Exposé XXVI. Parabolic subgroups of reductive groups

by M. Demazure

¹

This Exposé studies the parabolic subgroups of an $S$-reductive group $G$. Its essential result is the conjugacy theorem (5.4). The essential tool is the notion of transversal position of two parabolic subgroups, a notion studied systematically in §4. Another fact plays an important role: the decomposition of the unipotent radical $ra{d}^u(P)$ of a parabolic subgroup $P$ as successive extensions of vector groups (2.1)².

Various schemes associated with $G$ are studied in §3; §6 deals with the split subtori³ of $G$ and their relations with parabolic subgroups.

Finally, in §7, we briefly expound how, over a semi-local base, one formulates the "relative theory" of reductive groups as presented over fields in the article of A. Borel and J. Tits, Groupes réductifs, Publications Mathématiques de l'IHÉS, no. 27. In that article, cited [BT65] in what follows, the reader will moreover find, in the case of a base field, other results that have not been touched on here.

1. Recollections. Levi subgroups

Definition 1.1. Let $S$ be a scheme, $G$ an $S$-reductive group, $P$ a sub-$S$-group-scheme of $G$. One says that $P$ is a parabolic subgroup of $G$ if

(i) $P$ is smooth over $S$,

(ii) for each $s\in S$, the $s$-quotient-scheme $G_s/P_s$ is proper (i.e. Bible, §6.4, Th. 4 (= [Ch05], §6.5, Th. 5), $P_s$ contains a Borel subgroup of $G_s$).

Proposition 1.2 (Exp. XXII, 5.8.5). Let $S$ be a scheme, $G$ an $S$-reductive group, $P$ a parabolic subgroup of $G$. Then $P$ is closed in $G$, with connected fibers, and

$$P=Nor{m}_G(P).$$

Moreover, the quotient sheaf $G/P$ is representable by an $S$-scheme that is smooth and projective over $S$.

Proposition 1.3 (Exp. XXII, 5.3.9 and 5.3.11). Let $S$ be a scheme, $G$ an $S$-reductive group, $P$ and $P^{\prime }$ two parabolic subgroups of $G$. The following conditions are equivalent:

(i) $P$ and $P^{\prime }$ are conjugate in $G$, locally for the étale (resp. (fpqc)) topology.

(ii) For each $s\in S$, $P_s$ and $P_s^{\prime }$ are conjugate by an element of $G(s)$.

(iii) The strict transporter $Trans{t}_G(P,P^{\prime })$ of $P$ into $P^{\prime }$ (defined by

Transt_G(P, P')(S') = { g ∈ G(S') | int(g) P_{S'} = P'_{S'} }

for every $S^{\prime }\to S$) is a closed subscheme of $G$, smooth and of finite presentation over $S$, which is a principal homogeneous bundle on the right under $P$ and on the left under $P^{\prime }$.

Proposition 1.4. Let $S$ be a nonempty scheme, $(G,T,M,R)$ an $S$-split reductive group, $R^{\prime }$ a subset of $R$. The following conditions on $R^{\prime }$ are equivalent:

(i) $g_{R^{\prime }}=t\oplus {⨁}_{\alpha \in R^{\prime }}{g}^{\alpha }$ is the Lie algebra of a parabolic subgroup of $G$ containing $T$ (necessarily unique, Exp. XXII, 5.3.5).

(ii) $R^{\prime }$ is of type (R) (Exp. XXII, 5.4.2) and contains a system of positive roots.

(iii) $R^{\prime }$ is a closed subset of $R$ and satisfies: if $\alpha \in R-R^{\prime }$, then $-\alpha \in R^{\prime }$ (that is, $R=R^{\prime }\cup (-R^{\prime })$).

(iv) There exist a system of simple roots $\Delta$, and a subset $A$ of $\Delta$ such that $R^{\prime }$ is the union of the set of positive roots and the set of negative roots that are linear combinations of the elements of $A$.

(v) $R^{\prime }$ contains a system of simple roots of $R$; moreover, if $\Delta \subset R^{\prime }$ is a system of simple roots of $R$ and one sets

$$A=(-R^{\prime })\cap \Delta ,$$

then $R^{\prime }$ is the union of the set of positive roots and the set of negative roots that are linear combinations of the elements of $A$.

One has (i) ⇔ (ii) by Exp. XXII, 5.4.5 (ii) and 5.5.1. One has (iii) ⇒ (ii) by Exp. XXI, 3.3.6 and Exp. XXII, 5.4.7. One trivially has (v) ⇒ (iv) ⇒ (iii). One has (iii) ⇒ (v) by Exp. XXI, 3.3.6 and 3.3.10.

It therefore remains to prove that (i) implies that $R^{\prime }$ is a closed subset of $R$. But this last assertion can be verified on any geometric fiber; one may therefore assume that $S$ is the spectrum of an algebraically closed field.

Let $P$ be the parabolic subgroup of $G$ containing $T$ whose Lie algebra is $g_{R^{\prime }}$. Since the Borel subgroups of $P$ are the Borel subgroups of $G$ contained in $P$, it follows from Bible, §12.3, Th. 1, and from Exp. XXII, 5.4.5 (i), that if one denotes by $U$ the unipotent radical of $P$, then $T\cdot U$ is the subgroup of $G$ containing $T$ whose Lie algebra is

g_{R''} = t ⊕ ⨁_{α ∈ R''} g^α,

where R'' is the intersection of the systems of positive roots of $R$ contained in $R^{\prime }$. In particular, it follows that R'' is closed and that $R^{\prime \prime }\cap (-R^{\prime \prime })=\varnothing$. On the other hand, the group $H=P/U$ is reductive, the canonical image $\bar{T}$ of $T$ is a maximal torus of it ($T\to \bar{T}$ is an isomorphism), and one has an isomorphism of $\bar{T}$-modules, i.e. of $M$-graded vector spaces,

Lie(H) ≃ t ⊕ ⨁_{α ∈ R_s} g^α,

where $R_s$ is the complement of R'' in $R^{\prime }$. It follows that $R_s$ is naturally identified with the set of roots of $H$ relative to $\bar{T}$, and in particular satisfies $R_s=-R_s$. One immediately deduces that

R'' = { α ∈ R' | −α ∉ R' },        R_s = { α ∈ R' | −α ∈ R' }.

Let us now show that $R^{\prime }$ is closed. Let $\alpha ,\beta \in R^{\prime }$ such that $\alpha +\beta \in R$; let us prove that $\alpha +\beta \in R^{\prime }$. If $\alpha ,\beta \in R^{\prime \prime }$, then $\alpha +\beta \in R^{\prime \prime }$ because R'' is closed. If $\alpha \in R_s$, $\beta \in R^{\prime \prime }$, and if $\alpha +\beta \notin R^{\prime }$, then $\alpha +\beta \in -R^{\prime \prime }$, and one has $-\alpha =-(\alpha +\beta )+\beta \in R^{\prime \prime }$ because R'' is closed, which entails $-\alpha \in R^{\prime \prime }\cap R_s$ and contradicts the fact that $R_s\cap R^{\prime \prime }=\varnothing$. It therefore remains to study the case where $\alpha ,\beta \in R_s$. If $\alpha +\beta \notin R^{\prime }$, then $\alpha +\beta \in -R^{\prime \prime }$. But, as $\alpha +\beta \ne 0$, there exists a system of positive roots of the root system $R_s$ containing $\alpha$ and $\beta$, hence a Borel subgroup of $H=P/U$ containing the canonical image of $U_{\alpha }$ and $U_{\beta }$. Its inverse image in $P$ is a Borel subgroup containing $U_{\alpha }$, $U_{\beta }$ and $U$, hence $U_{\alpha }$, $U_{\beta }$ and $U_{-(\alpha +\beta )}$, which is impossible.

Corollary 1.5. A parabolic subgroup of a reductive group is of type (RC) (Exp. XXII, 5.11.1).

Proposition 1.6 (Exp. XXII, 5.11.4). Let $S$ be a scheme, $G$ an $S$-reductive group, $P$ a subgroup of $G$ of type (RC)⁴.

(i) $P$ possesses a largest invariant sub-group-scheme that is smooth and of finite presentation over $S$, with connected unipotent geometric fibers. It is a characteristic subgroup of $P$, called the unipotent radical of $P$, denoted $ra{d}^u(P)$. The quotient sheaf $P/ra{d}^u(P)$ is representable by an $S$-reductive group.

(ii) If $T$ is a maximal torus of $P$, $P$ possesses a reductive subgroup $L$ containing $T$ such that:

(a) Every reductive subgroup of $P$ containing $T$ is contained in $L$.

(b) $P$ is the semidirect product $L\cdot ra{d}^u(P)$, i.e. the canonical morphism $L\to P/ra{d}^u(P)$ is an isomorphism.

Moreover, $L$ is the unique subgroup (resp. reductive subgroup) of $P$ containing $T$ and satisfying (b) (resp. (a)). Finally, one has

Norm_P(L) = L,        Norm_P(T) = Norm_L(T).

1.7.

A subgroup $L$ of $P$ satisfying condition (b) above is called a Levi subgroup of $P$. It is a maximal reductive subgroup of $P$; indeed, it is reductive, since isomorphic to $P/ra{d}^u(P)$. Let us show that it is maximal for this property. Let $L^{\prime }$ be a reductive subgroup of $P$ containing $L$; to prove that $L^{\prime }=L$, one may reason locally for the (fpqc) topology, and hence assume that $L$ has a maximal torus $T$, and one is reduced to 1.6 (ii).

If $L$ and $L^{\prime }$ are two Levi subgroups of $P$, then $L$ and $L^{\prime }$ are conjugate in $P$, locally for the (fpqc) topology. Indeed, locally for this topology, one may assume that $L$ (resp. $L^{\prime }$) has a maximal torus $T$ (resp. $T^{\prime }$); since $T$ and $T^{\prime }$ are conjugate in $P$ locally for the (fpqc) topology, one may assume $T=T^{\prime }$, and one then has $L=L^{\prime }$, by 1.6 (ii). But since $P=L\cdot ra{d}^u(P)$ and $Nor{m}_P(L)=L$, one immediately deduces:

Corollary 1.8. Let $P$ be a subgroup of type (RC)⁵ of the $S$-reductive group $G$. If $L$ and $L^{\prime }$ are two Levi subgroups of $P$, there exists a unique $u\in ra{d}^u(P)(S)$ such that $int(u)L=L^{\prime }$.

Let us denote by $Lev(P)$ the functor of Levi subgroups of $P$: for $S^{\prime }\to S$, $Lev(P)(S^{\prime })$ is the set of Levi subgroups of $P_{S^{\prime }}$. From 1.8 one deduces:

Corollary 1.9. Let $P$ be a subgroup of type (RC)⁵ of the $S$-reductive group $G$. Then $Lev(P)$ is a principal homogeneous bundle under the $S$-group $ra{d}^u(P)$, and in particular is representable by an $S$-scheme that is smooth and affine over $S$, with integral geometric fibers.

It follows immediately from 1.6:

Corollary 1.10. Let $P$ be a subgroup of type (RC)⁵ of the $S$-reductive group $G$. The functor $Tor(P)$ of maximal tori of $P$ is representable by an $S$-scheme that is smooth and affine, the "relation $L\supset T$" defines a morphism

$$Tor(P)⟶Lev(P),$$

the fiber of which over $L\in Lev(P)(S)$ is identified with $Tor(L)$ (Exp. XXII, 5.8.3).

The first assertion of 1.10 is a consequence of the other two and of Exp. XXII, 5.8.3.

Definition 1.11. Let $S$ be a nonempty scheme, $G$ an $S$-reductive group, $P$ a parabolic subgroup of $G$, $E=(T,M,R,\Delta ,(X_{\alpha }{)}_{\alpha \in \Delta })$ a pinning of $G$. One says that $E$ is adapted to $P$, or that $E$ is a pinning of the pair $(G,P)$, if $P\supset T$ and if the Lie algebra of $P$ is of the form $t\oplus {⨁}_{\alpha \in R^{\prime }}{g}^{\alpha }$, where $R^{\prime }$ is a subset of $R$ containing $R^{+}$.

In particular, if $T\subset B$ is the Killing pair defined by the pinning, one has $T\subset B\subset P$.

Under the preceding conditions, one sets $\Delta (P)=\Delta \cap (-R^{\prime })$; then, by Exp. XXII, 5.4.3, one has:

α ∈ Δ(P)  ⇔  α ∈ Δ and U_{−α} ⊂ P  ⇔  α ∈ Δ and U_{−α} ∩ P ≠ e.

It follows immediately from 1.4 (v) and Exp. XXII, 5.11.3 and 5.10.6:

Proposition 1.12. Let $S$ be a scheme, $G$ an $S$-reductive group, $P$ a parabolic subgroup of $G$, $(T,M,R,\Delta ,(X_{\alpha }{)}_{\alpha \in \Delta })$ a pinning of $G$ adapted to $P$, $\Delta (P)$ the subset of $\Delta$ defined above.

(i) The unipotent radical $ra{d}^u(P)$ of $P$ is none other than

U_{R''} = ∏_{α ∈ R''} U_α,

where R'' is the set of positive roots that, in their decomposition along $\Delta$, involve at least one element of $\Delta -\Delta (P)$⁶ with a nonzero coefficient.

(ii) The unique Levi subgroup $L$ of $P$ containing $T$ is none other than

$$Z_{\Delta (P)}=Cent{r}_G(T_{\Delta (P)}),$$

where $T_{\Delta (P)}$ is the maximal torus of ${\bigcap }_{\alpha \in \Delta (P)}Ker(\alpha )$; moreover, one has $T_{\Delta (P)}=rad(L)$.

Corollary 1.13. Every Levi subgroup $L$ of the parabolic subgroup $P$ of the reductive group $G$ is a critical subgroup of $G$, i.e. satisfies (Exp. XXII, 5.10.4):

$$L=Cent{r}_G(rad(L)).$$

This follows immediately from 1.12 and from the following lemma, which is contained in 1.4 and Exp. XXII, 5.4.1:

Lemma 1.14. Locally for the étale topology, every pair $(G,P)$, where $P$ is a parabolic subgroup of the reductive group $G$, may be pinned (1.11).

Let us note:

Proposition 1.15. Let $S$ be a scheme, $G$ an $S$-reductive group, $P$ a parabolic subgroup of $G$, $E$ and $E^{\prime }$ two pinnings of $G$ adapted to $P$. The unique inner automorphism of $G$ over $S$ that transforms $E$ into $E^{\prime }$ (Exp. XXIV, 1.5) comes from $P$, via the morphism

P ⟶ P/Centr(P) = P/Centr(G) ⟶ G/Centr(G).

Indeed, it suffices to reason as in Exp. XXIV, 1.5, using:

Lemma 1.16 (Exp. XXII, 5.3.14 and 5.2.6). The maximal tori (resp. Borel subgroups, resp. Killing pairs) of a parabolic subgroup $P$ of the $S$-reductive group $G$ are conjugate in $P$, locally for the étale topology.

Proposition 1.17. Let $S$ be a scheme, $G$ an $S$-reductive group, $P$ and $P^{\prime }$ two parabolic subgroups of $G$, $B$ a Borel subgroup contained in $P$ and $P^{\prime }$. If $P$ and $P^{\prime }$ are conjugate in $G$ locally for the étale topology, then $P=P^{\prime }$.

Indeed, one may assume that there exists $g\in G(S)$ such that $int(g)P=P^{\prime }$. Then $B$ and $int(g{)}^{-1}B$ are two Borel subgroups of $P$. Possibly after extending $S$, by 1.16 one may assume that there exists $p\in P(S)$ such that $int(p)int(g^{-1})B=B$. Then

p g^{-1} ∈ Norm_G(B)(S) = B(S),

and $g\in B(S)\cdot p\subset P(S)$, hence $P^{\prime }=int(g)P=P$.

Remark 1.18. If $P$ and $P^{\prime }$ are two parabolic subgroups of $G$ containing a common Borel subgroup, then $P\cap P^{\prime }$ is again a parabolic subgroup of $G$. Indeed, it is smooth along the unit section (Exp. XXII, 5.4.5), and it contains a Borel subgroup.

Proposition 1.19. Let $S$ be a scheme, $G$ an $S$-reductive group, $G^{\prime }$ its derived group (Exp. XXII, 6.2.1).

(i) The maps

P ↦ P' = P ∩ G'    and    P ↦ P' · rad(G) = Norm_G(P')

are mutually inverse bijections between the set of parabolic subgroups of $G$ and the set of parabolic subgroups of $G^{\prime }$. One has $ra{d}^u(P)=ra{d}^u(P^{\prime })$.

(ii) Let $P$ be a parabolic subgroup of $G$ and $P^{\prime }=P\cap G^{\prime }$. The maps

L ↦ L' = L ∩ G' = L ∩ P'
L' ↦ L' · rad(G) = Centr_G(rad(L'))

are mutually inverse bijections between the set of Levi subgroups of $P$ and the set of Levi subgroups of $P^{\prime }$. Moreover, one has $rad(L^{\prime })=(rad(L)\cap G^{\prime }{)}^0$.

The proof (by reduction to the split case, for example) is straightforward and is left to the reader, together with that, immediate, of:

Proposition 1.20. Let $S$ be a scheme, $G$ an $S$-reductive group, $P$ a parabolic subgroup of $G$, $L$ a Levi subgroup of $P$. The maps

Q ↦ Q ∩ L = Q',     Q' ↦ Q' · rad^u(P)

are mutually inverse bijections between the set of parabolic subgroups of $G$ contained in $P$ and the set of parabolic subgroups of $L$. Moreover, the Levi subgroups of $Q^{\prime }$ are the Levi subgroups of $Q$ contained in $L$.

One may complete 1.6 as follows:

Proposition 1.21. Let $S$ be a scheme, $G$ an $S$-reductive group, $P$ a parabolic subgroup of $G$.

(i) $P$ possesses a largest invariant subgroup that is smooth and of finite presentation over $S$, with connected solvable geometric fibers. It is a characteristic subgroup of $P$, called the radical of $P$, and denoted $rad(P)$. The quotient sheaf $P/rad(P)$ is representable by an $S$-semisimple group.

(ii) If $L$ is a Levi subgroup of $P$, $rad(P)$ is the semidirect product of $ra{d}^u(P)$ and $rad(L)$; one has $rad(L)=L\cap rad(P)$, hence $L=Cent{r}_G(L\cap rad(P))$, and $P/rad(P)\simeq L/rad(L)$.

Indeed, assertion (i) being local, one may assume that $G$ has a Levi subgroup $L$, and one is reduced to proving that $R=ra{d}^u(P)\cdot rad(L)$ has the properties announced in (i), which is immediate. For (ii), it only remains to show that $rad(L)=L\cap rad(P)$, which follows immediately from the fact that $L\cap rad(P)$ is smooth and has connected fibers, $L$ being the centralizer of a torus⁷.

2. Structure of the unipotent radical of a parabolic subgroup

Proposition 2.1. Let $S$ be a scheme, $G$ an $S$-reductive group, $P$ a parabolic subgroup of $G$, $ra{d}^u(P)$ its unipotent radical. There exists a sequence of sub-group-schemes of $ra{d}^u(P)$

rad^u(P) = U_0 ⊃ U_1 ⊃ U_2 ⊃ · · · ⊃ U_n ⊃ · · ·

possessing the following properties:

(i) Each $U_i$ is smooth, with connected fibers, characteristic and closed in $P$. The commutator of a section of $U_i$ and a section of $U_j$ is a section of $U_{i+j+1}$ (over a varying $S^{\prime }\to S$).

(ii) For each $i⩾0$, there exists a locally free O_S-module $E_i$ and an isomorphism of $S$-group sheaves

$$U_i/U_{i+1}\stackrel{\sim }{\to }W(E_i).$$

Moreover, the automorphisms of $P$ (over a varying $S^{\prime }\to S$) act linearly on $W(E_i)$.

(iii) For every $s\in S$, one has $U_{n,s}=e$ for $n>\dim (ra{d}^u(P{)}_s)$.

2.1.1.

Suppose first that the pair $(G,P)$ is pinnable. Let $(T,M,R,\Delta ,\cdots )$ be a pinning of $G$ adapted to $P$; let $\Delta (P)$ be the part of $\Delta$ defined by $P$. Let ${\alpha }_1,\cdots ,{\alpha }_p$ be

the elements of $\Delta (P)$, and ${\beta }_1,\cdots ,{\beta }_q$ the elements of $\Delta -\Delta (P)$. Every root $\gamma \in R$ is written uniquely as

γ = a_1 α_1 + · · · + a_p α_p + b_1 β_1 + · · · + b_q β_q.

Set

a(γ) = b_1 + · · · + b_q. [^N.D.E-XXVI-7]

It follows immediately from the definitions that the following properties hold (cf. 1.12):

(i) $U_{\gamma }\subset P\iff a(\gamma )⩾0$.

(ii) $U_{\gamma }\subset ra{d}^u(P)\iff a(\gamma )>0$.

(iii) $a(n\gamma +m{\gamma }^{\prime })=na(\gamma )+ma({\gamma }^{\prime })$ for all $n,m\in \mathbb{Z}$.

For $i>0$, let $R_i$ be the set of roots $\gamma \in R$ such that $a(\gamma )>i$. Each $R_i$ is a closed set of roots satisfying $R_i\cap (-R_i)=\varnothing$. Consider (Exp. XXII, 5.6.5) the $S$-group

U_i = U_{R_i} = ∏_{γ ∈ R_i} U_γ.

It is a closed sub-group-scheme of $G$, smooth over $S$, with connected fibers.

Let $\alpha ,\beta \in R$; consider the commutation relation of Exp. XXII, 5.5.2

p_α(x) p_β(y) p_α(−x) = p_β(y) ∏_{n, m ∈ ℕ^*} p_{n α + m β}(C_{n, m, α, β} x^n y^m),

where each $p_{\gamma }$ is an isomorphism of vector groups $G_{a,S}\stackrel{\sim }{\to }{U}_{\gamma }$. Let us first remark that if $a(\alpha )>i$ and $a(\beta )>j$, one has

a(n α + m β) = n a(α) + m a(β) ⩾ n(i + 1) + m(j + 1) > i + j + 1

when $n$ and $m$ are > 0. It follows that the commutator of a section of $U_{\alpha }$ and a section of $U_{\beta }$ is a section of $U_{i+j+1}$ (over a varying $S^{\prime }\to S$), which indeed entails $(U_i,U_j)\subset U_{i+j+1}$. For each $i⩾0$, the quotient $U_i/U_{i+1}$ is therefore commutative; it is naturally identified with

U_i / U_{i+1} ≃ ∏_{a(γ) = i+1} U_γ ≃ W(E_i),

where $E_i$ is the direct sum of the $g^{\gamma }$ for $a(\gamma )=i+1$.

Let us return to the above commutation formula, and suppose $a(\alpha )⩾0$, $a(\beta )>i$. If $n,m\in {\mathbb{N}}^{\ast }$,

– either $a(n\alpha +m\beta )>i+1$,

– or $a(n\alpha +m\beta )=i+1$, in which case one necessarily has $m=1$.

This proves first that $int(p_{\alpha }(x))$ respects $U_i$ (hence also $U_{i+1}$), and then that, in the expression of $int(p_{\alpha }(x))p_{\beta }(y)$, only terms of the form $p_{n\alpha +\beta }(C_{n,1,\alpha ,\beta }{x}^{n}y)$ occur modulo $U_{i+1}$, which are therefore linear in $y$. It follows that the inner automorphisms defined by sections of $U_{\alpha }$ act linearly on the quotient $U_i/U_{i+1}$ identified with $W(E_i)$. As this is also trivially true for the inner automorphisms defined by sections of $T$, and as $P$ is generated by $T$ and the $U_{\alpha }$, $a(\alpha )⩾0$, one deduces that:

(i) each $U_i$ is invariant in $P$,

(ii) the inner automorphisms defined by sections of $P$ act linearly on $U_i/U_{i+1}\simeq W(E_i)$.

2.1.2.

Now let $(T^{\prime },M^{\prime },R^{\prime },{\Delta }^{\prime },\cdots )$ be a new pinning of $G$ adapted to $P$.

By 1.15, there exists an inner automorphism of $G$ coming from $P$ that transforms the old pinning into the new one. Possibly after extending $S$, one may assume that this inner automorphism is of the form $int(p)$, $p\in P(S)$. If one takes up the preceding constructions using the new pinning, it is clear that the groups $U_i^{\prime }$ and the isomorphisms $U_i^{\prime }/U_{i+1}^{\prime }\simeq W(E_i^{\prime })$ obtained are deduced from $U_i$ and $U_i/U_{i+1}\simeq W(E_i)$ by transport of structure via $int(p)$. It follows from remarks (i) and (ii) above that one therefore has $U_i^{\prime }=U_i$, and that the two vector structures constructed on $U_i/U_{i+1}=U_i^{\prime }/U_{i+1}^{\prime }$ coincide.

This shows that the groups $U_i$ and the vector structures on the quotients $U_i/U_{i+1}$ are independent of the pinning considered (and in particular invariant under every automorphism of $P$, as one sees easily).

We have therefore proved the proposition when the pair $(G,P)$ is pinnable (part (iii) is trivial, since by Exp. XXI 3.1.2, the set $a(\gamma )|\gamma \in R$ is an interval of $\mathbb{Z}$, hence one cannot have $\dim (U_{i,s})=\dim (U_{i+1,s})$ unless $U_{i,s}=e$).

2.1.3.

In the general case, there exists a covering family for the (fpqc) topology $S_j\to S$ such that each pair $(G_{S_j},P_{S_j})$ is pinnable (1.14). By the preceding, one has descent data on the $ra{d}^u(P_{S_j}{)}_i$, compatible with the vector structures of the quotients, and one concludes by descent of closed subschemes (resp. locally free modules)⁸.

Corollary 2.2. Let $S$ be an affine scheme, $G$ an $S$-reductive group, $P$ a parabolic subgroup of $G$. One has

$$H^1(S,ra{d}^u(P))=0,$$

i.e. every principal homogeneous bundle under $ra{d}^u(P)$ is trivial.

Indeed, $S$ decomposes as a sum of subschemes on each of which $ra{d}^u(P)$ is of constant relative dimension. One may

therefore, by (iii), assume that there exists $n$ such that $U_n=e$. Since $H^1(S,U_i/U_{i+1})=H^1(S,W(E_i))=0$ by TDTE I, B, 1.1 (or SGA 1, XI 5.1), one concludes at once.

Corollary 2.3. Under the preceding conditions, $P$ possesses a Levi subgroup $L$. If $L$ is a Levi subgroup of $P$, the canonical map

H^1(S, L) ⟶ H^1(S, P)

is bijective (cf. the introduction of Exp. XXIV for the definition of $H^1(S,-)$).

The first assertion follows from 2.2 and 1.9. The canonical map $H^1(S,L)\to H^1(S,P)$ is surjective, because $P$ is the semidirect product $L\cdot ra{d}^u(P)$. To prove that it is injective, it suffices to see that for every principal homogeneous bundle $Q$ under $L$, one has $H^1(S,ra{d}^u(P{)}_Q)=0$, where the subscript $Q$ denotes twisting by the $L$-bundle $Q$. This can be proved in two ways: one may take up the proof of 2.2, using the fact that the vector structures on the $U_i/U_{i+1}$ are invariant under $L$; one may also remark that $ra{d}^u(P{)}_Q$ is identified with the unipotent radical of the parabolic subgroup P_Q of G_Q, and apply 2.2 to P_Q.

Corollary 2.4. Let $S$ be a semi-local scheme, $G$ an $S$-reductive group, $P$ a parabolic subgroup of $G$. There exists a maximal torus $T$ of $G$ contained in $P$.

Indeed, in view of 2.3, $P$ has a Levi subgroup $L$, and it suffices to prove that $L$ has a maximal torus, which follows from Exp. XIV, 3.20.

Corollary 2.5. Let $S$ be an affine scheme, $G$ an $S$-reductive group, $P$ a parabolic subgroup of $G$. There exists a locally free O_S-module $E$ such that $ra{d}^u(P)$ is isomorphic as an $S$-scheme to $W(E)$.

Indeed, let us prove by induction on $i$ that one has an isomorphism of $S$-schemes

rad^u(P) / U_i ≃ W(E_0 ⊕ E_1 ⊕ · · · ⊕ E_{i−1}).

This is clear for $i=0$. Assume $i>0$; then $ra{d}^u(P)/U_i$ is a principal homogeneous bundle with base $ra{d}^u(P)/U_{i-1}$, under the group

(U_{i−1} / U_i)_{rad^u(P) / U_{i−1}} ≃ W(E_{i−1} ⊗ O_{rad^u(P) / U_{i−1}}).

Now the base is affine (e.g. by the induction hypothesis), so this bundle is trivial (TDTE I or SGA 1 XI, loc. cit.), and there exists an isomorphism of $S$-schemes $ra{d}^u(P)/U_i\simeq (ra{d}^u(P)/U_{i-1}){\times }_S(U_{i-1}/U_i)$, which completes the proof.

Corollary 2.6. Let $S$ be a semi-local scheme, $s_i$ the set of its closed points, $G$ an $S$-reductive group, $P$ a parabolic subgroup of $G$. The canonical map

rad^u(P)(S) ⟶ ∏_i rad^u(P)(Spec κ(s_i))

is surjective.

Indeed, if $S=\mathrm{Spec}(A)$, $\kappa (s_i)=A/p_i$, and if $E$ is given by the projective (hence flat) module $E$, we must prove that the map

E ⟶ ∏_i E ⊗_A A/p_i

is surjective. It suffices to do this when $E=A$, in which case it is well known (cf. Bourbaki, Alg. Comm. Chap. II, §1, no. 2, Proposition 5).

Corollary 2.7. Let $k$ be an infinite field, $G$ a $k$-reductive group, $P$ a parabolic subgroup of $G$; then $ra{d}^u(P)(k)$ is dense in $ra{d}^u(P)$.

Corollary 2.8. Let $S$ be a semi-local scheme, $s_i$ the set of its closed points, $G$ an $S$-reductive group, $P$ a parabolic subgroup of $G$, and $L_i$ a Levi subgroup of $P_{s_i}$ for each $i$. There exists a Levi subgroup $L$ of $P$ inducing $L_i$ for each $i$.

Indeed, let L_0 be a Levi subgroup of $P$ (2.3). For each $i$, let $u_i\in ra{d}^u(P)(\mathrm{Spec}(\kappa (s_i)))$ be such that $int(u_i)L_{0,s_i}=L_i$ (1.8); if $u\in ra{d}^u(P)(S)$ induces $u_i$ for each $i$ (2.6), then $L=int(u)L_0$ answers the question.

Corollary 2.9. In the situation of 2.1, let moreover $H$ be a sub-group-scheme of $G$, smooth and of finite presentation over $S$, with connected fibers, such that $P\cap H$ contains locally for the (fpqc) topology a maximal torus of $G$. Then for each $i⩾0$, there exists a locally direct factor submodule $F_i$ of $E_i$ such that the isomorphism $U_i/U_{i+1}\stackrel{\sim }{\to }W(E_i)$ induces an isomorphism of groups

(U_i ∩ H) / (U_{i+1} ∩ H) ⥲ W(F_i).

Indeed, $H$ is a subgroup of type (R) of $G$ (Exp. XXII, 5.2.1). On the other hand, the assertion to be proved is local for the (fpqc) topology, and one may assume $G$ split relative to a maximal torus of $P\cap H$; one may even reduce to the situation of 2.1.1, with $H$ defined by a subset $R^{\prime }$ of $R$. Taking up the notation of loc. cit., one sees by Exp. XXII, 5.6.7 (ii) that $U_i\cap H={\prod }_{\alpha \in R_i\cap R^{\prime }}{U}_{\alpha }$, hence that $(U_i\cap H)/(U_{i+1}\cap H)$ is identified with ${\prod }_{\alpha \in R^{\prime },a(\alpha )=i+1}{U}_{\alpha }$, which yields the result.

Corollary 2.10. In the situation of 2.9, the conclusions of 2.2, 2.5, 2.6, 2.7 are also valid when $ra{d}^u(P)$ is replaced by $ra{d}^u(P)\cap H$.

Corollary 2.11.⁹ Let $G$ be an $S$-reductive group, $P$ a parabolic subgroup, $H$ a subgroup of type (RC) of $P$ such that $ra{d}^u(H)=ra{d}^u(P)\cap H$. Then statements 2.2 to 2.8 are also valid when $P$ is replaced by $H$.

3. Scheme of parabolic subgroups of a reductive group

3.1.

Let $E$ be a finite twisted constant $S$-scheme (Exp. X, 5.1). Consider the $S$-functor $Of(E)$, where $Of(E)(S^{\prime })$ is the set of open and closed subschemes of $E_{S^{\prime }}$ (or, equivalently, the set of open and closed subsets of $E_{S^{\prime }}$); then $Of(E)$ is representable by a finite twisted constant $S$-scheme. Indeed, if $E=A_S$, where $A$ is a finite set, one has immediately $Of(E)\simeq P(A{)}_S$ (where $P(A)$ denotes the power set of $A$), and one concludes by descent of open and closed subschemes. One trivially has:

Of(E_{S'}) = Of(E)_{S'},        Of(E ×_S E') = Of(E) ×_S Of(E').

3.2.

Let $S$ be a scheme, $G$ an $S$-reductive group. The functor $Par(G)$ of parabolic subgroups of $G$ is defined by

Par(G)(S') = the set of parabolic subgroups of G_{S'}.

In particular, $G\in Par(G)(S)$, $Bor(G)\subset Par(G)$. We propose to define a morphism

$$t:Par(G)⟶Of(Dyn(G))$$

possessing the following properties:

(i) $t$ is functorial in $G$ (with respect to isomorphisms) and commutes with base change.

(ii) If $(T,M,R,\Delta ,\cdots )$ is a pinning of $G$ adapted to the parabolic subgroup $P$ (1.11), the canonical isomorphism $Dyn(G)\simeq {\Delta }_S$ (Exp. XXIV, 3.4 (iii)) transforms $t(P)$ into $\Delta (P{)}_S$ (notations of 1.11, 1.12).

Let first $P$ be a parabolic subgroup of $G$ and $(T,M,R,\Delta ,\cdots )$ a pinning of $G$ adapted to $P$. One defines $t(P)$ by (ii); the subscheme $t(P)$ of $Dyn(G)$ thus constructed is independent of the pinning chosen. Indeed, if $(T^{\prime },M^{\prime },R^{\prime },{\Delta }^{\prime },\cdots )$ is another pinning of $G$ adapted to $P$, the unique inner automorphism of $G$ transforming the first pinning into the second comes from $P$ (1.15); the canonical isomorphism $\Delta \stackrel{\sim }{\to }{\Delta }^{\prime }$ therefore transforms $\Delta (P)$ into ${\Delta }^{\prime }(P)$, which entails the announced result.

If now one no longer assumes $(G,P)$ to be pinnable, it follows immediately from 1.14 and the definition of $Dyn(G)$ (Exp. XXIV, 3.3) that one may define by descent an open and closed subscheme $t(P)$ of $Dyn(G)$, unique, such that for every $S^{\prime }\to S$ for which $(G,P{)}_{S^{\prime }}$ is pinnable, one has $t(P{)}_{S^{\prime }}=t(P_{S^{\prime }})$.

Theorem 3.3. Let $S$ be a scheme, $G$ an $S$-reductive group,

$$t:Par(G)⟶Of(Dyn(G))$$

the morphism defined above.

(i) For two parabolic subgroups $P$ and $P^{\prime }$ of $G$ to be conjugate locally for the (fpqc) topology (cf. 1.3), it is necessary and sufficient that $t(P)=t(P^{\prime })$.

(ii) $Par(G)$ is representable, and the morphism $t$ is smooth, projective, with integral geometric fibers.

By 3.2 (i), and the fact that inner automorphisms of $G$ act trivially on $Dyn(G)$ (Exp. XXIV, 3.4 (iv)), one indeed has $t(P)=t(P^{\prime })$ when $P$ and $P^{\prime }$ are conjugate. Conversely, let $P$ and $P^{\prime }$ be two parabolic subgroups of $G$ such that $t(P)=t(P^{\prime })$; let us prove that $P$ and $P^{\prime }$ are conjugate in $G$, locally for the (fpqc) topology; one may first

assume that the pairs $(G,P)$ and (G, P') are pinnable (1.14); by conjugacy of pinnings in $G$ (Exp. XXIV, 1.5), one may assume that there exists a pinning $(T,M,R,\Delta )$ of $G$ adapted to $P$ and $P^{\prime }$. Then $t(P)=t(P^{\prime })$ implies $\Delta (P)=\Delta (P^{\prime })$, hence $P=P^{\prime }$ (cf. 1.4 (v)). We have therefore proved (i). To demonstrate (ii), let us take up the notation of Exp. XXII, 5.11.5¹⁰.

We have a canonical morphism $Par(G)\to H_c$, and it is clear (e.g. by reduction to the pinned case) that it fits into a cartesian square (where the vertical arrows are monomorphisms)

Par(G) ────────t──────→ Of(Dyn(G))
   │                          │
   │                          │
   ▼                          ▼
   H_c ────────cℓ─────→ Cℓ_c.

Now (loc. cit.) $H_c$ is representable and the morphism $c\ell$ is smooth, quasi-projective, of finite presentation, with integral geometric fibers, hence the same holds for $t$.

It remains to prove that $t$ is proper; but this is now a local assertion for the (fpqc) topology, and one may reduce to the pinned case $G=(G,T,M,R,\Delta ,\cdots )$. One then has $Dyn(G)\simeq {\Delta }_S$, and it suffices to prove that for every subset ${\Delta }_1$ of $\Delta$, the $S$-scheme $t^{-1}(({\Delta }_1{)}_S)$ is proper over $S$. Now if P_1 is the parabolic subgroup of $G$ containing $T$ such that $\Delta (P_1)={\Delta }_1$, it follows from (i) that the morphism $G\to Par(G)$ defined set-theoretically by $g\mapsto int(g)P_1$ induces an isomorphism of $G/Nor{m}_G(P_1)$ onto $t^{-1}(({\Delta }_1{)}_S)$. Now, by 1.2, $G/Nor{m}_G(P_1)=G/P_1$ is projective over $S$.

Definition 3.4. $Of(Dyn(G))$ is called the scheme of types of parabolics of $G$; $t(P)$ is called the type of $P$.

Corollary 3.5. The $S$-functor $Par(G)$ is representable by an $S$-scheme that is smooth and projective over $S$. The decomposition

$$Par(G)⟶Of(Dyn(G))⟶S$$

is the Stein factorization (EGA III, 4.3.3) of the structural morphism $Par(G)\to S$.

Corollary 3.6. For each $t\in Of(Dyn(G))(S)$, the $S$-scheme

$$Pa{r}_t(G)=t^{-1}(t)$$

of parabolic subgroups of $G$ of type $t$ is smooth and projective over $S$, homogeneous under $G$. If $P$ is a parabolic subgroup of $G$, one has a canonical isomorphism $G/P\stackrel{\sim }{\to }Pa{r}_{t(P)}(G)$. One has $Pa{r}_{\varnothing }(G)=Bor(G)$, $Pa{r}_{Dyn(G)}(G)\stackrel{\sim }{\to }S$.

Remark 3.7. The $S$-scheme $Of(Dyn(G))$ is equipped with a natural order structure (the relation of domination, here set-theoretic inclusion, between subschemes). This order structure is a lattice; in particular, the infimum of two open and closed subschemes of $Dyn(G{)}_{S^{\prime }}$ is obviously their intersection. Let us moreover remark that if $B$ is a Borel subgroup of $G$, one may define the functor $X$ of parabolic subgroups of $G$ containing $B$. The morphism $X\to Of(Dyn(G))$ induced by $t$ is an isomorphism (for the structure of "ordered scheme"), by virtue of the assertion $P\subset Q\Rightarrow t(P)\subset t(Q)$ and of:

Lemma 3.8. Let $S$ be a scheme, $G$ an $S$-reductive group, $P$ a parabolic subgroup of $G$, $t^{\prime }$ a section of $Of(Dyn(G))$ over $S$, such that $t(P)\subset t^{\prime }$. There exists a unique parabolic subgroup $P^{\prime }$ of $G$, containing $P$, and such that $t(P^{\prime })=t^{\prime }$.

Possibly after extending the base, one may assume that $P$ contains a Borel subgroup $B$ of $G$. The uniqueness of $P^{\prime }$ then follows from 1.17. To demonstrate existence, one may place oneself in the split case, in which case the assertion is evident, cf. §1.

Remarks 3.8.1. (i) The assertion analogous to 3.8 obtained by reversing the inclusions is obviously false. It would, for instance, entail that every group of type A_1 has a Borel subgroup, which is not the case, cf. Exp. XX, §5.

(ii) It follows immediately from the preceding that $t(P)\subset t(Q)$ means that locally for (fpqc) or (ét), $P$ is conjugate to a subgroup of $Q$ (it suffices moreover to verify the assertion on geometric fibers). Moreover, one shall see in §5 that the étale topology may be replaced by the Zariski topology.

3.9.

The preceding discussions may be taken up in the case of critical subgroups. Let us recall (Exp. XXII, 5.10.4 and 5.10.5) that a reductive subgroup $H$ of the reductive group $G$ is critical if $H=Cent{r}_G(rad(H))$, that a subtorus $Q$ of $G$ is a C-critical torus if $Q=rad(Cent{r}_G(Q))$, and that critical subgroups and C-critical tori¹¹ are in bijective correspondence (via $H\mapsto rad(H)$ and $Q\mapsto Cent{r}_G(Q)$).

If $(G,T,M,R)$ is a split $S$-group, the subgroup of $G$ containing $T$ corresponding to the part $R^{\prime }$ of $R$ (Exp. XXII, 5.4.2) is critical if and only if $R^{\prime }$ is "vectorial" (that is, the intersection of $R$ with a vector subspace of $M\otimes \mathbb{Q}$), cf. Exp. XXII, 5.10.6.

If $G$ is an arbitrary $S$-reductive group, one defines, as in Exp. XXII, 5.11.5, an étale finite $S$-scheme $C{\ell }_{crit}$, which in the split case will be the constant scheme associated with the set of vectorial parts of $R$ modulo the action of the Weyl group. If $Crit(G)$ denotes the "functor of critical subgroups" of $G$, one has a canonical morphism $Crit(G)⟶C{\ell }_{crit}$, which fits into a cartesian diagram¹²

Crit(G) ────cℓ───→ Cℓ_{crit}
   │                   │
   │                   │
   ▼                   ▼
   H_c ────cℓ─────→ Cℓ_c.

Proposition 3.10. Let $S$ be a scheme, $G$ an $S$-reductive group, $Crit(G)$ the functor of its critical subgroups, $C{\ell }_{crit}$ and $c\ell :Crit(G)\to C{\ell }_{crit}$ the étale finite $S$-scheme and the morphism defined above.

(i) For critical subgroups $H$ and $H^{\prime }$ of $G$ to be conjugate (locally for the (fpqc) topology), it is necessary and sufficient that $c\ell (H)=c\ell (H^{\prime })$.

(ii) $Crit(G)$ is representable and the morphism $c\ell$ is smooth, affine, with integral geometric fibers.

This is proved as 3.3, with the exception of the assertion "$c\ell$ is affine". It suffices to prove that $Crit(G)$ is affine over $S$. Now $Crit(G)$ is naturally identified with the $S$-functor of critical tori of $G$, and one therefore has a canonical monomorphism

$$Crit(G)⟶M$$

where $M$ is the scheme of subgroups of multiplicative type of $G$ (Exp. XI, 4.1). To prove that $Crit(G)$ is affine over $S$, it suffices, by Exp. XII 5.3, to show that this morphism is an open and closed immersion, or equivalently, by making the base change $M\to S$, to prove the following assertion: if $Q$ is a subgroup of multiplicative type of the reductive group $G$, the $S^{\prime }\to S$ such that $Q_{S^{\prime }}$ is a critical torus of $G_{S^{\prime }}$ are those that factor through a certain open and closed subscheme of $S$. Now to say that $Q$ is a critical torus is to say:

(1) that $Q$ is a torus,

(2) $Q$ being a torus, that $rad(Cent{r}_G(Q))$, which is also a torus, is of the same relative dimension as $Q$.

These two conditions are indeed of the type envisaged.

Corollary 3.11. The $S$-functor $Crit(G)$ is representable by an $S$-scheme that is smooth and affine over $S$.

Corollary 3.12. Let $H$ be a critical subgroup of the $S$-reductive group $G$. Then $G/Nor{m}_G(H)$ and $G/H$ are representable by $S$-schemes that are affine and smooth over $S$.

The first assertion follows from 3.11, the second from the first and from Exp. XXII, 5.10.2.

Corollary 3.13. Let $Q$ be a subtorus of the $S$-reductive group $G$. Then $G/Cent{r}_G(Q)$ is representable by an $S$-scheme that is smooth and affine over $S$. The same holds for $G/Nor{m}_G(Q)$ if $Q$ is a critical subtorus of $G$.

Indeed, $H=Cent{r}_G(Q)$ is critical (Exp. XXII, 5.10.5), and one has $Nor{m}_G(H)=Nor{m}_G(Q)$ if $Q$ is critical (loc. cit. 5.10.8).

3.14.

By the conjugacy of Levi subgroups of parabolic subgroups of $G$, there exists a unique morphism

$$u:Of(Dyn(G))⟶C{\ell }_{crit}$$

such that for every parabolic subgroup $P$ of $G$, and every Levi subgroup $L$ of $P$, one has $c\ell (L)=u(t(P))$, and such that this holds after every base change.

3.15.

Let $S$ be a scheme, $G$ an $S$-reductive group. Consider the $S$-functors¹³:

PL(S')  = { pairs P ⊃ L, P parabolic of G_{S'}, L Levi subgroup of P };
PT(S')  = { pairs P ⊃ T, P parabolic of G_{S'}, T maximal torus of P };
CT(S')  = { pairs C ⊃ T, C critical subgroup of G_{S'}, T maximal torus of C };
PLT(S') = { triples P ⊃ L ⊃ T, (P, T) ∈ PT(S'), L Levi subgroup of P }.

One has evident morphisms between these functors and the functors $Par(G)$, $Crit(G)$,

$Tor(G)$ already introduced, and one has a commutative diagram in the form of a truncated cube (see the following figure):

                        g (ét.)
            CT  ←─────────────  PLT
           ╱   ╲                  ╲
        a╱       ╲b                ╲ (aff.)
        ╱          ╲                ╲
       ╱             ╲                ╲
      ╱  (aff.)    (ét.) k             ╲
     ╱       f (ét.)                    ╲
 Crit(G) ←─────────  PL                  PT     (e isom.)
     │                │                  │
     │ cℓ (aff.)      │                  │
     ▼                ▼                  ▼
  Cℓ_{crit}        Tor(G) ←──── h (ét.) ─ PT
         ╲          │                    ╱
       q ╲   (ét.)   │    d (aff.)      ╱
           ╲          │                ╱
       u ╲    │ r (aff.)             ╱  c (aff.)
              ▼                    ╱
       u (ét.) S                  ╱
           ╲                    ╱
              ╲ p (ét.)        ╱
                              ╱
                     t (proj.)
            Of(Dyn(G)) ────── Par(G)

Figure 3.15.1.

Theorem 3.16. (cf. Figure 3.15.1).

(i) All the morphisms of the diagram are smooth, surjective, and of finite presentation.

(ii) All the morphisms of the diagram, with the exception of $t$, are affine; the morphism $t$ is projective.

(iii) All the morphisms of the diagram are either étale finite or with integral geometric fibers: the morphisms $f$, $g$, $h$, $k$, $p$, $q$ and $u$ are étale finite, the morphisms $a$, $b$, $c$, $d$, $r$ and $c\ell$ are with integral geometric fibers, the morphism $e$ is an isomorphism.

(iv) The square $(a,b,f,g)$ is cartesian.

Proof. It is first clear that $e$ is an isomorphism, by 1.6 (ii). On the other hand, (iv) is evident.

The morphism $a$ is smooth, affine, with integral geometric fibers: indeed, by base change $Crit(G)\to S$, it suffices to verify that the morphism $Tor(L_0)\to S$, where L_0 is the universal critical subgroup, has these properties; but L_0 is reductive (by definition), and one is reduced to Exp. XXII, 5.8.3. The morphism $b$ therefore has the same properties, by virtue of (iv).

The morphism $d$ is also smooth, affine, with integral geometric fibers, by 1.9; the same therefore holds for $c=db{e}^{-1}$. The morphism $r$ has these same properties (Exp. XXII, 5.8.3), as does the morphism $c\ell$ (3.10).

On the other hand, we have already proved that the morphisms $p$ and $q$ are étale finite surjective (3.1 and 3.9). If we prove that $f$ and $k$ are étale finite surjective, the same properties will hold for $g$ (by (iv)) and for $h$ (since $h=kg{e}^{-1}$); as the properties stated for $t$ were established in 3.3 (ii), it therefore only remains to prove that $f$ (resp. $k$) is étale finite surjective; let us give the proof for $k$, that for $f$ being analogous.

It suffices to prove that if $T$ is a maximal torus of $G$, the functor $C$ of critical subgroups of $G$ containing $T$ is representable by an étale finite $S$-scheme with non-empty fibers; one may assume $G$ split with respect to $T$; let then $E$ be the set of vectorial parts of $R$ (root system of the splitting); $C$ is representable by E_S (3.9), which completes the proof.

Corollary 3.17. All the functors of the preceding diagram are representable by $S$-schemes that are smooth over $S$, and they are all affine over $S$, with the exception of $Par(G)$.

Remark 3.18. (i) The fact that the morphism $f:PL\to Crit(G)$ is étale surjective implies that a subgroup of $G$ is critical if and only if it is, locally for the étale topology, a Levi subgroup of a parabolic subgroup of $G$.

On the other hand, one should not believe that in general the map $f(S):PL(S)\to Crit(G)(S)$ is surjective: it may very well happen that a critical subgroup $C$ of $G$ does not come on $S$ from a parabolic subgroup of $G$; for example, a maximal torus is not always contained in a Borel subgroup (example: a non-split form of SL_2, cf. Exp. XX, §5).

(ii) Similarly, it may happen that the morphism $u:Of(Dyn(G))\to C{\ell }_{crit}$ is not an isomorphism: two parabolic subgroups of distinct types may have Levi subgroups of the same type; for example: in a group of type A_2, there are two types of parabolic subgroups whose Levi subgroups are of semisimple rank 1 (corresponding to the two vertices of the diagram), whereas there is only one type of critical subgroup of rank 1.

The analogous example with a group of type A_3 shows that, even over an algebraically closed field, non-isomorphic parabolic subgroups may have Levi subgroups of the same type.¹⁴

Let us close this section with an application to the theory of principal bundles.

Lemma 3.20.¹⁵ Let $S$ be a scheme, $G$ an $S$-reductive group, $F$ a principal homogeneous bundle under $G$, G_F the corresponding twisted form of $G$. Identify $Dyn(G)$ and $Dyn(G_F)$ (Exp. XXIV, 3.5). Let $P$ be a parabolic subgroup of $G$. One has a canonical isomorphism

$$F/P\stackrel{\sim }{\to }Pa{r}_{t(P)}(G_F).$$

In particular, for the structural group of $F$ to reduce to $P$, it is necessary and sufficient that G_F have a parabolic subgroup of type $t(P)$.

The proof goes exactly as in Exp. XXIV, 4.2.1.

3.21.

If $S$ is a scheme, $G$ an $S$-reductive group, and if $t\in Of(Dyn(G))(S)$, one denotes by $H_t^1(S,G)$ the subset of $H^1(S,G)$ formed of classes of principal bundles $F$ under $G$ such that the associated group G_F has a parabolic subgroup of type $t$. If $G$ itself has a parabolic subgroup $P$ of type $t$, $H_t^1(S,G)$ is none other than the image of $H^1(S,P)$ in $H^1(S,G)$, an image which therefore does not depend on the $P$ chosen.

4. Relative position of two parabolic subgroups

4.1. A preliminary result

Lemma 4.1.1. Let $k$ be a field, $G$ a $k$-reductive group, $P$ and $P^{\prime }$ two parabolic subgroups of $G$.

(i) Then $P\cap P^{\prime }$ is smooth, of the same reductive rank as $G$, and contains a maximal torus $T$ of $G$.

(ii) The set $R^{\prime }$ of roots of $P\cap P^{\prime }$ with respect to $T$ is a closed subset of the set $R$ of roots of $G$ with respect to $T$.¹⁶

Suppose first $k$ algebraically closed. Let $B$ (resp. $B^{\prime }$) be a Borel subgroup

of $P$ (resp. $P^{\prime }$). One knows that there exists $g\in G(k)$ such that $int(g)B=B^{\prime }$. On the other hand, if T_0 is a maximal torus of $B$ and one sets $N=Nor{m}_G(T_0)$, one knows (Bruhat's theorem, Bible, §13.4, Cor. 1 to Th. 3) that $G(k)=B(k)N(k)B(k)$. One therefore sees that there exist $b,b_1\in B(k)$ and $n\in N(k)$ such that $g=bn{b}_1$, hence $int(b)int(n)B=B^{\prime }$. One then has

P ∩ P' ⊃ B ∩ B' = int(b)(B ∩ int(n) B') ⊃ int(b) T_0.

Now suppose $k$ arbitrary. Applying the preceding result, one sees that $P_{\bar{k}}\cap P_{\bar{k}}^{\prime }$ contains a maximal torus of $G_{\bar{k}}$; by Exp. XXII, 5.4.5, one deduces that $(P\cap P^{\prime }{)}_{\bar{k}}$ is smooth "along the unit section", hence smooth since one is over a field (Exp. VI_A 1.3.1), hence $P\cap P^{\prime }$ is smooth. By Exp. VI_A 2.3.1, the neutral component $(P\cap P^{\prime }{)}^0$ of $P\cap P^{\prime }$ is therefore an open subgroup of $P\cap P^{\prime }$, smooth over $S$. One may then apply Exp. XIV, 1.1 to it, whence (i).

Finally, the set R_P (resp. $R_{P^{\prime }}$) of roots of $P$ (resp. $P^{\prime }$) with respect to $T$ is closed and one has $R^{\prime }=R_P\cap R_{P^{\prime }}$, whence (ii).

Remark 4.1.2. One may prove ([BT65], 4.5) that $P\cap P^{\prime }$ is connected;¹⁷ this will be used in 4.5.1.

Remark 4.1.3. The preceding lemma is not true over an arbitrary scheme. Indeed, let $G$ be, for example, a reductive group over an algebraically closed field $k$, and let $B$ be a Borel subgroup of $G$. Take $S=X$ as base and consider the Borel subgroups B_1 and B_2 of G_X, where $B_1=B_X$ and $B_2=int(g_0)B_1$, $g_0$ being the canonical (diagonal) section of G_X. For each $g\in X(k)$, the fiber of $B_1\cap B_2$ at $g$ is none other than $B\cap int(g)B$. If one assumes $B\ne G$, the dimension of this fiber varies with $g$, hence $B_1\cap B_2$ cannot be smooth over $X$.

4.2. Transversal position

Theorem 4.2.1. Let $S$ be a scheme, $G$ an $S$-reductive group, $P$ and $Q$ two parabolic subgroups of $G$. The following conditions on the pair $(P,Q)$ are equivalent:

(i) Lie(P/S) + Lie(Q/S) = Lie(G/S).

(ii) The canonical morphism $P{\times }_{S}Q\to G$ is smooth.

(ii') The canonical morphism $P\to G/Q$ is smooth.

(iii) The canonical morphism $P{\times }_{S}Q\to G$ is open.

(iii') The canonical morphism $P\to G/Q$ is open.

(iv) The canonical morphism $P{\times }_{S}Q\to G$ is fiberwise dominant.

(iv') The canonical morphism $P\to G/Q$ is fiberwise dominant.

(v) For every $s\in S$, "$P_s\cap Q_s$ is of minimum dimension", i.e. one has

dim(P_s ∩ Q_s) = dim P_s + dim Q_s − dim G_s.

(vi) There exists a covering family for the étale topology $S_i\to S$, and for each $i$ a Borel subgroup $B_i$ of $P_{S_i}$ and a Borel subgroup $B_i^{\prime }$ of $Q_{S_i}$, such that $B_i\cap B_i^{\prime }$ is a maximal torus of $G_{S_i}$.

(vii) There exists a covering family for the étale topology $S_i\to S$, and for each $i$ a splitting $(T_i,\cdots ,R_i)$ of $G_{S_i}$ and a system of positive roots $R_i^{+}$ of $R_i$, such that $P_{S_i}$ (resp. $Q_{S_i}$) is the subgroup of type (R) of $G_{S_i}$ containing $T_i$ and defined by a subset $R_i^{(1)}$ (resp. $R_i^{(2)}$) of $R$ containing $R_i^{+}$ (resp. $-R_i^{+}$) (see Exp. XXII, 5.4.2 and 5.2.1 for the definitions).

Proof. We shall prove the theorem following the logical diagram

(i) ⇔ (vii) ⇔ (vi)
                  ╲
                   ╲
(ii') ⇒ (iii') ⇒ (iv') ⇒ (v)
                  ╱
                 ╱
(ii)  ⇒ (iii)  ⇒ (iv)

One trivially has (ii) ⇒ (iii) and (ii') ⇒ (iii'). If (iii) holds, the set-theoretic image of the morphism $P{\times }_{S}Q\to G$ is an open set of $G$ containing the unit section; since the fibers of $G$ are connected, this image is dense on each fiber, which proves (iv). One similarly has (iii') ⇒ (iv').

One has (ii') ⇒ (ii), by the cartesian diagram

P ×_S Q ─────→ G
   │           │
 pr_1│         │
   ▼           ▼
   P ───────→ G/Q.

On the other hand (iv) or (iv') implies (v), by the theory of dimension (cf. EGA IV₂, 5.6.6). One notes that one may indeed assume $S=\mathrm{Spec}(k)$, $k$ an algebraically closed field, and that every non-empty fiber of the morphism (iv), resp. (iv'), at a point of $G(k)$, resp. of $(G/Q)(k)$, is isomorphic to $P\cap Q$ (as an immediate computation shows).

One has (vi) ⇒ (vii), by Exp. XXII, 5.5.1 (iv) and 5.9.2.

One has (vii) ⇒ (i), because to verify that Lie(P) + Lie(Q) = Lie(G), one may reason locally for (fpqc); thus if (vii) is satisfied one may assume $G$ split, $P\supset B_{R^{+}}$ and $Q\supset B_{-R^{+}}$ (usual notations), in which case one already has

Lie(B_{R^+}) + Lie(B_{−R^+}) = Lie(G).

Let us prove that (i) implies (ii').

Let $u:P\to G/Q$ be the canonical morphism; to prove that $u$ is smooth, it suffices to do so on the geometric fibers of $u$, since $P$ and $G/Q$ are smooth over $S$, and one may therefore assume that $S$ is the spectrum of an algebraically closed field. As the morphism $u$ is compatible with the obvious action of $P$ (one has $u(p{p}^{\prime })=pu(p^{\prime })$) it suffices, by a translation argument (cf. the proof of VI_B 1.3), to verify that $u$ is smooth at $e\in P(k)$, i.e. (SGA 1, II 4.7) that the tangent map to $u$ at $e$ is surjective; but the latter is naturally

identified with the canonical map Lie(P) → Lie(G)/Lie(Q), which is surjective if (i) is satisfied.

It therefore only remains to verify the last assertion, namely (v) ⇒ (vi). Suppose first that $S$ is the spectrum of an algebraically closed field. By 4.1.1, there exists a maximal torus $T$ contained in $P$ and $Q$; let $R$ (resp. R_1, resp. R_2) be the set of roots of $G$ (resp. $P$, resp. $Q$) relative to $T$.

One has:

dim(G) = dim(T) + Card(R),         dim(P) = dim(T) + Card(R_1),
dim(Q) = dim(T) + Card(R_2),       dim(P ∩ Q) = dim(T) + Card(R_1 ∩ R_2),

by Exp. XXII, 5.4.4 and 5.4.5 for example. The condition of (v) is therefore equivalent to

Card(R_1 ∩ R_2) = Card(R_1) + Card(R_2) − Card(R),

that is $R_1\cup R_2=R$. To demonstrate (vi), it suffices, by Exp. XXII, 5.9.2 and 5.4.5, to prove that $R_1\cap -R_2$ contains a system of positive roots of $R$. We are therefore reduced to proving:

Lemma 4.2.2. Let $R$ be a "root system" (e.g. the set of roots of a root datum in the sense of Exposé XXI). Let R_1 and R_2 be two closed subsets of $R$ each containing a system of positive roots. If $R_1\cup R_2=R$, then $R_1\cap -R_2$ contains a system of positive roots.

Indeed, since $R_1\cap -R_2=R_3$ is evidently closed, and by Exp. XXI, 3.3.6, it suffices to show that $R_3\cup -R_3=R$. Now one knows that $R_1\cup -R_1=R=R_2\cup -R_2$, and one concludes thanks to the following elementary fact: if A, A', B, B' are four subsets of a set $E$, and if $A\cup A^{\prime }=B\cup B^{\prime }=A\cup B=A^{\prime }\cup B^{\prime }=E$, then $(A\cap B^{\prime })\cup (A^{\prime }\cap B)=E$.

This completes the proof of (v) ⇒ (vi) in the case where the base is the spectrum of an algebraically closed field. Let

us now return to the general case and assume (v) is satisfied. Let $s\in S$; by the preceding, one may find a Borel subgroup $\bar{B}$ (resp. ${\bar{B}}^{\prime }$) of $P_s$ (resp. $Q_s$) such that $\bar{B}\cap {\bar{B}}^{\prime }$ is a maximal torus of $G_s$.

Since the $S$-scheme Bor(P) ≃ Bor(P/rad^u(P)) of Borel subgroups of $P$ is smooth, one may, applying "Hensel's lemma" (cf. Exp. XI 1.10) and reasoning locally for the étale topology (i.e. replacing $S$ by an $S^{\prime }\to S$ that is étale and covering $s$, and $s$ by a point of its fiber in $S^{\prime }$), assume that there exists a Borel subgroup $B$ of $P$ projecting onto $\bar{B}$; one may similarly assume that there exists a Borel subgroup $B^{\prime }$ of $Q$ projecting onto ${\bar{B}}^{\prime }$. Since $B_s\cap B_s^{\prime }$ is a maximal torus of $G_s$, there exists an open neighborhood $U$ of $s$ in $S$ such that $B_U\cap B_U^{\prime }$ is a maximal torus of G_U (Exp. XXII, 5.9.4), which proves (vi). QED.

Definition 4.2.3. A pair $(P,Q)$ satisfying the equivalent conditions (i) to (vii) of Theorem 4.2.1 is said to be in transversal position. One also says that $P$ is in transversal position relative to $Q$, or, by abuse of language, that $P$ and $Q$ are in (mutual) transversal position.

In view of (vi), this definition coincides in the case of Borel subgroups with that of Exp. XXII, 5.9.1.

Corollary 4.2.4. Let $S$ be a scheme, $G$ an $S$-reductive group, $P$ and $Q$ two parabolic subgroups of $G$.

(i) For $(P,Q)$ to be in transversal position, it is necessary and sufficient that for each point $s$ of $S$, the pair $(P_s,Q_s)$ be in transversal position; if $S^{\prime }\to S$ is a surjective morphism, and if $(P_{S^{\prime }},Q_{S^{\prime }})$ is in transversal position, then $(P,Q)$ is in transversal position.

(ii) There exists an open subscheme $U$ of $S$ having the following property: for a morphism $S^{\prime }\to S$ to factor through $U$, it is necessary and sufficient that $(P_{S^{\prime }},Q_{S^{\prime }})$ be in transversal position.

(iii) Consider the subfunctors

Gen(G)   ⊂ Par(G) ×_S Par(G),
Gen(/Q)  ⊂ Par(G),
Gen(P/Q) ⊂ G

defined as follows: for $S^{\prime }\to S$, $Gen(G)(S^{\prime })$ is the set of pairs of parabolic subgroups of $G_{S^{\prime }}$ in transversal position, $Gen(/Q)(S^{\prime })$ is the set of parabolic subgroups of $G_{S^{\prime }}$ in transversal position relative to $Q_{S^{\prime }}$, $Gen(P/Q)(S^{\prime })$ is the set of $g\in G(S^{\prime })$ such that $int(g)P_{S^{\prime }}$ is in transversal position relative to $Q_{S^{\prime }}$.

Each of these functors is representable by a universally schematically dense open subscheme over $S$¹⁸ (cf. Exp. XVIII §1) of the corresponding $S$-scheme $Par(G){\times }_{S}Par(G)$, resp. $Par(G)$, resp. $G$.

Assertions (i) follow at once from the description 4.2.1 (i) of the term "transversal position". To demonstrate (ii), one takes $U=S-Supp(Cokeru)$ where $u$ is the canonical morphism Lie(P) ⊕ Lie(Q) → Lie(G).

Since one has cartesian diagrams

G ──f──→ Par(G)        Par(G) ──f'──→ Par(G) ×_S Par(G)
↑          ↑              ↑                  ↑
│          │              │                  │
Gen(P/Q) → Gen(/Q)       Gen(/Q) ─────→ Gen(G)

(where $f(g)=int(g)P$ and $f^{\prime }(R)=(R,Q)$), it suffices to verify (iii) in the case of $Gen(G)$.

Let then P_0 be the canonical parabolic subgroup of $G_{Par(G)}$; set

X = Par(G) ×_S Par(G),     P = pr_1^*(P_0),     Q = pr_2^*(P_0);

applying assertion (ii) to the parabolic subgroups $P$ and $Q$ of G_X, one constructs an open subscheme $U$ of $X$, which, as one verifies at once, is indeed identified with $Gen(G)$. It remains to verify the density assertion, which can be done on geometric fibers¹⁹; one may therefore assume $S=\mathrm{Spec}(k)$, $k$ an algebraically closed field. As $Par(G)$ is smooth, it suffices to verify that $Gen(G)$ meets each irreducible component of $Par(G){\times }_{S}Par(G)$; in other words, by 3.3, it suffices to see that if $t,t^{\prime }\in Of(Dyn(G))(S)$, there exists a pair (P, P') in transversal position with $t(P)=t$, $t(P^{\prime })=t^{\prime }$. Now this is immediate: one chooses a pair (B, B') of Borel subgroups of $G$ such that $B\cap B^{\prime }$ is a maximal torus (one splits $G$ and applies Exp. XXII, 5.9.2) then one applies 3.8 to construct $P\supset B$ and $P^{\prime }\supset B^{\prime }$, with $t(P)=t$, $t(P^{\prime })=t^{\prime }$; $P$ and $P^{\prime }$ are of the desired types and are in transversal position by 4.2.1 (vi).