WHITEHEAD GROUPS OF LOOP GROUP SCHEMES OF NULLITY ONE

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

Abstract. We define and study the Whitehead group of isotropic (almost) simple simply connected group schemes over Laurent polynomial rings k[t^{±1}], where k is a field of characteristic 0. Our motivation for doing this comes from infinite dimensional Lie theory. Keywords: Reductive group scheme, Kac-Moody groups, Whitehead groups, Laurent polynomials, Kneser-Tits problem, buildings. MSC 2000 11E72, 14L30, 14E20.

1. Introduction

Let L be an isotropic absolutely almost simple simply connected algebraic group over a field k. Soulé and Margaux [Mg, Cor. 3.6] have given a precise description of the Whitehead group of the constant group scheme L ×_{k} 𝔸_{k}^{1} over an affine line 𝔸_{k}^{1}. The main purpose of this note is to replace the affine line by the punctured affine line Spec(R) with R = k[t^{±1}], and then study the structure of the Whitehead group of an arbitrary simple simply connected group schemes over R. We emphasize that our groups need not be constant, i.e. that they need not come from k-groups by base change to R.

Recall that for such an L, the Whitehead group W (k, L) is the quotient W (k, L) = L(k)/L(k)^{+} where L(k)^{+} stands for the subgroup of L(k) generated by the k-points of the unipotent radicals R_{u}(Q)(k) and R_{u}(Q^{−})(k) of two opposite proper parabolic subgroups Q and Q^{−} of L. It is worth noticing that L(k)^{+} is automatically a normal subgroup of L(k), hence W (k, L) carries a natural group structure. Furthermore, it is known that W (k, L) does not depend on the choice of a pair of opposite parabolic subgroups Q and Q^{−}.

Our motivation for looking at this problem for group schemes over R comes from infinite dimensional Lie theory (where groups and algebras over Laurent polynomial rings play a prominent role. See [CGP1], [CGP2], [GP1] and [GP2] for details). Recall that for proving conjugacy of Cartan subalgebras in extended affine Lie algebras (which are infinite dimensional Lie algebras over a ground field k) it is important to describe their automorphism groups.

Date: November 16, 2013.

V. Chernousov was partially supported by the Canada Research Chairs Program and an NSERC research grant.

P. Gille was supported by the Romanian IDEI project PCE_{−}2012-4-364.

A. Pianzola wishes to thank NSERC and CONICET for their continuous support.

Given an extended affine Lie algebra E we can consider its corresponding centreless core Ẽ (from which E can be reconstructed). We can view Ẽ as either a Lie algebra (infinite dimensional) over k or over its centroid R. It is known that Ẽ is a “simple” Lie algebra over R^{1} and that R is isomorphic to a Laurent polynomial ring in finitely many variables. One can prove that there exists a natural homomorphism Aut(E) → Aut_{k}(Ẽ) and one would like to show that it is surjective. The group Aut_{k}(Ẽ) is generated by its subgroups Aut_{R}(Ẽ) and Aut_{k}(R). The connected component of the group Aut_{R}(Ẽ) is a simple adjoint group scheme G over R. Elements of unipotent radicals of parabolic subgroups of G can be written using exponential map and the same exponential map provides us with their liftings to Aut(E). This is why it is important to understand how far the subgroup of G(R) generated by unipont elements is from the group G(R) itself.

The concept of Whitehead group for arbitrary (isotropic simple simply connected) group schemes has not been defined in the literature yet. Defining this object (under certain assumptions) is our first task. Consider a connected ring R and a simple simply connected group scheme 𝔊 over R.^{2} We will throughout assume that 𝔊 is isotropic, i.e. that it has a closed subgroup isomorphic to the multiplicative group G_{m,R} over R, or equivalently it has a proper parabolic subgroup.

We admit that, even under these assumptions, it is not clear a priori what the “correct” definition of Whitehead groups should be. Mimicking the field case, we could take a pair (𝔓, 𝔓^{−}) of opposite proper parabolic subgroup schemes of 𝔊 and define W_{𝔓}(R, 𝔊) as before, i.e. as the set of left (right) cosets of 𝔊(R) modulo the “elementary” subgroup E_{𝔓}(R) of 𝔊(R) generated by R_{u}(𝔓)(R) and R_{u}(𝔓^{−})(R). Note that W_{𝔓}(R, 𝔊) depends only on 𝔓, but not on the choice of an opposite parabolic subgroup 𝔓^{−} [PS, §1], and that W_{𝔓}(R, 𝔊) has no group structure in general case [Su].

Since we wish to work with a normal subgroup of 𝔊(R), we will consider the normal subgroup E_{𝔓}^{st}(R) of 𝔊(R) generated by E_{𝔓}(R) and we call the quotient group

W_{𝔓}^{st}(R, 𝔊) = 𝔊(R)/E_{𝔓}^{st}(R)

the stable Whitehead group of 𝔊 relative to 𝔓.

If 𝔊 contains a split torus G_{m,R}^{2} = G_{m,R} ×G_{m,R} or, more generally, if the fibers 𝔊 ×_{R} (R/𝔪) are of relative (split) rank ≥ 2 for all maximal ideals 𝔪 of R it is known that E_{𝔓}(R) = E_{𝔓}^{st}(R) and that this normal subgroup does not depend on the choice of 𝔓 (Petrov-Stavrova [PS, Theorem 1]). Hence in this case the stable Whitehead group is an invariant of 𝔊.

1This is true except for a well-understood family of absolute type A given by quantum tori with “generic” entries.

2We remind the reader that this means that, for all x ∈ Spec(R), the geometric fiber 𝔊\overline{x} of 𝔊 is an (almost) simple and simply connected algebraic group over the corresponding algebraic closure of residue field \overline{k(x)}. Because the base is assumed to be connected, the type of this simply connected group is unique (i.e. independent of x).

We assume henceforth that k is a field of characteristic 0, and let R = k[t^{±1}] be the corresponding ring of Laurent polynomials. Let G be a split almost simple simply connected algebraic group over k and let 𝔊 be a twisted form of the R-group G ×_{k} R, namely 𝔊 is a group scheme over R such that 𝔊 ×_{R} S ≃ G ×_{k} S for some faithfully flat and finitely presented extension S/R. Thus 𝔊 corresponds to a torsor over R under Aut(G). Since Aut(G) is smooth, we may assume that S/R is étale.

Recall that according to [GP2, Theorem 5.1] and (ibid, Cor. 6.3) any such 𝔊 is loop reductive. For the definitions of loop cocycles, loop group schemes and their properties we refer to § 2. Here we recall only that 𝔊 can be realized as a twist of G ×_{k} R by a loop cocycle η and that η gives rise to a connected reductive algebraic group H over k and a closed immersion H ×_{k} R ↪ 𝔊. In many cases one can think of H as being a “maximal” constant subgroup scheme of 𝔊.

We can now state our main result.

1.1. Theorem. (1) The group 𝔊(R) is generated by H(k) and E_{𝔓}^{st}(R). (2) Assume that 𝔊 is quasi-split. Then W_{𝔓}^{st}(R, 𝔊) = 1.

The interest of the result is for twisted non-split group schemes. Indeed the split case follows from a general result of Steinberg on Chevalley groups over euclidean rings [St1, Cor. 3, p. 115].

Under the conditions of the Petrov–Stavrova’s result quoted above we have a stronger result.

1.2. Corollary. Assume that for each closed point s of Spec(R), the k(s)– algebraic group 𝔊 ×_{R} k(s) is of relative rank ≥ 2.

(i) 𝔊(R) = H(k) · E_{𝔓}(R);

(ii) If 𝔊 is quasi-split, then E_{𝔓}(R) = 𝔊(R).

(iii) If k is algebraically closed, then E_{𝔓}(R) = 𝔊(R).

Note that if k is algebraically closed, it is known [P2] that 𝔊 is quasisplit. Assertion (iii) is then a consequence of (ii). In turn, assertion (ii) is a consequence of Theorem 1.1, part (2) and the above quoted result of Petrov–Stavrova asserting that E_{𝔓}(R) = E_{𝔓}^{st}(R).

For k non algebraically closed, our result stated in Theorem 1.1 is not enough to compute precisely the stable Whitehead group W_{𝔓}^{st}(R, 𝔊) in general, but we have some conjectures as to its nature. Let R = k[t^{±1}].

1.3. Conjecture. (1) E_{𝔓}(R) = E_{𝔓}^{st}(R);

(2) The natural maps

W_{𝔓}^{st}(R, 𝔊) → W (K, 𝔊 ×_{R} K) → W (F, 𝔊 ×_{R} F)

are group isomorphisms where K = k(t) and F = k((t)) is the completion of K at zero.

(3) The map H(k) → 𝔊(R) induces an isomorphism H(k)/R ≃ W_{𝔓}^{st}(R, 𝔊).

Assertion (1) can be rephrased by saying that E_{𝔓}(R) is a normal subgroup of 𝔊(R). In assertion (3), R stands for the R-equivalence relation for the k–group H.

We can see then the quasi-split case as providing some evidence for the conjecture (see Theorem 6.3).

Notation and conventions. Throughout this work k denotes a field of characteristic 0 and \overline{k} = k_{s} an algebraic (separable ) closure of k. We also set R = k[t^{±1}], the ring of Laurent polynomials attached to k, and let K = k(t) denote its fraction field.

By a k–group we will simply mean a group scheme over Spec(k). Similarly for an R–group.

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. The (relative) rank of a reductive R-group 𝔊 is the maximum of the ranks of its split subtori 𝔗 ⊂ 𝔊

2. The loop setting

For the reader’s convenience we recall the definition of loop group schemes. Throughout this section 𝔛 will denote a connected noetherian scheme over k, and G a k–group which is locally of finite presentation.^{3}

2.1. 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 automorphisms 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}, and the canonical map f : \underrightarrow{lim} Hom_{𝔛}(𝔛_{i}, 𝔛′) → F (𝔛′) is bijective.

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

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 group 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}} \overline{𝔛} → Spec(k)]. Then by [SGA1, Théorème IX.6.1]

π_{1}( Spec(k), b) ≃ Gal(k) := Gal(\overline{k}/k)

and the sequence

(2.1.1)  1 → π_{1}(\overline{𝔛}, \overline{a}) → π_{1}(𝔛, a) → Gal(k) → 1

is exact.

2.2. Example. Assume that 𝔛 = Spec (R) where R = k[t^{±1}] is the Laurent polynomial ring with coefficients in k. The simply connected cover R^{sc} of R is

R_{∞} = \underrightarrow{lim} \overline{R}_{m}

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

(2.2.1)  π_{1}(𝔛, a) = ℤ̂(1) ⋊ 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).

2.3. Loop torsors and groups. Because of the universal nature of 𝔛^{sc} we have a natural group homomorphism

(2.3.1)  G(\overline{k}) ⟶ G(𝔛^{sc}).

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). This action is continuous, and together with (2.3.1) yields a map

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

2.4. 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.

2.5. Geometric and arithmetic part of a loop cocycle. We assume henceforth that our geometric point a lies above a k-rational point of 𝔛. This provides (see §3.3 of [GP2] for details) an action of Gal (k) on π_{1}(\overline{𝔛}, \overline{a}) and natural splitting of the exact sequence (2.1.1). Thus

π_{1}(𝔛, a) = π_{1}(\overline{𝔛}, \overline{a}) ⋊ Gal (k).

By means of this decomposition we can think of loop cocycles as being comprised of a geometric and an arithmetic part, as we now explain.

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}).

2.6. 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}( Gal(k), 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 [GP2]. □

2.7. Remark. Assume that R = k[t^{±1}] and 𝔛 = Spec(R). It is easy to verify that η^{geo} arises from a unique k–group homomorphism

∞μ μ μ = \underleftarrow{lim}μ μ μ_{m} → _{z}G

2.8. Loop reductive groups. Let ℌ be a reductive group scheme over 𝔛. 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).

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

2.10. Example. Let R = k[t^{±1}]. According to [GP2] every reductive group scheme 𝔊 over R is loop reductive. Thus 𝔊 is isomorphic to the twist η(G ×_{k} R) of its Chevalley form G by a cocycle

η : π_{1}(R) → Aut(G)(R^{sc})

which takes values in the subgroup Aut(G)(k_{s}) of Aut(G)(R^{sc}).

3. Preliminaries

We keep the notation of the previous section. In particular, R = k[t^{±1}] is the ring of Laurent polynomials over a field k and 𝔊 is a twisted R–form of an almost simple split simply connected group G by a loop cocycle η.

3.1. The subgroup H. We fix a Killing couple (B, T) of G and denote by Δ the associated Dynkin diagram. For each subset I of Δ, we let T_{I} =

( ⋂_{α∈I} ker(α))^{0}, L_{I} = Z_{G}(T_{I}) and we denote by P_{I} and P_{I}^{−} the standard

parabolic subgroups attached to I.

Let I_{t} be the Tits index of 𝔊. By the Witt-Tits decomposition [GP2, § 8.2], we can assume additionally that η takes value in

Aut_{I_{t}}(G)(k_{s}) := Aut(G, P_{I_{t}}, L_{I_{t}})(k_{s}).

Then the twisted subgroup scheme 𝔓 = _{η}(P_{I_{t}} ×_{k} R) is a minimal parabolic subgroup of 𝔊 and we may consider the above defined subgroup E_{𝔓}(R) of 𝔊(R).

Furthermore, since loop cocycles define “toral classes” [GP2, §6.1], we can also assume that η takes values in

Aut_{I_{t}}(G, T)(k_{s}) := Aut(G, P_{I_{t}}, L_{I_{t}}, T)(k_{s}).

Any such cocycle factorizes at finite level: there exists an integer m ≥ 1, a finite Galois extension k̃ of k containing μ μ μ_{m}(k_{s}) such that η is represented by a cocycle

Γ → Aut_{I_{t}}(G, T)(k̃),

still denoted by η, where

Γ := Gal(k̃[t^{± \frac{1}{m}}]/R) = μ μ μ_{m}(k̃) ⋊ Gal(k̃/k).

Let Y be a quasi-projective k–variety equipped with a left action of the k–group Aut_{I_{t}}(G, T). Then we can twist the k-variety Y by η^{ar} [Se1, I.5.3] and the R-scheme Y_{R} = Y ×_{k} R by η [M, I.5.3] The composition of natural maps

η^{geo} : μ μ μ_{m} ⟶ Aut_{I_{t}}(_{η^{ar}} G,_{η^{ar}}T) ⟶ Aut(_{η^{ar}}Y)

gives a natural action of μ μ μ_{m} on the variety _{η^{ar}}Y and we denote by (_{η^{ar}}Y)^{η^{geo}} the corresponding subvariety consisting of fixed points. It follows immediately from the twisting procedure that we now have a natural closed immersion

j : (_{η^{ar}}Y)^{η^{geo}} ×_{k} R ↪ _{η}(Y_{R}).

This observation applied to Y = G gives rise to the k-algebraic group H = (_{η^{ar}}G)^{η^{geo}} and the closed immersion j : H ×_{k} R ↪ 𝔊. Note that according to Steinberg’s connectedness theorem [St2, Theorem 8.1], H is a (connected) reductive group. Thus we have natural embeddings H(k) ↪ H(R) ↪ 𝔊(R).

3.2. Example. Assume that 𝔊 is a quasi-split form of G = SL_{n+1} which does not come from k. The corresponding twisting cocycle η is determined by the following data: a quadratic étale extension S = R(\sqrt{u}), u ∈ R^{×}, of R over which 𝔊 becomes split, and an outer automorphism σ of SL_{n+1} of order 2. Since 𝔊 is not obtained from a k-group by base change we have u = at for some a ∈ k^{×}, hence η^{ar} = 1. It follows that H = SL_{n+1}^{⟨σ⟩}.

We now note that the twisting data is determined uniquely (up to equivalence) by the quadratic extension S/R. Indeed, any other choice of an outer automorphism of SL_{n+1} of order 2 gives rise to a loop cocycle η′ which is equivalent to η (because any two quasi-split R-forms of SL_{n+1} which are split over S are isomorphic over R (see [SGA3, XXIV 3.11])).

It is known that if n + 1 is even, say 2l, then over the algebraic closure \overline{k} there are two conjugacy classes of outer automorphisms of order two. They are of the form x → (x^{−1})^{τ}, x ∈ SL_{n+1} where τ is either a symplectic involution of the matrix algebra M_{n+1} or an orthogonal involution. In the first case H ≃ Sp_{l} and in the second case H ≃ SO_{n+1}. If n + 1 is odd then τ is automatically an orthogonal involution and hence H ≃ SO_{n+1}. Note that in the orthogonal case we may choose τ to be split, hence H = SO_{n+1} is also split.

3.3. Parabolic subgroups of 𝔊. We now come to the description of parabolic subgroups of our R–group 𝔊. Recall that by our construction 𝔓 = η(P_{I_{t}} ×_{k} R) is a minimal parabolic subgroup of 𝔊. The k–group

Aut_{I_{t}}(G, T) = Aut(G, P_{I_{t}}, L_{I_{t}}, T)

is a subgroup of Aut(G, T) and hence we have a natural mapping

Aut_{I_{t}}(G, T) → Aut(Δ).

Pushing the 1-cocycle η by this map, we get a homomorphism

η_{Δ} : Γ → Aut(Δ)

called sometimes the “star action”. As in the field case, the parabolic subgroups of 𝔊 containing 𝔓 are of the form _{η}(P_{I} ×_{k} R) for I running over the subsets of Δ containing I_{t} and stable under the action η_{Δ}. These are called the “standard parabolic subgroups” of 𝔊.

3.4. Proposition. Let 𝔔 be a parabolic subgroup of 𝔊. Then 𝔔 is 𝔊(R)– conjugate to a unique standard parabolic subgroup of 𝔊.

Proof. The unicity is clear since the standard parabolic subgroups provide distinct conjugacy classes.

Assume first that the parabolic subgroup 𝔔 is minimal and choose a Levi R–subgroup 𝔐 of 𝔔 (its existence is granted by [SGA3, XXVI.2.3]). Then 𝔐 is a loop reductive group scheme by [GP2, Theorem 5.1]. Furthermore Theorem 15.1 of [CGP2] asserts that the couple (𝔔, 𝔐) is 𝔊(R)-conjugate to (𝔓 = _{η}(P_{I_{t}} ×_{k} R),_{η}(L_{I_{t}} ×_{k} R)).

In the general case, let 𝔔_{min} ⊂ 𝔔 be a minimal parabolic subgroup of 𝔊. By the preceding case, we can assume that 𝔔_{min} = 𝔓, so that 𝔔 is a standard parabolic subgroup of 𝔊. □

3.5. Corollary. 𝔊(K) = 𝔓(K) 𝔊(R).

Proof. The set (𝔊/𝔓)(R) parametrizes R–parabolic subgroups of 𝔊 which are locally conjugate for the étale topology to 𝔓 [SGA3, XXVI.3.20]. Proposition 3.4 says that 𝔊(R) acts transitively on (𝔊/𝔓)(R), hence we have a natural bijection

𝔊(R)/𝔓(R) ⟶^{∼} (𝔊/𝔓)(R).

Similarly, by a theorem of Borel-Tits [BT65, th. 4.13] we have

𝔊(K)/𝔓(K) ⟶^{∼} (𝔊/𝔓)(K).

But dim(R) = 1 and 𝔊/𝔓 is a projective R–scheme, hence (𝔊/𝔓)(R) = (𝔊/𝔓)(K). This implies that the natural embedding 𝔊(R) ↪ 𝔊(K) induces a bijection

𝔊(R)/𝔓(R) ⟶^{∼} 𝔊(K)/𝔓(K)

and the result follows. □

3.6. Subgroups attached to roots. By our construction _{η}(T_{I_{t}} ×_{k} R) is the centre of the Levi subgroup _{η}(L_{I_{t}} ×_{k} R) of the minimal parabolic subgroup 𝔓 of 𝔊. Let 𝔖 be the maximal split subtorus of _{η}(T_{I_{t}} ×_{k} R) and 𝔖̂ be its character group. The torus 𝔖 is also maximal split in 𝔊, since otherwise 𝔓 would not be a minimal parabolic subgroup of 𝔊. Note that 𝔖_{F} is still maximal split over F [GP2, Cor. 7.4.3]. The torus 𝔖 acts by the adjoint representation on 𝔤 = Lie(𝔊)(R) and we have the decomposition [SGA3, XXIV.6]

𝔤 = 𝔤_{0} ⊕ ⨁_{α∈Ψ(𝔊,𝔖)} 𝔤_{α}

where Ψ(𝔊, 𝔖) ⊂ 𝔖̂ is the set of roots of 𝔊 with respect to 𝔖. This decomposition can be extended to F and since 𝔖_{F} is maximal split over F, Ψ(𝔊, 𝔖) is nothing but the relative root system of 𝔊_{F} with respect to 𝔖_{F}.

The minimal parabolic subgroup 𝔓 of 𝔊 defines a basis Δ_{0} of the root system Ψ(𝔊, 𝔖) and an order on Ψ(𝔊, 𝔖). We denote by 𝔚 = N_{𝔊}(𝔖)/ℨ_{𝔊}(𝔖) the relative Weyl group. It is a finite constant R–group, say 𝔚 ≃ W_{R}, and for each root α ∈ Ψ(𝔊, 𝔖) we denote by w_{α} ∈ W = 𝔚(R) the associated reflection.

For each simple root α ∈ Δ_{0} we denote by 𝔓_{α} = 𝔓_{α}^{+} (resp. 𝔓_{α}^{−}) the parabolic subgroup scheme [SGA3, XXVI.6.1] of 𝔊 such that its Lie algebra Lie(𝔓_{α}^{+})(R) (resp. Lie(𝔓_{α}^{−})(R)) is generated by 𝔤_{0}, 𝔤_{β} for all positive (resp. negative) roots β and all β ∈ Ψ(𝔊, 𝔖)∩ℚ_{<0}·α (resp. β ∈ Ψ(𝔊, 𝔖)∩ℚ_{>0}·α). These two parabolic subgroups are opposite and share the common Levi subgroup ℨ_{α} = 𝔓_{α} ∩ 𝔓_{α}^{−}. We denote by 𝔊_{α} = D(ℨ_{α}) the derived subgroup of ℨ_{α} [SGA3, XXII.6]. It is well-known that 𝔊_{α} is a semisimple R–group scheme.

3.7. Lemma. (1) 𝔖_{α} = (𝔖 ∩ 𝔊_{α})^{0} is a maximal split torus of the R–group 𝔊_{α}. Furthermore, 𝔖_{α} is of rank one.

(2) If 𝔊 is simply connected (resp. quasi-split), so is 𝔊_{α}.

Proof. (1) The F –group 𝔊_{α,F} is of relative rank one and 𝔖_{α,F} is its maximal split subtorus over F. A fortiori, 𝔖_{α} is a maximal split torus of 𝔊_{α} of rank 1.

(2) The fact that the derived group of a Levi subgroup of a parabolic subgroup scheme is simply connected can be checked on the geometric fibers. The problem is thus reduced to the case when the base scheme is (the spectrum of) an algebraically closed field. We can now apply [SS, Cor. 5.8].

Next we assume that 𝔊 is quasi-split. Then 𝔓 is a Borel subgroup of 𝔊. Since ℨ_{α} is a Levi subgroup of 𝔓_{α}, the intersection ℨ_{α} ∩ 𝔓 is a Borel subgroup of ℨ_{α} [SGA3, XXVI.1.20]. Thus ℨ_{α} is quasi-split and so is its derived group 𝔊_{α}. □

4. Proof of Theorem 1.1 (1)

Recall that 𝔊 is assumed to be almost simple simply connected. The proof that we provide is a refinement of the methods used by Soulé and Margaux. It is based on a delicate analysis of group actions on Bruhat-Tits euclidean (affine) buildings. At first glance, since Spec(k[t^{±1}]) is obtained from the projective line ℙ^{1} by deleting 0 and ∞, it would seem more natural to look at the action of 𝔊(R) on the twin building B(𝔊_{k((t))}) × B(𝔊_{k((t^{−1}))}) as it was done in [CGP1] and [GP2, §5.1]. This action is indeed well-understood in the split case (Abramenko [A]) but, as it turns out, it would appear to be ill-suited to deal with the twisted case. We work then separately with the building at infinity B_{−} = B(𝔊_{k((t^{−1}))}) and the building at zero B_{+} = B(𝔊_{k((t))}).

4.1. Buildings. Let Õ_{−} = k̃[[t^{− \frac{1}{m}}]], F̃_{−} = k̃((t^{− \frac{1}{m}})) and F_{−} = k((t^{−1})) We denote by B̃_{−} the Bruhat–Tits euclidean building of 𝔊F̃_{−} = G ×_{k} F̃_{−} [BT2]. It is equipped with a natural action of

Γ = Gal(k̃[t^{± \frac{1}{m}}]/k[t^{±1}]) = Gal(F̃_{−}/F_{−})

and Aut(G(F̃_{−})). This allows us to consider the twisted action of Γ on B̃_{−} given by

γ ⋆ x = η(γ) · γ(x) γ ∈ Γ, x ∈ B̃_{−}.

There exists a natural embedding B_{−} ↪ B̃_{−} and the Bruhat-Tits-Rousseau’s descent theorem [Ro, V.1] states that B_{−} ⟶^{∼} (B̃_{−})^{_{⋆}Γ} where the notation _{⋆}Γ means that we take the fixed points with respect to the twisted action of Γ.

The hyperspecial group G(Õ_{−}) of G(F̃_{−}) fixes a unique point ϕ̃_{−} of B̃_{−} [BT1, §9.1.9.c]. This point ϕ̃_{−} is _{⋆}Γ–stable because η takes values in Aut(G(k_{s})), hence it descends to a point ϕ_{−} of B_{−}.

Similarly, we set Õ_{+} = k̃[[t^{\frac{1}{m}}]], F̃_{+} = k̃((t^{\frac{1}{m}})) and the corresponding buildings (resp. origins) will be denoted B_{+} and B̃_{+} (resp. ϕ_{+} and ϕ̃_{+}).

The first step of the proof of Theorem 1.1 (1) is to see why it suffices to deal with points of 𝔊(R) having only a pole at infinity.

4.2. Reduction to a single pole. We shall use here the preliminaries considered in § 3.1. Consider the twisted action of Γ on the abstract groups G(k̃[t^{\frac{1}{m}}]) and G(k̃) by η. Since

G(k̃) ⊂ G(k̃[t^{\frac{1}{m}}]) ⊂ G(k̃[t^{± \frac{1}{m}}])

it follows that

H(k) = H^{0}(Γ,_{η}G(k̃)) ⊂ H^{0}(Γ,_{η}G(k̃[t^{\frac{1}{m}}])) ⊂ H^{0}(Γ,_{η}G(k̃[t^{± \frac{1}{m}}])) = 𝔊(R).

Clearly, the evaluation of t^{\frac{1}{m}} at 0

ev_{0} : G(k̃[t^{\frac{1}{m}}]) → G(k̃)

is a Γ-equivariant map (for the twisted action). It induces a map

H^{0}(Γ,_{η}G(k̃[t^{\frac{1}{m}}])) → H^{0}(Γ,_{η}G(k̃)) = H(k).

The composition of the natural mappings

H(k) ↪ H^{0}(Γ,_{η}G(k̃[t^{\frac{1}{m}}])) ⟶ H^{0}(Γ,_{η}G(k̃)) = H(k)

is the identity map, hence

H^{0}(Γ,_{η}G(k̃[t^{\frac{1}{m}}])) = Θ ⋊ H(k),

where

Θ = Ker(H^{0}(Γ,_{η}G(k̃[t^{\frac{1}{m}}])) → H(k))

which can also be thought of as H^{0}(Γ,_{η}Θ̃) with

Θ̃ = Ker(G(k̃[t^{\frac{1}{m}}]) ⟶^{ev_{0}} G(k̃)).

4.3. Lemma. (1) 𝔊(k((t)))^{+} is an open (in the t-adic topology) normal subgroup of 𝔊(k((t))).

(2) 𝔊(k((t))) = H(k) · 𝔊(k((t)))^{+}.

(3) 𝔊(R) = H(k) Θ E_{𝔓}(R).

Proof. (1) The group 𝔊(k((t)))^{+} is normal non-central in 𝔊(k((t))), it is then open by a result of Riehm, see [PR, §3.1, th. 3]. The above reference is for p-adic fields but it also works here since the implicit function theorem holds [Se2, LG III, §10.2].

(2) Let ℌ be the canonical Bruhat-Tits k[[t]]-group scheme associated to ϕ_{+} [BT2]. Its generic fiber is 𝔊_{k((t))} = _{η}(G ×_{k} k((t))) and we have

ℌ(k[[t]]) = Stab_{𝔊(k((t)))}(ϕ_{+}).

By Galois descent, this is also

Stab_{𝔊(k((t)))}(ϕ_{+}) = Stab_{G(k̃((t^{\frac{1}{m}})))}(ϕ̃_{+})^{_{∗}Γ} = G(k̃[[t^{\frac{1}{m}}]])^{_{∗}Γ}.

In particular, arguing as above we obtain that the specialization map at t^{\frac{1}{m}} = 0 defines a surjective map ev_{0} : ℌ(k[[t]]) → H(k) and the decomposition

(4.3.1)  ℌ(k[[t]]) = J ⋊ H(k)

where J is the kernel of ev_{0}.

4.4. Claim. 𝔊(k((t))) = ℌ(k[[t]]) · 𝔊(k((t)))^{+}.

We need to recall the construction of ℌ done in [BT2, §5] which starts with the construction of a smooth k[[t]]-group scheme ℨ with generic fiber Z = Z_{𝔊_{k((t))}}(𝔖_{k((t))}) [BT2, §5.2.1] such that

ℨ(k[[t]]) = {g ∈ Z(k((t))) | χ(g) ∈ k[[t]]^{×} ∀χ ∈ Hom_{k((t))−gp}(Z, 𝔾_{m,k((t))}}.

The group ℨ is a k[[t]]-subgroup scheme of ℌ and is connected (ibid, 5.2.3 and 5.2.5). Hence ℨ(k[[t]]) is a subgroup of ℌ(k[[t]]), and it is enough to show that ℨ(k[[t]]) generates the quotient 𝔊(k((t)))/𝔊(k((t)))^{+}.

Let C_{+} be a chamber of the apartment A(𝔖_{k((t))}) associated to the maximal split torus 𝔖_{k((t))} of 𝔊_{k((t))}. We know that C_{+} is a fundamental simplicial domain for the action of 𝔊(k((t))) on B_{+} ([BT2, prop. 5.2.12] and [BT1, cor. 2.1.6]), so that 𝔊(k((t))) is generated by the Stab_{𝔊(k((t)))}(x) for

x running over the points of the closure of C_{+} according to [So1].

Each point x ∈ A(𝔖_{k((t))}) defines a canonical k[[t]]-group scheme 𝔊_{x} such that 𝔊_{x}(k[[t]]) = Stab_{𝔊(k((t)))}(x). Since 𝔊_{k((t))} is simply connected,

we know that 𝔊_{x} is connected [BT2, 5.2.9], hence the group 𝔊_{x}(k[[t]]) is generated by its subgroup ℨ(k[[t]]) and 𝔊_{x}(k[[t]]) ∩ 𝔊(k((t)))^{+} (ibid, 5.2.4). In other words, the images of 𝔊_{x}(k[[t]]) and ℨ(k[[t]]) coincide in the quotient 𝔊(k((t)))/𝔊(k((t)))^{+}. Given that the above holds for all x belonging to the closure of C_{+}, we get that ℨ(k[[t]]) generates 𝔊(k((t)))/𝔊(k((t)))^{+}, whence the Claim.

It remains then to show that the image of the subgroup J of ℌ(k[[t]]) in W (k((t)), 𝔊) is also trivial. This group had been looked at in great detail in the appendix of [CGP2] where we called it the pro-unipotent radical of the group ℌ(k[[t]]). We showed that it is an open subgroup of ℌ(k[[t]]) and then also of 𝔊(k((t))). By means of the congruence filtration, it admits a fundamental system of neighbourhoods (J_{n}) where the J_{n}’s are closed and open normal subgroups such that J_{n}/J_{n+1} is a finite dimensional k-vector space for each n ≥ 0.

By Part (1), there exists an integer N ≥ 1 such that J_{N} ⊂ 𝔊(k((t)))^{+}. On the other hand there exists an integer d such that each element of W (k((t)), 𝔊_{k((t))}) ≃ 𝔊(k((t)))/R is of exponent dividing d [G, Rem. 7.6]. Here 𝔊(k((t)))/R is the group of R-equivalence classes. Arguing by induction on n we easily get that J^{d} maps onto J/J_{n} where J^{d} stands for the (normal) subgroup of J generated by the d-powers. Therefore the image of J = J^{d} · J_{N} in W (k((t)), 𝔊_{k((t))}) is trivial. In other words J ⊂ 𝔊(k((t)))^{+}, hence decomposition (4.3.1) yields

𝔊(k((t))) = H(k) · 𝔊(k((t)))^{+}.

(3) Let g ∈ 𝔊(R). Viewing g as an element of 𝔊(k((t))) and multiplying it on the left by a suitable element of H(k), we may assume without loss of generality that g ∈ 𝔊(k((t)))^{+}. Let u_{1}, …, u_{n} ∈ R_{u}(𝔓)(k((t))) and v_{1}, …, v_{n} ∈ R_{u}(𝔓^{−})(k((t))) be such that g = u_{1} v_{1} u_{2} v_{2} … u_{n} v_{n}. Recall that the group

ℌ(k[[t]]) = H^{0}(Γ,_{η}G(k̃[[t^{\frac{1}{m}}]]))

is an open (in the analytic topology) subgroup of

H^{0}(Γ,_{η}G(k̃((t^{\frac{1}{m}})))) = 𝔊(k((t))).

Since the unipotent radicals R_{u}(𝔓) and R_{u}(𝔓^{−}) of 𝔓 and 𝔓^{−} are split R_{u}(𝔓)(R) and R_{u}(𝔓^{−})(R) are dense in R_{u}(𝔓)(k((t))) and R_{u}(𝔓^{−})(k((t))) respectively. Choose

u′_{1}, …, u′_{n} ∈ R_{u}(𝔓)(R) and v′_{1}, …, v′_{n} ∈ R_{u}(𝔓^{−})(R)

such that

g_{0} := (u_{1} v_{1} u_{2} v_{2} … u_{n} v_{n})(u′_{1} v′_{1} u′_{2} v′_{2} … u′_{n} v′_{n})^{−1} ∈ H^{0}(Γ,_{η}G(k̃[[t^{\frac{1}{m}}]])).

Then g = g_{0} u′_{1} v′_{1} u′_{2} v′_{2} … u′_{n} v′_{n}. Evidently

g_{0} ∈ H^{0}(Γ,_{η}G(k̃[t^{\frac{1}{m}}])) = Θ ⋊ H(k) = H(k) · Θ

(because g and all u′_{i}, v′_{i} are in 𝔊(R)). Thus g ∈ H(k) · Θ · E_{𝔓}(R). □

4.5. Buildings and fundamental domains. We work in this subsection with the buildings B̃_{−} and B_{−} at infinity.

Let Ã_{−} = ϕ̃_{−} + (T̂)^{0} ⊗_{ℤ} ℝ be the apartment of B̃_{−} associated to the maximal split torus T ×_{k} F̃_{−} of G ×_{k} F̃_{−}. The canonical pairing ⟨ , ⟩ : (T̂)^{0} × T̂ → ℤ extends to

⟨ , ⟩ : ((T̂)^{0} ⊗_{ℤ} ℝ) × (T̂ ⊗_{ℤ} ℝ) → ℝ.

We then define the sector (quartier)

Q̃ := ϕ̃_{−} + D̃ where D̃ := {v ∈ T̂^{0} ⊗_{ℤ} ℝ | ⟨v, b⟩ ≥ 0 ∀ b ∈ Δ}.

The following is a quite formal consequence of Soulé’s theorem [So2].

4.6. Lemma. (a) For each point x ∈ Q̃, the decomposition G(k̃[t^{\frac{1}{m}}]) = Θ̃ ⋊ G(k̃) induces a decomposition

StabΘ̃(x) ⋊ Stab_{G(k̃)}(x) ≃ Stab_{G(k̃[t^{\frac{1}{m}}])}(x)

(b) The set G(k̃) · Q̃ is a simplicial fundamental domain for the action of the group Θ̃ on B̃_{−}.

Proof. (a) We need to show that the natural injective map

StabΘ̃(x) ⋊ Stab_{G(k̃)}(x) → Stab_{G(k̃[t^{\frac{1}{m}}])}(x)

is onto. If x = ϕ̃_{−}, we have seen that

G(k̃) = Stab_{G(k̃)}(x) = Stab_{G(k̃[t^{\frac{1}{m}}])}(x),

so we may assume that x ≠ ϕ̃_{−}. We recall that Stab_{G(k̃[t^{\frac{1}{m}}])}(x) is also

the isotropy (or fixator) of the half line [x[ in Q̃ of direction \overrightarrow{ϕ̃_{−}x}. Write x = ϕ̃_{−} + d with d ∈ D̃ and put

I_{x} = {α ∈ Δ | ⟨ d, α ⟩ = 0}.

¿From [Mg, Prop. 2.5.(3)], we know that

Fix_{G(k̃[t^{\frac{1}{m}}])}([x[) ⊂ U_{I_{x}}(k̃[t^{\frac{1}{m}}]) ⋊ L_{I_{x}}(k̃).

Since L_{I_{x}}(k̃) ⊂ Fix_{G(k̃[t^{\frac{1}{m}}])}([x[) (ibid, lemma 2.3.(1)), it suffices to look at

the group Stab_{U_{Ix}(k̃[t^{\frac{1}{m}}])}(x).

4.7. Claim. U_{I_{x}}(k̃) fixes x.

The claim yields that Stab_{U_{Ix}(k̃[t^{\frac{1}{m}}])}(x) is generated by Stab_{U_{Ix}(k̃)}(x)

and StabΘ̃(x) ∩ U_{I_{x}}(k̃[t^{\frac{1}{m}}]). Let us prove the Claim. The group U_{I_{x}}(k̃) is generated by the U_{α}(k̃) with α ∈ Δ satisfying ⟨ d, α ⟩ = 0. But U_{α}(k̃) fixes [x[ for each such α (subgroups of type (II) in [Mg, page 396]) whence the Claim.

(b) Soulé’s theorem states that Q̃ is a simplicial fundamental domain for the action of the group G(k̃[t^{\frac{1}{m}}]) on B̃_{−}. In particular, we have

B̃_{−} = G(k̃[t^{\frac{1}{m}}]) · Q̃ = (Θ̃ ⋊ G(k̃)) · Q̃ = Θ̃ · (G(k̃) · Q̃).

It remains to show that two points of G(k̃) · Q̃ which are in the same orbit of Θ̃ are necessary equal. Let y_{1} = g_{1} x_{1}, y_{2} = g_{2} x_{2} ∈ G(k̃) · Q̃ be such that gg_{1}x_{1} = g_{2}x_{2} with g ∈ Θ̃. By Soulé’s theorem we have x_{1} = x_{2}. Let x = x_{1} = x_{2}. Then

g_{2}^{−1}gg_{1} = (g_{2}^{−1}gg_{2})(g_{2}^{−1}g_{1}) ∈ Stab_{G(k̃[t^{\frac{1}{m}}])}(x).

Note that Θ̃ is a normal subgroup in G(k̃[t^{\frac{1}{m}}]). By applying (a) we get that g_{2}^{−1}g_{1} ∈ Stab_{G(k̃)}(x). We conclude that

y_{1} = g_{1} · x = (g_{2} (g_{2}^{−1} g_{1})) · x = g_{2} · x = y_{2},

as required. □

The major reason to use the groups Θ and Θ̃ is because they afford the following very precise control on their subgroups stabilizing points on the building B̃_{−}.

4.8. Lemma. Let x be a point of B̃_{−}.

(1) There exists a split unipotent k̃–group V such that V(k̃) = StabΘ̃(x).

(2) Assume that x ∈ B_{−}. Then H^{1}(Γ, _{η}(StabΘ̃(x))) = 1.

Proof. (1) By Lemma 4.6 we can assume that x = q ∈ Q̃. Let q = ϕ̃_{−} + d with d ∈ D̃ and put

I_{q} = {α ∈ Δ | ⟨ d, α ⟩ = 0}.

¿From [Mg, Prop. 2.5.(3)], we know that

Stab_{G(k̃[t^{\frac{1}{m}}])}(q) ⊂ U_{I_{q}}(k̃[t^{\frac{1}{m}} ]) ⋊ L_{I_{q}}(k̃) ⊂ P_{I_{q}}(k̃[t^{\frac{1}{m}} ])

where U_{I_{q}} is the unipotent radical of P_{I_{q}}. Since

Θ̃ = Ker(G(k̃[t^{\frac{1}{m}}]) ⟶^{ev_{0}} G(k̃))

it follows that

(4.8.1)  StabΘ̃(q) ⊂ U_{I_{q}}(k̃[t^{\frac{1}{m}}]).

Let Φ_{I_{q}}^{+} = {α ∈ Φ(G, T) | ⟨ d, α ⟩ > 0}. Clearly, every α ∈ Φ_{I_{q}}^{+} is a positive

root (because q ∈ Q̃) and the corresponding root subgroup U_{α} is contained in U_{I_{q}}. By [So2, end of §1.1] there exist non-negative integers m_{α,q}, α ∈ Φ_{I_{q}}^{+} such that

Stab_{U_{Iq} (k̃[t^{\frac{1}{m}}])}(q) = ⟨U_{α}(f) | α ∈ Φ_{I_{q}}^{+}, f ∈ k̃[t^{\frac{1}{m}}], deg(f) ≤ m_{α,q}⟩.

Therefore Stab_{U_{Iq} (k̃[t^{\frac{1}{m}}])}(q) is the group of k̃-points of a connected split

unipotent group, say W. But

StabΘ̃(q) = ker(Stab_{U_{Iq} (k̃[t^{\frac{1}{m}}])}(q) → U_{I_{q}}(k̃)),

hence StabΘ̃(q) = V(k̃) where V is a subgroup of W given by the additional condition f(0) = 0 . Evidently V is split unipotent (because char(k) = 0).

(2) Let x ∈ B_{−}. Since it is fixed by the twisted action of Γ, V(k̃) is Γ–stable with respect to the twisted action of Γ by η. So we may view V(k̃) as a Γ-module. The descending central sequence for V provides a filtration

1 ⊂ V_{n} ⊂ ⋯ ⊂ V_{1} ⊂ V_{0} = V

where V_{i} are k̃–subgroups of V such that V_{i}/V_{i+1} is isomorphic to G_{a,k̃}^{r_{i}}.

Taking k̃-points we get a filtration

1 ⊂ V_{n}(k̃) ⊂ ⋯ ⊂ V_{1}(k̃) ⊂ V_{0}(k̃) = V(k̃)

which is the descending central sequence of V(k̃) = StabΘ̃(x) satisfying V_{i}(k̃)/V_{i+1}(k̃) ≅ (k̃)^{r_{i}} for i = 0, .., n − 1. All terms of the last filtration are characteristic subgroups, so that they are stable with respect to the twisted action of Γ on V(k̃). Since char(k̃) = 0 the abelian group V_{i}(k̃)/V_{i+1}(k̃) is infinitely divisible. It follows that

H^{1}(Γ, _{η}(V_{i}(k̃)/V_{i+1}(k̃))) = 0

for i = 0, ..., n − 1. By dévissage, we get

1 = H^{1}(Γ, _{η}(V(k̃))) = H^{1}(Γ, _{η}(StabΘ̃(x))),

as required. □

4.9. Descent argument. Since η takes value in Aut_{I_{t}}(G, T)(k̃), it follows that G(k̃) and G(k̃) · Q̃ are stable under the twisted action of Γ. We put

J = (G(k̃) . Q̃)^{_{⋆}Γ}.

4.10. Lemma. (1) J is a contractible subcomplex of B_{−}.

(2) J is a fundamental simplicial domain for the action of Θ on B_{−}.

Proof. (1) We are given a point x ∈ J and have to show that the facet F_{x} in B_{−} associated to x is included in J. By descent theory of Bruhat-Tits building [Ro, V.1] we have F_{x} = (F̃_{x})^{_{⋆}Γ} = F̃_{x} ∩ B_{−} where F̃_{x} stands for the facet of B̃_{−} attached to x. Since G(k̃) · Q̃ is a subcomplex of B̃_{−}, we have F̃_{x} ⊂ G(k̃) · Q̃. Taking the Γ–invariants we conclude F_{x} ⊂ J.

The contractibility can be established as follows. Given x ∈ J, the segment [ϕ_{−}, x] is included in J. So the restriction of the standard contraction of B̃_{−} to ϕ̃_{−} [BT1, Prop. 7.4.20.(v)] induces a contraction of J to ϕ_{−}.

(2) According to Lemma 4.6, G(k̃) · Q̃ is a fundamental domain for the action of the group Θ̃ on B̃_{−}, hence two distinct points of J are not in the same orbit of Θ̃. It remains to show that any point x ∈ B_{−} is conjugate under Θ to a point of J. Again by Lemma 4.6, there exists a unique point q ∈ G(k̃) · Q̃ and g ∈ Θ̃ such that x = g · q. Since x, Θ̃ and G(k̃) · Q̃ are Γ–stable, by unicity we get that q ∈ J. For each γ ∈ Γ, from γ ⋆ x = x we have

x = γ ⋆ x = (γ ⋆ g) · q = g · q.

Hence γ ↦ a_{γ} = g^{−1} (γ ⋆ g) is a 1-cocycle for Γ with value in _{η}(StabΘ̃(q)).

According to Lemma 4.8 (2), we have H^{1}(Γ,_{η}(StabΘ̃(q)) = 1. Hence there exists g_{0} ∈ StabΘ̃(q) such that

a_{γ} = g^{−1} (γ ⋆ g) = g_{0}^{−1} (γ ⋆ g_{0})

for each γ ∈ Γ. We put g′ = gg_{0}^{−1} ∈ H^{0}(Γ, _{η}Θ̃) = Θ. Then

x = g · q = g′ · (g_{0} · q) = g′ · q ∈ Θ · J

as desired. □

Thus, the subspace J of B_{−} is contractible, hence connected and simply connected. Since it is a fundamental simplicial domain for the action of Θ on B_{−}, it follows that Θ is generated by the subgroups Stab_{Θ}(x) for x running over J [So1]. Recall that according to Lemma 4.3 (2) we have 𝔊(R) = H(k) Θ E_{𝔓}(R). The following lemma completes the proof of Theorem 1.1 (1).

4.11. Lemma. Let x ∈ J. Then there exists g_{x} ∈ 𝔊(R) such that

Stab_{Θ}(x) ⊂ g_{x} R_{u}(𝔓)(R) g_{x}^{−1}.

Proof. We write x ∈ J ⊂ G(k̃)·Q̃ in the form x = g·q where q = ϕ_{−}+d ∈ Q̃, d ∈ D̃ and g ∈ G(k̃). If q = ϕ_{−}, then x = ϕ_{−} and Stab_{G(k̃[t^{\frac{1}{m}}])}(ϕ) = G(k̃).

This implies that StabΘ̃(x) = 1 and a fortiori Stab_{Θ}(x) = 1 and there is nothing to do.

We now assume that d ≠ 0 and as in Lemma 4.8 (1) consider the set of roots

I_{q} = {α ∈ Δ | ⟨ d, α ⟩ = 0}.

Recall that by (4.8.1)

StabΘ̃(q) ⊂ U_{I_{q}}(k̃[t^{\frac{1}{m}}]) ⊂ P_{I_{q}}(k̃[t^{\frac{1}{m}}]).

Since x = g · q, we get

(4.11.1) StabΘ̃(x) ⊂ (gU_{I_{q}}g^{−1})(k̃[t^{\frac{1}{m}}]) ⊂ (gP_{I_{q}}g^{−1})(k̃[t^{\frac{1}{m}}]) = Q(k̃[t^{\frac{1}{m}}]).

where Q stands for the k̃–parabolic subgroups Q = gP_{I_{q}}g^{−1} of Gk̃.

We next want to show that Q is stable with respect to the twisted action of Γ.

4.12. Claim. η(Γ) ⊂ Aut(G, Q)(k̃).

We need here the link L_{ϕ_{−}} which is nothing but the (combinatorial) spherical building S(Gk̃) (see [Mg, §2.4]) and which is equipped with the twisted action of Γ (because ϕ_{−} is Γ–stable). For n >> 0 the point q_{n} = ϕ_{−} + \frac{d}{n} belongs to Lϕ̃_{−} and defines then a k̃–parabolic subgroup of G which turns

out to be P_{I_{q},k̃} because I_{q} = I_{q_{n}}. Similarly the point x_{n} = ϕ_{−} + \frac{g . d}{n} of Lϕ̃_{−}

defines the k̃–parabolic subgroups Q.

Note that by construction x = g · q = ϕ_{−} + g · d is _{⋆}Γ–stable. Since ϕ_{−} is also _{⋆}Γ–stable, so is g · d and hence so is x_{n}. Therefore the k̃–parabolic subgroup Q of G is preserved by the twisted action of Γ, whence the claim.

Recall that 𝔊 is the twist of G_{R} = G ×_{k} R by the 1-cocycle η. By the claim, we see that the parabolic group Q ×_{k} R of G_{R} defines by twisting an parabolic subgroup 𝔔 of 𝔊. By Lemma 3.4, there exists g_{x} ∈ G(R) such that 𝔓 ⊂ g_{x}^{−1} 𝔔 g_{x} of 𝔊. Clearly, g_{x}^{−1} R_{u}(𝔔) g_{x} ⊂ R_{u}(𝔓). The inclusion (4.11.1) reads then as

StabΘ̃(x) ⊂ R_{u}(𝔔)(k̃[t^{\frac{±1}{m}}]) ⊂ g_{x}R_{u}(𝔓)(k̃[t^{± \frac{1}{m}}])g_{x}^{−1}. (4.12.1)

Taking the fixed points under _{⋆}Γ yields the inclusion

Stab_{Θ}(x) ⊂ R_{u}(𝔔)(R) ⊂ g_{x} R_{u}(𝔓)(R) g_{x}^{−1}.

This concludes the proof. □

5. Steinberg’s method

We maintain the notation introduced in § 3.6. To finish the proof of part (2) of Theorem 1.1 we first reduce it to the case of groups of (relative) rank 1 and then we use the decomposition of 𝔊(R) in part (1). Our reduction to quasi-split groups of rank 1 is based, to some extent, on Steinberg’s method in [St2, § 8].

5.1. Proposition. Let w ∈ W and write w = w_{α_{1}} ⋯ w_{α_{l}} with α_{1}, ⋯, α_{l} ∈ Δ_{0}. Then

𝔓(K)w𝔓(K) ⊂ 𝔓(K) 𝔊_{α_{1}}(R) ⋯ 𝔊_{α_{l}}(R).

Proof. We reason by induction on l. Let first l = 1. Then we have

𝔓(K)w_{α}𝔓(K) ⊂ 𝔓(K)w_{α}𝔓(K) ∪ 𝔓(K) = 𝔓(K) ⟨w_{α}⟩ 𝔓(K) = 𝔓_{α}(K)

according to Tits’ system properties [Bor, §21.16]. We denote by rad^{u}(𝔓_{α}) the unipotent radical of 𝔓_{α}. According to [SGA3, XXVI.1.20], we have 𝔓 = rad^{u}(𝔓_{α}) ⋊ 𝔔 where 𝔔 is a parabolic subgroup of ℨ_{α}. Since 𝔓_{α} = rad^{u}(𝔓_{α}) ⋊ ℨ_{α}, it follows that we have an isomorphism ℨ_{α}/𝔔 ⟶^{∼} 𝔓_{α}/𝔓 (both quotients are representable). Then (ℨ_{α}/𝔔)_{K} ⟶^{∼} (𝔓_{α}/𝔓)_{K} is the K– variety of parabolic subgroups of (ℨ_{α})_{K} of the same type as 𝔔_{K}. According to [SGA3, XXVI.1.19], it is also the K–variety of parabolic subgroups of the derived group (𝔊_{α})_{K} of (ℨ_{α})_{K} of the same type as the K–parabolic subgroup 𝔔_{K} ∩ (𝔊_{α})_{K} of (𝔊_{α})_{K}. Now we apply Borel-Tits’ theorem [BT65, th. 4.13] which states that 𝔊_{α}(K) acts transitively on the K-points of that variety. In other words, the map 𝔊_{α}(K) → (ℨ_{α}/𝔔)(K) ⟶^{∼} (𝔓_{α}/𝔓)(K) is surjective, so that 𝔓_{α}(K) = 𝔊_{α}(K) · 𝔓(K). By taking the opposite decomposition we conclude that

𝔓(K)w_{α}𝔓(K) ⊂ 𝔓_{α}(K) ⊂ 𝔓(K) 𝔊_{α}(K).

It remains to check that 𝔓(K) 𝔊_{α}(K) ⊂ 𝔓(K) 𝔊_{α}(R). Let 𝔔_{α} = 𝔓∩𝔊_{α}. It is a parabolic subgroup of the R–group 𝔊_{α}. According to Proposition 3.5, we have 𝔊_{α}(K) = 𝔔_{α}(K) 𝔊_{α}(R). Therefore

𝔓(K) 𝔊_{α}(K) = 𝔓(K) 𝔔_{α}(K) 𝔊_{α}(R) = 𝔓(K) 𝔊_{α}(R).

Let now l ≥ 2 and write w = w_{α_{1}} w′. Then we have

𝔓(K) w 𝔓(K) = 𝔓(K) w_{α_{1}}w′ 𝔓(K)
⊂ (𝔓(K) w_{α_{1}}) (𝔓(K) w′ 𝔓(K))
⊂ 𝔓(K) w_{α_{1}} 𝔓(K) 𝔊_{α_{2}}(R) ⋯ 𝔊_{α_{l}}(R) [ Induction]
⊂ 𝔓(K) 𝔊_{α_{1}}(R) 𝔊_{α_{2}}(R) ⋯ 𝔊_{α_{l}}(R) [ Case l = 1]

as required. □

¿From the Bruhat decomposition 𝔊(K) = ⋃_{w∈W} 𝔓(K)w𝔓(K), we get the

following.

5.2. Corollary. 𝔊(R) is generated by 𝔓(R) and the 𝔊_{α}(R) for all roots α ∈ Δ_{0}.

Proof. Let g ∈ 𝔊(R). Viewing g as an element of 𝔊(K) we can write it in the form g = p_{1}wp_{2} where p_{1}, p_{2} ∈ 𝔓(K) and w ∈ W. By Proposition 5.1 we have g = p_{1}wp_{2} = p_{3}g_{1} ⋯ g_{l} where p_{3} ∈ 𝔓(K) and g_{1}, …, g_{l} ∈ 𝔊_{α_{i}}(R), α_{i} ∈ Δ_{0}. Since all g_{i} and g are in 𝔊(R) we conclude that p_{3} ∈ 𝔓(R). □

5.3. Application to the quasi-split case. We assume here additionally that 𝔊 is quasi-split. In this case 𝔅 = 𝔓 is a Borel subgroup of 𝔊.

5.4. Lemma. 𝔅(R) ⊂ ⟨E_{𝔅}(R), 𝔊_{α}(R)⟩ where α runs over Δ_{0}.

Proof. Recall that 𝔅 = R_{u}(𝔅) ⋊ 𝔗, where 𝔗 is the twist of T by η, so that we need to deal with 𝔗(R) only. But 𝔗 has a decomposition [SGA3, XXIV.3.13]

𝔗 ⟶^{∼} ∏_{α∈Δ_{0}} ∏_{R_{α}/R}(G_{m,R_{α}})

where the R_{α} are connected étale covers of R. It remains to note that each summand ∏_{R_{α}/R}(G_{m,R_{α}}) is a maximal torus of 𝔊_{α}, hence the statement follows. □

We can now easily finish the proof of Theorem 1.1 (2). By Lemma 3.7, 𝔊_{α} is semisimple simply connected and quasi-split of relative rank 1. Hence according to Lemma 5.4 and Corollary 5.2 we may assume without loss of generality that 𝔊 = 𝔊_{α} is a quasi-split group scheme of relative rank 1. The following cases can occur. 𝔊_{α} is of absolute type A_{1}. Then 𝔊_{α} = SL_{2}. Since R is euclidean 𝔊_{α}(R) is generated by “elementary matrices”. 𝔊_{α} is of absolute type A_{1}^{2}. Then 𝔊_{α} = ∏_{R′/R} SL_{2,R′} where R′/R is the unique

(connected) quadratic étale extension of R which splits 𝔊_{α}. It is known that R′ is a Laurent polynomial ring [GP2, Lemma 2.8]. Since 𝔊_{α}(R) = SL_{2}(R′) we are reduced to the previous case. 𝔊_{α} is of absolute type A_{1}^{3}. The argument is similar. 𝔊_{α} is of absolute type A_{2}. Let S/R be the quadratic (connected) étale extension splitting 𝔊_{α}.

Denote by τ the non-trivial automorphism of S over R. Since 𝔊_{α} is quasisplit over R and split over S it admits a realization 𝔊_{α} ≃ SU(f) where f = xτ(y) + zτ(z), x, y, z ∈ S, is a 3-dimensional hermitian form on V = S ⊕ S ⊕ S. Let W ⊂ V be the submodule spanned by first two components of V. Obviously, SU(f|_{W}) ≃ SL_{2}. Therefore SU(f|_{W})(R) is contained in the subgroup E_{𝔅}(R) of SU(f)(R). In particular, T(R) ⊂ E_{𝔅}(R) where T ≃ G_{m,R} is a split R–torus consisting of matrices of the form

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

Consider now the group H. By Example 3.2, H ≃ SO_{3}. Let T′ ⊂ H be a split maximal k-torus. Since the simply connected covering of H is SL_{2} the group of k-points of H is generated by unipotent elements and T′(k). The two maximal split tori T′_{R} and T of the R–group 𝔊_{α} are conjugate under the action of 𝔊_{α}(R) by [CGP2]. Since T(R) ⊂ E_{𝔅}(R), we get H(k) ⊂ H(R) ⊂ E_{𝔅}^{st}(R). The application of part (1) of Theorem 1.1 completes the proof of Theorem 1.1 (2).

6. Remarks on Conjecture 1.3

The following lemma shows that in parts (2) and (3) of Conjecture 1.3 one needs to prove injectivity only.

6.1. Lemma. (1) The map W_{𝔓}^{st}(R, 𝔊) → W (F, 𝔊_{F}) is surjective.

(2) The map W_{𝔓}^{st}(R, 𝔊) → W (K, 𝔊_{K}) is surjective.

Proof. (1) This follows from Lemma 4.3 (2) and Theorem 1.1 (1).

(2) This is an argument of strong approximation. Let g ∈ 𝔊(K) and let Σ be the set of closed points of Spec(R) where g is not regular. For each point x ∈ Σ, we denote by R_{x} the local ring at x; R_{x} is a DVR and we denote by R̂_{x} its completion and by K̂_{x} its fraction field. By [G, Lemme 4.5], we have 𝔊(K̂_{x}) = 𝔊(K̂_{x})^{+} · 𝔊(R̂_{x}) for each x ∈ Σ. In particular, we can decompose

g = g_{x}^{+} g_{x}, g_{x} ∈ 𝔊(R̂_{x}), g_{x}^{+} ∈ 𝔊(K̂_{x})^{+},

for each x ∈ Σ. Let Spec(R′) = Spec(R) \ Σ. By (loc. cit., Lemme 4.6), the group 𝔊(R′) ∩ 𝔊(K)^{+} is dense in ∏_{x∈Σ} 𝔊(K̂_{x})^{+}. Since ∏_{x∈Σ} 𝔊(R̂_{x}) is open in ∏_{x∈Σ} 𝔊(K̂_{x}), there exists g′ ∈ 𝔊(R′) ∩ 𝔊(K)^{+} such that

(g′)^{−1} g_{x}^{+} ∈ 𝔊(R̂_{x}), ∀x ∈ Σ.

It follows that (g′)^{−1} g ∈ 𝔊(R̂_{x}) for each x ∈ Σ, so that (g′)^{−1} g ∈ 𝔊(R). Thus [g] ∈ W (K, G_{K}) is in the image of the map W_{𝔓}^{st}(R, 𝔊) → W (K, 𝔊_{K}).

□

6.2. Remarks. (a) If Conjecture 1.3 (2) holds, the stable Whitehead group W_{𝔓}^{st}(R, 𝔊) has finite exponent.

(b) Since W (K, 𝔊 ×_{R} K) → 𝔊(K)/R is an isomorphism by [G, 7.2] we have well-defined maps

H(k)/R → H(K)/R → 𝔊(K)/R.

So if part (2) of the conjecture holds, it would imply that the map H(k) → W_{𝔓}^{st}(R, 𝔊) induces a well defined map H(k)/R → W_{𝔓}^{st}(R, 𝔊).

We summarize here cases that support our conjecture.

6.3. Theorem. Parts (2), (3) of Conjecture 1.3 hold in the following cases. (i) k is algebraically closed. (ii) 𝔊 is constant, i.e. η^{geo} = 1.

Proof. Case (i). By [P2], 𝔊 is quasi-split, hence we have

W_{𝔓}^{st}(R, 𝔊) = W (K, 𝔊 ×_{k} K) = W (F, 𝔊 ×_{k} F) = 1.

Also, H(k)/R = 1 and the assertion is therefore clear. Case (ii). We have H = _{η^{ar}}G and 𝔊 = H ×_{k} R. Then we have a welldefined map W (k, H) → W_{𝔓}^{st}(R, 𝔊). It fits in a sequence of maps

W (k, H) → W_{𝔓}^{st}(R, 𝔊) → W (K, 𝔊_{K}) → W (F, 𝔊_{F}).

The first map is surjective by Theorem 1.1 (1), the second and the third are surjective as well by Lemma 6.1. But the composite map W (k, H) → W (F, H) ≅ W (F, 𝔊_{F}) is an isomorphism [G, Theorem 7.3], so all the maps above are isomorphisms as well. Thus part (2) of the conjecture holds for 𝔊. Part (3) follows from the fact that the natural map W (k, H) → H(k)/R is an isomorphism (ibid, Theorem 7.2). □

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

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

UMR 5208 du CNRS - Institut Camille Jordan - Université Claude Bernard Lyon 1, 43 boulevard du 11 novembre 1918, 69622 Villeurbanne cedex - France.

Institute of Mathematics Simon Stoilow of the Romanian Academy, Calea Grivitei 21, RO-010702 Bucharest, Romania.

   E-mail address: gille@math.univ-lyon1.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@gmail.com