March 19, 2000 Preliminary Version

INVOLUTIONS, CLASSICAL GROUPS AND BUILDINGS

Ju-Lee Kim and Allen Moy

Abstract. In [CLT], Curtis-Lehrer-Tits construct a variation of the Tits building. The Curtis-Lehrer-Tits building ℒ(G, k) of a connected reductive k-group G has the important feature that it is a functor from the category of reductive groups defined over a field k and monomorphisms to the category of topological spaces and inclusions. An important consequence derived by Curtis-Lehrer-Tits from the functorial nature of the Curtis-Lehrer-Tits building ℒ(G, k) is that if s is a semisimple element of the group G(k) of k-rational points, and G′ is the connected component group of the centralizer of s, then the fixed point set ℒ(G, k)^{s} of s in ℒ(G, k) is the Curtis-Lehrer-Tits building ℒ(G′, k). We generalize this result to arbitrary involutions of Aut_{k}(G), and we also prove an analogue in the context of affine buildings.

1. Introduction

1.1. Let k be a field and G a connected reductive group defined over k. The Tits or spherical combinatorial building Δ(G, k) of G with respect to k is the simplicial complex whose simplices are the proper parabolic k-subgroups of G, reverse ordered by inclusions. A parabolic k-subgroup P is an r-simplex if and only if there are (r+1) distinct P_{0}, P_{1}, …, P_{r} maximal parabolic k-subgroups of G so that P = P_{0} ∩P_{1} ∩⋯∩P_{r}. The group 𝒢 := G(k) of k-rational points of G operates simplicially on Δ(G, k). In [CLT], Curtis-Lehrer-Tits construct a variation of the Tits building. The Curtis-Lehrer-Tits building ℒ(G, k) of G has the important feature that it is a functor from the category of connected reductive groups defined over k and monomorphisms to the category of topological spaces and inclusions. That is, given a monomorphism f : G → H of reductive groups defined over k, Curtis-Lehrer-Tits naturally associate an embedding of topological spaces ℒ(f) : ℒ(G, k) → ℒ(H, k). An important consequence derived by Curtis-Lehrer-Tits from the functorial nature of the Curtis-Lehrer-Tits building ℒ(G, k) is the following: Suppose s is a semisimple element of 𝒢. Let G′ := (G^{s})^{∘} denote the connected centralizer of s in G. Then, the fixed point set ℒ(G, k)^{s} of s in ℒ(G, k) is the Curtis-Lehrer-Tits building ℒ(G′, k). One of our main goals (Theorem 3.3.1) is to establish, when char(k) ≠ 2, a generalization of the Curtis-Lehrer-Tits result to arbitrary involutions in Aut_{k}(G).

1.2. Suppose k is a nonarchimedean local field, and G is a connected semisimple kgroup. Suppose further that τ is a k-automorphism of G so that the k-group G′ := (G^{τ})^{∘} is semisimple. Let ℬ(G, k) and ℬ(G′, k) denote the Bruhat–Tits affine buildings of G and G′ respectively. It is natural to ask what is the relation between ℬ(G, k)^{τ} and ℬ(G′, k). Under the assumption that the residual characteristic of k is odd, G is a special linear group G and τ ∈Aut_{k}(G) is an involution defining a classical group, we show (Theorem 6.7.3) ℬ(G, k)^{τ} can be identified with ℬ(G′, k). This type of result is related to the question of when Bruhat–Tits buildings are functorial [L].

The authors thank Anne-Marie Aubert for some useful comments. The authors were supported in part by the National Science Foundation grants DMS-9970454 and DMS-9801264.

2. Preliminaries on the Curtis-Lehrer-Tits building.

2.1. Let G be a connected reductive group defined over a field k and let 𝒢 denote the group of k-rational points G(k). We review the construction of the Curtis-Lehrer-Tits (spherical) building ℒ(G, k) of G in [CLT]. Let S be a maximal k-split torus of G and let Φ(S, G) be the k-roots of G with respect to S. Denote the k-cocharacters Hom_{k}(GL(1), S) of S as X_{∗}(S, k). The space ℒ(S, k) is defined as the sphere whose points represent rays in the real vector space X_{∗}(S, k) ⊗_{ℤ} ℝ. To any point b ∈ ℒ(S, k), i.e. a ray in X_{∗}(S, k) ⊗_{ℤ} ℝ, we associate the parabolic k-subgroup P(b) defined as the largest closed subgroup of G containing S and whose Lie algebra contains the roots ψ ∈ Φ(S, G) whose inner product with b is non-negative.

2.2. Denote as ℒ_{1} the disjoint union of the spheres ℒ(S, k) as S runs over all the maximal k-split tori of G. The group Aut_{k}(G) of k-automorphisms of G obviously acts on ℒ_{1}. In particular, 𝒢 acts on ℒ_{1} by inner automorphisms. We define an equivalence on points in ℒ_{1} as follows. Given b ∈ ℒ(S, k) and b′ ∈ ℒ(S′, k), we define b ∼ b′ if P(b) = P(b′) and there is a g ∈ P(b) conjugating S to S′ so that b′ = Ad(g)(b). The Curtis-Lehrer-Tits (spherical ) building ℒ(G, k) of G with respect to k is defined as the quotient of ℒ_{1} by the equivalence relation ∼. It is immediate from (2.1) that b ∈ ℒ(G, k) determines unambiguously a parabolic k-subgroup P(b) of G. The building ℒ(G, k) inherits an action by Aut_{k}(G); in particular an action of 𝒢.

For our purposes, the field k is fixed. Therefore, in order to simplify notation, we shall often use X_{∗}(S), ℒ(S), and ℒ(G) to denote X_{∗}(S, k), ℒ(S, k), and ℒ(G, k) respectively.

2.2.1. Lemma. (Lemma 2.2 of [CLT])

i) The stabilizer of a point b ∈ ℒ(G) in 𝒢 is P(b)(k).

ii) Let S be a maximal k-split torus in G. The canonical projection ℒ_{1} → ℒ(G) restricts to an injection of ℒ(S) into ℒ(G).

The image of ℒ(S) in ℒ(G) is called the (spherical ) apartment associated with S in ℒ(G). For convenience, we identify ℒ(S) with its corresponding apartment. A point b lies in an apartment ℒ(S) precisely when S ⊂ P(b).

2.2.1. Lemma. (statements (2.3) and (2.4) of [CLT]) iii) Any two points of ℒ(G) belong to an apartment. iv) If S and S′ are two maximal k-split tori, there is an element g ∈ 𝒢 conjugating S to

S′ (hence g takes ℒ(S) to ℒ(S′)), which fixes all points in ℒ(S) ∩ ℒ(S′).

2.3. Let S′ be any k-split torus and let S be a maximal k-split torus containing S′. The canonical injection X_{∗}(S′) → X_{∗}(S) induces an injection ℒ(S′) → ℒ(S) and hence an injection of ℒ(S′) into ℒ(G). As a map into ℒ(G), this injection is independent of the torus S. In particular, any point b ∈ ℒ(S) determines a parabolic k-subgroup P(b) ⊂ G.

2.3.1. Lemma. (Lemma 1.2 of [CLT]) Let S be a k-split torus in G.

i) For b ∈ ℒ(S), we have P(b) ⊃ C_{G}(S).

ii) For b ∈ ℒ(S), let S′ be the intersection of S with the radical of P(b). Then b ∈ ℒ(S′).

2.4. Suppose G′ is a connected reductive k-group and f : G′ → G is a k-monomorphism. Let ℒ′_{1} be the disjoint union of all ℒ(S′) with S′ a maximal k-split torus of G′. The map f : G′ → G induces a mapping ℒ′_{1} → ℒ(G). It is shown in [CLT:§4] that two points b, b′ ∈ ℒ′_{1} are equivalent under the equivalence relation ∼_{G}_{′} described by G′ if and only if they have the same image in ℒ(G). Thus, the map f : G′ → G induces an injective map ℒ(f) : ℒ(G′) → ℒ(G); or equivalently the Curtis-Lehrer-Tits building is a functor from the category of connected reductive k-groups with k-monomorphisms into the category of sets with injections.

2.5. As mentioned in the introduction, an important consequence of the functorial nature of the Curtis-Lehrer-Tits building is the following proposition.

2.5.1. Proposition. (Proposition 5.1 of [CLT]) Let s be a k-rational semisimple element of a connected reductive k-group G and denote the connected component group of the centralizer of s as G′ = C_{G}(s)^{∘}. Then, ℒ(G′, k) is the fixed point set ℒ(G, k)^{s} of s acting on ℒ(G, k).

3. Involutions and spherical buildings

3.1. Suppose τ ∈ Aut_{k}(G) is an involution, i.e. of order two. From now on, we assume char(k) ≠ 2. Under this assumption, the induced action of τ on Lie(G) is semisimple, and therefore Lie(G^{τ}) and G^{τ} are reductive. For notational ease, we let G′ denote the connected component group (G^{τ})^{∘}. Our main goal in this section is to prove

ℒ(G, k)^{τ} = ℒ(G′, k).

3.2. We begin with a preliminary lemma.

3.2.1. Lemma.

i) If S is a maximal k-split torus in a linear algebraic k-group Q, then the intersection S′ = S ∩ R(Q) of S with the radical R(Q) of Q is a maximal k-split torus in R(Q).

ii) In the situtation of i), any two maximal k-split tori in R(Q) are conjugate by an

element u ∈ R_{u}(Q)(k), where R_{u}(Q) is the unipotent radical of Q. iii) Suppose char(k) ≠ 2. If U is a unipotent k-group then if u ∈ U(k), there is a unique

v ∈ U(k) such that u = v^{2}.

3.2.2. Remark. A trivial modification of the proof of Lemma 3.2.1 iii), shows under the assumption ℓ is a prime and char(k) ≠ ℓ, if u ∈ U(k), then there is a unique v ∈ U(k), so that u = v^{ℓ}.

Proof. To prove (i), let T′ be a maximal split k-torus in R(Q) and T′ ⊂ T a maximal split k-torus in Q. There exists g ∈ Q(k) such that gTg^{−1} = S. Thus, S′ = S ∩ R(Q) = gTg^{−1} ∩ R(Q) = g(T ∩ R(Q))g^{−1} = gT′g^{−1} must be a maximal split k-torus in R(Q). Statement (ii) is [B1:11.23]. To prove statement (iii), let U = U_{0} ⊃ U_{1} ⊃ ⋯ ⊃ U_{r−1} ⊃ U_{r} = {0} be the descending central series of U, i.e. for i ≥ 1, U_{i} is defined as the group U_{i} := [U, U_{i−1}] generated by the commutators of elements in U with those in U_{i−1}. We note that U(k)/U_{i}(k) = (U/U_{i})(k). Define 𝒰 := U(k) and 𝒰_{i} := U_{i}(k). For 1 ≤ i ≤ r, we claim there is a unique sequence of elements \overline{v}_{i} ∈ 𝒰/𝒰_{i}, so that u𝒰_{i} = \overline{v}_{i}^{2} in 𝒰/𝒰_{i}. When i = 1, this assertion reduces to the statement that 𝒰/𝒰_{1} is a vector space over k. In particular, the condition on the characteristic of k is now clear. Suppose now by way of induction that the assertion is true for i, i.e. there is a unique \overline{v}_{i} ∈ 𝒰/𝒰_{i} so that u𝒰_{i} = \overline{v}_{i}^{2} in 𝒰/𝒰_{i}. Pick a representative v_{i} of \overline{v}_{i}. Then v_{i}^{−2}u ∈ 𝒰_{i}. But again, 𝒰_{i}/𝒰_{i+1} is a vector space over k so there is y_{i} ∈ 𝒰_{i} which is unique module 𝒰_{i+1} so that u = v_{i}^{2}y_{i}^{2} modulo 𝒰_{i+1}. But v_{i}^{2}y_{i}^{2} = v_{i}y_{i}y_{i}^{−1}v_{i}y_{i}y_{i} = v_{i}y_{i}(y_{i}^{−1}v_{i}y_{i}v_{i}^{−1})v_{i}y_{i} = (v_{i}y_{i})^{2} modulo 𝒰_{i+1}. Thus \overline{v}_{i+1} = v_{i}y_{i} modulo 𝒰_{i+1} has the property u = \overline{v}_{i+1}^{2} modulo 𝒰_{i+1}. Uniqueness of \overline{v}_{i+1} follows from the combination of uniqueness of \overline{v}_{i} and y_{i} in 𝒰/𝒰_{i} and 𝒰_{i}/𝒰_{i+1} respectively and that 𝒰_{i+1} ⊂ [𝒰, 𝒰_{i}]. □

3.3. We now prove this section’s main result.

3.3.1. Theorem. Suppose char(k) ≠ 2, G is a connected reductive k-group, τ ∈ Aut_{k}(G) is an involution and G′ = (G^{τ})^{∘}. Then

ℒ(G, k)^{τ} = ℒ(G′, k).

Proof. Let S be a maximal k-split torus. Suppose b ∈ ℒ(S) ⊂ ℒ(G). Let P(b)(k) be the parabolic subgroup which fixes b. By Lemma 3.2.1 (i), the split k-torus S′ = S ∩ R(P(b)) is a maximal split k-torus in R(P(b)), and by Lemma 2.3.1 (ii), we have b ∈ ℒ(S′). Of course, τ(S′) is a maximal split k-torus in R(τ(P(b))) and τ(b) ∈ ℒ(τ(S′)). If b is fixed by τ, then P(b) is τ invariant and therefore S′ and τ(S′) are maximal k-split tori in R(P(b)). By Lemma 3.2.1 (ii), there is a u ∈ R_{u}(P(b))(k) so that τ(S′) = uS′u^{−1}. If we apply τ to this last equality, we obtain S′ = τ(u)τ(S′)τ(u^{−1}) = τ(u)uS′u^{−1}τ(u^{−1}). Clearly R_{u}(P(b)) is τ-invariant, so τ(u)u ∈ R_{u}(P(b)) and τ(u)u normalizes S′. Since S′ is a maximal split ktorus in R(P(b)), we conclude τ(u)u = 1. By Lemma 3.2.1 (iii), there is a unique element v ∈ R_{u}(P(b))(k) so that u = v^{2}. Therefore v^{2} = u = τ(u^{−1}) = τ(v^{−1})^{2}, i.e. v and τ(v^{−1}) are square roots of u in R_{u}(P(b))(k), and therefore by uniqueness of square roots, τ(v^{−1}) = v. Consider the maximal split k-torus in R(P(b)) which is T = vS′v^{−1}. Then

τ(T) = τ(v)τ(S′)τ(v^{−1}) = v^{−1}uS′u^{−1}v = vS′v^{−1} = T.

That is, T is a τ-stable torus. Clearly b ∈ ℒ(T). So, b is a ray in X_{∗}(T) ⊗_{ℤ} ℝ which is τ invariant. This means b is a ray in (X_{∗}(T) ⊗_{ℤ} ℝ)^{τ} = X_{∗}(T^{τ}) ⊗_{ℤ} ℝ. A ray in X_{∗}(T^{τ}) ⊗_{ℤ} ℝ is precisely a point in ℒ(T^{τ}) ⊂ ℒ(G′, k). Thus, ℒ(G, k)^{τ} = ℒ(G′, k). □.

3.4. The next result shall be used when we discuss involutions and affine buildings. Geometrically, it implies that when k is a finite field of odd order and τ is a nontrivial involution of G, then any point b ∈ ℒ(G, k)^{τ} lies in a τ-invariant apartment of ℒ(G, k).

3.4.1. Proposition. Suppose k is a finite field of odd order q, G is a connected reductive k-group, τ ∈ Aut_{k}(G) is an involution and G′ = (G^{τ})^{∘}. If T is a k-split torus in G′, then there exists a maximal k-split torus S in G which contains T and is τ-invariant.

Proof. Let M = C_{G}(T) be a Levi k-subgroup of G. We make three trivial observations.

i) The group M is τ-invariant.

ii) Any maximal k-split torus in M must contain the torus T. iii) The reductive k-rank of M and G are equal. In particular, a maximal split k-torus in

M is also a maximal split k-torus in G. Denote the rank by r.

Let B (resp. U) be a Borel k-subgroup of M (resp. the unipotent radical of B). Then, the number of maximal split k-tori in B is equal to the order of the group U(k). The k-subgroup τ(B) is obviously also a Borel subgroup of M. Let Ω denote the set of maximal split k-tori in B ∩ τ(B). Obviously, since B ∩ τ(B) is a τ-invariant group, the set Ω is a τ-invariant set. The intersection of any two minimal parabolic k-subgroups must contain a minimal Levi k-subgroup. Thus, the intersection B ∩ τ(B) contains a minimal Levi k-subgroup and so a k-torus of dimension r, i.e. Ω is nonempty. Any two elements in Ω are conjugate in B ∩ τ(B). We conclude Ω has order a power of q = |k|, in particular has odd order. Thus, there must be a τ-fixed point S in Ω. The torus S is our desired torus. □

4. Simple algebras, involutions and classical groups

4.1. In the previous section, under the assumption that k is a field of odd characteristic, G is connected reductive k-group, τ ∈ Aut_{k}(G) is an involution and G′ := (G^{τ})^{∘}, we showed ℒ(G, k)^{τ} = ℒ(G′, k). In this section, we recall the relationship between the simple classical groups and simple algebras with involutions.

Let k be a field and suppose A is a finite dimensional simple algebra over k. A fundamental result of Weddeburn asserts that A is isomorphic as a k-algebra to a matrix algebra M_{n}(D) of a finite dimensional division algebra D over k. Furthermore, if D_{1} and D_{2} are two finite dimensional algebras over k and M_{n_{1}}(D_{1}) and M_{n_{2}}(D_{2}) are isomorphic as k-algebras, then n_{1} = n_{2} and D_{1} and D_{2} are isomorphic k-algebras.

4.2. A k-involution of a k-algebra A is an invertible k-linear map J : A → A satisfying

J^{2} = I (Identity) , and J(ab) = J(b)J(a).  (4.2.1)

An element a ∈ A is J-symmetric if J(a) = a and J-skew if J(a) = −a. The sets of Jsymmetric and J-skew elements are trivially k-linear subspaces of A. If the characteristic of k is not two, then A is the additive direct sum of these two linear subspaces.

4.3. Involutions fall into two distinct types. Suppose J is a k-involution of A. Since J is surjective, it must preserve the center C of the algebra A. Since we are assuming A is a simple algebra, the center C must be a finite dimensional field extension of k. The involution J is said to be of the first or second kind depending on whether C lies in the symmetrical elements of J. That is, J is of the first kind if the center C is elementwise fixed by J, and J is of the second kind if there are elements in the center C which are not J symmetric. When J is an involution of the second kind, the symmetric elements C^{J} in the center C is a finite extension of k so that [C : C^{J}] = 2.

Let C be the center of a simple finite dimensional k-algebra A and suppose S is a ksubfield of C. We call a k-involution J of A an involution over S of A if the J-symmetric elements in C is precisely S. We remark that C is either equal to S or a quadratic extension of S.

4.3.1. Theorem. (Theorem 11 page 154 of [A]) Let C be the center of a simple finite dimensional k-algebra A. Suppose S is a k-subfield of C and J is an involution of A over S. Then, a k-anti-isomorphism T : A → A is a k-involution over S if and only if there exists a J-symmetric or J-skew invertible element y ∈ A, i.e. J(y) = ±y, such that

T (a) = y^{−1} J(a) y a ∈ A.

4.3.2. Let D be a finite dimensional dvision algebra over k and let A = M_{n}(D). Denote the identity matrix in A as I. We identify d ∈ D with the matrix dI ∈ A. The center C of D becomes the center of A. For 1 ≤ i, j ≤ n, let e_{i,j} be the matrix whose r, s entry is δ_{i,r}δ_{j,s} (Kronecker delta).

4.3.3. Theorem. (Theorem 12 page 156 of [A]) Let D be a finite dimensional division algebra over k and set A = M_{n}(D). Let C be the center of D and S a k-subfield of C. Then, A has a k-involution over S if and only if there exists a k-involution of D over S. In this case, A has a k-involution J over S such that J(D) = D and J(e_{i,j}) = e_{j,i}.

4.4. Suppose D is a finite dimensional dvision algebra over k. For a ∈ M_{n}(D), denote by a^{t} the transpose of a. Also, if J is a k-anti-isomorphism of D, and a = (a_{i,j}) ∈ A set J(a) := (J(a_{i,j})). As a consequence of Theorems 4.3.1 and 4.3.3, we conclude that any k-involution T of A = M_{n}(D) over S ⊂ C is has the form

T (a) = y^{−1} J(a^{t}) y a ∈ A  (4.4.1)

where J is a k-involution of D over S and y ∈ A^{×} satisfies J(y^{t}) = ±y.

Let T be an involution of A = M_{n}(D). Write T as in (4.4.1). Set G := A^{×} = GL_{n}(D) and define τ : G → G by

τ(a) := T (a^{−1}).  (4.4.2)

Set

G^{τ} := { a ∈ G | τ(a) = a }.  (4.4.3)

Let V := D^{n} be the (A, D)-bimodule of size n column vectors with entries in D. We use the element y, which is either J-symmetric or J-skew, to define a form ⟨ , ⟩ on V:

⟨v, w⟩ := J(w^{t}) y v  (4.4.4)

In particular, if d_{1}, d_{2} ∈ D, we have

⟨v d_{1}, w d_{2}⟩ = J(d_{1}) ⟨v, w⟩ d_{2} .

Define ϵ ∈ {±1} by J(y^{t}) = ϵy. The form ⟨ , ⟩ is ϵ-hermitian, i.e.

J(⟨v, w⟩) = ϵ⟨w, v⟩ .  (4.4.5)

In terms of the form ⟨ , ⟩, the group G^{τ} has the description

\begin{aligned}G^{τ} & = \{ a ∈ G | τ(a) = a \} \\ & = \{ a ∈ G | ⟨av, aw⟩ = ⟨v, w⟩ ∀ v, w ∈ V \}. \end{aligned}  (4.4.6)

The two groups G and G^{τ} are the k-rational points of two reductive k-groups Γ and Γ^{τ}. Their explicit description as functors is the following: if K is an extension field of k then

Γ(K) = (A ⊗_{k} K)^{×}  (4.4.7)

and

Γ^{τ} (K) = { g ∈ Γ(K) | τ(g) = g }.  (4.4.8)

The k-group Γ (resp. Γ^{τ}) is of course a general linear (resp. classical) group. Recall the reduced norm is a k-group homomorphism

N_{D/k} : Γ ⟶ GL(1).

Define G := ker ( N_{D/k}) and G^{τ} := ker ( N_{D/k}|_{Γ^{τ}} ). Then,

G(K) = { g ∈ Γ(K) | N_{D/k}(g) = 1 }  (4.4.9)

and

G^{τ} (K) = G(K) ∩ Γ^{τ} (K).  (4.4.10)

5. Involutions and affine buildings I

5.1. Our goal in this and the next section is to prove an analogue of Theorem 3.3.1 for affine buildings. Let k be a nonarchimedean local field with ring of integers 𝔬 and residue field 𝔣 of odd characteristic. Suppose G is a connected semisimple group defined over k. Set 𝒢 = G(k) and let ℬ = ℬ(G, k) denote the Bruhat-Tits building of G with respect to k.

We set some notation. Let r denote the k-rank of G. A face E of a (closed) chamber C of ℬ is the closure of a (r − 1)-facet of C. If x ∈ ℬ, denote by G_{x} the reductive 𝔣-group which is the reduction modulo 𝔭 (the prime ideal of 𝔣) of the 𝔬-group scheme associated to the parahoric subgroup fixing the point x. Let Star(x) denote the union of the (closed) chambers of ℬ containing the point x. If S is a maximal k-split torus, let A(S, k) denote the apartment of S. Like our previous situation for the Curtis-Lehrer-Tits building, we will often drop k in our notation. Thus, we shorten ℬ(G, k) to ℬ(G) and A(S, k) to A(S). Suppose A(S) is an apartment containing the point x. Then S determines a 𝔬-scheme whose reduction modulo 𝔭 is a maximal 𝔣-split torus S_{x} in G_{x}. Every maximal 𝔣-split torus in G_{x} is obtained as an S_{x}. If S and S′ are two maximal k-split tori with x ∈ A(S) and x ∈ A(S′), then S_{x} = S′_{x} precisely when A(S) ∩ Star(x) = A(S′) ∩ Star(x).

5.2. Let ℒ(x) denote the set of geodesic rays in Star(x) which begin at the point x. This set can be identified with the Curtis-Lehrer-Tits building of G_{x} as follows. Given l ∈ ℒ(x), let S be a maximal k-torus with l ⊂ A(S). The ray l determines a unique ray in Hom_{k}(GL(1), S) ⊗_{ℤ} ℝ ≅ Hom_{𝔣}(GL(1), S_{x}) ⊗_{ℤ} ℝ, i.e. a point θ_{S_{x}}(l) in ℒ(S_{x}, 𝔣) ⊂ ℒ(G_{x}, 𝔣). If S′ is another k-torus with l ⊂ A(S′), there exists g ∈ 𝒢_{x} fixing l so that gSg^{−1} = S′. That g fixes l means the image of g in G_{x}(𝔣) lies in P(l) and conjugates S_{x} to S′_{x}. So, θ_{S_{x}}(l) = θ_{S}′_{x}(l). We drop the subscript and merely write θ : ℒ(x) → ℒ(G_{x}, 𝔣). We leave it to the reader to show that θ is a bijection (see [T: §3.5.4]).

5.3. Suppose τ ∈Aut_{k}(G) is a nontrivial involution. The associated action of τ on ℬ is a polysimplicial isometry. It is elementary ℬ^{τ} is nonempty. The set ℬ^{τ} is also obviously convex and so it has a well-defined dimension. Let m denote its dimension. Suppose x ∈ ℬ^{τ}. The following are elementary:

i) τ acts on Star(x) and ℒ(x).

ii) The parahoric subgroup fixing x is stable under τ. iii) τ induces an involution of the 𝔣-group G_{x}. For ease of notation, we also denote this

induced involution by τ. iv) θ ∘ τ = τ ∘ θ.

Let rk_{𝔣}((G_{x})^{τ}) denote the (reductive) 𝔣-rank of (G_{x})^{τ}. The dimension of ℒ(G_{x}, 𝔣)^{τ} equals rk_{𝔣}((G_{x})^{τ}) − 1. Since we have seen ℒ(G_{x}, 𝔣)^{τ} can be identified with the rays in Star(x)^{τ} starting from x, we also have dim(ℒ(G_{x}, 𝔣)^{τ}) equals m − 1, and thus m =rk_{𝔣}((G_{x})^{τ}).

5.4. Proposition. Suppose x ∈ ℬ^{τ}. Then, there exists a τ-invariant maximal k-split torus S ⊂ G so that

i) x ∈ A(S)^{τ}.

ii) dim(A(S)^{τ}) = m.

5.5. Preliminary remarks on the proof of Proposition 5.4. We begin by outlining our proof. By Proposition 3.4.1, there exists a τ-invariant maximal 𝔣-split torus T_{0} in G_{x}, so that T_{0}^{τ} is a maximal 𝔣-split torus in (G_{x})^{τ}. In particular, dim_{𝔣}(T_{0}^{τ}) = m. Let T be a maximal k-split torus so that x ∈ A(T) and T_{0} = T_{x}. The torus T is not necessarily τ-invariant. However, the τ-invariance of T_{0} implies that A(T) ∩ Star(x) is τ-invariant. Furthermore, dim(A(T) ∩ Star(x))^{τ} = m.

If A is an apartment containing A(T, k) ∩ Star(x), set

A_{τ} := { x ∈ A | τ(x) ∈ A }.

It is elementary that A_{τ} is τ-invariant, convex and a union of (closed) chambers of A. To prove Proposition 5.4, our goal is to begin with the apartment A_{0} := A(T) and show there is a sequence of apartments A_{i} containing A(T) ∩ Star(x) with the property that the sets (A_{i})_{τ} increase and converge to a τ-invariant apartment A. The k-torus S corresponding to A satisfies the conclusions i) and ii) of Proposition 5.4.

If A is an apartment and E ⊂ A is a face of a chamber, let H_{E} denote the affine root hyperplane in A which contains E. Suppose E′ is another (possibly the same) face of H_{E}. The intersection of all apartments containing the two faces E and E′ is a convex (r − 1)-dimensional set which we denote by 𝒞(E, E′). Obviously, 𝒞(E, E′) is contained in H_{E}. Define 𝒦 := 𝒦(E, E′) to be the collection of the sets Ω satisfying the following properties:

i) Ω contains 𝒞(E, E′), ii) Ω is convex and a union of chambers, iii) Ω is minimal (under inclusion) among all sets satisfying i) and ii).

5.5.1. Lemma. Suppose H_{ψ} is an affine root hyperplane in an apartment A and E, E′ are two faces in H_{ψ}.

i) If D is a chamber with E as a face, then there exists Ω ∈ 𝒦(E, E′) such that D ⊂ Ω. ii) If Ω_{1}, Ω_{2} ∈ 𝒦(E, E′) share a chamber D, then Ω_{1} = Ω_{2}.

Proof. We first consider the situation when 𝒞(E, E′) and D all belong to the apartment A. The existence of an Ω ∈ 𝒦 reduces to the existence of a union of chambers in A which contains 𝒞(E, E′), is convex and is minimal under inclusions. In the latter situation, if we take Ω to be the intersection of all half apartments in A which contain D and 𝒞(E, E′), then clearly Ω is minimal under inclusions. This implies existence. Uniqueness of Ω is a consequence of the property that Ω must contain the union of all chambers (in A) whose interiors meet any geodesic segment with one endpoint in the interior of the chamber D and the other endpoint in 𝒞(E, E′). To treat the general situation, we observe that the affine root group 𝒰_{ψ} acts on the set 𝒦. Indeed, the 𝒰_{ψ} action on the chambers containing E has two orbits (one for each of the two chambers in A containing E). Parts i) and ii) follow. □

5.5.2. Corollary. Let p = char(k) ≠ 2. Then, |𝒦| has the form 1 + p^{t}.

Proof. Let C_{1} and C_{2} denote the two chambers of A containing the face E and let Ω(C_{1}) and Ω(C_{2}) be the elements of 𝒦 which contain C_{1} and C_{2} respectively. As stated in the proof of Lemma 5.5.1, the affine root group 𝒰_{ψ} acts on 𝒦 with two orbits – the orbits of Ω(C_{1}) and Ω(C_{2}). It is clear Ω(C_{1}) is a singleton orbit and the size of the 𝒰_{ψ}-orbit of Ω(C_{2}) is a power of p. The corollary follows. □

5.6. Completion of the proof of Proposition 5.4. We define the sequence A_{i} inductively. If at any stage (A_{i})_{τ} = A_{i}, i.e. the apartment A_{i} is τ-invariant, we take A to be A_{i} and we are done. Therefore, we shall always suppose (A_{i})_{τ} ≠ A_{i}. We can choose two chambers C, D ⊂ A_{i} so that:

i) F = C ∩ D is a face of C (hence D).

ii) C ⊂ (A_{i})_{τ}, D ⊈ (A_{i})_{τ}.

We descriptively refer to the face F as a boundary face of (A_{i})_{τ}. We can and do assume (for the purpose of causing convergence in the compact open topology) that the distance of the boundary face F to the point x is minimal for all boundary faces of (A_{i})_{τ}. Denote as H_{F} the affine root hyperplane in A_{i} determined by F. Let J denote the half apartment of A_{i}\H_{F} containing C. Our proof divides into the following two cases: Case i) H_{F} ≠ H_{τ(F)}. Case ii) H_{F} = H_{τ(F)}.

Case i) H_{F} ≠ H_{τ(F)}. Let J′ denote (open) half apartment of A_{i}\H_{τ(F)} containing τ(C). The set (A_{i})_{τ} is contained in J ∩ J′. Let ψ denote affine root with the property H_{τ(F)} = H_{ψ} and the property that the affine root group 𝒰_{ψ} fixes τ(C). The group 𝒰_{ψ} fixes the half apartment J′ and permutes the chambers in the interior of Star(τ(F))\τ(C) transitively. Choose u ∈ 𝒰_{ψ} so that τ(D) ⊂ uA_{i}. Our hypothesis that H_{F} ≠ H_{τ(F)} implies D and C, hence D and (A_{i})_{τ}, lie on the same side of H_{τ(F)}. We conclude from this that u fixes D and (A_{i})_{τ}. Set A_{i+1} := uA_{i}. Then (A_{i})_{τ} ∪ D ⊂ A_{i+1} and indeed (A_{i})_{τ} ∪ D ⊂ (A_{i+1})_{τ}.

Case ii) H_{F} = H_{τ(F)}. Since J contains (A_{i})_{τ} ⊃ C and (A_{i})_{τ} is τ-invariant, we have τ(C) ⊂ J. The two sets 𝒞(F, τ(F)) and 𝒦(F, τ(F)) are τ-invariant. Since we are assuming the residual characteristic of k is odd, we conclude from Corollary 5.5.2 that |𝒦(F, τ(F))| is even. The unique element of 𝒦(F, τ(F)) containing C is obviously τ-invariant; hence there must also exists Ω ∈ 𝒦(F, τ(F)) which is τ-invariant and does not contain C. Choose u ∈ 𝒰_{ψ} so that uD ⊂ Ω and set A_{i+1} = uA_{i}. Then, (A_{i+1})_{τ} ⊃ (A_{i})_{τ} ∪ Ω. This completes our induction step.

As already mentioned, in our construction of the sequence A_{i}, at each stage, if there exists boundary faces on (A_{i})_{τ} we choose a boundary face F whose distance to x is minimal among boundary faces. Then, F ⊂ (A_{i+1})_{τ} and is no longer a boundary face. It follows the sequence of apartments A_{i} converge in the compact open topology to an apartment A which is τ-invariant. Hence the maximal k-split torus S associated to A is τ-invariant, and clearly i) x ∈ A and ii) dim(A^{τ}) = m. □

5.7. Corollary.

i) Suppose G is a reductive k-group, τ ∈ Aut_{k}(G) is an involution and T is a τ-invariant k-split torus. Then, there exists a τ-invariant maximal k-split torus S containing T.

ii) Suppose G is a semisimple k-group, τ ∈ Aut_{k}(G) is an involution and G′ := (G^{τ})^{∘}. Then, dim(ℬ^{τ}) = rk_{k}(G′).

Proof. To prove i), let L := L(T) denote the Levi k-subgroup which is the centralizer of T, and let Z := Z(L) denote the center of L. The quotient M := L/Z is a (semisimple) k-group and τ induces a k-involution in Aut_{k}(M), which we shall for ease of notation also denote by τ. By Proposition 5.4, there is a τ-invariant maximal k-split torus \overline{S} in M. Let C denote the preimage in L of \overline{S}. We can take S to be the maximal k-split torus of C.

To prove ii), suppose T is a maximal k-split torus of G^{τ}. By i), there exists a τ-invariant maximal k-split torus S of G containing T. We have

dim(ℬ^{τ}) ≥ dim(A(S)^{τ}) ≥ dim(T) = rk_{k}(G′).

Also, by Proposition 5.4, there exists a τ-invariant maximal k-split torus V so that

dim(ℬ^{τ}) = dim A(V)^{τ} = dim_{k}(V^{τ}) ≤ rk_{k}(G′)

So dim(ℬ^{τ}) = rk_{k}(G′). □

6. Involutions and affine buildings II

6.1. We continue with the assumptions of section 5: k is a nonarchimedean local field with ring of integers 𝔬 and residue field 𝔣 of odd characteristic, G is a connected semisimple group defined over k and τ ∈ Aut_{k}(G) is a nontrivial involution. Let G′ denote the connected reductive group (G^{τ})^{∘} and set 𝒢′ := G′(k) ⊂ 𝒢 = G(k). If ℋ is a subgroup of 𝒢, let 𝒞_{𝒢} (ℋ) and 𝒩_{𝒢} (ℋ) denote the centralizer and normalizer of ℋ respectively. We use similar notation for subgroups of 𝒢^{τ} and 𝒢′.

The main result in this section is Theorem 6.7.3. It states under the assumption G and τ ∈ Aut_{k}(G) are as in section 4 and G′ := (G^{τ})^{∘} is semisimple that ℬ′ := ℬ(G′, k) = ℬ(G′) identifies naturally with ℬ(G)^{τ}. We outline our plan to accomplish this. i) Suppose T is a maximal k-split torus of G′. Set 𝒯 := T(k). Let A(T, G′) denote the

apartment in ℬ(G′) associated to T. We find a canonical affine subspace A(T, G) of ℬ^{τ} which we eventually identify with the apartment A(T, G′).

ii) We show the restriction of the action of 𝒢 on ℬ to the group 𝒩_{𝒢′} (𝒯) ⊂ 𝒢 yields an action on the affine subspace A(T, G) which is equivalent to the action of 𝒩_{𝒢′} (𝒯) on A(T, G′) coming from the action of 𝒢′ on ℬ′.

iii) Suppose x ∈ A(T, G) and S is a τ-stable maximal k-split torus of G containing T. Set 𝒮 := S(k). Let 𝒱_{x} (resp. 𝒰_{x}) denote the subgroup generated by the affine 𝒯root groups (resp. affine 𝒮-root groups) of 𝒢′ (resp. 𝒢) fixing the point x. We show 𝒱_{x} ⊂ 𝒰_{x} ∩ 𝒢′.

We then identify ℬ^{τ} with ℬ′ so that the restriction action of 𝒢′ ⊂ 𝒢 on ℬ^{τ} is equivalent to its action on ℬ′.

6.2. Although we shall eventually assume G and τ ∈ Aut_{k}(G) are as in section 4, our preliminary results (Propositions 6.2.1 and 6.3.1) hold in the generality of G an arbitrary semisimple k-group and τ ∈ Aut_{k}(G) an arbitrary involution.

6.2.1. Proposition. Suppose T is a maximal k-split torus of G′.

i) There exists a τ-invariant maximal k-split torus S of G containing T.

ii) If S_{1} and S_{2} are two τ-invariant maximal k-split tori of G containing T, then

A(S_{1})^{τ} = A(S_{2})^{τ} .

Proof. Assertion i) is Corollary 5.7 i). To prove assertion ii), we have by Corollary 5.7 ii) that dim(ℬ(G)^{τ}) = rk_{k}(G′) = dim(A(S_{i})^{τ}). Let L denote the Levi k-subgroup C_{G}(T). Set

ℬ_{L} := union of the apartments A(S) ⊂ ℬ(G) with S containing T.

It is obvious ℬ_{L} is convex, τ-stable. The dimension of the convex set (ℬ_{L})^{τ} satisfies

dim(ℬ(G)^{τ}) ≥ dim((ℬ_{L})^{τ}) ≥ rk_{k}(G′) = dim(ℬ(G)^{τ}),

so dim((ℬ_{L})^{τ}) = rk_{k}(G′). The vector space V (T) := X_{∗}(T) ⊗_{ℤ} ℝ acts naturally on (ℬ_{L})^{τ}. Let (ℬ_{L})^{τ} /V (T) denote the quotient space. We have

dim((ℬ_{L})^{τ}) = rk_{k}(T) + dim((ℬ_{L})^{τ} /V (T)).  (6.2.2)

So, dim((ℬ_{L})^{τ} /V (T)) = 0 and therefore the convex set (ℬ_{L})^{τ} /V (T) must be a singleton point. This in particular means (A(S_{1}))^{τ} = (A(S_{2}))^{τ} □

As a consequence of this proposition, any maximal k-split torus T of G′ determines a canonical affine subspace of ℬ(G)^{τ} of dimension rk_{k}(G′). We denote this set as A(T, G).

6.3. The maximal k-split tori of G^{τ} and G′ := (G^{τ})^{∘} obviously coincide. Suppose T is such a maximal k-split torus of G′ and 𝒯 := T(k). Let 𝒩_{𝒢} (𝒯) (resp. 𝒞_{𝒢} (𝒯)) denote the normalizer (resp. centralizer) of 𝒯 in 𝒢. All these groups are τ-stable. The group 𝒩_{𝒢} (𝒯) acts on the space ℬ_{L} of (6.2.2); thus 𝒩_{𝒢} (𝒯)^{τ} = 𝒩_{𝒢^{τ}} (𝒯) acts on A(T, G). The groups 𝒞_{𝒢^{τ}} (𝒯) = 𝒞_{𝒢} (𝒯)^{τ} and 𝒞_{𝒢′} (𝒯) ⊂ 𝒞_{𝒢^{τ}} (𝒯) are compact modulo 𝒯 and so they have unique maximal bounded subgroups (𝒞_{𝒢^{τ}} (𝒯))^{b} and (𝒞_{𝒢′} (𝒯))^{b} respectively.

6.3.1. Proposition. Suppose T is a maximal k-split torus of G′. The maximal bounded subgroup (𝒞_{𝒢} (𝒯)^{τ})^{b} acts trivially on A(T, G).

Proof. The compact group (𝒞_{𝒢} (𝒯)^{τ})^{b} acts isometrically on the Euclidean space A(T, G); hence there is a fixed point x ∈ A(T, G). Any s ∈ 𝒩 = 𝒩_{𝒢} (𝒯)^{τ} normalizes (𝒞_{𝒢} (𝒯)^{τ})^{b} and hence (𝒞_{𝒢} (𝒯)^{τ})^{b} must fix sx. This means (𝒞_{𝒢} (𝒯)^{τ})^{b} must fix all the points 𝒩x and in particular all the points 𝒯x. But 𝒯 acts on A(T, G) cocompactly, so we conclude (𝒞_{𝒢} (𝒯)^{τ})^{b} must acts trivially on the points of a cocompact lattice of A(T, G) and thus (𝒞_{𝒢} (𝒯)^{τ})^{b} must fix A(T, G). □

6.4. It follows immediately from Proposition 6.3.1 that the action of 𝒩_{𝒢} (𝒯)^{τ} on A(T, G) factors to an action of 𝒩_{𝒢} (𝒯)^{τ} /(𝒞_{𝒢} (𝒯)^{τ})^{b} on A(T, G). Our next step is to establish the compatibility of the action of 𝒩_{𝒢} (𝒯)^{τ} on A(T, G) with the action of 𝒩_{𝒢′} (𝒯) on A(T, G′). Due to present limitations in our proof, we shall only be able to accomplish this when G is a special linear group and τ ∈ Aut_{k}(G) is an involution as in section 4.

As a first step in this direction, we consider the action of 𝒞_{𝒢′} (𝒯) on the affine spaces A(T, G′) and A(T, G). Define the homomorphism ν : 𝒞_{𝒢′} (𝒯) → X_{∗}(T)⊗_{ℤ}ℝ ⊂ X_{∗}(S)⊗_{ℤ}ℝ as in [SS: §1.1]. Then, the action of an element v ∈ 𝒯 on both A(T, G′) and A(T, G) ⊂ A(S, G) is a translation

tx = x + ν(v).

6.4.1. Suppose now that Γ, G and τ ∈ Aut_{k}(Γ) are as defined in section 4, more specifically by (4.4.8) and (4.4.2). Recall also G = Γ(k) = GL_{D}(V) where V = D^{n}. Let ⟨ , ⟩ be the form (4.4.4). Set G′ := (G^{τ})^{∘}.

6.4.2 Lemma. In the notation of (6.4.1), suppose T is a maximal k-split torus in G′ and suppose S̃ is a τ-stable maximal k-split torus in Γ containing T. Let S̃ be S̃(k). Then

(𝒩_{G^{τ}} (𝒯) ∩ 𝒩_{G}(S̃)) 𝒞_{G^{τ}} (𝒯)^{b} = 𝒩_{G^{τ}} (𝒯).  (6.4.3)

Proof. Let T and S̃ be as in the lemma. Let m be the Witt index of ⟨ , ⟩ and m_{0} = n−2m. By the Witt basis theorem, we have a basis {e_{i}, f_{j} | i = ±1, ⋯ ± m, j = 1, ⋯ m_{0}} which consists of eigenspaces of 𝒯 with ⟨e_{i}, e_{−j}⟩ = δ_{ij} = ϵ⟨e_{−j}, e_{i}⟩ for i > 0, ⟨e_{i}, f_{j}⟩ = 0, and ⟨f_{i}, f_{j}⟩ = 0 for i ≠ j. Let V_{s} be a subspace of V generated by {e_{i} | i = ±1, ⋯, ±m} over D and let V_{0} be generated by {f_{j} | j = 1, ⋯, m_{0}}. Then V = V_{s} ⊕ V_{0} is an orthogonal decomposition with respect to ⟨ , ⟩. We can naturally identify GL_{D}(V_{s}), GL_{D}(V_{0}) and GL_{D}(V_{s}) × GL(V_{0}) as subgroups of G = GL_{D}(V). Let G(V_{s}) = G^{τ} ∩ GL_{D}(V_{s}) and G(V_{0}) = G^{τ} ∩ GL_{D}(V_{0}). Then we have a natural embedding of G(V_{s}) × G(V_{0}) into G^{τ} and of GL_{D}(V_{s}) × GL_{D}(V_{0}) into G. Via these embeddings, we have 𝒯 ⊂ G(V_{s}) × G(V_{0}) and S̃ ⊂ GL_{D}(V_{s})×GL_{D}(V_{0}). Moreover, 𝒩_{G^{τ}} (𝒯) can be decomposed into a product of 𝒩_{G^{τ}} (𝒯)∩ G(V_{s}) and G(V_{0}). In (6.4.3), ⊂ is obvious. Since (𝒩_{G^{τ}} (𝒯) ∩ 𝒩_{G}(S̃)) ⊃ (𝒩_{G^{τ}} (𝒯) ∩ G(V_{s})) and 𝒞_{G^{τ}} (𝒯)^{b} ⊃ G(V_{0}), ⊃ also follows. □

Since 𝒞_{𝒢′} (𝒯)^{b} is not contained in 𝒞_{𝒢} (𝒮)^{b} in general, it is not obvious that there is a well defined monomorphism 𝒩_{𝒢′} (𝒯)/𝒞_{𝒢′} (𝒯)^{b} ⟶ 𝒩_{𝒢} (𝒮)/𝒞_{𝒢} (𝒮)^{b}. In the following corollary, we define such a map by Lemma 6.4.2. Note that there are four different groups 𝒢′, 𝒢, G^{τ} and G involved in the proof.

6.4.4. Corollary. We keep the notation from (6.4.1) and let S = S̃ ∩ G. If n𝒞_{𝒢′} (𝒯)^{b} is a coset of 𝒞_{𝒢′} (𝒯)^{b} in 𝒩_{𝒢′} (𝒯), let n′ ∈ n𝒞_{𝒢′} (𝒯)^{b} ∩ 𝒩_{𝒢} (𝒮). The element n′ is unique modulo the subgroup 𝒞_{𝒢} (𝒮)^{b} and the map

\begin{matrix}ϕ : 𝒩_{𝒢′} (𝒯)/𝒞_{𝒢′} (𝒯)^{b} & ⟶ 𝒩_{𝒢} (𝒮)/𝒞_{𝒢} (𝒮)^{b} \\ ϕ(nC_{𝒢′} (𝒯)^{b}) & = n′C_{𝒢} (𝒮)^{b} \end{matrix}

is a monomorphism of 𝒩_{𝒢′} (𝒯)/𝒞_{𝒢′} (𝒯)^{b} into 𝒩_{𝒢} (𝒮)/𝒞_{𝒢} (𝒮)^{b}.

Proof. Observe that 𝒩_{G}(𝒮) = 𝒩_{G}(S̃) and that there are natural monomorphisms

\begin{matrix}ϕ_{1} : 𝒩_{𝒢′} (𝒯)/𝒞_{𝒢′} (𝒯)^{b} ⟶ 𝒩_{G^{τ}} (𝒯)/𝒞_{G^{τ}} (𝒯)^{b}, \\ ϕ_{2} : 𝒩_{𝒢} (𝒮)/𝒞_{𝒢} (𝒮)^{b} ⟶ 𝒩_{G}(S̃)/𝒞_{G}(S̃)^{b}. \end{matrix}

By the above lemma, we can also define a monomorphism

ϕ_{3} : 𝒩_{G^{τ}} (𝒯)/𝒞_{G^{τ}} (𝒯)^{b} ⟶ 𝒩_{G}(S̃)/𝒞_{G}(S̃)^{b}

such that for m𝒞_{G^{τ}} (𝒯)^{b}, a coset of 𝒞_{G^{τ}} (𝒯)^{b} in 𝒩_{G^{τ}} (𝒯) and m′ ∈ m𝒞_{G^{τ}} (𝒯)^{b} ∩ 𝒩_{G}(S̃), ϕ_{3}(m𝒞_{G^{τ}} (𝒯)^{b}) = m′𝒞_{G}(S̃)^{b}. Then the image of ϕ_{3} ∘ ϕ_{1} factors through 𝒩_{𝒢} (𝒮)/𝒞_{𝒢} (𝒮)^{b}, that is, we have ϕ_{3} ∘ ϕ_{1} = ϕ_{2} ∘ ϕ and the corollary is proved. □

6.5. We introduce some notation. Suppose T is a maximal k-split torus in G′. If Y is a subset of the apartment A(T, G′), and w ∈ X_{∗}(T) ⊗_{ℤ} ℝ, set

\begin{aligned}Y + w & := \{ x + w | x ∈ Y \} \\ Y (w) & := \{ x + tw | x ∈ Y and t ≥ 0\} \end{aligned}

If ψ is a nonconstant affine root on A(T, G′) with gradient α ∈ Φ(G′, T) and H_{ψ} is the vanishing hyperplane of ψ, then H_{ψ}(α) is a half space in A(T, G′). We use similar notation for G.

Suppose α ∈ Φ(G′, T) and u is a nonidentity element of the α root group 𝒱_{α} ⊂ 𝒢′. The unique element in 𝒱_{−α}u𝒱_{−α} ∩ 𝒩_{𝒢′} (𝒯) is expressible [BT:§4.1.5.2] as m(u) = vuv, where v ∈ 𝒱_{−α}. The element m(u) acts on A(T, G′) as a reflection across a hyperplane H = H_{u}. The elements v and u, as elements in 𝒢′, fix the half-spaces H(−α) and H(α) of A(T, G′) respectively. We compare this to the situation where we view v and u as elements of 𝒢 acting on ℬ(G). It follows from the Witt basis in Lemma 6.4.2 and Corollary 6.4.4, that m(u) is reflection across a hyperplane h_{u} in A(T, G). We intend to identify A(T, G) and A(T, G′) so that the hyperplane h_{u} is identified with the hyperplane H = H_{u}. As a step towards this goal, we now show that in A(T, G) the elements v and u fix the affine half-space h_{u}(−α) and h_{u}(α) respectively.

Choose a τ-invariant maximal k-split torus S in G containing T. When viewed as an element in 𝒢, u is a product of elements in affine root groups 𝒰_{ϕ} (of G with respect to S) satisfying

grad(ϕ|_{A(T,G)}) is a multiple of α.  (6.5.1)

Suppose ϕ satisfies (6.5.1). Then, the affine root group 𝒰_{ϕ} fixes the half-space H_{ϕ}(α) and so fixes the affine half-subspace h_{u}(α) + Rα provided R ≥ 0 is sufficiently large. Since u lies in a product of affine root groups satisfying (6.5.1), it follows there exists an R_{u} ≥ 0 so that u fixes the affine half-subspace h_{u}(α) + R_{u}α. This in combination with convexity of the u-fixed point A(T, G)^{u} implies either A(T, G)^{u} equals A(T, G) or is a half-space. We shall see soon that A(T, G)^{u} is a half-space.

In a similar fashion, there is an R_{v} ≥ 0 so that v fixes the affine half-subspaces h_{u}(−α)− R_{v}α, and the fixed point set is either a half-space or all of A(T, G). Suppose either A(T, G)^{u} or A(T, G)^{v} is all of A(T, G). From m(u) = vuv, we conclude that m(u) must fix a half-space of A(T, G). This is clearly a contradiction to the action of m(u) being a reflection across the hyperplane h_{u}. Hence u and v fix half-spaces in A(T, G). Let b(u) and b(v) denote the boundary hyperplanes of the two half-spaces A(T, G)^{u} and A(T, G)^{v}. Heuristically, the action of u on A(T, G) is that it fixes pointwise the half-space A(T, G)^{u} and ‘folds along the hyperplane b(u)’ the closure of the complementary half-space ‘away’ from A(T, G). The situation for v is similar.

If the intersection of the two half-spaces A(T, G)^{u} and A(T, G)^{v} contains an open neighborhood U, then U must be fixed by the reflection m(u), an impossibility. Hence, either the intersection of the two half-spaces is empty or a hyperplane, which subsequently must be h_{u}. To rule out the intersection being empty, we argue by contradiction. Assume the intersection is empty. Choose a point x in the hyperplane b(v). Then, the empty intersection hypothesis implies

i) A(T, G)^{u} is contained in A(T, G) − A(T, G)^{v}, ii) the geodesic segment [x, ux] meets A(T, G)^{v} precisely at the point x, iii) there exists y in the open interval (x, ux) so that (x, y] lies in A(T, G) − A(T, G)^{v}.

Now, v[x, ux] = [vx, m(u)x] lies in A(T, G) since its two endpoints lie in A(T, G). We conclude that the geodesic interval [x, y] is sent by v to another geodesic interval inside A(T, G). But, since [x, y] meets A(T, G)^{v} in the point x, and v folds the half-space which is the closure of A(T, G) − A(T, G)^{u} off A(T, G) along the hyperplane b(u), it is impossible for v[x, y] to be in A(T, G). This is a contradiction. We conclude that v and u must fix a common point in A(T, G), from which we deduce that b(u) = b(v) = h_{u} is the common fixed set of u and v in A(T, G). Thus, v fixes the affine half-subspace h_{u}(−α), and u fixes h_{u}(α).

6.6. Let ι_{α} : A(T, G′) → A(T, G) be any isometry, i.e. a 𝒞_{𝒢′} (𝒯)-equivariant map, which takes h_{u} to H_{u}. The map ι_{α} is determined up to a translation by a vector parallel to H_{u}. Our discussion implies

i) ι_{α} commutes with the actions of m(u) on A(T, G′) and A(T, G).

ii) The element v fixes the affine half-subspace H_{u}(−α) = ι_{α}(h_{u}(−α)), and u fixes H_{u}(α) = ι_{α}(h_{u}(α)).

The above discussion is for a single root α ∈ Φ(G′, T). If we consider all roots, the assumption that G′ is semisimple, i.e. X_{∗}(T) ⊗_{ℤ} ℝ is spanned by the simple roots, implies there is a unique 𝒞_{𝒢′} (𝒯)-equivariant map ι : A(T, G′) → A(T, G) so that

i) For any root α ∈ Φ(G′, T) and u ∈ 𝒱_{α}\{1}, the map ι takes h_{u} to H_{u} and ι commutes with the actions of m(u) = vuv on A(T, G) and A(T, G′).

ii) The element v fixes the affine half-subspace H_{u}(−α) = ι(h_{u}(−α)), while u fixes H_{u}(α) = ι(h_{u}(α)).

For x ∈ A(T, G′), set

𝒱_{x} := subgroup of 𝒢′ generated by all the affine root groups

𝒱_{ψ} with respect to 𝒯 which fix the point x

and

𝒰_{ι(x)} := subgroup of 𝒢 generated by all the affine root groups

𝒰_{ϕ} with respect to 𝒮 which fix the point ι(x)

From the above, we see

𝒱_{x} ⊂ 𝒰_{ι(x)}.

6.7. We are now ready to establish the existence of an injective map π : ℬ(G′) → ℬ(G)^{τ}. To do this, we choose a maximal k-split torus T of G′ and a τ-stable maximal k-split torus S of G which contains T. Recall that any point z ∈ ℬ(G′) is expressible as z = gx, where x ∈ 𝒢′ and x ∈ A(T, G′). This expression is not unique, but if z = hy is a second representation of z, with h ∈ G′ and y ∈ A(T, G′), then there exists an n ∈ 𝒩_{𝒢′} (𝒯) so that i) y = nx, and ii) g^{−1}hn ∈ 𝒱_{x}. Define π : ℬ(G′) → ℬ(G)^{τ} as follows: If z is expressible as z = gx, set

π(z) = gι(x)  (6.7.1)

Here, ι(x) ∈ A(T, G) and the action of g on ι(x) is by viewing g as being in 𝒢.

6.7.2. Proposition. Let π be defined as in (6.7.1).

i) π is a well defined map.

ii) π is an injection.

Proof. To prove i), suppose z = gx and z = hy are two expressions for the point z, i.e. g, h ∈ 𝒢′ and x, y ∈ A(T, G′). Choose n′ ∈ 𝒩_{𝒢′} (𝒯) so that y = n′x, and g^{−1}hn ∈ 𝒱_{x}. According to Lemma 6.4.2 and Corollary 6.4.4, we can choose n ∈ n′𝒞_{𝒢′} (𝒯)^{b} so that ι(y) = nι(x). Since 𝒱_{x} ⊂ 𝒰_{ι(x)}, we have g^{−1}hn ∈ 𝒰_{ι(x)}. So, ι(x) = g^{−1}hnι(x), i.e. gι(x) = hnι(x) = hι(y). Hence, π is well-defined.

To prove ii), suppose z_{1} = gx and z_{2} = hy map to the same point under π, i.e. gι(x) = hι(y). The Bruhat decomposition 𝒢′ = 𝒱_{x}𝒩_{𝒢′} (𝒯)𝒱_{y}, allows us to express the element g^{−1}h as

g^{−1}h = u_{x}n^{−1}u_{y} with u_{x} ∈ 𝒱_{x} ⊂ 𝒰_{ι(x)}, u_{y} ∈ 𝒱_{y} ⊂ 𝒰_{ι(y)} and n ∈ 𝒩_{𝒢′} (𝒯).

Upon substitution, the equality ι(x) = g^{−1}hι(y) becomes ι(x) = u_{x}n^{−1}u_{y}ι(y) from which we conclude that y = nx and g^{−1}hn = u_{x}n^{−1}u_{y}n ∈ 𝒱_{x}. So z_{1} = gx = hy = z_{2}, and therefore π is injective. □

6.7.3. Theorem. Suppose k is a nonarchimedean local field with odd residual characteristic. Suppose G and τ ∈ Aut_{k}(G) are as in (4.4.10) and (4.4.2) and G′ := (G^{τ})^{∘} is semisimple. Then π is a G′(k)-equivariant isometry of ℬ(G′) onto ℬ(G)^{τ}.

Proof. Normalize distance on A(T, G) so that the map π|_{A(T,G′)} : A(T, G′) → A(T, G) is an isometry. For g ∈ 𝒢′, and z ∈ ℬ(𝒢′), we have gπ(z) = π(gz). Given any two points z_{1}, z_{2} ∈ ℬ(𝒢′), there exist points x_{1}, x_{2} ∈ A(T, G′) and a g ∈ 𝒢′ so that z_{i} = gx_{i}, thus π(z_{i}) = π(gx_{i}) = gπ(x_{i}). Then, dist(π(z_{1}), π(z_{2})) = dist(π(x_{1}), π(x_{2})) = dist(x_{1}, x_{2}) = dist(z_{1}, z_{2}). So π is an isometry. The final assertion that π is onto is Proposition 5.4 and Corollary 5.7. □

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