[PDF] Integration questions in separably good characteristics

Abstract

Let G be a reductive group over an algebraically closed field k of separably good characteristic p>0 for G. Under these assumptions a Springer isomorphism from the reduced nilpotent scheme of the Lie algebra of G to the reduced unipotent scheme of G always exists. This allows to integrate any p-nilpotent element of Lie(G) into a unipotent element of G. One should wonder whether such a punctual integration can lead to a systematic integration of p-nil subalgebras of Lie(G). We provide counterexamples of the existence of such an integration in general as well as criteria to integrate some p-nil subalgebras of Lie(G) (that are maximal in a certain sense). This requires to generalise the notion of infinitesimal saturation first introduced by P. Deligne and to extend one of his theorem on infinitesimally saturated subgroups of G to the previously mentioned framework.

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IIntegration questions in separably good characteristics

Marion JeanninFebruary 26, 2021

Abstract

Let G be a reductive group over an algebraically closed ﬁeld k of separably good charac-teristic p > G . Under these assumptions a Springer isomorphism φ : N red ( g ) → V red ( G )always exists, allowing to integrate any p -nilpotent elements of g into a unipotent elementof G . One should wonder whether such a punctual integration can lead to a systematicintegration of p -nil subalgebras of g . We provide counter-examples of the existence of suchan integration in general as well as criteria to integrate some p -nil subalgebras of g (that aremaximal in a certain sense). This requires to generalise the notion of inﬁnitesimal saturationﬁrst introduced by P. Deligne and to extend one of his theorem on inﬁnitesimally saturatedsubgroups of G to the previously mentioned framework. Let k be an algebraically closed ﬁeld and G be a reductive k -group. We denote by g its Liealgebra. When k is of characteristic 0, the classical theory comes with the well deﬁned expo-nential map which allows to integrate any nilpotent subalgebra u ⊆ g into a unipotent smoothconnected subgroup U ⊆ G (this means that Lie( U ) = u ). This deﬁnes an equivalence betweenthe category of k -nilpotent Lie algebras and the category of k -unipotent algebraic groups (seefor example [9, IV, §2, n ◦

4, Corollaire 4.5]).If now k is of characteristic p >

0, it is no longer possible to systematically associate aunipotent element of G to a nilpotent element of g . Such a punctual integration is ensured bythe existence of an isomorphism of reduced schemes: φ : N red ( g ) → V red ( G ), called a Springerisomorphism, between the reduced nilpotent variety of g and the reduced unipotent variety of G . Using [24] and [25] one can show that such an isomorphism always exists in separably goodcharacteristics for G (this has been observed by P. Sobaje in [32]). Unfortunately, this existenceis not suﬃcient to ensure a priori that any p -nil subalgebra can be integrated into a unipotentsmooth connected subgroup of G . This actually requires even stronger conditions on p : in [1, §6]V. Balaji, P. Deligne and A. J. Parameswaran show a result that implies the existence of suchan integration if p > h( G ) for h( G ) being the Coxeter number of G . Note that this integrationis then actually induced by the classical exponential map truncated at the p -power: in [23]G. McNinch indeed showed that if p > h( G ) then any p -nilpotent element has 1 as order of p -nilpotency. The authors attribute the result to J-P. Serre (see [31]). One could then expectthat such an integration would induce, as in the characteristic 0 framework, an equivalence ofcategories, this time between the category of p -nil Lie algebras and the category of unipotentalgebraic groups. This can actually been shown to break down and justiﬁes to introduce thenotion of inﬁnitesimal saturation deﬁned by P. Deligne in [8] and attributed to J-P. Serre by theauthor. Actually, if p > h( G ) the exponential map induces a bijective correspondence betweenthe p -nil Lie subalgebras of g and the unipotent algebraic subgroups of G . All this content isexplained in more details in subsection 3.1.In this article we focus on integration of p -nil subalgebras of g when p is separably good for G . As we will show in section 3 and 4 the fppf formalism introduced in [10, VIB proposition 7.1and remark 7.6.1] provides a way of associating a smooth connected unipotent subgroup J u ⊂ G to any p -nil subalgebra u ⊆ g . Unfortunately we will see that even if this subgroup is a naturalcandidate to integrate u it is in general too big, as one can only expect the following inclusion1 a r X i v : . [ m a t h . R T ] F e b o hold true u ⊆ j u := Lie( J u ). We actually provide in subsection 3.3 some counter-examples tothe existence of a general integration of p -Lie algebras under the separably good characteristicassumption.Notwithstanding this observation this technique still allow us to integrate some p -nil Liealgebras, as for example the p -radicals of Lie algebras whose normalisers are φ -inﬁnitesimallysaturated, for φ a Springer isomorphism for G . The notion of φ -inﬁnitesimal saturation here ex-tends the notion of inﬁnitesimal saturation when the punctual integration comes from a Springerisomorphism that is not necessarily the truncated exponential map (which happens for smallseparably good characteristics for G ).In section 4 we show how this extended notion together with the aforementioned fppf-formalism allows us to generalise a theorem of P. Deligne on the reduced part of inﬁnitesimallysaturated subgroups to the separably good characteristic framework. More precisely we showthe following statement: Theorem 1.1.

Let G be a reductive group over an algebraically closed ﬁeld k of characteristic p > which is assumed to be separably good for G . Let φ : N red ( g ) → V red ( G ) be a Springerisomorphism for G and let N ⊆ G be a φ -inﬁnitesimally saturated subgroup. Then:1. the subgroups N and Rad U ( N ) are normal in N . Moreover, the quotient N/N is a k -group of multiplicative type;2. in addition, if the reduced subgroup N is reductive, there exists in N a central subscheme M of multiplicative type such that if we denote by µ the kernel of M × N → N , then ( M × N ) /µ ∼ = N . Let us ﬁnally stress out that even if the existence of an integration seems to be satisﬁed onlyfor very speciﬁc and restrictive conditions on the restricted p -nil p -subalgebras, this still allowsto extend results classically known in characteristic zero to the characteristic p framework, suchas Morozov theorem (see [21], this also will be developed in more details in a future article).This is what motivated at ﬁrst the questions studied in this paper. Most of the content of thisarticle comes from the author’s PhD manuscript [21]. Let k be a ﬁeld of characteristic p > G be a reductive k group. We refer the reader to [34]for a deﬁnition and an exhaustive list of torsion characteristic for G , the notion of characteristicis discussed for example in [33, §0.3], a summary of all the conditions on the characteristicrelated to the studied group can be found in the preamble of [22] or in [14, §2]. We only recallhere some useful facts.In what follows k is assumed to be algebraically closed. When G is a semisimple k -groupthe following statement is a consequence of [22, theorem 2.2 et remark a)] : Corollary 2.1 (de [22, theorem 2]) . Let G be sa semisimple group over be an algebraically closedﬁeld k of characteristic p > which is not of torsion for G . Let u ⊆ g be a p -nil subalgebra (see6.3). Then there exists a Borel subgroup B ⊂ G such that u is a subalgebra of the Lie algebra of B , denoted by b . Remark 2.2.

1. The subalgebra u is actually contained in the unipotent radical of a Borelsubgroup B ⊆ G . Indeed b := Lie( B ) is nothing but the semidirect sum of rad u ( B ) (theLie algebra of the unipotent radical of B ) and t (the Lie algebra of a maximal torus of G ). This last factor being a torus it contains no p -nilpotent element (see for example thepreamble of the subsection 6.3) whence the inclusion u ⊂ rad u ( B ).2. The ﬁrst point of this remark actually allows to generalise the corollary to any reductive k -group G , for k an algebraically closed ﬁeld of characteristic p >

0, such that p is not a2orsion integer for G : let π : G → G := G/Z ( G ) = G/ Rad( G ) be the quotient map andset u := Lie( π )( u ). As rad( g ) is the Lie algebra of a torus, it has no nilpotent element,thus one has u ∼ = u . By what precedes there exists a Borel subgroup B ⊂ G such that u ⊆ rad u ( B ) ⊂ b . Let B = π − ( B ) be the preimage of B . As rad u ( B ) ∼ = rad u ( B ) onecan always assume that u is the subalgebra of the Lie algebra of the unipotent radical ofa Borel subgroup of b ⊆ g .We recall the notion of separably good characteristics, as deﬁned by J. Pevtsova and J. Starkin [27, deﬁnition 2.2] : when G is semisimple, the characteristic p is separably good for G if p isgood for G and the morphism G sc → G , for G sc the simply connected cover of G , is separable.When G is reductive, the characteristic p is separably good for G if it is separably good forthe derived group [ G, G ]. As underlined by the two authors if p is very good for G , it is alsoseparably good. Nevertheless, this last condition is restrictive for type A , which is the only typefor which very good and separably good characteristics do not coincide. fppf -formalism p -nilpotent elements: a starting point In what follows k is an algebraically closed ﬁeld of characteristic p > G is a reductive k -group. Denote by N red ( g ) the reduced nilpotent scheme of g ,namely, the reduced subscheme of g whose k -points are the p -nilpotent elements of g . Similarly denote by V red ( g ) reduced unipotentscheme of G , namely, the reduced subscheme of G whose k -points are the unipotent elementsof G . When G is a simply connected k -group and p is good for G , T. A. Springer establishesin [33, theorem 3.1] the existence of G -equivariant isomorphisms φ : N red ( g ) → V red ( G ) calledSpringer isomorphisms (we remind the reader that k is assumed to be algebraically closed here).This result has been studied and reﬁned by many mathematicians among those P. Bardsley andR. W. Richardson who extend in [2, 9.3.2] the existence of such isomorphisms to any reductive k -group which satisﬁes the standard hypotheses as deﬁned by J. C. Jantsen (see [20, deﬁnition2.11]). Let us also mention here the work of S. Herpel who shows in [14] the existence ofSpringer isomorphisms for any reductive k -groups in pretty good characteristics. This mainlyuses previous results from G. McNinch and D. Testerman (see [25, theorem 3.3]).Let us stress out that if G is any reductive k -group over an algebraically closed ﬁeld of char-acteristic p > h( G ) (for h( G ) being the Coxeter number of G ), J-P. Serre shows in [30, Part II,Lecture 2, Theorem 3] that there exists a unique Springer isomorphim exp : N red ( g ) → V red ( G )which is nothing but the classical exponential map truncated at the power p (as under thisassumption, any p -nilpotent element has p -nilpotency order equals to 1, see [23, 4.4 Corollary]).He moreover shows that for any Borel subgroup B ⊂ G this unique Springer isomorphism in-duces an isomorphism of algebraic groups exp B : ( rad u ( B ) , ◦ ) → Rad U ( B ) where rad u ( B ) is theLie algebra of the unipotent radical of B , endowed with the group structure induced by theBaker–Campbell–Hausdorﬀ law, and Rad U ( B ) is the unipotent radical of B . The reader mightalso a be referred to a recent article from V. Balaji, P. Deligne and A. J. Parameswaran [1,§6] for a detailed construction of this group isomorphism. This, combined with corollary 2.1actually allows to integrate restricted p -nil Lie subalgebras of g : Proposition 3.1.

Let G be a reductive k -group over an algebraically closed ﬁeld of characteristic p > h( G ) . Let u ⊂ g be a restricted p -nil p -Lie subalgebra. Then u can be integrated into asmooth, connected, unipotent subgroup of G (that is, there exists a smooth, connected, unipotentsubgroup U ⊂ G such that Lie( U ) ∼ = u as Lie algebras).Proof. According to [30, II, Lecture 2, Theorem 3], if p > h( G ) the Lie algebra of the unipotentradical of any Borel subgroup B ⊂ G is endowed with a group structure induced by the Baker–Campbell–Hausdorﬀ law (which has p -integral coeﬃcients as shown for example in [31, 2.2,propositon 1]). This law being deﬁned with iterated Lie brackets, it reduces to any subalgebraof rad u ( B ), endowing it with a group structure. As by assumption one has p > h( G ), the3haracteristic of k is not of torsion for G , thus there exists a Borel subgroup B ⊂ G such that u is a Lie subalgebra of rad u ( B ) (according to corollary 2.1). Hence what precedes in particularimplies that u is an algebraic group for the Baker–Campbell–Hausdorﬀ law.The isomorphism of groups exp b : rad u ( B ) → Rad U ( B ) deﬁned by J-P. Serre in [30, Part II,Lecture 2, Theorem 3] thus restricts to u . Denote by U the image of such restricted morphism.It is a smooth, connected unipotent subgroup of G .It remains to show that Lie( U ) ∼ = u . The notations used here are those of [10, II, Déﬁnition4.6.1]: let S be a scheme, for any O S -module F denote by W ( F ) the following contravariantfunctor over the category of S -schemes: W ( F )( S ) := Γ( S , F ⊗ O S O S )where Γ identiﬁes with the set of F ⊗ O S O S -sections over S . By [10, II, Lemme 4.11.7] one has u = W ( u ). In particular u is smooth and connected.The algebraic groups u and U being smooth, the isomorphism of algebraic groups (exp b ) | u induces an isomorphism Lie( U ) ∼ = Lie( u ) (see [10, VIIA proposition 8.2]). As u is a vector spaceover a ﬁeld one has Lie( u ) ∼ = u , hence Lie( U ) ∼ = u as Lie algebras. In other words the map exp b induces the identity on tangent spaces and the restricted p -nil p -subalgebra u thus integratesinto a smooth, connected unipotent subgroup U of G .Let us emphasize that contrary to what happens in characteristic 0 (see [9, IV,§2, 4.5])and as mentioned in the introduction, this integration does no longer induces an equivalence ofcategories, this time between the category of p -nil Lie algebras and the category of unipotentalgebraic groups, as shown in the following remark: Remark 3.2.

Let U be a unipotent subgroup of G and denote by u := Lie( U ) its Lie algebra.The ﬁeld k being algebraically closed, it is perfect, thus the subgroup U is k -embeddable intothe unipotent radical of a Borel subgroup B ⊂ G .Let log B : Rad U ( B ) → rad u ( B ) be the inverse isomorphism of algebraic groups of themorphism exp b , see [1, section 6] for an explicit construction. As for exp b , it is induced byisomorphism of reduced k -schemes log : V red ( G ) → N red ( g ).It is worth noting that in general one can not expect the integrated group exp( u ) to be thestarting group U . Equivalently, the equality log B ( U ) = u needs not being satisﬁed a priori. Thismeans that the truncated exponential map does not lead to a bijective correspondence betweenrestricted p -nil subalgebras of g and smooth connected unipotent subgroups of G .For example let G = SL and p >

3. We consider the the unipotent connected smoothsubgroups of G generated respectively by  t

00 1 00 0 1  and  t t p  . As U = U , necessarilythe restricted p -nil algebras log( U ) and log( U ) do not coincide, though the Lie algebras u = u being the same (it is the restricted p -nil p -Lie algebra generated by   .Finally, in [24, 10, Appendix], J-P. Serre shows that even if for smaller characteristics Springerisomorphisms are no longer unique, they all induce the same bijection between the G -orbits of N red ( g ) and those of V red ( G ).When G is simple, P. Sobaje recalls in G [32, Theorem 1.1 and Remark 2] that Springerisomorphisms exist for G , and that one can always ﬁnd such an isomorphism φ : N red ( g ) →V red ( G ) that restricts to an isomorphism of reduced schemes rad u ( B ) → Rad U ( B ). The authorthen precises that the diﬀerential of this restriction at 0 is a scalar multiple of the identity (thisdoes not depend on the considered Borel subgroup). He attributes this result to G. McNinch et D.Testerman (see [26, Theorem E]). This can be generalised to any reductive k -group in separablygood characteristic. Indeed the properties required for φ are preserved under separable isogeniesfor G (see [14, corollary 5.5] and [24, proposition 9]). This allows to consider semisimple groups4ather than simple one. The reductive case follows as “Springer isomorphisms are insensitive tothe center” (according to lemma 6.12 and remark 6.4 the radical of g is the Lie algebra of thecenter of G , thus is toral and does not contain any p -nilpotent elements). Note that for p < h( G )the Baker–Campbell–Hausdorﬀ law does no longer make sense, thus the aforementioned inducedmorphism rad u ( B ) → Rad U ( B ) is really an isomophism of schemes (and no longer of groupschemes, as there is no appropriate group law on rad u ( B ) here). Let G be a reductive group over an algebraically closed ﬁeld k of characteristic p > G . Let φ : N red ( g ) → V red ( G ) be a Springer isomorphismfor G such that for any Borel subgroup B ⊂ G the diﬀerential of φ restricted to rad u ( B ) is theidentity at 0 (this last assumption is allowed by [32, theorem 1.1 and remark 2] as explained inthe previous subsection 3.1). It deﬁnes, for any p -nilpotent element of g , a t -power map φ x : G a → Gt φ x ( t ) . Let u be a restricted p -nil p -Lie subalgebra of g . The t -power map then induces the followingmorphism: ψ u : W ( u ) × G a → G ( x, a ) φ x ( a ) , where the notations are those of [10, I.4.6], see also [10, II, lemme 4.11.7]. Denote by J u thesubgroup of G generated by ψ u as fppf-sheaf (see [10, VIB proposition 7.1 and remark 7.6.1]).This is a connected subgroup by [10, VIB Corollaire 7.2.1] as W ( u ) is geometrically reduced andgeometrically connected ; and reduced (hence smooth since k is algebraically closed), as it is theimage of a smooth (hence reduced) k -group. We will see in section 4 that it is also unipotent.One thus needs to compare u with the Lie algebra of J u , denoted by j u . We will show thatwhen the p -restricted p -nil Lie subalgebra u ⊆ g satisﬁes some maximality properties (as the onerequired in the statements of lemmas 5.1 and 5.3) it is integrable and actually integrated by J u .Before going any further let us stress out that when this integration holds true the normalisers N G ( J u ) and N G ( u ) turn out to be the same. More precisely: Lemma 3.3.

When J u integrates u (which in particular holds true when u is a subalgebra of g made of the p -nilpotent elements of the radical of N g ( u ) ), then N G ( J u ) = N G ( u ) .Proof. By lemma 3.4 one only needs to show the inclusion N G ( J u ) ⊆ N G ( u ) which is directaccording to lemma 6.5 as the equality Lie( J u ) = u is satisﬁed by assumption. Lemma 3.4.

The subgroup N G ( u ) normalises J u .Proof. First remind that φ being a Springer isomorphism it is G -equivariant, thus so is φ x .Hence the morphism ψ u is compatible with the G -action on u : in other words, for any g ∈ G and any ( x, t ) ∈ u × G a one has the following equality Ad( g ) ψ u ( x, t ) = ψ u (Ad( g ) x, t ).Let R be a k -algebra, for any g ∈ N G ( u )( R ) and h ∈ J u ( R ). By deﬁnition of J u there existsan fppf-covering S → R such that h S = ψ u ( x , s ) × · · · × ψ u ( x n , s n ) for x i ∈ u R ⊗ R S and s i ∈ S .But then one has(Ad( g ) h ) S = n Y i =1 Ad( g S ) ψ u ( x i , s i ) = n Y i =1 ψ u (Ad( g S ) x i , s i ) ∈ J u ( S ) ∩ G ( R ) = J u ( R ) , where the equality J u ( S ) ∩ G ( R ) = J u ( R ) follows from the fact that J u is generated by ψ u as afppf-sheaf. The second equality comes from the G -equivariance of φ x . Thus we have shown thatAd( g ) h ∈ J u ( R ) for all g ∈ N g ( u )( R ), in other words N G ( u )( R ) ⊆ N G ( J u )( R ) for any k -algebra R . Yoneda’s lemma then leads to the desired inclusion: N G ( u ) ⊆ N G ( J u ).5 .3 A counter-example to the existence of an integration for embedded re-stricted p -nil p -subalgebras Let G be a reductive group over an algebraically closed ﬁeld of characteristic p which is assumedto be separably good for G . In this section we provide an example for which a morphism ofintegrable restricted p -nil p -Lie subalgebras of g is not integrable into a morphism of unipotentgroups of G . In other words let u and v are two restricted p -nil p -Lie subalgebras of g that areintegrable into unipotent subgroups of G (denoted respectively by U and V ), if f : u → v isa morphism of p -Lie algebras, it is not true in general that there exists a morphism of groups U → V such that Lie( φ ) = f , namely the map Hom( G, G a ) → Hom p − Lie ( g , k ) is not surjective.This will in particular illustrates that a restricted p -nil p -subalgebra of an integrable restricted p -nil p -Lie algebra is not necessarily integrable.Before going any further, let us stress out that only the restricted p -nil p -subalgebras h ⊂ g can pretend to derive from an algebraic group (as this last property automatically imply that h is endowed with a p -structure inherited from the group see for example [9, II, §7, n ◦

3, Propo-sition 3.4]). Moreover, as underlined by the example presented in remark 3.2 the integration ofrestricted p -nil p -subalgebras of g does no longer induce a bijective correspondence with unipo-tents subgroups of G . This in particular implies that the integration of morphisms of p -restrictedLie algebras depends on the integration of the Lie algebras one start with.The following lemma make a connection between integration of morphisms and integrationof subalgebras: Lemma 3.5.

Let G and H be two algebraic smooth k -groups with respective Lie algebras g :=Lie( G ) , respectively h := Lie( H ) . Assume that f : g → h is an integrable morphism of restricted p -nil p -Lie algebras and denote by φ : G → H the resulting integrated morphism. Then f ( g ) isintegrable into an algebraic smooth connected k -group.Proof. Denote by v := f ( g ) the image of the morphism f which is assumed to be integrable intoa smooth morphism φ : G → H . One can a priori only expect the following inclusions to holdtrue f ( g ) ⊆ Lie( φ ( G )) ⊆ h , but as k is a ﬁeld and since the morphism φ is smooth as well as thegroups H and G the morphism f = Lie( φ ) : g → h is surjective (see [9, II, §5, proposition 5.3]),whence the following equalities v = h = Lie( H ). In particular the Lie algebra v is integrableinto an algebraic smooth connected k -group. Remark 3.6.

Let φ : G → H be a smooth morphism of algebraic k -groups. Assume that thederived morphism Lie( φ ) : g → h has a splitting s : h → g which is also a morphism of restricted p -Lie algebras. It is worth noting that this splitting does not necessarily lift to a smooth splitting σ : H → G of algebraic groups: according to [13, Corollaire 17.12.3] the integrated morphismmight only be a quasi-regular immersion as φ is smooth.This, in particular, tells us that even if G and H are smooth there is a priori no good reasonsfor s to be integrable. Moreover, if s is integrable into a smooth morphism σ : H → G , thenthe induced exact sequence of restricted p -Lie algebras is trivial: indeed set n := Ker( f ), as k isa ﬁeld, if σ is smooth its derived morphism is surjective, hence we have dit Lie( σ ( h )) = g and n = 0. Example 3.7.

Let k be a perfect ﬁeld of characteristic p > G a W G a , G a W = ( W × G a ) / h i ( x ) − i ( x p ) | x ∈ G a i G a . i ri r Frob = where Frob is the absolute Frobenius automorphism and the central term of the lower sequenceis the pushout of the morphisms i and Frob. The group G a being smooth, the exactness of the6wo horizontal sequences is preserved by derivation des deux suites horizontales est préservéepar dérivation (see [10, II, §5, proposition 5.3] and [9, II, §7, n ◦

3, proposition 3.4]). This leadsto the following commutative diagram of restricted p -Lie algebras and p -morphisms:0 Lie( G a ) w := Lie( W ) Lie( G a ) 0 , G a ) w Lie( G a ) 0 . Lie( i ) Lie( r )Lie( i ) Lie( r )Lie(Frob) = 0 π = As Lie(Frob) = 0 the p -morphism w → k is split (as a p -morphism). Let s : Lie( G a ) → w bethe resulting splitting.Even though Lie( G a ) and w are integrable, this splitting does not lift into a morphism ofalgebraic k -groups. The ﬁeld k being perfect one only needs to check it on k points. The verticalmorphisms inducing the identity morphism on k -points, if the lifting s : Lie( G a ) → w wereintegrable into a morphism of algebraic groups σ : G a ( k ) → W ( k ) such that Lie( σ ) k = s k , thelower exact sequence of the above commutative diagram of algebraic groups would be split (theLie-functor being left exact). According to the previous remark on k -points, the following exactsequence would then also be split:0 G a ( k ) | {z } = k W ( k ) k G a has p -torsion while W has none (seefor example [9, 5, §1, n ◦

1, corollaire 1.8]).

Remark 3.8.

Let u ⊂ g be a restricted p -nil p -subalgebra which is integrable into a unipotentsmooth connected subgroup U ⊆ G . This example together with the lemma 3.5 also shows thatnot any restricted p -nil p -subalgebra v ⊆ u of a restricted p -nil p -Lie algebra is integrable into asmooth connected unipotent group V tel que Lie( V ) = v . φ - inﬁnitesimal saturation and proof of theorem 1.1 φ -inﬁnitesimal saturation In what follows G is a reductive group over an algebraically closed ﬁeld k of characteristic p > G . Let φ : N red ( g ) → V red ( G ) be a Springerisomorphism for G , that is a G -equivariant morphism of reduced k -schemes. Deﬁnition 4.1.

A subgroup G ⊆ G is φ -inﬁnitesimally saturated if for any p -nilpotent element x ∈ g := Lie( G ) the t -power map: φ x : G a → G,t φ ( tx ) , factorises through G . In other words we ask for the following diagram to commute: G a G,G . φ x ∃

7t follows from the deﬁnition that the group G is itself φ -inﬁnitesimally saturated. Let usstress out that there non trivial examples of φ -inﬁnitesimally saturated subgroups of G namely: Lemma 4.2.

Any parabolic subgroup of G is φ -inﬁnitesimally saturated.Proof. In order to show this result we make use of the dynamic method introduced in [6, 4]and [7, §2.1]. Let T ⊂ P ⊂ G be respectively a maximal torus and a parabolic subgroup of G .As k is a ﬁeld there exists λ : G m → G a non-necessarily unique cocharacter of T such that P = P G ( λ ) (see [7, Proposition 2.2.9]). We aim to show that for any p -nilpotent element x ∈ p the image of the t -power map φ x belongs to P = P G ( λ ). The ﬁeld k being algebraically closed,this is enough to show it on k -points. As a reminder, the k -points of P when P is of the form P G ( λ ) is nothing but the set P G ( λ )( k ) = { g ∈ G ( k ) | lim s → λ ( s ) · g exists } , hence one only needs to prove that lim s → λ ( s ) · φ ( tx ) exists. This can be done by making useof the G -equivariance of φ : this leads to the following equality λ ( s ) · φ ( tx ) = φ ( λ ( s ) · tx ) , andsince x ∈ p g ( λ ) := Lie( P G ( λ )), the limit lim s → λ ( s ) · x exists by deﬁnition. We deduce from theabove equality that lim s → λ ( s ) · φ ( tx ) exists so φ ( tx ) ∈ P G ( λ ) = P , whence the result. Remark 4.3.

When p > h( G ) J-P. Serre shows in [29, Part 2, Lecture 2, Theorem 1.3] thatthere actually exists a unique Springer isomorphism for G which is nothing but the classicalexponential map truncated at the power p (as it follows from [23, 4.4, Corollary] that any p -nilpotent element x ∈ g satisﬁes the following identity x [ p ] = 1 when p > h( G )). In thisframework, being φ -inﬁnitesimally saturated is nothing but being exp-saturated, i.e. beinginﬁnitesimally saturated as deﬁned by P. Deligne in [8, Déﬁnition 1.5]).This being introduced we can show the following lemma which states that the generatedsubgroup J u seems to be the good candidate to integrate u in general. Lemma 4.4.

Let u ⊆ g be a restricted p -nil p -sub-algebra. Then the generated subgroup J u isunipotent and one has the following inclusion: u ⊆ Lie( J u ) := j u . Proof.

The Lie algebra u being a restricted p -nil p -sub-algebra of g and p being separably good for G (thus not of torsion) the corollary 2.1 allows to embed u into the Lie algebra of the unipotentradical of a Borel subgroup B ⊂ G . The subgroup J u obtained by considering the fppf-sheafgenerated by the image of the morphism induced by the t -power map is thus k -embeddableinto the unipotent radical of a Borel subgroup (as B is φ -inﬁnitesimally saturated according tolemma 4.2). In other words J u is unipotent (see for example [9, IV, §2, n ◦

2, proposition 2.5(vi)]).We still denote by φ the restriction of the Springer isomorphism to rad u ( B ). Remind that(d φ ) = id by assumption. The subgroup J u being generated by the images of the t -power maps φ x for all x ∈ u , the Lie algebra j u contains the diﬀerential in 0 of all such maps, hence theexpected inclusion.It will be shown in section 5 that J u does actually integrate u when the latter satisﬁes somemaximality hypotheses.The notion of φ -inﬁnitesimal saturation introduced here also allows us to generalise theo-rems [8, Théorème 1.7] and [1, Theorem 2.5] to φ -inﬁnitesimally reductive k -groups N over analgebraically closed ﬁeld k of characteristic p > G .This is the point of theorem 1.1. Let us ﬁrst remark that points (i) and (iii) of [8, Lemme 2.3]are still valid in the aforementioned framework and allows us to reduce ourselves to show theresult for connected N . More precisely: 8 emma 4.5.

1. If N is φ -inﬁnitesimally saturated in G then so is N ,2. if the reduced part N of N and its unipotent radical Rad U ( N ) are normal subgroupsof N then they are normal in N .Proof. See [8, Lemme 2.3] for a proof as the notion of φ -inﬁnitesimal saturation is nothing but ageneralisation of those of inﬁnitesimal saturation to the framework described above (see remark4.3).In what follows the φ -inﬁnitesimally saturated group N is thus assumed to be connected. Let H ⊆ N be a maximal connected subgroup of multiplicative type of N . The k -group H isthe direct product of a k -torus T together with a diagonalisable k -group D which is a product ofsubgroups of the form µ p i , with i ∈ N . Moreover the k -torus T is the nothing but the intersection H ∩ N red and it is a maximal torus of N and N red .Let Z := Z N red ( T ) be the connected centraliser of T for the action of N red and set W = Z/T .This is a unipotent subgroup of N (the reasoning is the same as the one of [8, §2.5]): accordingto [11, XVII, Proposition 4.3.1 iv)], the ﬁeld k being algebraically closed one only needs to showthat this quotient has no subgroup of µ p -type. But this is clear: if such a factor would exist itsinverse image in Z would be an extension of µ p by T in N red , hence of multiplicative type. Butthis is absurd as the maximal connected subgroups of multiplicative type of a smooth algebraicgroup over a ﬁeld are the maximal tori (by 6.1). Moreover, as T and Z are smooth so is W according to [9, II,§5, n ◦ k being perfect [11, exposé XVII, théorème 6.1.1] holds true and implies the exactnessof the following exact sequence:1 T Z W . To summarize we have an isomorphism Z N red ( T ) ∼ = T × W . Let X be the reduced k -sub-schemeof p -nilpotent elements of n = W ( n ) = Lie( Z N red ( T )). Lemma 4.6.

The centraliser Z N red ( T ) is the subgroup of N generated by T and the morphism ψ X : X × G a → G ( x, t ) φ ( tx ) . It is normalised by H .Proof. Let J X ⊆ G be the subgroup of N generated by the image of ψ X . The subgroup N being φ -inﬁnitesimally saturated, the t -power map induced by φ maps any p -nilpotent elementof Lie( Z N ( T )) to N . Thus ψ X factorises through N . Moreover J X being the image of a reduced k -scheme it is reduced, hence smooth (as k is algebraically closed). Thus the inclusion J X ⊂ N red holds true. Finally, as φ is a Springer isomorphism, it is G -equivariant. This implies that theimage of ψ X commute with any element of T . We just have shown that J X ⊂ Z N red ( T ).Let also E T,J X be the subgroup generated by T and J X as a fppf-sheaf. It is a smoothsubgroup of Z N red ( T ) (as T ∈ Z N red ( T ) is smooth and as J X ⊂ Z N red ( T ) by what precedes),which is also connected (by [10, VIB. corollaire 7.2.1] the torus T being geometrically connectedand geometrically reduced). Thus it is actually contained in the identity component of thereduced centraliser. At the Lie algebras level this leads to the following inclusion Lie( E T,J X ) ⊆ Lie( Z N red ( T )) = Lie( Z N red ( T )). 9s W is a unipotent subgroup of Z N red ( T ) ∼ = T × W , the Lie algebra w := Lie( W ) is arestricted p -nil p -sub-algebra of Z N red ( T ), hence is contained in the reduced sub-scheme X . Thelatter being the set of p -nilpotent elements of Lie( Z N ( T )) it is contained in the set of all p -nilpotent elements of g , which coincide with rad p ( g ) by lemma 6.20 (which holds true as here p ≥ p = 2 the conditions deﬁned in remark 6.16 are satisﬁed). As p is not of torsionfor G , corollary 2.1 holds true and allows to embed rad p ( g ), thus X , into the Lie algebra ofthe unipotent radical of a Borel subgroup B ⊆ G . Remember that the diﬀerential in 0 of therestriction of φ to this sub-algebra satisﬁes (d φ ) = id. The group J X being generated by theimage of ψ X , this property ensures that the diﬀerential at 0 of any φ ( tx ), for any t ∈ G a and x ∈ X belongs to j X := Lie( J X ) = Lie( J X ) . In other words one has the following inclusions w ⊆ X ⊆ Lie( J X ). Moreover the inclusion T ⊆ E T,J X , induces an inclusion of Lie algebras t := Lie( T ) ⊆ Lie( E T,J X ) = Lie( E T,J X ).As one has Z N red ( T ) ∼ = T × W , what precedes leads to the following inclusionLie( Z N red ( T )) = Lie( Z N red ( T )) ⊆ Lie( E T,J X ) , thus to the equality Lie( Z N red ( T )) = Lie( E T,J X ). As the groups involved here are smooth andconnected, this equality of Lie algebras lifts to the group level according to [9, II, §5 n ◦ Z N red ( T ) = E T,J X .It then remains to show that E T,J X is normalised by H . Reminds that it is the subgroupgenerated by T ⊆ H (which is normal in H ) and J X (which is characteristic in J X , see [9,II, §5, n ◦ J X is H -stable. First remark that X isstabilised by H (as the latter stabilises Lie( Z N ( T )) and the p -nilpotency is preserved by theadjoint action). The G -equivariance of φ (thus its H -equivariance) then allows to conclude: let R be a k -algebra, for any j ∈ J X ( R ) and h ∈ H ( R ), there is an fppf-covering S → R such that j S = ψ X ( x , t ) × · · · × ψ X ( x n , t n ) where x i ∈ X R ⊗ R S and s i ∈ S . But then(Ad( h ) j ) S = n Y i =1 Ad( h S ) ψ X ( x i , s i ) = n Y i =1 ψ X (Ad( h S ) x i , s i ) ∈ J X ( S ) ∩ G ( R ) = J X ( R ) , and by Yoneda lemma E T,J X is stable under the H -action. Lemma 4.7.

The restricted p -Lie algebra n red := Lie( N red ) is an ideal of n acted on by H .Proof. According to the proof of [1, lemma 2.14] the morphism of k -schemes N red × H → N is faithfully ﬂat. This being said N appears as the fppf-sheaf generated by N red and H . Thusone only needs to show that n red is H -stable, to show that n red is actually N -stable. The torus T = H ∩ N red acts on N red , respectively on N , leading to the following decompositions: n red = Lie( N red ) = Lie( Z N red ( T )) ⊕ M α ∈ X ( T ) ∗ n α red , and Lie( N ) = Lie( Z N ( T )) ⊕ M α ∈ X ( T ) ∗ n α , where X ( T ) ∗ stands for the group of non trivial characters of T . The torus T being normalin H , any factor in the decomposition of Lie( N ) is stable for H . We need to show that so isany factor of the decomposition of n red . Let us ﬁrst study positive weight spaces. The group N being generated as a fppf-sheaf by N red and the subgroup of multiplicative type H (whose Liealgebra is toral) the p -nilpotent elements of Lie( N ) are the p -nilpotent elements of Lie( N red ).This being observed, as for any α = 0 the weight space n α has only p -nilpotent elements (sincewe consider the action of a torus here) the following equality n α red := n α ∩ n red = n α is satisﬁed,whence the desired H -stability.It remains to show that Lie( Z N red ( T )) is H -stable. According to lemma 4.6 the subgroup H normalise Z N red ( T ) , thus the stability of Lie( Z N red ( T ) ) = Lie( Z N red ( T )).According to what precedes n red is stable for the action of H on n , hence this sub-algebra isinvariant for the action of N . Reasoning on the R [ (cid:15) ]-points for any k -algebra R , one can showsthat n red is an ideal of n . 10he proof of the following lemma is the same as the proof of [8, lemma 2.22] as relaxing thehypotheses had no consequences on the involved arguments. We reproduce the proof here toensure a consistency in notations. Lemma 4.8 (P. Deligne, [8, 2.22]) . Let V be the unipotent radical of N red . The action of H on Lie( N red ) = n red left Lie( V ) := v invariant.Proof. The torus T acts on n red thus on v . The Lie algebras n red and v have a weight spacedecomposition for this action, namely n red = n ⊕ L α ∈ X ( T ) ∗ n α red and v = v ⊕ L α ∈ X ( T ) ∗ v α .According to the proof of lemma 4.7 the decomposition of n red is H -stable. It remains to showthat so is any v α .Consider the following commutative diagram (as a reminder as Z = T × W and T and Z arenormal in N red , so is the subgroup W ⊆ N red , which is also unipotent, smooth and connected,hence contained in V := Rad U ( N red )) :1 W Z T Q , V N red Q. Let us ﬁrst study the H -stability of the weight-zero part of v . The diagram above being cartesianone has v = n ∩ v = z ∩ v = w . But w is H -stable as the subgroups T and Z are (for Z thishas been shown in lemma 4.6) and the sequence is split.Let us now focus on the positive weights, let q be the Lie algebra of the reductive quotient N red /V . The torus T acts on this Lie algebra which writes q = q ⊕ L α ∈ X ∗ ( T ) q α . There aretwo possible situations:• either α is not a weight of T on q . Then one has v α = n α red whence the H -stability of v α ;• or α is a non trivial weight of T on q . Then the weight spaces q α and q − α are of dimension1 (according to [12, XIX, proposition 1.12 (iii)]). As p > G ), the pairing q α × q − α → q := Lie( T Q )( X α , X − α ) [ X α , X − α ]induced by the bracket on q is non-degenerate (see [12, XXIII, corollaire 6.5]), thus mapsto a 1-dimensional subspace h α .Likewise, the bracket on n red induces a non-degenerate pairing of n α red and n − α red , and onehas the following commutative diagram: n α red × n − α red n , q α × q − α n / w ∼ = t q = t / w . Denote by d the image of the pairing of n α red and n − α red composed with the projection n → n / w . According to what precedes this is a line of n / w .The situation can be summarized in the commutative diagram below:11 v ± α n ± α red q ± α , Hom( n ∓ α red , d ) . In other words one has v ± α = ker (cid:16) v ± α → Hom( n ∓ α red , d ) (cid:17) and v ± α is a sub-representationof the representation deﬁned by the action of H on n red , thus it is H -stable.Combining lemmas 4.7 and 4.8 one can show an inﬁnitesimal version of [1, Théorème 2.5],namely: Proposition 4.9.

Let G be a reductive group over an algebraically closed ﬁeld k of characteristic p > which is assumed to be separably good for G . Let φ : N red ( g ) → V red ( G ) be a Springerisomorphism for G . If N ⊆ G is a φ -inﬁnitesimally saturated subgroup, then:1. the Lie algebra n red is an ideal of n ,2. the Lie algebra of the unipotent radical of N red is an ideal of n .Proof. The ﬁrst point of proposition is provided by lemma 4.7. The second point follows from adirect application of lemma 4.8 combined with [1, lemma 2.14]: the subgroup N being generatedas an fppf-sheaf by H and N red , one only needs to show that rad u ( N red ) is H -stable. This hasbeen shown by the aforementioned lemma. A reasoning on R [ (cid:15) ]-points, for any k -algebra R thenleads to the desired result, namely that rad u ( N red ) is an ideal of n . We can now show theorem 1.1.We start by showing that N red is a normal subgroup of N . The latter being generated by H and N red as an fppf-sheaf, one actually needs to show that N red is H -stable. The reasoningfollows the proof of lemma 4.7: we consider the subgroup E Z ,J n α generated by Z := Z N red ( T )and J n α , for α ∈ X ( T ) ∗ , these being themselves the subgroups generated as fppf-sheaves by theimage of the morphisms ψ α : W ( n α ) × G a → G ( x, t ) φ ( tx ) . Note that ψ α is well deﬁned for any α ∈ X ( T ) ∗ : any weight space n α consists in p -nilpotentelements (as we consider the action of a torus), they are geometrically reduced and geometricallyconnected (as the n α are vector spaces). Thus the groups J n α are smooth and connected (thislast point being ensured by [10, VIB, corollaire 7.2.1]).The arguments of the proof of lemma 4.6 apply and allow to show that the k -subgroup E Z ,J n α is contained in N (this subgroup being φ -inﬁnitesimally saturated), and even in N red asit is smooth. Moreover, as p ≥ p -nil, these latter are all embeddable into the Lie algebra of the unipotent radical of aBorel subgroup B ⊆ G . The diﬀerential at 0 of the restriction of φ to the Lie algebra of theunipotent radical of any Borel subgroup being the identity the weight spaces n α are all containedin Lie( E Z ,J n α ) =: e . The Lie algebra Lie( Z ) also satisﬁes this inclusion as Z ⊂ E Z ,J n α .To summarize one has n red = Lie( Z ) ⊕ L α ∈ X ( T ) ∗ n α ⊆ e . The group involved here beingsmooth and connected, the equality of the Lie algebras lifts to an equality of groups (see [9, II,§5, n ◦ E Z ,J n α = N . 12hus the problem restricts to showing the H -stability of E Z ,J n α . By lemma 4.6 the cen-traliser Z is H -invariant, thus one only has to show the H -stability of the J n α ’s. As H normalise T and φ is G -equivariant, any n α is H -invariant. Thus N red is a normal subgroup of N .Remind that in the preamble in subsection 4.2 we have explained that H actually writes H = T × D (for D a k -diagonalisable group). To prove that N/N red is of multiplicative type weshow that it is isomorphic to the group D . As H normalise N red (which is normal in N ) and as HN red = N and N red ∼ = H ∩ N red , one has an isomorphism of fppf-sheaves which turns out tobe an isomorphism of algebraic groups: H/H red ∼ = N/N red ∼ = D. To end the proof of the ﬁrst point of the theorem it remains to show that the unipotent radicalof N red , denoted by V , is normal in N . Once again, the fppf-formalism, reduces the problemto showing the H -invariance of V (the unipotent radical Rad U ( N red ) being normal in N red ).The reasoning follows the proof of the normality of N red in N : we consider the subgroup E W,J α v generated by W and J v α , for α ∈ X ( T ) ∗ . As W and J v α are normal in N red , the subgroup E W,J v α is a unipotent smooth connected normal subgroup of N , thus it is contained in the unipotentradical of N red .Moreover, for any non zero weight, the corresponding weight space can be embedded intothe Lie algebra of the unipotent radical of a Borel subgroup (as p ≥ p -nil). Once again we make use of the properties of the diﬀerential of φ at 0 to concludethat v = w ⊕ L α ∈ X ( T ) ∗ v α is contained in Lie( E W,J α v ). This implies the equality V = E W,J α v for the same reasons as above. This equality being satisﬁed the result follows from stabilityproperties established in the proof of lemma 4.8. Indeed we have shown then that W , as well asany J α v for non trivial α , is H -stable. Combining this with the G -equivariance of φ leads to theconclusion that V is a normal subgroup of N .It remains to show the last point of Theorem 1.1, which is a generalised version of [8, theorem1.7 iii)] (see also [1, theorem 2.5 ii)]). A careful reading of the proof of these two statementsshows that it does not depends on the additional hypotheses made by the authors. The proofis hence the same as the one provided by P. Deligne in the framework of [8, 2.25] (see also [1,corollary 2.15]) and is reproduced here only for sake of clarity.The reduced part N red ⊆ N is thus now assumed to be reductive. We show that theconnected component of the identity M of M = ker( H → Aut( N red ) is the central connectedsubgroup of multiplicative type we are seeking. It is clearly of multiplicative type as it is aclosed subgroup of H (see [9, IV §1 Corollaire 2.4 a)]). Thus we need to show that it is centraland that M × N red → N is an epimorphism. The ﬁrst assertion is clear as M centralises N red and N is generated by H and N red as a fppf-sheaf (as shown previously). To show that M × N red → N is an epimorphism, one proves that N is generated by M and N red as afppf-sheaf. We already know that it is generated by N red and H . To conclude we show that M is generated by M red ⊂ N red and M and that H is generated by M and T .The assertion for M is the consequence of structural properties of groups multiplicative type:the ﬁeld k being algebraically closed groups of multiplicative type are diagonalisable, hence M is a isomorphic to a product of G m , µ q and µ p i (for ( q, p ) = 1) (see the proof of [10, VIII,proposition 2.1]). Its reduced component being smooth of multiplicative type, the order of itstorsion part is coprime with p (see [10, VIII, Proposition 2.1], hence M/M red is a product ofgroups of the form µ p i for i ∈ N . Conversely, the quotient M/M is a product of µ q with( p, q ) = 1, hence the result.One still have to show that H is generated by M and T . Remind that we have shownpreviously that N red is stable under the action of H -conjugation on N . Note that this actionﬁxes T , hence we have the following diagram (see [12, XXIV, Proposition 2.11], the group N red being reductive): 13 Aut( N red ) ,T Ad N Ad red . Thus the action of H on N red factors through T Ad . Thus we have the following exactsequence: 1 M H T Ad . Hence H is generated as a fppf-sheaf by M and T Ad , whence by M and T as T Ad is a quotientof T . p -nil p -sub-algebras g Let us start with the very speciﬁc case which has motivated our interest in the questions studiedin this article: assume u ⊆ g to be a restricted p -sub-algebra which is the set of p -nilpotentelements of rad( N g ( u )). Note that N g ( u ) is a restricted p -algebra (as it derives from an algebraic k -group, namely N G ( u ) according to lemma 6.6). Moreover u is a restricted a p -nil p -sub-algebraof g . Lemma 5.1.

Let G be a reductive group over a ﬁeld k of characteristic p > which is assumedto be separably good for G and let u ⊆ g be a sub-algebra. If u is the set of p -nilpotent elementsof the radical of its normaliser in g , denoted by N g ( u ) , then the sub-algebra u is integrable.Proof. According to lemma 4.4, there is a unipotent smooth connected subgroup J u ⊂ G suchthat the following inclusion holds true u ⊆ j u := Lie( J u ). Moreover, as according to lemma 6.5one has J u ⊆ N G ( J u ) ⊆ N G ( j u ), at the Lie algebra level the following inclusions hold true: u ⊂ j u ⊆ Lie( N G ( J u )) ⊆ N g ( j u ) . Assume the inclusion u ⊂ j u to be strict. Then u is a proper sub-algebra of its normaliser in j u (this is a corollary of Engel theorem, see for example [5, §4 n ◦ u (cid:40) N j u ( u ) := j u ∩ N g ( u ). But J u is normalised by N G ( u ) according to lemma3.4, hence N j u ( u ) ⊆ N g ( u ) is an ideal of N g ( u ). It is a restricted p -algebra (as it derives from analgebraic group according to lemma 6.6), a restricted p -ideal (as the restriction of the p -structureof N g ( u ) coincide with the one inherited from N J u ( u )), and even a p -ideal p -nil (as j u is p -nilaccording to lemma 6.19). In particular, it is a solvable ideal of N g ( u ) whence the inclusion N j u ( u ) ⊆ rad( N g ( u )). To summarize, the set u of p -nilpotent elements of rad( N g ( u )) is containedin a p -nil ideal of this radical, namely N j u ( u ), hence it is equal to the latter. This contradictsthe strictness of the inclusion, thus one as u = j u . This in particular means that there exists aunipotent, smooth, connected subgroup J u ⊆ G such that Lie( J u ) = u , which means that u isintegrable.Let now h ⊆ g be a sub-algebra, and denote by u the p -radical of N g ( h ). The p -radical of N g ( h ) being a restricted p -nil p -ideal, the preamble of this section allows to associate to u aunipotent, smooth, connected subgroup J u ⊂ G . Lemma 5.2.

The subgroup N G ( h ) normalises J u .Proof. One only needs to apply verbatim the proof of lemma 3.4 as by assumption u is an idealof N g ( h ). 14 emma 5.3. Let G be a reductive group over a ﬁeld k of characteristic p > which is assumedto be separably good for G and let h ⊆ g be a sub-algebra such that the normaliser N G ( h ) is φ -inﬁnitesimally saturated. If u := rad p ( N g ( h )) then the sub-algebra u is integrable.Proof. Remind that one has the following inclusion u ⊆ j u according to lemma 4.4. More-over N G ( h ) being φ -inﬁnitesimally saturated, the group J u is a subgroup of N G ( h ) (as u :=rad p ( N g ( h )) is a restricted p -nil p -ideal, the t -power map of any p -nilpotent element of u factorsthrough N G ( h )). At the Lie algebra level this leads to: u ⊂ j u ⊆ Lie( N G ( h )) = N g ( h ) . Assume the inclusion u (cid:40) j u to be strict. Then u is a proper sub-algebra of its normaliserin j u (according to [5, §4 n ◦ u (cid:40) N j u ( u ) := j u ∩ N g ( u ) = j u ∩ N j u ( h ). But according to lemma 5.2 the subgroup J u is normalised by N G ( h ), hence N j u ( u ) ⊆ N g ( h ) is an ideal of N g ( h ). The same arguments of the one developed in the proof of5.1 allows us to show that it is a restricted p -nil p -ideal of N g ( h ) such that N j u ( u ) ⊆ rad( N g ( h )).This leads to the equality N j u ( h ) = u as u is nothing but the set of p -nilpotent elements ofrad( N g ( h )). This contradicts the strictness of the inclusion, whence the equality u = j u . Thisin particular means that u is integrable. Remark 5.4.

In the particular case where h = u , namely when u := rad p (N g ( u )) is the p -radicalof its normaliser in g , the φ -inﬁnitesimal saturation is superﬂuous as in this case the inclusion J u ⊆ N G ( J u ) is clear. k -groups of multiplicative type Corollary 6.1 (of [7, Proposition A.2.11]) . Let k be a ﬁeld and G be an aﬃne smooth algebraic k -group. The maximal connected subgroups of multiplicative type of G are the maximal tori of G .Proof. Without loss of generality one can assume G to be connected (as any maximal connectedsubgroup of G is contained in the identity component G ). Let H ⊂ G be a maximal connectedsubgroup of multiplicative type.Note that, as explained in the proof of [1, corollaire 3.3], the connected centraliser of H in G , denoted by Z G ( H ), is a smooth subgroup of G . This is an immediate consequence of thesmoothness theorem for centralisers (see for example [9, II, §5, 2.8]) : the group G being smooth,the set of H -ﬁxed points of G (for the H -conjugation) is smooth over k .We proceed by induction on the dimension of G , the case of dimension 0 being trivial.If now the group G is of strictly positive dimension then:1. either the inclusion Z G ( H ) ⊂ G is strict and then H is a maximal connected subgroup of Z G ( H ) of multiplicative type, thus is a k -torus (of Z G ( H ), hence of G ) by induction ;2. or Z G ( H ) = G and H is central in G . Then, by [7, proposition A.2.11] (applied to G ) onehas the following exact sequence:1 G t G V , where V is a unipotent smooth connected group and G t is the k -subgroup of G generatedby the k -tori of G . The subgroup H ⊆ G being maximal, connected of multiplicativetype in G , it fulﬁlls the same conditions in G t (note that the quotient G/G t = U being15nipotent, the subgroup of multiplicative type H intersects U trivially, in other words itis included in G t ). If V = 1 then H is a k -torus by induction. Otherwise G t = G and if T is a k -torus of G the subgroup H · T ⊂ G is connected of multiplicative type and contains H , thus equals H (as H is assumed to be maximal). Thus one actually has T ⊂ H , hence G t ⊂ H and we have shown that H = G t . This in particular implies the smoothness of H ,which turns out to be a k -torus. The formalism used in this section is developed in [9, II, §4], we especially refer the reader to [9,II, §4, 3.7] for notations. Let A be a ring and G be an aﬃne A -group functor. As a reminder:1. If R is a A -algebra R , we denote by R [ t ] the algebra of polynomials in t and by (cid:15) the imageof t via the projection R [ t ] → R [ t ] / ( t ) =: R [ (cid:15) ] . We associate to G a functor in Lie algebras denoted by Lie ( G ), which is the kernel of thefollowing exact sequence:1 Lie ( G )( R ) G ( R [ (cid:15) ]) G ( R ) 1 .pi For any y ∈ Lie ( G )( R ) we denote by e (cid:15)y the image of y in G ( R [ (cid:15) ]). In what follows thenotation Lie ( G )( R ) refers to the kernel of p as well as its image in G ( R [ (cid:15) ]). The Lie-algebraof G is given by the k -algebra Lie ( G )( A ) and is denoted by Lie( G ) := g . According to[9, II, §4, n ◦ G is smooth or A is a ﬁeld and G is locally of ﬁnitepresentation over A , the equality Lie( G ) ⊗ A R = Lie ( G )( A ) ⊗ A R = Lie ( G )( R ) = Lie( G R )holds true for any A -algebra R (these are suﬃcient conditions). When the aforementionedequality is satisﬁed the A -functor Lie ( G ) is representable by W ( g ) where for any A -module M and any A -algebra R we set W ( M )( R ) := M ⊗ A R ;2. for any A -algebra R we use the additive notation to describe the group law of Lie ( G )( R );3. the A -group functor G acts on Lie ( G ) as follows: for any A -algebra R the induced mor-phism is the following:Ad R : G R → Aut(

Lie ( G ))( R ) ,g Ad R ( g ) : Lie ( G )( R ) → Lie ( G )( R ) : x i ( g ) xi ( g ) − . When G is smooth (in particular when Lie ( G ) is representable) the G -action on Lie ( G )ﬁnes a linear representation G → GL( g ) (see [9, II, §4, n ◦ CentralisersLemma 6.2.

Let A be a ring and set S = Spec( A ) , if G is a smooth aﬃne S -group scheme forany subspace h ⊂ g the following equality is satisﬁed: Lie( Z G ( h )) = Z g ( h ) . roof. By deﬁnition one has:Lie( Z G ( h )) = g ∩ Z G ( h )( A [ (cid:15) ])= { g ∈ g | Ad( g A [ (cid:15) ] )( x ) = x, ∀ x ∈ h ( A [ (cid:15) ]) } . In G ( A [ (cid:15), (cid:15) ]) the last identity can be rewritten as e (cid:15)g e (cid:15) x e − (cid:15)g e − (cid:15) x = e (cid:15)(cid:15) [ g,x ] = 1, whence thevanishing of the Lie bracket [ g, x ] (which is a condition in G ( A [ (cid:15) ])). This leads to the followingequalities: Lie( Z G ( h )) = { g ∈ g | [ g, x ] = 0 , ∀ x ∈ h ( A [ (cid:15) ]) } = Z g ( h ) . Remark 6.3.

Even for algebraically closed ﬁelds k , the equality Lie( Z G ( h ) red ) = Z g ( h ) is apriori not satisﬁed, for Z G ( h ) red being the reduced part of the centraliser (see for example [20,2.3]). Remark 6.4.

Let S := Spec( A ) be an aﬃne scheme and G be a S -group scheme. Assume Z G tobe representable (this condition is in particular satisﬁed when G is locally free and separated (see[9, II, §1, n ◦ Z G ) := Lie ( Z G )( A )is a sub-algebra of z g .According to [11, XII Théorème 4.7 d) and Proposition 4.11] when G is smooth, aﬃne ofconnected ﬁbers and of zero unipotent rank over S , the center of G is the kernel of the adjointrepresentation Ad : G → GL( g ). Under these assumptions the equality Lie( Z G ) = z g holds true:indeed the following exact sequence of algebraic groups1 Z G G GL( g ) , Adinduces by derivation an exact sequence (see [9, II, §4, n ◦ Z G ) g End( g ) . ad := Lie(Ad)The desired equality follows as by deﬁnition z g := ker(ad). Let us emphasize that this inparticular applies to any reductive S -group G and to any parabolic subgroup P ⊆ G (as anyCartan subgroup of P is a Cartan subgroup of G ). Normalisers

Let S = Spec( A ) be an aﬃne scheme and G be a smooth S -group scheme of ﬁnite presentation.In what follows H ⊆ G is a closed locally free subgroup. Let us stress out that under theseconditions the normaliser N G ( H ) is representable by a closed group-sub-functor of G accordingto [9, II, §1 n ◦

3, Théorème 3.6 b)]. Moreover if H is smooth the aforementioned theorem providesthe representability of N G ( Lie ( H )) = N G ( h ) as then Lie ( H ) is representable by W ( h ) which islocally free. Lemma 6.5. If H ⊆ G is a closed subgroup then the inclusion N G ( H ) ⊆ N G ( Lie ( H )) is satisﬁed.In particular if H is smooth this leads to the inclusion N G ( H )( R ) ⊆ N G ( h R ) for any A -algebra R . roof. Let us remind that G acts on Lie ( G ) via the adjoint representation, namely for any A -algebra R one has:Ad R : G R → GL(

Lie ( G ))( R ) ,g Ad R ( g ) : Lie ( G )( R ) → Lie ( G )( R ) : x i ( g ) xi ( g ) − . Let g ∈ N G ( H )( R ) := { g ∈ G ( R ) | Ad( g )( H ⊗ A R ) = H ⊗ A R } (see for example [9, II, §1,n ◦ x ∈ Lie ( H )( R ) one hasAd( g )( x ) = i ( g ) xi ( g ) − ∈ H ( R [ (cid:15) ]) ∩ Lie ( H )( R ) , hence the inclusion N G ( H ) ⊆ N G ( Lie ( H )).If now H is smooth then Lie ( H ) is representable by a A -functor in Lie algebras and one has Lie ( H )( A ) ⊗ A R = Lie ( H )( R ) = Lie( H R ) := h R for any A -algebra R . Lemma 6.6.

Let h ⊆ g be a Lie sub-algebra then one has Lie(N G ( h )) = N g ( h ) .Proof. By deﬁnitionLie( N G ( h )) = g ∩ N G ( h )( A [ (cid:15) ])= { g ∈ g | Ad( g A [ (cid:15) ] )( x ) ∈ h A [ (cid:15) ] , ∀ x ∈ h ( A [ (cid:15) ]) } . In G ( A [ (cid:15), (cid:15) ]) as (cid:15) = 0, the last relation writes:Ad( e (cid:15)g ) e (cid:15) x = e (cid:15) x e (cid:15) (cid:15) [ g,x ] = e (cid:15) ( x + (cid:15) [ g,x ]) ∈ h R ∩ G ( A [ (cid:15), (cid:15) ]) . In other words one has:Lie( N G ( h )) = { g ∈ g | x + (cid:15) [ g, x ] ∈ h A [ (cid:15) ] , ∀ x ∈ h ( A [ (cid:15) ]) } = { g ∈ g | (cid:15) [ g, x ] ∈ h A [ (cid:15) ] , ∀ x ∈ h ( A [ (cid:15) ]) } = { g ∈ g | [ g, x ] ∈ h A [ (cid:15) ] , ∀ x ∈ h ( A [ (cid:15) ]) } = N g ( h ) . The second part of the following lemma is shown in the proof of [7, proposition 3.5.7] when k is a separably closed ﬁeld. The study of the proof shows that one actually only needs H ( k )to be Zariski-dense in H for the result to hold true. Let us stress out that this is especially truewhen:1. the ﬁeld k is perfect and the subgroup H is connected (see [3, corollary 18.2]),2. the ﬁeld k is inﬁnite and the subgroup H is reductive (see [3, corollary 18.2]),3. the subgroup H is unipotent, smooth, connected and split. Indeed, under these assump-tions H is isomorphic to a product of G a . These conditions are especially satisﬁed when k is perfect and H is unipotent, smooth and connected (which is a special case of 1.). Lemma 6.7.

Let H ⊆ G be a closed and smooth subgroup, then:1. in general only the inclusion Lie( N G ( H )) ⊆ N g ( h ) holds true,2. if H ( k ) is Zariski-dense in H then Lie( N G ( H )) = { x ∈ g | Ad( h )( x ) − x ∈ h ∀ h ∈ H ( k ) } . roof. The inclusion N G ( H ) ⊆ N G ( Lie ( H )) is provided by lemma 6.5. Combining this togetherwith the equality obtained in lemma 6.7 one obtains:Lie( N G ( H )) ⊆ Lie( N G ( h )) = N g ( h ) . As already mentioned, the second assertion of the lemme is shown in [7, proposition 3.5.7].

Remarks 6.8.

The ﬁrst point of the above lemma provides a strict inclusion of Lie algebras inthe general case. This is actually a positive characteristic phenomenon (see [17, 10.5 CorollaryB] and the remark that follows corollary B) :1. when k is of characteristic 0 the aforementioned inclusion is always an equality (see [17,13. Exercise 1]),2. when k is of characteristic p > p = 2, set G = SL and consider the Borel sub-group B of upper triangular matrices. The group B being parabolic it is its self normaliser,in other words N G ( B ) = B . However, at the Lie algebra level one has Lie( N G ( B )) = g (as k is of characteristic 2). Indeed sl is generated by ! , ! , ! and oneonly needs to show that the bracket of the following two matrices ! and ! still belongs to b . One has: " ! , ! = id ∈ b . p -restricted Lie algebras Let g be a restricted p -nil Lie algebra over k . In what follows we denote by [ p ] the p -structurefor g . According to [9, II, §7, n ◦ k -group scheme G is endowed witha p -structure. Moreover, for any algebraic subgroup H ⊂ G , the [ p ]-structure on Lie( H ) := h inherited from the group is compatible with the one on g . In other words h is a restricted p -sub-algebra of g . We refer the reader to [35, §2 Déﬁnition] for general theory of restricted p -Liealgebra.Let k be a ﬁeld and let g be a k -Lie algebra. As a reminder:1. the solvable radical (or radical) of g denoted by rad( g ) is the biggest solvable ideal of g (see [35, §1.7, deﬁnition]),2. the nilradical of g , denoted by Nil( g ), is the biggest nilpotent ideal of g (so all its elementsare ad-nilpotent, it is a corollary of Engel theorem, see for example [5, §4 n ◦ k is of characteristic 0, this is nothing but the set of ad-nilpotent elementsof the radical of g (see [35, §1, Corollary 3.10] and [5, §5, corollaire 7]). Let us stressout that the equality Nil( g / Nil( g )) = 0 is not satisﬁed in general (see [35, p. 20] for acounter-example).3. A sub-algebra h ⊆ g is nil if any element of h is ad-nilpotent for the bracket on g . Any niland ﬁnite dimensional k -Lie algebra is nilpotent. Corollary 6.9.

Let h be a restricted p -Lie algebra over k then rad( h ) is a restricted p -sub-algebraof h .Proof. This is enough to consider the following morphism of Lie algebras h (cid:16) h / rad( h ). Asrad( h / rad( h )) = 0 according to [35, 1, §7, Theorem 7.2], the center z rad( h / rad( h )) is trivial. By[35, 2.3, Exercise 7], the radical of h is a p -Lie sub-algebra.19ssume the Lie algebra g derives from an aﬃne algebraic k -group. Let ρ : G → GL( V ) be afaithful representation of ﬁnite dimension. An element x ∈ g is nilpotent, we write g -nilpotent,if Lie( ρ )( x ) is a nilpotent element of gl ( V ) (let us stress out that Lie( ρ ) is still injective, the Liefunctor being left exact (see [9, II,§4, 1.5])). On the same way, an element x ∈ g is semi-simpleif Lie( ρ )( x ) is a semi-simple element of gl ( V ). These notions are independent from the choiceof the faithful representation ρ (see [3, I.4.4, Theorem]). When k is perfect any est x ∈ g has aJordan decomposition in g (see for example [3, I.4.4, Theorem]). More generally, if g is a semi-simple Lie algebra over a ﬁeld of characteristic 0 (we do not assume here that g derives from analgebraic group), any element x ∈ g has a unique Jordan decomposition (see for example [5, §6n ◦ k is a perfect ﬁeld of characteristic p > g is a p -restricted Lie algebra, such a decomposition x = x s + x n (with x s semi-simple and x n nilpotent) always exists, with the additional condition for the nilpotent part tobe p -nilpotent. This means that there exists an integer m ∈ N such that x [ p m ] n = 0. We saythat m is the order of p -nilpotency of x n . In this framework, an element x ∈ g is p -semi-simpleif x belongs to the restricted p -Lie algebra generated by x [ p ] . An element x ∈ g is toral if x [ p ] = x . According to [35, §2 proposition 3.3] and the remark that follows this proposition bothdeﬁnitions of semi-simplicity are equivalent. In what follows an element is said to be p -semi-simple (respectively p -nilpotent) if it is semi-simple (respectively g -nilpotent). This equivalenceof deﬁnitions is a consequence of Iwasawa theorem (see [18]) which ensures that any Lie sub-algebra of ﬁnite dimension over a ﬁeld of characteristic p > p -Lie algebras with the additional constraint, thatthe involved representation is compatible with the p -structure.Let k be a ﬁeld of characteristic p >

0. Let h be a restricted p -algebra (this is in particularthe case if h derives from a subgroup H ⊂ G see for example [9, II, §7, n ◦ p -sub-algebra h is p -nilpotent if there exists an integer n ∈ N such that h [ p n ] = 0. When g is ofﬁnite dimension any restricted p -sub-algebra which is p -nilpotent is also p -nil (that is, any of itselements are p -nilpotent).It is worth noting that the study of ideals that consists only in semisimple elements of g canalso be very instructive. Let us remind the following result as an illustration (see [4, Proposition2.13]): if g = Lie( G ) is the Lie algebra of a reductive k -group, let j ⊆ g be an ideal stable for theaction of G by conjugation, then j consists only in semi-simple elements if and only if j ⊆ z g .In positive characteristic, the nilradical of a restricted p -algebra is well deﬁned, but doesnot satisﬁes longer the properties it had in the characteristic 0 settings. This justiﬁes to intro-duce the following object which appears to be, under certain hypotheses, the good analogue incharacteristic p > Deﬁnition 6.10.

Let h be a restricted p -algebra. The p -radical of h , denoted by rad p ( h ), is themaximal p -nilpotent p -ideal of h (such an object exists according to [35, 2.1, corollary 1.6] byexample).Let us also stress out that the Lie algebra of the unipotent radical of a connected algebraicgroup H , denoted by rad u ( H ), is an ideal of Nil( h ) (as U is a unipotent normal subgroup ofRad( H )). We aim to compare these diﬀerent objects: Lemma 6.11.

Let h be a restricted p -algebra. Then:1. the following inclusions are satisﬁed: rad p ( h ) ⊆ Nil( h ) ⊆ rad( h ) ,

2. the p -radical of h is a subset of the set of all p -nilpotent elements of rad( h ) ,3. the equality rad p ( h ) = Nil( h ) holds true if and only if the inclusion z h ⊆ rad p ( h ) is satisﬁed,where z h is the center of h . roof. The inclusion rad p ( h ) ⊆ Nil( h ) is clear as rad p ( h ) is a nil ideal of h (as it is p -nil). Henceit is a nilpotent ideal of h (the Lie algebras involved here being of ﬁnite dimension).The second inclusion is also direct as any nilpotent ideal is in particular solvable (see forexample [35, §1.5 Remark]). Hence the ﬁrst point of the lemma is shown.This last inclusion being satisﬁed and rad p ( h ) being p -nil, the restricted p -ideal is necessarilycontained in the set of all p -nilpotent elements of rad( h ). This ends the proof of 2.Let us show 3 . : the center of h being an abelian ideal of h , it is contained in the nilradicalof h . Thus if one has the equality Nil( h ) = rad p ( h ) one also has the inclusion z h ⊆ rad p ( h ).Reciprocally, if z h ⊆ rad p ( h ), let us show that any x ∈ Nil( h ) is p -nilpotent: it is ad-nilpotentaccording to corollary [5, §4 n ◦ h ) is nilpotent. The Liealgebra h being endowed with a p -structure there exists an integer n such that ad( x ) p n = 0 =ad( x [ p n ] ). In other words x [ p n ] belongs to the center of h . As we assumed z h to belong to rad p ( h )(which is an ideal p -nil), it is actually p -nilpotent. Hence there exists an integer m such that( x [ p n ] ) [ p m ] = ( x [ p n + m ] ) = 0, whence the p -nilpotency of any element of Nil( h ). This implies thatNil( h ) is a restricted p -ideal p -nil of h , the nilradical of h being a restricted p -ideal according tolemma [35, 2.3, Exercise 5d]. This leads to the desired equality.When g derives from a smooth, connected, algebraic k -group G these objects should becompared with the Lie algebra of the radical, respectively of the unipotent radical of G . Lemma 6.12.

Let k be a ﬁeld of characteristic p ≥ and G be a reductive k -group, then thefollowing equalities hold true: z g = rad( g ) = Nil( g ) . Remark 6.13.

The assumption on the characteristic allows a uniform proof of the above lemma.Notwithstanding this point, it is worth noting that the characteristic 2 case can be handled bya case-by-case analysis by making use of [16, table 1]. Moreover, lemma 6.17 below provides theequality z g = Nil( g ) in any characteristic p >

0, which is a weaker result. This last statementappears as a corollary of [37, Lemma 2.1].The following lemma will be useful in the proof of lemma 6.12:

Lemma 6.14.

Let S e G G ι π be a central exact sequence of algebraic groups, for e G and G two reductive k -groups. Let e T bea maximal k -torus of e G , and set T := e T /S . Then

Lie( π )( e g ) is an ideal of g and the quotient g / Lie( π )( e g ) is isomorphic to t / Lie( π )( e t ) as k -Lie algebras. In particular if k is of characteristic p > , the restricted p -Lie algebra g / Lie( π )( e g ) is toral.Proof. According to [12, XXII, Corollaire 4.1.6] (for example) the center of a reductive groupis a diagonalisable subgroup. The exact sequence of the lemma being central, the k -group S isdiagonalisable (as any subgroup of diagonalisable group deﬁned over a ﬁeld is diagonalisable,see [11, IX, Proposition 8.1]). Let E be a k -torus such that S ⊆ E . Let us stress out that suchan object always exists, the maximal connected subgroups of multiplicative type of a reductivegroup over a ﬁeld are the maximal tori (see lemma 6.1). Consider the following commutativediagram of algebraic k -groups: 21 1 G rm G rm E G G S e G G , π π where G is deﬁned for the lower left square to be commutative. It induces a commutativediagram of Lie algebras: 0 k r k r k r g g , s e g g π )Lie( π )the exactness of the second line coming from the smoothness of Ker( π ) (see [9, II, §5, n ◦

5, Propo-sition 5.3]). We show that Lie( π )( e g ) is ideal of g : let y ∈ g and x ∈ e g be such that Lie( π )( x ) = y ,and pick g ∈ g . As Lie( π ) is surjective there is exists g ∈ g such that Lie( π )( g ) = g . Thisprovides the equality [ y, g ] = [Lie( π )( x ) , Lie( π )( g )] = [Lie( π )( x ) , Lie( π )( g )] = Lie( π )([ x, g ]).The Lie algebra e g being the kernel of g → k r , it is an ideal of g . The bracket [ x, g ] hencebelongs to e g . In other words one has [ y, g ] ∈ Lie( π )( e g ) and Lie( π )( e g ) is an ideal of g .Let us stress out that it only remains to prove that the inclusion Lie( π )( e t ) ⊆ Lie( π )( e g ) ∩ t is actually an equality. This being established, one will only need to apply [4, corollaire 2.17]to end the proof, as this result states that t (cid:16) g / Lie( π )( e g ) is surjective. Let us then showthe aforementioned equality. It comes from the study of the right lower squares of the abovecommutative diagrams: the morphism π being surjective with toric kernel E , the group T is theimage of a torus T ⊆ G (by [11, IX, Proposition 8.2 (ii)]). Hence we have T = T /E = e T /S .The following square G G e G G,i π π i ( e T ) is contained in T . Thus the exact sequence1 e G G G rm i induces an exact sequence of tori (the subgroup T being diagonalisable according to [11, IX,Proposition 8.1] and smooth by [1, II, §5, n ◦

5, Proposition 5.3 (ii)]) :1 e T T T .i The exactness is here preserved by derivation as e T is smooth. Let us now consider the rightlower square of the above commutative diagram of Lie algebras:0 k r k r g g . e g g i ) =Lie( π )Lie( π )The kernel E being smooth the derived morphism Lie( π ) is still surjective, hence one still has t = t /k r . According to what precedes any y ∈ Lie( π )( e g ) ∩ t is the image of a certain x ∈ e g suchthat Lie( i )( x ) ∈ t . This combined with the exactness of the following derived exact sequence0 e t t t i )allows to conclude. Indeed it follows from the exactness that x ∈ e t and since y = Lie( π )( x ) =Lie( π )( i ( x )) ∈ Lie( π )( e t ), the expected inclusion, thus the equality, is obtained. Proof of lemma 6.12.

The center z g being a nilpotent ideal of g it is in particular solvable. Theinclusions z g ⊆ Nil( g ) ⊆ rad( g ) follow. One then only needs to show that rad( g ) ⊆ z g . Theinvolved objects being all compatible with base chance, we can without lost of generality assume k to be algebraically closed.A dévissage argument allows to reduce ourselves to prove the statement for G connected andsemi-simple: the reductive case can be deduced from the simply connected one, which is ruledby the semi-simple and simply connected case:1. if the k -group G is semi-simple and simply connected it decomposes into a product ofalmost simple groups (see [36, 3.1.1, p. 55]) and one can assume without lost of generality23hat G is almost simple. According to [15, Haupsatz], apart from the type G case incharacteristic 3, the quotient g / z g is a simple G -module, hence the radical rad( g / z g ) istrivial. Let us then focus on the pathological case: assume G to be k -group of type G and k to be of characteristic 3. According to [16, table 1] there are only two possibilitiesfor rad( g ): it is either trivial or the Lie algebra of a PGL factor. The Lie algebra pgl isnot solvable, so this last option cannot occur and one can conclude that rad( g ) = 0.2. Assume now that G is semi-simple. It then admits a universal covering, denoted by G sc (seefor example [36, 1.1.2, Theorem 1, p. 43]), and one can consider the following associatedcentral extension:1 µ G sc G .π Let T sc be a maximal k -torus of G sc and set T = T sc /µ (the corresponding Lie algebraswill be denoted by t sc , respectively t ). The above lemma ensures that Lie( π )(Lie( G sc )) isan ideal of g and one has the following exact sequence of restricted p -Lie algebras:0 Lie( µ ) Lie( G sc ) g g / Lie( π )(Lie( G sc )) 0 . t / t sc Lie( π ) ∼ = The extension being central, the preimage of rad( g ) is a solvable ideal of Lie( G sc ) (thisis a consequence of [35, 1.5, Theorem 5.1 (2)]). Hence it is contained in the radicalof Lie( G sc ) = z Lie( G sc ) , according to what precedes. Composing with Lie( π ), one canthen deduce that rad( g ) ∩ Lie( π )(Lie( G sc )) ⊆ z g , whence the desired equality rad( g ) ∩ Lie( π )(Lie( G sc )) = z g ∩ Lie( π )(Lie( G sc )).The above exact sequence hence induces the following one0 z g ∩ Lie( π )(Lie( G sc )) rad( g ) h . Lie( π )where h is a restricted p -sub-algebra of t / Lie( π )( t sc ), which is toral. Thus the p -nilpotentelements of rad( g ) are trivial and rad( g ) only has semi-simple elements. This will leadsto the desired inclusion rad( g ) ⊆ z g according to [4, Proposition 2.13] once we will haveshown that that N G (rad( g )) = G . Note that all the other assumptions of the Propositionare trivially satisﬁed as rad( g ) is a proper ideal of g as G is a reductive k -group. Letus then show the desired equality for the normaliser: according to [9, II,§5, n ◦ k -points, the group G being smooth and of ﬁnite presentation and the Liealgebra rad( g ) being reduced and closed in g . But this is clear as rad( g )(¯ k ) is stable underconjugation: the image of rad( g )(¯ k ) by G (¯ k )-conjugation is a solvable ideal of g (¯ k ), itsmaximality can be deduced by applying the inverse morphism.3. If G is any reductive k -group, the following exact sequence (see for example [12, XXIIDéﬁnition 4.3.6]) allows to reduce ourselves to the preceding cases:24 ( Z G ) red = Rad( G ) G G/ ( Z G ) red := G ss .π Indeed, the subgroup Rad( G ) being smooth this exact sequence induces after derivationan exact sequence of Lie algebras (see [9, §5, n ◦

5, Proposition 5.3])0 Lie(Rad( G )) g Lie( G ss ) 0 . Lie( π )The morphism Lie( π ) being surjective its image Lie( π )(rad( g )) is a solvable ideal ofLie( G ss ), hence is contained in the center of Lie( G ss ) by what precedes. Let x ∈ rad( g ),as k may be assumed to be algebraically closed, it admits a Jordan decomposition (forthe existence of such see for example [35, 2.3 Theorem 3.5]), say x = x s + x n with x s semi-simple and x n a p -nilpotent element of rad( g ). As π ( x ) ∈ z g one necessarily has π ( x n ) = 0, meaning that x n ∈ Lie(( Z G ) red ) which is toral, hence x n = 0. So rad( g ) onlyhas semi-simple elements and according to [4, Proposition 2.13] rad( g ) being a proper G -sub-module of g , we just have shown that rad( g ) ⊆ z g . Remark 6.15.

Lemma 6.12 in particular allows to measure the potential lack of smoothness ofthe center of G . More precisely one has:Lie( Z G ) / Lie(( Z G ) red ) ∼ = z g / Lie ( Z G ) red ∼ = rad( g ) / Lie(Rad( G )) , where the ﬁrst isomorphism comes from the remark 6.4, and this quotient is a restricted toral p -algebra according to the proof of lemma 6.12. Remark 6.16.

A careful study of the proof shows that the only diﬃculty one would have whentrying to extend the above result to the characteristic 2 relies on the fact that the G -module g / z g might not be simple. For an algebraically closed ﬁeld k of characteristic 2, this is no longeran issue if the root system of G only has irreducible components of A n -type according to [15,Haupsatz], which is always the case in this article.As mentioned in remark 6.13 the following result allows to reﬁne a bit the hypotheses on p in the study of the nilradical of the Lie algebra of a reductive group. Lemma 6.17 (Corollary of [37, lemma 2.1]) . Let G be a reductive k -group. If k is of charac-teristic assume that G Ad k s has no direct factor G isomorphic to SO n +1 for an integer n > .Under these assumptions Nil( g ) is the center of g .Proof. One inclusion is clear and does not require any additional assumption on the characteristicof k : the center of g being a nilpotent ideal of g it is contained in the nilradical of g which isthe maximal nilpotent ideal of g .To show the reverse inclusion one only needs to prove that Nil( g ) / z g = 0. Indeed thisvanishing condition implies that Nil( g ) is contained in z g , which is itself a nilpotent ideal of g .The equality Nil( g ) = z g follows.One still needs to show that Nil( g / z g ) = 0. The inclusion g / z g ⊆ Lie( G Ad ) is provided bythe exact sequence of Lie algebras of remark 6.4:0 z g = Lie( Z G ) g End( g )ad25ssume that Nil( g ) / z g = 0, we show that then [37, lemma 2.1] holds true which lead to acontradiction (as it would imply p = 2 and G be such as excluded in the assumptions).We thus have to check on the one hand that Nil( g ) / z g is a G Ad -sub-module of Lie( G Ad ), andon the other hand that for any maximal torus T Ad ⊆ G Ad the intersection Nil( g ) / z g ∩ Lie( T Ad )is trivial.To check that the ﬁrst condition is satisﬁed remark that Nil( g / z g ) = Nil( g ) / z g . Indeed, thepreimage of Nil( g / z g ) is a nilpotent ideal of g , the considered extension of Lie algebras0 z g = Lie( Z G ) g g / z g g ) / z g is a nilpotent ideal of g / z g , it is contained in Nil( g / z g ) hence theequality. Thus we are reduced to show that Nil( g / z g ) is a G Ad -sub-module of Lie( G Ad ), or inother words that N G Ad (Nil( g / z g )) = G Ad . Once again by [9, II,§5, n ◦ k -points as G Ad is smooth of ﬁnite presentation and g / z g is reduced and closed inLie( G Ad ). But this is clear Nil( g / z g (¯ k )) being stable for the adjoint action. Indeed the image ofNil( g / z g )(¯ k ) under the G Ad (¯ k )-conjugation is a nilpotent ideal of g / z g (¯ k ), its maximality followsby considering the reverse morphism.To check that the second condition is indeed satisﬁed, ﬁrst notice that any maximal torus T Ad ⊂ G Ad comes from a maximal torus T ⊂ G . At the Lie algebras level one can summarizethe situation with the following commutative diagram:Nil( g ) / z g g / z g Lie( G Ad ) , T ) / z g Lie( T Ad ) . ⊆⊆ ⊆ Assume that the intersection Nil( g ) / z g ∩ Lie( T Ad ) is not trivial. This in particular implies thatthe intersection Nil( g ) / z g ∩ Lie( T ) / z g is not trivial, as any element of the ﬁrst intersection occursas an element of the image of g → g / z g . Remember that we already have shown that z g iscontained in Nil( g ). But according to remark 6.4 it is nothing but the Lie algebra of Z G , whencethe inclusion z g ⊂ Lie( T ).Thus assuming the non-triviality of the intersection Nil( g ) / z g ∩ Lie( T ) / z g is equivalent tosuppose that the inclusion z g (cid:40) Lie( T ) ∩ Nil( g ) is strict, whence a contradiction. Indeed anyelement of the nilradical is ad-nilpotent (REF ?) and ad-nilpotent element of the Lie algebra of atorus is central. To see this, let n be the order of ad-nilpotency of x ∈ Lie( T ) and y ∈ g . Passingto the algebraic closure of k if necessary, the Lie algebra g has a weight space decompositionfor the action of the maximal torus T (which is locally splittable). Let R be an associated rootsystem, one has: g = t ⊕ L α ∈ R g α . Thus y writes y = y + P α ∈ R y α for y ∈ t and y α ∈ g α ,with α ∈ R . This leads to:0 = ad n ( x )( y ) = ad n ( x )( y + X α ∈ R y α ) = X α ∈ R α n ( x ) y α , where we have made use of the vanishing condition ad( x )( y ) = 0 as x ∈ t . This equality beingsatisﬁed if and only if ad( x )( y α ) = 0 for any α ∈ R , this implies that x ∈ z g .26 emark 6.18. In this article we always assumed a minima that p is not of torsion for G . Thisin particular implies that p is strictly greater than 2 if G has any factor of B n type. The abovelemma then tells us that in this article the equality Nil( g ) = z g is always satisﬁed. Lemma 6.19.

Let U be a unipotent algebraic k -group, then its Lie algebra u is p -nil. Inparticular, the Lie algebra of the unipotent radical of a smooth connected k -group G is a restricted p -nil p -ideal of g .Proof. As k is a ﬁeld it follows from [9, IV, §2, n ◦ k -group U is embeddable into the subgroup of upper triangular matrices U n,k of GL n for a certain n ∈ N . This leads to the following inclusion of restricted p -Lie algebras (all of them coming fromalgebraic k -group) u ⊆ u n,k . Note that the p -structure on u n,k is given by taking the p -power ofmatrices, this makes u n,k into a restricted p -nil p -sub-algebra, so is u .If now U is the unipotent radical of a smooth connected k -group G , its Lie algebra is anideal of g (as it is the Lie algebra of a normal subgroup of G ). As it derives from an algebraic k -subgroup if G it is endowed with a p -structure compatible with the p -structure of G (see forexample [9]). Hence it is a restricted p -ideal of g . It is p -nil by what precedes. Lemma 6.20.

Let k be a perfect ﬁeld and H be a smooth connected algebraic k -group. Then:1. if the reductive k -group H := H/ Rad U ( H ) satisﬁes the conditions of lemma 6.17 the Liealgebra of the unipotent radical of H is the p -radical of h , in other words the followingequality rad u ( H ) = rad p ( h ) holds true,2. if k is of characteristic p ≥ the p -radical of h is the set of p -nilpotent elements of rad( h ) . Remark 6.21.

In particular, let k is a perfect ﬁeld, consider a reductive k -group G and aparabolic subgroup P of G . If P is such that the Levi subgroups it deﬁnes satisﬁes the as-sumptions of lemma 6.17, the Lie algebra of its unipotent radical is the p -radical of p := Lie( P )and it is the set of p -nilpotent elements of rad( p ). As a reminder (see [12, proposition 1.21ii)]), if L ⊆ P is a Levi subgroup, the solvable radical Rad( P ) is the semi-direct product of theunipotent radical of P with the radical of L . This in particular implies the following inclusionLie( Z L ) ⊆ rad( p ) for Z L being the center of L . Proof.

We start by showing the ﬁrst point of the lemma. An implication is clear: according tothe lemma 6.19 the Lie algebra rad u ( H ) is a restricted p -nil p -ideal, in particular the inclusion rad u ( H ) ⊆ rad p ( h ) holds true.Let us show the reverse inclusion. The radical of H being a smooth subgroup the followingexact sequence of algebraic k -groups:1 Rad u ( H ) H H/

Rad U ( H ) =: H ,π induces an exact sequence of k -Lie algebras (see [9, II, §5, n ◦ rad u ( H ) h h / rad u ( H ) =: h . Lie( π )This is an exact sequence of restricted p -Lie algebras (see [9, II, §7 n ◦ ◦ p -structure). The derived morphism Lie( π ) being surjective the image ofNil( h ) under Lie( π ) still is an ideal. It is nilpotent as Lie( π ) is a morphism of restricted p -Liealgebras, whence the inclusion Lie( π )(Nil( h )) ⊆ Nil( h ). As h derives from a reductive k -group27hich does not ﬁt into the pathological case studied by A. Vasiu in [37] the lemma 6.17 applies.This leads to the following equality Nil( h ) = z h thus Lie( π )(Nil( h )) = z h .But according to lemma 6.11 1. one has rad p ( h ) ⊆ Nil( h ), hence any x ∈ rad p ( h ) is mappedto the center of z h . The restricted p -ideal rad p ( h ) being p -nil, the element x is p -nilpotent, so isLie( π )( x ) (as Lie( π ) is compatible with the p -structures of h and h ). In other words Lie( π )( x )is a p -nilpotent elements of z h , which is, according to remark 6.4, the Lie algebra of the centerof the reductive k -group H . Thus this center is a toral restricted p -sub-algebra (see for example[12, XXII, Corollaire 4.1.7]) hence Lie( π )( x ) = 0. In other words x belongs to rad u ( H ) whencethe equality rad p ( h ) = rad u ( H ). This concludes the proof of 1 . Let us then show the second point of the statement. One again an inclusion is clear: the p -radical rad p ( h ) being a restricted p -nil p -ideal of h , it is contained in the set of all p -nilpotentelements of h . Let us show the converse inclusion: let x ∈ rad( h ), the morphism Lie( π ) beingsurjective Lie( π ( x )) belongs to rad( h ) which is the center of h according to lemma 6.12 (whichholds true as p ≥ π )( x ) is also a p -nilpotent element, it necessarily vanishes (the centerof h being toral). In other words x belongs to rad u ( h ) which is the p -radical of h according tothe ﬁrst point of the lemma. Hence any p -nilpotent of rad( h ) belongs to rad p ( h ), whence thedesired equality. Acknowledgments:

The author would like to thank Philippe Gille for all the fruitful dis-cussions they had which go beyond the framework of this paper, his availability, his multiplereviews and corrections of this article; Benoît Dejoncheere for his supportive help, his severalreviews and useful advice; as well as both reviewers of her PhD manuscript: Anne-Marie Aubertand Vikraman Balaji for their corrections and remarks. Any critical remark must be exclusivelyaddressed to the author of this paper.

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