The structure of the group G(k[t]): Variations on a theme of Soulé

Benedictus Margaux

Abstract. Following Soulé’s ideas [14] we give a presentation of the abstract group G(k[t]) for any semisimple (connected) simply connected absolutely almost simple k–group G. As an application, we give a description of G(k[t]) in terms of direct limits, and show that the Whitehead group and the naïve group of connected components of G coincide.

Keywords: Bruhat-Tits buildings, arithmetic groups, simplicial fundamental domain. MSC: 20G10, 22

1 Introduction

Let k be a field and let G be a semisimple simply connected absolutely almost simple k–group. For G split, Soulé [14] has given a presentation of the group G(k[t]), thus extending a theorem of Nagao [8] for SL_{2} (see also [11, II.1.6]). The goal of this note is to provide a presentation of G(k[t]) in the general case.

We will follow Soulé’s original ideas and study the action of G(k[t]) on the Bruhat-Tits building [4] of G corresponding to the field K = k((\frac{1}{t})), where K is viewed as the completion of k(t) with respect to the valuation at ∞. As an application, we show that the Whitehead group of G coincides with the naïve group of connected components of G.

2 Structure of the group G(k[t])

Throughout k and G will be as above. For convenience the group G(k[t]) will be denoted by Γ.

2.1 Notation and statement of the main Theorem

Let S be a maximal k–split torus of G, and T be a maximal torus of G containing S. Recall that S_{K} is a maximal K-split torus of G_{K}. Let k̃/k a finite Galois extension which splits T (hence also G). Set G = Gal(k̃/k) and T̃ = T ×_{k} k̃.

Let G̃ = G ×_{k} k̃ and S̃ = S ×_{k} k̃. We choose compatible orderings on the root

systems Φ = Φ(G, S) and Φ̃ = Φ(G̃, T̃) (see [1]). We then have a set Δ of relative simple roots and a set Δ̃ of absolute simple roots.

It will be convenient to essentially maintain the same notation than in Soulé’s paper, namely:

• A = k[t], K = k((\frac{1}{t})), G = G(K);

• ω the valuation defined on K the valuation on K at ∞, that is, the valuation on K having O = k[[\frac{1}{t}]] as its ring of integers. We also have the analogous to the above objects for k̃, namely

• Ã = k̃[t], K̃ = k̃((\frac{1}{t})), Γ̃ = G(Ã), and Õ = k̃[[\frac{1}{t}]]. At the level of buildings we set.

• T the (affine) Bruhat-Tits building of the K–group G_{K} := G ×_{k} K and T̃ the Bruhat-Tits building of the K̃–group GK̃ := G ×_{k} K̃ [4, §4.2].^{1} We recall that both T and T̃ have a natural simplicial complex structure [4, §4.2.23].

Recall that T is equipped with an action of G(K) and that T̃ is equipped with an action of G(K̃) ⋊ G. We have an isometric embedding j : T → T̃ which identifies T with T̃^{G}. The hyperspecial group G(Õ) of G(K̃) fixes a unique point ϕ̃ of T̃ [3, §9.1.9.c]. This point descends to a point ϕ of T .

We denote by A the standard apartment of T associated to S (this is a real affine space) and similarly by Ã the standard apartment associated to T̃. The point ϕ̃ belongs to Ã (ibid.). Since Hom_{k−gr}(G_{m}, S) ⊗_{ℤ} ℝ ≅ Hom_{k−gr}(G_{m}, T) ⊗_{ℤ} ℝ ≅ (Hom_{k̃−gr}(G_{m,k̃}, T̃) ⊗_{ℤ} ℝ)^{G} [4, §4.2], we have j(A) = Ã^{G} so ϕ belongs to A and

A = ϕ + Hom_{k−gr}(G_{m}, S) ⊗_{ℤ} ℝ.

By means of the canonical pairing ⟨ , ⟩ : Hom_{k−gr}(S, G_{m}) × Hom_{k−gr}(S, G_{m}) → ℤ we can then define the sector (quartier)

Q := ϕ + D where D := {v ∈ Hom_{k−gr}(S, G_{m}) ⊗_{ℤ} ℝ | ⟨b, λ⟩ ≥ 0 ∀ b ∈ Δ}.

1Since G ×_{K} K̃ is split, the assumptions of [4, §5.1.1.1] are satisfied. This allows us to do away with the “standard” assumption that the base field k be perfect.

The following result generalizes Soulé’s theorem [14].

Theorem 2.1. The set Q is a simplicial fundamental domain for the action of G(k[t]) on T . In other words, any simplex of T is equivalent under the action of G(k[t]) to a unique simplex of Q.

2.2 Buildings and valuations

Let P be the minimal parabolic k–subgroup of G defined by S and Δ. We denote by U = R_{u}(P) the unipotent radical of P.

We denote by Ũã the split unipotent subgroup associated to a root ã ∈ Φ̃, and by ã^{∨} : SL_{2} → G the corresponding standard homomorphism (see [12, §2.2]).

The set of positive (resp. negative) roots with respect to the basis Δ of Φ will be denoted by Φ^{+} (resp. Φ^{−}). Given b ∈ Φ, the subset of absolute roots

Φ̃^{b} := {ã ∈ Φ̃ | ã_{|S×_{k}k̃} = b or 2b}

is positively closed in Φ̃. It defines then a split k̃-unipotent subgroup Ũ_{b} of G̃ which descends to a split k–unipotent subgroup U_{b} of G. As is [3], we make the convention that U_{2b} = 1 if 2b ∉ Φ.

For I ⊂ Δ, we define along standard lines

S_{I} = (⋂_{b∈I} ker(b))^{0} ⊂ S, L_{I} = Z_{G}(S_{I}) and P_{I} = U_{I} ⋊ L_{I}.

Thus P_{I} is the standard parabolic subgroup of G of type I and L_{I} its standard Levi subgroup (see [1, §21.11]). Recall that the root system Φ(L_{I}, S) = [I] is the subroot system of Φ consisting of roots which are linear combinations of I; the split unipotent k-group U_{I} is the subgroup of U generated by the U_{b} with b running over Φ^{+} \ [I].

Given ã ∈ Φ̃, the group Ũã := Ũã(K̃) = K̃ is equipped with the valuation ω, which we denote by φ̃_{a} : Ũ_{a} → ℝ ∪ {∞}. This defines the Chevalley-Steinberg “donnée radicielle valuée” (T (K̃), (Ũã, Mã)_{ã∈Φ̃}) where

Mã = T (K̃) ã^{∨}([ \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ])

[3, example 6.2.3.b], and also a filtration (Ũ_{ã,m})_{m∈ℤ} of Ũã where

Ũ_{ã,m} := φ̃^{−1}([m, +∞[).

Note that Ũ_{ã,0} = Ũã(Õ).

A crucial point of Bruhat-Tits theory is the descent of this data to G = G(K) [4, §5.1]. Given b ∈ Φ, the commutative group U_{b} := U_{b}(K) is equipped with the descended valuation φ_{b} : U_{b} → ℝ ∪ {∞}. The definition of φ_{b} is delicate, and is given as follows [4, §5.1.16]. Define

Ũ_{b,m} := ∏_{ã∈Φ̃^{b}, ã_{|S×_{k}k̃}=b} Ũ_{ã,m} . ∏_{ã∈Φ̃^{b}, ã_{|S×_{k}k̃}=2b} Ũ_{ã,2m} (m ∈ ℝ).

Then U_{b} is a subgroup of U_{b}(K̃) = Ũ_{b} = ⋃_{m∈ℝ} Ũ_{b,m} and the descended valuation is

defined by

φ_{b}(u) := Sup{m ∈ ℝ | u ∈ Ũ_{b,m}}.

Note that^{2} Θ_{b} := φ_{b}(U_{b} \ \{e\}) is either ℤ or \frac{1}{2}ℤ. As above, it gives rises to a filtration (U_{b,m})_{m∈Θ_{b}} of U_{b} such that U_{b,0} = U_{b}(O).

Again me make the convention that U_{2b} = 1 if 2b ∉ Φ.

2.3 Description of the isotropy group of a vertex

Given Ω ⊂ Q, we denote by Γ_{Ω} the corresponding isotropy subgroup, namely the elements of Γ that fix all elements of Ω. We introduce an analogous definition and notation for j(Ω) ∈ Ã. By Galois descent we have

Γ_{Ω} = (Γ̃_{j(Ω)})^{G}. (2.1)

In particular, since Γ̃ϕ̃ = G(Õ)∩Γ̃ = G(k̃) [13, §1.1], we have Γ_{ϕ} = (Γ̃ϕ̃)^{G} = G(k̃)^{G} = G(k).

If x ∈ Q \ {ϕ} and if [x[ is the half-line of origin x and direction ϕx^{⟶}, we claim that

Γ_{x} = Γ_{[x[}.

If G is split, this is proven in Soulé’s paper by reduction to the case of SL_{n}. By applying the identity (2.1) to x and [x[, our claim now readily follows from the absolute case.

The isotropy of [x[ in G = G(K) is the Bruhat-Tits abstract parahoric group P_{[x[} [3, § 7.1]. We have

P_{[x[} = U_{[x[}.H

2We use the notation Θ_{b}, rather than the more standard Γ_{b} found in [3], to avoid any possible confusion with the notation used in Soulé’s paper.

where H = Fix_{G}(A). By [4, §5.2.2], we have

H = Z_{G}(S)(O).

The group U_{[x[} is defined by means of the function [3, §6.4.2]

f_{[x[} : Φ → ℝ ∪ {∞}, b ↦ inf{s ∈ ℝ | b(y) + s ≥ 0 for all y ∈ [x[}.

Hence

f_{[x[}(b) = \{ \begin{matrix}0 & if b(x) = 0, \\ −b(x) & if b(x) > 0, \\ ∞ & if b(x) < 0. \end{matrix}

The group U_{[x[} ⊂ G is then the subgroup of G generated by the U_{b,m} for b ∈ Φ^{+} and m ≥ −b(x) (m ∈ Θ_{b}), together with the U_{b}(O) for b ∈ Φ^{−} such that b(x) = 0. In other words, by distinguishing positive roots which vanish at x, we see that U_{[x[} is the subgroup of G generated by subgroups of the following three “shapes”:

(I) U_{b,m} for b ∈ Φ^{+} such that b(x) > 0 and m ∈ Θ_{b} such that m ≥ −b(x);

(II) U_{b}(O) for b ∈ Φ^{+} such that b(x) = 0;

(III) U_{b}(O) for b ∈ Φ^{−} such that b(x) = 0.

Define U_{[x[}^{±} := U_{[x[} ∩ U^{±}(K) as in [3, §6.4.2]. By definition U_{[x[}^{+} and U_{[x[}^{−} generate U_{[x[}. On the other hand, U_{[x[}^{+} (resp. U_{[x[}^{−}) is the subgroup of U_{[x[} generated by the subgroups of type (I) and (II) (resp.(III)) [3, prop. 6.4.9]. Define the subset of roots

I_{x} = { b ∈ Δ | b(x) = 0 }.

This definition makes sense if x is an element of A, and we then have I_{ϕ} = Δ.

Lemma 2.2. We have

(2.2)  [I_{x}] ∩ Φ^{+} = { b ∈ Φ^{+} | b(x) = 0 },
(2.3)  Φ^{+} \ [I_{x}] = { b ∈ Φ^{+} | b(x) > 0} and
(2.4)  [I_{x}] ∩ Φ^{−} = { b ∈ Φ^{−} | b(x) = 0 }.

Proof. Observe that if b ∈ [I_{x}], b is a linear combination of elements of I_{x}, hence b(x) = 0. This implies that [I_{x}] ∩ Φ^{+} ⊂ { b ∈ Φ^{+} | b(x) = 0 }. Conversely, let b be a positive root such that b(x) = 0. Then b = ∑_{c∈Δ} n_{c} c where the n_{c}’s are non-negative integers. Hence ∑_{c∈Δ} n_{c} c(x) = 0. Since x ∈ Q, we have c(x) ≥ 0. Therefore n_{c} c(x) = 0 and b is a linear combination of elements of I_{x}. This shows (2.2). Since

{ b ∈ Φ^{+} | b(x) ≠ 0} = { b ∈ Φ^{+} | b(x) > 0},

we get also (2.3). Similar considerations apply to (2.4).

It follows from (2.2) and (2.4) respectively that the subgroups of shape (II) and (III) are subgroups of L_{I_{x}}(O). Furthermore, (2.3) shows that the subgroups of shape (I) are subgroups of U_{I_{x}}(K). Hence we get the following inclusion

U_{[x[} ⊂ (U_{[x[} ∩ U_{I_{x}}(K)) ⋊ L_{I_{x}}(O) ⊂ P_{I_{x}}(K). (2.5)

Lemma 2.3. 1. L_{I_{x}}(O) ⊂ P_{[x[} ⊂ U_{I_{x}}(K) ⋊ L_{I_{x}}(O) ⊂ P_{I_{x}}(K);

2. U_{I_{x}}(K) ∩ P_{[x[} ⊂ U_{[x[}^{+};

3. ⋃_{z≥1}(U_{[zx[}^{+} ∩ U_{I_{x}}(K)) = U_{I_{x}}(K).

Proof. Let I = I_{x}.

(1) Since U_{[x[} ⊂ U_{I}(K) ⋊ L_{I}(O) and Z_{G}(S) ⊂ L_{I} it follows that P_{[x[} = U_{[x[}.H = U_{[x[}.Z_{G}(S)(O) is a subgroup of U_{I}(K) ⋊ L_{I}(O).

Let us show that L_{I}(O) ⊂ P_{[x[}. Let V_{I} be the unipotent radical of the minimal standard parabolic subgroup of L_{I}, namely the k–subgroup of U generated by the U_{b} such that b ∈ Φ^{+} and b(x) = 0. We have [10, th. XXVI.5.1]

⋃_{g∈V_{I}(k)} gΩ = L_{I}

where Ω stands for the big cell V_{I}^{−} ×_{k} Z_{G}(S) ×_{k} V_{I} of L_{I}. Since O is local, it follows that

L_{I}(O) = V_{I}(k).Ω(O) = V_{I}(k).V_{I}^{−}(O) . H . V_{I}(O).

We conclude that L_{I}(O) ⊂ P_{[x[}.

(2) We claim that U(K) ∩ P_{[x[} = U_{[x[}^{+}. Note that this establishes (2) since U_{I}(K) ⊂ U(K). To prove the claim it we need to show that U(K) ∩ P_{[x[} ⊂ U_{[x[}^{+} (the reversed inclusion is obvious). With the notations of [3, §7], we have U(K) = U_{D}^{+} where D is the direction of the sector Q. By loc. cit. 7.1.4, we have

P_{[x[} ∩ U(K) = U_{[x[+D}

where U_{[x[+D} is the subgroup of G(K) attached to the subset [x[+D = x + D of A. This group is defined by means of the function [3, §6.4.2]

f_{x+D} : Φ → ℝ ∪ {∞}, b ↦ inf{s ∈ ℝ | b(y) + s ≥ 0 for all y ∈ x + D}.

Hence

f_{x+D}(b) = \{ \begin{matrix}−b(x) & if b > 0, \\ ∞ & if b < 0, \end{matrix}

so U_{x+D} = U_{[x[}^{+} as desired.

(3) If b ∈ Φ^{+} satisfies b(x) > 0, then the number

Inf{m ∈ Θ_{b} | m + b(zx) ≥ 0}

tends to −∞ as z tends to ∞. This readily yields that ⋃_{z≥1}(U_{[zx[}^{+} ∩ U_{I}(K)) = U_{I}(K).

Remark 2.4. Geometrically speaking, the K–parabolic P_{I_{x}} ×_{k} K is attached to the extremity of the half line [x[ in the spherical building at infinity [6, § 16.9]. Since P_{[x[} is the isotropy group of the half line [x[, it fixes its extremity. This point of view yields another way to prove the inclusion P_{[x[} ⊂ P_{I_{x}}(K) which is part of Lemma 2.3(1).

Given b ∈ Φ we set

m_{x}(b) := Inf{m ∈ Θ_{b} | m + b(x) ≥ 0}.

Since Γ_{x} = P_{[x[} ∩ Γ, we have the inclusion

(2.6)  ⟨(U_{b,m_{x}(b)}.U_{2b,m_{x}(2b)}) ∩ Γ, b ∈ Φ, b(x) ≥ 0⟩ ⊂ Γ_{x}.

Proposition 2.5. 1. Γ_{x} = (Γ_{x} ∩ U_{I_{x}}(K)) ⋊ L_{I_{x}}(k);

2. Γ_{x} = ⟨(U_{b,m_{x}(b)}.U_{2b,m_{x}(2b)}) ∩ Γ, b(x) > 0⟩ ⋊ L_{I_{x}}(k);

3. ⋃_{z≥1} Γ_{zx} = U_{I_{x}}(k[t]) ⋊ L_{I_{x}}(k).

Proof. To cut down on the notation we set I = I_{x}.

(1) According to Lemma 2.3.(1) L_{I}(k) = Γ ∩ L_{I}(O) fixes the point x. Hence the inclusion

(Γ_{x} ∩ U_{I}(K)) ⋊ L_{I}(k) ⊂ Γ_{x}.

To prove the reverse inclusion we make use of the projection P_{I}(K) → L_{I}(K). The image of Γ_{x} inside L_{I}(K) is a subgroup of L_{I}(A). On the other hand, by Lemma 2.3.(1), the image of P_{x} inside L_{I}(K) is the subgroup L_{I}(O) . Hence the image of Γ_{x} inside L_{I}(K) is a subgroup of L_{I}(A) ∩ L_{I}(O) = L_{I}(k). We thus have an exact sequence

1 → (Γ_{x} ∩ U_{I}(K)) → Γ_{x} → L_{I}(k)

which is a split surjection.

(2) Put V := ⟨(U_{b,m_{x}(b)}.U_{2b,m_{x}(2b)}) ∩ Γ, b ∈ Φ, b(x) > 0⟩. This is a subgroup of Γ_{x} (2.6) and of U_{I}(K) (2.5). So V ⊂ Γ_{x} ∩ U_{I}(K). For showing the reverse inclusion, it suffices to show that

(2.7)  Γ_{x} ∩ U_{I}(K) ⊂ ⟨(U_{b,m_{x}(b)}.U_{2b,m_{x}(2b)}) ∩ Γ, b(x) ≥ 0⟩.

From Lemma 2.3.(3) we have

Γ_{x} ∩ U_{I}(K) ⊂ Γ ∩ U_{[x[}^{+}.

Accordingly, it will suffice to show that Γ_{x} ∩ U_{[x[}^{+} is a subgroup of the right handside of (2.7). Let Φ_{red}^{+} = {b_{1}, ...., b_{N}} be the subset of reduced positive roots (with an

arbitrary order). The product induces a isomorphism of k–varieties ∏_{j=1}^{N} U_{b_{j}} ⟶^{∼} U [1,

prop. 21.9]. In particular, we have compatible bijections

∏_{j=1}^{N} U_{b_{j}}(K) ⟶^{∼} U(K)
(2.8)  ∪  ∪
∏_{j=1}^{N} U_{b_{j}}(A) ⟶^{∼} U(A).

By comparing these with the bijection [3, §6.4.9]

∏_{j=1}^{N} U_{b_{j},m_{x}(b_{j})}.U_{2b_{j},m_{x}(2b_{j})} ⟶^{∼} U_{[x[}^{+},

we see that Γ_{x} ∩ U_{I}(K) ⊂ U_{[x[}^{+} ∩ U(A) consists of products of elements (U_{b_{j},m_{x}(b_{j})}.U_{2b_{j},m_{x}(2b_{j})}) ∩ Γ with b_{j}(x) ≥ 0.

(3) This follows from (1) and Lemma 2.3.(3).

2.4 Action on the star of certain points

We will now make use of the spherical building B(G) of G [16, §5]. Recall that B(G) is a simplicial complex whose simplex are the k-parabolic subgroups of G. If Q is such a parabolic subgroup, the faces of its associated simplex are the simplexes associated to the maximal proper k-parabolic subgroups of Q. The standard apartment 𝔄 of B(G) is the subcomplex of k-parabolic subgroups containing S and the standard chamber ℭ is the simplex associated to the minimal k-parabolic subgroup P. We denote by W = N_{G}(S)/Z_{G}(S) the relative Weyl group of G.

If x ∈ T , we denote by L_{x} the star of x (étoile in french) of x,^{3} i.e. the subspace of T consisting of facets F such that x ∈ \overline{F} [4, §4.6.33].

We denote by S_{∗} = Hom_{k−gr}(G_{m}, S) the group of cocharacters of S. Inside the apartment A = ϕ + S_{∗} ⊗_{ℤ} ℝ, this corresponds to the lattice of points having type 0, i.e. the type of ϕ. The action of S(K) on T preserves A. More precisely, the element s ∈ S(K) acts on A as the translation by the vector v_{s} defined by the property [4, §5.1.22]

⟨v_{s}, b⟩ = −ω(b(s)) ∀b ∈ Φ. (2.9)

We denote by C ⊂ S_{∗}⊗_{ℤ}ℝ the vector chamber such that ϕ+C is the unique chamber of the sector Q which contains the special point ϕ in its adherence [3, §1.3.11].

Lemma 2.6. Let x be a point of S_{∗} ∩Q. Then the chambers of L_{x} ∩Q are the x+wC for w ∈ W(k) satisfying I_{x} ⊂ w.Φ^{+}.

Proof. Set I = I_{x}. The chambers of L_{x} are the x + wC with w ∈ W(k). Let y ∈ C. If x + wC ⊂ Q, then

b(x + w.y) = b(x) + (w^{−1}.b)(y) ≥ 0 ∀b ∈ Δ.

It follows that if b ∈ I, i.e. b(x) = 0, then (w^{−1}.b)(y) ≥ 0, and therefore b ∈ w(Φ_{+}). Conversely, if w ∈ W(k) satisfies I ⊂ w(Φ_{+}), then the above inequality holds for ϵy for all b ∈ Δ for ϵ > 0 small enough. Thus x + w . (ϵy) ∈ Q and x + wC ⊂ Q.

Lemma 2.7. Let I be a subset of Δ, and set W_{I} := N_{L_{I}}(S)/Z_{G}(S). Let 𝔄_{I} be the union of the w\overline{ℭ} for w running over the elements of W(k) satisfying I ⊂ w.Φ^{+}.

1. W_{I}(k) . 𝔄_{I} = 𝔄.

2. P_{I}(k) . 𝔄_{I} = B(G).

Proof. (1) We reason by induction on the cardinality of I. If I = ∅, then 𝔄_{I} = 𝔄 and there is nothing to prove. Assume that I = I′ ∪ {b}. We are given a chamber wℭ of 𝔄 with w ∈ W(k). We want to show that wℭ is equivalent under W_{I}(k) to a chamber of 𝔄_{I}. Since W_{I′}(k) ⊂ W_{I}(k), we can assume by the induction hypothesis that I′ ⊂ w.Φ^{+}. If b ∈ w.Φ^{+}, we have I ⊂ w.Φ^{+}. The other case is when −b ∈ w.Φ^{+}. Let s_{b} ∈ W_{I}(k) the reflexion associated to b. Then s_{b}(b) = −b, hence b ⊂ s_{b}w.Φ^{+}. For b′ ∈ I′, we have s_{b}(b′) = b′ + mb where m is non-negative. Therefore

3The terminology link is also used in the literature.

b′ = s_{b}^{2}(b′) = s_{b}(b′ + mb) = s_{b}(b′) − mb ∈ s_{b}w.Φ^{+}.

We conclude that I ⊂ s_{b}w.Φ^{+} and s_{b}.(wℭ) ⊂ 𝔄_{I}.

(2) Again it suffices to prove that any chamber of B(G) is equivalent under P_{I}(k) to a chamber of 𝔄_{I}. Let ℭ′ be a chamber of B(G). Let P′ be the underlying minimal k–parabolic subgroup. By [2, prop. 4.4.b] P_{I} ∩ P′ contains a maximal k-split torus of P_{I}. Since maximal k–split tori of P_{I} are conjugate under U_{I}(k), it follows that there exists u ∈ U_{I}(k) such that uSu^{−1} ⊂ P_{I} ∩ P′, hence S ⊂ u^{−1}P′u. So we can assume that S ⊂ P′, i.e. that ℭ′ ⊂ 𝔄. Then ℭ′ = wℭ for some w ∈ W(k). By 1), ℭ′ is then equivalent under W_{I}(k) to a chamber of 𝔄_{I}. Since N_{L_{I}}(S)(k) maps onto W_{I}(k), we conclude that ℭ′ is then equivalent under P_{I}(k) to a chamber of 𝔄_{I}.

We come now to the following important step in Soulé’s proof.

Lemma 2.8. Let x ∈ S_{∗} ∩ Q. Then Γ_{x}.(L_{x} ∩ Q) = L_{x}.

Proof. We will make use of the canonical smooth model 𝔓_{x}/O of the parahoric subgroup associated to x [4, §5.2]. As an O–group scheme 𝔓_{x} is isomorphic to G ×_{k} O, and we have an identification 𝔓_{x}(O) = P_{x}. The star L_{x} is the spherical building of 𝔓_{x} ×_{O} k ≅ G [4, §5.1.32]. Set for convenience I = I_{x}. By Lemma 2.6, L_{x} ∩ Q is identified with 𝔄_{I} in the spherical building B(G). Furthermore, the chamber x + C identifies with ℭ.

The inclusion Γ_{x}.(L_{x} ∩ Q) ⊂ L_{x} is clear. Let us prove the reverse inclusion. By definition, there exists λ ∈ S_{∗}∩Q such that x = λ. Define g_{λ} = λ(\frac{1}{t})^{−1} = λ(t) ∈ S(K). Since x = g_{λ}.ϕ by (2.9) above, we have

P_{x} = g_{λ} P_{ϕ} g_{λ}^{−1}. (2.10)

Thus

𝔓_{x}(O) ≅ P_{x} = g_{λ} G(O) g_{λ}^{−1} ⊂ G(K).

In view of Lemma 2.7.2, it will suffice to establish the following.

Claim 2.9. The image of the composite map

Γ_{x} ⊂ P_{x} ⟶ (𝔓_{x} ×_{O} k)(k) ≅ G(k)

contains P_{I}(k).

The group L_{I}(k) commutes with g_{λ} inside G(k(t)), and it is therefore included in the image in question (as we have already observed in Proposition 2.5). So it is enough to check that g_{λ}U(k)g_{λ}^{−1} ⊂ Γ_{x}, or equivalently that g_{λ}U(k)g_{λ}^{−1} ⊂ Γ. This can be verified by working over the field k̃ and checking the inclusion for the subgroups U_{b}(k̃) of U(k̃) for b ∈ Φ^{+}. To verify this we use that the product map induces a decomposition (with the notation of §2.2)

∏_{ã∈Φ̃^{b}, ã_{|S×_{k}k̃}=b} Ũã(k̃) . ∏_{ã∈Φ̃^{b}, ã_{|S×_{k}k̃}=2b} Ũã(k̃) ⟶^{∼} U_{b}(k̃).

For ã ∈ Φ̃^{b} and s ∈ k̃, we have

g_{λ} Uã(s) g_{λ}^{−1} = \{ \begin{matrix}Ũã(t^{⟨b,λ⟩} s) & if ã_{|S×_{k}k̃} = b, \\ Ũã(t^{2 ⟨b,λ⟩} s) & if ã_{|S×_{k}k̃} = 2b. \end{matrix}

Hence g_{λ} Uã(s) g_{λ}^{−1} ⊂ Γ̃. This establishes Claim 2.9. The proof of Lemma 2.8 is now complete

2.5 End of the proof of Theorem 2.1

We now finish the proof of the main Theorem.

Two distinct points of Q are not equivalent under Γ. Since two different points of Q̃ are not equivalent under Γ̃ [14, 1.3], it follows that two distinct points in Q are not equivalent under Γ. A point of T of type 0 is equivalent to a point of Q. We denote by M ⊂ S(K) = S_{∗} ⊗ K^{×} the subgroup generated by the λ(t) for λ running over S_{∗}. We denote by M_{+} ⊂ M the semigroup generated by the λ(t) for λ satisfying ⟨b, λ⟩ ≥ 0 for all b ∈ Δ. By a result of Raghunathan [9, th. 3.4],^{4} we have the decomposition

G(K) = Γ . M . G(O).

Again, since N_{G}(S)(k) maps onto W(k) and W(k).M_{+} = M, we have actually a decomposition

G(K) = Γ . M_{+} . G(O).

Since G(K)/G(O) is the set of points of type 0 of T , this shows that every such point of T is Γ–conjugated to a point of M.ϕ. But M_{+}.ϕ ⊂ Q, so we conclude that every such point of T is Γ–conjugated to a point of Q.

4This reference presupposes that the base field k is infinite, but this assumption is not necessary (see [7, III.3.4.2] for details).

Every point of T is equivalent to a point of Q. Let y be a point of T . Let F be a chamber of T containing y. Then \overline{F} contains a (unique) point x whose type is that of ϕ. By the preceding step, we can assume that x ∈ Q. Then y belongs to L_{x} and Lemma 2.8 shows that y is equivalent under Γ to a point of Q.

¿From the above it follows that T = Γ.Q, as it is stated in the Theorem.

3 Applications

We give two applications of the main Theorem. The notation and assumptions are as in the previous section. We begin by recalling some basic facts about direct limits of groups.

3.1 Direct limits of groups

Direct limits of groups occur in geometric group theory [11]. In what follows we will repeatedly encounter the following situation: We are given a family of subgroups (H_{λ})_{λ∈Λ} of a group H (indexed by some set Λ) and we wish to consider the group which is the direct limit of the groups (H_{λ}, H_{λ} ∩ H_{μ})_{λ,μ∈Λ} where the only transition maps are the inclusions H_{λ} ∩ H_{μ} ⊂ H_{λ} and H_{λ} ∩ H_{μ} ⊂ H_{μ}. We call the resulting group the direct limit of the family (H_{λ})_{λ∈Λ} with respect to their intersections.^{5}

Let T be an abstract simplicial complex, E the set of its vertices, and Φ the set of its simplexes. Denote by X the geometric realization of T. Let H be a group which acts in a simplicial way on T, and for which there exists a simplicial fundamental domain T′. Recall that T′ is a subcomplex of T such that if E′ (resp. Φ′) denotes the set of vertices (resp. simplexes) of T′, then for every s ∈ Φ, there exists a unique s′ ∈ Φ′ such that s ∈ H.s′.

The isotropy subgroup of H corresponding to an element z (respectively a subset M) of either T or X will be denoted by H_{z} (respectively H_{M}).

Theorem 3.1. (Soulé, [13]) Let T, X, H, T′ be as above. Assume that X is connected and simply connected and that the geometric realization X′ of T′ is connected. Then the group H is the direct limit of the family of isotropy subgroups (H_{M})_{M∈E′} with respect to their intersections.

Higher dimensional generalizations of this result have been established by Chebotarev [5]. As pointed out by one of the referees, when X has additional structures there are other presentations which are useful in practice.

5Another terminology, which is a slight abuse of language, is that H is the sum of the H_{M} amalgamated over their intersections [11, II.1.7].

Proposition 3.2. Under the hypothesis of Theorem 3.1, assume that X is equipped with a distance d such that

i) for any two points x, y, there is a unique geodesic linking x and y;

ii) for any x ∈ X, there exists an open neighbourhood D_{x} of x such that for any simplex F of X, D_{x} ∩ F ≠ ∅ ⟹ x ∈ \overline{F};

iii) H acts isometrically on X. Furthermore, we assume that

iv) for each simplex F of X the stabilizer of F (as a set) coincides with the isotropy group (pointwise stabilizer) of \overline{F} . Then

1. The group H is the direct limit of the family (H_{M} ∩ H_{N})_{M,N∈E′} with transition maps H_{M} ∩ H_{N} → H_{M} and H_{M} ∩ H_{N} → H_{N} for M, N belonging to an edge of X′.^{6}

2. The group H is the direct limit of the family of isotropy subgroups (H_{x})_{x∈X′} with respect to their intersections.

Note that when X is a tree the first statement of the Proposition allows us to recover a classical result [11, §4.5, th. 10].

Remark 3.3. Note that the first statement of the Proposition is different than that of Theorem 3.1. The point is that two vertices of X′ do not necessarily belong to a common edge. In other words, the presentation of H given by Proposition 3.2.(1) has less relations than the one given by Theorem 3.1.

Proof. We prove both statements at the same time. We denote by H^{†} the first limit and by H^{♯} the second one. We have an obvious surjective map H^{†} → H, while the inclusion E′ ⊂ X gives rise to a map H → H^{♯}. We denote by ξ : H^{†} → H → H^{♯} the composition of these two maps. It is enough then to show that H → H^{♯} is surjective, and to produce a section θ : H^{♯} → H^{†} of ξ.

If x ∈ X, we denote by F_{x} ⊂ X the (open) simplex attached to x. Since every F_{x} contains in its closure a vertex M, our hypothesis on stabilizers implies that H_{x} ⊂ H_{M}. It follows that H → H^{♯} is surjective.

To define the splitting θ : H^{♯} → H^{†} we proceed as follows. We are given x ∈ X, and M ∈ E′ such that M ∈ \overline{F}_{x}. Since the action is simplicial, we have H_{x} = H_{F_{x}}. By our hypothesis on the stabilizers, we have then the inclusion H_{x} ⊂ H_{M} ⊂ H.

6By taking M = N in E′ we see that the groups H_{M} are part of our family. Observe that if M, N are vertices of a common edge F, then H_{N} ∩ H_{M} is nothing but the isotropy group of \overline{F}.

Step 1 : The composite map

θ_{x,M} : H_{x} → H_{M} → H^{†}

does not depend of the choice of M:

We note that two distinct choices M, N of vertices of \overline{F}_{x} define an edge of X′, so that the maps H_{x} → H_{M} → H^{†} and H_{x} → H_{M} → H^{†} agree since they agree on H_{M} ∩ H_{N}. This establishes Step 1, and defines a map θ_{x} : H_{x} → H^{♯}. Step 2: If y ∈ \overline{F}_{x}, then θ_{x} and θ_{y} agree on the subgroup H_{x} of H_{y} :

Since \overline{F}_{y} ⊂ \overline{F}_{x}, we can pick a vertice M ∈ \overline{F}_{y}. By definition θ_{x,M} and θ_{y,M} agree on H_{y}. Hence θ_{x} and θ_{y} agree on H_{y} by the first step. Step 3: Connexity argument. We are given x, y ∈ X and we want to show that θ_{x} and θ_{y} agree on H_{x} ∩ H_{y}. Since H_{x} ∩ H_{y} acts trivially on the geodesic [x, y], we have H_{x} ∩ H_{y} ⊂ H_{z} for all z ∈ [x, y]. We consider then the restrictions Θ_{z} : H_{x} ∩ H_{y} ⊂ H_{z} → H^{†} of θ_{z} to H_{x} ∩ H_{y} for z running over [x, y].

Recall that D_{z} stands for the open neighbourhood of z ∈ X given by the hypothesis ii). Step 4: If z ∈ [x, y], then Θ_{z} = Θ_{z′} for all z′ ∈ D_{z} ∩ [x, y]. Since z′ ∈ F_{z′} ∩ D_{z}, assumption ii) implies that z ∈ \overline{F}_{z′}. Step 2 shows that θ_{z} and θ_{z′} agree on H_{z′} ⊂ H_{z}, hence Θ_{z} = Θ_{z′}.

We now finish the proof of the Proposition. Since the D_{z} ∩ [x, y] define an open covering of the connected space [x, y], Step 3 implies that Θ_{z} does not depend on z. In particular θ_{x} and θ_{y} agree on H_{x} ∩ H_{y}. By the universal property defining H^{♯}, we obtain a map θ : H^{♯} → H^{†}. By construction θ ∘ ξ = id_{H^{†}}.

For future use we record the following.

Lemma 3.4. Let H be a group which is the direct limit of a family of subgroups (H_{α})_{α∈Λ} of H with respect to their intersections.

1. Let Λ′ ⊂ Λ be a directed subset, i.e. for all α, β ∈ Λ′, there exists γ ∈ Λ′ such that H_{α} ⊂ H_{γ}, H_{β} ⊂ H_{γ}. Then the direct limit of the family (H_{α})_{α∈Λ′} with respect to their intersections is canonically isomorphic to the subgroup ⋃_{α∈Λ′} H_{α}

of H.

2. Let Λ = ⊔_{j∈J}Λ_{j} be a partition of Λ in directed subsets. For j ∈ J, denote by

H_{j} := ⋃_{α∈Λ_{j}} H_{α}

the subgroup of H associated to Λ_{j}. Then H is the direct limit of the family of subgroups (H_{j})_{j∈J} of H with respect to their intersections.

Proof. (1) Note that ⋃_{α∈Λ′} H_{α} is a subgroup of H since Λ′ is directed. For any group

M we have

Hom_{gr}(H′, M) = \underleftarrow{lim}_{α∈Λ′} Hom_{gr}(H_{α}, M),

whence the statement.

(2) Denote by H̃ the direct limit of the family of subgroups (H_{j})_{j∈J} of H with respect to their intersections. The inclusion maps H_{j} ⊂ H agree over their intersections, hence give rise to a natural map ξ : H̃ → H. For defining the reverse map, denote by α ↦ j(α) the map Λ → J which maps α to the unique index j such that α ∈ Λ_{j}. We then get maps

H_{α} ↪ H_{j(α)} → H̃ (α ∈ Λ).

Since these maps agree over their intersections they yield a map η : H → H̃. Given that the images of the H_{α} generate H (resp. H̃), we get that η ∘ ξ = idH̃ and ξ ∘ η = id_{H}.

3.2 The group G(k[t]) as a direct limit

Theorem 3.1 yields.

Corollary 3.5. Let V be the set of vertices of Q. The group Γ = G(k[t]) is the direct limit of the family (Γ_{x})_{x∈V} with respect to their intersections.

¿From the Corollary we see that Γ is generated by the Γ_{x}. By Proposition 2.5.(1) Γ_{x} consists of products of elements of G(k) and elements of U(k[t]), where U stands for the unipotent radical of the minimal parabolic subgroup attached to S and Δ. From this we obtain.

Corollary 3.6. G(k[t]) = ⟨ G(k), U(k[t]) ⟩.

Another presentation of Γ is given by means of Proposition 3.2.(2).

Corollary 3.7. The group Γ = G(k[t]) is the direct limit of the family (Γ_{x})_{x∈Q} with respect to their intersections.

Proof. We have to check that hypothesis (i) through (iv) of Proposition 3.2 are satisfied for the action of Γ on the Bruhat-Tits building T , which is a metric space.

(i) Two points of T are linked by a unique geodesic [3, §2.5]. (ii) Lemma 2.5.11 of loc. cit. guarantees that for any x ∈ X, there exists an open ball D_{x} of center x such that for any simplex F of X, D_{x} ∩ F ≠ ∅ ⟹ x ∈ \overline{F}. (iii) The group G(K) acts isometrically on T (ibid).

(iv) Since G is simply connected, the stabilizer of a simplex F of T (or facet with the terminology of Bruhat-Tits) under Γ ⊂ G(K) is also its pointwise stabilizer [4, prop. 4.6.32] and also of \overline{F} [3, prop. 2.4.13].

The Corollary now follows from Proposition 3.2.

We shall now give a nicer presentation of Γ. Given a subset I ⊂ Δ, define

Q_{I} := {x ∈ Q | I_{x} = I}.

It is a subcone of Q, i.e. zQ_{I} ⊂ Q_{I} for all z > 0. Define the subgroup Γ_{I} = U_{I}(k[t]) ⋊ L_{I}(k).

Lemma 3.8. 1. The (Γ_{x})_{x∈Q_{I}} form a directed family of subgroups of Γ.

2. Γ_{I} is the direct limit of the Γ_{x} for x ∈ Q_{I}.

Proof. (1) The sector Q is equipped with the partial order x ≤ y if y − x ∈ Q. By restriction, we get a partial order on Q_{I} which is directed. Indeed, given x, y ∈ Q_{I}, we have x + y ∈ Q_{I} and x + y ≥ x and x + y ≥ y.

Let x, y be elements of Q_{I} such that x ≤ y. Then b(y) ≥ b(x) for all b ∈ [I]^{+}, hence m_{y}(b) ≤ m_{x}(b) for all b ∈ [I]^{+}. It follows that for b ∈ [I]^{+} we have

U_{b,m_{x}(b)}.U_{2b,m_{x}(2b)} ⊂ U_{b,m_{y}(b)}.U_{2b,m_{y}(2b)}.

Now Proposition 2.5.(2) shows that Γ_{x} ⊂ Γ_{y}. Since Q_{I} is a directed subset of Q, we conclude that the (Γ_{x})_{x∈Q_{I}} form a directed family of subgroups of Γ.

(2) By Lemma 3.4.(1), it is enough to show that

(3.1)  ⋃_{x∈Q_{I}} Γ_{x} = Γ_{I}.

Proposition 2.5.1 shows that the inclusion ⊂ holds. Conversely, suppose that we are given an element g ∈ Γ_{I}. Let x ∈ Q_{I}. By Proposition 2.5.(3) there exists a real number z ≥ 1 such that g ∈ Γ_{zx}. Since zx ∈ Q_{I}, g belongs to the left handside of 3.1.

Theorem 3.9. The group Γ = G(k[t]) is the direct limit of the family of subgroups (Γ_{I})_{I⊂Δ} with respect to their intersections

Proof. Lemma 3.8.(2) shows that Γ_{I} is the limit of the directed family of subgroups (Γ_{x})_{x∈Q_{I}}. To finish the proof we apply Lemma 3.4.(2) to the decomposition

Q = ⊔_{I⊂Δ} Q_{I}

of Q into directed subsets.

3.3 Application to Whitehead groups

Let G(k)^{+} be the (normal) subgroup of G(k) generated by the (R_{u}P)(k) for P running over all parabolic k–subgroups of G. If card(k) ≥ 4, Tits has shown that every proper normal subgroup of G(k)^{+} is central [15]. The quotient W (k, G) = G(k)/G(k)^{+} is the Whitehead group of G [17]. By Tits’ result this group detects whether G(k) is projectively simple.

It turns out that the Whitehead group admits another characterization. Denote by HG(k) the (normal) subgroup of G(k) which consists in elements g ∈ G(k) for which there exists an element h ∈ Γ = G(k[t]) such that h(0) = e and h(1) = g. We denote by π_{0}(k, G) = G(k)/HG(k) this naïve group of connected components of G.

Theorem 3.10. There is a canonical isomorphism W (k, G) ⟶^{∼} π_{0}(k, G).

Proof. The unipotent radical V of a k–parabolic subgroup Q of G is a split unipotent group, so it satisfies H(V)(k) = V(k). Hence we have G(k)^{+} ⊂ HG(k) and a surjection G(k)/G(k)^{+} → π_{0}(k, G) = G(k)/HG(k). It remains to show that HG(k) ⊂ G(k)^{+}. Let g ∈ HG(k), and choose h ∈ G(k[t]) satisfying h(0) = e and h(1) = g. According to Corollary 3.6, the element h can be written in the form

h = g_{1}u_{1} g_{2} u_{2} ⋯ g_{n}u_{n}

with g_{i} ∈ G(k) and u_{i} ∈ U(k[t]) where U is the unipotent radical of a minimal parabolic k–subgroup of G. We can assume that u_{i}(0) = e, so the condition h(0) = e reads g_{1} ⋯ g_{n} = e. It follows that

h = g′_{1} u_{1} g′_{1}^{−1} ⋯ g′_{n} u_{n} g′_{n}^{−1}

with g′_{1} = g_{1}, g′_{2} = g_{1}g_{2}, ..., g′_{n} = g_{1} ⋯ g_{n} = e ∈ G(k). Hence g = h(1) = g′_{1} u_{1}(1) g′_{1}^{−1} ⋯ g′_{n} u_{n}(1) g′_{n}^{−1} ∈ G(k)^{+} as desired.

Acknowledgements: The author wishes to express his sincere gratitude to Gopal Prasad and the referees for their useful comments.

References

[1] A. Borel, Linear Algebraic Groups (Second enlarged edition), Graduate text in Mathematics 126 (1991), Springer.

[2] A. Borel and J. Tits, Groupes réductifs, Inst. Hautes Études Sci. Publ. Math. 27 (1965), 55–150.

[3] F. Bruhat and J. Tits, Groupes réductifs sur un corps local. I, Inst. Hautes Etudes Sci. Publ. Math. 41 (1972), 5–251.

[4] F. Bruhat and J. Tits, Groupes réductifs sur un corps local. II. Schémas en groupes. Existence d’une donnée radicielle valuée, Inst. Hautes Etudes Sci. Publ. Math. 60 (1984), 197–376.

[5] N. G. Chebotarëv, The decomposition of groups acting on complexes with a fundamental domain. (Russian) Uspekhi Mat. Nauk 37 (1982), no. 4(226), 191– 192, english translation in Russian Math Surveys 37 (1982), no. 4, 103-105.

[6] P. Garrett, Buildings and classical groups, Chapman and Hall, London, 1997.

[7] P. Gille, Torseurs sur la droite affine et R-équivalence, thèse de doctorat (Orsay, 1994), home page of the author.

[8] H. Nagao, On GL(2, K[x]), J. Inst. Polytech. Osaka City Univ. Ser. A 10 (1959), 117–121.

[9] M. S. Raghunathan, Principal bundles admitting a rational section, Invent. math. 116 (1994), 409-423.

[10] Séminaire de Géométrie algébrique de l’I.H.E.S., 1963-1964, schémas en groupes, dirigé par M. Demazure et A. Grothendieck, Lecture Notes in Math. 151-153. Springer (1970).

[11] J.-P. Serre, Arbres, amalgames, SL_{2}, rédigé avec la collaboration de Hyman Bass, Astérisque 46 (1977).

[12] T. A. Springer, Reductive groups, Automorphic forms, representations and Lfunctions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 1, pp. 3–27, Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979.

[13] C. Soulé, Groupes opérant sur un complexe simplicial avec domaine fondamental, C. R. Acad. Sci. Paris Sér. A-B 276 (1973), 607–609.

[14] C. Soulé, Chevalley groups over polynomial rings, Homological group theory (Proc. Sympos., Durham, 1977), 359–367, London Math. Soc. Lecture Note Ser. 36 (1979), Cambridge Univ. Press.

[15] J. Tits, Algebraic and abstract simple groups, Ann. of Math. 80 (1964), 313–329.

[16] J. Tits, Buidings of spherical type and finite BN-pairs, Lect. Notes in Math. 396 (1973), Springer.

[17] J. Tits, Groupes de Whitehead de groupes algébriques simples sur un corps (d’après V. P. Platonov et al.), Séminaire Bourbaki, 29e année (1976/77), Exp. No. 505, pp. 218–236, Lecture Notes in Math. 677 (1978).

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