Buildings of classical groups and centralizers of Lie algebra elements

P. Broussous and S. Stevens

Février 2004

Abstract

Let F_{o} be a non-archimedean locally compact field of residual characteristic not 2. Let G be a classical group over F_{o} (with no quaternionic algebra involved) which is not of type A_{n} for n > 1. Let β be an element of the Lie algebra 𝔤 of G that we assume semisimple for simplicity. Let H be the centralizer of β in G and 𝔥 its Lie algebra. Let I and I_{β}^{1} denote the (enlarged) Bruhat-Tits buildings of G and H respectively. We prove that there is a natural set of maps j_{β} : I_{β}^{1} → I which enjoy the following properties: they are affine, H-equivariant, map any apartment of I_{β}^{1} into an apartment of I and are compatible with the Lie algebra filtrations of 𝔤 and 𝔥. In a particular case, where this set is reduced to one element, we prove that j_{β} is characterized by the last property in the list. We also prove a similar characterization result for the general linear group.

Introduction

In this paper we establish new functoriality properties between affine Bruhat-Tits buildings of classical reductive groups over local fields. More precisely let F_{o} be a nonarchimedean local field of residual characteristic not 2 and G be the group of F_{o}-rational points of a classical group defined over F_{o}. We assume that G is the isometry group of an ε-hermitian form over an F -vector space, where F is a (commutative) extension of F_{o} of degree less than 2. We denote by 𝔤 the Lie algebra of G and by I its affine building. Let β be an element of 𝔤 that we assume to be semisimple for simplicity. Let H be the centralizer of β in G. Then H is the group of F_{o}-rational points of a product of groups of the form Res_{E_{o}/F_{o}}H_{i}, where E_{o}/F_{o} is a field extension and where Res denotes the functor of restriction of scalars. Here i runs over a finite set J̃. Each H_{i} is either a classical group as above or a general linear group. We denote by J_{+} ⊂ J̃ the (possibly empty) subset of indices corresponding to linear groups. We denote by 𝔥 the Lie algebra of H and by I_{β}^{1} its (enlarged) affine building. Then there is a natural set of maps j_{β} : I_{β}^{1} → I which depend on identifications of the enlarged buildings of H_{i}, i ∈ J_{+}, with certain sets of lattice functions (see §4 below). In particular, when J_{+} = ∅, there is a natural choice of j_{β}. The maps j_{β} enjoys the following properties:

a) They are affine.

b) They are H-equivariant.

c) They map any apartment of I_{β}^{1} into an apartment of I.

d) They are compatible with the Lie algebra filtrations of 𝔤 and 𝔥 (cf. §9). In [BL] it was proved that when G is the general linear group and β is an elliptic

element then, replacing the buildings by the non-enlarged buildings, there is such a natural map j_{β} satisfying the conditions above. It is actually characterized by properties a) and b). However in the case of a classical group (and assuming that J_{+} = ∅) it is no longer true that properties a), b) and c) characterize j_{β}. The simplest counter-example is the following. Consider the case of G = Sp_{2}(F_{o}) = SL(2, F_{o}). One may choose β is such a way that H is E^{1}, the group of norm 1 elements of a ramified quadratic extension E/F_{o}. Then I_{β}^{1} is reduced to a point and fixing a map j_{β} satisfying a) b) and c) amounts to choosing a point in I fixed by the torus E^{1}. But E^{1} is contained in an Iwahori subgroup of G and therefore fixes a chamber of I.

We prove that, in the case of a general linear group and of an elliptic element β, the map j_{β} of [BL] is actually characterized by property d). In the case of a classical group, we also prove that if J_{+} = ∅ and if a technical condition on β is satisfied then j_{β} is characterized by condition d). We conjecture that when J_{+} = ∅ then j_{β} is indeed characterized by property d).

In this work, we do not actually assume β to be semisimple but only to satisfy a weaker assumption (see hypothesis (H1) of §5). Such elements naturally appear in the generalization of the theory of strata due to Bushnell and Kutzko [BK] to the case of classical groups (see the work of the second author [S1], [S2]. Even though the work of the second author does not use the theory of affine buildings in a straightforward way (it uses the equivalent language of hereditary orders), the existence and properties of the maps j_{β} are applied to the representation theory of G, particularly in [S2].

The paper is organized as follows. In §2 we recall the structure of the maximal split tori of G. In §3,4, using ideas of Bruhat and Tits, we give a model of the affine building of G in terms of “self-dual lattice functions”. In §5 we study the centralizers in 𝔤 and G of the Lie algebra element β. The construction of the maps j_{β} is done in §6 and their properties are established in §7,8 and 9. In §10 We prove the uniqueness result for the general linear group and finally §11 is devoted to the uniqueness result in the classical group case.

We thank G. Henniart, B. Lemaire, G. Prasad and J.-K. Yu for stimulating discussions.

1. Notation Here F_{o} is the ground field; it is assumed to be non-archimedean, locally compact

and equipped with a discrete valuation v normalized in such a way that v(F_{o}^{×}) is the additive group of integers. We assume that the residual characteristic of F_{o} is not 2. We fix a Galois extension F/F_{o} such that [F : F_{o}] ⩽ 2 and set σ_{F} = id_{F} if F = F_{o} and take σ_{F} to be the generator of Gal(F/F_{o}) in the other case. We still denote by v the unique extension of v to F. We fix ε ∈ {±1} and a finite dimensional left F -vector space V. Recall that a σ_{F} -skew form h on V is a ℤ-bilinear map V × V → F such that

h(λx, μy) = λ^{σ_{F}} μh(x, y) , λ, μ ∈ F, x, y ∈ V .

Such a form is called ε-hermitian if h(y, x) = εh(x, y)^{σ_{F}} for all x, y ∈ V. From now on we fix such an ε-hermitian form on V and we assume it is non-degenerate (the orthogonal of V is {0}).

For a ∈ End_{F} (V), we denote by a^{σ_{h}} = a^{σ} the adjoint of a with respect to h, i.e. the unique F -endomorphism of V satisfying h(ax, y) = h(x, a^{σ}y) for all x, y ∈ V.

We denote by G the simple algebraic F_{o}-group whose set of F_{o}-rational points G is formed of the g ∈ GL_{F} (V) satisfying g.h = h (it is not necessarily connected). Here g.h is the form given by g.h(x, y) = h(gx, gy), x, y ∈ V.

We know ([Sch](6.6), page 260) that in the case σ_{F} ≠ id_{F}, we may reduce to the case ε = 1. So we have three possibilities:

σ_{F} = id_{F} and ε = 1, the orthogonal case;

σ_{F} = id_{F} and ε = −1, the symplectic case;

σ_{F} ≠ id_{F} and ε = 1, the unitary case.

We abbreviate G̃ = GL_{F} (V) and 𝔤̃ = End_{F} (V).

2. The maximal split tori of G Recall that a subspace W ⊂ V is totally isotropic if h(W, W) = 0 and that maximal

such subspaces have the same dimension r, the Witt index of h. Set I = {±1, ±2, …, ±r} and I_{o} = {(0, k) ; k = 1, …, n − 2r}. We fix a Witt decomposition of V, that is two maximal totally isotropic subspaces V_{+} and V_{−}, bases (e_{i})_{i=1,...,r}, (e_{−i})_{i=1,...,r}, (e_{i})_{i∈I_{o}} of V_{+}, V_{−} and V_{o} := (V_{+} + V_{−})^{⊥}, such that h(e_{i}, e_{i}) = 0, i ∈ I, h(e_{i}, e_{j}) = 0, for i, j ∈ I with j ≠ −i or i ∈ I, j ∈ I_{o}, h(e_{i}, e_{−i}) = 1, for i ∈ I with i > 0, h(x, x) ≠ 0, for x ∈ V_{o} and x ≠ 0.

The Witt decomposition gives rise to a maximal F_{o}-split torus S whose group of F_{o}-rational points is

S = {s ∈ G ; se_{i} ∈ F_{o}e_{i} , i ∈ I and (s − Id)V_{o} = 0} .

It has dimension r, the F_{o}-rank of G. Conversely any maximal F_{o}-split torus of G is obtained from a Witt decomposition as above. The centralizer Z of S in G has for F_{o}-rational points

Z = {z ∈ G ; ze_{i} ∈ F e_{i} , i ∈ I and zV_{o} = V_{o}} .

For each i ∈ I, we have a morphism of algebraic groups a_{i} : Z → Res_{F/F_{o}}(G_{m}) given by ze_{i} = a_{i}(z)e_{i}. Note that a_{−i}(z) = a_{i}(z)^{−σ}. We also denote by a_{i} : S → G_{m /F_{o}} the character obtained by restriction. We have a_{i} = −a_{−i} in X^{∗}(S), the ℤ-module of rational characters of S. The a_{i}, i ∈ I, i > 0, form a basis of X^{∗}(S).

The normalizer N of Z in G is the sub-algebraic group whose F_{o}-rational points are the elements of G which stabilize X_{o} and permute the lines V_{i} = F e_{i}, i ∈ I. The group N = N(F_{o}) is the semidirect product of Z by the subgroup N′ formed of the elements which permute the ±e_{i}, i ∈ I.

3. MM-norms and self-dual lattice-functions

We keep the notation as in the previous sections.

Recall that a norm on V is a map α : V → ℝ ∪ {∞} satisfying:

i) α(x + y) ⩾ Inf(α(x), α(y)), x, y ∈ V, ii) α(λx) = v(λ) + α(x), λ ∈ F, x ∈ V, iii) α(x) = ∞ if and only if x = 0. We denote by Norm^{1}(V) the set of norms on V.

Definition 3.1. (cf. [BT](2.1)) Let α ∈ Norm^{1}(V). We say that α is dominated by h if

α(x) + α(y) ⩽ v(h(x, y)) for all x, y ∈ V .

We say that α is an MM-norm for h ( maximinorante in french), if α is a maximal element of the set of norms dominated by h.

In [BT](2.5) an involution ¯ is defined on Norm^{1}(V) in the following way. If α ∈ Norm^{1}(V), then

ᾱ(x) = inf_{y∈V} [v(h(x, y)) − α(y)] , x ∈ V .

We then have Proposition 3.2. (cf. [BT](Prop. 2.5)) An element α of Norm^{1}(V) is an MM-norm if and only if ᾱ = α.

We are going to describe the set Norm_{h}^{1}(V) of MM-norms in terms of self-dual lattice-functions. Recall [BL] that a lattice-function in V is a function Λ which maps a real number to an 𝔬_{F} -lattice in V and satisfies:

i) Λ(r) ⊂ Λ(s) for r ⩾ s, r, s ∈ ℝ,

ii) Λ(r + v(π_{F})) = 𝔭_{F} Λ(r), r ∈ ℝ,

iii) Λ is left-continuous. Here 𝔬_{F} denotes the ring of integers of F, 𝔭_{F} the maximal ideal of 𝔬_{F} and π_{F} a uniformizer of F. As in [BL], we denote by Latt_{𝔬_{F}}^{1}(V) (or by Latt^{1}(V) when no confusion may occur) the set of 𝔬_{F} -lattice-functions in V. Recall [BL] that Norm^{1}(V) and Latt^{1}(V) may be canonically identified in the following way. To α ∈ Norm^{1}(V), we attach the function Λ = Λ_{α} given by

Λ(r) = {x ∈ V ; α(x) ⩾ r} , r ∈ ℝ .

Conversely a lattice-function Λ corresponds to the norm α given by

α(x) = sup{r ; x ∈ Λ(r)} , x ∈ V .

For a Λ ∈ Latt^{1}(V) and r ∈ ℝ, set

Λ(r+) = ⋃_{s>r} Λ(s) .

For an 𝔬_{F} -lattice L in V, we define its dual L^{♯} = L^{♯_{h}} by

L^{♯} = {x ∈ V ; h(x, L) ⊂ 𝔭_{F} } .

Finally, we define the dual Λ^{♯} = Λ^{♯_{h}} of a lattice-function Λ by

Λ^{♯}(r) = [Λ((−r)+)]^{♯} , r ∈ ℝ .

We say that a lattice function Λ is self dual if Λ^{♯} = Λ and we denote by Latt_{h}^{1}(V) the corresponding set.

Proposition 3.3. Given a norm α ∈ Norm^{1}(V), we have Λᾱ = Λ_{α}^{♯}. Corollary 3.4. Let α be a norm on V. Then α is an MM-norm if and only if the attached lattice-function Λ is self-dual. Proof of Proposition. Let x ∈ V and r ∈ ℝ. Then the fact that x ∈ Λᾱ(r)\Λᾱ(r+) is equivalent to the following points:

i) ᾱ(x) = r; ii) there exists y ∈ V such that v(h(x, y)) − α(y) = r, and for all y ∈ V, we have v(h(x, y)) − α(y) ⩾ r; iii) there exists y ∈ V such that v(h(x, y)) = 0 and α(y) = −r, and for all y ∈ V such that α(y) > −r, we have v(h(x, y)) > 0 (scale by a suitable power of a uniformizer π_{F}); iv) there exists y ∈ Λ_{α}(−r)\Λ_{α}(−r+) such that h(x, y) ∈ 𝔬_{F} \𝔭_{F}, and for all y ∈ Λ_{α}(−r+) we have h(x, y) ∈ 𝔭_{F};

v) x ∈ Λ_{α}^{♯}(r)\Λ_{α}^{♯}(r+). This proves that the two lattice-functions Λᾱ and Λ_{α}^{♯} share the same discontinuity points and that at those points they take the same values; so there are equal.

Let Norm^{2}𝔤̃ (resp. Latt^{2}𝔤) denote the G̃-set of square norms in 𝔤̃ (resp. of square lattice-functions in 𝔤̃; see [BT1] and [BL]). Recall that a lattice-function Λ^{2} in the F - vector space 𝔤̃ is square if there exists Λ ∈ Latt^{1}(V) such that Λ^{2} = End(Λ), where

End(Λ)(r) = {a ∈ 𝔤̃ ; aΛ(s) ⊂ Λ(s + r), s ∈ ℝ}, r ∈ ℝ .

An additive norm on 𝔤̃ is square if the corresponding lattice function is square. Recall [BT1] that Norm^{1}(V) and Norm^{2}𝔤̃ (and therefore Latt^{1}(V) and Latt^{2}𝔤̃ by transfer of structure) are endowed with affine structures : the barycenter of two points with positive weights is defined.

The involution σ acts on Norm^{2}𝔤̃ via

α^{σ}(a) = α(a^{σ}), a ∈ 𝔤̃, α ∈ Norm^{2}𝔤̃ .

By transfer of structure, σ acts on Latt^{2}𝔤̃ via

Λ^{σ}(r) = [Λ(r)]^{σ}, Λ ∈ Latt^{2}𝔤̃, r ∈ ℝ .

A square norm α (resp. a square lattice function Λ) is said to be self-dual if α = α^{σ} (resp. Λ = Λ^{σ}). We denote by Norm_{σ}^{2}𝔤̃ and Latt_{σ}^{2}𝔤̃ the corresponding sets.

Now, in terms of lattice functions, Corollary 2 of [BT2], page 163, writes: Lemma 3.5 The map Λ ↦ End(Λ) induces a bijection from the set of self-dual lattice functions in V to the set of self-dual square lattice functions in 𝔤̃.

In other words, for any Λ ∈ Latt_{σ}^{2}𝔤̃, there exists a unique Λ^{2} = Λ_{h}^{2} ∈ Latt_{h}^{1}(V) such that End(Λ) = Λ^{2}.

Note that the sets Latt_{h}^{1}(V), Norm_{h}^{1}(V), Latt_{σ}^{2}𝔤̃ and Norm_{σ}^{2}𝔤̃ are G-sets and that the various identifications among them are G-equivariant.

Let u ∈ F^{×} and assume that uh is still an ε-hermitian form with respect to σ_{F}. Then the involution σ of 𝔤̃ corresponding to uh remains the same and defines the same unitary group G ⊂ G̃. For Λ ∈ Latt^{1}(V) and s ∈ ℝ, we denote by Λ + s the lattice function given by (Λ + s)(r) = Λ(s + r), r ∈ ℝ. Lemma 3.6. Let Λ^{2} ∈ Latt_{σ}^{2}𝔤̃ and Λ_{h}^{2} (resp. Λ_{uh}^{2}) be the unique element of Latt_{h}^{1}(V) (resp. of Latt_{uh}^{1}(V)) satisfying End(Λ_{h}^{2}) = Λ^{2} (resp. End(Λ_{uh}^{2}) = Λ^{2}). Then Λ_{uh}^{2} = Λ_{h}^{2} − v(u)/2, that is Λ_{uh}^{2}(r) = Λ_{h}^{2}(r − v(u)/2), r ∈ ℝ. Proof. We easily check that for Λ ∈ Latt^{1}(V) and s ∈ ℝ, we have

Λ^{♯_{uh}} = u^{−σ}Λ^{♯_{h}} and (Λ + s)^{♯_{h}} = Λ − s .

We certainly have End(Λ_{h}^{2} − v(u)/2) = End(Λ_{h}^{2}) = Λ^{2}. So by a unicity argument, we must prove that Λ_{h}^{2} − v(u)/2 ∈ Latt_{uh}^{1}(V). But

(Λ_{h}^{2} − v(u)/2)^{♯_{uh}} = u^{−σ}(Λ_{h}^{2} − v(u)/2)^{♯_{h}}
= u^{−σ}(Λ_{h}^{2} + v(u)/2) = Λ_{h}^{2} + v(u)/2 − v(u^{σ}) = Λ_{h}^{2} − v(u)/2 ,

as required.

4. The building as a set of self-dual lattice-functions

Let I denote the building of the standard valuated root datum of G introduced in [BT2] and A denote the apartment of I attached to S. Write V^{∗} = X^{∗}(S ⊗ ℝ); this is an ℝ-vector space with basis (a_{i})_{i=1,...,r}. Let V denote the linear dual of V^{∗}. We identify A with V.

To a point p ∈ A ≃ V, we attach the norm α_{p} on V defined by

α_{p}(∑_{i∈I} λ_{i}e_{i} + x_{o}) = inf[ω(x_{o}), inf_{i∈I}(v(λ_{i}) − a_{i}(p))], x_{o} ∈ V_{o}, λ_{i} ∈ F for i ∈ I .

Here ω(x_{o}) = \frac{1}{2}v(h(x_{o}, x_{o})), x_{o} ∈ V_{o}.

Here are two important facts from [BT2]. Proposition 4.1. ([BT2](Prop. 2.9, 2.11(i))) The map p ↦ α_{p} is a bijection from A to the set of MM-norms on V which split in the decomposition V = ⊕_{i∈I}F e_{i} ⊕ V_{o}. It is N-equivariant.

For the notion of splitting for norms, see [BT1](1.4). Proposition 4.2. ([BT2](2.12)) i) The map p ↦ α_{p} extends in a unique way to a Gequivariant and affine bijection j_{h} : I → Norm_{h}^{1}(V) (in particular Norm_{h}^{1}(V) is a convex subset of Norm^{1}(V)). ii) The map j_{h} is the unique affine and G-equivariant map I → Norm_{h}^{1}(V).

From §3, we get a unique affine and G-equivariant map I → Latt_{h}^{1}(V) that we still denote by j_{h}.

For r ∈ ℝ, let 𝒱_{o}^{r} be the lattice of V_{o} given by {x_{o} ∈ V_{o} ; ω(x_{o}) ⩾ r}. For x ∈ ℝ, let ⌈x⌉ denote the least integer greater than or equal to x. Then the map j_{h} : I → Latt_{h}^{1}(V) is given on A by j_{h}(p) = Λ_{p}, where

Λ_{p}(r) = 𝒱_{o}^{r} ⊕ ⨁_{i∈I} 𝔭_{F}^{⌈r+a_{i}(p)⌉}e_{i} , r ∈ ℝ .

Let u be an element of F^{×} such that uh remains ε-hermitian with respect to σ_{F}. It follows from the proof of Lemma (3.6) that if Λ ∈ Latt^{1}(V), we have Λ ∈ Latt_{h}^{1}(V) if, and only if, Λ − v(u)/2 ∈ Latt_{uh}^{1}(V). Since End(Λ + s) = End(Λ), for Λ ∈ Latt^{1}(V) and s ∈ ℝ, the bijective map j_{σ} : I → Latt_{σ}^{2}(V), given by j_{σ} = End ∘ j_{h}, does not depend on the choice of the form h, the involution σ being fixed. By construction it is affine and G-equivariant. It is uniquely determined by these two properties. Indeed if j′_{σ} : I → Latt_{σ}^{2}(V) is affine and G-equivariant, so is (j′_{σ})^{−1} ∘ j_{σ} : I → I. But such a map must be the identity map.

We also recall here the description of the enlarged building I^{1} of G̃ = GL_{F} (V) in terms of lattice functions. Proposition 4.3. ([BT1](2.11)) i) There is a G̃-equivariant and affine bijection j : I^{1} → Norm^{1}(V). ii) If we have another affine and G̃-equivariant map j′ : I^{1} → Norm^{1}(V) then there exists r ∈ ℝ such that, for all α ∈ Norm^{1}(V), j′(α) = j(α) + r.

From [BL] Proposition 2.4, for each j as in Proposition 4.3, we get an affine and G̃-equivariant map I^{1} → Latt^{1}(V) that we also denote by j.

5. Centralizers of Lie algebra elements

We denote by 𝔤 the Lie algebra of G:

𝔤 = {a ∈ 𝔤̃ ; a + a^{σ} = 0} .

We consider an element β of 𝔤 satisfying

(H1)  The F -algebra E := F [β] ⊂ 𝔤̃ is a direct sum of fields.

We write 𝔥̃ (resp. 𝔥) for the centralizer of β in 𝔤̃ (resp. in 𝔤) and H̃ (resp. H) for the fixator of β in G̃ (resp. in G) for the adjoint action.

Since σ(β) = −β, we have easily that E ⊂ 𝔤̃ is σ-stable. We write

E = ⨁_{i=1,...,t}(E_{i} ⊕ E_{−i}) ⊕ ⨁_{k=1,...,s} E_{(0,k)},

where, for each i in J = {±1, …, ±t} or J_{o} = {(0, k) : k = 1, …, s}, E_{i} is a field extension of F, and we have labeled the components such that, for each i ∈ J_{o} ∪ J,

(H2)  σ(E_{i}) = E_{−i},

with the understanding that i = −i, for i ∈ J_{o}. We remark that the torus E ∩ G in G is anisotropic (modulo the centre) if and only if J = ∅ and that every maximal anisotropic torus in G takes this form (see [Mor] Proposition 1.3).

For each i ∈ J_{o}, we set E_{i}^{o} = {a ∈ E_{i} ; a = a^{σ}}, so that E_{i}/E_{i}^{o} is a Galois extension of degree ⩽ 2 and a generator of Gal(E_{i}/E_{i}^{o}) is σ_{E_{i}} := σ_{|E_{i}}. For i ∈ J_{o} ∪ J, let 1_{i} be the idempotent of E attached to E_{i}; from (H2), we have σ(1_{i}) = 1_{−i}. We have the decomposition

V = ⨁_{i∈J_{o}∪J} V_{i} , V_{i} = 1_{i}V .

Note that, if i ≠ −k, v ∈ V_{i} and w ∈ V_{k}, we have h(v, w) = h(1_{i}v, w) = h(v, 1_{i}w) = 0 so, for i ∈ J_{o} ∪ J,

V_{i}^{⊥} = ⨁_{k≠−i} V_{k}.

For i ∈ J_{o} ∪ J, V_{i} is naturally an E_{i}-vector space and we have obvious isomorphisms of algebras and groups respectively:

𝔥̃ ≃ ∏_{i∈J_{o}∪J} End_{E_{i}}V_{i} ,

H̃ ≃ ∏_{i∈J_{o}∪J} Aut_{E_{i}}V_{i} .

The involution σ stabilizes 𝔥̃ ⊂ 𝔤̃ and, for each i, σ(End_{E_{i}}V_{i}) = End_{E_{−i}}V_{−i}. For i ∈ J_{o}, we write σ_{i} = σ_{|End_{E_{i}}V_{i}}. Let us fix i ∈ J_{o}. The map σ_{i} is an involution of the central simple E_{i}-algebra End_{E_{i}}V_{i}. By a classical theorem ([Inv] Theorem 4.2), there exists ε_{i} ∈ {±1} and a non-degenerate ε_{i}-hermitian form h_{i} on V_{i} relative to σ_{E_{i}} such that σ_{i} is the involution attached to h_{i}. Of course h_{i} is only defined up to a scalar in E_{i}^{×}. Let

H_{i} = {g ∈ Aut_{E_{i}}V_{i} ; gg^{σ_{i}} = 1}

be the unitary group attached to h_{i}. On the other hand, for i ∈ J, we put

H_{i} = Aut_{E_{i}}V_{i},

so that σ(H_{i}) = H_{−i} and H_{i} is isomorphic to {g ∈ H_{i} × H_{−i} : gg^{σ} = 1} by h ↦ (h, h^{−σ}). Then, putting J_{+} = {1, …, t}, we have a natural group isomorphism

H ≃ ∏_{i∈J_{o}∪J_{+}} H_{i} .

We may actually require a compatibility relation between the forms h_{i}, i ∈ J_{o} and the form h. Let us fix i ∈ J_{o}. Let λ_{i} : E_{i} → F be any σ-equivariant non-zero F -linear form. Such forms exist. Indeed choose a non-zero linear form λ_{i}^{o} : E_{i}^{o} → F_{o}. If F = F_{o} then we put λ = λ_{i}^{o} ∘ Tr_{E/E_{i}^{o}}. Otherwise E_{i} = F E_{i}^{o} and we can extend λ_{i}^{o} by linearity to get the required map λ_{i}. In all cases we have:

(5.1)  λ_{i}^{o} ∘ Tr_{E_{i}/E_{i}^{o}} = Tr_{F/F_{o}} ∘ λ .

We still write h for the restriction of h to V_{i}. Lemma 5.2. Let i ∈ J_{o}. There exists a unique ε-hermitian form h_{i} : V_{i} × V_{i} → E_{i} relative to σ_{E_{i}} such that

(5.3)  h(v, w) = λ_{i}(h_{i}(v, w)), for all v, w ∈ V_{i} .

It is non-degenerate. Proof. Since we have the orthogonal decomposition

V = V_{i} ⊥ ⨁_{k≠i} V_{k} ,

the restriction h_{|V_{i}} is non-degenerate.

The F -linear map Hom_{E_{i}}(V_{i}, E_{i}) → Hom_{F} (V_{i}, F), φ ↦ λ_{i} ∘ φ is an isomorphism of F -vector space. Indeed if φ lies in the kernel, we have Im(φ) ⊂ Ker(λ_{i}), a strict subspace of E_{i}, and φ must be trivial. Moreover the two dual spaces have the same F -dimension. For v ∈ V_{i} let h_{v} be the element of Hom_{F} (V_{i}, F) given by h_{v}(w) = h(v, w). There exists a unique φ_{w} ∈ Hom_{E_{i}}(V_{i}, E_{i}) such that h_{v} = λ_{i} ∘ φ_{w}. It is now routine to check that h_{i}(v, w) := φ_{v}(w), v, w ∈ V_{i}, has the required properties.

We easily check that if h_{i} is as in the lemma, then the corresponding involution on End_{E_{i}}V_{i} is σ_{i}. In the following we assume that the forms h_{i}, i ∈ J_{o}, satisfy (5.3).

For technical reasons, we need one more assumption on the λ_{i}, i ∈ J_{o}. We fix i again. Let

ℐ = {e ∈ E_{i}^{o} ; λ_{i}^{o}(e𝔬_{E_{i}^{o}}) ⊂ 𝔭_{F_{o}}} .

This is an 𝔬_{E_{i}^{o}}-lattice in E_{i}^{o} and must have the form t𝔭_{E_{i}^{o}}, for some t ∈ (E_{i}^{o})^{×}. So replacing λ_{i} by e ↦ λ_{i}(tx), we may assume that ℐ = 𝔭_{E_{i}^{o}}. In the following we assume that the linear forms λ_{i}, i ∈ J_{o}, have this property. Lemma 5.4. Fix i ∈ J_{o}. Let λ_{i}^{1}, λ_{i}^{2} : E_{i} → F be two linear forms as above and let h_{i}^{1}, h_{i}^{2} be the corresponding ε-hermitian forms on V_{i} (i.e. h_{i}^{1} and h_{i}^{2} satisfy (5.3)). Then there exists u ∈ 𝔬_{E_{i}^{o}}^{×} such that h_{i}^{2} = uh_{i}^{1}.

Proof. Since h_{i}^{1} and h_{i}^{2} induce the same involution on End_{E_{i}}V_{i}, there exists u ∈ E_{i}^{×} such that h_{i}^{2} = uh_{i}^{1}. The fact that h_{i}^{1} and h_{i}^{2} are both ε-hermitian with respect to σ_{E_{i}} implies that u lies in E_{i}^{o}. Condition (5.3) writes

h(v, w) = λ_{i}^{1}(h_{i}^{1}(v, w)) = λ_{i}^{2}(uh_{i}^{1}(v, w)) , v, w ∈ V_{i} .

So λ_{i}^{1}(e) = λ_{i}^{2}(ue), e ∈ E_{i}. By applying Tr_{F/F_{o}} to this equality, we get λ_{i}^{o,1}(e) = λ_{i}^{o,2}(ue), e ∈ E_{i}^{o}. Hence

𝔭_{E_{i}^{o}} = {e ∈ E_{i}^{o} ; λ_{i}^{o,1}(e𝔬_{E_{i}^{o}}) ⊂ 𝔭_{F_{o}}}
= {e ∈ E_{i}^{o} ; λ_{i}^{o,2}(ue𝔬_{E_{i}^{o}} ⊂)𝔭_{F_{o}}} = u^{−1}𝔭_{E_{i}^{o}} .

So u ∈ 𝔬_{E_{i}^{o}}^{×} as required.

Let us fix i. Let L be an 𝔬_{E_{i}^{o}}-lattice in V_{i}. Then L has a dual L^{♯} relative to the form h_{|V_{i}} and a dual L^{♯_{i}} relative to the form h_{i}. Lemma 5.5. The lattices L^{♯} and L^{♯_{i}} coincide. Proof. We have L^{♯} = {v ∈ V_{i} ; h(v, L) ⊂ 𝔭_{F} }

= {v ∈ V_{i} ; Tr_{F/F_{o}}h(v, L) ⊂ 𝔭_{F_{o}}}

= {v ∈ V_{i} ; λ_{o} ∘ Tr_{E_{i}/E_{i}^{o}}h_{i}(v, L) ⊂ 𝔭_{F_{o}}}

= {v ∈ V_{i} ; Tr_{E_{i}/E_{i}^{o}}h_{i}(v, L) ⊂ 𝔭_{E_{i}^{o}}}

= {v ∈ V_{i} ; f(v, L) ⊂ 𝔭_{E_{i}}}

= L^{♯_{i}}, where the second and fifth equalities hold because F/F_{o} and E_{i}/E_{i}^{o} are at worst tamely ramified.

6. Embedding the building of the centralizer

We keep the notation as in the previous section. Assume for a moment that the extensions E_{i}/F, i ∈ J_{o} ∪ J, are separable. Then the group H is naturally the group of rational points of a reductive F -group H. Indeed each H_{i}, i ∈ J_{o} ∪ J, is naturally the group of rational points of a classical E_{i}-group H_{i} (we do not need E_{i}/F -separable here) and

H ≃ ∏_{i∈J_{o}∪J_{+}} Res_{E_{i}/F} H_{i} .

The (enlarged) affine building of H, I_{β}^{1} := I^{1}(H, F), is the cartesian product of the (enlarged) affine buildings I^{1}(Res_{E_{i}/F} H_{i}, F), i ∈ J_{o} ∪ J_{+}. For all i, the (enlarged) buildings I^{1}(Res_{E_{i}/F} H_{i}, F) and I^{1}(H_{i}, E_{i}) identify canonically. Note also that, for i ∈ J_{o}, the centre of H_{i} is compact so the enlarged building is also the non-enlarged building; in particular, if J = ∅ then all the buildings involved are non-enlarged.

Since we do not want any restriction on the extensions E_{i}/F, we shall take as a definition of the (enlarged) building I_{β}^{1} attached to the group H:

(6.1)  I_{β}^{1} := ∏_{i∈J_{o}∪J_{+}} I^{1}(H_{i}, E_{i})

We abbreviate I_{i}^{1} = I^{1}(H_{i}, E_{i}), i ∈ J_{o} ∪ J_{+}.

We are going to construct a map j_{β} : I_{β}^{1} → I. We normalize the lattice-functions in Latt_{𝔬_{E_{i}}}^{1}(V_{i}) by Λ_{i}(r+v_{i}(π_{i})) = 𝔭_{E_{i}}Λ_{i}(r), r ∈ ℝ, where, for each i, π_{i} denotes a uniformizer of E_{i} and v_{i} the unique extension of v to a valuation of E_{i}. It is straightforward that we have a well defined map

j̃_{β} : ∏_{i∈J_{o}∪J} Latt_{𝔬_{E_{i}}}^{1}(V_{i}) ⟶ Latt^{1}(V)

(Λ_{i})_{i∈J_{o}∪J} ↦ ⨁_{i∈J_{o}∪J} Λ_{i}

where (⨁_{i∈J_{o}∪J} Λ_{i}) (r) = ⨁_{i∈J_{o}∪J} Λ_{i}(r), for r ∈ ℝ. This map is clearly injective and equivariant for the action of group ∏_{i∈J_{o}∪J} Aut_{E_{i}}V_{i} ⊂ Aut_{F} V.

For i ∈ J_{o}, we denote by ♯_{i} the involution on Latt_{𝔬_{E_{i}}}^{1}(V_{i}) attached to h_{i}, and by Latt_{𝔬_{E_{i}},h_{i}}^{1}(V_{i}) ⊂ Latt_{𝔬_{E_{i}}}^{1}(V_{i}) the set of fixed points. For i ∈ J, we denote be ♯_{i} the map Latt_{𝔬_{E_{i}}}^{1}(V_{i}) → Latt_{𝔬_{E_{−i}}}^{1}(V_{−i}) given by

Λ_{i}^{♯_{i}}(r) = {v ∈ V_{−i} ; h(v, Λ_{i}(−r+)) ⊂ 𝔭_{F} } .

for Λ_{i} ∈ Latt_{𝔬_{E_{i}}}^{1}(V_{i}).

We define an involution b on ∏_{i∈J_{o}∪J} Latt_{𝔬_{E_{i}}}^{1}(V_{i}) by

(Λ_{i})_{i∈J_{o}∪J}^{b} = (Λ_{−i}^{♯_{−i}})_{i∈J_{o}∪J} ,

Then we have a bijection

ι_{h} : ∏_{i∈J_{o}} Latt_{𝔬_{E_{i}},h_{i}}^{1}(V_{i}) × ∏_{i∈J_{+}} Latt_{𝔬_{E_{i}}}^{1}(V_{i}) → ( ∏_{i∈J_{o}∪J} Latt_{𝔬_{E_{i}}}^{1}(V_{i}))^{b} ,

given by (Λ_{i})_{i∈J_{o}∪J_{+}} ↦ (Λ_{i})_{i∈J_{o}∪J}, with Λ_{−i} = Λ_{i}^{♯_{i}}, for i ∈ J_{+}.

Lemma 6.2. For x ∈ ∏_{i∈J_{o}∪J} Latt_{𝔬_{E_{i}}}^{1}(V_{i}), we have j̃_{β}(x^{b}) = j̃_{β}(x)^{♯_{h}}. In particular j̃_{β} ∘ ι_{h}

maps ∏_{i∈J_{o}} Latt_{𝔬_{E_{i}},h_{i}}^{1}(V_{i}) × ∏_{i∈J_{+}} Latt_{𝔬_{E_{i}}}^{1}(V_{i}) into Latt_{h}^{1}(V).

Proof. Fix (Λ_{i})_{i∈J_{o}∪J} ∈ ∏_{i∈J_{o}∪J} Latt_{𝔬_{E_{i}}}^{1}V_{i} and set Λ = j̃_{β} ((Λ_{i})_{i∈J_{o}∪J_{+}}). We have

Λ^{♯_{h}}(r) = Λ(−r+)^{♯_{h}} = {v ∈ V ; h(v, Λ(−r+)) ⊂ 𝔭_{F} } , r ∈ ℝ .

Fix r ∈ ℝ. We have

Λ(−r+) = ⨁_{i∈J_{o}∪J} Λ_{i}(−r+) .

Let v = ∑_{i∈J_{o}∪J} v_{i}, with v_{i} ∈ V_{i}, be an element of V. Since V_{i}^{⊥} = ⨁_{k≠−i} V_{k}, we have v ∈ Λ^{♯_{h}}(r) if and only if h(v_{−i}, Λ_{i}(−r+)) ⊂ 𝔭_{F}, for all i, that is if v_{−i} ∈ Λ_{i}^{♯_{i}}(r), for all i (by lemma (5.5) for i ∈ J_{o} or by definition for i ∈ J); the lemma follows.

With the notation of §4, for each set {j_{i}}_{i∈J_{+}} of maps j_{i} : I_{i}^{1} → Latt_{𝔬_{E_{i}}}^{1}(V_{i}) given by

Proposition 4.3, we define a map j_{β} : ∏_{i∈J_{o}∪J_{+}} I_{i}^{1} → I by

j_{β} = j_{h}^{−1} ∘ j̃_{β} ∘ ι_{h} ∘ (∏_{i∈J_{o}} j_{h_{i}} × ∏_{i∈J_{+}} j_{i}) .

These maps depend a priori on the forms h, and h_{j}, j ∈ J_{o}. Theorem 6.3. Each map j_{β} is injective, H-equivariant. The set of such maps (as {j_{i}}_{i∈J_{+}} varies) depends only on the involution σ.

In particular, if J = ∅ then there is a unique map j_{β}, depending only on the involution σ.

Proofs. The first two properties are straightforward. Assume that h′ = uh, u ∈ F^{×}, is another ε-hermitian form on V, with respect to σ_{F}, defining the same involution σ on 𝔤̃. Then u ∈ F_{o}. For i ∈ J_{o}, let h′_{i} be an ε-hermitian form on V_{i} satisfying

uh(v, w) = λ′_{i}(h′_{i}(v, w)) v, w ∈ V_{i} ,

where the λ′_{i} : E_{i} → F are linear forms as above. Then by lemma (5.4), for all i ∈ J_{o}, there exists u′_{i} ∈ 𝔬_{E_{i}^{o}}^{×} such that u^{−1}h′_{i} = u′_{i}h_{i}, that is h′_{i} = uu′_{i}h_{i}. Let {j_{i}}_{i∈J_{+}} be as above; we show that, for a suitable choice of {j′_{i}}_{i∈J_{+}}, we have

j_{h}^{−1} ∘ j̃_{β} ∘ ι_{h} ∘ j_{1} = j_{h′} ∘^{−1} j̃_{β} ∘ ι_{h′} ∘ j′_{1},

and the result follows. By Lemma (3.6), for i ∈ J_{+}, for all x_{i} ∈ I_{i}^{1}, we have j_{h′_{i}}(x_{i}) = j_{h_{i}}(x_{i}) − v(uu′_{i})/2 = j_{h_{i}}(x_{i}) − v(u)/2. For i ∈ J_{+}, we choose j′_{i} such that j′_{i}(x) = j_{i}(x) − v(u)/2 for x ∈ I_{i}^{1}, that is j′_{i} ∘ j_{i}^{−1}(Λ_{i}) = Λ_{i} − v(u)/2 for Λ_{i} ∈ Latt_{𝔬_{E_{i}}}^{1}(V_{i}). We abbreviate

j = ∏_{i∈J_{o}} j_{h_{i}} × ∏_{i∈J_{+}} j_{i}, j′ = ∏_{i∈J_{o}} j_{h′_{i}} × ∏_{i∈J_{+}} j′_{i};

then, for (Λ_{i})_{i∈J_{o}∪J_{+}} ∈ ∏_{i∈J_{o}} Latt_{𝔬_{E_{i}},h_{i}}^{1}(V_{i}) × ∏_{i∈J_{+}} Latt_{𝔬_{E_{i}}}^{1}(V_{i}), we have

j′ ∘ j^{−1} ((Λ_{i})_{i∈J_{o}∪J_{+}}) = (Λ_{i} − v(u)/2)_{i∈J_{o}∪J_{+}} .

It is also straightforward to check that

ι_{h′} ((Λ_{i} − v(u)/2)_{i∈J_{o}∪J_{+}}) = ι_{h} ((Λ_{i})_{i∈J_{o}∪J_{+}}) − v(u)/2,

for (Λ_{i})_{i∈J_{o}∪J_{+}} ∈ ∏_{i∈J_{o}} Latt_{𝔬_{E_{i}},h_{i}}^{1}(V_{i}) × ∏_{i∈J_{+}} Latt_{𝔬_{E_{i}}}^{1}(V_{i}). Then we have

j̃_{β} ∘ ι_{h′} ∘ j′ ∘ j^{−1} ((Λ_{i})_{i∈J_{o}∪J_{+}}) = j̃_{β} ∘ ι_{h′} ((Λ_{i} − v(u)/2)_{i∈J_{o}∪J_{+}})
= j̃_{β} (ι_{h} ((Λ_{i})_{i∈J_{o}∪J_{+}}) − v(u)/2)
= j̃_{β} ∘ ι_{h} ((Λ_{i})_{i∈J_{o}∪J_{+}}) − v(u)/2.

By Lemma (3.6) again, we have j_{h′}(x) = j_{h}(x) − v(u)/2, x ∈ I, that is Λ − v(u)/2 = j_{h′} ∘ j_{h}^{−1}(Λ), Λ ∈ Latt_{h}^{1}(V). So

j_{h′} ∘ j_{h}^{−1} ∘ j̃_{β} ∘ ι_{h} = j̃_{β} ∘ ι_{h′} ∘ j′ ∘ j^{−1} ,

and the lemma follows.

7. Affine structures

We keep the notation as in the previous sections. For x = (x_{i})_{i∈J_{o}∪J_{+}}, y = (y_{i})_{i∈J_{o}∪J_{+}} in I_{β}^{1} = ∏_{i∈J_{o}∪J_{+}} I_{i}^{1} and t ∈ [0, 1], we define the barycenter tx + (1 − t)y to be

(tx_{i} + (1 − t)y_{i})_{i∈J_{o}∪J_{+}} .

We define the barycenter of two points in ∏_{i∈J_{o}∪J_{+}} Latt_{𝔬_{E_{i}}}^{1}(V_{i}) in a similar way. Since, for

i ∈ J_{o}, Latt_{𝔬_{E_{i}},h_{i}}^{1}(V_{i}) is convex in Latt_{𝔬_{E_{i}}}^{1}(V_{i}), the subset ∏_{i∈J_{o}} Latt_{𝔬_{E_{i}},h_{i}}^{1}(V_{i})× ∏_{i∈J_{+}} Latt_{𝔬_{E_{i}}}^{1}(V_{i})

of ∏_{i∈J_{o}∪J_{+}} Latt_{𝔬_{E_{i}}}^{1}(V_{i}) is convex also.

Proposition 7.1. Let β be as in §5. Then each map j_{β} is affine: for all x, y ∈ I_{β}^{1}, t ∈ [0, 1], we have

j_{β}(tx + (1 − t)y) = tj_{β}(x) + (1 − t)j_{β}(y) .

Proof. By construction it suffices to prove that the maps j̃_{β} and ι_{h} are affine. We begin with j̃_{β}. Let (Λ_{i})_{i∈J_{o}∪J}, (M_{i})_{i∈J_{o}∪J} be elements of ∏_{i∈J_{o}∪J} Latt_{𝔬_{E_{i}}}^{1}(V_{i}). We must prove that

⨁_{i∈J_{o}∪J}(tΛ_{i} + (1 − t)M_{i}) = t ( ⨁_{i∈J_{o}∪J} Λ_{i}) + (1 − t) ( ⨁_{i∈J_{o}∪J} M_{i}) .

Let us recall the construction of the barycenter of two lattice functions (we do it for Latt^{1}(V)). Let Λ, M ∈ Latt^{1}(V). There exists an F -basis (e_{1}, …, e_{n}) of V which splits both Λ and M : there exist constants λ_{1}, …, λ_{n}, μ_{1}, …, μ_{n} in ℝ such that

Λ(r) = ⨁_{k=1,...,n} 𝔭_{F}^{⌈r+λ_{k}⌉}e_{k} , M(r) = ⨁_{k=1,...,n} 𝔭_{F}^{⌈r+μ_{k}⌉}e_{k} , r ∈ ℝ .

Then for t ∈ [0, 1], tΛ + (1 − t)M is given by

(tΛ + (1 − t)M)(r) = ⨁_{k=1,...,n} 𝔭_{F}^{⌈r+tλ_{k}+(1−t)μ_{k}⌉}e_{k} , r ∈ ℝ .

The proof that j̃_{β} is affine is then to construct a common splitting basis for ⨁_{i∈J_{o}∪J} Λ_{i} and ⨁_{i∈J_{o}∪J} M_{i} from bases ℬ_{i} of V_{i}, i ∈ J_{o} ∪ J, where ℬ_{i} splits Λ_{i} and M_{i}. We leave this easy exercise to the reader.

Now we turn to ι_{h}. Suppose i ∈ J_{+} and Λ_{i} ∈ Latt_{𝔬_{E_{i}}}^{1}(V_{i}), and let (e_{1}, …, e_{n}) be an E_{i}-basis of V_{i} which splits Λ_{i}. Let (e_{−1}, …, e_{−n}) be the dual E_{−i}-basis of V_{−i}, such that h(e_{−k}, e_{l}) = δ_{kl}, for 1 ≤ k, l ≤ n. It is straightforward to check that this basis splits Λ_{i}^{♯_{i}} and that,

(7.2)  if Λ_{i}(r) = ⨁_{k=1,...,n} 𝔭_{E_{i}}^{⌈r+λ_{k}⌉}e_{k} then Λ_{i}^{♯_{i}}(r) = ⨁_{k=1,...,n} 𝔭_{E_{−i}}^{⌈r−λ_{k}⌉}e_{−k}.

To show that ι_{h} is affine, we just need to check that, for i ∈ J_{+}, Λ_{i}, M_{i} ∈ Latt_{𝔬_{E_{i}}}^{1}(V_{i}) and t ∈ [0, 1], we have

(tΛ_{i} + (1 − t)M_{i})^{♯_{i}} = tΛ_{i}^{♯_{i}} + (1 − t)M_{i}^{♯_{i}}.

The details of the proof – which is to choose an E_{i}-basis of V_{i} which splits both Λ_{i} and M_{i}, take its dual basis and then use (7.2) – are again left to the reader.

8. The image of an apartment

We keep the notation of the previous sections. We will show that the image of an apartment of I_{β}^{1} under each map j_{β} is contained in an apartment of I.

Given a Witt decomposition V = V_{+} ⊕ V_{o} ⊕ V_{−}, with basis (e_{l})_{l=1,...,r} of V_{+} and the dual basis (e_{−l})_{l=1,...,r} of V_{−} (as in §2), we get a (self-dual) decomposition

V = ⨁_{l=1}^{r} V^{l} ⊕ V_{o} ⊕ ⨁_{l=1}^{r} V^{−l},

where V^{l} = F e_{l} = (⨁_{k≠−l} V^{l} ⊕ V_{o})^{⊥}. Such a decomposition (which we will also call a Witt decomposition) corresponds to the choice of an apartment 𝒜 in I: in terms of lattice functions, j_{h}(𝒜) is the set of self-dual lattice functions Λ such that

Λ(s) = ⨁_{l=1}^{r}(V^{l} ∩ Λ(s)) ⊕ (V_{o} ∩ Λ(s)) ⊕ ⨁_{l=1}^{r}(V^{−l} ∩ Λ(s)), for all s ∈ ℝ,

that is, Λ is split by the decomposition (cf. Proposition 4.1).

Similarly, the choice of an (enlarged) apartment 𝒜^{1} in I_{β}^{1} = ∏_{i∈J_{o}∪J_{+}} I_{i}^{1} is given by

similar E_{i}-decompositions of V_{i} for i ∈ J_{o} and (without the self-duality restriction) i ∈ J_{+}.

Proposition 8.1. Let 𝒜^{1} be an (enlarged) apartment of I_{β}^{1}. Then there is an apartment 𝒜 of I such that j_{β}(𝒜^{1}) ⊂ 𝒜.

Proof. We write 𝒜^{1} = ∏_{i∈J_{o}∪J_{+}} 𝒜_{i}^{1}, with 𝒜_{i}^{1} an (enlarged) apartment in I_{i}^{1}.

As above, for each i ∈ J_{o}, the apartment 𝒜_{i}^{1} corresponds to a Witt E_{i}-decomposition of V^{i}

V_{i} = ⨁_{l=1}^{r_{i}} V_{i}^{l} ⊕ V_{i,o} ⊕ ⨁_{l=1}^{r_{i}} V_{i}^{−l},

with V_{i}^{l} = (⨁_{k≠−l} V_{i}^{l} ⊕ V_{i,o})^{⊥}, dim_{E_{i}} V_{i}^{l} = 1 and r_{i} the (E_{i}-)Witt index of V_{i}. We write

Latt_{𝔬_{E_{i}}}^{𝒜^{1}}(V_{i}) for the set of lattice functions split by this decomposition, and Latt_{𝔬_{E_{i}},h_{i}}^{𝒜^{1}}(V_{i})

for the subset of self-dual lattice functions, so that j_{h_{i}}(𝒜_{i}^{1}) = Latt_{𝔬_{E_{i}},h_{i}}^{𝒜^{1}}(V_{i}).

Also, for each i ∈ J_{+}, the apartment 𝒜_{i}^{1} corresponds to a decomposition of V_{i} as a sum of 1-dimensional E_{i}-subspaces,

V_{i} = ⨁_{l=1}^{r_{i}} V_{i}^{l},

with r_{i} = dim_{E_{i}} V_{i}. As above, j_{i}(𝒜_{i}^{1}) = Latt_{𝔬_{E_{i}}}^{𝒜^{1}}(V_{i}), the set of lattice functions split by this decomposition.

We also take the dual splitting of V_{−i} as a sum of 1-dimensional E_{−i}-subspaces,

V_{−i}^{l} = (⨁_{k≠l} V_{i}^{k})^{⊥} .

We remark that, if Λ ∈ Latt_{𝔬_{E_{i}}}^{𝒜^{1}}(V_{i}) then Λ_{i}^{#_{i}} is split by this decomposition.

Now, for i ∈ J_{o} ∪ J_{+} and 1 ≤ l ≤ r_{i}, we decompose V_{i}^{l} as a sum of 1-dimensional F -subspaces as follows: fix v ∈ V_{i}^{l}, v ≠ 0, and let ℬ_{i} be an F -basis for E_{i} which splits the 𝔬_{F} -lattice sequence s ↦ 𝔭_{E_{i}}^{⌈s/e(E_{i}/F)⌉}; then we take the decomposition

V_{i}^{l} = ⨁_{b∈ℬ_{i}} F bv.

Note that any 𝔬_{E_{i}}-lattice sequence in V_{i}^{l} is split by this decomposition. For i ∈ J_{o}, we also take the dual decomposition of V_{i}^{−l} and, for i ∈ J_{+}, the dual decomposition of V_{−i}^{l}.

Now we need to decompose the anisotropic parts W := ⊕_{i∈J_{o}}V_{i,o} suitably, for which we cheat. Let G_{o} denote the classical group associated to the restriction of the form h to W and, for i ∈ J_{o}, let H_{i,o} denote the group associated to the restriction of the form h_{i} to V_{i,o}. Note that the groups H_{i,o} are compact so the building I_{β,o}^{1} := I^{1}(H_{i,o}, E_{i}) is reduced to a point.

Now, our constructions in §6 give an embedding of I_{β,o}^{1} in the building I_{o}^{1} := I^{1}(G_{o}, F) and the image is certainly contained in some apartment. Hence there is a Witt F - decomposition of W which splits the (unique) self-dual lattice sequence in W corresponding to I_{β,o}^{1}, and this is the decomposition we take.

Altogether, we have described a Witt F -decomposition of V, which corresponds to an apartment 𝒜 of I. We denote by Latt_{𝔬_{F} ,h}^{𝒜}(V) the set of self-dual lattice functions in V which are split by this splitting, so that j_{h}(𝒜) = Latt_{𝔬_{F} ,h}^{𝒜}(V).

Finally, by construction it is clear that j̃_{β} ∘ι_{h} maps ∏_{i∈J_{o}} Latt_{𝔬_{E_{i}},h_{i}}^{𝒜^{1}}(V_{i})× ∏_{i∈J_{+}} Latt_{𝔬_{E_{i}}}^{𝒜^{1}}(V_{i})

into Latt_{𝔬_{F} ,h}^{𝒜}(V) so j_{β}(𝒜^{1}) ⊂ 𝒜, as required.

9. Compatibility with Lie algebra filtrations

In this section, we fix H_{k}-equivariant identifications j_{k} : I^{1}(H_{k}, E_{k}) → Latt_{𝔬_{E_{k}}}^{1}(V_{k}), k ∈ J^{+}. They give rise to the map j_{β} : I_{β}^{1} → I(G, H) defined in §6.

Let x ∈ I(G, F) = I^{1}(G, F), that we see as a self-dual lattice function Λ in Latt_{h}^{1}(V). To x we can associate a filtration (𝔤_{x,r})_{r∈ℝ} of the Lie algebra 𝔤 as follows. First x defines a filtration (𝔤̃_{x,r})_{r∈ℝ} of 𝔤̃ by

𝔤̃_{x,r} = {a ∈ 𝔤̃ ; aΛ(s) ⊂ Λ(s + r), s ∈ ℝ}, r ∈ ℝ .

We then define

𝔤_{x,r} := 𝔤̃_{x,r} ∩ 𝔤 = {a ∈ 𝔤 ; aΛ(s) ⊂ Λ(s + r), s ∈ ℝ}, r ∈ ℝ .  (1)

Similarly a point x of I_{β}^{1} defines a filtration (𝔥_{x,r})_{r∈ℝ} of 𝔥. Write x = (x_{k})_{k∈J∪J_{o}}, x_{k} ∈ I^{1}(H_{k}, E_{k}); each x_{k} corresponding to a lattice function Λ_{k} of Latt_{𝔬_{E_{k}}}(V_{k}) (with Λ_{k}^{♯_{k}} = Λ_{−k}, k ∈ J ∪ J_{o}). We then define

(2)  𝔥_{x,r} := ⨁_{k∈J^{+}∪J_{o}} 𝔥_{x_{k},r}^{k}, r ∈ ℝ,

where

𝔥_{x_{k},r}^{k} = {a ∈ Lie(H_{k}) ; aΛ_{k}(s) ⊂ Λ_{k}(s + r), s ∈ ℝ}, r ∈ ℝ, k ∈ J^{+} ∪ J_{o} .

The filtration (𝔥_{x,r})_{r∈ℝ} only depends on the image x̄ of x in the non-enlarged building I_{β}. One can prove that for x ∈ I(G, F), (𝔤_{x,r})_{r∈ℝ} is the filtration of 𝔤 attached to x defined by Moy and Prasad [MP]. Similarly, when β is semisimple and x ∈ I^{1}(H, F), (𝔥_{x,r})_{r∈ℝ} is the filtration of 𝔥 attached to x̄ defined in loc. cit. The proof of this fact is announced by B. Lemaire and J.-K. Yu [BY].

Lemma 9.1. Let us see 𝔥 as being canonically embedded in 𝔥̃ = End_{E}V = ⨁_{k∈J∪J_{o}} End_{E_{k}}V_{k}

via

(a_{k})_{k∈J^{+}∪J_{o}} ↦ (b_{k})_{k∈J∩J_{o}} ,

where b_{k} = a_{k}, k ∈ J_{o}, and b_{−k} = −a_{k}^{σ}, k ∈ J^{+}. Fix x ∈ I_{β}^{1} as before and consider the 𝔬_{F} -lattice function in V given by

Λ = ⨁_{k∈J∪J_{o}} Λ_{k} (notation of §6) .

For r ∈ ℝ, let

𝔥̃_{x,r} = {a ∈ 𝔥̃ ; aΛ(s) ⊂ Λ(s + r), s ∈ ℝ}, r ∈ ℝ .

Then we have 𝔥_{x,r} = 𝔥̃_{x,r} ∩ 𝔥, r ∈ ℝ. Proof. Indeed, for all a = (a_{k})_{k∈J∪J_{o}} ∈ End_{E}V, we have a ∈ 𝔥̃_{x,r} ∩ 𝔥 if and only if a + a^{σ} = 0 and aΛ(s) ⊂ Λ(s + r), s ∈ ℝ, i.e.

a_{k}Λ_{k}(s) ⊂ Λ_{k}(s + r), s ∈ ℝ, k ∈ J ∪ J_{o} .

For k ∈ J_{o}, these conditions can be rewritten a_{k} ∈ Lie(H_{k}) and a_{k}Λ_{k}(s) ⊂ Λ_{k}(s + r), s ∈ ℝ, that is a_{k} ∈ 𝔥_{x,r}^{k}, as required. For k ∈ J, these conditions can be rewritten a_{−k} = −a_{k}^{σ} and

a_{k}Λ_{k}(s) ⊂ Λ_{k}(s + r), s ∈ ℝ  (a)
−a_{k}^{σ}Λ_{k}^{♯_{k}}(s) ⊂ Λ_{k}^{♯_{k}}(s + r), s ∈ ℝ .  (b)

So we must prove that conditions (a) and (b) are equivalent. By symmetry we only prove one implication. Applying the duality ♯_{k} on lattices of V_{k} to inclusion (b), we obtain

Λ_{k}((−s − r)+) ⊂ [a_{k}^{σ}Λ_{k}^{♯_{k}}(s)]^{♯_{k}}, s ∈ ℝ,

with

[a_{k}^{σ}Λ_{k}^{♯_{k}}(s)]^{♯_{k}} = {v ∈ V_{k} ; a_{k}v ∈ Λ_{k}((−s)+)}, s ∈ ℝ .

So we have

a_{k}Λ_{k}((−s − r)+) ⊂ Λ_{k}((−s)+) ⊂ Λ_{k}(−s), s ∈ ℝ ,

that is

a_{k}Λ(s+) ⊂ Λ_{k}(s + r), s ∈ ℝ .

On each open interval (u, v) where Λ_{k} is constant, we have

a_{k}Λ_{k}(s+) = a_{k}Λ_{k}(s) ⊂ Λ_{k}(s + r) ,

and (a) is true for s ∈ (u, v). Finally if s_{o} is a jump of Λ_{k} with Λ_{k} constant on (t, s_{o}], we have

a_{k}Λ_{k}(s_{o}) = a_{k}Λ_{k}(s+) ⊂ Λ_{k}(s + r), s ∈ (t, s_{o}) .

So

a_{k}Λ_{k}(s_{o}) ⊂ ⋂_{s∈(t,s_{o})} Λ_{k}(s + r) = Λ_{k}(s_{o} + r) ,

Λ_{k} being left continuous, and (a) is then true for all s ∈ ℝ.

Proposition 9.2. Let x ∈ I_{β}^{1}. Then we have

𝔤_{j_{β}(x),r} ∩ 𝔥 = 𝔥_{x,r}, r ∈ ℝ .

Proof. Indeed, with the notation of (9.1) and by definition of j_{β}, we easily see that

𝔤̃_{j_{β}(x),r} ∩ 𝔥̃ = 𝔥̃_{x,r} .

So our result is now a corollary of (9.1) since 𝔥 = 𝔤 ∩ 𝔥̃.

10. A unicity result for the general linear group

As in [BL]§I.2, we define an equivalence relation ∼ on Latt^{1}(V) by Λ_{1} ∼ Λ_{2} if there exists s ∈ ℝ such that Λ_{1}(s) = Λ_{2}(r+s), s ∈ ℝ. Then ∼ is compatible with the G̃-action and the quotient Latt_{𝔬_{F}}(V) := Latt^{1}(V)/ ∼ is naturally a G̃-set. We shall denote by Λ̄ an element of Latt_{𝔬_{F}}(V), where Λ is a representative in Latt^{1}(V). As a consequence of [BL]§I.2 and [BT1], there is a unique affine and G̃-equivariant map j : Ĩ → Latt_{𝔬_{F}}(V), where Ĩ denotes the non-enlarged building of G̃.

We fix an element β of 𝔤̃ satisfying

(H)  E : F [β] is a field .

As in §5 we denote by 𝔥̃ = End_{E}V the centralizer of β in 𝔤̃ and by H̃ = Aut_{E}V its centralizer in G̃. There is a canonical identification of the non-enlarged affine building Ĩ_{β} of H̃ with the H̃-set Latt_{𝔬_{E}}(V). Here we normalize the lattice functions of Latt_{𝔬_{E}}^{1}(V) by the condition Λ(s + v(π_{E})) = π_{E}Λ(s), s ∈ ℝ, where π_{E} is a uniformizer of E.

Any Λ̄ ∈ Latt_{𝔬_{F}}(V) defines a filtration (𝔤̃_{Λ̄,r})_{r∈ℝ} by

𝔤̃_{Λ̄,r} = {a ∈ End_{F} V ; aΛ(s) ⊂ Λ(r + s), s ∈ ℝ} .

Then the map End(Λ̄) : r ↦ 𝔤̃_{Λ̄,r} is an element of Latt^{1}𝔤̃. The map Λ̄ ↦ End(Λ̄), Latt_{𝔬_{F}}V → Latt^{1}𝔤̃ is a G̃-equivariant injection (cf. [BL]§4) for the action of G on Latt^{1}𝔤̃ by conjugation. Its image is Latt^{2}𝔤̃. From now on we shall canonically identify Ĩ (resp. Ĩ_{β} with Latt^{2}𝔥̃).

Let us recall the main result of [BL]. Theorem 10.1. There exists a unique affine and H̃-equivariant map j̃_{β} : Ĩ_{β} → Ĩ. It is injective, maps any apartment into an apartment and is compatible with the Lie algebra filtrations in the following sense:

(10.2)  𝔤̃_{j̃_{β},r} ∩ 𝔥̃ = 𝔥̃_{x,r}, x ∈ Ĩ_{β}, r ∈ ℝ .

Let us recall how j̃_{β} is constructed. If x ∈ Ĩ_{β} corresponds to End(Λ̄) ∈ Latt^{2}𝔥̃, then j̃(x) simply corresponds to End(Λ̄), where Λ, an 𝔬_{E}-lattice function in V, is now considered as an 𝔬_{F} -lattice function. Theorem 10.3. Let x ∈ Ĩ_{β} and y ∈ Ĩ satisfying

𝔤̃_{y,r} ∩ 𝔥̃ ⊃ 𝔥̃_{x,r}, r ∈ ℝ .

Then y = j̃_{β}(x). As a consequence the map j̃_{β} is characterized by property (10.2). Proof. Assume that x and y correspond to elements Λ̄_{x} and Λ̄_{y} of Latt_{𝔬_{E}}(V) and Latt_{𝔬_{F}}(V) respectively. Lemma 10.4. Under the assumption of (10.2), Λ_{y} is an 𝔬_{E}-lattice function. Proof. To prove that Λ_{y} is an 𝔬_{E}-lattice function we must prove that it is normalized by E^{×} = ⟨π_{E}⟩𝔬_{E}^{×}, or equivalently:

(10.5)  x𝔤̃_{y,r}x^{−1} = 𝔤̃_{y,r}, x ∈ E^{×}, r ∈ ℝ .

We first notice than 𝔬_{E} ⊂ 𝔥̃_{x,0} ⊂ 𝔤̃_{y,0}, so that 𝔬_{E}^{×} ⊂ 𝔤̃_{y,0}^{×} and (10.5) is true for x ∈ 𝔬_{E}^{×}. We are reduced to proving (10.5) when x = π_{E}.

We have π_{E} ∈ 𝔥̃_{x,1/e} ⊂ 𝔤̃_{y,1/e} and π_{E}^{−1} ⊂ 𝔥̃_{x,−1/e} ⊂ 𝔤̃_{y,−1/e}, where e = e(E/F). It follows that

(10.6)  π_{E}𝔤̃_{y,r}π_{E}^{−1} ⊂ 𝔤̃_{y,1/e}𝔤̃_{y,r}𝔤̃_{y,−1/e} ⊂ 𝔤̃_{y,r}, r ∈ ℝ .

Consider the duality “∗” on subsets of 𝔤̃ given by

S^{∗} = {a ∈ 𝔤̃ ; Tr(aS) ⊂ 𝔭_{F} }, S ⊂ 𝔤̃,

where Tr is the trace map. Recall ([BL](6.3)) that (𝔤̃_{y,r})^{∗} = 𝔤̃_{y,(−r)+}, r ∈ ℝ. Using a well known property of the trace map, we observe that

(π_{E}𝔤̃_{y,r}π_{E}^{−1})^{∗} = π_{E}(𝔤̃_{y,r})^{∗}π_{E}^{−1}, r ∈ ℝ .

So applying the duality to (10.6), we obtain

𝔤̃_{y,(−r)+} ⊂ π_{E}𝔤̃_{y,(−r)+}π_{E}^{−1}, r ∈ ℝ .

We have proved that on each open interval (r_{1}, r_{2}) where the lattice function (𝔤̃_{y,r})_{r∈ℝ} is constant, we have both containments

π_{E}𝔤̃_{y,r}π_{E}^{−1} ⊂ 𝔤̃_{y,r} and π_{E}𝔤̃_{y,r}π_{E}^{−1} ⊂ 𝔤̃_{y,r}, r ∈ ℝ .

So by continuity we have π_{E}𝔤̃_{y,r}π_{E}^{−1} = 𝔤̃_{y,r}, for all r, as required.

Let us return to the proof of (10.3). Since Λ_{y} is an 𝔬_{E}-lattice function, we have

𝔤̃_{y,r} ∩ 𝔥̃ = 𝔥̃_{x′,r}, r ∈ ℝ,

where x′ ∈ Ĩ_{β} is attached to Λ̄_{y}, Λ_{y} being seen as an 𝔬_{E}-lattice function. So by injectivity of the map Latt_{𝔬_{E}}^{1}(V) → Latt^{2}𝔥̃, we have Λ̄_{x} = Λ̄_{y} and y = j̃_{β}(x) by definition.

11. A unicity result in the 1-block case and a conjecture

With the notation of §5, we consider an element β ∈ 𝔤 satisfying:

(11.1)  E := F [β] ⊂ 𝔤̃ is a field and β ≠ 0 .

We fix an ε-hermitian form h_{E} on the E-vector space V relative to σ_{E} and we assume that it satisfies (5.3) as well as the condition J = 𝔭_{E^{o}} of §5. This allows us to identify I_{β}^{1} with Latt_{h_{E}}^{1}(V). Identifying I with Latt_{h}(V), the map j_{β} of §6 is simply given by

j_{β}(Λ) = Λ, Λ ∈ Latt_{h_{E}}^{1}(V),

where on the right hand side Λ is considered as an 𝔬_{F} -lattice function. Theorem 11.2. Under the assumption (11.1), let x ∈ I_{β}^{1} and y ∈ I satisfying

(11.3)  𝔤_{y,r} ∩ 𝔥 = 𝔥_{x,r}, r ∈ ℝ .

Then y = j_{β}(x). In particular the map j_{β} is characterized by compatibility with the Lie algebra filtrations. Proof. The point x (resp. y) corresponds to a self-dual lattice function Λ_{x} ∈ Latt_{h_{E}}^{1}(V) (resp. Λ_{y} ∈ Latt_{h}^{1}(V)). We may see x and y as points of Latt_{𝔬_{E}}^{1}(V) and Latt_{𝔬_{F}}^{1}(V) respectively and they give rise to filtrations of 𝔥̃ and 𝔤̃ as in §9: (𝔥̃_{x,r})_{r∈ℝ} and (𝔤̃_{y,r})_{r∈ℝ}. Write

𝔤_{y,r}^{+} = {a ∈ 𝔤̃_{y,r} ; a = a^{σ}}, r ∈ ℝ

and

𝔥_{x,r}^{+} = {a ∈ 𝔥̃_{x,r} ; a = a^{σ}}, r ∈ ℝ

Since 2 is invertible in 𝔬_{F}, we have:

𝔤̃_{y,r} = 𝔤_{y,r} ⊕ 𝔤_{y,r}^{+} and 𝔥̃_{y,r} = 𝔥_{x,r} ⊕ 𝔥_{x,r}^{+}, r ∈ ℝ .

Write

r_{o} = v_{Λ_{x}}(β) := Sup{r ∈ ℝ ; β ∈ 𝔥̃_{x,r}} .

Since β ∈ E^{×}, it normalizes Λ_{x} so that β𝔥̃_{x,r} = 𝔥̃_{x,r+r_{o}}, r ∈ ℝ. Moreover since β is central in 𝔥̃, we easily have that 𝔥_{x,r}^{+} = β𝔥_{x,r−r_{o}}, r ∈ ℝ. Hence, for r ∈ ℝ, we have

𝔥_{x,r}^{+} = β(𝔤_{y,r−r_{o}} ∩ 𝔥) = β(𝔤_{y,r−r_{o}} ∩ 𝔥̃) ⊂ 𝔤_{y,r} ∩ 𝔥̃ .

It follows that, for x ∈ ℝ, we have:

𝔥̃_{x,r} = 𝔥_{x,r} ⊕ 𝔥_{x,r}^{+} ⊂ 𝔤_{y,r} ∩ 𝔥̃ ⊕ 𝔤_{y,r}^{+} ∩ 𝔥̃ ⊂ 𝔤̃_{y,r} ∩ 𝔥̃ .

By applying (10.3), we obtain Λ̄_{y} = j̃_{β}(Λ̄_{x}), that is Λ̄_{y} = Λ̄_{x}. In particular we have End(Λ_{x}) = End(Λ_{y}) ∈ Latt_{σ}^{2}𝔥̃. But by (3.5) we have Λ_{x} = Λ_{y}, as required.

Let us give an example. Assume that G = Sp_{2}(F) = SL(2, F) (here F = F_{o}) and take β ∈ 𝔤 such that E/F is quadratic and ramified. Then H is the group E^{1} of norm 1 elements in E. The building of H is reduced to a point {x}. The group E^{×} fixes a unique chamber C of I and H ⊂ E^{×} fixes C pointwise. There are infinitely many maps j : I_{β}^{1} → I which are affine and G-equivariant; indeed j(x) can be any point of C. On the other hand there is a unique map j : I_{β}^{1} → I which is compatible with the Lie algebra filtrations: it maps x to the isobarycenter of C.

We conjecture that when J = ∅ (notation of §5) then the map j_{β} of §6 is characterized by condition (11.3). We may address the more general (but more informal) question. Being given two F -reductive groups H and G, as well as a morphism of algebraic groups φ : H → G, is there an affine and H(F)-equivariant map I(H, F) → I(G, F) which is compatible with the Lie algebra filtrations defined by Moy and Prasad. When is it characterized by this last property?

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[S1] S. Stevens, Semisimple strata for p-adic classical groups, Ann. Sci. École Norm. Sup. (4), Vol. 35, No. 3, 2002, pp. 423–435.

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Université de Poitiers School of Mathematics Département de Mathématiques University of East Anglia UMR 6086 du CNRS Norwich NR4 7TJ Bd Marie et Pierre Curie United Kingdom Téléport 2 – BP 30179 86962 Futuroscope Chasseneuil Cedex France