Pure elements and intertwining classes of simple strata in local

central simple algebras

February 1999

by Paul Broussous^{1} and Martin Grabitz^{2}

Introduction

Let G = GL_{m}(D) be the general linear group over a division algebra D with centre a p-adic field F. It has been conjectured that the smooth dual of G can be constructed as Bushnell and Kutzko did for GL_{N}(F) [BK2]. Checking such a conjecture means a lot of constructions and computations. The aim of this paper is to establish some fundamental results on simple strata: the “intertwining implies conjugacy” property and the existence of a simple stratum equivalent to a pure one. This virtually enables us to define simple types for supercuspidal representations.

Let us be more precise. We want to produce a list of pairs (J, λ) formed of a compact mod center subgroup J and an irreducible smooth representation λ of J, with at least the two following properties:

(a) any supercuspidal representation contains a pair (J, λ) and is then isomorphic to the compactly induced representation c-Ind_{J}^{G}λ;

(b) If two pairs are contained in the same irreducible representation, they are conjugate under the action of G.

The first step towards proving (a) was done in [Br2]: it was shown that any irreducible positive level supercuspidal representation contains the character attached to a simple stratum [𝔄, n, n − 1, β] (see [BK2] or [Br2] for the missing definitions). Recall [Br2] that a simple stratum is a quadruple [𝔄, n, q, β], where 𝔄 is a hereditary 𝔬_{F} -order in the F -algebra A = M_{m}(D), n and q are integers and β is an element of A, satisfying a certain list of properties. In particular the stratum must be pure (see the definition below) and in the case q = n − 1, the character attached to the stratum has a nice intertwining formula.

1The first author was supported by TMR grant ERBFMBICT961029 from the European Community.

2The second author was mostly supported by the “Graduiertenkolleg Geometrie und nichtlineare Analysis” of the Humboldt University Berlin and temporarily by the “Max-Planck-Institut für Mathematik” Bonn.

Recall that a stratum [𝔄, n, q, β] is pure if E = F [β] is a subfield of A whose multiplicative group normalizes 𝔄 (we then say that β is 𝔄-pure) and if β has 𝔄-valuation −n (see below). We prove (theorem (5.1) below) that any such pure stratum is equivalent to a simple one. This virtually enables us to construct the list of pairs (J, λ) by an inductive procedure (see [BK2]). Of course, a lot of things have to be checked but the case m = 1 ([Br3]) shows that this is reasonable.

The results of [BH] and our theorem (5.1) also enable us to compare a metric on certain sets of pure elements for A with a metric on pure elements for an arbitrary F -division algebra Δ of suitable index. In the work [Z2], it was shown that the latter metric actually agrees with a certain polynomial metric on the the set of minimal polynomials of the pure elements in Δ. So our proposition (7.5) actually shows that our metric (introduced in definition (7.4)) is in some sense independent of our algebra A of fixed reduced degree N = md (where d is the index of D). We conjecture that this can be used to introduce parameters for simple types of different central simple algebras of fixed reduced degree, as it was done in [Z2] for a F -division algebra of index N and GL_{N}(F).

Property (b) corresponds to a certain rigidity of simple strata which we also prove in this paper as (theorem (4.1.2)). The situation is here more complicated than in the split case of [BK2]. We first have to define a “type of embedding” for the 𝔄-pure elements. The type of embedding of β describes how the field extension F [β]/F, seen as an abstract field extension, is embedded in A with respect to 𝔄. We show the following fundamental results:

(i) the type of embedding can be described by a finite number of numerical invariants; (ii) if two 𝔄-pure elements are conjugate under G and have the same type of embedding, they are

conjugate under the normalizer of 𝔄;

(iii) given an irreducible polynomial in F [X], there is an effective criterion to decide when it has a root in A which is 𝔄-pure and has a prescribed type of embedding.

All of this relies on the papers [Br1] and [G] where pure elements are studied in detail. We define the type of embedding by using an equivalence relation on the set of pure elements (more precisely, on the corresponding “pure extensions” F [β]/F). Generalizing an idea of Zink [SZ], we show that in each equivalence class there is an element which is effectively described (a “pearl embedding”). We then use [Br1] to attach numerical invariants to this special element.

The rigidity theorem (4.1.2) takes the following form. Let [𝔄, n, q, γ_{i}], i = 1, 2, be two simple strata where γ_{1} and γ_{2} have the same type of embedding and assume they are intertwined by an element of G. Then they are conjugate by an element of the normalizer of 𝔄 in G. To prove this, we will strongly use a descent lemma (theorem (4.3.9)), comparing intertwinings in A with intertwinings in A ⊗_{F} L, for a finite unramified extension L/F: any pair of strata in A can be lifted to strata in A ⊗_{F} L; then if the latter strata are conjugate (under the normalizer of the order), the former strata are conjugate also. We will apply this to simple strata where the field extensions are totally ramified: they remain pure under unramified base change. To reduce to this case we will prove that we can assume that the unramified parts of the field extensions are the same and then work in the centralizer of this common unramified part. All these procedures introduce a change of ground field so we will need some results from [BH] which give the base change theory for simple strata and characters.

The paper is organized as follows. In §1, we recall some standard notation. In §2, we study the problem of embedding field extensions with respect to hereditary orders and show how to attach to them numerical invariants. We then give a numerical criterion to decide when a field extension can be embedded with prescribed invariants. In §3, we use our results to define a new notion of purity and we prove property (ii) above.

In §4 we establish the “intertwining implies conjugacy” property for simple strata and in §5 we prove the existence of a simple representative for a pure stratum. In §6 we prove the descent lemma relative to unramified base change and intertwining. Finally in §7 we give a short outlook on the possibility of comparing pure elements from different simple algebras of the same reduced degree, in the spirit of [Z2].

We are grateful to Guy Henniart and Ernst-Wilhelm Zink for their encouragements and valuable discussions. Proposition (4.1.3) below, which allows us to prove the rigidity theorem, was at first Zink’s conjecture. The first author would like to thank Guy Henniart for pointing out the importance of the unramified parts of the pure extensions in all these problems and for suggesting the use of cohomology in §6. Finally we both thank Shaun Stevens and Pierre Torasso for proofreading earlier versions of this manuscript.

1 Notation

For a ring R with unit element and a subset S ⊂ R, we denote by C_{R}(S) = {x ∈ R |: xs = sx, s ∈ S} the centralizer of S in A and by N_{R^{×}}(S) = {x ∈ R^{×} | xSx^{−1} = S} the normalizer of S in R^{×}, where R^{×} denotes the group of units of R. We denote by rad(R) the Jacobson radical of R, which is the intersection of all maximal left (resp. right) ideals of R and by R^{op} its opposite ring. We denote by M_{m}(R) the set of all m × m matrices, by GL_{m}(R) the subset of all invertible ones, and by I_{m} the unit matrix in M_{m}(R).

Let F be a non-archimedean local field and A a finite dimensional central simple F -algebra. Let V be a simple left A-module; then, by Wedderburn’s structure theorem A ≃ End_{D}(V) ≃ M_{m}(D), for a natural number m, where D^{op} = End_{A}(V) is a central F -division algebra. Moreover dim_{F} (D) = d^{2} is a square and its square root d is called the index of the algebra. The square root N of dim_{F} (A) = m^{2}d^{2} is called the reduced degree of A.

If E/F is a finite field extension embedded in A, then B := C_{A}(E) is a central simple E-algebra. Moreover for an arbitrary field extension E/F, we denote f(E/F) the inertia degree and e(E/F) the ramification index.

For a F -division algebra D, we denote by 𝔬_{D} its ring of integers, by 𝔭_{D} the unique maximal ideal of 𝔬_{D} and by 𝔨_{D} := 𝔬_{D}/𝔭_{D} the residual class field. We denote by ♯M the cardinality of a set M and let q = p^{k} := ♯𝔨_{F}, where p is the residual characteristic of F.

An 𝔬_{F} -order 𝔄 in A is a subring of A which contains the same unit element and is a complete 𝔬_{F} -lattice in A. In this situation we set K_{𝔄} := N_{A^{×}}(𝔄). Moreover 𝔄 is called hereditary, if it is hereditary as a ring. For standard results about hereditary orders, we refer to [R] and [CR]. For non-zero integers k_{1}, …, k_{u}, we denote by ⟨k_{1}, …, k_{u}⟩ their greatest common positive divisor.

For an element h of a group H, we denote by ⟨h⟩ the cyclic group generated by h.

2 Hereditary orders and embeddings

2.1 Preparations

Let us fix an isomorphism A ≃ M_{m}(D) and identify A with M_{m}(D). Let 𝔄 be a hereditary 𝔬_{F} -order in A. We denote by 𝔓 its Jacobson radical and by K_{𝔄} its normalizer. We know, from [BF] that there exists an 𝔬_{D}-lattice chain {X_{k}}_{k∈ℤ} in V, unique up to a translation of the index, such that

𝔄 = End_{𝔬_{D}}{X_{k}}_{k∈ℤ} = {x ∈ A | xX_{k} ⊂ X_{k}, k ∈ ℤ}.

Furthermore,

𝔄/𝔓 ≃ ⨁_{i=1}^{r} End_{𝔨_{D}}(X_{i−1}/X_{i}) ≃ M_{n_{1}}(𝔨_{D}) × … × M_{n_{r}}(𝔨_{D})

for some numbers n_{1}, …, n_{r}; the conjugacy class of 𝔄 determines them uniquely up to cyclic permutation and they are called the invariants of 𝔄.

These numbers also come from a normal form:

( 2.1.1 )

g𝔄g^{−1} = ( \begin{matrix}(𝔬_{D}) , & (𝔭_{D}) & ⋯, & (𝔭_{D}) \\ (𝔬_{D}) , & (𝔬_{D}) & ⋯, & ⋯ \\ ⋯, & ⋯ & ⋯, & (𝔭_{D}) \\ (𝔬_{D}) , & ⋯ & ⋯, & (𝔬_{D}) \end{matrix} )^{\{n_{1},...,n_{r}\}}

for a suitable element g ∈ A^{×}. Here (𝔬_{D}) (resp. (𝔭_{D})) means n_{i} × n_{j}-matrices with entries in 𝔬_{D} (resp. in 𝔭_{D}). The number r is uniquely determined and called the period of 𝔄.

We also have:

g𝔓g^{−1} = ( \begin{matrix}(𝔭_{D}) , & (𝔭_{D}) , & ⋯, & (𝔭_{D}) , & (𝔭_{D}) \\ (𝔬_{D}) , & (𝔭_{D}) , & ⋯, & ⋯, & ⋯ \\ ⋯, & (𝔬_{D}) , & ⋯, & ⋯, & ⋯ \\ (𝔬_{D}) , & ⋯, & ⋯, & (𝔭_{D}) , & ⋯ \\ (𝔬_{D}) , & ⋯, & ⋯, & (𝔬_{D}) , & (𝔭_{D}) \end{matrix} )^{\{n_{1},...,n_{r}\}} .

Let E/F be a field extension of F in A, such that E^{×} ⊂ K_{𝔄}. Let B = C_{A}(E) be its centralizer in A. From the theory of the Brauer group it is well known that we have B ≃ M_{m′}(D′), for D′ a central E-division algebra of index d′ = \frac{d}{⟨[E:F],d⟩} and m′ = \frac{⟨[E:F],d⟩m}{[E:F]} (see for example [Z](proposition 1)). From [Br1](1.3) we know, that 𝔅 = 𝔄 ∩ B is a hereditary 𝔬_{E}-order in B and, of course, K_{𝔄} ∩ B ⊂ K_{𝔅}. If in addition K_{𝔄} ∩ B ⊃ K_{𝔅}, 𝔄 is called a continuation of 𝔅. The pair (E, 𝔄) is called an embedding (with respect to 𝔄). We have the following notion from [Br1].

( 2.1.2 ) Definition (i) An embedding (E, 𝔄) is called good if 𝔄 is a continuation of 𝔅 as above. (ii) More generally, we call (K_{𝔅} : K_{𝔄} ∩ B) the index of the embedding.

From [G](2.2)(ii), we know that all embeddings (E, 𝔄) with ⟨f(E/F), d⟩ = 1 are good and that the associated continuation is unramified in the sense of pseudovaluations (cf. [Be]). More precisely, let ν_{𝔄} : A → ℤ ∪ {∞} be defined by ν_{𝔄}(x) := max{j ∈ ℤ | x ∈ 𝔓^{j}} for x ≠ 0 and ν_{𝔄}(0) := ∞. From [BF](1.2.8)(i) we know that ν_{𝔄} |_{K_{𝔄}} is a group homomorphism from K_{𝔄} to ℤ with kernel 𝔄^{×}; hence, ν_{𝔄}(K_{𝔄}) = tℤ, where t is a positive integer. The same holds true for 𝔅 and we similarly define ν_{𝔅}. By [G](2.2)(i) we have ν_{𝔅}(K_{𝔅}) = tℤ as well and for all j ∈ ℤ, 𝔓^{j} ∩ B = 𝔔^{j}, where 𝔔 is the Jacobson radical of 𝔅. Therefore there exists h ∈ K_{𝔅} such that ν_{𝔄}(h) = t; hence, K_{𝔄} = ⟨h⟩𝔄^{×} and K_{𝔅} = ⟨h⟩𝔅^{×}. This is similar to the theory of valuation of fields and h acts here in the function of the uniformizer. In the case of unramified continuation ν_{𝔄} and ν_{𝔅} have a common uniformizer.

2.2 Equivalence classes of embeddings

In this chapter, we use the case ⟨f(E/F), d⟩ = 1 to reduce the problem of studying embedding to the ‘division algebra part’ of the embedding, which will be defined below. We split the set of all embeddings into equivalence classes, such that the condition of good embedding, or more generally the index of the embedding, depends only on the equivalence class.

( 2.2.1 ) Definition Embeddings (E′,𝔄′) and (E,𝔄) are called equivalent, denoted by (E′,𝔄′)∼(E,𝔄), iff ⟨f(E/F), d⟩ = ⟨f(E′/F), d⟩ =: f and, letting E′_{d}, E_{d} be the unique unramified subextensions of F in E′, E respectively with [E′_{d} : F] = [E_{d} : F] = f, then there exists g ∈ A^{×} such that (gE′_{d}g^{−1}, g𝔄′g^{−1}) = (E_{d}, 𝔄).

We call f the degree and E_{d} (resp. E′_{d}) the division algebra part of the embedding (E, 𝔄) (resp. (E′, 𝔄′)).

We easily check:

( 2.2.2 ) Lemma Equivalence of embeddings with respect to hereditary orders is an equivalence relation.

( 2.2.3 ) Definition We denote by [E, 𝔄] the equivalence class of embeddings which contains (E, 𝔄) and by E the set of all equivalence classes of embeddings with respect to hereditary orders.

( 2.2.4 ) Corollary Every equivalence class of embeddings is of the form [E, 𝔄] with E/F unramified and [E : F] | d.

Proof.

Let [E, 𝔄] ∈ E and let E_{d} be the division algebra part of (E, 𝔄); then clearly (E, 𝔄) ∼ (E_{d}, 𝔄). □

( 2.2.5 ) Proposition The index of an embedding depends only on its equivalence class.

Proof.

Let (E′, 𝔄′) and (E, 𝔄) be two equivalent embeddings and E_{d}, E′_{d} be their division algebra parts. There exists g ∈ A^{×} such that (gE′_{d}g^{−1}, g𝔄′g^{−1}) = (E_{d}, 𝔄). Put C′ := C_{A}(E′_{d}), C := C_{A}(E_{d}), ℭ := 𝔄 ∩ C and ℭ′ := 𝔄′ ∩ C′. Then it is easy to see that K_{𝔄} = gK_{𝔄′}g^{−1}, C = gC′g^{−1} and K_{ℭ} = gK_{ℭ′}g^{−1}. Therefore we have (K_{ℭ} : K_{𝔄} ∩ C) = (gK_{ℭ}g^{−1} : gK_{𝔄} ∩ Cg^{−1}) = (K_{ℭ′} : K_{𝔄′} ∩ C′).

Let B := C_{A}(E), B′ := C_{A}(E′), 𝔅 := ℭ ∩ B = 𝔄 ∩ B and 𝔅′ := ℭ′ ∩ B′ = 𝔄′ ∩ B′. Because ⟨f(E/E_{d}), d′⟩ = ⟨f(E′/E′_{d}), d′⟩ = 1 and d′ := \frac{d}{[E_{d}:F]} = \frac{d}{[E′_{d}:F]} is the index of C resp. C′, it follows from [G](2.2)(i) that ℭ is an unramified continuation of 𝔅 and ℭ′ an unramified continuation of 𝔅′ (see the explanations following (2.1.2)). So there exists h ∈ K_{𝔅} such that K_{ℭ} = ⟨h⟩ℭ^{×} and K_{𝔅} = ⟨h⟩𝔅^{×}. If k is the least positive integer such that h^{k} ∈ K_{𝔄}, then (K_{𝔅} : K_{𝔄} ∩ B) = k = (K_{ℭ} : K_{𝔄} ∩ C). Similarly one shows (K_{𝔅′} : K_{𝔄′} ∩ B′) = (K_{ℭ′} : K_{𝔄′} ∩ C′); hence, (K_{𝔅} : K_{𝔄} ∩ B) = (K_{𝔅′} : K_{𝔄′} ∩ B′) as required.

□

( 2.2.6 ) Definition The index of an equivalence class of embeddings is by definition the index of a representative. Similarly we will say that a class is good if one representative is good.

2.3 Pearl-embeddings and numerical invariants of equivalence classes

Now we introduce a system of representatives for our equivalence classes of embeddings.

( 2.3.1 ) Definition

(i) An embedding datum is an element λ = (λ^{1}, …, λ^{r}) of ((ℕ ∪ {0})^{f})^{r}, where f, r ∈ ℕ and λ^{i} = (λ_{1}^{i}, …, λ_{f}^{i}) ∈ (ℕ ∪ {0})^{f} for i = 1, …, r, such that n_{1} + … + n_{r} = m, (n_{1}, ..., n_{r}) ∈ ℕ^{r} and

∑_{j=1}^{f} λ_{j}^{i} = n_{i} for i = 1, …, r.

(ii) We call f the degree of λ and (n_{1}, …, n_{r}) the partition associated to the datum λ as in (i). (iii) Let Φ be the set of all embedding data and λ ∈ Φ of degree f and with associated parti-

tion (n_{1}, ..., n_{r}) as in (i). We have a permutation L_{f} on (ℕ ∪ {0})^{f} given by L_{f}((m_{1}, …, m_{f})) := (m_{f}, m_{1}, …, m_{f−1}) and set Z(λ) := (L_{f}(λ^{r}), λ^{1}, …, λ^{r−1}). This definition gives a permutation Z of Φ. Let I := ⟨Z⟩\Φ the set of all orbits in Φ under the action of the subgroup of the group of all permutations of the set Φ generated by Z. Clearly I can be considered as a set of equivalence classes of Φ, where λ, η ∈ Φ are called equivalent, denoted λ ∼ η, if they are in the same orbit under ⟨Z⟩. Denote by λ̄ the equivalence class of λ.

(iv) For a datum λ of degree f and with associated partition (n_{1}, ..., n_{r}), we construct an embedding as follows: Choose (and fix) a maximal unramified extension L/F in D (thus [L : F] = d). For every divisior f | d we have a unique intermediate field L/L_{f}/F such that [L_{f} : F] = f. Let π be a prime element of 𝔬_{D} such that πLπ^{−1} = L and let σ ∈ Gal(L/F) be the map D → D, x ↦ πxπ^{−1} restricted to L. Denote by 𝔄_{n} the hereditary 𝔬_{F} -order in A given in normal form (2.1.1) with invariant tuple n = (n_{1}, …, n_{r}) (appearing in this order!). Let L_{f} ↪ A be embedded over F by

( 2.3.2 ) x ↦ diag(diag(σ^{ν_{1,1}}(x), …, σ^{ν_{1,n_{1}}} (x)) × … × diag(σ^{ν_{r,1}}(x), …, σ^{ν_{r,nr}} (x))),

where diag(. × … × .) means ‘diagonal block matrix’, diag(., …, .) means ‘diagonal matrix’ and the first λ_{1}^{i} ν_{i,j}s are zero, the next λ_{2}^{i} ν_{i,j}s are one, …, until the last λ_{f}^{i} ν_{i,j}s are f − 1 in each block i = 1, …, r. We write (λ) := (L_{f}, 𝔄_{n}), [λ] := [L_{f}, 𝔄_{n}] and call (λ) a standard pearl-embedding. (Note that with these conventions, L_{f}^{×} ⊂ ⟨π_{F} ⟩(𝔬_{L}^{×} × … × 𝔬_{L}^{×}) ⊂ K_{𝔄_{n}}, where π_{F} is a prime element of F and (𝔬_{L}^{×} × … × 𝔬_{L}^{×}) means the set of all diagonal matrices with entries in 𝔬_{L}^{×}.)

( 2.3.3 ) Lemma Embedding data λ, η are equivalent if and only if the associated embeddings (λ), (η) are equivalent.

Proof.

First, suppose λ ∼ η. Then there exists k ∈ ℤ such that Z^{k}(λ) = η. Let λ be of degree f with associated partition (n_{1}, ..., n_{r}) as in definition (2.3.1)(i). Let π be as in definition (2.3.1)(iii),

h_{i} := ( \begin{matrix}0 , & πI_{n_{i}} \\ I_{m−n_{i}} , & 0 \end{matrix} )

for i = 1, …, r, and 𝔄_{n} the hereditary order in normal form (2.1.1) such that the invariants (n_{1}, …, n_{r}) appear in the normal form in that order as in definition (2.3.1). Then h_{r}𝔄_{n}h_{r}^{−1}, h_{r−1}h_{r}𝔄_{n}h_{r}^{−1}h_{r−1}^{−1}, …, h_{2} … h_{r}𝔄_{n}h_{r}^{−1} … h_{2}^{−1} are the hereditary orders in normal form (2.1.1) such that the invariants n_{1}, …, n_{r} appear in the cyclic permuted orders n_{r}, n_{1}, …, n_{r−1}; n_{r−1}, n_{r}, n_{1}, …, n_{r−2}; …; n_{2}, …, n_{r}, n_{1} respectively.

Consider the operation of the h_{r} on (λ) = (L_{f}, 𝔄_{n}). The embedding L_{f} ↪ A is given by (2.3.2), so the embedding (h_{r}L_{f}h_{r}^{−1}, h_{r}𝔄_{n}h_{r}^{−1}) is given by

( 2.3.4 )

x ↦ diag(diag(σ^{ν_{r,1}+1}(x), …, σ^{ν_{r,nr} +1}(x)) × diag(σ^{ν_{1,1}}(x), …, σ^{ν_{1,n_{1}}} (x)) × … ×diag(σ^{ν_{(r−1),1}}(x), …, σ^{ν_{(r−1),n_{r−1}}} (x))),

where the notations are the same as in definition (2.3.1)(iv).

We can choose an n_{r} × n_{r}-permutation matrix P_{r} such that for p := diag(P_{r} × I_{1} × … × I_{r−1}) the embedding (ph_{r}L_{f}h_{r}^{−1}p^{−1}, ph_{r}𝔄_{n}h_{r}^{−1}p^{−1}) = (ph_{r}L_{f}h_{r}^{−1}p^{−1}, h_{r}𝔄_{n}h_{r}^{−1}) takes a standard form, which means that P_{r} permutes the σ^{ν_{r,k}+1}(x) (for k = 1, …, n_{r}) in the first block of (2.3.4) to an increasing order of the associated exponents. Then we can associate to this embedding a datum λ′. Hence, because f ≡ 0(mod f), we obtain (λ′) = (Z(λ)) = (ph_{r}L_{f}h_{r}^{−1}p^{−1}, ph_{r}𝔄_{n}h_{r}^{−1}p^{−1}) ∼ (L_{f}, 𝔄_{n}) = (λ).

Let S be the set of all standard pearl-embeddings. Because λ was arbitrarily chosen, we have shown that the map S → S given by (λ) ↦ (Z(λ)) is a bijection which preserves equivalence classes of embeddings with respect to hereditary orders. From the construction it is clear that the inverse of this mapping is given by (λ) ↦ (Z^{−1}(λ)) and also preserves equivalence classes of embeddings with respect to hereditary orders. Hence (η) = (Z^{k}(λ)) ∼ (λ), as required.

Conversely, suppose (λ) ∼ (η). Let λ be of degree f, with associated partition n = (n_{1}, ..., n_{r}) as in definition (2.3.1)(i), and η = (η^{1}, …, η^{r′}) = ((η_{1}^{1}, …, η_{f′}^{1}), …, (η_{1}^{r′}, …, η_{f′}^{r′})) be of degree f′ with associated partition n′ = (n′_{1}, …, n′_{r′}). Then clearly f = f′. Since 𝔄_{n} and 𝔄_{n′} are conjugate, we have r = r′ and (n′_{1}, …, n′_{r}) is a cyclic permutation of (n_{1}, ..., n_{r}). Then because (Z^{k}(λ)) ∼ (λ), as shown above, we can assume without loss of generality that n = n′.

By definition (2.2.1), there exists g ∈ A^{×} such that (gL_{f,λ}g^{−1}, g𝔄_{n}g^{−1}) = (L_{f,η}, 𝔄_{n}), where L_{f,λ}, L_{f,η} denote the images of L_{f} under the embeddings (λ), (η) respectively. Here in particular we have that g ∈ K_{𝔄_{n}}.

Let θ ∈ 𝔬_{L_{f}} be a generator of the extension L_{f}/F, then the image of θ in L_{f,λ} under the standard pearl-embedding attached to λ as in definition (2.3.1)(iii) is θ_{λ} := diag(diag(σ^{ν_{1,1}}(θ), …, σ^{ν_{1,n_{1}}} (θ)) × … × diag(σ^{ν_{r,1}}(θ), …, σ^{ν_{r,nr}} (θ))), where ♯{j ∈ {1, …, n_{i}} | ν_{i,j} = k − 1} = λ_{k}^{i} for k = 1, …, f and i = 1, …, r. The image of θ in L_{f,η} under the standard pearl-embedding attached to η as in definition (2.3.1)(iii) is θ_{η} := diag(diag(σ^{υ_{1,1}}(θ), …, σ^{υ_{1,n_{1}}} (θ)) × … × diag(σ^{υ_{r,1}}(θ), …, σ^{υ_{r,nr}} (θ))), where ♯{j ∈ {1, …, n_{i}} | υ_{i,j} = k − 1} = η_{k}^{i} for k = 1, …, f and i = 1, …, r. Clearly gθ_{λ}g^{−1} is a generator of the extension L_{f,η}/F; hence, it is the image of a generating element θ′ of L_{f}. Because θ and θ′ are Galois conjugate and conjugation by π induces the generator σ of Gal(L/F), there exists k ∈ ℤ such that π^{k}θ′π^{−k} = σ^{k}(θ′) = θ. Since πI_{m} normalizes 𝔄_{n} and L_{f,η}, we can assume (replacing g by (π^{k}I_{m})g) that gθ_{λ}g^{−1} = θ_{η}.

We know (see for example [G] proposition 1.2 and equation (2) of its proof) that K_{𝔄_{n}} = ⟨x_{s}⟩𝔄_{n}^{×} = 𝔄_{n}^{×}⟨x_{s}⟩, where

x_{s} := ( \begin{matrix}0 , & πI_{s} \\ I_{m−s} , & 0 \end{matrix} )

and where s = n_{1} + … + n_{t} and t is the smallest period of the numbers n_{1}, …, n_{r}. Hence there exists k′ ∈ ℤ such that gx_{s}^{−k′} ∈ 𝔄_{n}^{×}. Since x_{s} = h_{t} … h_{1} (with the notation of the proof of the first part of this proposition) and by iteration of the process of the first part, we see that there exists a permutation matrix p ∈ 𝔄_{n}^{×} (whence gx_{s}^{−k′}p^{−1} ∈ 𝔄_{n}^{×}) such that (px_{s}^{k′}E_{f,λ}x_{s}^{−k′}p^{−1}, px_{s}^{k′}𝔄_{n}x_{s}^{−k′}p^{−1}) = (Z^{k′t}(λ)). Therefore we can reduce to the case where g ∈ 𝔄_{n}^{×}.

Let ḡ := (g (mod 1 + 𝔓_{n})) and identify ḡ with its image under the isomorphism

( 2.3.5 ) 𝔄_{n}/𝔓_{n} ≃ M_{n_{1}}(𝔨_{D}) × … × M_{n_{r}}(𝔨_{D}).

Moreover, let ḡ_{i} ∈ GL_{n_{i}}(𝔨_{D}), for i = 1, …, r,, be its projections to the components on the right side of the isomorphism (2.3.5). Let θ_{λ}^{i}, θ_{η}^{i} be the images of θ_{λ}, θ_{η} under the reduction map from 𝔬_{L_{f,λ}}, 𝔬_{L_{f,η}} to 𝔨_{L_{f,λ}}, 𝔨_{L_{f,η}} respectively, followed by the canonical embeddings into 𝔄_{n}/𝔓_{n} of the residual fields 𝔨_{L_{f,λ}}, 𝔨_{L_{f,η}} respectively and finally projected to a single component i (in (2.3.5)). Then we have θ_{λ}^{i} = diag(σ̄^{ν_{i,1}}(θ̄), …, σ̄^{ν_{i,n_{i}}} (θ̄)) (in M_{n_{i}}(k_{d})) and θ_{η}^{i} := diag(σ̄^{υ_{i,1}}(θ̄), …, σ̄^{υ_{i,n_{i}}} (θ̄)) (in M_{n_{i}}(k_{D})) for i = 1, …, r, where σ̄ is the image of σ under the canonical isomorphisms Gal(L/F) ≃ Gal(𝔨_{L}/𝔨_{F}) ≃ Gal(𝔨_{D}/𝔨_{F}) and θ̄ is the canonical image of θ in 𝔨_{D}.

Since ḡ_{i}θ̄_{λ}^{i}ḡ_{i}^{−1} = θ̄_{η}^{i}, the characteristic polynomials of θ̄_{λ}^{i} and θ̄_{η}^{i} agree, so that all roots have the same multiplicities; subsequently, λ_{k}^{i} = η_{k}^{i} for k = 1, …, f and i = 1, …, r.

So we have reduced to the case where λ = η.

□

To show that our system of representatives is complete, we first need a generalization of a lifting lemma of Fröhlich ([F] 3(II)).

( 2.3.6 ) Lemma Let K/F be an unramified extension in A and 𝔄 a hereditary order in A with Jacobson radical 𝔓 such that 𝔬_{K} ⊂ 𝔄. Let j : 𝔬_{K} ↪ 𝔄 be another 𝔬_{F} -embedding. Suppose now that there exist g_{1} ∈ K_{𝔄} such that for all x ∈ 𝔬_{K}, g_{1}^{−1}(j(x))g_{1} ≡ x(mod 𝔓). Then there exists g ∈ K_{𝔄} such that g ≡ g_{1}(mod 1 + 𝔓) and, for all x ∈ 𝔬_{K} : g^{−1}(j(x))g = x.

Proof.

The idea of the proof is the same as in the case of a principal order in [F](3(II)). We construct a sequence {g_{n}}_{n∈ℕ} such that, for all n > 1 and all x ∈ 𝔬_{K}, g_{n−1} ≡ g_{n}(mod 1 + 𝔓^{n−1}) and g_{n}^{−1}(j(x))g_{n} ≡ x(mod 𝔓^{n}).

Once such a sequence is constructed, since g_{1}^{−1}g_{n} ∈ 𝔄^{×} and 𝔄^{×} is compact ), g := lim_{n∈ℕ^{⟶}} g_{n} exists and lies in K_{𝔄}.

g_{1} exists by the assumption. Let n > 1 and assume that g_{1}, …, g_{n−1} have already been constructed. For all x ∈ 𝔬_{K}, we have

( 2.3.7 ) g_{n−1}^{−1}(j(x))g_{n−1} ≡ x + δ(x)(mod 𝔓^{n}),

where δ : 𝔬_{K} → 𝔓^{n−1} is a function whose values are determined modulo 𝔓^{n}.

For all x, y ∈ 𝔬_{K}, we have g_{n−1}^{−1}(j(xy))g_{n−1} ≡ xy + δ(xy) (mod 𝔓^{n}) and g_{n−1}^{−1}(j(xy))g_{n−1} = (g_{n−1}^{−1}(j(x))g_{n−1}g_{n−1}^{−1}(j(y))g_{n−1}) ≡ xy + xδ(y) + δ(x)y (mod 𝔓^{n}) and therefore

( 2.3.8 ) δ(xy) ≡ xδ(y) + δ(x)y (mod 𝔓^{n}).

From (2.3.7) (and since 𝔬_{F} is in the center of A), we have δ(𝔬_{F}) ≡ 0, and from π_{F} 𝔓^{n−1} ⊂ 𝔓^{n} (π_{F} a prime element of 𝔬_{F}), 𝔭_{K} = π_{F} 𝔬_{K} and (2.3.8), we also obtain δ(𝔭_{K}) ≡ 0. Thus δ is in fact a 𝔨_{F} -derivation δ : 𝔨_{K} → 𝔓^{n−1}/𝔓^{n}, where 𝔓^{n−1}/𝔓^{n} is endowed with the (𝔨_{K}, 𝔨_{K})-bimodule structure induced by the (𝔬_{K}, 𝔬_{K})-bimodule structure of 𝔓^{n−1}.

Since 𝔨_{K} ⊗_{𝔨_{F}} 𝔨_{K}^{op} is a semisimple 𝔨_{F} -algebra, one knows from cohomology theory (see e.g. [CE](4.1;7.9)), that δ must then split: there exists u ∈ 𝔓^{n−1}, such that for all x ∈ 𝔬_{K} ( 2.3.9 ) δ(x) ≡ ux − xu (mod 𝔓^{n}).

Defining g_{n} := g_{n−1}(1 − u), the congruence g_{n} ≡ g_{n−1} (mod 𝔓^{n−1}) holds and for all x ∈ 𝔬_{K}, g_{n}^{−1}(j(x))g_{n} = (1 − u)^{−1}g_{n−1}^{−1}(j(x))g_{n−1}(1 − u) ≡ (1 + u)(x + δ(x))(1 − u) (mod 𝔓^{n}) ≡ (1 + u)(x + xu − ux)(1 − u) (mod 𝔓^{n}) ≡ x (mod 𝔓^{n}) as required.

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( 2.3.10 ) Proposition For [E, 𝔄] ∈ E there exists λ ∈ Φ such that (E, 𝔄) ∼ (λ).

Proof.

By corollary (2.2.4), we can assume that E/F is unramified and [E : F] | d. By conjugation with an element g ∈ A^{×}, we can reduce to the case where 𝔄 is given in normal form (2.1.1).

To determine λ ∈ Φ, set f := [E : F] and let r, n = (n_{1}, …, n_{r}) be the invariants of 𝔄 as in (2.1.1). The canonical embedding

( 2.3.11 ) 𝔨_{E} ↪ 𝔄/𝔓

induced by the inclusion 𝔬_{E} ⊂ 𝔄 splits into r embeddings via the projection onto one component with respect to the isomorphism 𝔄/𝔓 ≃ M_{n_{1}}(𝔨_{D}) × … × M_{n_{r}}(𝔨_{D}). The resulting embeddings

( 2.3.12 ) 𝔨_{E} ↪ M_{n_{i}}(𝔨_{D})

restricted to 𝔨_{E}^{×} lead to faithful representations

( 2.3.13 ) 𝔨_{E}^{×} ↪ GL_{n_{i}}(𝔨_{D})

of the finite cyclic group 𝔨_{E}^{×} over the finite field 𝔨_{D} of characteristic p. The representations have the property that the subgroup 𝔨_{F}^{×} is represented by the scalar matrices 𝔨_{F}^{×}I_{n_{i}}, via x ↦ xI_{n_{i}} for i = 1, …, r.

Because ♯𝔨_{E}^{×} = q^{f} − 1 = p^{kf} − 1, we have char(𝔨_{D}) = p ∤ ♯𝔨_{E}^{×}. Therefore Maschke’s proposition (see [M] 3.1) applies and the representations split into a direct sum of irreducibles. 𝔨_{E}^{×} is a group of exponent q^{f} − 1 and 𝔨_{D}^{×} is a cyclic group of order q^{d} − 1. Because f | d, we have q^{f} − 1 | q^{d} − 1, and therefore 𝔨_{D} contains a primitive (q^{f} − 1) − th root of unity. Consequently, by [M] 3.13, the representations split into direct sums of linear characters, where, as we noted before, each of them represents the elements of 𝔨_{F}^{×} identically.

Interpreting the embeddings (2.3.11),(2.3.12),(2.3.13) as inclusions, for i = 1, …, r, there exists, h̄_{i} ∈ GL_{n_{i}}(𝔨_{D}) such that the conjugated embedding h̄_{i}𝔨_{E}h̄_{i}^{−1} ↪ M_{n_{i}}(𝔨_{D}) (that is the embedding (2.3.12) followed by the adjoint action of h̄_{i}) restricted to h̄_{i}𝔨_{E}^{×}h̄_{i}^{−1} takes the form ( 2.3.14 ) x ↦ diag(χ_{1}^{i}(x), …, χ_{n_{i}}^{i}(x)),

where the χ_{j}^{i}’s are linear characters which represent 𝔨_{F}^{×} identically for j = 1, …, n_{i} and i = 1, …, r.

Coming back to h̄_{i}𝔨_{E}h̄_{i}^{−1} ↪ M_{n_{i}}(𝔨_{D}) for i = 1, …, r, we see that this embedding must be diagonal. By projection onto one diagonal component j of (2.3.14), this clearly means that each χ_{j}^{i} lifts to an embedding of fields θ_{j}^{i} : 𝔨_{E} ↪ 𝔨_{D} over 𝔨_{F}. Each of the θ_{j}^{i} is given by x ↦ σ̄^{ν_{i,j}}(x), where ν_{i,j} ∈ {0, .., f − 1} for j = 1, …, n_{i} and σ̄ ∈ Gal(𝔨_{D}/𝔨_{F}) is the image of σ as in definition (2.3.1)(iii) under the canonical isomorphisms Gal(L/F) ≃ Gal(𝔨_{L}/𝔨_{F}) ≃ Gal(𝔨_{D}/𝔨_{F}).

Changing the h̄_{i} by certain permutation matrices from GL_{n_{i}}(𝔨_{D}), we can assume that ν_{i,1} ≤ … ≤ ν_{i,n_{i}} for i = 1, …, r. Fitting the elements h̄_{i} together, we get an element h̄ of 𝔄^{×}/(1+𝔓). Choose h ∈ 𝔄^{×} such that h̄ = (h (mod 1+𝔓)). Then, we can replace E by hEh^{−1} without leaving the equivalence class of (E, 𝔄) and so that the assumptions of our further reductions hold.

We are thus reduced to the case where (E, 𝔄) has a canonical embedding of the residual field 𝔨_{E} ↪ 𝔄/𝔓 ≃ M_{n_{1}}(𝔨_{D}) × … × M_{n_{r}}(𝔨_{D}) given by x ↦ diag(σ̄^{ν_{1,1}}(x), …, σ̄^{ν_{1,n_{1}}} (x)) × … × diag(σ̄^{ν_{r,1}}(x), …, σ̄^{ν_{1,nr}} (x)), where ν_{i,j} ∈ {0, …, f − 1} for j = 1, …, n_{i} and ν_{i,1} ≤ … ≤ ν_{i,n_{i}} for i = 1, …, r.

Let λ_{k}^{i} := ♯{j ∈ {1, …, n_{i}} | ν_{i,j} = k − 1} for i = 1, …, r and k = 1, …, f. Then λ = ((λ_{1}^{1}, …, λ_{f}^{1}), …, (λ_{1}^{r}, …, λ_{f}^{r})) is an embedding datum, and, by definition, (λ) has the same canonical embedding of its residual field as (E, 𝔄), whence (E, 𝔄) ∼ (λ) follows easily from lemma (2.3.6).

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( 2.3.15 ) Theorem The map I → E given by λ̄ ↦ [λ] is a canonical bijection; i.e. it is independent of our fixed identification A ≃ M_{m}(D) and of the choices in definition (2.3.1)(iv).

Proof.

By lemma (2.3.3) the map is well-defined and injective. By proposition (2.3.10) it is surjective. Two isomorphisms A ≃ M_{m}(D) differ by the adjoint action of some g ∈ A^{×}, whence leave all

classes of E stable (by definition (2.2.1)). Note that if A ≃ M_{m}(D′) then D′ ≃ D; hence we can assume that D = D′.

Without loss of generality we can fix our identification A ≃ M_{m}(D). Another extension L′/F differs from L/F only by conjugation of an element x ∈ D^{×}; hence, for λ ∈ Φ the pearl-embedding (λ)′ with respect to L′/F differs from the pearl-embedding (λ) with respect to L/F only by conjugation with xI_{m} in A^{×} = GL_{m}(D), whence lies in the same equivalence class (again by definition (2.2.1)).

Thus we can fix the extension L/F in D and just look at a different prime element π′ of 𝔬_{D} which normalizes L. But it is well known that π′ and π induce the same automorphism σ ∈ Gal(L/F), which implies that they lead to the same standard pearl-embeddings.

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2.4 An explicit version of the criterion of [Br1]

In this section we apply the results of [Br1] to decide which classes of embeddings with respect to hereditary orders are good and which are “bad”. To do this, we first have to explain the concept of [Br1] and then apply it to the special case of pearl-embeddings.

( 2.4.1 ) Definition Let γ = (γ_{j})_{j∈ℤ} be a sequence with γ_{j} ∈ ℕ ∪ {0} for all j ∈ ℤ and let γ′ = (γ_{i_{j}})_{j∈ℤ} with (i_{j})_{j∈ℤ} be a ℤ-valued sequence such that i_{k} < i_{l} if k < l and such that γ′ is the sequence of all non-zero numbers from the sequence γ.

Let t a period of the sequence γ′. As in [Br1], we call the sequence γ t-zero-periodic, if the number of zeros occurs t-periodically in γ, which means that for all j ∈ ℤ, i_{j+1} − i_{j} = i_{j+t+1} − i_{j+t} (note that i_{j+1} − i_{j} − 1 is the number of zeros in the sequence γ between the non-zero numbers γ_{i_{j}} and γ_{i_{j+1}}).

Let V be a simple A module, L/F as in definition (2.3.1), and consider the central simple L-algebra A ⊗_{F} L ≃ M_{md}(L). V is also a simple A ⊗_{F} L module with the action (a ⊗ l)v := avl (note that V is a right D-vector space and L ⊂ D), and, with this action, we have A ⊗_{F} L ≃ End_{L}(V).

Let (E, 𝔄) be an arbitrary embedding in A and consider the algebra E ⊗_{F} L ⊂ A ⊗_{F} L. This is a semisimple commutative E-algebra; hence,

( 2.4.2 ) E ⊗_{F} L ≃ ⨁_{i=1}^{l} K_{i}

for some field extensions K_{i}/E in some closure of E and where E ⊗_{F} L acts on each K_{i} irreducibly for i = 1, …, r (this just means in this context that the algebra isomorphism (2.4.2) identifies the K_{i}’s with the simple components of E ⊗_{F} L).

From [Br1] chap. 2 we know that, if 1_{A} = 1_{E⊗_{F} L} = e_{1} + … + e_{l} is the representation of the 1 as a sum of central simple idempotents with respect to (2.4.2), then e_{i} ∈ 𝔬_{E} ⊗_{𝔬_{F}} 𝔬_{L} for i = 1, …, l. Moreover V^{i} := e_{i}V is a simple e_{i}(B ⊗_{F} L)e_{i}-module (where B = C_{A}(E)) and j↦X_{j}^{i} := e_{i}X_{j} (j ∈ ℤ) is a lattice sequence in the sense of [BK3] such that the inclusion ordered set {X_{j}^{i} | j ∈ ℤ} becomes a lattice chain in V^{i} for the hereditary e_{i}(𝔬_{E} ⊗_{𝔬_{F}} 𝔬_{L})e_{i}-orders e_{i}(𝔅 ⊗_{𝔬_{F}} 𝔬_{L})e_{i} for i = 1, …, l, where 𝔅 = 𝔄 ∩ B.

From [Br1] we derive:

( 2.4.3 ) Proposition Let (E′, 𝔄) be an arbitrary embedding in A with the same canonical embedding of residual field as (E, 𝔄) (or, more precisely, their images agree). For i = 1, …, l let γ^{(i)} := (dim_{𝔨_{L}}(X_{j}^{i}/X_{j+1}^{i}))_{j∈ℤ} and let γ^{′(i)} = (γ_{j}^{′(i)})_{j∈ℤ} be the subsequence of γ^{(i)} consisting of all non-zero members of γ^{(i)} as in definition (2.4.1); let t_{i} be the smallest period of γ^{′(i)}. Then:

(i) The embedding (E′, 𝔄) is good if and only if the sequence γ^{(i)} is t_{i}-zero-periodic for one (equivalently for all) i = 1, …, l.

(ii) Let i ∈ {1, …, l}. ℭ = 𝔄 ∩ C is a hereditary 𝔬_{E′}-order in C = C_{A}(E′), and for all j ∈ ℤ the sequence γ_{j+1}^{′(i)}, γ_{j+2}^{′(i)}, …, γ_{j+r′}^{′(i)} are the invariants of ℭ, where r′ ∈ ℕ is the unique number such that for all j ∈ ℤ : γ_{j+1}^{′(i)} + γ_{j+2}^{′(i)} + … + γ_{j+r′}^{′(i)} = m′, where m′ = \frac{⟨[E′:F],d⟩m}{[E′:F]} (in particular, t_{i} = t_{j}, for 1 ≤ i, j ≤ l).

(iii) Let i ∈ {1, …, l} and use the assumptions and notation of (ii). Let υ ∈ ℕ be the smallest number such that γ^{(i)} is t_{i}υ-zero-periodic and z ∈ K_{ℭ} such that K_{ℭ} = ⟨z⟩ℭ^{×}. Then K_{𝔄} ∩ C = ⟨z^{υ}⟩ℭ^{×} (and in fact υ is independent of i).

Proof.

Let E′ ⊗_{F} L ≃ ⨁_{i=1}^{l} K′_{i} be the decomposition into simple components as in (2.4.2) and 1_{A} = e′_{1} + … + e′_{l} the decomposition of 1_{A} as a sum of central simple idempotents with respect to the latter decomposition. Let ′X_{j}^{i} := e′_{i}X_{j} for i = 1, …, l and j ∈ ℤ. By the proof of [Br1](proposition 4.2.), we have γ^{(i)} = (dim_{𝔨_{L}}(′X_{j}^{i}/′X_{j+1}^{i}))_{j∈ℤ} for suitable choices of the indices. The assertions (i) and (ii) are reformulations of [Br1](theorem 1.3., proposition 3.2., proposition 4.2., lemma 4.3. and corollary 4.4.). The assertion (iii) is a generalization of [Br1](lemma 4.3.) and can be seen if we take into account that K_{𝔄} ∩ C = K({′X_{j}^{i}}_{j∈ℤ}) := {x ∈ C^{×} | there exists ν ∈ ℤ such that for all j ∈ ℤ: x′X_{j}^{i} = ′X_{j+ν}^{i}}.

Clearly we have ℭ^{×} ⊂ K({′X_{j}^{i}}_{j∈ℤ}) and this containment is proper because F^{×} ⊂ K({′X_{j}^{i}}_{j∈ℤ}), whence K({′X_{j}^{i}}_{j∈ℤ}) = ⟨z^{υ}⟩ℭ^{×} for some υ ∈ ℤ\{0}. Because all powers of elements of K({′X_{j}^{i}}_{j∈ℤ}) are elements of the latter group as well, υ is just the smallest power of z such that there exists ν ∈ ℤ such that for all j ∈ ℤ we have z^{υ}′X_{j}^{i} = ′X_{j+ν}^{i}. But ν exists if and only if the repetitions in the lattice sequence are such that, for all j ∈ ℤ the modules′X_{j}^{i} and z^{υ}′X_{j}^{i} appear with the same muliplicity in the sequence {′X_{j}^{i}}_{j∈ℤ}. Moreover, by [G](proposition 1.2(i)), t_{i} is the partition period of e′_{i}(ℭ ⊗_{𝔬_{F}} 𝔬_{L})e′_{i} ≃ ℭ ⊗_{𝔬_{F}} 𝔬_{K′_{i}}; this order has the same partition period as ℭ and K({′X_{j}^{i}}_{j∈ℤ}) ⊂ K_{e′_{i}(ℭ⊗_{𝔬_{F}} 𝔬_{L})e′_{i}}. Hence z^{υ} has period υt_{i} on the lattice chain of e′_{i}(ℭ ⊗_{𝔬_{F}} 𝔬_{L})e′_{i} in the simple e′_{i}(C ⊗_{F} L)e′_{i}-module e′_{i}V so the repetitions have to appear υt_{i}-periodically. This leads to the asserted condition for the sequence γ^{(i)}.

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( 2.4.4 ) Corollary Let λ = ((λ_{1}^{1}, …, λ_{f}^{1}), …, (λ_{1}^{r}, …, λ_{f}^{r})) ∈ Φ be an arbitrary embedding datum of degree f, with associated partition (n_{1}, …, n_{r}), as in definition (2.3.1)(i). Let r′ be the number of couples (i, j) such that λ_{j}^{i} ≠ 0, n′_{1}, …, n′_{r′} be the subsequence of non-zero numbers of λ_{1}^{1}, …, λ_{1}^{r}, …, λ_{f}^{1}, …, λ_{f}^{r} (in that order!), t(λ) the smallest period of the sequence n′_{1}, …, n′_{r′}, υ(λ) ∈ ℕ the smallest number such that the sequence …, λ_{f}^{r}, λ_{1}^{1}, …, λ_{1}^{r}, λ_{2}^{1}, …, λ_{2}^{r}, …, λ_{f}^{1}, …, λ_{f}^{r}, λ_{1}^{1}, … is υ(λ)t(λ)-zero-periodical and d′ := \frac{d}{f}.

Then:

(i) υ(λ) is the index of [λ].

(ii) If (E, 𝔄) ∈ [λ] is an arbitrary member of this equivalence class, then the 𝔬_{E}-order 𝔅 := 𝔄 ∩ B (with B = C_{A}(E)) is hereditary and has invariants n̄_{1}, …, n̄r̄, r̄, with r̄ = \frac{⟨e(E/F),d′⟩r′}{e(E/F)} and n̄_{i} = \frac{n′_{i}}{f(E/E_{d})} for i = 1, …, r̄, where E_{d} is the division algebra part of E.

(iii) Let E/F be a field extension and 𝔄 a hereditary 𝔬_{F} -order in A with invariants n_{1}, …, n_{r}. Then there exists an embedding i : E ↪ A such that i(E^{×}) ⊂ K_{𝔄} and (i(E), 𝔄) ∈ [λ] if and only if the following conditions hold:

(a) [E_{d} : F] = f,

(b) f(E/E_{d}) | ⟨λ_{1}^{1}, …, λ_{1}^{r}, …, λ_{f}^{1}, …, λ_{f}^{r}⟩ and

(c) υ(λ)e(E/F) | \frac{r′}{t(λ)}d′,

where E_{d} is the division algebra part of E.

Proof.

Note that A has been identified with M_{m}(D). Set V = D^{m} (column vectors) and D′ := C_{D}(L_{f}) (where L_{f} is as in definition 2.3.1(iv) and f is the degree of λ). Using the notation from the beginning of this chapter for the special case (E, 𝔄) = (λ), it is easy to see that in (2.4.2), we have l = f and K_{i} ≃ L for i = 1, …, f.

Because K_{i} ≃ L, for i = 1, …, f the K_{i} form a system of representatives of the simple E ⊗_{F} Lmodules and there are f isomorphism classes of them. But L, Lπ, …, Lπ^{f−1} ⊂ D is also a set of pairwise non-isomorphic simple E ⊗_{F} L-modules, where π is as in definition (2.3.1)(iii) and E ⊗_{F} L acts on D by (e ⊗ l)d := edl. Thus the e_{i} can be indexed in such a way that

( 2.4.5 ) e_{i} acts on D (under this action of E ⊗_{F} L) as a projector on the isotypic component of Lπ^{i−1} for i = 1, …, f.

Since D = L ⊕ Lπ ⊕ … ⊕ Lπ^{d−1} and D′ = L ⊕ Lπ^{f} ⊕ … ⊕ Lπ^{f( \frac{d}{f} −1)}, D may be written as D = D′ ⊕ D′π ⊕ … ⊕ D′π^{f−1}. Moreover with the above action of E ⊗_{F} L on D, we have Lπ^{i−1} ≃ Lπ^{j−1} iff i ≡ j (mod f) and with the indexing (2.4.5) e_{i} acts as a projector to D′π^{i−1} for i = 1, …, f.

Consider more generally the action of E ⊗_{F} L on D given by (e ⊗ l)d := σ^{j−1}(e)dl for j = 1, …, f and σ as in definition (2.3.1)(iii). Denote the module with the underlying set D and the latter action by (D)^{σ^{j−1}}. Denote the action of E ⊗_{F} L in (D)^{σ^{j−1}} by ⋆^{σ^{j−1}}.

Now let d = ∑_{ν=1}^{d} l_{ν}π^{ν−1} be an element of the set D and define a map ρ^{σ^{j−1}} : D → D by ρ^{σ^{j−1}}(d) := ∑_{ν=1}^{d} σ^{1−j}(l_{ν})π^{ν−j}. This map is clearly an F -vectorspace isomorphism. Moreover for ∑_{i=1}^{d} e_{i} ⊗ l′_{i} ∈ E ⊗_{F} L we have

ρ^{σ^{j−1}}((∑_{i=1}^{d} e_{i} ⊗ l′_{i}) ⋆^{σ^{j−1}} (∑_{ν=1}^{d} l_{ν}π^{ν−1})) = ρ^{σ^{j−1}}(∑_{i=1}^{d} ∑_{ν=1}^{d} σ^{j−1}(e_{i})l_{ν}π^{ν−1}l′_{i}) =
ρ^{σ^{j−1}}(∑_{i=1}^{d} ∑_{ν=1}^{d} σ^{j−1}(e_{i})l_{ν}σ^{ν−1}(l′_{i})π^{ν−1}) = ∑_{i=1}^{d} ∑_{ν=1}^{d} e_{i}σ^{1−j}(l_{ν})σ^{ν−j}(l′_{i})π^{ν−j} =
∑_{i=1}^{d} ∑_{ν=1}^{d} e_{i}σ^{1−j}(l_{ν})π^{ν−j}l′_{i} = (∑_{i=1}^{d} e_{i} ⊗ l′_{i}) ⋆^{id} (∑_{ν=1}^{d} σ^{1−j}(l_{ν})π^{ν−j}) =
(∑_{i=1}^{d} e_{i} ⊗ l′_{i}) ⋆^{id} ρ^{σ^{j−1}}(∑_{ν=1}^{d} l_{ν}π^{ν−1}).

Hence, in fact ρ^{σ^{j−1}} is an isomorphism of E ⊗_{F} L-modules (D)^{σ^{j−1}} ≃ D. If we denote the projection induced by the multiplication of e_{1} in (D)^{σ^{j−1}} by e_{1}^{σ^{j−1}} and in D (= (D)^{id}) by e_{1}^{id}, then we obtain an commutative diagramm:

(D)^{σ^{j−1}}  →^{ρ^{σ^{j−1}}} D
↓ e_{1}^{σ^{j−1}}  ↓ e_{1}^{id}
(D)^{σ^{j−1}}  →^{ρ^{σ^{j−1}}} D.

The map ρ^{σ^{j−1}} maps the subset D′π^{j−1} of D onto D′, the map e_{1}^{id} is a projector onto D′ and the map (ρ^{σ^{j−1}})^{−1} maps the subset D′ of D onto D′π^{j−1}. Therefore the map e_{1}^{σ^{j−1}} = (ρ^{σ^{j−1}})^{−1}∘e_{1}^{id}∘ρ^{σ^{j−1}} is a projetor to D′π^{j−1} for j = 1, …, f.

We conclude that

( 2.4.6 ) e_{1} acts under the module structure (D)^{σ^{j−1}} as an projector to D′π^{j−1} for j = 1, …, f.

Because of the definition of the action of A ⊗_{F} L on V, and since the embedding of E in A is given by equation (2.3.2) in definition (2.3.1), we have that E ⊗_{F} L acts componentwise on V = D^{m}. Furthermore, if we index the components of V in the same way as the diagonal elements of A in the embedding (2.3.2), then E ⊗_{F} L acts on the (i, j)-component on the underlying set D with the module structure of (D)^{σ^{ν_{i,j}}} for j = 1, …, n_{i} and i = 1, …, r.

By (2.4.6)

( 2.4.7 ) e_{1} acts on the (i, j)-component as a projection onto D′π^{ν_{i,j}}

for j = 1, …, n_{i} and i = 1, …, r.

Let ϵ_{i,j} : D → D′π^{ν_{i,j}} → D′ be the associated projection from D to the (L)^{σ^{ν_{i,j}}} -isotypic component followed by the map given by d′π^{ν_{i,j}} ↦ d′ for j = 1, …, n_{i} and i = 1, …, r.

Since 𝔄_{n} is given in the normal form 2.1.1, for a suitable choice of the index, the associated chain {X_{j}}_{j∈ℤ} takes the form: ^{t}X_{0} = 𝔬_{D} ⊕𝔬_{D} ⊕…⊕𝔬_{D} (sum of m copies of 𝔬_{D}); ^{t}X_{1} = 𝔭_{D} ⊕…⊕𝔭_{D} ⊕𝔬_{D} ⊕ …⊕𝔬_{D} (sum of n_{1} copies of 𝔭_{D} and of m−n_{1} copies of 𝔬_{D}); …; ^{t}X_{r−1} = 𝔭_{D} ⊕…⊕𝔭_{D} ⊕𝔬_{D} ⊕…⊕𝔬_{D} (sum of n_{1} + n_{2} + … + n_{r−1} copies of 𝔭_{D} and of n_{r} = m − n_{1} − … − n_{r−1} copies of 𝔬_{D}); ^{t}X_{r} = 𝔭_{D} ⊕ … ⊕ 𝔭_{D} = ^{t}(X_{0}𝔭_{D}), where ^{t}(.) means a transposed vector. Therefore

( 2.4.8 ) dim_{𝔨_{D′}} (X_{ν}^{1}/X_{ν+1}^{1})  =  ∑_{i=1}^{r} ∑_{j=1}^{n_{i}} dim_{𝔨_{D′}} (ϵ_{i,j}(𝔭_{D}^{k+δ_{i,l}})/ϵ_{i,j}(𝔭_{D}^{k+δ_{i,l+1}}))  =

∑_{j=1}^{n_{l+1}} dim_{𝔨_{D′}} (ϵ_{l+1,j}(𝔭_{D}^{k})/ϵ_{l+1,j}(𝔭_{D}^{k+1}))

for all ν ∈ ℤ, where δ_{s,t} := \{ \begin{matrix}1 & for s ≤ t \\ 0 & otherwise \end{matrix} and k, l ∈ ℤ are the unique integers such that ν = rk + l

and 0 ≤ l ≤ r − 1.

We have …; 𝔭_{D}^{−1} = 𝔬_{D′} ⊕ … ⊕ 𝔬_{D′}π^{f−2} ⊕ 𝔭_{D′}^{−1}π^{f−1}; 𝔬_{D} = 𝔬_{D′} ⊕ … ⊕ 𝔬_{D′}π^{f−1};

𝔭_{D} = 𝔭_{D′} ⊕ 𝔬_{D′}π ⊕ … ⊕ 𝔬_{D′}π^{f−1}; …; 𝔭_{D}^{f} = 𝔭_{D′} ⊕ … ⊕ 𝔭_{D′}π^{f−1}; 𝔭_{D}^{f+1} = 𝔭_{D′}^{2} ⊕ 𝔭_{D′}π ⊕ … ⊕ 𝔭_{D′}π^{f−1};

…. Hence by (2.4.7), for k ∈ ℤ, 0 ≤ l ≤ r − 1 and u the unique integer 0 ≤ u ≤ f − 1 such that u ≡ k (mod f), we have

( 2.4.9 ) dim_{𝔨_{D′}} (ϵ_{l+1,j}(𝔭_{D}^{k})/ϵ_{l+1,j}(𝔭_{D}^{k+1})) = \{ \begin{matrix}1 & for ν_{l+1,j} = u \\ 0 & otherwise \end{matrix}

for j = 1, …, n_{l+1}.

Therefore, for k ∈ ℤ and 0 ≤ l ≤ r − 1, we have

( 2.4.10 ) ∑_{j=1}^{n_{l+1}} dim_{𝔨_{D′}} (ϵ_{l+1,j}(𝔭_{D}^{k})/ϵ_{l+1,j}(𝔭_{D}^{k+1})) = ♯{j | j ∈ {1, …, n_{l+1}} ∧ ν_{l+1,j} = u} = λ_{u+1}^{l+1},

.

Putting (2.4.8),(2.4.9) and (2.4.10) together, for all ν ∈ ℤ, we have

( 2.4.11 ) dim_{𝔨_{D′}} (X_{ν}^{1}/X_{ν+1}^{1}) = λ_{u+1}^{l+1},

where u, k, l are the unique integers such that 0 ≤ l ≤ r − 1, k ∈ ℤ, ν = rk + l, 0 ≤ u ≤ f − 1 and u ≡ k (mod f).

Now note that C′ := C_{A}(E) ≃ M_{m}(D̃), where D̃ is a central E-division algebra of index d′ := \frac{d}{f} and D̃ ≃ D′ if we choose a suitable isomorphism E ≃ L_{f} and view D′ as an E-algebra. Moreover, set ℭ′ := 𝔄_{n} ∩ C.

(i) Follows now from (2.4.11) and proposition (2.4.3)(iii).

(ii) If (E, 𝔄) ∼ (λ) =: (E_{f}, 𝔄_{n}), there exists g ∈ A^{×} such that (gE_{d}g^{−1}, g𝔄g^{−1}) = (gE_{d}g^{−1}, 𝔄_{n}) = (E_{f}, 𝔄_{n}). Subsequently, by proposition (2.4.3)(ii), [Br1] (theorem (1.1)) and (2.4.11), ℭ′ = 𝔄_{n} ∩ C_{A}(E_{f}) is a hereditary order with invariants n′_{1}, …, n′_{r′}.

We have C := g^{−1}C_{A}(E_{f})g = C_{A}(E_{d}), ℭ := 𝔄 ∩ C = g^{−1}ℭ′g, K_{ℭ} = g^{−1}K_{ℭ′}g and K_{𝔄} = g^{−1}K_{𝔄_{n}}g; hence E^{×} ⊂ K_{𝔄} ∩ C ⊂ K_{ℭ} and ℭ is a hereditary 𝔬_{E_{d}}-order in C with invariants n′_{1}, …, n′_{r′}.

Since C ≃ M_{m}(D̂) for some central E_{d}-division algebra D̂ of index d′ and clearly ⟨f(E/E_{d}), d′⟩ = 1,

[G] (theorem (2.2)(i)(a),(ii)) applies. We conclude that (E, ℭ) is a good embedding and 𝔅 = ℭ ∩ B is a hereditary 𝔬_{E}-order in B with invariants n̄_{1}, …, n̄r̄, as required.

(iii) (⇒) (a) follows from definition (2.2.1) and (b) from part (ii). Because π_{E} ∈ K_{𝔄} ∩ C, where C = C_{A}(E_{d}), by proposition (2.4.3)(iii) there exists k ∈ ℤ and ϵ ∈ ℭ^{×} (ℭ = 𝔄∩C) such that z^{kυ(λ)}ϵ = π_{E}, where π_{E} is a prime element of 𝔬_{E} and z a uniformizer for ℭ. Therefore by [G](proposition 1.2(i)) we get υ(λ)t(λ) | \frac{r′d′}{e(E/E_{d})} and (c) follows (because E_{d}/F is unramified).

(⇐) Because of (a) we have E_{d} ≃ L_{f}, where L_{f} is as in definition (2.3.1)(iii) and we can embed E_{d}/F via (λ). From part (ii) and its proof we obtain (by identifying E_{d} with its image in A) that ℭ′ := 𝔄_{n} ∩ C′ is a hereditary 𝔬_{E_{d}}-order in C′ := C_{A}(E_{d}) with invariants n′_{1}, …, n′_{r′} continued by 𝔄_{n} to A.

As in the proof of (i), we have C′ ≃ M_{m}(D̃) for some central E_{d}-division algebra D̃ of index d′ and clearly ⟨f(E/E_{d}), d′⟩ = 1. Thus we may conclude by (b),(c) [G](theorem 2.2 (ii)), that there exists an embedding E ↪ C′ such that E^{×} ⊂ K_{ℭ′}.

Clearly 𝔬_{E}^{×} ⊂ ℭ^{′×} ⊂ 𝔄_{n}^{×} ⊂ K_{𝔄_{n}} and by (c), [G](proposition 1.2(i)) and proposition (2.4.3)(iii), we get π_{E} ∈ K_{𝔄_{n}} ∩ C′. Consequently, there exists an embedding (E, 𝔄_{n}) ∼ (λ) and because 𝔄 and 𝔄_{n} have the same invariants, whence are conjugate, we can find g ∈ A^{×} such that (gEg^{−1}, g𝔄_{n}g^{−1}) = (gEg^{−1}, 𝔄).

Thus (gEg^{−1}, 𝔄) ∼ (λ), as required.

□

\underline{Remark} Since the bijection in theorem (2.3.15) is canonical, the results of corollary (2.4.4) are also independent of all choices made.

3 Pure elements

We now apply our results to get a new notion of purity.

( 3.1 ) Definition (i) Let λ ∈ Φ be an embedding datum for a hereditary order 𝔄_{n} (whence λ̄ ∈ I is its equivalence class) of degree f and with associated partition n = (n_{1}, …, n_{r}). Let f′ be a divisor of f and denote by (λ)_{f′} the embedding of L_{f′} obtained by restricting the standard pearl-embedding (λ) of L_{f} to L_{f′} (with notations of definition (2.3.1)(iii)). We say that (E, 𝔄) is of type λ̄ if there exists a divisor f′ of f such that (E, 𝔄) ∼ (λ)_{f′}.

(ii) An equivalence class of embedding data τ̄, where τ is an embedding datum of maximal degree d, is called a type of embedding.

(iii) Let τ̄ be a type of embedding, then α ∈ A is called (𝔄, τ̄)-pure if (a) E := F [α] is a field with E^{×} ⊂ K_{𝔄} and

(b) (E, 𝔄) is of type τ̄.

( 3.2 ) Lemma Let τ̄ a type of embedding, α, β ∈ A be (𝔄, τ̄)-pure and suppose that there exists g ∈ A^{×} such that α = gβg^{−1}. Then there exists h ∈ K_{𝔄} such that α = hβh^{−1}.

Proof.

Let E^{α} := F [α], E^{β} := F [β] and let E_{d}^{α}, E_{d}^{β} be the division algebra parts of E^{α}, E^{β} respectively; then α = gβg^{−1} implies [E_{d}^{α} : F] = [E_{d}^{β} : F] =: f, hence there exists an embedding datum λ of degree f and with associated partition n = (n_{1}, …, n_{r}), where n_{1}, …, n_{r} are the invariants of 𝔄, such that (E_{d}^{α}, 𝔄) ∼ (E^{α}, 𝔄) ∼ (λ) ∼ (E^{β}, 𝔄) ∼ (E_{d}^{β}, 𝔄).

Therefore, there exists h_{1} ∈ K_{𝔄} such that h_{1}E_{d}^{β}h_{1}^{−1} = E_{d}^{α} and we can (replacing β by h_{1}βh_{1}^{−1} and g by gh_{1}^{−1}) reduce to the case where E_{d} := E_{d}^{α} = E_{d}^{β}.

Now choose z ∈ A^{×}, such that z𝔄z^{−1} = 𝔄_{n} where 𝔄_{n} is the hereditary order given in normal form (2.1.1) with invariants n = (n_{1}, …, n_{r}). The embedding (L_{f}, 𝔄_{n}) := (λ) is then given by x ↦ diag(diag(σ^{ν_{1,1}}(x), …, σ^{ν_{1,n_{1}}} (x)) × … × diag(σ^{ν_{r,1}}(x), …, σ^{ν_{r,nr}} (x))), using the notation of definition (2.3.1)(iii).

Conjugating 𝔄_{n} by x_{λ} := diag(diag(π^{ν_{1,1}}, …, π^{ν_{1,n_{1}}} ) × … × diag(π^{ν_{r,1}}, …, π^{ν_{r,nr}})) (with π as in definition (2.3.1)(iii)) and letting 𝔄_{n,λ} := x_{λ}𝔄_{n}x_{λ}^{−1}, we obtain

( 3.3 ) 𝔄_{n,λ} = (𝔭_{D}^{a_{i,j}})_{i,j=1,...,m},

where the last notation means the set of m × m-matrices with entries in 𝔭_{D}^{a_{i,j}} in the (i, j) place for suitable a_{i,j} ∈ ℤ, i = 1, …, m, j = 1, …, m.

( 3.4 ) The embedding (x_{λ}L_{f}x_{λ}^{−1}, 𝔄_{n,λ}) is then given by x → diag(x, …, x) = xI_{m}.

Because K_{x_{λ}z𝔄z^{−1}x_{λ}^{−1}} = x_{λ}zK_{𝔄}z^{−1}x_{λ}^{−1}, we can assume (replacing 𝔄 by 𝔄_{n,λ}, α by x_{λ}zαz^{−1}x_{λ}^{−1}, β by x_{λ}zβz^{−1}x_{λ}^{−1} and g by x_{λ}zgz^{−1}x_{λ}^{−1}) that

( 3.5 ) 𝔄 = 𝔄_{n,λ}.

The equivalences (E_{d}, 𝔄) ∼ (λ) ∼ (x_{λ}L_{f}x_{λ}^{−1}, 𝔄) imply that there exists h_{2} ∈ K_{𝔄} such that h_{2}E_{d}h_{2}^{−1} = x_{λ}L_{f}x_{λ}^{−1}. We may thus assume (replacing α by h_{2}αh_{2}^{−1}, β by h_{2}βh_{2}^{−1} and g by h_{2}gh_{2}^{−1}) that

( 3.6 ) E_{d} = x_{λ}L_{f}x_{λ}^{−1}.

Now let 𝔅_{α} := 𝔄∩C_{A}(α), 𝔅_{β} := 𝔄∩C_{A}(β) and ℭ := 𝔄∩C_{A}(E_{d}). By [Br1](theorem (1.1)), e(E^{α}/F) = e(E^{β}/F), f(E^{α}/E_{d}^{α}) = f(E^{β}/E_{d}^{β}), and, by corollary (2.4.4)(ii), we have:

( 3.7 ) 𝔅_{α} and 𝔅_{β} are hereditary orders with the same invariants.

Since C_{A}(E_{d}) ≃ M_{m}(D̃), for some central E_{d}-division algebra D̃ of index d′ = \frac{d}{f} and since ⟨f(E/E_{d}), d′⟩ = 1 and E^{α×} ⊂ K_{𝔄} ∩ C_{A}(E_{d}) ⊂ K_{ℭ}, E^{β×} ⊂ K_{𝔄} ∩ C_{A}(E_{d}) ⊂ K_{ℭ}, we get by [G](theorem 2.2(ii)) that the embeddings (E^{α}, ℭ), (E^{β}, ℭ) are good. Moreover, by [G](theorem 2.2(i)), we conclude in particular that

( 3.8 ) ℭ is the unique continuation of 𝔅_{α} from C_{A}(α) to C_{A}(E_{d}).

By [G](theorem 2.2 (i)(a)(b)),

( 3.9 ) there exists z ∈ K_{𝔅_{β}} such that K_{ℭ} = ℭ^{×}⟨z⟩,

which just means that the continuation of 𝔅_{β} by ℭ is unramified in the sense of the associated pseudo-valuations.

Since g𝔄g^{−1}∩C_{A}(α) = g𝔄g^{−1}∩C_{A}(gβg^{−1}) = g𝔅_{β}g^{−1} and because of (3.7), there exists y ∈ C_{A}(α) such that

( 3.10 ) yg𝔅_{β}g^{−1}y^{−1} = 𝔅_{α}.

The field ygE_{d}g^{−1}y^{−1}/F is an unramified subextension of degree f of E^{α}/F, whence ygE_{d}g^{−1}y^{−1} = E_{d} and therefore

( 3.11 ) yg ∈ N_{A^{×}}(E_{d}),

where N_{A^{×}} means the A^{×}-normalizer. .

Letting ℭ′ := yg𝔄g^{−1}y^{−1} ∩ C_{A}(E_{d}) and applying (3.11), we have ygC_{A}(E_{d})g^{−1}y^{−1} = C_{A}(ygE_{d}g^{−1}y^{−1}) = C_{A}(E_{d}). Therefore ℭ′ = ygℭg^{−1}y^{−1}; hence, from (3.10) and since K_{ygℭg^{−1}y^{−1}} = ygK_{ℭ}g^{−1}y^{−1}, we get that ℭ′ and ℭ are both continuations of 𝔅_{α}. As a consequence of (3.8), ℭ′ = ℭ and

( 3.12 ) yg ∈ N_{A^{×}}(ℭ).

From (3.11), there exists k ∈ ℤ such that, for all e ∈ E_{d}, ygeg^{−1}y^{−1} = σ^{k}(e) (where σ is as in definition (2.3.1)(iii)); hence by reductions (3.6) and (3.4)

( 3.13 ) π^{k}yg ∈ C_{A}(E_{d}).

By reductions (3.5) and (3.3), we also have ℭ = 𝔄 ∩ C_{A}(E_{d}) = (𝔭_{D}^{a_{i,j}} ∩ C_{D}(E_{d}))_{i,j=1,...,m} and π ∈ N_{A^{×}}(E_{d}); hence π ∈ N_{A^{×}}(ℭ). But this, together with (3.12) and (3.13), leads to π^{k}yg ∈ N_{A^{×}}(ℭ) ∩ C_{A}(E_{d}) = K_{ℭ}. By (3.9) there exists x ∈ K_{𝔅_{β}} such that π^{k}ygx ∈ ℭ^{×} ⊂ 𝔄^{×} ⊂ K_{𝔄}. Because of the reductions (3.5) and (3.3), we have π ∈ K_{𝔄}; hence ygx ∈ K_{𝔄}.

Setting h := ygx, we get hβh^{−1} = ygxβx^{−1}g^{−1}y^{−1} = ygβg^{−1}y^{−1} = yαy^{−1} = α (because x ∈ C_{A}(β) and y ∈ C_{A}(α)), as required.

□

( 3.14 ) Definition (i) For a type of embedding τ̄, we denote by A(𝔄, τ̄) the set of all (𝔄, τ̄)-pure elements and by Ad_{K_{𝔄}}\A(𝔄, τ̄) the set of all K_{𝔄}-orbits of A(𝔄, τ̄).

(ii) Let τ̄ be a type of embedding for a hereditary order 𝔄 and, for each divisor f of d, fix an embedding datum λ_{f} such that (τ)_{f} ≃ (λ_{f}). Let t(λ_{f}), υ(λ_{f}) be as in the assumptions of corollary (2.4.4) for λ_{f} and 𝔄. Denote by F_{𝔄,τ̄,f}[T] the set of all irreducible polynomials f(T) over F such that for E := F [T]/(f(T)) the numerical conditions (a)-(c) in corollary (2.4.4)(iii) are fulfilled. Set

F_{𝔄,τ̄}[T] := ⋃_{f|d} F_{𝔄,τ̄,f}[T].

( 3.15 ) Theorem Let τ̄ be a type of embedding for a hereditary order 𝔄. The map which takes each pure element to its minimal polynomial over F induces a bijection

Ad_{K_{𝔄}}\A(𝔄, τ̄) ↔ F_{𝔄,τ̄}[T].

Proof.

Clearly this is a well-defined map from Ad_{K_{𝔄}}\A(𝔄, τ̄) to a subset of all irreducible polynomials defined over F. By lemma (3.2) it is injective and by corollary (2.4.4)(iii) it is surjective.

□

( 3.16 ) Corollary Let τ̄ be a type of embedding for a hereditary order 𝔄 and α ∈ A be (𝔄, τ̄)-pure.

(i) Let F [α]/E/F be an arbitrary intermediate extension; then each β ∈ A such that F [β] = E is (𝔄, τ̄)-pure.

(ii) Let f be a divisor of f(F [α]/F) and e a divisor e(F [β]/F). Then for each irreducible polynomial p(T) over F such that f(K(p)/F) = f and e(K(p)/F) = e, where K(p) := F [T]/(p(T)), there exists an (𝔄, τ̄)-pure element β with minimal polynomial p(T).

Proof.

Let λ be an embedding datum such that (λ) ∼ (F [α], 𝔄) and let f′ := ⟨f(F [α]/F), d⟩. Clearly (λ) is of type τ̄.

(i) Let f̄:= ⟨f(E/F), d⟩, then f̄ | f′ and it is easy to see that (λ)f̄ is of type τ̄ as well.

Let F [α]_{d}, E_{d} be the division algebra parts of F [α], E respectively; then [F [α]_{d} : F] = f′, [E_{d} : F] = f̄ and F [α]_{d}/E_{d} is an unramified extension of degree \frac{f′}{f̄}. Let M be the compositum of E and F [α]_{d} in F [α]; then, because ⟨f(E/E_{d}), [F [α]_{d} : E_{d}]⟩ = 1, we have f(M/E) = [M : E] = \frac{f′}{f̄}. Let M_{d} be the division algebra part of M; then M_{d} = F [α]_{d}. Hence (M, 𝔄) ∼ (F [α]_{d}, 𝔄) ∼ (λ).

Let g ∈ A^{×} such that (gM_{d}g^{−1}, g𝔄g^{−1}) = (L_{f′}, 𝔄_{n}) := (λ); then L_{f′}/gE_{d}g^{−1}/F is the unique unramified intermediate extension of degree f̄ over F. Hence, (gE_{d}g^{−1}, 𝔄_{n}) = (λ)f̄. This implies (E, 𝔄) ∼ (λ)f̄ and hence (E, 𝔄) is of type τ̄ and each β ∈ A such that F [β] = E is (𝔄, λ̄)-pure, as required.

(ii) For this part let M/K(p) be an unramified extension of degree \frac{f(F [α]/F)}{f(K(p)/F)}, then we have f(M/F) = f(F [α]/F) and e(M/F) = e(K(p)/F). Because F [α]/F fulfills all the relations of (2.4.4)(iii)(a)-(c) with respect to λ, we have that M/F fulfills those relations as well. Therefore there exists an embedding M ↪ A over F such that, if M′ is the image of M under this embedding, (M′, 𝔄) is an embedding equivalent to (λ). Let α′ ∈ A be such that F [α′] = M′; then α′ is (𝔄, τ̄)- pure. Let K′ be the image of K(p) of our embedding in M′. Then clearly, there exists β ∈ A such that F [β] = K′ and β has minimal polynomial p(T). Now the first part applies and shows that β is (𝔄, τ̄)-pure as required.

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4 Intertwining implies conjugacy

4.1 The result

We first recall some notions from [Br2] (see §1.1 and §4.1). In this section and in sections 5, 6, the symbol q does not denote the cardinal of the residue field of F.

A stratum in A is a quadruple σ = [𝔄, n, q, β], where 𝔄 is an hereditary 𝔬_{F} -order in A, n, q are integers satisfying q < n, and β is an element of 𝔓^{−n}, 𝔓 = rad(𝔄). We say that σ is pure if β is 𝔄-pure and β𝔄 = 𝔓^{−n}; in particular E = F [β] is a field and E^{×} is contained in K, the normalizer of 𝔄 in G.

Let Ã = End_{F} (V). For each hereditary order 𝔄 in A (attached to an 𝔬_{D}-lattice chain {X_{k}} in V), define a hereditary order in Ã by:

𝔄̃ = End_{𝔬_{F}} {X_{k}} = {a ∈ Ã; aX_{k} ⊂ X_{k}}.

We easily check that 𝔓̃^{k} ∩ A = 𝔓^{k}, k ∈ ℤ, and K̃ ∩ A = K, where 𝔓̃ = rad(𝔄̃) and K̃ is the normalizer of 𝔄̃ in Ã^{×}. If r is the period of (X_{k}), then the period of 𝔄̃ is e(𝔄̃ | 𝔬_{F}) = rd, i.e. π_{F} 𝔄̃ = 𝔓̃^{rd} for any uniformizer π_{F} of F. Moreover, for all a ∈ A, we have ν𝔄̃(a) = ν_{𝔄}(a).

To any stratum σ = [𝔄, n, q, β], we attach the stratum σ̃ = [𝔄̃, n, q, β] of Ã. Then σ is pure if and only if σ̃ is. We say that σ is simple if σ̃ is simple in the sense of [BK2](1.5.5).

Two strata σ_{i} = [𝔄, n, q, γ_{i}], i = 1, 2, are said to be equivalent if γ_{1} ≡ γ_{2} mod 𝔓^{−q}. We say that they intertwine in G if there exists g in G such that

g(γ_{1} + 𝔓^{−q})g^{−1} ∩ (γ_{2} + 𝔓^{−q}) ≠ ∅.

Finally we say that a pure stratum [𝔄, n, q, β] is τ̄-pure, for a type of embedding τ̄, if β is (𝔄, τ̄)- pure.

Suppose for a moment that A is split. Then we have the following fundamental result ([BK2] theorem (2.6.1)):

( 4.1.1 ) Theorem (Bushnell-Kutzko) Let [𝔄, n, q, γ_{i}], i = 1, 2, be two simple strata and assume they intertwine in G. Then there exists k ∈ 𝔄^{×} such that kγ_{1}k^{−1} − γ_{2} ∈ 𝔓^{−q}.

When A is not split, this result is no longer true. However in this section we are going to prove the following generalization.

( 4.1.2 ) Theorem Let τ be a type of embedding and [𝔄, n, q, γ_{i}], i = 1, 2, be two τ̄-pure simple strata. Assume that they intertwine in G. Then there exists k ∈ K such that kγ_{1}k^{−1} − γ_{2} ∈ 𝔓^{−q}.

Remark. If q = n−1, then (fixing a smooth character of F with conductor 𝔭_{F}) the strata [𝔄, n, q, γ_{i}] give rise to characters ψ_{γ_{1}}, ψ_{γ_{2}} of the group U^{n}(𝔄) := 1 + 𝔓^{n} ⊂ G (see [Br2](1.1.2)). As a corollary of the theorem, we have that if the ψ_{γ_{i}} intertwine in G, they are actually conjugate by an element of K.

Note that if g ∈ G intertwines the strata [𝔄, n, q, γ_{i}], i = 1, 2, it intertwines the strata [𝔄̃, n, q, γ_{i}], i = 1, 2 as well. Indeed we have the containment:

g(γ_{1} + 𝔓^{−q})g^{−1} ∩ (γ_{2} + 𝔓^{−q}) ⊂ g(γ_{1} + 𝔓̃^{−q})g^{−1} ∩ (γ_{2} + 𝔓̃^{−q}).

Therefore our result follows from the following one:

( 4.1.3 ) Proposition With the assumptions of (4.1.2), assume that the simple strata [𝔄̃, n, q, γ_{i}], i = 1, 2, intertwine in Ã^{×}. Then there exists k ∈ K such that kγ_{1}k^{−1} − γ_{2} ∈ 𝔓^{−q}.

4.2 Simple pairs and change of base field

Before proving (4.1.3), we need to recall some concepts from [BH]. Let 𝔘 be a hereditary order in some split F -algebra U. Let β be a 𝔘-pure element of U and write E = F [β] ⊂ U. Define e_{β}(𝔘) to be the integer e(𝔘 | 𝔬_{F})/e(E/F). Then we have

ν_{𝔘}(β) = e_{β}(𝔘)ν_{E}(β),

where ν_{E} is the normalized valuation of E.

Assume that [𝔘, n, q, β] is a simple stratum in U. Then the simple pair attached to this stratum is the pair [m, P], where m = [q/e_{β}(𝔘)] ∈ ℤ and P is the minimal polynomial of β over F. A simple pair is any pair [m, P] arising from a simple stratum in some split simple algebra; such a stratum is called a realization of [m, P]. Actually in [BH] simple pairs are defined in an intrinsic way that we do not need for our purposes. Moreover, our language differs a little bit but the reader will check that it is rigorously equivalent.

If [𝔘, n, q, β] is a pure stratum such that [[q/e_{β}(𝔘)], P] is a simple pair, P being the minimal polynomial of β over F, then it is automatically simple ([BH]p.115), i.e. it is a realization of [[q/e_{β}(𝔘)], P].

The following result will be crucial for our proof.

( 4.2.1 ) Theorem ([BH](1.10)) Let [m_{i}, P_{i}], i = 1, 2 be two simple pairs. Assume that they have some respective realizations [𝔘, n_{i}, q_{i}, β_{i}], i = 1, 2, which intertwine in U^{×}. Then any other pair of realizations with a common order intertwine as well.

Let us fix a split algebra U, a hereditary order 𝔘 ⊂ U as well as a finite, Galois, tamely ramified extension K/F ⊂ U, such that K^{×} normalizes 𝔘. Let W be the centralizer of K in U and 𝔚 = 𝔘∩W. Then 𝔚 is a hereditary 𝔬_{K}-order in the K-algebra W. Let [𝔚, n, q, β_{i}], i = 1, 2, be simple strata in W. This automatically implies that the strata [𝔘, n, q, β_{i}] are at least pure (cf. [BH] §2). Let N_{𝔘}(K) be the set of elements in U^{×} which normalize 𝔘 and K.

( 4.2.2 ) Proposition ([BH](6.7)) Assume that the strata [𝔘, n, q, β_{i}], i = 1, 2, are simple and intertwine in U^{×}. Then there exists x ∈ N_{𝔘}(K) such that [𝔚, n, q, x^{−1}β_{1}x] is equivalent to [𝔚, n, q, β_{2}].

Finally we will need:

( 4.2.3 ) Proposition ([BH](4.3)) With the notation as before, assume we are given a pure stratum [𝔚, n, q, β] (so that [𝔘, n, q, β] is automatically pure). Then it is equivalent to a simple stratum [𝔚, n, q, γ] such that [𝔘, n, q, γ] is simple as well.

4.3 Proof of the theorem

We start by making some technical reductions. Since the strata [𝔄̃, n, q, γ_{i}] are simple and intertwine, they intertwine in K̃: there exists k ∈ K̃ such that [𝔄̃, n, q, kγ_{1}k^{−1}] is equivalent to [𝔄̃, n, q, γ_{2}]. We can then apply [BK2] theorem (2.4.1) to get:

e(F [γ_{1}]/F) = e(F [γ_{2}]/F), f(F [γ_{1}]/F) = f(F [γ_{2}]/F).

Let F [γ_{i}]^{unr} be the unramified part of F [γ_{i}]/F, i = 1, 2. By assumption the extensions F [γ_{i}]^{unr}/F, i = 1, 2, are (𝔄, τ̄)-pure. Moreover they are conjugate under A^{×} since they have the same degree over F. We can therefore write F [γ_{i}]^{unr} = F [ζ_{i}], i = 1, 2, with ζ_{1} = gζ_{2}g^{−1} for some g ∈ G. By applying lemma (3.2), we have that the ζ_{i}, whence the F [γ_{i}]^{unr}, are conjugate under K. Therefore:

( 4.3.1 ) By conjugating γ_{1} by an element of K, we may and will assume that F [γ_{1}]^{unr} = F [γ_{2}]^{unr} =: K.

Let B̃ be the centralizer of K in Ã, 𝔅̃ be the hereditary 𝔬_{K}-order B̃ ∩ 𝔄̃ and 𝔔̃ = rad(𝔅̃). The algebras K[γ_{i}] = F [γ_{i}], i = 1, 2, are fields and K[γ_{i}]^{×} ⊂ K̃ ∩ B̃ = K𝔅̃ (see [BK2]§ 1.2). Therefore the strata [𝔅̃, n, q, γ_{i}] are pure. Moreover by [BH] theorem (2.4), these strata are simple because the [𝔄̃, n, q, γ_{i}] are so.

Now since the [𝔄̃, n, q, γ_{i}] intertwine, we can apply (4.2.2) to get an element x ∈ N𝔄̃(K) such that

x^{−1}γ_{1}x − γ_{2} ∈ 𝔔̃^{−q},

where N𝔄̃(K) is the normalizer of K in K̃.

By conjugation, x induces on K an element σ of Gal(K/F): σ(u) = xux^{−1}, u ∈ K. Pick elements u_{i} ∈ K, i = 1, 2, such that K = F [u_{i}] and such that σ is given by σ(u_{1}) = u_{2}. Since the u_{i} are conjugate under A^{×} and are (𝔄, τ̄)-pure, there exists x_{o} ∈ K such that u_{2} = x_{o}u_{1}x_{o}^{−1}. It follows that x^{−1}x_{o} is an element of K̃ which centralizes K, i.e this is an element of K𝔅̃. Moreover (x^{−1}x_{o})(x_{o}^{−1}γ_{1}x_{o})(x^{−1}x_{o})^{−1} − γ_{2} ∈ 𝔔̃^{−q}; we still have F [x_{o}^{−1}γ_{1}x_{o}]^{unr} = K and the stratum [𝔄, n, q, x_{o}^{−1}γ_{1}x_{o}] remains simple because it is conjugate under K to a simple stratum. Therefore:

( 4.3.2 ) By conjugating γ_{1} by an element of K, we may and will assume that the strata [𝔅̃, n, q, γ_{i}], i = 1, 2, intertwine in B̃^{×}.

Recall that e_{γ_{i}}(𝔄̃) = e(𝔄̃ | 𝔬_{F})/e(F [γ_{i}]/F) = rd/e(F [γ_{i}]/F); this is an integer independent of i. If q_{o} = e_{γ_{i}}(𝔄̃)[q/e_{γ_{i}}(𝔄̃)], then [𝔄̃, n, q_{o}, γ_{i}] is simple because this is a realization of the simple pair of [𝔄̃, n, q, γ_{i}]. Moreover by (4.2.1), the simple strata [𝔄̃, n, q_{o}, γ_{i}], i = 1, 2, intertwine. If we prove that the [𝔄, n, q_{o}, γ_{i}], i = 1, 2, intertwine in K, we will be done. Indeed if kγ_{1}k^{−1} − γ_{2} ∈ 𝔓^{−q_{o}} for some k in K, then k intertwines the [𝔄, n, q, γ_{i}], i = 1, 2, for 𝔓^{−q_{o}} ⊂ 𝔓^{−q} (q_{o} ≤ q). Therefore:

( 4.3.3 ) We may and will assume that q is a multiple of e_{γ_{i}}(𝔄̃).

This restriction will be useful for reasons which will become clear.

Let B be the centralizer of K in A and 𝔅 = B ∩ 𝔄. According to [Br1] theorem (1.3), 𝔅 is a hereditary 𝔬_{K}-order in B. We certainly have K ∩ B ⊂ K_{𝔅} (the normalizer of 𝔅 in B^{×}) and the elements γ_{i}, i = 1, 2, are pure with respect to 𝔅. Put 𝔔 = rad(𝔅).

( 4.3.4 ) Lemma Write K = ⟨h⟩𝔄^{×} for some h ∈ K, so that h𝔄 = 𝔓^{t}, where ν_{𝔄}(K) = tℤ. Let ρ ∈ ℕ be the smallest positive integer such that there exists ϵ ∈ 𝔄^{×} with h^{ρ}ϵ ∈ K_{𝔅}. Then:

(i) 𝔓^{tρk} ∩ B = 𝔔^{tρkr′d′/rd}, k ∈ ℤ, where r′ is the period of 𝔅 and d′ is the index over E of a K-division algebra Δ such that B ≃ M_{m′}(Δ).

(ii) We have tρ | e_{γ_{i}}(𝔄̃) = rd/e(F [γ_{i}]/F). In particular, we have tρ | q.

Proof. (i) Write z = h^{ρ}ϵ, ϵ ∈ 𝔄^{×}, z ∈ K_{𝔅}. we have 𝔓^{tρk} ∩ B = (z^{k}𝔄) ∩ B = z^{k}(𝔄 ∩ B) = 𝔔^{ν_{𝔅}(z^{k})} = 𝔔^{kν_{𝔅}(z)}. For all u ∈ K ∩ B, we have ν_{𝔅}(u) = Cν_{𝔄}(u), where C does not depend on u. We have π_{F} 𝔄 = 𝔓^{rd}, for any uniformizer π_{F} of F, and, since K/F is unramified, we have π_{F} 𝔅 = 𝔔^{r′d′}. Therefore C = r′d′/rd and (i) follows from the equality ν_{𝔄}(z) = ρt.

(ii) Let π_{i} be a uniformizer of F [γ_{i}]; then there exists ϵ ∈ 𝔬_{F [γ_{i}]}^{×} ⊂ 𝔅^{×} such that π_{i}^{e(F [γ_{i}]/F)}ϵ = π_{F} is a uniformizer of F. Then π_{i} ∈ K ∩ B and ν_{𝔄}(π_{i}) = ν_{𝔄}(π_{F})/e(F [γ_{i}]/F) = rd/e(F [γ_{i}]/F) = e_{γ_{i}}(𝔄̃). Since all elements of K ∩ B have as 𝔅-valuation a multiple of ν_{𝔅}(z), we have ρt(r′d′/rd) | rd/e(F [γ_{i}]/F)(r′d′/rd), i.e. ρt | rd/e_{γ_{i}}(𝔄̃), as required.

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Put n′ = n(r′d′/rd) and q′ = qr′d′/rd. Then the strata [𝔅, n′, q′, γ_{i}] are pure and 𝔔^{−q′} ⊂ 𝔓^{−q} by the previous lemma. We are actually going to prove that these strata intertwine in 𝔅^{×}. This will give the existence of k ∈ 𝔅^{×} ⊂ 𝔄^{×} such that kγ_{1}k^{−1} − γ_{2} ∈ 𝔔^{−q′}, whence kγ_{1}k^{−1} − γ_{2} ∈ 𝔓^{−q} and we will be done.

Let L/K be an unramified extension which splits the K-algebra B. Put B̄ = B ⊗_{K} L. For each 𝔬_{K}-submodule (resp. 𝔬_{K}-order) X of B, we see X ⊗_{𝔬_{K}} 𝔬_{L} as an 𝔬_{L}-submodule (resp. 𝔬_{L}-order) of B̄ in a canonical way. In particular 𝔅̄ = 𝔅 ⊗_{𝔬_{K}} 𝔬_{L} is a hereditary 𝔬_{L}-order in B̄. Let 𝔔̄ = rad(𝔅̄). Then for all k ∈ ℤ, we have 𝔔̄^{k} = 𝔔^{k} ⊗_{𝔬_{K}} 𝔬_{L}.

( 4.3.5 ) Lemma The elements γ_{i} ⊗ 1 ∈ B̄ are pure with respect to 𝔅̄. Moreover ν𝔅̄(γ_{i} ⊗ 1) = n′, for i = 1, 2.

Proof. Fix i. Put E := L[γ_{i} ⊗ 1] = K[γ_{i}] ⊗_{K} L. By construction, the extension K[γ_{i}]/K is totally ramified and is therefore linearly disjoint from L. It follows that E is a field (and that the minimal polynomial of γ_{i} ⊗ 1 over L is that of γ_{i} over K). Moreover, since L/F is unramified, we have 𝔬_{E} = 𝔬_{K[γ_{i}]} ⊗_{𝔬_{K}} 𝔬_{L}. Since γ_{i} is 𝔅-pure, we have 𝔬_{K[γ_{i}]} ⊂ 𝔅 and therefore 𝔬_{E} ⊂ 𝔅̄, as well as 𝔬_{E}^{×} ⊂ K𝔅̄. Let π be a uniformizer of K[γ_{i}]; then π ⊗ 1 is a uniformizer of E. Since π normalizes 𝔅, we have that π ⊗ 1 normalizes 𝔅̄. We have obtained that E^{×} = ⟨π ⊗ 1⟩𝔬_{E}^{×} is contained in K𝔅̄; as required.

We have (γ_{i} ⊗ 1)𝔅̄ = (γ_{i}𝔅) ⊗_{𝔬_{K}} 𝔬_{L} = 𝔔^{n′} ⊗_{𝔬_{K}} 𝔬_{L} = 𝔔̄^{n′}, whence ν𝔅̄(γ_{i} ⊗ 1) = n′.

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Similarly we can consider the L-algebra C̃ = B̃ ⊗_{K} L and the hereditary 𝔬_{L}-order ℭ̃ = 𝔅̃ ⊗_{𝔬_{K}} 𝔬_{L}. By arguments similar to those of the previous lemma, we have that the strata [ℭ̃, n, q, γ_{i} ⊗ 1], i = 1, 2, are pure.

( 4.3.6 ) Lemma The strata [ℭ̃, n, q, γ_{i} ⊗ 1] are simple.

Proof. Let ℭ̃_{1} be the centralizer of γ_{1} ⊗ 1 in ℭ̃ and 𝔅̃_{1} the centralizer of γ_{1} in 𝔅̃. We observe that ℭ̃_{1} is stable under the action of Gal(L/K) and that the set of fixed points is 𝔅̃_{1} ⊗ 1. It follows (see e.g. [BH] (11.15)) that ℭ̃_{1} = 𝔅̃_{1} ⊗_{𝔬_{K}} 𝔬_{L}. Put 𝔑_{k}(γ_{1} ⊗1, ℭ̃) = {x ∈ ℭ̃; x(γ_{1} ⊗1)−(γ_{1} ⊗1)x ∈ rad(ℭ̃)^{k}} and 𝔑_{k}(γ_{1}, 𝔅̃) = {x ∈ 𝔅̃; xγ_{1} − γ_{1}x ∈ rad(𝔅̃)^{k}}, for k ∈ ℤ. Similarly we check that 𝔑_{k}(γ_{1} ⊗ 1, ℭ̃) = 𝔑_{k}(γ_{1}, 𝔅̃) ⊗_{𝔬_{K}} 𝔬_{L}.

Put k_{0}(γ_{1} ⊗ 1, ℭ̃) = max{k ∈ ℤ; 𝔑_{k}(γ_{1} ⊗ 1, ℭ̃) ⊄ ℭ̃_{1} + rad(ℭ̃)}, and k_{0}(γ_{1}, 𝔅̃) = max{k ∈ ℤ; 𝔑_{k}(γ_{1}, 𝔅̃) ⊄ 𝔅̃_{1} + rad(𝔅̃)}. Then, by definition ([BK2](1.5.5)), the fact that [𝔅̃, n, q, γ_{i}] is simple means that −q > k_{0}(γ_{1}, 𝔅̃). Let us prove that k_{0}(γ_{1}⊗1, ℭ̃) = k_{0}(γ_{1}, 𝔅̃); this will imply that [ℭ̃, n, q, γ_{1}⊗ 1] is simple as well. For all k ∈ ℤ, we have k ≥ k_{0}(γ_{1} ⊗ 1, ℭ̃) if and only if 𝔑_{k}(γ_{1}, 𝔅̃) ⊗_{𝔬_{K}} 𝔬_{L} ⊂ (𝔅̃_{1} + rad(𝔅̃)) ⊗_{𝔬_{K}} 𝔬_{L}, which is equivalent to 𝔑_{k}(γ_{1}, 𝔅̃) ⊂ 𝔅̃_{1} + rad(𝔅̃), i.e. k ≥ k_{0}(γ_{1}, 𝔅̃). Hence k_{0}(γ_{1} ⊗ 1, ℭ̃) = k_{0}(γ_{1}, 𝔅̃), as required.

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Let us recall that the simple strata [𝔅̃, n, q, γ_{i}], i = 1, 2, intertwine in K𝔅̃. It follows that the strata [ℭ̃, n, q, γ_{i} ⊗ 1], i = 1, 2, intertwine as well. Indeed, if kγ_{1}k^{−1} − γ_{2} ∈ rad(𝔅̃)^{−q}, for some k ∈ K𝔅̃, then (k ⊗ 1)(γ_{1} ⊗ 1)(k ⊗ 1)^{−1} − (γ_{2} ⊗ 1) ∈ rad(𝔅̃)^{−q} ⊗ 1 ⊂ rad(ℭ̃)^{−q} and k ⊗ 1 ∈ Kℭ̃.

Let [t, P_{i}] be the simple pair attached to [ℭ̃, n, q, γ_{i} ⊗ 1], i = 1, 2. Recall that P_{i} is the minimal polynomial of γ_{i} ⊗ 1 over L (that is, the minimal polynomial of γ_{i} over K) and t = [q/e_{γ_{i}⊗1}(ℭ̃)]. We check that e(ℭ̃ | 𝔬_{L}) = e(𝔅̃ | 𝔬_{K}). Moreover e(𝔅̃ | 𝔬_{K}) = e(𝔄̃ | 𝔬_{F}), because K/F is unramified. Since e(𝔄̃ | 𝔬_{F}) = rd and e(L[γ_{i} ⊗ 1]/L) = e(F [γ_{i}]/F), we get t = [qe(F [γ_{i}]/F)/rd] = qe(F [γ_{i}]/F)/rd (by assumption rd/e(F [γ_{i}]/F) divides q).

( 4.3.7 ) Lemma For i = 1, 2, the pure stratum [𝔅̄, n′, q′, γ_{i} ⊗ 1] is a realization of the simple pair [t, P_{i}].

Proof. The minimal polynomial of γ_{i} ⊗ 1 viewed as an element of B̄ = B ⊗_{K} L, or as an element of C̃ = B̃ ⊗_{K} L, is the minimal polynomial of γ_{i} over K (viewed as an element of B ⊂ B̃). Hence we just have to check that [q/(νe_{γ_{i}⊗1}(𝔅̄))] = [q/e_{γ_{i}⊗1}(ℭ̃)], for i = 1, 2, where ν = rd/r′d′. But this follows from e(𝔅̄ | 𝔬_{L}) = r′d′ and e(ℭ̃ | 𝔬_{L}) = rd.

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( 4.3.8 ) Corollary The strata [𝔅̄, n′, q′, γ_{i} ⊗ 1], for i = 1, 2, are simple and intertwine in 𝔅̄^{×}.

Proof. Indeed, the strata are simple because they are realizations of simple pairs. They intertwine since these simple pairs intertwine when realized as [ℭ̃, n, q, γ_{i} ⊗ 1], for i = 1, 2, in C̃. Using (4.1.1), we have, moreover, that they intertwine in 𝔅̄^{×}.

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Now the fact that the strata [𝔅, n, q, γ_{i}], for i = 1, 2, intertwine in 𝔅^{×} follows from the following descent lemma whose proof is postponed to §6.

( 4.3.9 ) Theorem Let E be a non-archimedean local field, H a central simple E-algebra and ℌ a hereditary 𝔬_{E}-order in H. Let Ē/E be a finite unramified extension and put H̄ = H ⊗_{E} Ē, ℌ̄ = ℌ ⊗_{𝔬_{E}} 𝔬Ē. Let 𝔊 (resp. 𝔊̄) be the radical of ℌ (resp. ℌ̄). Finally let x, y ∈ H, u ∈ ℤ. Assume that g(x ⊗ 1)g^{−1} − (y ⊗ 1) ∈ 𝔊̄^{u}, for some g ∈ KH̄ (resp. g ∈ ℌ̄^{×}). Then there exists h ∈ K_{ℌ} (resp. h ∈ ℌ^{×}) such that hxh^{−1} − y ∈ 𝔊^{u}.

We apply the theorem to H = B, ℌ = 𝔅, x = γ_{1}, y = γ_{2} and u = −q′. We get that the strata [𝔅, n′, q′, γ_{i}], i = 1, 2, intertwine in 𝔅^{×} as required. This achieves the proof of proposition (4.1.3) and theorem (4.1.2).

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5 Existence of a simple stratum equivalent to a pure one

The aim of this section is to prove the following generalization of [BK2] theorem (2.4.1)(i).

( 5.1 ) Theorem (i) Let [𝔄, n, q, β] be a pure stratum and assume that β is (𝔄, τ̄)-pure for some type of embedding τ̄. Then there exists a simple stratum [𝔄, n, q, γ] equivalent to [𝔄, n, q, β]; moreover, for any choice of this simple stratum, γ is (𝔄, τ̄)-pure.

(ii) If a simple stratum [𝔄, n, q, γ] is equivalent to a pure stratum [𝔄, n, q, β] then we have e(F [γ]/F) | e(F [β]/F) and f(F [γ]/F) | f(F [β]/F). Conversely, if among the pure strata equivalent to a given pure stratum [𝔄, n, q, β], [𝔄, n, q, γ] has the property that [F [γ] : F] is minimal, then it is necessarily simple.

Let us first explain how (ii) follows from (i). With the notation of (ii), we have that [𝔄̃, n, q, γ] is simple and equivalent to [𝔄̃, n, q, β]. According to the split case, the invariants of F [γ]/F divide the invariants of F [β]/F and this gives the first assertion. Now let [𝔄, n, q, γ] be a pure stratum equivalent to [𝔄, n, q, β] with [F [γ] : F] minimal. For a contradiction assume that it is not simple. Let [𝔄, n, q, δ] be a simple stratum equivalent to [𝔄, n, q, γ]. Since [𝔄̃, n, q, δ] is simple and [𝔄̃, n, q, γ] is not, we must have ([BK2](2.4.1)) [F [δ] : F] < [F [γ] : F], which gives the contradiction.

We need the following result.

( 5.2 ) Lemma Let [𝔄, n, q, γ_{i}], i = 1, 2, be two equivalent simple strata. Then γ_{1} and γ_{2} have the same type of embedding.

Proof. The strata [𝔄̃, n, q, γ_{i}], i = 1, 2, are simple and equivalent. According to the split case ([BK2] lemma (2.4.12)(i)), the rings of integers 𝔬_{F [γ_{i}]}, i = 1, 2 have the same image in 𝔄̃/𝔓̃, whence in 𝔄/𝔓 (seeing 𝔄/𝔓 as being canonically embedded in 𝔄̃/𝔓̃, these images actually lie in 𝔄/𝔓). Our result follows then from the definition of the type of embedding and lemma (2.3.6).

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As a consequence, in theorem (5.1)(i) it suffices to prove the existence of a simple stratum equivalent to [𝔄, n, q, β] and with the type of embedding τ̄.

Let us then show that we can reduce to the case where e_{β}(𝔄̃) | q. Let q_{o} = e_{β}(𝔄̃)[q/e_{β}(𝔄̃)] and assume that we have constructed a simple stratum [𝔄, n, q_{o}, γ] equivalent to [𝔄, n, q_{o}, β]. We have γ − β ∈ 𝔓^{−q_{o}} ⊂ 𝔓^{−q}, since q_{o} ≤ q. Moreover, [𝔄, n, q, γ] is simple. Indeed, we know that this is the case if and only if the invariant k_{o}(γ, 𝔄̃) is strictly less than −q ([BK2] Def. (1.5.5)). We know ([BK2](1.4.13)) that k_{o}(γ, 𝔄̃) is divisible by e_{γ}(𝔄̃) = e(𝔄̃ | 𝔬_{F})/e(F [γ]/F), whence by e_{β}(𝔄̃) = e(𝔄̃ | 𝔬_{F})/e(F [β]/F) , since e(F [γ]/F) divides e(F [β]/F). Since [𝔄, n, q_{0}, γ] is simple, we have −q_{o} > k_{o}(γ, 𝔄̃), i.e. −q_{o} ≥ k_{o}(γ, 𝔄̃) + e_{β}(𝔄̃) by the previous remark. On the other hand we have −q > −q_{o} − e_{β}(𝔄̃) by definition of q_{0}, whence −q > k_{o}(γ, 𝔄̃). This means that [𝔄̃, n, q, γ] is simple as required. Therefore

We may and will assume that e_{β}(𝔄̃) divides q.

Let K/F be the unramified part of F [β]/F, B the centralizer of K in A, 𝔅 = B ∩ 𝔄 and 𝔔 = rad(𝔅). With the arguments and notation of (4.3.4), we have that β𝔅 = 𝔔^{−n′}, where n′ = nr′d′/rd, that q′ = qr′d′/rd is an integer and that 𝔔^{−q′} ⊂ 𝔓^{−q}. Since K[β] = F [β] is a field satisfying K[β]^{×} ⊂ K ∩ B ⊂ K_{𝔅}, the stratum [𝔅, n′, q′, β] is pure.

Let L/K be an unramified extension which splits B. Set B̄ = B ⊗_{K} L and 𝔅̄ = 𝔅 ⊗_{𝔬_{K}} 𝔬_{L} ⊂ B̄. Identify B̄ with End_{L}W for some L-space W and put B̃ = End_{F} W. Let 𝔅̃ be the hereditary 𝔬_{F} - order of B̃ attached to the same lattice chain as 𝔅̄. Note that K[β]/K is totally ramified of degree e(K[β]/K) = e(F [β]/F). As in (4.3.7), the stratum [𝔅̄, n′, q′, β ⊗ 1] is pure. By applying (4.2.3), it is equivalent to a simple stratum [𝔅̄, n′, q′, γ] such that [𝔅̃, n′, q′, γ] is simple as well. The integer e(L[γ]/L) divides e(L[β ⊗ 1]/L) = e(K[β]/K) and f(L[γ]/L) = 1.

( 5.3 ) Lemma The minimal polynomial μ_{L} of γ over L and its minimal polynomial μ_{K} over K coincide. Moreover, e(K[γ]/K) | e(K[β]/K) and f(K[γ]/K) = 1.

Proof. Let 𝔅̂ be the hereditary 𝔬_{K}-order of End_{K}W attached to the same lattice chain as 𝔅̄. We can form the equivalent pure strata [𝔅̂, n′, q′, β ⊗ 1] and [𝔅̂, n′, q′, γ]. By [BH] theorem (2.4), since [𝔅̃, n′, q′, γ] is simple so is [𝔅̂, n′, q′, γ]. This proves the second assertion of the lemma.

We can decompose L[γ]/L in two towers. The first one is L[γ]/L/K, with L[γ]/L totally ramified and L/K unramified. The second one is L[γ]/K[γ]/K, with K[γ]/K totally ramified and L[γ]/K[γ] unramified, since this extension is generated over K[γ] by the roots of unity prime to p = char(k_{F}) in L^{×}. This automatically implies that e(L[γ]/K) = [L[γ] : L] = [K[γ] : K]. Therefore μ_{L} and μ_{K} have the same degree and, since μ_{L} | μ_{K}, we must have μ_{L} = μ_{K}.

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Thanks to the previous lemma, we can apply Corollary (3.16)(ii), to get a 𝔅-pure element γ_{o} in B with minimal polynomial μ_{K} = μ_{L}. The two elements of B̄, γ_{o} ⊗ 1 and γ, have the same minimal polynomial over L (namely μ_{K} = μ_{L}) and are both 𝔅̄-pure. By [BH] lemma (1.6), they are conjugate under 𝔅̄^{×}: there exists k ∈ 𝔅̄^{×} such that k(γ_{o} ⊗ 1)k^{−1} − β ⊗ 1 ∈ rad(𝔅̄)^{−q′}. We can now apply theorem (4.3.9) to get an element l of 𝔅^{×} such that lγ_{o}l^{−1} − β ∈ 𝔔^{−q′}. Write γ_{1} = lγ_{o}l^{−1}.

( 5.4 ) Lemma The stratum [𝔄, n, q, γ_{1}] is simple and equivalent to [𝔄, n, q, β].

Proof. We need to prove that the stratum [𝔄, n, q, γ_{1}] is pure. We first note that the stratum [𝔅, n′, q′, γ_{1}] is simple. Indeed the stratum [𝔅̂, n′, q′, γ_{1} ⊗ 1] is simple because [𝔅̂, n′, q′, γ] is simple and γ, γ_{1} ⊗ 1 have the same minimal polynomial over K. Write B = End_{D′}V′, for some K-division algebra D′ and some right D′-module V′, and 𝔅 = End_{𝔬_{D′}}^{0} (Y_{k}) for some 𝔬_{D′}-lattice chain (Y_{k})_{k∈ℤ} in V′. Consider the stratum [ℭ, n′, q′, γ_{1}], where ℭ = End_{𝔬_{K}}^{0} (Y_{k}) ⊂ End_{K}V′ (ℭ is attached to 𝔅 in the same way that 𝔄̃ is attached to 𝔄). We easily check that this stratum is a realization of the simple pair of [𝔅̂, n′, q′, γ]. Hence [𝔅, n′, q′, γ_{1}] is simple by definition.

Here we need a generalization of a lemma of Zink that we are going to state for the algebra A.

( 5.5 ) Lemma Let [𝔄, n, q, δ] be a simple stratum equivalent to a pure one [𝔄, n, q, α]. Then we have F [δ]^{×}(1 + 𝔓) ⊂ F [α]^{×}(1 + 𝔓).

Proof. By [Z2](1.6.1), this lemma is true when A is split. Therefore the equivalence [𝔄̃, n, q, δ] ∼ [𝔄̃, n, q, α] gives F [δ]^{×}(1 + 𝔓̃) ⊂ F [α]^{×}(1 + 𝔓̃). We conclude by taking the intersection with A.

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Since F [γ_{1}] ⊂ K[γ_{1}], F [γ_{1}] is necessarily a field. By applying the previous lemma to the equivalence [𝔅, n′, q′, γ_{1}] ∼ [𝔅, n′, q′, β], we have that F [γ_{1}]^{×}(1 + 𝔔) ⊂ K[γ_{1}]^{×}(1 + 𝔔) ⊂ K[β]^{×}(1 + 𝔔). Moreover K[β]^{×} = F [β]^{×} is contained in K, and so is 1 + 𝔔, since it is contained in 𝔅^{×}. Therefore F [γ_{1}]^{×}(1 + 𝔔) ⊂ K, i.e. F [γ_{1}]^{×} ⊂ K. Since both β and γ_{1} have 𝔅-valuation −n′, we have γ_{1}β^{−1} ∈ 𝔅^{×} ⊂ 𝔄^{×}. Hence ν_{𝔄}(γ_{1}) = ν_{𝔄}(β) = −n′. We have proved that [𝔄, n, q, γ_{1}] is pure. Since 𝔔^{−q′} ⊂ 𝔓^{−q}, this pure stratum is, moreover, equivalent to [𝔄, n, q, β].

To prove that [𝔄, n, q, γ_{1}] is simple, let us check that [𝔄̃, n, q, γ_{1}] is a realization of the simple pair attached to [𝔅̃, n′, q′, γ]. First, the minimal polynomial μ_{F} of γ over F is uniquely determined by its minimal polynomial μ_{K} over K. Indeed, we have μ_{K} | μ_{F} and μ_{F} is the unique irreducible polynomial of F [X] with leading coefficient 1 which is a multiple of μ_{K}: μ_{F} is the product of the (distinct) elements of the set {μ_{K}^{σ} | σ ∈ Gal(K/F)}. A similar remark for γ_{1} proves that γ and γ_{1} have the same minimal polynomial over F. It remains to prove that [q′/e_{γ}(𝔅̃)] = [q/e_{γ_{1}}(𝔄̃)], but this follows from q′ = r′d′/rd. This completes the proof of lemma (5.4).

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Finally we must check that γ_{1} is (𝔄, τ̄)-pure. For this it is enough to show that the unramified part F [γ_{1}]^{unr} of F [γ_{1}] is contained in the unramified part K of F [β]. By construction K[γ_{1}]/K is totally ramified and K is therefore the maximal unramified part of K[γ_{1}]/F. Now, since F ⊂ F [γ_{1}] ⊂ K[γ_{1}], we must have F [γ_{1}]^{unr} ⊂ K, as required. This complete the proof of (5.1).

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6 A descent lemma for the intertwining

We are going to prove theorem (4.3.9). Our proof uses classical ideas from Galois cohomology. This type of argument has already been used by Henniart in his thesis.

We give up the notation used to state theorem (4.3.9). In particular, 𝔄 is a given hereditary 𝔬_{F} -order in our algebra A, 𝔓 denotes its radical. Let L/F be any finite unramified extension and put Ā = A ⊗_{F} L, 𝔄̄ = 𝔄 ⊗_{𝔬_{F}} 𝔬_{L}. We have 𝔓̄^{k} = 𝔓^{k} ⊗_{𝔬_{F}} 𝔬_{L}, k ∈ ℤ, where 𝔓̄ = rad(𝔄̄). As usual, K denotes the normalizer of 𝔄 in A^{×} and K̄ denotes the normalizer of 𝔄̄ in Ā^{×}.

Let x, y ∈ A, m ∈ ℤ and k ∈ K̄ be such that k(x ⊗ 1)k^{−1} − (y ⊗ 1) ∈ 𝔓̄^{m}. We have to prove that there exists l ∈ K such that lxl^{−1} − y ∈ 𝔓^{m}. Moreover we have to check that one can choose l in 𝔄^{×} whenever k is in 𝔄̄^{×}.

The powers 𝔓̄^{k}, k ∈ ℤ are stable under the action of Γ = Gal(L/F), so is the group K̄, for if a ∈ Ā^{×} satisfies a𝔄̄a^{−1} = 𝔄̄, we have a^{s}𝔄̄^{s}a^{−s} = 𝔄̄^{s}, i.e. a^{s}𝔄̄a^{−s} = 𝔄̄. In particular, for all s ∈ Γ, k^{s}(x ⊗ 1)(k^{s})^{−1} − (y ⊗ 1) ∈ 𝔓̄^{m}, therefore k^{s}(x ⊗ 1)(k^{s})^{−1} ≡ k(x ⊗ 1)k^{−1} mod 𝔓̄^{m} and it follows that (k^{−1}k^{s})(x ⊗ 1)(k^{−1}k^{s})^{−1} − (x ⊗ 1) ∈ 𝔓̄^{m}, since K normalizes 𝔓̄^{m}. For all s, k and k^{s} have the same 𝔄̄-valuation and k^{−1}k^{s} is therefore in 𝔄̄^{×}.

Put I = I^{0} = {u ∈ 𝔄̄^{×} | u(x ⊗ 1)u^{−1} − (x ⊗ 1) ∈ 𝔓̄^{m}}. This set is a group: this is the normalizer of the coset (x ⊗ 1) + 𝔓̄^{m} under the action by conjugation of 𝔄̄^{×} on Ā/𝔓̄^{m}. Moreover, we easily check that this is a Γ-set, whose Γ-action is compatible with the group structure. The map s ↦ k^{−1}k^{s} is then a 1-cocycle of Γ in I.

( 6.1 ) Lemma We have H^{1}(Γ, I) = 1 (non abelian cohomology).

Proof. Put I^{n} = I ∩ (1 + 𝔓̄^{n}), n ≥ 1. Then, for all n ≥ 1, I^{n} is a Γ-stable normal subgroup of I. By a classical filtration argument, we are reduced to proving that for all n ≥ 0, H^{1}(Γ, I^{n}/I^{n+1}) = 1. For all n ≥ 0, put J̄^{n} = {h ∈ 𝔓̄^{n} / h(x ⊗ 1) − (x ⊗ 1)h ∈ 𝔓̄^{m}}. The J̄^{n} are 𝔬_{L}-lattices in Ā. The identity (hg)(x ⊗ 1) − (x ⊗ 1)(hg) = h(g(x ⊗ 1) − (x ⊗ 1)g) + (h(x ⊗ 1) − (x ⊗ 1)h)g implies that, for n, n′ ≥ 0, we have J̄^{n}J̄^{n′} ⊂ J̄^{n+n′}. It follows that J̄^{0} is a ring for which J̄^{1} is a two sided ideal; moreover I^{0} is the multiplicative group (J̄^{0})^{×}. We have a group homomorphism I^{0} ⟶ (J̄^{0}/J̄^{1})^{×}, given by (x ⊗ 1) ↦ (x ⊗ 1) + J̄^{1}. It is onto, for an element of J̄^{0} is invertible if and only if it is invertible mod J̄^{1}. Its kernel is I^{1}, whence we have an isomorphism: I^{0}/I^{1} ≃ (J̄^{0}/J̄^{1})^{×}.

The J̄^{n} are stable under Γ and we have J̄^{n} = J^{n}⊗_{𝔬_{F}} 𝔬_{L}, where J^{n} = J̄^{n Γ} = {h ∈ 𝔓^{n} / hx−xh ∈ 𝔓^{m}}. The J^{n} enjoy properties similar to those of the J̄^{n}. We have 𝔭_{L}J̄^{n} ⊂ J̄^{n+1} and the quotients J̄^{n}/J̄^{n+1} are k_{L}-vector spaces. Similarly the J^{n}/J^{n+1} are k_{F} -vector spaces. Since 𝔬_{L} is a flat 𝔬_{F} -module, we have isomorphisms of Γ-modules: J̄^{n}/J̄^{n+1} ≃ J^{n}/J^{n+1} ⊗_{𝔬_{F}} 𝔬_{L} ≃ J^{n}/J^{n+1} ⊗_{k_{F}} k_{L}.

When n = 0, we get (J̄^{0}/J̄^{1})^{×} ≃ [(J^{0}/J^{1}) ⊗_{k_{F}} k_{L}]^{×}, where J^{0}/J^{1} is a finitely generated k_{F} - algebra. The triviality of H^{1}(Γ, I/I^{1}) follows then from [S] chap. X, §1, ex. 2: if E/K is a finite Galois extension and H a finitely generated K-algebra, then H^{1}(Gal(E/K), [H ⊗ E]^{×}) = 1.

If n > 1, the map I^{n}/I^{n+1} ⟶ J̄^{n}/J̄^{n+1}, induced by 1+u ↦ u, u ∈ J^{n}, is a group homomorphism since J̄^{n}J̄^{n} ⊂ J̄^{n+1}. We easily check that this is an isomorphism. Moreover, we have I^{n}/I^{n+1} ≃ (J^{n}/J^{n+1}) ⊗_{k_{F}} k_{L} and the triviality of H^{1}(Γ, I^{n}/I^{n+1}) follows from the triviality of H^{1}(Γ, k_{L}).

Let us complete the proof of the theorem. By lemma (6.1), the cocycle s ↦ k^{−1}k^{s} is a coboundary: there exists k_{0} ∈ I such that k^{−1}k^{s} = k_{0}^{−1}k_{0}^{s} for all s ∈ Γ. This means that kk_{0}^{−1} ∈ K̄^{Γ} = K. Consequently we have (kk_{0}^{−1})(x ⊗ 1)(kk_{0}^{−1})^{−1} = k(k_{0}^{−1}(x ⊗ 1)k_{0})k^{−1} ≡ k(x ⊗ 1)k^{−1} (mod 𝔓̄^{m}), since k_{0} ∈ I. Therefore (kk_{0}^{−1})(x ⊗ 1)(kk_{0}^{−1})^{−1} ≡ y ⊗ 1 (mod 𝔓̄^{m}). By putting l ⊗ 1 = kk_{0}^{−1}, we have that lxl^{−1} − y ∈ 𝔓̄^{m} ∩ A = 𝔓^{m}, as required. Of course, if k ∈ 𝔄̄^{×} then l ∈ 𝔄^{×}, since k_{0} lies in I ⊂ 𝔄̄^{×}.

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7 Comparison of pure elements for different algebras

Let 𝔖 be a central F -division algebra of dimension (Nd)^{2}, where N is the reduced degree and d the index of our central simple F -algebra A. Each element x ∈ 𝔖^{×} generates a field extension and is pure for the unique hereditary 𝔬_{F} -order 𝔬_{𝔖} in 𝔖, which has normalizer 𝔖^{×}. Moreover in terms of (3.1) there is only one type of embedding and all non-zero elements are pure for this type. We have a canonical bijection

( 7.1 ) Ad_{𝔖^{×}}\𝔖 ↔ F_{Nd}^{irr}[T],

where F_{Nd}^{irr}[T] is the set of all irreducible polynomials of degree dividing Nd and the map is given by associating to each (pure) element its minimal polynomial over F. On the other hand, for a given hereditary order 𝔄 with period r in A and an associated type of embedding τ̄, (3.15) leads to an injection

( 7.2 ) Ad_{K_{𝔄}}\A(𝔄, τ̄) ↪ F_{Nd}^{irr}[T].

Combining (7.1) and (7.2), this leads to an injection

( 7.3 ) Ad_{K_{𝔄}}\A(𝔄, τ̄) ↪ Ad_{𝔖^{×}}\𝔖.

( 7.4 ) Definition (i) Let ω_{𝔄} : A → ℤ ∪ {∞} be the pseudo valuation (cf. [Be]) associated to the hereditary order 𝔄; hence, for x ∈ A\{0}, we put ω_{𝔄}(x) := max{j ∈ ℤ | x ∈ 𝔓^{tj}}, where t is the smallest positive power of 𝔓 which is a principal ideal of 𝔄, and ω_{𝔄}(0) = ∞.

(ii) Let \underline{x}, \underline{y} ∈ Ad_{K_{𝔄}}\A(𝔄, τ̄), then write

ν_{F}^{A(𝔄,τ̄)}(\underline{x}, \underline{y}) := \frac{t}{rd}max\{ω_{𝔄}(x − y) | x ∈ \underline{x}, y ∈ \underline{y}\}.

(iii) Let \underline{β}, \underline{γ} ∈ Ad_{𝔖^{×}}\𝔖, then write ν_{F}^{𝔖}(\underline{β}, \underline{γ}) := \frac{1}{Nd}max\{ν_{𝔖}(β − γ) | β ∈ \underline{β}, γ ∈ \underline{γ}\}.

( 7.5 ) Proposition For all \underline{x}, \underline{y} ∈ Ad_{K_{𝔄}}\A(𝔄, τ̄) we have ν_{F}^{A(𝔄,τ̄)}(\underline{x}, \underline{y}) = ν_{F}^{𝔖}(x̄, ȳ), where x̄, ȳ are the images of \underline{x}, \underline{y} under the injection (7.3).

Proof.

Let 𝔄̃ be the hereditary order in Ã, as in the beginning of (4.1). Fix a simple A-module V and an 𝔬_{D}-lattice chain {X_{j}}_{j∈ℤ} in V, such that 𝔄 = End_{𝔬_{D}}^{0}{X_{j}} and 𝔄̃ = End_{𝔬_{D}}^{0}{X_{j}}. Let t be as in (7.4). Put 𝔄̃_{1} := End_{𝔬_{F}}^{0} \{X_{tj}\}_{j∈ℤ}; then 𝔄̃_{1} is a canonical principal order of 𝔄̃ as in [G](1.2) with period \frac{rd}{t}.

We also have a natural injection

( 7.6 ) Ad_{K𝔄̃_{1}} \Ã(𝔄̃_{1}) ↪ F_{Nd}^{irr}[T];

hence, using (7.1) an injection

( 7.7 ) Ad_{K𝔄̃_{1}} \A(𝔄̃_{1}) ↪ Ad_{𝔖^{×}}\𝔖,

where Ad_{K𝔄̃_{1}} \A(𝔄̃_{1}) denotes the set of conjugacy classes of 𝔄̃_{1}-pure elements. We note that we only have one type of embedding with respect to 𝔄̃_{1}, because Ã is a split algebra. For \underline{μ}, \underline{η} ∈ Ad_{KÃ_{1}} \Ã(𝔄̃_{1}), we put ν_{F}^{Ã(𝔄̃_{1})}(\underline{μ}, \underline{η}) := \frac{t}{rd}max\{ν𝔄̃_{1}(μ − η) | μ ∈ \underline{μ}, η ∈ \underline{η}\}.

To describe the image of (7.2), let λ vary over all embedding indices such that (λ) is of type τ̄. Then an irreducible polynomial p over F belongs to the image, if and only if for E := F [T]/(p(T)), there is an embedding i : E ↪ A, with i(E)^{×} ⊂ K_{𝔄}, such that (i(E), 𝔄) is an embedding which belongs to the equivalence class of (λ), whence the extension E/F have to fulfill the relations of (2.4.4)(a)-(c). Clearly, if f is the degree of λ, we have ⟨λ_{1}^{1}, …, λ_{1}^{r}, …, λ_{f}^{1}, …, λ_{f}^{r}⟩f | \frac{Nt}{r}. Moreover, if C := C_{A}(E_{d}), ℭ := 𝔄∩C and K_{ℭ} = ⟨z⟩ℭ^{×}, then in the language of (2.4.4), we have \frac{r′d′}{t(λ)υ(λ)} = \frac{ν_{ℭ}(π_{F})}{ν_{ℭ}(z^{υ(λ)})} = \frac{ν_{𝔄}(π_{F})}{ν_{𝔄}(z^{υ(λ)})} = \frac{rd}{ν_{𝔄}(z^{υ(λ)})} divides \frac{rd}{t}.

But the image of (7.6) consists of all irreducible polynomials p over F such that, for K := F [T]/(p(T)), we have f(K/F) | \frac{Nt}{r} and e(K/F) | \frac{rd}{t}, whence it contains the image of (7.2). Therefore, the image of (7.3) is contained in the image of (7.7) and [Z2](theorem 1.8(i)(ii)) applies. We deduce that ν_{F}^{𝔖}(x̄, ȳ) = ν_{F}^{Ã(𝔄̃_{1})}(x̃, ỹ), where x̃, ỹ are the preimages of x̄, ȳ under the map (7.7).

( 7.8 ) Therefore, it is enough to show that ν_{F}^{A(𝔄,τ̄)}(\underline{x}, \underline{y}) = ν_{F}^{Ã(𝔄̃_{1})}(x̃, ỹ).

We can view A ≃ End_{D}(V) as being canonically embedded in Ã ≃ End_{F} (V). With this convention, we have

( 7.9 ) K_{𝔄} ⊂ K𝔄̃ ⊂ K𝔄̃_{1}.

Let us choose x ∈ \underline{x}, y ∈ \underline{y} and assume first that ν_{𝔄}(x) ≠ ν_{𝔄}(y), then ν𝔄̃_{1}(x) ≠ ν𝔄̃_{1}(y) and ν_{F}^{A(𝔄,τ̄)}(\underline{x}, \underline{y}) = \frac{t}{rd}min\{ω_{𝔄}(x), ω_{𝔄}(y)\} = \frac{t}{rd}min\{ν𝔄̃_{1}(x), ν𝔄̃_{1}(y)\} = ν_{F}^{Ã(𝔄̃_{1})}(x̃, ỹ). Hence, our result follows in this case.

So let us assume that ω_{𝔄}(x) = ω_{𝔄}(y) =: −n; hence ν𝔄̃_{1}(x) = ν𝔄̃_{1}(y) = −n. Then ν_{F}^{A(𝔄,τ̄)}(\underline{x}, \underline{y}), ν_{F}^{𝔄(𝔄̃_{1})}(x̃, ỹ) ≥ \frac{−nt}{rd}.

Clearly we are done if ν_{F}^{A(𝔄,τ̄)}(\underline{x}, \underline{y}) = \frac{−nt}{rd} = ν𝔄̃_{1} ; F(x̃, ỹ).

Assume first that ν_{F}^{A(𝔄,τ̄)}(\underline{x}, \underline{y}) = \frac{−qt}{rd} > \frac{−nt}{rd}, then the strata [𝔄, nt, qt, x], [𝔄, nt, qt, y] are pure and intertwine in K_{𝔄}; hence by (7.9) and

( 7.10 ) 𝔓^{−qt} ⊂ 𝔓̃^{−qt} ⊂ 𝔓̃_{1}^{−q}

(where 𝔓̃_{1} is the Jacobson radical of 𝔄̃_{1}), the strata [𝔄̃_{1}, n, q, x], [𝔄̃_{1}, n, q, y] intertwine in K𝔄̃_{1} as well. Therefore ν_{F}^{𝔄(𝔄̃_{1})}(x̃, ỹ) ≥ ν_{F}^{A(𝔄,τ̄)}(\underline{x}, \underline{y}) in this case.

On the other hand, if ν_{F}^{Ã(𝔄̃_{1})}(x̃, ỹ) = \frac{−qt}{rd} > \frac{−nt}{rd}, then the strata [𝔄̃_{1}, n, q, x], [𝔄̃_{1}, n, q, y] are pure and intertwine in K𝔄̃_{1}. The strata [𝔄, nt, qt, x], [𝔄, nt, qt, y] are pure as well and, by (5.1), are equivalent to simple strata [𝔄, nt, qt, x′], [𝔄, nt, qt, y′] respectively; hence, because of (7.10), the strata [𝔄̃_{1}, n, q, x], [𝔄̃_{1}, n, q, y] are equivalent to [𝔄̃_{1}, n, q, x′], [𝔄̃_{1}, n, q, y′] respectively. By definition, the strata [𝔄̃, nt, qt, x′], [𝔄̃, nt, qt, y′] are simple. Hence the strata [𝔄̃_{1}, n, q, x′], [𝔄̃_{1}, n, q, y′] are also simple, because it is easily seen that they are realizations of the same simple pair. Clearly the strata [𝔄̃_{1}, n, q, x′], [𝔄̃_{1}, n, q, y′] intertwine in K𝔄̃_{1} and using the same argument again, by (4.2.1), the strata [𝔄̃, nt, qt, x′], [𝔄̃, nt, qt, y′] intertwine; hence, by (4.1.3), the strata [𝔄, nt, qt, x′], [𝔄, nt, qt, y′] intertwine in K_{𝔄}. We deduce that the strata [𝔄, nt, qt, x], [𝔄, nt, qt, y] intertwine in K_{𝔄} as well and therefore ν_{F}^{A(𝔄,τ̄)}(\underline{x}, \underline{y}) ≥ ν_{F}^{𝔄(𝔄̃_{1})}(x̃, ỹ) in this case.

Altogether we obtain the equality (7.8), as required.

□

\underline{Concluding remark:} With the last proposition (7.5) and [Z2](theorem 1.8) we can also compare our metric on pure elements of (7.4)(i) and the polynomial metric of [Z2](definition 1.8). Moreover, using corollary 3.16 and imitating the procedure of [Z2](1.11), we can construct a fundamental domain for A(𝔄, τ̄) as in the remark following [Z2](definition/theorem 1.8).

8 References

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Université de Poitiers Mathématiques, SP2MI Boulevard 3 - Téléport 2 - BP 179 86960 Futuroscope Cedex broussou@wallis.univ-poitiers.fr

Humboldt-Universität Reine Mathematik Unter den Linden 6 10099 Germany grabitz@mathematik.hu-berlin.de