On Weil restriction of reductive groups and a theorem of Prasad

DOI: 10.1007/s002090100339

Math. Z. 248, 449–457 (2004)

Mathematische Zeitschrift

On Weil restriction of reductive groups and a theorem

of Prasad

Richard Pink

Departement Mathematik, ETH Zentrum, CH-8092 Z¨urich, Switzerland (e-mail:pink@math.ethz.ch)

Received: 19 October 2000; in final form: 3 January 2001 / Published online: 31 August 2004 – © Springer-Verlag 2004

Abstract. LetGbe a connected simple semisimple algebraic group over a local fieldF of arbitrary characteristic. In a previous article by the author the Zariski dense compact subgroups ofG(F )were classified. In the present paper this infor-mation is used to give another proof of a theorem of Prasad [8] (also proved by Margulis [3]) which asserts that, ifGis isotropic, every non-discrete closed sub-group of finite covolume contains the image ofG(F )˜ , whereG˜denotes the universal covering ofG. This result played a central role in Prasad’s proof of strong approx-imation. The present proof relies on some basic properties of Weil restrictions over possibly inseparable field extensions, which are also proved here.

Mathematics Subject Classification (2000): 20G25, 14L15

1. Weil restriction of linear algebraic groups

LetF be a field andFa subfield such that [F /F]<∞. In this section we discuss some properties of the Weil restrictionR_{F /F}GwhereGis a linear algebraic group overF. We are interested particularly in the case thatF /Fis inseparable, where the Weil restriction involves some infinitesimal aspects. Thus the natural setting is that of group schemes. We assume thatGis a connected affine group scheme of finite type that is smooth overF. The smoothness condition is equivalent to saying thatGis reduced and “defined overF” in the terminology of [11] Ch.11.

Throughout, we will speak of a scheme over a ringR when we really mean a scheme over SpecR. Similarly, for any ring homomorphismR → Rand any scheme X over R we will abbreviateX ×R R := X ×SpecR SpecR. The

basic facts on Weil restrictions that we need are summarized in [4] Appendix 2–3. Throughout the following we abbreviate

By [4] A.3.2, A.3.7 this is a connected smooth affine group scheme overF. The universal property of the Weil restriction identifiesG(F)withG(F ).

Next, we fix an algebraic closureEofFand abbreviateE:=F⊗FE. With

:=Hom_F(F, E)there is then a unique decompositionE=_σ∈Eσ, where

eachEσis a local ring with residue fieldEand the composite mapF →Eσ −→→E

is equal toσ. The Weil restriction from any finite dimensional commutativeE -algebra down toEis defined, and by [4] A.2.7–8 we have natural isomorphisms

G×FE∼=RE/E(G×F E) =RE/E σ∈ G×F Eσ ∼ = σ∈ Gσ (1.1) with Gσ :=REσ/E(G×F Eσ).

These isomorphisms are functorial inGand equivariant under Aut(E/F), which acts on the right hand side by permuting the factors according to its action on. Next, for everyσ ∈we fix a filtration ofEσ by ideals

Eσ Iσ,1. . .Iσ,q−1Iσ,q =0

with subquotients of length 1. Hereqis the degree of the inseparable part ofF /F. We also choose a basis of every successive subquotient. For every 1≤i≤qthere is a natural homomorphism Gσ =REσ/E(G×F Eσ)−→R(Eσ/Iσ,i)/E G×F (Eσ/Iσ,i) .

LetGσ,idenote its kernel. By [4] A.3.5 we find that eachGσ,iis smooth overF

and there are canonical isomorphisms

Gσ/Gσ,1∼=G×F,σE (1.2)

and

Gσ,i/Gσ,i+1∼=LieG⊗F,σGa,E (1.3)

for all 1≤i≤q−1, whereG_adenotes the additive group of dimension 1. More-over, this description is functorial inG. Namely, letH be another smooth group scheme overF and defineH:=R_{F /F}H,Hσ andHσ,iin the obvious way. Then

any homomorphismϕ: H → Ginduces homomorphismsR_{F /F}ϕ: H → G, Hσ → Gσ andHσ,i → Gσ,iand the resulting homomorphisms on subquotients

are just

ϕ×id :H×F,σ E−→G×F,σE (1.4)

and

Recall that an isogeny of algebraic groups is a surjective homomorphism with finite kernel. An isogenyϕis separable if and only if its derivativedϕ is an iso-morphism.

Proposition 1.6. Letϕ:H→Gbe a homomorphism of connected smooth linear algebraic groups overF.

(a) IfF /Fis separable, thenR_{F /F}ϕ:H→Gis an isogeny if and only ifϕis an isogeny.

(b) IfF /Fis inseparable, thenR_{F /F}ϕ:H →Gis an isogeny if and only ifϕ

is a separable isogeny.

Proof. In the separable case we haveE−−→∼ Eσ, and assertion (a) follows directly

from the decomposition 1.1 and the functoriality 1.4. So assume thatF /Fis insep-arable, i.e., thatq > 1. First note that dimH =[F /F]·dimH and dimG = [F /F]·dimG, by the successive extension above or by [4] A.3.3. Thus if eitherϕ orR_{F /F}ϕis an isogeny, we must have dimH =dimG.

IfR_{F /F}ϕis an isogeny, its kernel is finite; hence so is the kernel of its restriction Hσ,q−1→Gσ,q−1. By 1.5 this means thatdϕis injective. For dimension reasons

it follows thatdϕis an isomorphism; henceϕis a separable isogeny, as desired. Conversely, suppose thatϕis a separable isogeny. Then all the homomorphisms on subquotients 1.4 and 1.5 induced by R_{F /F}ϕ are surjective. Using the snake lemma inductively one deduces that R_{F /F}ϕ itself is surjective. For dimension reasons it is therefore an isogeny, as desired.

Theorem 1.7. IfGis reductive andFinfinite, thenG(F)is Zariski dense inG. Proof. If F /F is separable, the isomorphism 1.1 shows thatG is reductive. In that case the assertion is well-known: see [11] Cor.13.3.12 (i).

We will adapt the argument to the general case.

Assume first thatG=T is a torus. Choose a finite separable extensionF1/F

which splitsT, and fix an isomorphismGn_m,F1 −−→∼ T ×FF1, whereGmdenotes

the multiplicative group of dimension 1. Combining this with the norm map yields a surjective homomorphism

RF1/FGnm,F1 −→RF1/F(T ×FF1)−−−→Nm T .

SinceF1/F is separable, this morphism is smooth. By [4] A.2.4, A.2.12 it induces

a smooth homomorphism

RF1/FGnm,F1 ∼=RF /FRF1/FGnm,F1 −→RF /FT .

In particular, this morphism is dominant. On the other hand we have an open embed-dingGn_m,F1 →An_F1and hence, by [4] A.2.11, an open embeddingR_F1/FGn_m,F1 → RF1/FAn_F1. It is trivial to show thatRF1/FAn_F1 ∼= And_F, whered = [F1/F]. It

follows that theF-rational points inR_F1/FGn_m,F1 are Zariski dense, and so their images form a Zariski dense set ofF-rational points inR_{F /F}T, proving the the-orem in this case.

IfGis arbitrary letT be a maximal torus ofG. AsRF /FT is commutative, it

Lemma 1.8.R_{F /F}T is the centralizer ofTinG.

Proof. If F /F is separable, this follows from the fact that R_{F /F}T is a maxi-mal torus ofG. So assume thatF /F is inseparable of characteristic p. Since (RF /FT )/Tis unipotent, we haveT =(RF /FT )p

n

for suitablen0. AsT is smooth and the rational points ofR_{F /F}T are Zariski dense, the centralizer of Tis equal to the centralizer of(R_{F /F}T )(F)pn. Note that the universal property of the Weil restriction identifies(R_{F /F}T )(F)withT (F ).

Consider a schemeS overF and anS-valued pointϕ: S → G. Via the universal property of the Weil restrictionϕ corresponds to an S×F F-valued

pointϕ:S×FF →G. We have seen thatϕfactors through the centralizer ofT

if and only if it commutes with(R_{F /F}T )(F)pn. This is equivalent to saying that ϕcommutes withT (F )pn. AsT is a torus andF infinite, the subgroupT (F )pnis Zariski dense inT. The condition therefore amounts to saying thatϕfactors through the centralizer ofT. But this centralizer is equal toT. Therefore, translated back toG, the condition says thatϕfactors throughR_{F /F}T. This proves the lemma.

By Lemma 1.8 the subgroup R_{F /F}T is the centralizer of a maximal torus ofG, i.e., it is a Cartan subgroup ofG. Thus [11] Cor.13.3.12 implies thatG(F) is Zariski dense inG, proving Theorem 1.7.

Remark 1.9. IfFis a non-discrete complete normed field, Theorem 1.7 is true for arbitrary connected smooth algebraic groupsG. This is an easy consequence of the implicit function theorem.

Next we turn to simple groups. To fix ideas, a smooth linear algebraic group over a field will be called simple if it is non-trivial and possesses no non-trivial proper connected smooth normal algebraic subgroup. It is called absolutely simple if it remains simple over the algebraic closure of the base field.

IfGis simply connected semisimple and simple overF, it is isomorphic to RF1/FG1for an absolutely simple simply connected semisimple groupG1 over

some finite separable extensionF1/F (cf. [11] Ex.16.2.9). From [4] A.2.4 we then

deduce thatG∼=RF1/FG1. In this way questions aboutGcan be reduced to the

case thatGis absolutely simple.

Theorem 1.10. Assume thatGis simply connected semisimple and simple overF. ThenGis simple overF.

Proof. By the above remarks we may assume thatGis absolutely simple. Consider a non-trivial connected smooth normal algebraic subgroupH⊂G. Let

¯ H⊂

σ∈

G×F,σ E (1.11)

denote the image ofH×FEunder the composite of the natural maps

G×FE 1.1 ∼ = σ∈ Gσ −→→ σ∈ Gσ/Gσ,1 1.2 ∼ = σ∈ G×F,σE.

SinceHis non-trivial and “defined overF”, by [11] Cor.12.4.3 we haveH¯=1. Since H ⊂ G is a connected normal subgroup, so is H¯ in 1.11. It is there-fore equal to the product of some of the factors on the right hand side. AsH¯is non-trivial, it contains at least one of these factors. But by construction it is also invariant under Aut(E/F), which permutes the factors transitively. We deduce that the inclusion 1.11 is in fact an equality. Now the following lemma implies that H×FE=G×FE; and henceH=G, as desired.

Lemma 1.12. In the situation of Theorem 1.10, every normal algebraic subgroup H ⊂G×FEwhich surjects to

σ∈G×F,σEis equal toG×FE. Proof. Using descending induction oniwe will prove thatGσ,i⊂Hfor allσ ∈

and 1 ≤ i ≤ q. For i = q the assertion is obvious, becauseGσ,q = 1. Let us

assume the inclusion forGσ,i+1and abbreviate

gr_iHσ :=H∩Gσ,i Gσ,i+1 ⊂ Gσ,i Gσ,i+1 1_.3 ∼ =LieG⊗F,σ Ga,E. (1.13)

By functoriality of the isomorphism 1.3, the conjugation action ofG(E)onGσ,i

corresponds to the adjoint representation ofG×F,σ Eon the right hand side. As

H is a normal subgroup, all commutators betweenH andGσ,imust lie inH. It

follows that

(Ad_h−id)(LieG)⊗F,σGa,E ⊂griHσ (1.14)

for everyh ∈H (E). SinceGis simply connected, it is known that the space of coinvariants of its adjoint representation is trivial (cf. [1], [2], or [5] Prop.1.11). On the other handEis algebraically closed, so by assumptionH (E)maps to a Zariski dense subgroup ofG×F,σ E. Thus, ash varies, the subgroups in 1.14

generate LieG⊗F,σ Ga,E. The inclusion in 1.13 is therefore an equality, and so

we haveGσ,i⊂H.

At the end of the induction we have Gσ,1 ⊂ H for all σ ∈ .

Combin-ing this with the fact thatH surjects to_σ∈Gσ/Gσ,1, we finally deduceH =

G×F E, as desired. This proves Lemma 1.12 and thereby finishes the proof of

Theorem 1.10.

Remark 1.15. The analogue of Theorem 1.10 fails ifGis not simply connected and bothF /F and the universal central extensionπ: G˜ → Gare inseparable. The reason is that by Proposition 1.6 (b) the homomorphismR_{F /F}ϕ:R_{F /F}G˜ →G is not surjective, so its image is a subgroup that makesGnot simple.

Corollary 1.16. IfGis semisimple and simply connected, thenGis perfect. Proof. We may assume thatGis simple. ThenGis connected and non-commuta-tive; hence so isG. The commutator group ofGis therefore non-trivial connected and normal, and by [11] Cor.2.2.8 it is “defined over F” and thus smooth. By Theorem 1.10 it is therefore equal toG, as desired.

Theorem 1.17. IfGis simple isotropic and simply connected andF is infinite, then

Proof. By assumption there exists a closed embeddingG_m,F×_FF ∼=Gm,F →

G. The homomorphismGm,F →Gcorresponding to it by the universal property

of the Weil restriction is again non-trivial; henceGcontains a non-trivial split torus. The algebraic subgroup ofG that is generated by all split tori inGis therefore non-trivial. By construction it is normalized by G(F), so by Theorem 1.7 it is normal inG. Being generated by smooth connected subgroups, it is itself smooth and connected by [11] Prop.2.2.6 (iii). By Theorem 1.10 it is therefore equal toG, as desired.

2. Main results

In the following we consider a connected semisimple groupGover a local fieldF. Letπ: G˜ → Gdenote its universal central extension. The commutator pairing

˜

G× ˜G→ ˜Gfactors through a unique morphism [, ]∼:G×G→ ˜G.

For any closed subgroup⊂G(F )we let˜denote the closure of the subgroup ofG(F )˜ that is generated by the set of generalized commutators [, ]∼. Theorem 2.1. LetF be a local field, and letGbe an isotropic connected simple semisimple group overF. Let⊂G(F )be a non-discrete closed subgroup whose covolume for any invariant measure is finite. Then˜is open inG(F )˜ .

Before proving this, we note the following consequence (cf. [8], [3]).

Corollary 2.2. Under the assumptions of Theorem 2.1 we have˜ = ˜G(F ). In particular,containsπG(F )˜ .

Proof. SinceG(F )is not compact andis a subgroup of finite covolume, this sub-group is not compact. Thus˜is normalized by an unbounded subgroup ofG(F ), and it is open inG(F )˜ by Theorem 2.1. As in [6] Thm.2.2 one deduces from this that˜is unbounded. LetG(F )˜ +denote the subgroup ofG(F )˜ that is generated by the rational points of the unipotent radicals of all rational parabolic subgroups. The Kneser-Tits conjecture, which is proved in this case (see [7] Thm. 7.6 or [10]), asserts thatG(F )˜ + = ˜G(F ). On the other hand, a theorem of Tits [9] states that every unbounded open subgroup of G(F )˜ + is equal to G(F )˜ +. Altogether this implies˜= ˜G(F ), as desired.

Proof of Theorem 2.1. In the case char(F ) = 0 the proof in [8] §2 cannot be improved. It covers in particular the archimedean case. We will give a unified proof in the non-archimedean case, beginning with a few reductions.

Letaddenote the image ofin the adjoint groupGadofG. Then˜depends only onad. On the other hand, all the assumptions in 2.1 are still satisfied for ad ⊂ Gad(F ). Namely, since the homomorphismG(F ) → Gad(F )is proper with finite kernel, the subgroupadis still non-discrete and closed. On the other hand, as the image ofG(F )inGad(F )is cocompact, the covolume ofadinGad(F )

is again finite. To prove the theorem, we may therefore replaceGbyGad and byad. In other words, we may assume thatGis adjoint.

Next, sinceGis connected simple and adjoint, it is isomorphic toR_F1/FG1for

some absolutely simple connected adjoint groupG1over a finite separable extension

F1/F. IfG˜1denotes the universal covering ofG1, we then haveG˜ ∼=RF1/FG˜1. By

the definition of Weil restriction we haveG(F )∼=G1(F1)andG(F )˜ ∼= ˜G1(F1);

and sinceGis isotropic, so isG1. Thus after replacingF byF1andGbyG1we

may assume thatGis absolutely simple.

For the next preparations note thatF is non-archimedean, soG(F )possesses an open compact subgroup. Its intersection withis an open compact subgroup of ; let us call it. Let˜denote the closure of the subgroup ofG(F )˜ that is generated by the set of generalized commutators [, ]∼.

We will study the relation between these subgroups and various Weil restric-tions ofG. Consider any closed subfieldF ⊂F such that [F /F] is finite. Note that in the case char(F )=0 there is a unique smallest suchF, namely the closure ofQ. But in positive characteristic the extensionF /Fmay be arbitrarily large and, what is worse, it may be inseparable.

SetG :=R_{F /F}GandG˜ :=R_{F /F}G˜, and letπ:G˜ → G be the homo-morphism induced byπ. From Proposition 1.6 we know thatπis not necessarily an isogeny. IdentifyingG(F )withG(F)via the universal property of the Weil restriction, we can viewas a non-discrete closed subgroup of finite covolume of G(F). Similarly, we can view˜as a subgroup ofG˜(F).

Lemma 2.3.˜is Zariski dense inG˜.

Proof. LetH⊂GandH˜⊂ ˜Gbe the Zariski closures ofand˜, respectively. By [11] Lemma 11.2.4 (ii) these groups are “defined overF”, i.e., smooth overF. The intersection ofwith the identity component ofHis open inand thus again an open compact subgroup of. After shrinkingwe may therefore assume that His connected. For anyγ ∈ the subgroupγ γ−1is again an open compact subgroup of, so it is commensurable with. Thusγ Hγ−1is commensurable withH. SinceHis connected, they must be equal; henceHis normalized by. It is therefore also normalized by the Zariski closure of.

Under the assumptions of 2.1, a theorem of Wang [12] implies that the Zariski closure ofinG contains all split tori ofG. Thus, in particular, it contains the images underπof all split tori inG˜. SinceGis simple isotropic, so isG; hence by˜ Theorem 1.17 these tori generateG˜. It follows thatHis normalized by the image ofG˜. By constructionH˜is the algebraic subgroup ofG˜that is generated by the image of the connected varietyH×F Hunder [, ]∼. It is therefore connected

and normalized byG˜.

Since is non-discrete, the group is not finite, and soH is non-trivial. LetH denote the image ofH×F F under the canonical adjunction morphism

G×F F → G. By constructionH is just the Zariski closure ofinG, so by

the above arguments in the caseF=F it is normalized by the image ofG˜. But π: G˜ →Gis surjective, soH is a non-trivial connected normal subgroup ofG. AsGis absolutely simple, this impliesH = G. AsGis perfect, it follows that

˜

All in all we now deduce thatH˜ is a non-trivial connected smooth normal algebraic subgroup ofG˜. By Theorem 1.10 this impliesH˜= ˜G, as desired.

Note that Lemma 2.3 in the caseF =F says that˜is Zariski dense inG.˜ In particularis compact and Zariski dense inG, so we can apply [5] Main The-orem 0.2. It follows that there exists a closed subfieldE⊂F such that [F /E] is finite, an absolutely simple and simply connected semisimple algebraic groupH˜ overE, and an isogenyϕ˜:H˜ ×EF → ˜Gwith non-vanishing derivative, such that

˜

is the image underϕ˜of an open subgroup ofH (E).˜ Lemma 2.4.E=F.

Proof. Via the universal property of the Weil restriction the isogenyϕ˜corresponds to a homomorphismϕ˜:H˜ →RF /EG, which satisfies˜

˜

⊂ ˜ϕ(H (E))˜ ⊂(RF /EG)(E)˜ = ˜G(F ).

By Lemma 2.3 in the caseF=Ewe know that˜is Zariski dense inR_{F /E}G. It˜ follows thatϕ˜is dominant. This implies

dimH˜ ≥dimR_{F /E}G˜ =[F /E]·dimG˜ =[F /E]·dimH˜; hence [F /E]=1, as desired.

Lemma 2.5.ϕ˜is an isomorphism.

Proof. Asϕ˜is an isogeny between simply connected groups, it is an isomorphism if and only if it is separable. In characteristic zero this is automatically the case. (Sincedϕ˜ = 0, this is actually true whenever char(F ) =2,3 (cf. [5] Thm.1.7), but we do not need that fact.) So for the rest of the proof we may suppose that p :=char(F )is positive. SetF := {xp | x ∈ F}; thenF /Fis an inseparable extension of degreep. Consider the induced homomorphism

˜ ψ:=R_{F /F}ϕ˜:R_{F /F}H˜ −→RF /FG.˜ By construction it satisfies ˜ ⊂ ˜ψ(RF /FH )(F˜ ) ⊂(RF /FG)(F˜ ) ˜ ϕH (F )˜ ⊂ G(F )˜ .

Since˜is Zariski dense inR_{F /F}G˜by Lemma 2.3, we deduce thatψ˜ is dominant. So for dimension reasons it is an isogeny. Proposition 1.6 (b) now shows thatϕ˜is separable, as desired.

Combining Lemmas 2.4 and 2.5, we now deduce that˜is open inG(F ). Thus˜ ˜

is open inG(F ), completing the proof of Theorem 2.1.˜

Acknowledgements. The author would like to thank Gopal Prasad and Marc Burger for interesting conversations, and especially the former for suggesting a combination of the methods of [8] and [5] to obtain another proof of strong approximation in arbitrary characteristic.

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