Conjugacy theorems for loop reductive group schemes and Lie algebras

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 J. Tits on buildings


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 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 [PK]. 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 as higher nullity analogues of the affine algebras. 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.
V. Chernousov was partially supported by the Canada Research Chairs Program and an NSERC research grant.
A. Pianzola wishes to thank NSERC and CONICET for their continuous support.

1
(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 [P1]. 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. It builds a bridge between "Cartan subalgebras" of twisted forms of semisimple Lie algebras over a normal ring R (viewed as infinite dimensional Lie algebras over the base field), and split tori of the corresponding reductive group schemes over R.
Using the natural one-to-one correspondence between "Cartan subalgebras" and maximal split tori in question shown in Theorem 9.1 we establish their conjugacy in Theorem 15.1. Fundamental applications to the structure theory of extended affine Lie algebras are given in §16. The main ingredient of the proof of conjugacy is the classification of loop reductive torsors over Laurent polynomial rings given by Theorem 17.1, a result that we believe is of its own interest.
2. Generalities and statement of the fundamental objective 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 1 , . . . , t ±1 n ], K n = k(t 1 , . . . , t n ), F n = k((t 1 )) · · · ((t n )), and R n,m = k[t 1 )) · · · ((t 1 m n )). 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.

2.2.
Forms. Le 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 ⊗ k R (or simply a form of g) if there exists a faithfully flat and finitely presented R-algebra R such that (2.2 where all the above are isomorphisms of Lie algebras over R. Since g is finite dimensional the assumption that R/R be finitely presented is superflous whenever R is noetherian. The set of isomorphism classes of such forms is measured by the pointed set 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 where G ad 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 follow standard practice and for convenience denote H 1 f ppf Spec(R), Aut(g) R simply by H 1 f ppf R, Aut(g) when no confusion is possible. Similarly for the Zariski andétale topologies, as well as for k-groups other than Aut(g).

Remark. Since Aut(g) is smooth and affine over Spec(R)
H 1 et R, Aut(g) ≃ H 1 fppf R, Aut(g) . 2.4. Remark. Let R = R n be as in §2.1. By the Isotriviality Theorem of [GP2] the trivializing algebra R in (2.2.1) may be taken to be of the form for some m and some Galois extensionk of k containing all m-th roots of unity of k. The extension R/R is Galois (see §2.5 below) 2.5. Multiloop algebras. Assume now that k is algebraically closed. We fix a compatible set of primitive m-th roots of unity ξ m , namely such that ξ e me = ξ m for all e > 0. Let R = R n and R = R n,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 = (e 1 , . . . , e n ) ∈ Z n the corresponding element e = (e 1 , · · · , e n ) ∈ Gal( R/R) acts on R via e (t 1 m i ) = ξ e i m t 1 m i . 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 σ σ σ = (σ 1 , . . . , σ n ) of commuting elements of Aut k (g) satisfying σ m i = 1. For each n-tuple (i 1 , . . . , i n ) ∈ Z n we consider the simultaneous eigenspace g i 1 ...in = {x ∈ g : σ j (x) = ξ i j m x for all 1 ≤ j ≤ n}.
Then g = g i 1 ...in , and g = g i 1 ...in if we restrict the sum to those n-tuples (i 1 , . . . , i n ) for which 0 ≤ i j < m j , where m j is the order of σ j .
The multiloop algebra based on g corresponding to σ σ σ, commonly denoted by L(g, σ σ σ), is defined by L(g, σ σ σ) = (i 1 ,...,in)∈Z n m n ]. 1 Note that L(g, σ σ σ), which does not depend on the choice of common period m, is not only a k-algebra (in general infinite dimensional), but also naturally an R-algebra. A rather simple calculation shows that L(g, σ σ σ) ⊗ R R ≃ g ⊗ k R ≃ (g ⊗ k R) ⊗ R R. Thus L(g, σ σ σ) corresponds to a torsor over Spec(R) under Aut(g) (see §4 for details). The crucial point in the classification of forms of g ⊗ k R by cohomological methods is the exact sequence of pointed sets obtained from (2.2.2) (2.5.1) Grothendieck's theory of the algebraic fundamental group allows us to identify H 1 et R, Out(g) with the set of conjugacy classes of n-tuples of commuting elements of the corresponding finite (abstract) group Out(g) (recall that k is algebraically closed). This will be explained when we introduce loop torsors in §6.1. This conjugacy class is an important cohomological invariant attached to any twisted form of g ⊗ k R.
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 9.1 a bridge between these two very different and contrasting kinds of mathematical worlds.

Some terminology
If G is an X-group, the pointed set of non-abelianČech cohomology on the flat (resp.étale, resp. Zariski) site of X with coefficients in G, is denoted by H 1 f ppf (X, G) [resp. H 1 et (X, G), resp. H 1 Zar (X, G)]. These pointed sets measure the isomorphism classes of sheaf torsors over R under G with respect to the chosen topology (see [M1,Ch. IV §1] and [DG] for basic definitions and references). If X = Spec(R), following customary usage and depending on the context, we also use the notation H 1 f ppf (R, G) instead of H 1 f ppf (X, G). Similarly for theétale and Zariski site. If G is flat and locally of finite presentation over X, then G is necessarily smooth over X. 2 If furthermore G is affine over X, by faithfully flat descent all of our sheaf torsors are representable. They are thus torsors in the usual sense. Furthermore, the smoothness of G yields that all torsors are locally trivial for theétale topology. In particular, H 1 et (X, G) = H 1 f ppf (X, G). These assumptions on G hold in most of the situations that arise in our work. 1 The ring R∞ is a useful artifice that allows us to see all multiloop algebras based on a given g as subalgebras of one Lie algebra. 2 Since X is of "characteristic zero", the geometric fibers of G are smooth by Cartier's theorem [DG,II.6.1.1]. The smoothness criterion on fibers [EGA4,17.8.2], shows that G is indeed smooth over X.
Given an X-group G and a morphisms Y → X of k-schemes, we let G Y denote the Y-group G × X Y obtained by base change. For convenience, we will under these circumstances denote most of the times G). If S = Spec(L) where L is a field extension of k and G is an algebraic k-group we we shall always write H 1 (L, G) instead of H 1 et (L, G). This is the "usual" Galois cohomology of the field L and group G.
The expression linear algebraic group (defined) over k, is to be understood in the sense of Borel [Bor]. For a k-group G, this is equivalent to requiring that G be affine of finite type ( [SGA3,VI B ,11.11]) because such a group is smooth by Cartier's theorem. The connected component of the identity of G, will be denoted by G • . If R is an object in k-alg we will denote the corresponding multiplicative and additive groups by G m,R and G a,R .
As it has probably become evident to the reader by now, 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.
A reductive X-group is to be understood in the sense of [SGA3]. 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.
3.1. Definition. 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. We say that G is isotropic if G admits a subgroup isomorphic to G m,X . 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 [SGA3,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}.

3.2.
Remark. 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 Par t (G) of parabolic subgroups of G of type t is also smooth and projective over X (ibid. cor.3.6).

Preliminary I: The algebraic fundamental group.
The following is a brief but nonetheless necessary summary of (mostly background) material from [SGA1] mentioned in [GP3]. Through this section X will denote a connected and locally noetherian scheme over k. The generalities about the fundamental group are followed by a detailed analysis of the case of X = Spec(R n ). This knowledge is essential for the concept of loop reductive groups which is central to the present work. 4.1. The algebraic fundamental group. Fix a geometric point a of X i.e. a morphism a : Spec(Ω) → X where Ω is an algebraically closed field.
Let X f et be the category of finiteétale covers of X, and F the covariant functor from X f et to the category of finite sets given by That is, F (X ′ ) consists of all morphisms a ′ : Spec (Ω) → X ′ for which the diagram.
The group of automorphism of the functor F is called the algebraic fundamental group of X at a, and is denoted by π 1 (X, a). The functor F is pro-representable: There exists a directed set I, objects ( Since the X i are finite andétale over X the morphisms ϕ ij are affine. Thus the inverse limit X sc = lim ←− X i exists in the category of schemes over X [EGA4] §8.2. For any scheme X ′ over X we thus have a canonical map In computing X sc = lim ←− X i we may replace (X i ) i∈I by any cofinal family. This allows us to assume that the X i are (connected) Galois, i.e. the X i are connected and the (left) action of Aut X (X i ) on F (X i ) is transitive. We then have Thus π 1 (X, a) can be identified with the group lim ←− Aut X (X i ) opp . Each Aut X (X i ) is finite, and this endows π 1 (X, a) with the structure of a profinite topological group.

4.2.
Case of a normal integral scheme. Assume now that our X is normal and integral. Denote by k(X) its fraction field and let k(X) sep be a fixed separable closure of k(X). The structure of the fundamental group in this case goes back to Lang and Serre [LS]. A connected finiteétale cover Y of X is normal [SGA1,9.11] and integral (ibid, 10.1). According to [EGA4,18.1.12], the functor Y → k(Y) provides an equivalence between the categories of finiteétale connected covers of X and finite separable field extensions of k(X) inside k(X) sep which are unramified over X (i.e. the normalization of X in L is unramified over X); the quasi-inverse functor maps such field extension L/k(X) to the integral closure of X in L.
The passage to the limit is done in [SGA1,V.8.2]. The finite separable extensions of k(X) inside k(X) sep which are unramified over X form a distiguished class of subextensions so it makes sense to talk about the maximal subextension LS X of k(X) sep /k(X) which is unramified over X. The field LS X is the union of all finite subextensions unramified over X. By passing to the limit, the simply connected covering of X is the integral closure of X in LS X . We take as base point a : Spec k(X) sep → Spec k(X) . Then the profinite group π 1 (X, a) = Gal(LS X /k(X)) occurs as a quotient of Gal k(X) sep /k(X) . Note that in the case X = Spec(R) with R local the simply connected covering X sc of X is Spec(R sh ) where R sh is the strict henselisation of R (see [Ra,§X.2]).

4.3.
The algebraic fundamental group of R n . We look in detail at an example that is of central importance to this work, namely the case when 1 , . . . , t ±1 n ] is the Laurent polynomial ring in n-variables with coefficients in k. We refer the reader to [GP2] and [GP3] for details.
The simply connected cover R sc n of R n is The "evaluation at 1" provides a geometric point that we denote by a. The algebraic fundamental group is best described as where Z(1) denotes the abstract group lim ←−m µ µ µ m (k) equipped with the natural action of the absolute Galois group Gal(k).

Preliminaries II: Reductive group schemes
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 ([SGA3, XII, Définition 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 R is noetherian. A good example to clarify the difference of these two notions is any reductive k-group which is not split. Of course any split maximal torus is a maximal split torus (but not conversely, as we have just observed).
If S < G are X-groups and s ⊂ g are their respective Lie algebras 3 , we will denote by Z G (S) [resp. Z g (s)] the centralizer of S in G [resp. of s in g].
We now recall and establish for future reference some basic useful facts. 5.1. Lemma. Let H be a reductive X-group and T a maximal torus of H.
(1) If X is connected then T contains a unique maximal split subtorus T d . Moreover, if T is isotrivial then for T d to be non-trivial it is necessary and sufficient that there exists a non-trivial group homomorphism T → G m,X .
(2) Let S ⊂ H be a split torus and let C = Z H (S) be its centralizer in H. Then C is a closed reductive subgroup of H.
(3) Let C = Z H (S) be as in (2). Then there exists a parabolic subgroup P ⊂ H such that C is a Levi subgroup in P.

5.2.
Remark. Let C = Z H (S) be as above. Let T be the radical of C. 4 Assume that X is connected, and let T d ⊂ T be the maximal split subtorus of T. Since S is split we have S ⊂ T d . Note that if we are given additionally that S is a maximal split torus then S = T d .

5.3.
Lemma. Let S be a split torus of H, and let T be the radical torus of the reductive group C = Z H (S). Assume that X is connected . Then Z H (T d ) = C.
Proof. Since T is the centre of C we have C ⊂ Z H (T). Also, the inclusions Altogether, we obtain a) The following are equivalent: 3 For a discussion of Lie algebras see §8. 4 Recall that the radical of a reductive X-group is the unique maximal torus of its centre [SGA3,XXII,4.3.6].

1) S is maximal split in H;
2) S is maximal split in Z H (S); 3) The reductive group scheme Z H (S)/S is anisotropic.
b) The following are equivalent: 4) The reductive group scheme Z H (S) has no proper parabolic subgroups. 5) P is a minimal parabolic subgroup of H. Note that Lemma 5.3 will allows us to assume in practice that we are under case d) of the Proposition, so that all five conditions are equivalent.
(2) =⇒ (3). Let T 0 be a split subtorus of Z H (S)/S. Then its preimage in Z H (S) is a split subtorus of Z H (S), so is S by the hypothesis. Thus T 0 is trivial and Z H (S)/S is anisotropic.
(3) =⇒ (2). Assume that S is not maximal, so that there exists a split subtorus S 0 properly containing S. Then S 0 /S is a a non-trivial split subtorus of the group scheme Z H (S)/S which is therefore isotropic. b) According to [SGA3,XXVI.1.20], there is a one-to-one correspondence Thus the left handside consists of one element if and only if the right handside consists of one element, which shows that (4) and (5) are equivalent. c) If Z H (S)/S is anisotropic it is is a fortiori irreducible. But then Z H (S) is irreducible as well. d) Now we assume that S is the maximal split subtorus of the radical of Z H (S) and we shall prove (4) =⇒ (3). Our assumption is the irreduciblity of Z H (S), or equivalently, that of Z H (S)/S . Let T 0 be a split subtorus of Z H (S)/S. Since the semisimple part of Z H (S)/S is anisotropic T 0 is contained in its radical. Then its preimage in Z H (S) is a split torus of the radical of Z H (S) which contains S, so it is S. Thus T 0 = 1 and Z H (S)/S is anisotropic as desired. 5.5. Proposition. Let H be a reductive group scheme over X. Assume X is connected. If H contains a split subtorus S with the property that the fiber S x is a non-central torus of H x for some x of X, then H contains a proper parabolic subgroup.
Proof. Let S be a split torus of H as in the Proposition, and let C = Z H (S) be its centralizer in H. By Lemma 5.1(2),(3) C is a closed reductive subgroup of H and there exists a parabolic subgroup P ⊂ H such that C is a Levi subgroup of P. Since S x is non-central, we necessarily have C = H. By Remark 3.2 P is proper. 5.6. Proposition. Let G be an affine and smooth group scheme over X, S a subtorus of G, and let g and s denote their respective Lie algebras.
(1) There is a natural inclusion Z G (S) ⊂ Z G (s). Both of these functors commute with base change.
(2) Z G (S) is a smooth group scheme and Z G (s) a closed subgroup of G (in particular a group scheme).
(4) If G is reductive and X is connected then Z G (S) = Z G (s) and both of these are Levi subgroups of G.
(2) Z G (S) is smooth by [SGA3,XI cor 5.3]. Since S is smooth we can apply [SGA3,II cor. 4.11.8(ii] to conclude that s is locally a direct summand of g. By [DG,II §2 prop 1.4 (3) This is a particular case of [SGA3,Exp.II theo. 5.3.1(i)].
(4) We first look at the case of a base field. X = Spec(k ) and G simply connected : Then this is a result of Steinberg. See [St75,3.3 and 3.8] and [St75,0.2]. X = Spec(k ) and G reductive: Embed G into SL n for a suitable n. Then and we are reduced to the previous case In general, because of (1) and (2), we can proceed byétale descent. This reduces the problem to the case S ⊂ T ⊂ G where G is a Chevalley group, T its standard split maximal torus and S is split. This sequence is obtained by base change to X from a similar sequence over k by [SGA3,VII cor. 1.6]. Over k our equality holds. Since both centralizers commute with base change the result follows.

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. 5 6.1. Loop torsors. Because of the universal nature of X sc we have a natural group homomorphism (6.1.1) G(k)−→G(X sc ).
5 The case most relevant to our work is that of the group of automorphism of a reductive k-group.
The group π 1 (X, a) acts on k, hence on G(k), via the group homomorphism π 1 (X, a) → Gal (k) of (4.1.1). This action is continuous, and together with (6.1.1) yields a map where we remind the reader that these H 1 are defined in the "continuous" sense (see Remark 6.2 immediately below). On the other hand, by [GP3,prop.2.3] and basic properties of torsors trivialized by Galois extensions, we have Here and elsewhere when a profinite group A acts discretely on a module M the corresponding cohomology H 1 (A, M ) is the continuous cohomology as defined in [Se1]. Similarly, if a group H acts in both A and M, then Hom H (A, M ) stands for the continuos group homomorphism of A into M that commute with the action of H.
By means of the foregoing observations we make the following. G) belongs to the image of the composite map We will denote by H 1 loop (X, G) the subset of H 1 et (X, G) consisting of classes of loop torsors. They are given by (continuous) cocycles in the image of the natural map Z 1 π 1 (X, a), G(k) → Z 1 et (X, G), which we call loop cocycles. This fundamental concept is used in the definition of loop reductive groups which we will recall momentarily. As we shall see, loop reductive groups play a central role in our conjugacy theorem. 6 The following examples illustrate the immensely rich class of objects that fit within the language of loop torsors. 6.4. Examples. (a) If X = Spec (k) then H 1 loop (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 π 1 (X, a) on G(k) is trivial, so that H 1 π 1 (X, a), G(k) = Hom π 1 (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 π 1 (X, a), G(k) . To be precise, Int(g)(φ) : x → g −1 φ(x)g 6 The concept of loop torsors and loop reductive group were introduced in [GP3] among other things to gain better understanding of the "right" concept of EALAs over non closed fields.
(b2) π 1 (X, a) = Z(1) n : In this case H 1 π 1 (X, a), G(k) is the set of conjugacy classes of n-tuples σ σ σ = (σ 1 , . . . , σ n ) of commuting elements of finite order of G(k). 7 This last example is exactly the setup of multiloop algebras, and the motivation for the "loop torsor" terminology. 6.5. Geometric and arithmetic part of a loop cocycle. By means of the decomposition (4.1.1) we can think of loop cocycles as being comprised of a geometric and an arithmetic part, as we now explain. This material will be needed to establish the "density results" used in the proof of the main conjugacy theorem. It is included to facilitate the reading of the paper. 8 Let η ∈ Z 1 π 1 (X, a), G(k) . The restriction η |Gal(k) is called the arithmetic part of η and it is denoted by η ar . It is easily seen that η ar is in fact a cocycle in Z 1 Gal(k), G(k) . If η is fixed in our discussion, we will at times denote the cocycle η ar by the more traditional notation z. In particular, for s ∈ Gal(k) we write z s instead of η ar s . Next we consider the restriction of η to π 1 (X, a) that we denote by η geo and called the geometric part of η.
We thus have a map The group Gal(k) acts on π 1 (X, a) by conjugation. On G(k), the Galois group Gal(k) acts on two different ways. There is the natural action arising for the action of Gal(k) on k, and there is also the twisted action given by the cocycle η ar = 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).
We finish this section by recalling some basic properties of the twisting bijection (or torsion map) τ z : H 1 (X, G) → H 1 (X, z G). Let η ∈ Z 1 π 1 (X, a), G(k) and consider its corresponding pair Θ(η) = (z, η geo ). We can apply the same construction to the twisted k-group z G. This would lead to a map Θ z that will attach to a cocycle η ′ ∈ Z 1 π 1 (X, a), z G(k) a pair (z ′ , η ′ geo ) along the lines explained above. 6.8. Lemma. Let η ∈ Z 1 π 1 (X, a), G(k) . With the above notation, the inverse of the twisting map [Se1] τ −1 z : satisfies Θ z • τ −1 z (η) = (1, η geo ). 6.9. Remark. The notion of loop torsor behaves well under twisting by a Galois cocycle z ∈ Z 1 Gal(k), G(k) . Indeed the torsion map τ −1 z : maps loop classes to loop classes. 6.10. Loop reductive groups. Let H be a reductive group scheme overX. Since X is connected, for all x ∈ X the geometric fibers H x are reductive group schemes of the same "type" [SGA3,XXII,2.3]. By Demazure's theorem there exists a unique split reductive group H 0 over k such that H is a twisted form (in theétale topology of X) of H 0 = H 0 × k X. We will call H 0 the Chevalley k-form of H. The X-group H corresponds to a torsor E over X under the group scheme Aut(H 0 ), namely E = Isom gr (H 0 , H). We recall that Aut(H 0 ) is representable by a smooth and separated group scheme over X by [SGA3,XXII,2.3]. It is well-known that H is then the contracted product E ∧ Aut(H 0 ) H 0 (see [DG] III §4 n o 3 for details). Given H, in what follows we may denote H 0 simply by H.
We now recall one of the central concepts needed for our work.
6.11. Definition. We say that a group scheme H over X is loop reductive if it is reductive and if E is a loop torsor.

Preliminaries III: Reductive group schemes over a normal noetherian base
We begin with a useful variation of Lemma 5.3 under some extra assumptions on our connected base k-scheme X. Let P ⊂ H be a parabolic subgroup which admits a Levi subgroup L ⊂ P. 9 As above we denote by T the radical of L and by T d ⊂ T the maximal split subtorus of T. 7.1. Lemma. Assume that X is normal noetherian and integral. Let H be a reductive X-group. Then there exists anétale cover (U i ) i=1,..,l → X such that : Proof. Since X is normal noetherian, H is a locally isotrivial group scheme [SGA3,XXIV.4.1.6]. We can thus cover X by affine Zariski open subsets X 1 , . . . , X l such that there exists a finiteétale cover V i → X i for i = 1, .., l which splits H. For each i, choose a connected component U i of V i . According to the classification ofétale maps over X (see [EGA4,18.10.12] and also §4.2), we know that U i is a finiteétale cover of X i and that we may assume that l = 1. 7.3. Proposition. Assume that X is normal and noetherian. Let H be a reductive X-group, P ⊂ H be a parabolic subgroup and L ⊂ P a Levi subgroup. Let T be the radical of L and T d ⊂ T its maximal split subtorus.
By the Lemma below and by [SGA3,XXVI,prop. 6.8] the above inclusion is an equality locally in the Zariski topology, whence equal globally. 7.4. Lemma. Assume X = Spec(R) is affine and as in the Proposition. 10 Let x ∈ X and consider the localized ring R x . Then (T d ) Rx is the maximal split subtorus of T Rx . In particular, if K denotes the quotient field of R then T d × R K is the maximal split subtorus of T × R K.
Proof. It suffices to show that (T d ) K is the maximal split subtorus of T K . We may assume the R is local. By Remark 7.2 there exists a Galois extension R/R that splits T. Recall that T is determined by its lattice of characters X(T) equipped with an action of Gal ( R/R), and that T d corresponds to the maximal sublattice in X(T) stable (elementwise) with respect to Gal ( R/R). Similar considerations apply to T K . It remains to note that T K and T have the same lattices of characters and that Gal ( 7.5. Proposition. Let G be a reductive group over a normal ring R containing a proper parabolic subgroup P. Then G contains a split non-central subtorus G m,R .
Proof. We may assume that G is semisimple. Since the base is affine P contains a Levi subgroup L as we have already observed.
Let T be the radical of L. By [SGA3, XXVI, lemme 6.7] there exists a nontrivial morphism T → G m . Since R is normal ibid. lemme 6.6 shows that the torus T contains a split subtorus G m,R . 7.6. Corollary. For a reductive group scheme G over a normal ring R to contain a proper parabolic subgroup it is necessary and sufficient that it contain a non-central split subtorus.
Proof. This follows from Propositions 5.5 and 7.5. 7.7. Lemma. Let W be a finiteétale R-group with R normal. Let K be the field of quotients of R. Then (1) The canonical map with S a connected finite Galois extension of R. Let Γ = 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 If L denotes the field of quotients of S then L/K is also Galois with Galois group naturally isomorphic to Γ as explained in [Bbk, Ch.
is obtained by the base change K/R. By the above considerations the problem reduces to the study of the map when passing to the limit over S. Since R is normal by [EGA4,18.10.8 and 18.10.9] we have S ⊗ R K = L. On the other hand for S sufficiently large W(S) = W = W(L). The compatibility of the two Galois actions gives the desired injectivity.
(2) It is clear that H 1 Zar (R, W) is in the kernel of χ. 7.8. Remark. Recall that if Pic(R) = 1 a reductive group scheme G over R of rank ℓ is split if and only if it contains a split torus of rank ℓ by [SGA3,XXII,prop. 2.2]. 7.9. Proposition. Let G be a split reductive group scheme over R. Assume that R has trivial Picard group. Then any two split maximal tori T and T ′ of G are conjugate under G(R).
Proof. Consider the transporter functor Trans G (T, T ′ ). It is represented by a closed subscheme of G ([SGA3, XXII, theo. 5.3.9]) which is a N G (T)torsor ([SGA3, XXII, cor 5.3.11]). Let ξ ∈ H 1 (R, N G (T)) be the corresponding element and let W = N G (T)/T (the Weyl group, which is a finite constant R-group since T is split). Locally (for the Zariski topology) T and T ′ are conjugate by [SGA3,XXVI,prop. 6.16]. Hence ξ is locally trivial. By Lemma 7.7 its image under the canonical map H 1 Zar (R, W) is trivial. Hence ξ comes from H 1 (R, T). Since Pic(R) = 1 and since T is R-split we conclude ξ = 1. It follows that Trans G (T, T ′ )(R) = ∅.
7.10. Proposition. Let G be a split reductive R-group. Assume that R has the property that H 1 Zar (R, L) = 1 for all split reductive groups L. 11 Let S ⊂ G be a split torus. Then there exist split maximal torus of G containing S.
Proof. Let T ⊂ G be a split maximal torus and let C = Z G (S). We know from Lemma 5.1(2),(3) that C is a Levi subgroup of a parabolic subgroup, say P. From Lemma 5.3 it follows that there exists a subtorus T ′ ⊂ T such that Z G (T ′ ) is a Levi subgroup of a standard parabolic subgroup, say P ′ , containing T and having the same type as P.
We first claim that P and P ′ are conjugate under G(R). Indeed, the functor Trans G (P ′ , P) is represented by a scheme which is a principal homogeneous space under N G (P ′ ) = P ′ , hence it corresponds to an element SGA3,XXVI,cor. 5.5]) for every prime ideal p ⊂ R after the base change R → R p the groups P ′ p and P p are conjugate by an element in G(R p ). This implies that [ξ] ∈ H 1 Zar (R, P ′ ). Since P ′ is the semidirect product of its (split) unipotent radical and the Levi subgroup L = Z G (T ′ ), we conclude that ξ is equivalent to a locally trivial torsor under L. Since L is a split reductive group we have H 1 Zar (R, L) = 1 by assumption, hence the claim.
Without loss of generality we may thus assume that S ⊂ P ′ . It then follows that Z G (S) ⊂ P ′ is a Levi subgroup in P ′ . By [SGA3, XXVI, cor. 1.8], Z G (S) and Z G (T ′ ) are conjugate under G(R). Thus, up to conjugacy, we may assume that S is a central subtorus of Z G (T ′ ). This of course implies that S is contained in T.

AD and MAD subalgebras
Let R be an object in k-alg and G be an R-group, i.e a group scheme over R. Recall (see [DG] II §4.1) that to G we can attach an R-functor on Lie algebras Lie(G) which attaches to an object S of R-alg the kernel of the . This is an R-Lie algebra that will be denoted by g in what follows. 8.1. Remark. If G is smooth, the additive group of Lie (G) represents Lie (G), that is Lie(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 ∈ G(S) and x ∈ Lie(G)(S), then gxg −1 ∈ Lie(G)(S) This last product is computed in the group G(S[ǫ]) where g is viewed as an element of G(S[ǫ]) by functoriality. The above defines an action of G on Lie(G)(S), called the adjoint action and denoted by g → Ad(g). This action in fact induces an R-group homomorphism Ad : G → Aut (Lie (G) whose kernel is the centre of G.
Given a k-subspace V of g consider the R-group functor Z G (V ) defined by . 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 g admits a k-basis consisting of simultaneous eigenvectors of m, i.e. there exists a family (λ i ) of functionals λ i ∈ m * , and a k-basis A maximal AD subalgebra of g, namely one which is not properly included in any other AD subalgebra of g is called a MAD of g. 12 8.3. Example. 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 (G R Note that m is not its own normalizer. Indeed N g (m) = Z g (m) = h ⊗ k R. 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 type 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 MADs. 8.4. Remark. Let s be an abelian Lie subalgebra of g. Let m 1 and m 2 be two subalgebras of s which are AD subalgebras of g. Because s is abelian their sum m 1 + m 2 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 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).
8.5. Remark. Let m be an AD subalgebra of g. Then for any extension Thus g ⊗ R S is spanned as a k-space by common eigenvectors of m ⊗ 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 R n . 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.
This is a particular case of Theorem 1 of [P1] by taking Cor 2.3 of [GP2] 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. 8.7. Lemma. Let G be a semisimple algebraic group over a field L of characteristic 0. Let T ⊂ G be a torus and T d be the (unique) maximal split subtorus of T. Set g = Lie (G), t = Lie (T) and t d = Lie (T d ). Then (i) The adjoint action of T d on g is L-diagonalizable. In particular, t d is an AD subalgebra of g.
(ii) t d is the largest subalgebra of t satisfying the condition given in (i).
Proof. Part (i) is clear. As for (ii) we may assume that G is semisimple adjoint. Let T a be the largest anisotropic subtorus of T. The product . We must show that t a does not contain any nonzero element whose adjoint action on g is L-diagonalizable. Let h be such an element.
By means of this basis we identify GL (g) with GL n,L . Consider the adjoint representation diagrams Since G is of adjoint type Ad is injective, so that we can identify T with a subtorus, say T, of GL n,L . Similarly for T d and T a . Since T ≃ T we see that T d and T a are the maximal split and anisotropic parts of T.
Let D n be the diagonal subgroup of GL n,L . By construction we see that this last by [Hu,Theorem 12.5] since k is of characteristic 0. Thus D n ∩ T a has dimension > 0. But then the connected component of the identity of D n ∩ T a is a non-trivial split torus which contradicts the fact that T a is anisotropic.
9. The correspondence between MADs and maximal split tori Throughout this section R will denote an object of k-alg such that X = Spec(R) is integral and noetherian. The purpose of this section is to establish the following fundamental correspondence. 9.1. Theorem. Let G be a semisimple simply connected R-group and g its Lie algebra. Assume that R is normal integral and noetherian.
(1) Let m be a MAD subalgebra of g. Then Z G (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 8.4). 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.
is the cocharacter group of S. As we shall see in the proof of Lemma 9.5 The proof of the Theorem will be given at the end of this section after a long list of preparatory results.
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. 9.3. Lemma. Let m be an AD subalgebra of g. Then (1) dim k (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 = dim k (m).
Similarly if we replace R x by K or 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 7.2 there exists a finiteétale extension R/R with the following properties (i) R is integral and normal, Note the following facts: 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 [P1,Prop. 4]-lies in the fact that the image m of m under the injection sits inside a split Cartan subalgebra H of the split semisimple K-algebra g ⊗ k K. Consider the basis {ω 1 , . . . ,ω ℓ } of H consisting of fundamental coweights for a base α 1 , . . . , α ℓ of the root system of (g ⊗ k K, H). Let 1 ≤ n ≤ m be such that { v 1 , . . . v n } is a maximal set of K-linearly independent elements of g( K). To establish (2) and (3) it will suffice to show that n = m.
Assume on the contrary that n < m. Write v i = c jiωj with c 1i , . . . c ℓi in K. The fact that the eigenvalues of ad g( K) ( v i ) belongs to k show that the c ji necessarily belong to k. Indeed v i acts on g( K) α j as multiplication by the scalar c ji . Let Write K = k ⊕ W as a k-space and use this decomposition to write At the Lie algebra level the situation is as follows. Let s = Lie (S) ⊂ g. Then s ⊂ S(R[ε]). We avail ourselves of the useful convention that if s ∈ s then to view s as an element of S(R[ε]) we write e sε . There exist unique R-linear functionals dλ i : s → R such that Then for s ∈ s and v ∈ g λ i we have the following equality in g 9.5. Lemma. Consider the restriction Ad S : S → Gl (g) of the adjoint representation of G to S. There exists a finite number of characters λ 1 , . . . , λ l of S such that The λ i are unique and and Lie(S) = Rm(S).
Proof. We appeal to the explanation given in Remark 9.4. Let Then (9.4.1) shows not only that n ⊂ s is an AD subalgebra of g, but in fact that m(S) ⊂ 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 ≃ G n m,R and the character lattice X(S) of S is generated by the projections π i : G n m,R → G m,R . Since the kernel of the adjoint representation of G is finite the sublattice of X(S) generated by λ 1 , . . . , λ ℓ has finite index; in particular every character π of S can be written as a linear combination π = a 1 λ 1 + · · · + a ℓ λ ℓ with rational coefficients a 1 , . . . , a ℓ and hence dπ = a 1 dλ 1 + · · · + a n dλ ℓ . Similarly π can be written as π = a 1 π 1 + · · · + a n π n with a 1 , . . . , a n ∈ Z and we then have dπ = a 1 dπ 1 + · · · + a n dπ n . It follows that hence the last assertions follow immediately. 9.6. Proposition. Let m be an AD subalgebra of g. Then the submodule Rm is a direct summand of g.
Proof. Let M = g/Rm. Assume for a moment that M is a projective Rmodule. Then the exact sequence 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 R x -module M x is free. Since localization is a left exact functor, and by Lemma 9.3 we have (Rm) and the module g x satisfy the variation of Nakayama's lemma stated in [Lam] Cor 1.8. Hence R x m is a direct summand of g and this implies that M x is free. 9.7. Proposition. Let m be an AD subalgebra of g. Then Z G (m) is an affine R-group whose geometric fibres are (connected) reductive groups.
Proof. By Proposition 9.6 Rm is a direct summand of g. It follows from [DG] II §Prop.1.4 that Z G (Rm) = Z G (m) is a closed subgroup of G. In particular, Z G (m) is an affine scheme which is of finite type over Spec(R). Let x ∈ Spec(R) be a point and let k(x) be an algebraic closure of k(x). Since the functor Z G (m) = Z G (Rm) commutes with base change to verify the nature of its geometric fibers Z G (m)(x) we may look at Thus we may assume without loss of generality that the ground ring is a field. By results of Steinberg ([St75, 3.3 and 3.8] and [St75,0.2]) we conclude that Z G (m)(x) is connected and reductive. 9.8. Flatness of Z G (m). Fix a split Cartan subalgebra h of g. With respect to the adjoint representation ad : g → End k (g) we have the weight space decomposition g = ⊕ α∈Σ g α where α : h → k is a linear function such that the corresponding eigenspace g α is non-zero. The kernel of the adjoint representation is trivial, dim g α = 1 if α = 0 and g 0 = h. 9.9. Lemma. Let a ⊂ h be a subalgebra. Then: (1) The centralizer Z g (a) is a reductive Lie algebra whose centre is contained in h.
(2) If a ∈ a is in generic position then Z g (a) = Z g (a). Proof.
(1) The centralizer of a is generated by h and those g α for which α(x) = 0 for every x ∈ a. It is a well-known fact that this algebra is reductive.
(2) The inclusion ⊂ is obvious. Conversely, the centralizer of a is generated by h and those g α for which α(a) = 0. Since a is generic all such roots α also satisfy α(x) = 0 for all x ∈ a. 9.10. Lemma. Let a α ∈ k, α ∈ Σ. Then there exists at most one element h ∈ h such that α(h) = a α .
Proof. Since the kernel of the adjoint representation of g is trivial the result follows.
9.11. Lemma. Let S be an object of k-alg. Let v ∈ h ⊗ k S be an ad kdiagonalizable element of g ⊗ k S. If S is an integral domain then v ∈ h.
Proof. Let F be a field of quotients of S and view v as an element of g ⊗ k F. The eigenvalues a α of v with respect to the adjoint representation are a α = α(v). By assumption they all belong to k. Thus the nonhomogeneous linear system α(x) = a α , α ∈ Σ, 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 9.3(2)]. By Lemma 9.10 such a solution is unique, hence v ∈ h. 9.12. Remark. The Lemma fails if S is not connected (hence not integral), but we do not know if connected instead of integral is the right hypothesis to make. Consider for example g = sl 2 and S = k × k. Let h, e, f be the standard generators of sl 2 . Then v = h ⊗ (1, 0) is an ad k-diagonalizable element of h S which is not in h ≃ h ⊗ (1, 1).
Fix an arbitrary element h ∈ h. Recall that G acts on g by conjugation and it is known that the orbit O h = G · h is a Zariski closed subset of g (because h is a semisimple element). Let L ⊂ G be the isotropy subgroup of h in G. It is known that L is a reductive subgroup and we have an exact sequence in the fppf topology of X = Spec(R).
The algebraic k-varieties O h and G/L have as distinguished points h and the coset e = 1 · L respectively. The group G acts on both O h and G/L in a natural way and there exists a natural G-equivariant isomorphism λ : O h ≃ G/L which takes h into e (see [Bor] for details). Hence if R is an object in k-alg and x ∈ O h (R), then x and h are conjugate by an element in G(R) if and only if λ(x) ∈ G(R) · e.
We now return to our simply connected semisimple R-group G and its Lie algebra g 9.13. Lemma. Let m be an AD subalgebra of g. The affine scheme Z G (m) is flat over Spec(R).
Proof. That Z G (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 7.1 provides a finiteétale connected cover R/R which splits G. By replacing R by R we reduce the problem to the split case. Summarizing, without loss of generality we may assume that G = G× k R, g = g⊗ k R := g R and R is a local domain.
As observed in Lemma 9.3 m is contained in a split Cartan subalgebra H of g ⊗ k K := g K . Fix a generic vector v ∈ m ⊂ g K . Let { a α , α ∈ Σ } be the family of all eigenvalues of v with respect to the adjoint representation of g K . Clearly, a α ∈ k for every α ∈ Σ (because m is an AD subalgebra of g R ).
9.14. Sublemma. There exists a unique vector h ∈ h whose eigenvalues with respect to the adjoint representation are { a α , α ∈ Σ }. Moreover if v and h are viewed as elements of g K , then they are conjugate under G(K).
Proof. Uniqueness follows from Lemma 9.10. As for existence, we note that H and h K are conjugate over K, hence h K clearly contains an element with the prescribed property. By Lemma 9.11 this element is contained in h. The conjugacy assertion is by the very definition of h.
We now come back to the G-orbit O h of h. We remind the reader that this is a closed subvariety of g.
Proof. The element v ∈ g R can be viewed as a morphism The image of the generic point Spec(K) → Spec(R) → g is contained in O h for v and h are conjugate over K. Since O h is a closed subvariety in g and since Spec(R) is irreducible it follows that φ v factors through the embedding O h ֒→ g.
To finish the proof of Lemma 9.13 we first consider the particular case when m is contained in h. Then Z G (m) is obtained from the variety Z G (m) by the base change R/k so that flatness is clear.
In the general case, let h ∈ h be the element provided by Sublemma 9.14. By Sublemma 9.15 we have v ∈ O h (R) = (G/L)(R). Denote by R sh the strict henselisation of the local ring R, that is the simply connected cover of R attached to a separable closure K s of K as outlined in section 4.2. Since the map p : G → G/L is smooth and surjective, Hensel's lemma [M1, §4] shows that G(R sh ) → (G/L)(R sh ) is surjective. But R sh is the inductive limit of the finite (connected) Galois covers of R, so there exists one such cover R ′ and a point g ′ ∈ 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 Z g R (h) = Z g R (m) since h = v ∈ m is a generic vector. Since the center of Z g R (h) is contained in h R and since m is contained in the center of its centralizer we have m ⊂ h R . Applying Lemma 9.11 then shows that m ⊂ h. Thus we have reduced the general case to the previous one.
9.16. Proposition. If m is an AD subalgebra of g then Z G (m) is a reductive R-group.
Proof. Since Z G (m) is flat and also finitely presented over R the differential criteria for smoothness shows that Z G (m) is in fact smooth over R because of Lemma 9.7. Thus Z G (m) is affine and smooth over R with geometric fibers which are (connected) reductive groups in the usual sense (this last again by Lemma 9.7). By definition Z G (m) is a reductive R-group.
Proof of Theorem 9.1. (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 G (m). By Remark 8.4 the Lie algebra of S contains a unique maximal subalgebra m(S) which is an AD-subalgebra of g. By definition S < H = Z G (Rm). Let us as before denote Lie(S) by s. Recall that K denotes the quotient field of R. By Lemma 9.5 we have dim (m(S)) = rank (S), so that to establish that m(S) = m it will suffice to show that rank (S) ≥ dim k (m), or what is equivalent, that dim K (S K ) ≥ dim k (m) where as usual S K = S × R K.
We have H K = Z G K (Rm) = Z G 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 S K is the maximal split torus of T K = rad (H K ) by Lemma 7.4. We also have Since Km is in the centre of Z g K (Km) = Lie (H K ) and the centre of Lie (H K ) coincides with Lie (T K ) we conclude that Km ⊂ Lie (T K ). On the other hand Km is an AD subalgebra of g K , so that by Lemma 8.7 Km ⊂ Lie (S K ). This shows that dim K (Km) ≤ dim K (S K ). But by Lemma 9.3(3) we have dim k (m) = dim K (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 G. Indeed, if S is contained in a split torus S ′ of larger rank then m(S) ⊂ m(S ′ ) is a proper subalgebra which contradicts to the fact that m = m(S) is a MAD.
(2) Let S be a maximal split torus of G, and let s = Lie (S) be its Lie algebra. By Remark 8.4 s contains a unique maximal subalgebra m(S) = m which is an AD-subalgebra of g. This algebra, which was denoted by m(S) will for the remaining proof of (2) be denoted by m. We have by Lemma 9.5 that Rm = Lie (S). Thus, appealing to Proposition 5.6 and Lemma 9.3(1) we obtain We claim that m is maximal. Assume otherwise. Then by Lemma 9.3(1) m is properly included in a MAD subalgebra m ′ of g. We have By Proposition 9.16 H ′ and H are reductive R-groups. Let T ′ and T be their radicals and let T ′ d , T d be their maximal split tori. We have S ⊂ T ⊂ T ′ and hence S ⊂ T d ⊂ T ′ d . But S is a maximal split torus in G. Therefore S = T ′ d = T d and this implies m = m(S) = m(T d ) = 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 S(m) is the maximal split torus of the radical of H = Z G (Rm). The proof of (1) shows that the MAD subalgebra corresponding to S(m) is precisely m.
Conversely, if S is a maximal split torus of G then the maximal split torus corresponding to m(S) is the maximal split torus of the radical of the reductive group Z G (Rm(S)) = Z G (s) = Z G (S) as explained in the proof of (1). Clearly S is inside the radical of Z G (S). Since S is maximal split in G it is maximal split in the radical of Z G (S). Thus S = S ′ .
(4) Follows from the construction and functoriality in the definition of the adjoint action at the Lie algebra and group level.

A Cohomological obstruction to conjugacy
In this section R denotes a normal noetherian and integral domain and K its field of quotients. Let G be a reductive group scheme over R. We say that a maximal split torus S of G is generically maximal split if S K is a maximal split torus of G K .
10.1. Proposition. Let S be a generically maximal split torus of G. If Proof. (i) If S Rp is not maximal split then neither is S K = S Rp ⊗ Spec(Rp) Spec(K).
(ii) By [SGA3, XI, 6.11 (a)], τ S,S ′ is a closed subscheme of G. It is clearly a right (formal) torsor under the affine R-group N G (S). Since S Rp and S ′ Rp are maximal split tori of G Rp they are conjugate under G(R p ) by [SGA3,XXVI,6.16]. Thus τ S,S ′ is an N G (S)-torsor which is locally trivial (i.e. there exists a Zariski open cover X = ∪ X i such that τ S,S ′ (X i ) = ∅).
Proof of the Proposition. Let S ′ be a generically maximal split torus of G. The transporter τ S,S ′ yields according to Lemma 10.2 an element α ∈ H 1 Zar (R, N G (S)). Our aim is to show that α is trivial. Consider the exact sequence (on Xé t ) of R-groups Then W is a finiteétale group over R (see [SGA3,XI,5.9]). By Lemma 7.7(2) the image of α in H 1 et (R, W), which we know lies in H 1 Zar (R, W), is trivial. Thus we may assume α ∈ H 1 et R, Z G (S) . To finish the proof we need to show that . For this it suffices to show that the image α p of α in Since S is generically maximal split, S Rp is a maximal split torus of G Rp . Similarly for S ′ Rp . Now by [SGA3,XXVI] S Rp and S ′ Rp are conjugate under G Rp (R p ) = G(R p ), Thus the image of α under the composition of the natural maps Let P be a parabolic subgroup of G Rp containing Z G R p (S Rp ) as a Levi subgroup (see Lemma 5.1). Then (ibid.) we have . By (10.2.1) it now follows that α p is trivial.

A Counter-example to conjugacy for multiloop algebras
Let G and g be as in Theorem 9.1. We know that the conjugacy of two MADs 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 = R 2 = k[t ±1 1 , t ±1 2 ] with generators T 1 , T 2 and relations T 2 1 = t 1 , T 2 2 = t 2 and T 2 T 1 = −T 1 T 2 and let A = M 2 (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 SL 4,R . It contains a split torus S whose R-points are matrices of the form It is well-known that this torus is maximal (one can for example see this by passing to F 2 ).
Consider now the D-linear map f : V = D ⊕ D → D given by Let L be its kernel. It is shown in [GP1] that this sequence is split 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 ∈ R × 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 subspace L in V is isomorphic to one of the components of V = D ⊕ D, in particular L is free -a contradiction.

The nullity one case
In this section we look in detail at the case n = 1, i.e. R = k[t ±1 ], where k is assumed to be algebraically closed. The twisted forms of g ⊗ k R are nothing but the derived algebra of the affine Kac-Moody Lie algebras modulo their centres [P2].
We maintain all of our previous notation, except for the fact that now we specify that n = 1.
12.1. Lemma. 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 S K is a maximal split torus of the algebraic K-group G K .
We consider the reductive R-group H = Z G (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 where m is the multiplication of H and µ µ µ is a finite group of multiplicative type. The computation of centralizers, derived groups and radicals are compatible with base change,which gives the exact sequence of algebraic K-groups Since S is central in H it lies inside rad (H), hence it is a maximal split torus of rad (H). If S K is not a maximal split torus of G K , there exists a split torus S ′ of H K such that S ′ is not a subgroup of rad (H K ). Thus if we set (S ′ ∩ H ′ K ) • = T then T is a non-trivial split torus of H ′ K . Then Z H ′ K (T) is a Levi subgroup of a proper parabolic subgroup P of H ′ K . Let t = type (P) be the type of P. Let Par t (H ′ ) be the R-scheme of parabolic subgroups of H ′ of type t. Then Par t (H ′ )(K) = ∅. Since Par t (H ′ ) is proper and R is regular of dimension 1, it follows that Par t (H ′ )(R) = ∅. Let P ′ be a parabolic subgroup of P ′ of type t. Then P ′ is a proper subgroup, so that by Proposition 7.5 P ′ contains a copy of G m,R . But then m : S×G m,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.

12.2.
Theorem. In nullity one all MADs of g are conjugate under the adjoint action of G(R).
Proof. In view of the last Lemma and Proposition 10.1 it will suffice to show that if S is a maximal split torus of G, then H 1 Zar (R, Z G (S)) = 1. Since Z G (S) is a reductive R-group one in fact has a much stronger result, namely that H 1 et (R, Z G (S)) = 1 (see [P2,Theorem 3.1]). 12.3. Remark. Let G be the "simply connected" Kac-Moody (abstract) group corresponding to g (see [PK], and also [Kmr] and [MP] for details). We have the adjoint representation Ad : G → Aut k−Lie (g). The celebrated Peterson-Kac conjugacy theorem [PK] for symmetrizable Kac-Moody (ap-plied to the affine case) asserts that all MADs of g are conjugate under the adjoint action of the group Ad (G) on g, while our result gives conjugacy under the image of G(R), where the image is that of the adjoint representation Ad : G → 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 [Kmr]). The untwisted case appears to remain unstudied.

A density property for points of loop groups
In this section X = Spec(R n ). For a description of π 1 (X, a) see 4.3. Let G be a linear algebraic k-group. Let η ∈ Z 1 π 1 (X, a), G(k) be a loop cocycle and recall the decomposition η = (η geo , z) into geometric and arithmetic parts described in Lemma 6.6. Recall that we may view η geo as a k-group homomorphism ∞ µ µ µ → z G. We denote below by ( z G) η geo the centralizer k-subgroup of z G with respect to the above group homomorphism.
13.1. Remark. By continuity there exists m and a Galois extensionk of k such that η factors through where Γ n,m := Gal(R n,m ⊗ kk /R n ) = µ µ µ n m (k) ⋊ Gal( k/k) where m > 0 and k/k is a finite Galois extension containing all m-roots of unity in k. In this way η can be viewed as a Galois cocycle in Z 1 Γ n,m , G(R n,m ⊗ kk ) . We can thus twist G Rn by η. We call this procedure "reasoning at the finite level".
Recall that an abstract group M is pro-solvable if it admits a filtration · · · ⊂ M n+1 ⊂ M n ⊂ · · · ⊂ M 0 = M by normal subgroups such that ∩ M n = 1 and M n /M n+1 is abelian for all n ≥ 0. If there exists a filtration such that M n /M n+1 are k-vector spaces, we say that M is pro-solvable in k-vector spaces.
13.2. Theorem. Let G be a linear algebraic k-group such that G • is reductive. Let η ∈ Z 1 π 1 (X, a), G(k) be a loop cocycle such that the twisted R n -group H = η (G Rn ) is anisotropic. There exists a family of pro-solvable groups in k-vector spaces (J i ) i=1,..,n such that Proof. We may assume by twisting by z that z is trivial. We note that since H is anisotropic the algebraic F n -group H Fn is also anisotropic by [GP3] cor. 7.4.3. It is convenient to work at a finite level, namely with a cocycle η : Γ n,m → G(k) as in Remark 13.1. We may also assume thatk splits a given chosen maximal torus of the k-group G • .
We proceed by induction on n ≥ 0; the case n = 0 being obvious. We reason by means of a building argument and we view F n,m and its subfield F n = ( F n,m ) Γn,m as local complete fields with the residue fields F n−1,m and F n−1 respectively.
We consider the (enlarged) Bruhat-Tits building B n = B (G Fn,m ) of the F n,m -group G Fn,m [Ti,§2.1]. It is equipped with a natural action of G( F n,m )⋊ Γ n,m . It is shown in [GP3] Theorem 7.9 that the building of H Fn inside B n consists of a single point φ whose stabilizer is We next decompose µ µ µ n m = µ µ µ n−1 m × µ µ µ m . Here the second component is a finite k-group of multiplicative type and it acts on G via η geo . We let G n−1 denote the k-subgroup of G which is the centralizer of this action [DG,II 1.3.7]. The connected component of the identity of G n−1 is reductive according to [Ri]. Since the action of µ µ µ n−1 m on G given by η geo commutes with that of µ µ µ m the k-group morphism η geo : µ µ µ n m → G factors through G n−1 .
The crucial point for the induction argument is the fact that η geo n−1 : µ µ µ n−1 m → G n−1 is anisotropic so that the twisted F n−1 -group η n−1 G n−1 is anisotropic. This is established just as in [GP3,theo. 7.9]. We look now at the specialization map Let P be the parahoric subgroup of H • (F n ) attached to the point φ. Since the building of H Fn consists of the single point we have P = H • (F n ).
Recall that the notation P * stands for the "pro-unipotent radical" of P as defined in §18.4 of the Appendix.
13.3. Claim. We have P * = ker(sp n ) and the image of sp n is H n−1 (F n−1 ).
Because G is a k-group it is clear that the kernel of the specialization by the very definition of Q * . Hence ker(sp n ) = P ∩ Q * = P * by Corollary 18.9 applied to the point φ.
The group H n−1 (F n−1 ) is a subgroup of H n (F n ) which maps identically to itself by sp n , so we have to verify that the specialization h n−1 of an element h ∈ H(F n ) belongs to H n−1 (F n−1 ). Specializing (13.2.1) at t n = 0, we get We now apply the relation (13.3.1) to the generator τ n of the Galois group Gal F n,m / F n−1,m ((t n )) ; it yields where η(τ n ) ∈ G( k), so that h n−1 ∈ G n−1 (F n−1,m ). Furthermore, the equality (13.3.1) restricted toΓ n−1,m shows that h n−1 ∈ H n−1 (F n−1 ). This establishes the Claim.
We can now finish the induction process. The group H n−1 (F n−1 ) is a subgroup of H(F n ), so where J n := ker(sp n ) is the "pro-unipotent radical" and hence it is prosolvable in k-spaces. By using the induction hypothesis, we have as desired.
We have G η geo (k) ⊂ H(R n ), so we get the second identity as well.
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 η : π 1 (R n ) → Aut(H)(k). We have the following commutative diagram of torsion bijections (H) . The vertical maps are bijective by [Gir,III 2.5.4] and Remark 6.9, while the bottom map is bijective by [GP3,theorem 8.1]. We thus have a bijection The exact sequence 1 → H → Aut(H) → Out(H) → 1 gives rise to the commutative diagram of exact sequence of pointed sets Let v ∈ H 1 et (R n , H) be a toral class mapping to 1 ∈ H 1 (F n , H). Since ψ is bijective there exists u ∈ Out(H)(R n ) such that v = ϕ(u) and u ∈ Im γ. Since Out(H)(R n ) is a finite group, the Density Theorem 13.2 shows that Aut(H)(R n ) and Aut(H)(F n ) have the same image in Out(H)(F n ). So u ∈ Im δ, which implies that γ = 1 ∈ H 1 et (R n , H). Case 2: H is irreducible. Set Z = Z(H); it is an R n -group of multiplicative type and we have an exact sequence of R n -group schemes Here the adjoint group H ad is anisotropic since H is irreducible. This exact sequence gives rise to the diagram Note that the second vertical map is bijective by [GP2,Prop. 3.4. (3)] since Z is of finite type ([SGA3, XII, §3]).
Let v ∈ H 1 et (R n , H) be a toral class mapping to 1 ∈ H 1 (F n , H). Taking into account the adjoint anisotropic case, a diagram chase provides an element u ∈ H 1 et (R n , Z) such that v = i * (u) and u belongs to the image of the characteristic map ϕ Fn . Since H 1 et (R n , Z) is an abelian torsion group, the Density Theorem 13.2 shows that H ad (F n ) and H ad (R n ) have the same image in H 1 et (R n , Z). So u belongs to the image of ϕ Rn , and this implies that v = i * (u) = 1 ∈ H 1 et (R n , H) as desired.
15. Conjugacy of certain parabolic subgroup schemes and maximal split tori 15.1. Theorem. Let H be a loop reductive group scheme over R n . Then there exists a unique H(R n )-conjugacy class of (a) Couples (L, P) where P is a minimal parabolic R n -subgroup scheme of H and L is a Levi subgroup of P such that L is a loop reductive group scheme.
(b) Maximal split subtorus S of H such that Z H (S) is a loop reductive group scheme.

15.2.
Remark. The counter-example in §11 shows that the assumption that L and Z H (S) be loop reductive group schemes is not superflous.
15.3. Lemma. The proof of Theorem 15.1.a) reduces to the semisimple simply connected case.
Proof. Denote by H sc the simply connected covering of the derived group scheme of H, and denote by E the radical torus of H. We assume that Theorem 15.1.a) holds for H sc . There is a canonical central isogeny [H, §1.2] Let (L, P) where P is a parabolic subgroup of H containing a Levi subgroup L. Then where P sc is a minimal parabolic subgroup of the R n -group H sc and L sc is a Levi subgroup of P sc . Similarly for (L ′ , P ′ ). Conversely, from a couple (M, Q) for H sc , we can define a couple (M × E)/C, (Q × E)/C) for H. By [GP3,cor. 6.3], loop group schemes are exactly those carrying a maximal torus. Since this last property is insensitive to central extensions [SGA3,XII.4.7], the correspondence described above exchanges loop objects L with loop objects L sc . Also it exchanges minimal parabolics of H with minimal parabolics of H sc .
Existence: By the simply connected case, we know that there exists a couple (M, Q) for H sc such that Q is a minimal parabolic subgroup of H sc and M is a Levi subgroup which is loop reductive. Then (L, P) := (M × E)/C, (Q × E)/C) is as desired for H.

Conjugacy:
We are given another couple (L ′ , P ′ ). Since (L ′ ) sc is loop reductive, the simply connected case yields that (L sc , P sc ) and (L ′ sc , P ′ sc ) are H sc (R n )-conjugate. By applying f , we conclude that (L, P) and (L ′ , P ′ ) are H(R n )-conjugate.
Proof of Theorem 15.1 15.4. Existence. (a) Lemma 15.3 enables to assume that H is semisimple simply connected. There exists a split semisimple simply connected k-group H (namely the Chevalley k-form of H) and a loop cocycle η : π 1 (R n ) → Aut(H)(k) such that H = η (H Rn ). Let (T, B) be a Killing couple of H and Π ⊂ ∆(H, T) is the base of the finite root system associated to (T, B). We denote by H ad the adjoint group of H and by (T ad , B ad ) the corresponding Killing couple. We have Aut(H) = Aut (H ad ).
For each I ⊂ ∆, we have the standard parabolic subgroup P I of H and its Levi subgroup L I , as well as P I,ad and L I,ad for H ad .
Let I be the subset of uncircled vertices in the Witt-Tits diagram of H Fn . The version of the "Witt-Tits decomposition" given in [GP3,Cor. 8.4] applied to Aut (H ad Thus we may assume that η has values in Aut(H, P I , L I )(k) = Aut(H ad , P I,ad , L I,ad )(k). The twisted R n -group schemes P = η (P I ) and L = η (L I ) are as desired since P Fn is a minimal F n -parabolic subgroup of H Fn by the definition of the Witt-Tits index.
(b) Let S be the maximal split subtorus of the radical of L, with L as in (a) By Proposition 7.3 we have Z H (S) = L. This implies that Z H (S) is a loop reductive group. Now S is a maximal split torus of H (hence has both required properties) because it is so over F n . To see this note that by our construction the semisimple part of the centralizer of S is anisotropic over F n . So if S ⊂ S ′ is a proper inclusion over F n , then S ′ sits inside the radical, say T, of the centralizer of S. But by Lemma 7.4, the torus S is still maximal split in T over K n and hence over F n because T is split over a Galois extension R n,m /R n for some integer m.
15.5. Conjugacy. Let (L, P) be a couple as prescribed by Theorem 15.1. Recall that S is the maximal split subtorus of the radical of L.
(a) Let (M, Q) be another couple (satisfying the same conditions). Consider the R n -scheme Y = H/P of parabolic subgroups of type t(P) [SGA3,XXVI]. The exact sequence ofétale R n -sheaves 1 → P → H f −→ Y → 1 induces exact sequences of pointed sets [Gir,III.3 P) is a bijection by [SGA3,XXVI,3.2]) and by base change

.2.2]
The R n -parabolic subgroup Q defines a point y ∈ Y(R n ).
15.6. Claim. ϕ(y) ∈ H 1 toral (R n , L) ≃ H 1 toral (R n , P). Indeed ϕ(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 isomorphic 13 to the twist F P, and F L is a Levi subgroup of the R n -group F P. Since Levi subgroups of F P are conjugate under R u ( F P)(R n ) [SGA3, XXVI, 1.8], it follows that F L is R n -isomorphic to M. The group scheme F L carries then a maximal torus and the claim is proved.
On the other hand, we have already observed before in §15.4 that P Fn and Q Fn are minimal parabolic subgroups of H Fn , hence they are conjugate under H(F n ). In other words, y viewed as an element of Y(F n ) is in the image of ψ Fn , hence ϕ Fn (y) = 1. It follows that ϕ(y) belongs to the kernel of H 1 toral (R n , L) irr → H 1 (F n , L) which is trivial by Theorem 14.1. Thus y ∈ Im ψ, i.e. P and Q are H(R n )conjugate and so are the couples (L, P) and (M, Q).
(b) Let S ′ be a maximal split subtorus of H such that its centralizer L ′ = Z H (S ′ ) is a loop reductive group scheme. By Lemma 5.1, Z H (S ′ ) is a Levi subgroup of a parabolic subgroup of P ′ of H. By Proposition 5.4.c, P ′ is a minimal parabolic subgroup of H. By (1), the couple (L ′ , P ′ ) is conjugate under H(R n ) to (L, P). We may thus assume that L = L ′ , i.e. Z H (S) = Z H (S ′ ). It follows S ′ is a central split subtorus of L, hence that S ′ ⊂ S. But S ′ is a maximal split subtorus of H, so we conclude that S = S ′ as desired.
We record further properties of the couples considered in the Theorem. 15.7. Corollary. Let (P, L) be as in Theorem 15.1.
(1) If S is the maximal split central subtorus of L, S is maximal in H and L = Z H (S).
(2) P Fn is a minimal parabolic subgroup of H Fn .
(5) S Fn is maximal split in H Fn .
13 Surprisingly enough, this compatibility is not in Giraud's book. A proof can be found in [De,lemme 4.2.33 page 175]. Proof.
( 5) The key observation is that S is a maximal torus of H hence equal to the maximal central split subtorus of Z H (S). Proposition 5.4.a applied to the anisotropic group (Z H (S)/S) Fn shows that S Fn is a maximal R n -torus of H Fn .

Applications to infinite dimensional Lie Theory
This section should be considered as an outline of the main application of our conjugacy theorem to infinite dimensional Lie theory. A detailed version of these results, as well as others, will be made available in a forthcoming paper intended for specialists in the area. *** Throughout this section we assume that k is algebraically closed of characteristic zero. G will denote a simple simply connected Chevalley group over k, and g its Lie algebra. We fix integers n ≥ 0, m > 0 and an n-tuple σ σ σ = (σ 1 , . . . , σ n ) of commuting elements of Aut k (g) satisfying σ m i = 1. Let m n ]. Recall that R/R is Galois and that we can identify Gal( R/R) with (Z/mZ) n via our choice of compatible roots of unity. Recall also from the Introduction the multiloop algebra based on g corresponding to σ σ σ, is It is a twisted form of the R-Lie algebra g ⊗ k R which is split by R : The R/R form L(g, σ σ σ) is given by a natural loop cocycle (denoted by z in [GP1,§5]) Since Aut(g) ≃ Aut(G) we can also consider by means of η the twisted R-group G = η G R . As before we denote the Lie algebra of G by g. It is well known, and in any case easy to verify (see for example the proof of [GP1,prop 4.10]) that the determination of Lie algebras commutes with the twisting process. Thus g ≃ L(g, σ σ σ) 16.1. Borel-Mostow MADS. By a Theorem of Borel and Mostow [BM] there exists a Cartan subalgebra h of g that is stable under the action of σ σ σ (by which we mean that each of the σ i stabilizes h). By restricting σ σ σ to h we can consider the loop algebra based on h with respect to σ σ σ, Let T be the maximal torus of G corresponding to h. Denote by T σ σ σ (resp. h σ σ σ ) the fixed point subgroup of T (resp. subalgebra of h ) under σ σ σ, i.e the elements of T (resp. h) that are fixed by each of the σ i . Since the torus T is also σ σ σ-stable and, just as above, we can consider its twisted form T = η T R and corresponding Lie algebra h = η h R . The same formalism already mentioned yields that h ≃ L(h, σ σ σ).
It is not difficult to see that h is a direct summand of g as an R-module whose geometric fibers h(x) are Cartan subalgebras of g(x) for all x ∈ X = Spec(R). Thus h is a Cartan subalgebra of g in the sense of [SGA3].
Let T d be the maximal split torus of T. It is easy to see that According to Remark 8.4 its Lie algebra t d contains a unique maximal subalgebra m which is an AD-subalgebra of g. The description of this algebra is quite simple: m = h 0,...,0 ⊗ k 1 ⊂ L(g, σ σ σ) ≃ g.
Note that m is precisely h σ σ σ ⊗ k 1. By Theorem 9.1 m is a MAD if and only if T d is a maximal split torus of G, in which case m = m(T d ). We will call MADs of a multiloop algebra which are of this form Borel Mostow MADs of g.
Let ∆ σ σ σ = ∆ σ σ σ (g, h) be the subset of ∆(g, h) consisting of those roots that vanish on h σ σ σ = h 0,...,0 . We have Because m ⊂ g 0,...,0 it is straighforward to verify that Z g (m) is precisely the multiloop algebra L Z g (h σ σ σ ), σ σ σ . Note that Z g (h σ σ σ ) is the Lie algebra of the reductive k-group H := Z G (h σ σ σ ) = Z G (T σ σ σ ) and hence by twisting we conclude that Z g (m) is the Lie algebra of (2) m is a MAD if and only if the dimension of h 0,...,0 is maximal among the Cartan subalgebras of g normalised by σ σ σ. In particular, Borel-Mostow MADs exist. Proof.
(1) We have explained above that Z G (T d ) ≃ η H R . This last group is loop reductive by definition since η is a loop cocycle. (2) We know that all MADs in g and all maximal split tori in G have the same dimension, say r. Since m is an AD-subalgebra we have dim k m = dim k (h 0,...,0 ) ≤ r and hence m is a MAD if and only if dim k (h 0,...,0 ) = r. It is then enough to show that there exists a Borel-Mostow AD of rank r, that is we need to find a Cartan subalgebra h ′ of g normalized by σ σ σ such that dim k (h ′ 0,...,0 ) = r. Up to conjugating by an element of G(k), we can assume that σ σ σ normalises in an irreducible way a standard parabolic group P I [GP1,3.4] and also the standard Levi subgroup L I by complete reductivity [Mt]. Then the twist z (P I ) R is still minimal parabolic of G over the field F n . It follows that if S is the torus consisting of the fixed point subgroup of the radical of L I under σ σ σ then S R ֒→ z (L R ) ⊂ z G R is the maximal split torus in the radical of z (L I ) R and hence a maximal split torus in G; in particular dim k S = r.
Let s be the Lie algebra of S. We have dim k s = dim k S = r and by our construction σ σ σ acts trivially on s. The reductive subalgebra Z g (s) is stable under σ σ σ, so the application of Borel-Mostow's theorem provides a Cartan subalgebra h ′ of Z g (s) stable under σ σ σ.
Now we have a Lie algebras s over k of dimension r which is contained in h ′ and is inside of s ′ = (h ′ ) σ . If s ′ is strictly larger than s then it gives rise to a MAD of dimension > r, which is impossible. Thus s is the subalgebra of h ′ consisting of the fixed point under σ σ σ.
According to our Conjugacy Theorem all Borel-Mostow MADs of a multiloop algebra are conjugate under G(R). There is a very important class of multiloop algebras, the so called Lie tori, where Borel-Mostow MADs play a crucial role. We now turn our attention to them. 14 16.3. Applications to EALAs. As in [AABGP] and [N1] it will be convenient for us to work with root systems that contain 0. So by a finite irreducible root system we will mean a finite subset ∆ of a finite dimensional vector space V over k such that 0 ∈ ∆ and ∆ × := ∆ \ {0} is a finite irreducible root system in V in the usual sense; see [Bbk1,chap. VI,§1,déf. 1]. Note that ∆ × is not assumed to be reduced.
We will use the following notation for the root system ∆. Let be the root lattice of ∆. Let , : V × V * → k denote the natural pairing of V with V * . If α ∈ ∆ × , α ∨ will denote the coroot of α in V * ; that is, α ∨ is the unique element of V * such that α, α ∨ = 2 and the map r α : β → β − β, α ∨ α stabilizes ∆.
14 Lie tori were introduced by Y. Yoshii [Y1, Y2] and further studied by E. Neher in [N1, N2]. 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.

Let
denote the set of indivisible roots in ∆, and let ∆ ind := ∆ × ind ∪ {0}. Lie tori are Lie algebras that are graded by groups of the form Q × Λ and satisfy some additional axioms that we are going to recall. For our purpose we may assume that Λ ≃ Z n for some n ≥ 0 called the nullity of the Lie torus.
Before proceeding with the definition and basic properties of Lie tori let us recall some basic notation on Q × Λ-graded algebras.
Let L = ⊕ (α,λ)∈Q×Λ L λ α be a Q × Λ-graded algebra. Then L = ⊕ λ∈Λ L λ is Λ-graded and L = ⊕ α∈Q L α is Q-graded algebra, with L λ = ⊕ α∈Q L λ α for λ ∈ Λ and L α = ⊕ λ∈Λ L λ α for α ∈ Q, and we have L λ α = L α ∩ L λ . Conversely if L has a Q-grading and a Λ-grading that are compatible (which means that each L α is a Λ-graded subspace of L or equivalently that each L λ is a Q-graded subspace of L), then L is Q × Λ-graded with L λ α = L α ∩ L λ . Hence, a Q × Λ-graded algebra L has three different associated support sets, namely the Q × Λ-support, the Qsupport and the Λ-support denoted respectively by supp Q×Λ (L), supp Q (L) and supp Λ (L). Of course a degree belongs to one of these supports if the corresponding homogeneous subspace is non-zero.
We now recall the standard definition of a Lie torus. Afterwards we will adapt it to our purposes.

Definition.
A Lie Λ-torus of (relative) type ∆ is a Q × Λ-graded Lie algebra L over k which (with the notation as above) satisfies: and (16.4.1) [ (LT3) L is generated as an algebra by the spaces L α , α ∈ ∆ × . (LT4) supp Λ (L) = Λ. Q is the root grading group and we call the Q-grading of L the root grading. Similarly, Λ is the external grading group and we call the Λ-grading of L the external grading.
16.5. Definition. Let L be a Lie Λ-torus of type ∆. Following [N1], we define the root grading pair for L to be the pair (s, t) of subalgebras of L where s is the subalgebra of L generated by {L 0 α } α∈∆ × and t = α∈∆ × [L 0 α , L 0 −α ]. Notice that t is a subalgebra of s, s ⊆ L • and t ⊆ L 0 0 . We will see below in (16.6.2) that the subalgebra t completely determines the root grading of L.
The next proposition summarizes the basic properties of the root grading pair of a Lie torus. Parts (i)-(v) were announced in [N1, §3]. See [ABFP2] for details. 16.6. Proposition. Let L be a centreless Lie Λ-torus of type ∆ with root grading pair (s, t).
(i) If α ∈ ∆ × ind , then Λ 0 2α = 0. (ii) s is a finite dimensional split simple Lie algebra and t is a split Cartan subalgebra of s.
(iii) There is a unique linear isomorphism α →α of V onto t * such that ∆ ind = ∆(s, t) and (v) If (α, λ) ∈ supp Q×Λ (L) and α ∈ ∆ × then (16.6.3) [e λ α , f λ α ] =α ∨ (vi) We have L 0 = s and L 0 0 = t. 16.7. Remark. We see that a Lie torus is really two things, more precisely a pair L = (L, D) consisting of a k-Lie algebra L together with an "external root data" D comprised of a finite irreducible root system ∆ and a Q × Λgrading on L satisfying certain axioms. This is reminiscent of the concept of "epinglage" in the finite dimensional case.
The concept of isomorphism also deserves some comments. On the one hand we have isomorphisms of Lie tori as Lie algebras. On the other we have an obvious concept of isomorphism of what we've called external root data, and we can define isomorphism that preserve this information (they are called bi-graded isomorphism in [ABFP2]). This is the strongest form of isomorphism, for it preserves the structure of the full root system. There is an intermediate concept, that of isotopy. The interested reader can look up §2 of [ABFP2] for details.
For our purpose we want to observe that given an automorphism θ of L we can construct a new data, call it D θ , in an obvious way so that L θ = (L, D θ ) is a Lie torus. ∆, Q, Λ do not change, nor do the supports. As the reader may have guessed, all we do is to set (L θ ) λ α = θ(L λ α ), and use the elements θ(e λ α ) and θ(f λ α ) to satisfy axiom LT2(ii). By (vi) of the last Proposition we see that the grading pair of L θ is θ(s), θ(t) . Clearly the isomorphism θ : L → L θ preserves the external root data.
The relevance of centreless Lie tori is that they sit at the "bottom" of every EALA (see [AABGP], [N1] and [N2]). A good example is provided by the affine Kac-Moody Lie algebras. They are of the form (see [Kac]) where L is a loop algebra of the form L(g, π) for some (unique) g and some (unique up to conjugacy) diagram automorphism π 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 = h π + kc + kd plays the role of the Cartan subalgebra for E.
The infinite dimensional Lie algebra E admits an invariant non-degenerate bilinear form whose restriction to H is non-degenerate. With respect to H our algebra E admits a root space decomposition. The roots are of two types: anisotropic (real) or isotropic (imaginary). This terminology comes from transferring the form to H * and computing the "length" of the roots.
The core of E is the subalgebra generated by all the anisotropic roots. The correct way to recover L inside E is as its core (which is L ⊕ kc) modulo its centre (which is kc). 15 It is known that EALAs of nullity 1 are precisely the affine Kac-Moody Lie algebras [ABGP]. Neher has shown that this "pattern" of realizations holds in all nullities. Loosely speaking an EALA is always of the form where L is a Lie torus, C is central and D is a space of derivations. Given L, the recipes for all possibles C is central and D are completely understood. Thus many questions about EALAs (e.g. their classification) come down to analogous questions about Lie tori.
With the above in mind as motivation, we now return to our paper. The definition of Lie tori has as a central ingredient the relative type ∆, and the double grading by Q × Λ with its corresponding root grading pair (s, t). On the other hand L is a Lie algebra, and it is natural to ask (and essential for the classification) whether ∆ and the root grading pairs are invariants of L. In other words.
15 In nullity one the core coincides with the derived algebra, but this is not necessarilty true in higher nullities.
We shall see as an application of our Conjugacy Theorem that the answer to this question is affirmative whenever L is finitely generated as a module over its centroid, which we henceforth assume. 16 That Λ ≃ Λ ′ is an assertion about the nullity being an invariant of L. It is well know that this is true. For example, the nullity is the transcendence degree of the field of quotients of the centroid of L [ABP2.5, cor.6.4].
To address the remaining questions we turn to the realization theorem of [ABFP2]. The finiteness assumption and the realization theorem tells us that the Lie tori are multiloop algebras. 17 This is the key that allows us to bring conjugacy into the picture.
We now use φ • φ ′ −1 to put two Λ Lie tori structures of relative type ∆ and ∆ ′ on L = L(g, σ σ σ) (see Remark 16.7). The original structure has root grading pair (s, t) where s = g σ σ σ and t is a Borel-Mostow MAD. About the second pair (s ′′ , t ′′ ) what we do know is s ′ ≃ s ′′ , t ′ ≃ t ′′ and that Z G (t ′′ ) is a loop reductive group. By the Conjugacy Theorem the exists an element θ ∈ G(R) such that θ(t ′ ) = t. This leads to yet another Lie Λ torus of type ∆ ′ on L with grading pairs (s ′′′ , t) with s ′′′ ≃ s ′′ . The root system of (s ′′′ , t) is of type ∆ ′ while that of (s, t) is of type ∆. Since t determines the Q-grading we conclude that ∆ ≃ ∆ ′ , hence that s ≃ s ′ . This completes the proof that our Question has an affirmative answer.
As a consequence we see that the (relative) type ∆ is an invariant of a Lie torus. 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 type. More generally, it is the analogue of the fact that the relative type of a finite dimensional simple Lie algebra (in characteristic 0) or of a simple algebraic group is an invariant of the algebra or group in question.
The invariance of the relative type was established in [Als] 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.

Acyclicity, II
17.1. Theorem. Let H be a loop reductive group scheme over R n . Then the natural map H). is bijective.

17.2.
Remark. The theorem generalizes (in characteristic 0) our main result in [CGP]. 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 H 1 G) is bijective and that every G-torsor is toral. The Theorem also generalizes the Acyclicity result of [GP3], 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 13.2 to the case of arbitrary loop reductive group schemes, not necessary anisotropic.
17.3. Theorem. Let H be a linear algebraic k-group whose connect component of the identity is reductive. Let η : π 1 (R n ) → H(k) be a loop cocycle and consider the loop reductive R n -groups H = η H Rn and H • = η H • Rn . Let (P, L) be a couple given by Theorem 15.1 for H • . Then there exists a normal subgroup J of L(F n ) which is a quotient of a group admitting a composition serie whose quotients are pro-solvable groups in k-vector spaces such that where H(F n ) + stands for the normal subgroup of H(F n ) generated by one parameter additive F n -subgroups.

17.4.
Remark. If H is semisimple simply connected, isotropic and F nsimple, we know that H(F n )/H(F n ) + ∼ = H(F n )/R [G, 7.2] so that the group H(F n )/H(F n ) + has finite exponent (ibid, 7.6). In this case, the decomposition reads H(F n ) = H(R n ), H(F n ) + .
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 = G n m (the case n = 1 follows from the identity . Since all finite connectedétale coverings of R n are also Laurent polynomial rings over field extensions of k [GP3, 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 that v = i * (u) and u Fn belongs to the image of the characteristic map ϕ Fn . Since H 1 et (R n , C) is an abelian torsion group, the Density Theorem 17.3 shows that H ad (F n ) and H ad (R n ) have the same images in H 1 et (R n , C). So u belongs to the image of ϕ Rn . Hence v = i * (u) = 1 ∈ H 1 et (R n , H). Surjectivity: Follows by a simple chasing in the diagrams above.
Question. Assume that H is loop semisimple simply connected, isotropic and F n -simple, Let H(R n ) + ⊂ H(R n ) be the (normal) subgroup generated by the R u (P)(R n ) where P runs over the set of parabolic subgroups of H considered in Theorem 15.1. Is the map an isomorphism? Note that the map is surjective by Remark 17.4. The question is then all about the injectivity of the map in question.

Appendix: Greenberg functors, Bruhat-Tits theory and pro-unipotent radicals
We are given a complete discrete valuation field K of valuation ring O = O K and of the perfect residue field k = O/πO. Here π ∈ O is a uniformizer. In the inequal characteristic case denote by e 0 the absolute ramification index of O, i.e. p = uπ e 0 for a unit u ∈ O where p = char(k); in the equal characteristic case, put e 0 = 1. We denote by O sh the strict henselization of O, or in other words, its maximal unramified extension.
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 Λ and r ≥ 0, we denote by W r (Λ) the group of Witt vectors of length r and by W (Λ) = lim ←− W r (Λ) the ring of Witt vectors (see [Se2,§II.6]). There exists a unique ring homomorphism W (k) → O commuting with the projection on k = W 0 (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 Λ → S(W r (Λ)) is representable by an affine k-scheme Green r (S). The projective limit We have G r (X)(k) = X(O/p r O) and G(X)(k) = X(O). We have G(Spec(O)) = Spec(k); if X is a O-group scheme, then G(X) and the G r (X) carry a natural k-group structure [B, 4.1].
18.2. Lemma. Let L/K be a finite extension, O L the valuation ring of L and l/k the corresponding residue extension. Let Y/O L be an affine scheme. Let H/l be the relative Greenberg functor of Y with respect to W (l). Then we have natural isomorphisms of k-schemes (for all r ≥ 1) In particular if k = l then we have G r Proof. We have a commutative square So by the functorial properties of the Weil restriction, we have Y W (l) ⊗ W (k) W r (Λ) .
Since W r (Λ) is a W r (k)-module, we have W (l) ⊗ W (k) W r (Λ) = W r (l) ⊗ Wr(k) W r (Λ) = W r (Λ ⊗ k l) by [I, 1.5.7]. Hence Y (W r (Λ ⊗ k l)) = R l/k H r (Λ) as desired. By passing to the limit, we get the second identity.
where x is viewed as an element in B(G, K sh ) via the canonical mapping B(G, K) ֒→ B (G, K sh  The application of the relative Greenberg functor to the smooth affine group scheme P x defines a projective system of affine k-groups P x,n (n ≥ 1) such that P x,n (k) = P x (O/π ne 0 O). The P x,n are smooth according to [B,Lemme 4.1.1]. The kernel P x,n+1/n of the transition maps P x,n+1 → P x,n are k-unipotent abelian groups which are successive extensions of the vector group of Lie(P x ) ⊗ O k (ibid. or [M2,III.4.3]).
For each n ≥ 1, we denote by R n,x := R u (P x,n ) the unipotent radical of P x,n ; since k is perfect, it is defined over k and split [DG,IV.2.3.9]. The quotient M x of P x,n by R x,n is independent of n. It is nothing but the quotient of the special fiber of P x by its k-unipotent radical R x . The k-group M • x is reductive according to [BT2,4.6.12]. We consider the "maximal pro-unipotent normal subgroup" P * x := ker P x (O) → M x (k) which is of analytic nature. Denote by P x /k := lim ←− n≥1 P x,n and by P * x /k = ker P x → M x ). By construction we have P * x = P * x (k). 18.5. Lemma. For each n ≥ 1, there is a short exact sequence of affine k-groups 1 → ker(P x → P x,n ) → P * x → R x,n → 1. Proof. Apply the snake lemma to the commutative diagram of k-groups 18.6. Lemma. The k-group P * x is the unique maximal split pro-unipotent closed normal subgroup of the pro-algebraic affine k-group P x .
Proof. Since ker(P x → P x,1 ) = lim ←− n ker(P x,n → P x,1 ) is pro-unipotent, the above exact sequence shows that P * x is pro-unipotent. Let U x be a pro-unipotent normal closed subgroup of P x . The image of U x by the map P x → M x is a normal unipotent connected k-subgroup. Since M • x is reductive, its image is trivial. Therefore U x ⊂ P * x which completes the proof. 18.7. Behaviour under a Galois extension. Just as does the whole theory, the construction of P * x 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 Γ = Gal(L/K) acts on the building B (G, L). The Bruhat-Tits-Rousseau theorem ( [Ro,§5], see also [Pr]) states that the natural map For σ ∈ Γ, we have σ(Q z ) = Q σ(z) . Hence for the canonical group schemes over O L attached to z and σ(z), there is a natural cartesian square Put y = j(x) ∈ B(G, L) Γ . and we have then an O-action of Γ on the scheme Q y . We note that P x = G(K) ∩ Q y = G(L) Γ ∩ Q y = Q Γ y . (18.7.1) As above we consider the groups Q y,n and their projective limit Q y . Since k is the residue field of O L , all Q y,n and Q y are k-groups. The action of Γ on Q y induces its action on Q y,n , hence on M y where M y stands for the reductive k-group attached to y, and on their projective limit Q y . By Lemma 18.6, Q * y is a characteristic k-subgroup of Q y , hence Γ also acts on the pro-algebraic k-group Q * y . Our goal is to prove the following fact: 18.8. Proposition. There is a natural closed embedding P x → Q y and we have P * x = P x ∩ Q * y . This gives rise to an isomorphism M x ∼ −→ M Γ y . By taking k-points we get the following wished compatibility, namely.
18.9. Corollary. We have Consider the Weil restriction J x := Π O L /O Q y and recall it is a smooth O-scheme [Yu,§2.5]. Let N be the kernel of the natural map P x → J x , its generic fiber is trivial. As above, applying the Greenberg functors to the O-schemes J x and N we get k-groups J x,n , J x and N n , N.
Since the Greenberg functor is left exact, we get an exact sequence 1 → N → P x → J x .
Since N K = 1, we have N = 1 according to Lemma 18.3 (2). Hence we may view P x as a closed subgroup of J x . But according to Lemma 18.2, J x,n is nothing but Q y,n . This implies J x is isomorphic in a natural way to Q y . Thus we have constructed a natural closed embedding P x → Q y . Define the k-subgroups Q Γ y := lim ←− n Q Γ y,n and (Q * y ) Γ = Q Γ y ∩ Q * y of Q y and Q * y respectively. 18.10. Lemma. (1) If k ′ /k is a finite extension of fields, the projective system Q Γ y,n (k ′ ) n≥1 has surjective transitions maps. Therefore the projective system of k-groups Q Γ y,n n≥1 has surjective transitions maps. (2) If k ′ /k is a field finite extension, we have an exact sequence 1 → (Q * y ) Γ (k ′ ) → Q Γ y (k ′ ) → M Γ y (k ′ ) → 1; hence the sequence of the pro-algebraic k-groups 1 → (Q * y ) Γ → Q Γ y → M Γ y → 1 is also exact.
(3) The algebraic k-group M Γ y 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 that k = k ′ without loss of generality. Since Q y,n+1/n is a k-split unipotent group, we have an exact sequence 1 → Q y,n+1/n (k) → Q y,n+1 (k) → Q y,n (k) → 1.
Since Q y,n+1/n (k) admits a characteristic central composition serie in kvector spaces and the order of Γ is invertible in k, the right hand side is trivial. A fortiori, the system (Q Γ y,n ) of k-groups is surjective (because Q Γ y,n (k) = Q y,n (k) Γ ). (2) By part (1), the map Q Γ y (k) → (Q y,1 ) Γ (k) is surjective. The same argument as in (1) shows that (Q y,1 ) Γ (k) → M Γ y (k) is also surjective. By taking the composition of these maps we conclude the map Q Γ y (k) → M Γ y (k) is surjective whence the desired exactness of both sequences.
(3) The group Γ may be viewed as a finite abelian constant group scheme whose order is invertible in k. Hence Γ is also a (smooth) k-group of multiplicative type. Since M y is affine and smooth, Grothendieck's theorem of smoothness of centralizers [SGA3,XI,5.3] shows that M Γ y is smooth. Its connected component of the identity is reductive by a result of Richardson [Ri,prop. 10.1.5].
We can now proceed to the proof of Proposition 18.8.
Proof. We have to show that our closed embedding P x → Q y which we constructed above induces an isomorphism P * x ∼ −→ P x ∩ Q * y . Since P x ∩ Q * y is a normal closed split pro-unipotent subgroup of P x it is contained in P * x . Hence it remains only to show that P * x ⊂ Q * y . We now recall from (18.7.1) that P x = Q Γ y and Q Γ y (k) = Q y (k) Γ = Q Γ y . By Lemma 18.10, Q Γ y (k) projects onto M Γ y (k), so the composite map P x = P x (k) → Q Γ y (k) → M Γ y (k) is surjective. Since this is true for all finite extensions of k, the homomorphism of k-algebraic groups P x → M Γ y is surjective. But (M Γ y ) • is reductive, hence this map is trivial on the pro-unipotent radical P * x . We get then a surjective map M x → M Γ y and also a homomorphism P * x → (Q * y ) Γ ⊂ Q * y as required.