Some Endoscopic Properties of The

Essentially Tame Jacquet-Langlands

Correspondence

Kam-Fai Tam

Received: September 28. 2015

Communicated by Takeshi Saito

Abstract. Let F be a non-Archimedean local field of characteristic 0 and G be an inner form of the general linear group G^{∗} = GL_{n} over F. We show that the rectifying character appearing in the essentially tame Jacquet-Langlands correspondence of Bushnell and Henniart for G and G^{∗} can be factorized into a product of some special characters, called zeta-data in this paper, in the theory of endoscopy of Langlands and Shelstad. As a consequence, the essentially tame local Langlands correspondence for G can be described using admissible embeddings of L-tori.

2010 Mathematics Subject Classification: Primary 22E50; Secondary 11S37, 11F70. Keywords and Phrases: essentially tame Jacquet-Langlands correspondence, inner forms, admissible pairs, zeta-data, endoscopy, admissible embeddings

Contents

1 Introduction  346
1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347
1.2 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348
1.3 Relation with the previous results . . . . . . . . . . . . . . . . . 350
1.4 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352

2 Some basic setups  352
2.1 Root system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352
2.2 Galois groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354
2.3 Division algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . 355
2.4 Hereditary orders in central simple algebra . . . . . . . . . . . 356
2.5 Embedding conditions . . . . . . . . . . . . . . . . . . . . . . . 357
2.6 Characters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358
3 The essentially tame Jacquet-Langlands correspondence 358
3.1 Admissible characters . . . . . . . . . . . . . . . . . . . . . . . 358
3.2 The correspondences . . . . . . . . . . . . . . . . . . . . . . . . 359
3.3 Some subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . 361
3.4 Simple characters . . . . . . . . . . . . . . . . . . . . . . . . . . 362
3.5 Local constructions of attached pairs . . . . . . . . . . . . . . . 363
3.6 Finite symplectic modules . . . . . . . . . . . . . . . . . . . . . 364
3.7 Invariants of finite symplectic modules . . . . . . . . . . . . . . 366
3.8 Values of rectifiers . . . . . . . . . . . . . . . . . . . . . . . . . 367
4 Finite symplectic modules  368
4.1 Standard modules of central simple algebra . . . . . . . . . . . 368
4.2 Complete decomposition of finite symplectic modules . . . . . . 371
4.3 Some properties of parities of jumps . . . . . . . . . . . . . . . 373
4.4 Symmetric submodules . . . . . . . . . . . . . . . . . . . . . . . 373
4.4.1 Case when f is odd . . . . . . . . . . . . . . . . . . . . 373
4.4.2 Case when f is even . . . . . . . . . . . . . . . . . . . . 374
4.4.3 Case when both f and m are even . . . . . . . . . . . . 374
4.4.4 Case when f is even and m is odd . . . . . . . . . . . . 374
4.5 t-factors of isotypic components . . . . . . . . . . . . . . . . . . 375
5 Zeta-data  377
5.1 Admissible embeddings of L-tori . . . . . . . . . . . . . . . . . 377
5.2 Symmetric unramified zeta-data . . . . . . . . . . . . . . . . . 379
5.3 Zeta-data associated to admissible characters . . . . . . . . . . 380
5.3.1 Case when f is odd . . . . . . . . . . . . . . . . . . . . 381
5.3.2 Case when f is even . . . . . . . . . . . . . . . . . . . . 382
5.4 The main theorem . . . . . . . . . . . . . . . . . . . . . . . . . 383
5.5 Functorial property . . . . . . . . . . . . . . . . . . . . . . . . . 384

1 Introduction

Let G^{∗} be a general linear group over a non-Archimedean local field of characteristic 0, and G be an inner form of G^{∗}. In this paper, we refine the results of the rectifying characters in the context of the essentially tame Jacquet-Langlands correspondence [BH11] by proving that each rectifying character admits a factorization into a product of characters called ζ-data, defined similarly to χ-data in [LS87], which are significant in describing the essentially tame local Langlands correspondence for G. We know from [BH11] that the rectifying characters are quadratic characters that measure the difference between two correspondences for essentially tame supercuspidal representations of G and G^{∗}: the representation theoretic one by matching the maximal simple types of the two groups, and the functorial one from the Jacquet-Langlands correspondence. On the representation theoretic side, the maximal simple types of G can be constructed using certain characters of its elliptic maximal tori, while on the functorial side, the Langlands parameters for G can be functorially lifted from the parameters of the same collection of characters. Combining these results with our result on rectifying characters, we show that the essentially tame local Langlands correspondence for G can be described completely by admissible embeddings, defined in [LS87], of the L-groups of elliptic maximal tori into the L-group of G, generalizing an analogous description proved by the author [Tam] in the split case (when G = G^{∗}).

1.1 Background

Let F be a non-Archimedean local field, G^{∗} be the group GL_{n} defined over F, and G be an inner form of G^{∗} defined over F. The set of F -points G(F) of G is therefore isomorphic to GL_{m}(D) as a group, where D is a central division algebra over F of dimension d^{2} and m = n/d. Let A_{m}^{2}(D) (resp. A_{n}^{2}(F)) be the discrete series of G(F) (resp. G^{∗}(F)), i.e., the set of equivalence classes of irreducible admissible representations that are essentially square integrable mod-center. The Jacquet-Langlands correspondence asserts a bijection

JL : A_{n}^{2}(F) → A_{m}^{2}(D)

determined by a character relation (see (3.5)) between a representation in A_{n}^{2}(F) and its image in A_{m}^{2}(D). The existence of this bijection is known, starting from the case n = 2 [JL70], when G(F) is the multiplicative group of the quaternion algebra over F. For arbitrary n, when G(F) is the multiplicative group of a division algebra, the existence is proved by [Rog83]. The general situations are treated by [DKV84] in the characteristic zero case and by [Bad02], [BHL10] in the positive characteristic case. Bushnell and Henniart describe in [BH11, (2.1)] the image of JL when it is restricted to the subset A_{n}^{0}(F) of supercuspidal representations. The image is the subset of representations in A_{m}^{2}(D), each of whose parametric degree is equal to n. We do not need the full definition of the parametric degree of a representation, so we only refer to [BH11, Section 2.7] for details. We only need to know that

• the parametric degree of a representation in A_{m}^{2}(D) is a positive integer divisor of n, • a representation in A_{m}^{2}(D) is supercuspidal if its parametric degree is n;

the converse is true in the split case (when G = G^{∗}) but not in general, and

• the parametric degree is preserved under JL.

Furthermore, we can describe the image of JL of each representation π ∈ A_{n}^{0}(F) when π is essentially tame, a notion we will explain in Section 3.2. More precisely, if we let A_{m}^{et}(D) (resp. A_{n}^{et}(F)) be the set of essentially tame representations in A_{m}^{2}(D) (resp. A_{n}^{2}(F)) whose parametric degree is n, then we can describe completely the essentially tame Jacquet-Langlands correspondence:

JL : A_{n}^{et}(F) → A_{m}^{et}(D),

as in [BH11, Theorem A]. To explain the theorem and describe JL completely, we require the notion of admissible characters from [How77]. In Section 3.1, we define the set P_{n}(F) of (equivalence classes of) admissible pairs (E/F, ξ) ∈ P_{n}(F), where E/F is a tamely ramified extension of degree n and ξ is a character of E^{×} admissible over F. This set bijectively parametrizes both A_{n}^{et}(F) and A_{m}^{et}(D) explicitly [BH11], using the theory of simple types of G(F) developed in [BF85], [Gra07], [Séc04], [Séc05a], [Séc05b], [SS08], [BSS12] which generalizes the corresponding theory in the split case [BK93], [BH96] and the division algebra case [Zin92], [Bro96]. If we denote by

F Π : P_{n}(F) → A_{n}^{et}(F), (E/F, ξ) ↦ _{F} Π_{ξ}  (1.1)

and

DΠ : P_{n}(F) → A_{m}^{et}(D), (E/F, ξ) ↦ _{D}Π_{ξ}  (1.2)

the above bijections, then Bushnell and Henniart proved in [BH11] that the composition

ν : P_{n}(F) ⟶^{_{F} Π} A_{n}^{et}(F) ⟶^{JL} A_{m}^{et}(D) ⟶^{_{D}Π^{−1}} P_{n}(F)

maps an admissible pair (E/F, ξ) ∈ P_{n}(F) to another pair of the form (E/F, ξ · Dν_{ξ}), where _{D}ν_{ξ} is a tamely ramified character _{D}ν_{ξ} of E^{×} depending on ξ. We borrow the terminology from [BH10] and call the character _{D}ν_{ξ} the rectifier of ξ for the essentially tame Jacquet-Langlands correspondence.

1.2 Main results

The main result of this paper is to relate the rectifier _{D}ν_{ξ} with a special set of characters, called ζ-data in this paper, introduced in the theory of endoscopy of Langlands and Shelstad [LS87]. The significance of ζ-data will be explained in the next section, together with a brief summary of the previous results of the author [Tam].

To describe the main result, we first assume that char(F) = 0 (see Remark 1.3 about this assumption). Let T be the F -torus such that T (F) = E^{×}. We regard T as a maximal torus embedded in G. In contrast to the split case, we have to carefully choose the embedding T → G relative to an hereditary 𝔬_{F} -order in G(F) associated to ξ. This will be explained in Section 2.5. Given this embedding, let Φ = Φ(G, T) be the root system, which is invariant under the action of the absolute Galois group Γ_{F} of F if we view Φ as a subset of the character group of T. For each root λ ∈ Φ, we denote by E_{λ} the fixed field of the stabilizer of λ in Γ_{F}, so that E_{λ} is a field extension of a Γ_{F} -conjugate of E. We recall from [LS87, Corollary 2.5.B] that ζ-data is a set of characters

{ζ_{λ}} = {ζ_{λ}}_{λ∈Γ_{F} \Φ},

where each ζ_{λ} is a character of E_{λ}^{×} satisfying the conditions in loc. cit. (and will be recalled in Section 5.1). Here λ ranges over a suitable subset of roots in Φ, denoted by Γ_{F} \Φ for the moment, representing the Γ_{F} -orbits of Φ and such that E_{λ} is a field extension of E (but not just its conjugate). The following theorem restates the main result, Theorem 5.5, in a simpler way.

Theorem 1.1. Given a character ξ of E^{×} admissible over F.

(i) There exists a set of ζ-data {ζ_{λ,ξ}}_{λ∈Γ_{F} \Φ} such that

Dν_{ξ} = ∏_{λ∈Γ_{F} \Φ} ζ_{λ,ξ}|_{E^{×}}.

(ii) The values of each ζ_{λ,ξ} can be expressed in terms of certain invariants, called t-factors in this paper, of the corresponding component in the complete symmetric decomposition of the finite symplectic modules associated to ξ (see the notations and definitions in Sections 4.2 and 3.7).

We explain statement (ii) of the above theorem. The finite symplectic modules appear in the respective constructions of the extended maximal simple types inducing _{F} Π_{ξ} and _{D}Π_{ξ} in (1.1) and (1.2) (see [BH11, (2.5.4)], or the summary in Section 3.5). Each of these modules admits an orthogonal decomposition, called a complete symmetric decomposition in this paper (Proposition 4.4), whose components are parametrized by the same set Γ_{F} \Φ parameterizing the factors in (i) of the Theorem. The t-factors are, roughly speaking, defined by the symplectic signs attached to these components. When proving Theorem 1.1, we pick a choice of characters {ζ_{λ,ξ}}_{λ∈Γ_{F} \Φ}, where each character ζ_{λ,ξ} has values in terms of the t-factors of the corresponding component, the one indexed by λ. We then show that these characters constitute a set of ζ-data. Moreover, using the multiplicativity of t-factors, the product of these ζ-data, when restricted to E^{×}, is equal to the rectifier _{D}ν_{ξ}, whose values are given in the First and Second Comparison Theorems of [BH11]. Hence our result refines the one in loc. cit.. While the finite symplectic modules and their decompositions are also studied by the author in the split case [Tam], there are extra conditions on the components of these modules in the general case. These conditions come from the extra ramifications of the related compact subgroups in constructing the extended maximal simple types. The degrees of these ramifications depend on the residue degree f(E/F) and, with other conditions similar to those in the split case, determine whether each component is trivial or not. This new phenomenon will be fully studied in Section 4. In particular, when E/F is totally ramified and ξ is fixed, the finite symplectic modules are isomorphic to each other for all inner forms of G^{∗}, a fact already known in [BH11, Proposition 5.6].

1.3 Relation with the previous results

The significance of the factorization of_{D}ν_{ξ} in Theorem 1.1(i) comes from [Tam], which proves an analogous factorization of the rectifier _{F} μ_{ξ} for the essentially tame local Langlands correspondence [BH05a]. We first recall from loc. cit. that the rectifier _{F} μ_{ξ} measures the difference between the “naïve correspondence” and the essentially tame Langlands correspondence for G^{∗}; more precisely, the Langlands parameter of _{F} Π_{ξ} defined in (1.1) is the induced representation

Ind_{W_{E}}^{W_{F}}(ξ ·_{F} μ_{ξ})  (1.3)

of the Weil group W_{F} of F, where ξ ·_{F} μ_{ξ} is regarded as a character of W_{E} by class field theory [Tat79]. In [Tam, Theorem 1.1], the author proved that the rectifier _{F} μ_{ξ} admits a factorization

F μ_{ξ} = ∏_{λ∈Γ_{F} \Φ} χ_{λ,ξ}|_{E^{×}},

where {χ_{λ,ξ}}_{λ∈Γ_{F} \Φ} is a set of χ-data, consisting of characters of E_{λ}^{×} satisfying the conditions similar to those of ζ-data (see Section 5.1). With a collection of χ-data, we follow [LS87, Section 2.6] to construct an admissible embedding

I_{{χ_{λ,ξ}}} : ^{L}T → ^{L}G

of the L-group ^{L}T of the maximal torus T into the L-group ^{L}G of G^{∗}. (For convenience, we call ^{L}T an L-torus in this paper.) Let ξ̃ : W_{F} → ^{L}T be an L-homomorphism whose class is the parameter of the character ξ of E^{×} = T (F) under the local Langlands correspondence of the torus T, i.e., the Artin reciprocity for E^{×} [Tat79]. In a previous result of the author [Tam, Corollary 1.2], the Langlands parameter (1.3) of _{F} Π_{ξ} is isomorphic to the composition

I_{{χ_{λ,ξ}}} ∘ ξ̃: W_{F} → ^{L}T → ^{L}G ⟶^{natural proj.} GL_{n}(ℂ)

as a representation of W_{F}. In other words, the essentially tame local Langlands correspondence for G^{∗} can be described by admissible embeddings of L-tori. A set of ζ-data is the ‘difference’ of two sets of χ-data, in the sense that, given a set {χ_{λ}} of χ-data, we have

{χ′_{λ}} is another set of χ-data ⇔ {χ_{λ}(χ′_{λ})^{−1}} is a set of ζ-data.

If we define the local Langlands correspondence for G as the composition of the local Langlands correspondence for G^{∗} and the Jacquet-Langlands correspondence JL, then we can express the Langlands parameter of _{D}Π_{ξ} using an admissible embedding of L-tori, as follows.

Corollary 1.2. Let {χ_{λ,ξ}} and {ζ_{λ,ξ}} be respectively the χ-data and the ζdata associated to an admissible character ξ. The Langlands parameter

Ind_{W_{E}}^{W_{F}}(ξ ·_{F} μ_{ξ} ·_{D}ν_{ξ})

of _{D}Π_{ξ} is isomorphic to

I_{{χ_{λ,ξ}·ζ_{λ,ξ}}} ∘ ξ̃: W_{F} → ^{L}T → ^{L}G ⟶^{natural proj.} GL_{n}(ℂ)

as a representation of W_{F}.

Hence analogously we can describe the essentially tame local Langlands correspondence for G by admissible embeddings of L-tori. As a consequence, we show in Proposition 5.7 that the factorization of _{D}ν_{ξ} in Theorem 1.1(i) is functorial, in the following sense. Let K/F be an intermediate extension of E/F, so that if the pair (E/F, ξ) is admissible over F, then (E/K, ξ) is admissible over K by definition. We denote the centralizer of K^{×} in G(F) by GL_{m_{K}} (D_{K}), where D_{K} is a K-division algebra and m_{K} a positive integer. If {ζ_{λ,ξ}} is the set of ζ-data associated to ξ, then the partial product

∏_{λ∈Γ_{F} \Φ, λ|_{K^{×}} ≠1} ζ_{λ,ξ}|_{E^{×}}.

(a product similar to Theorem 1.1.(i), with factors ranging over the characters being non-trivial on K^{×}) is the rectifier _{D_{K}} ν_{ξ} of ξ over K.

Remark 1.3. We would like to remark on the condition of the characteristic char(F) = 0, as we also did in [Tam, Remark 1.3]. The readers should be aware that the works of [JL70], [DKV84], [Bad02], [BH11] make the Jacquet-Langlands correspondence valid for local fields of arbitrary characteristic. In our paper, we apply the condition char(F) = 0 only because we refer to the theory of endoscopy from [LS87], [KS99]. However, we do not actually need this condition for the part of the theory that we allude to, which is about the admissible embeddings of L-tori. In [Tam, Section 6] (or rather [LS87, Section 2.5]), we see that these kind of embeddings can be defined without any condition of char(F). Therefore, the condition char(F) = 0 in this paper should be treated as a mild condition.

1.4 Notations

Throughout the paper, F denotes a non-Archimedean local field of characteristic 0. Its ring of integers is 𝔬_{F} with the maximal ideal 𝔭_{F}. The residue field k_{F} = 𝔬_{F} /𝔭_{F} has q elements and is of characteristic p. We denote by v_{F} : F^{×} → ℤ the discrete valuation on F. We denote by Γ_{F} the absolute Galois group of F, and by W_{F} the Weil group of F. The multiplicative group F^{×} decomposes into a product of subgroups

⟨ϖ_{F} ⟩ × μ_{F} × U_{F}^{1} .

They are namely the group generated by a prime element ϖ_{F}, the group μ_{F} of roots of unity of order prime to p, and the 1-unit group U_{F}^{1} := 1 + 𝔭_{F}. We will identify μ_{F} with k_{F}^{×} in the natural way. We then write U_{F} = U_{F}^{0} := μ_{F} × U_{F}^{1} and U_{F}^{i} := 1 + 𝔭_{F}^{i} for each positive integer i. Let μ_{n} be the group of nth roots of unity in the algebraic closure F̄ of F, and z_{n} be a choice of primitive nth root in μ_{n}. The F -level of a character ξ of F^{×} is the smallest integer a ≥ −1 such that ξ|_{U_{F}^{a+1}} is trivial. A character ξ of F^{×} is called unramified if ξ|_{U_{F}} is trivial, or equivalently, if its F -level is −1. It is called tamely ramified if ξ|_{U_{F}^{1}} is trivial, or equivalently, if its F -level is 0. Given a field extension E/F, we denote its ramification index by e = e(E/F) and its residue degree by f = f(E/F). We also denote by tr_{E/F} and norm N_{E/F} the trace and norm respectively. We fix an additive character ψ_{F} of F of level 0, which means that ψ_{F} is trivial on 𝔭_{F} but is non-trivial on 𝔬_{F}. Hence ψ_{F} |_{𝔬_{F}} induces a non-trivial character of k_{F}. We write ψ_{E} = ψ_{F} ∘ tr_{E/F}. Suppose that A is a central simple algebra over F. We denote the reduced trace by trd_{A/F} and the reduced norm by Nrd_{A/F}. Given a set X, we denote its cardinality by #X. If H is a group and X is a H-set, then we denote the action of h ∈ H on x ∈ X by x ↦ ^{h}x. The set of H-orbits is denoted by H\X. If π is a representation of H (over a given field), we denote its equivalence class by (H, π).

2 Some basic setups

2.1 Root system

Given a field extension E/F of degree n, we let T be the induced torus Res_{E/F} 𝔾_{m}. We embed T into G as an elliptic maximal torus, and denote the image still by T. The choice of this embedding will be specific in Section 2.5, but at this moment this choice is irrelevant. Let Φ = Φ(G, T) be the root system of T in G. Following [Tam, Section 3.1], we can denote each root in Φ by [ \begin{matrix}g \\ h \end{matrix}] where g = gΓ_{E} and h = hΓ_{E} are distinct cosets in Γ_{F} /Γ_{E}. (We use the same notation g for an element in Γ_{F} and its Γ_{E}-cosets, for notation convenience.) The Γ_{F} -action on Φ is given by x · [ \begin{matrix}g \\ h \end{matrix}] = [ \begin{matrix}xg \\ xh \end{matrix}] . For each root λ ∈ Φ, we denote by [λ] its Γ_{F} -orbit W_{F} λ. Each Γ_{F} -orbit contains a root of the form [ \begin{matrix}1 \\ g \end{matrix} ] for some non-trivial coset g ∈ Γ_{F} /Γ_{E}. For each root λ ∈ Φ, we denote the stabilizers {g ∈ Γ_{F} |^{g}λ = λ} and {g ∈ Γ_{F} |^{g}λ = ±λ} by Γ_{λ} and Γ_{±λ} respectively and their fixed fields by E_{λ} and E_{±λ} respectively. We call a root λ symmetric if [E_{λ} : E_{±λ}] = 2, and asymmetric otherwise. Equivalently, λ is symmetric if and only if λ and −λ are in the same Γ_{F} -orbit. Note that the symmetry of Φ is preserved by the Γ_{F} -action. Let

(i) Γ_{F} \Φ_{sym} be the set of Γ_{F} -orbits of symmetric roots,

(ii) Γ_{F} \Φ_{asym} be the set of Γ_{F} -orbits of asymmetric roots, and

(iii) Γ_{F} \Φ_{asym/±} be the set of equivalence classes of asymmetric Γ_{F} -orbits by identifying [λ] and [−λ].

We denote by (Γ_{E}\Γ_{F} /Γ_{E})′ the collection of non-trivial double cosets, and by [g] the double coset Γ_{E}gΓ_{E}. We can deduce the following proposition easily.

Proposition 2.1. The map

Γ_{F} \Φ → (Γ_{E}\Γ_{F} /Γ_{E})′, [λ] = W_{F} [ \begin{matrix}1 \\ g \end{matrix} ] ↦ [g],

is a bijection between the set Γ_{F} \Φ of Γ_{F} -orbits of the root system Φ and the set (Γ_{E}\Γ_{F} /Γ_{E})′ of non-trivial double cosets.

We can therefore call g ∈ Γ_{F} symmetric if [g] = [g^{−1}], and asymmetric otherwise, so that the bijection in Proposition 2.1 preserves symmetries on both sides. Let

(i) (Γ_{E}\Γ_{F} /Γ_{E})_{sym} be the set of symmetric non-trivial double cosets,

(ii) (Γ_{E}\Γ_{F} /Γ_{E})_{asym} be the set of asymmetric non-trivial double cosets, and

(iii) (Γ_{E}\Γ_{F} /Γ_{E})_{asym/±} be the set of equivalence classes of (Γ_{E}\Γ_{F} /Γ_{E})_{asym} by identifying [g] with [g^{−1}].

We choose subsets D_{sym} and D_{asym/±} of representatives in Γ_{F} /Γ_{E} of (Γ_{E}\Γ_{F} /Γ_{E})_{sym} and (Γ_{E}\Γ_{F} /Γ_{E})_{asym/±} respectively, and write

D_{asym} = D_{asym/±} ⊔ {g^{−1}|g ∈ D_{asym/±}}.

We also choose subsets R_{sym} and R_{asym/±} of representatives in Φ of orbits in Γ_{F} \Φ_{sym} and Γ_{F} \Φ_{asym/±} respectively such that every root λ ∈ R_{±} := R_{sym} ⊔ R_{asym/±} is of the form [ \begin{matrix}1 \\ g \end{matrix} ] for some g ∈ D_{±} := D_{sym} ⊔ D_{asym/±}, and write

R_{asym} = R_{asym/±} ⊔ (−R_{asym/±}).

Hence R_{sym}, R_{asym}, and R_{asym/±} correspond bijectively to D_{sym}, D_{asym}, and D_{asym/±} respectively by the identification in Proposition 2.1. Denote E_{g} := E_{λ} and E_{±g} := E_{±λ}. Notice that E_{g} = E(^{g}E), composite field of E and ^{g}E.

2.2 Galois groups

Let E/F be a field extension of degree n. In most of the paper, we assume that E/F is tamely ramified, which means that p is coprime to e. By [Lan94, II.§5], we can choose ϖ_{E} and ϖ_{F} such that

ϖ_{E}^{e} = z_{E/F} ϖ_{F}, for some z_{E/F} ∈ μ_{E}.  (2.1)

Choose in F̄^{×} a primitive eth root of unity z_{e} and an eth root z_{E/F,e} of z_{E/F}. (We do not require that z_{E/F,e}^{a} = z_{e}, if a is the multiplicative order of z_{E/F}.) Denote L = E[z_{e}, z_{E/F,e}] and l = [L : E]. With the choices of ϖ_{F} and ϖ_{E} as in (2.1), we define the following F -automorphisms on L.

(i) ϕ : z ↦ z^{q}, for all z ∈ μ_{L}, and ϕ : ϖ_{E} ↦ z_{ϕ}ϖ_{E}.

(ii) σ : z ↦ z, for all z ∈ μ_{L}, and σ : ϖ_{E} ↦ z_{e}ϖ_{E}.

Here z_{ϕ} lies in μ_{E} satisfying (z_{ϕ}ϖ_{E})^{e} = z_{E/F}^{q} ϖ_{F}. More generally, we write

ϕ^{i}ϖ_{E} = z_{ϕ^{i}}ϖ_{E} where z_{ϕ^{i}} = z_{ϕ}^{1+q+···+q^{i−1}} is an eth root of z_{E/F}^{q^{i}−1}. Therefore, Γ_{L/F} = ⟨σ⟩ ⋊ ⟨ϕ⟩ with relation ϕ ∘ σ ∘ ϕ^{−1} = σ^{q}. Suppose that Γ_{L/E} = ⟨σ^{c}ϕ^{f}⟩ for some integer c satisfying the condition:

e divides c (\frac{q^{fl} − 1}{q^{f} − 1}) .

We can choose

{σ^{i}ϕ^{j}|i = 0, …, e − 1, j = 0, …, f − 1}

as coset representatives for the quotient Γ_{E/F} = Γ_{F} /Γ_{E}. Moreover, elements in a fixed double coset are of the form [σ^{i}ϕ^{j}] with a fixed j mod f.

Proposition 2.2 ([Tam, Proposition 3.3]). The double coset [g] = [σ^{i}ϕ^{j}] is symmetric only if j = 0 or, when f is even, j = f/2.

We call those symmetric [σ^{i}] ramified and those symmetric [σ^{i}ϕ^{f/2}] unramified, and denote by (Γ_{E}\Γ_{F} /Γ_{E})_{sym−ram} and (Γ_{E}\Γ_{F} /Γ_{E})_{sym−unram} respectively the collections of symmetric ramified and symmetric unramified double cosets. We provide several useful results concerning the parity of certain subsets in Γ_{E}\Γ_{F} /Γ_{E}.

Proposition 2.3 ([Tam, Propositions 3.4 and 3.5]). (i) If [g] is symmetric unramified, then the degree [E_{g} : E] is odd.

(ii) The parity of #(Γ_{E}\Γ_{F} /Γ_{E})_{sym−unram} is equal to that of e(f − 1).

Lemma 2.4. Suppose that f is even. The following are equivalent.

(i) There exists σ^{i}ϕ^{f/2} ∈ W_{F [ϖ_{E}]} for some i.

(ii) z_{ϕ^{f/2}} is an eth root of unity.

(iii) z_{E/F} ∈ K_{+}, where K_{+}/F is unramified of degree f/2.

(iv) f_{ϖ} := [E : F [ϖ_{E}]] is even.

Proof. (i) is equivalent to (ii) since ^{σ^{i}ϕ^{f/2}}ϖ_{E} = z_{e}^{i}z_{ϕ^{f/2}}ϖ_{E}. To show that (iii)

implies (ii), we recall that z_{ϕ^{f/2}} is an eth root of z_{E/F}^{q^{f/2}−1}. If z_{E/F} ∈ K_{+}, then

z_{E/F}^{q^{f/2}−1} = 1 and z_{ϕ^{f/2}} is an eth root of unity. The converse is similar. To show the equivalence of (iii) and (iv), we notice that f(F [ϖ_{E}]/F) = f(F [z_{E/F}]/F) = f/f_{ϖ}. Hence that F [z_{E/F}] ⊆ K_{+} is equivalent to that f_{ϖ} is even.

Lemma 2.5. Suppose that g = σ^{i}ϕ^{f/2} satisfies the conditions in Lemma 2.4.

(i) The double coset [σ^{i}ϕ^{f/2}] is automatically symmetric.

(ii) The set (Γ_{E}\Γ_{F [ϖ_{E}]}/Γ_{E})_{sym−unram} = (Γ_{E}\Γ_{F} /Γ_{E})_{sym−unram} ∩ (Γ_{E}\Γ_{F [ϖ_{E}]}/Γ_{E}) contains a single element [σ^{i}ϕ^{f/2}].

Proof. For (i), we consider the actions of σ^{i}ϕ^{f/2} and its inverse (σ^{i}ϕ^{f/2})^{−1} on E. We certainly have ^{σ^{i}ϕ^{f/2}}ϖ_{E} = ^{(σ^{i}ϕ^{f/2})^{−1}}ϖ_{E} = ϖ_{E} by definition. We also have ^{σ^{i}ϕ^{f/2}}z = ^{(σ^{i}ϕ^{f/2})^{−1}}z = z^{q^{f/2}} for all z ∈ μ_{E}. Therefore σ^{i}ϕ^{f/2}Γ_{E} = (σ^{i}ϕ^{f/2})^{−1}Γ_{E} and in particular [σ^{i}ϕ^{f/2}] = [(σ^{i}ϕ^{f/2})^{−1}]. For (ii), we know by Lemma 2.4.(ii) that the double coset is the one containing σ^{i}ϕ^{f/2} where z_{ϕ^{f/2}} = z_{e}^{−i}.

Proposition 2.6. When f is even, the parity of the cardinality of (Γ_{E}\Γ_{F} /Γ_{E})_{sym−unram} − Γ_{F [ϖ_{E}]} is equal to e + f_{ϖ} − 1.

Proof. Recall

(i) by Lemma 2.4 that there exists σ^{i}ϕ^{f/2} ∈ W_{F [ϖ_{E}]} if and only if f_{ϖ} is even, and

(ii) by Proposition 2.3 that the parity of the number of symmetric [σ^{i}ϕ^{f/2}] is the same as that of e.

By combining these facts, we have the assertion.

2.3 Division algebra

Let D be a division algebra over F of dimension n^{2}. Denote its unique maximal order by 𝔬_{D} and the maximal ideal of 𝔬_{D} by 𝔭_{D}. Suppose that the Hasseinvariant of D is h = h(D), so that gcd(n, h) = 1. By [Rei03, (14.5) Theorem], we can choose a primitive (q^{n} − 1)th root z of unity in D and a uniformizer ϖ_{D} such that

ϖ_{D}^{n} = ϖ_{F} and ϖ_{D}zϖ_{D}^{−1} = z^{q^{h}}.  (2.2)

We write K_{n} = F [z] and μ_{D} = ⟨z⟩, then K_{n} is a maximal unramified extension (of degree n) in D, and μ_{D} is a group of roots of unity of order q^{n} − 1, both defined up to conjugacy by D^{×}. Therefore, the conjugation of ϖ_{D} acts on K_{n} as the hth power of the Frobenius automorphism, i.e.,

ϖ_{D}uϖ_{D}^{−1} = ^{ϕ^{h}}u for all u ∈ K_{n}.

We write U_{D}^{i} := 1 + 𝔭_{D}^{i} for all positive integer i. The multiplicative subgroup D^{×} hence decomposes into a semi-direct product

(⟨ϖ_{D}⟩ ⋉ μ_{D}) ⋉ U_{D}^{1}.

Let E/F be a tamely ramified field extension of degree n, and let K be the maximal unramified sub-extension in E/F. We assume that K ⊆ K_{n} and that the uniformizers ϖ_{E} and ϖ_{F} satisfy ϖ_{E}^{e} = z_{E/F} ϖ_{F} as in (2.1) for some z_{E/F} ∈ μ_{E}. If we define z_{D/E} ∈ μ_{D} = μ_{K_{n}} to be a solution of

N_{K_{n}/K}(z_{D/E}) = z_{E/F} ,  (2.3)

then we may take ϖ_{E} = ϖ_{D}^{f}z_{D/E} and this defines an embedding of E into D over F. Note that from (2.2)

zϖ_{D}^{i}z^{−1} = z^{1−q^{hi}}ϖ_{D}^{i}  (2.4)

for all z ∈ μ_{E} = μ_{K} and all i ∈ ℤ.

2.4 Hereditary orders in central simple algebra

If G is an F -inner form of G^{∗} = GL_{n}, then G(F) = A^{×}, where A be a central simple algebra over F. By Wedderburn Theorem [Rei03, (7.4)Theorem], A is isomorphic to Mat_{m}(D), where D is a division algebra of F -dimension d^{2} and md = n. Therefore, G(F) ≅ GL_{m}(D). Any field extension of degree n can be embedded into A as a maximal subfield in A, and any two such embeddings are conjugate under G(F). Let 𝔄 be an 𝔬_{F} -hereditary order in A, 𝔓_{𝔄} be its Jacobson radical, and 𝔎_{𝔄} be the G(F)-normalizer of 𝔄^{×}. If 𝔄 is principal, in the sense that there exists ϖ_{𝔄} ∈ 𝔎_{𝔄} such that ϖ_{𝔄}𝔄 = 𝔄ϖ_{𝔄} = 𝔓_{𝔄}, then the valuation v_{𝔄} : 𝔎_{𝔄} → ℤ is defined by x𝔄 = 𝔄x = 𝔓_{𝔄}^{v_{𝔄}(x)} for all x ∈ 𝔎_{𝔄}. We also write U_{𝔄} = U_{𝔄}^{0} = 𝔄^{×}, U_{𝔄}^{i} = 1 + 𝔓_{𝔄}^{i} for each positive integer i, U_{𝔄}^{x} = U_{𝔄}^{⌈x⌉} for all x ∈ ℝ_{≥0}, and

U_{𝔄}^{x+} = ⋃_{x∈ℝ_{≥0}, y>x} U_{𝔄}^{y}.

Suppose that E_{0} is a subfield in A and 𝔄 is E_{0}-pure, i.e., E_{0}^{×} ⊆ 𝔎_{𝔄}, then we define the ramification index e(𝔄/𝔬_{E_{0}}) to be the integer e satisfying v_{𝔄}|_{E_{0}}^{×} = ev_{E_{0}}. We therefore have

𝔭_{E_{0}}^{i}𝔓_{𝔄}^{j} = 𝔓_{𝔄}^{ie+j}

and

𝔭_{E_{0}}^{i} ∩ 𝔓_{𝔄}^{j} = 𝔭_{E_{0}}^{max{i,j/e}}

for all i, j ∈ ℤ. In the split case G = G^{∗}, i.e., when A = A^{∗} = Mat_{n}(F), we denote the hereditary order 𝔄 by 𝔄^{∗}.

2.5 Embedding conditions

Suppose we fix an F -embedding E_{0} ↪ A, let E_{0} and 𝔄 be as in the previous section, and write A_{0} the centralizer of E_{0} in A. We can restrict the embedding to E_{0}^{×} ↪ G(F), and denote the centralizer by A_{0}^{×} = Z_{G(F)}(E_{0}^{×}). Under the above setup, there are many choices of 𝔄 among its Z_{G(F)}(E_{0}^{×})-conjugacy class. In this paper, we assume the conditions (i)-(iii) below, all adopted from [BH11], to fix a unique 𝔄.

(i) [BH11, Section 3.2] If E/E_{0} is an unramified extension in A_{0} such that [E : F] = n, then we require that 𝔄 is E-pure, i.e., E^{×} ⊆ 𝔎_{𝔄}.

Let 𝔄_{0} be the centralizer of E_{0} in 𝔄, i.e., 𝔄_{0} = 𝔄 ∩ A_{0}, which is a hereditary 𝔬_{E_{0}}-order in A_{0}, with Jacobson radical 𝔓_{𝔄_{0}} = 𝔓_{𝔄} ∩ A_{0}.

(ii) [BH11, (2.3.1)(2)] There exists a fixed integer e(𝔄/𝔄_{0}) ≥ 1 such that

𝔓_{𝔄}^{k} ∩ A_{0} = 𝔓_{𝔄_{0}}^{k/e(𝔄/𝔄_{0})} for every k ∈ ℤ.  (2.5)

We say that 𝔄 is the canonical continuation of 𝔄_{0} in A. Under (ii), we have moreover 𝔎_{𝔄} ∩ A_{0} = 𝔎_{𝔄_{0}}.

(iii) [BH11, (2.3.2)] 𝔄_{0} is a maximal hereditary 𝔬_{E_{0}}-order in A_{0}, i.e., e(𝔄_{0}/𝔬_{E_{0}}) = 1.

Under these conditions, 𝔄 is the unique E_{0}-pure hereditary order in A such that 𝔄 ∩ A_{0} = 𝔄_{0}. Moreover, 𝔄 is maximal among all E_{0}-pure hereditary orders in A, and both 𝔄 and 𝔄_{0} are principal (by [BH11, the remark after (2.3.2)]). By [Zin99, 0. Theorem], if the 𝔬_{D}-period of 𝔄 is denoted by r = r(𝔄) = e(𝔄/𝔬_{D}), i.e. ϖ_{D}𝔄 = 𝔓_{𝔄}^{r}, then we have an isomorphism

𝔄/𝔓_{𝔄} ≅ Mat_{s}(k_{D})^{r},  (2.6)

where s = s(𝔄) = f/e(𝔄/𝔬_{E}). Once A (and hence D) is fixed, the integers r and s depend only on E; indeed s = s(E/F) = gcd(f, m) and r = r(E/F) = e/ gcd(d, e) = m/s. If K is an intermediate subfield in E/F, we write f_{K} = f(E/K) and e_{K} = e(E/K). By [Zin99, 1. Prop.] the centralizer A_{K} = Z_{A}(K) is isomorphic to Mat_{m_{k}}(D_{K}), where D_{K} is a division algebra over K of degree d_{K}^{2}, with

d_{K} = \frac{d}{gcd(d, n(K/F))} and m_{K} = gcd(m, n(E/K)).

Documenta Mathematica 21 (2016) 345–389

Let 𝔄_{K} be the centralizer of K in 𝔄. It is routine to check that (E_{0}K, 𝔄_{K}) is a canonical continuation of 𝔄_{E_{0}K} and is also E-pure, i.e., (i)-(iii) are satisfied when we change our base field from F to K. We therefore have an isomorphism

𝔄_{K}/𝔓_{𝔄_{K}} ≅ Mat_{s_{K}} (k_{D_{K}})^{r_{K}} ,  (2.7)

where r_{K} = e_{K}/ gcd(d_{K}, e_{K}) and s_{K} = gcd(f_{K}, m_{K}).

2.6 Characters

Let Nrd_{A/F} : A^{×} → F^{×} be the reduced norm of G(F) = A^{×}. By [Rei03, (33.4)Theorem] and [NM43, Satz 2], Nrd_{A/F} is surjective and its kernel is the commutator subgroup of G(F). Therefore, any character of G(F) factors through Nrd_{A/F}. We define the 𝔄-level of a character ξ of A^{×} as the smallest integer a ≥ −1 such that ξ|_{U_{𝔄}^{a+1}} is trivial. This is analogous to the F -level of a character of F^{×} defined in Section 1.4.

Proposition 2.7. If the F -level of a character ξ of F^{×} is r, then the 𝔄-level of the character ξ ∘ Nrd_{A/F} is r · e(𝔄/𝔬_{F}).

Proof. It suffices to show that

Nrd_{A/F} (U_{𝔄}^{r·e(𝔄/𝔬_{F})}) = U_{F}^{r} for all r ∈ ℝ_{≥0}.

This follows from [BF85, (2.8.3)].

3 The essentially tame Jacquet-Langlands correspondence

3.1 Admissible characters

We recall the definition of admissible characters in [How77], [Moy86]. Given a tamely ramified finite extension E/F, let ξ be a character of E^{×}. We call a pair (E/F, ξ) admissible over F if ξ is an admissible character over F, i.e., for every intermediate subfield K between E/F,

(i) if ξ factors through the norm N_{E/K}, then E = K;

(ii) if ξ|_{U_{E}^{1}} factors through N_{E/K} then E/K is unramified.

Two admissible pairs (E/F, ξ) and (E′/F, ξ′) are called F -equivalent if there is g ∈ Γ_{F} such that ^{g}E = E′ and ^{g}ξ = ξ′. Let P_{n}(F) be the set of F -equivalence classes of the admissible pair (E/F, ξ), with each class in P_{n}(F) still denoted by (E/F, ξ) for convenience. Every admissible character ξ admits a factorization (see [How77, Corollary of Lemma 11] or [Moy86, Lemma 2.2.4])

ξ = ξ_{−1}(ξ_{0} ∘ N_{E/E_{0}}) ⋯ (ξ_{t} ∘ N_{E/E_{t}})(ξ_{t+1} ∘ N_{E/F}),  (3.1)

where, in the notations above, the decreasing sequence of fields

E = E_{−1} ⊇ E_{0} ⊋ E_{1} ⊋ ⋯ ⊋ E_{t} ⊋ E_{t+1} = F.

and the increasing E-levels a_{−1} = 0 < a_{0} < a_{1} < ⋯ < a_{t} ≤ a_{t+1} of the characters ξ_{k} ∘ N_{E/E_{k}}, k = 0, …, t + 1, are uniquely determined. We call the E-levels a_{k} the jumps of ξ and call the collection {E_{k}, a_{k}|k = 0, …, t} the jump data of ξ. By convention, when E_{0} = E, we replace (ξ_{0} ∘ N_{E/E_{0}})ξ_{−1} by ξ_{0} and assume that ξ_{−1} is trivial; otherwise, ξ_{−1} is tamely ramified and E/E_{0} is unramified [Moy86, Defnition 2.2.3]. There are certain generic conditions imposed on the jump data of the character by its admissibility, but we do not need them fully in this paper. We refer the interested reader to [Moy86, Definition 2.2.3] and [BH10, Section 8.2] for these conditions (and when E/F is totally ramified, see also [BH05b, Section 1]), and only use one of their consequences in (3.11). We fix a (non-canonical) choice of ξ_{−1} in the factorization (3.1) as follows. We fix a choice of the wild component ξ_{w} of ξ to be the product

(ξ_{0} ∘ N_{E/E_{0}}) ⋯ (ξ_{t} ∘ N_{E/E_{t}})(ξ_{t+1} ∘ N_{E/F})

which satisfies

ξ_{w}(ϖ_{F}) = 1 and ξ_{w} has a p-power order  (3.2)

(see [BH11, Lemma 1 of Section 4.3]), and define the tame component of ξ to be ξ_{−1} = ξξ_{w}^{−1}. We write

Ξ = Ξ(ξ) = ξ_{0}(ξ_{1} ∘ N_{E_{0}/E_{1}}) ⋯ (ξ_{t+1} ∘ N_{E_{0}/F}),  (3.3)

such that ξ_{w} = Ξ ∘ N_{E/E_{0}}. Suppose that E_{0}/F is a tamely ramified extension of degree dividing n and E/E_{0} is unramified. Let Ξ be a character of U_{E_{0}}^{1}. Following [BH05a, Section 1.3], we call (E_{0}/F, Ξ) an admissible 1-pair over F if Ξ does not factor through any norm N_{E_{0}/K} with F ⊆ K ⊊ E_{0}. We denote by P_{n}^{1}(F) the set of F - equivalence classes of these pairs. Therefore, the map

P_{n}(F) → P_{n}^{1}(F), (E/F, ξ) ↦ (E_{0}/F, Ξ(ξ)|_{U_{E_{0}}^{1}} ),

is well-defined and surjective. Notice that we can define the jump-data of a 1-pair, such that the jump-data of an admissible pair is the same as that of its associated 1-pair.

3.2 The correspondences

Let G^{∗} be GL_{n} defined over F, and G be an inner form of G^{∗} whose F -point is isomorphic to GL_{m}(D) for some central F -division algebra D of dimension d^{2} and n = md. Let A_{n}^{2}(F) (resp. A_{m}^{2}(D)) be the collection of equivalence classes of irreducible smooth representations of G^{∗}(F) (resp. G(F)) which are essentially square-integrable modulo center. The Jacquet-Langlands correspondence is a canonical bijection

JL : A_{n}^{2}(F) → A_{m}^{2}(D)  (3.4)

between the two collections determined by a character relation between π ∈ A_{n}^{2}(F) and its image JL(π) ∈ A_{m}^{2}(D): for every pair of semi-simple elliptic regular elements (g, g^{∗}), where g ∈ G(F) and g^{∗} ∈ G^{∗}(F), with the same reduced characteristic polynomial, we have [BH11, Section 1.4]

(−1)^{n−m}Θ_{π}(g^{∗}) = Θ_{JL(π)}(g),  (3.5)

where Θ_{π} (resp. Θ_{JL(π)}) is the character of π (resp. JL(π)). For each representation π ∈ A_{m}^{2}(D), let

(i) f(π) be the number of unramified characters χ of F^{×} that χ ⊗ π ≅ π (here χ is regarded as a representation of G(F) by composing with the reduced norm map Nrd : G(F) → F^{×}), and

(ii) δ(π) be the parametric degree of π (we do not require its full definition, so we only refer to [BH11, Section 2.7] for details).

It is known that δ(π) is a positive integer and is a multiple of f(π). Moreover, π is supercuspidal if δ(π) = n, while the converse is only true in the split case (when G = G^{∗}). Recall that we denote by A_{n}^{0}(F) the set of supercuspidal representations of G^{∗}(F). The correspondence (3.4) restricts to a bijection

JL : A_{n}^{0}(F) → {π ∈ A_{m}^{2}(D)|δ(π) = n}.

We call π essentially tame if p does not divide δ(π)/f(π). Let A_{m}^{et}(D) be the set of isomorphism classes of irreducible representations in A_{m}^{2}(D) which are essentially tame and satisfy δ(π) = n. Therefore A_{n}^{et}(F) is the same collection defined in [BH05a]. Since the Jacquet-Langlands correspondence in (3.4) preserves the invariants δ(π) and f(π), we have the following theorem [BH11, 2.8. Corollary 2].

Theorem 3.1 (Essentially tame Jacquet-Langlands correspondence). The restriction of the Jacquet-Langlands correspondence induces a bijection

JL : A_{n}^{et}(F) → A_{m}^{et}(D).

This bijection preserves the central characters on both sides.

Bushnell and Henniart described this bijection explicitly in a way parallel to [BH05a], [BH05b], [BH10]. We recall the results briefly as follows. On the one hand, we have the bijection

F Π : P_{n}(F) → A_{n}^{et}(F), (E/F, ξ) ↦ _{F} Π_{ξ},  (3.6)

generalizing the construction of Howe [How77]. On the other hand, there is an analogous bijection

DΠ : P_{n}(F) → A_{m}^{et}(D), (E/F, ξ) ↦ _{D}Π_{ξ},  (3.7)

using the constructions in [Séc04], [Séc05a], [Séc05b], [SS08]. In fact, what is constructed in [BH11] is the inverse of (3.7), using the method called ‘attachedpairs’, since the construction parallel to (3.6) exhibits some ‘novel technical difficulties’ as mentioned in [BH11, Introduction 4.]. In the split case, the attached-pair method yields the inverse of (3.6) (see 4.4 of [BH11]). The composition of the bijection in (3.6), the correspondence in Theorem 3.1, and the inverse of (3.7),

Dν : P_{n}(F) ⟶^{_{F} Π} A_{n}^{et}(F) ⟶^{JL} A_{m}^{et}(D) ⟶^{_{D}Π^{−1}} P_{n}(F),  (3.8)

determines a tamely ramified quadratic character_{D}ν_{ξ} of E^{×} for each admissible character ξ of E^{×}, depending only on the wild part of ξ, such that for each admissible pair (E/F, ξ), the pair (E/F,_{D}ν_{ξ} · ξ) is also admissible and

Dν(E/F, ξ) = (E/F,_{D}ν_{ξ} · ξ).  (3.9)

We call this character _{D}ν_{ξ} the rectifier of ξ (for the Jacquet-Langlands correspondence). Using the First and Second Comparison Theorems of [BH11], we can compute the values of _{D}ν_{ξ}. To express these values, we need the knowledge of certain invariants of finite symplectic modules, which will be described in Section 3.8. Finally, with the expression of _{D}ν_{ξ}, we see that we can describe the correspondence in Theorem 3.1 explicitly, using (3.6), (3.7), and (3.8).

3.3 Some subgroups

We recall certain subgroups of G(F). Suppose that the jump data {E_{k}, a_{k}|k = 0, …, t} are defined by the factorization (3.1) of an admissible pair (E/F, ξ), or equivalently, of its associated 1-pair (E_{0}/F, Ξ). We require that (E_{0}, 𝔄) satisfy the conditions in Section 2.5. We write A_{k} the centralizer of E_{k} in A and 𝔄_{k} = A_{k} ∩ 𝔄. We can then define 𝔓_{𝔄_{k}}, U_{𝔄_{k}}, U_{𝔄_{k}}^{x}, and U_{𝔄_{k}}^{x+} for x ∈ ℝ_{≥0} analogously as in Section 2.4. Following [Gra07, Definition 4.1], we construct the pro-p subgroups

\begin{matrix}H^{1}(Ξ, 𝔄) = U_{𝔄_{0}}^{1}U_{𝔄_{1}}^{(a_{0}e(𝔄_{1}/𝔬_{E})/2)+} ⋯ U_{𝔄_{t}}^{(a_{t−1}e(𝔄_{t}/𝔬_{E})/2)+}U_{𝔄}^{(a_{t}e(𝔄/𝔬_{E})/2)+} and \\ J^{1}(Ξ, 𝔄) = U_{𝔄_{0}}^{1}U_{𝔄_{1}}^{a_{0}e(𝔄_{1}/𝔬_{E})/2} ⋯ U_{𝔄_{t}}^{a_{t−1}e(𝔄_{t}/𝔬_{E})/2}U_{𝔄}^{a_{t}e(𝔄/𝔬_{E})/2}. \end{matrix} ; (3.10)

We also construct the subgroups

J(Ξ, 𝔄) = U_{𝔄_{0}}U_{𝔄_{1}}^{a_{0}e(𝔄_{1}/𝔬_{E})/2} ⋯ U_{𝔄_{t}}^{a_{t−1}e(𝔄_{t}/𝔬_{E})/2}U_{𝔄}^{a_{t}e(𝔄/𝔬_{E})/2} and
J(Ξ, 𝔄) = E^{×}J(Ξ, 𝔄) = E_{0}^{×}J(Ξ, 𝔄).

We abbreviate these groups by H^{1}, J^{1}, J and J if the admissible character ξ is fixed. Notice that H^{1}, J^{1}, J are compact subgroups of G(F) and J is a compact-mod-center subgroup of G(F). In [Séc04, Section 3.1], these subgroups are defined based on a simple stratum [𝔄, −v_{𝔄}(β), 0, β], where β is a suitable element in E_{0}, depending on Ξ and such that

−v_{𝔄}(β) = the E-level of Ξ ∘ N_{E/E_{0}}.

The group H^{1}(Ξ, 𝔄) is denoted by H^{1}(β, 𝔄) in loc. cit. (and similarly for the other subgroups). This construction is an obvious generalization of [BK93, Section 3.1] (see also [BH11, Section 2.5] and the Comment therewithin).

3.4 Simple characters

Given an admissible 1-pair (E_{0}/F, Ξ) and a finite unramified extension E/E_{0}, we define H^{1}(Ξ, 𝔄) as in (3.10). Using the idea of [Moy86, Section 3.2] (see also [Séc04, Definitions 3.22, 3.45, Proposition 3.47]), we attach to (E_{0}/F, Ξ) a simple character (H^{1}(Ξ, 𝔄), θ_{Ξ,E}) as follows. Suppose that Ξ admits a factorization of the form (3.3), with each ξ_{k} ∘ N_{E_{0}/E_{k}}, where k = 0, …, t + 1, a character of U_{E_{0}}^{1}. The generic conditions on the factorization imply that for each ξ_{k} there is c_{k} ∈ E_{k} ∩ 𝔭_{E}^{−a_{k}} such that E_{k+1}[c_{k}] = E_{k} and

ξ_{k} ∘ N_{E/E_{k}}(1 + x) = ψ_{F} (tr_{E/F} (c_{k}x)) for all x ∈ 𝔭_{E}^{(a_{k}/2)+}.  (3.11)

Note that the element c_{k} can be chosen mod 𝔭_{E}^{−a_{k}/2}. We denote the character on the right side of (3.11) by ψ_{c_{k}}. We define a character θ_{Ξ,E} of the subgroup H^{1}(Ξ, 𝔄) in (3.10) by the following inductive procedure. We first define a character θ_{t+1} on the subgroup U_{𝔄_{t+1}}^{(a_{t}e(𝔄_{t+1}/𝔬_{E})/2)+} (note that indeed E_{t+1} = F and 𝔄_{t+1} = 𝔄) by

ξ_{t+1} ∘ Nrd_{A/F} on U_{𝔄_{t+1}}^{(a_{t}e(𝔄_{t+1}/𝔬_{E})/2)+}.

Inductively, suppose θ_{k+1} is defined, we construct θ_{k} on the subgroup

U_{𝔄_{k}}^{(a_{k−1}e(𝔄_{k}/𝔬_{E})/2)+}U_{𝔄_{k+1}}^{(a_{k}e(𝔄_{k+1}/𝔬_{E})/2)+} ⋯ U_{𝔄}^{(a_{t}e(𝔄/𝔬_{E})/2)+}

by

(ξ_{k} ∘ Nrd_{A_{k}/E_{k}}) ⋯ (ξ_{t+1} ∘ Nrd_{A/F}) on U_{𝔄_{k}}^{(a_{k−1}e(𝔄_{k}/𝔬_{E})/2)+} and
(ψ_{c_{k}} ∘ trd_{𝔄_{k}/E_{k}})θ_{k+1} on U_{𝔄_{k+1}}^{(a_{k}e(𝔄_{k+1}/𝔬_{E})/2)+} ⋯ U_{𝔄}^{(a_{t}e(𝔄/𝔬_{E})/2)+}.

(3.12)

On the intersection U_{𝔄_{k}}^{(a_{k}e(𝔄_{k}/𝔬_{E})/2)+}, we have

θ_{k+1} = (ξ_{k+1} ∘ Nrd_{A_{k+1}/E_{k+1}}) ⋯ (ξ_{t+1} ∘ Nrd_{A/F})

and

ψ_{c_{k}} ∘ trd_{𝔄_{k}/E_{k}} = ξ_{k} ∘ Nrd_{A_{k}/E_{k}}.

By Proposition 2.7 and (3.11), the two characters in (3.12) agree on the intersection, and so θ_{k} is defined. Finally, we define θ_{Ξ,E} = θ_{0} on H^{1}(Ξ, 𝔄). By construction, this character θ_{Ξ,E} is normalized by 𝔎_{𝔄_{0}}, hence it is a simple character by [Gra07, 5.3 Definition.(i)]. To get back our character Ξ from θ_{Ξ,E}, just notice that θ_{Ξ,E}|_{U_{E}^{1}} factors through N_{E/E_{0}}, hence there is a unique character Ξ of U_{E_{0}}^{1} such that Ξ ∘ N_{E/E_{0}} = θ_{Ξ,E}|_{U_{E}^{1}}.

3.5 Local constructions of attached pairs

We briefly summarize the construction in [BH11, Sec. 4] of the bijection _{D}Π in (3.7). We will distinguish between the ‘level-zero’ case and the ‘positive-level’ case (we refer to [BH11, Sections 2.4 and 2.6] the definition of these cases ). As we remarked in the Introduction, it is indeed the inverse of _{D}Π that we are going to describe. In the level-zero case, the construction is similar to the one in the split case. Each level zero π ∈ A_{m}^{et}(D) contains a representation (GL_{m}(𝔬_{D}), λ), called a maximal simple type of level zero, inflated from an irreducible cuspidal representation (GL_{m}(k_{D}), λ̄). This representation λ̄ corresponds, via Green’s parametrization [Gre55], to a k_{E}/k_{F} -regular character ξ̄ of k_{E}^{×}, where E/F is the unramified extension of degree n. We define the character ξ of E^{×} such that ξ|_{𝔬_{E}^{×}} is the inflation of ξ̄ and ξ|_{F^{×}} is the central character of π. By [BH11, 4.2. Proposition], the attached pair (E/F, ξ) is admissible and the correspondence

A_{m}^{et}(D)_{level−0} → P_{n}(F)_{level−0}, π ↦ (E/F, ξ),  (3.13)

is bijective. We can show that π = cInd_{J}^{G(F)}Λ, where the condition δ(π) = n implies that J = F^{×}GL_{m}(𝔬_{D}), and (J, Λ) is defined by the conditions

Λ|_{GL_{m}(𝔬_{D})} = λ and Λ|_{F^{×}} is a multiple of ξ|_{F^{×}}.  (3.14)

The representation (J, Λ) is called an extended maximal simple type of level zero. In the positive level case, we first recall the construction of a extended maximal simple type in general. Suppose we have a simple character (H^{1}, θ). For example, we can construct a simple character θ = θ_{Ξ,E} as in Section 3.4 using an admissible pair (E/F, ξ). We notice that the commutator subgroup [J^{1}, J^{1}] lies in H^{1} [BK93, (3.1.15)]. By [Séc04, Théorème 3.52], the above simple character θ induces an non-degenerate alternating bilinear form

h_{θ}(x, y) = θ([1 + x, 1 + y]), for all 1 + x, 1 + y ∈ J^{1},  (3.15)

on the 𝔽_{p}-vector space

A𝔙_{ξ} := J^{1}/H^{1}.

The classical theory of Heisenberg representation implies that there is a unique representation η̄ of J^{1}/ ker θ containing the character θ of H^{1}/ ker θ as a central character. We then define η as the inflation of η̄ to J^{1}. By [Séc05a, Théorème 2.28], there exists a unique irreducible representation (J, κ), which is called a β-extension (or wide extension in [BH11]) of η and satisfies certain conditions on its intertwining in G(F) (see [BH11, (2.5.5)]). We now choose a maximal simple type (GL_{m_{0}}(𝔬_{D_{0}}), σ) of A_{0}^{×} = Z_{G(F)}(E_{0}^{×}) of level zero and inflate it to a representation (J, σ), since we know that J = GL_{m_{0}}(𝔬_{D_{0}})J^{1}. We obtain a maximal simple type (J, λ), where λ = κ ⊗ σ. By [Séc05b, Théorème 5.2], there exists an irreducible representation Λ of J = E_{0}^{×}J^{1} (by the condition δ(π) = n), extending λ and whose compact-induction to G(F) is irreducible and supercuspidal. The representation (J, Λ) is called an extended maximal simple type. By [BH11, Lemma 2 of Section 4.3], we can fix a unique extended type containing (J, λ) and satisfying the (non-canonical) conditions:

ϖ_{F} ∈ ker Λ and det Λ has a p-power order.

Following [BH11, Section 3 and 4], we have to approach indirectly to describe the inverse of _{D}Π. Suppose that we are given a representation π ∈ A_{m}^{et}(D) of positive level. By [SS08, Théorème 5.21], it contains an extended maximal simple type (J, Λ) of the above form, such that Λ|_{H^{1}} is a multiple of a simple character (H^{1}, θ). There is a unique character ξ_{w} of E^{×}, depending on θ|_{U_{𝔄_{0}}^{1}} , satisfying the conditions in (3.2). In particular, we have

θ|_{U_{𝔄_{0}}^{1}} = Ξ ∘ Nrd_{A_{0}/E_{0}}, such that ξ_{w}|_{U_{E}^{1}} = Ξ ∘ N_{E/E_{0}}.

By the discussion of the previous paragraph, we can attach to ξ_{w} an extended maximal simple type (J, Λ_{w}) such that Λ ≅ Λ_{−1}⊗Λ_{w} for a uniquely determined extended maximal simple type (J, Λ_{−1}) of level zero. Attached to (J, Λ_{−1}) is a level zero character ξ_{−1} of E^{×} admissible over E_{0}, as mentioned in the level zero case. Finally, by [BH11, 4.3. Proposition], the attached pair (E/F, ξ), where ξ = ξ_{−1}ξ_{w}, is admissible and independent of the various choices above. We call (E/F, ξ) a pair attached to π. The technical part is to show that the attaching map

A_{m}^{et}(D) → P_{n}(F), π ↦ (E/F, ξ)

is well-defined and injective. This is done in the Parametrization Theorem of [BH11, Section 6]. The composition (3.8) is then injective (since the maps F Π and JL are known to be bijective) and preserves the restriction of each character to the subgroup F^{×}U_{E}^{1}, which is of finite index of E^{×}. Therefore, the map in (3.8) and hence _{D}Π in (3.7), is bijective.

3.6 Finite symplectic modules

Since the group J normalizes the subgroups H^{1}, J^{1}, and the simple character θ of H^{1}, it acts on the finite quotient _{A}𝔙_{ξ} := J^{1}/H^{1}. This quotient is denoted by _{A^{∗}}𝔙_{ξ} in the split case A = A^{∗}, which is studied in [BH10] and [Tam]. Notice that the quotient is clearly a finite dimensional 𝔽_{p}-vector space. The action of J induces a symplectic 𝔽_{p}J-module structure on this space with respect to the non-degenerate alternating form h_{θ} in (3.15). We have a decomposition of

A𝔙_{ξ} = _{A}𝔙_{ξ,0} ⊕ ⋯ ⊕_{A}𝔙_{ξ,t},  (3.16)

into 𝔽_{p}J-submodules, where

\begin{aligned}_{A}𝔙_{ξ,k} & = \frac{U_{𝔄_{k+1}}^{a_{k}e(𝔄_{k+1}/𝔬_{E})/2}}{U_{𝔄_{k}}^{a_{k}e(𝔄_{k}/𝔬_{E})/2}U_{𝔄_{k+1}}^{(a_{k}e(𝔄_{k+1}/𝔬_{E})/2)+}} \\ & = \frac{𝔓_{𝔄_{k+1}}^{a_{k}e(𝔄_{k+1}/𝔬_{E})/2}}{𝔓_{𝔄_{k}}^{a_{k}e(𝔄_{k}/𝔬_{E})/2} + 𝔓_{𝔄_{k+1}}^{(a_{k}e(𝔄_{k+1}/𝔬_{E})/2)+}} \end{aligned}  (3.17)

for k = 0, …, t. In this paper, We call this decomposition the coarse decomposition of _{A}𝔙_{ξ}. By [Séc04, Proposition 3.9], the decomposition (3.16) is orthogonal. In the sequel, we will be interested in the adjoint action of E^{×} on_{A}𝔙_{ξ} restricted from that of J, which factors through the finite group Ψ_{E/F} := E^{×}/F^{×}(E^{×} ∩ J^{1}). The following Proposition appears in [BH11, Proposition 5.6]. We re-interpret its proof here.

Proposition 3.2. If E/F is totally ramified, then _{A}𝔙_{ξ} ≅ _{A^{∗}}𝔙_{ξ}.

Proof. From the proofs of Propositions 4.1 and 4.2 (which are purely algebra and do not require the knowledge of this section), we see that the totally ramified condition implies that

𝔓_{𝔄_{k}^{∗}}^{j}/𝔓_{𝔄_{k}^{∗}}^{j+1} ≅ 𝔓_{𝔄_{k}}^{j}/𝔓_{𝔄_{k}}^{j+1} ≅ Ind_{1}^{Ψ_{E/E_{k}}} k_{F}  (3.18)

as a k_{F} Ψ_{E/E_{k}}-module, for all j ∈ ℤ and k = 1, …, t + 1. Moreover, we know that in the split case the index e(𝔄_{k}^{∗}/𝔬_{E}) (appearing in the powers in (3.17), when A = A^{∗}) is always 1, and in general each e(𝔄_{k}/𝔬_{E}) (again appearing in (3.17)) divides f(E/E_{k}) (remember that f(E/E_{k})/e(𝔄_{k}/𝔬_{E}) is the integer s_{E_{k}} appearing in (2.7), when K = E_{k}), which is equal to 1 in the totally ramified case. Hence from the expression in (3.17) and using (3.18), we see that both A𝔙_{ξ,k} and _{A^{∗}}𝔙_{ξ,k} are isomorphic to

\frac{𝔓_{𝔄_{k+1}}^{a_{k}/2}}{𝔓_{𝔄_{k}}^{a_{k}/2} + 𝔓_{𝔄_{k+1}}^{(a_{k}/2)+}} .

Hence their sums _{A}𝔙_{ξ} and _{A^{∗}}𝔙_{ξ} are also isomorphic.

Documenta Mathematica 21 (2016) 345–389

3.7 Invariants of finite symplectic modules

Let Γ be a finite cyclic group whose order is not divisible by p. We call a finite 𝔽_{p}Γ-module V symplectic if there is a non-degenerate alternating form h : V × V → 𝔽_{p} which is Γ-invariant, in the sense that

h(^{γ}v_{1},^{γ}v_{2}) = h(v_{1}, v_{2}), for all γ ∈ Γ, v_{1}, v_{2} ∈ V.

The simple module V_{λ} corresponding to a character λ ∈ Hom(Γ, 𝔽̄_{p}^{×}) is the field 𝔽_{p}[λ(Γ)] generated over 𝔽_{p} by the image λ(Γ), with Γ-action

γv = λ(γ)v, for all γ ∈ Γ, v ∈ V_{λ}.

Its 𝔽_{p}-linear dual V_{λ}^{∗} = Hom(V_{λ}, 𝔽_{p}) is isomorphic to V_{λ^{−1}} by the map

V_{λ^{−1}} → V_{λ}^{∗}, v ↦ (w ↦ tr_{𝔽_{p}[λ(Γ)]/𝔽_{p}}(wv)),

such that the canonical pairing ⟨·, ·⟩ : V_{λ} × V_{λ^{−1}} → 𝔽_{p} is Γ-invariant. We recall some basic facts from [BF83, (8.2.3)] and [BH10, Sec. 3, Prop. 4].

Proposition 3.3. (i) An indecomposable symplectic 𝔽_{p}Γ-module is isomorphic to either one of the following two kinds,

(a) a hyperbolic module of the form V_{λ} = V_{λ} ⊕ V_{λ^{−1}} such that either λ^{2} = 1 or V_{λ} ≇ V_{λ^{−1}}, with the alternating form

h_{V_{λ}}((v_{1}, v_{1}^{∗}), (v_{2}, v_{2}^{∗})) = ⟨v_{1}, v_{2}^{∗}⟩ − ⟨v_{2}, v_{1}^{∗}⟩ ,

for all (v_{1}, v_{1}^{∗}), (v_{2}, v_{2}^{∗}) ∈ V_{λ};

(b) an anisotropic module of the form V_{λ} with λ^{2} ≠ 1 and V_{λ} ≅ V_{λ^{−1}}. In this case, [𝔽_{p}[λ(Γ)] : 𝔽_{p}] is even and the alternating form h_{V_{λ}} is defined, up to Γ-isometry, by

(v_{1}, v_{2}) ↦ tr_{𝔽_{p}[λ(Γ)]/𝔽_{p}}(αv_{1}v̄_{2}), for all v_{1}, v_{2} ∈ V_{λ},

where v ↦ v̄ is the 𝔽_{p}-automorphism of 𝔽_{p}[λ(Γ)] of order 2 and α ∈ 𝔽_{p}[λ(Γ)]^{×} satisfies ᾱ = −α.

(ii) If V_{λ} is anisotropic and 𝔽_{p}[λ(Γ)]_{±} denotes the subfield of 𝔽_{p}[λ(Γ)] such that 𝔽_{p}[λ(Γ)]/𝔽_{p}[λ(Γ)]_{±} is quadratic, then λ(Γ) is a subgroup of ker(N_{𝔽_{p}[λ(Γ)]/𝔽_{p}[λ(Γ)]_{±}})

(iii) The Γ-isometry class of a symplectic 𝔽_{p}Γ-module (V, h) is determined by the underlying 𝔽_{p}Γ-module V.

Part (iii) is particularly useful because, when we talk about invariants of Γisometry classes of symplectic 𝔽_{p}Γ-modules, we do not have to write down the alternating forms explicitly.

Given a finite symplectic 𝔽_{p}Γ-module 𝔙, we attach a sign t_{Γ}^{0}(𝔙) ∈ {±1} and a quadratic character t_{Γ}^{1}(𝔙) of Γ. We also set

t_{Γ}(𝔙) = t_{Γ}^{0}(𝔙)t_{Γ}^{1}(𝔙)(γ),

where γ is any generator of Γ. We call these t-factors of 𝔙. We recall from [BH10, Section 3] the definition the t-factors.

(i) If Γ acts on 𝔙 trivially, then

t_{Γ}^{0}(𝔙) = 1 and t_{Γ}^{1}(𝔙) ≡ 1.

(ii) Let 𝔙 be an indecomposable symplectic 𝔽_{p}Γ-module.

(a) If 𝔙 = 𝔙_{λ} ⊕ 𝔙_{λ^{−1}} is hyperbolic, then

t_{Γ}^{0}(𝔙) = 1 and t_{Γ}^{1}(𝔙) = sgn_{λ(Γ)}(𝔙_{λ}).

Here sgn_{λ(Γ)}(𝔙_{λ}) : Γ → {±1} is the character whose image γ ↦ sgn_{λ(γ)}(𝔙_{λ}) is the signature of the multiplication by λ(γ) as a permutation of the set 𝔙_{λ}.

(b) If 𝔙 = 𝔙_{λ} is anisotropic, then

t_{Γ}^{0}(𝔙) = −1 and

t_{Γ}^{1}(𝔙)(γ) = (\frac{γ}{ker(N_{𝔽_{p}[λ(Γ)]/𝔽_{p}[λ(Γ)]_{±}})}) for any γ ∈ Γ.

Here (\frac{·}{·}) is the symbol defined as follows: for every finite cyclic group H,

(\frac{x}{H} ) = \begin{cases}1 & if x ∈ H^{2}, \\ −1 & otherwise. \end{cases}

(iii) If 𝔙 decomposes into an orthogonal sum 𝔙_{1}⊥ ⋯ ⊥𝔙_{t} of indecomposable symplectic 𝔽_{p}Γ-modules, then

t_{Γ}^{i}(𝔙) = t_{Γ}^{i}(𝔙_{1}) ⋯ t_{Γ}^{i}(𝔙_{t}) for i = 0, 1.

Notice that when p = 2, the order of Γ is odd. In this case, t_{Γ}^{1}(𝔙) is always trivial, because all signature characters and symbols (\frac{·}{·}) are trivial.

3.8 Values of rectifiers

Given a tamely ramified extension E/F and an F -admissible character ξ of E^{×}, let _{D}ν_{ξ} be the rectifier of ξ defined in (3.9). To describe the values of _{D}ν_{ξ}, we need to impose a condition on ϖ_{E} defined in (2.1):

ϖ_{E} ∈ E_{0},  (3.19)

where E_{0} be the first field appearing in the factorization (3.1) of ξ. This condition is the same as in the Second Comparison Theorem of [BH11, Section 7], where we further require that ϖ_{E}^{r} ∈ F for some integer r coprime to p. Indeed, from the assumption in (2.1) this extra requirement is automatic in our situation. Under (3.19), the roots of unity z_{E/F}, z_{ϕ^{i}} (defined in Section 2.2), and others related to ϖ_{E} in later sections all depend on the first field E_{0} in the jump data of ξ. The values of the rectifier _{D}ν_{ξ} depends on the t-factors

t_{μ}^{1}(_{A}𝔙_{ξ}), t_{μ}^{1}(_{A^{∗}}𝔙_{ξ}), t_{⟨ϖ⟩}(_{A}𝔙_{ξ}) and t_{⟨ϖ⟩}(_{A^{∗}}𝔙_{ξ}),

where μ = μ_{E/F} and ⟨ϖ⟩ = ⟨ϖ⟩_{E/F} abbreviate the following subgroups of Ψ_{E/F} = E^{×}/F^{×}U_{E}^{1},

μ := μ_{E}/μ_{F} and

⟨ϖ⟩ := the subgroup generated by the image of ϖ_{E}.

(3.20)

By the First and Second Comparison Theorems of [BH11], the rectifier _{D}ν_{ξ} has values

\begin{matrix}_{D}ν_{ξ}|_{μ_{E}} = t_{μ}^{1}(_{A}𝔙_{ξ})t_{μ}^{1}(_{A^{∗}}𝔙_{ξ}) and \\ _{D}ν_{ξ}(ϖ_{E}) = (−1)^{n−m+f_{ϖ}−m_{ϖ}}t_{⟨ϖ⟩}(_{A}𝔙_{ξ})t_{⟨ϖ⟩}(_{A^{∗}}𝔙_{ξ}), \end{matrix}  (3.21)

where f_{ϖ} := [E : F [ϖ_{E}]] = f(E/F [ϖ_{E}]) and m_{ϖ} = gcd(m, f_{ϖ}). In the case when E/F is totally ramified, Proposition 3.2 implies that

Dν_{ξ} is unramified and _{D}ν_{ξ}(ϖ_{E}) = (−1)^{n−m},

as stated in [BH11, 5.3.Theorem].

4 Finite symplectic modules

4.1 Standard modules of central simple algebra

Let 𝔄 be the hereditary E-pure order in A, as discussed in Sections 2.4 and 2.5. The isomorphism (2.6) implies that 𝔓_{𝔄}^{j}/𝔓_{𝔄}^{j+1} ≅ Mat_{s}(k_{D})^{r} for all j ∈ ℤ, where s = gcd(f, m) and r = e/ gcd(d, e). We denote this quotient by (Mat_{s}(k_{D})^{r})_{j} when we want to emphases the index j. Notice that as k_{F} Ψ_{E/F} -modules, all (Mat_{s}(k_{D})^{r})_{j}, for j ranges over all ℤ, are isomorphic to each other. When 𝔄 = 𝔄^{∗} and j = 0, we know that 𝔘_{𝔄^{∗}} := 𝔄^{∗}/𝔓_{𝔄^{∗}} ≅ Mat_{n}(k_{F}) admits a root-space decomposition

𝔘_{𝔄^{∗}} ≅ 𝔘_{0} ⨁_{[λ]∈Γ_{F} \Φ} 𝔘_{[λ]},

where 𝔘_{0} ≅ 𝔬_{E}/𝔭_{E} on which Ψ_{E/F} acts trivially, and 𝔘_{[λ]} is the k_{F} -subspace on which Ψ_{E/F} acts by the character λ. Note that the equivalence class of the k_{F} Ψ_{E/F} -module 𝔘_{[λ]} depends only on the Γ_{F} -orbit of λ.

For future computation, we rewrite the above decomposition as

𝔘_{𝔄^{∗}} ≅ ⨁_{[g]∈Γ_{E}\Γ_{F} /Γ_{E}} 𝔘_{[g]}  (4.1)

using the identification in Proposition 2.1. Here 𝔘_{[g]} ≅ k_{E(^{g}E)} as a k_{F} -vector space for each [g] ∈ Γ_{E}\Γ_{F} /Γ_{E}, and the Ψ_{E/F} -action on each v ∈ 𝔘_{[g]} is given as follows: if [g] = [σ^{i}ϕ^{j}], then

zv = (z^{q^{j}−1})^{−1}v for all z ∈ μ_{E} and ^{ϖ_{E}}v = (z_{e}^{i}z_{ϕ^{j}})^{−1}v,  (4.2)

where z_{e} and z_{ϕ^{j}} are defined in Section 2.2. For general inner form G, we first consider a simple case when A is a division algebra D. We write (k_{D})_{j} := 𝔓_{D}^{j}/𝔓_{D}^{j+1} for each j ∈ ℤ.

Proposition 4.1. For each j ∈ ℤ, the k_{F} Ψ_{E/F} -module (k_{D})_{j} is isomorphic to

⨁_{[σ^{i}ϕ^{hj}]∈Γ_{E}\Γ_{F} /Γ_{E}} 𝔘_{[σ^{i}ϕ^{hj}]}.  (4.3)

Proof. Recall that, if we denote by K the maximal unramified extension in D (of degree n over F), then

𝔓_{D}^{i} = ⋯ ⊕ 𝔭_{K}ϖ_{D}^{i} ⊕ 𝔬_{K}ϖ_{D}^{i+1} ⊕ · · · .

Hence we can use xϖ_{D}^{i}, with x ∈ μ_{K} ∪ {0}, as a representative in 𝔓_{D}^{i} of an element in (k_{D})_{j}. We regard (k_{D})_{j} as a k_{E}-vector space of dimension e such that z ∈ μ_{E} acts on each piece (k_{E})_{j} by the character [ \begin{matrix}1 \\ ϕ^{hj} \end{matrix} ] (z) = z^{1−q^{hj}} as in (2.4). Therefore, a Frobenius reciprocity argument (which is still valid when p does not divide #Ψ_{E/F}) implies that

(k_{D})_{j} ≅ Ind_{μ_{E}/μ_{F}}^{Ψ_{E/F}} (k_{E})_{j};

more precisely, the action of ϖ_{E} has eigenvalues (z_{e}^{i}z_{ϕ^{hj}})^{−1}, i = 0, …, e −

1, where we recall that z_{ϕ^{hj}} is an eth root of z_{E/F}^{q^{hj}−1}. Hence we have the decomposition (4.3) with the Ψ_{E/F} -action on each component 𝔘_{[σ^{i}ϕ^{hj}]} as in (4.2) for each fixed j.

For the general A, if we write (Mat_{s}(k_{D})^{r})_{j′} := 𝔓_{𝔄}^{j′}/𝔓_{𝔄}^{j′+1} for each j′ ∈ ℤ, then we have the following result.

Proposition 4.2. For each j′ ∈ ℤ, we have a decomposition

(Mat_{s}(k_{D})^{r})_{j′} ≅ ⨁_{\begin{matrix}[σ^{i}ϕ^{j}]∈Γ_{E}\Γ_{F} /Γ_{E} \\ j≡hj′ mod f/s \end{matrix}} 𝔘_{[σ^{i}ϕ^{j}]}.

as a Ψ_{E/F} -module. (Recall that f/s = d/ gcd(d, e) = e(𝔄/𝔬_{E}).)

Proof. Denote by E_{D} = E ∩ D the maximal subfield contained in both E and D, then [k_{E_{D}} : k_{F}] = f/s and [k_{D} : k_{E_{D}}] = gcd(d, e) = e/r. By Proposition 4.1, we know that z ∈ μ_{E_{D}} acts on (Mat_{s}(k_{D})^{r})_{j′} as a sum of (k_{E_{D}})_{hj′}, i.e., z acts by the character [ 1 ; ϕ̄^{hj′} ] (z) = z^{1−q^{hj′}}, where ϕ̄ is the image of ϕ under the natural projection

Γ_{k_{E}/k_{F}} → Γ_{k_{E_{D}} /k_{F}} .  (4.4)

(Note that the arguments above concerning Proposition 4.1 still valid even though E_{D} may not be a maximal subfield of D.) We now consider the k_{E_{D}}- embeddings (where all choices are conjugate to each other)

k_{E_{D}} ⟶ k_{E} ⟶ Mat_{s}(k_{E_{D}}) ⟶ Mat_{s}(k_{D}),

Notice that k_{E} is a maximal subfield of Mat_{s}(k_{E_{D}}). By the “twisted group ring decomposition”, we know that z ∈ μ_{E} acts on Mat_{s}(k_{E_{D}}) as a sum of ((k_{E})_{hj′})^{s}, i.e., z acts on each of the s summands of k_{E} by the character [ \begin{matrix}1 \\ ϕ^{j} \end{matrix} ] (z) = z^{1−q^{j}}, for ϕ^{j} ranges over the s pre-images of ϕ̄^{hj′} under the natural projection (4.4). We denote this μ_{E}-module by Mat_{s}(k_{E_{D}})_{j′}. Finally, since the relative degree of (Mat_{s}(k_{D}))^{r} over Mat_{s}(k_{E_{D}}) is e = [Ψ_{E/F} : μ_{E}/μ_{F}], a Frobenius reciprocity argument (which is still valid when p does not divide #Ψ_{E/F}) implies that

(Mat_{s}(k_{D})^{r})_{j′} ≅ Ind_{μ_{E}/μ_{F}}^{Ψ_{E/F}} Mat_{s}(k_{E_{D}})_{j′};

Therefore, we have obtained the desired decomposition and proved the proposition. The following Corollary is a direct consequence of Proposition 4.2.

Corollary 4.3. The graded algebra

𝔘_{𝔄} := ⨁_{j′=0}^{f/s−1}(Mat_{s}(k_{D})^{r})_{j′},

is isomorphic to 𝔘_{𝔄^{∗}} as a Ψ_{E/F} -module.

We provide some notations for later use. We write

𝔘_{sym} := ⨁_{[g]∈(Γ_{E}\Γ_{F} /Γ_{E})_{sym}} 𝔘_{[g]}

and also 𝔘_{sym−ram} and 𝔘_{sym−unram} analogously. Given intermediate extensions F ⊆ K ⊆ L ⊆ E, we write

𝔘_{K/L} := ⨁_{[g]∈Γ_{E}\(Γ_{L}−Γ_{K})/Γ_{E}} 𝔘_{[g]}

We also define the symmetric module associated to 𝔘_{[g]} (or 𝔘_{[g^{−1}]}) by

𝔘_{[g]} := \begin{cases}𝔘_{[g]} ⊕ 𝔘_{[g^{−1}]} & if [g] is asymmetric, \\ 𝔘_{[g]} & if [g] is symmetric, \end{cases}  (4.5)

and call

𝔘_{𝔄} ≅ 𝔘_{0} ⨁_{[g]∈(Γ_{E}\Γ_{F} /Γ_{E})_{sym}⊔(Γ_{E}\Γ_{F} /Γ_{E})_{asym/±}} 𝔘_{[g]}  (4.6)

the complete symmetric decomposition of 𝔘_{𝔄}. If 𝔙 is a submodule of 𝔘_{𝔄}, we also use the same convention to denote its submodules, for example, 𝔙_{K/L} = 𝔘_{K/L} ∩ 𝔙 and 𝔙_{[g]} = 𝔘_{[g]} ∩ 𝔙, and also call

𝔙 ≅ (𝔘_{0} ∩ 𝔙) ⨁_{[g]∈(Γ_{E}\Γ_{F} /Γ_{E})_{sym}⊔(Γ_{E}\Γ_{F} /Γ_{E})_{asym/±}} 𝔙_{[g]}

the complete symmetric decomposition of 𝔙

4.2 Complete decomposition of finite symplectic modules

We are interested in the adjoint action of E^{×} on _{A}𝔙_{ξ} restricted from that of J, which factors through the finite group E^{×}/F^{×}(E^{×}∩J^{1}) ≅ Ψ_{E/F}. We also know that this action preserves the symplectic structure h_{θ} (3.15) on _{A}𝔙_{ξ}. Hence A𝔙_{ξ} is moreover a finite symplectic 𝔽_{p}Γ-module for each cyclic subgroup Γ of Ψ_{E/F}. We denote the 𝔘_{[g]}-isotypic component in _{A}𝔙_{ξ} by _{A}𝔙_{ξ,[g]}, and obtain the decompositions

A𝔙_{ξ} = ⨁_{[g]∈(Γ_{E}\Γ_{F} /Γ_{E})′} _{A}𝔙_{ξ,[g]} = ⨁_{[g]∈(Γ_{E}\Γ_{F} /Γ_{E})_{sym}⊔(Γ_{E}\Γ_{F} /Γ_{E})_{asym/±}} _{A}𝔙_{ξ,[g]}

(4.7)

inherited from (4.1) and (4.6) respectively. These decomposition are finer than the one in (3.16). Indeed, it is easy to see that

A𝔙_{ξ,k} := _{A}𝔙_{ξ,E_{k}/E_{k+1}} = ⨁_{[g]∈Γ_{E}\(Γ_{E_{k+1}} −Γ_{E_{k}})/Γ_{E}} _{A}𝔙_{ξ,[g]}

for k = 0, …, t.

Proposition 4.4. The complete symmetric decomposition of _{A}𝔙_{ξ} is orthogonal with respect to the alternating form h_{θ}.

Proof. Since we know that the Ψ_{E/F} -components of _{A}𝔙_{ξ} consist of those in the standard module 𝔘_{𝔄}, which is isomorphic to the standard one 𝔘_{𝔄^{∗}} in the split case, the proof of the assertion is just analogous to the one in the split case [Tam, (5.10)], based on the argument of [BF83, (8.2.3),(8.2.4)].

We would like to describe the isotypic component appearing in the complete decomposition (4.7) of _{A}𝔙_{ξ}. We first write e(𝔄/𝔄_{k+1}) := e(𝔄/𝔬_{E})/e(𝔄_{k+1}/𝔬_{E}).

Proposition 4.5. The quotient 𝔓_{𝔄}^{j} ∩ A_{k+1}/𝔓_{𝔄}^{j+} ∩ A_{k+1} is non-trivial if and only if j ∈ e(𝔄/𝔄_{k+1})ℤ.

Proof. Since 𝔓_{𝔄}^{j} ∩ A_{k+1} = 𝔓_{𝔄_{k+1}}^{j/e(𝔄/𝔄_{k+1})} for all j ∈ ℤ, the assertion follows directly.

We now specify j = j_{k} = e(𝔄/𝔬_{E})a_{k}/2 for some integer a_{k}, and so 𝔓_{𝔄}^{j_{k}} ∩A_{k+1} = 𝔓_{𝔄_{k+1}}^{a_{k}e(𝔄_{k+1}/𝔬_{E})/2}, such that the index on the right side is the one appearing in the group J^{1} (3.10). The condition in Proposition 4.5 is satisfied if and only if a_{k} is even or e(𝔄_{k+1}/𝔬_{E}) is even, in which case

A𝔙_{ξ,k} ≅ \frac{𝔓_{𝔄_{k+1}}^{a_{k}e(𝔄_{k+1}/𝔬_{E})/2}}{𝔓_{𝔄_{k+1}}^{a_{k}e(𝔄_{k+1}/𝔬_{E})/2+} + 𝔓_{𝔄_{k}}^{a_{k}e(𝔄_{k}/𝔬_{E})/2}}

≅ \begin{cases}Mat_{s_{k+1}}(k_{D_{k+1}})^{r_{k+1}}/Mat_{s_{k}}(k_{D_{k}})^{r_{k}} & when a_{k}e(𝔄_{k}/𝔬_{E})/2 ∈ ℤ, \\ Mat_{s_{k+1}}(k_{D_{k+1}})^{r_{k+1}} & otherwise, \end{cases}

where r_{k} and s_{k} are the invariants of 𝔄_{k} analogous to r and s of 𝔄. To summarize, _{A}𝔙_{ξ,k} is isomorphic to

0 when a_{k} is odd and e(𝔄_{k+1}/𝔬_{E}) is odd,
Mat_{s_{k+1}}(k_{D_{k+1}})^{r_{k+1}}
when a_{k} is odd, e(𝔄_{k}/𝔬_{E}) is odd, and e(𝔄_{k+1}/𝔬_{E}) is even,
Mat_{s_{k+1}}(k_{D_{k+1}})^{r_{k+1}}/Mat_{s_{k}}(k_{D_{k}})^{r_{k}}
when a_{k} is even or e(𝔄_{k}/𝔬_{E}) is even.

The action of Ψ_{E/F} on 𝔓_{𝔄}^{j_{k}} ∩ A_{k+1} is given by σ^{i}ϕ^{jh} ∈ Γ_{E_{k+1}}, where j has image hj_{k} in the natural projection ℤ/fℤ → ℤ/e(𝔄/𝔬_{E})ℤ. Therefore, directly from the description of _{A}𝔙_{ξ,k} above, we have the following decompositions.

Proposition 4.6. The complete decomposition of the component _{A}𝔙_{ξ,k} is given as follows.

(i) When a_{k} is odd and e(𝔄_{k+1}/𝔬_{E}) is odd, then _{A}𝔙_{ξ,k} is trivial.

(ii) When a_{k} is odd, e(𝔄_{k}/𝔬_{E}) is odd, and e(𝔄_{k+1}/𝔬_{E}) is even, then

A𝔙_{ξ,k} ≅ ⨁_{\begin{matrix}[g]=[σ^{i}ϕ^{j}]∈Γ_{E}\Γ_{E_{k+1}} /Γ_{E} \\ j≡hj_{k} mod e(𝔄/𝔬_{E}) \end{matrix}} 𝔘_{[g]}.

(iii) When a_{k} is even or e(𝔄_{k}/𝔬_{E}) is even, then

A𝔙_{ξ,k} ≅ ⨁_{\begin{matrix}[g]=[σ^{i}ϕ^{j}]∈Γ_{E}\(Γ_{E_{k+1}} −Γ_{E_{k}})/Γ_{E} \\ j≡hj_{k} mod e(𝔄/𝔬_{E}) \end{matrix}} 𝔘_{[g]}.

4.3 Some properties of parities of jumps

Let R be the index when f(E/E_{R}) is odd and f(E/E_{R+1}) is even.

Lemma 4.7. We have (Γ_{E_{R+1}}/Γ_{E})_{sym−unram} = ((Γ_{E_{R+1}} −Γ_{E_{R}})/Γ_{E})_{sym−unram}.

Proof. Recall Proposition 2.2 that every symmetric unramified [g] are of the form [σ^{i}ϕ^{f/2}], so there is no coset of the form σ^{i}ϕ^{f/2} belonging to Γ_{E_{R}}.

Let Q be the index when e(𝔄_{Q}/𝔬_{E}) is odd and e(𝔄_{Q+1}/𝔬_{E}) is even.

Lemma 4.8. Suppose that f is even. We always have R ≤ Q. If moreover m is odd, then Q = R.

Proof. We know that e(𝔄_{k}/𝔬_{E}) divides f(E/E_{k}), so that if Q < R, then the even number e(𝔄_{Q+1}/𝔬_{E}) divides f(E/E_{Q+1}), which divides the odd number f(E/E_{R}). This is a contradiction. Hence R ≤ Q. When R ⪇ Q, then

e(𝔄_{R+1}/𝔬_{E}) is odd and f(E/E_{R+1}) is even.  (4.8)

Since

e(𝔄_{R+1}/𝔬_{E}) = f_{E_{R+1}}/s_{E_{R+1}} = \frac{f(E/E_{R+1})}{gcd(f(E/E_{R+1}), gcd(m, n(E/E_{R+1})))}, (4.9)

the statement (4.8) is equivalent to saying that

the 2-powers of the numerator and

the denominator on the right side of (4.9) are equal.

(4.10)

This power is greater than 0. Hence (4.10) is equivalent to that

(the 2-power of m) ≥ (the 2-power of f(E/E_{R+1})) ⪈ 0.  (4.11)

If m is odd, then (4.11) is a contradiction.

4.4 Symmetric submodules

We write _{A}𝔙_{ξ,sym} = _{A}𝔙_{ξ} ∩ 𝔘_{sym} and _{A}𝔙_{ξ,sym−ram} and _{A}𝔙_{ξ,sym−unram} analogously.

4.4.1 Case when f is odd

From Proposition 3.2, we always have

A𝔙_{ξ,sym} = _{A}𝔙_{ξ,sym−ram} ≅ _{A^{∗}}𝔙_{ξ,sym−ram} = _{A^{∗}}𝔙_{ξ,sym}.  (4.12)

4.4.2 Case when f is even

Notice that the natural projection ℤ/fℤ → ℤ/e(𝔄/𝔬_{E})ℤ maps

f/2 ↦ \begin{cases}0 & if e(𝔄/𝔬_{E}) divides f/2, \\ e(𝔄/𝔬_{E})/2 ≠ 0 & otherwise \end{cases}  (4.13)

The condition that e(𝔄/𝔬_{E}) divides f/2 is equivalent to that s is even. When f is even, then s = gcd(f, m) is even if and only if m is even. We hence separate the cases according to the parity of m.

4.4.3 Case when both f and m are even

In this case, f/2 is mapped to 0 by ℤ/fℤ → ℤ/e(𝔄/𝔬_{E})ℤ. We separate the cases according to the parity of the jump a_{k}. When a_{k} is odd, neither 0 or f/2 is mapped to hj_{k} ≠ 0 ∈ ℤ/e(𝔄/𝔬_{E})ℤ, and so _{A}𝔙_{ξ,k} is trivial. When a_{k} is even, both 0 and f/2 are mapped to hj_{k} = 0 by (4.13), and so _{A}𝔙_{ξ,k,sym} ≅ 𝔘_{k,sym}. We also recall that

A^{∗}𝔙_{ξ,k} ≅ \begin{cases}0 & if a_{k} is odd, \\ 𝔘_{k,sym} & if a_{k} is even. \end{cases}

Whatever the parity of a_{k} is, we always have _{A}𝔙_{ξ,sym} ≅ _{A^{∗}}𝔙_{ξ,sym}.

4.4.4 Case when f is even and m is odd

In this case, notice that e(𝔄/𝔬_{E}) must be even, and

j_{k} = e(𝔄/𝔬_{E})a_{k}/2 ≡ \begin{cases}0 \\ e(𝔄/𝔬_{E})/2 \end{cases} mod e(𝔄/𝔬_{E}) \{ \begin{matrix}if a_{k} is even, \\ if a_{k} is odd. \end{matrix}

Therefore,

A𝔙_{ξ,k,sym} = \begin{cases}_{A}𝔙_{ξ,k,sym−ram} & if a_{k} is even, \\ _{A}𝔙_{ξ,k,sym−unram} & if a_{k} is odd. \end{cases}

Using Proposition 4.6, we find that when a_{k} is even,

A𝔙_{ξ,k,sym} = 𝔘_{E_{k}/E_{k+1},sym−ram} = ⨁_{[σ^{i}]∈(Γ_{E}\(Γ_{E_{R+1}} −Γ_{E_{R}})/Γ_{E})_{sym}} 𝔘_{[σ^{i}]},

Here the index R is defined in Section 4.3. When a_{k} is odd, _{A}𝔙_{ξ,k,sym} is trivial when k < R, and is isomorphic to

𝔘_{E/E_{R+1},sym−unram} = ⨁_{[σ^{i}ϕ^{f/2}]∈(Γ_{E}\Γ_{E_{R}} /Γ_{E})_{sym}} 𝔘_{[σ^{i}ϕ^{f/2}]}

when k = R, and to

𝔘_{E_{k}/E_{k+1},sym−unram} = ⨁_{[σ^{i}ϕ^{f/2}]∈(Γ_{E}\(Γ_{E_{k+1}} −Γ_{E_{k}})/Γ_{E})_{sym}} 𝔘_{[σ^{i}ϕ^{f/2}]}

when k > R. We observe that, whether a_{k} is odd or even, the symmetric unramified part of A𝔙_{ξ} and _{A^{∗}}𝔙_{ξ} are complementary, in the sense that

A𝔙_{ξ,k,sym−unram} ⊕_{A^{∗}}𝔙_{ξ,k,sym−unram} = 𝔘_{E_{k}/E_{k+1},sym−unram}

for all k = 0, …, t We summarize the above in the following.

Proposition 4.9. We always have _{A}𝔙_{ξ,sym−ram} ≅ _{A^{∗}}𝔙_{ξ,sym−ram} and

(i) when f is odd, or when both f and m are even, then _{A}𝔙_{ξ,sym−unram} ≅ A^{∗}𝔙_{ξ,sym−unram};

(ii) when f is even and m is odd, then _{A}𝔙_{ξ,sym−unram} ⊕ _{A^{∗}}𝔙_{ξ,sym−unram} = 𝔘_{sym−unram}.

4.5 t-factors of isotypic components

We recall the values of the t-factors t_{Γ}^{i}(𝔙), i = 0, 1, when Γ is one of the cyclic subgroups μ and ⟨ϖ⟩ of Ψ_{E/F} defined in (3.20), and 𝔙 is a symmetric module 𝔘_{[g]} defined in (4.5). The following Proposition describes all t_{Γ}^{i}(𝔘_{[g]}) except when [g] = [σ^{e/2}].

Proposition 4.10 ([Tam, Proposition 4.9]). (i) If [g] = [σ^{i}ϕ^{j}] is asymmetric, then

t_{μ}^{0}(𝔘_{[g]}) = 1, t_{μ}^{1}(𝔘_{[g]}) : z ↦ sgn_{z^{q^{i}−1}}(𝔘_{[g]}),
t_{⟨ϖ⟩}^{0}(𝔘_{[g]}) = 1, and t_{⟨ϖ⟩}^{1}(𝔘_{[g]})(ϖ_{E}) = sgn_{z_{e}^{i}z_{ϕ^{j}}} (𝔘_{[g]}).

(ii) If [g] = [σ^{i}] is symmetric and not equal to [1] or [σ^{e/2}], then

\begin{matrix}t_{μ}^{0}(𝔘_{[g]}) = 1, t_{μ}^{1}(𝔘_{[g]}) ≡ 1, \\ t_{⟨ϖ⟩}^{0}(𝔘_{[g]}) = −1, and t_{⟨ϖ⟩}^{1}(𝔘_{[g]}) : ϖ_{E} ↦ (\frac{z_{e}^{i}}{ker(N_{𝔽_{p}[z_{e}^{i}]/𝔽_{p}[z_{e}^{i}]_{±}})}) . \end{matrix}

(iii) If [g] = [σ^{i}ϕ^{f/2}] is symmetric, then

t_{μ}^{0}(𝔘_{[g]}) = −1, t_{μ}^{1}(𝔘_{[g]}) is quadratic,

and

Documenta Mathematica 21 (2016) 345–389

(I) if z_{e}^{i}z_{ϕ^{f/2}} = 1, then t_{⟨ϖ⟩}^{0}(𝔘_{[g]}) = 1 and t_{⟨ϖ⟩}^{1}(𝔘_{[g]}) ≡ 1;

(II) if z_{e}^{i}z_{ϕ^{f/2}} = −1, then t_{⟨ϖ⟩}^{0}(𝔘_{[g]}) = 1 and t_{⟨ϖ⟩}^{1}(𝔘_{[g]})(ϖ_{E}) =

(−1)^{\frac{1}{2} (q^{f/2}−1)};
(III) if z_{e}^{i}z_{ϕ^{f/2}} ≠ ±1, then t_{⟨ϖ⟩}^{0}(𝔘_{[g]}) = −1 and

t_{⟨ϖ⟩}^{1}(𝔘_{[g]}) : ϖ_{E} ↦ (\frac{z_{e}^{i}z_{ϕ^{f/2}}}{ker(N_{𝔽_{p}[z_{e}^{i}z_{ϕ^{f/2}} ]/𝔽_{p}[z_{e}^{i}z_{ϕ^{f/2}} ]_{±}})}) .

In the exceptional case, when [g] = [σ^{e/2}], we have μ_{E_{g}} = μ_{E}. To unify notation, we define

t_{μ}^{1}(𝔘_{[σ^{e/2}]}) : μ_{E} → \{±1\}, x ↦ (\frac{x}{μ_{E}} ) .  (4.14)

The 𝔽_{p}⟨ϖ⟩-module structure of 𝔘_{[σ^{e/2}]} does not concern us (see the explanation after Formula (5.5)). The following properties concerning symmetric double cosets are useful when computing the above t-factors.

Proposition 4.11. Suppose that [g] is symmetric.

(i) If [g] is ramified (resp. unramified), then [𝔘_{[g]} : k_{E}] is even (resp. odd).

(ii) Let 𝔽_{p}[[ \begin{matrix}1 \\ g \end{matrix} ] (ϖ_{E})] be the field extension of 𝔽_{p} generated by the image of [ \begin{matrix}1 \\ g \end{matrix} ] (ϖ_{E}) in k̄_{F}^{×}. If [g] ≠ [σ^{e/2}], then the degree [𝔘_{[g]} : 𝔽_{p}[[ \begin{matrix}1 \\ g \end{matrix} ] (ϖ_{E})]] is odd.

Proof. The first statement for ramified [g] is a simple calculation, and that for unramified [g] is a consequence of Proposition 2.3. The second statement is proved in [Tam, Lemma 4.8]. We would like to extend our definition of the t-factors t_{μ}^{i}(𝔘_{[g]}), with i = 0, 1, from μ to μ_{g} = μ_{E_{g}}/μ_{F}. We define

t_{μ_{g}}^{0}(𝔘_{[g]}) := t_{μ}^{0}(𝔘_{[g]})

and for all z ∈ μ_{g},

t_{μ_{g}}^{1}(𝔘_{[g]}) : z ↦ \begin{cases}sgn_{[ \begin{matrix}1 \\ g \end{matrix} ](z)}(𝔘_{[g]}) & if [g] is asymmetric, \\ (^{\frac{[ \begin{matrix}1 \\ σ^{i}ϕ^{f[𝔘_{[g]}:k_{E}]/2} \end{matrix} ](z)}{ker N_{k_{Eg} /k_{E_{±g}}}}} ) & if [g] is symmetric. \end{cases}

Proposition 4.12. The restriction t_{μ_{g}}^{1}(𝔘_{[g]}) to μ is t_{μ}^{1}(𝔘_{[g]}) .

Documenta Mathematica 21 (2016) 345–389

Proof. For asymmetric [g], the result is immediate by definition. For symmetric ramified [g], the restriction of the root [ \begin{matrix}1 \\ σ^{i}ϕ^{f[𝔘_{[g]}:k_{E}]} \end{matrix} ] to μ = μ_{E}/μ_{F} is trivial, so the assertion is again true. When [g] is symmetric unramified, we

have to show that the restriction of (_{\frac{·}{ker N_{k_{Eg} /k_{E_{±g}}}}} ) to μ is (\frac{·}{ker N_{k_{E} /k_{E±}}} ), or

equivalently, to show that the index of the subgroup ker N_{k_{E}/k_{E±}} ≅ μ_{q^{f/2}+1} of ker N_{k_{E}/k_{E±}} ≅ μ_{q^{f[𝔘_{[g]}:k_{E}]/2}+1} is odd, which follows from Proposition 4.11.

5 Zeta-data

5.1 Admissible embeddings of L-tori

As mentioned in Section 1.3, to understand ζ-data, it is better to first understand χ-data, which is motivated by constructing admissible embeddings of L-tori [LS87, Section 2.6]. We take T to be an elliptic torus of G isomorphic to Res_{E/F} 𝔾_{m}. Its dual torus T̂ is Ind_{E/F} (ℂ^{×}), which is isomorphic to (ℂ^{×})^{n} as a group. It is equipped with the induced action of the Weil group W_{F}, which factors through the action of the Galois group Γ_{F}. We define the L-torus ^{L}T := T̂ ⋊ W_{F} as the L-group of T. We assume that the dual torus T̂ is embedded into the L-group ^{L}G = Ĝ × W_{F} of G, where Ĝ = GL_{n}(ℂ), with image T . For convenience, we simply denote the image of t ∈ T̂ by the embedding T̂ → T ⊂ Ĝ also by t ∈ T . This embedding should be defined using the chosen splittings of G and Ĝ. As we do not need the full detail of the definition of this embedding, we only refer to [LS87, Section 2.5] for details (or, when (G, T) = (GL_{n}, Res_{E/F} 𝔾_{m}), see [Tam, Section 6.1]). All we need to know is that we can always assume that the image T is the diagonal subgroup of Ĝ. With the embedding T̂ → T chosen, an admissible embedding from ^{L}T to ^{L}G is a morphism of groups I : ^{L}T → ^{L}G of the form

I(t ⋊ w) = tI(1 ⋊ w) for all t ⋊ w ∈ ^{L}T.

Note that an admissible embedding maps W_{F} into NĜ(T ), i.e., the factor I(1 ⋊ w) above lies in NĜ(T ). Two admissible embeddings I_{1}, I_{2} are called Int(T )-equivalent if there is t ∈ T such that

I_{1}(w) = tI_{2}(w)t^{−1} for all w ∈ W_{F} .

By [LS87, Section 2.6], admissible embeddings exist, and the collection of these embeddings can be described as follows.

Proposition 5.1. The set of admissible embeddings from ^{L}T to ^{L}G is a Z^{1}(W_{F} , T̂)-torsor, and the set of the Int(T )-equivalence classes of these embeddings is an H^{1}(W_{F} , T̂)-torsor.

The idea in [LS87, Section 2.5] of constructing an admissible embedding is to choose a set of characters

{χ_{λ}}_{λ∈Φ}, where χ_{λ} : E_{λ}^{×} → ℂ^{×},

called χ-data, such that the following conditions hold.

Definition 5.2. (i) For each λ ∈ Φ, we have χ_{−λ} = χ_{λ}^{−1} and χ_{^{w}λ} = ^{w}χ_{λ} for all w ∈ W_{F}.

(ii) If λ is symmetric, then χ|_{E_{±λ}^{×}} equals the quadratic character δ_{E_{λ}/E_{±λ}} attached to the extension E_{λ}/E_{±λ}.

Remember that, in Section 2.1, we choose a subset R_{±} = R_{sym} ⊔ R_{asym/±} of Φ representing the orbits W_{F} \Φ_{sym} and W_{F} \Φ_{asym/±}. Hence, by condition (i), the set of χ-data depends completely on the subset {χ_{λ}}_{λ∈R_{±}}. We still call such a subset a set of χ-data. Moreover, using Artin reciprocity [Tat79], we may regard each χ_{λ} as a character of the Weil group W_{E_{λ}}. Following the recipe in [LS87, Section 2.5], we can define an admissible embedding

I_{{χ_{λ}}} : ^{L}T → ^{L}G

depending on a given set of χ-data. In our present situation, we can describe the admissible embedding I_{{χ_{λ}}} in Proposition 5.3 below. We first recall the Langlands correspondence for the torus T = Res_{E/F} 𝔾_{m}, which is a bijection

Hom(T (F), ℂ^{×}) → H^{1}(W_{F} , T̂).  (5.1)

Given a character ξ of T (F) = E^{×}, we denote by ξ̃ a 1-cocycle in Z^{1}(W_{F} , T̂) whose class is the image of ξ under (5.1). Given χ-data {χ_{λ}}_{λ∈R_{±}}, we define

μ := μ_{{χ_{λ}}} = ∏_{λ∈R} Res_{E^{×}}^{E_{λ}^{×}}χ_{λ},

where R = R_{±} ⊔(−R_{asym/±}) is a subset representing Γ_{F} \Φ. It is easy to check that the product of the restricted characters is independent of representatives in R, so we usually write

μ = ∏_{[λ]∈Γ_{F} \Φ} Res_{E^{×}}^{E_{λ}^{×}}χ_{λ}.

Proposition 5.3 ([Tam, Proposition 6.5]). For every character ξ of E^{×}, the composition

I_{{χ_{λ}}} ∘ ξ̃: W_{F} → ^{L}T → ^{L}G ⟶^{proj} GL_{n}(ℂ)

is isomorphic to Ind_{E/F} (ξ · μ_{{χ_{λ}}}) as a representation of W_{F}.

We now define a analogous set of characters

{ζ_{λ}}_{λ∈Φ}, where ζ_{λ} : E_{λ}^{×} → ℂ^{×},

called ζ-data, such that the following conditions hold.

Definition 5.4. (i) For each λ ∈ Φ, we have ζ_{−λ} = ζ_{λ}^{−1} and ζ_{^{w}λ} = ^{w}ζ_{λ} for all w ∈ W_{F}.

(ii) If λ is symmetric, then ζ|_{E_{±λ}^{×}} is trivial.

We can view a set of ζ-data as the difference of two sets of χ-data. Motivated from Propositions 5.1 and 5.3, the product character

ν := ν_{{ζ_{λ}}} = ∏_{[λ]∈Γ_{F} \Φ} Res_{E^{×}}^{E_{λ}^{×}}ζ_{λ}.

can be viewed as measuring the difference of two admissible embeddings. Recall that, similar to choosing R_{±}, we can also choose D_{±} = D_{sym} ⊔ D_{asym/±} to be a subset of Γ_{F} /Γ_{E} consisting of representatives of (Γ_{E}\Γ_{F} /Γ_{E})_{sym} and (Γ_{E}\Γ_{F} /Γ_{E})_{asym/±} respectively, and obtain a bijection from Proposition 2.1,

R_{±} = R_{sym} ⨆ R_{asym/±} → D_{±} = D_{sym} ⨆ D_{asym/±}, λ = [ \begin{matrix}1 \\ g \end{matrix} ] ↦ g.

We usually denote by E_{g} and E_{±g} the fields E_{λ} and E_{±λ} respectively, if g ∈ D_{±} corresponds to λ ∈ R_{±}. We also denote by ζ_{g} the character ζ_{λ}, and write

ν := ν_{{ζ_{g}}} = ∏_{[g]∈(W_{E}\W_{F} /W_{E})′} Res_{E^{×}}^{E_{g}^{×}}ζ_{g}.

5.2 Symmetric unramified zeta-data

We choose a specific ζ-data ζ_{g} for each [g] = [σ^{i}ϕ^{f/2}] ∈ (W_{E}\W_{F} /W_{E})_{sym−unram}, base on the results from the χ-datum χ_{g}. Notice that, since E_{g}/E_{±g} is quadratic unramified, the norm group N_{E_{g}/E_{±g}}(E_{g}^{×}) has a decomposition

μ_{E_{±g}} × ⟨z_{e}^{i}z_{ϕ^{f/2}}ϖ_{E}^{2}⟩ × U_{E_{±g}}^{1}

and we take a root of unity z_{0} ∈ μ_{E_{g}} such that

z_{0}ϖ_{E} ∈ E_{±g}^{×} − N_{E_{g}/E_{±g}}(E_{g}^{×}).

We only consider tamely ramified χ-data and ζ-data, i.e., we require that

χ_{g}|_{U_{Eg}^{1}} ≡ 1 and ζ_{g}|_{U_{Eg}^{1}} ≡ 1.

Therefore, the Definition 5.2.(ii) of χ-data is explicitly (see [Tam, (7.6)])

χ_{g}(μ_{E_{±}g}) = 1, χ_{g}(z_{e}^{i}z_{ϕ^{f/2}}ϖ_{E}^{2}) = 1, and χ_{g}(z_{0}ϖ_{E}) = −1.  (5.2)

Hence, given a χ-datum χ_{g}, we can obtain a ζ-datum ζ_{g} easily by requiring

ζ_{g}|_{μ_{Eg}} = χ_{g}|_{μ_{Eg}} and ζ_{g}(ϖ_{E}) = −χ_{g}(ϖ_{E}).

In [Tam, Section 7.4], in the cases when 𝔙_{ξ,[g]} ≅ 𝔘_{[g]} is non-trivial, we construct a χ-datum

χ_{g}|_{μ_{Eg}} = t_{μ_{g}}^{1}(𝔘_{[g]}) and χ_{g}(ϖ_{E}) = \begin{cases}−t_{⟨ϖ⟩}(𝔘_{[g]}) & if ^{σ^{i}ϕ^{f/2}}ϖ_{E} = ϖ_{E}, \\ t_{⟨ϖ⟩}(𝔘_{[g]}) & otherwise. \end{cases}

In other words, the character χ_{g} satisfies the conditions in (5.2). Hence the character

ζ_{g}|_{μ_{Eg}} = t_{μ_{g}}^{1}(𝔘_{[g]}) and ζ_{g}(ϖ_{E}) = \begin{cases}t_{⟨ϖ⟩}(𝔘_{[g]}) & if ^{σ^{i}ϕ^{f/2}}ϖ_{E} = ϖ_{E}, \\ −t_{⟨ϖ⟩}(𝔘_{[g]}) & otherwise \end{cases}

is a ζ-datum. This ζ-datum will be used in the next section.

5.3 Zeta-data associated to admissible characters

Given an admissible character ξ of E^{×} over F, we first assign, for each [g] ∈ (W_{E}\W_{F} /W_{E})_{asym/±}, the values of the ζ-data to

ζ_{g,ξ}|_{μ_{Eg}} = sgn_{μ_{Eg}} (_{A}𝔙_{[g]})sgn_{μ_{Eg}} (_{A^{∗}}𝔙_{[g]}) = t_{μ_{g}}^{1}(_{A}𝔙_{ξ,[g]})t_{μ_{g}}^{1}(_{A^{∗}}𝔙_{ξ,[g]}).

In this way, the product of the characters

ζ_{g,ξ}ζ_{g^{−1},ξ} = ζ_{g,ξ} (ζ_{g,ξ}^{g})^{−1} = ζ_{g,ξ} ∘ [ \begin{matrix}1 \\ g \end{matrix} ]

has values

\begin{aligned}(ζ_{g,ξ} ∘ [ \begin{matrix}1 \\ g \end{matrix} ]) |_{μ_{E}}(z) & = sgn_{μ_{E}}(_{A}𝔙_{ξ,[g]})([ \begin{matrix}1 \\ g \end{matrix} ] (z))sgn_{μ_{E}}(_{A^{∗}}𝔙_{ξ,[g]})([ \begin{matrix}1 \\ g \end{matrix} ] (z)) \\ & = t_{μ}^{1}(_{A}𝔙_{ξ,[g]})t_{μ}^{1}(_{A^{∗}}𝔙_{ξ,[g]}) \end{aligned}

and

\begin{aligned}(ζ_{g,ξ} ∘ [ \begin{matrix}1 \\ g \end{matrix} ]) (ϖ_{E}) & = ζ_{g,ξ}|_{μ_{E}} ([ \begin{matrix}1 \\ g \end{matrix} ] (ϖ_{E})) \\ & = sgn_{[ \begin{matrix}1 \\ g \end{matrix} ](ϖ_{E})}(_{A}𝔙_{ξ,[g]})sgn_{[ \begin{matrix}1 \\ g \end{matrix} ](ϖ_{E})}(_{A^{∗}}𝔙_{ξ,[g]}) \\ & = t_{⟨ϖ⟩}^{1}(_{A}𝔙_{ξ,[g]})(ϖ_{E})t_{⟨ϖ⟩}^{1}(_{A^{∗}}𝔙_{ξ,[g]})(ϖ_{E}) \\ & = t_{⟨ϖ⟩}(_{A}𝔙_{ξ,[g]})t_{⟨ϖ⟩}(_{A^{∗}}𝔙_{ξ,[g]}). \end{aligned}  (5.3)

We then assign, for each [g] ∈ (W_{E}\W_{F} /W_{E})_{asym/±}, arbitrary values to ζ_{g,ξ}(ϖ_{E}) and ζ_{g^{−1},ξ}(ϖ_{E}), as long as the product satisfies (5.3). (This phenomenon is comparable to [LS87, Lemma 3.3.A], as explained in [Tam, Remark 7.2].) It is routine to check that each ζ_{g,ξ} is a ζ-datum. Indeed, this checking is exactly the same as that in the χ-data case (see [Tam, Section 7.2]), since Definition 5.2.(i) is the same as that of χ-data. We then assign values to the ζ-data for each [g] ∈ (W_{E}\W_{F} /W_{E})_{sym} case by case.

5.3.1 Case when f is odd

Recall from (4.12) that

t_{Γ}^{i}(_{A}𝔙_{ξ,sym}) = t_{Γ}^{i}(_{A^{∗}}𝔙_{ξ,sym}), for i = 0, 1 and Γ = μ, ⟨ϖ⟩.  (5.4)

We assign the ζ-data to the following values. If e is odd (so that m is odd since m divides e), we assign all ζ_{g,ξ} to be trivial. If e is even, then we just take all ζ_{g,ξ}, [g] ≠ [σ^{e/2}], to be trivial and

ζ_{σ^{e/2},ξ}|_{μ_{E}} ≡ 1 and ζ_{σ^{e/2},ξ}(ϖ_{E}) = (−1)^{m}.

To show that ζ_{σ^{e/2},ξ} is a ζ-datum, notice that since N_{E/E_{±σ^{e/2}}} (ϖ_{E}) = −ϖ_{E_{±σ^{e/2}}} = −ϖ_{E}^{2}, and since

ζ_{σ^{e/2},ξ}(ϖ_{E})^{2} = ζ_{σ^{e/2},ξ}(−1)ζ_{σ^{e/2},ξ}(−ϖ_{E}^{2}) = (1)(1) = 1,

we can assign χ_{σ^{e/2},ξ}(ϖ_{E}) to either 1 or −1 to obtain a ζ-datum. By (5.4), we can rewrite our assigned ζ-data as

\begin{matrix}ζ_{g,ξ}|_{μ_{E}} = t_{μ}^{1}(_{A}𝔙_{ξ,[g]})t_{μ}^{1}(_{A^{∗}}𝔙_{ξ,[g]}) \\ ζ_{g,ξ}(ϖ_{E}) = \begin{cases}t_{⟨ϖ⟩}(_{A}𝔙_{ξ,[g]})t_{⟨ϖ⟩}(_{A^{∗}}𝔙_{ξ,[g]}) & if g ≠ σ^{e/2}, \\ (−1)^{m}t_{⟨ϖ⟩}(_{A}𝔙_{ξ,[g]})t_{⟨ϖ⟩}(_{A^{∗}}𝔙_{ξ,[g]}) & if g = σ^{e/2}. \end{cases} \end{matrix}  (5.5)

Note that t_{⟨ϖ⟩}(_{A}𝔙_{ξ,[σ^{e/2}]}) is not defined (see the paragraph containing Formula (4.14)). In fact, we just take

t_{⟨ϖ⟩}(_{A}𝔙_{ξ,[σ^{e/2}]}) = t_{⟨ϖ⟩}(_{A^{∗}}𝔙_{ξ,[σ^{e/2}]}) = 1,

since _{A}𝔙_{ξ,[σ^{e/2}]} ≅ _{A^{∗}}𝔙_{ξ,[σ^{e/2}]} by Proposition 3.2, and it is shown in [Tam, Proposition 5.3] that _{A^{∗}}𝔙_{ξ,[σ^{e/2}]} is always trivial. The product of ζ-data is equal to

∏_{[g]∈(W_{E}\W_{F} /W_{E})′} ζ_{g,ξ}(ϖ_{E}) ≡ 1

and ∏_{[g]∈(W_{E}\W_{F} /W_{E})′} ζ_{g,ξ}(ϖ_{E}) = \begin{cases}(−1)^{m} & if e is even, \\ 1 & if e is odd, \end{cases}

which is rewritten as

∏_{[g]∈(W_{E}\W_{F} /W_{E})′} ζ_{g,ξ}|_{μ_{E}} ≡ t_{μ}^{1}(_{A}𝔙_{ξ})t_{μ}^{1}(_{A^{∗}}𝔙_{ξ})

and

∏_{[g]∈(W_{E}\W_{F} /W_{E})′} ζ_{g,ξ}(ϖ_{E}) = \begin{cases}(−1)^{m}t_{⟨ϖ⟩}(_{A}𝔙_{ξ})t_{⟨ϖ⟩}(_{A^{∗}}𝔙_{ξ}) & if e is even, \\ t_{⟨ϖ⟩}(_{A}𝔙_{ξ})t_{⟨ϖ⟩}(_{A^{∗}}𝔙_{ξ}) & if e is odd. \end{cases}

The product is equal to the rectifier given in (3.21),

Dν_{ξ}|_{μ_{E}} ≡ 1 and _{D}ν_{ξ}(ϖ_{E}) = (−1)^{m(d−1)} = (−1)^{e−m},

when E/F is totally ramified.

5.3.2 Case when f is even

Let K be the maximal unramified extension of E/F. If we define _{D_{K}} ν_{ξ} to be the ramified part of _{D}ν_{ξ}, which is also the rectifier corresponding to the admissible pair (E/K, ξ), then we have

D_{K} ν_{ξ} = ∏_{[g]∈(W_{E}\W_{K}/W_{E})′} ζ_{g,ξ}|_{E^{×}},

as when f is odd, and in particular

D_{K} ν_{ξ}(ϖ_{E}) = (−1)^{e−m_{K}} ,

where m_{K} = gcd(e, m). Therefore, our plan is to distribute the sign

(−1)^{e−m_{K}} (−1)^{n−m+f_{ϖ}−m_{ϖ}} = \begin{cases}1 & if m is even, \\ (−1)^{e+f_{ϖ}+1} & if m is odd, \end{cases}  (5.6)

to each ζ_{g,ξ}(ϖ_{E}), where [g] is symmetric unramified, multiplying the product of t-factors t_{⟨ϖ⟩}(_{A}𝔙_{ξ,[g]})t_{⟨ϖ⟩}(_{A^{∗}}𝔙_{ξ,[g]}). As before, we separate the cases according to the parity of m. When m is even, recall from Proposition 4.9 that we have either

both _{A}𝔙_{ξ,k,sym} and _{A^{∗}}𝔙_{ξ,k,sym} are trivial,

or

both _{A}𝔙_{ξ,k,sym} and _{A^{∗}}𝔙_{ξ,k,sym} are isomorphic to 𝔘_{k,sym}.

We assign the trivial ζ-data for all [g] ∈ (W_{E}\W_{F} /W_{E})_{sym−unram}, so that

ζ_{g,ξ}|_{μ_{E}} = t_{μ}^{1}(_{A}𝔙_{ξ,[g]})t_{μ}^{1}(_{A^{∗}}𝔙_{ξ,[g]}) and ζ_{g,ξ}(ϖ_{E}) = t_{⟨ϖ⟩}(_{A}𝔙_{ξ,[g]})t_{⟨ϖ⟩}(_{A^{∗}}𝔙_{ξ,[g]}).

The product of ζ_{g,ξ}(ϖ_{E}) is trivial, or we can write

\begin{aligned}& ∏_{[g]∈(W_{E}\W_{F} /W_{E})′} ζ_{g,ξ}|_{μ_{E}} = t_{μ}^{1}(_{A}𝔙_{ξ})t_{μ}^{1}(_{A^{∗}}𝔙_{ξ}) \\ and & ∏_{[g]∈(W_{E}\W_{F} /W_{E})′} ζ_{g,ξ}(ϖ_{E}) = t_{⟨ϖ⟩}(_{A}𝔙_{ξ})t_{⟨ϖ⟩}(_{A^{∗}}𝔙_{ξ}). \end{aligned}

Note that in the second product, the sign without t-factors is equal to (5.6), which is just 1. When m is odd, we have

A𝔙_{ξ,sym−unram} ⊕_{A^{∗}}𝔙_{ξ,sym−unram} = 𝔘_{sym−unram}.

We then assign the ζ-data to be

\begin{aligned}& ζ_{g,ξ}|_{μ_{E}} = t_{μ}^{1}(_{A}𝔙_{ξ,[g]})t_{μ}^{1}(_{A^{∗}}𝔙_{ξ,[g]}) = t_{μ}^{1}(𝔘_{[g]}) \\ and & ζ_{g,ξ}(ϖ_{E}) = −t_{⟨ϖ⟩}(_{A}𝔙_{ξ,[g]})t_{⟨ϖ⟩}(_{A^{∗}}𝔙_{ξ,[g]}) = −t_{⟨ϖ⟩}(𝔙_{[g]}). \end{aligned}

for all symmetric unramified [g] except the one which stabilizes ϖ_{E}, in which we assign

\begin{aligned}& ζ_{g,ξ}|_{μ_{E}} = t_{μ}^{1}(_{A}𝔙_{ξ,[g]})t_{μ}^{1}(_{A^{∗}}𝔙_{ξ,[g]}) = t_{μ}^{1}(𝔘_{[g]}) \\ and & ζ_{g,ξ}(ϖ_{E}) = t_{⟨ϖ⟩}(_{A}𝔙_{ξ,[g]})t_{⟨ϖ⟩}(_{A^{∗}}𝔙_{ξ,[g]}) = t_{⟨ϖ⟩}(𝔙_{[g]}). \end{aligned}

In Section 5.2, we checked that the above characters give rise to ζ-data. The product of ζ is hence

\begin{aligned}& ∏_{[g]∈(W_{E}\W_{F} /W_{E})′} ζ_{g,ξ}|_{μ_{E}} = t_{μ}^{1}(_{A}𝔙_{ξ})t_{μ}^{1}(_{A^{∗}}𝔙_{ξ}) \\ and & ∏_{[g]∈(W_{E}\W_{F} /W_{E})′} ζ_{g,ξ}(ϖ_{E}) = (−1)^{e+f_{ϖ}+1}t_{⟨ϖ⟩}(_{A}𝔙_{ξ})t_{⟨ϖ⟩}(_{A^{∗}}𝔙_{ξ}), \end{aligned}

by Proposition 2.6. Again in the second product, the sign without t-factors is equal to (5.6).

5.4 The main theorem

To summarize, we verified the following theorem.

Theorem 5.5. Let ξ be an admissible character of E^{×} over F.

(i) Let _{A}𝔙_{ξ} (resp. _{A^{∗}}𝔙_{ξ}) be the finite symplectic module defined by ξ when G(F) = GL_{m}(D) (resp. when G^{∗}(F) = GL_{n}(F)). The following conditions define a collection of ζ-data {ζ_{g,ξ}}_{g∈D_{asym/±}⊔D_{sym}}.

(a) All ζ_{g,ξ} are tamely ramified.

(b) If g ∈ D_{asym/±}, then
ζ_{g,ξ}|_{μ_{Eg}} = t_{μ_{g}}^{1}(_{A}𝔙_{ξ,[g]})t_{μ_{g}}^{1}(_{A^{∗}}𝔙_{ξ,[g]})

and

ζ_{g,ξ}(ϖ_{E}) can be any value satisfying
ζ_{g,ξ}(ϖ_{E})ζ_{g^{−1},ξ}(ϖ_{E}) = t_{⟨ϖ⟩}(_{A}𝔙_{ξ,[g]})t_{⟨ϖ⟩}(_{A^{∗}}𝔙_{ξ,[g]}).
(c) If g ∈ D_{sym}, then
ζ_{g,ξ}|_{μ_{Eg}} = t_{μ_{g}}^{1}(_{A}𝔙_{ξ,[g]})t_{μ_{g}}^{1}(_{A^{∗}}𝔙_{ξ,[g]})

and

ζ_{g,ξ}(ϖ_{E}) = ϵ_{g}t_{⟨ϖ⟩}(_{A}𝔙_{ξ,[g]})t_{⟨ϖ⟩}(_{A^{∗}}𝔙_{ξ,[g]}),

where ϵ_{g} is equal to 1 if g ∈ (D_{sym−ram} − {σ^{e/2}}) ∪ W_{F [ϖ_{E}]} and is equal to (−1)^{m} if g ∈ (D_{sym−unram} − W_{F [ϖ_{E}]}) ∪ {σ^{e/2}}.

(ii) Let _{D}ν_{ξ} be the rectifier of ξ and {ζ_{g,ξ}}_{g∈D_{±}} be the ζ-data in (i), then

Dν_{ξ} = ∏_{[g]∈(W_{E}\W_{F} /W_{E})′} ζ_{g,ξ}|_{E^{×}}.

Remark 5.6. As long as the F -dimension of the division algebra D is fixed, the rectifier _{D}ν_{ξ} is independent of the Hasse-invariant h = h(D) of D, as stated in [BH11, Theorem C]. This is because the modules_{A}𝔙_{ξ}, where A = Mat_{n}(D) and D ranges over all division algebra with same F -dimension, are all isomorphic to each other. Similarly, the ζ-data {ζ_{g,ξ}} are independent of h(D).

5.5 Functorial property

Let K be an intermediate subfield in E/F, and write

n_{K} = n(E/K) = f_{K}e_{K} = f(E/K)e(E/K) and m_{K} = gcd(m, n_{K}).

Similar to Section 3.2, we have the Jacquet-Langlands correspondence

JL_{K} : A_{n_{K}}^{et} (K) → A_{m_{K}}^{et} (D_{K}),

between essentially tame supercuspidal representations of G(F)_{K} = GL_{m_{K}} (D_{K}) and its split inner form G^{∗}(F)_{K} = GL_{n_{K}} (K). We can parametrize both collections by the admissible pairs in P_{n_{K}} (K), and obtain the rectifier map

D_{K} ν : P_{n_{K}} (K) ⟶^{_{K}Π} A_{n_{K}}^{et} (K) ⟶^{JL_{K}} A_{m_{K}}^{et} (D_{K}) ⟶^{_{D_{K}} Π^{−1}} P_{n_{K}} (K),

such that

D_{K} ν(E/K, ξ) = (E/F, ξ ·_{D_{K}} ν_{ξ}).

for a tamely ramified character _{D_{K}} ν_{ξ} of E^{×} for each pair (E/K, ξ) ∈ P_{n_{K}} (K). With the embedding condition for (E_{0}K, 𝔄_{K}) as discussed in Section 2.5, we define the subgroups (see also [BH11, 3.2 Proposition])

H_{K}^{1} = H^{1}(Ξ, 𝔄) ∩ G(F)_{K} and J_{K}^{1} = J^{1}(Ξ, 𝔄) ∩ G(F)_{K}

Each subgroup above admits a similar factorization as in (3.10). We then obtain

A_{K} 𝔙_{ξ} = J_{K}^{1}/H_{K}^{1} = _{A}𝔙_{ξ} ∩ 𝔘_{E/K}

and similarly for _{A_{K}^{∗}} 𝔙_{ξ}. Denote Ψ_{E/K} = E^{×}/K^{×}U_{E}^{1}, and view _{A_{K}} 𝔙_{ξ} and _{A_{K}^{∗}} 𝔙_{ξ} as k_{K}Ψ_{E/K}- submodules of 𝔘_{E/K}. Denote the subgroups of Ψ_{E/K} by

μ_{E/K} = μ_{E}/μ_{K} and

⟨ϖ⟩_{E/K} = the subgroup generated by the image of ϖ_{E}.

Using the results in Section 3.8, with the base field changed from F to K, the values of _{D_{K}} ν_{ξ} is given by

\begin{aligned}& _{D_{K}} ν_{ξ}|_{μ_{E}} = t_{μ_{E/K}}^{1} (_{A_{K}} 𝔙_{ξ})t_{μ_{E/K}}^{1} (_{A_{K}^{∗}} 𝔙_{ξ}) \\ and & _{D_{K}} ν_{ξ}(ϖ_{E}) = (−1)^{n_{K}−m_{K}+f_{ϖ,K}−m_{ϖ,K}} t_{⟨ϖ⟩_{E/K}} (_{A_{K}} 𝔙_{ξ})t_{⟨ϖ⟩_{E/K}} (_{A_{K}^{∗}} 𝔙_{ξ}) \end{aligned}

for a prime element ϖ_{E} ∈ E_{0}K (see the beginning of Section 3.8). Here

f_{ϖ,K} = f(E/K[ϖ_{E}]) and
m_{ϖ,K} = gcd(m_{K}, f_{ϖ,K}) = gcd(m, n_{K}, f(E/K[ϖ_{E}])).

Now suppose that (E/F, ξ) ∈ P_{n}(F). By the definition of admissibility, we can regard ξ as an admissible character over K and form the pair (E/K, ξ) ∈ P_{n_{K}} (K).

Proposition 5.7. In this situation, we have

D_{K} ν_{ξ} = ∏_{[g]∈(W_{E}\W_{K}/W_{E})′} ζ_{g,ξ}|_{E^{×}}.

Proof. Notice that if 𝔙 is a k_{F} Ψ_{E/F} -submodule of 𝔘_{E/K}, we have

t_{μ_{E/K}}^{1} (𝔙) = t_{μ_{E/F}}^{1} (𝔙) and t_{⟨ϖ⟩_{E/K}} (𝔙) = t_{⟨ϖ⟩_{E/F}} (𝔙),

where μ_{E/F} and ⟨ϖ⟩_{E/F} are just μ and ⟨ϖ⟩ respectively considered in (3.20). Hence we have

D_{K} ν_{ξ}|_{μ_{E}} = ∏_{[g]∈(W_{E}\W_{K}/W_{E})′} ζ_{g,ξ}|_{μ_{E}}.

It remains to consider the values of both characters at ϖ_{E}. Notice that the left side has value

(−1)^{n_{K}−m_{K}+f_{ϖ,K}−m_{ϖ,K}} t_{⟨ϖ⟩_{E/K}} (_{A_{K}} 𝔙_{ξ})t_{⟨ϖ⟩_{E/K}} (_{A_{K}^{∗}} 𝔙_{ξ}),

while the right side has value

(a sign) · ∏_{[g]∈(W_{E}\W_{K}/W_{E})′} t_{⟨ϖ⟩}(_{A_{K}} 𝔙_{ξ,[g]})t_{⟨ϖ⟩}(_{A_{K}^{∗}} 𝔙_{ξ,[g]}).

The t-factors on both sides are clearly equal. We will recall, by Theorem 5.5, the values of the sign on the right side in different cases and show that, in each case, this sign is equal to the one on the left side.

We first consider when f_{K} is odd, which can be reduced to the case when E/K is totally ramified. We further separate into cases.

• When e is odd, or when e is even and e_{K} is odd, then m_{K} is also odd. The sign on the left is (−1)^{e_{K}−m_{K}} = 1, while that on the right is also 1 since σ^{e/2} ∉ W_{K}.

• When e_{K} is even, the sign on the left is (−1)^{e_{K}−m_{K}} = (−1)^{m} since m_{K} ≡ m mod 2, while that on the right is (−1)^{m} since σ^{e/2} ∈ W_{K}.

We then consider then f_{K} is even. Let L be the maximal unramified extension of E/K. We recall, after disregarding the symmetric ramified component (as we did at the beginning of Sub-section 5.3.2), the sign on the left is equal to (see (5.6))

(−1)^{e_{K}−m_{L}}(−1)^{n_{K}−m_{K}+f_{ϖ,K}−m_{ϖ,K}} = \begin{cases}1 & if m_{K} is even, \\ (−1)^{e_{K}+f_{ϖ,K}+1} & if m_{K} is odd. \end{cases}

Recall from Proposition 2.6 that the number e_{K}+f_{ϖ,K}+1 is just the cardinality of

(Γ_{E}\Γ_{K}/Γ_{E})_{sym−unram} − Γ_{K[ϖ_{E}]}.

Hence by Theorem 5.5, the sign on the right side is

(−1)^{m(e_{K}+f_{ϖ,K}+1)}.

By knowing that m_{K} ≡ m mod 2, the sign above is equal to the one on the left side.

Acknowledgements

This work was started when the author was receiving funding from the European Research Council under the European Community’s Seventh Framework Programme (FP7/2007-2013) / ERC Grant agreement no.290766 (AAMOT) during his visit in IMJ, Paris. It was then finished when he became a post-doc in McMaster University. He would like to thank Vincent Sécherre for a few discussions in Paris, and Shaun Stevens for reading the manuscript and his comments. He would also like to thank the referee for the careful reading.

References

[Bad02] Alexandru Ioan Badulescu, Correspondance de Jacquet-Langlands pour les corps locaux de caractéristique non nulle, Ann. Sci. École Norm. Sup. (4) 35 (2002), no. 5, 695–747. MR 1951441 (2004i:11048)

[BF83] Colin J. Bushnell and Albrecht Fröhlich, Gauss sums and p-adic division algebras, Lecture Notes in Mathematics, vol. 987, Springer-Verlag, Berlin-New York, 1983. MR 701540 (84m:12017)

[BF85] C. J. Bushnell and A. Fröhlich, Nonabelian congruence Gauss sums and p-adic simple algebras, Proc. London Math. Soc. (3) 50 (1985), no. 2, 207–264. MR 772712 (86g:11071)

[BH96] Colin J. Bushnell and Guy Henniart, Local tame lifting for GL(N). I. Simple characters, Inst. Hautes Études Sci. Publ. Math. (1996), no. 83, 105–233. MR 1423022 (98m:11129)

[BH05a] , The essentially tame local Langlands correspondence. I, J. Amer. Math. Soc. 18 (2005), no. 3, 685–710 (electronic). MR 2138141 (2006a:22014)

[BH05b] , The essentially tame local Langlands correspondence. II. Totally ramified representations, Compos. Math. 141 (2005), no. 4, 979– 1011. MR 2148193 (2006c:22017)

[BH10] , The essentially tame local Langlands correspondence, III: the general case, Proc. Lond. Math. Soc. (3) 101 (2010), no. 2, 497–553. MR 2679700

[BH11] , The essentially tame Jacquet-Langlands correspondence for inner forms of GL(n), Pure Appl. Math. Q. 7 (2011), no. 3, Special Issue: In honor of Jacques Tits, 469–538. MR 2848585 (2012f:22027)

[BHL10] Colin J. Bushnell, Guy Henniart, and Bertrand Lemaire, Caractère et degré formel pour les formes intérieures de GL(n) sur un corps local de caractéristique non nulle, Manuscripta Math. 131 (2010), no. 1-2, 11–24. MR 2574988 (2010m:22013)

[BK93] C.J. Bushnell and P.C. Kutzko, The admissible dual of GL(N) via compact open subgroups, Annals of mathematics studies, Princeton University Press, 1993.

[Bro96] Paul Broussous, Extension du formalisme de Bushnell et Kutzko au cas d’une algèbre à division, C. R. Acad. Sci. Paris Sér. I Math. 323 (1996), no. 3, 225–228. MR 1404764 (97f:22028)

[BSS12] P. Broussous, V. Sécherre, and S. Stevens, Smooth representations of GL_{m}(D) V: Endo-classes, Doc. Math. 17 (2012), 23–77. MR 2889743 [DKV84] P. Deligne, D. Kazhdan, and M.-F. Vignéras, Représentations des

algèbres centrales simples p-adiques, Representations of reductive groups over a local field, Travaux en Cours, Hermann, Paris, 1984, pp. 33–117. MR 771672 (86h:11044)

[Gra07] Martin Grabitz, Simple characters for principal orders in M_{m}(D), J. Number Theory 126 (2007), no. 1, 1–51. MR 2347425 (2008k:11125)

[Gre55] J. A. Green, The characters of the finite general linear groups, Trans. Amer. Math. Soc. 80 (1955), 402–447. MR 0072878 (17,345e)

[How77] Roger E. Howe, Tamely ramified supercuspidal representations of Gl_{n}, Pacific J. Math. 73 (1977), no. 2, 437–460. MR 0492087 (58 #11241)

[JL70] H. Jacquet and R. P. Langlands, Automorphic forms on GL(2), Lecture Notes in Mathematics, Vol. 114, Springer-Verlag, Berlin, 1970. MR 0401654 (53 #5481)

[KS99] Robert E. Kottwitz and Diana Shelstad, Foundations of twisted endoscopy, Astérisque (1999), no. 255, vi+190. MR 1687096 (2000k:22024)

[Lan94] Serge Lang, Algebraic number theory, second ed., Graduate Texts in Mathematics, vol. 110, Springer-Verlag, New York, 1994. MR 1282723 (95f:11085)

[LS87] R. P. Langlands and D. Shelstad, On the definition of transfer factors, Math. Ann. 278 (1987), no. 1-4, 219–271. MR 909227 (89c:11172)

[Moy86] Allen Moy, Local constants and the tame Langlands correspondence, Amer. J. Math. 108 (1986), no. 4, 863–930. MR 853218 (88b:11081)

[NM43] Tadasi Nakayama and Yozô Matsushima, Über die multiplikative Gruppe einer p-adischen Divisionsalgebra, Proc. Imp. Acad. Tokyo 19 (1943), 622–628. MR 0014081 (7,238a)

[Rei03] I. Reiner, Maximal orders, London Mathematical Society Monographs. New Series, vol. 28, The Clarendon Press Oxford University Press, Oxford, 2003, Corrected reprint of the 1975 original, With a foreword by M. J. Taylor. MR 1972204 (2004c:16026)

[Rog83] Jonathan D. Rogawski, Representations of GL(n) and division algebras over a p-adic field, Duke Math. J. 50 (1983), no. 1, 161–196. MR 700135 (84j:12018)

[Séc04] Vincent Sécherre, Représentations lisses de GL(m, D). I. Caractères simples, Bull. Soc. Math. France 132 (2004), no. 3, 327–396. MR 2081220 (2005f:22027)

[Séc05a] , Représentations lisses de GL(m, D). II. β-extensions, Compos. Math. 141 (2005), no. 6, 1531–1550. MR 2188448 (2006j:22018)

[Séc05b] , Représentations lisses de GL_{m}(D). III. Types simples, Ann. Sci. École Norm. Sup. (4) 38 (2005), no. 6, 951–977. MR 2216835 (2007a:22010)

[SS08] V. Sécherre and S. Stevens, Représentations lisses de GL(m, D). IV. Représentations supercuspidales, J. Inst. Math. Jussieu 7 (2008), no. 3, 527–574. MR 2427423 (2009d:22023)

[Tam] Kam-Fai Tam, Admissible embeddings of L-tori and the essentially tame local Langlands correspondence, to appear in International Mathematics Research Notices.

[Tat79] J. Tate, Number theoretic background, Automorphic forms, representations and L-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 2, Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 3–26. MR 546607 (80m:12009)

[Zin92] Ernst-Wilhelm Zink, Representation theory of local division algebras, J. Reine Angew. Math. 428 (1992), 1–44. MR 1166506 (93i:11141)

[Zin99] , More on embeddings of local fields in simple algebras, J. Number Theory 77 (1999), no. 1, 51–61. MR 1695699 (2001d:11116)

K.-F. Tam

Dept. of Math. & Stat. McMaster University 1280 Main Street West Hamilton, Ontario, Canada L8S 4K1 geotam@math.mcmaster.ca 390

Documenta Mathematica 21 (2016)