ON p-ADIC NORMS AND QUADRATIC EXTENSIONS, II

CHRISTOPHE CORNUT

Abstract. We investigate the graph theoretical aspects and combinatorics of the action of a unitary group in n variables over a local non-archimedian field on the Bruhat-Tits building of an orthogonal group in 2n + 1 variables.

1. Introduction

1.1. This paper is part of an ongoing project whose final aim is to construct and study a new Euler System for certain motives over totally real fields. These motives, which are pure of weight −1 and endowed with a symplectic duality, should occur in suitable twists of the middle cohomology of Shimura varieties attached to certain orthogonal groups. As for the Euler System, it should essentially be the image by suitable Abel-Jacobi maps of special cycles of unitary type in the ambient Shimura variety. The basic case is the Euler System of Heegner points, which is closely related to the spherical pair U(1) ⊂ SO(2, 1), giving rise to divisors on Shimura curves [2]. Our project involves the spherical pair U(1, n − 1) ⊂ SO(2, 2n − 1), which yields cycles of codimension n on Shimura varieties of dimension 2n − 1. One of the first step in this program is to obtain a reasonable understanding of the distribution relations that intertwine the Hecke and Galois actions on these cycles.

1.2. This mostly reduces to the following local question. Let E/F be a quadratic extension of p-adic local fields, and let (V, ϕ) = (W, tr_{E/F} ψ)⊥(D, ϕ) be a quadratic F -space which is the orthogonal sum of the quadratic space that underlies an Hermitian E-space (W, ψ) and an anisotropic F -line D. Put

SU = SU(W, ψ) ⊂ U = U(W, ψ) ⊂ SO = SO(V, ϕ)

and let K be a compact open subgroup of SO. Let S = ind_{SU}^{SO}1 be the space of all locally constant and compactly supported functions SU\SO → C, on which SO and T = U/SU act respectively by right and left multiplication. The local model of our space of cycles with its commuting Hecke and Galois actions turns out to be the space S^{K} of right K-invariant functions in S, with the commuting actions of the local Hecke algebra H = H(SO, K) and the torus T.

1.3. We thus want to have some understanding on the structure of this specific H[T]-module, especially in the case where E/F is unramified, SO is split, and K is hyperspecial. More generally, the results of this paper pertain to the case where E/F is unramified and K is the stabilizer of a vertex in the Bruhat-Tits building I of SO (with no further assumptions on the odd orthogonal group SO).

1.4. In a previous paper [1], we have described the orbits of U on the building I, merely viewing the latter as a set. We now also consider the simplicial structure of that building. The main result is a formula (or perhaps more something like an algorithm) that computes, for each vertex or edge in the building, the volume of its stabilizer in U (using section 5.4 and Proposition 4.4.1) and the image of this stabilizer in the compact torus T (Lemma 5.1.14).

1.5. Notation & Conventions.

1.5.1. We let F be a non-archimedian local field of odd residue characteristic p with ring of integers O, maximal ideal P, and finite residue field F = O/P of order q (a power of p). We let π be a uniformizing element of F, so that P = Oπ. We denote by |•| : F → R_{≥0} the normalized discrete valuation of F for which |π| = q^{−1}.

1.5.2. We fix once and for all an unramified quadratic extension E of F, with ring of integers O_{E}, maximal ideal PO_{E} and residue field E = O_{E}/PO_{E}, a quadratic extension of F. For n ≥ 0, we denote by O_{n} =^{def} O+P^{n}O_{E} the order of conductor n in O_{E} = O_{0}. It is a local, complete, Gorenstein, 1-dimensional ring with maximal ideal R_{n} = PO_{n−1} and residue field O_{n}/R_{n} ≃ F if n > 0 (and maximal ideal R_{0} = PO_{0} and residue field E = O_{0}/R_{0} if n = 0). We denote by λ ↦ \overline{λ} the non-trivial automorphism of E/F and let tr : E → F and N : E → F be the usual trace and norm maps. We let η be any generator of the free, rank one O-module O_{E} ∩ ker tr. Therefore η belongs to O_{E}^{×} , \overline{η} = −η, η^{2} ∈ O^{×} and O_{n} = O ⊕ Oπ^{n}η for all n ≥ 0.

1.5.3. An F -norm on an F -vector space V is a map α : V → R_{≥0} such that

α(x + y) ≤ max {α(x), α(y)} , α(ax) = |a| α(x) and α(x) = 0 ⟺ x = 0

for all x, y ∈ V and a ∈ F. If α is such a norm and λ ∈ R, we denote by

B^{0}(α, λ) =^{def} {x ∈ V : α(x) < q^{λ}} and B(α, λ) =^{def} {x ∈ V : α(x) ≤ q^{λ}}

its open and closed balls of radius q^{λ}. These are O-sublattices in V for which PB(α, λ) ⊂ B^{0}(α, λ) ⊂ B(α, λ). We abusively refer to the quotient F-vector space

S(α, λ) =^{def} B(α, λ)/B^{0}(α, λ)

as the sphere of radius q^{λ}.

2. The linear case

2.1. Classification.

2.1.1. Let N be the O-linear and additive category whose objects are pairs (V, α) where V is a finite dimensional E-vector space and α is an F -norm on V. A morphism f : (V_{1}, α_{1}) → (V_{2}, α_{2}) is an E-linear morphism f : V_{1} → V_{2} such that α_{2}(f(v)) ≤ α_{1}(v) for all v ∈ V_{1}. The category N has an internal Hom [1, §3.1.3], a tensor product [1, §3.5] and a duality [1, §3.3] which are compatible with the analogous operations on the category of E-vector spaces:

\underline{Hom} ((V_{1}, α_{1}), (V_{2}, α_{2})) = (Hom_{E}(V_{1}, V_{2}), Hom(α_{1}, α_{2}))
(V_{1}, α_{1}) ⊗ (V_{2}, α_{2}) = (V_{1} ⊗_{E} V_{2}, α_{1} ⊗ α_{2})
(V, α)^{⋆} = (V^{⋆}, α^{⋆})

2.1.2. Let (V, α) be a 1-dimensional object of N. There exists a non-zero element e of V and a pair (ρ, c) ∈ R × R_{≥0} such that for all λ ∈ E, α(λe) = q^{ρ} ∥λ∥_{c} where the F -norm ∥•∥_{c} : E → R_{≥0} is defined by

∀x, y ∈ F : ∥x + yη∥_{c} = max {q^{−c} |x| , |y|} .

We say that α has type (ρ, c) with respect to e. The possible types of α form a class [ρ, c] in R/Z × R_{≥0} and the map (V, α) ↦ [ρ, c] induces a bijection

L ∋ [V, α] ⟼^{≃} [ρ, c] ∈ R/Z × R_{≥0}

where L is the set of isomorphism classes of 1-dimensional objects in N.

2.1.3. For i ∈ {1, 2}, let (V_{i}, α_{i}) be a 1-dimensional object of type (ρ_{i}, c_{i}) with respect to some E-basis e_{i} of V_{i}. Then α_{1} ⊗ α_{2} is of type (ρ_{1} + ρ_{2}, min(c_{1}, c_{2})) with respect to the E-basis e_{1} ⊗ e_{2} of V_{1} ⊗ V_{2} [1, §3.5].

2.1.4. Similarly, if (V, α) is of type (ρ, c) with respect to some E-basis e of V, then (V, α)^{⋆} = (V^{⋆}, α^{⋆}) is of type (c − ρ, c) with respect to the dual E-basis e^{⋆} of V^{⋆} [1, §3.3]. This second formula suggests the change of variables

(\frac{θ+c}{2}, c) = (ρ, c) ↔ (θ, c) = (2ρ − c, c).

In these new (θ, c)-coordinates, we have L ≃ R/2Z × R_{≥0} and the formula for the duality becomes just (θ, c) ↦ (−θ, c).

2.1.5. Any object (V, α) of N decomposes (non-uniquely) as a direct sum of 1dimensional subobjects (V, α) = ⊕(V_{i}, α_{i}), and the map

(V, α) ↦ [V, α] =^{def} ∑[V_{i}, α_{i}]

defines a bijection between the set of isomorphism classes of objects in N and the commutative monoid N[L] of effective divisors on L.

2.1.6. For (V, α) ∈ N, we put det(V, α) = (det V, det α) where

det V =^{def} Λ_{E}^{n}V ⊂ V^{⊗n}  and  det α =^{def} α^{⊗n}| det V

with n = dim_{E} V. If (V, α) = ⊕_{i=1}^{n}(V_{i}, α_{i}) with α_{i} of type (ρ_{i}, c_{i}) with respect to some E-basis e_{i} of V_{i}, then det V is of type (∑ ρ_{i}, min{c_{i}}) with respect to the E-basis e_{1} ∧ ⋯ ∧ e_{n} = ∑_{σ∈𝔖_{n}} sign(σ)e_{σ(1)} ⊗ ⋯ ⊗ e_{σ(n)} of det V (by 2.1.3).

2.1.7. Fix a finite dimensional E-vector space V and let N (V) be the set of all F -norms on V: this is the Bruhat-Tits building of GL_{F} (V). Let n = dim_{E} V and

N[L]^{n} =^{def} {D ∈ N[L] : deg D = n}.

Then by 2.1.5, the map α ↦ [α] = [V, α] induces a bijection

[•] : GL_{E}(V)\N (V) ⟶^{≃} N[L]^{n}.

For [α] = ∑_{i=1}^{n}[ρ_{i}, c_{i}], we set c_{min}(α) = min{c_{i}} and c_{max}(α) = max{c_{i}}.

2.1.8. The following lemma should have been included in [1].

Lemma. Let GL(α) ⊂ GL_{E}(V) be the stabilizer of α ∈ N (V). Then

detGL(α) = O_{⌈c_{min}(α)⌉}^{×}.

Proof. Fix a decomposition (V, α) = ⊕(V_{i}, α_{i}) with (V_{i}, α_{i}) of type (ρ_{i}, c_{i}). Then

O_{⌈c_{i}⌉}^{×} = GL(α_{i}) = detGL(α_{i}) ⊂ detGL(α) ⊂ GL(det α) = O_{⌈c_{min}(α)⌉}^{×}

using 2.1.6 for the last equality. This proves our claim since c_{min}(α) = min{c_{i}}. □

2.2. Filtrations. Fix an F -norm α on a finite dimensional E-vector space V.

2.2.1. For any real number c ≥ 0, let α_{c} be the F -norm on V ≃ V ⊗_{E} E defined by α_{c} = α ⊗ ∥•∥_{c}, so that α_{c} = α for all c ≥ c_{max}(α) [1, §3.5]. Since the identity map on E is a morphism (E, ∥•∥_{c_{1}}) → (E, ∥•∥_{c_{2}}) for any c_{1} ≤ c_{2}, we actually obtain a family of compatible norms on V,

α_{0} → ⋯ → α_{c_{1}} → ⋯ → α_{c_{2}} → ⋯ → α.

Applying the same construction to the dual norm α^{⋆} on V^{⋆} [1, §3.3] and dualizing back to V, we obtain a dual compatible family of F -norms on V,

α → ⋯ → α^{c_{2}} → ⋯ → α^{c_{1}} → ⋯ → α^{0}

where α^{c} = ((α^{⋆})_{c})^{⋆} with again α^{c} = α for c ≥ c_{max}(α).

2.2.2. For any c ≥ 0 and λ ∈ R, we define F-subspaces of S(α, λ) by

S_{c}(α, λ) = ker (S(α, λ) → S(α^{c}, λ)) and S^{c}(α, λ) = im (S(α_{c}, λ) → S(α, λ)) .

Therefore S_{c}(α, λ) is decreasing from S_{0}(α, λ) to S_{∞}(α, λ) = 0, while S^{c}(α, λ) is increasing from S^{0}(α, λ) to S^{∞}(α, λ) = S(α, λ). From the study of the 1-dimensional case below (together with 2.1.5), we find that S_{c}(α, λ) ⊂ S^{c}(α, λ) for all c ≥ 0. We have thus obtained a canonical filtration of S(α, λ) by F-subspaces

0 ⊂ ⋯ ⊂ S_{c}(α, λ) ⊂ ⋯ ⊂ S_{0}(α, λ) ⊂ S^{0}(α, λ) ⊂ ⋯ ⊂ S^{c}(α, λ) ⊂ ⋯ ⊂ S(α, λ).

2.2.3. For any c ≥ 0 and λ ∈ R, we define a subquotient of S(α, λ) by

S(α, λ, c) = S^{c}(α, λ)/S_{c}(α, λ).

We view this subquotient as the middle term (that is, the image or coimage) in the canonical factorization of the composite map S(α_{c}, λ) → S(α, λ) → S(α^{c}, λ):

S(α, λ)
↗  ↘
S(α_{c}, λ) ↠ S(α, λ, c) ↪ S(α^{c}, λ)

Since multiplication by η on E is a morphism (E, ∥•∥_{c}) → (E, q^{−c} ∥•∥_{c}) in the category N, multiplication by η on V yields a commutative diagram

α_{c}  ⟶^{×1} α^{c}
×η ↓  ↓ ×η
q^{−c}α_{c} ⟶^{×1} q^{−c}α^{c}

Passing to the spheres, we obtain the outer commutative square in

S(α_{c}, λ) ↠ S(α, λ, c) ↪ S(α^{c}, λ)
[η] ↓ [η] ↓ [η] ↓
S(α_{c}, λ + c) ↠ S(α, λ + c, c) ↪ S(α^{c}, λ + c)

in which the middle vertical map is uniquely determined by the commutativity of the whole diagram. From the analysis of the 1-dimensional case below (together with 2.1.5), we find that for c > 0,

(2.1)  ker ([η] : S(α, λ, c) → S(α, λ + c, c)) = S_{−}^{c}(α, λ)/S_{c}(α, λ)
(2.2)  im ([η] : S(α, λ − c, c) → S(α, λ, c)) = S_{c}^{+}(α, λ)/S_{c}(α, λ)

where S_{−}^{c}(α, λ) = ∪_{d<c}S^{d}(α, λ) and S_{c}^{+}(α, λ) = ∩_{d<c}S_{d}(α, λ). For c = 0, [η] is an automorphism of S(α, λ, 0) which defines an E = F[η]-vector space structure on this F-vector space.

2.2.4. For all c ≥ 0 and λ ∈ R, we finally define

Gr^{c}S(α, λ) = S^{c}(α, λ)/S_{−}^{c}(α, λ) and Gr_{c}S(α, λ) = S_{c}^{+}(α, λ)/S_{c}(α, λ)

where S_{−}^{0}(α, λ) = S_{0}(α, λ) and S_{0}^{+}(α, λ) = S^{0}(α, λ), and still denote by

[η] : Gr^{c}S(α, λ) → Gr_{c}S(α, λ + c)

the isomorphism induced by [η] : S(α, λ, c) → S(α, λ + c, c).

2.2.5. Suppose that dim_{E} V = 1 and α has type (ρ, d) with respect to the E-basis e of V. Then α^{⋆} is of type (d − ρ, d) with respect to the dual basis e^{⋆} of V^{⋆}. For c ≥ d, α_{c} = α = α^{c} and thus for any λ ∈ R,

S_{c}(α, λ) = {0} ⊂ S^{c}(α, λ) = S(α, λ) = S(α, λ, c).

For c < d, α_{c} is of type (ρ, c) with respect to e, (α^{⋆})_{c} is of type (d − ρ, c) with respect to e^{⋆} and α^{c} = ((α^{⋆})_{c})^{⋆} is of type (ρ + c − d, c) with respect to e. We have

S(α, λ) = 0 unless λ ≡ ρ − d or ρ − ⌈d⌉ mod Z
S(α_{c}, λ) = 0 unless λ ≡ ρ − c or ρ − ⌈c⌉ mod Z
S(α^{c}, λ) = 0 unless λ ≡ ρ − d or ρ + c − d − ⌈c⌉ mod Z

Put n = ⌈d⌉. Elementary computations lead to the following table, where the indicated subquotients X of S(α, λ) are described as A/B if X = Ae/Be.

c < d  d = n  d < n
λ  ρ − d  ρ − d  ρ − n
S(α, λ) O_{n}/PO_{n} O_{n}/R_{n} R_{n}/PO_{n}
S_{c}(α, λ) R_{n}/PO_{n}  0  R_{n}/PO_{n}
S^{c}(α, λ) R_{n}/PO_{n}  0  R_{n}/PO_{n}
S(α, λ, c)  0  0  0

Given the table, it is also easily verified that the map

[η] : S(α, λ, c) → S(α, λ + c, c)

is trivial unless c = d and λ ≡ ρ − d mod Z, in which case it has kernel S_{−}^{d}(α, λ) and image S_{d}^{+}(α, λ + c) if c = d > 0, and yields the obvious E = F[η]-vector space structure on S(α, λ, 0) = S(α, λ) = P^{ρ−λ}O_{E} · e/P^{ρ−λ+1}O_{E} · e ≃ E if c = d = 0.

2.2.6. Returning to the general case, the above table shows that the multiplicity m_{[ρ,c]}(α) of [ρ, c] ∈ L in [α] ∈ N[L] is given by

m_{[ρ,c]}(α) = \begin{cases}dim_{E} S(α, ρ, 0) & if c = 0, \\ dim_{F} Gr^{c}S(α, ρ − c) = dim_{F} Gr_{c}S(α, ρ) & if c ≠ 0. \end{cases}

3. Buildings of orthogonal Groups

Let V be a finite dimensional F -vector space with a non-degenerate symmetric pairing ϕ : V ×V → F and let G = O(V, ϕ) be the corresponding orthogonal group.

3.1. The dual of a lattice L in V is the lattice L̃ = {v ∈ V |ϕ(v, L) ⊂ O}. We say that L is almost self-dual if and only if P L̃ ⊂ L ⊂ L̃. The dual of a π-periodic flag of lattices B in V is the π-periodic flag of lattices B̃ = {L̃|L ∈ B}. We say that B is self -dual if and only if B̃ = B. The dual of a norm α : V → R is the norm α̃ : V → R defined by α̃(v) = sup_{w∈V} \frac{|ϕ(v,w)|}{α(w)}. We say that α is self-dual if and only if α̃ = α. These various concepts are related as follows.

3.2. For a self-dual π-periodic flag of lattices B in V, the almost self-dual lattices in B form a non-empty finite segment |B| = {L_{0} ⊊ L_{1} ⊊ ⋯ ⊊ L_{r}} of

B = {⋯ ⊊ PL̃_{0} ⊂ L_{0} ⊊ L_{1} ⊊ ⋯ ⊊ L_{r} ⊂ L̃_{r} ⊊ L̃_{r−1} ⋯ ⊊ L̃_{0} ⊂ P^{−1}L_{0} ⊊ ⋯ } .

The map B ↦ |B| yields a G-equivariant bijection between the set of all π-periodic self-dual flags of lattices in V and the set of all non-empty finite chains of almost self-dual lattices in V.

3.3. For a self-dual norm α on V, the flag of balls B(α) = {B(α, λ)|λ ∈ R} is a self-dual π-periodic flag of lattices in V and |B(α)| = \{B(α, λ)|λ ∈ [0, \frac{1}{2}[\}. Indeed

(3.1) { \atop B(α, λ)} = B^{0}(α, 1 − λ) and { \atop B^{0}(α, λ}) = B(α, 1 − λ) for all λ ∈ R.

The map α ↦ B(α) is a G-equivariant surjection between the set of all self-dual norms on V and the set of all π-periodic self-dual flags of lattices in V.

3.4. We say that two self-dual norms α and β on V belong to the same facet F if and only if B(α) = B(β) = B(F). This defines a partition of the set I of all selfdual norms on V into facets, which may be identified with π-periodic self-dual flags of lattices in V, and also with non-empty finite chains of almost self-dual lattices in V. We define a G-equivariant partial order on the set of facets by

F ≺ G ⟺ B(F) ⊂ B(G) ⟺ |B(F)| ⊂ |B(G)| .

3.5. The dimension and type of a facet F with |B(F)| = {L_{0} ⊊ L_{1} ⊊ ⋯ ⊊ L_{r}} are respectively defined by dim F = r and

type F = dim_{F} (L_{0}/PL̃_{0}, L_{1}/L_{0}, ⋯, L_{r}/L_{r−1}, L̃_{r}/L_{r}) ∈ N^{r+2}.

These invariants depend only upon the G-orbit of F, and we shall see that the type in fact classifies these G-orbits. The dimension and type of a self-dual norm α are those of its facet F(α).

3.6. Let V = V_{0}⊥H_{1}⊥ ⋯ ⊥H_{m} be a Witt decomposition: V_{0} is an anisotropic F -subspace of V and the H_{i}’s are hyperbolic F -planes in V. The apartment A ⊂ I defined by such a decomposition is the set of all self-dual norms α on V such that for all v_{0} ∈ V_{0} and h_{i} ∈ H_{i},

α (v_{0} + h_{1} + ⋯ + h_{m}) = max {α(v_{0}), α(h_{1}), ⋯, α(h_{m})} .

The choice of a Witt basis (e_{i}, e_{−i}) of H_{i} for each i defines an isometry

coord_{(e_{±i})} : A ⟶^{≃} R^{m} given by coord_{(e_{±i})}(α) = (λ_{1}, ⋯, λ_{m})

where α(e_{±i}) = q^{±λ_{i}} and for all v_{0} ∈ V_{0} and x_{±i} ∈ F,

α (v_{0} + ∑ (x_{i}e_{i} + x_{−i}e_{−i})) = max {|Q(v_{0})|^{1/2} , |x_{±i}| q^{±λ_{i}}} .

3.7. Any self-dual norm α is contained in some apartment A as above, and it is possible to choose the Witt basis (e_{i}, e_{−i}) of the H_{i}’s so that α has coordinates

coord_{(e_{±i})}(α) = (λ_{1}, ⋯, λ_{m}) with 0 ≤ λ_{1} ≤ λ_{2} ≤ ⋯ ≤ λ_{m} ≤ \frac{1}{2}.

These coordinates are then uniquely determined by the G-orbit of α. Indeed, let

δ_{0} = dim_{F} S(α|V_{0}, 0) and δ_{\frac{1}{2}} = dim_{F} S(α|V_{0}, \frac{1}{2}).

These integers depend only upon (V, ϕ) and sum to the F -dimension of V_{0}, the “anisotropic part” of V. For λ ∈ [0, \frac{1}{2}], put m(λ) = |\{i ∈ 1, ⋯, m|λ_{i} = λ\}|. Then

type F(α) = (δ_{0} + 2m(0), m(s_{1}), ⋯, m(s_{r}), δ_{\frac{1}{2}} + 2m(\frac{1}{2}))

where \{s_{1} < ⋯ < s_{r}\} = \{λ_{1}, ⋯, λ_{m}\}∩]0, \frac{1}{2}[= \{λ ∈]0, \frac{1}{2}[|S(α, λ) ≠ 0\}. We refer to

rtype F(α) =^{def} (m(0), m(s_{1}), ⋯, m(s_{r}), m(\frac{1}{2})) ∈ N^{r+2}

as the reduced type of the facet and call inv(α) =^{def} (λ_{1}, ⋯, λ_{n}) the invariant of α.

3.8. Since all data (V_{0}, (e_{±i})) as above are conjugated by G, we see that two selfdual norms belong to the same G-orbit if and only if they have the same invariant. The same argument also shows that two facets belong to the same G-orbit if and only if they have the same type (or reduced type). If the anisotropic part V_{0} of V is non-trivial, any self-dual norm α on V is fixed by an element g ∈ G with det g = −1 and we may replace G = O(V, ϕ) by SO(V, ϕ) in the above statements.

3.9. Let F be a facet of reduced type (m_{0}, m_{1}, ⋯, m_{r}, m_{r+1}). Let α be any self-dual norm in F and choose an apartment A containing α, together with a datum (V_{0}, (e_{±i})) as above such that coord_{(e_{i})}α = invα = (λ_{1}, ⋯, λ_{m}). Define Δ_{r} = \{0 < t_{1} < ⋯ < t_{r} < \frac{1}{2}\} and map each element of Δ_{r} to the norm γ ∈ A with coordinates (ν_{1}, ⋯, ν_{m}) characterized by 0 ≤ ν_{1} ≤ ⋯ ≤ ν_{m} ≤ \frac{1}{2} and

m_{0} = |\{i|ν_{i} = 0\}| , m_{r+1} = |\{i|ν_{i} = \frac{1}{2}\}| and m_{j} = |\{i|ν_{i} = t_{j}\}|

for all j ∈ {1, ⋯, r}. Let Δ_{r}((e_{±i})) ⊂ A be the image of this map. An elementary computation shows that for any norm γ in A, B(γ) = B(α) ⟺ γ ∈ Δ_{r}((e_{±i})). Now let β be any norm in F and choose g ∈ G so that γ = gβ lies in A with coordinates 0 ≤ ν_{1} ≤ ⋯ ≤ ν_{m} ≤ \frac{1}{2}. Since γ and α have the same type, it must be that γ ∈ Δ_{r}((e_{±i})), so that B(γ) = B(α). However B(γ) = g · B(β) and B(α) = B(β), so that g · B(β) = B(β), hence g · |B(β)| = |B(β)| and g actually fixes each of the balls of β, i.e. γ = gβ = β. This shows that F = Δ_{r}((e_{±i})).

3.10. A vertex is given by one of the following equivalent objects:

• a self-dual norm α on V with log_{q} α(V − \{0\}) ⊂ \frac{1}{2}Z,

• a 0-dimensional facet F,

• an almost self-dual lattice L in V.

The equivalences are given by F = {α} and |B(F)| = |B(α)| = {L}. We typically denote by α_{L} the self-dual norm with |B(α_{L})| = {L}. The type of the vertex is

dim_{F} (L/P L̃, L̃/L) = dim_{F} (S(α, 0), S(α, \frac{1}{2})) ∈ N^{2}.

We denote by I(\frac{1}{2}) the set of all vertices.

3.11. An edge is given by one of the following equivalent objects

• a self-dual norm α on V with log_{q} α(V − \{0\}) ⊂ \frac{1}{4}Z,

• a facet F of dimension ≤ 1,

• a pair of almost self-dual lattices (L_{0} ⊂ L_{1}) in V,

The equivalences are given by F = F(α), {L_{0}, L_{1}} = |B(F)| = |B(α)|, F is the segment [α_{L_{0}}, α_{L_{1}}] joining α_{L_{0}} and α_{L_{1}} in the building I, and α is the middle of that segment. We typically write α = α_{L_{0}L_{1}} for this middle point. By convention, the source and target of the edge are the vertices corresponding respectively to L_{0} and L_{1}. The type of the edge is the triple

dim_{F} (L_{0}/PL̃_{0}, L_{1}/L_{0}, L̃_{1}/L_{1}) = dim_{F} (S(α, 0), S(α, \frac{1}{4}), S(α, \frac{1}{2})) ∈ N^{3}.

We denote by I(\frac{1}{4}) the set of all edges, so that I(\frac{1}{2}) ⊂ I(\frac{1}{4}) : according to our definitions, the vertices are the degenerate edges {L_{0}, L_{1}} for which L_{0} = L_{1}.

3.12. For any self-dual norm α on V and λ ∈ R, the formula (3.1) shows that the pairing πϕ : B(α, λ) × B(α, 1 − λ) → O_{F} induces a perfect pairing

⟨•, •⟩_{λ} : S(α, λ) × S(α, 1 − λ) → F.

These pairings are symmetric and π-periodic in the following sense:

⟨x, y⟩_{λ} = ⟨y, x⟩_{1−λ} and ⟨[π]X, Y ⟩_{λ} = ⟨X, [π]Y ⟩_{1+λ}

for (x, y) ∈ S(α, λ) × S(α, 1 − λ) and (X, Y) ∈ S(α, 1 + λ) × S(α, 1 − λ). If λ ∈ \frac{1}{2}Z, the formula ϕ_{λ}(x, y) = ⟨x, [π^{2λ−1}]y⟩_{λ} thus defines a perfect symmetric pairing

ϕ_{λ} : S(α, λ) × S(α, λ) → F

and ϕ_{λ−1}([π]x, [π]y) = ϕ_{λ}(x, y).

3.13. With notations as above, an edge of reduced type (a, b, c) is also given by

• a pair (α_{0}, H_{0}) where α_{0} is a vertex of type (a, b + c) and H_{0} is a bdimensional totally isotropic F-subspace of (S(α_{0}, \frac{1}{2}), ϕ_{\frac{1}{2}} );

• a pair (α_{1}, H_{1}) where α_{1} is a vertex of type (a + b, c) and H_{1} is a bdimensional totally isotropic F-subspace of (S(α_{1}, 0), ϕ_{0});

They correspond to the edge F = [α_{0}, α_{1}]. If |B(F)| = {L_{0} ⊂ L_{1}}, then

H_{0} = L_{1}/L_{0} ⊂ L̃_{0}/L_{0} = S(α_{0}, \frac{1}{2}) and H_{1} = PL̃_{0}/PL̃_{1} ⊂ L_{1}/PL̃_{1} = S(α_{1}, 0).

3.14. We need a variant of the above notions involving 2-dimensional facets of reduced type (a, 1, b, c) or (a, b, 1, c), linking edges of reduced type ((a + 1, b, c) and (a, b + 1, c)) or ((a, b + 1, c) and (a, b, c + 1)) respectively. In view of 3.9, the distance between the related two edges (viewed as norms) will thus be equal to \frac{1}{4}. The following ad-hoc definition is hand-picked for our purposes. For any λ ∈ \frac{1}{4}Z, we define a λ-corner to be either one of the following equivalent objects:

• a pair (α_{1}, D_{1}) where α_{1} ∈ I(\frac{1}{4}) is an edge and D_{1} ⊂ S(α_{1}, λ + \frac{1}{4}) is an Fline such that D_{1} ⊂ D_{1}^{⊥} if λ+ \frac{1}{4} belongs to \frac{1}{2}Z, where D_{1}^{⊥} is the orthogonal complement of D_{1} for ϕ_{λ+ \frac{1}{4}} ;

• a pair (α_{2}, H_{2}) where α_{2} ∈ I(\frac{1}{4}) is an edge and H_{2} ⊂ S(α_{2}, λ) is an F-hyperplane such that H_{2}^{⊥} ⊂ H_{2} if λ belongs to \frac{1}{2}Z, where H_{2}^{⊥} is the orthogonal complement of H_{2} for ϕ_{λ}.

The equivalence is now given by the following rules. First of all, α_{1} and α_{2} have the same closed ball at λ + \frac{ϵ}{4}, where ϵ = 1 if λ ∈ \frac{1}{2}Z and ϵ = −1 otherwise. Let next B_{1}^{0} ⊂ B_{1} and B_{2}^{0} ⊂ B_{2} be the open and closed balls of α_{1} at λ + \frac{1}{4} and α_{2} at λ. Then B_{2}^{0} ⊂ B_{1}^{0} ⊊ B_{2} ⊂ B_{1} and

D_{1} = B_{2}/B_{1}^{0} ⊂ B_{1}/B_{1}^{0} = S(α_{1}, λ + \frac{1}{4})), H_{2} = B_{1}^{0}/B_{2}^{0} ⊂ B_{2}/B_{2}^{0} = S(α_{2}, λ).

We denote such a λ-corner by (α_{1}, D_{1}) ↦ (α_{2}, H_{2}).

4. The even orthogonal case

4.1. Classification. We recall the following results of [1, §4].

4.1.1. Let M be the additive category of triples (V, ψ, α) where (V, ψ) is a finite dimensional E-hermitian space and α is a self-dual F -norm on the underlying symmetric F -space (V, ϕ), with ϕ = trψ. A morphism f : (V_{1}, ψ_{1}, α_{1}) → (V_{2}, ψ_{2}, α_{2}) is an E-linear morphism f : V_{1} → V_{2} such that ψ_{2}(f(v), f(w)) = ψ_{1}(v, w) and α_{2}(f(v)) ≤ α_{1}(v) for all v, w ∈ V_{1}.

4.1.2. There are four isomorphism classes of anisotropic E-hermitian space, namely the classes A_{0}, A_{1}, A′_{1} and A_{2} which are respectively represented by

A_{0} = (0, 0), A_{1} = (E, \overline{x}y), A′_{1} = (E, π\overline{x}y) and A_{2} = A_{1}⊥A′_{1}.

Each of these spaces carries a unique self-dual F -norm, namely the F -norm defined by x ↦ |Q(x)|^{1/2} where Q(x) = \frac{1}{2}ϕ(x, x) = ψ(x, x) is the underlying quadratic form. The invariants of these norms are respectively equal to

a_{0} = 0, a_{1} = [0, 0], a′_{1} = [1, 0] and a_{2} = a_{1} + a′_{1} in N[L]

where we have used the (θ, c)-coordinates for a_{1} and a′_{1}. The pair of integers (δ_{0}, δ_{1/2}) attached to the underlying anisotropic symmetric spaces in section 3.7 equals 2(Δ_{0}, Δ_{2}) with (Δ_{0}, Δ_{2}) respectively given by

(0, 0), (1, 0), (0, 1) and (1, 1).

4.1.3. For any object (V, ψ, α) of M, there exists a Witt decomposition

(V, α) = (V_{0}, α_{0})⊥(V_{+}, α_{+}) ⊕ (V_{−}, α_{−})

where V_{0} and V_{±} are respectively anisotropic and totally isotropic E-subspaces of (V, ψ). For any such decomposition, α_{0} is a self-dual norm on V_{0} and x ↦ ηψ(x, •) defines an isomorphism between (V_{−}, α_{−}) and (V_{+}, α_{+})^{⋆}. The isomorphism class of (V, ψ, α) is uniquely determined by the isomorphism class of (V_{0}, ψ|V_{0}) (itself uniquely determined by the isomorphism class of (V, ψ)) together with the invariant

ω(α) =^{def} [V_{+}, α_{+}] = [V_{−}, α_{−}] ∈ N[\overline{L}]

where \overline{L} is the quotient of L induced by the duality ⋆ on N. Note that using the (θ, c)-coordinates, we have \overline{L} ≃ R/ ∼ ×R_{≥0} where the equivalence relation ∼ on R is defined by θ_{1} ∼ θ_{2} ⟺ θ_{1} = ±θ_{2} mod 2Z.

4.1.4. Fix a finite dimensional E-hermitian space (V, ψ) and let I(V, ϕ) be the set of all self-dual norms on the underlying symmetric F -space (V, ϕ): this is the Bruhat-Tits building of SO(V, ϕ). Let n = witt_{E}(V, ψ) be the Witt index and set

N[\overline{L}]^{n} =^{def} \{D ∈ N[\overline{L}] : deg D = n\} .

Then by 4.1.3, the map α ↦ ω(α) induces a bijection

ω : U(V, ψ)\I(V, ϕ) ⟶^{≃} N[\overline{L}]^{n}.

If ω(α) = ∑_{i=1}^{n}[θ_{i}, c_{i}], then c_{max}(α) = max{c_{i}} but

c_{min}(α) = \begin{cases}0 & if dim_{E} V ≠ 2n, \\ min\{c_{i}\} & if dim_{E} V = 2n. \end{cases}

4.1.5. For r ∈ N, let U_{r} = \{z/\overline{z} : z ∈ O_{r}^{×}\}. Thus U_{0} is the group of norm 1 elements in E^{×} and U_{r} ⊂ O_{r}^{×} ∩ U_{0}. In fact, U_{r} × {±1} = O_{r}^{×} ∩ U_{0} for all r > 0, but we will not use this equality.

Lemma. Let U(α) ⊂ U(V, ψ) be the stabilizer of α ∈ I(V, ϕ). Then

det U(α) = U_{r} with r = ⌈c_{min}(α)⌉ .

Proof. Fix a Witt decomposition (V, α) = (V_{0}, α_{0})⊥⊥_{i=1}^{n}(V_{i}, α_{i})⊕(V_{−i}, α_{−i,}) where V_{0} is an anisotropic E-subspace of V and the V_{±i}’s are isotropic E-lines, with α_{i} of type (ρ_{i}, c_{i}) on V_{i}. Then

det U(α_{0}) and det U(α_{i} ⊕ α_{−i}) ⊂ det U(α) ⊂ det GL_{E}(α) ∩ U_{0}

where GL_{E}(α) is the stabilizer of α in GL_{E}(V). Recall from lemma 2.1.8 that det GL_{E}(α) = O_{r}^{×}. On the other hand, it is easy to see that U_{r_{i}} ⊂ det U(α_{i} ⊕ α_{−i}) for r_{i} = ⌈c_{i}⌉ (i ∈ {1, ⋯, n}). Since U(α_{0}) = U(V_{0}, ψ|V_{0}), we also have

det U(α_{0}) = \begin{cases}\{1\} & if V_{0} = 0, \\ U_{0} & if V_{0} ≠ 0. \end{cases}

Therefore U_{r} ⊂ detU(α) ⊂ O_{r}^{×} ∩ U_{0}, which unfortunately only proves the lemma when r = 0. For the general case, see the appendix. □

4.2. Vertices and edges. Fix a finite dimensional E-hermitian space (V, ψ) with underlying symmetric form ϕ and let I = I(V, ϕ) be the building of SO(V, ϕ).

4.2.1. For any self-dual norm α on V with ω(α) = ∑_{i=1}^{n}[ρ_{i}, c_{i}] = ∑_{i=1}^{n}[θ_{i}, c_{i}],

α is a vertex ⇔ ∀i : ρ_{i}, c_{i} ∈ \frac{1}{2}Z ⇔ ∀i : θ_{i}, c_{i} ∈ \frac{1}{2}Z and θ_{i} ≡ c_{i} mod Z
α is an edge ⇔ ∀i : ρ_{i}, c_{i} ∈ \frac{1}{4}Z ⇔ ∀i : θ_{i}, c_{i} ∈ \frac{1}{4}Z and θ_{i} ≡ c_{i} mod \frac{1}{2}Z

In the following tables, we list and name the various elements of \overline{L} which may occur in the ω-invariant of vertices or edges. For each of them, we describe the almost self-dual balls L_{0} ⊂ L_{1} of a self-dual norm α on an hyperbolic E-plane H which has the prescribed ω-invariant. The norm α splits according to a Witt decomposition H = Ee_{+} ⊕ Ee_{−} with ηψ(e_{+}, e_{−}) = 1, and its restriction to Ee_{+} has the indicated type with respect to e_{+}. The parameter is an integer r ∈ N, and the ball L = A_{+}e_{+} ⊕ A_{−}π^{−r}e_{−} is described as A_{+} ⊕ A_{−}. The first three lines corresponds to vertices, and the remaining seven lines to edges.

Name  (θ, c)  L_{0}  L_{1}  L̃_{1}  L̃_{0}
0_{r}  (r, r)  O_{r} ⊕ O_{r}
1_{r+1} (r + \frac{1}{2} , r + \frac{1}{2} )  O_{r} ⊕ P^{−1}O_{r+1}  P^{−1}O_{r+1} ⊕ O_{r}
2_{r}  (r + 1, r)  PO_{r} ⊕ O_{r}  O_{r} ⊕ P^{−1}O_{r}
m_{r}  (r + \frac{1}{2} , r)  PO_{r} ⊕ O_{r}  O_{r} ⊕ O_{r}  O_{r} ⊕ P^{−1}O_{r}
01r+1 (r + \frac{1}{4} , r + \frac{1}{4} )  O_{r+1} ⊕ O_{r}  O_{r} ⊕ O_{r}  O_{r} ⊕ P^{−1}O_{r+1}
21r+1 (r + \frac{3}{4} , r + \frac{1}{4} )  PO_{r} ⊕ O_{r}  O_{r+1} ⊕ O_{r} O_{r} ⊕ P^{−1}O_{r+1}  O_{r} ⊕ P^{−1}O_{r}
02r+1  (r, r + \frac{1}{2} )  O_{r+1} ⊕ O_{r+1}  O_{r} ⊕ O_{r}  P^{−1}O_{r+1} ⊕ P^{−1}O_{r+1}
20r+1 (r + 1, r + \frac{1}{2} )  PO_{r} ⊕ O_{r}  O_{r+1} ⊕ P^{−1}O_{r+1}  O_{r} ⊕ P^{−1}O_{r}
10r+1 (r + \frac{3}{4} , r + \frac{3}{4} )  O_{r+1} ⊕ O_{r}  O_{r+1} ⊕ P^{−1}O_{r+1}  O_{r} ⊕ P^{−1}O_{r+1}
12r+1 (r + \frac{1}{4} , r + \frac{3}{4} ) O_{r+1} ⊕ O_{r+1} O_{r+1} ⊕ O_{r} O_{r} ⊕ P^{−1}O_{r+1} P^{−1}O_{r+1} ⊕ P^{−1}O_{r+1}

4.2.2. Let S be the monoid of all finitely supported functions s : N → N. Define

Ver = {0, 1, 2} and Edg = Ver ∪ {m, 01, 10, 02, 20, 12, 21}.

For any symbol X ∈ Edg and s ∈ S, define

s · X = \begin{cases}∑_{r≥0} s(r) · X_{r} & if X ∈ \{0, 1, 2, m\} \\ ∑_{r≥0} s(r) · X_{r+1} & if X ∈ \{01, 10, 02, 20, 12, 21\} \end{cases} in N[\overline{L}].

Let V and E be the set of all functions X ↦ s_{X} on respectively Ver and Edg with values in S, and view V and E as embedded in N[\overline{L}] by s_{⋆} ↦ ∑_{X} s_{X} · X. Define functions I : S → N, I : V → N and I : E → N by I(s) = ∑_{r≥0} s(r) and I(s_{⋆}) = ∑_{X} I(s_{X}). Let S^{m} ⊂ S, V^{m} ⊂ V and E^{m} ⊂ E be the fibers of I above m ∈ N. Define the reduced type of an element s_{⋆} ∈ V by the formula

rtype (s_{⋆}) = (I(2s_{0} + s_{1}), I(s_{1} + 2s_{2})) ∈ N^{2}.

Define also [1] : S → S by ([1]s)(0) = 0 and ([1]s)(r + 1) = s(r). Finally, define projections source, target : E → V by the following formulas:

source(s_{⋆}) ( \begin{matrix}0 \\ 1 \\ 2 \end{matrix} ) = ( \begin{matrix}s_{0} \\ s_{1} + s_{01} + s_{10} \\ s_{2} + s_{m} + s_{02}[1] + s_{20} + s_{12}[1] + s_{21} \end{matrix} )

target(s_{⋆}) ( \begin{matrix}0 \\ 1 \\ 2 \end{matrix} ) = ( \begin{matrix}s_{0} + s_{m} + s_{01} + s_{10}[1] + s_{02} + s_{20}[1] \\ s_{1} + s_{12} + s_{21} \\ s_{2} \end{matrix} )

Note that I ∘ source = I = I ∘ target : V → N.

4.2.3. Let U = U(V, ψ). Then the ω-invariant α ∈ I ↦ ω(α) ∈ N[\overline{L}] induces • a bijection between the U-orbits of vertices U\I(\frac{1}{2}) and V^{n},

• a bijection between the U-orbits of edges U\I(\frac{1}{4}) and E^{n}. Moreover, rtype (α) = rtype ∘ ω(α) for α in V^{n} and

U\I(\frac{1}{4}) ⟶^{source} U\I(\frac{1}{2}) ⟶^{ω} V^{n} = U\I(\frac{1}{4}) ⟶^{ω} E^{n} ⟶^{source} V^{n},
U\I(\frac{1}{4}) ⟶^{target} U\I(\frac{1}{2}) ⟶^{ω} V^{n} = U\I(\frac{1}{4}) ⟶^{ω} E^{n} ⟶^{target} V^{n}.

This follows from the above tables.

4.3. Filtrations. Fix a self-dual F -norm α on a finite dimension E-hermitian space (V, ψ) with underlying symmetric space (V, ϕ). We denote by (Δ_{0}, Δ_{2}) the pair of integers attached to the anisotropic part of (V, ψ) in section 4.1.2.

4.3.1. For c ≥ 0, the F -norms α_{c} and α^{c} are exchanged by the involution β ↦ β̃ of section 3.1, see [1, 4.1.2]. Thus for any λ ∈ R,

{ \atop B(α_{c}, λ)} = B^{0}(α^{c}, 1 − λ) and { \atop B^{0}(α_{c}, λ)} = B(α^{c}, 1 − λ).

The pairing πϕ : B(α_{c}, λ) × B(α^{c}, 1 − λ) → O_{F} therefore induces a perfect pairing on the corresponding spheres, which fits into the following commutative diagram:

S(α_{c}, λ) × S(α^{c}, 1 − λ) ⟶ F
↓  ↑  ∥
S(α, λ) × S(α, 1 − λ) ⟶^{⟨·,·⟩_{λ}} F

It follows that for any λ ∈ R and c ≥ 0,

S_{c}(α, λ)^{⊥} = S^{c}(α, 1 − λ) and S^{c}(α, λ)^{⊥} = S_{c}(α, 1 − λ)

where the orthogonal complements are taken with respect to ⟨•, •⟩_{λ}.

4.3.2. We thus obtain a family of derived perfect pairings

⟨•, •⟩_{λ,c} : S(α, λ, c) × S(α, 1 − λ, c) → F

which are symmetric and π-periodic in the sense that

⟨x, y⟩_{λ,c} = ⟨y, x⟩_{1−λ,c} and ⟨[π]X, Y ⟩_{λ,c} = ⟨X, [π]Y ⟩_{1+λ,c}

for all (x, y) ∈ S(α, λ, c) × S(α, 1 − λ, c) and (X, Y) ∈ S(α, 1 + λ, c) × S(α, 1 − λ, c). But now also for (x, y) ∈ S(α, λ, c) × S(α, 1 − λ − c, c),

⟨[η]x, y⟩_{λ+c,c} + ⟨x, [η]y⟩_{λ,c} = 0

because ϕ(ηx, y) + ϕ(x, ηy) = 0 since tr_{E/F} η = 0.

4.3.3. The formula [x, y]_{λ,c} = ⟨[η]x, y⟩_{λ−c,c} thus defines a family of perfect pairings

[•, •]_{λ,c} : Gr_{c}S(α, λ) × Gr_{c}S(α, 1 + c − λ) → F

which are now skew-symmetric and π-periodic in the sense that

[x, y]_{λ,c} + [y, x]_{1+c−λ,c} = 0 and [[π]X, Y]_{λ,c} = [X, [π]Y]_{1+λ,c}

for all x, y, X and Y in the relevant spaces.

4.3.4. If λ ∈ \frac{1}{2}Z, then S_{c}(α, λ)^{⊥} = S^{c}(α, λ) for the symmetric pairing ϕ_{λ} on S(α, λ). The S_{c}(α, λ)’s are therefore totally isotropic F-subspaces of (S(α, λ), ϕ_{λ}). The induced perfect symmetric pairings

ϕ_{λ,c} : S(α, λ, c) × S(α, λ, c) → F

may also be defined by ϕ_{λ,c}(x, y) = ⟨x, [π^{2λ−1}]y⟩_{λ,c}. If also c belongs to Z, then

ϕ_{λ,c}([π^{c}η]x, y) + ϕ_{λ,c}(x, [π^{c}η]y) = 0

for all x, y ∈ S(α, λ, c).

4.3.5. If similarly 2λ ∈ c + Z, we obtain a perfect symplectic pairing

θ_{λ,c} : Gr_{c}S(α, λ) × Gr_{c}S(α, λ) → F

defined by θ_{λ,c}(x, y) = [x, [π^{2λ−1−c}]y]_{λ,c}. If also c belongs to Z (i.e. λ ∈ \frac{1}{2}Z), the pairings ϕ_{λ,c} and θ_{λ,c} are related by the following commutative diagram:

ϕ_{λ,c} : Gr^{c}S(α, λ) × Gr_{c}S(α, λ) ⟶ F
[π^{c}η] ↓≃ ∥ ∥
θ_{λ,c} : Gr_{c}S(α, λ) × Gr_{c}S(α, λ) ⟶ F

For c = 0 and λ ∈ \frac{1}{2}Z, these pairings together define a perfect hermitian pairing

ψ_{λ,0} : S(α, λ, 0) × S(α, λ, 0) → E = F[η]

on the E-vector space S(α, λ, 0) by ψ_{λ,0}(x, y) = ϕ_{λ,0}(x, y) − θ_{λ,0}(x, y) · η.

4.3.6. Suppose now that α ∈ I(\frac{1}{2}), i.e. α is a vertex. Then Gr_{c}S(α, λ) = 0 unless (λ, c) ∈ \frac{1}{2}Z × \frac{1}{2}N, in which case S_{c}(α, λ)^{⊥} = S^{c}(α, λ) for the perfect symmetric pairing ϕ_{λ} on S(α, λ). If ω(α) = s_{⋆} ∈ V, then

Δ_{0} + 2s_{0}(0) = dim_{E} S(α, 0, 0) and Δ_{2} + 2s_{2}(0) = dim_{E} S(α, \frac{1}{2}, 0)

and for all r ∈ N,

s_{1}(r) = dim_{F} Gr_{r+ \frac{1}{2}} S(α, 0) = dim_{F} Gr_{r+ \frac{1}{2}} S(α, \frac{1}{2})

while for r > 0,

2s_{0}(r) = dim_{F} Gr_{r}S(α, 0) and 2s_{2}(r) = dim_{F} Gr_{r}S(α, \frac{1}{2}).

This follows from the definitions by inspection of the tables in 4.2.1 and 2.2.5.

4.3.7. Let still α ∈ I(\frac{1}{2}) be a vertex with ω(α) = t_{⋆} ∈ V, and recall from section 3.13 that the edges β ∈ I(\frac{1}{4}) with target α correspond to totally isotropic F-subspaces H of (S(α, 0), ϕ_{0}). We shall now relate ω(β) = e_{⋆} ∈ E to the relative positions of H and the above filtrations on S(α, 0). We already know that

t_{0} = e_{0} + e_{m} + e_{01} + e_{10}[1] + e_{02} + e_{20}[1]
t_{1} = e_{1} + e_{12} + e_{21}
t_{2} = e_{2}

For c ∈ \frac{1}{2}N, define

\begin{aligned}H_{c} & = & H ∩ S_{c}(α, 0) \\ H_{c}^{+} & = & H ∩ S_{c}^{+}(α, 0) \\ Gr_{c}H & = & H_{c}^{+}/H_{c} \end{aligned} and \begin{aligned}H^{c} & = & H ∩ S^{c}(α, 0) \\ H_{−}^{c} & = & H ∩ S_{−}^{c}(α, 0) \\ Gr^{c}H & = & H^{c}/H_{−}^{c} \end{aligned}

so that Gr_{c}H and Gr^{c}H are F-subspaces of Gr_{c}S(α, 0) and Gr^{c}S(α, 0) which are orthogonal with respect to ϕ_{0}^{c}, while H(c) = H^{c}/H_{c} is a totally isotropic F-subspace of (S(α, 0, c), ϕ_{0,c}). If moreover c = r > 0 is a positive integer, define orthogonal F-subspaces of the symplectic F-space (Gr_{r}S(α, 0), θ_{0,r}) by

t_{r}H = [π^{r}η]Gr^{r}H and b_{r}H = Gr_{r}H.

Let t_{r}^{⊥}H ⊃ b_{r}H and b_{r}^{⊥}H ⊃ t_{r}H be their orthogonal complements. Elementary computations then show that for all r ∈ N,

e_{21}(r) = dim_{F} Gr_{r+ \frac{1}{2}} H and e_{12}(r) = dim_{F} Gr^{r+ \frac{1}{2}} H

while for r > 0,

dim_{F} ( \begin{matrix}b_{r}H ∩ t_{r}H \\ b_{r}H ∩ b_{r}^{⊥}H \\ t_{r}H ∩ t_{r}^{⊥}H \\ b_{r}H \\ t_{r}H \end{matrix} ) = ( \begin{matrix}1 \\ 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0 & 2 \\ 1 & 0 & 1 & 0 & 2 \end{matrix} ) ( \begin{matrix}e_{m}(r) \\ e_{10}(r − 1) \\ e_{01}(r) \\ e_{20}(r − 1) \\ e_{02}(r) \end{matrix} ) .

and finally for r = 0,

dim_{E} ( \begin{matrix}H(0) ∩ ηH(0) \\ H(0) ∩ ηH(0)^{⊥} + ηH(0) ∩ H(0)^{⊥} \\ H(0) + ηH(0) \end{matrix} ) = ( \begin{matrix}1 \\ 1 & 1 \\ 1 & 1 & 2 \end{matrix} ) ( \begin{matrix}e_{m}(0) \\ e_{01}(0) \\ e_{02}(0) \end{matrix} ) .

In this last formula, the orthogonal complements are taken with respect to the symmetric form ϕ_{0,0} underlying the E-hermitian form ψ_{0,0} on S(α, 0, 0).

4.3.8. Suppose similarly that H is a totally isotropic F-subspace of (S(α, \frac{1}{2}), ϕ_{\frac{1}{2}} ) corresponding to an edge β ∈ I(\frac{1}{4}) with source α ∈ I(\frac{1}{2}), and put ω(β) = e_{⋆} and ω(α) = s_{⋆}, so that

s_{0} = e_{0}
s_{1} = e_{1} + e_{01} + e_{10}
s_{2} = e_{2} + e_{m} + e_{02}[1] + e_{20} + e_{12}[1] + e_{21}

Define H_{c}, H^{c}, Gr^{c}H, Gr_{c}H and H(c) just as above, with λ = 0 replaced by λ = \frac{1}{2}. For c = r > 0 a positive integer, put again t_{r}H = [π^{r}η]Gr^{r}H and b_{r}H = Gr_{r}H, which are orthogonal subspaces of the symplectic F-space (Gr_{r}S(α, \frac{1}{2}), θ_{\frac{1}{2} ,r}) with orthogonal complements t_{r}^{⊥}H ⊃ b_{r}H and b_{r}^{⊥}H ⊃ t_{r}H. Then for all r ∈ N,

e_{01}(r) = dim_{F} Gr_{r+ \frac{1}{2}} H and e_{10}(r) = dim_{F} Gr^{r+ \frac{1}{2}} H

while for r > 0,

dim_{F} ( \begin{matrix}b_{r}H ∩ t_{r}H \\ b_{r}H ∩ b_{r}^{⊥}H \\ t_{r}H ∩ t_{r}^{⊥}H \\ b_{r}H \\ t_{r}H \end{matrix} ) = ( \begin{matrix}1 \\ 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0 & 2 \\ 1 & 0 & 1 & 0 & 2 \end{matrix} ) ( \begin{matrix}e_{m}(r) \\ e_{12}(r − 1) \\ e_{21}(r) \\ e_{02}(r − 1) \\ e_{20}(r) \end{matrix} )

and finally for r = 0,

dim_{E} ( \begin{matrix}H(0) ∩ ηH(0) \\ H(0) ∩ ηH(0)^{⊥} + ηH(0) ∩ H(0)^{⊥} \\ H(0) + ηH(0) \end{matrix} ) = ( \begin{matrix}1 \\ 1 & 1 \\ 1 & 1 & 2 \end{matrix} ) ( \begin{matrix}e_{m}(0) \\ e_{21}(0) \\ e_{20}(0) \end{matrix} )

where the orthogonal complements are now taken with respect to ϕ_{\frac{1}{2} ,0}.

4.3.9. For all d ∈ N, define

π(d) = ∏_{i=1}^{d}(q^{i} − 1), σ(d) = ∏_{i=1}^{d}(q^{i} + 1) and τ(d) = ∏_{i=1}^{d}(q^{i} − (−1)^{i}).

For γ ∈ {π, σ, τ}, s ∈ S and s_{⋆} : X → S (where X is a finite set), define

γ(s) = _{∏r∈N}γ(s(r)) and γ(s_{⋆}) = _{∏x∈X}γ(s_{x}).

Proposition. Fix e_{⋆} ∈ E with source(e_{⋆}) = s_{⋆} and target(e_{⋆}) = t_{⋆} in V. For x ∈ {s, t}, let α_{x} be a vertex with ω(α_{x}) = x_{⋆}. Then:

(1) The number of edges α with target(α) = α_{t} and ω(α) = e_{⋆} equals

q^{Λ_{t}(e_{⋆})} · \frac{π(t_{⋆})}{π(e_{⋆})} · \frac{σ(t_{0})}{σ (e_{0}, e_{02}, e_{20})} · \frac{1}{σ (e_{m}(0))} · \frac{τ (Δ_{0} + 2t_{0}(0)) · π (e_{0}(0)) · σ (e_{0}(0))}{π (t_{0}(0)) · σ (t_{0}(0)) · τ (Δ_{0} + 2e_{0}(0))}

for some explicit constant Λ_{t}(e_{⋆}) ∈ N.

(2) The number of edges α with source(α) = α_{s} and ω(α) = e_{⋆} equals

q^{Λ_{s}(e_{⋆})} · \frac{π(s_{⋆})}{π(e_{⋆})} · \frac{σ(s_{2})}{σ (e_{2}, e_{02}, e_{20})} · \frac{1}{σ (e_{m}(0))} · \frac{τ (Δ_{2} + 2s_{2}(0)) · π (e_{2}(0)) · σ (e_{2}(0))}{π (s_{2}(0)) · σ (s_{2}(0)) · τ (Δ_{2} + 2e_{2}(0))}

for some explicit constant Λ_{s}(e_{⋆}) ∈ N.

Proof. These are the numbers of totally isotropic F-subspaces H of S = S(α_{t}, 0) and S = S(α_{s}, \frac{1}{2}) which satisfy the conditions of section 4.3.7 and 4.3.8, respectively. With the notations of these sections, let N(0) be the number of possible choices for H(0). For each c ∈ \frac{1}{2}N with c > 0, let N(c) be the number of possible choices for the pair (Gr^{c}H, Gr_{c}H). Put also

b(c) = dim_{F} Gr_{c}H, t(c) = dim_{F} Gr^{c}H, s(c) = dim_{F} Gr_{c}S = dim_{F} Gr^{c}S

and define e(c) by s(c) = b(c) + e(c) + t(c). Let finally

M(c) = dim_{F} H(c − \frac{1}{2}) and D(c) = dim_{F} H(c − \frac{1}{2})^{⊥}/H(c − \frac{1}{2})

where the orthogonal complement is taken within S(c − \frac{1}{2}), so that

M(c) = M(\frac{1}{2}) +_{∑d<c} (b(d) + t(d))
and D(c) = D(\frac{1}{2}) + 2_{∑d<c}e(d)

with M(\frac{1}{2}) = dim_{F} H(0) and D(\frac{1}{2}) = dim_{F} H(0)^{⊥}/H(0). Now given

Gr_{c}H ⊂ Gr_{c}S, H(c − \frac{1}{2}) ⊂ S(c − \frac{1}{2}) and Gr^{c}H ⊂ Gr^{c}S,

the number of possible choices for H(c) ⊂ S(c) equals q^{λ(c)} where

λ(c) = M(c) · e(c) + t(c) · (D(c) + M(c) + e(c) + \frac{t(c)−1}{2})

by lemma 4 of the appendix. The number of possible choices for H thus equals

q^{∑_{c>0} λ(c)} · ∏_{c≥0} N(c).

We now compute the above constants for the second formula, i.e. using the conditions of section 4.3.8 above. First of all, for r ∈ N and c = r + \frac{1}{2},

b(c) = e_{01}(r), t(c) = e_{10}(r), e(c) = e_{1}(r) and s(c) = s_{1}(r)

while for c = r > 0, s(c) = 2s_{2}(r) and

b(c) = e_{m}(r) + e_{12}(r − 1) + 2e_{02}(r − 1)
t(c) = e_{m}(r) + e_{21}(r) + 2e_{20}(r)
e(c) = 2e_{2}(r) + e_{12}(r − 1) + e_{21}(r)

and finally M(\frac{1}{2}) = 2e_{m}(0) + 2e_{20}(0) + e_{21}(0) and D(\frac{1}{2}) = 2Δ_{2} + 4e_{2}(0) + 2e_{21}(0), which gives M(c), D(c) and λ(c) as a function of e_{⋆} for all c ∈ \frac{1}{2}N − \{0\}. Now lemma 1, 2 and 3 of the appendix give respectively

\begin{matrix}N(0) = & \frac{q^{λ′(0)} · τ (Δ_{2} + 2s_{2}(0))}{π (e_{m}(0), e_{21}(0), e_{20}(0)) · σ (e_{m}(0), e_{20}(0)) · τ (Δ_{2} + 2e_{2}(0))} , \\ N(r + \frac{1}{2} ) = & \frac{π (s_{1}(r))}{π (e_{1}(r), e_{01}(r), e_{10}(r))} , \\ N(r) = & \frac{q^{λ′(r)} · π (s_{2}(r)) · σ (s_{2}(r))}{π (e_{m}(r), e_{12}(r − 1), e_{21}(r), e_{02}(r − 1), e_{20}(r), e_{2}(r)) · σ (e_{02}(r − 1), e_{20}(r), e_{2}(r))} \end{matrix}

for r ∈ N (and r ≠ 0 in the last formula) with

\begin{matrix}λ′(0) = & 2e_{20}(0) (e_{21}(0) + Δ_{2} + 2e_{2}(0)) + e_{20}(0) (e_{20}(0) − 1) + \frac{1}{2} e_{21}(0) (e_{21}(0) − 1) , \\ λ′(r) = & 2e_{02}(r − 1) (e_{21}(r) + e_{20}(r) + e_{2}(r)) + 2e_{20}(r) (e_{12}(r − 1) + e_{2}(r)) + e_{12}(r − 1)e_{21}(r) \end{matrix}

for 0 ≠ r ∈ N. This proves our second formula, with the exponent

Λ_{s}(e_{⋆}) = ∑_{c∈ \frac{1}{2} N,c>0} λ(c) + ∑_{r∈N} λ′(r).

The proof of the first one is entirely similar: simply switch 0 and 2 everywhere. □

4.4. Volumes. Let μ be a Haar measure on U(V, ψ). Let U(α) be the stabilizer of α ∈ I(V, ϕ) in U(V, ψ) and define μ(α) = μ(U(α)), so that μ(α) depends only on the U(V, ψ)-orbit of α.

4.4.1. For a suitable (but unique) choice of μ, the following proposition holds.

Proposition. For any edge α ∈ I(\frac{1}{4}) with e_{⋆} = ω(α) ∈ E,

μ(α) = q^{−Λ(e_{⋆})}·π (e_{⋆})·σ (e_{0}, e_{2}, e_{02}, e_{20})·σ (e_{m}(0))· \frac{τ(Δ_{0} + 2e_{0}(0), Δ_{2} + 2e_{2}(0))}{π(e_{0}(0), e_{2}(0)) · σ(e_{0}(0), e_{2}(0))}

for an exponent Λ(e_{⋆}) ∈ Z described in section 4.4.2 below.

Proof. Let μ̃(e_{⋆}) be the right hand side of the above equation. Since I is connected, we just have to show that for s_{⋆} = source(e_{⋆}) and t_{⋆} = target(e_{⋆}), the ratios μ̃(s_{⋆})/μ̃(e_{⋆}) and μ̃(t_{⋆})/μ̃(e_{⋆}) match the numbers found in proposition 4.3.9. This is obvious for the prime-to-q part of these ratios, but we need to verify that

Λ_{s}(e_{⋆}) = Λ(e_{⋆}) − Λ(s_{⋆}) and Λ_{t}(e_{⋆}) = Λ(e_{⋆}) − Λ(t_{⋆}).

This is a routine, but highly tedious task. □

4.4.2. A closed formula for Λ(e_{⋆}) is given by

Λ(e_{⋆}) = ∑_{0≠c∈ \frac{1}{4} N} X_{c}^{t} (A_{c}X_{c} + B_{c}Y_{c} + L_{c})

where for 0 ≠ c ∈ \frac{1}{4}N and any α ∈ I(\frac{1}{4}) with ω(α) = e_{⋆},

Y_{c} = ( \begin{matrix}dim_{F} S_{−}^{c}(α, 0)/S_{c}^{+}(α, 0) \\ 2 · dim_{F} S_{−}^{c}(α, \frac{1}{4})/S_{c}^{+}(α, \frac{1}{4}) \\ dim_{F} S_{−}^{c}(α, \frac{1}{2})/S_{c}^{+}(α, \frac{1}{2}) \end{matrix} ) = ( \begin{matrix}dim_{F} S(α, 0, c − \frac{1}{4}) \\ 2 · dim_{F} S(α, \frac{1}{4}, c − \frac{1}{4}) \\ dim_{F} S(α, \frac{1}{2}, c − \frac{1}{4}) \end{matrix} )

while the matrices X_{c}, A_{c}, B_{c} and L_{c} are given by the following table, in which k = ⌊c⌋ ∈ N and ((λ)) denotes a matrix all of whose coefficients are equal to λ (and whose size should be clear from the context).

\begin{matrix}c & X_{c} & A_{c} − ((2k)) & B_{c} − ((k)) & L_{c} + ((k)) \\ k + 0 & ( \begin{matrix}e_{0}(k) \\ e_{m}(k) \\ e_{2}(k) \end{matrix} ) & ( \begin{matrix}−1 & 0 & 0 \\ 0 & −\frac{1}{2} & 0 \\ 0 & 0 & −1 \end{matrix} ) & ( \begin{matrix}−1 & 0 & 0 \\ 0 & −\frac{1}{2} & 0 \\ 0 & 0 & −1 \end{matrix} ) & ( \begin{matrix}0 \\ −\frac{1}{2} \\ 0 \end{matrix} ) \\ k + \frac{1}{4} & ( \begin{matrix}e_{01}(k) \\ e_{21}(k) \end{matrix} ) & ( \begin{matrix}\frac{1}{2} & 0 \\ 0 & \frac{1}{2} \end{matrix} ) & ( \begin{matrix}0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix} ) & ( \begin{matrix}−\frac{1}{2} \\ −\frac{1}{2} \end{matrix} ) \\ k + \frac{1}{2} & ( \begin{matrix}e_{02}(k) \\ e_{1}(k) \\ e_{20}(k) \end{matrix} ) & ( \begin{matrix}1 & 1 & 0 \\ 1 & \frac{1}{2} & 1 \\ 0 & 1 & 1 \end{matrix} ) & ( \begin{matrix}1 & 0 & 0 \\ 0 & \frac{1}{2} & 0 \\ 0 & 0 & 1 \end{matrix} ) & ( \begin{matrix}−1 \\ −\frac{1}{2} \\ −1 \end{matrix} ) \\ k + \frac{3}{4} & ( \begin{matrix}e_{12}(k) \\ e_{10}(k) \end{matrix} ) & ( \begin{matrix}1 & \frac{3}{2} \\ \frac{3}{2} & 1 \end{matrix} ) & ( \begin{matrix}1 & \frac{1}{2} & 0 \\ 0 & \frac{1}{2} & 1 \end{matrix} ) & ( \begin{matrix}−1 \\ −1 \end{matrix} ) \end{matrix}

Note also that with these definitions,

Y_{c} = ( \begin{matrix}2Δ_{0} \\ 0 \\ 2Δ_{2} \end{matrix} ) + ∑_{d∈ \frac{1}{4} N,0≤d<c} T_{d}X_{d}

where T_{d} depends only upon d mod Z and

T_{0} = ( \begin{matrix}4 & 0 & 0 \\ 0 & 4 & 0 \\ 0 & 0 & 4 \end{matrix} ), T_{\frac{1}{4}} = ( \begin{matrix}2 & 0 \\ 2 & 2 \\ 0 & 2 \end{matrix} ), T_{\frac{1}{2}} = ( \begin{matrix}0 & 2 & 0 \\ 4 & 0 & 4 \\ 0 & 2 & 0 \end{matrix} ) and T_{\frac{3}{4}} = ( \begin{matrix}0 & 2 \\ 2 & 2 \\ 2 & 0 \end{matrix} ).

5. The odd orthogonal case

5.1. Classification. We recall the following results from [1, §5].

5.1.1. A quasi-hermitian E-space is a triple (\overline{V} , ψ, D) where (\overline{V} , ψ) is a finite dimensional hermitian E-space and D is an anisotropic F -line in the symmetric F -space (\overline{V} , ϕ) underlying (\overline{V} , ψ), with ϕ = trψ. Such a structure defines

\underline{V} = (E · D)^{⊥} ⊂ V = \underline{V} ⊥D ⊂ \overline{V} = V ⊥ηD

Thus (\underline{V} , ψ) is a non-degenerate E-hyperplane of (\overline{V} , ψ) while (V, ϕ) is a nondegenerate F -hyperplane in (\overline{V} , ϕ). The Witt indices are related by

witt_{F} (V, ϕ) = witt_{E}(\underline{V} , ψ) + witt_{E}(\overline{V} , ψ).

Note that this notion differs slightly from the one used in [1, §5].

5.1.2. A quasi-hermitian E-space (\overline{V} , ψ, D) induces a commutative diagram

\xymatrix{U(\underline{V} , ψ) \ar[d] \ar[rr] &  & U(\overline{V} , ψ) \ar[d] \\ SO(\underline{V} , ϕ) \ar[r] & SO(V, ϕ) \ar[r] & SO(\overline{V} , ψ)}

For X ∈ \{\underline{V} , V, \overline{V} \}, let I(X) be the set of self-dual F -norms on (X, ϕ): this is the Bruhat-Tits building of SO(X, ϕ). A self dual norm on \underline{V} or V extends uniquely to a self-dual norm on V = \underline{V} ⊥D or \overline{V} = V ⊥ηD. This yields embeddings

I(\underline{V}) ↪ I(V) and I(V) ↪ I(\overline{V})

which are respectively SO(\underline{V} , ϕ) and SO(V, ϕ)-equivariant, respect the metrics and map each building onto a convex subset of the next one. Convex projection defines equivariant retractions of these embeddings, see [1, §2.2.8]. We denote by

\underline{α} ∈ I(\underline{V}) and \overline{α} ∈ I(\overline{V})

the (restriction of) convex projection and unique self-dual extension of α ∈ I(V).

5.1.3. Let I_{0}(V) be the image of I(\underline{V}) in I(V) and define more generally

I_{s}(V) =^{def} {α ∈ I(V) : d(α) = s} where d(α) =^{def} dist(α, I_{0}(V)) = dist(α, α_{0})

for all s ∈ R_{≥0}, where α_{0} ∈ I_{0}(V) is the convex projection of α ∈ I(V) (and thus also the unique self-dual extension of \underline{α} ∈ I(\underline{V})). For d(α) = s and t ∈ [0, s], we let α_{t} be the point at distance t from α_{0} on the segment joining α_{0} and α_{s} = α in I(V), i.e. \{α_{t}\} = I_{t}(V)∩[α_{0}, α_{s}]. The map α_{s} ↦ α_{t} thus defines an SO(\underline{V} , ϕ)-equivariant map from I_{s}(V) to I_{t}(V).

5.1.4. For any non-degenerate E-subspace W of \underline{V} with orthogonal complement \overline{V}′ in \overline{V}, the triple (\overline{V}′, ψ, D) is a quasi-hermitian E-subspace of (\overline{V} , ψ, D) for which

\underline{V} = W ⊥\underline{V}′, V = W ⊥V′, and \overline{V} = W ⊥\overline{V}′.

In this situation, for any self-dual F -norm α on V,

α|W ∈ I(W) ⟺ α|V′ ∈ I(V′) ⟺ α = α|W ⊥α|V′

in which case also for β = α|W and α′ = α|V′, d(α) = d(α′) = s and

\overline{α} = β⊥\overline{α′}, \underline{α} = β⊥\underline{α′} and α_{t} = β⊥α′_{t} for all t ∈ [0, s].

We say that α is special if and only if it does not admit such a splitting with W ≠ 0. Any self-dual F -norm α on V therefore does have such a splitting for which α′ is a special self-dual F -norm on V′.

5.1.5. A special basis B of V is an orthogonal F -basis

B = (v_{0}, w_{1}, v_{1}, w_{2}, ⋯, w_{n}, v_{n})

of V such that F v_{n} = D, v_{i−1} = ηw_{i} and Q(v_{i}) + Q(w_{i}) = 0 for all 1 ≤ i ≤ n. It defines two orthogonal decompositions of V, namely

V = Ew_{1}⊥ ⋯ ⊥Ew_{n}⊥D = F v_{0}⊥H_{1}⊥ ⋯ ⊥H_{n}

where H_{i} = F w_{i}⊥F v_{i} is an hyperbolic F -plane whose isotropic F -lines are spanned by e_{±i} = \frac{1}{2}(v_{i} ± w_{i}). The latter decomposition yields an apartment A(B) of I(V) and the special basis B of V gives an isometry coord_{B} : A(B) ⟶^{≃} R^{n} mapping α ∈ A(B) to the n-tuple (s_{1}, ⋯, s_{n}) ∈ R^{n} defined by

α(e_{±i}) = q^{r_{0}±s_{i}} with |Q(v_{i})| = |Q(w_{i})| = |ϕ(e_{i}, e_{−i})| = q^{2r_{0}}.

The apartment A(B) is stable under the projection α ↦ \underline{α}, which corresponds to the projection (s_{1}, ⋯, s_{n}) ↦ (s_{1}, ⋯, s_{n−1}, 0). Therefore s_{n} = d(α) and for all t ∈ [0, s_{n}], α_{t} also belongs to A(B) with coord_{B}(α_{t}) = (s_{1}, ⋯, s_{n−1}, t). Note that special bases do exist if and only if V is split, i.e. dim_{F} V = 2witt_{F} V + 1.

5.1.6. We consider n-tuples (s_{1}, ⋯, s_{n}) ∈ R^{n} satisfying one of the conditions (SP) 0 < s_{1} < s_{3} < ⋯ < s_{2⌊\frac{n−1}{2}⌋+1} and 0 < s_{2} < s_{4} < ⋯ < s_{2⌊\frac{n}{2}⌋}, (Sp) 0 ≤ s_{1} < s_{3} < ⋯ < s_{2⌊\frac{n−1}{2}⌋+1} and 0 < s_{2} < s_{4} < ⋯ < s_{2⌊\frac{n}{2}⌋}, (sp) 0 ≤ s_{1} ≤ s_{3} ≤ ⋯ ≤ s_{2⌊\frac{n−1}{2}⌋+1} and 0 ≤ s_{2} ≤ s_{4} ≤ ⋯ ≤ s_{2⌊\frac{n}{2}⌋}.

The formulas s_{0} = 0 and c_{i} = s_{i} +s_{i−1} for i = 1, ⋯, n define a bijective correspondence between such n-tuples (s_{1}, ⋯, s_{n}) and sequences (c_{1}, ⋯, c_{n}) for which (SP) 0 < c_{1} < c_{2} < ⋯ < c_{n}, (Sp) 0 ≤ c_{1} < c_{2} < ⋯ < c_{n}, (sp) 0 ≤ c_{1} ≤ c_{2} ≤ ⋯ ≤ c_{n}. For i = 1, ⋯, n, we put θ_{i} = (−1)^{i}(s_{i} − s_{i−1}) ∈ R and define

[s_{1}, ⋯, s_{n}]_{D} = ∑_{i=1}^{n}[θ_{i} + Q(D), c_{i}] in N[\overline{L}]

where we have used the (θ, c)-coordinates on \overline{L} ≃ R/ ∼ ×R_{≥0}. Here

Q(D) = log_{q} Q(D − {0}) = 2Z or 1 + 2Z.

We also define

⌈s_{1}, ⋯, s_{n}⌉_{D} = ∑_{i≡n mod 2}[θ_{i} + Q(D), c_{i}],
and ⌊s_{1}, ⋯, s_{n}⌋_{D} = ∑_{i≢n mod 2}[θ_{i} + Q(D), c_{i}].

These two elements of N[\overline{L}] therefore sum to [s_{1}, ⋯, s_{n}]_{D}, from which the sequences (c_{1}, ⋯, c_{n}) and (s_{1}, ⋯, s_{n}) are easily retrieved.

5.1.7. Any element a of N[\overline{L}] has a unique decomposition a = 2a_{reg} + a_{sp} with a_{reg} ∈ N[\overline{L}] and a_{sp} = ∑_{x∈X} x for a finite subset X of \overline{L}. We say that a_{reg} and a_{sp} are the regular and special parts of a and define a subset N[\overline{L}]_{D} of N[\overline{L}] by

a ∈ N[\overline{L}]_{D} ⟺ ∃(s_{1}, ⋯, s_{m}) satisfying (Sp) such that a_{sp} = [s_{1}, ⋯, s_{m}]_{D} .

Note that if (s_{1}, ⋯, s_{m}) merely satisfies (sp), then still [s_{1}, ⋯, s_{m}]_{D} ∈ N[\overline{L}]_{D}.

5.1.8. For any self-dual F -norm α on V, we put

ω(α) =^{def} ω(\underline{α}) + ω(\overline{α}) in N[\overline{L}]

and let ω_{reg} and ω_{sp} be the corresponding regular and special parts, so that

ω(α) = 2ω_{reg}(α) + ω_{sp}(α).

This defines a map ω : U(\underline{V} , ψ)\I(V) → N[\overline{L}]^{n} where n = witt_{F} V. We shall see below that the latter is a bijection onto N[\overline{L}]_{D}^{n} = N[\overline{L}]_{D} ∩ N[\overline{L}]^{n}.

5.1.9. By Theorem 5.2 of [1], a self-dual norm α on V is special if and only if there exists a special basis B of V such that α ∈ A(B) with coord_{B}(α) = (s_{1}, ⋯, s_{n}) satisfying (SP) – in particular, special norms do exists precisely when V is split. On the other hand, Proposition 5.1 of [1] asserts that for any self-dual norm α ∈ A(B) for which coord_{B}(α) = (s_{1}, ⋯, s_{n}) merely satisfies (sp),

ω(\underline{α}) = ⌊s_{1}, ⋯, s_{n}⌋_{D} and ω(\overline{α}) = ⌈s_{1}, ⋯, s_{n}⌉_{D} .

Then ω(α) = [s_{1}, ⋯, s_{n}]_{D}, from which the sequences (c_{1}, ⋯, c_{n}) and (s_{1}, ⋯, s_{n}) are easily retrieved. Therefore (s_{1}, ⋯, s_{n}) is a well-defined invariant attached to such norms.

5.1.10. Let now α be any self-dual norm on V and fix a splitting α = β⊥α′ corresponding to V = W ⊥V′ where α′ = α|V′ is a special self-dual norm on V′. Let n = witt_{F} V, m = witt_{F} V′ and r = witt_{E}W, so that n = 2r + m + δ for some δ ∈ {0, 1}. Let (s_{1}, ⋯, s_{m}) be the invariant of α′ on V′, as defined above (satisfying (SP)). Fix also Witt decompositions

(W, β) = (W_{0}, β_{0})⊥ ((W_{+}, β_{+}) ⊕ (W_{−}, β_{−}))
(\underline{V}′, \underline{α}′) = (\underline{V} ′_{0}, \underline{α}′_{0})⊥ ((\underline{V} ′_{+}, \underline{α}′_{+}) ⊕ (\underline{V} ′_{−}, \underline{α}′_{−}))
(\overline{V}′, \overline{α}′) = (\overline{V} ′_{0}, \overline{α}′_{0})⊥ ((\overline{V} ′_{+}, \overline{α}′_{+}) ⊕ (\overline{V} ′_{−}, \overline{α}′_{−}))

where for X ∈ \{W, \underline{V}′, \overline{V}′\}, X_{0} and X_{±} are suitable E-subspaces of X which are respectively anisotropic and totally isotropic. Then

ω(\underline{α}) = ω(β) + ⌊s_{1}, ⋯, s_{m}⌋_{D} + \underline{Ω} and ω(\overline{α}) = ω(β) + ⌈s_{1}, ⋯, s_{m}⌉_{D} + \overline{Ω}

where \underline{Ω} = ω(β_{0}⊥\underline{α}′_{0}) and \overline{Ω} = ω(β_{0}⊥\overline{α}′_{0}). Let A ∈ \{A_{1}, A′_{1}\} be the isomorphism class of (E · D, ψ), so that A = A_{1} if Q(D) = 2Z while A = A′_{1} if Q(D) = 1 + 2Z. The definition of a special basis implies that

\begin{cases}\underline{V} ′_{0} = 0 and (\overline{V} ′_{0}, ψ) ∈ A & if m is even, \\ (\underline{V} ′_{0}, ψ) ∈ A and \overline{V} ′_{0} = 0 & if mis odd. \end{cases}

Therefore \underline{Ω} = 0 if m is even, \overline{Ω} = 0 if m is odd, and \underline{Ω} + \overline{Ω} = Ω is the ω-invariant of the canonical norm x ↦ |Q(x)|^{1/2} on W_{0}⊥E · D, namely

Ω = \begin{cases}0 & if δ = 0 \\ [Q(D), 0] & if δ = 1 \end{cases} in N[\overline{L}] with \overline{L} ≃ R/ ∼ ×R_{≥0}

since indeed δ = witt_{E} (W_{0}⊥E · D). We thus obtain

(ω(\underline{α}), ω(\overline{α})) = (ω(β), ω(β)) + \begin{cases}(⌊s_{1}, ⋯, s_{m}⌋_{D} , ⌈s_{1}, ⋯, s_{m}⌉_{D}) & if n ≡ m mod 2 \\ (⌊0, s_{1}, ⋯, s_{m}⌋_{D} , ⌈0, s_{1}, ⋯, s_{m}⌉_{D}) & if n ≢ m mod 2 \end{cases}

so that ω(α) = 2ω_{reg}(α) + ω_{sp}(α) with ω_{reg}(α) = ω(β) and

ω_{sp}(α) = \begin{cases}[s_{1}, ⋯, s_{m}]_{D} & if n ≡ m mod 2, \\ [0, s_{1}, ⋯, s_{m}]_{D} & if n ≢ m mod 2. \end{cases}

It follows that ω(α) indeed belongs to N[\overline{L}]_{D}.

5.1.11. In these constructions, we may scale the special basis B′ = (v_{0}, ⋯, v_{m}) of V′ so that its last vector is equal to a fixed non-zero vector v_{m}^{0} ∈ D. All such normalized sequences B′ lie in the same orbit for the diagonal action of U(\underline{V} , ψ) on V^{2m+1}. It follows that the ω-invariant defines a bijection (cf. Theorem 5.3 of [1])

ω : U(\underline{V} , ψ)\I(V, ϕ) ⟶^{≃} N[\overline{L}]_{D}^{n}.

5.1.12. If ω_{sp}(α) = [s_{1}, ⋯, s_{m}]_{D}, then d(α) = s_{m} and for all t ∈ [s_{m−2}, s_{m}],

ω(α_{t}) = 2ω_{reg}(α) + [s_{1}, ⋯, s_{m−1}, t]_{D} .

Therefore ω_{reg}(α_{t}) = ω_{reg}(α) and ω_{sp}(α_{t}) = [s_{1}, ⋯, s_{m−1}, t]_{D} if t > s_{m−2} while

ω_{reg}(α_{s_{m−2}}) = ω_{reg}(α) + [(−1)^{m}(s_{m−2} − s_{m−1}) + Q(D), s_{m−1} + s_{m−2}],
and ω_{sp}(α_{s_{m−2}}) = [s_{1}, ⋯, s_{m−2}]_{D} .

Note also that

ω(\underline{α}) = ω_{reg}(α) + ⌊s_{1}, ⋯, s_{m}⌋_{D} and ω(\overline{α}) = ω_{reg}(α) + ⌈s_{1}, ⋯, s_{m}⌉_{D}

5.1.13. If ω(α) = ∑_{i=1}^{n}[θ_{i}, c_{i}] with ω_{sp}(α) = [s_{1}, ⋯, s_{m}]_{D}, we set

c_{min}(α) =^{def} \begin{cases}min\{c_{i}\} & if V splits \\ 0 & otherwise \end{cases} ≤ c_{max}^{sp}(α) =^{def} s_{m−1} + s_{m} ≤ c_{max}(α) =^{def} max\{c_{i}\}

with the convention that s_{0} = 0 = s_{−1} (for m = 0 or 1).

5.1.14. Recall from section 4.1.5 that U_{r} = \{z/\overline{z} : z ∈ O_{r}^{×}\} for r ∈ N.

Lemma. Let U(α) be the stabilizer of α in U(\underline{V} , ψ). Then

det U(α) = U_{r} with r = ⌈c_{min}(α)⌉ .

Proof. With the notations of section 5.1.10, we have

U(β) × U(α′) ⊂ U(α) ⊂ U(\underline{α}) and U(\overline{α})

so that

det U(β) · det U(α′) ⊂ det U(α) ⊂ det U(\underline{α}) ∩ det U(\overline{α}).

On the other hand, a case by case study gives

min \{c_{min}(β), c_{min}(α′)\} = c_{min}(α) = max \{c_{min}(\underline{α}), c_{min}(\overline{α})\} .

It is therefore sufficient to show that

det U(β) = Ur̃, det U(α′) = U_{r′}, det U(\underline{α}) = U\underline{r} and det U(\overline{α}) = U\overline{r}

where r̃ = ⌈c_{min}(β)⌉, r′ = ⌈c_{min}(α′)⌉, \underline{r} = ⌈c_{min}(\underline{α})⌉ and \overline{r} = ⌈c_{min}(\overline{α})⌉. The first, third and fourth of these equalities readily follow from lemma 4.1.5. As for the second one, it is itself the special case of the lemma. So suppose that α = α′ is special and let B = (v_{0}, w_{1}, v_{1}, w_{2}, ⋯, w_{n}, v_{n}) be a special basis of V such that α ∈ A(B) with coord_{B}(α) = (s_{1}, ⋯, s_{n}) satisfying (SP). Thus r = ⌈s_{1}⌉ ≥ 1, and given the argument above, we just have to show that U_{r} ⊂ det U(α). If V′ = Ew_{1}⊥F v_{1} and α′ = α|V′, then α′ is still a special self-dual F -norm on V′ and U(α′) ⊂ U(α). We may thus also assume that n = 1.

We need to show that for all v ∈ V and u ∈ U_{r}, α(uv) ≤ α(v). Let |Q(v_{1})| = q^{2r_{0}} and e_{±} = \frac{1}{2}(v_{1} ± w_{1}), so that for all x, y, z ∈ F,

α(xv_{0} + ye_{+} + ze_{−}) = max {|x| q^{r_{0}}, |y| q^{r_{0}+s_{1}}, |z| q^{r_{0}−s_{1}}} .

For u = a + ηb ∈ E (with a, b ∈ F) acting as (u, Id) on V′ = Ew_{1} ⊕ F v_{1}, we have

uv_{0} = av_{0} + bη^{2}(e_{+} − e_{−}) and λe_{±} = \frac{1}{2} (±bv_{0} + (1 ± a)e_{+} + (1 ∓ a)e_{−}) .

Therefore α(ux) ≤ α(x) for all x ∈ V if and only if

\{ \begin{matrix}max \{|b| q^{r_{0}}, |1 ± a| q^{r_{0}+s_{1}}, |1 ∓ a| q^{r_{0}−s_{1}}\} & ≤ & q^{r_{0}±s_{1}} \\ and  max \{|a| q^{r_{0}}, |b| q^{r_{0}+s_{1}}\} & ≤ & q^{r_{0}} \end{matrix}

i.e. |a| ≤ 1, |1 − a| ≤ q^{−2s_{1}} and |b| ≤ q^{−s_{1}}. These conditions are met by any u ∈ U_{r}. Indeed, for u = λ/\overline{λ} with λ = a′ + ηb′ ∈ O_{r}^{×}, we have a′ ∈ O^{×} and b′ ∈ P^{r}. Therefore if c = a^{′−1}b′, then c ∈ P^{r} and

u = \frac{1 + ηc}{1 − ηc} = (1 + ηc) · (1 + ηc + (ηc)^{2} + ⋯) = a + ηb

with a = 1 + 2 ((ηc)^{2} + (ηc)^{4} + ⋯) and b = 2c (1 + (ηc)^{2} + (ηc)^{4} + ⋯). □

5.1.15. We define D(α, λ) = S(α|D, λ) ⊂ S(α, λ). Thus D(α, λ) = 0 unless λ ∈ d(α) + \frac{1}{2}Q(D), in which case it is an F-line in S(α, λ). If moreover d(α) ∈ \frac{1}{2}N, so that λ ∈ \frac{1}{2}Z, then D(α, λ) is anisotropic for ϕ_{λ} if d(α) = 0, and isotropic otherwise. For all c ≥ 0 and λ ∈ R, we put S_{c}(α, λ) = S_{c}(\overline{α}, λ)∩S(α, λ). Note that for c ≥ c_{max}^{sp}(α), the subspace S_{c}(\overline{α}, λ) of S(\overline{α}, λ) is already contained in S(α, λ).

5.2. Vertices and edges. Fix a quasi-hermitian space (\overline{V} , ψ, D) with underlying symmetric form ϕ = tr_{E/F} ψ. The pair of integers (δ_{0}, δ_{\frac{1}{2}} ) of section 3.7 equals

(δ_{0}, δ_{\frac{1}{2}} ) = 2(Δ_{0}, Δ_{2}) ± \begin{cases}(1, 0) & if Q(D) = 2Z \\ (0, 1) & if Q(D) = 1 + 2Z \end{cases}

where (Δ_{0}, Δ_{2}) ∈ {(0, 0), (0, 1), (1, 0), (1, 1)} is the pair attached to the hermitian E-space (\underline{V} , ψ) in section 4.1.2, and where the sign is uniquely determined by the conditions 0 ≤ δ_{0}, δ_{\frac{1}{2}} ≤ 2. Put I = I(V, ϕ), n = witt_{F} (V, ϕ) and U = U(\underline{V} , ψ).

5.2.1. If α ∈ I is special and ω(α) = [s_{1}, ⋯, s_{n}]_{D} = ∑_{i=1}^{n}[θ_{i}, c_{i}], then

α is a vertex ⇔ ∀i : s_{i} ∈ \frac{1}{2}N ⇔ ∀i : θ_{i}, c_{i} ∈ \frac{1}{2}Z and θ_{i} ≡ c_{i} mod Z
α is an edge ⇔ ∀i : s_{i} ∈ \frac{1}{4}N ⇔ ∀i : θ_{i}, c_{i} ∈ \frac{1}{4}Z and θ_{i} ≡ c_{i} mod \frac{1}{2}Z

Suppose that α is an edge with source α\{\frac{1}{2}\} and target α\{0\}. Then for x ∈ \{0, \frac{1}{2}\},

ω(α\{x\}) = [s_{1}\{x\}, ⋯, s_{n}\{x\}]_{D} ∈ N[\overline{L}]_{D}^{n}

where for s ∈ \frac{1}{4}N, we denote by s\{x\} the unique element of x + \frac{1}{2}Q(D) which is closest to s if s ∉ \frac{1}{2}N and put s\{x\} = s otherwise. This follows from 5.1.9 by explicit computations in the apartment A(B) of I defined by a special basis B of V for which α belongs to A(B) with coord_{B}α = (s_{1}, ⋯, s_{n}). Note that while (s_{1}, ⋯, s_{n}) does satisfy (SP), (s_{1}{x}, ⋯, s_{n}{x}) may only satisfy (sp). And indeed, the source and target of a special edge need not be special anymore.

5.2.2. The ω-invariant α ∈ I ↦ ω(α) ∈ N[\overline{L}]_{D}^{n} therefore induces

• a bijection between U\I(\frac{1}{2}) and V_{D}^{n} = V^{n} ∩ V_{D} where V_{D} = V ∩ N[\overline{L}]_{D},

• a bijection between U\I(\frac{1}{4}) and E_{D}^{n} = E^{n} ∩ E_{D} where E_{D} = E ∩ N[\overline{L}]_{D}.

Moreover, one checks using 5.2.1 that the source and target maps E → V of section 4.2.2 are compatible with the actual source and target maps I(\frac{1}{4}) → I(\frac{1}{2}) under the above identifications, which means that they map E_{D} onto V_{D} and satisfy

U\I(\frac{1}{4}) ⟶^{source} U\I(\frac{1}{2}) ⟶^{ω} V_{D}^{n} = U\I(\frac{1}{4}) ⟶^{ω} E_{D}^{n} ⟶^{source} V_{D}^{n},
U\I(\frac{1}{4}) ⟶^{target} U\I(\frac{1}{2}) ⟶^{ω} V_{D}^{n} = U\I(\frac{1}{4}) ⟶^{ω} E_{D}^{n} ⟶^{target} V_{D}^{n}.

5.3. Projections and λ-corners. For s ∈ \frac{1}{4}N, we put I_{s}(\frac{1}{4}) = I_{s} ∩ I(\frac{1}{4}) and denote by pr_{s} : I_{s+ \frac{1}{4}} (\frac{1}{4}) → I_{s}(\frac{1}{4}) the map induced by the projection I_{s+ \frac{1}{4}} → I_{s} of section 5.1.3. It is computed as follows.

5.3.1. Suppose that α_{1} belongs to I_{s+ \frac{1}{4}} (\frac{1}{4}) and use 5.1.9 to pick a decomposition (V, α_{1}) = (W, β)⊥(V′, α′_{1}) and a special basis B = (v_{0}, w_{1}, v_{1}, ⋯, w_{m}, v_{m}) of V′ such that α′_{1} = α|V′_{1} belongs to the apartment A(B) defined by

V′ = F v_{0}⊥H_{1}⊥ ⋯ ⊥H_{m} where H_{i} = F e_{i} ⊕ F e_{−i}, e_{±i} = \frac{1}{2}(v_{i} ± w_{i})

with coord_{B}(α′_{1}) = (s_{1}, ⋯, s_{m−1}, s + \frac{1}{4}) satisfying SP. Then for α_{2} = pr_{s}α_{1}, we have (V, α_{2}) = (W, β)⊥(V′, α′_{2}) with α′_{2} in the same apartment and given by coord_{B}(α′_{2}) = (s_{1}, ⋯, s_{m−1}, s) (satisfying sp). Note that m ≥ 1 since s + \frac{1}{4} > 0.

5.3.2. Let α′′_{i} be the restriction of α′_{i} to the hyperbolic F -plane H_{m} and denote by α′′ the common restriction of α′_{1} and α′_{2} to the orthogonal complement V′′ of H_{m} in V′. Thus for i ∈ {1, 2} and any λ ∈ R,

(V, α_{i}) = (W, β)⊥(V′′, α′′)⊥(H_{m}, α′′_{i}),
B(α_{i}, λ) = B(β, λ)⊥B(α′′, λ)⊥B(α′′_{i}, λ),
B^{0}(α_{i}, λ) = B^{0}(β, λ)⊥B^{0}(α′′, λ)⊥B^{0}(α′′_{i}, λ),
and S(α_{i}, λ) = S(β, λ) ⊕ S(α′′, λ) ⊕ S(α′′_{i}, λ).

The latter decomposition is compatible with the pairings ⟨•, •⟩_{λ}. Since

α′′_{1}(e_{±m}) = q^{r_{0}±(s+ \frac{1}{4} )} and α′′_{2}(e_{±m}) = q^{r_{0}±s} with q^{2r_{0}} = |Q(v_{m})| ,

we find that for λ = r_{0} + s, B(α_{1}, λ + \frac{ϵ}{4}) = B(α_{2}, λ + \frac{ϵ}{4}) where ϵ = 1 if λ ∈ \frac{1}{2}Z and ϵ = −1 otherwise, while

B^{0}(α_{2}, λ) ⊂ B^{0}(α_{1}, λ + \frac{1}{4}) = B(α_{1}, λ) ⊊ B(α_{2}, λ) ⊂ B(α_{1}, λ + \frac{1}{4})

with dim_{F} B(α_{2}, λ)/B(α_{1}, λ) = 1. Moreover,

D_{1} =^{def} B(α_{2}, λ)/B^{0}(α_{1}, λ + \frac{1}{4}) ⊂ S(α′′_{1}, λ + \frac{1}{4}) ⊂ S(α_{1}, λ + \frac{1}{4})

is the F-line spanned by the image of e_{m} ∈ B(α′′_{1}, λ + \frac{1}{4}) in S(α′′_{1}, λ + \frac{1}{4}). It is isotropic for ϕ_{λ+ \frac{1}{4}} if λ + \frac{1}{4} belongs to \frac{1}{2}Z. And indeed since e_{−m} = \frac{1}{2}(v_{m} − w_{m}) belongs to B^{0}(α′′_{1}, λ + \frac{1}{4}) while e_{m} = \frac{1}{2}(v_{m} + w_{m}) with v_{m}, w_{m} ∈ B(α′′_{1}, λ + \frac{1}{4}), this F-line D_{1} equals D(α_{1}, λ + \frac{1}{4}). Also,

H_{2} =^{def} B^{0}(α_{1}, λ + \frac{1}{4})/B^{0}(α_{2}, λ) ⊃ S(β, λ) ⊕ S(α′′, λ)

is an F-hyperplane of S(α_{2}, λ), a complement of which is the F-line D_{2} spanned by the image of e_{m} ∈ B(α_{2}, λ) in S(α_{2}, λ), and also the canonical F-line D(α_{2}, λ) (which is spanned by the image of v_{m} ∈ B(α_{2}, λ), and equals D_{2} if s > 0). If λ ∈ \frac{1}{2}Z, then H_{2}^{⊥} ⊂ H_{2} for the pairing ϕ_{λ} on S(α_{2}, λ).

5.3.3. For i ∈ \{1, 2\}, we also have (\overline{V} , \overline{α}_{i}) = (W, β)⊥(\overline{V}′, \overline{α}′_{i}). Thus for all c ≥ 0,

S(\overline{α}_{i}, λ) = S(β, λ) ⊕ S(\overline{α}′_{i}, λ) and S_{c}(\overline{α}_{i}, λ) = S_{c}(β, λ) ⊕ S_{c}(\overline{α}′_{i}, λ).

Lemma. For the F-subspace H_{2} of S(\overline{α}_{2}, λ) ⊃ S(α_{2}, λ) and any c ≥ 0,

S_{c}(\overline{α}_{2}, λ) ⊂ H_{2} ⟺ c ≥ s_{m−1} + s.

Proof. For all c ≥ 0, S_{c}(β, λ) ⊂ S(β, λ) ⊂ H_{2}. Since ω(\overline{α}′_{2}) = ⌈s_{1}, ⋯, s_{m−1}, s⌉_{D} by 5.1.9, c_{max}(\overline{α}′_{2}) = s_{m−1} + s and S_{c}(\overline{α}′_{2}, λ) = 0 ⊂ H_{2} for all c ≥ s_{m−1} + s. In view of (2.2), we thus have to show that

H_{2} + im ([η] : S(\overline{α}′_{2}, λ′) → S(\overline{α}′_{2}, λ)) = S(\overline{α}_{2}, λ)

where λ′ = λ − s_{m−1} − s = r_{0} − s_{m−1}. If m = 1, s_{m−1} = 0 and we use

v_{0} ∈ B(\overline{α}′_{2}, λ′) with ηv_{0} = η^{2}w_{1} ∈ B(\overline{α}′_{2}, λ) − B^{0}(\overline{α}′_{1}, λ + \frac{1}{4}).

If m > 1, we consider instead e_{−(m−1)} ∈ B(\overline{α}′_{2}, λ′), for which

ηe_{−(m−1)} = \frac{1}{2}(η^{2}w_{m} − v_{m−2}) ∈ B(\overline{α}′_{2}, λ) − B^{0}(\overline{α}′_{1}, λ + \frac{1}{4}).

This proves our claim in both cases. □

5.3.4. These computations show that the edges α_{1} and α_{2} = pr_{s}α_{1} are related by a λ-corner (cf. section 3.14) of the type considered in the following lemma.

Lemma. Fix λ ∈ \frac{1}{4}Z and s ∈ \frac{1}{4}N. Let (α_{1}, D_{1}) ↦ (α_{2}, H_{2}) be a λ-corner. The following conditions are equivalent:

(1) d(α_{1}) = s + \frac{1}{4} and D_{1} = D(α_{1}, λ + \frac{1}{4}),

(2) d(α_{2}) = s and H_{2} ⊕ D(α_{2}, λ) = S(α_{2}, λ).

If so, then λ ∈ s + \frac{1}{2}Q(D), pr_{s}(α_{1}) = α_{2} and

c_{max}^{sp}(α_{1}) = \frac{1}{4} + min \{c ≥ c_{max}^{sp}(α_{2}) : S_{c}(α_{2}, s) ⊂ H_{2}\} .

Proof. Write B_{1}^{0} ⊂ B_{1} and B_{2}^{0} ⊂ B_{2} for the open and closed balls of α_{1} at λ + \frac{1}{4} and α_{2} at λ, so that B_{2}^{0} ⊂ B_{1}^{0} ⊊ B_{2} ⊂ B_{1} with D_{1} = B_{2}/B_{1}^{0} and H_{2} = B_{1}^{0}/B_{2}^{0}. Note that

D(α_{1}, λ + \frac{1}{4}) = D ∩ B_{1} + B_{1}^{0}/B_{1}^{0} and D(α_{2}, λ) = D ∩ B_{2} + B_{2}^{0}/B_{2}^{0}.

Therefore D_{1} = D(α_{1}, λ + \frac{1}{4}) ⟺ S(α_{2}, λ) = H_{2} ⊕ D(α_{2}, λ), in which case

λ + \frac{1}{4} ∈ d(α_{1}) + \frac{1}{2}Q(D) and λ ∈ d(α_{2}) + \frac{1}{2}Q(D).

Since also |d(α_{1}) − d(α_{2})| ≤ dist(α_{1}, α_{2}) = \frac{1}{4}, this shows that (1) ⟺ (2). The remaining assertions follow from the above computations of pr_{s}α_{1}. □

5.4. The fibers of convex projection. Fix ω = 2ω_{reg} + [s_{1}, ⋯, s_{m}]_{D} ∈ V_{D}^{n} – with (s_{1}, ⋯, s_{m}) satisfying Sp. Let β ∈ I(\underline{V}) be any self-dual F -norm on \underline{V} for which ω(β) = ω_{reg} + ⌊s_{1}, ⋯, s_{m}⌋_{D}. We shall now compute the number

N(ω) = |\{α ∈ I(\frac{1}{4}) : ω(α) = ω and \underline{α} = β\}| .

5.4.1. Fix α ∈ I(\frac{1}{4}) with ω(α) = ω and \underline{α} = β. Put s = d(α) = s_{m} and for t ∈ [0, s], let {α_{t}} = [α_{0} : α_{s}] ∩ I_{t} where α_{0} is the convex projection of α_{s} = α to I_{0}(V). Put s_{0} = s_{−1} = 0. Then for t ∈]s_{i−2}, s_{i}] with 1 ≤ i ≤ m and i ≡ m mod 2,

ω_{sp}(α_{t}) = [s_{1}, ⋯, s_{i−1}, t]_{D} and c_{max}^{sp}(α_{t}) = s_{i−1} + t

With these notations, N(ω) = ∏_{t∈[0,s[∩ \frac{1}{4} N}N(ω, t) where

N(ω, t) = |\{α′ ∈ I_{t+ \frac{1}{4}} (\frac{1}{4}) : ω(α′) = ω(α_{t+ \frac{1}{4}} ) and pr_{t}(α′) = α_{t}\}| .

5.4.2. Fix λ(0) ∈ \frac{1}{2}Q(D) and set λ(t) = t + λ(0). For t ∈ [0, s[∩\frac{1}{4}N, we know from section 5.3 that N(ω, t) is also the number of F-hyperplane complements H of D(α_{t}, λ(t)) in S(α_{t}, λ(t))such that H^{⊥} ⊂ H for the symmetric pairing ϕ_{t+r_{0}} on S(α_{t}, λ(t)) if λ(t) ∈ \frac{1}{2}Z, and such that

min \{c ≥ c_{max}^{sp}(α_{t}) : S_{c}(α_{t}, λ(t)) ⊂ H\} = c_{max}^{sp}(α_{t+ \frac{1}{4}} ) − \frac{1}{4}.

5.4.3. For all t ∈ [0, s[, there is a unique 1 ≤ i ≤ m with i ≡ m mod 2 such that t ∈ [s_{i−2}, s_{i}[. Put c(t) = s_{i−1} + t and define

ν(t) = dim_{F} S(β, λ(t)) − dim_{F} S_{c(t)}(β, λ(t)) − \begin{cases}0 & if λ(t) ∉ Z \\ 1 & if λ(t) ∈ Z \end{cases}

and R(t) = \begin{cases}1 & if t ≠ s_{i−2}, \\ 1 − q^{− dim_{F} Gr_{c(t)}S(β,λ(t))} & if t = s_{i−2} and c(t) ≠ 0, \\ 1 − (−q)^{− dim_{E} S(β,λ(0),0)} & if c(t) = 0. \end{cases}

Note that c(t) = 0 implies t = s_{i−1} = s_{i−2} = 0. Note also that for all t ∈ [0, s[∩\frac{1}{4}N, c(t) = c_{max}^{sp}(α_{t+ \frac{1}{4}} ) − \frac{1}{4} by 5.4.1. We will prove that

Proposition. For all t ∈ [0, s[∩\frac{1}{4}N, N(ω, t) = q^{ν(t)}R(t).

5.4.4. Suppose first that t ∈]s_{i−2}, s_{i}[∩\frac{1}{4}N. Then N(ω, t) is just the number of F-hyperplane complements H of D(α_{t}, λ(t)) in S(α_{t}, λ(t)) with S_{c(t)}(α_{t}, λ(t)) ⊂ H and H^{⊥} ⊂ H if t ∈ \frac{1}{2}N, namely

N(ω, t) = q ∧ (dim_{F} S(α_{t}, λ(t)) − dim_{F} S_{c(t)}(α_{t}, λ(t)) − \{ \begin{matrix}2 & if t ∈ \frac{1}{2}N \\ 1 & if t ∉ \frac{1}{2}N \end{matrix} \})

by lemma 7 and 5 of the appendix, respectively. Using explicit decompositions of (V, α) as we did already several times, we find that

dim_{F} S(α_{t}, λ(t)) = dim_{F} S(β, λ(t)) + \begin{cases}1 & if t ∈ N \\ 2 & if t ∈ \frac{1}{2}N − N \\ 1 & if t ∈ \frac{1}{4}N − \frac{1}{2}N \end{cases}

and dim_{F} S_{c(t)}(α_{t}, λ(t)) = dim_{F} S_{c(t)}(β, λ(t))

since also c(t) = c_{max}^{sp}(α_{t}). Therefore N(ω, t) = q^{ν(t)}R(t) in this case.

5.4.5. Suppose next that t = s_{i−2} ≠ 0. Thus N(ω, t) now equals q^{ν(t)} minus

q ∧ (dim_{F} S(α_{t}, λ(t)) − dim_{F} S_{c(t)− \frac{1}{4}} (α_{t}, λ(t)) − \{ \begin{matrix}2 & if t ∈ \frac{1}{2}N \\ 1 & if t ∉ \frac{1}{2}N \end{matrix} \})

But c(t) − \frac{1}{4} ≥ c_{max}^{sp}(α_{t}), so that again

dim_{F} S_{c(t)− \frac{1}{4}} (α_{t}, λ(t)) = dim_{F} S_{c(t)− \frac{1}{4}} (β, λ(t))

and N(ω, t) = q^{ν(t)}R(t). If t = 0 but c(0) ≠ 0, we obtain N(ω, 0) = q^{ν(0)}R(0) similarly, using Lemma 6 instead of 7 in the appendix.

5.4.6. Suppose finally that t = 0 = c(0). Then lemma 6 below shows that

N(ω, 0) = q^{ν(0)}(1 − ϵq^{\frac{1}{2} dim_{F} S(β,λ(0),0)})

where ϵ = 1 if S(β, λ(0)) is split and −1 otherwise. Note that S(β, λ(0)) splits if and only if S(β, λ(0), 0) does, and recall from section 4.3.5 that the latter is the F-quadratic space underlying an E-hermitian space. Therefore

ϵq^{\frac{1}{2} dim_{F} S(β,λ(0),0)} = (−q)^{dim_{E} S(β,λ(0),0)}

and N(ω, 0) = q^{ν(0)} · R(0), which finishes the proof of proposition 5.4.3.

6. Appendix

6.1. Proof of Lemma 4.1.5. Let (V, ψ) be a finite dimensional E-hermitian space which is split and let α be an F -norm on V which is self-dual with respect to the underlying symmetric F -form ϕ = trψ. Fix a Witt decomposition

(V, α) = ⊥_{i=1}^{n}(V_{i}, α_{i}) ⊕ (V_{−i}, α_{−i})

in which all the V_{±i}’s are isotropic E-lines in V, so that ηψ induces isomorphims (V_{i}, α_{i}) ≃ (V_{−i}, α_{−i})^{⋆} = (V_{−i}^{⋆}, α_{−i}^{⋆}). We let (ρ_{±i}, c_{±i}) be the type of (V_{±i}, α_{±i}) with respect to some E-basis e_{±i} of V_{±i}, chosen such that ηψ(e_{−i}, e_{i}) = 1 (and thus ρ_{−i} = c_{i} − ρ_{i} and c_{−i} = c_{i}).

For i and j in {±1, ⋯, ±n}, we denote by u_{i,j} : V_{j} → V_{i} the (i, j)-component of an E-linear endomorphism u : V → V. Thus for the adjoint endomorphism u^{⋆} : V → V (with respect to ψ), we have

(u^{⋆})_{i,j} : V_{j} → V_{i}  =  (u_{−j,−i})^{⋆} : V_{−j}^{⋆} → V_{−i}^{⋆}

under the above identifications V_{i} ≃ V_{−i}^{⋆} and V_{j} ≃ V_{−j}^{⋆}. In the sequel, we will describe certain endomorphisms of V by specifying only some of their components, in matrix form: the unspecified u_{i,j}’s will always be given by Kronecker’s symbol, namely u_{i,j} = 0 if i ≠ j and u_{i,j} = Id : V_{i} → V_{i} if i = j.

We denote by U(α) the stabilizer of α in U(V, ψ). Note that a given element u ∈ U(V, ψ) fixes α if and only if it also belongs to End_{N} (V, α) which amounts to requiring that all of its components u_{j,i} : V_{j} → V_{i} are actually morphisms (V_{j}, α_{j}) → (V_{i}, α_{i}) in the category N. Indeed and more generally, an element g ∈ O(V, ϕ) fixes α if and only if it is compatible with the norm in the weak sense that α(gv) ≤ α(v) for all v ∈ V, because then also

α(v) = sup \frac{|ϕ(v, w)|}{α(w)} = sup \frac{|ϕ(gv, gw)|}{α(w)} ≤ sup \frac{|ϕ(gv, gw)|}{α(gw)} = α(gv).

This being said, we may construct some obvious families of elements in U(α):

• For 1 ≤ i ≤ n and γ = γ_{i} : (V_{i}, α_{i}) → (V_{i}, α_{i}) an isomorphism,

t(γ_{i}) = ( \begin{matrix}γ \\ & γ^{⋆−1} \end{matrix} ) in U(V_{i} ⊕ V_{−i}).

• For 1 ≤ i ≠ j ≤ n and x = x_{j,i} : (V_{i}, α_{i}) → (V_{j}, α_{j}) a morphism,

u(x_{j,i}) = ( \begin{matrix}Id \\ x & Id \\ & & Id & −x^{⋆} \\ & & & Id \end{matrix} ) in SU(V_{i} ⊕ V_{j} ⊕ V_{−i} ⊕ V_{−j})

• For 1 ≤ i ≠ j ≤ n and x = x_{−j,i} : (V_{i}, α_{i}) → (V_{−j}, α_{−j}) a morphism,

u(x_{−j,i}) = ( \begin{matrix}Id \\ & Id \\ & −x^{⋆} & Id \\ x & & & Id \end{matrix} ) in SU(V_{i} ⊕ V_{j} ⊕ V_{−i} ⊕ V_{−j})

• For 1 ≤ i ≠ j ≤ n and x = x_{j,−i} : (V_{−i}, α_{−i}) → (V_{j}, α_{j}) a morphism,

u(x_{−j,i}) = ( \begin{matrix}Id & & & −x^{⋆} \\ & Id & x \\ & & Id \\ & & & Id \end{matrix} ) in SU(V_{i} ⊕ V_{j} ⊕ V_{−i} ⊕ V_{−j})

• For 1 ≤ i ≤ n and x = x_{−i,i} : (V_{i}, α_{i}) → (V_{−i}, α_{−i}) a morphism such that x^{⋆} = −x,

u(x_{−i,i}) = \binom{Id}{x Id} in SU(V_{i} ⊕ V_{−i})

• For 1 ≤ i ≤ n and x = x_{i,−i} : (V_{−i}, α_{−i}) → (V_{i}, α_{i}) a morphism such that

x^{⋆} = −x,

u(x_{i,−i}) = \binom{Id x}{Id} in SU(V_{i} ⊕ V_{−i})

• For 1 ≤ i ≠ j ≤ n and θ = θ_{j,i} : (V_{i}, α_{i}) → (V_{j}, α_{j}) an isomorphism,

w(θ_{j,i}) = ( \begin{matrix}& θ^{−1} \\ θ \\ & & & θ^{⋆} \\ & & θ^{⋆−1} \end{matrix} ) in SU(V_{i} ⊕ V_{j} ⊕ V_{−i} ⊕ V_{−j})

• For 1 ≤ i ≠ j ≤ n and θ = θ_{−j,i} : (V_{i}, α_{i}) → (V_{−j}, α_{j}) an isomorphism,

w(θ_{−j,i}) = ( \begin{matrix}& & & θ^{−1} \\ & & θ^{⋆−1} \\ & θ^{⋆} \\ θ \end{matrix} ) in SU(V_{i} ⊕ V_{j} ⊕ V_{−i} ⊕ V_{−j})

• For 1 ≤ i ≤ n and θ = θ_{−i,i} : (V_{i}, α_{i}) → (V_{−i}, α_{−i}) an isomorphism,

w(θ_{−i,i}) = ( \begin{matrix}& θ^{⋆−1} \\ θ \end{matrix} ) in U(V_{i} ⊕ V_{−i})

Lemma. These elements span U(α).

Proof. Without loss of generality, we may assume that c_{1} ≥ c_{2} ≥ ⋯ ≥ c_{n}. Let g ∈ U(α). By corollary 3.5 of [1], there exists j ∈ {±1, ⋯, ±n} such that g_{j,1} : (V_{1}, α_{1}) → (V_{j}, α_{j}) is an isomorphism. Replacing g by t(g_{1,1}^{−1}) ∘ g (if j = 1) or ω(g_{j,1}) ∘ g (if j ≠ 1), we may assume that j = 1 and g_{1,1} = Id : V_{1} → V_{1}. Replacing g by u(−g_{±i,1}) ∘ g, we may also assume that g_{±i,1} = 0 for all 1 < i ≤ m. Then g_{−1,1}^{⋆} = −g_{−1,1}, and thus replacing g by u(−g_{−1,1}) ∘ g, we may finally also assume that g_{−1,1} = 0. Multiplying now g on the right by similar unipotent elements, we arrive at a situation where g_{1,i} = g_{i,1} = 0 for all i ∈ {−1, ±2, ⋯, ±n}, with g_{1,1} = Id. Then g|V_{1} ⊕ V_{−1} is the identity, and g′ ∈ U(α′) where g′ and α′ are the restrictions of g and α to the orthogonal complement V′ = ⊥_{i=2}^{n}(V_{i} ⊕ V_{−i}) of V_{1} ⊕ V_{−1}. We conclude by induction on n. □

We may now prove lemma 4.1.5. Of the above generators, only the first and last families contribute to the determinant. An isomorphism γ_{i} : (V_{i}, α_{i}) → (V_{i}, α_{i}) is the scalar multiplication by some element γ_{i} ∈ O_{r_{i}}^{×} with r_{i} = ⌈c_{i}⌉, and det t(γ_{i}) thus equals γ_{i}/\overline{γ_{i}} ∈ U_{r_{i}}. On the other hand, let θ_{−i,i} : (V_{i}, α_{i}) → (V_{−i}, α_{−i}) be an isomorphism. Then ρ_{−i} ≡ ρ_{i} mod Z, i.e. c_{i} −2ρ_{i} = n_{i} ∈ Z and θ_{−i,i}(e_{i}) = π^{n_{i}}γ_{i}e_{−i} for some γ_{i} ∈ O_{r_{i}}^{×} (with r_{i} = ⌈c_{i}⌉ as above). But then θ_{−i,i}^{⋆}(e_{i}) = −π^{n_{i}} \overline{γ}_{i}e_{−i} and thus again det w(θ_{−i,i}) = γ_{i}/\overline{γ}_{i} ∈ U_{r_{i}}, QED.

6.2. Counting Lemmas. Recall from section 4.3.9 the functions

π(r) = (q^{r} − 1)(q^{r−1} − 1) ⋯ (q − 1),
σ(r) = (q^{r} + 1)(q^{r−1} + 1) ⋯ (q + 1),
τ(r) = (q^{r} − (−1)^{r})(q^{r−1} − (−1)^{r−1}) ⋯ (q + 1).

The following formulas are classical

|GL(r, F)| = q^{\frac{(r−1)r}{2}} π(r), |U(r, F)| = q^{\frac{(r−1)r}{2}} τ(r), |Sp(2r, F)| = q^{r^{2}}σ(r)π(r).

Lemma 1. Let (V, ψ) be a E-hermitian space of E-dimension d = 2a + 2b + 2c + e. The number of totally isotropic F-subspaces H of V for which

dim_{E} ( \begin{matrix}H ∩ ηH \\ H ∩ ηH^{⊥} + ηH ∩ H^{⊥} \\ H + ηH \end{matrix} ) = ( \begin{matrix}1 \\ 1 & 1 \\ 1 & 1 & 2 \end{matrix} ) ( \begin{matrix}a \\ b \\ c \end{matrix} )

equals

q^{2c(b+e)+c(c−1)+ \frac{b(b−1)}{2}} · \frac{τ(d)}{π(a)π(b)π(c)σ(a)σ(c)τ(e)}

Proof. We consider the following diagram

\xymatrix{ &  & H \ar@{-->}[dr] &  &  & H_{1} = H ∩ ηH \\  & H′ \ar@{-->}[ur] \ar@{-->}[dr] &  & \overline{H} \ar[dr] &  &  & H_{2} = H ∩ ηH^{⊥} + H^{⊥} ∩ ηH \\ H_{1} \ar@{-->}[ur] \ar@{-->}[dr] &  & H_{2} \ar@{-->}[ur] \ar@{-->}[dr] &  & H_{3} & H_{3} = H + ηH \\  & ηH′ \ar@{-->}[ur] \ar@{-->}[dr] &  & η\overline{H} \ar@{-->}[ur] &  & H′ = H ∩ H_{2} \\  &  & ηH \ar@{-->}[ur] &  &  & \overline{H} = H + H_{2}}

We choose in this order (1) the totally isotropic E-subspace H_{1} of V, (2) the totally isotropic E-subspace H_{2} = H_{2}/H_{1} of V_{1} = H_{1}^{⊥}/H_{1}, (3) the non-degenerate E-subspace V_{3} = H_{3}/H_{2} of V_{2} = H_{2}^{⊥}/H_{2}, (4) the totally isotropic F-subspace H_{4} = \overline{H}/H_{2} of V_{3} such that H_{4} ⊕ ηH_{4} = V_{3}, (5) the F-subspace H_{5} = H′/H_{1} of H_{2} such that H_{5} ⊕ ηH_{5} = H_{2}, and finally (6) the complement H_{6} = H/H′ of H_{2}/H′ in the F-space \overline{H}/H′. We now compute the number of possible choices at each step. For (1) and (2), [3, 9.4.1] gives respectively

\frac{τ(d)}{π(a)σ(a)τ(d − 2a)} and \frac{τ(d − 2a)}{π(b)σ(b)τ(d − 2a − 2b)}

For (3): the subspaces V_{3} are all conjugated under U(V_{2}), which gives

\frac{|U(d − 2a − 2b)|}{|U(2c)| · |U(d − 2a − 2b − 2c)|} = q^{2c(d−2a−2b−2c)} \frac{τ(d − 2a − 2b)}{τ(2c) · τ(d − 2a − 2b − 2c)}

possible choices for V_{3}. For (4), we introduce the symplectic form θ = tr_{E/F}ηψ on V_{3}. If (e_{±i}) is a Witt E-basis of V_{3} (with ηψ(e_{i}, e_{−i}) = 1), then (e_{±i}) spans over F a totally isotropic F-subspace H_{4} of V_{3} such that H_{4} ⊕ ηH_{4} = V_{3}, and (e_{±i}) is a Witt F-basis for (H_{4}, θ|H_{4}). Conversely, for any totally isotropic F-subspace H_{4} of V_{3} such that H_{4} ⊕ ηH_{4} = V_{3}, the restriction of θ to H_{4} is non-degenerate and for any Witt F-basis (e_{±i}) of (H_{4}, θ|H_{4}), (e_{±i}) is also a Witt E-basis of V_{3}. This gives

\frac{|U(2c)|}{|Sp(2c)|} = q^{c(c−1)} · \frac{τ(2c)}{π(c)σ(c)}

possible choices for H_{4}. For (5), a similar argument gives

\frac{|GL(b, E)|}{|GL(b, F)|} = q^{\frac{(b−1)b}{2}} · σ(b)

possible choices for H_{5}. Finally for (6), there are q^{2bc} possible complements H/H′ of H_{2}/H′ in \overline{H}/H′. □

Lemma 2. Let V_{1} and V_{2} be F-vector spaces of dimension d = a + b + c with a perfect pairing ⟨•, •⟩ : V_{1} × V_{2} → F. The number of pairs (H_{1}, H_{2}) of F-subspaces H_{1} ⊂ V_{1} and H_{2} ⊂ V_{2} of F-dimensions a and b such that ⟨H_{1}, H_{2}⟩ = 0 equals

\frac{π(d)}{π(a)π(b)π(c)}.

Proof. We choose first H_{1} in V_{1} and then H_{2} in H_{1}^{⊥} ⊂ V_{2}. There are

\frac{π(d)}{π(a) · π(d − a)} and \frac{π(d − a)}{π(b)π(d − a − b)}

choices at each step. □

Lemma 3. Let (V, θ) be a symplectic F-vector space of F-dimension 2d with

d = a + b + c + b′ + c′ + e.

The number of pairs of F-subspaces (B, C) of V such that θ(B, C) = 0 and

dim_{F} ( \begin{matrix}B ∩ C \\ B ∩ B^{⊥} \\ C ∩ C^{⊥} \\ B \\ C \end{matrix} ) = ( \begin{matrix}1 \\ 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0 & 2 \\ 1 & 0 & 1 & 0 & 2 \end{matrix} ) ( \begin{matrix}a \\ b \\ c \\ b′ \\ c′ \end{matrix} )

equals

q^{2b′(e+c′+c)+2c′(b+e)+bc} \frac{π(d)σ(d)}{π(a)π(b)π(c)π(b′)π(c′)π(e)σ(b′)σ(c′)σ(e)}

Proof. We consider the following diagram

\xymatrix{ &  & B \ar[dr] &  &  & A_{1} = B ∩ C \\  & B′ \ar@{-->}[ur] \ar@{-->}[dr] &  & \overline{B} \ar@{-->}[dr] &  &  & A_{2} = B ∩ B^{⊥} + C ∩ C^{⊥} \\ A_{1} \ar@{-->}[ur] \ar@{-->}[dr] &  & A_{2} \ar@{-->}[ur] \ar@{-->}[dr] &  & A_{3} & A_{3} = B + C \\  & C′ \ar@{-->}[ur] \ar[dr] &  & \overline{C} \ar@{-->}[ur] &  & X′ = X ∩ X^{⊥} \\  &  & C \ar@{-->}[ur] &  &  & \overline{X} = A_{2} + X}

for X ∈ {B, C}. We choose in this order: (1) the totally isotropic subspace A_{1} of V, (2) the totally isotropic subspace A_{2} = A_{2}/A_{1} of V_{1} = A_{1}^{⊥}/A_{1}, (3) the nondegenerate subspace A_{3} = A_{3}/A_{2} of V_{2} = A_{2}^{⊥}/A_{2}, (4) the orthogonal decomposition \overline{B}⊥\overline{C} = A_{3} where \overline{X} = \overline{X}/A_{2} for X ∈ \{B, C\}, (5) the decomposition B′ ⊕ C′ = A_{2} where X′ = X′/A_{1} for X ∈ {B, C}, and finally (6) the complement X = X/X′ of A_{2}/X′ in \overline{X}/X′ for X ∈ \{B, C\}. We now compute the number of possible choices at each step. For (1) and (2), [3, 9.4.1] gives respectively

\frac{π(d)σ(d)}{π(a)π(d − a)σ(d − a)} and \frac{π(d − a)σ(d − a)}{π(b + c)π(d − a − b − c)σ(d − a − b − c)}.

For (3) and (4) together, note that all pairs (\overline{B}, \overline{C}) are conjugated by Sp(V_{2}). Thus

\frac{|Sp(V_{2})|}{|Sp(\overline{B})| · |Sp(\overline{C})| · |Sp(\overline{D})|} = q^{2(b′+c′)e+2b′c′} · \frac{π(d − a − b − c)σ(d − a − b − c)}{π(b′)σ(b′)π(c′)σ(c′)π(e)σ(e)}

where \overline{D}⊥\overline{B}⊥\overline{C} = V_{2}. For (5), we obtain

\frac{|GL(b + c, F)|}{|GL(b, F)| · |GL(c, F)|} = q^{bc} · \frac{π(b + c)}{π(b) · π(c)}

and finally for (6), q^{2b′c+2bc′}. □

Lemma 4. Let (V, ϕ) be a symmetric F-space. Suppose that we are given a sequence of totally isotropic subspaces B ⊂ T^{⊥} ⊂ S ⊂ M of V, and let

e = dim_{F} T^{⊥}/B, t = dim_{F} S/T^{⊥}, m = dim_{F} M/S and d = dim_{F} M^{⊥}/M.

The number of totally isotropic F-subspaces H of V for which

H ∩ S = B, H ∩ S^{⊥} + S = M and H + S^{⊥} = T

equals q^{me+t(d+m+e+ \frac{1}{2} (t−1))}.

Proof. We consider the following diagram

\xymatrix{ &  & H \ar@{-->}[dr] &  &  & H′ = H ∩ S^{⊥} \\  & H′ \ar@{-->}[ur] \ar[dr] &  & \overline{H} \ar@{-->}[dr] &  & \overline{H} = H + T^{⊥} \\ B \ar@{-->}[ur] \ar@{-->}[dr] &  & K \ar@{-->}[ur] \ar[dr] &  & L & K = H′ + T^{⊥} \\  & T^{⊥} \ar@{-->}[ur] \ar@{-->}[dr] &  & M \ar@{-->}[ur] &  & L = H + S \\  &  & S \ar@{-->}[ur]}

We choose in this order (1) the complement H′/B of S/B in M/B, which then gives K = H′ + T^{⊥}, (2) the complement L/M of M^{⊥}/M in K^{⊥}/M, which gives the non-degenerate subspace L/K of K^{⊥}/K, (3) the totally isotropic F-subspace \overline{H}/K of L/K such that \overline{H}/K ⊕ M/K = L/K, and (4) the complement H/H′ of K/H′ in \overline{H}/H′. For (1), (2) and (4), there are respectively q^{m(e+t)}, q^{dt} and q^{et} possible choices. For (3), there are q^{\frac{t(t−1)}{2}} choices by a simple case of [3, 9.4.2]. □

Lemma 5. Let V be an F-vector space of F-dimension d ≥ 1 + a with an F-line D and an F-subspace A ⊅ D of F-dimension a. The number of F-hyperplanes H of V such that A ⊂ H and D ⊕ H = V equals q^{d−a−1}.

Proof. Easy. □

Lemma 6. Let (V, ϕ) be an odd-dimensional symmetric F-space, D an anisotropic F-line in V and S a totally isotropic F-subspace of W = D^{⊥}. Put a = dim_{F} S and 2d = dim_{F} W. The number of F-hyperplanes H of V such that

H^{⊥} ⊂ H, S ⊂ H and H ⊕ D = V

equals q^{d−1}(q^{d−a} − 1) if V is split and q^{d−1}(q^{d−a} + 1) otherwise.

Proof. This is also the number of isotropic F-lines L = H^{⊥} in S^{⊥} ⊕ D such that L ⊄ S^{⊥}, where S^{⊥} is the orthogonal complement of S in W. Let L be the image of L in V = W ⊕ D where W = S^{⊥}/S. This is an isotropic F-line of V which is not contained in W. By [3, 9.4.1], the number of such lines equals q^{d−a−1}(q^{d−a} − 1) if V is split and q^{d−a−1}(q^{d−a} + 1) otherwise. For each L = L + S/S, there are exactly q^{a} complements L of S in L + S. □

Lemma 7. Let (V, ϕ) be a symmetric F-space, D an isotropic F-line in V and S a totally isotropic subspace of V such that D ⊄ S but D ⊂ S^{⊥}. Write d = dim_{F} V and a = dim_{F} S. The number of F-hyperplanes H in V such that H^{⊥} ⊂ H, S ⊂ H and H ⊕ D = V is equal to q^{d−a−2}.

Proof. This is also the number of isotropic F-lines L = H^{⊥} in S^{⊥} such that L ⊄ D^{⊥}. Let L and D be the images of L and D in V = S^{⊥}/S. Then D is an isotropic F-line in V and L is an isotropic F-line in V such that L ⊄ D^{⊥}. By [3, 9.4.2] or lemma 4 above, the number of all such lines L equals q^{d−2a−2}. For each L = L + S/S, there are q^{a}-choices for the complement L of S in L + S. □

References

[1] C. Cornut, Normes p-adiques et extensions quadratiques. Ann. Inst. Fourier (Grenoble) 59 (2009), no. 6, pp. 2223–2254.

[2] B.H. Gross, Heegner points and representation theory. Heegner points and Rankin L-series, pp. 37–65, Math. Sci. Res. Inst. Publ., 49, Cambridge Univ. Press, Cambridge, 2004.

[3] A.E Brouwer, A.M Cohen and A. Neumaier, Distance-regular Graphs. Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, Band 18, Springer-Verlag.

CNRS - Institut de Mathématiques de Jussieu. UMR 7586, Case 7012, Bâtiment Chevaleret, 75205 Paris CEDEX 13.

   E-mail address: cornut@math.jussieu.fr