Simple characters for principal orders and their matching

by Martin Grabitz, Berlin

Autumn, 1999

Introduction

In this paper we want to explain how one can obtain a simple character from a simple stratum, which is founded on a principal order, for an arbitrary local central simple algebra. Moreover, we want to match these characters among various algebras. Our way to construct the simple characters is a mixture of the approach of [Z7] and [BK3]. From [Z7] we use the results about finite nilpotent rings and by using the methods of [Br3] we combine it with the results of [BK3]. We use the results of [BG] to obtain so called ‘special approximation sequences’, which can be used for the final matching of simple characters. Finally, we are able to extend the definition of a ps-character due to [BH] to the context of arbitrary simple algebras. I am grateful to E.-W. Zink for proofreading and valuable discussions.

0 Notation and preparations

Let F be a non-archimedean local field and A a finite-dimensional central simple algebra over F. Let V be a left simple A-module. V is a right D = End_{A}(V)-vector space and

A ≃ End_{D}(V) ≃ M_{m}(D)

(where M_{m}(D) denotes the ring of m × m matrices in D). Denote by d the index of the division algebra D (that is, dim_{F} (D) = d^{2}), fix an isomorphism A ≃ M_{m}(D), and identify A with M_{m}(D). Denote by 𝔬_{F}, 𝔬_{D}, … the rings of integers of F, D, …, by 𝔭_{F}, 𝔭_{D}, … the respective unique prime ideals and by 𝔨_{F}, 𝔨_{D}, … the associated residual fields.

( 0.1 ) An 𝔬_{F} -order 𝔄 in A is a subring with the same unit element as A, which is a complete 𝔬_{F} - lattice in A. Here complete 𝔬_{F} -lattice means that 𝔄 is a finitely generated free 𝔬_{F} -module and contains an F -basis of the F -vectorspace A. Let 𝔄 be a hereditary 𝔬_{F} -order in A, i. e., an 𝔬_{F} -order which is (left- or right-) hereditary as a ring. We denote by 𝔓 its Jacobson radical and by K_{𝔄} its normalizer. It is well known (see [R](39.12)) that 𝔄 is hereditary iff 𝔓 is a projective left or right 𝔄-module, which is also the case iff 𝔓 is a fractional invertible ideal of 𝔄. Hence, there exists an (𝔄,𝔄)-bisubmodule 𝔓^{−1} of A which is a complete 𝔬_{F} -lattice in A such that 𝔓𝔓^{−1} = 𝔓^{−1}𝔓 = 𝔄.

For an hereditary order 𝔄 we set U^{j}(𝔄) := 1 + 𝔓^{j} for all j ∈ ℕ. These groups lead to a filtration K_{𝔄} ⊃ 𝔄^{×} ⊃ U^{1}(𝔄) ⊃ U^{2}(𝔄) ⊃ ..., where K_{𝔄} denotes the A^{×}-normalizer of 𝔄.

An 𝔬_{D}-lattice chain in V is a family {X_{j}}_{j∈ℤ} of complete 𝔬_{F} -lattices in V that are right 𝔬_{D}-modules such that

a) ∀j ∈ ℤ, X_{j} ⊋ X_{j+1},

b) ∃r ∈ ℕ, ∀j ∈ ℤ, X_{j}𝔭_{D} = X_{j+r}.

( 0.2 ) One knows from [BF] that 𝔄 is also hereditary iff there exists a (unique up to a translation of the index) 𝔬_{D}-lattice chain {X_{k}}_{k∈ℤ} in V such that

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

( 0.3 ) We have

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

such that the ordered numbers n_{1},…,n_{r} provide numerical invariants, up to a cyclic permutation, of the conjugacy class of 𝔄. These numbers also come from a normal form (see [R] 39.14)

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}\}} ,

where (𝔬_{D}) (and (𝔭_{D}), respectively) mean n_{i} × n_{j}-matrices, where the lower-diagonal and diagonal blocks have entries in 𝔬_{D} and the upper-diagonal blocks have entries in 𝔭_{D} (note that n_{1}+…+n_{r} = m) and g is a suitable element of A^{×}. Here the number r is uniquely determined and called the period of 𝔄 and {X_{j}}_{j∈ℤ}, respectively. The numbers n_{1}, …, n_{r} are unique up to a cyclic permutation.

The modules X_{0}, …, X_{r−1} of the lattice chain of 𝔄 in V (as above) are a system of representatives of the isomorphy classes of direct indecomposable right 𝔄-modules

(1)

(see [BF](1.2.8)(ii) and [R](39.23)).

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}\}} .

If 𝔄 is given in normal form 0.3, then, for the simple A-module V := D^{m} (column vectors), we can choose the numbering of the chain {X_{j}}_{j∈ℤ} such that ∀j ∈ ℤ, ^{t}X_{j} =

𝔭_{D}^{l}((⨁_{i=1}^{k}( ⨁_{μ=1}^{n_{i}} 𝔭_{D})) ⨁( ⨁_{i=k+1}^{r}( ⨁_{μ=1}^{n_{j}} 𝔬_{D}))), where k, l ∈ ℤ such that j = lr + k, 0 ≤ k < r, and ^{t}(.)

means the set of transposed vectors.

An order 𝔄 is called principal if its Jacobson radical 𝔓 is a principal ideal. By 0.1 such an 𝔄 is hereditary, too.

( 0.4 ) From [BF](1.3.2) we know that a hereditary order 𝔄 with invariants n_{1}, …, n_{r} is a principal order if and only if n_{1} = … = n_{r} = s. In this situation we call s the invariant of 𝔄.

1 Strata and duality

Let ψ_{F} be a character of F^{+} with conductor 𝔭_{F} and let ψ_{A} := ψ_{F} ∘ T rd_{A/F}, where T rd denotes the reduced trace. From [BF] we know that the pairing A × A → ℂ^{×}, (x, y) ↦ ψ_{A}(xy) allows us to identify A with its Pontrjagin dual. Let us define the dual of any subset S of A by

( 1.1 ) S^{∗} := {a ∈ A | ψ_{A}(sa) = 1, for all s ∈ S}.

If u ∈ ℤ, we have

( 1.2 ) (𝔓^{u})^{∗} = 𝔓^{1−u}

(see [BF] and take into account that ψ_{F} has conductor 𝔭_{F}). For all integers n ≥ 1 and n > q > [\frac{n}{2}], where [.] denotes the least integer function, we have a canonical isomorphism of groups:

( 1.3 ) U^{q+1}(𝔄)/U^{n+1}(𝔄) → 𝔓^{q+1}/𝔓^{n+1},

induced by the map 1 + x ↦ x. In particular, the quotient U^{q+1}(𝔄)/U^{n+1}(𝔄) is finite abelian. Its dual is isomorphic to 𝔓^{−n}/𝔓^{−q} via the map:

( 1.4 ) b + 𝔓^{−q} ↦ ψ_{b}^{×} ∈ { \atop U^{q+1}(𝔄)/U^{n+1}(𝔄)}, ψ_{b}^{×}(1 + x) := ψ_{A}(bx).

Moreover, for n > q ≥ 0, we have a canonical isomorphism of groups:

( 1.5 ) 𝔓^{−n}/𝔓^{−q} → { \atop 𝔓^{q+1}/𝔓^{n+1}}

via the map

( 1.6 ) b + 𝔓^{−q} ↦ ψ_{b}^{+}, ψ_{b}^{+}(x) := ψ_{A}(bx).

( 1.7 ) Notation For later purposes we set S^{⊥ψ_{b}^{+}} := {x ∈ 𝔓 | ∀s ∈ S : ψ_{b}^{+}(xs − sx) = 1} for any subset S of 𝔓.

As in [Br2](1.1.3) we define:

( 1.8 ) Definitions (i) A stratum in A is a quadruple [𝔄, n, q, b], where

(a) 𝔄 is a hereditary 𝔬_{F} -order in A;

(b) n and q are integers with q ≤ n; hence, 𝔓^{−q} ⊂ 𝔓^{−n};

(c) b ∈ 𝔓^{−n}.

(ii) Two strata [𝔄, n, q, b_{i}], i = 1, 2, are said to be equivalent if b_{1} − b_{2} ∈ 𝔓^{−q}.

From the isomorphism (1.4), it is clear that the equivalence classes of strata of the form [𝔄, n, q, b] parametrize the dual of U^{q+1}(𝔄)/U^{n+1}(𝔄) if n ≥ 1 and n > q > [\frac{n}{2}]. For G := A^{×}, we define the formal intertwining of the stratum [𝔄, n, q, b] as the set I_{G}[𝔄, n, q, b] := \{x ∈ G | x^{−1}(b + 𝔓^{−q})x ∩ (b + 𝔓^{−q}) ≠ ∅\}, which in the case of n > q > [\frac{n}{2}] is the same as the intertwinig I_{G}(ψ_{b}^{×}) of our character ψ_{b}^{×}. We need the following important lemma, which is a generalization of [BK3](2.4.11):

( 1.9 ) Lemma Consider a stratum [𝔄, n, n−1, b] and suppose that I_{G}[𝔄, n, n−1, b] = G, then (b+𝔓^{1−n})∩F ≠ ∅. Moreover, if χ is a continuous character of U^{ν+1}(𝔄), where ν ≥ 0 and where χ is intertwined by

each element of G, then χ factors through the reduced norm Nrd_{A|F} : A^{×} → F^{×}.

\underline{Proof:}

As our first assertion is invariant under conjugation, we can assume that our hereditary order 𝔄 is given in normal form (0.3), where we identify A with M_{m}(D). Taking into account the explicit descriptions of the powers of 𝔓, we may assume that there exists an integer t with 0 ≤ t ≤ r − 1 and

( 1.10 )

b ≡ ( \begin{matrix}0 , & ⋯, & 0 , & 0 , & b_{1} , & 0 , & ⋯, & 0 \\ ⋯, & ⋯, & ⋯, & ⋯, & ⋯, & ⋯, & ⋯, & ⋯ \\ 0 , & ⋯, & ⋯, & ⋯, & 0 , & 0 , & b_{r−t−1} , & 0 \\ 0 , & ⋯, & ⋯, & ⋯, & ⋯, & ⋯, & 0 , & b_{r−t} \\ b_{r−t+1} , & 0 , & ⋯, & ⋯, & ⋯, & ⋯, & ⋯, & 0 \\ 0 , & b_{r−t+2} , & 0 , & ⋯, & ⋯, & ⋯, & ⋯, & 0 \\ ⋯, & ⋯, & ⋯, & ⋯, & ⋯, & ⋯, & ⋯, & ⋯ \\ 0 , & ⋯, & 0 , & b_{r} , & 0 , & 0 , & ⋯, & 0 \end{matrix} ) mod 𝔓^{1−n},

where r is the period of 𝔄. We use the block description as in (0.3). To be more precise, t is the unique integer with −n = kr + t and 0 ≤ t ≤ r − 1, where k is also an appropriate integer. Here b_{i} is an n_{i}×n_{τ_{r}^{t}(i)}-matrix, where τ_{r} is the usual cyclic permutation of the numbers {1, …, r} and b_{i} can be chosen with entries in a system of representatives of 𝔭_{D}^{k}/𝔭_{D}^{k+1} if it is on or above the block diagonal, or in a system of representatives of 𝔭_{D}^{k+1}/𝔭_{D}^{k+2} if it is below the block diagonal.

If b is in 𝔓^{1−n}, we have 0 ∈ (b + 𝔓^{1−n}) ∩ F and there is nothing to show for the first assertion. We may assume that one of the b_{i} is nonzero. So let i be such that b_{i} ≠ 0 and let x := diag(I_{n_{1}}, …, I_{n_{i−1}}, π_{F}^{2} I_{n_{i}}, I_{n_{i+1}}, …, I_{n_{r}}), where diag(., …, .) means a block diagonal matrix and I_{n} the n × n identity matrix. Then x(b + 𝔓^{1−n}) ∩ (b + 𝔓^{1−n})x ≠ ∅ by assumption. The elements in the second set have an entry in the (i, τ_{r}^{t}(i))’s block which is in 𝔭_{D}^{k}\𝔭_{D}^{k+2}, but if we have t ≠ 0, then, for the elements in the first set, all entries in the same block lie in 𝔭_{D}^{k+2d}, which is impossible.

Therefore, we can assume that

( 1.11 ) b ≡ diag(b_{1}, …, b_{r}) mod 𝔓^{1−n},

and, in particular, −n = kr. Let b′ := diag(b′_{1}, …, b′_{r}) be a matrix such that b ≡ π_{D}^{k}b′ mod 𝔓^{1−n}, where the matrices b′_{1}, …, b′_{r} have entries in a system of representatives of 𝔬_{D}/𝔭_{D} ≃ 𝔨_{D}.

Consider x ∈ 𝔬_{D}^{×}I_{m} ⊂ 𝔄^{×}, then, by assumption, x(b + 𝔓^{1−n})x^{−1} ∩ (b + 𝔓^{1−n}) ≠ ∅, which is the same as

( 1.12 ) π_{D}^{−k}xπ_{D}^{k}b′x^{−1} ≡ b′ mod 𝔓,

because 𝔄 is normalized by π_{D} (by (0.3)), π_{F} = π_{D}^{d} (for an appropriate choice of π_{D}) and π_{F} 𝔄 = 𝔓^{rd}.

Let x̄ := (x(mod 𝔓)) and b̄ := (b′(mod 𝔓)), then we can view x̄ as an element of 𝔨_{D}^{×} and b̄ as an element of M_{n_{1}}(𝔨_{D}) × … × M_{n_{r}}(𝔨_{D}), where 𝔨_{D} is viewed as being embedded diagonally into the center of the latter algebra. The equation (1.12) now says that σ^{k}(x̄)b̄x̄^{−1} = b̄, where σ is a generator of Gal(𝔨_{D}/𝔨_{F}) induced by the adjoint action of π_{D} on a suitable maximal unramified extension field L/F in D (note that for suitable L we have 𝔬_{D} = 𝔬_{L} + 𝔬_{L}π_{D} + … + 𝔬_{L}π_{D}^{d−1} and 𝔭_{D} = 𝔭_{L} + 𝔬_{L}π_{D} + … + 𝔬_{L}π_{D}^{d−1}); hence, because b̄ ≠ 0 and x̄ centralizes b̄, we obtain σ^{k}(x̄) = x̄. As x̄ ∈ 𝔨_{D}^{×} was arbitrarily chosen, we conclude that d must be a divisor of k. As π_{D}^{k} = πF^{\frac{k}{d}} ∈ F^{×}, replacing b by a multiple with a central element in A (namely π_{F}^{− \frac{k}{d}}), it is sufficient to prove our first assertion under the assumption that n = 0 and b = b′.

Consider x = π_{D}^{ν}I_{m} for an arbitrary integer ν. In this case our assumption implies that σ^{ν}(b̄) = b̄, where σ acts on each entry of the matrix b̄ ∈ M_{n_{1}}(𝔨_{D}) × … × M_{n_{r}}(𝔨_{D}). Therefore b_{1}, …, b_{r} can be chosen with entries in a system of representatives of 𝔬_{F} /𝔭_{F}.

Consider an arbitrary invertible x of the same kind as b, then our assumption leads to x̄^{−1}b̄x̄ = b̄; hence, we can assume that b_{i} = a_{i}I_{n_{i}}, where a_{i} is chosen from a system of representatives of 𝔬_{F} /𝔭_{F}. Finally, consider x as an arbitrary permutation matrix of A = M_{m}(D), then we conclude that a_{1} = a_{2} = … = a_{r} =: a and we obtain (b + 𝔓) = (a + 𝔓).

This leads to (b + 𝔓^{1−n}) ∩ F ≠ ∅ as required.

To prove now the second assertion, we can assume the first assertion in full generality. So let χ be a continuous character of U^{ν+1}(𝔄) that is intertwined by the whole of G, then also χ_{|U^{n}(𝔄)} is intertwined by the whole of G for all n ≥ ν + 1. Let n be the least integer such that χ_{|U^{n+1}(𝔄)} ≡ 1, then there exists a stratum [𝔄, n, n−1, b] such that χ_{|U^{n}(𝔄)} = ψ_{b}^{×}, and by the first part, we can choose b ∈ F. Moreover, if we follow the arguments of the proof, we conclude that dr is a divisor of n.

Let L/F be an unramified field extension which splits A. Then, clearly, the stratum [𝔄 ⊗_{𝔬_{F}} 𝔬_{L}, n, n − 1, b ⊗ 1] is intertwined by the whole of (A ⊗_{F} L)^{×} and we conclude from the split case of [BK3](2.4.11) and its proof that ψ_{b⊗1}^{×} = ψ_{L,b}^{×} ∘ det_{A⊗_{F} L/L}, where ψ_{L,b}^{×}(x) := ψ_{L}(b(x − 1)) for x ∈ U^{\frac{n}{rd}} (𝔬_{L}) and ψ_{L} is an additive character of L with conductor 𝔭_{L} and restriction ψ_{F} to F (also ψ_{b⊗1}^{×} is defined by ψ_{L}, similarly as ψ_{b}^{×} by ψ_{F}). Hence, ψ_{b}^{×} = ψ_{F,b}^{×} ∘ Nrd_{A/F} by the definition of the reduced norm, where ψ_{F,b} is the restriction of ψ_{L,b⊗1} to U^{\frac{n}{rd}} (𝔬_{F}). Let χ_{0} := ψ_{F,b}^{×} ∘ Nrd_{A/F}, then χ′ := χ_{0}^{−1}χ is a character of U^{ν+1}(𝔄) which is intertwined by the whole of G and which is trivial on U^{n}(𝔄). Moreover, we have χ = χ′χ_{0}.

By induction on n we can realize χ as a product of characters that factor through the reduced norm; hence, so does χ.

□

2 Sound elements and their canonical mappings

We want to give a brief description of the connection between the general theory of pure elements explained in [BG] and sound embeddings as introduced in [F]. For this purpose we study the embedding data as defined in [BG](2.3.1) for sound embeddings.

To abbreviate the assertions we introduce the following notation as given in [BG] above [BG](2.1.2):

( 2.1 ) Notation Let 𝔄 be a hereditary 𝔬_{F} -order in A and E/F an extension in A such that E^{×} ⊂ K_{𝔄}, then the pair (E, 𝔄) is called an embedding (with respect to 𝔄).

( 2.2 ) Proposition If 𝔄 is a principal order with invariant s and period r in A = M_{m}(D) given in normal form (0.3) and (L, 𝔄) an embedding, where L/F is an unramified extension of degree f dividing d, then the following assertions are equivalent:

(i) 𝔄 ∩ C_{A}(L) is a principal order of C_{A}(L) continued by 𝔄 in the sense of [G](0.6)(i.e., K_{𝔄} ∩ C_{A}(L) = K_{𝔄∩C_{A}(L)}).

(ii) There exists a divisor d_{1} of f that is prime to r such that λ[d_{1}] := ((λ_{1}^{1}, …, λ_{f}^{1}), …, (λ_{1}^{r}, …, λ_{f}^{r})), with

λ_{j}^{i} := \{ \begin{matrix}\frac{sd_{1}}{f} & if there exists an n ∈ ℤ such that i ≡ 1 + nd_{1} (mod r) and j ≡ 1 + [\frac{nd_{1}}{r}] (mod f) \\ 0 & else, \end{matrix}

is an embedding datum in the sense of [BG](2.3.1) (in particular \frac{f}{d_{1}} is a divisor of s) such that the embedding (L, 𝔄) is equivalent in the sense of [BG](2.2.1) to the embedding (λ[d_{1}]) defined in [BG](2.3.1).

(iii) The embedding (L, 𝔄) is sound in the sense of [F](2.11a)(2.11b).

If the equivalent conditions (i)-(iii) are fulfilled, the sound-embedding L^{×} ⊂ K_{𝔄} in the sense of [F](2.11a)(2.11b) has invariant d_{1} = d^{(1)} as defined in [F](2.12) and 𝔄∩C_{A}(L) has invariant \frac{sd_{1}}{f} and period \frac{rf}{d_{1}} . The invariants d_{1} and f determine the embedding (L, 𝔄) up to the equivalence relation in [BG](2.2.1). Finally, we want to note that if f is a divisor of d and d_{1} a divisior of f that is prime to r and such that \frac{f}{d_{1}} divides s, then λ as defined in (ii) is always an embedding datum in the sense of [BG](2.3.1).

\underline{Proof:}

(i) ⇒ (ii) : Clearly, there exist an embedding datum λ of degree f such that (L, 𝔄) is equivalent to (λ) = ((λ_{1}^{1}, …, λ_{f}^{1}), …, (λ_{1}^{r}, …, λ_{f}^{r})). If 𝔄 ∩ C_{A}(L) is a principal order continued by 𝔄, then all λ_{j}^{i} must either be 0 or equal to the invariant s′ of 𝔄 ∩ C_{A}(L). Therefore, by

∑_{j=1}^{f} λ_{j}^{i} = s,  (2)

we conclude that s′ | s. By sr = m = s′r′ we have \frac{s}{s′} = \frac{r′}{r} and, as 𝔄 is a continuation of 𝔄 ∩ C_{A}(L), [BG](2.4.4) applies and λ is 1-zero-periodical in the sense of [BG](2.4.1) (i.e., in the infinite sequence of numbers …, λ_{f}^{r}, λ_{1}^{1}, …, λ_{1}^{r}, …, λ_{f}^{1}, …, λ_{f}^{r}, λ_{1}^{1}, … the number of zeros between two non-zero terms is always the same). Since we have r′ non-zero numbers in the sequence

λ_{1}^{1}, …, λ_{1}^{r}, …, λ_{f}^{1}, …, λ_{f}^{r}  (3)

of rf numbers, we conclude that r′ must be a divisor of rf; hence, \frac{r′}{r} is a divisor of f.

Let d_{1} := \frac{f}{( \frac{r′}{r})} = \frac{f}{( \frac{s}{s′})} and conclude that s′ = \frac{sd_{1}}{f} and r′ = \frac{rf}{d_{1}} , then up to equivalence we can assume that λ_{1}^{1} = \frac{sd_{1}}{f} and the number between two non-zero numbers in the sequence (3) is d_{1} − 1;

hence, they are the λ_{τ_{f}^{[ \frac{nd_{1}}{r} ]}(1)}^{τ_{r}^{nd_{1}−[ \frac{nd_{1}}{r} ]r}(1)} = λ_{τ_{f}^{[ \frac{nd_{1}}{r} ]}(1)}^{τ_{r}^{nd_{1}} (1)} for n ∈ ℤ, where τ_{r} ∈ S_{r} is the cyclic permutation of

{1, …, r} and τ_{f} ∈ S_{f} is the cyclic permutation of {1, …, f}. Finally, again by (2), we conclude that exactly \frac{f}{d_{1}} non-zero terms must appear in a sequence λ_{1}^{i}, …, λ_{f}^{i}. For i = 1 we obtain that λ_{j}^{1} ≠ 0 iff \frac{r}{⟨d_{1},r⟩} | n and j ≡ [\frac{nd_{1}}{r}] (mod f); hence, j = k \frac{d_{1}}{⟨d_{1},r⟩} for k ∈ \{1, …, \frac{f⟨d_{1},r⟩}{d_{1}}\}. We deduce that ⟨d_{1}, r⟩ = 1 and we are done with the first direction.

(ii) ⇒ (i) : This direction follows immediately by [BG](2.4.4) and it also implies that 𝔄 ∩ C_{A}(L) has invariant \frac{sd_{1}}{f} and period \frac{rf}{d_{1}} .

By [F](Theorem 2) we also have (i) ⇔ (iii) and we have finished the proof of the first assertion. If (i)-(iii) are fulfilled, then again the first of the latter assertions follows by [F](Theorem 2). For the

very last assertion we have to show that ∑_{j=1}^{f} λ_{j}^{i} = s for i = 1, …, r, under the asserted assumptions i.e., there are exactly \frac{f}{d_{1}} different numbers j ∈ \{1, …, f\} such that there exists an integer n with j ≡ 1 + [\frac{nd_{1}}{r}] (mod f) and i ≡ 1 + nd_{1} (mod r) for fixed i. As d_{1} is prime to r, the latter congruence has a solution n′ and clearly all numbers translated by a multiple of r fulfil the same congruence. Therefore, the numbers j ∈ \{1, …, f\} are the representatives of the classes 1 + [\frac{n′d_{1}}{r}] + kd_{1} (mod f) for k ∈ ℤ and, obviously, there are exactly \frac{f}{d_{1}} different ones of them.

□

\underline{Remark:} It is clear that all sound embeddings M^{×} ⊂ K_{𝔄}, where M/L is an unramified extension of degree prime to \frac{d}{f}, are in the same equivalence class as (L, 𝔄) and have invariant d^{(1)} = d_{1} in the sense of [F](2.12), and the latter assertion provides the connection between all sound embeddings in the sense of [F](2.11a)(2.11b) and sound pearl-embeddings. □

( 2.3 ) Proposition Let 𝔄 be a principal 𝔬_{F} -order in A, (E, 𝔄) an embedding and B := C_{A}(E), then the following assertions are equivalent:

(i) K_{𝔄} ∩ B = K_{𝔄∩B} and 𝔄 ∩ B is a principal order;

(ii) for the maximally unramified subextesion K/F in E/F the embedding (K, 𝔄) is sound in the sense of [F](2.11a)(2.11b);

(iii) for the maximally unramified subextension L/F of E/F with a degree dividing d (i.e., [L : F] = ⟨f(E/F), d⟩) the embedding (L, 𝔄) is sound in the sense of [F](2.11a)(2.11b);

(iv) there exists an embedding (M, 𝔄) such that M/F is a maximal extension in A and M/E an extension in B (i.e., the embedding (E, 𝔄) extends to an embedding of a maximal field).

\underline{Proof:}

(i) ⇒ (ii) : If 𝔄 ∩ B is a principal order, then, by [F](Corollary 1) there exists an embedding (M, 𝔄 ∩ B) such that the extension M/E is maximal in B; hence, by definition and (i) (M, 𝔄) is a maximal embedding which extends the embedding (K, 𝔄). By [F](Theorem 2(i)) the embedding (K, 𝔄) is sound.

(ii) ⇒ (iii) : For the embedding (L, 𝔄) the extension L/F is a subextension of (K, 𝔄); hence, again by [F](Theorem 2(i)) the embedding (L, 𝔄) is sound.

(iii) ⇒ (iv) : Again, by [F](Theorem 2(i)) or (2.2) 𝔄 ∩ C_{A}(L) is principal and continued by 𝔄. As C_{A}(L) has index \frac{d}{f} and ⟨f(E/L), \frac{d}{f} ⟩ = 1, we deduce by [G](Theorem 2.2(ii)) that the order 𝔄 ∩ B is principal and continued by 𝔄. Now again by [F](Corollary 1) there exists an embedding (M, 𝔄 ∩ B) such that the extension M/E is maximal in B; hence, by definition and because 𝔄 continues 𝔄 ∩ B the embedding (M, 𝔄) is well defined and a maximal embedding which extends the embedding (E, 𝔄).

(iv) ⇒ (i) : As before we observe that the field extension M/E is maximal in B; hence, our conclusion is given by [Z1](Theorem 2).

□

\underline{Remark:} In the split case d = 1 every embedding (E, 𝔄) is sound because condition (iii) is allways fulfilled. □

Now assume again that 𝔄 is a principal order with invariant s and period r (sr = m). Then the last Proposition enables us to extend the definition of a sound embedding of [F] from unramified extensions to arbitrary extensions.

( 2.4 ) Definition (i) An embedding (E, 𝔄) is called sound if it fulfils the equivalent conditions of (2.3).

(ii) Similarly as in [BG] β ∈ A is called 𝔄-pure if E := F [β] is a field with E^{×} ⊂ K_{𝔄}.

(iii) Moreover, β as in (ii) is called 𝔄-sound iff the embedding (F [β], 𝔄) is sound in the sense of (i).

Let β and E := F [β] be as in Definition (2.4)(iii), then there exists a field extension M/E such that the extension M/F is maximal in A/F and M^{×} ⊂ K_{𝔄}. Let B := C_{A}(E) be the centralizer of E in A and 𝔅 := 𝔄 ∩ B. By [Z1](Theorem 2) 𝔅 is a principal order and β is (e(M/F), f(M/F))-pure in the sense of [Z1](Definition 4). Let 𝔔 be the Jacobson radical of 𝔅, then by [Z1](Theorem 2) for all j ∈ ℤ we have 𝔓^{j} ∩ B = 𝔔^{[ \frac{j+e_{𝔄/𝔅}−1}{e_{𝔄/𝔅}}]}, where e_{𝔄/𝔅} := ⟨f(E/F), \frac{f(M/F)}{s}⟩. Clearly, e_{𝔄/𝔅} just depends on the embedding of E and not on the special choice of M.

( 2.5 ) Notation

We call e_{𝔄/𝔅} the ramification exponent of 𝔄 over 𝔅 and define a function σ_{𝔄/𝔅} : ℤ → ℤ by σ_{𝔄/𝔅}(j) := [\frac{j+e_{𝔄/𝔅}−1}{e_{𝔄/𝔅}}] for all j ∈ ℤ.

( 2.6 ) Lemma (i) For all j ∈ ℤ we have σ_{𝔄/𝔅}(1 − j) = 1 − σ_{𝔄/𝔅}(j).

(ii) For all j ∈ ℤ such that e_{𝔄/𝔅} | j, we have σ_{𝔄/𝔅}([\frac{j}{2}] + 1) = [\frac{σ_{𝔄/𝔅}(j)}{2}] + 1.

(iii) If r′ is the period of 𝔅, r the period of 𝔄 and d′ is the index of B, then we have e_{𝔄/𝔅} = \frac{rd}{r′d′e(E/F)}.

\underline{Proof:}

(i) Let us write j = e_{𝔄/𝔅}k − t, where 0 ≤ t < e_{𝔄/𝔅}, then σ_{𝔄/𝔅}(1 − j) = [\frac{1−e_{𝔄/𝔅}k+t+e_{𝔄/𝔅}−1}{e_{𝔄/𝔅}}] = [\frac{e_{𝔄/𝔅}(1−k)+t}{e_{𝔄/𝔅}}] = 1 − k = 1 − [\frac{e_{𝔄/𝔅}k−t+e_{𝔄/𝔅}−1}{e_{𝔄/𝔅}}] = 1 − σ_{𝔄/𝔅}(j).

(ii) Let us write j = e_{𝔄/𝔅}k, then we have two cases:

(a) We have 2 | k; hence, we can write k = 2l and obtain σ_{𝔄/𝔅}([\frac{j}{2}] + 1) = [\frac{e_{𝔄/𝔅}l+1+e_{𝔄/𝔅}−1}{e_{𝔄/𝔅}}] =

l + 1 = [^{\frac{( \frac{j}{e_{𝔄/𝔅}})}{2}}] + 1 = [\frac{σ_{𝔄/𝔅}(j)}{2}] + 1.

(b) We have 2 ∤ k; hence, we can write k = 2l + 1 and obtain σ_{𝔄/𝔅}([\frac{j}{2}] + 1) = [\frac{e_{𝔄/𝔅}l+[ \frac{e_{𝔄/𝔅}}{2} ]+1+e_{𝔄/𝔅}−1}{e_{𝔄/𝔅}}] = l + 1 = [^{\frac{( \frac{j}{e_{𝔄/𝔅}})}{2}}] + 1 = [\frac{σ_{𝔄/𝔅}(j)}{2}] + 1.

(iii) The proof is essentially the same as for [BG](4.3.4), where we just resign the assumption that E/F is unramified and note that then ν_{𝔅}(π_{F}) = r′d′e(E/F).

□

The second main interest of this section is to introduce two canonical mappings associated to an element which generates a field extension in A/F. As in [BK3](1.3)(1.4) we define a tame co-restriction and the adjoint map for β:

( 2.7 ) Definition Let β be an element which generates a field extension E/F in A and let B := C_{A}(E).

(i) For all x ∈ A let a_{β}(x) := βx − xβ and we call the map a_{β} : A → A the adjoint map associated to β.

(ii) A tame co-restriction 𝔰_{β} on A with respect to β is a (B, B)-bimodule homomorphism 𝔰_{β} : A → B such that for all principal orders 𝔄 for which β is 𝔄-sound we have 𝔰_{β}(𝔄) = 𝔄 ∩ B.

As in [BK3](1.3.4) one can prove (using Lemma (2.6)(i) and (1.2) above):

( 2.8 ) Proposition(i) Let ψ_{E}, ψ_{F} be continuous additive characters of E, F with conductors 𝔭_{E} and 𝔭_{F}, respectively (E := F [β]). Define additive characters ψ_{B}, ψ_{A} of B, A as above in section (1) for A. There exists a unique map 𝔰_{β} : A → B such that for all a ∈ A, b ∈ B we have ψ_{A}(ab) = ψ_{B}(𝔰_{β}(a)b) and this map is a tame co-restriction with respect to β.

(ii) For 𝔄 as in this section (i.e., 𝔄 is a principal order), β 𝔄-sound and all j ∈ ℤ we have: 𝔰_{β}(𝔓^{j}) = 𝔔^{σ_{𝔄/𝔅}(j)}.

□

Moreover, as in [BK3] we obtain an infinite exact sequence:

( 2.9 ) … →^{𝔰_{β}} A →^{a_{β}} A →^{𝔰_{β}} A →^{a_{β}} …

3 Simple strata and embeddings

Let us fix a simple left A-module V and identify A with End_{D}(V), where V is a right D ≃ End_{A}(V)- vectorspace. Again let 𝔄 be a principal 𝔬_{F} -order in A as in the last section. Let {X_{j}}_{j∈ℤ} be the lattice chain of 𝔄 in V as in (0.2), which means that 𝔄 = End_{𝔬_{D}}({X_{j}}_{j∈ℤ}). Let Ã := End_{F} (V) and 𝔄̃ := End_{𝔬_{F}} ({X_{j}}_{j∈ℤ}), then we have canonical embeddings A ↪ Ã and 𝔄 ↪ 𝔄̃. Similarly as in [Br2](1.1.7) we define:

( 3.1 ) Definitions (i) A stratum [𝔄, n, q, β] is said to be pure if β is 𝔄-pure and n = −ν_{𝔄}(β). (ii) A pure stratum as in (i) is said to be sound if β is 𝔄-sound. (iii) A stratum as in (ii) is said to be simple if the associated stratum [𝔄̃, n, q, β] is simple in the

sense of [BK3](1.5.5).

In fact, one can define a simple stratum in the same way as in [BK](1.5.5) if one generalizes the notion of the critical exponent k_{0}(β, 𝔄):

( 3.2 ) Definition (i) For all j ∈ ℤ we set (as in [BK3](1.4.3)) 𝔑_{j}(β, 𝔄) := {x ∈ 𝔄 | a_{β}(x) ∈ 𝔓^{j}}; (ii) for an 𝔄-sound element β we set k_{0}(β, 𝔄) := max{k ∈ ℤ | 𝔑_{k}(β, 𝔄) ⊈ 𝔅 + 𝔓}, where

𝔅 := 𝔄 ∩ C_{A}(E).

To prove that the latter generalization makes sense we have to make clear more of the details with respect to the connection of the orders 𝔄 and 𝔄̃. For this purpose we set B̃ := CÃ(E) and 𝔅̃ := 𝔄̃ ∩ B̃. Moreover, let 𝔔̃ be the Jacobson radical of 𝔅̃. The following lemma is a slight modification of [Br4](I,4.2):

( 3.3 ) Lemma Let A = M_{m}(D), and V = D^{m} (column vectors) be the standard simple A-module, then the isomorphism M_{m}(D) ≃ End_{D}(V) enables us to identify A via the canonical embedding into Ã = End_{F} (V) and the canonical isomorphism Ã = End_{F} (V) ≃ End_{D}(V) ⊗_{F} D^{op} (where (.)^{op} denotes the opposite algebra structure of a given algebra) with the subset A ⊗ 1 of A ⊗_{F} D^{op}. In the same manner we have B̃ = End_{E}(V) ≃ B ⊗ D^{op} and we can identify B with the subset B ⊗ 1. Moreover, these canonical isomorphisms restrict for all j ∈ ℤ to isomorphisms

(i) 𝔓̃^{j} ≃ ⊕_{k=0}^{d−1}𝔓^{j+kr} ⊗_{𝔬_{F}} π_{D}^{−k}𝔬_{L} and

(ii) 𝔔̃^{j} ≃ ⊕_{k=0}^{d−1}𝔔^{σ_{𝔄/𝔅}(j+kr)} ⊗_{𝔬_{F}} π_{D}^{−k}𝔬_{L},

where L/F is an unramified field extension in D of degree d (i.e., L splits D).

( 3.4 ) Lemma In the notation of (3.3) we have 𝔑_{j}(β, 𝔄̃) = ⊕_{k=0}^{d−1}𝔓^{kr} ∩ 𝔑_{j+kr}(β, 𝔄) ⊗_{𝔬_{F}} π_{D}^{−k}𝔬_{L}. In particular we have 𝔑_{j}(β, 𝔄̃) ∩ A = 𝔑_{j}(β, 𝔄).

\underline{Proof:}

Let x ∈ 𝔄̃, then by (3.3) we have x = ⊕_{k=0}^{d−1}x_{k} ⊗ π_{D}^{−k}l_{k}, where x_{k} ∈ 𝔓^{kr} and l_{k} ∈ 𝔬_{L} for k = 0, …, d − 1 and 𝔓̃^{j} = ⊕_{k=0}𝔓^{j+kr} ⊗_{𝔬_{F}} π_{D}^{−k}𝔬_{L}. Therefore a_{β⊗1}(x) ∈ 𝔓̃^{j} if and only if a_{β}(x_{k}) ∈ 𝔓^{j+kr} for k = 0, …, d − 1. Hence, x ∈ 𝔑_{j}(β, 𝔄̃) if and only if x_{k} ∈ 𝔓^{kr} ∩ 𝔑_{j+kr}(β, 𝔄) as required.

□

We need a lemma which generalizes [BK3](2.1.1):

( 3.5 ) Lemma If ν, j ∈ ℤ, then 𝔓^{ν} ∩ 𝔑_{ν+j}(β, 𝔄) = 𝔔^{σ_{𝔄/𝔅}(ν)}𝔑_{j}(β, 𝔄).

\underline{Proof:}

It is easy to see that the right-hand side is contained in the left-hand side. By [BK](2.1.1) we have 𝔓̃^{ν} ∩ 𝔑_{ν+j}(β, 𝔄̃) = 𝔔̃^{ν}𝔑_{j}(β, 𝔄̃). If we use the notions of (3.3) and identify A with its image A ⊗ 1, we deduce that 𝔓^{ν} ∩ 𝔑_{ν+j}(β, 𝔄) = 𝔓̃^{ν} ∩ 𝔑_{ν+j}(β, 𝔄̃) ∩ A = (𝔔̃^{ν}𝔑_{j}(β, 𝔄̃)) ∩ A = (3.3(ii)+3.4) 𝔔^{σ_{𝔄/𝔅}(ν)}𝔑_{j}(β, 𝔄) + π_{F}^{−1} ∑_{k=1}^{d−1} 𝔔^{σ_{𝔄/𝔅}(ν+kr)}𝔑_{j+(d−k)r}(β, 𝔄) = 𝔔^{σ_{𝔄/𝔅}(ν)}𝔑_{j}(β, 𝔄), where the latter equality comes from the fact that the last summands are contained in the first one.

□

Now we see that the use of the two different kinds of critical exponents actually leads to the same result:

( 3.6 ) Proposition We have k_{0}(β, 𝔄) = k_{0}(β, 𝔄̃).

\underline{Proof:}

In the notations of (3.3) and the definition of critical exponents we have to show that 𝔑_{j}(β, 𝔄) ⊂ (𝔅 + 𝔓) if and only if 𝔑_{j}(β, 𝔄̃) ⊂ (𝔅̃ + 𝔓̃).

First assume that 𝔑_{j}(β, 𝔄) ⊂ (𝔅+𝔓), then 𝔑_{j}(β, 𝔄̃) = (3.4) ⊕_{k=0}^{d−1}𝔓^{kr} ∩𝔑_{j+kr}(β, 𝔄)⊗_{𝔬_{F}} π_{D}^{−k}𝔬_{L} = (Lemma 3.5) ⊕_{k=0}^{d−1}𝔔^{σ_{𝔄/𝔅}(kr)}𝔑_{j}(β, 𝔄) ⊗_{𝔬_{F}} π_{D}^{−k}𝔬_{L} ⊂ (by assumption) ⊕_{k=0}^{d−1}𝔔^{σ_{𝔄/𝔅}(kr)}(𝔅 + 𝔓) ⊗_{𝔬_{F}} π_{D}^{−k}𝔬_{L} ⊂ ⊕_{k=0}^{d−1}𝔔^{σ_{𝔄/𝔅}(kr)} ⊗_{𝔬_{F}} π_{D}^{−k}𝔬_{L} + ⊕_{k=0}^{d−1}𝔓^{1+kr} ⊗_{𝔬_{F}} π_{D}^{−k}𝔬_{L} = (3.3((ii)+(i))) 𝔅̃ + 𝔓̃.

Conversely, assume that 𝔑_{j}(β, 𝔄̃) ⊂ (𝔅̃ + 𝔓̃), then it follows that 𝔑_{j}(β, 𝔄) = (3.4) 𝔑_{j}(β, 𝔄̃) ∩ A ⊂ (by assumption) (𝔅̃ + 𝔓̃) ∩ A = (3.3((ii)+(i))) 𝔅 + 𝔓.

So we are done.

□

As in the split case one can introduce the notion of a simple pair due to [BH](1.5):

( 3.7 ) Definition A simple pair is a pair [q^{0}, β′] of an integer q^{0} and an element β′ from an algebraic closure of F such that q^{0} < min{−ν_{E}(β′), −k_{0}(β′, 𝔄(E))}, where E := F [β′] and 𝔄(E) is the principal order in A(E) := End_{F} (E) with 𝔬_{F} -lattice chain {𝔭_{E}^{j}}_{j∈ℤ} in the simple A(E)-module E.

As in the split case the stratum [𝔄, n, q, β] is simple for all 𝔄-sound embeddings β of β′ to A (where we must assume now that [E : F] | md) and all q ∈ ℤ with q^{0} = [\frac{e(E/F)q}{rd}], where n = −ν_{𝔄}(β) and r is the period of 𝔄. Conversely, a simple stratum [𝔄, n, q, β] leads to a simple pair [[\frac{e(E/F)q}{rd}], β]. We also note that k_{0}(β, 𝔄) = k_{0}(β, 𝔄(E))\frac{rd}{e(E/F)} and ν_{𝔄}(β) = ν_{E}(β′)\frac{rd}{e(E/F)}.

From [BG](5.1) and its proof it is straightforward that each sound stratum [𝔄, n, q, β] is equivalent to a simple stratum [𝔄, n, q, γ] in the sense of (3.1). In fact, one can see from the proof of [BG](5.1)(5.4) that it could have been shown that:

( 3.8 ) Proposition Let [𝔄, n, q, β] be a sound stratum, K/F the maximally unramified subextension of the extension E/F. Let M/K be an unramified extension in A such that M[β] is a field, and M^{×} ⊂ K_{𝔄} is sound embedded in the sense of (2.4). Let ℭ := 𝔄 ∩ C_{A}(M), then there exists a ℭ-sound element γ such that the stratum [𝔄, n, q, γ] is simple and equivalent to [𝔄, n, q, β]. Moreover, we have F [γ]^{ur} ⊂ K and finally the stratum [ℭ, n′, q′, γ] is simple as well as equivalent to [ℭ, n′, q′, β], where n′ := σ_{𝔄/ℭ}(n) and q′ := σ_{𝔄/ℭ}(q).

\underline{Proof:}

Because of 𝔬_{E} ⊂ 𝔄, 𝔬_{M} ⊂ 𝔄 and because there is an integral basis of 𝔬_{M[β]} | 𝔬_{M} which is contained in 𝔬_{F [β]} we also have 𝔬_{M[β]} ⊂ 𝔄. Since the extension M[β]/F [β] is unramified, we conclude that M[β]^{×} ⊂ K_{𝔄}. Of course, we also have M[β]^{×} ⊂ K_{ℭ} and the stratum [ℭ, n′, q′, β] is sound.

As in the beginning of the proof of [BG](5.1) one reduces to the case of \frac{rd}{e(E/F)} | q and then, by (2.6)(iii), we have σ_{𝔄/ℭ}(q) = q \frac{r̄d̄}{rd}, where r̄ is the period of ℭ and d̄ is the index of C_{A}(M).

If we now pay attention to the changed notation and if we follow the proof of [BG](5.1) where we replace K by M with the above assumptions, then we find a ℭ-sound element γ such that the stratum [ℭ, n′, q′, γ] (in [BG] denoted by [𝔅, n′, q′, γ_{1}]) is simple and equivalent to [ℭ, n′, q′, β] (in [BG] denoted by [𝔅, n′, q′, β]). The argument F [β] = K[β] used following [BG](5.5) is not necessary for proving F [γ]^{×} ⊂ K_{𝔄} (in the notation of [BG] F [γ_{1}]^{×} ⊂ K), because, as we still remarked above, M[β]^{×} ⊂ K_{𝔄}. Therefore, we can follow the proof of [BG](5.1) as before to show that the stratum [𝔄, n, q, γ] (in the notation of [BG] [𝔄, n, q, γ_{1}]) is pure (in the sense of [BG]) and equivalent to [𝔄, n, q, β]. Clearly in our situation the ℭ-soundness of β implies its 𝔄-soundness and we have finished the main part of our proposition. The last assertion is straightforward.

□

\underline{Remark:} (i) From [BG](5.1)(ii) we deduce that all pure strata [𝔄, n, q, γ] that are equivalent to [𝔄, n, q, β] such that [F [γ] : F] has minimal degree among all strata fulfilling the same property are necessarily simple. Moreover, in this situation we have f(F [γ]/F) | f(F [β]/F) and e(F [γ]/F) | e(F [β]/F).

(ii) From (2.6)(iii) and in the notation of (2.5),(3.8) one deduces that if \frac{rd}{e(E/F)} | q, then we have q′ = \frac{q}{e_{𝔄/ℭ}}. □

The last proposition makes it possible to define the notion of an approximation sequence:

( 3.9 ) Definition

(A) Let [𝔄, n, l, β] be a sound stratum, then the last jump of the stratum is the smallest integer q that is greater than or equal to l and smaller than or equal to n such that there exists a sound stratum [𝔄, n, q, γ] equivalent to [𝔄, n, q, β] with the property that [F [γ] : F] < [F [β] : F].

(B) Let [𝔄, n, 0, β] be a simple stratum, then the last proposition enables us to find for each integer q, where 0 ≤ q ≤ n, a simple stratum [𝔄, n, q, γ^{(q)}] that is equivalent to [𝔄, n, q, β] and, of course, [𝔄, n, q_{2}, γ^{(q_{2})}] is equivalent to [𝔄, n, q_{2}, γ^{(q_{1})}] if q_{2} ≥ q_{1}. An approximation sequence for [𝔄, n, 0, β] is a sequence of simple strata [𝔄, n, q_{i}, γ_{i}], i = 1, …, μ, μ ∈ ℕ, where

(i) q_{1} = 0 and γ_{1} = β,

(ii) q_{μ} = n and γ_{μ} = 0,

(iii) q_{i+1} > q_{i} is the last jump for the stratum [𝔄, n, q_{i}, γ_{i}] in the sense of (A) and, finally, (iv) [𝔄, n, q_{i+1}, γ_{i+1}] is a simple stratum equivalent to [𝔄, n, q_{i+1}, γ_{i}] for i = 1, …, μ − 1. The numbers q_{1}, …, q_{μ} called the jumps of the approximation sequence. (C) A sequence [𝔄, n, q_{i}, γ_{i}], i = 1, …, μ, where we have q_{1} < q_{2} < … < q_{μ} and μ ∈ ℕ and which

fulfils (A)(i)(ii)(iv) and contains an approximation sequence for the simple stratum [𝔄, n, 0, β] in the sense of (B) is called a weak approximation sequence for the simple stratum [𝔄, n, 0, β] if we have \frac{rd}{e(F [γ_{i−1}]/F)} | q_{i} for all i = 2, …, μ. Then we also call the numbers q_{1}, …, q_{μ} the jumps of the weak approximation sequence.

\underline{Remark:} From [BG](Theorem (5.1)(ii)) and the construction it is clear that (if n > 0) such an approximation sequence for [𝔄, n, 0, β] always exists and that the sequence of numbers q_{1}, …, q_{μ} is the same for all approximation sequences. In fact, we have q_{i+1} = −k_{0}(γ_{i}, 𝔄) for i = 1, …, μ − 2. In particular, all approximation sequences are weak approximation sequences.

( 3.10 ) Definition A simple stratum [𝔄, n, 0, β] with n > 0 that has an approximation sequence of length μ = 2 is said to be minimal over F.

If [𝔄, n, q_{i}, γ_{i}], i = 1, …, μ, where n > 0, is an approximation sequence to a simple stratum, then it is easy to see that the stratum [𝔄, n, 0, γ_{j}] is simple and has an approximation sequence [𝔄, n, q_{k}, γ_{k}], k = j, …, μ for j = 1, …, μ − 1. In particular, the stratum [𝔄, n, 0, γ_{μ−1}] is minimal over F. These facts enable us to introduce inductive arguments along an approximation sequence as in [BK3].

The main statement of this section is a consequence of the stronger version (3.8) of [BG](5.1):

( 3.11 ) Theorem

Let [𝔄, n, 0, β] be a simple stratum and K be the maximal unramified part of the extension E/F, where E := F [β]. Let C := C_{A}(K) be the centralizer of K in A and let ℭ := 𝔄 ∩ C. By assumption the embedding K^{×} ⊂ K_{𝔄} is sound and, therefore, ℭ is a principal order. Moreover, the stratum [ℭ, σ_{𝔄/ℭ}(n), 0, β] is simple and in this situation, there exists an approximation sequence [𝔄, n, q_{i}, γ_{i}], i = 1, …, μ, for [𝔄, n, 0, β] with the following properties:

(i) K = F [γ_{1}]^{ur} ⊃ F [γ_{2}]^{ur} ⊃ … ⊃ F [γ_{μ}]^{ur}, where F [γ_{i}]^{ur} is the maximal unramified part of the extension F [γ_{i}]/F for i = 1, …, μ and

(ii) [ℭ, σ_{𝔄/ℭ}(n), σ_{𝔄/ℭ}(q_{i}), γ_{i}], i = 1, …, μ, is a weak approximation sequence for [ℭ, σ_{𝔄/ℭ}(n), 0, β].

\underline{Proof:}

First choose an arbitrary approximation sequence with jumps q_{1} < … < q_{μ} and apply (3.8) to the stratum [𝔄, n, q_{2}, β] with M = K to obtain the first two special approximations [𝔄, n, q_{1}, γ_{1}] = [𝔄, n, 0, β] and [𝔄, n, q_{2}, γ_{2}] = [𝔄, n, q_{2}, γ]. Clearly, K[γ_{2}] is a field and we have performed the beginning of an induction over μ.

So let us assume that we have approximations [𝔄, n, q_{1}, γ_{1}], …, [𝔄, n, q_{i−1}, γ_{i−1}] such that K[γ_{1}], …, K[γ_{i−1}] are fields, then (3.8) applies again and the obtained approximation [𝔄, n, q_{i}, γ_{i}] has also the property that K[γ_{i}] is a field.

Now we obtain the approximation sequence inductively with the above properties as required. □ This special kind of approximations is not necessary for introducing simple characters, but one can

use such approximations for the later matching of simple characters among various simple algebras.

( 3.12 ) Definition An approximation sequence as in (3.11) is said to be special.

4 Orbitfilters of additive characters and their stabilizers

Let 𝔄 be a principal order and [𝔄, n, 0, β] be a simple stratum. Moreover, use the conventions for E, B and 𝔅 from section (2). We define ψ_{β}^{+} as in section (1). In the next section (5) we want to construct certain admissible pairs to representation filters of the desending series U^{1}(𝔄) ⊃ U^{2}(𝔄) ⊃ U^{3}(𝔄) ⊃ … (see [Z2]) and in section (6) we want to match them with similar admissible pairs in other simple algebras with the later aim of generalizing the results of [Z3] and [Z4]. The admissible pairs generate stable representation filters, and as in [Z5] (or [Z7]) we want to obtain the associated groups as stabilizers of certain orbitfilters associated with ψ_{β}^{+}. In [BK3] the underlying Heisenberg characters are called ‘simple characters’ and we want to use the same conventions. As in [Br3] we restrict ourselves to the split case by using similar constructions for the canonical splittings, where we use the conventions of section (3) above for Ã and 𝔄̃.

Let us consider the family of orbits F(ψ_{β}^{+}) := {(U^{j}(𝔄)) ∘ ψ_{β|𝔓^{j}}^{+}}_{j∈ℕ}, where U^{j}(𝔄) ∘ ψ_{β|𝔓^{j}}^{+} means the character orbit of ψ_{β|𝔓^{j}}^{+} under the adjoint action of U^{j}(𝔄) and ψ_{β|𝔓^{j}}^{+} is the restriction of ψ_{β}^{+} to 𝔓^{j}.

Let us fix a weak approximation sequence [𝔄, n, q_{i}, γ_{i}], i = 1, …, μ for [𝔄, n, 0, β].

( 4.1 ) Definition (i) Set J^{1}(ψ_{β}^{+}) := Stab_{U^{1}(𝔄)}(F(ψ_{β}^{+})), where U^{1}(𝔄) acts again by conjugation, (ii) let 1 + 𝔍^{1}(ψ_{β}^{+}) := J^{1}(ψ_{β}^{+}), (iii) let ℌ^{1}(ψ_{β}^{+}) := 𝔍^{1}(ψ_{β}^{+})^{⊥ψ_{β}^{+}}, (iv) let H^{1}(β, 𝔄) := 1 + ℌ^{1}(ψ_{β}^{+}),

(v) let 𝔍^{1}(β, 𝔄) := ∑_{i=1}^{μ} 𝔓^{[ \frac{q_{i}+1}{2} ]} ∩ C_{𝔓}(γ_{i}),

(vi) let J^{1}(β, 𝔄) := 1 + 𝔍^{1}(β, 𝔄) and, finally,

(vii) let ℌ^{1}(β, 𝔄) := ∑_{i=1}^{μ} 𝔓^{[ \frac{q_{i}}{2} ]+1} ∩ C_{𝔓}(γ_{i}).

Clearly, by definition, 𝔍^{1}(ψ_{β}^{+}) and ℌ^{1}(ψ_{β}^{+}) are rings that are normalized by K_{𝔅}. From (1.2) we conclude that we have (𝔓^{i})^{⊥ψ_{β}^{+}} = {x ∈ 𝔓 | ∀y ∈ 𝔓^{i} : ψ_{β}^{+}(xy − yx) = ψ_{a_{β}(x)}^{+}(y) = 1} = {x ∈ 𝔓 | a_{β}(x) ∈ 𝔓^{−i+1}} = 𝔓 ∩ 𝔑_{−i+1}(β, 𝔄) for all i ∈ ℤ; hence, from [Z7](1.8) we obtain:

( 4.2 ) Lemma (i) 𝔍^{1}(ψ_{β}^{+}) = ∑_{i=1}^{∞} 𝔓^{i} ∩ 𝔑_{−i}(β, 𝔄) and (ii) ℌ^{1}(ψ_{β}^{+}) = ∑_{i=1}^{∞} 𝔓^{i} ∩ 𝔑_{−i+1}(β, 𝔄).

□

From these descriptions of the rings 𝔍^{1}(ψ_{β}^{+}), ℌ^{1}(ψ_{β}^{+}) and the definition of a weak approximation sequence ((β − γ_{i}) ∈ 𝔓^{−q_{i}} and −a_{γ_{i}}(C_{A}(γ_{i})) = {0} for i = 1, …, μ) one easily deduces the inclusions 𝔍^{1}(β, 𝔄) ⊂ 𝔍^{1}(ψ_{β}^{+}) and ℌ^{1}(β, 𝔄) ⊂ ℌ^{1}(ψ_{β}^{+}).

To see the opposite inclusion, let us consider the same situation in the split algebra Ã. For that purpose let ψ̃_{β}^{+} be the additive character of the Jacobson radical 𝔓̃ of 𝔄̃ defined similarly as ψ_{β}^{+} of 𝔓. By Lemma (4.2)(ii) and [BK3]((3.1.9)(i) with t = 2i − 1 for i = 1, 2, …, its proof, induction over the weak approximation sequence, and (3.1.2)) we have ℌ^{1}(β, 𝔄̃) ⊃ ℌ^{1}(ψ̃_{β}^{+}). By Lemma (4.2)(i) and [BK3]((3.1.10)(i) with t = 2i for i = 1, 2, …, its proof, induction over the weak approximation sequence and (3.1.2)) we have 𝔍^{1}(β, 𝔄̃) ⊃ 𝔍^{1}(ψ̃_{β}^{+}). As the opposite inclusions hold by the same argument as above, we deduce that ℌ^{1}(β, 𝔄̃) ∩ A = ℌ^{1}(ψ̃_{β}^{+}) ∩ A and 𝔍^{1}(β, 𝔄̃) ∩ A = 𝔍^{1}(ψ̃_{β}^{+}) ∩ A. Similarly as in [Br3] we conclude:

( 4.3 ) Proposition We have ℌ^{1}(β, 𝔄) = ℌ^{1}(ψ_{β}^{+}) and 𝔍^{1}(β, 𝔄) = 𝔍^{1}(ψ_{β}^{+}). In particular, H^{1}(β, 𝔄) and J^{1}(β, 𝔄) are groups that are normalized by K_{𝔅}. Moreover, the definitions of the latter groups are, in fact, independent of the special choice of the fixed weak approximation sequence [𝔄, n, q_{i}, γ_{i}], i = 1, …, μ for [𝔄, n, 0, β].

Another important property follows easily by the explicit description of Lemma (4.2):

( 4.4 ) Lemma (i) ℌ^{1}(β, 𝔄) is a two-sided ideal in 𝔍^{1}(β, 𝔄) and (ii) ℌ^{1}(β, 𝔄), 𝔍^{1}(β, 𝔄) are (𝔅, 𝔅)-bimodules.

□

As an example of an inductive argument along the weak approximation sequence for [𝔄, n, 0, β], we can generalize the Propositions [BK3](3.1.16) and [BK3](3.1.22). Following the proofs of [BK3] argument by argument without great change and using arguments as in [Br3] to obtain group inclusions for subgroups of A from the split case in Ã, we obtain:

( 4.5 ) Proposition Let [𝔄, n, 0, β] be a simple stratum with weak approximation sequence [𝔄, n, q_{i}, γ_{i}], i = 1, …, μ, where μ > 1, then we have:

(i) For 0 ≤ ν ≤ q_{2} − 1 we have an exact sequence

𝔓^{q_{2}−ν} ∩ 𝔑_{−ν}(β, 𝔄) + 𝔓^{[ \frac{q_{2}+1}{2} ]} ∩ 𝔍^{1}(β, 𝔄) →^{a_{β}} (𝔓^{ν+1} ∩ ℌ^{1}(β, 𝔄))^{∗} →^{𝔰_{β}} 𝔔^{σ_{𝔄/𝔅}(−ν)} → 0.

(ii) For 1 ≤ ν ≤ q_{2} we have an exact sequence

𝔓^{1+q_{2}−ν} ∩ 𝔑_{1−ν}(β, 𝔄) + 𝔓^{[ \frac{q_{2}}{2} ]+1} ∩ ℌ^{1}(β, 𝔄) →^{a_{β}} (𝔍^{ν}(β, 𝔄))^{∗} →^{𝔰_{β}} 𝔔^{σ_{𝔄/𝔅}(1−ν)} → 0.

(iii) In particular, for 1 ≤ ν ≤ [\frac{q_{2}+1}{2}], we have an exact sequence

0 → 𝔔^{σ_{𝔄/𝔅}([ \frac{q_{2}}{2} ]+1)} → 𝔓^{[ \frac{q_{2}}{2} ]+1} ∩ ℌ^{1}(β, 𝔄) →^{a_{β}} (𝔍^{ν}(β, 𝔄))^{∗} →^{𝔰_{β}} 𝔔^{σ_{𝔄/𝔅}(1−ν)} → 0.

□

5 Simple characters for principal orders

In this section we want to construct a K_{𝔅}J^{1}(β, 𝔄) stable character θ_{β} of H^{1}(β, 𝔄) with the property that we have θ_{β}((1 + x)(1 + y)(1 + x)^{−1}(1 + y)^{−1}) = ψ_{β}^{+}(xy − yx) for all 1 + x, 1 + y ∈ J^{1}(β, 𝔄). By construction it is clear that the right-hand side of the equation defines a non-degenerate alternating form 𝔍^{1}(β, 𝔄)/ℌ^{1}(β, 𝔄)×𝔍^{1}(β, 𝔄)/ℌ^{1}(β, 𝔄) → ℂ^{×}; hence, the left-hand side defines a non-degenerate alternating form J^{1}(β, 𝔄)/H^{1}(β, 𝔄)×J^{1}(β, 𝔄)/H^{1}(β, 𝔄) → ℂ^{×}, too. Consequently, θ_{β} determines a K_{𝔅}J^{1}(β, 𝔄) stable Heisenberg representation η_{β} of J^{1}(β, 𝔄), which is the admissible pair, and which we referred to at the beginning of the previous section (4).

Let us be more definite now. Again, let [𝔄, n, 0, β] be a simple stratum and let us fix a weak approximation sequence [𝔄, n, q_{i}, γ_{i}], i = 1, …, μ. Further, let us use the notation for 𝔓, E, B, 𝔅 and 𝔔 as above. If n = 0, we set η_{β} = θ_{β} ≡ 1. Otherwise we have μ ≥ 2 and can construct our simple character inductively along the weak approximation sequence.

For this purpose we first assume that [𝔄, n, 0, β] is minimal over F. In this case H^{1}(β, 𝔄) = 1 + 𝔔 + 𝔓^{[ \frac{n}{2} ]+1} = U^{1}(𝔅)U^{[ \frac{n}{2} ]+1}(𝔄) and ψ_{β}^{×} is a character of U^{[ \frac{n}{2} ]+1}(𝔄)/U^{n+1}(𝔄). Clearly, we can restrict ψ_{β}^{×} to U^{[ \frac{n}{2} ]+1}(𝔄) ∩ B = U^{σ_{𝔄/𝔅}([ \frac{n}{2} ]+1)}(𝔅) and using the duality (1.4) and Lemma (1.9) we see that this restriction of ψ_{β}^{×} factors through the reduced norm Nrd_{B/E}. Therefore it makes sense to define the following:

( 5.1 ) Definition If the stratum [𝔄, n, 0, β] is minimal over F, then a simple character θ_{β} is a character chosen as the unique character, which is defined by ψ_{β}^{×} on U^{[ \frac{n}{2} ]+1}(𝔄) and by λ_{β} ∘Nrd_{B/E} on U^{1}(𝔅), where λ_{β} is a character of U^{1}(𝔬_{E}) such that (λ_{β} ∘ Nrd_{B/E})_{|U^{σ_{𝔄/𝔅}([ \frac{n}{2} ]+1)}(𝔅)} = (ψ_{β}^{×})_{|U^{σ_{𝔄/𝔅}([ \frac{n}{2} ]+1)}(𝔅)}.

From the latter definition it is clear that, in the minimal case, θ_{β} is normalized by K_{𝔅}.

Now let [𝔄, n, 0, β] be an arbitrary simple stratum with n > 0 again. We can choose a simple character θ_{γ_{μ−1}} for the minimal stratum [𝔄, n, 0, γ_{μ−1}] and define a simple character θ_{β} by an inductive procedure along the weak approximation sequence for [𝔄, n, 0, β].

Clearly we do not lose anything of the generality of our arguments if we assume that a simple character is already defined for γ_{2}, … γ_{μ−1} and define θ_{β} under the assumption of the knowledge of a simple character θ_{γ}, where γ := γ_{2}, because this is the inductionstep we are looking for. Throughout this section let q := q_{2}. Note that since 𝔓^{[ \frac{q}{2} ]+1} ∩ ℌ^{1}(γ, 𝔄) is a (𝔅, 𝔅)-bimodule normalized by K_{𝔅} and because of Lemma (4.2), we have H^{1}(β, 𝔄) = U^{1}(𝔅)(U^{[ \frac{q}{2} ]+1}(𝔄) ∩ H^{1}(γ, 𝔄)) and each factor of the right-hand side is normalized by K_{𝔅}. What we need for the inductive definition is the proof of the following Lemma:

( 5.2 ) Lemma Assume that we have a simple character θ_{γ} of H^{1}(γ, 𝔄), then the character (ψ_{β−γ}^{×}θ_{γ})_{|U^{[ \frac{q}{2} ]+1}(𝔄)∩H^{1}(γ,𝔄)} is normalized by K_{𝔅} and, if restricted to U^{[ \frac{q}{2} ]+1}(𝔄) ∩ H^{1}(γ, 𝔄) ∩ B, it factors through the reduced norm Nrd_{B/E}.

Note that by (1.9) this Lemma holds true if the stratum [𝔄, n, 0, β] is minimal over F, if we set θ_{γ_{μ}} ≡ 1 and H^{1}(γ_{μ}, 𝔄) := U^{1}(𝔄), where we already know that q_{μ} = n and γ_{μ} = 0. Thus, under the assumption that Lemma (5.2) holds true in general, the following definition (similar to [BK3](3.2.3)) makes sense:

( 5.3 ) Definition (i) Suppose that a simple character θ_{γ} for the simple stratum [𝔄, n, 0, γ] is already chosen, then a simple character θ_{β} for [𝔄, n, 0, β] is a character chosen as the unique character, which is defined by (ψ_{β−γ}^{×}θ_{γ})_{|U^{[ \frac{q}{2} ]+1}(𝔄)∩H^{1}(γ,𝔄)} on U^{[ \frac{q}{2} ]+1}(𝔄) ∩ H^{1}(γ, 𝔄) and by λ_{β}∘Nrd_{B/E} on U^{1}(𝔅), where λ_{β} is a character of U^{1}(𝔬_{E}) such that (λ_{β} ∘ Nrd_{B/E})_{|U^{σ_{𝔄/𝔅}([ \frac{q}{2} ]+1)}(𝔅)} = (ψ_{β−γ}^{×}θ_{γ})_{|U^{σ_{𝔄/𝔅}([ \frac{q}{2} ]+1)}(𝔅)}. We also demand from a simple character that it is normalized by the whole of K_{𝔅}.

(ii) Denote the set of all simple characters for [𝔄, n, 0, β] by C(β, 𝔄)

From each step of the inductive definition of a simple character θ_{β} along a weak approximation sequence [𝔄, n, q_{i}, γ_{i}], i = 1, …, μ we obtain a uniquely defined character λ_{γ_{i}}, i = 1, …, μ, and conversely from the sequence {(γ_{i}, λ_{γ_{i}})}_{i=1,...,μ} we can recover simple characters θ_{γ_{μ−1}}, …, θ_{γ_{2}} and finally θ_{γ_{1}} = θ_{β}.

( 5.4 ) Definition (i) A sequence {(γ_{i}, λ_{γ_{i}})}_{i=1...,μ} as we obtain from the inductive definition of a simple character θ_{β} is called a defining sequence for θ_{β}.

(ii) We denote by D(β, 𝔄) the set of all defining sequences for simple characters for the stratum [𝔄, n, 0, β], where we also vary over all possible weak approximation sequences.

Clearly, a defining sequence for a simple character is not unique, as is a weak approximation sequence [𝔄, n, 0, γ_{i}], i = 1, …, μ for a simple stratum [𝔄, n, 0, β]. Until proving Corollary (5.14)(5.15) we have to fix our weak approximation sequence.

To prove our main Lemma (5.2), we first need a result about the intertwining of the simple characters that already exist. Then the property that θ_{β} is normalized by the whole of K_{𝔅} will also follow, because K_{𝔅} normalizes J^{1}(β, 𝔄). To simplify notation, we sometimes assume that there already exists a simple character for [𝔄, n, 0, β], if it is allowed according to our induction argument.

( 5.5 ) Notation For x, y ∈ 𝔓 set [1 + x, 1 + y] := (1 + x)(1 + y)(1 + x)^{−1}(1 + y)^{−1}.

First we prove the main property of simple characters which is a generalization of [Z7](4.3 miii):

( 5.6 ) Proposition Let θ_{β} be a simple character, then for all x, y ∈ 𝔍^{1}(β, 𝔄) we have

θ_{β}([1 + x, 1 + y]) = ψ_{β}^{+}(xy − yx).

\underline{Proof:}

First let us assume that [𝔄, n, 0, β] is minimal over F. We write 1 + x, 1 + y ∈ J^{1}(β, 𝔄) in the form (1 + x) = (1 + x_{1})(1 + x_{2}) and (1 + y) = (1 + y_{1})(1 + y_{2}), where 1 + x_{1}, 1 + y_{1} ∈ U^{1}(𝔅) and 1 + x_{2}, 1 + y_{2} ∈ U^{[ \frac{n+1}{2} ]}(𝔄). With these conventions we obtain θ_{β}([1 + x, 1 + y]) =

θ_{β}([(1 + x_{1})(1 + x_{2}), (1 + y_{1})(1 + y_{2})]) =  θ_{β}(^{(1+x_{1})}[1 + x_{2}, 1 + y])θ_{β}([1 + x_{1}, 1 + y]) =

θ_{β}([1 + x_{2}, 1 + y])θ_{β}([1 + x_{1}, 1 + y]), because, by construction, θ_{β} is invariant under the adjoint action of K_{𝔅}. Moreover, [1 + x_{1}, 1 + y] = 1 + (x_{1}y − yx_{1})(1 + x_{1})^{−1}(1 + y)^{−1} ∈ U^{[ \frac{n}{2} ]+1}(𝔄); hence, θ_{β}([1 + x_{1}, 1 + y]) = ψ_{β}^{×}([1 + x_{1}, 1 + y]) = ψ_{β}^{+}((x_{1}y − yx_{1})(1 + x_{1})^{−1}(1 + y)^{−1}) = ψ_{β}^{+}(x_{1}y − yx_{1}) = 1, because ψ_{β}^{+} is trivial on 𝔓^{n+1} and β commutes with x_{1}. With the same kind of arguments we have θ_{β}([1 + x_{2}, 1 + y]) = θ_{β}([1 + x_{2}, 1 + y_{1}])θ(^{(1+y_{1})}[1 + x_{2}, 1 + y_{2}]) =

θ_{β}([1 + x_{2}, 1 + y_{2}]) = ψ_{β}^{×}(1 + (x_{2}y_{2} − y_{2}x_{2})(1 + x_{2})^{−1}(1 + y_{2})^{−1}) = ψ_{β}^{+}(x_{2}y_{2} − y_{2}x_{2}) = ψ_{β}^{+}(xy − yx),

where the latter equation again holds true because x_{1}, y_{1} commute with β and 2[\frac{n+1}{2}] + 1 ≥ n + 1.

So let us turn to the general case and assume that our assertion is true for θ_{γ} and, as above, we can set 1 + x = (1 + x_{1})(1 + x_{2}) and

1 + y = (1 + y_{1})(1 + y_{2}),  where  x_{1}, y_{1} ∈ 𝔔  and  x_{2}, y_{2} ∈ 𝔓^{[ \frac{q+1}{2} ]} ∩ 𝔍^{1}(β, 𝔄).  As
above  we  obtain  θ_{β}([1 + x, 1 + y]) =  θ_{β}(^{(1+x_{1})}[1 + x_{2}, 1 + y])θ_{β}([1 + x_{1}, 1 + y]) =
θ_{β}([1 + x_{2}, 1 + y])θ_{β}([1 + x_{1}, 1 + y]) = θ_{β}([1 + x_{2}, 1 + y_{1}])θ(^{(1+y_{1})}[1 + x_{2}, 1 + y_{2}])θ_{β}([1 + x_{1}, 1 + y]) =

θ_{β}([1 + x_{2}, 1 + y_{1}])θ_{β}([1 + x_{2}, 1 + y_{2}])θ_{β}([1 + x_{1}, 1 + y]). Again one has to use similar arguments to prove the triviality of θ_{β}([1 + x_{2}, 1 + y_{1}]) and θ_{β}([1 + x_{1}, 1 + y]). As an example, we show that θ_{β}([1 + x_{2}, 1 + y_{1}]) = 1. By Lemma (4.4) [1 + x_{2}, 1 + y_{1}] = 1 + (x_{2}y_{1} − y_{1}x_{2})(1 + x_{2})^{−1}(1 + y_{1})^{−1} ∈ (𝔓^{[ \frac{q}{2} ]+1} ∩ ℌ^{1}(β, 𝔄)) = (𝔓^{[ \frac{q}{2} ]+1} ∩ ℌ^{1}(γ, 𝔄)); hence, by induction we have θ_{β}([1 + x_{2}, 1 + y_{1}]) = ψ_{β−γ}^{×}(1 + (x_{2}y_{1} − y_{1}x_{2})(1 + x_{2})^{−1}(1 + y_{1})^{−1})θ_{γ}([1 + x_{2}, 1 + y_{1}]) = ψ_{β−γ}^{+}(x_{2}y_{1} − y_{1}x_{2})ψ_{γ}^{+}(x_{2}y_{1} − y_{1}x_{2}) = ψ_{β}^{+}(x_{2}y_{1} − y_{1}x_{2}) = 1. Here we also have to note that ψ_{β−γ}^{+} is trivial on 𝔓^{q+1}. Finally, we obtain θ_{β}([1 + x_{2}, 1 + y_{2}]) = ψ_{β−γ}^{+}(x_{2}y_{2} − y_{2}x_{2})ψ_{γ}^{+}(x_{2}y_{2} − y_{2}x_{2}) = ψ_{β}^{+}(xy − yx), where the last equation again holds because x_{1} + x_{2} + x_{1}x_{2} ≡ x_{2} (mod ℌ^{1}(β, 𝔄)) and y_{1} + y_{2} + y_{1}y_{2} ≡ y_{2} (mod ℌ^{1}(β, 𝔄)) and Definition (4.1)(i)(ii)(iii).

□

As another example of an inductive argument along the weak approximation sequence for [𝔄, n, 0, β], we can generalize the Propositions [BK3](3.2.6) and [BK3](3.2.8)(3.2.11). Following the proofs of [BK3] argument by argument without great change and using arguments as in [Br3] to obtain group inclusions for subgroups of A from the split case in Ã, we obtain:

( 5.7 ) Proposition Let θ_{β} be a simple character and [\frac{q}{2}] ≤ ν ≤ q − 1. Let k, l ≥ 1 be integers such that k + l ≥ ν + 1 and k + 2l ≥ q + 1. If, moreover, x ∈ 𝔓^{k} ∩ 𝔑_{−q+k}(β, 𝔄) and y ∈ 𝔓^{l} ∩ 𝔑_{−q+l}(β, 𝔄), then the commutator [1+x, 1+y] is in U^{ν+1}(𝔄)∩H^{1}(β, 𝔄) and θ_{β}([1+x, 1+y]) = ψ_{(1+x)^{−1}β(1+x)−β}^{×}(1+ y).

□

( 5.8 ) Proposition Let [\frac{q}{2}] ≤ ν ≤ q − 1 and θ_{β} be a simple character. Let k, l ≥ 1 be integers such that k + l ≥ ν + 1 and k + 2l ≥ q + 1. If, moreover, x ∈ 𝔓^{k} ∩ 𝔑_{−r+k}(β, 𝔄) and y ∈ 𝔓^{l} ∩ 𝔍^{1}(β, 𝔄), then [1 + x, 1 + y] is in U^{ν+1}(𝔄) ∩ H^{1}(β, 𝔄) and θ_{β}([1 + x, 1 + y]) = ψ_{(1+x)^{−1}β(1+x)−β}^{×}(1 + y).

□

( 5.9 ) Corollary Let [\frac{q}{2}] ≤ ν ≤ q − 1 and θ_{β} be a simple character. Let k, l ≥ 1 be integers such that k + l ≥ ν + 1 and k + 2l ≥ q + 1. If, moreover, x ∈ 𝔓^{k} ∩ 𝔑_{−r+k}(β, 𝔄) and y ∈ 𝔓^{l} ∩ ℌ^{1}(β, 𝔄), then [1 + x, 1 + y] is in U^{ν+1}(𝔄) ∩ H^{1}(β, 𝔄) and θ_{β}([1 + x, 1 + y]) = ψ_{(1+x)^{−1}β(1+x)−β}^{×}(1 + y).

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( 5.10 ) Proposition If θ_{β} is a simple character for [𝔄, n, 0, β] and 0 ≤ ν ≤ q − 1, then (θ_{β})_{|U^{ν+1}(𝔄)∩H^{1}(β,𝔄)} is normalized by J^{1}(β, 𝔄).

\underline{Proof:}

Follows from the fact that J^{1}(β, 𝔄) normalizes U^{ν+1}(𝔄) ∩ H^{1}(β, 𝔄), Proposition (5.6) and the definition of 𝔍^{1}(β, 𝔄), ℌ^{1}(β, 𝔄), respectively.

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The result on the intertwining of simple characters, which we need to prove our main Lemma (5.2), is a generalization of [BK3](3.3.2) and [Z7](4.5):

( 5.11 ) Theorem Let θ_{β} be a simple character and 0 ≤ ν ≤ q − 1, then the intertwining of (θ_{β})_{|U^{ν+1}(𝔄)∩H^{1}(β,𝔄)} consists of the union of all double classes

B^{×}//(1 + 𝔓^{q−ν} ∩ 𝔑_{−ν}(β, 𝔄) + 𝔓^{[ \frac{q+1}{2} ]} ∩ 𝔍^{1}(β, 𝔄)).

\underline{Proof:}

The proof is in fact the same as in [BK3](3.3.2). One has to use [Br2](4.1.1) instead of [BK3](1.5.8), [Br2](4.3.3) instead of [BK3](1.4.16), Proposition (4.5)(i) instead of [BK3](3.1.16), Proposition (5.7) instead of [BK3](3.2.6), Proposition (5.8) instead of [BK3](3.2.8) and Proposition (5.10) instead of [BK3](3.3.1). Moreover, one has to use arguments as in [Br3] to obtain inclusions of subrings of A from the split case in Ã. For example, to generalize Lemma [BK3](3.3.16) one has to use [BK3](1.3.16) in the split case and will obtain for x ∈ B^{×}: (𝔓^{ν+1} +x^{−1}𝔓^{ν+1}x)∩B = (𝔓̃^{ν+1} +x^{−1}𝔓̃^{ν+1}x)∩B̃ ∩A = [BK3](1.3.16) (𝔔̃^{ν+1} + x^{−1}𝔔̃^{ν+1}x) ∩ A = (3.3(ii)) 𝔔^{σ_{𝔄/𝔅}(ν+1)} + x^{−1}𝔔^{σ_{𝔄/𝔅}(ν+1)}x, where B̃ = CÃ(E) and 𝔔̃ = 𝔓̃ ∩ B̃.

Within the proof we also obtain a generalization of [BK3](3.3.9):

( 5.12 ) Proposition Let [𝔄, n, 0, β] be a simple non-minimal stratum and for a fixed l, 0 ≤ l ≤ q = q_{2} (q the last jump of our weak approximation sequence as above) let [𝔄, n, l, γ] be simple and equivalent to [𝔄, n, l, β]. Further, let t be the last jump of a weak approximation sequence for [𝔄, n, 0, γ] and let [\frac{l}{2}] ≤ ν ≤ q − 1. If θ_{β} is a simple character for [𝔄, n, 0, β], which is defined depending on a simple character θ_{γ} for [𝔄, n, 0, γ] as in (5.3), and if x is an element of a double class x̄ ∈ C_{A}(γ)^{×}//(1 + 𝔓^{t−l} ∩ 𝔑_{−t}(β, 𝔄) + 𝔓^{[ \frac{t+1}{2} ]} ∩ 𝔍^{1}(γ, 𝔄)), then we have for all h ∈ (x^{−1}(U^{ν+1}(𝔄) ∩ H^{1}(β, 𝔄))x ∩ (U^{ν+1}(𝔄) ∩ H^{1}(β, 𝔄)))

θ_{β}^{x}(h) = θ_{β}(h)ψ_{x^{−1}βx−β}^{×}(h),
θ_{γ}^{x}(h) = θ_{γ}(h)ψ_{x^{−1}γx−γ}^{×}(h).

\underline{Proof:}

In the split case (see [BK](3.3.9)) the proposition was formulated only for l = q, where the notation q = r was used. However, for the purposes of [BK3](3.3.20) the case of l < q has to be dealt with too, which is in fact implicitly given in the proofs of [BK](3.3.5) and [BK](3.3.9). Analogously, to prove (5.14) below, we need to show (5.12) for any l ≤ q. Proceeding as in [BK3] one has to replace [BK3](3.1.16) by our Proposition (4.5)(i).

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As a corollary we can now prove our main lemma:

Proof of Lemma (5.2):

Using [Br2](4.1.1) instead of [BK3](1.5.8) and Proposition (5.12) instead of [BK3](3.3.9) we can use the same arguments as in [BK3] to generalize [BK3](3.3.18) and obtain:

( 5.13 ) Corollary Let [\frac{q}{2}] ≤ ν ≤ q − 1, and assume that q < n. Let [𝔄, n, q, γ] be simple and equivalent to [𝔄, n, q, β], then the map

C(β, 𝔄)_{|U^{ν+1}(𝔄)∩H^{1}(β,𝔄)} → C(γ, 𝔄)_{|U^{ν+1}(𝔄)∩H^{1}(γ,𝔄)},

(θ_{β})_{|U^{ν+1}(𝔄)∩H^{1}(β,𝔄)} ↦ (θ_{β})_{|U^{ν+1}(𝔄)∩H^{1}(γ,𝔄)}(ψ_{β−γ}^{×})^{−1}

is bijective, where C(β, 𝔄)_{|U^{ν+1}(𝔄)∩H^{1}(β,𝔄)} and C(γ, 𝔄)_{|U^{ν+1}(𝔄)∩H^{1}(γ,𝔄)} denote the sets of restricted characters.

In fact, our lemma (5.2) is just a reformulation of (5.13).

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Following the arguments of [BK3], using Proposition (5.12) instead of [BK3](3.3.9) and [Br2](4.1.1) instead of [BK3](1.5.8), we can generalize [BK3](Corollary (3.3.20)) without great changes:

( 5.14 ) Corollary (i) The definition of the set C(β, 𝔄) is, in fact, independent of the special choice of a weak approximation sequence for [𝔄, n, 0, β].

(ii) Let 0 ≤ l ≤ q − 1, and let [𝔄, n, l, β′] be a simple stratum equivalent to [𝔄, n, l, β], then C(β′, 𝔄)_{|U^{[ \frac{l}{2} ]+1}(𝔄)∩H^{[ \frac{l}{2} ]+1}(β′,𝔄)} = C(β, 𝔄)_{|U^{[ \frac{l}{2} ]+1}(𝔄)∩H^{[ \frac{l}{2} ]+1}(β,𝔄)}ψ_{β′−β}^{×}, where C(β′, 𝔄)_{|U^{[ \frac{l}{2} ]+1}(𝔄)∩H^{[ \frac{l}{2} ]+1}(β′,𝔄)}, C(β, 𝔄)_{|U^{[ \frac{l}{2} ]+1}(𝔄)∩H^{[ \frac{l}{2} ]+1}(β,𝔄)} are the sets of restricted characters.

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Note that in the case l = 0, i.e. β − β′ ∈ 𝔄, we have ψ_{β′−β}^{×} ≡ 1. In particular, the set C(β, 𝔄) depends only on the equivalence class of the simple stratum [𝔄, n, 0, β]. As a reformulation of the last corollary we obtain:

( 5.15 ) Corollary (i) Let θ_{β} be a fixed simple character for the simple stratum [𝔄, n, 0, β], then, for each weak approximation sequence [𝔄, n, q_{i}, γ_{i}], i = 1, …, μ, we can find a defining sequence {(γ_{i}, λ_{γ_{i}})}_{i=1,...,μ} for θ_{β}.

(ii) The above inductive procedure of defining a simple character as in (5.1)(5.3) leads to a well defined surjective map D(β, 𝔄) → C(β, 𝔄) from defining sequences (see (5.4)) to simple characters.

For later purposes we need a proposition of [BF] which was published with a misprint. The version of [BF](2.8.3) that we use here is:

( 5.16 ) Lemma Let 𝔄 be a principal order with period r in a simple algebra A of index d. Let e_{𝔄/𝔬_{F}} := rd and let σ_{𝔄/𝔬_{F}} : ℤ → ℤ be given by σ_{𝔄/𝔬_{F}} (j) := [\frac{j+e_{𝔄/𝔬_{F}} −1}{e_{𝔄/𝔬_{F}}}] for j ∈ ℤ, then the restrictions of the reduced norm mapping Nrd_{A/F} induce surjective mappings U^{j}(𝔄) → U^{σ_{𝔄/𝔬_{F}} (j)}(𝔬_{F}) for all j ∈ ℕ.

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\underline{Remark:} Because the definition of σ_{𝔄/𝔬_{F}} and σ_{𝔄/𝔅} is analogous, also the analogue of Lemma (2.6)(i)(ii) holds true for e_{𝔄/𝔬_{F}} and σ_{𝔄/𝔬_{F}}. □

The following proposition makes the relationship between defining sequences and simple characters more obvious:

( 5.17 ) Proposition If {λ_{γ_{i}}}_{i=1,...,μ} is a defining sequence for a simple character θ_{β} ∈ C(β, 𝔄), then θ_{β} determines the characters λ_{γ_{i}} on the groups U^{s_{i}+1}(𝔬_{F [γ_{i}]}) with s_{i} = [\frac{σ_{𝔄∩C_{A}(γ_{i})/𝔬_{F [γ_{i}]}} (σ_{𝔄/𝔄∩C_{A}(γ_{i})}(q_{i}))}{2}] for i = 1, …, μ and, conversely, the restrictions of the λ_{γ_{i}} to the latter groups for i = 1, …, μ determine θ_{β}. Moreover, each sequence of continuations {λ′_{γ_{i}}}_{i=1,...,μ} of these restrictions to the whole groups U^{1}(𝔬_{F [γ_{i}]}) for i = 1, …, μ is a defining sequence for θ_{β}.

\underline{Proof:}

In the minimal case there is nothing to show because q_{1} = 0 and λ_{γ_{2}} ≡ 1; hence, θ_{β} determines its defining sequence uniquely. Suppose now that the assertion holds true for all simple characters θ_{γ} ∈ C(γ, 𝔄), where γ = γ_{2} is the last approximation of the simple stratum [𝔄, n, 0, β] of our given weak approximation sequence.

By (5.13) the defining character θ_{γ} of the inductive definition of θ_{β} is uniquely determined on U^{[ \frac{q}{2} ]+1}(𝔄) ∩ H^{1}(γ, 𝔄), where we have q = q_{2}. As H^{1}(γ, 𝔄) = U^{1}(𝔄 ∩ C_{A}(γ))(U^{[ \frac{q}{2} ]+1}(𝔄) ∩ H^{1}(γ, 𝔄)) and θ_{γ} is given on U^{1}(𝔄∩C_{A}(γ)) by λ_{γ} ∘Nrd_{A∩C_{A}(γ)/F [γ]}, we deduce from (5.16) that λ_{γ} is determined by θ_{β} on U^{[^{\frac{σ_{𝔄∩C_{A}(γ)/𝔬_{F [γ]}} (σ_{𝔄/𝔄∩C_{A}(γ)}(q))}{2}} ]+1}(𝔬_{F [γ]}), whereas a twist of λ_{γ} by an arbitrary character of U^{1}(𝔬_{F [γ]})/U^{[^{\frac{σ_{𝔄∩C_{A}(γ)/𝔬_{F [γ]}} (σ_{𝔄/𝔄∩C_{A}(γ)}(q))}{2}} ]+1}(𝔬_{F [γ]}) shifts θ_{γ} within C(γ, 𝔄) but has no influence on θ_{β}. Now our assertion follows by induction along the approximation sequence.

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6 Matching of simple characters among various algebras

In this section let n > 0 again. To prove the main result of this section, we need two propositions:

( 6.1 ) Proposition Let [𝔄, n, 0, β] be a simple stratum in a central simple algebra A/F and E := F [β]. Let K/F be the maximal unramified part of the extension E/F, C := C_{A}(K) and ℭ := 𝔄 ∩ C, then the stratum [ℭ, σ_{𝔄/ℭ}(n), 0, β] is simple and H^{1}(β, 𝔄) ∩ C ⊃ H^{1}(β, ℭ). Moreover, we assume that the simple characters C(β, ℭ) are defined with respect to the additive character ψ_{K} := ψ_{F} ∘ T r_{K/F} (which has conductor 𝔭_{K}). The map C(β, 𝔄) → C(β, ℭ) induced by restriction, is then well defined and injective. If [𝔄, n, q_{i}, γ_{i}], i = 1, …, μ, is a \underline{special} approximation sequence for [𝔄, n, 0, β] and θ_{β,𝔄} ∈ C(β, 𝔄) has a defining sequence {(γ_{i}, λ_{γ_{i}})}_{i=1,...,μ}, then {(γ_{i}, λ_{γ_{i}} ∘ Nrd_{K[γ_{i}]/F [γ_{i}]})}_{i=1,...,μ} is a defining sequence for the restriction θ_{β,ℭ} ∈ C(β, ℭ) of θ_{β,𝔄}. Finally, the image of the map is characterized by the property that for a defining sequence {(γ_{i}, λ′_{γ_{i}})}_{i=1,...,μ} of an element of the image, the characters λ′_{γ_{i}} factor through the norm Nrd_{K[γ_{i}]/F [γ_{i}]} for i = 1, …, μ.

\underline{Proof:}

By the remark following (3.8) we know that e_{𝔄/ℭ} | q_{i} for i = 1, …, μ; hence, from (4.1)(vii) and (2.6)(ii) we obtain ℌ^{1}(β, 𝔄) ∩ C = (∑_{i=1}^{μ} 𝔓^{[ \frac{q_{i}}{2} ]+1} ∩ C_{𝔓}(γ_{i})) ∩ C ⊃ ∑_{i=1}^{μ}(𝔓^{[ \frac{q_{i}}{2} ]+1} ∩ C_{𝔓}(γ_{i}) ∩ C) = ∑_{i=1}^{μ} ℜ^{σ_{𝔄/ℭ}([ \frac{q_{i}}{2} ]+1)} ∩ C_{ℜ}(γ_{i}) = ∑_{i=1}^{μ} ℜ^{[ \frac{σ_{𝔄/ℭ}(q_{i})}{2} ]+1} ∩ C_{ℜ}(γ_{i}) = ℌ^{1}(β, ℭ), where ℜ is the Jacobson radical of ℭ, as required. Let θ_{β,𝔄} be a simple character of H^{1}(β, 𝔄) and θ_{β,ℭ} its restriction to H^{1}(β, ℭ).

To prove the assertion about simple characters, let us use the method of induction along the weak approximation sequence:

To start the induction correctly, we have to assume the same situation for β as for a first approximation γ_{μ−1}; hence, that K ⊃ F [β]^{ur}, where F [β]^{ur} is the maximal unramified subextension of F [β]/F, that K[β] is a field not necessarily equal to E and, finally, that the embedding K[β]^{×} ⊂ K_{𝔄} is sound (i.e. β is ℭ-sound). If [𝔄, n, 0, β] is minimal over F, we have H^{1}(β, 𝔄) = U^{1}(𝔅)U^{[ \frac{n}{2} ]+1}(𝔄) and H^{1}(β, ℭ) = U^{1}(𝔅 ∩ C_{B}(K[β]))U^{[ \frac{σ_{𝔄/ℭ}(n)}{2} ]+1}(ℭ). If θ_{β,𝔄} is given on U^{1}(𝔅) by λ_{β} ∘ Nrd_{B/E}, then θ_{β,ℭ} on U^{1}(𝔅 ∩ C_{B}(K[β])) by (λ_{β} ∘ Nrd_{B/E})_{|C_{B}(K[β])} = λ_{β} ∘ Nr_{K[β]/E} ∘ Nrd_{C_{B}(K[β])/K[β]}. Moreover, because of (T rd_{A/F})_{|C} = T r_{K/F} ∘ T rd_{C/K} and ψ_{K} = ψ_{F} ∘ T r_{K/F}, we have that θ_{β,ℭ} is given on U^{[ \frac{σ_{𝔄/ℭ}(n)}{2} ]+1}(ℭ) by ψ_{β,K}^{×}, where ψ_{β,K}^{×} is defined similarly with respect to ψ_{K} as ψ_{β}^{×} with respect to ψ_{F}. It follows that θ_{β,ℭ} is a simple character. As the norm mapping Nr_{K[β]/E} : U^{1}(𝔬_{K[β]}) → U^{1}(𝔬_{E}) is surjective (see for example [Se](V,3,Prop. 3)), as the reduced norm mapping Nrd_{B/E} : U^{1}(𝔅) → U^{1}(𝔬_{E}) is surjective by (5.16) and as, again by (5.16), the reduced norm mapping Nrd_{C_{B}(K[β])/K[β]} : U^{1}(𝔅 ∩ C_{B}(K[β])) → U^{1}(𝔬_{K[β]}) is surjective too, we can recover θ_{β,𝔄} from θ_{β,ℭ}.

If, conversely, we start from a simple character θ_{β,ℭ} with defining sequence {(β, λ′_{β})} and assume that it has the form λ′_{β} = λ_{β} ∘ Nr_{K[β]/F [β]}, then, by Lemma 1.9, ψ_{β}^{×} is a character that factors on U^{[ \frac{σ_{𝔄/𝔅}(n)}{2} ]+1}(𝔅) through the reduced norm Nrd_{B/E} and its restriction to U^{[ \frac{σ_{𝔄/𝔅∩C_{B} (K[β])}(n)}{2} ]+1}(ℭ ∩ C_{A}(K[β])) agrees with ψ_{β,K}^{×}; hence, with the restriction of λ_{β} ∘Nr_{K[β]/F [β]} ∘ Nrd_{C_{A}(K[β])/K[β]}.

Again we have (Nrd_{B/E})_{|C_{B}(K[β])} = Nr_{K[β]/E} ∘ Nrd_{C_{B}(K[β])/K[β]}, (T rd_{A/F})_{|C} = T r_{K/F} ∘ T rd_{C/K} and the maps Nr_{K[β]/E} : U^{i}(𝔬_{K[β]}) → U^{i}(𝔬_{E}) are surjective for all i ≥ 1 (see [Se](V, 3, Prop. 3)). By (5.16), its concluding remark, e_{𝔅∩C_{B}(K[β])/𝔬_{K[β]}} | σ_{𝔄/𝔅∩C_{B}(K[β])}(n) (by (2.6)(ii)) and e_{𝔅/𝔬_{E}} | σ_{𝔄/𝔅}(n) (again by (2.6)(ii)), we have surjective maps Nrd_{C_{B}(K[β])/K[β]} : U^{[ \frac{σ_{𝔄/𝔅∩C_{B} (K[β])}(n)}{2} ]+1}(𝔅 ∩ C_{A}(K[β])) → U^{[^{\frac{σ_{𝔅∩C_{B} (K[β])/𝔬_{K[β]}} (σ_{𝔄/𝔅∩C_{B} (K[β])}(n))}{2}} ]+1}(𝔬_{K[β]}) and Nrd_{B/E} : U^{[ \frac{σ_{𝔄/𝔅}(n)}{2} ]+1}(𝔅) → U^{[ \frac{σ_{𝔅/𝔬_{E}} (σ_{𝔄/𝔅}(n))}{2} ]+1}(𝔬_{E}). Because of σ_{𝔅∩C_{B}(K[β])/𝔬_{K[β]}}(σ_{𝔄/𝔅∩C_{B}(K[β])}(n)) = −ν_{K[β]}(β) = −ν_{F [β]}(β) = σ_{𝔅/𝔬_{E}}(σ_{𝔄/𝔅}(n)) and U^{[ \frac{σ_{𝔄/𝔅}(n)}{2} ]+1}(𝔅) ∩ C_{B}(K[β]) = U^{[ \frac{σ_{𝔄/𝔅∩C_{B} (K[β])}(n)}{2} ]+1}(𝔅 ∩ C_{B}(K[β])), we conclude that ψ_{β}^{×} can be given on U^{[ \frac{σ_{𝔄/𝔅}(n)}{2} ]+1}(𝔅) only by the restriction of λ_{β} ∘ Nrd_{B/E} and θ_{β,ℭ} lifts to a simple character θ_{β,𝔄} of H^{1}(β, 𝔄) with defining sequence {(β, λ_{β})} as required. Therefore the assertion is true in the minimal case.

For the induction step let γ := γ_{2} be the last approximation of β and q := q_{2} be the last jump. Furthermore, assume that K ⊃ F [β]^{ur} and that K[β] is a field, which is not nessecarily equal to E. By the induction hypothesis we assume that the restriction θ_{γ,ℭ} of a simple character θ_{γ,𝔄} of H^{1}(γ, 𝔄) to H^{1}(γ, ℭ) is again simple and that one can recover θ_{γ,𝔄} from θ_{γ,ℭ}. We have H^{1}(β, 𝔄) = U^{1}(𝔅)(U^{[ \frac{q}{2} ]+1}(𝔄) ∩ H^{1}(γ, 𝔄)) and H^{1}(β, ℭ) = U^{1}(𝔅∩C_{B}(K[β]))(U^{[ \frac{σ_{𝔄/ℭ}(q)}{2} ]+1}(ℭ)∩H^{1}(γ, ℭ)). Again, as in the minimal case, if θ_{β,𝔄} is given on U^{1}(𝔅) by λ_{β} ∘ Nrd_{B/E}, then θ_{β,ℭ} on U^{1}(𝔅 ∩ C_{B}(K[β])) by λ_{β} ∘ Nrd_{K[β]/E} ∘ Nrd_{C_{B}(K[β])/K[β]}. Also θ_{β,ℭ} is given on U^{[ \frac{σ_{𝔄/ℭ}(q)}{2} ]+1}(ℭ) ∩ H^{1}(γ, ℭ) by ψ_{β−γ,K}^{×}θ_{γ,ℭ}; hence, θ_{β,ℭ} is a simple character. By induction and the same argument as in the minimal case, we can recover θ_{β,𝔄} from θ_{β,ℭ} and obtain a well defined injective map. Assuming the equation σ_{𝔅∩C_{B}(K[β])/𝔬_{K[β]}}(σ_{𝔄/𝔅∩C_{B}(K[β])}(q)) = σ_{𝔅/𝔬_{E}}(σ_{𝔄/𝔅}(q)), also the characterization of the image of the map holds true by the same argument as in the minimal case if we use (5.2) and the induction assumption that a simple character θ_{γ,ℭ} with defining sequence {(γ_{i}, λ_{γ_{i}} ∘ Nr_{K[γ_{i}]/F [γ_{i}]})}_{i=2,...,μ} lifts to a simple character θ_{γ,𝔄} with defining sequence {(γ_{i}, λ_{γ_{i}})}_{i=2,...,μ}.

For the required equation we just have to note that by (2.6)(iii) and e(K[β]/F) = e(E/F), we have e_{𝔅∩C_{B}(K[β])/𝔬_{K[β]}}e_{𝔄/𝔅∩C_{B}(K[β])} = \frac{rd}{e(E/F)} = e_{𝔅/𝔬_{E}}e_{𝔄/𝔅}; hence, we are done.

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( 6.2 ) Lemma Let [𝔄, n, q, β] be a sound stratum in a central simple algebra A/F and E := F [β]. Let L/F be an unramified extension such that E ⊗_{F} L is again a field, then the stratum [𝔄, n, q, β] in A is simple if and only if the stratum [𝔄 ⊗_{𝔬_{F}} 𝔬_{L}, n, q, β ⊗ 1] in A ⊗_{F} L is simple.

\underline{Proof:}

By definition (3.1)(iii) and [BK3](1.5.5) the stratum [𝔄, n, q, β] is simple if and only if −k_{0}(β, 𝔄̃) > q, where 𝔄̃ is the principal order attached to the same lattice chain in a simple A-module that is also a simple Ã-module and Ã the canonical splitting of A as described at the beginning of section (3). By Proposition (3.6) we have k_{0}(β, 𝔄) = k_{0}(β, 𝔄̃) and with similar notations we have k_{0}(β⊗1, 𝔄⊗_{𝔬_{F}} 𝔬_{L}) = k_{0}(β ⊗ 1, { \atop 𝔄 ⊗_{𝔬_{F}} 𝔬_{L}}). Now, clearly, our assertion follows from k_{0}(β, 𝔄) = k_{0}(β ⊗ 1, 𝔄 ⊗_{𝔬_{F}} 𝔬_{L}), which can be deduced from straightforward computations using the fact that if 𝔓 is the Jacobson radical of 𝔄, then 𝔓 ⊗_{𝔬_{F}} 𝔬_{L} is the Jacobson radical of 𝔄 ⊗_{𝔬_{F}} 𝔬_{L}.

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( 6.3 ) Lemma Let ψ_{F} be an additive character of F with conductor 𝔭_{F} and L/F an unramified extension, then there exists an extension ψ_{L} of ψ_{F} to L^{+} with conductor 𝔭_{L}.

\underline{Proof:}

The character ψ_{F} can be identified with the sequence {ψ_{F,i}}_{i=0,1,2,...} in the projective limit ←^{lim}_{i=0,1,2,...} { \atop 𝔭_{F}^{−i}/𝔭_{F}}, where ψ_{F,i} := (ψ_{F})_{|𝔭_{F}^{−i}/𝔭_{F}} for i = 0, 1, 2, . . .. Here the limit is to be under-

stood in such a way that a sequence \{χ_{i}\}_{i=0,1,2,...} ∈ ∏_{i=0,1,2,...} { \atop 𝔭_{F}^{−i}/𝔭_{F}} is in ←^{lim}_{i=0,1,2,...} { \atop 𝔭_{F}^{−i}/𝔭_{F}} if and

only if we have (χ_{k})_{|𝔭_{F}^{−l}/𝔭_{F}} = χ_{l} for all l < k.

In the same way we define ←^{lim}_{i=0,1,2,...} { \atop 𝔭_{L}^{−i}/𝔭_{L}}, and finding an extension ψ_{L} of ψ_{F} with conductor

𝔭_{L} is the same as finding a sequence \{ψ_{L,i}\}_{i=0,1,2,...} ∈←^{lim}_{i=0,1,2,...} { \atop 𝔭_{L}^{−i}/𝔭_{L}} such that (ψ_{L,i})_{|𝔭_{F}^{−i}/𝔭_{F}} = ψ_{F,i}

for i = 0, 1, 2, …, where we regard 𝔭_{F}^{−i}/𝔭_{F} as a subgroup of 𝔭_{L}^{−i}/𝔭_{L} via the natural embedding induced by the inclusions 𝔭_{F}^{−i} ⊂ 𝔭_{L}^{−i} for all i ≥ 0 and 𝔭_{F} ⊂ 𝔭_{L} (here we have to use the assumption that the extension L/F is unramified).

We construct ψ_{L} inductively. First we extend ψ_{F,0} to 𝔬_{L}/𝔭_{L} to obtain ψ_{L,0}, which provides the beginning of our induction. Now assume that for i ≥ 0 we have constructed extensions ψ_{L,k} of the characters ψ_{F,k} for 0 ≤ k ≤ i such that we have (ψ_{L,k})_{|𝔭_{L}^{−l}/𝔭_{L}} = ψ_{L,l} for l ≤ k and l, k ∈ {0, …, i}.

Now let ψ′ be an extension of ψ_{L,i} to 𝔭_{L}^{−i−1}/𝔭_{L}, then ψ′_{|𝔭_{F}^{−i−1}/𝔭_{F}} differs from ψ_{F,i+1} by a character χ of 𝔭_{F}^{−i−1}/𝔭_{F}^{−i}, which can be extended to a character χ′ of 𝔭_{L}^{−i−1}/𝔭_{L}^{−i}. Now ψ_{L,i+1} = ψ′χ′ extends ψ_{L,i} as well as ψ_{F,i+1} as required.

□

( 6.4 ) Proposition Let [𝔄, n, 0, β] be a simple stratum in a central simple algebra A/F and E := F [β]. Assume that E/F is totally ramified. Let L/F be an unramified splitting field of A, then the stratum [𝔄 ⊗_{o_{F}} o_{L}, n, 0, β ⊗ 1] is simple, ℌ^{1}(β ⊗ 1, 𝔄 ⊗_{𝔬_{F}} 𝔬_{L}) = ℌ^{1}(β, 𝔄) ⊗_{𝔬_{F}} 𝔬_{L}. Moreover, we assume that the simple characters C(β ⊗ 1, 𝔄 ⊗_{𝔬_{F}} 𝔬_{L}) are defined with respect to an additive character ψ_{L} with conductor 𝔭_{L} of L, which extends ψ_{F} (which exists by the above Lemma (6.3)). The map C(β ⊗ 1, 𝔄 ⊗_{𝔬_{F}} 𝔬_{L}) → C(β, 𝔄), induced by restricting a character of H^{1}(β ⊗ 1, 𝔄 ⊗_{𝔬_{F}} 𝔬_{L}) to the subset 1 ⊗ 1 + ℌ^{1}(β, 𝔄) ⊗ 1 ≃ H^{1}(β, 𝔄), is then well defined and surjective. If [𝔄, n, q_{i}, γ_{i}], i = 1, …, μ, is a weak approximation sequence for [𝔄, n, 0, β], then [𝔄 ⊗_{𝔬_{F}} 𝔬_{L}, n, q_{i}, γ_{i} ⊗ 1], i = 1, …, μ, is a weak approximation sequence for [𝔄 ⊗_{𝔬_{F}} 𝔬_{L}, n, 0, β ⊗ 1], and if θ_{β⊗1} ∈ C(β ⊗ 1, 𝔄 ⊗_{𝔬_{F}} 𝔬_{L}) has a defining sequence {(γ_{i} ⊗ 1, λ_{γ_{i}⊗1})}_{i=1,...,μ}, then {(γ_{i}, (λ_{γ_{i}⊗1})_{|U^{1}(𝔬_{F [γ_{i}]})}}_{i=1,...,μ} is a defining sequence for the restriction θ_{β} ∈ C(β, 𝔄) of θ_{β⊗1}. Moreover, to each fixed simple character θ_{β}, each fixed defining sequence {λ_{γ_{i}}}_{i=1,...,μ} to θ_{β} and each fixed preimage θ_{β⊗1}, we can find a defining sequence {λ_{γ_{i}⊗1}}_{i=1,...,μ} for θ_{β⊗1} such that {(λ_{γ_{i}⊗1})_{|U^{1}(𝔬_{F [γ_{i}]})}}_{i=1,...,μ} = {λ_{γ_{i}}}_{i=1,...,μ}.

\underline{Proof:}

Since E/F is totally ramified, E⊗_{F} L is a field and Lemma (6.2) applies to our weak approximation sequences for [𝔄, n, 0, β] and for [𝔄 ⊗_{𝔬_{F}} 𝔬_{L}, n, 0, β ⊗ 1]. By (4.1)(vii) and as 𝔓 ⊗_{𝔬_{F}} 𝔬_{L} is the Jacobson radical of 𝔄 ⊗_{𝔬_{F}} 𝔬_{L}, the equation ℌ^{1}(β ⊗ 1, 𝔄 ⊗_{𝔬_{F}} 𝔬_{L}) = ℌ^{1}(β, 𝔄) ⊗_{𝔬_{F}} 𝔬_{L} is trivial. Let Ā := A ⊗_{F} L, 𝔄̄ := 𝔄 ⊗_{𝔬_{F}} 𝔬_{L}, B̄ := CĀ(E ⊗_{F} L) and 𝔅̄ := 𝔄̄ ∩ B̄.

The proof again argues by induction. So first, we assume that [𝔄, n, 0, β] is minimal over F. We have H^{1}(β ⊗ 1, 𝔄̄) = U^{1}(𝔅̄)U^{[ \frac{n}{2} ]+1}(𝔄̄) and H^{1}(β, 𝔄) = U^{1}(𝔅)U^{[ \frac{n}{2} ]+1}(𝔄). If θ_{β⊗1} is a simple character of H^{1}(β ⊗1, 𝔄̄), then it is given by a character λ_{β⊗1} ∘det_{B̄/E⊗_{F} L} on U^{1}(𝔅̄) and by ψ_{β⊗1}^{×} on U^{[ \frac{n}{2} ]+1}(𝔄̄), where for all x ∈ U^{[ \frac{n}{2} ]+1}(𝔄̄) the character ψ_{β⊗1}^{×} is given by ψ̄_{β⊗1}^{×}(x) := ψ_{L} ∘ T r_{Ā/L}(β ⊗ 1(x − 1)). By the definition of a reduced norm and a reduced trace, the restriction of θ_{β⊗1} to H^{1}(β, 𝔄) is given on U^{1}(𝔅) by (λ_{β⊗1})_{|U^{1}(𝔬_{E})} ∘Nrd_{B/E} and on U^{[ \frac{n}{2} ]+1}(𝔄) by (ψ̄_{β⊗1}^{×})_{|U^{[ \frac{n}{2} ]+1}(𝔄)} = ψ_{β}^{×}. Hence, the restriction of θ_{β⊗1} is a simple character as asserted above.

Conversely, if θ_{β} is a simple character of H^{1}(β, 𝔄) given by λ_{β} ∘ Nrd_{B/E} on U^{1}(𝔅) and by ψ_{β}^{×} on U^{[ \frac{n}{2} ]+1}(𝔄), then it can clearly be extended to U^{1}(𝔅)U^{[ \frac{n}{2} ]+1}(𝔄̄) by ψ̄_{β⊗1}^{×} on U^{[ \frac{n}{2} ]+1}(𝔄̄). Since, by Lemma 1.9, (ψ̄_{β⊗1}^{×})_{|U^{[ \frac{n}{2} ]+1}(𝔄̄)∩B̄} factors through the determinant map det_{B̄|E⊗_{F} L} and, by the definition of the reduced norm, λ_{β} extends to a character λ̄_{β} of U^{1}(𝔬_{E⊗_{F} L}) such that (λ̄_{β} ∘ det_{B̄/(E⊗_{F} L)})_{|U^{[ \frac{n}{2} ]+1}(𝔅̄)} = (ψ̄_{β⊗1}^{×})_{|U^{[ \frac{n}{2} ]+1}(𝔅̄)}. Therefore θ_{β} extends to a simple character of H^{1}(β ⊗ 1, 𝔄̄) and we have shown that our restriction map is surjective in the minimal case.

So let us assume now that [𝔄, n, 0, β] has a weak approximation sequence with the final approximation [𝔄, n, q, γ]. It is well known from [BG] that f(F [γ]/F) | f(E/F) and e(F [γ]/F) | e(E/F) in this situation; hence, the extension F [γ]/F is again totally ramified. We have H^{1}(β ⊗ 1, 𝔄̄) = U^{1}(𝔅̄)(U^{[ \frac{q}{2} ]+1}(𝔄̄) ∩ H^{1}(γ ⊗ 1, 𝔄̄)) and H^{1}(β, 𝔄) = U^{1}(𝔅)(U^{[ \frac{q}{2} ]+1}(𝔄) ∩ H^{1}(γ, 𝔄)). If θ_{β⊗1} is a simple character of H^{1}(β ⊗ 1, 𝔄̄), then it is given by a character λ_{β⊗1} ∘ det_{B̄/E⊗_{F} L} on U^{1}(𝔅̄) and by ψ_{(β−γ)⊗1}^{×}θ_{γ⊗1} on (U^{[ \frac{q}{2} ]+1}(𝔄̄) ∩ H^{1}(γ ⊗ 1, 𝔄̄)), where θ_{γ⊗1} is a simple character of H^{1}(γ ⊗ 1, 𝔄̄) and ψ̄_{(β−γ)⊗1}^{×} is defined as in the minimal case. By induction hypothesis, we can assume that the restriction θ_{γ} of θ_{γ⊗1} to H^{1}(γ, 𝔄) is again a simple character and fulfils the assertion about the defining sequences. The restriction of θ_{β⊗1} to H^{1}(β, 𝔄) is given on U^{1}(𝔅) by (λ_{β⊗1})_{|U^{1}(𝔬_{E})} ∘ Nrd_{B/E} and on (U^{[ \frac{q}{2} ]+1}(𝔄) ∩ H^{1}(γ, 𝔄)) by (ψ̄_{(β−γ)⊗1}^{×}θ_{γ⊗1})_{|(U^{[ \frac{q}{2} ]+1}(𝔄)∩H^{1}(γ,𝔄))} = (ψ_{β−γ}^{×}θ_{γ})_{|(U^{[ \frac{q}{2} ]+1}(𝔄)∩H^{1}(γ,𝔄))}. Therefore, the restriction of θ_{β⊗1} is a simple character as asserted above.

Conversely, if θ_{β} is a simple character of H^{1}(β, 𝔄) given by λ_{β} ∘ Nrd_{B/E} on U^{1}(𝔅) and by ψ_{β−γ}^{×}θ_{γ} on (U^{[ \frac{q}{2} ]+1}(𝔄) ∩ H^{1}(γ, 𝔄)), where θ_{γ} is a simple character of H^{1}(γ, 𝔄), then it can clearly be extended to U^{1}(𝔅)(U^{[ \frac{q}{2} ]+1}(𝔄̄) ∩ H^{1}(γ ⊗ 1, 𝔄̄)) by ψ̄_{(β−γ)⊗1}^{×}θ_{γ⊗1} on (U^{[ \frac{q}{2} ]+1}(𝔄̄) ∩ H^{1}(γ ⊗ 1, 𝔄̄)), where θ_{γ⊗1} is a simple extension of θ_{γ}, which exists by induction hypothesis. By Lemma 5.2, (ψ̄_{(β−γ)⊗1}^{×}θ_{γ⊗1})_{|U^{[ \frac{q}{2} ]+1}(𝔅̄)} factors through the determinant map det_{B̄|E⊗_{F} L} and, by the definition of the reduced norm, λ_{β} extends to a character λ̄_{β} of U^{1}(𝔬_{E⊗_{F} L}) such that (λ̄_{β} ∘ det_{B̄/(E⊗_{F} L)})_{U^{[ \frac{q}{2} ]+1}(𝔅̄)} = (ψ̄_{(β−γ)⊗1}^{×}θ_{γ⊗1})_{U^{[ \frac{q}{2} ]+1}(𝔅̄)}. Hence, θ_{β} extends to a simple character of H^{1}(β ⊗ 1, 𝔄̄) and by induction we have shown that our restriction map is surjective in the general case.

Moreover, if θ_{β⊗1} is an arbitrary but fixed preimage of θ_{β} and {λ_{γ_{i}}}_{i=1,...,μ} is a fixed defining sequence of θ_{β}, then the charcters λ′_{γ_{i}⊗1} are uniquely determined on U^{s̄_{i}+1}(𝔬_{F [γ_{i}]⊗_{F} L}) with s̄_{i} = [\frac{σ_{𝔄∩C_{A}(γ_{i})⊗𝔬_{F} 𝔬_{L}/𝔬_{F [γ_{i}]⊗_{F} L}} (q_{i})}{2}], by (5.17) and because of e_{𝔄/𝔄∩C_{A}(γ_{i})} = 1 (see [G] Theorem 2.2(i)) for each defining sequence {λ′_{γ_{i}⊗1}}_{i=1,...,μ} with respect to our given approximation sequence [𝔄 ⊗_{𝔬_{F}} 𝔬_{L}, n, q_{i}, γ_{i} ⊗ 1] i = 1, …, μ and, conversely, these restrictions determine θ_{β⊗1}. Obviously, as θ_{β⊗1} is a preimage of θ_{β}, we have (λ′_{γ_{i}⊗1})_{|U^{s_{i}+1}(𝔬_{F [γ_{i}]})} = (λ_{γ_{i}})_{|U^{s_{i}+1}(𝔬_{F [γ_{i}]})}, where s_{i} = [\frac{σ_{𝔄∩C_{A}(γ_{i})/𝔬_{F [γ_{i}]}} (q_{i})}{2}]. Therefore its possible to extend the characters (λ′_{γ_{i}⊗1})_{|U^{s̄_{i}+1}(𝔬_{F [γ_{i}]⊗_{F} L})} to U^{1}(𝔬_{F [γ_{i}]})U^{s̄_{i}+1}(𝔬_{F [γ_{i}]⊗_{F} L}) and, finally, to characters λ_{γ_{i}⊗1} of U^{1}(𝔬_{F [γ_{i}]⊗_{F} L}) such that they coincide with the λ_{γ_{i}} on U^{1}(𝔬_{F [γ_{i}]}) for i = 1, …, μ. Again by (5.17) the sequence {λ_{γ_{i}⊗1}}_{i=1,...,μ} is a defining sequence for θ_{β⊗1} as required.

□

For the rest of this section, let β be an element of an algebraic closure of F which generates a field extension E/F of finite degree, where we set E := F [β]. As in [BK3] let A(E) := End_{F} (E) and 𝔄(E) := End_{𝔬_{F}} ({𝔭_{E}^{j}}_{j∈ℤ}). Further, assume that [0, β] is a simple pair, i.e., 0 > ν_{E}(β), k_{0}(β, 𝔄(E)). Moreover, let K/F be the maximal unramified subextension of E/F, let C(E) := End_{K}(E) = C_{A(E)}(K) and ℭ(E) := End_{𝔬_{K}} ({𝔭_{E}^{j}}_{j∈ℤ}) = 𝔄(E) ∩ C(E). Let n_{0} := −ν_{E}(β), then the strata [𝔄(E), n_{0}, 0, β], [ℭ(E), n_{0}, 0, β] are simple. The main result of this section is a generalization of [BK3](3.6.13):

( 6.5 ) Theorem Let A be a central simple algebra over F with the property that [E : F] | N, where N := \sqrt{dim_{F} (A)} is the reduced degree of A. Moreover, let d be the index of A and write N = md. Let 𝔄 be a principal 𝔬_{F} -order in A with invariant s and period r such that e(E/F) | rd and f(E/F) | d̈s, where d̈ is the greatest divisor of d that is prime to r, i.e., E/F has an embedding in A/F which is 𝔄-sound (see [G](1.9)(iii) with t = 1).

(i) Let ι : E ↪ A be an F -embedding of E with E^{×} ⊂ K_{𝔄} which is sound. Moreover, let n = n_{0} \frac{rd}{e(E/F)} and if q_{0,i}, i = 1, …, μ, are the unique numbers coming from an approximation sequence [𝔄(E), n_{0}, q_{0,i}, γ_{i}], i = 1, …, μ for [𝔄(E), n_{0}, 0, β], then set q_{i} := q_{0,i} \frac{rd}{e(E/F)} for i = 1, …, μ. There exist embeddings ι_{1}, …, ι_{μ} such that [𝔄, n, q_{i}, ι_{i}(γ_{i})], i = 1, …, μ, is a \underline{special} approximation sequence for [𝔄, n, 0, ι(β)] and ι_{1} = ι.

(ii) There exists a unique bijection τ_{𝔄(E),𝔄,β,ι} : C(β, 𝔄(E)) → C(ι(β), 𝔄)) with the following property:

If we are given a simple character θ_{β} of H^{1}(β, 𝔄(E)), fix a defining sequence {(γ_{i}, λ_{γ_{i}})}_{i=1,...,μ}, which comes from a \underline{special} approximation sequence [𝔄(E), n_{0}, q_{0,i}, γ_{i}] for [𝔄(E), n_{0}, 0, β] and fix embeddings ι_{i} : F [γ_{i}] → A, i = 1, …, μ, as in (i), then τ_{𝔄(E),𝔄,β,ι}(θ_{β}) is a simple character with defining sequence {(ι_{i}(γ_{i}), λ_{γ_{i}} ∘ ι_{i}^{−1})}_{i=1,...,μ}.

\underline{Proof:}

(i) By [BG]((5.1) and its proof) and Theorem (3.11), respectively, we can find a \underline{special} approximation sequence [𝔄, n, q_{i}, γ′_{i}], i = 1, …, μ for [𝔄, n, 0, ι(β)]. Following the arguments of the proof of [BG](5.1) between [BG](5.2) and [BG](5.3), choosing a (W, E)-decomposition for 𝔅̃ as in [BK3](1.2.8) and using intertwining-implies-conjugacy arguments, we can choose γ′_{2} in such a way that it has the same minimal polynomial over F as γ_{2}. Going along the approximation sequence inductively, we obtain a choice of the γ′_{1}, …, γ′_{μ} with the same minimal polynomials as the γ_{1}, …, γ_{μ} over F; hence, the maps induced by mapping γ_{i} at γ′_{i} for i = 1, …, μ lead to the embeddings ι_{1}, …, ι_{μ} as required.

(ii) If the map exists, then, obviously, it is uniquely determined by the property. For the existence we have to note that by Proposition (6.1) there exists a map C(β, 𝔄(E)) → C(β, ℭ(E)) with a well known description of simple characters in terms of defining sequences and being injective. By Proposition (6.4) there exists a surjective map C(β ⊗ 1, ℭ(E) ⊗_{𝔬_{K}} 𝔬_{L}) → C(β, ℭ(E)), where L/K is an unramified splitting field of C. The latter map gives us the possibility to obtain a defining sequence of an image from the defining sequence of a preimage.

If we choose to each image of the first map an arbitrary but fixed preimage with respect to the second map, we obtain an injective map C(β, 𝔄(E)) → C(β ⊗ 1, ℭ(E) ⊗_{𝔬_{K}} 𝔬_{L}) with the property that, to a defining sequence {(γ_{i}, λ_{γ_{i}})}_{i=1,...,μ} of a preimage (coming from a special approximation sequence for [𝔄(E), n_{0}, 0, β]), we obtain a defining sequence {(γ_{i} ⊗ 1, λ_{γ_{i}⊗1})}_{i=1,...,μ} of the image such that (λ_{γ_{i}⊗1})_{|U^{1}(𝔬_{F [γ_{i}]})} = λ_{γ_{i}} ∘ Nr_{K[γ_{i}]/F [γ_{i}]} for i = 1, …, μ.

Now the embeddings ι_{1}, …, ι_{μ} induce embeddings ι′_{1}, …, ι′_{μ} of K[γ_{1} ⊗ 1]/K, …, K[γ_{μ} ⊗ 1]/K such that [ℭ ⊗_{𝔬_{F}} 𝔬_{L}, σ_{𝔄/ℭ}(n), σ_{𝔄/ℭ}(q_{i}), ι′_{i}(γ_{i} ⊗ 1)], i = 1, …, μ, is a weak approximation sequence for [ℭ ⊗_{𝔬_{F}} 𝔬_{L}, σ_{𝔄/ℭ}(n), 0, ι′_{1}(β ⊗ 1)].

By [BK3](3.6.13) we have a canonical bijection

τ_{ℭ(E)⊗_{𝔬_{K}} 𝔬_{L},ℭ⊗_{𝔬_{K}} 𝔬_{L},β,0} :C(β ⊗ 1, ℭ(E) ⊗_{𝔬_{K}} 𝔬_{L}) → C(ι(β) ⊗ 1, ℭ ⊗_{𝔬_{K}} 𝔬_{L})

such that for each defining sequence {(γ_{i} ⊗ 1, λ_{γ_{i}⊗1})}_{i=1,...,μ} of a fixed preimage the image is described by the defining sequence {(γ_{i} ⊗ 1, λ_{γ_{i}⊗1} ∘ ι_{i}^{′−1})}_{i=1,...,μ}.

Using Proposition (6.4) again, we obtain a simple character of H^{1}(β, ℭ) with defining sequence {(γ_{i}, (λ_{γ_{i}⊗1} ∘ Nr_{K[γ_{i}⊗1]/F [γ_{i}⊗1]} ∘ ι_{i}^{′−1})_{|U^{1}(𝔬_{K[γ_{i}]})})}_{i=1,...,μ} and because of (λ_{γ_{i}⊗1} ∘ Nr_{K[γ_{i}⊗1]/F [γ_{i}⊗1]} ∘ ι_{i}^{′−1})_{|U^{1}(𝔬_{K[γ_{i}]})}) = λ_{γ_{i}} ∘ ι_{i}^{−1} ∘ Nr_{K[ι_{i}(γ_{i})]/F [ι_{i}(γ_{i})]} for i = 1, …, μ also Proposition (6.1) applies again. By taking a preimage under the map C(ι(β), 𝔄) → C(ι(β), ℭ), we finally obtain a character of H^{1}(β, 𝔄) with defining sequence {(ι_{i}(γ_{i}), λ_{γ_{i}} ∘ ι_{i}^{−1})}_{i=1,...,μ} as required.

Again, the last two steps do not depend on the special choice of embeddings and defining sequences; hence, following the diagram

C(β, 𝔄(E)) ↪^{res(6.1)} C(β, ℭ(E)) ↞^{res(6.4)} C(β ⊗ 1, ℭ(E) ⊗_{𝔬_{K}} 𝔬_{L})
↕ [BK3]
C(ι(β), 𝔄) ↪^{res(6.1)} C(ι(β), ℭ) ↞^{res(6.4)} C(ι(β) ⊗ 1, ℭ ⊗_{𝔬_{K}} 𝔬_{L}),

we obtain a map τ_{𝔄(E),𝔄,β,ι} with the asserted properties with respect to the embeddings and the defining sequences. As the situation is completely symmetric, we can show the bijectivity by constructing the inverse map. If we fix special defining sequences and embeddings, the last assertion of (6.4) makes obvious that our map cannot depend on the special choices of the preimages with respect to the horizontal surjective maps of our diagram; hence, we are done.

□

The latter result enables us to generalize Definition [BH](8.2…):

( 6.6 ) Definition To a simple pair [0, β] a ps-character is defined as a pair (θ, β), where θ = {(A, ι, 𝔄) ↦ θ(A, ι, 𝔄) ∈ C(ι(β), 𝔄)} is a simple character valued function as follows: to each triple (A, ι, 𝔄) fulfilling the assumptions of Theorem (6.5), θ associates a simple character of H^{1}(ι(β), 𝔄) such that τ_{𝔄(E),𝔄,β}(θ(A(E), κ, 𝔄(E)) = θ(A, ι, 𝔄), where κ is the canonical embedding of E to A(E) = End_{F} (E).

Clearly, a ps-character is uniquely determined by a value on one triple and we have:

( 6.7 ) Corollary Let θ_{β} be a simple character of H^{1}(β, 𝔄(E)), then there exists a unique pscharacter (θ, β) such that θ(A(E), κ, 𝔄(E)) = θ_{β}, where κ is the canonical embedding of E to A(E) = End_{F} (E).

□

7 References

[BF] C. J. Bushnell and A. Fröhlich: Non-abelian congruence Gauss sums and p-adic simple algebras, Proc. London Math. Soc. (3) 50, 1985, 207-264

[BG] P. Broussous and M. Grabitz: Pure elements and intertwining classes of simple strata in local central simple algebras, preprint, 1998

[BH] C. J. Bushnell and G. Henniart: Local Tame Lifting for Gl(N), preprint King’s College London, 1997

[BK1] C. J. Bushnell and P. C. Kutzko: Semisimple types in Gl(N), preprint, 1996

[BK2] C. J. Bushnell and P. C. Kutzko: Simple types in Gl(N), Computing conjugacy classes, Contemporary Mathematics Vol. 177, 1994

[BK3] C. J. Bushnell and P. C. Kutzko: The admissible dual of Gl(N) via compact open subgroups, Annals of Mathematics Studies Number 129, Princeton University Press 1993

[Br1] P. Broussous: Hereditary orders and embeddings of local fields, Journal of Algebra 204, 1998, 324-336

[Br2] P.Broussous: Minimal Strata for Gl(m, D), preprint King’s College London, 1997

[Br3] P. Broussous: Extension du formalisme de Bushnell et Kutzko au cas d’une algèbre à division, Proceed. London Math. Soc., Vol. 77, Part 2, 292-326, 1998

[Br4] P. Broussous: Strates simples dans une algébre simple I,II, discussion paper, Strasbourg, 1996

[F] A. Fröhlich: Principal orders and embeddings of local fields in algebras, Proc. London Math. Soc. (3) 54, 1987, 247-266

[G] M. Grabitz: Continuation of hereditary orders in local central simple algebras, preprint, Max-Planck-Institut für Mathematik (Bonn, Germany) 1998 (MPI 98-95)

[R] I. Reiner: Maximal Orders, Academic Press, New York, 1975

[Se] J.-P. Serre, Local fields, Springer Verlag, New York, GTM 67, 1979

[SZ] A. Silberger and E.-W. Zink: An explicit matching theorem for level zero discrete series of unit groups of p-adic simple algebras, preprint, 1997

[Z1] E.-W. Zink: More on embeddings of local fields in simple algebras, Max-Planck-Institut für Mathematik (Bonn, Germany) (MPI 96-155), 1996

[Z2] E.-W. Zink: Representation filters and their application in the theory of local fields, Journal für die reine und angewandte Mathematik 387, 182-208, 1988

[Z3] E.-W. Zink: Comparison of Gl_{N} and division algebra representations - Some remarks on the local case, Mathematische Nachrichten 159, 47-72, 1992

[Z4] E.-W. Zink: Comparison of Gl_{N} and division algebra representations II, Max-Planck-Institut für Mathematik (Bonn, Germany) 1993 (MPI 93-49)

[Z5] E.-W. Zink: Representation theory of local division algebras, Journal für die reine und angewandte Mathematik 428, 1-44, 1992

[Z6] E.-W. Zink: ϵ-Factors of discrete series representations of central simple algebras, Max-Planck-Institut für Mathematik (Bonn, Germany) 1996 (MPI 96-17)

[Z7] E.-W. Zink: Representation theory of local division algebras I and II, Karl-Weierstass-Institut für Mathematik (Berlin, Germany), preprint 1990