CONJUGACY THEOREMS FOR LOOP REDUCTIVE GROUP SCHEMES AND LIE ALGEBRAS

V. CHERNOUSOV, P. GILLE, AND A. PIANZOLA

Abstract. The conjugacy of split Cartan subalgebras in the finite dimensional simple case (Chevalley) and in the symmetrizable Kac-Moody case (Peterson-Kac) are fundamental results of the theory of Lie algebras. Among the Kac-Moody Lie algebras the affine algebras stand out. This paper deals with the problem of conjugacy for a class of algebras –extended affine Lie algebras– that are in a precise sense higher nullity analogues of the affine algebras. Unlike the methods used by Peterson-Kac, our approach is entirely cohomological and geometric. It is deeply rooted on the theory of reductive group schemes developed by Demazure and Grothendieck, and on the work of J. Tits on buildings. Keywords: Reductive group scheme, torsor, Laurent polynomials, nonabelian cohomology, building MSC 2000 11E72, 14L30, 14E20.

1. Introduction

Let g be a split simple finite dimensional Lie algebra over a field k of characteristic 0. From the work of Cartan and Killing one knows that g is determined by its root system. The problem, of course, is that a priori the type of the root system may depend on the choice of 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.

(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 \overline{k} an algebraic closure of k. For integers n ≥ 0 and m > 0 we set

R_{n} = k[t_{1}^{±1}, …, t_{n}^{±1}], K_{n} = k(t_{1}, …, t_{n}), F_{n} = k((t_{1})) ⋯ ((t_{n})),

and

R_{n,m} = k[t_{1}^{± \frac{1}{m}} , …, t_{n}^{± \frac{1}{m}}], K_{n,m} = k(t1^{\frac{1}{m}} , …, tn^{\frac{1}{m}}), F_{n,m} = k((t1^{\frac{1}{m}})) ⋯ ((tn^{\frac{1}{m}})).

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 R̃ such that

(2.2.1)  L ⊗_{R} R̃ ≃ (g ⊗_{k} R) ⊗_{R} R̃ ≃ g ⊗_{k} R̃,

where all the above are isomorphisms of Lie algebras over R̃. Since g is finite dimensional the assumption that R̃/R be finitely presented is superflous whenever R is noetherian. The set of isomorphism classes of such forms is measured by the pointed set

H_{fppf}^{1}( Spec(R), Aut(g)_{R})

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

(2.2.2)  1 ⟶ G_{ad} ⟶ Aut(g) ⟶ Out(g) ⟶ 1

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_{fppf}^{1}( Spec(R), Aut(g)_{R}) simply by H_{fppf}^{1}(R, Aut(g)) when no confusion is possible. Similarly for the Zariski and étale topologies, as well as for k–groups other than Aut(g).

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

H_{ét}^{1}(R, Aut(g)) ≃ H_{fppf}^{1} (R, Aut(g)).

2.4. Remark. Let R = R_{n} be as in §2.1. By the Isotriviality Theorem of [GP2] the trivializing algebra R̃ in (2.2.1) may be taken to be of the form

R̃ := R_{n,m} ⊗_{k} k̃ = k̃[t_{1}^{± \frac{1}{m}} , …, t_{n}^{± \frac{1}{m}}]

for some m and some Galois extension k̃ of k containing all m-th roots of unity of \overline{k}. The extension R̃/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_{me}^{e} = ξ_{m} for all e > 0. Let R = R_{n} and R̃ = R_{n,m}. Then R̃/R is Galois. Via our choice of roots of unity, we can identify Gal(R̃/R) with (ℤ/mℤ)^{n} as follows: For each e = (e_{1}, …, e_{n}) ∈ ℤ^{n} the corresponding element \overline{e} =

(\overline{e}_{1}, ⋯, \overline{e}_{n}) ∈ Gal(R̃/R) acts on R̃ via \overline{e}(ti^{\frac{1}{m}}) = ξ_{m}^{e_{i}}ti^{\frac{1}{m}} .

The primary example of forms L of g⊗_{k}R which are trivialized by a Galois extension R̃/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_{i}^{m} = 1. For each n–tuple (i_{1}, …, i_{n}) ∈ ℤ^{n} we consider the simultaneous eigenspace

g_{i_{1}...i_{n}} = {x ∈ g : σ_{j}(x) = ξ_{m}^{i_{j}}x for all 1 ≤ j ≤ n}.

Then g = ∑ g_{i_{1}...i_{n}}, and g = ⨁ g_{i_{1}...i_{n}} 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},...,i_{n})∈ℤ^{n}} g_{i_{1}...i_{n}} ⊗ t1^{\frac{i_{1}}{m}} … tn^{\frac{in}{m}} ⊂ g ⊗_{k} R̃ ⊂ g ⊗_{k} \overline{R}_{∞}

where \overline{R}_{∞} = \underrightarrow{lim}\overline{k}[t_{1}^{± \frac{1}{m}} , …, t_{n}^{± \frac{1}{m}}].^{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} R̃ ≃ g ⊗_{k} R̃ ≃ (g ⊗_{k} R) ⊗_{R} R̃.

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)  H_{ét}^{1}(R, G_{ad}) → H_{ét}^{1}(R, Aut(g)) → H_{ét}^{1}(R, Out(g)) → 1

Grothendieck’s theory of the algebraic fundamental group allows us to identify H_{ét}^{1}(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.

3. Some terminology

If 𝔊 is an 𝔛–group, the pointed set of non-abelian Čech cohomology on the flat (resp. étale, resp. Zariski) site of 𝔛 with coefficients in 𝔊, is denoted by H_{fppf}^{1}(𝔛, 𝔊) [resp. H_{ét}^{1}(𝔛, 𝔊), resp. H_{Zar}^{1}(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_{fppf}^{1}(R, 𝔊) instead of H_{fppf}^{1}(𝔛, 𝔊). Similarly for the étale 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 the étale topology. In particular, H_{ét}^{1}(𝔛, 𝔊) = H_{fppf}^{1}(𝔛, 𝔊). These assumptions on 𝔊 hold in most of the situations that arise in our work.

1The ring \overline{R}_{∞} is a useful artifice that allows us to see all multiloop algebras based on a given g as subalgebras of one Lie algebra.

2Since 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 H_{ét}^{1}(𝔜, 𝔊_{𝔜}) by H_{ét}^{1}(𝔜, 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_{ét}^{1}(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).

4. 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_{fet} be the category of finite étale covers of 𝔛, and F the covariant functor from 𝔛_{fet} to the category of finite sets given by

F (𝔛′) = {geometric points of 𝔛′ above a).

That is, F (𝔛′) consists of all morphisms a′ : Spec (Ω) → 𝔛′ for which the diagram.

\xymatrix{ & 𝔛′ \ar[d] \\ Spec (Ω) \ar[ur]^{a′} \ar[r]^{a} & 𝔛}

commutes. 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 (𝔛_{i})_{i∈I} of 𝔛_{fet}, surjective morphisms φ_{ij} ∈ Hom_{𝔛}(𝔛_{j}, 𝔛_{i}) for i ≤ j and geometric points a_{i} ∈ F (𝔛_{i}) such that

a_{i} = φ_{ij} ∘ a_{j},
The canonical map f : \underrightarrow{lim} Hom_{𝔛}(𝔛_{i}, 𝔛′) → F (𝔛′) is bijective.

Since the 𝔛_{i} are finite and étale over 𝔛 the morphisms φ_{ij} are affine. Thus the inverse limit

𝔛^{sc} = \underleftarrow{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

Hom_{P ro−𝔛}(𝔛^{sc}, 𝔛′) =^{def} \underrightarrow{lim} Hom_{𝔛}(𝔛_{i}, 𝔛′) ≃ F (𝔛′)

obtained by considering the canonical morphisms φ_{i} : 𝔛^{sc} → 𝔛_{i}.

In computing 𝔛^{sc} = \underleftarrow{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

F (𝔛_{i}) ≃ Hom_{P ro−𝔛}(𝔛^{sc}, 𝔛_{i}) ≃ Hom_{𝔛}(𝔛_{i}, 𝔛_{i}) = Aut_{𝔛}(𝔛_{i}).

Thus π_{1}(𝔛, a) can be identified with the group\underleftarrow{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.

Suppose now that our 𝔛 is a geometrically connected k–scheme. We will denote 𝔛 ×_{k} \overline{k} by \overline{𝔛}. Fix a geometric point \overline{a} : Spec(\overline{k}) → \overline{𝔛}. Let a (resp.

b) be the geometric point of 𝔛 [resp. Spec(k)] given by the composite maps a : Spec(\overline{k}) →^{\overline{a}} \overline{𝔛} → 𝔛 [resp. b : Spec(\overline{k}) →^{\overline{a}} 𝔛 → Spec(k)]. Then by [SGA1, théorème IX.6.1] π_{1}( Spec(k), b) ≃ Gal(k) := Gal(\overline{k}/k) and the sequence

(4.1.1)  1 → π_{1}(\overline{𝔛}, \overline{a}) → π_{1}(𝔛, a) → Gal(k) → 1

is exact.

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 finite étale 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 finite étale 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 𝔛 = Spec (R_{n}) where R_{n} = k[t_{1}^{±1}, …, t_{n}^{±1}] 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_{n}^{sc} of R_{n} is

\overline{R}_{n,∞} = \underrightarrow{lim} \overline{R}_{n,m}

with \overline{R}_{n,m} = \overline{k}[t_{1}^{± \frac{1}{m}} , …, t_{n}^{± \frac{1}{m}}]. The “evaluation at 1” provides a geometric point that we denote by a. The algebraic fundamental group is best described as

(4.3.1)  π_{1}(𝔛, a) = ℤ̂(1)^{n} ⋊ Gal (k).

where ℤ̂(1) denotes the abstract group \underleftarrow{lim}_{m} μ μ μ_{m}(\overline{k}) equipped with the natural action of the absolute Galois group Gal(k).

5. 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.

Proof. (1) This is [SGA3, XXVI, 6.5, 6.6].

(2) See [SGA3, XIX, 2.2].

(3) See [SGA3, XXVI, cor. 6.2]. □

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 𝔖 ⊂ 𝔗_{d} ⊂ 𝔗 yield

Z_{ℌ}(𝔗) ⊂ Z_{ℌ}(𝔗_{d}) ⊂ Z_{ℌ}(𝔖) = ℭ.

Altogether, we obtain

ℭ ⊂ Z_{ℌ}(𝔗) ⊂ Z_{ℌ}(𝔗_{d}) ⊂ ℭ,

whence the result. □

5.4. Proposition. Let ℌ be a reductive group scheme over 𝔛. Assume 𝔛 is connected. Let 𝔖 be a split subtorus of ℌ and let 𝔓 be a parabolic subgroup of ℌ containing Z_{ℌ}(𝔖) as Levi subgroup.

a) The following are equivalent:

3For a discussion of Lie algebras see §8.

4Recall 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.

c) We have (3) ⟹ (4).

d) If 𝔖 is the maximal split subtorus of the radical of Z_{ℌ}(𝔖), we have (1) ⟺ (2) ⟺ (3) ⟺ (4) ⟺ (5).

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.

Proof. a) (1) ⟹ (2) is obvious.

(2) ⟹ (1). Let 𝔖_{0} be a split subtorus of ℌ containing 𝔖. Then 𝔖 ⊂ 𝔖_{0} ⊂ Z_{ℌ}(𝔖), hence 𝔖 = 𝔖_{0}.

(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

{ parabolics 𝔔 of ℌ included in 𝔓 } < −− > { parabolics 𝔐 of Z_{ℌ}(𝔖)}

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).

(3) Lie(Z_{𝔊}(𝔰)) = Z_{𝔤}(𝔰).

(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.

Proof. (1) This is easy to verify. See [SGA3, I and II 5.3.3].

(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] Z_{𝔊}(𝔰) is a closed subgroup of 𝔊.

(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

Z_{𝔊}(𝔖) = 𝔊 ∩ Z_{SL_{n}}(𝔖) and Z_{𝔊}(𝔰) = 𝔊 ∩ Z_{SL_{n}}(𝔰).

and we are reduced to the previous case

In general, because of (1) and (2), we can proceed by étale 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. □

6. 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(\overline{k})⟶G(𝔛^{sc}).

5The 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 \overline{k}, hence on G(\overline{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

H^{1}(π_{1}(𝔛, a), G(\overline{k})) → H^{1}(π_{1}(𝔛, a), G(𝔛^{sc})),

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

H^{1}(π_{1}(𝔛, a), G(𝔛^{sc})) = \underrightarrow{lim} H^{1}(Aut_{𝔛}(𝔛_{i}), G(𝔛_{i}))
= \underrightarrow{lim} H_{ét}^{1}(𝔛_{i}/𝔛, G) ⊂ H_{ét}^{1}(𝔛, G).

6.2. Remark. 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.

6.3. Definition. A torsor 𝔈 over 𝔛 under G is called a loop torsor if its isomorphism class [𝔈] in H_{ét}^{1}(𝔛, G) belongs to the image of the composite map

H^{1}(π_{1}(𝔛, a), G(\overline{k})) → H^{1}(π_{1}(𝔛, a), G(𝔛^{sc})) ⊂ H_{ét}^{1}(𝔛, G).

We will denote by H_{loop}^{1}(𝔛, G) the subset of H_{ét}^{1}(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(\overline{k})) → Z_{ét}^{1}(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_{loop}^{1}(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(\overline{k}) is trivial, so that

H^{1}(π_{1}(𝔛, a), G(\overline{k})) = Hom (π_{1}(𝔛, a), G(\overline{k}))/Int G(\overline{k})

where the group Int G(\overline{k}) of inner automorphisms of G(\overline{k}) acts naturally on the right on Hom (π_{1}(𝔛, a), G(\overline{k})). To be precise, Int(g)(ϕ) : x → g^{−1}ϕ(x)g for all g ∈ G(\overline{k}), ϕ ∈ Hom (π_{1}(𝔛, a), G(\overline{k})) and x ∈ π_{1}(𝔛, a). Two particular cases are important:

6The 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.

(b1) G abelian: In this case H^{1}(π_{1}(𝔛, a), G(\overline{k})) is just the group of continuous homomorphisms from π_{1}(𝔛, a) to G(\overline{k}).

(b2) π_{1}(𝔛, a) = ℤ̂(1)^{n} : In this case H^{1}(π_{1}(𝔛, a), G(\overline{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(\overline{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(\overline{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(\overline{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_{s}^{ar}.

Next we consider the restriction of η to π_{1}(\overline{𝔛}, \overline{a}) that we denote by η^{geo} and called the geometric part of η.

We thus have a map

Θ : Z^{1}(π_{1}(𝔛, a), G(\overline{k})) ⟶ Z^{1}( Gal(k), G(\overline{k})) × Hom (π_{1}(\overline{𝔛}, \overline{a}), G(\overline{k}))
η  ↦  ( η^{ar} , η^{geo} )

The group Gal(k) acts on π_{1}(\overline{𝔛}, \overline{a}) by conjugation. On G(\overline{k}), the Galois group Gal(k) acts on two different ways. There is the natural action arising for the action of Gal(k) on \overline{k}, and there is also the twisted action given by the cocycle η^{ar} = z. Following standard practice to view the abstract group G(\overline{k}) as a Gal(k)–module with the twisted action by z we write _{z}G(\overline{k}).

6.6. Lemma. The map Θ described above yields a bijection between Z^{1}(π_{1}(𝔛, a), G(\overline{k})) and couples (z, η^{geo}) with z ∈ Z^{1}(π_{1}(𝔛, a), G(\overline{k})) and η^{geo} ∈ Hom_{Gal(k)} (π_{1}(\overline{𝔛}, \overline{a}),_{z}G(\overline{k})).

Proof. See Lemma 3.7 of [GP3]. □

6.7. Remark. Assume that 𝔛 = Spec(R_{n}). It is easy to verify that η^{geo} arises from a unique k–group homomorphism

∞μ μ μ = (\underleftarrow{lim}μ μ μ_{m})^{n} → _{z}G

7That the elements are of finite order follows from the continuity assumption.

8The reader is referred to [GP3] for more details

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(\overline{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(\overline{k})) a pair (z′, η′^{geo}) along the lines explained above.

6.8. Lemma. Let η ∈ Z^{1}(π_{1}(𝔛, a), G(\overline{k})). With the above notation, the inverse of the twisting map [Se1]

τ_{z}^{−1} : Z^{1}(π_{1}(𝔛, a), G(\overline{k})) ⟶^{∼} Z^{1}(π_{1}(𝔛, a),_{z}G(\overline{k}))

satisfies Θ_{z} ∘ τ_{z}^{−1}(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(\overline{k})). Indeed the torsion map τ_{z}^{−1} : H_{ét}^{1}(𝔛, G) → H_{et}^{1}(𝔛,_{z}G) 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\overline{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 the étale 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.

7. 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 an étale cover (𝔘_{i})_{i=1,..,l} → 𝔛 such that :

(i) ℌ ×_{𝔛} 𝔘_{i} is a split reductive 𝔘_{i}–group scheme,

(ii) 𝔘_{i} = Spec(R_{i}) with R_{i} a normal noetherian domain.

9The existence of 𝔏 is automatic if the base scheme is affine by [SGA3, XXVI.2.3]

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 finite étale 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 of étale maps over 𝔛 (see [EGA4, 18.10.12] and also §4.2), we know that 𝔘_{i} is a finite étale cover of 𝔛_{i} and that 𝔘_{i} = Spec(R_{i}) where R_{i} is a normal domain. Since R_{i} is finite over the noetherian ring H^{0}(𝔛_{i}, O_{𝔛}), it is noetherian as well. □

7.2. Remark. If 𝔛 is local, one single 𝔘_{i} suffices, i.e. 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. Then Z_{ℌ}(𝔗_{d}) = 𝔏.

Proof. The existence of 𝔗_{d} follows from Lemma 5.1(1). Since 𝔗 is the centre of 𝔏 we have 𝔏 ⊂ Z_{ℌ}(𝔗). The inclusion 𝔗_{d} ⊂ 𝔗 yields Z_{ℌ}(𝔗) ⊂ Z_{ℌ}(𝔗_{d}). Thus we have 𝔏 ⊂ Z_{ℌ}(𝔗_{d}). 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})_{R_{x}} is the maximal split subtorus of 𝔗_{R_{x}}. 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 R̃/R that splits 𝔗. Recall that 𝔗 is determined by its lattice of characters X(𝔗) equipped with an action of Gal (R̃/R), and that 𝔗_{d} corresponds to the maximal sublattice in X(𝔗) stable (elementwise) with respect to Gal (R̃/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 (R̃/R) ≃ Gal (K̃/K) by [Bbk, Ch5 §2.2 theo.2]). □

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}. □

10All of our normal rings are hereon assumed to be integral domains.

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

(1) The canonical map

χ : H_{ét}^{1}(R, 𝔚) ⟶ H^{1}(K, 𝔚_{K})

is injective.

(2) H_{Zar}^{1}(R, 𝔚) = 1.

Proof. (1) Because of the assumptions on 𝔚 we can compute H_{ét}^{1}(R, 𝔚) as the limit of H_{ét}^{1}(S/R, 𝔚) 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 H_{ét}^{1}(S/R, 𝔚) = H^{1}(Γ, 𝔚(S)) (see [SGA1, XI §5]).

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.5 §2.2 theo. 2].

Our map χ : H_{ét}^{1}(R, 𝔚) ⟶ H_{ét}^{1}(K, 𝔚_{K}) is obtained by the base change K/R. By the above considerations the problem reduces to the study of the map

χ : H^{1}(Γ, 𝔚(S)) ⟶ H^{1}(Γ, 𝔚(S ⊗_{R} K))

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_{Zar}^{1}(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_{Zar}^{1}(R, N_{𝔊}(𝔗)) → H_{Zar}^{1}(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_{Zar}^{1}(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 [ξ] ∈ H_{ét}^{1}(R, 𝔓′). By ([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_{Zar}^{1}(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_{Zar}^{1}(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. □

8. 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 natural map 𝔊(S[ϵ]) → 𝔊(S) where S[ϵ] is the algebra of dual number over S. Let Lie(𝔊) = 𝔏𝔦𝔢(𝔊)(R). 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[ϵ]) where g is viewed as an element of 𝔊(S[ϵ]) 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

11For example R = R_{n}. See [GP2, Cor. 2.3].

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

(8.1.1)  Z_{𝔊}(V) : S → {g ∈ 𝔊(S) : Ad(g)(v_{S}) = v_{S} for every v ∈ V }

for all S in R–alg, where v_{S} denotes the image of v in 𝔤 ⊗_{R} S.

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

8.2. Remark. Note that Z_{𝔊}(V) = Z_{𝔊}(RV). This follows from the fact that the adjoint action of 𝔊 on 𝔤 is “linear” (in a functorial way).

We now introduce some of the central concepts of this work.

A subalgebra 𝔪 of the k-Lie algebra 𝔤 is called an AD subalgebra if 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 { v_{i} }_{i∈I} of 𝔤 such that

[h, v_{i}] = ⟨ λ_{i}, h ⟩ v_{i} for all h ∈ 𝔪.

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}) = g ⊗_{k} R. Assume that R is connected. Then 𝔪 = h ⊗ 1 is a MAD of 𝔤 by [P1, cor. to theo.1(i)]. We have Z_{G_{R}}(𝔪) = T_{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).

12It is not difficult to see that any such 𝔪 is necessarily abelian, so AD can be thought as shorthand for abelian k–diagonalizable or ad k–diagonalizable.

8.5. Remark. Let 𝔪 be an AD subalgebra of 𝔤. Then for any extension S/R in k–alg the image 𝔪 ⊗ 1 of 𝔪 in 𝔤 ⊗_{R} S is an AD subalgebra of 𝔤 ⊗_{R} S. Indeed if x ∈ 𝔪 and v ∈ 𝔤 are such that [x, v] = λv for some λ ∈ k, then [x ⊗ 1, v ⊗ s] = v ⊗ λs = λ(v ⊗ s) for all s ∈ S. 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.

8.6. Theorem. All MADs of g ⊗_{k} R_{n} are conjugate to h ⊗ 1 under G(R_{n}). □

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 morphism T_{d} × T_{a} → T is a central isogeny, hence 𝔱 = 𝔱_{d} ⊕ 𝔱_{a} where 𝔱_{a} = Lie (T_{a}). 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. Fix a basis { v_{1}, …, v_{n} } of g and scalars λ_{i} ∈ L such that

[ h, v_{i}] = λ_{i}v_{i} ∀ 1 ≤ i ≤ n.

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

T ↪ G ⟶^{Ad} GL(𝔤) ≃ GL_{n,L}

and

𝔱 ↪ 𝔤 ⟶^{ad} 𝔤𝔩 (𝔤) ≃ 𝔤𝔩_{n,L}.

Since G is of adjoint type Ad is injective, so that we can identify T with a subtorus, say T̃, of GL_{n,L}. Similarly for T_{d} and T_{a}. Since T ≃ T̃ we see that T̃_{d} and T̃_{a} are the maximal split and anisotropic parts of T̃.

Let D_{n} be the diagonal subgroup of GL_{n,L}. By construction we see that

ad_{𝔤}(h) ∈ Lie (D_{n}) ∩ Lie (T̃_{a}) = Lie (D_{n} ∩ T̃_{a}),

this last by [Hu, Theorem 12.5] since k is of characteristic 0. Thus D_{n} ∩ T̃_{a} has dimension > 0. But then the connected component of the identity of D_{n} ∩ T̃_{a} is a non-trivial split torus which contradicts the fact that T̃_{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.

9.2. Remark. Since 𝔖 is split we have Lie (𝔖) = X(𝔖)^{o} ⊗_{ℤ} R where X(𝔖)^{o} is the cocharacter group of 𝔖. As we shall see in the proof of Lemma 9.5 𝔪(𝔖) = X(𝔖)^{o} ⊗_{ℤ} k.

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).

(3) Let {v_{1}, …, v_{m}} be a k–basis of 𝔪. For every x ∈ 𝔛 the elements v_{i} ⊗1 ∈ 𝔤 ⊗_{R} R_{x} are R_{x}–linearly independent. 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 finite étale extension R̃/R with the following properties

(i) R̃ is integral and normal,

(ii) 𝔊 ×_{R} R̃ ≃ GR̃ ,

Note the following facts:

(iii) the canonical map 𝔤 → 𝔤 ⊗_{R} R̃ ≃ g ⊗_{k} R̃ is injective,

(iv) if {v_{1}, …, v_{m}} are k–linearly independent elements of 𝔪 which are R–linearly dependent, then the image of the elements {v_{1}, …, v_{m}} on Lie(𝔊 ×_{R} R̃) ≃ g ⊗_{k} R̃ are k–linearly independent and are R̃–linearly dependent.

Let K̃ be the field of fractions of R̃. By Remark 8.5 the image of 𝔪 under the injection 𝔤 ↪ 𝔤 ⊗_{R} R̃ ≃ g ⊗_{k} R̃ is an AD subalgebra of g ⊗_{k} R̃. By [P1, theo.1.(i)] the dimension of 𝔪 is at most the rank of g. This establishes (1).

As for (2) and (3), the crucial point–as explained in [P1, Prop. 4]–lies in the fact that the image 𝔪̃ of 𝔪 under the injection

𝔤 ↪ 𝔤 ⊗_{R} R̃ ≃ g ⊗_{k} R̃ ↪ g ⊗ K̃

sits inside a split Cartan subalgebra H of the split semisimple K̃-algebra g ⊗_{k} K̃. Consider the basis {ω̌_{1}, …, ω̌_{ℓ}} of H consisting of fundamental coweights for a base α_{1}, …, α_{ℓ} of the root system of (g ⊗_{k} K̃, H). Let 1 ≤ n ≤ m be such that {ṽ_{1}, … ṽ_{n}} is a maximal set of K̃–linearly independent elements of g(K̃). To establish (2) and (3) it will suffice to show that n = m.

Assume on the contrary that n < m. Write ṽ_{i} = ∑ c_{ji}ω̌_{j} with c_{1i}, … c_{ℓi} in K̃. The fact that the eigenvalues of ad_{g(K̃)}(ṽ_{i}) belongs to k show that the

c_{ji} necessarily belong to k. Indeed ṽ_{i} acts on g(K̃)_{α_{j}} as multiplication by the scalar c_{ji}.

Let v = v_{n+1}. Write ṽ = ∑ a_{i}ṽ_{i} with a_{1}, …, a_{n} in K̃. Let c_{jn+1} = λ_{j}. Then < α_{j}, ṽ >= λ_{j} and

ṽ = ∑_{j}(∑_{i} a_{i}c_{ji})ω̌_{j} = ∑_{j} λ_{j}ω̌_{j}

and therefore for all 1 ≤ j ≤ ℓ

∑_{i} a_{i}c_{ji} = λ_{j}.

Write K̃ = k ⊕ W as a k–space and use this decomposition to write a_{i} = d_{i} + w_{i} . Then

∑_{i} d_{i}c_{ji} = λ_{j}.

A straightforward calculation shows that < α_{j}, ṽ − ∑_{i} d_{i}ṽ_{i} >= 0 for all j. This forces

v_{n+1} = v = ∑_{i} d_{i}v_{i}

which contradicts the linear independence of the v′_{i}s over k. □

9.4. Remark. Let 𝔖 < 𝔊 be a split torus. Then there exist characters λ_{i} : 𝔖 → G_{m,R} for 1 ≤ i ≤ l such that

𝔤 = ⨁_{i=1}^{l} 𝔤_{λ_{i}}

where

𝔤_{λ_{i}} = { v ∈ 𝔤 : Ad(g)v = λ_{i}(g)v ∀g ∈ 𝔖(R) }.

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

λ_{i}(e^{sε}) = 1 + dλ_{i}(s)ε ∈ R[ε]^{×} = G_{m,R}(R[ε]).

Then for s ∈ 𝔰 and v ∈ 𝔤_{λ_{i}} we have the following equality in 𝔤

(9.4.1)  [s, v] = dλ_{i}(s)v.

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

𝔤 = ⨁_{i=1}^{l} 𝔤_{λ_{i}}

The λ_{i} are unique and

𝔪(𝔖) = { s ∈ Lie (𝔖) ⊂ 𝔖 (R[ε]) : dλ_{i}(s) ∈ k }.

Furthermore

dim_{k}(𝔪(𝔖)) = rank(𝔖) = rank_{R−mod}(R𝔪(𝔖))

and Lie(𝔖) = R𝔪(𝔖).

Proof. We appeal to the explanation given in Remark 9.4. Let

𝔫 = { s ∈ 𝔰 : dλ_{i}(s) ∈ k ∀i }.

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_{m,R}^{n} and the character lattice X(𝔖) of 𝔖 is generated by the projections π_{i} : G_{m,R}^{n} → 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

𝔪(𝔖) = { s ∈ 𝔰 : dλ_{i}(s) ∈ k ∀i }
= { s ∈ 𝔰 : dπ(s) ∈ k ∀π ∈ X(𝔖) }
= { s ∈ 𝔰 : dπ_{i}(s) ∈ k ∀i }.

The identification 𝔖 ≃ G_{m,R}^{n} induces the identification 𝔰 ≃ G_{a,R}^{n}. The above equalities yield

𝔪(𝔖) ≃ { (s_{1}, …, s_{n}) : s_{i} ∈ k ∀i },

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 R– module. Then the exact sequence

0 ⟶ R𝔪 ⟶ 𝔤 ⟶ M ⟶ 0

is split and the Proposition follows.

Thus it remains to show that M is a projective R–module or, equivalently, that for every prime ideal x of R the localized R_{x}–module M_{x} is free. Since localization is a left exact functor, and by Lemma 9.3 we have (R𝔪)_{x} = R_{x}𝔪 the sequence

0 ⟶ R_{x}𝔪 ⟶ 𝔤_{R_{x}} ⟶ M_{x} ⟶ 0

is exact. By Lemma 9.3(3), the elements

v_{1} ⊗ 1, …, v_{m} ⊗ 1 ∈ R_{x}𝔪 ⊂ 𝔤 ⊗_{R} R_{x} = 𝔤_{x}

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(\overline{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)(\overline{x}) we may look at

Z_{𝔊}(R𝔪) ⊗_{R} k(\overline{x}) = Z_{𝔊(\overline{x})}(k(\overline{x}) 𝔪(\overline{x}))

where 𝔊(\overline{x}) = 𝔊 ⊗_{R} k(\overline{x}) and 𝔪(\overline{x}) is the image of 𝔪 under 𝔤 → 𝔤 ⊗_{R} k(\overline{x}). 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)(\overline{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 k– diagonalizable 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

1 ⟶ L ⟶ G ⟶^{ϕ} G/L ⟶ 1

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 finite étale connected cover R̃/R which splits 𝔊. By replacing R by R̃ 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.

9.15. Sublemma. v ∈ O_{h}(R).

Proof. The element v ∈ g_{R} can be viewed as a morphism

ϕ_{v} : Spec (R) → g.

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. Since 𝔰 ⊂ 𝔖(R[ε])) it follows that in 𝔤 we have [𝔰, R𝔪] = 0. In particular since 𝔪 (𝔖) ⊂ 𝔰 we have [ 𝔪(𝔖), 𝔪] = 0. But then by Remark 8.4 𝔪 + 𝔪(𝔖) is an AD subalgebra of 𝔤. Since 𝔪 is a MAD we necessarily have 𝔪(𝔖) ⊂ 𝔪 and now we are going to show that 𝔪(𝔖) = 𝔪.

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

Lie (ℌ_{K}) = Lie (Z_{𝔊_{K}}(R𝔪)) = Lie (Z_{𝔊_{K}}(K𝔪)) = Z_{𝔤_{K}}(K𝔪).

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

Z_{𝔊}(𝔪) = Z_{𝔊}(R𝔪) = Z_{𝔊}(𝔰) = Z_{𝔊}(𝔖).

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

ℌ′ := Z_{𝔊}(R 𝔪′) ⊂ ℌ := Z_{𝔊}(R𝔪) = Z_{𝔊}(𝔖).

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. □

10. 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

(10.1.1)  H_{Zar}^{1}(R, Z_{𝔊}(𝔖)) = 1

then all generically maximal split tori of 𝔊 are conjugate under 𝔊(R).

Proof. We begin with a Lemma.

10.2. Lemma. Let 𝔖 and 𝔖′ be generically maximal split tori of 𝔊. Then (i) 𝔖_{R_{𝔭}} is a maximal split torus of 𝔊_{R_{𝔭}} for all 𝔭 ∈ 𝔛 := Spec (R). (ii) The transporter τ_{𝔖,𝔖′} = Trans_{𝔊}(𝔖, 𝔖′) is a (Zariski) locally trivial

N_{𝔊}(𝔖)–torsor over R.

Proof. (i) If 𝔖_{R_{𝔭}} is not maximal split then neither is 𝔖_{K} = 𝔖_{R_{𝔭}} ⊗_{Spec(R_{𝔭})} 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_{R_{𝔭}} and 𝔖′_{R_{𝔭}} are maximal split tori of 𝔊_{R_{𝔭}} 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_{Zar}^{1}(R, N_{𝔊}(𝔖)). Our aim is to show that α is trivial.

Consider the exact sequence (on 𝔛_{ét}) of R–groups

1 ⟶ Z_{𝔊}(𝔖) ⟶ N_{𝔊}(𝔖) ⟶ 𝔚 ⟶ 1

with 𝔚 = N_{𝔊}(𝔖)/Z_{𝔊}(𝔖). Then 𝔚 is a finite étale group over R (see [SGA3, XI, 5.9]). By Lemma 7.7(2) the image of α in H_{ét}^{1}(R, 𝔚), which we know lies in H_{Zar}^{1}(R, 𝔚), is trivial. Thus we may assume α ∈ H_{ét}^{1}(R, Z_{𝔊}(𝔖)). To finish the proof we need to show that

α ∈ Im [H_{Zar}^{1}(R, Z_{𝔊}(𝔖)) ⟶ H_{ét}^{1}(R, Z_{𝔊}(𝔖))].

For this it suffices to show that the image α_{𝔭} of α in

H_{ét}^{1}(R_{𝔭}, Z_{𝔊}(𝔖) ×_{R} R_{𝔭}) = H_{ét}^{1}(R_{𝔭}, Z_{𝔊_{R𝔭}} (𝔖_{R_{𝔭}}))

is trivial for all 𝔭 ∈ 𝔛.

Since 𝔖 is generically maximal split, 𝔖_{R_{𝔭}} is a maximal split torus of 𝔊_{R_{𝔭}}. Similarly for 𝔖′_{R_{𝔭}}. Now by [SGA3, XXVI] 𝔖_{R_{𝔭}} and 𝔖′_{R_{𝔭}} are conjugate under 𝔊_{R_{𝔭}}(R_{𝔭}) = 𝔊(R_{𝔭}), Thus the image of α under the composition of the natural maps

(10.2.1) H_{ét}^{1}(R, N_{𝔊}(𝔖)) ⟶ H_{ét}^{1}(R_{P} , N_{𝔊_{R𝔭}} (𝔖_{R_{𝔭}})) ⟶ H_{ét}^{1}(R_{𝔭}, 𝔊_{R_{𝔭}})

is trivial.

Let 𝔓 be a parabolic subgroup of 𝔊_{R_{𝔭}} containing Z_{𝔊_{R𝔭}} (𝔖_{R_{𝔭}}) as a Levi subgroup (see Lemma 5.1). Then (ibid.) we have

H_{ét}^{1}(R_{𝔭}, Z_{𝔊_{R𝔭}} (𝔖_{R_{𝔭}})) ≃ H_{ét}^{1}(R_{𝔭}, 𝔓) ↪ H_{ét}^{1}(R_{𝔭}, 𝔊_{R_{𝔭}}).

By (10.2.1) it now follows that α_{𝔭} is trivial. □

11. 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_{2}^{±1}] with generators T_{1}, T_{2} and relations T_{1}^{2} = 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

( \begin{matrix}x & 0 \\ 0 & x^{−1} \end{matrix} )

where x ∈ R^{×}. 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

(u, v) → (1 + T_{1})u − (1 + T_{2})v.

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.

12. 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

1 ⟶ μ μ μ ⟶ rad (ℌ) ×_{R} ℌ′ ⟶^{m} ℌ ⟶ 1

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

1 ⟶ μ μ μ_{K} ⟶ rad (ℌ_{K}) ×_{K} ℌ′_{K} ⟶^{m} ℌ_{K} ⟶ 1

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_{Zar}^{1}(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_{ét}^{1}(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 (applied 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.

13. 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(\overline{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 extension k̃ of k such that η factors through

η : Γ̃_{n,m} → G(k̃)

where

Γ̃_{n,m} := Gal(R_{n,m} ⊗_{k} k̃/R_{n}) = μ μ μ_{m}^{n}(k̃) ⋊ Gal(k̃/k)

where m > 0 and k̃/k is a finite Galois extension containing all m–roots of unity in \overline{k}. In this way η can be viewed as a Galois cocycle in Z^{1}(Γ̃_{n,m}, G(R_{n,m}⊗_{k}k̃)). We can thus twist G_{R_{n}} 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(\overline{k})) be a loop cocycle such that the twisted R_{n}–group ℌ = _{η}(G_{R_{n}}) is anisotropic. There exists a family of pro-solvable groups in k-vector spaces (J_{i})_{i=1,..,n} such that

ℌ(F_{n}) = J_{n} ⋊ J_{n−1} ⋊ … ⋊ J_{1} ⋊ (_{z}G)^{η^{geo}}(k) = (J_{n} ⋊ J_{n−1} ⋊ … ⋊ J_{1}) · ℌ(R_{n}).

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_{F_{n}} is also anisotropic by [GP3] cor. 7.4.3. It is convenient to work at a finite level, namely with a cocycle

η : Γ̃_{n,m} → G(k̃)

as in Remark 13.1. We may also assume that k̃ 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 F̃_{n,m} and its subfield F_{n} = (F̃_{n,m})Γ̃_{n,m} as local complete fields with the residue fields F̃_{n−1,m} and F_{n−1} respectively.

We consider the (enlarged) Bruhat-Tits building B_{n} = B(GF̃_{n,m}) of the

F̃_{n,m}–group GF̃_{n,m} [Ti, §2.1]. It is equipped with a natural action of G(F̃_{n,m})⋊

Γ̃_{n,m}. It is shown in [GP3] Theorem 7.9 that the building of ℌ_{F_{n}} inside B_{n}

consists of a single point ϕ whose stabilizer is G(F̃_{n−1,m}[[tn^{\frac{1}{m}}]]). Since ℌ(F_{n}) stabilizes ϕ it follows that

(13.2.1) ℌ(F_{n}) = \{g ∈ G(F̃_{n−1,m}[[tn^{\frac{1}{m}}]]) | η(σ) σ(g) = g ∀σ ∈ Γ̃_{n,m}\}.

We next decompose μ μ μ_{m}^{n} = μ μ μ_{m}^{n−1} × 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_{m}^{n−1} on G given by η^{geo} commutes with that of μ μ μ_{m} the k–group morphism η^{geo} : μ μ μ_{m}^{n} → G factors through G_{n−1}.

Denote by η_{n−1}^{geo} the restriction of η^{geo} to the k–subgroup μ μ μ_{m}^{n−1} of μ μ μ_{m}^{n}. Set Γ̃_{n−1,m} := μ μ μ_{m}^{n−1}(k̃) ⋊ Gal(k̃/k) and consider the loop cocycle

η_{n−1} : Γ̃_{n−1,m} → G_{n−1}(k̃)

attached to (1, η_{n−1}^{geo}). We define

ℌ_{n−1,R_{n−1}} = _{η_{n−1}}(G_{n−1,R_{n−1}}).

The crucial point for the induction argument is the fact that η_{n−1}^{geo} : μ μ μ_{m}^{n−1} → 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

sp_{n} : ℌ(F_{n}) ↪ G(F̃_{n−1,m}[[tn^{\frac{1}{m}}]]) → G(F̃_{n−1,m}).

Let P be the parahoric subgroup of ℌ^{∘}(F_{n}) attached to the point ϕ. Since the building of ℌ_{F_{n}} 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

map G(F̃_{n−1,m}[[tn^{\frac{1}{m}}]]) → G(F̃_{n−1,m}) is contained in G^{∘}(F̃_{n−1,m}[[tn^{\frac{1}{m}}]]). Since (ℌ/ℌ^{∘})(F_{n}) injects into (ℌ/ℌ^{∘})(F̃_{n,m}) = (G/G^{∘})(F̃_{n,m}), the kernel of the specialization map sp_{n} is the same for ℌ^{∘}(F_{n}) and ℌ(F_{n}). The parahoric

subgroup of G^{∘}(F̃_{n,m}) attached to the point ϕ is Q = G^{∘}(F̃_{n−1,m}[[tn^{\frac{1}{m}}]]) and we have

Q^{∗} = ker(Q → G^{∘}(F̃_{n−1,m}))

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

(13.3.1)  η(γ) ^{γ}h_{n−1} = h_{n−1} ∀γ ∈ Γ̃_{n,m}.

We now apply the relation (13.3.1) to the generator τ_{n} of the Galois group Gal(F̃_{n,m}/F̃_{n−1,m}((t_{n}))); it yields

(13.3.2)  η(τ_{n}) h_{n−1} = h_{n−1},

where η(τ_{n}) ∈ G(k̃), so that h_{n−1} ∈ G_{n−1}(F̃_{n−1,m}). Furthermore, the equality (13.3.1) restricted to Γ̃_{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

ℌ(F_{n}) = J_{n} ⋊ ℌ_{n−1}(F_{n−1})

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

ℌ_{n−1}(F_{n−1}) = (J_{n−1} ⋊ ⋯ ⋊ J_{1}) ⋊ G_{n−1}^{η_{n−1}^{geo}}(k).

Since G_{n−1}^{η_{n−1}^{geo}} = G^{η^{geo}}, we conclude that

ℌ(F_{n}) = (J_{n} ⋊ ⋯ ⋊ J_{1}) ⋊ G^{η^{geo}}(k)

as desired.

We have G^{η^{geo}}(k) ⊂ ℌ(R_{n}), so we get the second identity as well. □

14. Acyclicity, I

Let ℌ be a loop reductive group scheme. We will denote by H_{toral}^{1}(R_{n}, ℌ) [resp. H_{toral}^{1}(R_{n}, ℌ)_{irr} ] the subset of H^{1}(R_{n}, ℌ) consisting of isomorphism classes of torsors 𝔈 such that the twisted R_{n}–group _{𝔈}ℌ admits a maximal torus [resp. admits a maximal torus and is irreducible].

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

H_{toral}^{1}(R_{n}, ℌ)_{irr} → H^{1}(F_{n}, ℌ).

is injective.

Proof. By twisting, it is enough to show that for an irreducible loop reductive group ℌ the canonical map H_{toral}^{1}(R_{n}, ℌ) → H^{1}(F_{n}, ℌ) has trivial kernel. Indeed reductive R_{n}–group schemes admitting a maximal torus are precisely the loop reductive groups [GP3, Theorem 6.1]. We now reason by succesive cases.

Case 1: ℌ is adjoint and anisotropic. We may view ℌ as a twisted form of a Chevalley group scheme H_{R_{n}} by a loop cocycle η : π_{1}(R_{n}) → Aut(H)(\overline{k}). We have the following commutative diagram of torsion bijections

H_{toral}^{1}(R_{n}, Aut(ℌ)) ⟶ H^{1}(F_{n}, Aut(ℌ))
τ_{η}↓^{≃}  ^{τ_{η}}↓^{≃}
H_{toral}^{1}(R_{n}, Aut(H)) ⟶ H^{1}(F_{n}, Aut(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

ψ : H_{toral}^{1}(R_{n}, Aut(ℌ)) ⟶^{∼} H^{1}(F_{n}, Aut(ℌ)).

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

Aut(ℌ)(R_{n}) ⟶^{δ} Out(ℌ)(R_{n}) ⟶^{φ} H_{ét}^{1}(R_{n}, ℌ) ⟶ H_{ét}^{1}(R_{n}, Aut(ℌ))
↓  ||  ↓  ↓^{ψ}
Aut(ℌ)(F_{n}) ⟶^{γ} Out(ℌ)(F_{n}) ⟶ H^{1}(F_{n}, ℌ) ⟶ H^{1}(F_{n}, Aut(ℌ)).

Let v ∈ H_{ét}^{1}(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_{ét}^{1}(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

1 → ℨ ⟶^{i} ℌ → ℌ_{ad} → 1.

Here the adjoint group ℌ_{ad} is anisotropic since ℌ is irreducible. This exact sequence gives rise to the diagram

ℌ_{ad}(R_{n}) ⟶^{φ_{Rn}} H_{ét}^{1}(R_{n}, ℨ) ⟶^{i_{∗}} H_{ét}^{1}(R_{n}, ℌ) ⟶ H_{ét}^{1}(R_{n}, ℌ_{ad})
↓  ↓^{≃}  ↓  ↓
ℌ_{ad}(F_{n}) ⟶^{φ_{Fn}} H^{1}(F_{n}, ℨ) ⟶ H^{1}(F_{n}, ℌ) ⟶ H^{1}(F_{n}, ℌ_{ad}).

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_{ét}^{1}(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_{ét}^{1}(R_{n}, ℨ) such that v = i_{∗}(u) and u belongs to the image of the characteristic map φ_{F_{n}}. Since H_{ét}^{1}(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_{ét}^{1}(R_{n}, ℨ). So u belongs to the image of φ_{R_{n}}, and this implies that v = i_{∗}(u) = 1 ∈ H_{ét}^{1}(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]

1 → ℭ → ℌ^{sc} × 𝔈 ⟶^{f} ℌ → 1.

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

f^{−1}(𝔓) = 𝔓^{sc} × 𝔈, f^{−1}(𝔏) = 𝔏^{sc} × 𝔈

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)(\overline{k}) such that ℌ = _{η}(H_{R_{n}}). Let (T, B) be a Killing couple of H and Π ⊂ Δ(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 ⊂ Δ, 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_{F_{n}}. The version of the “Witt-Tits decomposition” given in [GP3, Cor. 8.4] applied to Aut(H_{ad}) shows that

[η] ∈ Im(H_{loop}^{1}(R_{n}, Aut(H_{ad}, P_{I,ad}, L_{I,ad}))_{irr} → H_{loop}^{1}(R_{n}, Aut(H_{ad}))).

Thus we may assume that η has values in Aut(H, P_{I}, L_{I})(\overline{k}) = Aut(H_{ad}, P_{I,ad}, L_{I,ad})(\overline{k}). The twisted R_{n}–group schemes 𝔓 = _{η}(P_{I}) and 𝔏 = _{η}(L_{I}) are as desired since 𝔓_{F_{n}} is a minimal F_{n}–parabolic subgroup of ℌ_{F_{n}} 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 R̃_{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 of étale R_{n}–sheaves 1 → 𝔓 → ℌ ⟶^{f} 𝔜 → 1 induces exact sequences of pointed sets [Gir, III.3.2.2]

ℌ(R_{n}) ⟶^{ψ} 𝔜(R_{n}) ⟶^{φ} H_{ét}^{1}(R_{n}, 𝔓) ⟶ H_{ét}^{1}(R_{n}, ℌ)
↑^{≃}
H_{ét}^{1}(R_{n}, 𝔏)

(note that a natural mapping H_{ét}^{1}(R_{n}, 𝔏) → H_{ét}^{1}(R_{n}, 𝔓) is a bijection by [SGA3, XXVI, 3.2]) and by base change

ℌ(F_{n}) ⟶^{ψ_{Fn}} 𝔜(F_{n}) ⟶^{φ_{Fn}} H^{1}(F_{n}, 𝔓) ⟶ H^{1}(F_{n}, ℌ)
↑^{≃}
H^{1}(F_{n}, 𝔏)

The R_{n}–parabolic subgroup 𝔔 defines a point y ∈ 𝔜(R_{n}).

15.6. Claim. φ(y) ∈ H_{toral}^{1}(R_{n}, 𝔏) ≃ H_{toral}^{1}(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 𝔉𝔓. 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 𝔉𝔏 carries then a maximal torus and the claim is proved.

On the other hand, we have already observed before in §15.4 that 𝔓_{F_{n}} and 𝔔_{F_{n}} are minimal parabolic subgroups of ℌ_{F_{n}}, 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_{F_{n}}, hence φ_{F_{n}}(y) = 1. It follows that φ(y) belongs to the kernel of

H_{toral}^{1}(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_{F_{n}} is a minimal parabolic subgroup of ℌ_{F_{n}}.

(3) Z_{ℌ}(𝔖)_{F_{n}} is irreducible,

(4) (Z_{ℌ}(𝔖)/𝔖)_{F_{n}} is anisotropic.

(5) 𝔖_{F_{n}} is maximal split in ℌ_{F_{n}}.

13Surprisingly enough, this compatibility is not in Giraud’s book. A proof can be found in [De, lemme 4.2.33 page 175].

Proof. (1) This is established in the proof in §15.4.b).

(2) Since ℌ is a loop reductive, this follows from [GP3, cor. 7.4.(1)].

(3) The R_{n}–group Z_{ℌ}(𝔖) is irreducible by 5.4.a and it is also loop reductive. The statement follows again from [GP3, cor. 7.4.(1)].

(4) The R_{n}–group Z_{ℌ}(𝔖)/𝔖 is anisotropic by 5.4.c. The statement follows then from [GP3, cor. 7.4.(3)].

(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)_{F_{n}} shows that 𝔖_{F_{n}} is a maximal R_{n}–torus of ℌ_{F_{n}}.

□

16. 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_{i}^{m} = 1. Let

R = k[t_{1}^{±1}, …, t_{n}^{±1}] and R̃ = k[t_{1}^{± \frac{1}{m}} , …, t_{n}^{± \frac{1}{m}}].

Recall that R̃/R is Galois and that we can identify Gal(R̃/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

L(g,σ σ σ) = ⨁_{(i_{1},...,i_{n})∈ℤ^{n}} g_{i_{1}...i_{n}} ⊗ t1^{\frac{i_{1}}{m}} … tn^{\frac{in}{m}} ⊂ g ⊗_{k} R̃

It is a twisted form of the R–Lie algebra g ⊗_{k} R which is split by R̃ :

L(g,σ σ σ) ⊗_{R} R̃ ≃ g ⊗_{k} R̃ ≃ (g ⊗_{k} R) ⊗_{R} R̃.

The R̃/R form L(g,σ σ σ) is given by a natural loop cocycle (denoted by z in [GP1, §5])

η = η(σ σ σ) ∈ Z^{1}(Γ, Aut(g)(k)) ⊂ Z^{1}(Γ, Aut(g)(R̃)).

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,

L(h,σ σ σ) = ⨁_{(i_{1},...,i_{n})∈ℤ^{n}} h_{i_{1}...i_{n}} ⊗ t1^{\frac{i_{1}}{m}} … tn^{\frac{in}{m}} ⊂ h ⊗_{k} R̃

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(\overline{x}) are Cartan subalgebras of 𝔤(\overline{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

𝔗_{d} ≃ T_{R}^{σ σ σ} = _{η}(T_{R}^{σ σ σ}) ⊂ 𝔊 = _{η}G_{R}.

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 Δ_{σ σ σ} = Δ_{σ σ σ}(g, h) be the subset of Δ(g, h) consisting of those roots that vanish on h^{σ σ σ} = h_{0,...,0}. We have

Z_{g}(h^{σ σ σ}) = h ⨁ ⨁_{α∈Δ_{σ σ σ}} g^{α}.

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

Z_{𝔊}(𝔗_{d}) = Z_{𝔊}(T_{R}^{σ σ σ}) ≃ _{η}H_{R}.

16.2. Proposition. (1) Z_{𝔊}(𝔗_{d}) is a loop reductive group.

(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 Δ of a finite dimensional vector space V over k such that 0 ∈ Δ and Δ^{×} := Δ \ {0} is a finite irreducible root system in V in the usual sense; see [Bbk1, chap. VI, §1, déf. 1]. Note that Δ^{×} is not assumed to be reduced.

We will use the following notation for the root system Δ. Let

Q = Q(Δ) := Σ_{α∈Δ}ℤα ⊂ V

be the root lattice of Δ. Let ⟨ , ⟩ : V × V^{∗} → k denote the natural pairing of V with V^{∗}. If α ∈ Δ^{×}, 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 Δ.

14Lie 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

Δ_{ind}^{×} := \{α ∈ Δ^{×} : \frac{1}{2}α ∉ Δ\}

denote the set of indivisible roots in Δ, and let

Δ_{ind} := Δ_{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_{α}^{λ} 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_{α}^{λ} for λ ∈ Λ and L_{α} = ⊕_{λ∈Λ}L_{α}^{λ} for α ∈ Q,

and we have L_{α}^{λ} = 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_{α}^{λ} = 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.

16.4. Definition. A Lie Λ-torus of (relative) type Δ is a Q × Λ-graded Lie algebra L over k which (with the notation as above) satisfies:

(LT1) supp_{Q}(L) = Δ.

(LT2)

(i) (Δ_{ind}^{×}, 0) ⊂ supp_{Q×Λ}(L).

(ii) If (α, λ) ∈ supp_{Q×Λ}(L) and α ≠ 0, then there exist elements e_{α}^{λ} ∈ L_{α}^{λ} and f_{α}^{λ} ∈ L_{−α}^{−λ} such that

L_{α}^{λ} = ke_{α}^{λ}, L_{−α}^{−λ} = kf_{α}^{λ},

and

(16.4.1)  [[e_{α}^{λ}, f_{α}^{λ}], x_{β}] = ⟨β, α^{∨}⟩x_{β}

for x_{β} ∈ L_{β}, β ∈ Q.

(LT3) L is generated as an algebra by the spaces L_{α}, α ∈ Δ^{×}.

(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 Δ. 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_{α}^{0}}_{α∈Δ^{×}} and t = ∑_{α∈Δ^{×}}[L_{α}^{0}, L_{−α}^{0}]. 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 Δ with root grading pair (s, t).

(i) If α ∈ Δ_{ind}^{×}, then Λ_{2α}^{0} = 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 → α̃ of V onto t^{∗} such that Δ̃_{ind} = Δ(s, t) and

(16.6.1)  [e_{α}^{0}, f_{α}^{0}] = α̃^{∨}

for α ∈ Δ_{ind}^{×}. Here α̃^{∨} ∈ (t^{∗})^{∗} ≃ t.

(iv) If α ∈ Q then

(16.6.2)  L_{α} = {x ∈ L : [h, x] = α̃(h)x for h ∈ t}.

(v) If (α, λ) ∈ supp_{Q×Λ}(L) and α ∈ Δ^{×} then

(16.6.3)  [e_{α}^{λ}, f_{α}^{λ}] = α̃^{∨}

(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 Δ and a Q × Λ– grading 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. Δ, Q, Λ do not change, nor do the supports. As the reader may have guessed, all we do is to set (L_{θ})_{α}^{λ} = θ(L_{α}^{λ}), and use the elements θ(e_{α}^{λ}) and θ(f_{α}^{λ}) 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])

E = L ⊕ kc ⊕ kd

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

E = L ⊕ C ⊕ D

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 Δ, 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 Δ and the root grading pairs are invariants of L. In other words.

Question: Let L be a centreless Lie Λ-torus of type Δ with root grading pair (s, t). Assume that L admits the structure of a centreless Lie Λ′-torus of type Δ′ with root grading pair (s′, t′). Is Λ ≃ Λ′, Δ ≃ Δ′, s ≃ s′ and t ≃ t′?

15In 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.

The data (Δ, Λ) leads to a Lie algebra isomorphism ϕ : L → L(g,σ σ σ). Similarly we have ϕ′ : L → L(g′,σ σ σ′). By descent considerations it is clear that g ≃ g′: This is the Invariance of the Absolute Type. See for example [GP1, theo. 4.9] or [ABP2.5, theo 8.16.]. The explicit nature of the Realization Theorem shows that ϕ maps (s, t) to (g^{σ σ σ}, h_{t}) where h_{t} is a Cartan subalgebra of the simple Lie algebra g^{σ σ σ}. Furthermore h_{t} is a Borel-Mostow MAD h^{σ σ σ} of g^{σ σ σ}. Similar considerations apply to (Δ′, Λ′).

We now use ϕ ∘ ϕ′^{−1} to put two Λ Lie tori structures of relative type Δ and Δ′ 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 Δ′ on L with grading pairs (s′′′, t) with s′′′ ≃ s′′. The root system of (s′′′, t) is of type Δ′ while that of (s, t) is of type Δ. Since t determines the Q–grading we conclude that Δ ≃ Δ′, hence that s ≃ s′. This completes the proof that our Question has an affirmative answer.

As a consequence we see that the (relative) type Δ 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.

16The exceptions occur only on type A and they are completely classified.

17The converse is false in nullity > 1.

17. Acyclicity, II

17.1. Theorem. Let ℌ be a loop reductive group scheme over R_{n}. Then the natural map

H_{toral}^{1}(R_{n}, ℌ) → H^{1}(F_{n}, 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_{ét}^{1}(R_{1}, G) → H^{1}(F_{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(\overline{k}) be a loop cocycle and consider the loop reductive R_{n}–groups ℌ = _{η}H_{R_{n}} and ℌ^{∘} = _{η}H^{∘}_{R_{n}}. 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

ℌ(F_{n}) = ⟨ ℌ(R_{n}), J, ℌ(F_{n})^{+}⟩

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_{n}- simple, 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_{m}^{n} (the case n = 1 follows from the identity F_{1}^{×} = R_{1}^{×} . ker(k[[t_{1}]]^{×} → k^{×}). Since all finite connected étale 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

1 → 𝔖 ⟶^{i} 𝔈 ⟶^{f} 𝔗 → 1

of multiplicative R_{n}–group schemes. It gives rise to a commutative diagram

1 ⟶ 𝔖(R_{n}) ⟶^{i_{Rn}} 𝔈(R_{n}) ⟶^{f_{Rn}} 𝔗(R_{n}) ⟶^{φ_{Rn}} H_{ét}^{1}(R_{n}, 𝔖) ⟶ 1
↓  ↓  ↓  ^{≃}↓
1 ⟶ 𝔖(F_{n}) ⟶^{i_{Fn}} 𝔈(F_{n}) ⟶^{f_{Fn}} 𝔗(F_{n}) ⟶^{φ_{Fn}} H^{1}(F_{n}, 𝔖) ⟶ 1

with exact rows. Note that the right vertical map is an isomorphism by [GP2, Prop. 3.4] and that surjectivity on the right horizontal maps is due to the fact H_{ét}^{1}(R_{n}, 𝔈) = H^{1}(F_{n}, 𝔈) = 1. By diagram chasing we see that

𝔗(R_{n})/f_{F_{n}}(𝔈(R_{n})) ⟶^{∼} 𝔗(F_{n})/f_{R_{n}}(𝔈(F_{n})).

Therefore the case of the induced torus 𝔈 provides a suitable group J such that 𝔗(F_{n}) = 𝔗(R_{n}) · f_{F_{n}}(J).

Case (2): ℌ = 𝔏 is irreducible. Let ℭ be the radical torus of 𝔏. We have an exact sequence [SGA3, XXI, 6.2.4]

1 ⟶ μ μ μ ⟶^{i} D𝔏 ×_{R_{n}} ℭ ⟶^{f} 𝔏 ⟶ 1.

It gives rise to a commutative diagram of exact sequences of pointed sets

(D𝔏 × ℭ)(R_{n}) ⟶^{f_{Rn}} 𝔏(R_{n}) ⟶^{φ_{Rn}} H_{ét}^{1}(R_{n},μ μ μ) ⟶^{i_{∗,Rn}} H_{loop}^{1}(R_{n}, D𝔏 × ℭ)
↓  ↓  ^{≃}↓  ↓
(D𝔏 × ℭ)(F_{n}) ⟶^{f_{Fn}} 𝔏(F_{n}) ⟶^{φ_{Fn}} H^{1}(F_{n},μ μ μ) ⟶^{i_{∗,Fn}} H^{1}(F_{n}, D𝔏 × ℭ).

Note that the image of the map H_{ét}^{1}(R_{n},μ μ μ) → H_{ét}^{1}(R_{n}, D𝔏) is contained in H_{toral}^{1}(R_{n}, D𝔏)_{irr}. So taking into consideration Theorem 14.1 (applied to the irreducible loop reductive group scheme D𝔏 and chasing the above diagram we see that

𝔏(R_{n})/f_{R_{n}}((D𝔏)(R_{n}) × ℭ(R_{n})) ⟶^{∼} 𝔏(F_{n})/f_{F_{n}}((D𝔏)(F_{n}) × ℭ(F_{n})).

The case of D𝔏 done in Proposition 13.2 together with the case of the torus ℭ provide a suitable normal group J such that 𝔏(F_{n}) = 𝔏(R_{n}) · J.

Case (3). ℌ = ℌ^{∘}. Since ℌ is loop reductive by assumption it suffices to observe that ℌ(F_{n}) is generated by 𝔏(F_{n}) and ℌ^{+}(F_{n}) [BT73, 6.11].

Case (4). For the general case it remains to show that for an arbitrary element g ∈ ℌ(F_{n}) the coset gℌ^{∘}(F_{n}) contains at least one R_{n}–point of ℌ.

Let 𝔖 be the maximal split torus of the radical of 𝔏. The torus g𝔖_{F_{n}}g^{−1} ⊂ ℌ_{F_{n}}^{∘} is maximal split, hence there is g_{1} ∈ ℌ^{∘}(F_{n}) such that g𝔖_{F_{n}}g^{−1} = g_{1}𝔖_{F_{n}}g_{1}^{−1}. Thus replacing g by g_{1}^{−1}g if necessary, we may assume that g𝔖_{F_{n}}g^{−1} = 𝔖_{F_{n}}. Then we also have g(𝔏_{F_{n}})g^{−1} = 𝔏_{F_{n}}, so that g ∈ N_{ℌ}(𝔏)(F_{n}).

The torus 𝔖 is clearly normal in N_{ℌ}(𝔏). Hence we have an exact sequence

1 ⟶ 𝔖 ⟶ N_{ℌ}(𝔏) ⟶ ℌ′ := N_{ℌ}(𝔏)/𝔖 ⟶ 1.

Note that since H_{ét}^{1}(R_{n}, 𝔖) = 1, the natural maps N_{ℌ}(𝔏)(R_{n}) → ℌ′(R_{n}) and N_{ℌ}(𝔏)(F_{n}) → ℌ′(F_{n}) are surjective. Furthermore, ℌ′ satisfies all conditions of Theorem 13.2, so that the required fact follows immediately from that theorem applied to ℌ′ and from the surjectivity of the above maps. □

We can proceed to the proof of Theorem 17.1 which is very similar to that of Theorem 14.1.

Proof. Injectivity: By twisting, it is enough to show that the natural map H_{toral}^{1}(R_{n}, ℌ) → H^{1}(F_{n}, ℌ) has trivial kernel. We first assume that ℌ is adjoint. We may view ℌ as the twisted form of a Chevalley group scheme H_{R_{n}} by a loop cocycle η : π_{1}(R_{n}) → Aut(H(\overline{k})). The same reasoning given in Case 1 of the proof of Theorem 14.1 shows that we have a natural bijection

(17.4.1)  H_{toral}^{1}(R_{n}, Aut(ℌ)) ⟶^{∼} H^{1}(F_{n}, Aut(ℌ)).

The exact sequence

1 → ℌ → Aut(ℌ) → Out(ℌ) → 1

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

Aut(ℌ)(R_{n}) ⟶^{γ} Out(ℌ)(R_{n}) ⟶^{φ} H_{ét}^{1}(R_{n}, ℌ) ⟶ H_{ét}^{1}(R_{n}, Aut(ℌ))
↓  ||  ↓  ↓
Aut(ℌ)(F_{n}) ⟶^{ψ} Out(ℌ)(F_{n}) ⟶ H^{1}(F_{n}, ℌ) ⟶ H^{1}(F_{n}, Aut(ℌ)).

Let v ∈ H_{ét}^{1}(R_{n}, ℌ) be a toral class mapping to 1 ∈ H^{1}(F_{n}, ℌ). In view of bijection (17.4.1) there exists u ∈ Out(ℌ)(R_{n}) such that v = φ(u) and u belongs to the image of ψ. Since Out(ℌ)(R_{n}) is a finite group, the Density Theorem 17.3 shows that Aut(ℌ)(R_{n}) and Aut(ℌ)(F_{n}) have same image in Out(ℌ)(F_{n}). So u belongs to the image of γ, hence v = 1 ∈ H_{ét}^{1}(R_{n}, ℌ). Let now ℌ be an arbitrary reductive group. Set ℭ = Z(ℌ). This is an R_{n}–group of multiplicative type and we have an exact (central) sequence of R_{n}-group schemes

1 → ℭ ⟶^{i} ℌ → ℌ_{ad} → 1.

This exact sequence gives rise to the diagram of exact sequence of pointed sets

ℌ_{ad}(R_{n}) ⟶^{φ_{Rn}} H_{ét}^{1}(R_{n}, ℭ) ⟶^{i_{∗}} H_{ét}^{1}(R_{n}, ℌ) ⟶ H_{ét}^{1}(R_{n}, ℌ_{ad}) ⟶^{Δ} H_{ét}^{2}(R_{n}, ℭ)
↓  ↓^{≃}  ↓  ↓  ↓^{≃}
ℌ_{ad}(F_{n}) ⟶^{φ_{Fn}} H^{1}(F_{n}, ℭ) ⟶ H^{1}(F_{n}, ℌ) ⟶ H^{1}(F_{n}, ℌ_{ad}) ⟶^{Δ_{Fn}} H_{ét}^{2}(F_{n}, ℭ).

The isomorphisms H_{ét}^{i}(R_{n}, ℭ) ≅ H^{i}(F_{n}, ℭ) comes from [GP2, prop. 3.4.(3)] for i = 1, 2.

Let v ∈ H_{ét}^{1}(R_{n}, ℌ) be a toral class mapping to 1 ∈ H^{1}(F_{n}, ℌ). Taking into account the adjoint case, a diagram chase provides u ∈ H_{ét}^{1}(R_{n}, ℭ) such that v = i_{∗}(u) and u_{F_{n}} belongs to the image of the characteristic map φ_{F_{n}}. Since H_{ét}^{1}(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_{ét}^{1}(R_{n}, ℭ). So u belongs to the image of φ_{R_{n}}. Hence v = i_{∗}(u) = 1 ∈ H_{ét}^{1}(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

ℌ(R_{n})/ℌ(R_{n})^{+} → ℌ(F_{n})/ℌ(F_{n})^{+}

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.

18. 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.

18.1. Greenberg functor. We recall here basic facts, see the references [Gb], [M2, §III.4], [BLR], [B].

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) = \underleftarrow{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

Green (𝔖) : = \underleftarrow{lim}_{r}Green_{r}(𝔖)

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

\underline{G}_{r}(𝔛) := Green_{r}( ∏_{O/W (k)} 𝔛), \underline{G}(X) := Green ( ∏_{O/W (k)} 𝔛).

We have \underline{G}_{r}(𝔛)(k) = 𝔛(O/p^{r}O) and \underline{G}(𝔛)(k) = 𝔛(O). We have \underline{G}(Spec(O)) = Spec(k); if 𝔛 is a O–group scheme, then \underline{G}(𝔛) and the \underline{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 \underline{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)

\underline{G}_{r}( ∏_{O_{L}/O} 𝔜) ≃ ∏_{l/k} \underline{H}_{r}(𝔜), \underline{G}( ∏_{O_{L}/O} 𝔜) ≃ ∏_{l/k} \underline{H}(𝔜).

In particular if k = l then we have \underline{G}_{r}( ∏_{O_{L}/O} 𝔜) = \underline{H}_{r}(𝔜) and \underline{G}( ∏_{O_{L}/O} 𝔜) ≃

\underline{H}(𝔜).

Proof. We have a commutative square

O ⟶ O_{L}
↑  ↑
W (k) ⟶ W (l).

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

(18.2.1)  ∏_{O/W (k)} ∏_{O_{L}/O} 𝔜 = ∏_{O_{L}/W (k)} 𝔜 = ∏_{W (l)/W (k)} ∏_{O_{L}/W (l)} 𝔜.

Let Λ be a k–algebra. Using (18.2.1) and the definitions of the Greenberg functors, we have

\underline{G}_{r}( ∏_{O_{L}/O} 𝔜)(Λ) = Green_{r}( ∏_{O_{L}/W (k)} 𝔜)(Λ)

= ( ∏_{W (l)/W (k)} ∏_{O_{L}/W (l)} 𝔜)(W_{r}(Λ))

= ( ∏_{O_{L}/W (l)} 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) ⊗_{W_{r}(k)} W_{r}(Λ) = W_{r}(Λ ⊗_{k} l)

by [I, 1.5.7]. Hence

\underline{G}_{r}( ∏_{O_{L}/O} 𝔜)(Λ) = ( ∏_{O_{L}/W (l)} 𝔜)(W_{r}(Λ ⊗_{k} l)) = R_{l/k}(\underline{H}_{r})(Λ)

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

18.3. Lemma. (1) Let 𝔛/O be an affine scheme of finite type such that 𝔛_{K} = ∅. Then \underline{G}(𝔛) = ∅.

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

Proof. (1) We have 𝔛 = Spec (A) where A is an O/π^{d}O–algebra of finite type for d large enough Put r_{0} = d e_{0}. Then p^{r_{0}}A = 0. For a k–algebra Λ we have by definition

\underline{G}(𝔛)(Λ) = Hom_{O}(A, W (Λ) ⊗_{W (k)} O).

But W (Λ) ⊗_{W (k)} O is p–torsion free, so \underline{G}(𝔛) = ∅.

(2) We have N = Spec(B) and we have the decomposition B = O ⊕ I where I is the kernel of the co-unit of the corresponding Hopf algebra. The Omodule I is an ideal of B which is an O/π^{d}O-algebra of finite type. The same reasoning as above shows that

\underline{G}(𝔑)(Λ) = Hom_{O}(B, W (Λ) ⊗_{W (k)} O)
= Hom_{O}(O, W (Λ) ⊗_{W (k)} O)
= \underline{G}(Spec(O))(Λ).

Thus \underline{G}(𝔑) = \underline{G}(Spec(O)) which is nothing but that Spec(k) as reminded above. □

Secondly, assume that k and K have the same characteristic (0 or p > 0) and we still assume that k is perfect. Then k embeds in O (in an unique way, [EGA4, 21.5.3]) and for an O-scheme 𝔛 the functors

\underline{G}(𝔛) := ∏_{O|k} 𝔛 and \underline{G}_{r}(𝔛) := ∏_{O/π^{r}O | k}(𝔛 ×_{O} O/π^{r}O)

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

𝔛(O) = \underleftarrow{lim}_{r} 𝔛(O/π^{r}O) = \underleftarrow{lim}_{r} \underline{G}_{r}(𝔛)(k)

where the \underline{G}_{r}(𝔛) are k-schemes (by Weil restriction [BLR, §7.6]). The two lemmas are true as well.

18.4. Congruence filtration. Let G be a reductive K–group and denote by B = B(G, K) its (extended) Bruhat-Tits building. Let x be a point of B and denote by P_{x} the parahoric subgroup

P_{x} = { g ∈ G(K) | g(x) = x }.

Denote by 𝔓_{x} the canonical smooth group scheme over O defined by Bruhat-Tits [BT2, §5.1] with generic fiber G and such that 𝔓_{x}(O) = P_{x} or, more precisely,

𝔓_{x}(O^{sh}) = { g ∈ G(K^{sh}) | g(x) = x }

where x is viewed as an element in B(G, K^{sh}) via the canonical mapping B(G, K) ↪ B(G, K^{sh}). Since 𝔓_{x} is smooth we have

𝔓_{x}(O) = \underleftarrow{lim}_{n≥1}𝔓_{x}(O/π^{n}O)

and the transition maps 𝔓_{x}(O/π^{n+1}O) → 𝔓_{x}(O/π^{n}O) are surjective with kernel Lie(𝔓_{x}) ⊗_{O} k ([M2, III.4.3])

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 := \underleftarrow{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

1 ⟶ P_{x}^{∗} ⟶ P_{x} ⟶ M_{x} → 1
↓  ↓  ||
1 ⟶ R_{x,n} ⟶ P_{x,n} ⟶ M_{x} → 1.

□

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}) = \underleftarrow{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 = \overline{K} = \overline{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

j : B(G, K) → B(G, L)

induces a bijection B(G, K) ⟶^{∼} B(G, L)^{Γ}. For z ∈ B(G, L), we denote by Q_{z} the parahoric subgroup of G(L) and by 𝔔_{z} the canonical group scheme over O_{L} attached to the point z.

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

𝔔_{σ(z)}  ⟶^{f_{σ,z}}  𝔔_{z}
↓  ↓
Spec(O_{L}) ⟶^{(σ^{−1})^{∗}} Spec(O_{L}).

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

(18.7.1)  P_{x} = G(K) ∩ Q_{y} = G(L)^{Γ} ∩ Q_{y} = Q_{y}^{Γ}.

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_{y}^{Γ}.

By taking k–points we get the following wished compatibility, namely.

18.9. Corollary. We have

P_{x}^{∗} ⟶^{∼} P_{x} ∩ Q_{y}^{∗}.

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_{y}^{Γ} := \underleftarrow{lim}_{n}Q_{y,n}^{Γ} and (Q_{y}^{∗})^{Γ} = Q_{y}^{Γ} ∩ 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_{y,n}^{Γ}(k′))_{n≥1} has surjective transitions maps. Therefore the projective system of k-groups (Q_{y,n}^{Γ})_{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_{y}^{Γ}(k′) → M_{y}^{Γ}(k′) → 1;

hence the sequence of the pro-algebraic k-groups

1 → (Q_{y}^{∗})^{Γ} → Q_{y}^{Γ} → M_{y}^{Γ} → 1

is also exact.

(3) The algebraic k–group M_{y}^{Γ} 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.

It gives rise to the exact sequence of pointed sets

1 → Q_{y,n+1/n}(k)^{Γ} → Q_{y,n+1}(k)^{Γ} → Q_{y,n}(k)^{Γ} → H^{1}(Γ, Q_{y,n+1/n}(k)).

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_{y,n}^{Γ}) of k–groups is surjective (because Q_{y,n}^{Γ}(k) = Q_{y,n}(k)^{Γ}).

(2) By part (1), the map Q_{y}^{Γ}(k) → (Q_{y,1})^{Γ}(k) is surjective. The same argument as in (1) shows that (Q_{y,1})^{Γ}(k) → M_{y}^{Γ}(k) is also surjective. By taking the composition of these maps we conclude the map Q_{y}^{Γ}(k) → M_{y}^{Γ}(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_{y}^{Γ} 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_{y}^{Γ} and Q_{y}^{Γ}(k) = Q_{y}(k)^{Γ} = Q_{y}^{Γ}. By Lemma 18.10, Q_{y}^{Γ}(k) projects onto M_{y}^{Γ}(k), so the composite map

P_{x} = P_{x}(k) → Q_{y}^{Γ}(k) → M_{y}^{Γ}(k)

is surjective. Since this is true for all finite extensions of k, the homomorphism of k-algebraic groups P_{x} → M_{y}^{Γ} is surjective. But (M_{y}^{Γ})^{∘} is reductive, hence this map is trivial on the pro-unipotent radical P_{x}^{∗}. We get then a surjective map M_{x} → M_{y}^{Γ} and also a homomorphism P_{x}^{∗} → (Q_{y}^{∗})^{Γ} ⊂ Q_{y}^{∗} as required. □

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Department of Mathematics, University of Alberta, Edmonton, Alberta T6G 2G1, Canada

E-mail address: chernous@math.ualberta.ca

UMR 8553 du CNRS, École Normale Supérieure, 45 rue d’Ulm, 75005 Paris, France.

E-mail address: gille@ens.fr

Department of Mathematics, University of Alberta, Edmonton, Alberta T6G 2G1, Canada.

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

E-mail address: a.pianzola@math.ualberta.ca