TYPES AND REPRESENTATIONS OF p -ADIC SYMPLECTIC GROUPS

Submitted for the degree of Doctor of Philosophy

Shaun Ainsley Ross Stevens

Department of Mathematics King’s College London University of London

December 1998 2

ABSTRACT

Let F be a non-archimedean local field and let G = 𝔾(F) be the F -points of a reductive group defined over F. Bushnell and Kutzko have described a strategy to classify the representations of G via the theory of types, which associates to each inertial class in the Bernstein spectrum a pair (K, ρ) consisting of a compact open subgroup K of G and an irreducible representation ρ of K. We impose the restriction that the residual characteristic of F not be 2. In this thesis we begin the construction of types associated to certain discrete series (in particular, to supercuspidal) representations of G = Sp_{2N}(F) by transferring Bushnell and Kutzko’s construction for GL_{2N}(F) to our situation. Certain objects in the construction, in particular the simple characters, transfer simply by restriction. In a certain case, we complete the construction of the type (K, ρ) and hence construct new supercuspidal representations in the wildly ramified case. In this case, we are also able to describe a (tentative) transfer map from certain supercuspidal representations of GL_{2N}(F) to supercuspidal representations of Sp_{2N}(F), which associates to each representation π of GL_{2N}(F) a set Π(π) of representations of Sp_{2N}(F).

4

CONTENTS

ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1 PRELIMINARIES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.1 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.2 Hereditary orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.3 Parahoric subgroups and filtrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .19 1.4 Characters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .21

2 EXACTNESS AND INTERTWINING . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.1 Exact sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.2 Intertwining of simple strata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3 RESIDUAL SUBSPACES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3.1 Residual subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .30 3.2 Splittings of self-dual lattice chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 3.3 Residual maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4 REFINEMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 4.1 (W, E)-decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 4.2 Refinements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

5 STANDARD FORM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 5.1 Lifting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 5.2 Jordan form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 5.3 Non-split fundamental strata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 5.4 Pure is equivalent to simple . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

6 SIMPLE CHARACTERS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 6.1 The orders ℌ and 𝔍 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 6.2 Simple characters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 6.3 Intertwining . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 6.4 Heisenberg Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

7 A SPECIAL CASE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 7.1 Construction of types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 7.2 Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 6

ACKNOWLEDGEMENTS

I would like thank my supervisor, Colin Bushnell: for setting me off on this research problem; for keeping me on track when I needed it; and for having faith in me throughout. I would like especially to thank Paul Broussous for his tirelessness in answering questions, for proofreading the earlier versions of this thesis and for many useful suggestions. Most of all, I would like to thank Ginny for putting up with my moods and for her support throughout these last three years. The research for this thesis was funded by the King’s College London Association.

8

INTRODUCTION

The aim of this thesis is to begin the systematic examination of the admissible dual of the symplectic group Sp_{2N}(F) over a non-archimedean local field, of residual characteristic not 2, via the theory of types (see below).

Let F be a non-archimedean local field and let 𝔾 be a reductive group defined over F. Let G = 𝔾(F) be the F -points of 𝔾 and let ℜ(G) be the category of smooth representations of G. For i = 1, 2, let M_{i} be a Levi subgroup of G and let σ_{i} be an irreducible supercuspidal representation of M_{i}. We say that the pairs (M_{i}, σ_{i}) are inertially equivalent if there exist g ∈ G and χ an unramified quasicharacter of M_{2} such that M_{1} = gM_{2}g^{−1} and σ_{2} ⊗ χ ≃ σ_{1}^{g}. We write 𝔰 = [M_{1}, σ_{1}]_{G} for the inertial equivalence class of (M_{1}, σ_{1}) with respect to this equivalence relation and write B for the set of such equivalence classes (called the Bernstein spectrum). For each inertial equivalence class 𝔰 = [M, σ]_{G}, we can define a subcategory ℜ^{𝔰}(G) of ℜ(G), which is the full subcategory of representations π of G such that any irreducible subquotient of π is equivalent to an irreducible subquotient of the parabolically induced representation Ind_{L,P}^{G} τ, for some (L, τ) ∈ 𝔰, P a parabolic subgroup of G with Levi factor L. Then we have the Bernstein decomposition ℜ(G) = ∏_{𝔰∈B} ℜ^{𝔰}(G) ([B] or see [BK2]). In [BK2], Bushnell and Kutzko describe a strategy for classifying the representations of the group G via types: these are pairs (K, ρ) consisting of a compact open subgroup K and an irreducible representation ρ of K. The pair (K, ρ) is then called an 𝔰-type, for 𝔰 = [M, σ]_{G}, if, for π any irreducible representation of G, π contains ρ (i.e. Hom_{K}(ρ, π) ≠ 0) if, and only if, π ∈ |ℜ^{𝔰}(G)|. If this is the case, then we have an equivalence of categories

M_{ρ} : ℜ^{𝔰}(G) ⟶^{≃} H(G, ρ)-Mod,

where H(G, ρ) is the spherical Hecke algebra and H(G, ρ)-Mod is the category of left modules over H(G, ρ). So the strategy is first to construct types for each inertial equivalence class 𝔰 and then to describe H(G, ρ)-Mod. This has been done for G = GL_{N}(F) (see [BK], [BK1]) and, partially, for G a division algebra over F (see [Br1]) and for G = SL_{N}(F) (see [BK3], [BK4]).

If 𝔰 = [M, σ]_{G} and M is a proper Levi subgroup of G then [BK2] §8 describes a method for approaching the construction of an 𝔰-type via covers, if you already have an [M, σ]_{M}-type (i.e. a type for the decomposition of ℜ(M)). This has been used to construct types for certain classical groups (see [Bl], [Au]) and, in this way, types have been constructed for every [M, σ]_{G}, M ⪇ G = Sp_{4}(F) if the residual characteristic is not 2 ([BB]). Types have also been constructed for principal series representations of split groups ([Ro]). For 𝔰 = [G, σ]_{G} (i.e. for supercuspidal representations of G), types have been constructed in the tame case ([Ad], [Kim], [M2], [M3]).

In this thesis we look at the case G = Sp_{2N}(F), for F of residual characteristic not 2, and take some steps toward the construction of types in the arbitrarily ramified case. The construction procedure follows closely [BK] and we often call upon results from there. It would also be possible to extend the results here to all unitary groups (indeed, many of the results are given in this generality), a work which is postponed to a later date.

At this point, we should remark that, in the recent paper [Ka], Kariyama generalizes the methods of Carayol (in [Ca]) to the symplectic group, for arbitrarily ramified tori. We shall discuss the similarities and differences with this work at the end of this introduction.

Let F be a non-archimedean local field of residual characteristic not 2 and V a 2Ndimensional F -vector space, equipped with a nondegenerate alternating form h. This form then induces an adjoint involution on A = End_{F} (V). The symplectic group G can then be seen as the set of g ∈ A such that g\overline{g} = 1 and the symplectic Lie algebra 𝔤 = A_{−} as the set of x ∈ A such that x + \overline{x} = 0. A skew stratum in A is a quadruple [𝔄, n, r, β] which is a stratum in the sense of [BK] (1.5) (i.e. 𝔄 is a hereditary 𝔬_{F} -order in A, n > r are integers and β ∈ 𝔓^{−n}, where 𝔓 is the Jacobson radical of 𝔄) subject to the additional conditions that 𝔄 is stable under the involution and β ∈ A_{−}. In the case of a stratum of the form [𝔄, n, n − 1, β] for GL_{N}(F), the stratum corresponds to a character ψ_{β} of the group U^{n}(𝔄) which is trivial on U^{n+1}(𝔄), where U^{r}(𝔄), r ∈ ℤ is the standard filtration (by powers of the Jacobson radical) of the parahoric subgroup 𝔄^{×} of GL_{N}(F). If a representation π of GL_{N}(F) contains the representation ψ_{β} then we say that π contains the stratum. For the symplectic group G the situation is similar, with the parahoric subgroup P = 𝔄 ∩ G and the standard filtration P^{r}(𝔄) = U^{r}(𝔄) ∩ G. In fact, the character ψ_{β} of P^{n}(𝔄) is just the restriction of the character ψ_{β} of U^{n}(𝔄).

The construction procedure for simple types begins with a skew simple stratum [𝔄, n, 0, β] (see chapter 2 or [BK] (1.5.5) for the definition). These skew simple strata have the property that their intertwining is, in some sense, as small as possible. We then construct some special characters (called simple characters, see chapter 6 for definitions) of certain compact open subgroups H_{−}^{m+1}(β, 𝔄) ⊂ P^{m+1}(𝔄), for 0 ≤ m ≤ n − 1. In fact, these turn out to be nothing other than the restrictions of simple characters for GL_{2N}(F) ([BK] (3.2)). These simple characters also have a “good” intertwining formula.

Given a simple character θ of H_{−}^{1}(β, 𝔄), there exists a unique irreducible (Heisenberg) representation η of another compact open subgroup J_{−}^{1}(β, 𝔄) ⊃ H_{−}^{1}(β, 𝔄) which contains θ. This representation also has a “good” intertwining formula.

The final step then consists in extending the representation η to a representation κ of a compact open subgroup J_{−}(β, 𝔄) ⊃ J_{−}^{1}(β, 𝔄) which also has “small” intertwining (for GL_{2N}(F), such a κ is called a β-extension of η). We complete this step in the case where the field extension F [β]/F is maximal in A (i.e. it is of degree 2N).

The simple characters and the groups H_{−}^{k}(β, 𝔄), J_{−}^{k}(β, 𝔄) are defined by an inductive process. For [𝔄, n, 0, β] a (skew) simple stratum, there exists an integer r such that the stratum [𝔄, n, r − 1, β] is simple but [𝔄, n, r, β] is not - it is only pure (see chapter 2 or [BK] (1.5.5)).

The main result in [BK] which allows the induction procedure to be set up says that there is an element γ such that the stratum [𝔄, n, r, γ] is simple and equivalent to [𝔄, n, r, β] ([BK] (2.4.1)). Then the objects for the stratum [𝔄, n, 0, β] are defined in terms of those for [𝔄, n, 0, γ].

In our case, we must prove the analogous result: for [𝔄, n, r, β] a skew pure stratum there exists an element γ ∈ A_{−} such that the stratum [𝔄, n, r, γ] is skew simple and equivalent to [𝔄, n, r, β] (see (5.4.7)).

This construction procedure will not produce all types for Sp_{2N}(F) and we now discuss the reasons for this.

The idea for GL_{N}(F) is that, given a positive level representation π of GL_{N}(F), we can find a stratum which is contained in π and is fundamental (nondegenerate in [MP]). i.e. the coset β + 𝔓^{1−n} contains no nilpotent elements.

However, for the symplectic group (the situation is similar for other reductive groups) the results of Moy and Prasad ([MP]) require us to use more than just the standard filtrations of parahoric subgroups in order to obtain a similar result - in general the filtrations come from self-dual lattice functions (see [Br3]). Recent work [PY] has shown that for the symplectic group the only filtrations which are really needed are the standard ones and Morris’s “C-chain” filtrations (see also [M3]).

Here we consider only the standard filtrations; however, the results of chapter 1 were written by Morris for C-chain filtrations (indeed they are valid for any self-dual lattice function filtration) and the results of chapter 2 are, in a sense, generalizable to this case.

In searching for supercuspidal representations we next require the notion of a split stratum: if a stratum is split then we want any representation containing it to be non-supercuspidal (so any supercuspidal will contain a non-split stratum). In the case of GL_{N}(F) this is fairly simply expressed in terms of a characteristic polynomial of the stratum (see [BK] (2.3.3) or chapter 5 for the definition of characteristic polynomial) - a stratum is split if and only if its characteristic polynomial has at least two distinct irreducible factors. This definition is the correct one because every maximal torus T of GL_{N}(F) contained in no proper Levi subgroup is of the form T = E^{×} for E a field extension of F of degree N. Here we use the same definition of a split stratum but this will not be the correct one in general. As Morris remarked [M3] (1.3) (see also [Kim] (1.1)) the maximal tori in G = Sp_{2N}(F) contained in no proper Levi take the form

N_{1}(E_{1}) × ⋯ × N_{1}(E_{r}),

where E_{i}/F is a field extension of degree 2n_{i} contained in A, i = 1, …, r, ∑_{i=1}^{r} n_{i} = N, \overline{E}_{i} = E_{i} but is non-trivial on E_{i}, i = 1, …, r, and N_{1}(E_{i}) = \{e ∈ E_{i} : e\overline{e} = 1\} are the norm 1 elements of E_{i}. So we are restricting considerably the supercuspidals we hope to construct (even in Sp_{4}(F) we will miss some). However, we can hope methods similar to those of “semisimple types” [BK1] may allow us to obtain the remaining supercuspidals in the future. We are able to show (as a corollary to the results of chapter 5) that if an irreducible representation π contains a nonsplit fundamental skew stratum then it contains a skew simple stratum.

Note that there is a general definition of split stratum for any reductive group due to Lemaire [Le]. This definition is in terms of the building of G and a latticetheoretic translation has yet to be done.

There are several differences between this work and the paper [Ka]. In that paper, arbitrary maximal anisotropic tori are considered. This has the advantage that those strata which appear split (by our definition) are not excluded. However, considering only maximal tori will certainly not be enough - it is not so even for GL_{N}(F). Further, only elements which are a sum of minimal elements (see [BK] (1.4.14)) are considered, which again is insufficient even for GL_{N}(F). As in this thesis, only the standard filtrations of parahoric subgroups are considered.

We now give a brief summary of the contents of each chapter. In chapter 1 we present some preliminary results, most of which are due to Morris ([M1],[M2],[M3]). In chapter 2 we check that we have the exact sequences analogous to those in [BK] (1.4) and we prove the intertwining theorem for a skew simple stratum. In chapter 3 we introduce the notion of a residual subspace of a lattice chain 𝔏 (due to Bushnell) and prove some results about block decompositions for 𝔄. In chapter 4 we introduce the analogue of the (W, E)-decomposition [BK] (1.2.6) for the symplectic Lie algebra and use it to begin the refinement process. In chapter 5 we prove some results on Jordan decompositions and prove that a nonsplit fundamental skew stratum is “contained in” a skew simple stratum (cf.

[BK] (2.3.4)) and that a skew pure stratum is equivalent to a skew simple one (cf.

[BK] (2.4.1)). In chapter 6 we define the groups H_{−}^{m+1}(β, 𝔄) and J_{−}^{m+1}(β, 𝔄), define simple characters and calculate their intertwining. Finally, in chapter 7, we consider the case where the field extension associated to the skew simple stratum is a maximal extension of F in A; in this case we complete the construction of the type and examine a tentative transfer map from certain supercuspidals of GL_{2N}(F) to supercuspidals of G.

14

PRELIMINARIES

In this chapter we recall some results concerning self-dual lattice chains and parahoric subgroups of symplectic and unitary groups. Most of the results in this section can be found in Morris’s papers [M1], [M2], [M3].

(1.1) Notation

Let F be a non-archimedean field equipped with a Galois involution x ↦ \overline{x} with fixed field F_{0}. We allow the possibility that F_{0} = F. We will use the following notation throughout:

𝔬_{F} the discrete valuation ring in F;
𝔭_{F} the maximal ideal of 𝔬_{F};
k_{F} = 𝔬_{F} /𝔭_{F}, the residue class field of F;
p the residual characteristic of F;
q = p^{f} = #k_{F} .

(1.1.1) Assumption We assume throughout that p ≠ 2.

We have similar notation 𝔬_{0}, 𝔭_{0}, k_{0} for the same objects in F_{0}. We also use

ψ_{0} some fixed continuous character of the additive group of F_{0},

with conductor 𝔭_{0};

ψ_{F} = ψ_{0} ∘ tr_{F/F_{0}}, where tr_{F/F_{0}} denotes trace;
V an F -vector space of finite dimension N;
A = End_{F} (V).

Note that F/F_{0} is at worst tamely ramified, since the residual characteristic of F is not 2, so the character ψ_{F} of (F, +) has conductor 𝔭_{F}. If F/F_{0} is unramified, we put π_{F} = π_{0}, a uniformizer of F_{0}; if F/F_{0} is ramified, we choose π_{F} to be a uniformizer of F such that π_{F} + \overline{π_{F}} = 0 and put π_{0} = π_{F}^{2} . In either case we have π_{F} = ±\overline{π_{F}}.

Let h : V × V → F be a nondegenerate ϵ-hermitian form on V, with ϵ = ±1 (see [Sch] (7.1.2), for example). If F = F_{0} we exclude the case ϵ = 1 (i.e. we rule out orthogonal groups). For definiteness, assume that h is F -linear in the first variable. So

h(λv, w) = λh(v, w) = λϵ\overline{h(w, v)}, for v, w ∈ V, λ ∈ F.

Then h induces an adjoint involution on A, also denoted , given, for a ∈ A, by

h(av, w) = h(v, \overline{a}w), for all v, w ∈ V.

Note that for F embedded diagonally in A, the two involutions coincide. We set G = {g ∈ Aut_{F} (V) : h(gv, gw) = h(v, w) for all v, w ∈ V }

= \{g ∈ Aut_{F} (V) : g\overline{g} = 1\}.

This is the group of F_{0}-points of a unitary (or symplectic) group defined over F_{0}. For X an additive subgroup in A invariant under the involution, define

X_{−} = \{x ∈ X : \overline{x} = −x\}, the skew elements;

X_{+} = \{x ∈ X : \overline{x} = x\} , the symmetric elements.

These are both additive groups and if X is an 𝔬_{F} -lattice then

X = X_{+} ⊕ X_{−}
x =\frac{1}{2}(x + \overline{x}) + \frac{1}{2}(x − \overline{x})

since 2 ∈ 𝔬_{F}^{×}. Set tr_{0} = tr_{F/F_{0}} ∘ tr_{A/F} : A → F_{0} where tr denotes trace. Then this is an orthogonal direct sum decomposition with respect to tr_{0} since, for x ∈ X_{−}, y ∈ X_{+}, we have

tr_{0}(xy) = tr_{F/F_{0}}(tr_{A/F} (xy)) = tr_{F/F_{0}}(\overline{tr_{A/F} (\overline{xy})})
= tr_{F/F_{0}}(tr_{A/F} (\overline{y} \overline{x})) = −tr_{0}(yx) = −tr_{0}(xy),

and hence tr_{0}(xy) = 0. Note that A_{−} is just the Lie algebra of G. For L an 𝔬_{F} -lattice in V, define the dual lattice of L to be

L^{#} = {x ∈ V : h(x, L) ⊂ 𝔭_{F} }.

Then L^{##} = L and L^{#} can be identified with Hom(L, 𝔭_{F}) by the non-degeneracy of h. For 𝔏 = {L_{k} : k ∈ ℤ} an 𝔬_{F} -lattice chain, define the dual chain

𝔏^{#} = {L_{k}^{#} : k ∈ ℤ}.

We say that 𝔏 is self-dual if 𝔏^{#} = 𝔏. (1.1.2) Lemma Let 𝔏 be a self-dual lattice chain in V. Then there exists a unique d ∈ ℤ such that L_{k}^{#} = L_{d−k} for k ∈ ℤ. Proof: L_{k}^{#} = L_{k′} for some k′ = k′(k) ∈ ℤ and L_{k}^{##} = L_{k} so k ↦ k′ is an order-reversing bijection ℤ → ℤ. So it is of the form k ↦ d − k. ■ In fact, by changing the indexing of the lattices in 𝔏 (so that L_{[ \frac{d+1}{2} ]} becomes L_{0}), we may assume that d = 0 or −1.

(1.2) Hereditary orders

Let 𝔏 be a lattice chain in V and let 𝔄 = End_{𝔬_{F}}^{0} (𝔏) be the associated hereditary 𝔬_{F} -order in A (see [BK] (1.1)). Let 𝔓 be the Jacobson radical of 𝔄. Then 𝔓 is invertible as a fractional ideal of 𝔄 and the 𝔓^{n}, for n ∈ ℤ, give a filtration of A. There is also a valutation map ν_{𝔄} associated to the hereditary order 𝔄, given by

ν_{𝔄}(x) = max{n ∈ ℤ : x ∈ 𝔓^{n}}, x ∈ A,

with the understanding that ν_{𝔄}(0) = ∞.

(1.2.1) Lemma Let 𝔏 be a self-dual lattice chain in V and let 𝔄 be the associated hereditary 𝔬_{F} -order, with Jacobson radical 𝔓. Then we have \overline{𝔓^{n}} = 𝔓^{n}, for n ∈ ℤ.

Proof: Choose x ∈ 𝔓^{n} ; so xL_{k} ⊂ L_{k+n} for all k ∈ ℤ. Let d be the unique integer such that L_{k}^{#} = L_{d−k} for k ∈ ℤ given by (1.1.2). Fix k ∈ ℤ and let v ∈ L_{k}; then

h(\overline{x}v, L_{k+n}^{#}) = h(v, xL_{d−k−n}) ⊂ h(v, L_{d−k}) ⊂ 𝔭_{F}

so \overline{x}v ∈ L_{k+n}^{##} = L_{k+n}. This is true for all k ∈ ℤ so \overline{x} ∈ 𝔓^{n}. ■

In particular, in the situation of (1.2.1), we have \overline{𝔄} = 𝔄 and, for n ∈ ℤ, we may define the additive groups 𝔓_{−}^{n} and 𝔓_{+}^{n} as in (1.1) above; so 𝔓 = 𝔓_{−}^{n} ⊕ 𝔓_{+}^{n}.

(1.2.2) Lemma ([M3] (2.8)) Let 𝔄 be a hereditary 𝔬_{F} -order in A such that \overline{𝔄} = 𝔄 and let 𝔟 be a left ideal in 𝔄. Suppose that x ∈ 𝔟 satisfies \overline{x} = ηx for η = ±1. Then x = y + η\overline{y} for some y ∈ 𝔟.

Proof: 2 ∈ 𝔬_{F}^{×} so \frac{1}{2}.1_{A} ∈ 𝔄. Then set y = (\frac{1}{2}.1_{A})x ∈ 𝔟. ■

Let 𝔄 be a hereditary 𝔬_{F} -order in A such that \overline{𝔄} = 𝔄 and let 𝔟 ⊂ 𝔓 be a left ideal in 𝔄 such that 𝔟 = \overline{𝔟}. We now define the Cayley transform C on \{x ∈ A_{−} : det(1 − \frac{x}{2}) ≠ 0\} by

C(x) = (1 + \frac{x}{2})(1 − \frac{x}{2})^{−1}.

It is easy to check that im C ⊂ G. Moreover, if x ∈ 𝔟_{−} then C(x) exists since 𝔟^{n} ⊂ 𝔓^{n} ⊂ 𝔭_{F}^{j(n)}𝔄, for some j(n) ∈ ℤ, and j(n) → ∞ as n → ∞ so (1 − \frac{x}{2})^{−1} can be defined by the usual power series expansion; then also C(x) ∈ (1+𝔟)∩G. Note that this is not quite the same Cayley transformation as that used by Morris in [M1].

(1.2.3) Lemma (cf. [M1] (2.13)(c)) With notation as above, we have a bijection

𝔟_{−} → (1 + 𝔟) ∩ G
x ↦ C(x)

Proof: The map is certainly injective since, if u ∈ (1 + 𝔟) ∩ G then, for u = C(x), we must have x = −2(1 − u)(1 + u)^{−1}.

Now let u ∈ (1+𝔟)∩G; then u = 1+y, for some y ∈ 𝔟, so 1+u = 2+y = 2(1+ \frac{y}{2} ). Then 2 ∈ 𝔬_{F}^{×} implies \frac{y}{2} ∈ 𝔟 so 1 + u is indeed invertible and we have

−2(1 − u)(1 + u)^{−1} − 2\overline{(1 − u)(1 + u)^{−1}} = 0,
(1 + \overline{u})(1 − u) + (1 − \overline{u})(1 + u) = 0,
2 = 2\overline{u}u,
u\overline{u} = 1.

So x is indeed skew and x = y(1 − \frac{y}{2} + (\frac{y}{2} )^{2} − …) ∈ 𝔟; so x ∈ 𝔟_{−}. ■

We now give a multiplicative version of (1.2.2)

(1.2.4) Lemma Let 𝔄 be a hereditary 𝔬_{F} -order in A such that \overline{𝔄} = 𝔄 and let 𝔟 ⊂ 𝔓 be a left ideal in 𝔄. Let u ∈ 1 + 𝔟 ⊂ U^{1}(𝔄).

(i) If 𝔟 = \overline{𝔟} and u\overline{u} = 1 then there exists v ∈ 1 + 𝔟 such that u = v\overline{v}^{−1}. (ii) If u = \overline{u} then there exists v ∈ 1 + 𝔟 such that u = v\overline{v}.

Proof: (i) This follows from (1.2.3) since u ∈ (1 + 𝔟) ∩ G so u = C(x) for some x ∈ 𝔟_{−}. Then we put v = 1 + \frac{x}{2}.

(ii) For each n ∈ ℕ, we will find inductively v_{n} ∈ 1+𝔟 such that v_{n}−v_{n−1} ∈ 𝔟^{(n−1)} and v_{n}\overline{v}_{n} ≡ u mod 𝔟^{n}. Then, as 𝔄 is compact and 𝔟^{n} ⊂ 𝔓^{n}, there exists v ∈ 𝔄 such that v_{n} → v; then v ∈ 1 + 𝔟 and v\overline{v} = u as required. We can take v_{1} = 1, so assume we have found v_{n} as required, i.e. v_{n}^{−1}u\overline{v_{n}} ^{−1} ≡ 1 mod 𝔟^{n}. Write v_{n}^{−1}u\overline{v_{n}} ^{−1} = 1 + x, x ∈ 𝔟^{n}; then \overline{v_{n}^{−1}u\overline{v_{n}} ^{−1}} = v_{n}^{−1}u\overline{v_{n}} ^{−1} implies \overline{x} = x. By (1.2.2), there exists y ∈ 𝔟^{n} such that y + \overline{y} = x. We obtain

(1 − y)v_{n}^{−1}u\overline{v_{n}} ^{−1}(1 − \overline{y}) = 1 + x − y − \overline{y} + O(𝔟^{2n}) ∈ 1 + 𝔟^{(n+1)}

So we set v_{n+1} = v_{n}(1 − y)^{−1}. ■

In particular, we will apply the preceding two lemmas to 𝔟 = 𝔓^{n}. Before proceeding we give some further preliminary results which will be useful later.

(1.2.5) Lemma ([M4] (6.1)) Let X ⊃ Y be 𝔬_{F} -lattices in A which are stable under the involution, so that X/Y inherits an involution and (X/Y)_{−} is defined. Then

(i) the natural map I_{−} → (I/J)_{−} is surjective; x ↦ x + J

(ii) the map I_{−}/J_{−} → (I/J)_{−} is an isomorphism.

x + J_{−} ↦ x + J

Proof: (i) Let y + J ∈ (I/J)_{−}, so that y + \overline{y} = a ∈ J. Since a = \overline{a}, (1.2.2) implies that a = b + \overline{b} for some b ∈ J. Then set y′ = y − b. Then part (ii) is clear. ■

There is an analogous lemma in the case of (X/Y)_{+}. Further

X/Y =  (X/Y)_{−} ⊕ (X/Y)_{+}
x + Y =\frac{1}{2}(x − \overline{x}) + Y + \frac{1}{2}(x + \overline{x}) + Y

(1.2.6) Lemma Let X, Y be additive subgroups of A invariant under the involution. Then

(X + Y)_{−} = X_{−} + Y_{−}

Proof: We clearly have the containment ⊃. Let x + y ∈ (X + Y)_{−}; so x + \overline{x} = −(y + \overline{y}). We have x + \overline{x} ∈ (X ∩ Y)_{+} and, by (1.2.2), there exists z ∈ (X ∩ Y) such that x + \overline{x} = z + \overline{z}. Then set x′ = x − z, y′ = y + z. ■

We now give a multiplicative version of the previous lemma.

(1.2.7) Lemma Let K, K′ be subgroups of GL_{2N} such that K ∩ K′ = 1 + 𝔟, where 𝔟 is a left ideal in some hereditary 𝔬_{F} -order 𝔄, both invariant under the involution and with 𝔟 ⊂ 𝔓 = rad 𝔄. Then (K.K′) ∩ G = K ∩ G . K′ ∩ G.

Proof: Let kk′ ∈ (K_{1}K_{2}) ∩ G. Then k′\overline{k′} = k^{−1}\overline{k^{−1}} lies in K_{1} ∩ K_{2} and, by (1.2.4)(ii), there exists u ∈ K_{1} ∩ K_{2} such that k′\overline{k′} = u\overline{u}. Then we set k_{1} = ku, k′_{1} = u^{−1}k′. ■

(1.3) Parahoric subgroups and filtrations

Let 𝔄 be a hereditary 𝔬_{F} -order in A; then we have a parahoric subgroup in GL_{2N}(F) given by U(𝔄) = 𝔄^{×}, the groups of units in 𝔄. This has standard filtration given by

U^{n}(𝔄) = 1 + 𝔓^{n} for n ≥ 1.

Set P = P (𝔄) = 𝔄 ∩ G, a parahoric subgroup of G; this comes equipped with its standard filtration P^{n} = P^{n}(𝔄) = {x ∈ P : x ≡ 1 mod 𝔓^{n}} = U^{n}(𝔄) ∩ G, for n ≥ 1. This gives a filtration of P (𝔄) by normal open subgroups and [P^{n}, P^{m}] ⊆ P^{m+n} for m, n ∈ ℤ. Then we can apply (1.2.3) to give

(1.3.1) Proposition ([M1] (2.13)(c)) For each n ≥ 1, the Cayley map provides a bijection 𝔓_{−}^{n} → P^{n}; x ↦ C(x).

(1.3.2) Corollary ([M3] (2.1.4)(b)) If 2n ≥ m ≥ n ≥ 1, the map 𝔓_{−}^{n} → P^{n} in (1.3.1) induces an isomorphism of abelian groups

𝔓_{−}^{n}/𝔓_{−}^{m} ≃ P^{n}/P^{m}
x ↦ 1 + x

Proof: We only need to check that the map is indeed a group homomorphism but this follows since 2n ≥ m. ■

As 𝔓 is stable under the involution, 𝔄/𝔓 is a k_{F} -algebra which inherits an involution. Set

\overline{P} (𝔄) = \{x ∈ 𝔄/𝔓 : x\overline{x} = 1\}
= \{x + 𝔓 : x\overline{x} − 1 ∈ 𝔓\}

(1.3.3) Proposition ([M3] (2.11)) The natural map P → \overline{P} (𝔄) is surjective.

Proof: Suppose x ∈ 𝔄 and x + 𝔓 ∈ \overline{P} (𝔄). Let x_{1} = x. For each n, we will find inductively x_{n} ∈ 𝔄 such that x_{n} − x_{n−1} ∈ 𝔓^{n−1} and x_{n}\overline{x_{n}} ≡ 1 mod 𝔓^{n}. Then, as 𝔄 is compact, there exists t ∈ 𝔄 such that x_{n} → t; then t − x ∈ 𝔓 and t\overline{t} = 1, i.e. t ∈ G ∩ 𝔄 = P and t maps to x + 𝔓. We have x_{1} already so assume we have found x_{n} as required, for some n ≥ 1. Then \overline{x_{n}\overline{x_{n}} − 1} = x_{n}\overline{x_{n}} − 1 so, applying (1.2.2), there exists a ∈ 𝔓^{n} such that

x_{n}\overline{x_{n}} = 1 + a + \overline{a}.

Then (1 + a)(1 + \overline{a}) = x_{n}\overline{x_{n}} + a\overline{a} and a\overline{a} ∈ 𝔓^{2n} ⊂ 𝔓^{n+1} so

1 ≡ ((1 + a)^{−1}x_{n})\overline{((1 + a)^{−1}x_{n})} mod 𝔓^{n+1}.

So set x_{n+1} = (1 + a)^{−1}x_{n}. ■

(1.3.4) Corollary ([M3] (2.14)(a)) The natural map in (1.3.3) induces an isomorphism P/P^{1} ≃ \overline{P} (𝔄)

Proof: Consider the natural map φ : P ↪ 𝔄 → 𝔄/𝔓. By (1.3.3), im φ = \overline{P} (𝔄). Further

ker φ = {x ∈ P : x + 𝔓 = 1 + 𝔓}

= {x ∈ P : x ∈ 1 + 𝔓} = P^{1}. ■

Then we see that \overline{P} (𝔄) is the reductive quotient of the parahoric subgroup P.

(1.4) Characters

Recall that tr_{0} = tr_{F/F_{0}} ∘ tr_{A/F}. If X is an 𝔬_{F} -lattice in A (hence an 𝔬_{0}-lattice in A) define

X^{∗} = {a ∈ A : tr_{0}(aX) ⊂ 𝔭_{0}},

which is also an 𝔬_{F} -lattice. Since 2 ∈ 𝔬_{F}^{×} and F/F_{0} is of degree at most 2, F is at worst tamely ramified over F_{0} so

X^{∗} = {a ∈ A : tr_{A/F} (aX) ⊂ 𝔭_{F} }

which is the same as the usual definition of X^{∗} (as in [BK] (1.1.4)). In particular

(𝔓^{n})^{∗} = 𝔓^{1−n} , for all n ∈ ℤ

(see [Bu] p.190). If X is also stable under the involution, we can define

(X_{−})^{∗} = {a ∈ A_{−} : tr_{0}(aX_{−}) ⊂ 𝔭_{0}}.

Then, as the direct sum X = X_{−} ⊕ X_{+} is orthogonal with respect to tr_{0}, we have

(X_{−})^{∗} = (X^{∗})_{−}

and, in particular,

(𝔓_{−}^{n})^{∗} = 𝔓_{−}^{1−n}.

Recall that ψ_{0} is a character of the additive group of F_{0} with conductor 𝔭_{0}. As in [W] (II.5), the map

A_{−} → (A_{−})ˆ
x ↦ (y ↦ ψ_{0}(tr_{0}(xy)))

is an isomorphism of abelian groups, where ˆ denotes the Pontrjagin dual. Given an 𝔬_{0}-lattice L in A_{−}, set Lˆ = {χ ∈ (A_{−})ˆ : χ(L) ≡ 1}. Then the identification A_{−} → (A_{−})ˆ enables us to identify Lˆ with L^{∗}. Moreover, if L_{1} ⊃ L_{2} then (L_{1}/L_{2})ˆ ≃ L_{2}ˆ /L_{1}ˆ ≃ L_{2}^{∗}/L_{1}^{∗}. We have obtained:

(1.4.1) Lemma ([M2] (4.19)) If 2n ≥ m ≥ n ≥ 1 there is a P -equivariant isomorphism of abelian groups

𝔓_{−}^{1−m}/𝔓_{−}^{1−n} ˜→ (P^{n}/P^{m})ˆ
b + 𝔓_{−}^{1−n} ↦ ψ_{b}

where ψ_{b}(p) = ψ_{0}(tr_{0}(b(p − 1))) for p ∈ P^{n}.

Then ψ_{b} is the restriction to P^{n} of the character ψ_{b} of U^{n}(𝔄) defined in [BK] (1.1.6), since ψ_{0} ∘ tr_{0} = ψ_{F} ∘ tr_{A/F} and ψ_{F} has conductor 𝔭_{F}.

EXACTNESS AND INTERTWINING

In this chapter we show that the exact sequences of [BK] (1.4) restrict well to the symplectic Lie algebra. This will follow from the fact that if E/F is a subfield of A stable under the involution then there is a tame corestriction on A relative to E/F which commutes with the involution. (See below or [BK] (1.3) for definitions.). These exact sequences, together with the formula for the intertwining in GL_{2N} of simple strata [BK] (1.5.8), allow us to calculate the intertwining in G of skew simple strata, showing that it is, in some sense, as small as possible. Note that a similar intertwining theorem has been obtained in the case E/F tamely ramified by Morris ([M3] (3.13)).

(2.1) Exact Sequences

Let E/F be a subfield of A such that E = \overline{E} and let E_{0} be the fixed field of the involution. Let 𝔄 be a hereditary 𝔬_{F} -order in A which is invariant under the involution and suppose E^{×} normalizes 𝔄. Then we define

B = End_{E}(V), the A-centralizer of E;
𝔅 = 𝔄 ∩ B,  𝔔 = 𝔓 ∩ B.

Then, by [BK] (1.2.4), 𝔅 is a hereditary 𝔬_{E}-order with Jacobson radical 𝔔 and 𝔔^{n} = 𝔓^{n} ∩ B. Note also that E = \overline{E} implies that B = \overline{B}. Recall, from [BK] (1.3.3), that a tame corestriction on A (relative to E/F) is a (B, B)-bimodule homomorphism s : A → B such that s(𝔄) = 𝔄 ∩ B for any hereditary 𝔬_{F} -order 𝔄 in A which is normalized by E^{×}. By [BK] (1.3.2), a tame corestriction is uniquely determined up to multiplication by a unit u ∈ 𝔬_{E}^{×}. For E/F tamely ramified, we can take s to be orthogonal projection, relative to the pairing (x, y) ↦ tr_{A/F} (xy) (see [BK] (1.3.8)(ii)). We now prove that there is a tame corestriction which commutes with the involution on A.

(2.1.1) Lemma There exists a tame corestriction on A relative to E/F such that s(\overline{x}) = \overline{s(x)} for all x ∈ A.

Proof: Recall that we have the character ψ_{0} of the additive group of F_{0}, with conductor 𝔭_{0}; then ψ_{F} = ψ_{0} ∘tr_{F/F_{0}} is a character of the additive group of F, with conductor 𝔭_{F} since F/F_{0} is at worst tamely ramified. We put ψ_{A} = ψ_{F} ∘ tr_{A/F} = ψ_{0} ∘ tr_{0}, as in (1.1), where tr_{0} = tr_{F/F_{0}} ∘ tr_{A/F}. Similarly, let ψ_{E_{0}} be a character of the additive group of E_{0}, with conductor 𝔭_{E_{0}}; then ψ_{E} = ψ_{E_{0}} ∘ tr_{E/E_{0}} is a character of the additive group of E, with conductor 𝔭_{E}. We put ψ_{B} = ψ_{E} ∘ tr_{B/E} = ψ_{E_{0}} ∘ tr_{0}^{E}, where tr_{0}^{E} = tr_{E/E_{0}} ∘ tr_{B/E}.

As in [BK] (1.3.4), there exists a unique map s : A → B such that

ψ_{A}(ab) = ψ_{B}(s(a)b), a ∈ A, b ∈ B

and this is a tame corestriction on A relative to E/F. Then, for a ∈ A_{−}, b ∈ B_{+}, we have ψ_{B}(s(a)b) = ψ_{A}(ab) = ψ_{0}(tr_{0}(ab)) = ψ_{0}(0) = 1 as A_{−}, A_{+} are orthogonal with respect to tr_{0}. Hence s(a) is orthogonal to B_{+} with respect to tr_{0}^{E}, that is, s(a) ∈ B_{−}. Similarly, we have that s(A_{+}) ⊂ B_{+}. Now, for a ∈ A, a = a_{+} + a_{−} with a_{+} ∈ A_{+} and a_{−} ∈ A_{−} so we have

\overline{s(a)} = \overline{s(a_{+})} + \overline{s(a_{−})}
= s(a_{+}) − s(a_{−})
= s(a_{+} − a_{−}) = s(\overline{a})

as required. ■

From now on, let s be a tame corestriction given by (2.1.1); it is uniquely determined up to multiplication by u ∈ 𝔬_{E}^{×} such that u = \overline{u}. Then s splits as s_{−} : A_{−} → B_{−} and s_{+} : A_{+} → B_{+}.

Now let β ∈ A_{−} be such that the algebra E = F [β] is a field and consider the adjoint map

a_{β} : A → A
x ↦ βx − xβ

This is a (B, B)-bimodule homomorphism with kernel B. Then

\overline{a_{β}(x)} = \overline{βx − xβ} = \overline{x}\overline{β} − \overline{β}\overline{x} = −\overline{x}β + β\overline{x} = a_{β}(\overline{x})

So a_{β} also splits as a_{β}^{−} : A_{−} → A_{−} and a_{β}^{+} : A_{+} → A_{+}. In particular we have an infinite exact sequence

⋯ ⟶^{s} A ⟶^{a_{β}} A ⟶^{s} A ⟶^{a_{β}} ⋯

which splits as

⋯ ⟶^{s_{−}}A_{−} ⟶^{a_{β}^{−}}A_{−} ⟶^{s_{−}}A_{−} ⟶^{a_{β}^{−}} ⋯
⋯ ⟶^{s_{+}}A_{+} ⟶^{a_{β}^{+}}A_{+} ⟶^{s_{+}}A_{+} ⟶^{a_{β}^{+}} ⋯

Let β be as above and let k ∈ ℤ. Define

𝔑_{k} = 𝔑_{k}(β, 𝔄) = {x ∈ 𝔄 : a_{β}(x) ∈ 𝔓^{k}}.

Then \overline{𝔑_{k}} = 𝔑_{k} and 𝔑_{k} is an 𝔬_{F} -lattice in A. In particular, 𝔑_{k} also splits as 𝔑_{k}^{+} ⊕ 𝔑_{k}^{−}.

Let n = −ν_{𝔄}(β); then a_{β}(𝔄) ⊂ 𝔓^{−n} whence 𝔑_{k} = 𝔄 for k ≤ −n. Suppose E ≠ F. Define

k_{0} = k_{0}(β, 𝔄) = max{k ∈ ℤ : 𝔑_{k} ⊄ 𝔅 + 𝔓}

which exists, by [BK] (1.4.4), and we have k_{0} ≥ −n. If E = F then set k_{0}(β, 𝔄) = −∞.

(2.1.2) Remark Let e = e(E|F) be the ramification index and ν = ν_{E}(β). We say the element β is minimal if it satisfies

(i) gcd(ν, e) = 1; (ii) π_{F}^{−ν}.β^{e} + 𝔭_{E} generates the residue class field extension k_{E}/k_{F}. Then, by [BK] (1.4.15), k_{0}(β, 𝔄) = −n if, and only if, β is minimal. Moreover, we then have

𝔑^{r−n}(β, 𝔄) = 𝔅 + 𝔓^{r} for r ≥ 0

and, in particular, 𝔑_{k_{0}} = 𝔑_{−n} = 𝔄.

(2.1.3) Proposition ([BK] (1.4.7)) Let k, r ∈ ℤ and suppose k ≥ k_{0}(β, 𝔄) , r ≥ 1. Then the following sequences are exact:

0 → 𝔑_{k}/𝔑_{k+r} ⟶^{a_{β}} 𝔓^{k}/𝔓^{k+r} ⟶^{s} 𝔔^{k}/𝔔^{k+r} → 0

0 → 𝔑_{k}/𝔅 ⟶^{a_{β}} 𝔓^{k} ⟶^{s} 𝔔^{k} → 0

By (1.2.5) all the terms in the exact sequence are stable under the (induced) involution and split as a sum of skew elements and symmetric elements. Further, we have seen that a_{β} and s both preserve the skew elements and the symmetric elements. Therefore these exact sequences split in two. In particular, we have:

(2.1.4) Proposition Let k, r ∈ ℤ and suppose k ≥ k_{0}(β, 𝔄), r ≥ 1. Then the following sequences are exact:

0 → 𝔑_{k}^{−}/𝔑_{k+r}^{−} ⟶^{a_{β}^{−}} 𝔓_{−}^{k}/𝔓_{−}^{k+r} ⟶^{s_{−}} 𝔔_{−}^{k}/𝔔_{−}^{k+r} → 0

0 → 𝔑_{k}^{−}/𝔅_{−} ⟶^{a_{β}^{−}} 𝔓_{−}^{k} ⟶^{s_{−}} 𝔔_{−}^{k} → 0

Note also that we have

\overline{𝔔^{m}𝔑_{k}} = \overline{𝔑_{k}} \overline{𝔔^{m}} = 𝔑_{k}𝔔^{m}
= 𝔔^{m}𝔑_{k} by [BK] (1.4.8)

so, from [BK] (1.4.10) and (1.2.5), we also have

(2.1.5) Corollary For m, k ∈ ℤ , k ≥ k_{0}(β, 𝔄) the following sequences are exact:

0 → (𝔔^{m}𝔑_{k})_{−}/𝔔_{−}^{m} ⟶^{a_{β}^{−}} 𝔓_{−}^{m+k} ⟶^{s_{−}} 𝔔_{−}^{m+k} → 0

0 → (𝔔^{m}𝔑_{k})_{−}/(𝔔^{m}𝔑_{k+1})_{−} ⟶^{a_{β}^{−}} 𝔓_{−}^{m+k}/𝔓_{−}^{m+k+1} ⟶^{s_{−}} 𝔔_{−}^{m+k}/𝔔_{−}^{m+k+1} → 0

Note that if E/F is tamely ramified then, taking s to be the projection pr:A → B, all these exact sequences are split by the inclusion B ↪ A.

Recall from [BK] (1.3.10) that a lattice L is called E-exact if s(L) = L ∩ B.

(2.1.6) Lemma Let γ ∈ B, r ∈ ℤ; then L = 𝔓^{r} + γ𝔓^{r}γ^{−1} is E-exact.

Proof: From [BK] (1.3.16), the lattice Lγ = 𝔓^{r}γ + γ𝔓^{r} is E-exact and, further, an Ad (E^{×})-invariant (𝔄(E), 𝔬_{E})-bilattice. Then L = (Lγ)γ^{−1} is also an Ad (E^{×})- invariant (𝔄(E), 𝔬_{E})-bilattice, and hence is E-exact by [BK] (1.3.12). ■

In particular, if γ ∈ B ∩ G then (𝔓_{+}^{r} + γ𝔓_{+}^{r}γ^{−1}) ∩ B = (𝔓^{r} + γ𝔓^{r}γ^{−1})_{+} ∩ B = (𝔔^{r} + γ𝔔^{r}γ^{−1})_{+} = 𝔔_{+}^{r} + γ𝔔_{+}^{r}γ^{−1}.

Let k = k_{0}(β, 𝔄) and write 𝔑 for 𝔑_{k_{0}(β,𝔄)}. Finally in this section we present the multiplicative analogue of (𝔔^{m}𝔑)_{−} for the symplectic group G. For m ≥ 1 define

Q^{m} = (1 + 𝔔^{m}𝔑) ∩ G.

Then, by (1.2.3), there is a bijection (𝔔^{m}𝔑)_{−} → Q^{m} given by x ↦ C(x). Note that for β minimal over F, 𝔑 = 𝔄 so Q^{m} = P^{m}(𝔄).

(2.2) Intertwining of simple strata

Recall (from [BK] (1.5)) that a stratum in A is a 4-tuple [𝔄, n, r, b] consisting of a hereditary 𝔬_{F} -order 𝔄 in A, integers n ≥ r, and an element b ∈ A such that ν_{𝔄}(b) ≥ −n. If [𝔄_{i}, n_{i}, r_{i}, b_{i}] are strata in A, i = 1, 2, and 𝔓_{i} = rad(𝔄_{i}), we say they are equivalent, denoted

[𝔄_{1}, n_{1}, r_{1}, b_{1}] ≅ [𝔄_{2}, n_{2}, r_{2}, b_{2}], if
b_{1} + 𝔓_{1}^{−r_{1}} = b_{2} + 𝔓_{2}^{−r_{2}}.

If they are equivalent then, by [BK] (1.5.2), 𝔄_{1} = 𝔄_{2}, r_{1} = r_{2} and, if we have ν_{𝔄_{i}}(b_{i}) = −n_{i} for i = 1, 2, also n_{1} = n_{2}.

(2.2.1) Definition A stratum [𝔄, n, r, b] in A is called skew if b + \overline{b} = 0 and 𝔄 is invariant under the involution on A.

(2.2.2) Definition ([BK] (1.5.5)) Let [𝔄, n, r, β] be a stratum in A. It is pure if

(i) the algebra E = F [β] is a field, (ii) E^{×} ⊂ 𝔎(𝔄), (iii) ν_{𝔄}(β) = −n. It is called simple if, in addition, (iv) r < −k_{0}(β, 𝔄).

We now define the formal intertwining in G of a skew stratum [𝔄, n, r, b]

I_{G}[𝔄, n, r, b] = {x ∈ G : x^{−1}(b + 𝔓_{−}^{−r})x ∩ (b + 𝔓_{−}^{−r}) ≠ ∅}

(2.2.3) Theorem Let [𝔄, n, r, β] be a simple stratum with β ∈ A_{−} and 𝔄 stable under the involution . Write k = k_{0}(β, 𝔄) and 𝔑 = 𝔑_{k}(β, 𝔄) and, for m ≥ 1, define Q^{m} as in (2.1). Then

I = I_{G}[𝔄, n, r, β] = Q^{−(k+r)}(B ∩ G)Q^{−(k+r)}

Before proceeding with the proof, we remark that B ∩ G is a unitary group (we will examine it more closely in chapter 4). It will therefore be necessary for us to prove many results (concerning refinement of strata etc.) for unitary groups as well as for symplectic groups.

Proof: Let J = {x ∈ G : x^{−1}(b + 𝔓^{−r})x ∩ (b + 𝔓^{−r}) ≠ ∅}. This is the GL(2N)- intertwining of the stratum (in the sense of [BK] (1.5.7)) contained in G. Clearly I ⊂ J. Suppose, on the other hand, x ∈ J; then there exist b_{i} ∈ 𝔓^{−r}, i = 1, 2 such that

x(β + b_{1})x^{−1} = β + b_{2}.

We now write b_{i} = u_{i} + v_{i}, u_{i} ∈ 𝔓_{+}^{−r}, v_{i} ∈ 𝔓_{−}^{−r}, for i = 1, 2 so

x(β + v_{1})x^{−1} + xu_{1}x^{−1} = (β + v_{2}) + u_{2}.

Now xA_{−}x^{−1} ⊂ A_{−} and xA_{+}x^{−1} ⊂ A_{+} so A = A_{−} ⊥ A_{+} implies

x(β + v_{1})x^{−1} = β + v_{2},

i.e. x ∈ I. So we see that I = J = (1 + 𝔔^{d}𝔑)B^{×}(1 + 𝔔^{d}𝔑) ∩ G by [BK] (1.5.8), where d = −r − k > 0.

We continue with two lemmas:

(2.2.4) Lemma We have

(1 + 𝔔^{d}𝔑)B^{×}(1 + 𝔔^{d}𝔑) ∩ G ⊂ (1 + 𝔔^{d}𝔑)(B ∩ G)(1 + 𝔔^{d}𝔑).

Proof: Let t ∈ (1+𝔔^{d}𝔑)B^{×}(1+𝔔^{d}𝔑)∩G with ν_{𝔄}(t) = m. So t = (1+x_{1})^{−1}b(1+ y_{1}) with b ∈ B^{×}, x_{1}, y_{1} ∈ 𝔔^{d}𝔑 ⊂ 𝔓^{d}. Then we have

(1 + x_{1})t = b(1 + y_{1}),
(1 + x_{1})(1 + \overline{x_{1}}) = b(1 + y_{1})(1 + \overline{y_{1}})\overline{b},
b\overline{b} − 1 = x_{1} + \overline{x_{1}} + x_{1}\overline{x_{1}} − b(y_{1} + \overline{y_{1}} + y_{1}\overline{y_{1}})\overline{b}.

So b\overline{b} ≡ 1 (mod 𝔓_{+}^{d} + b𝔓_{+}^{d}\overline{b}). We will find inductively b_{n} ∈ U^{d}(𝔅)bU^{d}(𝔅) such that

(2.2.5)  b_{n}\overline{b_{n}} ≡ 1 (mod 𝔓_{+}^{nd} + b_{n}𝔓_{+}^{nd}\overline{b_{n}})

and b_{n+1} ≡ b_{n} (mod 𝔓^{m+nd}). Granting this, we have ν_{𝔄}(b_{n}) = ν_{𝔄}(t) = m so b_{n}\overline{b_{n}} ≡ 1 (mod 𝔓_{+}^{nd} + 𝔓_{+}^{nd+2m}). Then the sequence \{b_{n}\} converges to b′ ∈ U^{d}(𝔅)bU^{d}(𝔅) (since the double coset is compact) and b′\overline{b′} = 1. So t ∈ (1 + 𝔔^{d}𝔑)(B ∩ G)(1 + 𝔔^{d}𝔑) since 𝔔^{d} ⊂ 𝔔^{d}𝔑. We are reduced to proving (2.2.5). The case n = 1 is immediate: take b_{1} = b. So let n ≥ 1 and assume we have found the required element b_{n}. We write b_{n} = (1 + x)t(1 + y)^{−1} with x, y ∈ 𝔔^{d}𝔑 ⊂ 𝔓^{d}. Then, as above,

b_{n}\overline{b_{n}} − 1 = x + \overline{x} + x\overline{x} − b_{n}(y + \overline{y} + y\overline{y})\overline{b_{n}}

Now b_{n}\overline{b_{n}} − 1 ∈ (𝔓_{+}^{nd} + b_{n}𝔓_{+}^{nd}\overline{b_{n}}) ∩ B = 𝔔_{+}^{nd} + b_{n}𝔔_{+}^{nd}\overline{b_{n}} by the remark following (2.1.6). So there exist v′, w′ ∈ 𝔔_{+}^{nd} such that b_{n}\overline{b_{n}} − 1 = v′ − b_{n}w′\overline{b_{n}}. Then let v = \frac{1}{2}v′, w = \frac{1}{2}w′; v, w ∈ 𝔔_{+}^{nd} since 2 ∈ 𝔬_{F}^{×} and

b_{n}\overline{b_{n}} − 1 = v + \overline{v} − b_{n}(w + \overline{w})\overline{b_{n}} = x + \overline{x} + x\overline{x} − b_{n}(y + \overline{y} + y\overline{y})\overline{b_{n}}

We set b_{n+1} = (1 − v)b_{n}(1 − w)^{−1}. Then

v + \overline{v} − b_{n+1}(w + \overline{w})\overline{b_{n+1}}

=(v + \overline{v}) − (1 − v)b_{n}(1 − w)^{−1}(w + \overline{w})(1 − \overline{w})^{−1}\overline{b_{n}}(1 − \overline{v})
≡(1 − v)[(v + \overline{v}) − b_{n}(1 − w)^{−1}(w + \overline{w})(1 − \overline{w})^{−1}\overline{b_{n}}](1 − \overline{v}) (mod 𝔓_{+}^{2nd})
≡(1 − v)[v + \overline{v} − b_{n}(w + \overline{w})\overline{b_{n}}](1 − \overline{v})  (mod 𝔓_{+}^{2nd} + b_{n+1}𝔓_{+}^{2nd}\overline{b_{n+1}})
≡(1 − v)[x + \overline{x} + x\overline{x} − b_{n}(y + \overline{y} + y\overline{y})\overline{b_{n}}](1 − \overline{v})  (mod 𝔓_{+}^{2nd} + b_{n+1}𝔓_{+}^{2nd}\overline{b_{n+1}})
≡(1 − v)(x + \overline{x} + x\overline{x})(1 − \overline{v}) − (1 − v)b_{n}(1 − w)^{−1}(y + \overline{y} + y\overline{y})(1 − \overline{w})^{−1}\overline{b_{n}}(1 − \overline{v})
(mod 𝔓_{+}^{(n+1)d} + b_{n+1}𝔓_{+}^{(n+1)d}\overline{b_{n+1}})
≡x + \overline{x} + x\overline{x} − b_{n+1}(y + \overline{y} + y\overline{y})\overline{b_{n+1}}  (mod 𝔓_{+}^{(n+1)d} + b_{n+1}𝔓_{+}^{(n+1)d}\overline{b_{n+1}})

Now (1 − v)(1 + x)t = b_{n+1}(1 − w)(1 + y) so, misappropriating the O-notation in the obvious way, we get

(1 − v)(1 + x)(1 + \overline{x})(1 − \overline{v}) = b_{n+1}(1 − w)(1 + y)(1 + \overline{y})(1 − \overline{w})\overline{b_{n+1}},

hence 1 − v − \overline{v} + x + \overline{x} + x\overline{x} + O(𝔓_{+}^{(n+1)d}) =

b_{n+1}\overline{b_{n+1}} − b_{n+1}(−w − \overline{w} + y + \overline{y} + y\overline{y} + O(𝔓_{+}^{(n+1)d}))\overline{b_{n+1}}

and  b_{n+1}\overline{b_{n+1}} ≡ 1 (mod 𝔓_{+}^{(n+1)d} + b_{n+1}𝔓_{+}^{(n+1)d}\overline{b_{n+1}}).

This completes our induction and the proof of the lemma. ■

(2.2.6) Lemma Let t ∈ G; then

(1 + 𝔔^{d}𝔑)t(1 + 𝔔^{d}𝔑) ∩ G = Q^{d}tQ^{d}.

Proof: Let h ∈ (1 + 𝔔^{d}𝔑)t(1 + 𝔔^{d}𝔑) ∩ G and ν_{𝔄}(t) = m. We will find inductively t_{n} ∈ Q^{d}hQ^{d} such that

(2.2.7)  t_{n} ≡ t (mod 𝔔^{nd}𝔑t_{n} + t𝔔^{nd}𝔑).

Granting this, we have ν_{𝔄}(t_{n}) = ν_{𝔄}(t) = m so t_{n} ≡ t (mod 𝔓^{nd+m}). Then the sequence {t_{n}} converges to t. The double coset Q^{d}hQ^{d} is compact in G so we conclude that t ∈ Q^{d}hQ^{d}, or h ∈ Q^{d}tQ^{d}, as required. So we are reduced to proving (2.2.7). The case n = 1 is immediate; take t_{1} = h. So let n ≥ 1 and assume we have found the required element t_{n} satisfying (2.12). Then there exist x, y ∈ 𝔔^{nd}𝔑 such that t_{n} = t − xt_{n} + ty, i.e. (1 + x)t_{n} = t(1 + y), and we have

(1 + x)(1 + \overline{x}) = t(1 + y)(1 + \overline{y})\overline{t},
hence x − ty\overline{t} ≡ −(\overline{x} − t\overline{y}\overline{t}) (mod (𝔔^{2nd}𝔑)_{+} + t(𝔔^{2nd}𝔑)_{+}\overline{t}),

i.e. x − ty\overline{t} ∈ (𝔔^{nd}𝔑 + t𝔔^{nd}𝔑\overline{t})_{−} + (𝔔^{2nd}𝔑)_{+} + t(𝔔^{2nd}𝔑)_{+}\overline{t}. Now we have (𝔔^{nd}𝔑 + t𝔔^{nd}𝔑\overline{t})_{−} = (𝔔^{nd}𝔑)_{−} + t(𝔔^{nd}𝔑)_{−}\overline{t} so there exist v, w ∈ (𝔔^{nd}𝔑)_{−} such that

(2.2.8)  x − ty\overline{t} ≡ v − tw\overline{t} (mod 𝔔^{2nd}𝔑 + t𝔔^{2nd}𝔑\overline{t}).

We set t_{n+1} = C(v)t_{n}C(−w) ∈ Q^{d}hQ^{d}. Then, from (2.2.8), we have

(x − v)t ≡ t(y − w) (mod 𝔔^{2nd}𝔑t + t𝔔^{2nd}𝔑)
i.e. (x − v)(1 + x)C(−v)t_{n+1}C(w)(1 + y)^{−1} ≡ t(y − w)
(mod 𝔔^{2nd}𝔑t + t𝔔^{2nd}𝔑)
i.e. (x − v)(1 + x)C(−v)t_{n+1} ≡ t(y − w)(1 + y)C(−w)

(mod 𝔔^{2nd}𝔑t_{n+1} + t𝔔^{2nd}𝔑)

i.e. (x − v)t_{n+1} ≡ t(y − w) (mod 𝔔^{2nd}𝔑t_{n+1} + t𝔔^{2nd}𝔑)

Then (1 + x)C(−v)t_{n+1} = t(1 + y)C(−w) implies

(1 + (x − v) + 𝔔^{2nd})t_{n+1} = t(1 + (y − w) + 𝔔^{2nd})
and t_{n+1} ≡ t (mod 𝔔^{2nd}𝔑t_{n+1} + t𝔔^{2nd}𝔑).

Now n ≥ 1 so t_{n+1} ≡ t (mod 𝔔^{(n+1)d}𝔑t_{n+1} + t𝔔^{(n+1)d}𝔑) as required. ■

Returning to the proof of the theorem, we have

I = (1 + 𝔔^{d}𝔑)B^{×}(1 + 𝔔^{d}𝔑) ∩ G
= (1 + 𝔔^{d}𝔑)(B ∩ G)(1 + 𝔔^{d}𝔑) ∩ G,  by (2.2.4),
= Q^{d}(B ∩ G)Q^{d},  by (2.2.6). ■

Note that, following the proof of (2.2.6), we find

(2.2.9)  U^{n}(𝔄)tU^{n}(𝔄) ∩ G = P^{n}(𝔄)tP^{n}(𝔄),

for t ∈ G and n ≥ 1. Further, replacing A with B, 𝔄 with 𝔅 etc. we have

(2.2.10)  U^{n}(𝔅)tU^{n}(𝔅) ∩ G = P^{n}(𝔅)tP^{n}(𝔅),

for t ∈ B ∩ G, n ≥ 1 and where P^{n}(𝔅) = U^{n}(𝔅) ∩ G = P^{n}(𝔄) ∩ B. (2.2.11) Proposition Let t ∈ B ∩ G, n ≥ 1. Then

P^{n}(𝔄)tP^{n}(𝔄) ∩ B = P^{n}(𝔅)tP^{n}(𝔅).

Proof: P^{n}(𝔄)tP^{n}(𝔄) ∩ B = U^{n}(𝔄)tU^{n}(𝔄) ∩ G ∩ B,  by (2.2.9),

= U^{n}(𝔅)tU^{n}(𝔅) ∩ G,  by [BK] (1.6.1),
= P^{n}(𝔅)tP^{n}(𝔅),  by (2.2.10). ■

RESIDUAL SUBSPACES

In this chapter we introduce the notion of a residual subspace of a lattice chain in V. This is due to Bushnell, as are the results (3.1.1–9). These will be seen to correspond to subspaces of V which split 𝔏 in some sense. They will play a role similar to the ℤ/eℤ-graded algebras in [M3]. We can then apply the results obtained about residual subspaces to find “nice” block decompositions for the hereditary order associated to a self-dual lattice chain in V (cf. [BK] (2.5.1)). In particular, the involution on A will permute the blocks with respect to this decomposition. This chapter is given in the generality of unitary groups so, as in (1.1), h will be an ϵ-hermitian from on V.

(3.1) Residual subspaces

Let 𝔏 be an 𝔬_{F} -lattice chain in V of period e. Then we have an additive norm ν_{𝔏} (in the sense of [BT] (1.1)) on V associated to 𝔏 given by

ν_{𝔏}(v) = sup_{k∈ℤ}{k : v ∈ L_{k}}.

Then ν_{𝔏} satisfies the following properties:

ν_{𝔏}(xv) = eν_{F} (x) + ν_{𝔏}(v) for x ∈ F^{×}, v ∈ V;
ν_{𝔏}(v + w) ≥ inf(ν_{𝔏}(v), ν_{𝔏}(w)) for v, w ∈ V;
ν_{𝔏}(v) = ∞ ⟺ v = 0.

(3.1.1) Definition An 𝔬_{F} -basis of 𝔏 is a basis {v_{1}, …, v_{N}} of V which splits ν_{𝔏}, in the sense that

ν_{𝔏}(∑_{i=1}^{N} x_{i}v_{i}) = inf_{1≤i≤n} ν_{𝔏}(x_{i}v_{i}).

This is not the usual definition of an 𝔬_{F} -basis, as in [BK] (1.1.7), but is somewhat weaker. Indeed, for 0 ≤ i ≤ e − 1, let {v_{ij} : 1 ≤ j ≤ dim_{k_{F}} L_{i}/L_{i+1}} be a basis for L_{i}/L_{i+1}, and choose v̂_{ij} ∈ V such that v̂_{ij} + L_{i+1} = v_{ij}. Then {v̂_{ij}} is an 𝔬_{F} -basis of 𝔏 in the sense of [BK].

(3.1.2) Lemma With notation as above, {v̂_{ij}} is an 𝔬_{F} -basis of 𝔏 in the sense of (3.1.1).

Proof: Let v = ∑ x_{ij}v̂_{ij} and let l = inf ν_{𝔏}(x_{ij}v̂_{ij}). Then certainly ν_{𝔏}(v) ≥ l. Let {(i_{k}, j_{k}) : 1 ≤ k ≤ r} be the set of indices for which ν_{𝔏}(x_{ij}v̂_{ij}) = l. Since 0 ≤ ν_{𝔏}(v̂_{ij}) ≤ e − 1 for all i, j, we have

ν_{𝔏}(v̂_{i_{1}j_{1}}) = ⋯ = ν_{𝔏}(v̂_{i_{r}j_{r}});
ν_{F} (x_{i_{1}j_{1}}) = ⋯ = ν_{F} (x_{i_{r}j_{r}}).

So we may assume ν_{F} (x_{i_{k}j_{k}}) = 0 and l = ν_{𝔏}(v̂_{i_{k}j_{k}}) for 1 ≤ k ≤ r. However, by definition, the v̂_{i_{k}j_{k}} + L_{l+1} are linearly independent over k_{F} so v ∉ L_{l+1} and ν_{𝔏}(v) = l. ■

Conversely, an 𝔬_{F} -basis {v_{1}, …, v_{N}} of 𝔏 in the sense of (3.1.1) is an 𝔬_{F} -basis in the sense of [BK] if and only if 0 ≤ ν_{𝔏}(v_{i}) ≤ e − 1, for 1 ≤ i ≤ N.

(3.1.3) Definition Let 𝔏 be an 𝔬_{F} -lattice chain in V. A residual subspace of 𝔏 is a family V = {V_{k} : k ∈ ℤ} such that

(i) V_{k} is a k_{F} -subspace of L_{k}/L_{k+1} for each k ∈ ℤ (ii) for any x ∈ F^{×} with valuation ν = ν_{F} (x), we have xV_{k} = V_{k+eν}, where e = e(𝔏).

Since, by definition, the V_{k} are k_{F} -subspaces, it is enough to verify (ii) for π_{F}, rather than every x ∈ F^{×}. It is easy to produce these residual subspaces. Let W be an F -subspace of V, and define

𝔏(W)_{k} = \frac{W ∩ L_{k}}{W ∩ L_{k+1}} = \frac{(W ∩ L_{k}) + L_{k+1}}{L_{k+1}} , k ∈ ℤ.

The family {𝔏(W)_{k} : k ∈ ℤ} is then surely a residual subspace of 𝔏.

(3.1.4) Proposition Let {V_{k} : k ∈ ℤ} be a residual subspace of the lattice chain 𝔏. Then:

(i) there exists a subspace W of V such that 𝔏(W)_{k} = V_{k} for all k ∈ ℤ; (ii) for any subspace W satisfying (i), we have

dim_{F} (W) = ∑_{i=0}^{e−1} dim_{k_{F}} (V_{i});

(iii) let W^{1}, W^{2} be subspaces of V. Then 𝔏(W^{1})_{k} = 𝔏(W^{2})_{k}, for all k ∈ ℤ, if and only if there exists x ∈ U^{1}(𝔄) such that W^{2} = xW^{1}.

Proof: For each i, 0 ≤ i ≤ e − 1, choose a basis {v_{ij}} of V_{i} and let v̂_{ij} ∈ V be such that v̂_{ij} + L_{i+1} = v_{ij}. Let W be the F -span of the v̂_{ij}. Then {v̂_{ij}} is part of an 𝔬_{F} -basis of 𝔏 so V_{i} = 𝔏(W)_{i} and we have proved (i). (ii) comes from the fact that, for any subspace W of V, dim_{F} (W) = dim_{k_{F}} \frac{W ∩L_{0}}{W ∩L_{e}}.

(iii) If x ∈ U^{1}(𝔄) then xW^{1} ∩ L_{k} = x(W^{1} ∩ L_{k}) and x acts as the identity on L_{k}/L_{k+1} so 𝔏(W^{1})_{k} = 𝔏(xW_{1})_{k}. Conversely, suppose 𝔏(W^{1})_{k} = 𝔏(W^{2})_{k} for all k ∈ ℤ. Choose a basis {w_{ij}^{k}} for W^{k} such that {w_{ij}^{k}} (mod L_{i+1}) is a basis of 𝔏(W^{k})_{i}, for 0 ≤ i ≤ e − 1, k = 1, 2. There exists a subset B of V such that B ∪ {w_{ij}^{k}} is an 𝔬_{F} -basis of 𝔏 for both values of k. There exists x ∈ Aut_{F} (V) such that xw_{ij}^{1} = w_{ij}^{2} and such that xb = b for b ∈ B. Then x stabilises each L_{i} and acts as the identity on each quotient L_{i}/L_{i+1} so x ∈ U^{1}(𝔄). ■ In particular, the dimension of a residual subspace V is well defined: dim_{k_{F}} V = ∑_{i=0}^{e−1} dim_{k_{F}} (V_{i}), which is the dimension of any subspace W of V such that V = 𝔏(W).

(3.1.5) Proposition Let 𝔏 be a lattice chain in V, with residual subspaces V^{1}, V^{2}, such that

L_{i}/L_{i+1} = (V^{1})_{i} ⊕ (V^{2})_{i}

for all i ∈ ℤ. Let V^{1}, V^{2} be subspaces of V such that (V^{k})_{i} = 𝔏(V^{k})_{i} for all i, k. Then V = V^{1} ⊕ V^{2}, and the pair (𝔏(V^{1})_{i}, 𝔏(V^{2})_{i}) determines the pair (V^{1}, V^{2}) up to translation by an element of U^{1}(𝔄). Further, for any such pair (V^{1}, V^{2}), we have

L_{k} = (L_{k} ∩ V^{1}) ⊕ (L_{k} ∩ V^{2}), k ∈ ℤ

Proof: Choose bases {v_{ij}^{k}} of V_{i}^{k}, for 0 ≤ i ≤ e − 1, k = 1, 2 and let v̂_{ij}^{k} ∈ L_{i} be such that v̂_{ij}^{k} + L_{i+1} = v_{ij}^{k}. Let V^{k} be the F -span of the v̂_{ij}^{k}, for k = 1, 2. Then {v̂_{ij}^{k} : k = 1, 2} is an 𝔬_{F} -basis of 𝔏 so V^{k} = 𝔏(V^{k}), for k = 1, 2 and V = V^{1} ⊕ V^{2}. The uniqueness up to translation by an element of U^{1}(𝔄) follows as in (3.1.4). We clearly have (L_{k} ∩ V^{1}) ⊕ (L_{k} ∩ V^{2}) ⊂ L_{k} for k ∈ ℤ. The converse holds since {v̂_{ij}^{k} : k = 1, 2} is an 𝔬_{F} -basis of 𝔏. ■

(3.1.6) Proposition Let 𝔏 = {L_{k} : k ∈ ℤ} be a self-dual lattice chain in V. Let d ∈ ℤ be the unique integer such that L_{k}^{#} = L_{d−k} for all k ∈ ℤ, given by (1.1.2). Then for each i ∈ ℤ, h induces a nondegenerate k_{F} -sesquilinear pairing

(3.1.7)  h_{i} : \frac{L_{i}}{L_{i+1}} × \frac{L_{d−i−1}}{L_{d−i}} → k_{F}

Proof: For i ∈ ℤ, x ∈ L_{i}, y ∈ L_{d−i−1}, we have π_{F} x ∈ L_{i+1} so h(π_{F} x, y) ∈ 𝔭_{F}. Hence h(x, y) ∈ 𝔬_{F} and we can reduce modulo 𝔭_{F} to get the pairing h_{i}. For nondegeneracy we need that if y ∈ L_{d−i−1} satisfies h(L_{i}, y) ⊂ 𝔭_{F} then y ∈ L_{d−i}, which is immediate, and symmetrically. ■ Now let V = {V_{i}} be a residual subspace of our self-dual lattice chain 𝔏. We say that V is nondegenerate if the pairing

V_{i} × V_{d−i−1} → k_{F}

induced by (3.1.7) is nondegenerate. Irrespective of this condition, given a residual subspace V, we can define its orthogonal complement V^{⊥} = {V_{i}^{⊥} : i ∈ ℤ} by

V_{i}^{⊥} = {x ∈ L_{i}/L_{i+1} : h_{i}(x, v) = 0, v ∈ V_{d−i−1}}.

Then V^{⊥} is indeed a residual subspace of 𝔏 and we have the identity

V^{⊥⊥} = V.

In the usual way, V is nondegenerate if and only if V ∩V^{⊥} = {0}, or, more precisely,

L_{i}/L_{i+1} = V_{i} ⊕ V_{i}^{⊥},

for all i.

(3.1.8) Proposition (i) Let V be a nondegenerate residual subspace of the selfdual lattice chain 𝔏. Let W be a subspace of V such that 𝔏(W) = V. Then h|_{W ×W} is nondegenerate. (ii) Let W be any subspace of V such that 𝔏(W) is nondegenerate. Then 𝔏(W^{⊥}) is nondegenerate and

𝔏(W^{⊥}) = 𝔏(W)^{⊥}.

Proof: (i) Let w ∈ W be such that ν_{𝔏}(w) = i; so w ∈ L_{i} but w ∉ L_{i+1}. By hypothesis there exists w′ ∈ W ∩ L_{d−i−1} such that h_{i}(w + L_{i+1}, w′ + L_{d−i−1}) ≠ 0. Hence h(w, w′) ∉ 𝔭_{F} and, in particular, h(w, w′) ≠ 0. (ii) We certainly have 𝔏(W^{⊥})_{i} ⊂ 𝔏(W)_{i}^{⊥} for all i ∈ ℤ. The result then follows by comparing dimensions. ■

(3.1.9) Corollary Let V^{1}, V^{2} be nondegenerate residual subspaces of 𝔏 such that L_{i}/L_{i+1} = V_{i}^{1} ⊥ V_{i}^{2} for i ∈ ℤ. Then there exist subspaces V^{1}, V^{2} of V such that V = V^{1} ⊥ V^{2} and 𝔏(V^{k}) = V^{k} for k = 1, 2.

(3.1.10) Lemma Let W be a totally isotropic subspace of V. Then 𝔏(W^{⊥}) = 𝔏(W)^{⊥}.

Proof: The proof is the same as (3.1.8)(ii). ■

We call a residual subspace V of 𝔏 totally isotropic if h_{i}(V_{i}, V_{d−i−1}) ≡ 0, for all i ∈ ℤ.

(3.1.11) Proposition Let W be a totally isotropic subspace of 𝔏. Then there exists a totally isotropic subspace W of V such that W = 𝔏(W).

Proof: Put n = dim_{k_{F}} W. The idea of the proof is to split W into (totally isotropic) one-dimensional residual subspaces, contained in mutually orthogonal nondegenerate two-dimensional subspaces X^{i} of V, 1 ≤ i ≤ n. Hence we reduce to the case dim_{k_{F}} W = 1 and dim_{F} V = 2. We choose a non-zero w_{1} ∈ W_{r_{1}}, for some r_{1} ∈ ℤ. By the nondegeneracy of h_{r_{1}}, there exists v_{1} ∈ V_{d−r_{1}−1} such that h_{r_{1}}(w_{1}, v_{1}) = 1. Let Y^{1}, Z^{1} be the residual subspaces given by

Y_{k}^{1} = \{ \begin{matrix}π̃_{F}^{s}⟨w_{1}⟩_{k_{F}} & if k = r_{1} + se; \\ 0 & otherwise, \end{matrix}
Z_{k}^{1} = \{ \begin{matrix}π̃_{F}^{s}⟨v_{1}⟩_{k_{F}} & if k = d − r_{1} − 1 + se; \\ 0 & otherwise, \end{matrix}

where e = e(𝔏) is the 𝔬_{F} -period of 𝔏, ⟨v⟩_{k_{F}} = k_{F} v is the k_{F} -span of v and π̃_{F} is the k_{F} -linear isomorphism L_{k}/L_{k+1} → L_{k+e}/L_{k+e+1} induced by multiplication by π_{F} (see (3.3) for more details). We put X^{1} = Y^{1} + Z^{1}. The residual subspace X^{1} is nondegenerate so, by (3.1.8), there exists a subspace X^{1} of V such that 𝔏(X^{1}) = X^{1} and h_{X^{1}×X^{1}} is nondegenerate. Replacing V by X^{1}^{⊥} and W by W ∩ X^{1}^{⊥}, we may repeat the above to obtain, for 1 ≤ i ≤ n, residual vectors w_{i}, v_{i} and nondegenerate subspaces X^{i} of V such that X^{i} ⊥ X^{j}, for i ≠ j, dim_{F} X^{i}=2 and W is generated by ⟨w_{1}, ...w_{n}⟩_{k_{F}} and π̃_{F}.

For each i we will find ŵ_{i} ∈ X^{i} such that ŵ_{i}+L_{r_{i}+1} = w_{i} and such that h(ŵ_{i}, ŵ_{i}) = 0. Then we put W = ⟨ŵ_{i} : 1 ≤ i ≤ n⟩_{F} and we are done. We drop the subscript i. Let x ∈ X be such that x + L_{r+1} = w. Let s ∈ ℤ be such that r + se ≥ d − r − 1 > r + (s − 1)e so

L_{r+se+1} ⊆ L_{r}^{#} ⫋ L_{r+(s−1)e+1}.

Suppose first that r + se > d − r − 1; then π_{F}^{s} x ∈ L_{r+se} ⊆ L_{r}^{#} so h(x, x) ∈ 𝔭_{F}^{1−s}. Otherwise, r + se = d − r − 1; then π_{F}^{s} x ∈ L_{d−r−1}. Then h(π_{F}^{s} x, x) + 𝔭_{F} = h_{d−r−1}(π̃_{F}^{s}w, w) = 0 so h(x, x) ∈ 𝔭_{F}^{1−s} in this case also. For n ∈ ℕ we find, by induction x_{n} ∈ X such that w = x_{n} + L_{r+1}, x_{n+1} − x_{n} ∈ L_{r+(n−1)e+1} and h(x_{n}, x_{n}) ∈ 𝔭_{F}^{n−s}. Then we put ŵ = lim_{n→∞} x_{n} and we are done.

We may take x_{1} = x so assume we have found x_{n} as required. If h(x_{n}, x_{n}) ∈ 𝔭_{F}^{n+1−s} then we may take x_{n+1} = x_{n} so assume h(x_{n}, x_{n}) ∈ 𝔭_{F}^{n−s}\𝔭_{F}^{n+1−s}; say h(x_{n}, x_{n}) = α_{n}π_{F}^{n−s}, α_{n} ∈ 𝔬_{0}^{×}, π_{F}^{n−s} ∈ F_{0}. By the nondegeneracy of h_{r} : V_{r} × V_{d−r−1} → k_{F}, there exists y_{n} ∈ L_{d−r−1} such that h(x_{n}, y_{n}) + 𝔭_{F} = \frac{1}{2}α_{n}. Then h(y_{n}, y_{n}) ∈ 𝔭_{F}^{s} since L_{d−r−1} ⊂ L_{r+(s−1)e+1} = π_{F}^{s−1}L_{d−r−1}^{#} and π_{F}^{n−s}y_{n} ∈ L_{d−r−1+(n−s)e} ⊂ L_{r+(s−1)e+1+(n−s)e} = L_{(n−1)e+r+1}. Then we put x_{n+1} = x_{n} −π_{F}^{n−s}y_{n} and we have

h(x_{n+1}, x_{n+1}) = h(x_{n}, x_{n}) − π_{F}^{n−s}h(x_{n}, y_{n}) − π_{F}^{n−s}h(y_{n}, x_{n}) + π_{F}^{2(n−s)}h(y_{n}, y_{n})

= α_{n}π_{F}^{n−s} − \frac{1}{2}α_{n}π_{F}^{n−s} + O(𝔭_{F}^{n+1−s})

− \frac{1}{2}α_{n}π_{F}^{n−s} + O(𝔭_{F}^{n+1−s}) + O(𝔭_{F}^{2n−s})

∈ 𝔭_{F}^{n+1−s}, since n ≥ 1.

■

(3.2) Splittings of self-dual lattice chains

We continue with the notation of the previous section; in particular h is a nondegenerate ϵ-hermitian form on V. Let 𝔏 be a self-dual lattice chain in V.

(3.2.1) Definition A splitting of 𝔏 is a decomposition of V

V = V_{∞} ⊕ V_{−r} ⊕ ⋯ ⊕ V_{−1} ⊕ V_{0} ⊕ V_{1} ⊕ ⋯ ⊕ V_{r}

into F -subspaces such that the following hold:

(i) For all k ∈ ℤ we have L_{k} = ⨁_{ω∈Ω} L_{k} ∩ V_{ω}, where Ω is the set of ω ∈ {∞, −r, … − 1, 0, 1, …, r} such that V_{ω} ≠ 0; (ii) 𝔏^{ω} := {L_{k} ∩ V_{ω} : k ∈ ℤ is a lattice chain of period 1, for ω ∈ Ω; (iii) V_{ω}^{⊥} = ⨁_{v≠−ω} V_{v} for all ω ∈ Ω (with the understanding that −∞ = ∞).

(3.2.2) Proposition Let 𝔏 be a self-dual lattice chain in V. Then there exists a splitting of 𝔏.

Proof: Let d = 0 or −1 be the integer such that L_{k}^{#} = L_{d−k} for k ∈ ℤ, given by (1.1.2), and let e = e(𝔏) be the 𝔬_{F} -period of the lattice chain. Define residual subspaces V^{(i)}, for i ∈ ℤ, by

V_{k}^{(i)} = \{ \begin{matrix}L_{k}/L_{k+1} & i ≡ k (mod e) \\ 0 & otherwise. \end{matrix}

Then h induces a duality V^{(i)} ↔ V^{(d−i−1)} by (3.1.6). Set W^{(j)} = V^{(j−1)} + V^{(d−j)} for j ∈ ℤ; this is a direct sum if 2j ≢ d + 1(mod g), while V^{(j−1)} = V^{(d−j)} otherwise.

(3.2.3) Lemma The residual subspaces W^{(j)} are nondegenerate.

Proof: We have to show that the pairing

W_{k}^{(j)} × W_{d−k−1}^{(j)} → k_{F}

is nondegenerate. But if W_{k}^{(j)} ≠ 0 then, without loss of generality, V_{k}^{(d−j)} = L_{k}/L_{k+1} ≠ 0 so k ≡ d − j (mod g). Then d − k ≡ j (mod g) so W_{d−k}^{(j)} = V_{d−k}^{(j)} = L_{d−k}/L_{d−k+1}. Then by (3.1.6) the pairing is nondegenerate. ■

Set f = [\frac{e+1−d}{2}]; then L_{k}/L_{k+1} = W_{k}^{(d+1)} ⊕ W_{k}^{(d+2)} ⊕ ⋯ ⊕ W_{k}^{(d+f)} for all k ∈ ℤ. The residual subspaces W^{(d+1)}, …, W^{(d+f)} are orthogonal so, by (3.1.9), there exist subspaces W^{(d+1)}, …, W^{(d+f)} of V such that

V = W^{(d+1)} ⊥ ⋯ ⊥ W^{(d+f)}

and W^{(j)} = 𝔏(W^{(j)}) for d+1 ≤ j ≤ d+f. So we can write h = h_{d+1} ⊥ ⋯ ⊥ h_{d+f} with h_{j} a nondegenerate ϵ-hermitian form on W^{(j)}.

(3.2.4) Lemma 𝔏^{j} = {L_{k} ∩ W^{(j)} : k ∈ ℤ} is a self-dual lattice chain in W^{(j)}, d + 1 ≤ j ≤ d + f. Moreover,

e_{j} = e(𝔏_{j}) = \{ \begin{matrix}1  if 2j ≡ d + 1 (mod e) \\ 2  otherwise. \end{matrix}

Proof: That 𝔏^{j} is a lattice chain of period e_{j} in W^{(j)} is clear. For all k ∈ ℤ, (L_{k} ∩W^{(j)})^{#} = L_{k}^{#} +(W^{(j)})^{#} = L_{d−k} +W^{(1)} +⋯+W^{(j−1)} +W^{(j+1)} +⋯+W^{(f)}. Restricting the duality operation to W^{(j)}, this says that (L_{k} ∩ W^{(j)})^{#} is the projection in W^{(j)} of L_{d−k}, which is L_{d−k} ∩ W^{(j)}. ■

Note that all but at most 2 of the 𝔏^{j} have period 2. If d = −1 then e_{d+1} = 1 and put V_{d+1} = W^{(d+1)}, otherwise V_{d+1} = 0. If 2f = e + 1 − d then e_{d+f} = 1 and put V_{∞} = W^{(d+f)}, otherwise V_{∞} = 0. Now consider the case e_{j} = 2. Then V^{(j−1)} and V^{(d−j)} are totally isotropic residual subspaces of 𝔏^{j} so, by (3.1.11), there are totally isotropic subspaces V_{j} and V_{−j} of W^{(j)} such that 𝔏(V_{j}) = V^{(j−1)} and 𝔏(V_{−j}) = V^{(d−j)}. Then clearly W^{(j)} = V_{j} ⊕ V_{−j}.

Altogether we have

V = V_{∞} ⊕ V_{−r} ⊕ ⋯ ⊕ V_{−1} ⊕ V_{0} ⊕ V_{1} ⊕ ⋯ ⊕ V_{r}

as required, e = 2r, 2r + 1 or 2r + 2. So this completes the proof of (3.2.2). ■ We think of Ω ⊂ {∞, −r, ..., r} as an ordered set of indices. Also, ℤ/eℤ acts on Ω by cyclic permutation. For ω ∈ Ω, i ∈ ℤ, we will write ω + i for the translate by i + eℤ of ω.

We can push this further. We choose 𝔬_{F} -bases for V_{∞} ∩ L_{d−r−1}, V_{−j} ∩ L_{d−j} for r ≥ j ≥ 1, and V_{0} ∩ L_{d} and 𝔬_{F} -bases for V_{j} ∩ L_{j−1} for 1 ≤ j ≤ r. Then, with respect to this basis B, h has matrix

( \begin{matrix}J_{∞} & 0 & ⋯ & & & & ⋯ & 0 \\ 0 & 0 & ⋯ & & & ⋯ & 0 & I \\ ⋮ & ⋮ & & & & ... & ... & 0 \\ & & & & & I & ... & ⋮ \\ & & & & J_{0} \\ & ⋮ & ... & ϵI \\ ⋮ & 0 & ... & ... & & & & ⋮ \\ 0 & ϵI & 0 & ⋯ & & & ⋯ & 0 \end{matrix} )  where  I = ( \begin{matrix}& & & & 1 \\ & . & . & . \\ 1 \end{matrix} ) ,

J_{ω} is the matrix of h|_{V_{ω}×V_{ω}}, for ω = ∞, 0, and 𝔄 = 𝔄(𝔏) has the form

(3.2.5)  ( \begin{matrix}𝔬_{F} & 𝔬_{F} & ⋯ & 𝔬_{F} & 𝔬_{F} \\ 𝔭_{F} & 𝔬_{F} & ... & & 𝔬_{F} \\ ⋮ & ... & ... & ... & ⋮ \\ 𝔭_{F} & & 𝔭_{F} & 𝔬_{F} & 𝔬_{F} \\ 𝔭_{F} & 𝔭_{F} & ⋯ & 𝔭_{F} & 𝔬_{F} \end{matrix} ) .

So we have chosen a basis of 𝔏 in the sense of [BK] (1.1.7). Note that the above matrices are n-block matrices, where n = (n_{∞}, n_{−r}, …, n_{r}) is a vector of positive integers and n_{ω} = dimV_{ω}, for ω ∈ Ω. Putting A^{(ωω′)} = Hom(V_{ω′}, V_{ω}), for ω, ω′ ∈ Ω, we get a block decomposition A = ∐_{ω,ω′} A^{(ωω′)} with respect to which 𝔄 has the form above. Then, for a ∈ A^{(ωω′)}, we have h(V_{v′}, \overline{a}V_{v}) = h(aV_{v′}, V_{v}) which is non-zero if and only if v′ = ω′ and v = −ω. Hence \overline{a} ∈ A^{(−ω′,−ω)}. In particular, the involution on A fixes diagonal “bands” of blocks in A.

We now define a set of representatives for the cosets 𝔓^{m}/𝔓^{m+1}, for any integer m, as in [BK] (2.5.5). We define integers l, k by

(3.2.6)  0 ≤ l = m − ek ≤ e − 1.

Consider the set B_{l} = B_{l}(𝔄) of e-tuples of matrices

b = (b_{ω} : ω ∈ Ω)

where b_{ω} has entries in 𝔬_{F} and dimensions n_{ω} × n_{ω+l}. For b ∈ B_{l}, we define an n-block matrix r_{m}(b) over F by

r_{m}(b)_{ω,ω′} = \{ \begin{matrix}π_{F}^{k} b_{ω} & if ω ≤ r − l and ω = ω′ − l, \\ π_{F}^{k+1}b_{ω} & if ω > r − l and ω = ω′ − l, \\ 0 & otherwise. \end{matrix}

Then the map b → r_{m}(b) induces a bijection between B_{l}(𝔄) (modulo 𝔭_{F}) and 𝔓^{m}/𝔓^{m+1}. Moreover, since the involution fixes bands of blocks, \overline{r_{m}(b)} = r_{m}(b′), for some b′ ∈ B_{l}(𝔄). Then we can define an involution on B_{l}(𝔄) by putting \overline{b} = b′. (Note that, in the case F ≠ F_{0}, the involution on B_{l}(𝔄) is dependent on m.)

(3.2.7) Definition Let [𝔄, n, n − 1, b] be a stratum in A. It is in band form if b = r_{−n}(b), for some b ∈ B_{l}(𝔄), where l, k ∈ ℤ are such that 0 ≤ l = −n − ek ≤ e − 1.

Any stratum is equivalent to a stratum in band form and further, since the involution on A fixes bands of blocks, any skew stratum is equivalent to a skew stratum in band form.

It will sometimes be more convenient to choose a slightly different basis, which is a basis for 𝔏 in the sense of (3.1.1). We can choose bases for V_{∞} and V_{0} such that

(3.2.8)  J_{∞} = ( \begin{matrix}& & π_{F}^{−1}I \\ & K_{∞} \\ ϵπ_{F}^{−1}I \end{matrix} ) ,  J_{0} = ( \begin{matrix}& & I \\ & K_{0} \\ ϵI \end{matrix} ) ,

where K_{ω} is the matrix of the anisotropic part of h|_{V_{ω}×V_{ω}}, ω = ∞, 0. Then, rearranging the bases for V_{∞}, V_{−r}, …, V_{r}, the form h has matrix

(3.2.9)  ( \begin{matrix}K_{∞} \\ & & & I \\ & & K_{0} \\ & ϵI \end{matrix} ) .

Moreover, the matrix

K = ( \begin{matrix}K_{∞} \\ & K_{0} \end{matrix} )

is at most 2 × 2 since the anisotropic part of V with respect to h is at most 2-dimensional. From [M1] (1.8), the only possibilities for K are

K = ( δ) , ( δε) or ( \begin{matrix}−δε \\ & δ \end{matrix} )

where ε ∈ F_{0}, ε ∉ N_{F/F_{0}}(F) and δ = 1, if h is hermitian, δ ∈ F_{−}, if h is skewhermitian.

(3.3) Residual maps

In this section we look at maps between residual subspaces. Let 𝔏 be an 𝔬_{F} -lattice chain in V and 𝔄 the associated hereditary order in A. Let x ∈ A be such that ν_{𝔄}(x) = n. Then x induces maps

x̃_{k} : L_{k}/L_{k+1} → L_{k+n}/L_{k+n+1}
v + L_{k+1} ↦ xv + L_{k+n+1}

for k ∈ ℤ. Let V be the residual subspace 𝔏(V) of 𝔏, so V_{k} = L_{k}/L_{k+1}, for k ∈ ℤ. Then we say that x̃ : V → V is of degree n.

(3.3.1) Lemma Let 𝔄 be a hereditary 𝔬_{F} -order in A. Then x ∈ 𝔎(𝔄) if and only if x̃_{k} is an isomorphism for all k ∈ ℤ.

Proof: Suppose x ∈ 𝔎(𝔄) and put n = ν_{𝔄}(x); then ν_{𝔄}(x^{−1}) = −n so x^{−1} induces a map of degree −n, x̃^{−1} : V → V. Then, for k ∈ ℤ, we see that (x̃^{−1})_{k+n} ∘ x̃_{k} is the identity on L_{k}/L_{k+1}, since x̃_{k}(v + L_{k+1}) = xv + L_{k+n+1} = x(v + L_{k+1}). Conversely, suppose x ∉ 𝔎(𝔄) and put n = ν_{𝔄}(x). We have that xL_{k} ⫋ L_{k+n}, for some k ∈ ℤ. Then x̃_{k} is not surjective; in particular, it is not an isomorphism. ■ In particular, π_{F} induces an isomorphism π̃_{F} : V → V of degree e = e(𝔏), the 𝔬_{F} - period of 𝔏. Indeed, if W is any residual subspace of 𝔏 then (π̃_{F})_{k}W_{k} = W_{k+e}, for all k ∈ ℤ, so we have an isomorphism π̃_{F} : W → W. Let W^{1}, W^{2} be residual subspaces of 𝔏 and let ϱ : W^{1} → W^{2} be of degree n. i.e. ϱ is a set of k_{F} -linear maps {ϱ_{k} : k ∈ ℤ}, where ϱ_{k} : W_{k}^{1} → W_{k+n}^{2}. Then we say ϱ is a residual map if ϱ ∘ π̃_{F} = π̃_{F} ∘ ϱ, i.e. for all k ∈ ℤ

ϱ_{k+e} ∘ (π̃_{F})_{k} = (π̃_{F})_{k+n} ∘ ϱ_{k}.

(3.3.2) Lemma Let ϱ : V → V be a residual map of degree n. Then there exists x ∈ A, ν_{𝔄}(x) = n such that ϱ = x̃.

Proof: We may assume that 0 ≤ n ≤ e − 1 by scaling by π̃_{F}. For 0 ≤ k ≤ e − 1 choose a basis {v_{ik} : 1 ≤ i ≤ d_{i}} of V_{k} and choose v̂_{ik} ∈ V such that v̂_{ik} + L_{k+1} = v_{ik}. Taking the index k modulo e, we have

ϱ(v_{ik}) = ∑_{1≤i≤d_{n+k}} ϱ_{ik}v_{i,n+k}

for some ϱ_{ik} ∈ k_{F}. Choose x_{ik} ∈ 𝔬_{F} such that x_{ik} + 𝔭_{F} = ϱ_{ik} for each i, k and define x by

xv̂_{ik} = ∑_{1≤i≤d_{n+k}} x_{ik}v̂_{i,n+k}.

Clearly x ∈ A, ν_{𝔄}(x) = n and x̃ = ϱ. ■ We write End_{k_{F}}^{n} V for the k_{F} -module of residual maps of degree n.

(3.3.3) Corollary For each n ∈ ℤ, the natural map 𝔓^{n} → End_{k_{F}}^{n} V : x ↦ x̃ induces an isomorphism of k_{F} -spaces 𝔓^{n}/𝔓^{n+1} → End_{k_{F}}^{n} V.

Proof: The natural map is surjective by (3.3.2) and clearly the kernel is 𝔓^{n}. ■ Note also that if ϱ is a residual isomorphism (i.e. ϱ_{k} is an isomorphism for each k ∈ ℤ) then x ∈ 𝔎(𝔄).

Let now h : V × V → F be an ϵ-hermitian form as before and let 𝔏 be a self-dual lattice chain in V. Then, by (3.1.6) we have nondegenerate sesquilinear forms h_{k} : V_{k} × V_{d−k−1} → k_{F} for all k ∈ ℤ.

Let ϱ : V → V be a residual map of degree n. Then we define the adjoint residual map \overline{ϱ}, also of degree n, by

h_{k+n}(ϱ_{k}v, v′) = h_{k}(v, \overline{ϱ}_{d−k−1−n}v′)

for v ∈ V_{k}, v′ ∈ V_{d−k−1−n}. This is well-defined as h_{k} is nondegenerate for all k ∈ ℤ.

(3.3.4) Lemma Let x ∈ A; then \overline{x̃} = ˜ \overline{x}.

Proof: Put n = ν_{𝔄}(x) and choose any v ∈ V_{k}, v′ ∈ V_{d−k−1−n}. Choose also v̂, v̂′ ∈ V such that v̂ + L_{k+1} = v and v̂′ + L_{d−k−n} = v′. Then

h_{k+n}(x̃v, v′) = h(xv̂.v̂′) + 𝔭_{F}

= h(v̂, \overline{x}v̂′) + 𝔭_{F}

= h_{k}(v, ˜ \overline{x}v′). ■

A residual map ϱ is called skew if ϱ = −\overline{ϱ}.

(3.3.5) Lemma Let ϱ be a skew residual map of degree n. Then there exists x ∈ A_{−}, ν_{𝔄}(x) = n, such that ϱ = x̃.

Proof: Let y ∈ A, ν_{𝔄}(y) = n be such that ỹ = ϱ, by (3.3.2). Then { \atop (−\overline{y})} = −\overline{ỹ} = ϱ so, by (3.3.3), −\overline{y} ∈ y + 𝔓^{1+n}. Then, by (1.2.2), there exists u ∈ 𝔓^{1+n} such that y + \overline{y} = u + \overline{u} and we put x = y − u. ■

(3.3.6) Corollary Writing End_{k_{F} ,−}^{n}V for the skew residual maps of degree n, we have an isomorphism 𝔓_{−}^{n}/𝔓_{−}^{n+1} → End_{k_{F} ,−}^{n}V.

Put A = End_{k_{F}}^{0} V. Then the map

A → ∏_{i=0}^{e−1} End_{k_{F}} L_{i}/L_{i+1}
ϱ ↦ (ϱ_{0}, …, ϱ_{e−1})

is an isomorphism of k_{F} -algebras since ϱ_{re+i} = π̃_{F}^{r}ϱ_{i}π̃_{F}^{−r} for all r ∈ ℤ, 0 ≤ i ≤ e − 1. So we have notions of minimal polynomial (over k_{F}) and Jordan decomposition in A.

REFINEMENTS

In this chapter, h is an alternating form, i.e. this chapter is only for the symplectic group. In this chapter we look at the refinement process for skew simple strata (cf. [BK] (2.2)). In order to do this we must first adapt the notion of a (W, E)-decomposition [BK] (1.2) to our situation. We now recall this. Let E/F be a subfield of A and write A(E) = End_{F} (E), B = End_{E}(V). Let W be the F -span of an E-basis of our vector space V. Then the choice of W induces an embedding of algebras ι_{W} : A(E) ↪ A and an isomorphism

(4.0.1)  A(E) ⊗_{E} B ≃ A

of (A(E), B)-bimodules ([BK] (1.2.6)). Moreover, if 𝔄 is a hereditary order in A normalised by E then, by [BK] (1.2.8) we can choose W such that the isomorphism of (4.0.1) restricts to an isomorphism

(4.0.2)  𝔄(E) ⊗_{𝔬_{E}} 𝔅,

where 𝔄(E) is the unique hereditary 𝔬_{F} -order in A normalised by E ([BK] (1.2.7)) and 𝔅 = 𝔄 ∩ B. In our situation, E/F is a subfield of A stable under the involution but not fixed pointwise by it. Then E, considered as an F -vector space, has a nondegenerate alternating form on it (indeed we will need two different forms in general). Then A(E) has an involution and we would like A(E)_{−} to be embedded in A_{−} by ι_{W}. In general, we will need to take not the space A(E)_{−} but the space of elements skew according to both involutions (induced from the two different forms on E). This will allow us to prove that if [𝔄, n, r, β] is a skew simple stratum and b ∈ 𝔓_{−}^{−r} such that [𝔅, r, r −1, s(b)] is equivalent to a skew simple stratum then the stratum [𝔄, n, r − 1, β + b] is equivalent to a skew simple stratum (cf. [BK] (2.2.8)).

(4.1) (W , E)-decomposition

Let E/F be a subfield of A such that E = \overline{E} and let E_{0} be the fixed field of the involution, E_{0} ≠ E. Let e = e(E|F) be the ramification degree of E/F and let 𝔇^{−1} be the inverse different of E/F. Let B = End_{E}V be the A-centraliser of E so B = \overline{B}. There exists a hermitian form f : V × V → E by

h(ev, w) = tr_{E/F} (δef(v, w)) for all e ∈ E, v, w ∈ V

for some δ ∈ 𝔭_{E}^{e−1}𝔇^{−1}, δ ∉ 𝔭_{E}^{e+1}𝔇^{−1}, such that δ + \overline{δ} = 0. This is well-defined as tr_{E/F} is nondegenerate, and f itself is nondegenerate by the nondegeneracy of h. Then f determines an adjoint involution on B, which is in fact the involution on A restricted to B since, for x ∈ B and v, w ∈ V, e ∈ E, we have

tr_{E/F} (δef(xv, w)) = h(exv, w) = h(xev, w)
= h(ev, \overline{x}w) = tr_{E/F} (δef(v, \overline{x}w)).

As a direct consequence of this we have:

(4.1.1) Lemma The centraliser in G of E, B ∩ G, is the unitary group of Eautomorphisms of V which fix the hermitian form f.

From the theory of hermitian forms (see [Sch] (1.5.11), (7.1)), V then has a Witt decomposition as an E-space, V = V^{an} ⊥ V^{sp} where f|_{V^{an}×V^{an}} is anisotropic, dim_{E}V^{an} ≤ 2, and f|_{V^{sp}×V^{sp}} is split (i.e. it is a sum of hyperbolic planes). Let 𝔄 be an 𝔬_{F} -order in A, normalised by E, such that 𝔄 = \overline{𝔄} and let 𝔏 = 𝔏(𝔄) be the self-dual 𝔬_{F} -lattice chain associated to it. Let 𝔅 = 𝔄 ∩ B and 𝔔 = 𝔓 ∩ B, where 𝔓 is the Jacobson radical of 𝔄, as usual. Then 𝔏 is in fact a self-dual 𝔬_{E}-lattice chain in B since, for L ∈ 𝔏,

L^{#} = {v ∈ V : h(x, L) ⊂ 𝔭_{F} } = {v ∈ V : δf(v, L) ⊂ 𝔇^{−1}𝔭_{F} }
= \{ \begin{matrix}\{v ∈ V : f(v, L) ⊂ 𝔭_{E}\}  if δ ∉ 𝔭_{E}^{e}𝔇^{−1}; \\ \{v ∈ V : f(v, L) ⊂ 𝔬_{E}\}  if δ ∈ 𝔭_{E}^{e}𝔇^{−1}. \end{matrix}

Then by [M1] (1.7) we can choose a basis for the lattice chain such that, with respect to this basis, the hermitian form f has matrix:

( \begin{matrix}C_{1} & 0 & 0 & 0 \\ 0 & 0 & 0 & I \\ 0 & 0 & C_{2} & 0 \\ 0 & I & 0 & 0 \end{matrix} ) ,  where I = ( \begin{matrix}0 & & & & 1 \\ & . & . & . \\ 1 & & & & 0 \end{matrix} )  and C = ( \begin{matrix}C_{1} & 0 \\ 0 & C_{2} \end{matrix} )

and C is the matrix of f|_{V_{an}×V_{an}}.

The possibilities for C are:

dimV^{an} = 1  C = ( 1 ) or ( ε) ;
dimV^{an} = 0  C = ( \begin{matrix}−ε & 0 \\ 0 & 1 \end{matrix} );

where ε ∈ E_{0}, ε ∉ N_{E/E_{0}}(E). (In fact, in the case dimV^{an} = 1, we may always assume C = (1) by changing δ to δε and choosing a basis for the form given by this.) Note also that we may change ε by any norm so we will choose it carefully.

(4.1.2) Lemma In the situation above there exists ε ∈ E_{0}, ε ∉ N_{E/E_{0}}(E) such that ε^{n} ∈ F for some n ∈ ℤ, (n, p) = 1.

Proof: We first consider the case E/E_{0} ramified. Then let ε be a root of unity in E_{0} which is not a square. Then ε^{m} = 1 for some m ∈ ℤ, (m, p) = 1. Then this ε will do. For suppose ε is a norm; then any element of 𝔬_{0}^{×} is a norm since ε generates the group of roots of unity while every element of U^{1}(𝔬_{0}) is a square as p ≠ 2. Further π_{0} is also a norm, for some prime element π_{0}, since E/E_{0} is ramified. So every element of E_{0} is a norm, which is absurd. Now consider the case E/E_{0} unramified. Let E_{tr} be the maximal tamely ramified extension of F contained in E_{0} and let m = e(E_{tr}|F) be the ramification degree of E_{tr}/F, (m, p) = 1. Then we can choose a uniformizer π_{tr} of E_{tr} such that π_{tr}^{m} = ξπ_{F}, for ξ a (p prime) root of unity in E_{tr}. We let ε = π_{tr} and observe that ν_{E}(ε) is a power of p, since E_{0}/E_{tr} is wildly ramified, which is odd. However, all norms have even valuation so ε cannot be a norm. ■ So we will choose ε according to (4.1.2) and fix it.

Let W be the F -subspace of V spanned by the given E-basis so that V ≅ E ⊗_{F} W. Let W_{1} be the subspace of W spanned by the vector v in the basis such that f(v, v) = ±ε (if it exists, W_{1} = 0 otherwise) and let W_{2} be the subspace spanned by the remaining basis vectors. Set V_{i} = E ⊗_{F} W_{i}, i = 1, 2. Then, for i = 1, 2, we define symmetric bilinear forms f_{i} : W_{i} × W_{i} → F by

f_{1}(v_{1}, w_{1}) = ε^{−1}f(v_{1}, w_{1})  for v_{1}, w_{1} ∈ W_{1}
f_{2}(v_{2}, w_{2}) = f(v_{2}, w_{2})  for v_{2}, w_{2} ∈ W_{2}.

We also define two skew-symmetric bilinear forms g_{i} : E × E → F, i = 1, 2 by

g_{1}(e, e′) = tr_{E/F} εδe\overline{e′}  for e, e′ ∈ E
g_{2}(e, e′) = tr_{E/F} δe\overline{e′}  for e, e′ ∈ E

giving rise to involutions ^{1} and ^{2} on A(E) = End_{F} (E) respectively. These involutions both restrict to on E for E ↪ A(E). Note that, in general, the two involutions are not the same, i.e. there exists a ∈ A(E) such that a^{1} ≠ a^{2}. In fact,

for a ∈ A(E), ã^{1}^{2} = εaε^{−1}.

Now define ⨁_{i=1}^{2} g_{i} ⊗ f_{i} : V × V → F by

⨁_{i=1}^{2} g_{i} ⊗f_{i}(e⊗(w_{1} +w_{2}), e′ ⊗(w′_{1} +w′_{2})) = g_{1}(e, e′) f_{1}(w_{1}, w′_{1})+g_{1}(e, e′) f_{2}(w_{2}, w′_{2})

for e, e′ ∈ E, w_{i}, w′_{i} ∈ W_{i}, i = 1, 2. Then

⨁_{i=1}^{2} g_{i} ⊗ f_{i}(e⊗(w_{1} + w_{2}), e′ ⊗ (w′_{1} + w′_{2}))
= tr_{E/F} (εδe\overline{e′}) f_{1}(w_{1}, w′_{1}) + tr_{E/F} (δe\overline{e′}) f_{2}(w_{1}, w′_{1})
= tr_{E/F} δe\overline{e′}[εf_{1}(w_{1}, w′_{1}) + f_{2}(w_{2}, w′_{2})]
= tr_{E/F} δe\overline{e′}f(w_{1} + w_{2}, w′_{1} + w′_{2}), since W_{1} ⊥ W_{2},
= h(e ⊗ (w_{1} + w_{2}), e′ ⊗ (w′_{1} + w′_{2})).

So ⨁_{i=1}^{2} g_{i} ⊗ f_{i} = h.

Let A^{ij} = Hom(V_{i}, V_{j}), i, j = 1, 2, so A = ⨁_{i,j} A^{ij}. Let ι_{i} : A(E) ↪ A^{ii} = A(E) ⊗_{F} End_{F} (W_{i}) be the embedding given by ι_{i}(a) = a ⊗ 1, as in [BK] (1.2.5), and let ι : A(E) ↪ A^{11} ⊕ A^{22} ⊂ A be given by ι(a) = ι_{1}(a) + ι_{2}(a). Then for a ∈ A(E) we have

h(ι(a)(e(w_{1} + w_{2})), e′(w′_{1} + w′_{2})) = g_{1}(ae, e′)f_{1}(w_{1}, w′_{1}) + g_{2}(ae, e′)f_{2}(w_{2}, w′_{2})
= g_{1}(e, ẽ′^{1})f_{1}(w_{1}, w′_{1}) + g_{2}(e, ẽ′^{2})f_{2}(w_{2}, w′_{2})
= h(e(w_{1} + w_{2}), (ι_{1}(a^{1}) + ι_{2}(a^{2}))(e′(w′_{1} + w′_{2}))).

So \overline{ι(a)} = ι_{1}(a^{1}) + ι_{2}(a^{2}). We also have, from (4.0.1), that the isomorphism V ≅ E ⊗_{F} W gives rise to an isomorphism of (A(E), B)-bimodules,

(4.1.3)  A(E) ⊗_{E} B ≅ A

and, as in [BK2] (5.3), this restricts to isomorphisms

(4.1.4)  𝔄(E) ⊗_{𝔬_{E}} 𝔅 ≅ 𝔄  and  𝔄(E) ⊗_{𝔬_{E}} 𝔔^{n} ≅ 𝔓^{n}  for n ∈ ℤ

where 𝔄(E) = End_{𝔬_{E}}^{0}{𝔭_{E}^{i} : i ∈ ℤ} is the unique hereditary 𝔬_{F} -order in A(E) normalised by E and 𝔓(E) is its radical.

(4.1.5) Lemma The lattice chain {𝔭_{E}^{i} : i ∈ ℤ} is self-dual with respect to both

alternating forms, g_{1} and g_{2}. So { \atop 𝔄(E)}^{1} = { \atop 𝔄(E)}^{2} = 𝔄(E).

Proof: Consider the form g_{1} (for g_{2} it will be similar and simpler). Then

(𝔭_{E}^{i})^{#} = {x ∈ E : g_{1}(x, 𝔭_{E}^{i}) ⊂ 𝔭_{E}} = {x ∈ E : tr_{E/F} δεx𝔭_{E}^{i} ⊂ 𝔭_{F} }

= {x ∈ E : δεx𝔭_{E}^{i} ⊂ 𝔇^{−1}𝔭_{F} }

= \{ \begin{matrix}ε^{−1}𝔬_{E} & if δ ∉ 𝔭_{E}^{e}𝔇^{−1} \\ ε^{−1}𝔭_{E} & if δ ∈ 𝔭_{E}^{e}𝔇^{−1} \end{matrix}

so {𝔭_{E}^{i} : i ∈ ℤ} is self-dual. The second assertion now follows from (1.1.2). ■ In particular, we will be able to look at the exact sequences of (2.1). First we must check that the tame corestriction found in (2.1.1) will commute with both involutions.

(4.1.6) Lemma Let s_{0} be a tame corestriction on A(E) relative to E/F which commutes with ^{1}. Then it also commutes with ^{2}.

Proof: We have s(a^{2}) = s(εa^{1}ε^{−1}) = εs(a^{1})ε^{−1}, since ε ∈ B(E) = E, so s(a^{2}) =

εs̃(a)^{1}ε^{−1} = s̃(a)^{2}. ■

(4.1.7) Proposition (cf. [BK] (1.3.9)) Let s_{0} be a tame corestriction on A(E) relative to E/F which commutes with the involutions on A(E). Then the map s = s_{0} ⊗ 1_{B} is a tame corestriction on A relative to E/F which commutes with the involution on A.

Proof: By [BK] (1.3.9), s is certainly a tame corestriction on A relative to E/F. Moreover, for a ∈ A(E), b ∈ B, we have

s(\overline{a ⊗ b}) = s(\overline{ι(a)b}) = s(\overline{b} \overline{ι(a)})

= \overline{b}s(ι_{1}(a^{1}) + ι_{2}(a^{2})) = \overline{b}(ι_{1}(s_{0}(a^{1})) + ι_{2}(s_{0}(a^{2})))

= \overline{b}(ι_{1}({ \atop s_{0}(a)}^{1}) + ι_{2}({ \atop s_{0}(a)}^{2})) = \overline{b}(\overline{ι(s_{0}(a))})

= \overline{b}(\overline{s(ι(a))}) = \overline{s(ι(a))b} = \overline{s(ι(a)b)} = \overline{s(a ⊗ b)}. ■

Note then that any tame corestriction s on A relative to E/F which commutes with the involution on A takes the form s_{0} ⊗ 1_{B} as above, since s and s_{0} are uniquely determined up to multiplication by u ∈ 𝔬_{E}^{×} such that u = \overline{u}.

Now let E_{1}/E be a subfield of B such that \overline{E_{1}} = E_{1}. Then, as above, we have two involutions on A(E_{1}), ^{1} and ^{2}, and elements δ_{1}, ε_{1} ∈ E_{1}.

(4.1.8) Lemma Let s′ be a tame corestriction on A(E_{1}) relative to E/F which commutes with ^{1}. Then it also commutes with ^{2}.

Proof: Since ε_{1} ∈ E_{1} ⊂ End_{E}(E_{1}), the proof is the same as for (4.1.5). ■

(4.1.9) Lemma Let s be a tame corestriction on A relative to E/F which commutes with . Then s = s′ ⊗ 1_{B_{1}} for some tame corestriction s′ on A(E_{1}) relative to E/F which commutes with ^{1} and ^{2}.

Proof: As in the proof of [BK] (2.2.8), for s′ as above s′⊗1_{B_{1}} is a tame corestriction on A relative to E/F and, as in (4.1.7), it commutes with the involution. Then, changing s′ by some u ∈ 𝔬_{E}^{×} such that u = \overline{u}, we have s = s′ ⊗ 1_{B_{1}}. ■

Now let β ∈ A_{−} be such that E = F [β] is a field and put B = End_{E}(V) as usual. Let c ∈ B_{−} be such that E_{1} = E[c] is a field. Let ^{1}, ^{2} be the involutions on

A(E_{1}) = End_{F} (E_{1}) as above; recall that ã^{1}^{2} = ε_{1}aε_{1}^{−1}, where ε_{1} satisfies \overline{ε}_{1} = ε_{1}

and ε^{n_{1}} ∈ F for some n_{1} ∈ ℤ, (n_{1}, p) = 1. Then, writing σ_{1}(a) = ã^{1}^{2}, we have σ_{1}^{n_{1}} ≡ 1. Set Γ_{1} = ⟨σ_{1}⟩, the cyclic group generated by σ_{1}. The extension E/F is a subfield of A(E_{1}) so we may consider B(E_{1}), the centraliser of E in A(E_{1}). Then the tame corestriction s′ on A(E_{1}) relative to E/F given by (2.1.1) commutes with σ_{1} by (4.1.8), as does a_{β} since σ_{1}(β) = β. Recall that 𝔄(E_{1}) is the unique hereditary order in A(E_{1}) normalised by E_{1}, with Jacobson radical 𝔓(E_{1}). We put 𝔅(E_{1}) = 𝔄(E_{1}) ∩ B(E_{1}) and 𝔔(E_{1}) = 𝔓(E_{1}) ∩ B(E_{1}), the Jacobson radical of the hereditary 𝔬_{E}-order 𝔅(E_{1}). Put 𝔑_{k} = 𝔑_{k}(β, 𝔄(E_{1})). Then the exact sequence (from [BK] (1.4.10))

0 → 𝔔(E_{1})^{m}𝔑_{k}/𝔔(E_{1})^{m} → 𝔓(E_{1})^{m+k} → 𝔔(E_{1})^{m+k} → 0, k ≥ k_{0}(β, 𝔄(E_{1})),

is Γ_{1}-equivariant. So we have a long exact sequence in cohomology

0 → (𝔔(E_{1})^{m}𝔑_{k}/𝔔(E_{1})^{m})^{Γ_{1}} → (𝔓(E_{1})^{m+k})^{Γ_{1}} → (𝔔(E_{1})^{m+k})^{Γ_{1}}
→ H^{1}(Γ_{1},𝔔(E_{1})^{m}𝔑_{k}/𝔔(E_{1})^{m}) → H^{1}(Γ_{1}, 𝔓(E_{1})^{m+k}) → …

Since |Γ_{1}| = n_{1} and (n_{1}, p) = 1, we have H^{1}(Γ_{1}, 𝔔(E_{1})^{m}𝔑_{k}/𝔔(E_{1})^{m}) = 0, by [Bn] (10.2), and hence an exact sequence

(4.1.10)  (𝔓(E_{1})^{m+k})^{Γ_{1}} → (𝔔(E_{1})^{m+k})^{Γ_{1}} → 0

in A^{Γ_{1}} = {a ∈ A(E_{1}) : a^{1} = a^{2}}. Set

A_{−}^{Γ_{1}} = {a ∈ A^{Γ_{1}} : a^{1} = −a}
A_{+}^{Γ_{1}} = {a ∈ A^{Γ_{1}} : a^{1} = a}.

Then A^{Γ_{1}} = A_{−}^{Γ_{1}} ⊕ A_{+}^{Γ_{1}} and (4.1.10) splits into two sequences to give

(4.1.11)  (𝔓(E_{1})^{m+k})_{−}^{Γ_{1}} → (𝔔(E_{1})^{m+k})_{−}^{Γ_{1}} → 0

Note that c ∈ E_{1} ↪ A(E_{1}) satisfies c^{1} = c^{2} = \overline{c} = −c so c ∈ A_{−}^{Γ_{1}}.

(4.2) Refinements

Let [𝔄, n, r, β] be a skew simple stratum in A. As usual we write E = F [β], B for the A-centraliser of E, 𝔅 = 𝔄 ∩ B, and 𝔔 = 𝔓 ∩ B, the Jacobson radical of 𝔅. We also fix a tame corestriction s on A relative to E/F such that s commutes with the involution on A, as in (2.1.1). As in [BK] (2.2), a refinement of our given stratum is a stratum of the form [𝔄, n, r − 1, β + b], where b ∈ 𝔓^{−r}. We can then form the derived stratum [𝔅, r, r − 1, s(b)], which is stratum in B. A skew refinement is a refinement where we also have b + \overline{b} = 0. Then the derived stratum is a skew stratum.

(4.2.1) Lemma (cf. [BK] (2.2.1)) Let [𝔄, n, r, β] be a skew simple stratum in A. Let [𝔄, n, r − 1, β + b], [𝔄, n, r − 1, β + b′] be skew refinements of it. Write k = k_{0}(β, 𝔄), 𝔑 = 𝔑_{k}(β, 𝔄). Then the derived strata [𝔅, r, r−1, s(b)], [𝔅, r, r−1, s(b′)] in B are equivalent if and only if there exists y ∈ (𝔔^{−(r+k)}𝔑)_{−} such that

[𝔄, n, r − 1, C(y)^{−1}(β + b)C(y)] ∼ [𝔄, n, r − 1, β + b′].

Proof: We have [𝔅, r, r −1, s(b)] ∼ [𝔅, r, r −1, s(b′)] if and only if s(b−b′) ∈ 𝔔_{−}^{1−r}. By (2.1.5), this happens if and only if there exists y ∈ (𝔔^{−(r+k)}𝔑)_{−} such that b − b′ ≡ a_{β}(y) (mod 𝔓_{−}^{1−r}). Then, since −(r + k) ≥ 1, we have

C(y)^{−1}(β + b)C(y) ≡ β + b + a_{β}(y) (mod 𝔓^{1−r}),

since (C(y) − y − 1) ∈ 𝔔^{−2(r+k)}𝔑 so a_{β}(C(y) − y − 1) ∈ 𝔓^{−(r+k)−r} ⊂ 𝔓^{1−r}. ■

(4.2.2) Proposition (cf. [BK] (2.2.8)) Let [𝔄, n, r, β] be a skew simple stratum in A. Let B be the A-centraliser of E = F [β] and 𝔅 = 𝔄 ∩ B. Let b ∈ A with ν_{𝔄}(b) = −r and let s be a tame corestriction on A relative to E/F which commutes with the involution on A. Suppose that the stratum [𝔅, r, r − 1, s(b)] is equivalent to a skew simple stratum [𝔅, r, r − 1, c] in B. Then [𝔄, n, r − 1, β + b] is equivalent to a skew simple stratum [𝔄, n, r − 1, β_{1}]. Moreover, k_{0}(β_{1}, 𝔄) = max{k_{0}(β, 𝔄), k_{0}(c, 𝔅)}.

Proof: We follow the proof of [BK] (2.2.8). Let E_{1} = F [β, c], so E_{1} = \overline{E}_{1}, and let B_{1} denote the A-centraliser of the field E_{1}, 𝔅_{1} = 𝔄 ∩ B_{1}. Then we have a decomposition

𝔄 = 𝔄(E_{1}) ⊗_{𝔬_{E_{1}}} 𝔅_{1}

as in (4.1.4). By (4.1.9) our tame corestriction s on A relative to E/F takes the form s′ ⊗ 1_{B_{1}} for some tame corestriction s′ on A(E_{1}) relative to E/F, which commutes with the involutions on A(E_{1}). Let ℭ denote the hereditary 𝔬_{E}-order 𝔄(E_{1}) ∩ End_{E}(E_{1}). Then we have decompositions 𝔄(E_{1}) = 𝔄(E) ⊗_{𝔬_{E}} ℭ and 𝔅 = ℭ ⊗_{𝔬_{E_{1}}} 𝔅_{1}. Put e_{1} = e(𝔅_{1}|𝔬_{E_{1}}). Then the strata [𝔄(E_{1}), n/e_{1}, r/e_{1}, β], [ℭ, r/e_{1}, r/e_{1} − 1, c] are simple and k_{0}(β, 𝔄(E_{1})) = k_{0}(β, 𝔄)/e_{1} by [BK] (1.4.13). Now c ∈ A(E_{1})_{−}^{Γ_{1}} so, by (4.1.11), choose b_{1} ∈ 𝔄(E_{1})_{−}^{Γ_{1}} with ν_{𝔄(E_{1})}(b_{1}) = −r/e_{1} such that s′(b_{1}) = c. By [BK] (2.2.3), the stratum [𝔄(E_{1}), n/e_{1}, r/e_{1} − 1, β + b_{1}] is simple. Consider the stratum [𝔄, n, r − 1, ι(β + b_{1})]; this is a skew refinement of [𝔄, n, r, β], since

\overline{ι(β + b_{1})} = ι_{1}({ \atop β + b_{1}}^{1}) + ι_{2}({ \atop β + b_{1}}^{2}) = ι_{1}(−(β + b_{1})) + ι_{2}(−(β + b_{1})) = −ι(β + b_{1}),

and

[𝔅, r, r − 1, ι(s(b_{1}))] = [𝔅, r, r − 1, c]

as in [BK] (2.2.8). Then, by(4.2.1), we can find a conjugate of [𝔄, n, r−1, ι(β+b_{1})] equivalent to [𝔄, n, r − 1, β + b]. The rest of the proof is exactly the same as [BK] (2.2.8). ■

(4.2.3) Remark If E_{1} = E[c] is a maximal subfield of B then we can use [BK] (2.2.2) to deduce that [𝔅, r, r − 1, s(b)] is itself simple, with E[s(b)] a maximal subfield of B, and [BK] (2.2.3) to conclude that [𝔄, n, r − 1, β + b] is itself simple.

STANDARD FORM

The main result of this chapter is the following (cf. [BK] (2.4.1)), which we prove in (5.4): Let [𝔄, n, n − 1, β] be a skew pure stratum in A. Then it is equivalent to a skew simple stratum. This then allows us to use induction “along β” (i.e. on k_{0}(β, 𝔄)) as in [BK] to define simple characters and prove results concerning them (see chapter 6). In order to prove this result we need several results in the generality of unitary groups attached to a hermitian form so h will be:

ϵ-hermitian in sections (5.1–2); skew-hermitian in section (5.3).

(5.1) Lifting

In this section h : V × V → F will be an ϵ-hermitian form. Let 𝔄 be a maximal order associated to a self-dual lattice chain 𝔏 = {L_{i} : i ∈ ℤ} in V and we put 𝔓 =rad 𝔄, as usual. Let V = 𝔏(V) be the standard residual subspace of 𝔏 and put A = End_{k_{F}}^{0} V (≃ 𝔄/𝔭_{F} 𝔄 by (3.3.2)). For each k ∈ ℤ, V inherits from V a nondegenerate ϵ-sesquilinear form h_{k} : V_{k} × V_{d−k−1} → k_{F} as in (3.1.6), where d is the unique integer such that L_{k}^{#} = L_{d−k} for all k ∈ ℤ. Then we have an adjoint involution on A so we can define A_{−}, A_{+} to be the skew elements, the symmetric elements respectively. In fact A_{±} ≃ 𝔄_{±}/(𝔭_{F} 𝔄)_{±} by (1.2.5).

(5.1.1) Lemma Let η = ± be a sign. The natural map 𝔄_{η} → A_{η} is surjective.

For f(X) ∈ F [X], f(X) = a_{n}X^{n} + ⋯ + a_{0}, define \overline{f}(X) ∈ F [X] to be the polynomial \overline{f}(X) = \overline{a_{n}}X^{n} + ⋯ + \overline{a_{0}}. Also, for f(X) ∈ 𝔬_{F} [X], f(X) = a_{n}X^{n} + ⋯ + a_{0}, define f̃(X) ∈ k_{F} [X] to be the polynomial obtained by reducing the coefficients modulo 𝔭_{F}.

(5.1.2) Lemma Let γ, γ′ ∈ U(𝔄). Suppose γ, γ′ have the same minimal polynomial, that the reduction modulo 𝔭_{F} of this minimal polynomial is irreducible and that γ ≡ γ′(mod 𝔓^{n}). Then there exists v ∈ U^{n}(𝔄) such that γ = vγ′v^{−1}.

Proof: The elements γ, γ′ generate unramified extensions over F; in particular, they are minimal over F. Write B_{γ} for the centralizer of γ in A and 𝔅_{γ} = 𝔄 ∩ B_{γ}. By [BH1] (1.6), there exists u ∈ U(𝔄) such that γ = uγ′u^{−1}. Since γ − γ′ ∈ 𝔓^{n}, we have γu−uγ ∈ 𝔓^{n}. i.e. u ∈ 𝔑_{n}(γ, 𝔄). But γ is minimal so 𝔑_{n}(γ, 𝔄) = 𝔅+𝔓^{n}. Hence u = wv, for w ∈ U(𝔅) and v ∈ U^{n}(𝔄), and γ = vγ′v^{−1} as required. ■

We fix an irreducible polynomial ϕ(X) ∈ k_{F} [X] of degree N, ϕ(X) ≠ X. The following is a Hensel’s Lemma-type result.

(5.1.3) Proposition Let η = ± be a sign and let ϱ ∈ A_{η}. Suppose ϱ has irreducible characteristic polynomial ϕ(X) ∈ k_{F} [X]. Let Φ(X) ∈ 𝔬_{F} [X] be a polynomial such that Φ̃(X) = ϕ(X) and such that \overline{Φ}(ηX) = Φ(X). Then there exists γ ∈ 𝔄_{η} such that γ̃ = ϱ and Φ(γ) = 0.

Proof: By Hensel’s Lemma, there exists γ_{1} ∈ U(𝔄) such that γ̃_{1} = ϱ and Φ(γ_{1}) = 0. Also, \overline{γ_{1}} ≡ ηγ_{1} (mod 𝔓) since γ̃_{1} ∈ A_{η}. For n ≥ 1, we will find, by induction, γ_{n} ∈ U(𝔄) such that γ̃_{n} = ϱ, γ_{n} is conjugate to γ_{1} (in particular, Φ(γ_{n}) = 0), \overline{γ_{n}} ≡ ηγ_{n} (mod 𝔓^{n}) and γ_{n} ≡ γ_{n−1} (mod 𝔓^{n−1}). Granting this, let γ be the limit of the sequence \{γ_{n}\}. Then γ̃ = ϱ, \overline{γ} = ηγ and Φ(γ) = 0 as required. We have already found γ_{1} so assume we have found γ_{n} as required, for some n ≥ 1.

We have that Φ(γ_{n}) = 0 and also Φ(η\overline{γ_{n}}) = \overline{\overline{Φ}(ηγ_{n})} = \overline{Φ(γ_{n})} = 0 so γ_{n} and η\overline{γ_{n}} are conjugate. Moreover, γ_{n} ≡ η\overline{γ_{n}} (mod 𝔓^{n}) hence η\overline{γ_{n}} = u_{n}γ_{n}u_{n}^{−1}, for some u_{n} ∈ U^{n}(𝔄), by (5.1.2). Write u_{n} = 1 + x_{n}, x_{n} ∈ 𝔓^{n}, so that η\overline{γ_{n}} ≡ γ_{n} + x_{n}γ_{n} − γ_{n}x_{n} (mod 𝔓^{2n}). We put y_{n} = \frac{x_{n}+\overline{x_{n}}}{2} and v_{n} = 1 + y_{n} ∈ U^{n}(𝔄); then v_{n} = \overline{v}_{n} and η\overline{γ_{n}} ≡ v_{n}γ_{n}v_{n}^{−1} (mod 𝔓^{2n}). By (1.2.4)(ii), there exists w_{n} ∈ U^{n}(𝔄) such that v_{n} = \overline{w_{n}}w_{n}. Then we put γ_{n+1} = w_{n}γ_{n}w_{n}^{−1}. ■

We finish this section with some results which will prove important later.

(5.1.4) Lemma Let η = ± be a sign. Let γ, γ′ ∈ 𝔄_{η}^{×} be such that γ ≡ γ′ mod 𝔓^{r}, r ≥ 1. Then there exists u ∈ U^{r}(𝔄) such that γ ≡ uγ′\overline{u} (mod 𝔓^{2r}).

Proof: Let a = γ − γ′ ∈ 𝔓^{r} and put u = (1 + \frac{1}{2}aγ′^{−1}). ■

(5.1.5) Proposition With hypotheses as in (5.1.4), there exists u ∈ U^{r}(𝔄) such that γ = uγ′\overline{u}.

Proof: We find, by induction, u_{n} ∈ U^{r}(𝔄) such that

(5.1.6)  γ ≡ u_{n}γ′\overline{u_{n}} (mod 𝔓^{rn})

and u_{n+1} ≡ u_{n} (mod 𝔓^{rn}). Then {u_{n}} converges to u ∈ U^{r}(𝔄) such that γ = uγ′\overline{u}. We can take u_{1} = 1 so suppose we have found u_{n} ∈ U^{r}(𝔄) satisfying (5.1.6). Then γ and u_{n}γ′\overline{u_{n}} satisfy the conditions of (5.1.4) (with r replaced by rn). So there exists u′ ∈ U^{rn}(𝔄) such that γ ≡ u′u_{n}γ′\overline{u_{n}}\overline{u′} mod 𝔓^{2rn} and, in particular, this congruence holds (mod 𝔓^{(r+1)n}) since r ≥ 1. So we set u_{n+1} = u′u_{n}. ■

(5.1.7) Lemma Let Φ(X) ∈ 𝔬_{F} [X] be an irreducible polynomial of degree N such that \overline{Φ}(X) = Φ(X) and such that its reduction modulo 𝔭_{F}, Φ̃ is also irreducible. Let γ ∈ U(𝔄) have characteristic polynomial Φ(X). Let ∂ : 𝔄 → 𝔄 be the map given by ∂(x) = x\overline{γ} − γx, for x ∈ 𝔄. Suppose y ∈ Im (∂) ∩ 𝔓_{−}; then there exists ξ ∈ 𝔓_{+} such that y = ∂(ξ).

Proof: We need only show that for y ∈ Im (∂) ∩ 𝔓 there exists ξ ∈ 𝔓 such that y = ∂(ξ), since ∂ maps 𝔄_{+} to 𝔄_{−} and vice versa. Let E/F be a splitting field for Φ(X) and let ∂_{E} : 𝔄 ⊗_{𝔬_{F}} 𝔬_{E} → 𝔄 ⊗_{𝔬_{F}} 𝔬_{E} be the extended map. The result then holds for ∂ if and only if it holds for ∂_{E}. Let λ_{i}, i = 1, ..., n, be the eigenvalues of γ in E; these are also the eigenvalues of \overline{γ}. Further, the λ_{i} are distinct modulo 𝔭_{F} since γ generates a maximal extension of F in A. Hence the non-zero eigenvalues of ∂_{E}, which are λ_{i} −λ_{j} for 1 ≤ i ≠ j ≤ n, are ≢ 0 (mod 𝔭_{F}) and the result follows. ■

(5.1.8) Proposition Let α, β ∈ U(𝔄) ∩ A_{+} be such that γ := αβ generates a maximal (unramified) extension of F in A. Let ϕ(X) ∈ k_{F} [X] be the reduction modulo 𝔭_{F} of the characteristic polynomial of γ and suppose that \overline{ϕ}(X) = ϕ(X). Let Φ(X) ∈ 𝔬_{F} [X] be such that \overline{Φ}(X) = Φ(X) and such that Φ̃(X) = ϕ(X). Let γ′ ∈ U(𝔄) be such that γ′ ≡ γ (mod 𝔭_{F}) and such that Φ(γ′) = 0. (There exists such a γ′ by Hensel’s Lemma.) Then there exist α′ ∈ α + 𝔓_{+}, β′ ∈ β + 𝔓_{+} such that γ′ = α′β′.

Proof: Write γ′ = γ + δ with δ ∈ 𝔓. Note that α\overline{γ} = αβα = γα and also that γ′ generates a maximal unramified extension of F. We have α\overline{γ′} − γ′α = α\overline{δ} − δα ∈ 𝔓_{−} so, by (5.1.7), there exists ξ ∈ 𝔓_{+} such that ξ\overline{γ′} − γ′ξ = α\overline{γ′} − γ′α. Then we put α′ = α − ξ and β′ = α′^{−1}γ′. ■

(5.2) Jordan form

Let V be an N-dimensional k_{F} -vector space, equipped with a nondegenerate ϵhermitian form h : V × V → k_{F}. We put A = End_{k_{F}} V. Recall also that k_{0} is the residue class field of F_{0}. For W a subspace of V we define its orthogonal complement to be

W^{⊥} = {v ∈ V : h(v, W) ≡ 0}.

Then, by the nondegeneracy of h, W^{⊥⊥} = W.

(5.2.1) Lemma Let W be a totally isotropic residual subspace of 𝔏. Then h induces an ϵ-hermitian form h′ on W^{⊥}/W which is nondegenerate.

Proof: For v ∈ W^{⊥}, v′ ∈ W^{⊥}, we define h′ by h′(v+W, v′+W) = h(v, v′). This is well defined since v, v′ ∈ W^{⊥} and W is totally isotropic. Suppose now there exists v ∈ W^{⊥} such that h′(v + W, W^{⊥}/W) ≡ 0; then h(v, W^{⊥}) ≡ 0 so v ∈ W^{⊥⊥} = W as required. ■

Let y ∈ A be such that y = η\overline{y}, η = ±. Let the characteristic polynomial of y be ϕ(X)^{t} ∈ k_{F} [X], with ϕ(X) irreducible of degree N/t. Let y = y_{ss} + y_{np} be the Jordan decomposition of y. Then, by uniqueness of the decomposition, we have y_{ss} = η\overline{y}_{ss} and y_{np} = η\overline{y}_{np}. Then y_{ss} is elliptic with irreducible minimal polynomial ϕ(X) and 𝔩 = k_{F} [y_{ss}] is a field with y_{np}, y ∈ End_{𝔩}V. We also set 𝔩_{0} = \{l ∈ 𝔩 : \overline{l} = l\}; note that 𝔩 = 𝔩_{0} if, and only if, k_{F} = k_{0} and η = +. As tr_{𝔩/k_{F}} and h are nondegenerate, there exists a nondegenerate form f : V × V → 𝔩 such that, for v, v′ ∈ V,

h(lv, v′) = tr_{𝔩/k_{F}} (l f(v, v′)), for all l ∈ 𝔩.

The adjoint involution on End_{𝔩}V defined by f is precisely that defined by h. If 𝔩 = 𝔩_{0} then f is ϵ-bilinear while if 𝔩 ≠ 𝔩_{0} then f is ϵ-hermitian. In either case, V = V^{sp} ⊥ V^{an}, where f|_{V^{an}×V^{an}} is anisotropic and f|_{V^{sp}×V^{sp}} is split. In fact, if 𝔩 = 𝔩_{0} and ϵ = −1 then f is split so dim_{𝔩}V^{an} = 0; if 𝔩 = 𝔩_{0} and ϵ = +1 then dim_{𝔩}V^{an} ≤ 2; while if 𝔩 ≠ 𝔩_{0} then N_{𝔩/𝔩_{0}} is surjective so dim_{𝔩}V^{an} ≤ 1.

(5.2.2) Lemma (i) With notation as above, there exists a flag of 𝔩-subspaces of V, V = V^{0} ⊃ V^{1} ⊃ ⋯ ⊃ V^{s} = 0 such that V^{q}^{⊥} = V^{s−q} and y_{np}V^{q−1} ⊂ V^{q} and such that, moreover, dim_{𝔩}V^{q}/V^{q+1} = 1 for all q (so s = t) except in the case 𝔩 = 𝔩_{0}, ϵ = +1 and t even when, possibly, dim_{𝔩}V^{t/2−1}/V^{t/2} = 2 and s = t − 1. (ii) Given such a flag, there exist 𝔩-subspaces W^{q}, 1 ≤ q ≤ s, of V such that V^{q−1} = ⨁_{j=q}^{s} W^{j} and W^{q}^{⊥} = ⨁_{j≠s−q+1} W^{j} for 1 ≤ q ≤ s.

Proof: (i) We proceed by induction on t, the cases t = 0 and t = 1 being obvious. So assume t ≥ 2 and that we have the result for t − 2. Since y_{np} is nilpotent there exists v ∈ V such that y_{np}v = 0. Then for v′ ∈ V, f(y_{np}v′, v) = ηf(v′, y_{np}v) = 0 so y_{np}V ⊂ ⟨v⟩_{𝔩}^{⊥}. Suppose first there exists such a v such that f(v, v) = 0; then let V^{s−1} = ⟨v⟩_{𝔩}, V^{1} = ⟨v⟩_{𝔩}^{⊥}. Then, replacing V with V^{1}/V^{s−1} we have, by the inductive hypothesis, a flag

V^{1}/V^{s−1} = V′^{0} ⊃ V′^{1} ⊃ ⋯ ⊃ V′^{s−2} = 0

which we can lift to a flag as required. (Note that the form induced by f on V^{1}/V^{s−1} is indeed nondegenerate by (5.2.1).) Now suppose that for all v ∈ V such that y_{np}v = 0 we have f(v, v) ≠ 0. Since y_{np}V ⊂ ⟨v⟩_{𝔩}^{⊥} and t ≥ 2, there exists v′ ∈ ⟨v⟩_{𝔩}^{⊥} such that y_{np}v′ = 0. We deal first with the case 𝔩 ≠ 𝔩_{0}. Since N_{𝔩/𝔩_{0}}𝔩 = 𝔩_{0}, we may assume f(v, v) = −f(v′, v′) = 1. But then y_{np}(v + v′) = 0 and f(v + v′, v + v′) = 0, contradicting our assumption. Suppose now 𝔩 = 𝔩_{0}; then, since f(v, v) ≠ 0, we must have ϵ = +1. If t = 2 then we put V_{0} = V, V_{1} = 0 and we are done. Otherwise we have, as before, that y_{np}V ⊂ ⟨v, v′⟩_{𝔩}^{⊥} so there exists v′′ ∈ ⟨v, v′⟩_{𝔩}^{⊥} such that y_{np}v′′ = 0. But then f(v′′, v′′) = −l^{2}f(v, v) − l′^{2}f(v′, v′) for some l, l′ ∈ 𝔩 so, putting w = lv + l′v′ + v′ we have f(w, w) = 0 but y_{np}w = 0, contradicting the assumption. (ii) We proceed by induction on s. If s = 1 we are done, putting W^{1} = V so we assume s ≥ 2. Choose a non-zero v_{1} ∈ V^{s−1} and w ∈ V such that f(v_{1}, w) = 1. Put w_{1} = w − \frac{1}{2}f(w, w)v_{1}; then we put W^{1} = ⟨w_{1}⟩_{𝔩} and W^{s} = ⟨v_{1}⟩_{𝔩}. Suppose now that we have chosen W^{j} for 1 ≤ j ≤ q − 1, s − q + 2 ≤ j ≤ s. If s − q + 2 = q then we are done. Otherwise, choose v ∈ V^{s−q−1} such that v ∉ V^{s−q}. We then put v_{q} = v − ∑_{j=1}^{q−1} f(v, w_{j})v_{j} and set W^{s−q+1} = ⟨v_{q}⟩_{𝔩}. If s − q + 1 = q then we are done. Otherwise, choose w ∈ V^{q} such that f(v_{q}, w) = 1. Put w′ = w − \frac{1}{2}f(w, w)v_{q} and w_{q} = w′ − ∑_{j=1}^{q−1} f(w′, w_{j})v_{j}. Then we put W^{q} = ⟨w_{q}⟩_{𝔩}. ■

Before continuing, we examine the case dim_{𝔩}W^{t/2−1} = 2 of (5.2.2) in more detail. So k_{F} = k_{0}, η = +, ϵ = +1. Choose u′, u′′ ∈ W^{t/2−1} such that f(u′, u′′) = 0 and put U′ = ⟨u′⟩_{𝔩}, U′′ = ⟨u′′⟩_{𝔩}; then we have

U′^{⊥} = U′′ ⊕ ⨁_{i≠ \frac{t}{2} −1} W^{i},  y_{np}U′ ⊂ ⨁_{i> \frac{t}{2} −1} W^{i};

U′′^{⊥} = U′ ⊕ ⨁_{i≠ \frac{t}{2} −1} W^{i},  y_{np}U′′ ⊂ ⨁_{i> \frac{t}{2} −1} W^{i}.

Now suppose V = V_{1} ⊕ V_{2}, where V_{1}, V_{2} are maximal isotropic subspaces of V; in particular, f is split. Suppose further that yV_{i} ⊂ V_{i}, for i = 1, 2; then y_{ss} and y_{np} also fix V_{i}, i = 1, 2. Also, we can write t = 2t′, for t′ ∈ ℤ. Let b ∈ A be such that bV_{1} = V_{2}, bV_{2} = 0, b = −\overline{b}, and b commutes with y. Then b also commutes with y_{ss} and y_{np} so b ∈ End_{𝔩}V.

(5.2.3) Lemma (i) With notation as above, there exist flags of 𝔩-subspaces

V_{1} = V_{1}^{0} ⊃ V_{1}^{1} ⊃ ⋯ ⊃ V_{1}^{s} = 0
V_{2} = V_{2}^{0} ⊃ V_{2}^{1} ⊃ ⋯ ⊃ V_{2}^{s} = 0

such that (V_{1}^{q} ⊕ V_{2})^{⊥} = V_{2}^{s−q}, bV_{1}^{q} = V_{2}^{q} and y_{np}V_{1}^{q−1} ⊂ V_{1}^{q} for 1 ≤ q ≤ s and such that, moreover, dim_{𝔩}V_{1}^{q}/V_{1}^{q+1} = 1 for all q (so s = t′) except in the case 𝔩 = 𝔩_{0}, ϵ = −1, and t′ even when, possibly dim_{𝔩}V_{1}^{t′/2−1}/V_{1}^{t′/2} = 2 and s = t′ − 1. (ii) Given such flags, there exist residual 𝔩-subspaces W_{i}^{q}, 1 ≤ q ≤ s, of V_{i}, i = 1, 2, such that V_{i}^{q−1} = ⨁_{j=q}^{s} W_{i}^{j}, W_{1}^{q}^{⊥} = V_{1} ⊕ ⨁_{j≠s−q+1} W_{2}^{j} and W_{2}^{q} = bW_{1}^{q} for 1 ≤ q ≤ s, i = 1, 2.

Proof: We define a nondegenerate −ϵ-hermitian form f′ on V_{1} by f′(v, v′) = f(v, bv′). Then we apply (5.2.2) to (V_{1}, f′) and put V_{2}^{q} = bV_{1}^{q}. ■

We now examine the case dim_{𝔩}W_{1}^{t′/2−1} = 2 of (5.2.3) in more detail. So k_{F} = k_{0}, η = +, ϵ = −1.

Choose u′, u′′ ∈ W_{1}^{t′/2−1} such that f(bu′, u′′) = 0 and put U′ = ⟨u′⟩_{𝔩}, U′′ = ⟨u′′⟩_{𝔩}; then we have

U′^{⊥} = V_{1} ⊕ bU′′ ⊕ ⨁_{i≠ \frac{t′}{2} −1} W_{2}^{i},  y_{np}U′ ⊂ ⨁_{i> \frac{t′}{2} −1} W_{1}^{i};

U′′^{⊥} = V_{1} ⊕ bU′ ⊕ ⨁_{i≠ \frac{t′}{2} −1} W_{2}^{i},  y_{np}U′′ ⊂ ⨁_{i> \frac{t′}{2} −1} W_{1}^{i}.

(5.3) Non-split fundamental strata

In this section, h will be a skew-hermitian form. Let [𝔄, m, m − 1, b] be a stratum in A and set

y = y_{b} = π_{F}^{m/g}b^{e/g} ∈ 𝔄

where e = e(𝔄) is the 𝔬_{F} -period of 𝔄 and g = (m, e). Let Φ(X) ∈ 𝔬_{F} [X] be the characteristic polynomial of y and let ϕ_{b}(X) ∈ k_{F} [X] be its reduction modulo 𝔭_{F}. This polynomial ϕ_{b}(X) depends only on the equivalence class of the stratum, since it is also the characteristic polynomial of y considered as an element of 𝔄/𝔓, and is called the characteristic polynomial of the stratum.

(5.3.1) Definition Let [𝔄, m, m−1, b] be a stratum in A, ϕ_{b}(X) its characteristic polynomial.

(i) The stratum is called fundamental if ϕ_{b}(X) is not a power of X. (ii) The stratum is called split if ϕ_{b}(X) has (at least) two distinct prime factors.

In particular, a non-split fundamental stratum has characteristic polynomial ϕ(X)^{t} for ϕ(X) ∈ k_{F} [X] an irreducible monic polynomial of degree N/t, ϕ(X) ≠ X.

Now let [𝔄, m, m − 1, b] be a skew non-split fundamental stratum in A. We recall that π_{F} is a fixed uniformizer of F such that π_{F} = \overline{π_{F}} if F/F_{0} is unramified, while π_{F} = −\overline{π_{F}} if F/F_{0} is ramified. In particular, we have y = η\overline{y}, for η = ±, and \overline{ϕ}(ηX) = ϕ(X).

The main result of this section is the following:

(5.3.2) Theorem (cf. [BK] (2.3.4)) Let [𝔄, m, m − 1, b] be a skew non-split fundamental stratum in A. Then there exists a skew simple stratum [𝔄′, m′, m′ − 1, b′] in A such that

b + 𝔓^{1−m} ⊂ b′ + 𝔓′^{1−m′}.

Moreover, m′/e(𝔄′) = m/e(𝔄) and the lattice chain defining 𝔄′ contains that which defines 𝔄. In particular, [𝔄′, m′, m′ − 1, b] is a skew stratum and it is equivalent to [𝔄′, m′, m′ − 1, b′].

Proof: Put e = e(𝔄), g = (m, e) and y = π_{F}^{m/g}b^{e/g}, as above; so \overline{y} = ηy, η = ±. Let ϕ(X) ∈ k_{F} [X] be the unique (monic) irreducible factor of the characteristic polynomial of the stratum [𝔄, n, n − 1, b]. Let Φ(X) ∈ 𝔬_{F} [X] be such that Φ̃(X) = ϕ(X) and such that \overline{Φ}(ηX) = Φ(X). Then we will show that we can choose b′ such that the element y′ = π_{F}^{m/g}b′^{e/g} has minimal polynomial Φ(X). We now reduce to the case where the stratum takes a standard form (see (5.3.10) below). Let 𝔏 = {L_{k} : k ∈ ℤ} be the (self-dual) lattice chain associated to 𝔄 and, after renumbering, let d = 0 or −1 be the integer such that L_{k}^{#} = L_{d−k} for all k ∈ ℤ, given by (1.1.2). We define residual subspaces W^{(i)} of 𝔏, 0 ≤ i ≤ g − 1, by

W_{k}^{(i)} = \{ \begin{matrix}L_{k}/L_{k+1} & if k ≡ i (mod g) \\ 0 & otherwise \end{matrix}

Let f = [\frac{g+1−d}{2}]; for 1 ≤ j ≤ f, set

V^{(j)} = W^{(j+d−1)} + W^{(g−j)}

Note that if 2j ≢ 1−d (mod g) then the sum is direct, while if 2j ≡ 1−d (mod g) then V^{(j)} = W^{(j+d−1)}. Set

g_{j} = \{ \begin{matrix}1 & if 2j ≡ 1 − d (mod g) \\ 2 & otherwise. \end{matrix}

(5.3.3) Lemma The residual subspaces V^{(j)} are nondegenerate.

Proof: This is identical to (3.2.3). ■

The residual subspaces V^{(j)} are clearly orthogonal and L_{k}/L_{k+1} = V_{k}^{(1)} ⊕⋯⊕V_{k}^{(f)} for all k so we can apply (3.1.9) to get an orthogonal decomposition

V = V^{(1)} ⊥ ⋯ ⊥ V^{(f)}

such that V^{(j)} = 𝔏(V^{(j)}) for 1 ≤ j ≤ f. Then h decomposes as h = h_{1} ⊥ ⋯ ⊥ h_{f}, where h_{j} is a nondegenerate ϵ-hermitian form on V^{(j)}, 1 ≤ j ≤ f.

Write L_{k}^{(j)} = L_{k} ∩ V^{(j)} for k ∈ ℤ, 1 ≤ j ≤ f.

(5.3.4) Lemma The set 𝔏^{(j)} = {L_{k}^{(j)} : k ∈ ℤ} is a self-dual lattice chain in V^{(j)}, 1 ≤ j ≤ f. Moreover, the 𝔬_{F} -period e_{j} = e(𝔏^{(j)}) is \frac{e}{g} g_{j}.

Proof: This is identical to (3.2.4). ■ Putting A^{(ij)} = Hom(V^{(j)}, V^{(i)}), we get a “block decomposition”:

A = ∐_{1≤i,j≤f} A^{(ij)}.

Write 1^{(j)} for the projection V → V^{(j)} with kernel ∐_{i≠j} V^{(i)} so that 1^{(j)} is in fact the identity element of the algebra A^{(jj)} and A^{(ij)} = 1^{(i)}.A.1^{(j)}. Then 1^{(j)}L_{k} = L_{k}^{(j)} ⊂ L_{k} for all k ∈ ℤ so we have 1^{(j)} ∈ 𝔄, 1 ≤ j ≤ f. We denote ^{j} the adjoint involution on A^{(jj)} induced by the form h_{j}, 1 ≤ j ≤ f. Then set A_{−}^{(jj)} = \{x ∈ A^{(jj)} : \overline{x}^{j} = −x\}.

(5.3.5) Lemma Write b = ∑ b_{ij} with b_{ij} ∈ A^{(ij)}. Then

(i) b_{jj} ∈ A_{−}^{(jj)} for 1 ≤ j ≤ f; (ii) 𝔓^{−m} ∩ A^{(ij)} = 𝔓^{1−m} ∩ A^{(ij)} for i ≠ j; (iii) b ≡ ∑ b_{jj} (mod 𝔓^{1−m}).

Proof: (i) Let w, w′ ∈ V^{(j)}. Then we have

h_{j}(b_{jj}w, w′) = h(bw, w′) = h(w, −bw′) = h_{j}(w, −b_{jj}w′)

so \overline{b_{jj}}^{j} = −b_{jj}.

(ii) Let x ∈ 𝔓^{−m} ∩ A^{(ij)}, i ≠ j so xL_{k}^{(j)} ⊂ L_{k−m}^{(i)}. Note that

L_{k}^{(j)} ≠ L_{k+1}^{(j)} if and only if L_{k−m}^{(j)} ≠ L_{k−m+1}^{(j)}
and  L_{k}^{(j)} ≠ L_{k+1}^{(j)} implies L_{k}^{(i)} = L_{k+1}^{(i)}

for k ∈ ℤ. Then if L_{k}^{(j)} ≠ L_{k+1}^{(j)} we have x(L_{k}^{(j)}) ⊂ L_{k−m}^{(i)} = L_{k−m+1}^{(i)} while if L_{k}^{(j)} = L_{k+1}^{(j)} then x(L_{k}^{(j)}) = x(L_{k+1}^{(j)}) = L_{k−m+1}^{(i)}. So x ∈ 𝔓^{1−m} ∩ A^{(ij)}. (iii) is immediate from (ii). ■

Let 𝔄^{(j)} = 𝔄(𝔏^{(j)}) be the hereditary 𝔬_{F} -order in A^{(jj)} corresponding to 𝔏^{(j)}.

(5.3.6) Lemma With the notation above, we have 𝔄^{(j)} = 𝔄 ∩ A^{(jj)}.

Proof: Let x ∈ 𝔄 ∩ A^{(jj)}; then xL_{k} ⊂ L_{k} for k ∈ ℤ so xL_{k}^{(j)} ⊂ L_{k}^{(j)}. i.e. x ∈ 𝔄^{(j)}. Conversely, let x ∈ 𝔄^{(j)}; then xL_{k}^{(j)} ⊂ L_{k}^{(j)}. But x = 1^{(j)}x1^{(j)} so xL_{k} = 1^{(j)}x1^{(j)}L_{k} = 1^{(j)}xL_{k}^{(j)} ⊂ L_{k}^{(j)} ⊂ L_{k}. ■

Let 𝔓^{(j)} be the Jacobson radical of 𝔄^{(j)}.

(5.3.7) Lemma (i) 𝔓 ∩ A^{(jj)} = 𝔓^{(j)}; (ii) 𝔓^{−m} ∩ A^{(jj)} = (𝔓^{(j)})^{− \frac{m}{g} g_{j}}; (iii) 𝔓^{1−m} ∩ A^{(jj)} = (𝔓^{(j)})^{1− \frac{m}{g} g_{j}}.

Proof: The proof of (i) is similar to (5.3.6), as are the proofs of (ii) and (iii), having observed that L_{k}^{(j)} ≠ L_{k+1}^{(j)} if and only if k ≡ −j or j + d − 1 (mod g), so that

(𝔓^{(j)})^{g_{j}}L_{k}^{(j)} = L_{k+g}^{(j)}  for k ∈ ℤ;
(𝔓^{(j)})^{1+g_{j}}L_{k}^{(j)} = L_{k+g+1}^{(j)} for k ≡ −j or j + d − 1 (mod g). ■

In particular, we have ν_{𝔄^{(j)}}(b_{jj}) = −\frac{m}{g}g_{j} = −m_{j}. We consider the stratum

[𝔄^{(j)}, m_{j}, m_{j} − 1, b_{jj}] in A^{(jj)}. Set y_{j} = π_{F}^{m_{j}/g_{j}}b_{jj}^{e_{j}/g_{j}} = π_{F}^{m/g}b_{jj}^{e/g}. Then we have

y ≡ ∑_{j} y_{j} (mod 𝔓)

from(5.3.5)(iii), since b_{jj}b_{ii} = 0 for i ≠ j. The characteristic polynomial of y as an element of 𝔄/𝔓 = ∐ 𝔄^{(j)}/𝔓^{(j)} is the product of the characteristic polynomials of the y_{j} as elements of 𝔄^{(j)}/𝔓^{(j)}. In particular, [𝔄^{(j)}, m_{j}, m_{j} − 1, b_{jj}] is a non-split skew fundamental stratum with characteristic polynomial a power of ϕ(X). Each lattice chain 𝔏^{(j)} is given by

⋯ ⫌ π_{F}^{−1}L_{r_{j}}^{(j)} ⊃ L_{r_{j}}^{(j)}^{#} ⫌ ⋯ ⫌ L_{0}^{(j)}^{#} ⊃ L_{0}^{(j)} ⫌ ⋯ ⫌ L_{r_{j}}^{(j)} = π_{F} L_{r_{j}}^{(j)}^{#} ⫌ …

so e_{j} = 2r_{j}, 2r_{j} + 1 or 2r_{j} + 2. Note further that if g_{j} = 2 then, for k = j + d − 1, (L_{k}^{(j)})^{#} = L_{d−k}^{(j)} = L_{−j+1}^{(j)} = L_{k}^{(j)} as −j + 1 > −j. So if we renumber the lattices of 𝔏^{j} then (L_{0}^{(j)})^{#} = L_{0}^{(j)}.

Putting this together, we split the possibilities into five cases:

(5.3.8) (i) (e_{j}, m_{j}) = 2, L_{0}^{(j)} = L_{0}^{(j)}^{#} and L_{r_{j}}^{(j)} = π_{F} L_{r_{j}}^{(j)}^{#}; so e_{j} = 2r_{j}; (ii) (e_{j}, m_{j}) = 1, L_{0}^{(j)} ≠ L_{0}^{(j)}^{#} and L_{r_{j}}^{(j)} = π_{F} L_{r_{j}}^{(j)}^{#}; so e_{j} = 2r_{j} + 1; (iii) (e_{j}, m_{j}) = 1, L_{0}^{(j)} ≠ L_{0}^{(j)}^{#} and L_{r_{j}}^{(j)} ≠ π_{F} L_{r_{j}}^{(j)}^{#}; so e_{j} = 2r_{j} + 2; (iv) (e_{j}, m_{j}) = 1, L_{0}^{(j)} = L_{0}^{(j)}^{#} and L_{r_{j}}^{(j)} ≠ π_{F} L_{r_{j}}^{(j)}^{#}; so e_{j} = 2r_{j} + 1;

(v) (e_{j}, m_{j}) = 1, L_{0}^{(j)} = L_{0}^{(j)}^{#} and L_{r_{j}}^{(j)} = π_{F} L_{r_{j}}^{(j)}^{#}; so e_{j} = 2r_{j} + 1.

Suppose, in each of these cases, we can find a stratum [𝔄^{′(j)}, m′_{j}, m′_{j} − 1, b′_{jj}] such

that 𝔄^{′(j)} ⊂ 𝔄^{(j)}, y′_{jj} = π_{F}^{m′_{j}/g′_{j}}b′_{jj}^{e′_{j}/g′_{j}} has minimal polynomial Φ(X) and

b_{jj} + (𝔓^{(j)})^{1−m_{j}} ⊂ b′_{jj} + (𝔓′^{(j)})^{1−m′_{j}}.

Then put b′ = ∑ b′_{j}, m′ = ∑ m′_{j} and e′ = ∑ e′_{j}. Set

𝔏′ = {𝔬_{F} -lattices L ∈ V : L_{i} ⊇ L ⊇ L_{i+1} for some i ∈ ℤ and,
for 1 ≤ j ≤ f, L ∩ V^{(j)} = L′_{k_{j}}^{(j)} for some k_{j} ∈ ℤ}.

(5.3.9) Lemma With notation as above, 𝔏′ is a self-dual lattice chain in V and we have e(𝔏′) = e′ and b′𝔄′ = 𝔓′^{m′}, where 𝔄′ = 𝔄(𝔏′) and 𝔓′ is the radical of 𝔄′.

Proof: For each i ∈ ℤ there exists a unique j, 1 ≤ j ≤ f, such that L_{i}^{(j)} ≠ L_{i+1}^{(j)}. Hence 𝔏′ is a lattice chain and e(𝔏′) = e′. It is self-dual since each 𝔏^{′(j)} is. ■

Then F [b′] = F [b′_{jj}] for any j, 1 ≤ j ≤ f since b′_{ii}b′_{jj} = 0 for i ≠ j so F [b′] is a field and b′ is minimal over F. In particular, [𝔄′, m′, m′ − 1, b′] is a skew simple stratum in A. Moreover, we have

b + 𝔓^{1−m} = ∐_{j=1}^{f}(b_{jj} + 𝔓^{(j)}^{1−m_{j}}) ∐_{i≠j} 𝔓^{1−m} ∩ A^{(ij)}

⊂ ∐_{j=1}^{f}(b′_{jj} + 𝔓′^{(j)}^{1−m′_{j}}) ∐_{i≠j} 𝔓′^{1−m′} ∩ A^{(ij)} = b′ + 𝔓′^{1−m′},

since 𝔓^{1−m} ∩ A^{(ij)} = 𝔓′^{1−m′} ∩ A^{(ij)}, for i ≠ j, by counting lattices. Hence we are done.

Before completing the proof in the cases (i–v) of (5.3.8) we describe what we mean by standard form. We choose a splitting of 𝔏, V_{∞}⊕V_{−r}⊕⋯⊕V_{−1}⊕V_{0}⊕V_{1}⊕⋯⊕V_{r} (see (3.2.2)) with, possibly, V_{∞} = 0 and/or V_{0} = 0. Let Ω be the set of indices ω ∈ {∞, −r, ..., r} such that V_{ω} is non-trivial; then, as in (3.2), ℤ/eℤ acts on Ω by translation in the obvious way. For ω ∈ Ω, i ∈ ℤ, we will write ω + i for the translate by i + eℤ of ω. We also choose a basis B as in (3.2.5), such that the matrices J_{∞}, J_{0} take the form described in (3.2.8). Put δ = deg ϕ(X).

(5.3.10) Definition Let [𝔄, m, m − 1, b] be a non-split fundamental stratum in band form, b = r_{−m}(b), where b = (b_{ω} : ω ∈ Ω).

(a) Suppose first we are in one of (i–v) of (5.3.8). Then the stratum [𝔄, m, m−1, b] is in standard form with respect to B if the following holds: Each b_{ω} is block diagonal (in blocks of size δ × δ or, in cases (iv), (v), possibly one middle block of size 2δ × 2δ), except for the following which are upper block triangular modulo 𝔭_{F}:

case (i) b_{1} and b_{−1−m}; case (ii) b_{[ \frac{e}{2} ]m}; case (iii) b_{∞} and b_{∞−m}; case (iv) b_{∞+[ \frac{e}{2} ]m}; case (v) b_{\frac{e−m+1}{2}} .

(b) Suppose otherwise. Then the stratum [𝔄, m, m − 1, b] is in standard form with respect to B if each of the strata [𝔄^{(j)}, m_{j}, m_{j} − 1, b_{jj}] (as described above) is in standard form.

(5.3.11) Proposition Let [𝔄, m, m − 1, b] be a non-split fundamental stratum which takes the form of one of cases (i–v) of (5.3.8) and is in band form, b = r_{−m}(b), where b = (b_{ω} : ω ∈ Ω). Then there exists a self-dual basis B which matches the splitting such that [𝔄, m, m − 1, b] is in standard form with respect to B.

Proof: We treat the five cases separately, although there are many similarities.

Case (i)

The maps b̃_{k} and (π̃_{F})_{k} are isomorphisms for all k ∈ ℤ by (3.3.1) so, in particular, we have dim_{k_{F}} V_{k} = dim_{k_{F}} V_{k+2} for all k ∈ ℤ. Further, duality via h gives dim_{k_{F}} V_{k} = dim_{k_{F}} V_{−1−k} for all k ∈ ℤ, so dim_{k_{F}} V_{k} = \frac{N}{e} for all k ∈ ℤ.

Form y = π_{F}^{m/2}b^{e/2} and consider ỹ_{0}, an automorphism of V_{0}. It has characteristic polynomial ϕ(X)^{t}, for some t, so, using Jordan canonical form, we can find a flag of k_{F} -subspaces

V_{0} = V_{0}^{0} ⊃ V_{0}^{1} ⊃ ⋯ ⊃ V_{0}^{s} = 0

such that ϕ(ỹ_{0})V_{0}^{q−1} ⊂ V_{0}^{q} for 1 ≤ q ≤ s = t and which is stable under ỹ_{0}. So, as an automorphism of V_{0}^{q}/V_{0}^{q+1}, ỹ_{0} has (irreducible) characteristic polynomial ϕ(X). Then we obtain a flag in each V_{k} by translation by b̃, π̃_{F} and by duality:

V_{k} = V_{k}^{0} ⊃ V_{k}^{1} ⊃ ⋯ ⊃ V_{k}^{s} = 0

with

V_{k}^{q} = \{ \begin{matrix}π̃_{F}^{u} b̃^{ν}V_{0}^{q} & k ≡ 0 (mod 2) \\ π̃_{F}^{u} b̃^{ν}(V_{0}^{s−q})^{⊥} & k ≡ 1 (mod 2), \end{matrix}

where u, ν ∈ ℤ are such that

k = \{ \begin{matrix}ue + νm & if k ≡ 0 (mod 2) \\ ue + νm − 1 & if k ≡ 1 (mod 2) \end{matrix}

Note that this is well defined since the flags are ỹ-stable and that (V_{k}^{q})^{⊥} = V_{−1−k}^{s−q}. Now we must lift this structure from the residual level to V. Choose a decomposition V_{0} = ⨁_{j=1}^{s} W_{0}^{j} such that V_{0}^{q−1} = ⨁_{j=q}^{s} W_{0}^{j} for 1 ≤ q ≤ s. Put W_{−1}^{q} = ⋂_{j≠s−q+1}(W_{0}^{j})^{⊥} for 1 ≤ q ≤ s; then V_{−1}^{q−1} = ⨁_{j=q}^{s} W_{−1}^{j}. Then for i = 0, −1, 1 ≤ q ≤ s, let W^{i,q} be the residual subspace of 𝔏 given by

(W^{i,q})_{k} = \{ \begin{matrix}π̃_{F}^{u}W_{i}^{q} & if k = ue + i \\ 0 & otherwise. \end{matrix}

Recall that we have a splitting V_{−r} ⊕ ⋯ ⊕ V_{−1} ⊕ V_{1} ⊕ ⋯ ⊕ V_{r} of 𝔏 (so Ω = {−r, …, −1, 1, …, r} in this case). For 1 ≤ q ≤ s, there exist (totally isotropic) subspaces W_{1}^{q} of V_{1} such that

𝔏(W_{1}^{q}) = W^{0,q}.

We also put

𝔏(W_{−1}^{q}) = ⋂_{j≠s−q+1}(W_{1}^{q})^{⊥};

then 𝔏(W_{−1}^{q}) = W^{−1,q} and (W_{1}^{q})^{⊥} ∩ V_{−1} = ⨁_{j≠s−q+1} W_{−1}^{j}. Now, for 1 ≤ ν ≤ \frac{e}{2} − 1 and 1 ≤ q ≤ s, we put

W_{1−νm}^{q} = b^{ν}W_{1}^{q},
W_{−1+νm}^{q} = b^{−ν}W_{−1}^{q},

so that V_{ω} = W_{ω}^{1} ⊕⋯⊕W_{ω}^{q} for all ω ∈ Ω. Then b_{ω} is block diagonal for all ω ∈ Ω, except b_{1} and b_{−1−m} which are upper block triangular modulo 𝔭_{F} since y is.

Case (ii) As in case (i), dim_{k_{F}} V_{k} = \frac{N}{e} for all k ∈ ℤ. Form y = π_{F}^{m}b^{e} and consider ỹ_{−1} as an automorphism of V_{−1}. It has characteristic polynomial ϕ(X)^{t}, for some t. We also have the nondegenerate form h_{−1} : V_{−1} × V_{−1} → k_{F} and, for the involution associated to h_{−1}, we have \overline{ỹ_{−1}} = ±ỹ_{−1}. By (5.2.2)(i), applied to (V_{−1}, h_{−1}), we can find a flag of k_{F} -subspaces

V_{−1} = V_{−1}^{0} ⊃ V_{−1}^{1} ⊃ ⋯ ⊃ V_{−1}^{s} = 0

such that (V_{−1}^{q})^{⊥} = V_{−1}^{s−q} for 0 ≤ q ≤ s = t, ϕ(ỹ_{−1})V_{−1}^{q−1} ⊂ V_{−1}^{q} for 1 ≤ q ≤ s and which is stable under ỹ_{−1}. From this flag we can obtain a flag in each V_{k} by translating by b̃ and π̃_{F}:

V_{k} = V_{k}^{0} ⊃ V_{k}^{1} ⊃ ⋯ ⊃ V_{k}^{s} = 0

with

V_{k}^{q} = π̃_{F}^{u} b̃^{ν}V_{−1}^{q}

where u, ν ∈ ℤ are such that k + 1 = ue + νm. Note that this is well defined as the flag is ỹ-stable and that (V_{k}^{q})^{⊥} = V_{−2−k}^{s−q}.

By (5.2.2)(ii), there exists a decomposition V_{−1} = ⨁_{j=1}^{s} W_{−1}^{j} such that V_{−1}^{q−1} = ⨁_{j=q}^{s} W_{−1}^{j} and (W_{−1}^{j})^{⊥} = ⨁_{j≠s−q+1} W_{−1}^{j}, for 1 ≤ q ≤ s. Then for 1 ≤ q ≤ s, let W^{−1,q} be the residual subspace of 𝔏 given by

(W^{−1,q})_{k} = \{ \begin{matrix}π̃_{F}^{u}W_{i}^{q} & if k = ue − 1 \\ 0 & otherwise. \end{matrix}

Recall that we have a splitting V_{−r} ⊕ ⋯ ⊕ V_{−1} ⊕ V_{0} ⊕ V_{1} ⊕ ⋯ ⊕ V_{r} of 𝔏 (so Ω = {−r, …, 0, …, r}). For 1 ≤ q ≤ s, we will now choose subspaces W_{0}^{q} of V_{0} such that 𝔏(W_{0}^{q}) = W^{−1,q}.

If s is odd then W^{−1,[ \frac{s+1}{2} ]} is a nondegenerate residual subspace so there exists a nondegenerate subspace W_{0}^{[ \frac{s+1}{2} ]} of V_{0} such that 𝔏(W_{0}^{[ \frac{s+1}{2} ]}) = W^{−1,[ \frac{s+1}{2} ]}. Then we

replace V_{0} with (W_{0}^{[ \frac{s+1}{2} ]})^{⊥} ∩ V_{0} to reduce to the case where s is even. If s is even then we first choose totally isotropic subspaces W_{0}^{1}, W_{0}^{s} of V_{0} such that 𝔏(W_{0}^{q}) = W^{−1,q}, for q = 1, s. The subspace W_{0}^{1} ⊕ W_{0}^{s} is nondegenerate and, replacing V_{0} by (W_{0}^{1} ⊕ W_{0}^{s})^{⊥} ∩ V_{0}, we may continue inductively to choose W_{0}^{q}, for 1 ≤ q ≤ s. With these choices, W_{0}^{q} is totally isotropic if, and only if, W^{−1,q} is and, further, (W_{0}^{q})^{⊥} ∩ V_{0} = ⨁_{j≠s−q+1} W_{0}^{j}. Now, for 1 ≤ ν ≤ [\frac{e}{2}] and 1 ≤ q ≤ s, we put

W_{0−νm}^{q} = b^{ν}W_{0}^{q},
W_{0+νm}^{q} = b^{−ν}W_{0}^{q},

so that V_{ω} = W_{ω}^{1} ⊕⋯⊕W_{ω}^{q} for all ω ∈ Ω. Then b_{ω} is block diagonal for all ω ∈ Ω, except b_{m[ \frac{e}{2} ]}, which is upper block triangular modulo 𝔭_{F}.

Case (iii) We proceed as in case (ii) to obtain the k_{F} -spaces V_{k}^{q}, 1 ≤ q ≤ s, k ∈ ℤ, and hence subspaces W_{0}^{q} of V_{0}. (Note that here the splitting of 𝔏 takes the form V_{∞} ⊕ V_{−r} ⊕ ⋯ ⊕ V_{−1} ⊕ V_{0} ⊕ V_{1} ⊕ ⋯ ⊕ V_{r}, so Ω = {∞, −r, …, 0, …, r}.) Again, for 1 ≤ ν ≤ \frac{e}{2} − 1 and 1 ≤ q ≤ s, we put

W_{0−νm}^{q} = b^{ν}W_{0}^{q};
W_{0+νm}^{q} = b^{−ν}W_{0}^{q}.

(Note that we have not yet chosen subspaces of V_{∞}.) Then b_{ω} is block diagonal for ω ∈ Ω \ {∞, ∞ − m}. Now we must choose subspaces W_{∞}^{q} of V_{∞} such that b_{∞} is upper block triangular modulo 𝔭_{F}; for then b_{∞−m} must also be upper block triangular modulo 𝔭_{F}, since y is. (In fact, b_{∞−m} = (π_{F}^{−1}J_{∞}^{−1}I)^{†}b_{∞}, where J_{∞}, I are as in (3.2.5) and ^{†} represents transposition in the off-diagonal; so b_{∞−m} is upper block triangular modulo 𝔭_{F} if, and only if, b_{∞} is also, regardless of the element y.) We consider the k_{F} -space V_{r}, with form h′_{r} : V_{r} × V_{r} → k_{F} given by h′_{r}(v, v′) = h_{r}(v, π̃_{F}^{−1}v′), for v, v′ ∈ V_{r}. This form is still skew-hermitian if F/F_{0} is unramified, but is symmetric if F/F_{0} is ramified.

We have subspaces V_{r}^{q} ⊂ V_{r} such that (V_{r}^{q})^{⊥} = π̃_{F}^{−1}V_{r}^{s−q} (for ⊥ relative to h_{r}). So, for ⊥ relative to the form h′_{r}, we have (V_{r}^{q})^{⊥} = V_{r}^{s−q}. We may now apply (5.2.2)(ii) to obtain subspaces W_{r}^{q} of V_{r} such that V_{r}^{q−1} = ⨁_{j=q}^{s} W_{r}^{j} and (W_{r}^{j})^{⊥} = ⨁_{j≠s−q+1} W_{r}^{j}, for 1 ≤ q ≤ s.

Now we define residual subspaces W^{r,q} as for W^{−1,q} and find subspaces W_{∞}^{q} of V_{∞} as for W_{0}^{q} in case (ii). Then, since b̃_{r+m}V_{r+m}^{q} = V_{r}^{q}, for 1 ≤ q ≤ s, we have that b_{∞} : V_{∞+m} → V_{∞} is upper block triangular modulo 𝔭_{F}, as required.

Case (iv) We treat this case as case (ii) except that we take V_{r} instead of V_{−1}, with form h′_{r} : V_{r} × V_{r} → k_{F} given by h′_{r}(v, v′) = \overline{π_{F}}^{−1}h(v, v′) + 𝔭_{F} (= h_{r}(v, π̃_{F}^{−1}v′)) for v, v′ ∈ V_{r}. As in case (iii), this form is skew-hermitian if F/F_{0} is unramified, but symmetric if F/F_{0} is ramified. The rest is effectively identical to case (ii), with V_{0} replaced by V_{∞}, although it is possible to have a double-sized block from (5.2.2)(i) (in the case F/F_{0} ramified, m odd and t even). As in case (ii), there is only one b_{ω} which is upper block triangular modulo 𝔭_{F}, namely b_{∞+m[ \frac{e}{2} ]}.

Case (v) As in case (i), dimV_{k} = \frac{N}{e} for all k ∈ ℤ. Note that −m must be odd so we put −m = 2l + 1; then b induces the map b̃_{−l−1} : V_{−l−1} → V_{l}. Form y = π_{F}^{m}b^{e} and consider ỹ as an automorphism of V_{−l−1}⊕V_{l}, where V_{−l−1} and V_{l} are maximal isotropic subspaces. It has characteristic polynomial ϕ(X)^{2t}, some t. We also have the form h′ on V_{−l−1} ⊕ V_{l}, given by h′(v_{−l−1} + v_{l}, v′_{−l−1} + v′_{l}) = h_{−l−1}(v_{−l−1}, v′_{l}) + h_{l}(v_{l}, v′_{−l−1}), which is nondegenerate, since h_{l} is, and skewhermitian. For the involution associated to this form, we have \overline{ỹ} = ±ỹ. By (5.2.3)(i), applied to (V_{−l−1} ⊕ V_{l}, h′), we can find flags of k_{F} -subspaces

V_{−l−1} = V_{−l−1}^{0} ⊃ V_{−l−1}^{1} ⊃ ⋯ ⊃ V_{−l−1}^{s} = 0
V_{l} = V_{l}^{0} ⊃ V_{l}^{1} ⊃ ⋯ ⊃ V_{l}^{s} = 0

such that (V_{−l−1}^{q})^{⊥} = V_{l}^{s−q}, b̃V_{−l−1}^{q} = V_{l}^{q}, each flag is ỹ-stable and ϕ(ỹ)V_{l}^{q−1} ⊂ V_{l}^{q} for 1 ≤ q ≤ s, s = t or t − 1. (Note that s = t − 1 can only occur if F = F_{0}; for it occurs only if k_{F} = k_{0} and, if F ≠ F_{0}, then F/F_{0} must be ramified. But then \overline{y} = (−1)^{m/g}(−1)^{e/g}y = −y so η = −.) From this flag we can obtain a flag in each V_{k} by translating by b̃ and π̃_{F}:

V_{k} = V_{k}^{0} ⊃ V_{k}^{1} ⊃ ⋯ ⊃ V_{k}^{s} = 0

with

V_{k}^{q} = π̃_{F}^{u} b̃^{ν}V_{−1}^{q}

where u, ν ∈ ℤ are such that k + l + 1 = ue + νm. Note that this is well defined as the flag is ỹ-stable and that (V_{k}^{q})^{⊥} = V_{−1−k}^{s−q}.

By (5.2.3)(ii) there exist decompositions V_{i} = ⨁_{j=1}^{s} W_{i}^{j}, i = −l − 1, l, such that V_{i}^{q−1} = ⨁_{j=q}^{s} W_{i}^{j}, (W_{−l−1}^{q})^{⊥} = V_{−l−1} ⊕ ⨁_{j≠s−q+1} W_{l}^{j} and W_{l}^{q} = b̃W_{−l−1}^{q} for 1 ≤ q ≤ s,i = −l − 1, l. Then for i = −l − 1, l, 1 ≤ q ≤ s, let W^{i,q} be the residual subspace of 𝔏 given by

(W^{i,q})_{k} = \{ \begin{matrix}π̃_{F}^{u}W_{i}^{q} & if k = ue + i \\ 0 & otherwise. \end{matrix}

Recall that we have a splitting V_{−r} ⊕ ⋯ ⊕ V_{−1} ⊕ V_{1} ⊕ ⋯ ⊕ V_{r} of 𝔏 (so Ω = {−r, …, −1, 1, …, r}). Then there exist (totally isotropic) subspaces W_{−l−1}^{q} of V_{−l−1} such that

𝔏(W_{−l−1}^{q}) = W^{−l−1,q}.

We also put

W_{l+1}^{q} = ⋂_{j≠s−q+1}(W_{−l−1}^{j})^{⊥};

then 𝔏(W_{l+1}^{q}) = W^{l,q} and (W_{l+1}^{q})^{⊥} ∩ V_{−l−1} = ⨁_{j≠s−q+1} W_{−l−1}^{j}. Now, for 1 ≤ ν ≤ \frac{e}{2} − 1 and 1 ≤ q ≤ s, we put

W_{l+1−νm}^{q} = b^{ν}W_{l+1}^{q},
W_{−l−1+νm}^{q} = b^{−ν}W_{−l−1}^{q},

so that V_{ω} = W_{ω}^{1} ⊕⋯⊕W_{ω}^{q} for all ω ∈ Ω. Then b_{ω} is block diagonal for all ω ∈ Ω, except b_{\frac{e+m−1}{2}} which is upper block triangular modulo 𝔭_{F}.

This completes the proof of (5.3.11) in all cases. ■

(5.3.12) Remark In cases (i), (ii), (iii), (iv), it is possible for us to choose the bases for the W_{ω}^{q} such that all the b_{ω} are diagonal with 1s and −1s on the diagonal, excepting those blocks which are upper triangular by blocks modulo 𝔭_{F}. In case (v), we may choose the bases for the W_{ω}^{q} such that all the b_{ω} are 1 or −1, excepting b_{\frac{e+m−1}{2}} and b_{\frac{1−m}{2}} .

We now prove a result similar to [BK] (2.5.8) about the shapes of the blocks of a stratum in standard form which is equivalent to a simple stratum.

(5.3.13) Proposition Let [𝔄, m, m − 1, b] be a non-split fundamental stratum which takes the form of one of cases (i–v) of (5.3.8) and is in band form, b = r_{−m}(b), where b = (b_{ω} : ω ∈ Ω). Suppose [𝔄, m, m − 1, b] is equivalent to a simple stratum. Then there exists a self-dual basis B which matches the splitting such that [𝔄, m, m − 1, b] is in standard form with respect to B and b_{ω} is block diagonal modulo 𝔭_{F} for all ω ∈ Ω.

Proof: Put y = π_{F}^{m/g}b^{e/g} as usual; then y = r_{0}(y), for some y = (y_{ω}) ∈ B_{0}(𝔄). By [BK] (2.5.8), y (mod 𝔭_{F}) is semisimple so y_{ω} is block diagonal (mod 𝔭_{F}), for all ω ∈ Ω. Then, cases (ii), (iv), (v), where only one b_{ω} is upper triangular modulo 𝔭_{F}, we are done, as indeed we are in case (i) since 1 and −1 are incongruent modulo 2 in Ω.

Consider now case (iii) where b_{∞} and b_{∞+m} are upper block triangular modulo 𝔭_{F}. Write

b_{∞} = ( \begin{matrix}α & β & γ \\ 0 & θ & ζ \\ 0 & 0 & ξ \end{matrix} ) (mod 𝔭_{F}),

where the block sizes are [\frac{s}{2}]δ, 0 (respectively δ), [\frac{s}{2}]δ if s is even (respectively odd) and we have chosen a self-dual basis matching the decomposition (as in (3.2.9)). We also have that α and ξ are upper block triangular modulo 𝔭_{F}.

A simple matrix calculation shows that

b_{∞+m} = ( \begin{matrix}ξ^{∗} & −ζ^{∗}K & −γ^{∗} \\ 0 & −θ^{∗}K & −β^{∗} \\ 0 & 0 & −α^{∗} \end{matrix} ) (mod 𝔭_{F}),

where x^{∗} is the conjugate transpose of x (for transposition in the off-diagonal) and K is the central block of IJ_{∞} (notation as in (3.2)).

Let u = r_{0}(u), with u = (u_{ω}) ∈ B_{0}(𝔄), be the unipotent block diagonal matrix all of whose blocks are the identity except u_{∞} which is given by

( \begin{matrix}1 & (ζξ^{−1})^{∗}K & \frac{1}{2}(ζξ^{−1})^{∗}Kζξ^{−1} \\ 0 & 1 & ζξ^{−1} \\ 0 & 0 & 1 \end{matrix} ) .

Then (u^{−1}bu)_{∞} has matrix

( \begin{matrix}α & β′ & γ′ \\ 0 & θ & 0 \\ 0 & 0 & ξ \end{matrix} ) (mod 𝔭_{F}),

for some β′, γ,. So, changing the basis B, we may assume ζ = 0.

We have y = r_{0}(y), with y = (y_{ω}) ∈ B_{0}(𝔄), and

y_{∞+m} = ( \begin{matrix}ξ^{∗}α & ξ^{∗}β & ξ^{∗}γ − γ^{∗}ξ \\ 0 & −δ^{∗}Kδ & −β ∗ ξ \\ 0 & 0 & −α^{∗}ξ \end{matrix} ) Z (mod 𝔭_{F}),

where Z ∈ GL(sd, 𝔬_{F}) is block diagonal. Since y_{∞+m} is block diagonal modulo 𝔭_{F}, we have −ξ^{∗}β ≡ 0 (mod 𝔭_{F}) and ξ^{∗}γ − γ^{∗}ξ ≡ 0 (mod 𝔭_{F}). In particular, β ≡ 0 (mod 𝔭_{F}) and (γξ^{−1})^{∗} ≡ γξ^{−1} (mod 𝔭_{F}).

Put y = \frac{1}{2}(γ^{∗}ξ + ξ^{∗}γ). Let u′ = r_{0}(u′), with u′ = (u′_{ω}) ∈ B_{0}(𝔄), be the unipotent block diagonal matrix all of whose blocks are the identity except u′_{∞} which is given by

( \begin{matrix}1 & 0 & y \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} ) .

Then (u′^{−1}bu′)_{∞} has matrix

( \begin{matrix}α & 0 & 0 \\ 0 & θ & 0 \\ 0 & 0 & ξ \end{matrix} ) (mod 𝔭_{F}).

Hence, again changing the basis B, we may assume β = 0 and γ = 0. Let u′′ = r_{0}(u′′), with u′′ = (u′′_{ω}) ∈ B_{0}(𝔄), be the block diagonal matrix all of whose blocks are the identity except u_{∞} which is given by

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

Then (u′′^{−1}bu′′)_{∞} has matrix

( \begin{matrix}1 & 0 & 0 \\ 0 & θ & 0 \\ 0 & 0 & (α^{−1})^{∗}ξ \end{matrix} ) (mod 𝔭_{F}).

So, changing the basis B, we may assume α = 1; then ξ must be block diagonal modulo 𝔭_{F} since y is. ■

(5.3.14) Corollary Let [𝔄, m, m−1, b] be a non-split fundamental stratum in band form, b = r_{−m}(b), where b = (b_{ω} : ω ∈ Ω). Suppose [𝔄, m, m−1, b] is equivalent to a simple stratum. Then there exists a self-dual basis B which matches the splitting such that [𝔄, m, m − 1, b] is in standard form with respect to B and b_{ω} is block diagonal modulo 𝔭_{F} for all ω ∈ Ω.

We now return to the proof of (5.3.2), in cases (i–v) of (5.3.8), using notation from the proof of (5.3.11). We first define a self-dual lattice chain 𝔏′, using the residual subspaces V_{k}^{q} in the proof of (5.3.11). In all cases we have a flag of k_{F} -subspaces in each V_{k} and they satisfy the relation

(V_{k}^{q})^{⊥} = V_{d−k−1}^{s−q}.

Then for each k ∈ ℤ, 0 ≤ q ≤ s − 1 there exists a unique lattice L′_{sk+q} such that

L′_{sk+q}/L_{k+1} = V_{k}^{q}.

Then 𝔏′ = {L′_{i}; i ∈ ℤ} is a lattice chain in V, of period e′ = es.

(5.3.15) Lemma The lattice chain 𝔏′ is self-dual.

Proof: We show that (L′_{sk+q})^{#} = L′_{s(d−k)−q}.

Since L_{k} ⊃ L′_{sk+q} ⊃ L_{k+1}, we have L_{d−k} ⊂ (L′_{sk+q})^{#} ⊂ L_{d−k−1}. Then, for v ∈ L_{d−k−1}, we have

h(v, L′_{sk+q}) ⊂ 𝔭_{F} ⟺ h(v + L_{d−k}, V_{k}^{q}) ≡ 0
⟺ v + L_{d−k} ⊂ (V_{k}^{q})^{⊥} = V_{d−k−1}^{s−q}
⟺ v ∈ L′_{s(d−k−1)+(s−q)} = L′_{s(d−k)−q}.

■

Let 𝔄′ = 𝔄(𝔏′); then 𝔄′ ⊂ 𝔄 and \overline{𝔄′} = 𝔄′.

(5.3.16) Lemma With notation as above, we have b𝔄′ = 𝔓′^{−m′}, where m′ = ms.

Proof: We have b ∈ 𝔎(𝔄) by [BK] (2.5.6) so L_{k−m} = bL_{k} ⊃ bL′_{sk+q} ⊃ bL_{k+1} = L_{k−m+1}. Then, for v ∈ L_{k−m}, we have

v ∈ bL′_{sk+q} ⟺ v + L_{k−m+1} ∈ b̃V_{k}^{q} = V_{k−m}^{q}
⟺ v ∈ L′_{s(k−m)+q}.

So b ∈ 𝔎(𝔄′) and b𝔄′ = 𝔓′^{−ms}. ■

Now we find an element b′ ∈ b + 𝔓′^{1−m′} such that Φ(b′) = 0. We take a splitting for 𝔏′,

V = V′_{∞} ⊕ V′_{−r′} ⊕ ⋯ ⊕ V′_{−1} ⊕ V′_{0} ⊕ V′_{1} ⊕ ⋯ ⊕ V′_{r′}

with possibly V′_{∞} = 0 and/or V′_{0} = 0 and which is subordinate to the chosen splitting for 𝔏. We also put Ω′ = {ω′ : V′_{ω′} ≠ 0}. We have dim_{F} V′_{ω′} = δ or 2δ for all ω′ ∈ Ω′, where δ = deg ϕ(X). Moving to stratum equivalent to [𝔄′, m′, m′ − 1, b], we may assume b is in band form, b = r_{−m′}(b) for b ∈ B_{−m′}(𝔄′). Then y = π_{F}^{m′/g′}b^{e′/g′} is of the form y = r_{0}(y), y = (y_{ω′}) ∈ B_{0}(𝔄′), with

y_{ω′} = b_{ω′}b_{ω′−m} … b_{ω′−(e′/g′−1)m′} ∈ Hom(V′_{ω′}, V′_{ω′}).

Let C be a set of representatives for the equivalence classes of Ω′ under the action of ℤ/g′ℤ, so card(C) = g′. If V′_{0} ≠ 0 then we require that 0 ∈ C; if V′_{∞} ≠ 0 and ∞ ≢ 0 (mod g′), then we require that ∞ ∈ C. Also, if we are in case (v) and m′ is odd, then we require that \frac{1−m′}{2} ∈ C. Then we have four cases:

(a) ω′ ≠ 0, ∞, \frac{1−m′}{2} ;

(b) ω′ = 0;

(c) ω′ = ∞;

(d) ω′ = \frac{1−m′}{2} .

Case (a) In this case we have dimV′_{ω′} = δ. By Hensel’s Lemma, there exists x_{ω′} ≡ y_{ω′} (mod 𝔭_{F}) such that Φ(x_{ω′}) = 0. Then x_{ω′} = u_{ω′}y_{ω′} for some u_{ω′} ≡ 1 (mod 𝔭_{F}).

Case (b) In this case we have dimV′_{0} = δ.

By(5.1.3) there exists x_{ω′} ∈ A_{−}^{(ω′ω′)} such that x_{ω′} ≡ y_{ω′} (mod 𝔭_{F}) and Φ(x_{ω′}) = 0. Then, by (5.1.5), there exists u_{ω′} ≡ 1 (mod 𝔭_{F}) such that x_{ω′} = u_{ω′}y_{ω′} \overline{u_{ω′}}.

Case (c)

If dimV′_{∞} = δ then this is identical to case (b). So suppose we are in the case dimV′_{∞} = 2δ.

By the discussion following (5.2.2), we have 𝔏(V′_{∞})_{−r′−1} = U′ ⊥ U′′, with y_{−r′−1}U′ = U′ and y_{−r′−1}U′′ = U′′. Let U′, U′′ be orthogonal subspaces of V′_{∞} such that 𝔏(U′)_{−r′−1} = U′ and 𝔏(U′′)_{−r′−1} = U′′. Then y_{∞} ≡ y′_{∞} + y′′_{∞} (mod 𝔭_{F}), where y′_{∞} ∈ Hom(U′, U′) and y′′_{∞} ∈ Hom(U′′, U′′). Moving to an equivalent stratum if necessary, we assume that y_{∞} = y′_{∞} + y′′_{∞}.

The spaces U′, U′′ are equipped with nondegenerate skew-hermitian forms (by restriction of h). Then we apply case (b) to find x′_{∞}, x′′_{∞} and u′_{∞}, u′′_{∞}. Then we put x_{∞} = x′_{∞} + x′′_{∞} and u_{∞} = u′_{∞} + u′′_{∞}. In particular, we have Φ(x_{∞}) = Φ(x′_{∞}) + Φ(x′′_{∞}) = 0 and u_{∞} ≡ 1 (mod 𝔭_{F}).

Case (d)

Following the remark (5.3.12), we assume that b_{υ′} = ±1 for υ′ ≡ ω′ (mod g′), υ′ ≠ ω′, ω′ − r′ − 1.

We first treat the case dimV′_{ω′} = δ.

Write α = b_{ω′} and β = b_{ω′−r′−1}. A simple matrix calculation shows that b + \overline{b} = 0 if and only if α = α^{∗} and β = β∗, where x^{∗} is the conjugate transpose of x, for x ∈ 𝕄(δ, F) (for transposition in the off-diagonal). The map x ↦ x^{∗} is the adjoint involution of a hermitian form on 𝕄(δ, F).

By Hensel’s Lemma, there exists x_{ω′} ≡ y_{ω′} (mod 𝔭_{F}) such that Φ(x_{ω′}) = 0. By (5.1.8), there exist α′ ≡ α (mod 𝔭_{F}), β′ ≡ β (mod 𝔭_{F}) such that α′^{∗} = α′, β′^{∗} = β′ and x_{ω′} = α′β′. By (5.1.5), there exist u_{ω′}, u_{ω′−r′−1} ≡ 1 (mod 𝔭_{F}) such that α′ = u_{ω′}αu_{ω′}^{∗} and β′ = u_{ω′−r′−1}βu_{ω′−r′−1}^{∗}.

Now consider the case dimV′_{ω′} = 2δ

By the discussion following (5.2.3), we have 𝔏(V′_{ω′})_{ω′−1} = U′ ⊕ U′′ such that 𝔏(V′_{ω′−r′−1})_{ω′−r′−1} = b̃U′ ⊕ b̃U′′ and h(b̃U′, U′′) ≡ 0. Let U′ be a subspace of V_{ω′} ⊕ V_{ω′−r′−1} such that 𝔏(U′)_{ω′−1} = U′ and 𝔏(U′)_{ω′−r′−1} = b̃U′. Put U′′ = U′^{⊥} ∩ (V_{ω′} ⊕ V_{ω′−r′−1}).

The spaces U′, U′′ are equipped with nondegenerate skew-hermitian forms (by restriction of h). Then we apply the case dimV′_{ω′} = δ to find x′_{ω′}, x′′_{ω′} and u′_{ω′}, u′′_{ω′} and u′_{ω′−r′−1}, u′′_{ω′−r′−1}. Then we put x_{ω′} = x′_{ω′} + x′′_{ω′}, u_{ω′} = u′_{ω′} + u′′_{ω′} and u_{ω′−r′−1} = u′_{ω′−r′−1}+u′′_{ω′−r′−1}. In particular, we have Φ(x_{ω′}) = Φ(x′_{ω′})+Φ(x′′_{ω′}) = 0, u_{ω′} ≡ 1 (mod 𝔭_{F}) and u_{ω′−r′−1} ≡ 1 (mod 𝔭_{F}).

We set u_{ω′} = 1, if u′_{ω} is not defined, and put u = r_{0}(u) where u = (u_{ω′}) ∈ B_{0}(𝔄′). Put b′ = ub\overline{u} and y′ = π_{F}^{m′/g′}b′^{e′/g′} = r_{0}(y′). Then y′ = (y_{ω′}), with y′_{ω′} a conjugate of x_{υ′}, where υ′ ∈ C is such that ω′ ≡ υ′ (mod g′). In particular, we have Φ(y′) = 0.

(5.3.17) Proposition [𝔄′, m′, m′ − 1, b′] is a skew simple stratum in A with b + 𝔓^{1−m} ⊂ b′ + 𝔓′^{1−m′}.

Proof: y′ generates an unramified field extension over F and has normalized valuation 0 with respect to this extension. So, by [Br2] (3.2.11), [𝔄′, m′, m′ − 1, b′] is simple. Then b′ is skew, \overline{𝔄′} = 𝔄′ and b + 𝔓^{1−m} ⊂ b′ + 𝔓′^{1−m′} by construction. ■

This completes the proof of (5.3.2). ■

(5.4) Pure is equivalent to simple

We first recall an important result from [BK] (2.4).

(5.4.1) Theorem [BK (2.4.1)] (i) Let [𝔄, n, r, β] be a pure stratum in A. There exists a simple stratum [𝔄, n, r, γ] in A such that

[𝔄, n, r, γ] ∼ [𝔄, n, r, β].

For any simple stratum [𝔄, n, r, γ] satisfying this condition, e(F [γ]|F) divides e(F [β]|F) and f(F [γ]|F) divides f(F [β]|F). In particular, among all the pure strata [𝔄, n, r, β′] equivalent to the given stratum [𝔄, n, r, β], the simple ones are precisely those for which the field extension F [β′]/F has minimal degree. (ii) Let [𝔄, n, r, γ_{1}], [𝔄, n, r, γ_{2}] be simple strata in A which are equivalent to each other. Then

(a) k_{0}(γ_{1}, 𝔄) = k_{0}(γ_{2}, 𝔄);

(b) e(F [γ_{1}]|F) = e(F [γ_{2}]|F) and f(F [γ_{1}]|F) = f(F [γ_{2}]|F);

(c) Let s_{1} be a tame corestriction on A relative to F [γ_{1}]/F. Then there exists δ ∈ F [γ_{1}] such that

s_{1}(γ_{1} − γ_{2}) ≡ δ (mod 𝔓^{1−r})

where 𝔓 = rad(𝔄). (iii) Let [𝔄, n, r, β] be a pure stratum in A with r = −k_{0}(β, 𝔄). Let [𝔄, n, r, γ] be a simple stratum in A which is equivalent to [𝔄, n, r, β], let s_{γ} be a tame corestriction on A relative to F [γ]/F, let B_{γ} be the A-centralizer of γ, and 𝔅_{γ} = 𝔄 ∩ B_{γ}. Then [𝔅_{γ}, r, r − 1, s_{γ}(β − γ)] is equivalent to a simple stratum in B_{γ}.

Let h be a skew-hermitian form.

(5.4.2) Theorem Let [𝔄, n, n − 1, b] be a skew non-split fundamental stratum in band form. Suppose also that it is equivalent to some simple stratum. Then it is equivalent to a skew simple stratum.

Proof: We first choose a basis for V such that b is in standard form. By (5.3.13), each b_{j,j−n} is block diagonal (mod 𝔭_{F}). Let b′ be block diagonal such that [𝔄, n, n− 1, b] is equivalent to [𝔄, n, n − 1, b′]. Then, as in the discussion preceding (5.3.17), we can perturb to make b′ minimal. ■

(5.4.3) Corollary Let [𝔄, n, n−1, b] be a skew pure stratum. Then it is equivalent to a skew simple stratum.

Proof: Let [𝔄, n, n − 1, b′] be a skew equivalent stratum in band form. Then the result follows immediately from (5.4.2). ■

We now return to the case where h is an alternating form, as in chapter 4.

(5.4.4) Proposition Let [𝔄, n, r, β] be a simple stratum with β minimal and with β + \overline{β} ∈ 𝔓^{−r}. Then it is equivalent to a skew simple stratum [𝔄, n, r, γ].

(5.4.5) Remark The element γ given by (5.4.4) will in fact be minimal by (5.4.1)(ii)(a).

Proof: We prove by induction that for t ≥ r there exists a skew simple stratum [𝔄, n, t, γ_{t}] equivalent to [𝔄, n, t, β]. The case t = n − 1 is given by (5.4.2) so assume we have a stratum [𝔄, n, t, γ_{t}] as required, t > r. We drop the index t. Let E_{γ} = F [γ], let B_{γ} be the A-centralizer of γ, 𝔅_{γ} = 𝔄 ∩ B_{γ}, 𝔔_{γ} = 𝔓 ∩ B_{γ} and let s_{γ} be a tame corestriction relative to E_{γ}/F which commutes with the involution. By (5.4.1)(ii), there exists δ ∈ E_{γ} such that s_{γ}(β − γ) ≡ δ (mod 𝔔_{γ}^{1−t}). Now γ + \overline{γ} = 0 and β + \overline{β} ∈ 𝔓^{−r} ⊂ 𝔓^{1−t} so δ + \overline{δ} ∈ 𝔔_{γ}^{1−t}. Let ϵ = \frac{1}{2}(δ + \overline{δ}) ∈ 𝔔_{γ}^{1−t} ∩ E_{γ}; then δ − ϵ is skew. Setting b = β − γ, we have s_{γ}(b) ≡ δ − ϵ (mod 𝔔_{γ}^{1−t}). Then [𝔅_{γ}, t, t − 1, s_{γ}(b)] is equivalent to [𝔅_{γ}, t, t − 1, δ − ϵ] which is skew simple since k_{0}(δ − ϵ, 𝔄) = −∞. Then, by (4.2.2), there exists γ_{t−1} such that γ_{t−1} + \overline{γ_{t−1}} = 0 and [𝔄, n, t − 1, γ_{t−1}] is simple and equivalent to [𝔄, n, t − 1, β]. ■

(5.4.6) Proposition Let [𝔄, n, r, β] be a simple stratum with k_{0}(β, 𝔄) = −s and β + \overline{β} ∈ 𝔓^{−r}. Then there exists an equivalent skew simple stratum [𝔄, n, r, γ].

Proof: We proceed by induction on s. The case s = n is (5.4.4) so we assume we have the result for k_{0}(β, 𝔄) ≤ −(s + 1). By (5.4.1)(i), there exists a simple stratum [𝔄, n, s, β′] equivalent to the pure stratum [𝔄, n, s, β]. Further, β′ + \overline{β′} ∈ 𝔓^{−s} and k_{0}(β′, 𝔄) ≤ −(s + 1) so, by the induction hypothesis, there exists γ′ ∈ A_{−} such that [𝔄, n, s, γ′] is simple and equivalent to [𝔄, n, s, β′] and hence to [𝔄, n, s, β]. Let E_{γ′} = F [γ′], let B_{γ′} be the A-centralizer of γ′, 𝔅_{γ′} = 𝔄 ∩ B_{γ′}, 𝔔_{γ′} = 𝔓 ∩ B_{γ′} and let s_{γ′} be a tame corestriction relative to E_{γ′}/F. By (5.4.1)(iii), [𝔅_{γ′}, s, s − 1, s_{γ′}(β − γ′)] is equivalent to a simple stratum. Let b = β − γ′ so b + \overline{b} ∈ 𝔓^{−r} ⊂ 𝔓^{1−s}. Hence s_{γ′}(b) + \overline{s_{γ′}(b)} ∈ 𝔔_{γ′}^{1−s}. So [𝔅_{γ′}, s, s − 1, s_{γ′}(β − γ′)] is equivalent to a skew simple stratum by (5.4.2). Then, by (4.2.2), there exists γ_{o} ∈ A_{−} such that [𝔄, n, s−1, γ_{o}] is simple and equivalent to [𝔄, n, s− 1, γ′ + b] = [𝔄, n, s − 1, β]. We now prove, by induction on t, that for s − 1 ≥ t ≥ r there exists a skew simple stratum [𝔄, n, t, γ_{t}] equivalent to [𝔄, n, t, β]. This will then complete the induction on s and the proof. We have just done the case t = s − 1 and the rest of the proof is identical to that of (5.4.4). ■

(5.4.7) Theorem Let [𝔄, n, r, β] be a skew pure stratum in A. Then there exists a skew simple stratum [𝔄, n, r, γ] equivalent to it.

Proof: By (5.4.1)(i) there exists a simple stratum [𝔄, n, r, γ′] which is equivalent to [𝔄, n, r, β]. Then γ′ + \overline{γ′} ≡ 0 (mod 𝔓^{−r}) and k_{0}(γ′, 𝔄) = −s for some s > r. So, by (5.4.6), there exists a skew simple stratum [𝔄, n, r, γ] as required. ■

SIMPLE CHARACTERS

From this chapter onwards, h is an alternating form, i.e. we consider only the symplectic group. In particular, N is even.

In this chapter we define some orders ℌ(β, 𝔄), 𝔍(β, 𝔄) in A associated to a skew simple stratum [𝔄, n, 0, β] and hence some subgroups H_{−}^{m+1}(β, 𝔄), J_{−}^{m+1}(β, 𝔄) of G. We then define a set C_{−}(𝔄, m, β) of characters of H_{−}^{m+1}(β, 𝔄), called simple characters. These are defined analogously to the GL_{N}(F) case ([BK] (3.2)) and, in fact, the simple characters for G will be precisely the restrictions to G of simple characters for the groups H^{m+1}(β, 𝔄) in GL_{N}(F). This chapter relies heavily on [BK] chapter 3; many results are proved using the analogous results there.

(6.1) The orders ℌ and 𝔍

Throughout this section [𝔄, n, 0, β] will be a skew simple stratum in A and we set r = −k_{0}(β, 𝔄).

(6.1.1) Definition ([BK] (3.1.7)) (i) Suppose β is minimal over F; put

ℌ(β) = ℌ(β, 𝔄) = 𝔅_{β} + 𝔓^{[ \frac{n}{2} ]+1}.

(ii) Suppose that r < n and let [𝔄, n, r, γ] be a skew simple stratum equivalent to [𝔄, n, r, β]; put

ℌ(β) = ℌ(β, 𝔄) = 𝔅_{β} + ℌ(γ, 𝔄) ∩ 𝔓^{[ \frac{r}{2} ]+1}.

Note that, by [BK] (3.1.9), this inductive definition is independent of the choice of γ such that [𝔄, n, r, γ] is a skew simple stratum equivalent to [𝔄, n, r, β].

(6.1.2) Lemma Let [𝔄, n, 0, β] be a skew simple stratum in A; then we have \overline{ℌ(β, 𝔄)} = ℌ(β, 𝔄)

Proof: We proceed by induction along β. If β is minimal then ℌ(β) is the sum of 𝔅_{β} and 𝔓^{[ \frac{n}{2} ]+1}, both of which are invariant under the involution. If r < n and [𝔄, n, r, γ] is a skew simple stratum equivalent to [𝔄, n, r, β] then ℌ(β)) is the sum of 𝔅_{β} and ℌ(γ, 𝔄) ∩ 𝔓^{[ \frac{r}{2} ]+1}, both of which are invariant under the involution, the latter by induction. ■

(6.1.3) Definition ([BK] (3.1.8)) (i) Suppose β is minimal over F; put

𝔍(β) = 𝔍(β, 𝔄) = 𝔅_{β} + 𝔓^{[ \frac{n+1}{2} ]}.

(ii) Suppose that r < n and let [𝔄, n, r, γ] be a skew simple stratum equivalent to [𝔄, n, r, β]; put

𝔍(β) = 𝔍(β, 𝔄) = 𝔅_{β} + 𝔍(γ, 𝔄) ∩ 𝔓^{[ \frac{r+1}{2} ]}.

Again, this definition is independent of the choice of γ and, exactly as in (6.1.2) we have:

(6.1.4) Lemma Let [𝔄, n, 0, β] be a skew simple stratum in A; then we have \overline{𝔍(β, 𝔄)} = 𝔍(β, 𝔄)

So we can define ℌ(β)_{−} = ℌ(β) ∩ A_{−} and 𝔍(β)_{−} = 𝔍(β) ∩ A_{−}. We now set

ℌ_{−}^{k}(β) = ℌ(β)_{−} ∩ 𝔓^{k}, 𝔍_{−}^{k}(β) = 𝔍(β)_{−} ∩ 𝔓^{k}, k ≥ 0.

In particular, as in [BK] (3.1), we get that for β minimal over F,

ℌ_{−}^{k}(β) = \{ \begin{matrix}𝔔_{β}^{k} + 𝔓_{−}^{[ \frac{n}{2} ]+1} & for 0 ≤ k ≤ [\frac{n}{2}] \\ 𝔓_{−}^{k} & for k ≥ [\frac{n}{2}] + 1, \end{matrix}

and that in the general case

ℌ_{−}^{k}(β) = \{ \begin{matrix}𝔔_{β}^{k} + ℌ_{−}^{[ \frac{r}{2} ]+1}(γ) & for 0 ≤ k ≤ [\frac{r}{2}] \\ ℌ_{−}^{k}(γ) & for k ≥ [\frac{r}{2}] + 1, \end{matrix}

where [𝔄, n, r, γ] is a skew simple stratum equivalent to [𝔄, n, r, β]. Similar remarks apply to 𝔍. Now, as in [BK] (3.1.14), we define two families of compact open subgroups of GL_{N}(F) by

\begin{matrix}H^{m}(β, 𝔄) = ℌ(β, 𝔄) ∩ U^{m}(𝔄) \\ J^{m}(β, 𝔄) = 𝔍(β, 𝔄) ∩ U^{m}(𝔄) \end{matrix} \} for m ≥ 0.

Then we define two families of compact open subgroups of G by

\begin{matrix}H_{−}^{m}(β, 𝔄) = H^{m}(β, 𝔄) ∩ G \\ J_{−}^{m}(β, 𝔄) = J^{m}(β, 𝔄) ∩ G \end{matrix} \} for m ≥ 0.

Then, by (1.2.4) we have a bijection between ℌ_{−}^{m}(β) and H_{−}^{m}(β) given by the Cayley map.

(6.1.5) Proposition (cf. [BK] (3.1.15)) (i) For 0 ≤ m ≤ [\frac{r}{2}] + 1, we have

H_{−}^{m}(β) = P^{m}(𝔅_{β})H_{−}^{[ \frac{r}{2} ]+1}(β),

and, for 0 ≤ m ≤ [\frac{r+1}{2}],

J_{−}^{m}(β) = P^{m}(𝔅_{β})J_{−}^{[ \frac{r+1}{2} ]}(β).

(ii) For m ≥ 0, the groups H_{−}^{m}(β), J_{−}^{m}(β) are normalized by 𝔎(𝔅_{β}) ∩ G. (iii) J_{−}^{m}(β) ⊃ H_{−}^{m}(β) and H_{−}^{m+1}(β) is a normal subgroup of J_{−}(β), for all m ≥ 1. (iv) For k, l ≥ 1, [J_{−}^{k}(β), J_{−}^{l}(β)] ⊂ H_{−}^{k+l}(β).

Proof: By [BK] (3.1.15), H_{−}^{m}(β) = U^{m}(𝔅_{β})H^{[ \frac{r}{2} ]+1}β) ∩ G. But U^{m}(𝔅_{β}) ∩ H^{[ \frac{r}{2} ]+1}(β) = 1 + 𝔔_{β}^{m} ∩ ℌ^{[ \frac{r}{2} ]+1}(β) so, by (1.2.7) we have U^{m}(𝔅_{β})H^{[ \frac{r}{2} ]+1}β) ∩ G = P^{m}(𝔅_{β})H^{[ \frac{r}{2} ]+1}(β). The same argument for J completes (i). Then (ii), (iii) and (iv) follow directly from [BK] (3.1.15). ■

(6.1.6) Lemma (cf. [BK] (3.1.19)) For m ≥ −1, we have

(ℌ_{−}^{m+1}(β))^{∗} = a_{β}(𝔍_{−}^{[ \frac{r+1}{2} ]}) + 𝔓_{−}^{−m}.

Proof: By [BK] (3.1.19) and (1.2.6). ■

We now choose a tame corestriction s_{β} on A relative to F [β]/F.

(6.1.7) Proposition (cf. [BK] (3.1.16)) For −1 ≤ m ≤ r − 1, we have an exact sequence

(𝔔_{β}^{r−m}𝔑(β))_{−} + 𝔍_{−}^{[ \frac{r+1}{2} ]}(β) ⟶^{a_{β}^{−}} (ℌ_{−}^{m+1}(β))^{∗} ⟶^{s_{β}^{−}} 𝔔_{β}^{−m} → 0.

Proof: By [BK] (3.1.16) and (1.2.6). ■

(6.1.8) Corollary For 0 ≤ m ≤ [\frac{r}{2}] + 1, we have an exact sequence

0 → 𝔔_{β}^{[ \frac{r+1}{2} ]} → 𝔍_{−}^{[ \frac{r+1}{2} ]}(β) ⟶^{a_{β}^{−}} (ℌ_{−}^{m+1}(β))^{∗} ⟶^{s_{β}^{−}} 𝔔_{β}^{−m} → 0.

There are similar results for 𝔍 (see [BK] (3.1.21), (3.1.22)).

(6.2) Simple characters

(6.2.1) Definition (cf. [BK] (3.2.1)) Let β be skew and minimal over F, E = F [β]. For 0 ≤ m ≤ n − 1 let C_{−}(𝔄, m, β) denote the set of characters θ of H_{−}^{m+1}(β) such that

(i) θ|_{H_{−}^{m+1}(β)∩P^{[ \frac{n}{2} ]+1}(𝔄)} = ψ_{β}

(ii) θ|_{H_{−}^{m+1}(β)∩(B_{β}∩G)} factors through det_{B_{β}} : B_{β} ∩ G → N_{1}(E) where N_{1}(E) = \{e ∈ E : e\overline{e} = 1\}.

(6.2.2) Proposition In the situation of (6.2.1) we have

(i) C_{−}(𝔄, m, β) = \{ψ_{β}\} for [\frac{n}{2}] ≤ m ≤ n − 1; (ii) for all θ ∈ C_{−}(𝔄, m, β) there exists θ′ ∈ C(𝔄, m, β) such that θ′|_{H_{−}^{m+1}(β)} = θ; (iii) every θ ∈ C_{−}(𝔄, m, β) is normalized by 𝔎(𝔅_{β}) ∩ G.

Proof: (i) is clear from the definition and (iii) will follow from (ii) and [BK] (3.2.2)(ii). If m ≥ [\frac{n}{2}] then θ = ψ_{β} which extends to the character ψ_{β} ∈ C(𝔄, m, β). Suppose then that m < [\frac{n}{2}] and take θ ∈ C_{−}(𝔄, m, β); then θ|_{H_{−}^{m+1}(β)∩(B_{β}∩G)} = θ|_{P^{m+1}(𝔅_{β})} = χ ∘ det_{B_{β}}, for some character χ of the closed subgroup det_{B_{β}}(P^{m+1}(𝔅_{β})) of N_{1}(E). By the observation following [BK] (3.2.1), the character ψ_{β} of U^{[ \frac{n}{2} ]+1}(𝔅_{β}) also factors through the determinant, ψ_{β} = χ_{1} ∘ det_{B_{β}} and, moreover, χ_{1}|_{detP^{[ \frac{n}{2} ]+1}(𝔅_{β})} = χ|_{detP^{[ \frac{n}{2} ]+1}(𝔅_{β})} since θ|_{P^{[ \frac{n}{2} ]+1}(𝔅_{β})} = ψ_{β}. Then

we define a character χ′ of det_{B_{β}}(P^{m+1}(𝔅_{β})U^{[ \frac{n}{2} ]+1}(𝔅_{β})) by

χ′(pu) = χ(p)χ_{1}(u) p ∈ det_{B_{β}}(P^{m+1}(𝔅_{β})), u ∈ det_{B_{β}}(U^{[ \frac{n}{2} ]+1}(𝔅_{β})).

Extend this to a character of E^{×}, also denoted χ′. Now H_{−}^{m+1}(β) = P^{m+1}(𝔅_{β}). P^{[ \frac{n}{2} ]+1}(𝔄), where the first factor normalizes the second, and further H^{m+1}(β) = U^{m+1}(𝔅_{β}).U^{[ \frac{n}{2} ]+1}(𝔄). So we define θ′ by

θ′(uh) = χ′(det_{B_{β}}(u)).ψ_{β}(h), u ∈ U^{m+1}(𝔅_{β}), h ∈ U^{[ \frac{n}{2} ]+1}(𝔄).

Since U^{m+1}(𝔅_{β}) normalizes ψ_{β} on U^{[ \frac{n}{2} ]+1}(𝔄) by [BK] (3.2.2) and χ′ ∘ det_{B_{β}}, ψ_{β} agree on U^{[ \frac{n}{2} ]+1}(𝔅_{β}), this defines a character of H^{m+1}(β) and clearly θ′ ∈ C(𝔄, m, β). ■

(6.2.3) Remark The fibres of the restriction map C(𝔄, m, β) → C_{−}(𝔄, m, β) are of the form θ.X where θ ∈ C(𝔄, m, β) and X is the group of characters of the finite group U^{m+1}(𝔅_{β})/(P^{m+1}(𝔅_{β})U^{[ \frac{n}{2} ]+1}(𝔅_{β})) which factor through det_{B_{β}}.

(6.2.4) Definition (cf. [BK] (3.2.3)) Let [𝔄, n, r, γ] be a skew simple stratum equivalent to the skew pure stratum [𝔄, n, r, β]. Then, for 0 ≤ m ≤ r − 1, let C_{−}(𝔄, m, β) be the set of characters θ of H_{−}^{m+1}(β) such that

(i) θ|_{H_{−}^{m+1}(β)∩(B_{β}∩G)} factors through det_{B_{β}}; (ii) θ is normalized by 𝔎(𝔅_{β}) ∩ G; (iii) if m′ = max\{m, [\frac{r}{2}]\}, the restriction θ|_{H_{−}^{m′+1}(β)} is of the form θ_{0}ψ_{c} for some

θ_{0} ∈ C_{−}(𝔄, m′, γ), c = β − γ.

For m ≥ r we set C_{−}(𝔄, m, β) = C_{−}(𝔄, m, γ).

(6.2.5) Proposition (cf. [BK] (3.2.4)) For m ≥ [\frac{n}{2}] we have C_{−}(𝔄, m, β) = {ψ_{β}}.

(6.2.6) Proposition In the situation of (6.2.4), for all θ ∈ C_{−}(𝔄, m, β) there exists θ′ ∈ C(𝔄, m, β) such that θ′|_{H_{−}^{m+1}(β)} = θ.

Proof: We proceed by induction along β. The case where β is minimal is just (6.2.2)(ii). Let [𝔄, n, r, γ] be a skew simple stratum equivalent to [𝔄, n, r, β]. If m ≥ [\frac{r}{2}] then θ = θ_{0}ψ_{c} for some θ_{0} ∈ C_{−}(𝔄, m, γ). By induction, there exists θ′_{0} ∈ C(𝔄, m, γ) such that θ′_{0}|_{H_{−}^{m+1}(γ)} = θ′_{0}|_{H_{−}^{m+1}(β)} = θ_{0}; also ψ_{c} extends to the

character ψ_{c} of H^{m+1}(β). Then, by [BK] (3.3.18), θ′ = θ′_{0}ψ_{c} is as required. If m < [\frac{r}{2}] then θ|_{H_{−}^{m′+1}(β)} = θ_{0}ψ_{c} for some θ_{0} ∈ C_{−}(𝔄, [\frac{r}{2}], γ). Then, by induc-

tion, there exists θ′_{0} ∈ C(𝔄, [\frac{r}{2}], γ) such that θ′_{0}|_{H_{−}^{[ \frac{r}{2} ]+1}(γ)} = θ_{0}. Now H_{−}^{m+1}(β) =

P^{m+1}(𝔅_{β}).H_{−}^{[ \frac{r}{2} ]+1}(β), with the first factor normalising the second, and also H^{m+1}(β) = U^{m+1}(𝔅_{β}).H^{[ \frac{r}{2} ]+1}(β). Then θ_{H_{−}^{m+1}(β)∩(B_{β}∩G)} = θ|_{P^{m+1}(𝔅_{β})} = χ ∘

det_{B_{β}} for some character χ of the closed subgroup det_{B_{β}}(P^{m+1}(𝔅_{β})) of N_{1}(E). Further, θ′_{0}ψ_{c} ∈ C(𝔄, [\frac{r}{2}], β) by [BK] (3.3.18) so θ′_{0}ψ_{c}|_{U^{[ \frac{r}{2} ]+1}(𝔅_{β})} = χ_{1} ∘ det_{B_{β}} for

some character χ_{1} of det_{B_{β}}(U^{[ \frac{r}{2} ]+1}(𝔅_{β})). Moreover, we have χ_{1}|_{detP^{[ \frac{r}{2} ]+1}(𝔅_{β})} = χ|_{detP^{[ \frac{r}{2} ]+1}(𝔅_{β})} since θ_{0}ψ_{c}|_{P^{[ \frac{r}{2} ]+1}(𝔅_{β})} = θ. Then we define a character χ′ of

det_{B_{β}}(P^{m+1}(𝔅_{β})U^{[ \frac{r}{2} ]+1}(𝔅_{β})) by

χ′(pu) = χ(p)χ_{1}(u) p ∈ det_{B_{β}}(P^{m+1}(𝔅_{β})), u ∈ det_{B_{β}}(U^{[ \frac{r}{2} ]+1}(𝔅_{β})).

Extend this to a character of E^{×}, also denoted χ′. Then we define θ′ by

θ′(uh) = χ′(det_{B_{β}}(u)).θ′_{0}(h)ψ_{c}(h), u ∈ U^{m+1}(𝔅_{β}), h ∈ H^{[ \frac{r}{2} ]+1}(β).

Then U^{m+1}(𝔅_{β}) normalizes θ′_{0}ψ_{c} on H^{[ \frac{r}{2} ]+1}(β) = H^{[ \frac{r}{2} ]+1}(γ) since, as above, θ′_{0}ψ_{c} ∈ C(𝔄, [\frac{r}{2}], β) and χ′ ∘ det_{B_{β}}, θ′_{0}ψ_{c} agree on U^{[ \frac{n}{2} ]+1}(𝔅_{β}) so this defines a character of H^{m+1}(β) and θ′ ∈ C(𝔄, m, β). ■

(6.2.7) Corollary (cf. [BK] (3.2.5)) For 0 ≤ m ≤ [\frac{r}{2}], restriction induces a surjective map

C_{−}(𝔄, m, β) → C_{−}(𝔄, [\frac{r}{2}], β).

Note that, as in the proof of [BK] (3.2.5), the fibres are of the form θ.X for θ ∈ C(𝔄, m, β) and X is the group of characters of P^{m+1}(𝔅_{β})/P^{[ \frac{r}{2} ]+1}(𝔅_{β}) which factor through det_{B_{β}}.

By (6.2.6) we could have defined C_{−}(𝔄, m, β) to be the set C(𝔄, m, β) of characters of H^{m+1}(β) restricted to H_{−}^{m+1}(β). Indeed, from now on we will use this description.

(6.2.8) Corollary In the situation of (6.2.4), C_{−}(𝔄, m, β) is independent of the choice of the element γ.

Proof: By [BK](3.2.20)(i), C(𝔄, m, β) is independent of the choice of γ and C_{−}(𝔄, m, β) = {θ|H_{−}^{m+1}(β) : θ ∈ C(𝔄, m, β)}. ■ We now generalize(6.2.3) to describe the fibres of the restriction map C(𝔄, m, β) → C_{−}(𝔄, m, β) in all cases. First we give a result concerning the surjectivity of the determinant map. We put P (𝔬_{E}) = N_{1}(E) ⊂ U(𝔬_{E}), P^{n}(𝔬_{E}) = U^{n}(𝔬_{E}) ∩ N_{1}(E) for n ≥ 1.

(6.2.9) Lemma For n ≥ 0, we have det_{B/E}P^{n+1}(𝔅) = P^{[ \frac{n}{e}]+1}(𝔬_{E}).

Proof: We have det_{B/E}U^{n+1}(𝔅) = U^{[ \frac{n}{e}]+1}(𝔬_{E}) by [BF2](2.8.3). Then certainly det_{B/E}P^{n+1}(𝔅) ⊂ P^{[ \frac{n}{e}]+1}(𝔬_{E}) since det_{B/E} commutes with the involution . Suppose now we have X ∈ P^{[ \frac{n}{e}]+1}(𝔬_{E}); then, by (1.2.4)(i), X = U\overline{U}^{−1} for some U ∈ U^{[ \frac{n}{e}]+1}(𝔬_{E}). But we know U = det_{B/E}u for some u ∈ U^{n+1}(𝔅). Then, putting x = u\overline{u}^{−1}, we have x ∈ P^{n+1}(𝔅) and det_{B/E}x = X as required. ■

(6.2.10) Proposition Let [𝔄, n, 0, β] be a skew simple stratum with r = −k_{0}(β, 𝔄) and let [𝔄, n, r, γ] be a skew simple stratum equivalent to [𝔄, n, r, β]. Let 0 ≤ m ≤ r − 1. Then restriction induces a surjective map C(𝔄, m, β) → C_{−}(𝔄, m, β). If m ≥ [\frac{r}{2}] then the fibres are in bijection with the fibres of C(𝔄, m, γ) → C_{−}(𝔄, m, γ); if m < [\frac{r}{2}] then the fibres are of the form Θ.X ∘ det_{B_{β}/E_{β}}, where X is the group

of characters of U^{[ \frac{m}{e}]+1}(𝔬_{E_{β}})/(P^{[ \frac{m}{e}]+1}(𝔬_{E_{β}})U^{[ \frac{[ \frac{r}{2} ]}{e}]+1}(𝔬_{E_{β}})) and Θ is a fibre of C(𝔄, [\frac{r}{2}], γ) → C_{−}(𝔄, [\frac{r}{2}], γ).

Proof: As usual, we will work by induction along β. For β minimal this follows from (6.2.3) and (6.2.8). First we look at the case m ≥ [\frac{r}{2}]. Let θ′ ∈ C(𝔄, m, β); then θ′ = θ′_{0}ψ_{c}, for θ′_{0} ∈ C(𝔄, m, γ), c = β − γ and the result is clear. Now suppose m < [\frac{r}{2}] and let θ′ ∈ C(𝔄, m, β); then θ′|_{H^{[ \frac{r}{2} ]+1}(β)} = θ′_{0}ψ_{c}, for θ′_{0} ∈ C(𝔄, [\frac{r}{2}], γ), c = β − γ. Then the result follows from (6.2.3) and the case m = [\frac{r}{2}]. ■

We recall here a result of Glauberman ([G] or see [BH2] §A2). Let M be a finite group and let Γ be a subgroup of AutM such that |M|, |Γ| are relatively prime. We write ΓM for the semi-direct product Γ ⋉ M. The group Γ acts on the set Irr(M) of equivalence classes of irreducible representations of M; we denote the set of fixed points by Irr(M)^{Γ}. Let ρ ∈ Irr(M)^{Γ}. Then, by [G] Theorem 1, there exists a unique (up to equivalence) representation ρ̃ of ΓM such that ρ̃|_{M} ≃ ρ and detρ̃(γ) = 1 for all γ ∈ Γ.

(6.2.11) Proposition ([G] Theorem 3) Suppose, with notations as above, that the group Γ is cyclic. There is a canonical bijection

g_{Γ} : Irr(M)^{Γ} ⟶^{≃} Irr(M^{Γ}),

where M^{Γ} is the centralizer of Γ in M. Explicitly, for ρ ∈ Irr(M)^{Γ}, the representation ζ = g_{Γ}(ρ) is given as follows. There is a sign ϵ = ϵ(ρ, Γ) such that

tr ζ(x) = ϵtr ρ̃(γx),

for all generators γ of Γ and x ∈ M^{Γ}, where ρ̃ is the extension of ρ to ΓM as above.

We now apply this to our situation. Let [𝔄, n, 0, β] be a skew simple stratum. All simple characters θ ∈ C(𝔄, m, β) are trivial on the group U^{n+1}(𝔄) so they are, essentially, characters of the finite p-group H^{m+1}(β, 𝔄)/U^{n+1}(𝔄). Let σ : A^{×} → A^{×} be the involution given by σ(x) = \overline{x}^{−1}. Then σ acts on the group H^{m+1}(β, 𝔄)/U^{n+1}(𝔄), since both groups in this quotient are fixed by the involution . We put Γ = ⟨σ⟩, a cyclic group of order 2. Then we have H^{m+1}(β, 𝔄)^{Γ} = H_{−}^{m+1}(β, 𝔄) and U^{n+1}(𝔄)^{Γ} = P^{n+1}(𝔄); in particular, we have (H^{m+1}(β, 𝔄)/U^{n+1}(𝔄))^{Γ} = H_{−}^{m+1}(β, 𝔄)/P^{n+1}(𝔄). Since p is not 2, we apply Glauberman’s correspondence (6.2.11) to the groups M = H^{m+1}(β, 𝔄)/U^{n+1}(𝔄), Γ; this gives us a bijection between those equivalence classes of representations of H^{m+1}(β, 𝔄)/U^{n+1}(𝔄) which are stable by σ and the equivalence classes of irreducible representations of H_{−}^{m+1}(β, 𝔄)/P^{n+1}(𝔄). Moreover, the relationship between the characters of these representations implies that we have a bijection

(6.2.12)  C(𝔄, m, β)^{Γ} ⟶^{≃} C_{−}(𝔄, m, β),

given by restriction of characters. (Note that the character of H^{m+1}(β, 𝔄) corresponding to a given simple character of H_{−}^{m+1}(β, 𝔄) is indeed a simple character, since we can construct it explicitly as in (6.2.2)(ii),(6.2.6).)

(6.3) Intertwining

(6.3.1) Proposition Let θ ∈ C_{−}(𝔄, m, β), 0 ≤ m ≤ r − 1 and let j ∈ J_{−}(β). Then θ(jhj^{−1}) = θ(h) for all h ∈ H_{−}^{m+1}(β).

Proof: [BK] (3.3.1). ■

(6.3.2) Theorem (cf. [BK] (3.3.2)) Let [𝔄, n, 0, β] be a skew simple stratum in A, r = −k_{0}(β, 𝔄). Let 0 ≤ m ≤ r − 1 and θ ∈ C_{−}(𝔄, m, β). Then

I_{G}(θ|H_{−}^{m+1}(β))

= (1 + 𝔔_{β}^{r−m}𝔑(β) + 𝔍^{[ \frac{r+1}{2} ]}(β)) ∩ G . B_{β} ∩ G . (1 + 𝔔_{β}^{r−m}𝔑(β) + 𝔍^{[ \frac{r+1}{2} ]}(β)) ∩ G

(6.3.3) Remark We can write

(1 + 𝔔_{β}^{r−m}𝔑(β) + 𝔍^{[ \frac{r+1}{2} ]}(β)) ∩ G = ((1 + 𝔔_{β}^{r−m}𝔑(β)) . J^{[ \frac{r+1}{2} ]}(β)) ∩ G
= Q_{r−m} . J_{−}^{[ \frac{r+1}{2} ]}.

Moreover, when m ≤ [\frac{r}{2}], 𝔔_{β}^{r−m}𝔑(β) ⊂ 𝔍^{[ \frac{r+1}{2} ]} so the Theorem says

I_{G}(θ|H_{−}^{m+1}(β)) = J_{−}^{[ \frac{r+1}{2} ]}(β) . B_{β} ∩ G . J_{−}^{[ \frac{r+1}{2} ]}(β)

Proof: We proceed by induction along β, following the proof of [BK] (3.3.2). The case r = ∞ is trivial so we begin with r = n. Then β is minimal so 𝔍^{[ \frac{n+1}{2} ]} = 𝔓^{[ \frac{n+1}{2} ]} and 𝔔_{β}^{n−m}𝔑(β) = 𝔓^{n−m}. Let m′ = max\{m, [\frac{n}{2}]\}; so (1+𝔔_{β}^{r−m}𝔑(β)+𝔍^{[ \frac{r+1}{2} ]}(β))∩ G = P^{n−m′}(𝔄). Then

I_{G}(θ|H_{−}^{m+1}(β)) ⊂ I_{G}(θ|P^{m′+1}(𝔄)) = I_{G}[𝔄, n, m′, β]
= P^{n−m′}(𝔄) . B_{β} ∩ G . P^{n−m′}(𝔄)

by (2.2.3). But we also have

I_{G}(θ|H_{−}^{m+1}(β)) ⊃ I_{GL_{2N}} (θ|H^{m+1}(β)) ∩ G
= U^{n−m′}(𝔄)B_{β}^{×}U^{n−m′}(𝔄), by [BK] (3.3.2),
= P^{n−m′}(𝔄) . B_{β} ∩ G . P^{n−m′}(𝔄), by (2.2.3),

so we in fact have the equality required. Now we look at the general case where r < n. Let [𝔄, n, r, γ] be a skew simple stratum equivalent to [𝔄, n, r, β] and put s = −k_{0}(γ, 𝔄). We look first at the case m ≥ [\frac{r}{2}]. Thus H_{−}^{m+1}(β) = H_{−}^{m+1}(γ) and, for θ ∈ C_{−}(𝔄, m, β), we have I_{G}(θ) ⊂ I_{G}(θ|H_{−}^{r+1}(β)). Then, as in [BK] (3.3.2), the restriction of θ to H_{−}^{r+1}(β) = H_{−}^{r+1}(γ) lies in C(𝔄, r, γ). Put I^{r}(γ) = I_{G}(θ|H^{r+1}) so, by induction,

I^{r}(γ) = (1 + 𝔔_{γ}^{s−r}𝔑(γ) + 𝔍^{[ \frac{s+1}{2} ]}(γ)) ∩ G . B_{γ} ∩ G . (1 + 𝔔_{γ}^{s−r}𝔑(γ) + 𝔍^{[ \frac{s+1}{2} ]}(γ)) ∩ G

We define another set

I_{m}^{−}(β) = {x ∈ G : x^{−1}(β + ℌ_{−}^{m+1}(β)^{∗})x ∩ (β + ℌ_{−}^{m+1}(β)^{∗}) ≠ ∅},

i.e. the formal intertwining of the coset β + ℌ_{−}^{m+1}(β)^{∗}. Note that this is just I_{m}^{+}(β)∩G, where I_{m}^{+}(β) is the formal GL_{N}-intertwining of the coset β +ℌ^{m+1}(β)^{∗}, as in [BK].

(6.3.4) Lemma (cf. [BK] (3.3.5)) I_{G}(θ) = I_{m}^{−}(β) ∩ I^{r}(γ).

Proof: We first show that

J_{−}^{[ \frac{r+1}{2} ]}(β) . I_{m}^{−}(β) . J_{−}^{[ \frac{r+1}{2} ]}(β) = I_{m}^{−}(β).

We clearly have the containment ⊃. But

J_{−}^{[ \frac{r+1}{2} ]}(β) . I_{m}^{−}(β) . J_{−}^{[ \frac{r+1}{2} ]}(β) ⊂ J^{[ \frac{r+1}{2} ]}(β) . I_{m}^{+}(β) . J^{[ \frac{r+1}{2} ]}(β) ∩ G
= I_{m}^{+}(β) ∩ G, by [BK] (3.3.6),
= I_{m}^{−}(β).

We now take x ∈ I^{r}(γ); then, as all the sets are bi-invariant by J_{−}^{[ \frac{r+1}{2} ]}(β), we may assume that x ∈ (1 + 𝔔_{γ}^{s−r}𝔑(γ)) ∩ G . B_{γ} ∩ G . (1 + 𝔔_{γ}^{s−r}𝔑(γ)) ∩ G. Let h ∈ x^{−1}H_{−}^{m+1}(β)x ∩ H_{−}^{m+1}(β); then, as in [BK] (3.3.5),

θ^{x}(h)θ^{−1}(h) = ψ_{x^{−1}βx−β}(h).

So x intertwines θ if and only if ψ_{x^{−1}βx−β}(h) = 1 for all h ∈ x^{−1}H_{−}^{m+1}(β)x ∩ H_{−}^{m+1}(β), i.e. if and only if x intertwines ψ_{β}|H_{−}^{m+1}(β). But this is if and only if x^{−1}βx − β ∈ (x^{−1}ℌ_{−}^{m+1}(β)x ∩ ℌ_{−}^{m+1}(β))^{∗} = x^{−1}(ℌ_{−}^{m+1})^{∗}x + (ℌ_{−}^{m+1})^{∗}. i.e x ∈ I_{m}^{−}(β). ■ Now, as in [BK] (3.3.10), we have

I_{m}^{−}(β) ⊃ (1+𝔔_{β}^{r−m}𝔑(β)+𝔍^{[ \frac{r+1}{2} ]}(β))∩G . B_{β} ∩G . (1+𝔔_{β}^{r−m}𝔑(β)+𝔍^{[ \frac{r+1}{2} ]}(β))∩G.

Then, for x ∈ I_{m}^{−}(β), we have

x^{−1}(β + (ℌ_{−}^{m+1})^{∗})x ∩ (β + (ℌ_{−}^{m+1})^{∗}) ≠ ∅

so there exist δ_{1}, δ_{2} ∈ (ℌ_{−}^{m+1})^{∗} such that

(6.3.5)  x^{−1}(β + δ_{1} + 𝔓_{−}^{−m})x ∩ (β + δ_{2} + 𝔓_{−}^{−m}) ≠ ∅.

(6.3.6) Lemma (cf. [BK] (3.3.12)) Let δ ∈ (ℌ_{−}^{m+1}(β))^{∗}. Then there exists y ∈ 𝔍_{−}^{[ \frac{r+1}{2} ]}(β) such that

C(y)^{−1}(β + δ + 𝔓_{−}^{−m})C(y) = β + 𝔓_{−}^{−m}.

Proof: We show that for k ∈ ℤ, δ ∈ (ℌ_{−}^{m+1}(β)^{∗} ∩ 𝔓_{−}^{k}) + 𝔓_{−}^{−m}, there exists y ∈ 𝔍_{−}^{[ \frac{r+1}{2} ]}(β) such that

C(y)^{−1}(β + δ)C(y) ≡ β (mod (ℌ_{−}^{m+1}(β)^{∗} ∩ 𝔓_{−}^{k+1}) + 𝔓_{−}^{−m})

and the result will follow by induction. Let δ ∈ (ℌ_{−}^{m+1}(β)^{∗} ∩ 𝔓_{−}^{k}) + 𝔓_{−}^{−m}. By (6.1.6), there exists y ∈ 𝔍_{−}^{[ \frac{r+1}{2} ]}(β) such that δ + a_{β}(y) ∈ 𝔓_{−}^{−m}; then a_{β}(y) ∈ (ℌ_{−}^{m+1}(β)^{∗} ∩ 𝔓_{−}^{k}) + 𝔓_{−}^{−m}. We have C(y) = 1 + y′ for some y′ ∈ 𝔍^{[ \frac{r+1}{2} ]} so, as in the proof of [BK] (3.3.12),

C(y)^{−1}δC(y) ≡ δ (mod (ℌ^{m+1}(β)^{∗} ∩ 𝔓^{k+1}) + 𝔓^{−m}).

Further C(y)^{−1}βC(y) = β + (1 + y′)^{−1}a_{β}(y′). Now y′ is given by a power series in y and, for n ≥ 2, a_{β}(y^{n}) = ∑_{i=0}^{n} y^{i}a_{β}(y)y^{n−i} ∈ 𝔍^{[ \frac{r+1}{2} ]}(ℌ^{m+1}(β)^{∗} ∩ 𝔓^{k}) + 𝔓^{−m}) + (ℌ^{m+1}(β)^{∗} ∩𝔓^{k})+𝔓^{−m})𝔍^{[ \frac{r+1}{2} ]} ⊂ ℌ^{m+1}(β)^{∗} ∩𝔓^{k+1} +𝔓^{−m}. Hence a_{β}(y′) ≡ a_{β}(y) (mod (ℌ^{m+1}(β)^{∗} ∩𝔓^{k+1})+𝔓^{−m}). Then (1+y′)^{−1} = 1+y′′, for some y′′ ∈ 𝔍^{[ \frac{r+1}{2} ]}, so

(1 + y′)^{−1}a_{β}(y′) ≡ a_{β}(y) + y′′a_{β}(y) ≡ a_{β}(y) (mod (ℌ^{m+1}(β)^{∗} ∩ 𝔓^{k+1}) + 𝔓^{−m}).

Then altogether we have

C(y)^{−1}(β + δ)C(y) ≡ β + δ + a_{β}(y) (mod (ℌ^{m+1}(β)^{∗} ∩ 𝔓^{k+1}) + 𝔓^{−m})
≡ β (mod (ℌ^{m+1}(β)^{∗} ∩ 𝔓^{k+1}) + 𝔓^{−m}).

Then, since the elements on both sides of this congruence are skew, we have the assertion. ■ Then we can write in (6.3.5)

β + δ_{i} + 𝔓_{−}^{−m} = C(y_{i})^{−1}(β + 𝔓_{−}^{−m})C(y_{i}), i = 1, 2,

for some y_{1}, y_{2} ∈ 𝔍_{−}^{[ \frac{r+1}{2} ]}(β). Thus, if z = C(y_{1})xC(y_{2})^{−1}, we have

z^{−1}(β + 𝔓_{−}^{−m})z ∩ (β + 𝔓_{−}^{−m}) ≠ ∅.

Then, by (2.2.3), z ∈ (1 + 𝔔_{β}^{r−m}𝔑(β)) ∩ G . B_{β} ∩ G . (1 + 𝔔_{β}^{r−m}𝔑(β)) ∩ G and so x lies in

(1 + 𝔔_{β}^{r−m}𝔑(β) + 𝔍^{[ \frac{r+1}{2} ]}(β)) ∩ G . B_{β} ∩ G . (1 + 𝔔_{β}^{r−m}𝔑(β) + 𝔍^{[ \frac{r+1}{2} ]}(β)) ∩ G

Finally we prove

I_{m}^{−}(β) ⊂ I^{r}(γ)

and this, together with (6.3.4), will conclude the proof of the theorem in the case

m ≥ [\frac{r}{2}]. Since I^{r}(γ) is bi-invariant under multiplication by J_{−}^{[ \frac{r+1}{2} ]}(β) = J_{−}^{[ \frac{r+1}{2} ]}(γ) it is enough to show that

I^{r}(γ) ⊃ (1 + 𝔔_{β}^{r−m}𝔑(β)) ∩ G . B_{β} ∩ G . (1 + 𝔔_{β}^{r−m}𝔑(β)) ∩ G.

This is just the formal intertwining of β + 𝔓_{−}^{−m} so it is contained in the formal intertwining of β + 𝔓_{−}^{−r} = γ + 𝔓_{−}^{−r}, which is (1 + 𝔔_{γ}^{s−r}𝔑(γ)) ∩ G . B_{γ} ∩ G . (1 + 𝔔_{γ}^{s−r}𝔑(γ)) ∩ G and this is certainly contained in I^{r}(γ).

Now we assume that m ≤ [\frac{r}{2}]. Note that

𝔔_{β}^{r−[ \frac{r}{2} ]}𝔑(β) = 𝔔_{β}^{[ \frac{r+1}{2} ]}𝔑(β) ⊂ 𝔍^{[ \frac{r+1}{2} ]}(β)

by [BK] (3.1.10) so the assertion is

I_{G}(θ) = J_{−}^{[ \frac{r+1}{2} ]}(β) . B_{β} ∩ G . J_{−}^{[ \frac{r+1}{2} ]}(β), m < [\frac{r}{2}].

We certainly have

I_{G}(θ) ⊂ I_{G}(θ|H_{−}^{[ \frac{r}{2} ]+1}) = J_{−}^{[ \frac{r+1}{2} ]}(β) . B_{β} ∩ G . J_{−}^{[ \frac{r+1}{2} ]}(β),

by the case m = [\frac{r}{2}] above. But also

I_{G}(θ|H_{−}^{m+1}) ⊃ I_{GL_{2N}} (θ|H^{m+1}) ∩ G = J^{[ \frac{r+1}{2} ]}(β)B_{β}^{×}J^{[ \frac{r+1}{2} ]}(β) ∩ G
⊃ J_{−}^{[ \frac{r+1}{2} ]}(β) . B_{β} ∩ G . J_{−}^{[ \frac{r+1}{2} ]}(β)

So we in fact have equality and we have completed the proof of (6.3.2). ■

(6.3.7) Corollary (of the proof) In the situation of (6.3.2) let 0 ≤ m ≤ [\frac{r}{2}]. Then

J^{m}(β).B_{β}^{×}.J^{m}(β) ∩ G = J_{−}^{m}(β) . B_{β} ∩ G . J_{−}^{m}(β).

(6.4) Heisenberg Representations

(6.4.1) Theorem (cf. [BK] (3.4.1)) Let 1 ≤ m ≤ r and let θ ∈ C_{−}(𝔄, m−1, β). The pairing

k_{θ} : (j, k) ↦ θ[j, k] j, k ∈ J_{−}^{m}(β)

induces a nondegenerate alternating bilinear form

J_{−}^{m}(β)/H_{−}^{m}(β) × J_{−}^{m}(β)/H_{−}^{m}(β) → ℂ^{×}.

Proof: We have [J_{−}^{m}(β), J_{−}^{m}(β)] ⊂ H_{−}^{2m}(β) ⊂ H_{−}^{m}(β). Moreover, J_{−}^{m}(β) normalizes θ so ker(θ) is a normal subgroup of J_{−}^{m}(β) and H_{−}^{m}(β)/ker(θ) is central in J_{−}^{m}(β)/ker(θ). So k_{θ} defines an alternating bilinear form on J_{−}^{m}(β)/H_{−}^{m}(β). For nondegeneracy, we need

θ[j, k] = 1 for all k ∈ J_{−}^{m}(β) if and only if h ∈ H_{−}^{m}(β).

The implication ⇐ is immediate.

We deal first with the case m = [\frac{r+1}{2}]. Let j, k ∈ J_{−}^{[ \frac{r+1}{2} ]}(β), so j = C(x) = 1 + x′,

k = C(y) = 1 + y′ for some x, y ∈ 𝔍_{−}^{[ \frac{r+1}{2} ]}(β) and x′, y′ ∈ 𝔍^{[ \frac{r+1}{2} ]}(β). Then, by [BK] (3.2.12),

θ[j, k] = ψ_{(1+x′)^{−1}β(1+x′)−β}(1 + y′).

We also have (1 + x′)^{−1}β(1 + x′) − β = a_{β}(x′) − (1 + x′)^{−1}x′a_{β}(x′), and x′a_{β}(x′) ∈ 𝔍^{[ \frac{r+1}{2} ]}(ℌ^{[ \frac{r}{2} ]+1})^{∗} by [BK] (3.1.17) and this is contained in (𝔍^{[ \frac{r+1}{2} ]})^{∗}. Also, as in the proof of (6.3.6), a_{β}(x′) − a_{β}(x) ∈ 𝔍^{[ \frac{r+1}{2} ]}(ℌ^{[ \frac{r}{2} ]+1})^{∗} + (ℌ^{[ \frac{r}{2} ]+1})^{∗}𝔍^{[ \frac{r+1}{2} ]} ⊂ (𝔍^{[ \frac{r+1}{2} ]})^{∗}. Altogether, we have

(1 + x′)^{−1}β(1 + x′) − β ≡ a_{β}(x) (mod (𝔍^{[ \frac{r+1}{2} ]})^{∗})

and, since both sides are skew, this is in fact (mod (𝔍_{−}^{[ \frac{r+1}{2} ]})^{∗}). So θ[C(x), C(y)] = 1, for all y ∈ 𝔍^{[ \frac{r+1}{2} ]}, if and only if a_{β}(x) ∈ (𝔍^{[ \frac{r+1}{2} ]})^{∗}, which, by [BK] (3.1.22), is if

and only if x ∈ (B_{β} + ℌ^{[ \frac{r}{2} ]+1}) ∩ 𝔍^{[ \frac{r+1}{2} ]} ∩ A_{−} = ℌ_{−}^{[ \frac{r+1}{2} ]}.

Now suppose m < [\frac{r+1}{2}] so J_{−}^{m}(β) = P^{m}(𝔅_{β}).J_{−}^{[ \frac{r+1}{2} ]}(β). Since P^{m}(𝔅_{β}) normalizes

θ, the commutator subgroups [P^{m}(𝔅_{β}), P^{m}(𝔅_{β})] and [P^{m}(𝔅_{β}), J_{−}^{[ \frac{r+1}{2} ]}(β)] are

both contained in ker(θ). Take j = uj′ ∈ J_{−}^{m}(β) with u ∈ P^{m}(𝔅_{β}), j′ ∈ J_{−}^{[ \frac{r+1}{2} ]}. Then θ[j, J_{−}^{m}] = 1 if and only if θ[j′, J_{−}^{m}] = 1 which, by the first part, implies

j′ ∈ H_{−}^{[ \frac{r+1}{2} ]}. Then j ∈ P^{m}(𝔅_{β}).H_{−}^{[ \frac{r+1}{2} ]} = H_{−}^{m}(β). Now suppose m > [\frac{r+1}{2}]. If β is minimal over F, this means m ≥ [\frac{r}{2}]+1 and J_{−}^{m} = H_{−}^{m} = P^{m}(𝔄) so the assertion is trivial. Otherwise, let [𝔄, n, r, γ] be a skew simple stratum equivalent to [𝔄, n, r, β]. We have [J_{−}^{m}(β), J_{−}^{m}(β)] ⊂ H_{−}^{2m}(β) by (6.1.5) and H_{−}^{2m}(β) = H_{−}^{2m}(γ). Moreover, 2m ≥ r + 1 so θ|H_{−}^{2m}(β) ∈ C_{−}(𝔄, 2m − 1, β). The result now follows by induction along β, since H_{−}^{m}(γ) = H_{−}^{m}(γ). ■

(6.4.2) Proposition (cf. [BK] (5.1.1)) Let [𝔄, n, 0, β] be a skew simple stratum in A, θ ∈ C_{−}(𝔄, 0, β). There exists a unique irreducible representation η(θ) of J_{−}^{1}(β, 𝔄) such that η(θ)|H_{−}^{1}(β, 𝔄) contains θ. Moreover, η(θ)|H_{−}^{1}(β, 𝔄) is a multiple of θ and

dim(η(θ)) = (J_{−}^{1}(β, 𝔄) : H_{−}^{1}(β, 𝔄))^{\frac{1}{2}} .

The G-intertwining of η(θ) is J_{−}^{1}(β, 𝔄).B ∩ G.J_{−}^{1}(β, 𝔄).

Proof: Given (6.4.1), the proof is identical to that of [BK] (5.1.1). ■

A SPECIAL CASE

In this chapter we look at the case where the element β in our skew simple stratum [𝔄, n, 0, β] generates a maximal field extension E = F [β] over F in A. In this case we are able to complete the construction of the type and construct supercuspidal representations of G. In the case where E/F is wildly ramified, these are new supercuspidals.

(7.1) Construction of types

Let [𝔄, n, 0, β] be a skew simple stratum in A such that E = F [β] is a maximal field extension of F in A. Then the centralizer B of E in A is just the field E itself and B ∩ G = P (𝔬_{E}) = N_{1}(E) is the group of norm-1 elements of E (for the norm N_{E/E_{0}}, where E_{0} is the fixed field of the involution) and 𝔅 = 𝔄 ∩ B = 𝔬_{E}.

Put r = −k_{0}(β, 𝔄) and let [𝔄, n, r, γ] be a skew simple stratum equivalent to [𝔄, n, r, β]. Then we have

H_{−}^{1}(β) = P^{1}(𝔬_{E})H_{−}^{[ \frac{r}{2} ]+1}(γ);
J_{−}^{1}(β) = P^{1}(𝔬_{E})J_{−}^{[ \frac{r+1}{2} ]}(γ);
J_{−}(β) = P (𝔬_{E})J_{−}^{[ \frac{r+1}{2} ]}(γ).

In particular, J_{−}(β) = P (𝔬_{E})J_{−}^{1}(β) and J_{−}(β)/J_{−}^{1}(β) ≃ P (𝔬_{E})/P^{1}(𝔬_{E}) is a finite cyclic group. (It is isomorphic to P (k_{E}) = \{x ∈ k_{E}^{×} : x\overline{x} = 1\}, where is the involution on k_{E} induced by the involution on 𝔬_{E}. If E/E_{0} is ramified, this is just ℤ/2ℤ; if E/E_{0} is unramified, this is ℤ/(q_{0} + 1)ℤ, where q_{0} = #k_{E_{0}}.)

Let θ ∈ C_{−}(𝔄, 0, β) be a simple character and let η be the Heisenberg representation of J_{−}^{1}(β) containing θ given by (6.4.2). Hence the intertwining of η is I_{G}(η) = J_{−}^{1}(β).B ∩ G.J_{−}^{1}(β) = P (𝔬_{E})J_{−}^{1}(β) = J_{−}(β).

Now J_{−}(β)/J_{−}^{1}(β) is cyclic so we can extend η to a representation κ of J_{−}(β); indeed, all such extensions take the form κ⊗χ, for χ a character of P (𝔬_{E})/P^{1}(𝔬_{E}).

(7.1.1) Theorem With notation as above, put π = Ind_{J_{−}}^{G}κ ,where J_{−} = J_{−}(β). Then π is an irreducible supercuspidal representation of G. Moreover, (J_{−}, κ) is a [G, π]_{G}-type

Proof: J_{−} is a compact open subgroup of G so certainly compact mod centre and we clearly have I_{G}(κ) ⊂ I_{G}(η) = J_{−}. Then, by [Ca] (1.5), π is irreducible and supercuspidal. Finally, (J_{−}, κ) is a [G, π]_{G}-type by [BK2] (5.4). ■

(7.2) Transfer

We recall from (6.2) that we have the involution σ : A^{×} → A^{×} given by σ(x) = \overline{x}^{−1}. We put Γ = ⟨σ⟩, a cyclic group of order 2. Then, for 0 ≤ m ≤ n − 1, we have a bijection C(𝔄, m, β)^{Γ} ⟶^{∼} C_{−}(𝔄, m, β), given by restriction of characters, as in (6.2.12).

Let [𝔄, n, 0, β] be a skew simple stratum such that E = F [β] is a totally ramified maximal field extension of F in A. We fix a simple character θ ∈ C(𝔄, 0, β)^{Γ} and write θ_{−} = θ|_{H_{−}^{1}(β,𝔄)}.

By [BK] (5.1.1), there exists a unique irreducible representation η of J^{1} = J^{1}(β, 𝔄) which contains θ. Further, there exist precisely two extensions of η to J(β, 𝔄) which are fixed by σ; they are κ_{θ}^{1}, given by

κ_{θ}^{1}(ξj) = η(j), for ξ ∈ μ′_{p}(F), j ∈ J^{1},

where μ′_{p}(F) is the group of roots of unity in F of order prime to p; and κ_{θ}^{2}, given by

κ_{θ}^{2}(ξj) = χ(ξ)η(j), for ξ ∈ μ′_{p}(F), j ∈ J^{1},

where χ is the character of μ′_{p}(F) given by

χ(ξ) = \{ \begin{matrix}1 & if ξ is a square in F; \\ −1 & if ξ is a non-square in F. \end{matrix}

The situation in the symplectic group is similar. By (6.4.2), there exists a unique irreducible representation η_{−} of J_{−}^{1} = J_{−}^{1}(β, 𝔄) which contains θ_{−}. (In fact, this is just the unique irreducible component of the restriction of η to J_{−}^{1}.) Further, there exist precisely two extensions of η_{−} to J_{−}(β, 𝔄); they are κ_{θ,−}^{1}, given by

κ_{θ,−}^{1}(ξj) = η(j), for ξ = ±1, j ∈ J_{−}^{1};

and κ_{θ,−}^{2}, given by

κ_{θ,−}^{2}(ξj) = ξη(j), for ξ = ±1, j ∈ J_{−}^{1}.

Now let π be an irreducible supercuspidal representation of GL_{N}(F) = A^{×}. Then π contains a simple type (J, λ) which is unique up to conjugacy in GL_{N}(F). This type is built from a simple character θ, which again is unique up to conjugacy in GL_{N}(F). Suppose that there is a skew simple stratum [𝔄, n, 0, β] in A such that θ ∈ C(𝔄, 0, β) and such that E = F [β] is a totally ramified maximal extension of F in A. Then λ = κ, a β-extension of the unique irreducible representation η of J^{1} = J^{1}(β, 𝔄) which contains θ. Suppose that the (equivalence class of the) representation κ is fixed by the involution σ. In particular, this implies that θ ∈ C(𝔄, 0, β)^{Γ}. Then by the discussion above, κ = κ_{θ}^{i}, for i = 1 or 2; this is determined by the restriction of the central character ω_{π} of π to μ′_{p}(F). We put i(π) = 1 if ω_{π} is trivial on μ′_{p}(F), i(π) = 2 otherwise; then κ = κ_{θ}^{i(π)}.

Put θ_{−} = θ|_{H_{−}^{1}(β,𝔄)}. This character is not (in general) determined up to conjugacy in G, or even up to intertwining. Hence, in order to describe the representations of G obtained from π, we must consider all the simple types contained in π. Let K be the set of κ such that (J, κ) is a simple type contained in π, κ is fixed by σ and κ is built from a skew simple stratum [𝔄, n, 0, β]. Let Θ be the set of simple characters θ such that θ is contained in κ, for some κ ∈ K. We put Θ_{−} to be the set of restrictions θ|_{H_{−}^{1}} for θ ∈ Θ. The group G acts on Θ_{−} by conjugation. Let ϑ be a set of representatives for this action.

We now put 𝔨 = {κ_{θ,−}^{i(π)} : θ_{−} ∈ ϑ}. Each κ_{−} ∈ 𝔨 induces to an irreducible supercuspidal representation π_{κ_{−}} of G, by (7.1.1). Note that the representations π_{κ_{−}} are not necessarily inequivalent, since we do not have an intertwining implies conjugacy theorem. Finally, we put Π = Π(π) = {π_{κ_{−}} : κ_{−} ∈ 𝔨}. Hence we have associated to π a set of irreducible supercuspidal representations of 𝔾.

82

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