Conjugacy theorems for loop reductive group schemes and Lie algebrasAcademic research paper on "Mathematics"

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Academic research paper on topic "Conjugacy theorems for loop reductive group schemes and Lie algebras"

﻿Bull. Math. Sci. (2014) 4:281-324 DOI 10.1007/s13373-014-0052-8

Conjugacy theorems for loop reductive group schemes and Lie algebras

V. Chernousov • P. Gille • A. Pianzola

Received: 16 February 2014 / Revised: 10 March 2014 / Accepted: 7 June 2014 / Published online: 3 July 2014

Abstract The conjugacy of split Cartan subalgebras in the finite-dimensional simple case (Chevalley) and in the symmetrizable Kac-Moody case (Peterson-Kac) are fundamental results of the theory of Lie algebras. Among the Kac-Moody Lie algebras the affine algebras stand out. This paper deals with the problem of conjugacy for a class of algebras—extended affine Lie algebras—that are in a precise sense higher nullity analogues of the affine algebras. Unlike the methods used by Peterson-Kac, our approach is entirely cohomological and geometric. It is deeply rooted on the theory of reductive group schemes developed by Demazure and Grothendieck, and on the work of Bruhat-Tits on buildings. The main ingredient of our conjugacy proof is the classification of loop torsors over Laurent polynomial rings, a result of its own interest.

Communicated by Efim Zelmanov.

V. Chernousov was partially supported by the Canada Research Chairs Program and an NSERC research grant. P. Gille a bénéficié du soutien du projet ANR Gatho, ANR-12-BS01-0005. A. Pianzola wishes to thank NSERC and CONICET for their continuous support.

V. Chernousov (B) • A. Pianzola

Department of Mathematics, University of Alberta, Edmonton, AB T6G 2G1, Canada e-mail: vladimir@ualberta.ca

A. Pianzola

e-mail: a.pianzola@math.ualberta.ca P. Gille

UMR 5208 du CNRS, Institut Camille Jordan, Université Claude Bernard Lyon 1, 43 boulevard du 11 novembre 1918, 69622 Villeurbanne Cedex, France e-mail: gille@math.univ-lyon1.fr

A. Pianzola

Centro de Altos Estudios en Ciencia Exactas, Avenida de Mayo 866, 1084 Buenos Aires, Argentina

Keywords Loop reductive group scheme ■ Torsor ■ Laurent polynomials ■ Non-abelian cohomology ■ Conjugacy ■ Building

Mathematics Subject Classification (2010) 11E72 ■ 17B67 ■ 20G15 ■ 20G35 1 Introduction

Let g be a split simple finite-dimensional Lie algebra over a field k of characteristic 0. From the work of Cartan and Killing one knows that g is determined by its root system. The problem, of course, is that a priori the type of the root system may depend on the choice of a split Cartan subalgebra. One of the most elegant ways of establishing that this does not happen, hence that the type of the root system is an invariant of g, is the conjugacy theorem of split Cartan subalgebras due to Chevalley: all split Cartan subalgebras of g are conjugate under the adjoint action of G(k) where G is the split simply connected group corresponding to g.

Variations of this theme are to be found on the seminal work of Peterson and Kac on conjugacy of "Cartan subalgebras" for symmetrizable Kac-Moody Lie algebras [35]. Except for the toroidal case, nothing is known about conjugacy for extended affine Lie algebras (EALAs for short); a fascinating class of algebras which can be thought of as higher nullity analogues of the affine algebras.

The aim of this paper is two-fold. First, to show the existence and conjugacy of what we call Borel-Mostow subalgebras; an important class of "Cartan subalgebras" of multiloop algebras (Borel-Mostow subalgebras are rather special. A general conjugacy result fails, as we show in Sect. 9). As an application of conjugacy we show that the root system attached to a Lie torus is an invariant (see Theorem 13.2). Second, it turns out that to solve the conjugacy problem we are, out of necessity, faced with the classification problem of loop reductive group schemes over a Laurent polynomials ring Rn = k[t±lt±1]. Our second main result provides us with a local-global principle for classification of loop torsors over Rn, a result that we believe is of its own interest. The case n = 1 was done in our paper [13] and here we consider the general case. For details and precise statements we refer to Sects. 11 and 14. The philosophy that we follow is motivated by two assumptions:

(1) The affine Kac-Moody and extended affine Lie algebras are among the most relevant infinite-dimensional Lie algebras today.

(2) Since the affine and extended affine algebras are closely related to finite-dimensional simple Lie algebras, a proof of conjugacy ought to exist that is faithful to the spirit of finite-dimensional Lie theory.

That this much is true for toroidal Lie algebras (which correspond to the "untwisted case" in this paper) has been shown in [33]. The present work is much more ambitious. Not only it tackles the twisted case, but it does so in arbitrary nullity.

Some of the algebras covered by our result are related to extended affine Lie algebras, but our work depicts a more global point of view. For every k-algebra R which is a normal ring it builds a bridge between ad-k-diagonalizable subalgebras of twisted forms of semisimple Lie algebras over R (viewed as infinite-dimensional Lie algebras over the base field k), and split tori of the corresponding reductive group schemes over

R. Using this natural one-to-one correspondence, shown in Theorem 7.1, we are able to attach a cohomological obstacle to conjugacy which eventually leads to the proof of our main conjugacy result in Theorem 12.1. The main ingredient of the proof of conjugacy is the classification of loop reductive torsors over Laurent polynomial rings given by Theorem 14.1.

2 Generalities on multiloop algebras and forms

2.1 Notation and conventions

Throughout this work, with the exception of the Appendix, k will denote a field of characteristic 0 and k an algebraic closure of k. For integers n > 0 and m > 0 we set

Rn = k[t:±1,...,t±1], Kn = k(h,...,tn), Fn = k ((t1)) ■■■((tn)),

±1 ±1 _L1 11

Rn,m = k[t1 , ■ ■■,tn L Kn,m = k(t1 , ■ ■■,tn X Fn,m = k ((t1 )) ■■■ ((tn ))-

The category of commutative associate unital algebras over k will be denoted by k-alg. If X is a scheme over Spec (k), by an X-group we will understand a group scheme over X. When X = Spec(R) for some object R of k-alg, we use the expression R-group. If R is an object in k-alg we will denote the corresponding multiplicative and additive groups by Gm,R and Ga,R .

We will use bold roman characters, e.g. G, g to denote k-groups and their Lie algebras. The notation G and g will be reserved for R-groups (which are usually not obtained from a k-group by base change) and their Lie algebras.

2.2 Forms

Let g be a finite-dimensional split semisimple Lie algebra over k. Recall that a Lie algebra L over R is called a form of g ®kR (or^simply a form of g) if there exists a faithfully flat and finitely presented R-algebra R such that

L ®rR ~ (g ®kR) ®rR - g ®kR, (2.1)

where all the above are isomorphisms of Lie algebras over R. The set of isomorphism classes of such forms is measured by the pointed set

H1ppf(Spec(R), Aut(g)R)

where Aut(g)R is the R-group obtained by base change from the k-linear algebraic group Aut(g). We have a split exact sequence of k-groups

1 Gad Aut(g) Out(g) 1 (2.2)

where Gad is the adjoint group corresponding to g and Out(g) is the constant k-group corresponding to the finite (abstract) group of symmetries of the Coxeter-Dynkin diagram of g.

By base change we obtain an analogous sequence over R. In what follows we will denote Hf f (Spec(R), Aut(g)R) simply by Hfppf(R, Aut(g)) when no confusion is possible. Similarly for the Zariski and étale topologies, as well as for k-groups other than Aut(g).

Remark 2.1 Since Aut(g) is smooth and affine over Spec(R)

Hf ( R, Aut (g)) ~ Hfppf (R, Aut (g)).

Remark 2.2 Let R = Rn be as in Sect. 2.1. By the Isotriviality Theorem of [18] the trivializing algebra R in (2.1) may be taken to be of the form

_ _ _ ± l ± l R := Rn,m ®kk= k[tf m ,...,tnm ]

for some m and some Galois extension k of k containing all m-th roots of unity of k. The extension R/R is Galois.

2.3 Multiloop algebras

Assume now that k is algebraically closed. We fix a compatible set of primitive m-th

roots of unity fm, namely such that ^e = fm for all e > 0. Let R = Rn and R = Rn,m.

Then R/R is Galois. Via our choice of roots of unity, we can identify Gal(R /R) with

(Z/mZ)n as follows: For each e = (e1,...,en) e Zn the corresponding element ~ ~ _ 1 l e = (ex, ■■■ , en) e Gal( R/R) acts on R via e(tm) = fetm.

The primary example of forms L of g ®k R which are trivialized by a Galois

extension R /R as above are the multiloop algebras based on g. These are defined as

follows. Consider an n-tuple a = (o\,...,on) of commuting elements of Autk (g)

satisfying a™ = 1. For each n-tuple (i\,..., in) e Zn we consider the simultaneous

eigenspace

gi 1... in = {x e g : aj (x) = f^x for all 1 < j < n}.

Then g = ^ gi1...in, and g = 0 gi1...in if we restrict the sum to those n-tuples (ii,..., in) for which 0 < ij < mj, where mj is the order of aj.

The multiloop algebra based on g corresponding to a, commonly denoted by L (g, a), is defined by

^^ i! in, _ _

L (g, a) =@ gi1...in ® t1m ...tnm C g ®kR C g ®kR^

(i1,...,in )eZn

_ - ± 1 ± 1

where = limk[t1 m, ...,tn m ].1 Note that L (g, a), which does not depend on the

choice of common period m, is not only a k-algebra (in general infinite-dimensional),

1 The ring R(x> is a useful artifice that allows us to see all multiloop algebras based on a given g as subalgebras of one Lie algebra.

but also naturally an R-algebra. A rather simple calculation shows that L (g, ff) ®rR - g ®kR - (g ®k R) ®r R■

Thus L(g, a) corresponds to a torsor over Spec(R) under Aut(g).

It is worth to point out that the cohomological information is always about the twisted forms viewed as algebras over R (and not k). In practice, as the affine Kac-Moody case illustrates, one is interested in understanding these algebras as objects over k (and not R). We find in Theorem 7.1 a bridge between these two very different and contrasting kinds of mathematical worlds.

3 Preliminaries I: Reductive group schemes

3.1 Some terminology

Let X be a k-scheme. A reductive X-group is to be understood in the sense of [41]. In particular, a reductive k-group is a reductive connected algebraic group defined over k in the sense of Borel. We recall now two fundamental notions about reductive X-groups.

Definition 3.1 Let G be a reductive X-group. We say that G is reducible if G admits a proper parabolic subgroup P which has a Levi subgroup, and irreducible otherwise.

Definition 3.2 We say that G is isotropic if G admits a subgroup isomorphic to Gm,£. Otherwise we say that G is anisotropic.

We denote by Par(G) the X-scheme of parabolic subgroup of G. This scheme is smooth and projective over X [41, XXVI, 3.5]. Since by definition G is a parabolic subgroup of G, when X is connected, to say that G admits a proper parabolic subgroup is to say that Par(G)(X) = {G}.

Remark 3.3 If X is connected, to each parabolic subgroup P of G corresponds a "type" t = t(P) which is a subset of the corresponding Coxeter-Dynkin diagram. Given a type t, the scheme Part (G) of parabolic subgroups of G of type t is also smooth and projective over X (ibid. cor.3.6).

Let H denote a reductive X-group. If T is a subgroup of H the expression "T is a maximal torus of H" has a precise meaning ([41, XII, Definition 1.3]). A maximal torus may or may not be split. If it is, we say that T is a split maximal torus. This is in contrast with the concept of maximal split torus which we also need. This is a closed subgroup of H which is a split torus and which is not properly included in any other split torus of H. Note that split maximal tori (even maximal tori) need not exist, while maximal split tori always do exist if X is noetherian.

If S < H are X-groups and s c g are their respective Lie algebras we will denote by Zfi(&) [resp. Z0(s)] the centralizer of S in H [resp. of s in g]. If S c H is a split torus then Zfi(S) is a closed reductive subgroup (see [41, XIX, 2.2]). Also, if X is

connected and T a torus of H then T contains a unique maximal split subtorus Td (see [41, XXVI, 6.5, 6.6]).

We now recall and establish for future reference some basic useful facts.

Lemma 3.4 Let H be a reductive X-group and S c H a split torus. Then there exists a parabolic subgroup p c H such that Z h(S) is a Levi subgroup of p.

Proof See [41, XXVI, cor. 6.2]. □

Lemma 3.5 Let S be a split torus of H, and let T be the radical of the reductive group C = Zh(S).2 If X is connected then Z^(Td) = C.

Proof Since T is the centre of C we have C c Zh(T). Also, the inclusions S c Td c T yield

Z h (T) c Z fi(Td ) c Zh(S) = C, whence the result. □

Proposition 3.6 Let H be a reductive group scheme over X. Assume X is connected. Let S be a split subtorus of H and let p be a parabolic subgroup of H containing Zfi(S) as Levi subgroup. Then following are equivalent:

(1) The reductive group scheme Z^(&) has no proper parabolic subgroups.

(2) p is a minimal parabolic subgroup of H.

If S is the maximal split subtorus of the radical of Zh(S) these two conditions are equivalent to

(3) The reductive group scheme Z^(&)/S is anisotropic.

Proof According to [41, XXVI.1.20], there is a bijective correspondence

{parabolics Q of H included in p} < — > { parabolics M of Zh(S) }

Thus the left handside consists of one element if and only if so does the right handside.

Proposition 3.7 Let G be a reductive group scheme over a connected base scheme X, S a split subtorus of G, and let g and s denote their respective Lie algebras. Then

(1) Lie(Zg(s)) = Zg(s).

(2) Z g(S) is a Levi subgroup of G and Z&(S) = Z&(s). Proof (1) This is a particular case of [41, II théo. 5.3.1(i)].

(2) That Zg(S) is a Levi subgroups of G follows from Lemma 3.4. To establish the equality Z®(S) = Z&(s) we reason in steps.

(a) Assume X = Spec(k) and G simply connected: Then this is a result of Steinberg. See [44, 3.3 and 3.8] and [44, 0.2].

2 Recall that the radical of a reductive X-groupis the unique maximal torus of its centre [41, XXII, 4.3.6]. Springer

(b) Assume X = Spec(k) and G reductive: Embed G into SLn for a suitable n. Then

and we are reduced to the previous case. (c) In general, we proceed by étale descent. This reduces the problem to the case 6 c T c G where G is a Chevalley group and T its standard split maximal torus. This sequence is obtained by base change to X from a similar sequence over k by [41, VII cor. 1.6]. Over k our equality holds. Since both centralizers commute with base change the equality follows.

4 Loop torsors and loop reductive group schemes

Throughout this section X will denote a connected and noetherian scheme over k and G a k-group which is locally of finite presentation.3

4.1 The algebraic fundamental group

If X is a k-scheme and if a of is a geometric point of X i.e. a morphism a : Spec— X where ^ is an algebraically closed field, we denote the algebraic fundamental group of X at a by ^i(X, a) (see [40] for details).

Suppose now that our X is a geometrically connected k-scheme. We will denote X xk k by X. Fix a geometric point a : Spec(k) — X. Let a (resp. b) be the geometric

point of X (resp. Spec(k)) given by the composite maps a : Spec(k) -— X — X

(resp. b : Spec (k) -A X a Spec(k)). Then by [40, theo. IX.6.1] ^(Spec(k), b) ~ Gal(k) := Gal(k/k) and the sequence

1 — ^1(X, a) — ^1(X, a) — Gal(k) — 1 (4.1)

is exact.

4.2 The algebraic fundamental group of Rn

We refer the reader to [18,19] for details. The simply connected cover Xsc of X Spec (Rn) is Spec(Rn^ ) where

Rn^> = lim Rn,m

3 The case most relevant to our work is that of the group of automorphism of a reductive k-group.

- - ± -1 ± -1 with Rn,m = k[t1 m ,...,tn m ]. The "evaluation at 1" provides a geometric point that we denote by a. The algebraic fundamental group is best described as

ni (X, a) = Z(1)n x Gal (k). (4.2)

where Z(1) denotes the abstract group lim ¡im (k) equipped with the natural action

of the absolute Galois group Gal (k). 4.3 Loop torsors

Because of the universal nature of Xsc we have a natural group homomorphism

G(k)—> G(Xsc). (4.3)

The group n1(X, a) acts on k, hence on G(k), via the group homomorphism n1 (X, a) ^ Gal (k) of (4.1). This action is continuous, and together with (4.3) yields a map

H 1(n1(X, a), G(k)) ^ H 1(n1(X, a), G(Xsc)),

where we remind the reader that these H1 are defined in the "continuous" sense. On the other hand, by [19, prop. 2.3] and basic properties of torsors trivialized by Galois extensions we have a natural inclusion

H 1(n1(X, a), G(Xsc)) c H}t (X, G).

By means of the foregoing observations we make the following.

Definition 4.1 A torsor E over X under G is called a loop torsor if its isomorphism class [E] in H1 (X, G) belongs to the image of the composite map

H 1(n1(X, a), G(k)) ^ H 1(n1(X, a), G(Xsc)) c H}t (X, G).

We will denote by H1oop (X, G) the subset of H1t (X, G) consisting of classes of loop torsors. They are given by (continuous) cocycles in the image of the natural map Z1 (n1 (X, a), G(k)) ^ Z1t (X, G), which we call loop cocycles.

This fundamental concept is used in the definition of loop reductive groups which we will recall momentarily. The following examples illustrate the immensely rich class of objects that fit within the language of loop torsors.

Examples 4.2 (a) If X = Spec (k) then H1oop (X, G) is nothing but the usual Galois

cohomology of k with coefficients in G. (b) Assume that k is algebraically closed. Then the action of n1(X, a) on G(k) is trivial, so that

H 1(n1(X, a), G(k)) = Hom(n1(X, a), G(k))/ Int G(k)

where the group Int G(k) of inner automorphisms of G(k) acts naturally on the

right on Hom(^i(X, a), G(k)). Two particular cases are important:

(b1) G abelian: In this case H 1(n1 (X, a), G(k)) is just the group of continuous

homomorphisms from n1(X, a) to G(k). (b2) n1(X, a) = Z(1)n : In this case H 1(n1(X, a), G(k)) is the set of conjugacy classes of «-tuples a = (a1 ,...,an) of commuting elements of finite order of G(k ).4

This last example is exactly the setup of multiloop algebras, and the motivation for the "loop torsor" terminology.

4.4 Geometric and arithmetic part of a loop cocycle

By means of the decompositions (4.1) and (4.2) we can think of loop cocycles as being comprised of a geometric and an arithmetic part, as we now explain.

Let n e Z1 (n1 (X, a), G(k)). The restriction n|Gai(k) is called the arithmetic part of n and it is denoted by nar ■ It is easily seen that nar is in fact a cocycle in Z 1(Gal(k), G(k)). If n is fixed in our discussion, we will at times denote the cocycle nar by the more traditional notation z. In particular, for s e Gal(k) we write zs instead of n?. _

Next we consider the restriction of n to n1 (X, a) that we denote by ngeo and called the geometric part of n■ We thus have a map

© : Z1 (n1 (X, a), G(k)) -> Z 1(Gal(k), G(k)) x Hom(n1(X, a), G(k))

n ^ ( nar , ngeo)

The group Gal(k) acts on n1(X, a) by conjugation. On G(k), the Galois group Gal(k) acts on two different ways. There is the natural action arising from the action of Gal (k) on k, and there is also the twisted action given by the cocycle nar = z. Following standard practice to view the abstract group G(k) as a Gal(k)-module with the twisted action by z we write z G(k).

Lemma 4.3 The map © described above yields a bijection between Z 1(n1(X, a), G(k)) and couples (z, ngeo) with z e Z 1(Gal(k), G(k)) and ngeo e HomGal(k) (n1 (X, a), zG(k)).

Proof See [19, lemma 3.7]. □

Remark 4.4 Assume that X = Spec(Rn). It is easy to verify that ngeo arises from a unique k-group homomorphism

= (fim Pm)n ^ zG

We finish this section by recalling some basic properties of the twisting bijection (or torsion map) tz : H 1(X, zG) ^ H 1(X, G). Take a cocycle n e Z 1(n1(X, a), G(k))

4 That the elements are of finite order follows from the continuity assumption.

and consider its corresponding pair &(n) = (z, ngeo). We can apply the same construction to the twisted k-group z G. This would lead to a map &z that will attach to a cocycle n e Za), zG(k)) a pair (z', n'geo) along the lines explained above.

Lemma 4.5 Let n e Z 1(^1(X, a), G(k)). With the above notation, the inverse of the twisting map [42]

t"1 : Z 1(ni(X, a), G(k)) Z 1(ni(X, a), zG(k))

satisfies &z ◦ T-1(n) = (1, ngeo). □

Remark 4.6 The notion of loop torsor behaves well under twisting by a Galois cocycle z e Z 1(Gal(k), G(k)). Indeed the torsion map t-1 : H]t (X, G) ^ Hjt (X, zG) maps loop classes to loop classes.

4.5 Loop reductive groups

Let H be a reductive group scheme over X. Since X is connected, for all x e X the geometric fibers Hx are reductive group schemes of the same "type" [41, XXII, 2.3]. By Demazure's theorem there exists a unique split reductive group H over k such that H is a twisted form (in the étale topology of X) of H0 = H xk X. We will call H the Chevalley k-form of H ■ The X-group H corresponds to a torsor E over X under the group scheme Aut(H0), namely E = Isom^r (H0, H). We recall that Aut(H0) is representable by a smooth and separated group scheme over X by [41, XXII, 2.3]. It is well-known that H is then the contracted product E AAut(H°) H0 (see [15] III §4 no3 for details).

We now recall one of the central concepts needed for our work.

Definition 4.7 We say that a group scheme H over X is loop reductive if it is reductive and if E is a loop torsor.

5 Preliminaries II: Reductive group schemes over a normal noetherian base

We begin with a useful variation of Lemma 3.5 under some extra assumptions on our connected base k-scheme X.

Lemma 5.1 Assume that X is normal noetherian and integral. Let H be a reductive X-group. Then there exists an étale cover (Ui)i=i,.,i ^ X such that:

(i) H xx Ui is a split reductive U-group scheme,

(ii) Ui = Spec(Ri ) with Ri a normal noetherian domain.

(iii) If H is a torus and X = Spec( R) there exists a Galois extension Rf R that splits H.

Proof Since X is normal noetherian, H is a locally isotrivial group scheme [41, XXIV.4.1.6]. We can thus cover X by affine Zariski open subsets X1,...,Xl where Xi = Spec(Ai) and such that there exists a finite étale cover V ^ X; for i = 1,l

which splits Hxî • For each i, choose a connected component U of V;. According to the classification of étale maps over X (see [22, 18.10.12]) we know that U; is a finite étale cover of X; and that U; = Spec(R; ) where R; is a normal domain. Since R; is finite over the noetherian ring A;, it is noetherian as well.

(iii) By [41, X, théo. 5.16] there exists a finite étale extension of X that splits H. The result now follows by considering a connected component of this extension and basic properties of the algebraic fundamental group (see [45, 5.3.9]). □

Remark 5.2 If X is local, one single U; suffices.

Proposition 5.3 Let X be normal and noetherian. Let H be a reductive X-group, P c H be a parabolic subgroup and L c P a Levi subgroup.5 Let T be the radical of L and Td its maximal split subtorus. Then Zh(Td ) = L.

Proof Since T is the centre of L we have L c Zh(T). The inclusion Td c T yields Zfi(T) c Zfi(Td ). Thus we have L c Z^(Td ). By the Lemma below and by [41, XXVI, prop. 6.8] the above inclusion is an equality locally in the Zariski topology, hence globally. □

Lemma 5.4 Assume X = Spec(R) is affine and as in the Proposition. Let x e X and consider the localized ring Rx. Then (Td)Rx is the maximal split subtorus of TRx. In particular, if K denotes the quotient field of R then Td x R K is the maximal split subtorus of T x R K.

Proof It suffices to show that (Td )K is the maximal split subtorus of TK. Recall that T is determined by its lattice of characters X (T) equipped with an action of Gal (Rf R), and that Td corresponds to the maximal sublattice in X (T) stable (elementwise) with respect to Gal (Rf R). Similar considerations apply to TK.It remains to note that TK and T have the same lattices of characters and that Gal ( Rf R) ~ Gal ( KfK ) by [9, Ch5 §2.2 theo.2]). □

Proposition 5.5 Let G be a reductive group over a normal ring6 R. If G contains a proper parabolic subgroup P then it contains a split non-central subtorus Gm,R .

Proof We may assume that G is semisimple. Since the base is affine, P contains a Levi subgroup L. Let T be the radical of L and Td its maximal split subtorus. By Proposition 5.3, Z^(Td) = L. Hence Td = 1. □

Corollary 5.6 For a reductive group scheme G over a normal ring R to contain a proper parabolic subgroup it is necessary and sufficient that it contains a non-central split subtorus.

Let R be an object in k-alg and G be an R-group, i.e a group scheme over R. Recall (see [15] II §4.1) that to G we can attach an R-functor on Lie algebras Lie(G) which

5 The existence of L is automatic if the base scheme is affine by [41, XXVI.2.3].

6 All of our normal rings are hereon assumed to be integral and noetherian.

associates to an object S of R-alg the kernel of the natural map G(S[e]) ^ G(S) where S[e] is the algebra of dual numbers over S. Let Lie(G) = £ie(G)(R). This is an R-Lie algebra that will be denoted by g in what follows.

Remark 6.1 If G is smooth, the additive group of Lie(G) represents £ie(G), that is £ie(G)(S) = Lie(G) ®R S as S-Lie algebras (this equality is strictly speaking a functorial family of canonical isomorphisms).

If S is in R-alg, g e G(S) and x e £ie(G)(S), then gxg-1 e £ie(G)(S). This last product is computed in the group G(S[e]) where g is viewed as an element of G(S[e]) by functoriality. The above defines an action of G on £ie(G)(S), called the adjoint action and denoted by g ^ Ad(g). This action in fact induces an R-group homomorphism

whose kernel is the centre of G.

Given a k-subspace V of g consider the R-group functor Z&(V) defined by

ZG(V) : S ^{g e G(S) : Ad(g)(vs) = vs for every v e V} (6.1)

for all S in R-alg, where vs denotes the image of v in g ®R S.

We will denote by RV the R-span of V inside g, i.e. RV is the R-submodule of g generated by V.

Remark 6.2 Note that Z&(V) = Z&(RV). This follows from the fact that the adjoint action of G on g is "linear" (in a functorial way).

We now introduce some of the central concepts of this work. A subalgebra m of the k-Lie algebra g is called an AD subalgebra if the adjoint action of each element x e m on g is k-diagonalizable, i.e. g admits a k-basis consisting of eigenvectors of ad0(x). A maximal AD subalgebra of g, namely one which is not properly included in any other AD subalgebra of g is called a MAD subalgebra of g.7

Example 6.3 Let G be a semisimple Chevalley k-group and T its standard maximal split torus. Let h be the Lie algebra of T; it is a split Cartan subalgebra of g. For all R we have g := Lie(GR) = g ®k R. Assume that R is connected. Then m = h ® 1 is a MAD subalgebra of g by [33, cor. to theo.1(i)]. We have ZGr (m) = TR.

Note that m is not its own normalizer. Indeed N0(m) = Zfl(m) = h®kR.Thus h®1 is not a Cartan subalgebra of g in the usual sense. However, in infinite-dimensional Lie theory—for example, in the case of Kac-Moody Lie algebras—these types of subalgebras do play the role that the split Cartan subalgebras play in the classical theory. This is our motivation for studying conjugacy questions related to MAD subalgebras.

7 It is not difficult to see that any such m is necessarily abelian, so AD can be thought as shorthand for abelian k-diagonalizable or ad k-diagonalizable.

Remark 6.4 Let s be an abelian Lie subalgebra of g. Let m1 and m2 be two subalgebras of s which are AD subalgebras of g. Because s is abelian their sum m1 + m2 is also an AD subalgebra of g. By considering the sum of all such subalgebras we see that s contains a unique maximal subalgebra m(s) which is an AD subalgebra of g. Of course this AD subalgebra need not be a MAD subalgebra of g.

We will encounter this situation when s is the Lie algebra of a torus S inside a reductive group scheme G. In this case we denote m(s) by m(S).

Remark 6.5 Let m be an AD subalgebra of g. Then for any extension S/R in k-alg the image m ® 1 of m in g ®R S is an AD subalgebra of g ®R S. Indeed if x e m and v e g are such that [x, v] = Xv for some X e k, then [x ® 1,v ® s] = v ® Xs = X(v ® s) for all s e S. Thus g ®R S is spanned as a k-space by eigenvectors of adg®RS(x ® 1). Note that if the map g ^ g ®R S is injective, for example if S/R is faithfully flat, then we can identify m with m ® 1 and view m as an AD subalgebra of g ®r S.

The main thrust of this work is to investigate the question of conjugacy of MAD subalgebras of g when g is a twisted form of g ®k Rn. The result we aim for is in the spirit of Chevalley's work, as explained in the Introduction. In the "untwisted case" the result is as expected.

Theorem 6.6 Let g be a split finite-dimensional semisimple Lie algebra over k and G the corresponding simply connected Chevalley group. Then all MAD subalgebras of g ®k Rn are conjugate to h ® 1 under G(Rn).

This is a particular case of Theorem 1 of [33] by taking Cor 2.3 of [18] into consideration. The proof is cohomological in nature, which is also the approach that we will pursue here. As we shall see, the general twisted case holds many surprises in place.

We finish by stating and proving a simple result for future use.

Lemma 6.7 Let G be a semisimple algebraic group over a field L of characteristic 0. Let T c G be a torus and Td be the (unique) maximal split subtorus of T. Set g = Lie (G), t = Lie (T) and td = Lie (Td). Then

(i) The adjoint action of Td on g is L-diagonalizable. In particular, td is an AD subalgebra of g.

(ii) td is the largest subalgebra oft satisfying the condition given in (i).

Proof Part (i) is clear. As for (ii) we may assume that G is semisimple adjoint. Let Ta be the largest anisotropic subtorus of T. The product morphism Td x Ta ^ T is a central isogeny, hence t = td © ta where ta = Lie (Ta). We must show that ta does not contain any nonzero element whose adjoint action on g is L-diagonalizable. Let h be such an element. Fix a basis {v1,...,vn } of g and scalars Xi e L such that

[ h, vt ] = Xi vt V 1 < i < n.

By means of this basis we identify GL(g) with GLn,L. Consider the adjoint representation diagrams

T ^ G GL(g) ~ GLn,L

t ^ g ^ gl (g) - gln,L■

Since G is of adjoint type Ad is injective, so that we can identify Twith ajubtorus, say T, of GLn,L. Similarly for T and Ta. Since T - T we see that T and Ta are the maximal split and anisotropic parts of T.

Let Dn be the diagonal subgroup of GLn,L. By construction we see that

ads(h) e Lie (Dn) n Lie (Tfl) = Lie (Dn n Ta),

this last by [24, theo. 12.5] since char(k) = 0. Thus Dn n Ta has dimension > 0. But then the connected component^of the identity of Dn n Ta is a non-trivial split torus which contradicts the fact that Ta is anisotropic. □

7 The correspondence between MAD subalgebras and maximal split tori

Throughout this section R will denote an object of k-alg such that X = Spec(R) is normal integral and noetherian and K its fraction field. The purpose of this section is to establish the following fundamental correspondence.

Theorem 7.1 Let G be a semisimple simply connected R-group and g its Lie algebra.

(1) Let m be a MAD subalgebra of g. Then Z©(m) is a reductive R-group and its radical contains a unique maximal split torus S(m) of G.

(2) Let S is a maximal split torus of G, and let m(S) be the unique maximal subalgebra of Lie algebra Lie (S) which is an AD subalgebra of g (see Remark 6.4). Then m(S) is a MAD subalgebra of g.

(3) The process m ^ S(m) and S ^ m(S) described above gives a bijection between the set of MAD subalgebras of g and the set of maximal split tori of G.

(4) If m and m' are two MAD subalgebras of g, then for m and m' to be conjugate under the adjoint action of G(R) it is necessary and sufficient that the maximal split tori S(m) and S(m') be conjugate under the adjoint action of G(R) on g.

Remark 7.2 Since S is split we have Lie (S) = X(S)° R where X(S)° is the cocharacter group of S. As we shall see in the proof of Lemma 7.5 m(S) = X (Sf k.

The proof of the Theorem will be given at the end of this section after a long list of preparatory results. What is remarkable about this correspondence is that MAD subalgebras exist over k but not over R while, in general, the exact opposite is true for split tori of G. It is this correspondence that allows us to use the methods from [41] to the study of conjugacy questions.

We begin with some general observations and fixing some notation that will be used throughout the proofs of this section. Since X is connected all geometric fibers of G are of the same type. Let G be the corresponding Chevalley group over k and g its Lie algebra.

Lemma 7.3 Let m be an AD subalgebra of g. Then

(1) dimk (m) < rank (g). In particular any AD subalgebra of g is included inside a MAD subalgebra of g.

(2) The natural map m ®k R ^ R m is an R-module isomorphism. In particular R m is a free R-module of rank = dimk (m).

(3) Let {v1,...,vm} be ak-basis of m. For every x e X the elements vi ® 1 e g ® RRx are Rx-linearly independent. Similarly if we replace Rx by Kor any field extension of K.

Proof The three assertions are of local nature, so we can assume that R is local. We will establish the Lemma by first reducing the problem to the split case. According to Remark 5.2 there exists a finite étale extension R/R such that R is integral and normal and G x R R ~ G r\ Note that the canonical map g ^ g ®rR ~ g ®kR is injective and that if {v1,...,vm} are k-linearly independent elements of m which are R-linearly dependent, then the image of the elements {v1,..., vm} on Lie(G xR R) ~ g ®k R are k-linearly independent and are R-linearly dependent.

Let K be the field of fractions of R. By Remark 6.5 the image of m under the injection g ^ g ®R R ~ g ®k R is an AD subalgebra of g ®k R. By [33, theo.l.(i)] the dimension of m is at most the rank of g. This establishes (1).

As for (2) and (3), the crucial point—as explained in [33, prop. 4]—lies in the fact that the image K of m under the injection g ^ g ®k K sits inside a split Cartan subalgebra H of the split semisimple K-algebra g ®k K. Consider the basis {<y1;..., co¿} of H consisting of the fundamental coweights for a base a1,...,au of the root system of (g®k K, H).Let 1 < n < m be such that {K1,... Kn} is a maximal set of K-linearly independent elements of g(KK). To establish (2) and (3) it will suffice to show that n = m.

Assume on the contrary that n < m. Write vi = £ cjiVj with c1i,... cu in K. The fact that the eigenvalues of adg(K)(K) belong to k show that the cji necessarily belong to k. Indeed Vi acts on g(K)aj as multiplication by the scalar cji.

Let v = vn+1. Write v = 'Yn=1 a^i with a1,...,an in K. Let cjn+1 = X j. Then (aj ,v)=Xj and

j ™ j •

Therefore for all 1 < j < I we have Xi aicji = ^j■

Write K = k © W as a k-space and use this decomposition to write ai = di + wi. Then ^i dicji = X j ■ A straightforward calculation shows that (aj ,K - XidiK) = 0 for all j■ This forces

= v = di

which contradicts the linear independence of the v[s over k. □

Remark 7.4 Let S < G be a split torus. Then there exist characters Xi : S ^ Gm,R for 1 < i < l such that g = ©^=1 gx; where

gk = {v e g : Ad(g)v = kt (g)v Vg e S(R)}.

At the Lie algebra level the situation is as follows. Let s = Lie (S) c g. Then s c S(R[e]). We avail ourselves of the useful convention that if s e s then to view s as an element of S(R[e]) we write ess. There exist unique R-linear functionals dki : s ^ R such that

(ess) = 1 + dki(s)e e R[e]x = Gm,R(R[e]).

Then for s e s and v e gki we have the following equality in g

[s,v] = d ki (s )v. (7.1)

Lemma 7.5 Consider the restriction Ads : S ^ Gl (g) of the adjoint representation of G to S. There exists a finite number of characters k1,...,ki of S such that g = 0i=1 gki • The ki are unique and

m(S) = {s e Lie (S) c S (R[e]) : dk{ (s) e k}.

Furthermore

dimk (m(S)) = rank(S) = rank R-mod (Rm(S)) and Lie(S) = Rm(S).

Proof We appeal to the explanation given in Remark 7.4. Let n ={s e s : dki (s) e k Vi}.

Then (7.1) shows not only that n c s is an AD subalgebra of g, but in fact that m(S) c n. By maximality we have m(S) = n as desired.

We now establish the last assertions. Let n be the rank of S, so S ~ Gm R and the character lattice X (S) of S is generated by the projections : Gm R ^ Gm,R. Since the kernel of the adjoint representation of G is finite the sublattice of X (S) generated by ki,..., ki has finite index; in particular every character n of S can be written as

a linear combination n = a1k1 +-----+ agke with rational coefficients ai;ai and

hence dn = a1dk1 +-----+andki. Similarly n can be written as n = a1n1 +-----+annn

with a1,...,an e Z and we then have dn = a1dn1 +----+ andnn. It follows that

m(S) = {s e s = {s e s = {s e s

dki (s) e k Vi} dn(s) e k Vn e X(S)} dn (s) e k Vi}.

The identification S ~ Gm R induces the identification s ~ Gna R. The above equalities yield

m(S) ~ {(s1,sn) : s, e k Vi}, hence the last assertions follow immediately. □

Proposition 7.6 Let m be an AD subalgebra of g. Then the submodule Rm is a direct summandof g.

Proof Let M = g/Rm. Assume for a moment that M is a projective R-module. Then the exact sequence

0 —> Rm —> g —> M —> 0

is split and the Proposition follows.

Thus it remains to show that M is a projective R-module or, equivalently, that for every prime ideal x of R the localized Rx-module Mx is free. Since localization is a left exact functor, and by Lemma 7.3 we have (Rm)x = Rxm the sequence

0 —> Rx m —> gRx Mx 0 is exact. By Lemma 7.3(3), the elements

vi ® 1,...,vm ® 1 e Rx m c g ® r Rx = gx

and the module gx satisfy the variation of Nakayama's lemma stated in [28, cor. 1.8]. Hence Rx m is a direct summand of g and this implies that Mx is free. □

Proposition 7.7 Let m be an AD subalgebra of g. Then Z©(m) is an affine R-group whose geometric fibres are (connected) reductive groups.

Proof By Proposition 7.6 Rm is a direct summand of g. It follows from [15, IIprop.1.4] that Z ©( Rm) = Z©(m) is a closed subgroup of ©.In particular, Z ©(m) is an affine scheme which is of finite type over Spec(R)■

Let x e Spec(R) be a point and let k(x) be an algebraic closure of k(x)■ Since the functor Z©(m) = Z©(Rm) commutes with base change, to verify the nature of its geometric fibers Z©(m)(x) we may look at

where G(x) = © ®r k(x) and m(x) is the image of m under g ^ g ®R k(x). Thus we may assume without loss of generality that the ground ring is a field. By results of Steinberg ([44, 3.3 and 3.8] and [44, 0.2]) we conclude that Z©(m)(x) is connected and reductive. □

Fix a split Cartan subalgebra h of g. With respect to the adjoint representation ad : g ^ Endk (g) we have the weight space decomposition

where a : h ^ k is a linear function such that the corresponding eigenspace ga is non-zero. The kernel of the adjoint representation of g is trivial, dim ga = 1 if a = 0 and g0 = h.

Lemma 7.8 Let a c h be a subalgebra. Then:

(1) The centralizer Zg(a) is a reductive Lie algebra whose centre is contained in h.

(2) If a e a is in generic position then Zg (a) = Zg (a).

Proof (1) The centralizer of a is generated by h and those ga for which a(x) = 0 for

every x e a. It is a well-known fact that this algebra is reductive. (2) The inclusion c is obvious. Conversely, the centralizer of a is generated by h and those ga for which a(a) = 0. Since a is generic all such roots a also satisfy a(x) = 0 for all x e a.

Lemma 7.9 Let aa e k, a e £. Then there exists at most one element h e h such that a(h) = aa.

Proof Since the kernel of the adjoint representation of g is trivial the result follows.

Lemma 7.10 Let S be an object ofk-alg. Let v e h ®k S be an ad k-diagonalizable element of g ®k S. If S is an integral domain then v e h.

Proof Let F be a field of quotients of S and view v as an element of g ®k F. The eigenvalues aa of v with respect to the adjoint representation are aa = a(v). By assumption they all belong to k. Thus the nonhomogeneous linear system a(x) = aa, a e £, has a solution over F, namely v. Since the coefficients of this system of equations are in k it also has a solution over k [see the proof of Lemma 7.3(2)]. By Lemma 7.9 such a solution is unique, hence v e h. □

Fix an arbitrary element h e h. Recall that G acts on g by conjugation and it is known that the orbit Oh = G ■ h is a Zariski closed subset of g (because h is semisimple). Let L c G be the isotropy subgroup of h in G. As we saw above L is a reductive subgroup and we have an exact sequence

1 —> L —> G G/L 1.

The algebraic k-varieties Oh and G/L have the distinguished points h and the coset e = 1 ■ L respectively. The group G acts on both Oh and G/L in a natural way and there exists a natural G-equivariant isomorphism X : Oh ~ G/L which takes h into e (see [5, III §9] for details). Hence if R is an object in k-alg and x e Oh (R), then x and h are conjugate by an element in G(R) if and only if k(x) e G(R) ■ e.

We now return to our simply connected semisimple R -group G and its Lie algebra

Lemma 7.11 Let m be an AD subalgebra of g. The affine scheme Z ©(m) is flat over Spec(R).

Proof That Z©(m) is an affine scheme over R has already been established. Since flatness is a local property it will suffice to establish the result after we replace R by its localization at each element of X. Lemma 5.1 provides a finite étale connected

cover R/R which splits ©. By replacing R by R we reduce the problem to the split case. Summarizing, without loss of generality we may assume that © = G xk R, g = g R := gR and R is a local domain.

As observed in Lemma 7.3 m is contained in a split Cartan subalgebra H of g ®k K := gK. Fix a generic vector v e m c gK. Let {aa, a e S } be the family of all eigenvalues of v with respect to the adjoint representation of gK . Since m is an AD subalgebra of gR, we have aa e k for every a e S. □

Sublemma 7.12 There exists a unique vector h e h whose eigenvalues with respect to the adjoint representation are {aa, a e S }. Moreover if v and h are viewed as elements of gK, then they are conjugate under G(K).

Proof Uniqueness follows from Lemma 7.9. As for existence, we note that H and hK are conjugate over K, hence hK clearly contains an element with the prescribed property. By Lemma 7.10 this element is contained in h. The conjugacy assertion follows from the construction of h. □

We now come back to the G-orbit Oh of h. We remind the reader that this is a closed subvariety of g.

Sublemma 7.13 v e Oh (R).

Proof The element v e gR can be viewed as a morphism

: Spec (R) ^ g.

The image of the generic point Spec(K) ^ Spec(R) ^ g is contained in Oh for v and h are conjugate over K. Since Oh is a closed subvariety of g and since Spec(R) is irreducible it follows that factors through the embedding Oh g. □

To finish the proof of Lemma 7.11 we first consider the particular case when m is contained in h. Then Z©(m) is obtained from the variety ZG (m) by the base change R/k so that flatness is clear.

In the general case, let h e h be the element provided by Sublemma 7.12. By Sublemma 7.13 we have v e Oh (R) = (G/L)(R). Denote by Rsh the strict henselisation of the local ring R, that is the simply connected cover of R attached to a separable closure Ks of K (see [37, §X.2 ]). Since the map p : G ^ G/L is smooth and surjective, Hensel's lemma [29, §4] shows that G(Rsh) ^ (G/L)(Rsh) is surjective. But Rsh is the inductive limit of the finite (connected) Galois covers of R, so there exists one such cover R' and a point g' e G(R') such that v = g'.h. Up to replacing R by R' (which is a noetherian normal domain) we may assume that v = h.

We now recall that ZgR (h) = ZgR (m) since h = v e m is a generic vector. Since the center of ZgR (h) is contained in hR and since m is contained in the center of its centralizer we have m c hR .Applying Lemma 7.10 then shows that m c h. Thus we have reduced the general case to the previous one. □

Proposition 7.14 If m is an AD subalgebra of g then Z©(m) is a reductive R-group.

Proof Since Z©(m) is flat and also finitely presented over R the differential criteria for smoothness shows that Z©(m) is in fact smooth over R because of Proposition 7.7. Furthermore, geometric fibers of Z©(m) are (connected) reductive groups in the usual sense (this last again by Proposition 7.7). By definition Z©(m) is a reductive R-group. □

Proof of Theorem 7.1 (1) Let m be a MAD subalgebra of g, and let S denote the maximal split torus of the radical T of the reductive R-group Z©(m). By Remark 6.4 the Lie algebra of S contains a unique maximal subalgebra m(S) which is an AD-subalgebra of g. By definition S < H = Z©(Rm). Denote Lie(S) by s. Since s c S(R[e]) it follows that in g we have [s, Rm] = 0. In particular since m (S) c s we have [m(S), m ] = 0. But then by Remark 6.4 m + m(S) is an AD subalgebra of g. Since m is a MAD subalgebra we necessarily have m(S) c m and now we are going to show that m(S) = m.

Recall that K denotes the quotient field of R. By Lemma 7.5 we have dim (m(S)) = rank (S), so that to establish that m(S) = m it will suffice to show that rank (S) > dimk(m), or equivalently that dimK(SK) > dimk(m).

We have HK = Z©K (Rm) = Z©K (Km), as can be seen from the fact that the computation of the centralizer commutes with base change. Since S is the maximal split torus of T then SK is the maximal split torus of TK = rad (HK) by Lemma 5.4. We also have

Lie (Hk ) = Lie (Z ©K (Rm)) = Lie (Z©K (K m)) = Z^ (K m).

Since Km is in the centre of Z0K (Km) = Lie (HK) and the centre of Lie (HK) coincides with Lie (TK) we conclude that Km c Lie (TK). On the other hand Km is an AD subalgebra of gK, so that by Lemma 6.7 Km c Lie (SK). This shows that dimK(Km) < dimK(SK). But by Lemma 7.3(3) we have dimk(m) = dimK(Km). This completes the proof that m(S) = m.

Now it is easy to finish the proof that S is a maximal split torus in 6. If S is contained in a split torus S' of larger rank then m(S) c m(S') is a proper subalgebra which contradicts to the fact that m = m(S) is a MAD subalgebra.

(2) Let S be a maximal split torus of 6, and let s = Lie (S) be its Lie algebra. By Remark 6.4 s contains a unique maximal subalgebra m(S) = m which is an AD-subalgebra of g. We have by Lemma 7.5 that Rm = Lie (S). Thus, appealing to Proposition 3.7 and Lemma 7.3(1) we obtain

We claim that m is maximal. Assume otherwise. Then by Lemma 7.3(1) m is properly included in a MAD subalgebra m' of g. We have

By Proposition 7.14 H' and H are reductive R-groups.Let T' and T be their radicals and let T'd, Td be their maximal split tori. We have S c T c T' and hence S c Td c T'd. But S is a maximal split torus in 6. Therefore S = T'd = Td and this implies m = m(S) = m(Td) = m(T'd). Recall that in part (1) we showed that m(T'd) = m' and thus m = m' - a contradiction.

(3) If m is a MAD subalgebra of g, the corresponding maximal split torus 6(m) is the maximal split torus of the radical of H = Z&(Rm). The proof of (1) shows that the MAD subalgebra corresponding to 6 (m) is m.

Conversely, if 6 is a maximal split torus of G then the maximal split torus corresponding to m(6) is the maximal split torus of the radical of the reductive group Zg(Rm(6)) = Zg(s) = Zg(6) as explained in the proof of (1). Clearly 6 is inside the radical of Z® (6). Since 6 is maximal split in G it is maximal split in the radical of ZG(6). Thus 6 = 6'.

(4) Follows from the construction and functoriality in the definition of the adjoint action at the Lie algebra and group level. □

8 A sufficient condition for conjugacy

In this section R denotes a normal noetherian domain and K its field of quotients. Let G be a reductive group scheme over R. We say that a maximal split torus 6 of G is generically maximal split if 6K is a maximal split torus of GK.

Proposition 8.1 Let 6 be a generically maximal split torus of G. If

Hlar(R, Zg(6)) = 1 (8.1)

then all generically maximal split tori of G are conjugate under G (R). We begin with two preliminary results.

Lemma 8.2 Let W be a finite étale R-group with R normal. Let K be the field of quotients of R. Then

(1) The canonical map

x : Hi ( R, W) H \ K, Wk )

is injective.

(2) H1ar(R, W) = 1.

Proof (1) Because of the assumptions on W we can compute H 1t (R, W) as the limit of H1 (S/R, W) with S a connected finite Galois extension of R.

Let Y = Gal(S/R). It is well-known that W corresponds to a finite group W together with an action of the algebraic fundamental group of R, and that H]t ( S/R, W) = H x(r, W( S)) (see [40, XI §5]). If L denotes the field of quotients of S then L/K is also Galois with Galois group naturally isomorphic to Y as explained in [9, Ch.5 §2.2 theo. 2]. Our map x is obtained by the base change K/R. By the above considerations the problem reduces to the study of the map

X : HX(Y, W(S)) HX(Y, W(S ®r K))

when passing to the limit over S. Since R is normal by [22, 18.10.8 and 18.10.9] we have S ®rK = L. If S is sufficiently large, W(S) = W = W(L ). The compatibility of the two Galois actions gives the desired injectivity.

(2) It is clear that H\ (R, W) is in the kernel of x- □

Lemma 8.3 Let S and S' be genetically maximal split tori of G. Then the transporter Te,&' = Trans©(S, S') is a (Zariski) locally trivial N©(S)-torsor over R.

Proof By [41, XI, 6.11 (a)], ts s' is a closed subscheme of G. It is clearly a right (formal) torsor under the affine R-group N©(S). Since SRp and SRp are maximal split tori of GRp they are conjugate under G( Rp) by [41, XXVI, 6.16]. Thus Te, S' is an N©(S)-torsor which is locally trivial (i.e. there exists a Zariski open cover X = U X such that Te, e № ) = 0). □

Proof of Proposition 8.1 Let S' be a generically maximal split torus of G. The transporter Te, e' yields according to Lemma 8.3 an element a e HZar(R, N©(S)). Our aim is to show that a is trivial.

Consider the exact sequence (on Xe-t ) of R -groups

with W = N©(S)/Zg(S). Then W is a finite étale group over R (see [41, XI, 5.9]). By Lemma 8.2(2) the image of a in H1 (R , W), which we know lies in HZar (R, W), is trivial. Thus we may assume a e Hjt (R, Z©(S)). To finish the proof we need to show that

a e Im [HlZar(R, Z&(&)) H}t(R, Z©(S))].

For this it suffices to show that the image ap of a in

is trivial for all p e X.

Since S is generically maximal split, SRp is a maximal split torus of GRp. Similarly for S'Rp .Nowby[41, XXVI prop. 6.16] SRp and S'Rp are conjugate under GRp (Rp) = G(Rp), Thus the image of a under the composition of the natural maps

H1 ( R, Z ©(S)) ^ H1 ( R, N©(S)) ^ H1 ( Rp, N©Rp (Srp )) ^ Hlét ( Rp, Grp )

is trivial. Let P be a parabolic subgroup of GRp containing Z©Rp (SRp ) as a Levi subgroup (see Lemma 3.4). Then (see the proof of [41, XXVI cor. 5.10]) we have

H1 (Rp Z©Rp (SRp )) - H1 (Rp P) ^ H1 (Rp GRp ). It now follows that ap is trivial. □

8.1 A counter-example to conjugacy for multiloop algebras

Let G and g be as in Theorem 7.1. We know that the conjugacy of two MAD sub-algebras in g is equivalent to the conjugacy of the corresponding maximal split tori. The following example shows that in general maximal split tori are not necessarily conjugate.

Let D be the quaternion algebra over R = R2 = k [t±\ t^1] with generators T1, T2 and relations Tx2 = t\, T22 = t2 and T2Ti = — TiT2 and let A = M2(D). We may view A as the D-endomorphism algebra of the free right rank 2 module V = D © D over D. Let G = SL (1, A). This is a simple simply connected R-group of absolute type SL4, r . It contains a split torus S whose R-points are matrices of the form

(0x-1)

where x e Rx. It is well-known that this is a maximal split torus of G. Consider now the D-linear map f : V = D © D ^ D given by

(u, v) ^ (1 + Ti)u — (1 + T2)v.

Let L be its kernel. It is shown in [17] that f splits and that L is a projective D-module of rank 1 which is not free. Since f is split, we have another decomposition V ~ L© D. Let S' be the split torus of G whose R-points consist of linear transformations acting on the first summand L by multiplication x e Rx and on the second summand by x—1. As before, S' is also a maximal split torus of G.

We claim that S and S' are not conjugate under G(R). To see this we note that given S we can restore the two summands in the decomposition V = D © D as eigenspaces of elements S(R). Similarly, we can uniquely restore the two summands in the decomposition V = L © D out of S'. Assuming now that S and S' are conjugate by an element in G(R) we obtained immediately that the D-submodule L in V is isomorphic to one of the components of V = D © D, in particular L is free -a contradiction.

9 The nullity one case

In this section we look in detail at the case R = k [t±1] where k is assumed to be algebraically closed. It is known that twisted forms of g ®k R are nothing but the derived algebras of the affine Kac-Moody Lie algebras modulo their centres [34]. We maintain all of our previous notation, except for the fact that now we specify that n = 1.

Lemma 9.1 Every maximal split torus of G is generically maximal split.

Proof Let S be a maximal split torus of our simply connected R-group G. We must show that SK is a maximal split torus of GK. We consider the reductive R-group H = Zg(S), its derived (semisimple) group D (H) which we denote by H', and the radical rad (H) of H. Recall that rad (H) is a central torus of H and that we have an exact sequence of R-groups

1 —> /i —> rad (H) x r H —> H —> 1 where m is the multiplication and / is a finite group of multiplicative type.

Since S is central in H it lies inside rad (H), hence it is the maximal split torus of rad (H). Recall that by Lemma 5.4, SK is still the maximal split torus of rad (H)K. If SK is not a maximal split torus of 6K, there exists a split torus S' of Hk such that S' is not a subgroup of rad (HK). Thus if we set (S' n H'K)° = T then T is a non-trivial split torus of HK. Then Z h (T) is a Levi subgroup of a proper parabolic subgroup p

of HK.

Let t = type (P) be the type of p. Let Part (H') be the R-scheme of parabolic subgroups of H' of type t. Then Part (H')(K) = 0. Since Part (H') is proper and R is regular of dimension 1, it follows that Part(H')(R) = 0. Let p' be a parabolic subgroup H' of type t. It is a proper subgroup, so that by Proposition 5.5 p' contains a copy of Gm, R. But then m : S x Gm,R —^ H yields a split torus of H that properly contains S (since the multiplication map has finite kernel), which contradicts the maximality of S. □

Theorem 9.2 In nullity one all MAD subalgebras of g are conjugate under the adjoint action of G(R).

Proof In view of the last Lemma and Proposition 8.1 it will suffice to show that if 6 is a maximal split torus of 6, then H\ar( R, Z g(S)) = 1. Since Z®(S) is a reductive R-group one in fact has a much stronger result, namely that Hjt(R, Z®(S)) = 1 (see [34, theo. 3.1]). □

Remark 9.3 Let G be the "simply connected" Kac-Moody (abstract) group corresponding to g (see [PK], and also [27] and [MP] for details). We have the adjoint representation Ad : G — Autk-Lie(g). The celebrated Peterson-Kac conjugacy theorem [35] for symmetrizable Kac-Moody (applied to the affine case) asserts that all MAD subalgebras of g are conjugate under the adjoint action of the group Ad (G) on g, while our result gives conjugacy under the image of 6(R), where the image is that of the adjoint representation Ad : 6 — Aut (g) evaluated at R. In the untwisted case it is known that the two groups induce the same group of automorphisms of g (see for example [27]). The twisted case appears to remain unstudied.

10 A density property for points of loop groups

In this section X = Spec(Rn). For a description of n1(X, a) see Sect. 4.2.

Let G be a linear algebraic k-group. Let n e Zx(ni(X, a), G(k)) be a loop cocy-cle and recall the decomposition n = (ngeo, z) into geometric and arithmetic parts described in Lemma 4.3. Recall that we may view ngeo as a k-group homomorphism — z G. We denote below by (z G)ngeo the centralizer in z G of the group homomorphism ngeo. Thus defined (z G)ngeo is a k-subgroup of z G

Remark 10.1 By continuity there exists m and a Galois extension k of k such that n factors through

n : rn,m — G(k)

Tn,m := Gal(Rn,m ®k k/Rn) = ^ (k) * Gal(k/k)

where m > 0 and k/k is a finite Galois extension containing all m-roots of unity in k. By means of this interpretation n can be viewed as a Galois cocycle in Z 1(f n,m, G(Rn,m ®k k)). We call this procedure "reasoning at the finite level".

We say that an abstract group M is pro-solvable if it admits a filtration

••• C Mn+1 c Mn C ••• C M0 = M

by normal subgroups such that n Mn = 1 and Mn/Mn+1 is abelian for all n > 0. If there exists a filtration such that Mn /Mn+1 are k-vector spaces, we say that M is pro-solvable in k-vector spaces.

Theorem 10.2 Let G be a linear algebraic k-group such that G° is reductive. Let n e Z 1(^1(X, a), G(k)) be a loop cocycle such that the twisted Rn-group H = n(GRn) is anisotropic. There exists a family of pro-solvable groups in k-vector spaces (J )i=1,..,n such that

H(Fn) ~ Jn x Jn—1 x ... x J1 x (zG) ' (k) ~ (Jn x Jn—1 x ... x J1) ■ H(Rn)■

Proof Twisting by z we may assume that z is trivial. It is convenient to work at a finite level, namely with a cocycle n : fn,m ^ G(k) as in Remark 10.1.

We proceed by induction on n > 0; the case n = 0 being obvious. We reason by means of a building argument and we view Fnm and its subfield Fn = (Fnm)fn'm as local complete fields with the residue fields Fn—1m and Fn—1 respectively. Let Bn = B(Gknm) be the (enlarged) Bruhat-Tits building of the Fn,m -group Gkn m [1012,46, §2.1]. Recall that Bn is equipped with a natural action of G( Fn,m) x fn,m .Since H is anisotropic the algebraic Fn-group HFn is also anisotropic by [19, cor. 7.4.3]. It is shown in [19, theo. 7.9] that the building of H Fn inside Bn consists of a single point

\$ whose stabilizer is G(Fn—1,m[[tnm]]). Since H(Fn) stabilizes \$ it follows that

H(Fn) = {g e G(Fn—1,m[[tnm]]) I n(a)a(g) = g Va e fnm}. (10.1)

We next decompose //¡n = in-1 x lm. The second component is a finite k-group of multiplicative type acting on G via ngeo. We let Gn—1 denote the k-subgroup of G which is the centralizer of this action [15, II 1.3.7]. The connected component of Gn—1 is reductive according to [38]. Since the action of /m-1 on G given by ngeo commutes with that of /m the k-group morphism ngeo : im ^ G factors through Gn—1.

Denote by ng—1 the restriction of ngeo to the k-subgroup m-1 of .Set fn—1,m := /^nr—l(k) x Gal(k/k) and consider the loop cocycle

nn—1 : fn—1,m ^ Gn—1(^)

attached to (1, nf—1). We define

Hn—1,Rn—1 = nn—1 (Gn—1,Rn—1).

The crucial point for the induction argument is the fact that the twisted Fn-1-group nn-1Gn-1 is anisotropic. This is established just as in [19, theo. 7.9]. We look now at the specialization map

_ 1 ~ spn : H(Fn) — G(Fn-hm [[tnm ]]) — G(Fn-1,m).

Let P be the parahoric subgroup of H° (Fn) attached to the point 0. Since the building of HFn consists of the single point we have P = H°(Fn). Recall that the notation P* stands for the "pro-unipotent radical" of P as defined in Sect. 14 of the Appendix. □

Claim 10.3 We have P* = ker(spn) and the image of spn is Hn-1(Fn-1).

Because G is a k-group it is clear that the kernel of the specialization map _ 1 ~ _ 1 G( Fn-1,m [[tnm ]]) — _G( Fn-1,m ) is contained in G°( Fn-1,m [[tnm ]]). Since (H/H°)( Fn) injects into (H/H°)(Fn,m) = (G/G°)(Fnm), the kernel of the specialization map spn

is contained in H°(Fn). The parahoric subgroup of G°(F„,m) attached to the point 0 ~ 1 ' is Q = G°(Fn-1,m [[tnm ]]) and we have

Q * = ker( Q — G°( Fn-1,m ))

by the very definition of Q*. Hence ker(spn) = P n Q* = P* by Corollary 15.6 applied to the point 0.

The group Hn-1(Fn-1) is a subgroup of Hn(Fn) which maps identically to itself by spn, so we have to verify that the specialization hn-1 of an element h e H(Fn) belongs to Hn-1(Fn-1). Specializing (10.1) at tn = 0, we get

n(y)Yhn-1 = hn-1 Vy eVn,m. (10.2)

We now apply the relation (10.2) to the generator tn of the Galois group Gal(Fn,m/Fn-1,m((tn))); it yields

n(Tn) hn-1 = hn-1, (10.3)

where n(Tn) e G(k), so that hn-1 e Gn-1 (Fn-1,m). Furthermore, the equality (10.2) restricted to fn-1,m shows that hn-1 e Hn-1(Fn-1). This establishes the Claim.

We can now finish the induction process. The group Hn-1(Fn-1) is a subgroup of H(Fn), so

H(Fn) = Jn * Hn-1(Fn-1)

where Jn := ker(spn) is the "pro-unipotent radical" and hence it is pro-solvable in k-spaces. By using the induction hypothesis, we have

Hn-1(Fn-1) = (Jn-1 * ••• * J1) * Gn-1 (k).

n — 1 geo

Since Gn--11 = Gn , we conclude that

H(Fn) = (Jn x ...X Ji) x G^geo(k)

as desired.

We have Gn (k) c H(R ), so we get the second equality as well.

11 Acyclicity, I

Let H be a loop reductive group scheme. We will denote by Hforal(Rn, H) (resp. Htorai(Rn■ H)irr) the subset of H 1(Rn, H) consisting of isomorphism classes of H-torsors E such that the twisted Rn-group gH admits a maximal torus (resp. admits a maximal torus and is irreducible).

Theorem 11.1 Let H be a loop reductive group scheme. Then the natural map H}orai(Rn, H)irr — H 1(Fn, H)

is injective.

Proof By twisting, it is enough to show that for an irreducible loop reductive group H the canonical map Hforal(Rn, H) — H 1(Fn, H) has trivial kernel. Indeed reductive Rn-group schemes admitting a maximal torus are precisely the loop reductive groups [19, theo. 6.1]. We now reason by successive cases.

Case 1 H is adjoint and anisotropic. We may view H as a twisted form of a Chevalley group scheme H Rn by a loop cocycle n : Rn) — Aut(H)(k).Wehavethefollowing commutative diagram of torsion bijections

Htoral(Rn ■

Aut(H)) -► H 1(Fn■ Aut(H))

H1oral( Rn ■ Aut (H))

- H 1(Fn■ Aut(H)).

The vertical maps are bijective by [20, III 2.5.4] and Remark 4.6, while the bottom map is bijective by [19, theo. 8.1]. We thus have a bijection

f : H10ral(Rn■ Aut(H)) H1 (Fn■ Aut(H)).

The exact sequence 1 — H — Aut (H) — Out(H) — 1 gives rise to the commutative diagram of exact sequences of pointed sets

Aut (H)( Rn)

Aut (H)( Fn)

Out(H)( Rn) II

Out(H)( Fn)

H}f (Rn ■ H)

H 1( Fn ■ H)

Hi (Rn ■ Aut (H))

H 1( Fn ■ Aut (H)).

Let v e H1t (Rn, H) be a toral class mapping to 1 e H1 (Fn, H). Since f is bijective thereexists u e Out(H)(Rn) such that v = p(u) and u e Im y .Since Out(H)(Rn) isa

finite group, the Density Theorem 10.2 shows that Aut(H)(Rn) and Aut(H)(Fn) have the same image in Out(H)( Fn ).Sou e Im 5, which implies that y = 1 e H1 (Rn, H). Case 2 H is irreducible. Set 3 = Z (H); it is an Rn-group of multiplicative type and we have an exact sequence of Rn-group schemes

1 — 3 —- H — Had — 1-

Here the adjoint group Had is anisotropic since H is irreducible. This exact sequence gives rise to the diagram

H1 (Rn, 3) —-—- H1 (Rn, H)

> H!(Fn, 3)

- H!(Fn, H)

■ H H Fn, Had )-

Note that the second vertical map is bijective by [18, prop. 3.4.(3)] since 3 is of finite type ([41, XII, §3]).

Let v e H1t (Rn, H) be a toral class mapping to 1 e H 1(Fn, H). Taking into account the adjoint anisotropic case, a diagram chase provides an element u e Hjt (Rn, 3) such that v = i*(u) and u belongs to the image of the characteristic map yFn. Since H1t (Rn, 3) is an abelian torsion group, the Density Theorem 10.2 shows that Had(Fn) and Had(Rn) have the same image in Hjt (Rn, 3). So u belongs to the image of pRn, and this implies that v = i*(u) = 1 e H1t (Rn, H) as desired. □

12 Conjugacy of certain parabolic subgroup schemes and maximal split tori

Theorem 12.1 Let H be a loop reductive group scheme over Rn. There exists a unique H(Rn )-conjugacy class of

(a) Couples (L, P) where P is a minimal parabolic Rn-subgroup scheme of H and L is a Levi subgroup of P such that L is a loop reductive group scheme.

(b) Maximal split subtori S of H such that Z^(&) is a loop reductive group scheme.

Remark 12.2 The counter-example in Sect. 8.1 shows that the assumption that L and Zh(S) be loop reductive group schemes is not superflous.

Proof (i) Reduction to the semisimple simply connected case. Let Hsc be the simply connected covering of the derived group scheme of H, and let E be the radical torus of H. There is a canonical central isogeny [23, §1.2]

1 — /i — Hsc x E -— H — 1.

Let (L, P) be a pair where P is a parabolic subgroup of H containing a Levi subgroup L. Then

f-1(P) = Psc x E, f-1 (L) = Lsc x E

where Psc is a minimal parabolic subgroup of the Rn-group Hsc and Lsc is a Levi subgroup of Psc. Conversely, from a couple (M, Q) for Hsc, we can define a couple ((MxE)/^, (QxE)/^) for H.By [19, cor. 6.3], loop group schemes are exactly those carrying a maximal torus. Since the last property is insensitive to central extensions [41, XII.4.7], the correspondence described above exchanges loop objects L with loop objects Lsc. Also it exchanges minimal parabolics of H with minimal parabolics of Hsc. Thus without loss of generality we may assume that H is simply connected.

(ii) Existence(a). Let H betheChevalley k-formof H and let n : n1( Rn) — Aut (H)(k) be a loop cocycle such that H = n(HRn). Let (T, B) be a Killing couple of H and n c A(H, T) be the base of the root system associated to (T, B). We denote by Had the adjoint group of H and by (Tad, Bad) the corresponding Killing couple. We have Aut(H) = Aut(Had). For each I c n, we have the standard parabolic subgroup P/ of H and its Levi subgroup LI, as well as PI,ad and LI,ad for Had.

Let I c n be the subset of circled vertices in the Witt-Tits diagram of H Fn. The version of the "Witt-Tits decomposition" given in [19, cor. 8.4] applied to Aut(Had) shows that

[n] e Im(H1oop(R n ■ Aut (Had ■ PI,ad ■ L I,ad ))irr — Hloop( Rn ■ Aut (Had )))■ Thus we may assume that n has values in

Aut (H, Pi , LI )(k) = Aut (Had, Pi ,ad, LIM )(k).

The twisted Rn-group schemes P = n(PI) and L = V(LI) are as desired for pFn is a minimal Fn-parabolic subgroup of H Fn by the definition of the Witt-Tits index.

(iii) Existence (b). Consider the pair (L, P) constructed in (ii) and let S be the maximal split subtorus of the radical T of L. By Proposition 5.3 we have Zh(S) = L so that Zfi(&) is a loop reductive group. To show that S is a maximal split torus of H it suffices to establish that so is S Fn.

Assume that SFn c S' is a proper inclusion where S' is a split torus in H Fn. By construction PFn is a minimal parabolic subgroup over Fn. Hence LFn = C^Fn (SFn) = CfiFn (S'). This implies that S' is contained in the radical TFn of LFn. But by Lemma 5.4, S is still maximal split in T over Kn and hence over Fn because T is split over a Galois extension Rn,m/Rn for some integer m - a contradiction.

(iv) Conjugacy (a). Let (L, P) be the couple constructed in (ii). Consider the Rn-

scheme Y = H/P of parabolic subgroups of type t(P). The exact sequence 1 —

P — H —> Y — 1 induces exact sequences of pointed sets

H(Rn) Y(Rn) Hj.(Rn, P)-► Hj.(Rn, H)

H1 (Rn - L)

(note that the natural mapping Hjt (Rn, L) — Hjt (Rn, P) is a bijection by [41, XXVI, 3.2]) and by base change

H(Fn) --U Fn) H 1(Fn, P) -► H 1(Fn, H)

H 1( Fn, L)

Let (M, Q) be another couple satisfying the conditions of Theorem 12.1. By [19], QFn c HFn is still a minimal parabolic subgroup; in particular Q has the same type t(P) and hence it corresponds to a point y e Y(Rn). □

Claim 12.3 <p(y) e Hloral(r«, L) ~ H1oral(Rn, P).

Indeed, p(y) is the class of the p-torsor E := f-1( y). We can assume without loss of generality that E is obtained from an L-torsor F. Then Q is isomorphic8 to the twist FP, and fL is a Levi subgroup of the Rn-group fP. Since Levi subgroups of fP are conjugate under Ru (fP)(Rn) [41, XXVI, 1.8], it follows that fL is Rn-isomorphic to M. The group scheme fL carries then a maximal torus and the claim is proved.

On the other hand, since P Fn and Q Fn are minimal parabolic subgroups of H Fn they are conjugate under H(Fn). Then y viewed as an element of Y(Fn) is in the image of fFn, hence pFn (y) = 1. It follows that p(y) belongs to the kernel of

H1oral(Rn, L)irr ^ Hl(Fn, L)

which is trivial by Theorem 11.1. Thus y e Im f, i.e. P and Q are H(Rn)-conjugate and so are the couples (L, P) and (M, Q).

(v) Conjugacy (b). We still denote by (L, P) the couple constructed in (ii). Let ©' be a maximal split subtorus of H such that its centralizer L' = Zfi(&) is a loop reductive group scheme. By Lemma 3.4, Z^(&') is a Levi subgroup of a parabolic subgroup of P' of H. By Proposition 3.6 (c), P' is a minimal parabolic subgroup of H. By (iv), the couple (L', P') is conjugate under H (Rn) to (L, P). We may thus assume that L = L', i.e. Zfi(S>) = Zh(S'). It follows & is a central split subtorus of L, hence & c S. But & is a maximal split subtorus of H, so we conclude that & = S' as desired. □

13 Applications to infinite-dimensional Lie theory

Throughout this section we assume that k is algebraically closed of characteristic zero, G is a simple simply connected Chevalley group over k, and g its Lie algebra. We fix integers n > 0, m > 0 and an «-tuple a = (ai,...,an) of commuting elements of Autk (g) satisfying = 1. Let R = Rn and R = Rnm .Recall that R/R is Galois and that we can identify Gal(R/R) with (Z/mZ)n via our choice of compatible primitive roots of unity.

Recall also from the Introduction the multiloop algebra based on g corresponding to a, is

8 Surprisingly enough, this compatibility is not in Giraud's book. A proof can be found in [14, lemme 4.2.33].

'± in _

l(g,a)= 0 gii...i„®tm ...tnm cg®kR

(il,...,in )eZn

It is a twisted form of the R-Lie algebra g ®k R which is split by R. The R/R form L (g, a) is given by a natural loop cocycle

n = n(a) e Z 1(r, Aut(g)(k)) c Z 1(r, Aut(g)(R)).

Since Aut(g) — Aut(G) we can also consider by means of n the twisted R-group G = n Gr . As before we denote the Lie algebra of G by g. Clearly, g — L (g, a).

By a Theorem of Borel and Mostow [6] there exists a Cartan subalgebra h of g that is stable under the action of a (by which we mean that each of the ai stabilizes h). By restricting a to h we can consider the loop algebra based on h with respect to a,

yy i1 in _

L (h, a) = 0 hi!...i„ ® tm ...tnm c h ®k R

(i1,...,in )eZn

Let T be the maximal torus of G corresponding to h. Denote by Ta (resp. ha) the fixed point subgroup of T (resp. subalgebra of h) under a, i.e the elements of T (resp. h) that are fixed by each of the ai. Since the torus T is also a-stable, just as above, we can consider its twisted form T = nTR and the corresponding Lie algebra h = nhR. The same formalism already mentioned yields that h — L(h, a). Let Td be the maximal split torus of T. It is easy to see that

Td — TR = n(TR) c G = nGr.

According to Remark 6.4 its Lie algebra td contains a unique maximal subalgebra m which is an AD subalgebra of g. The description of this algebra is quite simple:

m = ha ®k 1 = ho,...,o ®k 1 C L (g, a) — g.

By Theorem 7.1 m is a MAD subalgebra if and only if Td is a maximal split torus of G, in which case m = m(Td). We will call MAD subalgebras of a multiloop algebra which are of this form Borel Mostow MAD subalgebras of g.

Clearly, Zg (m) is precisely the multiloop algebra L (Zg (ha), a). Note that by Propo-sition3.7, Zg (ha) is the Lie algebra of the reductive k-group H := ZG (ha) = ZG(Ta) and hence by twisting we conclude that Zfl(m) is the Lie algebra of Z&(Td) = ZG(TR) — n H R ■

Proposition 13.1 (1) ZG(Td) is a loop reductive group.

(2) m is a MAD subalgebra if and only if the dimension of h0,...,0 is maximal among the Cartan subalgebras of g normalised by a .In particular, Borel-Mostow MAD subalgebras exist.

Proof (1) We have explained above that Z©(Td) ~ nHR. This last group is loop

reductive by definition since n is a loop cocycle. (2) It follows from Theorem 12.1 that all maximal split tori in G whose centralizers are loop reductive groups and corresponding MAD subalgebras are conjugate, hence have the same dimension, say r, equal to the rank of G over Fn. Since m is an AD subalgebra whose centralizer is a multiloop algebra Theorem 12.1 applied to Z©(Td) shows that dimk m = dimk(h0,...,0) < r and hence m is a MAD subalgebra if and only if dimk(h0,...,0) = r. It is then enough to show that there exists a Borel-Mostow AD subalgebra of rank r, that is we need to find a Cartan subalgebra h' of g normalized by a such that dimk(h'0,...,0) = r.

If r = 0 there is nothing to prove. Assume that r > 0. Denote by I the type of minimal parabolic subgroups of G over Fn. Fix a Cartan subalgebra h0 c g, the corresponding maximal torus T0 c G and a basis of the root system £ (T0, G). In the course of the proof of Theorem 12.1 we showed that up to conjugacy by an element of G(k), we can assume that a normalizes the standard parabolic group PI and also the standard Levi subgroup LI .

Let S c T0 be the torus consisting of the fixed point subgroup of the radical of LI under a. Then SR n(LR) c nGR is the maximal split torus in the radical of n(LI)R. Since the twist n(PI)R ®R Fn is a minimal parabolic subgroup of G over Fn and n(L)I ®R Fn is its Levi subgroup it follows that SR ®R Fn is a maximal split torus of G over Fn; in particular dimk (S) = r.

Let s c h0 be the Lie algebra of S. We have dimk (s) = dimk (S) = r and by our construction a acts trivially on s. The reductive subalgebra Zg(s) is stable under a, so the application of Borel-Mostow's theorem provides a Cartan subalgebra h' of Zg(s) stable under a. Its fixed subalgebra has dimension < r and contains s, hence it coincides with s. □

According to our Conjugacy Theorem all Borel-Mostow MAD subalgebras of a multiloop algebra are conjugate under G(Rn). There is a very important class of multiloop algebras, the so-called Lie tori, where Borel-Mostow MAD subalgebras play a crucial role. We now turn our attention to them.9

Theorem 13.2 Let L be a centreless Lie torus which is finitely generated over its centroid. The (relative) type A is an invariant of L.

Proof After sorting through the several relevant definitions, the Theorem follows from our conjugacy of Borel-Mostow MAD subalgebras in view of the realization of the Lie tori in question as multiloop algebras as established in [3]. □

The spirit of this result should be interpreted as the analogue that on g we cannot choose two different Cartan subalgebras that will lead to root systems of different

9 Lie tori were introduced by Yoshii [47,48] and further studied by Neher in [31,32]. The terminology is consistent with that of tori in the theory of non-associative algebras, e.g. Jordan tori. But in the presence of algebraic groups, where tori are well defined objects, the terminology is a bit unfortunate.

type. More generally, it is the analogue of the fact that the relative type of a finite-dimensional simple Lie algebra (in characteristic o) or of a simple algebraic group is an invariant of the algebra or group in question.

The relevance of centreless Lie tori is that they sit at the "bottom" of every Extended Affine Lie Algebra (see [2,31,32]). A good example is provided by the affine Kac-Moody Lie algebras. They are of the form (see [26])

where L is a loop algebra of the form L (g, n) for some (unique) g and some (unique up to conjugacy) diagram automorphism n of g. The element c is central and d is a degree derivation for a natural grading of L. If h is the standard Chevalley split Cartan subalgebra of g, then H = hn + kc + kd plays the role of the Cartan subalgebra for E.

Remark 13.3 The invariance of the relative type was established in [1] by using strictly methods from EALA theory. Allison also showed that under the assumption that conjugacy (as established in this paper) holds, any isotopy between Lie tori necessarily preserves the external root data information. This is a very important result for the theory of EALAs for, together with conjugacy, it yields a very precise description of the group of automorphisms of Lie tori.

14 Acyclicity, II

Theorem 14.1 Let H be a loop reductive group scheme over Rn. Then the natural map

Htoral(Rn ■. H) ^ H1(Fn, H).

is bijective.

Remark 14.2 The theorem generalizes (in characteristic 0) our main result in [13]. Indeed, in that paper we showed that if n = 1 and G is a reductive group over an arbitrary field k of good characteristic then H1(Ri, G) ^ Hx(Fi, G) is bijective and that every G-torsor is toral. The Theorem also generalizes the Acyclicity result of [19], which is used in the present proof and covers the case when H is "constant".

The proof of the theorem is based on the following statement which generalizes the Density Theorem 1o.2 to the case of arbitrary loop reductive group schemes, not necessary anisotropic.

Theorem 14.3 Let H be a linear algebraic k-group whose connect component of the identity is reductive. Let n : n1 (Rn) ^ H(k) be a loop cocycle and consider the loop reductive Rn-groups H = nHR and H° = nH°R . Let (P, L) be a couple given by Theorem 12.1 for H°. Then there exists a normal subgroup J of L(Fn) which is a quotient of a group admitting a composition serie whose quotients are pro-solvable groups in k-vector spaces such that

H(Fn) = (H(Rn), J, H(Fn)+)

where H( Fn )+ stands for the normal subgroup of H( Fn ) generated by one parameter additive Fn-subgroups.

Remark 14.4 If H is semisimple simply connected, isotropic and Fn-simple we know that H(Fn)/H(Fn)+ = H(Fn)/R [16, 7.2], where R is an R-equivalence, so that the group H(Fn)/H(Fn)+ has finite exponent (ibid, 7.6). In this case, the decomposition readsH(Fn) = <H(Rn), HF)+).

Proof Case 1 H is a torus T. We leave it to the reader to reason by induction on n to establish the case of a split torus T = Gm (the case n = 1 follows from the identity Fj* = Rxx ■ ker(k[[t1]]x ^ kx). Since all finite connected étale coverings of Rn are also Laurent polynomial rings over field extensions of k [19, lemma 2.8] and the statement is stable under products, the theorem also holds for induced tori.

Let T be an arbitrary torus. Since T is isotrivial, it is a quotient of an induced torus E. We have then an exact sequence

1 ^ S —+ E f T ^ 1 of multiplicative Rn-group schemes. It gives rise to a commutative diagram

1 -► S(Rn) E(Rn) —T(Rn) —H). (Rn, S) -► 1

1 -► &(Fn) E(Fn) T(Fn) H!(Fn, S) -► 1

with exact rows. Note that the right vertical map is an isomorphism by [18, prop. 3.4] and that surjectivity on the right horizontal maps is due to the fact Hl{ (Rn, E) = Hx(Fn, E) = 1. By diagram chasing we see that

T(Rn)/fFn (E(Rn)) T(Fn)/fRn (E(Fn)). Therefore the case of the induced torus E provides a suitable group J such that T( Fn) =

T(Rn ) • fFn (J).

Case 2 H = L is irreducible. Let C be the radical torus of L. We have an exact sequence [41, XXI, 6.2.4]

1 -> f —DL x Rn C —^ L -> 1.

Here f is a natural multiplication map and f is its kernel. It gives rise to a commutative diagram of exact sequences of pointed sets

(DL x C)(Rn) — L( Rn) -—V Hjt (Rn, f) -——V Hi Rn, DL x C)

(DL x C)(Fn) —^ L(Fn) V Hl(Fn, f) ^ Hl(Fn, DL x C).

Note that the image of the map H1 (Rn, f) ^ H1 (Rn, DL) is contained in Hordi (Rn, DL). So taking into consideration Theorem 11.1 (applied to the irreducible loop reductive group scheme DL and chasing the above diagram we see that

L(Rn)/fRn ((DL)(Rn) x C(Rn)) L(Fn)/fFn((DL)(Fn) x C(Fn)).

The case of DL done in Proposition 10.2 together with the case of the torus C provide a suitable normal group J such that L(Fn) = L(Rn) ■ J.

Case 3 H = H°. Since H is loop reductive by assumption it suffices to observe that H(Fn) is generated by L(Fn) and H+(Fn) [7, 6.11].

Case 4 For the general case it remains to show that for an arbitrary element g e H( Fn) the coset gH°(Fn) contains at least one Rn-point of H.

Let S be the maximal split torus of the radical of L. The torus gSFng-1 c H°Fn is maximal split, hence gSFng-1 = g1SFng-1 for some g1 e H°(Fn). Thus replacing g by g-1 g if necessary, we may assume that gSFng-1 = SFn. Then we also have g(LFn)g-1 = Lf„, so that g e Nn(L)(Fn).

The torus S is clearly normal in Nfi(L). Hence we have an exact sequence

Nh(L) H := NH(L)/& 1.

Note that since Hjt(Rn, S) = 1, the natural maps NH(L)(Rn) ^ H'(Rn) and Nfi(£)(Fn) ^ H'(Fn) are surjective. Furthermore, H' satisfies all conditions of Theorem 10.2, so that the required fact follows immediately from that theorem applied to H' and from the surjectivity of the above maps. □

We can proceed to the proof of Theorem 14.1.

Proof Injectivity: By twisting, it is enough to show that the natural map H1oral (Rn, H) ^ H 1(Fn, H) has trivial kernel.

We first assume that H is adjoint. We may view H as the twisted form of a Chevalley group scheme HRn by a loop cocycle n : n\(Rn) ^ Aut(H(£)). The same reasoning given in Case 1 of the proof of Theorem 11.1 shows that we have a natural bijection

Htoral( Rn Aut(H)) H!(Fn■ Aut(H)).

(14.1)

The exact sequence

1 ^ H ^ Aut (H) ^ Out(H) ^ 1

gives rise to a commutative diagram of exact sequence of pointed sets

Aut(H)(Rn) Out(H)(Rn) H1 (Rn, H) -► H1t(Rn, Aut(H))

„ II | ^ Aut(H)(Fn) — Out(H)(Fn) -> H 1(Fn, H) -> H 1(Fn, Aut(H)).

Let v e Hjt(Rn, H) be a toral class mapping to 1 e H*(Fn, H). In view of bijec-tion (14.1) there exists u e Out(H)(Rn) such that v = (p(u) and u belongs to the image of f. Since Out(H)(Rn ) is a finite group, the Density Theorem 14.3 shows that Aut(H)(Rn) and Aut(H)(Fn) have same image in Out(H)(Fn). So u belongs to the image of y, hence v = 1 e H1(Rn, H).

Let now H be an arbitrary reductive group. Set C = Z (H). This is an Rn-group of multiplicative type and we have an exact (central) sequence of Rn -group schemes

1 ^ C H ^ Had ^ 1. This exact sequence gives rise to the diagram of exact sequences of pointed sets

Had(Rn) Hk(Rn■ C) —Îî-^ Hjt(Rn■ H) -► Hjt(Rn■ Had) —^ Hi(Rn, C)

Had (Fn) —-H 1(Fn, C) -^ H 1(Fn, H) -> H 1(Fn, Had) H* (Fn, C).

The isomorphisms Hlt (Rn, C) = Hl (Fn, C) comes from [18, prop. 3.4.(3)] for i = 1, 2.

Let v e H1t (Rn, H) be a toral class mapping to 1 e H 1(Fn, H). Taking into account the adjoint case, a diagram chase provides u e H1{(Rn, C) such that v = it.(u) and u Fn belongs to the image of the characteristic map yFn. Since H1t (Rn, C) is an abelian torsion group, the Density Theorem 14.3 shows that Had (Fn) and Had (Rn) have the same images in H1t(Rn, C). So u belongs to the image of pRn. Hence v = it.(u) = 1 e H1t (Rn, H). e

Surjectivity Follows by a simple chasing in the diagrams above. □

Question. Assume that H is loop semisimple simply connected, isotropic and Fn -simple. Let H(Rn)+ c H(Rn) be the (normal) subgroup generated by the Ru (P)(Rn) where P runs over the set of parabolic subgroups of H considered in Theorem 12.1. Is the map

H(Rn)/H(Rn)+ ^ H(Fn)/H(Fn)+

an isomorphism?

Note that the map is surjective by Remark 14.4.

Appendix: Greenberg functors, Bruhat-Tits theory and pro-unipotent radicals

We are given a complete discrete valuation field K of valuation ring O = OK and of perfect residue field k = O/n O. Here n e O is a uniformizer. In the inequal

characteristic case denote by e0 the absolute ramification index of O, i.e. p = uneo for a unit u e O where p = char(k); in the equal characteristic case, put e0 = 1. We denote by Osh the strict henselization of O, or in other words, its maximal unramified extension.

Greenberg functor

We recall here basic facts, see the references [21], [30, §III.4], [8], [4].

Assume first that we are in the unequal characteristic case, that is K is of characteristic 0 and k is of characteristic p > 0.

For each k-algebra A and r > 0, we denote by Wr (A) the group of Witt vectors of length r and by W(A) = lim Wr (A) the ring of Witt vectors (see [43, §II.6]). There exists a unique ring homomorphism W(k) ^ O commuting with the projection on k = W0(k) (ibid, II.5).

Let S be an affine W(k)-scheme. Recall that for each r > 0, the functor k-alg ^ Sets given by

A ^ S(Wr(A))

is representable by an affine k-scheme Greenr (S). The projective limit Green (S) : = limGreenr (S)

is a scheme which satisfies Green(S)(A) = S(W(A)). If X is an affine O-scheme, we deal also with the relative versions of the Greenberg functor

Gr(X) := Greenr ( XJ , G(X) := Green (

\O/ W(k) ) \O/ W(k)

We have Gr(X)(k) = X(O/prO) and G(X)(k) = X(O). We also have G(Spec(O)) = Spec(k); if X is a O-group scheme, then G(X) and the Gr (X) carry a natural k-group structure [4, 4.1].

Lemma 15.1 Let L /K be a finite extension, OL the valuation ring of L and l/k the corresponding residue extension. Let Y/OL be an affine scheme. Let H_/1 be the relative Greenberg functor of Y with respect to W (l). Then we have natural isomorphisms ofk-schemes (for allr > 1J

Gr (n YJ -nHr(Y), G (n YJ -nH(Y).

\ol/o ) l/k \ol/o ) l/k

In particular if k = l then we have Gr( n Y) = Hr (Y) and g( n Y) —

xol/o ' yOL/o '

H (Y).

Proof We have a commutative square

O -► Ol

W(k) -> W(l).

So by the functorial properties of the Weil restriction, we have

n n Y = n Y = n n Y. (151)

O/ W(k) OL/O OL/ W(k) W(l)/W(k) OL/ W(l)

Let A be a k-algebra. Using (15.1) and the definitions of the Greenberg functors, we have

Gr ( EI Y ) (A) = Greenr ( Y ) (A)

\Ol/O j \Ol / W (k) J

n n YJ (Wr (A)) VW(l)/ W(k) Ol/ W(l) J

= ( n YJ (W (l) ®W (k) Wr (A)).

\Ol / W (l) J Since Wr (A) is a Wr (k)-module, we have

W(l) ®w(k) Wr (A) = Wr (l) ®Vr (k) Wr (A) = Wr (A ®k l) by [25, 1.5.7]. Hence

Gr (n y) (A) = ( n y) (Wr(A ®kl)) = Rl/k(Hr)(A)

\Ol/O J \Ol / W (l) J

as desired. By passing to the limit, we get the second identity. □

Lemma 15.2 (1) Let X/O be an affine scheme of finite type such that = 0. Then G (X) = 0

(2) Let N/ O be an affine group scheme of finite type such that NK = Spec (K). Then G(N) = G (Spec (O)) = Spec(k).

Proof (1) We have X = Spec (A) where A is an O/nd O-algebra of finite type for d large enough Put r0 = d e0. Then pr0 A = 0. For a k-algebra A we have by definition

G(X)(A) = Homo(A, V(A) ®w(k) O). But W(A) ®W(k) O is p-torsion free, so G(X) = 0.

(2) We have N = Spec(B) and we have the decomposition B = O © I where I is the kernel of the co-unit of the corresponding Hopf algebra. The O-module I is an ideal of B which is an O/nd O-algebra of finite type. The same reasoning as above shows that

G(N)(A) = Homo(B, W(A) ®w(k) O) = Homo (O, W(A) ®w(k) O) = G (Spec( O ))(A).

Thus G(N) = G(Spec(O)) which is nothing but Spec(k) as explained above.

Secondly, assume that k and K have the same characteristic (0 or p > 0) and we still assume that k is perfect. Then k embeds in O (in a unique way, [22, 21.5.3]) and for an O -scheme X the functors

G(X) X and Gr (X) := (X xo O/nrO)

O |k O /nrO | k

play the desired role [8, §9.6] and allow us to write

X(O) = lim X(O/nrO) = lim Gr(X)(k)

where the Gr(X) are k-schemes (by Weil restriction considerations [8, §7.6]). The two lemmas are true as well.

Congruence filtration

Let G be a reductive K-group and denote by B = B(G, K) its (extended) Bruhat-Tits building. Let x be a point of B and denote by Px the parahoric subgroup

Px = {g e G(K) | g(x) = x}.

Denote by px the canonical smooth group scheme over O defined by Bruhat-Tits [11, §5.1] with generic fiber G and such that px (O) = Px or, more precisely,

Px(Osh) = {g e G(Ksh) | g(x) = x}

where x is viewed as an element in B(G, Ksh) via the canonical mapping B(G, K) ^ B(G, Ksh). Since px is smooth we have

Px (O) = lim Px (O /nnO)

and the transition maps px (O/nn+1 O) ^ px (O/nnO) are surjective with kernel Lie(Px) ®ok ([30, III.4.3])

The application of the relative Greenberg functor to the smooth affine group scheme px defines a projective system of affine k-groups Px,n (n > 1) such that

Px,n (k) = Px (O/nne0 O).

The Px,n are smooth according to [4, Lemme 4.1.1]. The kernel Px,n+1/n of the transition maps Px,n+1 ^ Pxn are k-unipotent abelian groups which are successive extensions of the vector group of Lie(px) ®O k (ibid. or [30, III.4.3]).

For each n > 1, we denote by Rn,x := Ru (Pxn) the unipotent radical of Px,n; since k is perfect, it is defined over k and split [15, IV.2.3.9]. The quotient Mx of Px,n by Rx,n is independent of n. It is nothing but the quotient of the special fiber of px by its k-unipotent radical Rx. The k-group M£ is reductive according to [11, 4.6.12]. We consider the "maximal pro-unipotent normal subgroup"

P* := ker(Px(O) ^ Mx (k))

which is of analytic nature. Denote by

Px / k := Km Px ,n n>1

and by P*/k = ker(Px ^ Mx). By construction we have P* = P* (k). Lemma 15.3 For each n > 1, there is a short exact sequence of affine k-groups 1 ^ ker(Px ^ Px ,n) ^ Px ^ Rx ,n ^ 1.

Proof Apply the snake lemma to the commutative diagram of k-groups

1 -► PX -► Px -► Mx ^ 1

1 -► Rx -► Px -► Mx ^ 1.

Lemma 15.4 The k-group PX is the unique maximal split pro-unipotent closed normal subgroup of the pro-algebraic affine k-group Px.

Proof Since

ker(Px ^ Px,1) = limker(Px n ^ Px,1)

is pro-unipotent, the above exact sequence shows that PX is pro-unipotent. Let Ux be a pro-unipotent normal closed subgroup of Px. The image of Ux by the map Px ^ Mx is a normal unipotent connected k-subgroup. Since M£ is reductive, its image is trivial. Therefore Ux c PX which completes the proof. □

Behaviour under a Galois extension

Just as does the whole theory, the construction of P* has a very nice behaviour with respect to unramified extensions of K. The behaviour under a given tamely ramified finite Galois field extension L/K is subtle. Since such an extension is a tower of an unramified extension and a totally ramified one, we may concentrate on the case when L/K is totally (tamely) ramified. Then L/K is cyclic of degree e invertible in k = K = L .The Galois group T = Gal(L / K) acts on the building B(G, L). The Bruhat-Tits-Rousseau theorem ([39, §5], see also [36]) states that the natural map

j : B(G, K) ^ B(G, L)

induces a bijection B(G, K) —> B(G, L)T. For z e B(G, L), we denote by Qz the parahoric subgroup of G(L) and by Qz the canonical group scheme over OL attached to the point z.

For a e T, we have a(Qz) = Qa(z). Hence for the canonical group schemes over OL attached to z and a(z), there is a natural cartesian square

Qa(z) f"'Z > Qz

Spec( Ol ) Spec( Ol ).

Put y = j (x) e B(G, L)r. We then have an O-action of V on the scheme Qy. We note that

Px = G(K) n Qy = G(L)V n Qy = QV. (15.2)

As above we consider the groups Qy,n and their projective limit Qy. Since k is the residue field of Ol, all Qy,n and Qy are k-groups. The action of V on Qy induces its action on Qy,n, hence on My where My stands for the reductive k-group attached to y, and on their projective limit Qy. By Lemma 15.4, Q* is a characteristic k-subgroup of Qy, hence V also acts on the pro-algebraic k-group Q*. Our goal is to prove the following fact:

Proposition 15.5 There is a natural closed embedding Px ^ Qy and we have

P* = Px n Q*.

This gives rise to an isomorphism Mx —> MV.

By taking k-points we get the following wished compatibility, namely. Corollary 15.6 We have

P* Px n Q y.

Consider the Weil restriction Jx := nOL/O (Qy) and recall it is a smooth O-scheme [49, §2.5]. Let N be the kernel of the natural map px ^ Jx, its generic fiber is trivial. As above, applying the Greenberg functors to the O-schemes Jx and N we get k-groups Jx,n, Jx and N„, N.

Since the Greenberg functor is left exact, we get an exact sequence

1 ^ n ^ Px ^ Jx •

Since N^ = 1, we have N = 1 according to Lemma 15.2 (2). Hence we may view Px as a closed subgroup of Jx. But according to Lemma 15.1, Jx,n is nothing but Qy,n. This implies Jx is isomorphic in a natural way to Qy. Thus we have constructed a natural closed embedding Px ^ Qy.

" riy,n and (Qy) = Qy 11 Qy of Qy and Qy

Define the k-subgroups Q^ := lim Q^« and (Q*)r = Q^ n Q* of Qy and QÎ

respectively.

Lemma 15.7 (1) If k'/ k is a finite extension of fields, the projective system

(qL (k'))n> i has surjective transitions maps. Therefore the projective system

ofk-groups (Qy,n)n>1 has surjective transitions maps.

(2) Ifk' / k is a field finite extension, we have an exact sequence

1 ^ (Q;)y(k') ^ Qy (k) ^ My (k) ^ 1;

hence the sequence of the pro-algebraic k-groups

1 ^ (q; )y ^ Qy ^ My ^ 1

is also exact.

(3) The algebraic k-group My is smooth and its connected component of the identity is reductive.

Proof (1) Since Bruhat-Tits theory is insensitive to finite unramified extensions, we may assume without loss of generality that k = k'. Since Qy,«+1/« is a k-split unipotent group, we have an exact sequence

1 ^ Qy,n+1/n (k) ^ Qy,n+1(k) ^ Qy,n (k) ^ 1.

It gives rise to the exact sequence of pointed sets

1 ^ Qy ,n+1/n (k)y ^ Qy,„+1(k)y ^ Qy,n(k)y ^ H 1(y, Qy , n +1 / n (k)).

Since Qy,n+1/n (k) admits a characteristic central composition serie in k-vector spaces and the order of y is invertible in k, the right hand side is trivial. A fortiori, the system (Qy,n) of k-groups is surjective (because Qy,n (k) = Qy,n (k)y). (2) By part (1), the map Qy (k) ^ (Qyi1)y(k) is surjective. The same argument as in (1) shows that (Qyi1)y(k) ^ My (k) is also surjective. By taking the composition of these maps we conclude the map Qy (k) ^ My (k) is surjective whence the desired exactness of both sequences.

(3) The group V may be viewed as a finite abelian constant group scheme whose order is invertible in k. Hence V is also a (smooth) k-group of multiplicative type. Since My is affine and smooth, Grothendieck's theorem of smoothness of centralizers [41, XI, 5.3] shows that M£ is smooth. Its connected component of the identity is reductive by a result of Richardson [38, prop. 10.1.5].

We can now proceed to the proof of Proposition 15.5.

Proof We have to show that our closed embedding Px ^ Qy which we constructed above induces an isomorphism P* —> Px 0 Q*. Since Px 0 Q* is a normal closed split pro-unipotent subgroup of Px it is contained in P*. Hence it remains only to show that P* c Q*.

We now recall from (15.2) that Px = Qy and Qy(k) = Qy (k)y = Qy. By Lemma 15.7, Qy (k) projects onto M£ (k), so the composite map

Px = Px (k) ^ Qy (k) ^ My (k)

is surjective. Since this is true for all finite extensions of k, the homomorphism of

k-algebraic groups Px ^ M£ is surjective. But (M£)° is reductive, hence this map is

trivial on the pro-unipotent radical P*. We get then a surjective map Mx ^ and

, ,'cO'

Cyj c Vy

also a homomorphism P' ^ (Qy )r c Qy as required.

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