Exposé XXII. Reductive groups: splittings, subgroups, quotient groups

by M. Demazure

¹ This Exposé consists of two parts. The first (1 through 5.5) gathers the technical results needed for the proof of the uniqueness and existence theorems. The second (5.6 to the end) will not be used in that proof; the end of section 5 will be used in particular in Exposé XXVI on parabolic subgroups; section 6 establishes, in the scheme-theoretic setting, the classical results on the derived group of a reductive group.

1. Roots and coroots. Split groups and root data

Theorem 1.1. Let $S$ be a scheme, $G$ an $S$-reductive group, $T$ a maximal torus of $G$, $\alpha$ a root of $G$ relative to $T$.

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

$$\exp \alpha :W(g\alpha )\to G$$

inducing on the Lie algebras the canonical morphism $g\alpha \to g$. This morphism is a closed immersion. The corresponding morphism

T ·α W(gα) → G

is also a closed immersion.

If $p\alpha :Ga,S\to G$ is a monomorphism normalized by $T$ with multiplier $\alpha$, there exists a unique $X\alpha \in \Gamma (S,g\alpha )\times$² such that $p\alpha (x)=\exp \alpha (xX\alpha )$; one has $Lie(p\alpha )(1)=X\alpha$, and the two preceding formulas set up a bijective correspondence between $\Gamma (S,g\alpha )\times$ and the set of monomorphisms $Ga,S\to G$ normalized by $T$ with multiplier $\alpha$.

(ii) There exists a unique duality (denoted $(X,Y)\mapsto XY$)

gα ⊗_{O_S} g_{-α} ⥲ O_S,

and a unique morphism of groups

$$\alpha \ast :Gm,S\to T,$$

such that formula (F) of Exp. XX 2.1 holds. One has

α ∘ α* = 2,     (-α)* = -α*,

and $\alpha \ast$ is given by the formula of Exp. XX 2.7.

Indeed, a morphism normalized by $T$ with multiplier $\alpha$ factors necessarily through the closed subgroup $Z\alpha =Cent{r}_G(T\alpha )$ of $G$ (cf. Exp. XIX 3.9). Now $(Z\alpha ,T,\alpha )$ is an $S$-elementary system (Exp. XX 1.4), and one is reduced to the results of Exposé XX (1.5, 2.1, and 5.9).

Remark 1.2. Part (i) of Theorem 1.1 remains valid if one only assumes that $\alpha$ is a character of $T$, non-trivial on each fiber. Indeed, one then has a decomposition $S=S^{\prime }\coprod S^{\prime \prime }$ such that $\alpha |S^{\prime }$ is a root of $G_{S^{\prime }}$ relative to $T_{S^{\prime }}$ and $g\alpha |S^{\prime \prime }=0$. If $S=S^{\prime }$, one is reduced to 1.1; if $S=S^{\prime \prime }$ the result is trivial; the general case follows immediately.

Notations 1.3. As in Exposé XX, we denote by $U\alpha$ the image of $W(g\alpha )$; it is a closed subgroup of $G$, equipped canonically with a vector-space structure. We shall call it the vector group associated with the root $\alpha$. We say that $\alpha \ast$ is the coroot associated with $\alpha$. Sections $X\alpha \in \Gamma (S,g\alpha )$ and $X_{-\alpha }\in \Gamma (S,g_{-\alpha })$ are said to be paired if $X\alpha \cdot X_{-\alpha }=1$. Then $X\alpha \in \Gamma (S,g\alpha )\times$ and likewise for $X_{-\alpha }$. The corresponding morphisms $p\alpha$ and $p_{-\alpha }$ are contragredient to one another in the sense of XX, 1.5,³ and one has

pα(x) p_{-α}(y) = p_{-α}(y / (1 + xy)) · α*(1 + xy) · pα(x / (1 + xy)).

Proposition 1.4. Under the conditions of 1.1, let $w\in Nor{m}_G(T)(S)$. Then $\beta =\alpha \circ int(w{)}^{-1}:T\to Gm,S$ is a root of $G$ relative to $T$, $\beta \ast =int(w)\circ \alpha \ast$ is the corresponding coroot, and the following diagram is commutative:

W(gα)  --expα-->  G
  |                |
Ad(w)            int(w)
  ↓                ↓
W(gβ)  --expβ-->  G.

Trivial: transport of structure.

Definitions 1.5. (a) Under the conditions of 1.1, we denote by $s\alpha$ the automorphism of $T$ defined by

$$s\alpha (t)=t\cdot \alpha \ast (\alpha (t){)}^{-1}.$$

We denote by $(,)$ the canonical pairing:

Hom_{S-gr.}(Gm,S, T) × Hom_{S-gr.}(T, Gm,S) → Hom_{S-gr.}(Gm,S, Gm,S) = ℤ_S.

Then $s\alpha$ operates on ${\mathrm{Hom}}_{S-gr.}(T,Gm,S)$, resp. ${\mathrm{Hom}}_{S-gr.}(Gm,S,T)$, by the following formulas, where $\chi$ (resp. $u$) denotes an arbitrary section of ${\mathrm{Hom}}_{S-gr.}(T,Gm,S)$ (resp. of ${\mathrm{Hom}}_{S-gr.}(Gm,S,T)$):

sα(χ) = χ − (α*, χ) α,
sα(u) = u − (u, α) α*.

One has $s\alpha \circ s\alpha =id$ and $s_{-\alpha }=s\alpha$.

(b) If $X\in \Gamma (S,g\alpha )\times$, then the inner automorphism $w\alpha (X)$ of $T$ defined by

wα(X) = expα(X) exp_{-α}(-X⁻¹) expα(X)

(cf. Exp. XX 3.1) coincides with $s\alpha$ (loc. cit.). One then concludes from 1.4:

Corollary 1.6. Let $S$ be a scheme, $G$ an $S$-reductive group, $T$ a maximal torus of $G$, $\alpha$ and $\beta$ two roots of $G$ relative to $T$. Then

sα(β) = β − (α*, β) α

is a root of $G$ relative to $T$, and the corresponding coroot is

sα(β)* = sα(β*) = β* − (β*, α) α*.

Corollary 1.7. Under the preceding conditions, $\alpha \ast =\beta \ast$ implies $\alpha =\beta$.

Indeed, if $\alpha \ast =\beta \ast$, one has (cf. XXI.1.4)

sβ(α) = α − 2β,    sα(β) = β − 2α,

and from this one deduces immediately

(sβ sα)ⁿ(α) = α + 2n(β − α).

If $\beta \ne \alpha$, there exists $s\in S$ such that ${\alpha }_s\ne {\beta }_s$. But then the preceding formula shows that there exist infinitely many distinct roots of $G_s$ relative to $T_s$, which is impossible.

Definitions 1.8.0.⁴ If $u:Gm,S\to T$ is a morphism of groups, we shall say that $u$ is a coroot of $G$ relative to $T$ if there exists a root $\alpha$ of $G$ relative to $T$ such that $\alpha \ast =u$. Consider the functor $R\ast$ of coroots of $G$ relative to $T$ defined as follows:

R*(S′) = the set of coroots of G_{S′} relative to T_{S′}.

If $R$ is the functor of roots of $G$ relative to $T$ (Exp. XIX 3.8), one has a canonical morphism $R\to R\ast$. By virtue of 1.7 and Exp. XIX 3.8, one has:

Corollary 1.8. The canonical morphism $R\to R\ast$ is an isomorphism. In particular, $R\ast$ is representable by a finite twisted constant $S$-scheme, which is an open and closed subscheme of ${\mathrm{Hom}}_{S-gr}(Gm,S,T)$.

This leads us to the following definition:

Definition 1.9. Let $S$ be a scheme, $T$ an $S$-torus. We call twisted root datum in $T$ the data:

(i) a finite subscheme $R$ of ${\mathrm{Hom}}_{S-gr.}(T,Gm,S)$,

(ii) a finite subscheme $R\ast$ of ${\mathrm{Hom}}_{S-gr.}(Gm,S,T)$,

(iii) an isomorphism $R\stackrel{\sim }{\to }R\ast$ denoted $\alpha \mapsto \alpha \ast$,

satisfying the following conditions:

(DR 1)  For every S′ → S and every α ∈ R(S′), one has α ∘ α* = 2.
(DR 2)  For every S′ → S and every α, β ∈ R(S′), one has
        α − (β*, α) β ∈ R(S′),    α* − (α*, β) β* ∈ R*(S′).

Moreover, if for $\alpha \in R(S^{\prime })$ ($S^{\prime }\ne \varnothing$) one has $2\alpha \notin R(S^{\prime })$, the root datum is said to be reduced.

Proposition 1.10. Let $S$ be a scheme, $G$ an $S$-reductive group, $T$ a maximal torus of $G$, $R$ (resp. $R\ast$) the scheme of roots (resp. of coroots) of $G$ relative to $T$. Then $(R,R\ast )$ is a reduced twisted root datum in $T$.

The only thing that remains to be checked is that this twisted root datum is reduced. This was done in Exp. XIX 3.10.

1.11.

Let $T=D_S(M)$ be a trivialized torus. If we denote by $M\ast$ the abelian group dual to $M$, we have canonical isomorphisms (cf. Exp. VIII 1.5):

$${\mathrm{Hom}}_{S-gr.}(T,Gm,S)\stackrel{\sim }{\to }{M}_S,{\mathrm{Hom}}_{S-gr.}(Gm,S,T)\stackrel{\sim }{\to }M{\ast }_S,$$

hence isomorphisms of groups:

Hom_{S-gr.}(T, Gm,S) → Hom_{loc.const.}(S, M),
Hom_{S-gr.}(Gm,S, T) → Hom_{loc.const.}(S, M*).

A character of $T$ (resp. a morphism of groups $Gm,S\to T$) will be called constant (relative to the given trivialization) if the preceding isomorphism transforms it into a constant map from $S$ to $M$ (resp. $M\ast$).

1.12.

With the same notations, let $(M,M\ast ,R,R\ast )$ be a root datum (Exp. XXI). Then $(R_S,R{\ast }_S)$ is a twisted root datum in $T$. Conversely, if $(R,R\ast )$ is a twisted root datum in a torus $T$, a splitting of this root datum is the data of an ordinary root datum $(M,M\ast ,R,R\ast )$ together with an isomorphism $T\simeq D_S(M)$ that transforms $(R,R\ast )$ into $(R_S,R{\ast }_S)$.

Definition 1.13. Let $S$ be a scheme, $G$ an $S$-reductive group, $T$ a maximal torus of $G$. We call splitting of $G$ relative to $T$ the data

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

(ii) of a system of roots $R$ of $G$ relative to $T$ (Exp. XIX 3.6),

satisfying the following two conditions:

(D₁) $S$ is non-empty and the roots $\alpha \in R$ (resp. the corresponding coroots) are identified with constant functions from $S$ to $M$ (resp. $M\ast$).

(D₂) The $g\alpha$ ($\alpha \in R$) are free O_S-modules.

We say that $G$ is splittable relative to $T$ if there exists a splitting of $G$ relative to $T$. By a splitting of $G$ we mean the data of a maximal torus $T$ of $G$ and a splitting of $G$ relative to $T$. We say that $G$ is splittable if there exists a splitting of $G$. By an $S$-split group we mean an $S$-reductive group equipped with a splitting; we denote it by a symbol of the form $(G,T,M,R)$, or simply $G$ when there is no risk of confusion.

Condition (D₁) entails that $R$ (resp. $R\ast$) is canonically identified with a subset of $M$ (resp. $M\ast$).

Proposition 1.14. Let $S$ be a scheme (non-empty), $(G,T,M,R)$ an $S$-split group. Then

R(G, T, M, R) = (M, M*, R, R*)

is a reduced root datum (Exp. XXI 1.1 and 2.1.3); it is a splitting of the twisted root datum of 1.10.

This is a trivial consequence of 1.10 and Exp. XIX 3.7.

We shall sometimes write, for simplicity, $R(G,T,M,R)=R(G)$. We shall systematically use the notations $V$, $V(R)$, $W$, … of Exp. XXI.

Remark 1.15. (a) If $S$ is connected non-empty (resp. if $\mathrm{Pic}(S)=0$), the condition (D₁) (resp. (D₂)) is automatically satisfied.

(b) If $(G,T,M,R)$ is an $S$-split group, then for every $S^{\prime }\to S$, $S^{\prime }\ne \varnothing$, $(G_{S^{\prime }},T_{S^{\prime }},M,R)$ is an $S^{\prime }$-split group, and $R(G,T,M,R)=R(G_{S^{\prime }},T_{S^{\prime }},M,R)$.

1.16.

Let $T=D_S(M)$ be a trivialized torus. The Lie algebra $t$ of $T$ is canonically identified (Exp. II 5.1.1) with

$$t\simeq M\ast \otimes O_S.$$

For every morphism of groups $u:T\to Gm,S$, $Lie(u)$ is a linear form

Lie(u) : t → O_S = Lie(Gm,S/S).

In particular, if $u$ is defined by an element $\alpha \in M$, then $Lie(u)$ is the linear form $\alpha$ on $M\ast \otimes O_S$ defined by $\alpha$:

α(m ⊗ x) = (m, α) x.

Symmetrically, for every morphism of groups $h:Gm,S\to T$, $Lie(h)$ is an O_S-morphism $O_S=Lie(Gm,S/S)\to t$, canonically defined by the section

H = Lie(h)(1) ∈ Γ(S, t).

In particular, if $h$ is defined by an element $m\in M\ast$, one has

$$H=Lie(h)(1)=m\otimes 1.$$

Comparing the two definitions, one finds in particular

α(H) = (h, α) · 1 ∈ Γ(S, O_S).

1.17.

These definitions apply in particular to the case where $T$ is the maximal torus of a split group. Each root $\alpha \in R$ defines an infinitesimal root $\alpha \in {\mathrm{Hom}}_{O_S}(t,O_S)$ with

α(m ⊗ x) = (m, α) x.

Each coroot $\alpha \ast \in R\ast$ defines an infinitesimal coroot

Hα ∈ Γ(S, t),    Hα = α* ⊗ 1.

For $\alpha ,\beta \in R$, one has the relation

$$\alpha (H\beta )=(\beta \ast ,\alpha )\cdot 1,$$

and in particular

$$\alpha (H\alpha )=2.$$

In particular, if 2 is invertible on $S$, then $\alpha$ and $H\alpha$ are non-zero on each fiber.

2. Existence of a splitting. Type of a reductive group

Proposition 2.1. Let $S$ be a scheme, $G$ an $S$-reductive group, $T$ a maximal torus of $G$. Suppose $T$ is split. Then $G$ is locally splittable relative to $T$: for every $s_0\in S$, there exists an open neighborhood $U$ of $s_0$ such that the $U$-group G_U is splittable relative to T_U.

Indeed, write $T\simeq D_S(M)$ and

g = ⨿_{m ∈ M} g^m.

Let $R=m\in M|m\ne 0,g^m(s_0)\ne 0$. Shrinking $S$ if necessary (replacing it by an open neighborhood of $s_0$), we may suppose that the $g\alpha$, $\alpha \in R$, are free, and that the $g^m$, $m\ne 0$, $m\notin R$, are zero. We then have

g = t ⨿ ⨿_{α ∈ R} gα,

with the $g\alpha$ free of rank 1. It follows that $R$ is a system of roots of $G$ relative to $T$ (Exp. XIX 3.6). The coroots $\alpha \ast$ corresponding to the $\alpha \in R$ are then identified with locally constant functions on $S$ with values in $M\ast$. Shrinking $S$ further, we may take them constant, and we are done.

Note that the proof gives immediately:

Proposition 2.2. Let $S$ be a non-empty connected scheme with $\mathrm{Pic}(S)=0$, for example $\mathrm{Spec}(\mathbb{Z})$ or a local scheme (in particular the spectrum of a field). If $G$ is an $S$-reductive group possessing a split maximal torus $T$, then $G$ is splittable relative to $T$.

We deduce immediately from 2.1 and the fact that a reductive group locally possesses maximal tori for the étale topology (Exp. XIX 2.5):

Corollary 2.3. Let $S$ be a scheme, $G$ an $S$-reductive group (resp. and $T$ a maximal torus of $G$). Then $G$ is locally splittable (resp. locally splittable relative to $T$) for the étale topology on $S$.

Corollary 2.4. Let $k$ be a field, $G$ a $k$-reductive group. There exists a finite separable extension $K/k$ such that G_K is splittable.

Remark 2.5. Using 2.1 and the remark Exp. XIX 2.9, one immediately proves the following result: let $G=(G,T,M,R)$ be an $S$-split group; there exists a cover of $S$ by open sets Uᵢ such that each split group $G_{U_i}$ arises by base change from a split group over a noetherian ring (in fact, a finitely generated $\mathbb{Z}$-algebra). We will furthermore prove that every split group over $S$ already arises from a $\mathbb{Z}$-split group (Exp. XXV).

2.6.

Let $k$ be an algebraically closed field and $G$ a $k$-reductive group. One knows (e.g. by 2.4) that there exist splittings of $G$. Let $(G,T,M,R)$ and $(G,T^{\prime },M^{\prime },R^{\prime })$ be two splittings of $G$; the root data $R(G,T,M,R)$ and $R(G,T^{\prime },M^{\prime },R^{\prime })$ are then isomorphic.

Indeed, one sees first that one can reduce to the case where $T=T^{\prime }$ (because there exists $g\in G(k)$ such that $T^{\prime }=int(g)T$, and one verifies easily that if one transports a splitting by an automorphism of $G$, one obtains a root datum isomorphic to the initial datum); but since $S=\mathrm{Spec}(k)$ is connected, the isomorphism $D_k(M)\stackrel{\sim }{\to }T\stackrel{\sim }{\to }{D}_k(M^{\prime })$ arises from a unique isomorphism $M\simeq M^{\prime }$; for the same reason, there is at most one system of roots of $G$ relative to $T$.

Definition 2.6.1.⁵ If $G$ is a $k$-reductive group ($k$ an algebraically closed field), we call type of $G$ the isomorphism class of the root datum defined by an arbitrary splitting of $G$; if $G$ is a torus, of type $M$ in the sense of Exp. IX 1.4, then the type of $G$ as reductive group is given by the trivial root datum $(M,M\ast ,\varnothing ,\varnothing )$.

By 1.15 (b),⁶ the type is invariant under (algebraically closed) extension of the base field.

Definition 2.7. If $G$ is an $S$-reductive group and $s\in S$, we call type of $G$ at $s$ the type of the $s$-reductive group $G_s$.

For every $S^{\prime }\to S$ and every $s^{\prime }\in S^{\prime }$ projecting to $s\in S$, the type of $G_{S^{\prime }}$ at $s^{\prime }$ is equal to the type of $G$ at $s$.

If $G$ is splittable, and if $(G,T,M,R)$ is a splitting of $G$, then the type of $G$ at $s$ is the isomorphism class of $R(G,T,M,R)$ by 1.15 (b).⁶ It then follows immediately from 2.3:

Proposition 2.8. Let $G$ be an $S$-reductive group ($S\ne \varnothing$). The function

s ↦ type of G at s

is locally constant on $S$. In particular, there is a partition of $S$ into non-empty open subschemes such that on each of them $G$ is of constant type. More precisely, let $E$ be the set of types of the fibers of $G$; for every $t\in E$, let $S_t$ be the set of points $s\in S$ where $G$ is of type $t$; then $(S_t{)}_{t\in E}$ is a partition of $S$ and each $S_t$ is open and closed (and non-empty).

3. The Weyl group

3.1.

Let $S$ be a scheme, $G$ an $S$-reductive group, $T$ a maximal torus of $G$. Then

$$W_G(T)=Nor{m}_G(T)/T$$

is a finite étale $S$-group (Exp. XIX 2.5). The morphism $n\mapsto int(n)$ induces, by passage to the quotient, a canonical monomorphism (which is in fact an open immersion):

$$W_G(T)\to {\mathrm{Aut}}_{S-gr.}(T).$$

3.2.

Suppose now that $G$ is splittable relative to $T$. Choose a splitting, say $(G,T,M,R)$. We then have a canonical isomorphism (Exp. VIII 1.5)

$${\mathrm{Aut}}_{S-gr.}(T)\simeq ({\mathrm{Aut}}_{gr.}(M){)}_S.$$

In particular, if $W$ is the Weyl group of the root datum $R(G)$ (Exp. XXI 1.1.8), we have a monomorphism

$$W_S\to {\mathrm{Aut}}_{S-gr.}(T).$$

3.3.

For each root $\alpha \in R$, the symmetry $s\alpha \in W$ operates on $M$ by

sα(x) = x − (α*, x) α,

hence on $T$ (via the preceding morphism) by

$$s\alpha (t)=t\cdot \alpha \ast (\alpha (t){)}^{-1}.$$

On the other hand, since $g\alpha$ is assumed free, there exists $X\in \Gamma (S,g\alpha )\times$. Consider $w\alpha (X)\in Nor{m}_G(T)(S)$ (Exp. XX 3.1). One has (loc. cit.)

$$int(w\alpha (X))(t)=s\alpha (t).$$

Since $W$ is generated by the $s\alpha$, $\alpha \in R$, it follows from the preceding remarks that if we regard $W$ and $Nor{m}_G(T)(S)/T(S)$ as groups of automorphisms of $T$, we have

$$W\subset Nor{m}_G(T)(S)/T(S)\subset W_G(T)(S).$$

By definition of the constant group W_S associated with $W$ (cf. I 1.8), we thus have a commutative diagram

W_S  ──────────→  W_G(T)
   ↘                ↙
       Aut_{S-gr.}(T).

Proposition 3.4. Let $S$ be a scheme, $(G,T,M,R)$ an $S$-split group, $W$ the Weyl group of the root datum $R(G)$. Then the canonical monomorphism

$$W_S\to W_G(T)=Nor{m}_G(T)/T$$

is an isomorphism.

These are both étale groups over $S$; it therefore suffices to check that for every $s\in S$, $W_S(s)\to W_G(T)(s)$ is an isomorphism.⁷ The latter assertion follows, for example, from Bible, § 11.3, th. 2.

Remark 3.5. Using 2.3, the preceding proposition gives a new proof of the fact that the Weyl group of a maximal torus of an $S$-reductive group $G$ is finite over $S$ (Exp. XIX 2.5 (ii)).⁸

3.6.

Under the conditions of 3.1, for every $w\in W_G(T)(S)$, we denote by $N_w$⁹ the fiber product of the following diagram:

N_w  ────→  Norm_G(T)
 |              |
 ↓              ↓ w
 S    ────→  W_G(T).

This is an open and closed subscheme of $Nor{m}_G(T)$, which is a principal homogeneous bundle under $T$ on the left (resp. on the right) by the law $(t,q)\mapsto tq$ (resp. $(q,t)\mapsto qt$). If $n\in N_w(S)$, one has

N_{ww′} = n · N_{w′},    N_{w′w} = N_{w′} · n.

3.7.

In particular, if $\alpha$ is a root of $G$ relative to $T$, $N_{s\alpha }$ is none other than what was denoted $N\times$ in Exp. XX 3.0. If $g\alpha$ is free on $S$, one then has $N_{s\alpha }(S)\ne \varnothing$.

By 3.4 and condition (D₂) of the splitting, we deduce:

Corollary 3.8. Under the conditions of 3.4, the morphism

Norm_G(T)(S) → W_G(T)(S) = Hom_{loc.cons.}(S, W)

is surjective. In particular, for every $w\in W$, there exists $n_w\in Nor{m}_G(T)(S)$ such that $int(n_w)|T=w$.

4. Homomorphisms of split groups

4.1. The "big cell"

4.1.1.

Let $(G,T,M,R)$ be a split $S$-reductive group. Choose a system of positive roots (Exp. XXI 3.2.1) $R+$ of the root datum $R(G)$. Set $R-=-R+$.

Choose a total ordering on $R+$ (resp. $R-$) and consider the morphism induced by the product in $G$

u : ∏_{α ∈ R−} Uα ×_S T ×_S ∏_{α ∈ R+} Uα → G.

This is an open immersion. Indeed, since both sides are flat and of finite presentation over $S$, it suffices to verify this on each geometric fiber (SGA 1, I 5.7 and VIII 5.4); one is thus reduced to the case where $S$ is the spectrum of an algebraically closed field; but, by Bible, § 13.4, cor. 2 to th. 3, $u$ is radicial and dominant; since the tangent map of $u$ at the origin is an isomorphism (definition of a system of roots), $u$ is birational; but $G$ being normal, one may apply Zariski's "Main Theorem" (EGA III₁, 4.4.9) and $u$ is an open immersion.

Let us show that the image $\Omega$ of this open immersion is independent of the ordering chosen on $R+$ (resp. $R-$). Since this is a question of comparing open subsets of $G$, one is reduced to proving that they have the same geometric points, so one may again assume that $S$ is the spectrum of an algebraically closed field. Then the assertion is none other than Bible, § 13, prop. 1 (c) and th. 1 (a).

We have thus proved:

Proposition 4.1.2. Let $(G,T,M,R)$ be an $S$-split group. Let $R+$ be a system of positive roots of $R$. There exists an open subset ${\Omega }_{R+}$ of $G$ such that for every total ordering on $R+$ (resp. $R-$), the morphism induced by the product in $G$

∏_{α ∈ R−} Uα ×_S T ×_S ∏_{α ∈ R+} Uα → G

is an open immersion with image ${\Omega }_{R+}$.

Remark 4.1.3. One can translate 4.1.2 as follows: choose, for every $\alpha \in R$, an isomorphism of vector groups $p\alpha :Ga,S\stackrel{\sim }{\to }U\alpha$ (cf. 1.19); then the morphism (set $N=Card(R+)=Card(R-)$)

Ga,S^N ×_S T ×_S Ga,S^N → G

defined set-theoretically by

((xα)_{α ∈ R−}, t, (xα)_{α ∈ R+}) ↦ ∏_{α ∈ R−} pα(xα) · t · ∏_{α ∈ R+} pα(xα)

is an open immersion whose image depends only on $R+$ (and not on the choice of the $p\alpha$ or the orderings on $R+$ and $R-$).

Notation 4.1.4. We write ${\Omega }_{R+}={\prod }_{\alpha \in R-}U\alpha \cdot T\cdot {\prod }_{\alpha \in R+}U\alpha$.¹⁰

Proposition 4.1.5. The scheme ${\Omega }_{R+}$ is of finite presentation over $S$ (hence retrocompact in $G$) and is universally schematically dense in $G$ relative to $S$ (cf. Exp. XVIII 1).

The first assertion is trivial. Then,¹¹ ${\Omega }_{R+}$ is flat and of finite presentation over $S$, and contains the unit section, hence meets each fiber of $G$ in a non-empty open subset, hence in a dense one; the second assertion follows from Exp. XVIII 1.3.¹²

Corollary 4.1.6. Let $(G,T,M,R)$ be a split $S$-reductive group. Then

Centr(G) = ⋂_{α ∈ R} Ker(α).

¹³

Consequently, $Centr(G)$ is representable by a closed diagonalizable subgroup of $G$.

The second assertion follows immediately from the first. To prove the latter, one may invoke Exp. XII 4.8 and 4.11; one may also proceed directly as follows.¹⁴

Let $S^{\prime }\to S$. If $t\in T(S^{\prime })$ and $\alpha (t)=1$ for every $\alpha \in R$, then $int(t)$ induces the identity on $T_{S^{\prime }}$ and on each $(U\alpha {)}_{S^{\prime }}$, $\alpha \in R$, hence also on $({\Omega }_{R+}{)}_{S^{\prime }}$, hence on $G_{S^{\prime }}$ by schematic density, whence $t\in Centr(G)(S^{\prime })$.

Conversely, since $Cent{r}_{G_{S^{\prime }}}(T_{S^{\prime }})=T_{S^{\prime }}$ (cf. Exp. XIX 2.8), if $g\in G(S^{\prime })$ centralizes $T_{S^{\prime }}$ and the $(U\alpha {)}_{S^{\prime }}$, it is a section of $T_{S^{\prime }}$ that annihilates the $\alpha \in R$.

Corollary 4.1.7. Let $S$ be a scheme, $G$ an $S$-reductive group. Then the center of $G$ is representable by a closed subgroup of $G$, of multiplicative type; it is also "the intersection of the maximal tori of $G$" in the following sense: for every $S^{\prime }\to S$, $Centr(G)(S^{\prime })$ is the set of $g\in G(S^{\prime })$ whose inverse image in $G(S^{\prime \prime })$, for every $S^{\prime \prime }\to S^{\prime }$, is contained in all the $T(S^{\prime \prime })$, where $T$ runs through the set of maximal tori of $G_{S^{\prime \prime }}$.

Taking into account 2.3, the first assertion follows from 4.1.6 by descent.¹⁵ Let us prove the second assertion. Let $H$ be "the intersection of the maximal tori of $G$" in the preceding sense. One obviously has $Centr(G)\subset H$.¹⁶ Then, by descent, it suffices to prove $Centr(G)=H$ in the case where $G$ is split. Since $H$ is contained in the intersection of the maximal tori of $G$ in the usual sense, this follows from the following remark: if $(G,T,M,R)$ is a splitting, $\alpha \in R$ and $X\in \Gamma (S,g\alpha )\times$, then $int(\exp \alpha (X))(T)\cap T=Ker(\alpha )$, as a trivial computation shows. (Cf. also Exp. XII 8.6 and 8.8 for a more general statement.)

Remark 4.1.8. In what follows, we shall systematically identify, in the split case, $T$ with $D_S(M)$. Then $Centr(G)$ is none other than $D_S(M/{\Gamma }_0(R))$, where ${\Gamma }_0(R)$ is the subgroup of $M$ generated by $R$ (cf. Exp. XXI 1.1.6). If ${\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_n$ is a system of simple roots of $R$, one immediately has (cf. Exp. XX 1.19):

Centr(G) = ⋂ Ker(αᵢ) = ⋂ Centr(Z_{αᵢ}).

Proposition 4.1.9. Let $S$ be a scheme, $(G,T,M,R)$ an $S$-split group, $Q$ an $S$-torus, ${\alpha }_0$ a character of $Q$, $L$ an invertible O_S-module,

f : Q → T,    p : W(L) → G

morphisms of groups satisfying the set-theoretic relation

p(α₀(q) x) = int(f(q)) · p(x),

for all $q\in Q(S^{\prime })$, $x\in W(L)(S^{\prime })$, $S^{\prime }\to S$. Suppose that $f$ separates the elements of $R$ in the following sense: if $\alpha ,{\alpha }^{\prime }\in R$ and $m,m^{\prime }\in \mathbb{Z}$, then $m\alpha \circ f=m^{\prime }{\alpha }^{\prime }\circ f$ implies $m\alpha =m^{\prime }{\alpha }^{\prime }$.¹⁷ Finally, let $s\in S$ be such that $({\alpha }_0{)}_s\ne e$ and $p_s\ne e$.

There then exist an open set $U$ of $S$ containing $s$, an integer $q>0$ such that $x\mapsto x^q$ is an endomorphism of Ga,U, a root $\alpha \in R$, and an isomorphism of O_U-modules

$$h:(L|U{)}^{\otimes q}\stackrel{\sim }{\to }g\alpha |U$$

such that

(i) $(\alpha \circ f)|U=(q{\alpha }_0)|U$,

(ii) $p(X)=\exp \alpha (h(X^q))$ for every $X\in W(L)(S^{\prime })$, $S^{\prime }\to U$.

Moreover, once $U$ is chosen, $q$, $\alpha$ and $h$ are uniquely determined.

Shrinking $S$ if necessary, we may suppose that ${\alpha }_0$ is non-zero on every fiber of $S$. Choose a system of positive roots $R+$ of $R$ and let $V=p^{-1}({\Omega }_{R+})$. This is an open subset of $W(L)$ containing the zero section and stable under multiplication by every ${\alpha }_0(q)$, $q\in Q(S^{\prime })$, $S^{\prime }\to S$. Since ${\alpha }_0$ is non-trivial on every fiber, it follows immediately that $V=W(L)$, hence that $p$ factors through ${\Omega }_{R+}$. Choose an arbitrary ordering on $R+$ and $R-$; all products will be taken in this ordering. We thus have unique morphisms

aα : W(L) → Uα,    α ∈ R,
b  : W(L) → T

such that

p(x) = ∏_{α ∈ R−} aα(x) · b(x) · ∏_{α ∈ R+} aα(x).

Writing the covariance condition under $Q$, one obtains immediately

aα(α₀(q) x) = α(f(q)) aα(x),    α ∈ R,
b(α₀(q) x)  = b(x)

for all $x\in W(L)(S^{\prime })$, $q\in Q(S^{\prime })$, $S^{\prime }\to S$. The second condition gives at once $b=e$.

Now let $\alpha \in R$ be such that $(a\alpha {)}_s\ne e$ (we know such an $\alpha$ exists, since $p_s$ is supposed $\ne e$). Applying Exp. XIX 4.12 (a), one deduces that there exists an integer $n>0$ such that $(\alpha \circ f{)}_s=(n{\alpha }_0{)}_s$. Shrinking $S$, one can assume $\alpha \circ f=n{\alpha }_0$ (Exp. IX 5.3). But then, for every ${\alpha }^{\prime }\in R$, ${\alpha }^{\prime }\ne \alpha$, one has $({\alpha }^{\prime }\circ f{)}_s\ne m{\alpha }_0$ for every integer $m>0$, by virtue of the hypothesis made on $f$ (and the fact that the only roots proportional to $\alpha$ are $\alpha$ and $-\alpha$). Applying again Exp. XIX 4.12 (a), this time to $a{\alpha }^{\prime }$, one deduces that $a{\alpha }^{\prime }$ is zero in a neighborhood of $S$; since $R$ is finite, one may, shrinking $S$ again, suppose the $a{\alpha }^{\prime }$ zero for ${\alpha }^{\prime }\in R$, ${\alpha }^{\prime }\ne \alpha$. One then has $p=a\alpha$, and one may apply Exp. XIX 4.12 (b), then (c), which gives the announced result (the uniqueness assertions are obvious).

Remark 4.1.10. The condition imposed on $f$ in 4.1.9 is satisfied in particular when $f$ is surjective (= faithfully flat).

Proposition 4.1.11. Let $(G,T,M,R)$ be an $S$-split group, $R+$ a system of positive roots of $R$, ${\Omega }_{R+}$ the corresponding "big cell".

(i) Let $H$ be a separated $S$-group functor¹⁸ for (fppf). If $f,g:G\Rightarrow H$ are two morphisms of groups that coincide on ${\Omega }_{R+}$, then $f=g$.

(ii) Let $H$ be an $S$-sheaf of groups for (fppf) and $f:{\Omega }_{R+}\to H$ an $S$-morphism satisfying the following condition: for every $S^{\prime }\to S$ and every $x,y\in {\Omega }_{R+}(S^{\prime })$ such that $xy\in {\Omega }_{R+}(S^{\prime })$, one has $f(xy)=f(x)f(y)$. There then exists a (unique, by (i)) morphism of groups $\bar{f}:G\to H$ extending $f$.

Indeed, by 4.1.5, (i) (resp. (ii)) follows immediately from Exp. XVIII 2.2 (resp. 2.3 and 2.4).

Remark 4.1.12. If $\alpha \in R+$, one has

(†)    Ω_{R+} ∩ Zα = U_{-α} · T · Uα.

¹⁹ Indeed, for every $S^{\prime }\to S$, if $g={\prod }_{\beta \in R-}p\beta (x\beta )\cdot t\cdot {\prod }_{\beta \in R+}p\beta (x\beta )$ is an element of ${\Omega }_{R+}(S^{\prime })$ and if $t^{\prime }\in T\alpha (S^{\prime \prime })$, then

t′ g t′⁻¹ = ∏_{β ∈ R−} pβ(β(t′) xβ) · t · ∏_{β ∈ R+} pβ(β(t′) xβ)

and since $\alpha$ and $-\alpha$ are the only two elements of $R$ that take the value 1 on $T\alpha$, we obtain that $g$ belongs to $Z\alpha =Centr(T\alpha )$ if and only if $x\beta =0$ for $\beta \ne \pm \alpha$.

By (†), one deduces from XX 2.1 that if $X\in \Gamma (X,g\alpha )$ and $Y\in \Gamma (S,g_{-\alpha })$, one has:

expα(X) exp_{-α}(Y) ∈ Ω_{R+}(S) ⇔ 1 + XY invertible.

4.2. Morphisms of split groups

Definition 4.2.1. Let $S$ be a scheme (non-empty), $(G,T,M,R)$ and $(G^{\prime },T^{\prime },M^{\prime },R^{\prime })$ two $S$-split groups. We say that the morphism of $S$-groups $f:G\to G^{\prime }$ is compatible with the splittings, or defines a morphism of split groups, if the restriction of $f$ to $T$ factors through a morphism $f_T:T\to T^{\prime }$ of the form $f_T=D_S(h)$, where $h:M^{\prime }\to M$ is a morphism of groups satisfying the following condition:

there exists a bijection $d:R\stackrel{\sim }{\to }{R}^{\prime }$²⁰ and for each $\alpha \in R$ an integer $q(\alpha )>0$ such that $x\mapsto x^{q(\alpha )}$ is an endomorphism of Ga,S and that

h(d(α)) = q(α) α,    ᵗh(α*) = q(α) d(α)*.

Remark 4.2.2. It is immediate that $h$, $d$, $q(\alpha )$ for $\alpha \in R$, are uniquely determined by $f$. One writes $h=R(f)$. The $q(\alpha )$ are the root exponents of $f$ (or of $h$).

Let $p$ be the prime number (if it exists) that is zero on $S$; set $p=1$ if there is no prime number zero on $S$. Then $R(f)$ is a $p$-morphism of reduced root data in the sense of Exp. XXI 6.8. One has thus defined a functor $R$ from the category of $S$-split groups to that of reduced root data (equipped with $p$-morphisms).

Proposition 4.2.3. Under the conditions of 4.2.1, one has the following properties:

(i) For every $\alpha \in R$, there exists a unique isomorphism of O_S-modules

$$f\alpha :(g\alpha {)}^{\otimes q(\alpha )}\stackrel{\sim }{\to }{g}_{d(\alpha )}^{\prime }$$

such that

$$f(\exp \alpha (X))={\exp }_{d(\alpha )}(f\alpha (X^{q(\alpha )}))$$

for every $X\in W(g\alpha )(S^{\prime })$, $S^{\prime }\to S$.

(ii) For every $\alpha \in R$, one has $q(-\alpha )=q(\alpha )$ and $f\alpha$ and $f_{-\alpha }$ are contragredient to one another.

(iii) For every $\alpha \in R$, every $Z\in W(g\alpha )\ast (S^{\prime })$, $S^{\prime }\to S$, one has

$$f(w\alpha (Z))=w_{d(\alpha )}(Z^{q(\alpha )}).$$

By hypothesis the diagram

Gm,S  --α*-->   T   --α-->   Gm,S
 |              |              |
 q(α)          f_T            q(α)
 ↓              ↓              ↓
Gm,S  --d(α)*-> T′ --d(α)--> Gm,S

is commutative. It follows that $f$ maps $Ker(\alpha )$ into $Ker(d(\alpha ))$, hence $T\alpha$ into $T_{d(\alpha )}^{\prime }$, hence $Z\alpha$ into $Z_{d(\alpha )}^{\prime }$. There is then nothing more to do than apply Exp. XX 3.10 and 3.11 to the groups $Z\alpha$ and $Z_{d(\alpha )}^{\prime }$.

Proposition 4.2.4. The morphism $f$ induces a morphism $f_N$ of $Norm(T)$ into $Nor{m}_{G^{\prime }}(T^{\prime })$, hence a morphism $f_W$ of $W_G(T)$ into $W_{G^{\prime }}(T^{\prime })$; the latter is an isomorphism. More precisely, if we denote by $d:W(R(G))=W\to W^{\prime }=W(R(G^{\prime }))$ the isomorphism extending $s\alpha \mapsto s_{d(\alpha )}$ (Exp. XXI 6.8.4), we have a commutative diagram of isomorphisms:

W_G(T)  --f_W-->  W_{G′}(T′)
   ↑                  ↑
   ≀                  ≀
   |                  |
  W_S  --d_S-->     W′_S.

This follows immediately from 3.4, Exp. XXI 6.8.4, and (iii) above.

Remark 4.2.5. With the notations of 4.2.3, the restriction of $f$ to ${\Omega }_{R+}$ (for a system of positive roots $R+$) is written explicitly: it maps ${\Omega }_{R+}$ to ${\Omega }_{d(R+)}^{\prime }$ ($d(R+)$ is a system of positive roots of $R^{\prime }$ by Exp. XXI 6.8.7) and is given by the set-theoretic formula:

f(∏_{α ∈ R−} expα(Xα) · t · ∏_{α ∈ R+} expα(Xα))
  = ∏_{α ∈ R−} exp_{d(α)}(fα(Xα^{q(α)})) · f_T(t) · ∏_{α ∈ R+} exp_{d(α)}(fα(Xα^{q(α)})).

Proposition 4.2.6. (i) $f$ is surjective (= faithfully flat in the present case, cf. VI_B 3.11) if and only if $f_T$ is.

(ii) One has $Ker(f)\subset {\Omega }_{R+}$.

Let us prove (i): if $f$ is surjective, then $f_T(T)=f(T)$ is a maximal torus of $G^{\prime }$ (indeed $f(T)$ is a subtorus of a maximal torus $T^{\prime }$ (Exp. IX 6.8); to verify that $f(T)=T^{\prime }$, one reduces to the case of an algebraically closed field, where this is Bible, § 7.3, th. 3 (a)).

If $f_T$ is surjective, the preceding formula shows that $f$ induces a surjection from $\Omega ={\Omega }_{R+}$ onto ${\Omega }^{\prime }={\Omega }_{d(R+)}^{\prime }$.²¹ Since the fibers of $G^{\prime }$ are connected, it follows (cf. Exp. VI_A, 0.5) that $f$ is surjective.

Let us prove (ii) and for this admit a result to be proved below (5.7.4): choose for each $w\in W$ an $n_w\in Nor{m}_G(T)(S)$ representing it; then the open sets $n_w\Omega$ ($w\in W$) form a cover of $G$. It is then enough to prove that $Ker(f)\cap n_w\Omega \ne \varnothing$ implies $w=1$. If $x\in \Omega (S^{\prime })$, $S^{\prime }\to S$ and $f(n_{w}x)=1$, then $f(x)=f(n_w{)}^{-1}$; but $f(x)\in {\Omega }^{\prime }(S^{\prime })$ and $f(n_w{)}^{-1}\in Nor{m}_{G^{\prime }}(T^{\prime })(S^{\prime })$. By 4.2.4, one is reduced to proving:

Lemma 4.2.7. Under the conditions of 4.1.2, one has $\Omega \cap Nor{m}_G(T)=T$.

Let

x = ∏_{α ∈ R−} pα(xα) · t · ∏_{α ∈ R+} pα(xα) = v t u ∈ Ω(S′).

If $x$ normalizes $T_{S^{\prime }}$, one has for every $t^{\prime }\in T(S^{\prime })$,

x t′ x⁻¹ = t′′ ∈ T(S′),

that is, $x{t}^{\prime }=t^{\prime \prime }x$, which is written

v (t t′) (t′⁻¹ u t′) = (t′′ v t′′⁻¹) (t′′ t) u,

which gives $t^{\prime -1}u{t}^{\prime }=u$, hence $u\in Cent{r}_G(T)(S^{\prime })=T(S^{\prime })$, that is $u=1$. Similarly $v=1$.

Corollary 4.2.8. One has

Ker(f) = ∏_{α ∈ R−} Kα · Ker(f_T) · ∏_{α ∈ R+} Kα,

where for each $\alpha \in R$, $K\alpha$ denotes the finite $S$-group

Kα = Ker(Uα → Uα^{⊗q(α)}) ≃ α_{q(α), S}.

To apply this corollary, let us set:

Definition 4.2.9. Let $S$ be a scheme, $G$ and $G^{\prime }$ two $S$-reductive groups. A morphism of $S$-groups $f:G\to G^{\prime }$ that is faithfully flat and finite (i.e. surjective with finite kernel over $S$) is called an isogeny. If moreover $Ker(f)$ is a central subgroup of $G$, one says that $f$ is a central isogeny.

Proposition 4.2.10. Let $f:G\to G^{\prime }$ be a morphism of split groups. For $f$ to be an isogeny (resp. a central isogeny) it is necessary and sufficient that $f_T$ be an isogeny, i.e. that $R(f)$ be injective with finite cokernel (resp. and that for every $\alpha \in R$, one has $q(\alpha )=1$).

Indeed, by 4.2.8, $Ker(f)$ is finite over $S$ if and only if $Ker(f_T)$ is finite over $S$, and $Ker(f)\subset T$ if and only if each $q(\alpha )=1$ ($Ker(f)$ is then central since of multiplicative type and invariant, cf. Exp. IX 5.5).

Remark 4.2.11. (a) One thus sees that $f:G\to G^{\prime }$ is a central isogeny if and only if $R(f):R(G^{\prime })\to R(G)$ is an isogeny in the sense of Exp. XXI 6.2; moreover one has in this case (with the notations of loc. cit.):

Ker(f) = D_S(K(R(f))),    K(R(f)) = Coker(R(f)).

(b) If $G$ and $G^{\prime }$ are semisimple, every morphism of split groups $f:G\to G^{\prime }$ is an isogeny.²²

(c) If $f:G\to G^{\prime }$ is faithfully flat and finite and if $G$ is reductive (resp. semisimple), then $G^{\prime }$ is also. It is indeed of finite presentation over $S$ (Exp. V 9.1), affine over $S$ (EGA II 6.7.1), smooth over $S$ (Exp. VI 9.2), with connected reductive (resp. semisimple) geometric fibers by Exp. XIX 1.7.

Definition 4.2.1 may seem arbitrary. It is justified by the following proposition (which we state, for simplicity, for semisimple groups).

We say that a morphism $f:G\to G^{\prime }$ of $S$-reductive groups is splittable if there exist splittings of $G$ and $G^{\prime }$ with which $f$ is compatible. One then has:

Proposition 4.2.12. Let $S$ be a scheme, $G$ and $G^{\prime }$ two semisimple $S$-groups, $f:G\to G^{\prime }$ a morphism of groups. Let $s\in S$. The following conditions are equivalent:

(i) $Ker{f}_s$ is finite (⇔ $e$ is isolated in Ker f(s)) and $f_s$ is surjective, i.e. $f_s$ is an isogeny.

(ii) $f_s$ is splittable.

(iii) There exists an étale morphism $S^{\prime }\to S$ covering $s$ such that $f_{S^{\prime }}:G_{S^{\prime }}\to G_{S^{\prime }}^{\prime }$ is splittable.

One has obviously (iii) ⇔ (ii); (ii) ⇒ (i) follows from 4.2.11 (b) (this is where the hypothesis that $G$ and $G^{\prime }$ are semisimple intervenes — the other implications are valid for reductive groups).

Let us now prove (i) ⇒ (iii). One may suppose $G$ and $G^{\prime }$ split in such a way that $f$ induces a morphism $f_T:T\to T^{\prime }$ (2.3 and Exp. XIX 2.8); shrinking $S$, one may suppose that $f_T=D_S(h)$, where $h$ is a morphism of groups $M^{\prime }\to M$. Let $\alpha \in R$, and consider the composite morphism

p : W(gα) --expα--> G --f--> G′.

Since $Ker(p_s)$ is finite, $p_s\ne e$. On the other hand $f_{T_s}$ is surjective; one may therefore apply 4.1.9, and there exist an open subset $V_{\alpha }$ of $S$ containing $s$, a root ${\alpha }^{\prime }\in R^{\prime }$, an integer $q(\alpha )$ such that $x\mapsto x^{q(\alpha )}$ is an endomorphism of $Ga,V_{\alpha }$, and an isomorphism of $O_{V_{\alpha }}$-modules

$$f\alpha :(g\alpha {)}^{\otimes q(\alpha )}|V_{\alpha }\stackrel{\sim }{\to }{g}_{{\alpha }^{\prime }}^{\prime }|V_{\alpha }$$

such that $f(\exp \alpha (X\alpha ))={\exp }_{d(\alpha )}(f\alpha (X{\alpha }^{q(\alpha )}))$ and ${\alpha }^{\prime }\circ f_T=h({\alpha }^{\prime })=q(\alpha )\alpha$. One may replace $S$ by the intersection of the $V_{\alpha }$, for $\alpha \in R$. Set ${\alpha }^{\prime }=d(\alpha )$. It is clear that $d:R\to R^{\prime }$ is a bijection, because the kernel of $h$ is finite ($f_{T_s}$ being surjective). It only remains to prove that $f_T\circ \alpha \ast =q(\alpha ){\alpha }^{\prime }\ast$, which is done by a trivial modification of the argument used in Exp. XX 3.11.

In any case, as one has seen in the course of the demonstration, one has (i) ⇒ (iii). Therefore:

Corollary 4.2.13. Let $S$ be a scheme, $f:G\to G^{\prime }$ an isogeny of reductive groups. Then $f$ is locally splittable for the étale topology.

4.3. Central quotients of reductive groups

Let us first consider a particular case.

Proposition 4.3.1. Let $S$ be a scheme, $(G,T,M,R)$ an $S$-split group, $N$ a subgroup of $M$ containing $R$, $Q=D_S(M/N)\subset Centr(G)$. Then:

(i) $G^{\prime }=G/Q$ is an $S$-reductive group, and $T^{\prime }=T/Q$ is a maximal torus of it;

(ii) if one identifies $T^{\prime }$ with $D_S(N)$, then $R\subset N$ is a system of roots of $G^{\prime }$ relative to $T^{\prime }$, $(G^{\prime },T^{\prime },N,R)$ is a splitting of $G^{\prime }$, and $R(G^{\prime })$ is canonically identified with the induced root datum (Exp. XXI 6.5) $R(G{)}^N$;

(iii) the canonical morphism $G\to G^{\prime }$ is compatible with the splittings, with root exponents 1, and gives by functoriality the canonical morphism (loc. cit.) $R(G{)}^N\to R(G)$.

One knows that $G^{\prime }=G/Q$ is representable by an affine group scheme over $S$ (Exp. VIII 5.7), smooth over $S$ (Exp. VI_B 9.2), with connected reductive geometric fibers (as quotients of reductive groups, cf. Exp. XIX 1.7); $G^{\prime }$ is therefore an $S$-reductive group.

It is clear that $T^{\prime }=T/Q\simeq D_S(N)$ is a maximal torus of it. Note next that, choosing a system of positive roots $R+$ of $R$, the open set ${\Omega }_{R+}$ of 4.2 is stable under $Q$ and that one has a canonical isomorphism

Ω_{R+}/Q ≃ ∏_{α ∈ R−} Uα ×_S (T/Q) ×_S ∏_{α ∈ R+} Uα,

and that ${\Omega }_{R+}/Q$ is an open subset of $G^{\prime }$ containing the unit section (cf. Exp. IV, 4.7.2 and 6.4.1).

It follows that, if one denotes by $g^{\prime }$ the Lie algebra of $G^{\prime }$ and by ᾱ the character of $T/Q$ induced by $\alpha$ (or, which amounts to the same, defined by $\alpha \in N$ in the identification $T/Q=D_S(N)$), the canonical morphism $g\to g^{\prime }$ induces for each $\alpha \in R$ an isomorphism

$$g\alpha \stackrel{\sim }{\to }{g}_{\bar{\alpha }}^{\prime }.$$

One has thus proved that $R$ is a system of roots of $G^{\prime }$ relative to $T^{\prime }$, and one finishes the proof without difficulty.

Corollary 4.3.2. Let $S$ be a scheme, $G$ an $S$-reductive group, $Q$ a normal subgroup of multiplicative type of $G$. Then $Q$ is central in $G$, the quotient $G/Q$ is representable by an $S$-reductive group, and the canonical morphism $G\to G/Q$ is locally splittable for the étale topology (with root exponents equal to 1).

The first assertion follows from Exp. IX 5.5; the others are local for the étale topology and one is reduced to 4.3.1.

Definition 4.3.3. Let $G$ be an $S$-reductive group. We say that $G$ is adjoint (resp. simply connected*) if for every $s\in S$, the type of $G$ at $s$ is given by an adjoint (resp. simply connected) root datum, i.e. (Exp. XXI 6.2.6) such that $M$ be generated by $R$ (resp. $M\ast$ generated by $R\ast$).*

Proposition 4.3.4. (i) An adjoint (resp. simply connected) reductive group is semisimple.

(ii) If $T$ is a maximal torus of the adjoint (resp. simply connected) reductive group $G$ and $\alpha$ is a root of $G$ relative to $T$, then the infinitesimal root $\alpha$ is non-zero on every fiber (resp. $\alpha \ast$ is a monomorphism and the infinitesimal coroot $H\alpha$ is non-zero on every fiber).

Indeed, (i) is trivial; (ii) is checked on geometric fibers and follows immediately from Exp. XXI 6.2.8.

Proposition 4.3.5. (i) For the reductive group $G$ to be adjoint, it is necessary and sufficient that $Centr(G)=e_S$.

(ii) For every reductive group $G$, the quotient group $G/Centr(G)$ is an adjoint reductive group.

Indeed, one may suppose $G$ split; then (i) is trivial (since $Centr(G)=D_S(M/{\Gamma }_0(R))$), and (ii) follows from 4.3.1.

Definition 4.3.6. Let $G$ be an $S$-reductive group. We call adjoint group of $G$ and denote by $ad(G)$ the group $G/Centr(G)$. We call radical of $G$ and denote by $rad(G)$ the maximal torus (unique by Exp. XII 1.12) of $Centr(G)$. We call semisimple group associated with $G$ the quotient $G/rad(G)$.

The preceding definitions are compatible with base change. If $s\in S$, $rad(G{)}_s$ is indeed the radical of $G_s$ in the usual sense (Exp. XIX 1.6).

4.3.7.

If $(G,T,M,R)$ is a split group, then $rad(G)=D_S(M/N)$, where $N=M\cap V(R)$, so the semisimple group associated with $G$ (and similarly the adjoint group of $G$) is equipped with a canonical splitting (4.3.1) and one has a diagram of split groups

$$G\to ss(G)\to ad(G)$$

corresponding to the canonical diagram of root data (Exp. XXI 6.5.5)

$$ad(R(G))\to ss(R(G))\to R(G).$$

Remark 4.3.8. Let $(G,T,M,R)$ be an $S$-split adjoint (resp. simply connected) group, $\Delta$ a system of simple roots of $R$. Then the family ${\alpha }_{\alpha \in \Delta }$, resp. ${\alpha \ast }_{\alpha \in \Delta }$, induces an isomorphism

T ⥲ (Gm,S)^Δ,    resp.    (Gm,S)^Δ ⥲ T.

Indeed, $M={\Gamma }_0(R)$ (resp. …) and $\Delta$ is a basis of the free abelian group ${\Gamma }_0(R)$ (Exp. XXI 3.1.8).

Remark 4.3.9. The radical of a reductive group is a characteristic subgroup (i.e. stable under ${\mathrm{Aut}}_{S-gr.}(G)$), by its very definition.

5. Subgroups of type (R)

We are especially interested in reductive groups, but some of the results we shall establish are valid more generally for a wider class of groups: groups of type (RR).

5.1. Groups of type (RR)

Definition 5.1.1. Let $S$ be a scheme, $G$ an $S$-group scheme. We say that $G$ is of type (RR) if it satisfies the following conditions:

(i) $G$ is smooth and of finite presentation over $S$ and has connected fibers.

(ii) $G$ locally possesses maximal tori for the (fpqc) topology.

(iii) For every $s\in S$, every maximal torus $T$ of $G_s$, and every root of $G_s$ relative to $T_s$ (Exp. XIX 1.10), $Lie(G_s{)}^{\alpha }$ is of rank 1 (as a vector space over $\kappa (s)$).

(iv) For every $s\in S$ and every maximal torus $T$ of $G_s$, let $R$ denote the set of roots of $G_s$ relative to $T$ and ${\Gamma }_0(R)$ the subgroup of ${\mathrm{Hom}}_{s-gr.}(T,Gm,s)$ generated by $R$; then the content²³ of every root $\alpha \in R$ in the free abelian group ${\Gamma }_0(R)$ (which is therefore a positive integer) is invertible on $S$.

Recall 5.1.1.1.²⁴ Let us recall that if $G$ is a smooth connected algebraic group over an algebraically closed field $k$, a Cartan subgroup of $G$ is the centralizer of a maximal torus of $G$ (XII 1.0), and such a subgroup is smooth and connected: for this, as well as for other characterizations of Cartan subgroups, see Bible, § 7.1, Th. 1 in the affine case and Exp. XII Th. 6.6 in the general case. If $S$ is an arbitrary scheme and $G$ an $S$-smooth group of finite type, a Cartan subgroup of $G$ is an $S$-smooth subgroup $C$ of $G$ such that, for every $s\in S$, $C_s$ is a Cartan subgroup of $G_s$ (XII Def. 3.1).

Remark 5.1.2. (a) By virtue of Exp. XII 7.1 (where the hypothesis that $G$ is separated holds here since $G$ has connected fibers, cf. Exp. VI_B 5.5), (i) and (ii) entail that $G$ locally possesses, for the étale topology, maximal tori (resp. Cartan subgroups), conjugate locally for the étale topology.

(b) The Cartan subgroups of $G$ have connected fibers (Exp. XII 6.6).

(c) If $G$ is affine over $S$, (i) and (ii) are respectively equivalent to

(i′) $G$ is smooth over $S$ with connected fibers.

(ii′) The reductive rank of the fibers of $G$ is locally constant (Exp. XII 1.7).

(d) By Exp. XII 8.8 (c) and (d), $G$ has a reductive center $Z$ and, for every $s\in S$, with the notations of (iv), one has²⁵ $Z_s={\bigcap }_{\alpha \in R}Ker(\alpha )$, whence

$${\mathrm{Hom}}_{s-gr.}((T/Z{)}_s,Gm,s)\simeq {\Gamma }_0(R).$$

(e) Condition (iv) holds in particular in the following two cases:

(1) $S$ is of characteristic 0;

(2) every root is an indivisible element of the group generated by the roots.

(f) Being of type (RR) is stable under base change and is local for the (fpqc) topology.

From remarks (c) and (e), one immediately deduces:

Proposition 5.1.3. A reductive group is of type (RR).

Proposition 5.1.4. Let $S$ be a scheme, $G$ an $S$-group of type (RR), $Q$ a central subgroup of $G$ of finite presentation over $S$ such that the quotient $G/Q$ is representable (for instance $G$ affine over $S$ and $Q$ of multiplicative type (IX 2.3) or else $S$ artinian (VI_A 3.3.2)); then $G/Q$ is an $S$-group of type (RR).

Indeed, $G/Q$ is smooth over $S$ (Exp. VI_B 9.2), of finite presentation over $S$ (Exp. V 9.1) and with connected fibers, so condition (i) holds. On the other hand, condition (ii) follows from Exp. XII 7.6; it remains to verify conditions (iii) and (iv).

Let $G^{\prime }=G/Q$, $u:G\to G^{\prime }$ the canonical morphism, $T^{\prime }=u(T)$ the maximal torus of $G^{\prime }$ image of $T$ (cf. Exp. XII 7.1 (e)); for each $\alpha \in R$, denote again by $\alpha$ the character of $T^{\prime }$ defined by $\alpha$ (one has $Q\cap T\subset {\bigcap }_{\alpha \in R}Ker(\alpha )$ according to 5.1.2 (d)). Let us first prove:

Lemma 5.1.5. Under the conditions of 5.1.4, let $T=D_S(M)$ be a trivialized maximal torus of $G$, and suppose that the decomposition of $g=Lie(G)$ under $Ad(T)$ is of the form

g = g⁰ ⨿ ⨿_{α ∈ R} gα,    R ⊂ M − {0},

where for every $s\in S$, $g\alpha (s)\ne 0$ for $\alpha \in R$ (so $g\alpha$ is an invertible O_S-module for every $\alpha \in R$ and $R$ is the set of roots of $G_s$ relative to $T_s$ for every $s\in S$).

Then the Lie algebra $g^{\prime }$ of $G^{\prime }$ decomposes under $Ad(T^{\prime })$ as follows:

g′ = g′⁰ ⨿ ⨿_{α ∈ R} g′^α,

and $Lie(u)$ induces an isomorphism of $g\alpha$ onto $g^{\prime \alpha }$.

Indeed, let $p=Lie(u):g\to g^{\prime }$. One immediately has $p(g\alpha )\subset g^{\prime \alpha }$ for every $\alpha \in R$, and $p(g^0)\subset g^{\prime 0}$. Since

Ker(p) = Lie(Q) ⊂ Lie(Centr_G(T)) = g⁰,

$p$ induces a monomorphism from $g\alpha$ into $g^{\prime \alpha }$, for every $\alpha \in R$.

To prove the lemma, it suffices to do so when $S$ is the spectrum of an algebraically closed field, and by virtue of the preceding remarks, it then suffices to prove that $rg(g^{\prime })=rg(g^{\prime 0})+Card(R)$. Now set $C=Cent{r}_G(T)$, $C^{\prime }=Cent{r}_{G^{\prime }}(T^{\prime })$; by Exp. XII 7.1 (e), $u$ induces a faithfully flat morphism $C\to C^{\prime }$ of kernel $Q$. One thus has

dim C′ + dim Q = dim C.

But $G$, $G^{\prime }$, $C$ and $C^{\prime }$ are smooth, so

dim G = rg(g) = rg(g⁰) + Card(R) = dim C + Card(R)
              = dim Q + dim C′ + Card(R) = dim Q + rg(g′⁰) + Card(R)
rg(g′) = dim G′ = dim G − dim Q

which entails

$$rg(g^{\prime })=rg(g^{\prime 0})+Card(R),$$

that is, the desired relation.

Let us return to the proof of 5.1.4; one may suppose that $S$ is the spectrum of an algebraically closed field. Let $T$ be a maximal torus of $G$; applying 5.1.5, one immediately has (iii) and (iv) for $G/Q$.

To use the preceding proposition, let us introduce a definition:

Definition 5.1.6. We say that the $S$-group scheme $G$ is of type (RA) if it is of type (RR) and if it further satisfies the following condition (iv′) (stronger than (iv)):

(iv′) For every $s\in S$ and every maximal torus $T$ of $G_s$, every root of $G_s$ relative to $T$ has a content in ${\mathrm{Hom}}_{s-gr.}(T,Gm,s)$ that is invertible on $S$.

Remarks 5.1.7. (a) An $S$-reductive adjoint group is of type (RA).

(b) If $S$ is of characteristic 0, every group of type (RR) is of type (RA).

(c) Being of type (RA) is stable under base change and is local for the (fpqc) topology.

Remark (a) above generalizes to:

Proposition 5.1.8. Let $S$ be a scheme, $G$ an $S$-group of type (RR), $Z$ its reductive center, suppose the quotient $G/Z$ representable (for instance $G$ affine over $S$, or $S$ artinian); then $G/Z$ is of type (RA).

Indeed, this follows immediately from 5.1.4, 5.1.5, and 5.1.2 (d).

Remark 5.1.9. If $G$ is of type (RR) and if $T$ is a maximal torus of $G$, one may apply Exp. XIX 6.3. In particular $W_G(T)$ is étale, quasi-finite and separated over $S$.

5.2. Subgroups of type (R)

Definition 5.2.1. Let $S$ be a scheme, $G$ a smooth $S$-group scheme of finite presentation with connected fibers,²⁶ $H$ a group subscheme of $G$. We say that $H$ is of type (R) if:

(i) $H$ is smooth, of finite presentation over $S$ and with connected fibers.²⁶

(ii) For every $s\in S$, $H_s$ contains a Cartan subgroup of $G_s$.

This notion is stable under base change and local for the (fpqc) topology.

Recall 5.2.2. (cf. Exp. XII 7.9) Under the preceding conditions:

(a) $H=Nor{m}_G(H{)}^0$.

(b) If $G$ locally possesses, for the étale topology, Cartan subgroups (resp. maximal tori), so does $H$, and the Cartan subgroups (resp. maximal tori) of $H$ are Cartan subgroups (resp. maximal tori) of $G$.

Examples 5.2.3. (a) Borel subgroups: a Borel subgroup of $G$ is a subgroup $H$ of type (R) whose geometric fibers are Borel subgroups of those of $G$.²⁷

(b) Parabolic subgroups: a parabolic subgroup of $G$ is a subgroup of type (R) whose geometric fibers contain Borel subgroups.

Other examples are given by the following propositions.

Proposition 5.2.4. Let $G$ be as in 5.2.1, $K\subset H$ two group subschemes of $G$, $H$ assumed to be of type (R). Then $K$ is a subgroup of type (R) of $H$ if and only if it is a subgroup of type (R) of $G$.

²⁸ Indeed, let $s\in S$. Since $H$ is of type (R), every maximal torus of $H_s$ is a maximal torus of $G_s$, and so likewise for Cartan subgroups.

Proposition 5.2.5. Let $G$ be as in 5.2.1, $T$ a maximal torus of $G$, $Q$ a subtorus of $T$, $Z=Cent{r}_G(Q)$. If $H$ is a subgroup of type (R) of $G$ containing $T$, then $H\cap Z$ is a subgroup of type (R) of $Z$.

Let us first recall that $Z$ is a closed group subscheme of $G$, of finite presentation (Exp. XI 6.11), with connected fibers (Exp. XII 6.6), smooth over $S$ (Exp. XI 2.4), so it satisfies the conditions imposed in Definition 5.2.1. Similarly, $H\cap Z$ is of finite presentation, smooth and has connected fibers (since $H\cap Z=Cent{r}_H(Q)$); moreover $H\cap Z\supset Cent{r}_G(T)$.

Proposition 5.2.6. Let $S$ be a scheme, $G$ an $S$-group of type (RR) (resp. (RA)), $H$ a subgroup of type (R) of $G$. Then $H$ is an $S$-group of type (RR) (resp. (RA)).