = p_α(x) p_{α + β}(y) p_{2α + β}(2 x y) p_{3α + β}(3 x^2 y) p_{3α + 2β}(3 x y^2),
p_{2α + β}(y) p_α(x) = p_α(x) p_{2α + β}(y) p_{3α + β}(3 x y),
p_{3α + β}(y) p_β(x) = p_β(x) p_{3α + β}(y) p_{3α + 2β}(-x y),
p_{2α + β}(y) p_{α + β}(x) = p_{α + β}(x) p_{2α + β}(y) p_{3α + 2β}(3 x y).

3.4.2.

The proof occupies Nos. 3.4.2 to 3.4.9. One has $({\beta }^{\ast },\alpha )=-1$, $({\alpha }^{\ast },\beta )=-3$, whence $\beta (t_{\alpha })=\alpha (t_{\beta })=-1$. Define $X_{\alpha +\beta }$, $X_{2\alpha +\beta }$, $X_{3\alpha +\beta }$ and $X_{3\alpha +2\beta }$ as in (ii). One has at once

Ad(w_β) X_{α + β} = α(t_β) X_α = -X_α,
Ad(w_α) X_{2α + β} = α(t_α) β(t_α) X_{α + β} = -X_{α + β},
Ad(w_α) X_{3α + β} = -β(t_α) X_β = X_β,
Ad(w_β) X_{3α + 2β} = α(t_β)^3 β(t_β) X_{3α + β} = -X_{3α + β}.

Finally, since $(3\alpha +2\beta )\pm \alpha$ and $(2\alpha +\beta )\pm \beta$ are not roots,

Ad(w_α) X_{3α + 2β} = X_{3α + 2β},    Ad(w_β) X_{2α + β} = X_{2α + β},

which completes the proof of (ii).

3.4.3.

By virtue of Exp. XXII, 5.5.2, there exist scalars⁹

A, B, C, D, E, F, G, H, J ∈ G_a(S),

such that

p_β(y) p_α(x) = p_α(x) p_β(y) p_{α + β}(A x y) p_{2α + β}(B x^2 y) p_{3α + β}(C x^3 y) p_{3α + 2β}(D x^3 y^2),   (1)
p_{α + β}(y) p_α(x) = p_α(x) p_{α + β}(y) p_{2α + β}(E x y) p_{3α + β}(F x^2 y) p_{3α + 2β}(G x y^2),           (2)
p_{2α + β}(y) p_α(x) = p_α(x) p_{2α + β}(y) p_{3α + β}(H x y),                                                  (3)
p_{3α + β}(y) p_β(x) = p_β(x) p_{3α + β}(y) p_{3α + 2β}(J x y).                                                 (4)

3.4.4.

Make $int(w_{\beta })$ act on (1), (3) and (4):

p_{-β}(-y) p_{α + β}(x) =
    p_{α + β}(x) p_{-β}(-y) p_α(-A x y) p_{2α + β}(B x^2 y) p_{3α + 2β}(C x^3 y) p_{3α + β}(-D x^3 y^2),       (5)
p_{2α + β}(y) p_{α + β}(x) = p_{α + β}(x) p_{2α + β}(y) p_{3α + 2β}(H x y),                                    (6)
p_{3α + 2β}(y) p_{-β}(-x) = p_{-β}(-x) p_{3α + 2β}(y) p_{3α + β}(-J x y).                                       (7)

Making $int(w_{\alpha })$ act on (1), one finds

p_{3α + β}(-y) p_{-α}(-x) =
    p_{-α}(-x) p_{3α + β}(-y) p_{2α + β}(A x y) p_{α + β}(-B x^2 y) p_β(C x^3 y) p_{3α + β}(D x^3 y^2).        (8)

3.4.5.

Write

w_β p_α(x) w_β^{-1} = p_{α + β}(x),

that is, $\alpha +2\beta$ not being a root,

p_β(1) p_α(x) p_α(-1) = p_{-β}(1) p_{α + β}(x) p_{-β}(-1).            (9)

Transforming the left member of (9) by (1), then (4), one obtains:

p_β(1) p_α(x) p_β(-1)
   = p_α(x) p_β(1) p_{α + β}(A x) p_{2α + β}(B x^2) p_{3α + β}(C x^3) p_{3α + 2β}(D x^3) p_β(-1)
   = p_α(x) p_β(1) p_{α + β}(A x) p_{2α + β}(B x^2) p_β(-1) p_{3α + β}(C x^3) p_{3α + 2β}((D - C J) x^3)
   = p_α(x) p_{α + β}(A x) p_{2α + β}(B x^2) p_{3α + β}(C x^3) p_{3α + 2β}((D - C J) x^3).            (10)

Transforming the right member of (9) by (5), then (7):

p_{-β}(1) p_{α + β}(x) p_{-β}(-1)
   = p_{α + β}(x) p_{-β}(1) p_α(A x) p_{2α + β}(-B x^2) p_{3α + 2β}(-C x^3) p_{3α + β}(-D x^3) p_{-β}(-1)
   = p_{α + β}(x) p_α(A x) p_{2α + β}(-B x^2) p_{3α + 2β}(-C x^3) p_{3α + β}((C J - D) x^3).            (11)

Using now (2), this right member becomes

p_α(A x) p_{α + β}(x) p_{2α + β}(A E x^2) p_{3α + β}(A^2 F x^3) p_{3α + 2β}(A G x^3) ×
    p_{2α + β}(-B x^2) p_{3α + 2β}(-C x^3) p_{3α + β}((C J - D) x^3)
= p_α(A x) p_{α + β}(x) p_{2α + β}((A E - B) x^2) p_{3α + β}((A^2 F + C J - D) x^3) p_{3α + 2β}((A G - C) x^3). (12)

So (9) rewrites:

p_α(x) p_{α + β}(A x) p_{2α + β}(B x^2) p_{3α + β}(C x^3) p_{3α + 2β}((D - C J) x^3) =
   p_α(A x) p_{α + β}(x) p_{2α + β}((A E - B) x^2) p_{3α + β}((A^2 F + C J - D) x^3) p_{3α + 2β}((A G - C) x^3),

which gives

A = 1,    E = 2 B,    C + D = F + C J,    F = G.

3.4.6.

Write now

w_α p_β(y) w_α^{-1} = p_{3α + β}(-y),

that is, $4\alpha +\beta$ not being a root,

p_α(1) p_β(y) p_α(-1) = p_{-α}(1) p_{3α + β}(-y) p_{-α}(-1).            (13)

Transform the left member by (1):

p_α(1) p_β(y) p_α(-1) = p_β(y) p_{α + β}(-A y) p_{2α + β}(B y) p_{3α + β}(-C y) p_{3α + 2β}(-D y^2).

Transform the right member of (13) successively by (8), (6) and (4):

p_{-α}(1) p_{3α + β}(-y) p_{-α}(-1)
   = p_{3α + β}(-y) p_{2α + β}(A y) p_{α + β}(-B y) p_β(C y) p_{3α + 2β}(D y^2)
   = p_{3α + β}(-y) p_{α + β}(-B y) p_{2α + β}(A y) p_{3α + 2β}(-A B H y^2) p_β(C y) p_{3α + 2β}(D y^2)
   = p_β(C y) p_{3α + β}(-y) p_{α + β}(-B y) p_{2α + β}(A y) p_{3α + 2β}((D - C J - A B H) y^2)
   = p_β(C y) p_{α + β}(-B y) p_{2α + β}(A y) p_{3α + β}(-y) p_{3α + 2β}((D - C J - A B H) y^2).

So (13) rewrites:

p_β(C y) p_{α + β}(-B y) p_{2α + β}(A y) p_{3α + β}(-y) p_{3α + 2β}((D - C J - A B H) y^2) =
   p_β(y) p_{α + β}(-A y) p_{2α + β}(B y) p_{3α + β}(-C y) p_{3α + 2β}(-D y^2),

whence

C = 1,    A = B,    D - C J - A B H = -D.

Taking into account the results already obtained:

A = B = C = 1,    E = 2,    F = G,    D + 1 = F + J,    2 D = H + J.

3.4.7.

Write

w_β p_{3α + β}(x) w_β^{-1} = p_{3α + 2β}(x),

that is,

p_β(1) p_{3α + β}(x) p_β(-1) = p_{-β}(1) p_{3α + 2β}(x) p_{-β}(-1).

Transforming the left member by (4), the right by (7), one obtains:

p_{3α + β}(x) p_{3α + 2β}(-J x) = p_{3α + 2β}(x) p_{3α + β}(-J x),

so $J=-1$.

3.4.8.

Write finally

w_α p_{α + β}(y) w_α^{-1} = p_{2α + β}(y),

that is,

p_α(1) p_{α + β}(y) p_α(-1) = p_{-α}(1) p_α(-1) p_{2α + β}(y) p_α(1) p_{-α}(-1).

Transforming the left member by (2), the right by (3), one obtains:

p_{α + β}(y) p_{2α + β}(-E y) p_{3α + β}(F y) p_{3α + 2β}(-G y^2) =
    p_{-α}(1) p_{2α + β}(y) p_{3α + β}(H y) p_{-α}(-1).

It is immediate to see that, if one makes $p_{-\alpha }(-1)$ commute with $p_{3\alpha +\beta }(Hy)$, then $p_{2\alpha +\beta }(y)$, one does not introduce in the right member any new terms in $p_{3\alpha +\beta }$. The latter therefore writes, denoting by empty parentheses quantities whose exact value does not matter to us:

p_{α + β}( ) p_{2α + β}( ) p_β( ) p_{3α + β}(H y) p_{3α + 2β}( ).

Comparing with the left member, one has at once $F=H$, whence by the previous results $2D=D+1$, so $D=1$, and finally $F=G=H=2D-J=3$, which completes the determination of the coefficients $A,\cdots ,J$ and the proof of (iii).