Simple characters for principal orders, part II

Martin Grabitz, Berlin

Spring, 2003

Introduction

In this paper, we extend some results on simple characters for central simple local algebras associated with a representation filter comming from a principal order. In particular, we give a general proof of rigidity and refinement of simple characters. We prove the “intertwining implies conjugacy” property in the tame case. We construct a list of hypothetical simple types and show for the case of Gl_{l}(D), where l is a prime number, that these types are types in the sense of [BK4]. Finally, we explain a well definied map between certain conjugacy classes of extended types for a division algebra D of index l · Ind(D) (Ind(D) the index of D) and conjugacy classes of simple types for tame Gl_{l}(D), i.e. we assume that l · Ind(D) is prime to the residual characteristic of our ground field F. This map preserves central character as well as Artin conductor of representations containing these types. It turns out that it maps finite packages of representations containing these types to finite packages of the same size. Modulo the evident conjecture that this map is injective, we can show that our construction of simple types are complete in the sense that each irreducible essentially square integrable representation of tame Gl_{l}(D) contains a simple type. In particular, it would imply that all supercuspidal representations of tame Gl_{l}(D) are induced from a compact modulo center subgroup. Our main reference is the first part of [G2]. In particular, we use the general notation of [G2](0).

We impose our description of simple characters by establishing an explicit product expansion with respect to a given special approximation sequence (see 3.1).

I am grateful to my teacher Ernst-Wilhelm Zink, who introduced me to the representation theory of central simple algebras, who helped me with many inspiring discussions, who proofread the paper and who pointed out several gaps. Special thanks also due to P. Broussous, G. Henniart, C. Kaiser, G. Kirchner, J. Schlippe, V. Secherre and U. Weselmann for valuable discussions. I thank the professors Bushnell, Faltings, Harder, Henniart, Manin and Colliot-Thélène for their invitations to guest-stays.

We hope to get the finish of the sketched proof from generalizations of considerations as done in [BH] to the mixed case of tame A = M_{m}(D). To get the type property for our hypothetical types, we mainly need to solve the problems concerning the Bruhat decomposition as concidered in the proof of 12.7, because the rest is mainly the same as in our special case or contained in a thesis of Vincent Sécherre [VS].

Contents

1 Rigidity and refinement of simple characters 2

2 Mixed groups and some group identities 11

3 A property of matching in a special situation 17 4 Simple characters and their extensions by transferring between orders 18

5 Reimann’s method to extend to a pro-p-Sylow subgroup (cf. [HR]) 19

6 A simplification of the problem of non-degeneracy of pairs 20

7 Some special cases where all pairs are non-degenerate 24

8 Extending to level 0 26

9 Another remark on the split case 29

10 A hypothetical list of simple types and the “level-0-case” 30

11 Maximal simple types 34

12 The case of tame Gl_{l}(D) with l prime 35

13 References 45

1 Rigidity and refinement of simple characters

Before studying the Heisenberg representations attached to simple characters, we start by asking for the parameters of a given character set. As we have already seen in [G2](5), the character set C(β, 𝔄) only depends on the equivalence class of a given simple stratum [𝔄, n, 0, β]. Conversely, we may recover the stratum [𝔄, n, [\frac{n}{2}], β] by restriction of simple characters. In this section, we generalize some further results of [BK1] to the case of an arbitrary simple algebra. First of all, we show that a character set can be recovered from a single simple character and we call this the rigidity property of a character set. The second property is the possibility to refine a simple character w.r.t. an enlarged approximation sequence. Conversely, the third property is the fact that a simple character, given at a certain level by a defining sequence depending on an approximation sequence, cannot intertwine with another simple character given by a shorter one and we call this the rigidity of the simple character itself. Before showing these properties, we need the corresponding properties for the simple strata. Finally, in the tame case we deduce from the three first properties a forth property, called “intertwining implies conjugacy”, a property that has already been proved for simple strata (see [BG]). The proofs follow the presentation of [BK1] in the split case, but in a crucial point we use a trick of Corwin’s [Co].

( 1.1 ) Proposition ([BK1](3.5.1),(3.5.4)) For i=1,2, let [𝔄, n, ν, β_{i}] be simple strata such that C(β_{1}, 𝔄)_{|U^{ν+1}(𝔄)∩H^{1}(β_{1},𝔄)} ∩ C(β_{2}, 𝔄)_{|U^{ν+1}(𝔄)∩H^{1}(β_{2},𝔄)} ≠ ∅ (in particular U^{ν+1}(𝔄) ∩ H^{1}(β_{1}, 𝔄) = U^{ν+1}(𝔄) ∩ H^{1}(β_{2}, 𝔄))), then [F [β_{1}] : F] = [F [β_{2}] : F] and both strata share the same sequence of jumps.

\underline{proof:}

We may argue as in the split case and use the formula [G2](5.11) for the intertwining of the restriction of a simple character θ of the above intersection of character sets and obtain the equality S(β_{1}, 𝔄)C_{A}(β_{1})^{×}S(β_{1}, 𝔄) = S(β_{2}, 𝔄)C_{A}(β_{2})^{×}S(β_{2}, 𝔄), where S(β_{i}, 𝔄) = (1 + 𝔓^{q−ν} ∩ 𝔑_{−ν}(β_{i}, 𝔄) + 𝔓^{[ \frac{q+1}{2} ]} ∩ 𝔍^{1}(β_{i}, 𝔄)).

Intersecting both sides by 𝔄^{×}, multiplying and factoring by (1 + 𝔓), we obtain (Gls̄(𝔨))r̄ ≃ (𝔄 ∩ C_{A}(β_{1}))^{×}/(1 + 𝔓 ∩ C_{A}(β_{1})) ≃ (𝔄 ∩ C_{A}(β_{2}))^{×}/(1 + 𝔓 ∩ C_{A}(β_{2})). The assertion now follows from \frac{N}{[F [β_{1}]:F]} = r̄s̄[𝔨 : 𝔨_{F}] = \frac{N}{[F [β_{2}]:F]}, where N = m · d is the reduced degree of A.

The second assertion follows now from the first one by varying the power ν, because the equality of jumps must follow from the equality of field degrees for the simple approximations. In particular, we have k_{0}(β_{1}, 𝔄) = k_{0}(β_{2}, 𝔄). The choice of simple approximations and a further restriction of θ yields the equality of all the other jumps.

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( 1.2 ) Lemma Let A/F be a split central simple algebra, let E = F [β] and F [γ]/F be totally ramified extensions in A and let [𝔄, n, ν, β] be a pure stratum with simple approximation [𝔄, n, ν, γ], where ν = −k_{0}(β, 𝔄). Then [𝔅_{γ}, ν, ν − 1, s_{γ}(β − γ)] is equivalent to a simple stratum [𝔅_{γ}, ν, ν − 1, δ], where F [γ, δ] is totally ramified over F [γ] and e(F [γ, δ]/F) | e(E/F). Here 𝔅_{γ} := 𝔄 ∩ C_{A}(γ) and s_{γ} : A → C_{A}(γ) is a tame corestriction with respect to the characters ψ_{F} and ψ_{F [γ]}.

\underline{Proof:}

The first assertion is a special case of [BK1](2.4.1)(iii). For the assertion about the invariants we follow the proof of [BK1](2.4.1)(iii) in detail and first observe that the assertion underlying [BK1](2.4.1)(i)(ii) is independent of the special choice of the approximation γ.

Consequently, let B := C_{A}(E), 𝔅 := 𝔄 ∩ B, A(E) := End_{F} (E) and 𝔄(E) := End_{o_{F}} {𝔭_{E}^{j}}_{j∈ℤ}, then the choice of a ‘(W, E)-decomposition’ according to [BK1](1.2.6) leads to an isomorphism A(E)⊗_{E}B ≃ A of (A(E), B)-bimodules and an isomorphism 𝔄(E) ⊗_{𝔬_{E}} 𝔅 ≃ 𝔄 of (𝔄(E), 𝔅)-bimodules. Now we choose a simple stratum [𝔄(E), n̄, ν̄, γ] that is equivalent to [𝔄(E), n̄, ν̄, β], where we identify the elements β, γ with the preimages in A, n̄ := −ν_{E}(β), ν̄ := [\frac{ν+e−1}{e}] and e := \frac{r(𝔄)}{e(E/F)}. The tame corestriction s_{γ} : A → C_{A}(γ) can be written in the form s_{γ} = s̄ ⊗ 1, where s̄: A(E) → C_{A(E)}(γ) is a tame corestriction attached to the same additive characters ψ_{F} and ψ_{F [γ]} as s_{γ} is.

Now we find a simple stratum [𝔅(E)_{γ}, ν̄, ν̄ − 1, δ] that is equivalent to the stratum [𝔅(E)_{γ}, ν̄, ν̄ − 1, s̄(β − γ)], where 𝔅(E)_{γ} := 𝔄(E) ∩ C_{A(E)}(γ).

The preimage of δ now fulfils the assumptions of the proposition, where we have to remark that 𝔄(E) is a minimal order in A(E) of period e(E/F), and e(F [γ, δ]/F [γ]) must be a divisor of the period of 𝔅(E)_{γ}, which is \frac{e(E/F)}{e(F [γ]/F)}. Also the degree of the unramified part of the extension must divide the invariant of 𝔅(E)_{γ}, which is 1; hence, F [γ, δ]/F [γ] is totally ramified.

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( 1.3 ) Proposition (Rigidity for simple strata) Let A be an arbitrary simple algebra and [𝔄, n, ν, β] be a sound stratum with special simple approximation [𝔄, n, ν, γ], where ν = −k_{0}(β, 𝔄), then [𝔅_{γ}, ν̄, ν̄ − 1, s_{γ}(β − γ)] is equivalent to a simple stratum [𝔅_{γ}, ν̄, ν̄ − 1, δ] such that F [γ, δ]^{ur} ⊂ K and e(F [γ, δ]/F) | e(E/F). Here 𝔅_{γ} := 𝔄 ∩ C_{A}(γ), s_{γ} : A → C_{A}(γ) is a tame corestriction with respect to the two additive characters ψ_{F} , ψ_{F [γ]}, ν̄ := σ_{𝔄/𝔅_{γ}} (ν), E := F [β] and K := E^{ur}.

\underline{Proof:}

As in the proof of the preceding lemma we first remark that the assertion is independent of the special choice of the special approximation sequence for [𝔄, n, ν, γ]. To see this we generalize [BK1](2.4.1)(ii) by using the canonical splitting Ã = A⊗_{F} D^{op}. For this purpose we observe that each tame corestriction s_{γ} : A → C_{A}(γ) leads to a tame corestriction s_{γ} ⊗ 1 : A ⊗_{F} D^{op} → C_{A}(γ) ⊗_{F} D^{op}.

Finally, we also want to find a reduction to the totally ramified case for the extension E/F. We set ψ_{K} := ψ_{F} ∘ tr_{K/F}, ψ_{K[γ]} := ψ_{F [γ]} ∘ tr_{K[γ]/F [γ]}, C := C_{A}(K), C_{γ} := C_{A}(K[γ]). Let s_{K} : A → C be the tame corestriction attached to ψ_{F} and ψ_{K}, s_{K[γ]} : C_{A}(γ) → C_{γ} be the tame corestriction attached to ψ_{F [γ]} and ψ_{K[γ]}. Finally, let s_{γ}^{K} : C → C_{γ} be the tame corestriction attached to ψ_{K} and ψ_{K[γ]}. By the definition of tame corestrictions, we have a commutative diagram:

A →^{s_{γ}} C_{A}(γ)
s_{K} ↓  ↓  s_{K[γ]}
C →^{s_{γ}^{K}} C_{γ}.

A simple calculation shows (cf. [BK1](1.3.8)(ii)) that the vertical mappings are projectors. Since s_{γ}(β −γ) commutes with γ, s_{γ} is a (C_{A}(γ), C_{A}(γ))-bimodule homomorphism, K commutes with β −γ and also K[γ] commutes with s_{γ}(β − γ), we obtain s_{K}^{γ}(s_{γ}(β − γ)) = s_{γ}(β − γ). As β − γ commutes with K, we obtain s_{K[γ]}(s_{K}(β − γ)) = s_{K[γ]}(β − γ). From the commutativity of the diagram above we obtain s_{γ}(β − γ) = s_{K[γ]}(β − γ).

If [𝔄, n, ν, γ] is a special approximation of [𝔄, n, ν, β], then [ℭ, ñ, ν̃, γ] is an approximation of [ℭ, ñ, ν̃, β], where ℭ := 𝔄 ∩ C_{A}(K), ñ := σ_{𝔄/ℭ}(n) and ν̃ := σ_{𝔄/ℭ}(ν). If we showed that the stratum [ℭ_{γ}, ν′, ν′ − 1, s_{K[γ]}(β − γ)] to be equivalent to a simple stratum [ℭ_{γ}, ν′, ν′ − 1, δ′], with e(K[δ′, γ]/K) | e(E/K), where ν′ := σ_{𝔄/ℭ_{γ}} (ν), then of course for any special simple approximation [𝔅_{γ}, ν̄, ν̄ − 1, δ] of the stratum [𝔅_{γ}, ν̄, ν̄ − 1, δ′], this stratum would also be equivalent to [𝔅_{γ}, ν̄, ν̄ − 1, s_{γ}(β − γ)], F [δ, γ]^{ur} ⊂ K and e(F [γ, δ]/F) | e(E/F).

Therefore, we restrict ourselves to the totally ramified case for the extension E/F. Consequently we assume that K = F and C = A. For uniformity of notation we write for the uniformity of the notation B_{γ} := C_{A}(γ) = C_{γ}. We also remark that in our case n̄ = n and ν̄ = ν.

We also choose an unramified splitting field L/F in D of degree d, where D is a division algebra of the same Brauer class of A; moreover, A ≃ M_{m}(D), and therefore we have A ⊗_{F} L ≃ M_{m·d}(L). The stratum [𝔄 ⊗_{𝔬_{F}} 𝔬_{L}, n, ν, β ⊗ 1] is pure and has the simple approximation [𝔄 ⊗_{𝔬_{F}} 𝔬_{L}, n, ν, γ ⊗ 1]. In the same manner we may consider the split algebra Ã := A ⊗_{F} D^{op}, which contains A ⊗_{F} L. As described in [G2](3) we form the canonical lift 𝔄̃ of our principal order and obtain A ⊗_{F} L = CÃ(L), 𝔄 ⊗_{𝔬_{F}} 𝔬_{L} = 𝔄̃ ∩ A ⊗_{F} L. Again the stratum [𝔄̃, n, ν, β ⊗ 1] is pure and has the simple approximation [𝔄̃, n, ν, γ ⊗ 1]. According to 1.2 the stratum [𝔅_{γ} ⊗_{𝔬_{F}} 𝔬_{L}, ν, ν − 1, s_{γ}(β − γ) ⊗ 1] is equivalent to a simple stratum [𝔅_{γ} ⊗_{𝔬_{F}} 𝔬_{L}, ν, ν − 1, δ_{0}], where L[δ_{0}, γ]/L is a totally ramified extension with e(L[δ_{0}, γ]/L) | e(E ⊗_{F} L/L).

We may also consider 𝔅̃_{γ} := 𝔄̃ ∩ CÃ(F [γ]) and, of course, the stratum [𝔅̃_{γ}, ν, ν − 1, s_{γ}(β − γ)] is then equivalent to [𝔅̃_{γ}, ν, ν − 1, δ_{0}]. We may choose a lattice chain for 𝔅_{γ} ⊗_{𝔬_{F}} 𝔬_{L} in a simple B̃_{γ}-module which is also for 𝔅̃_{γ} (note B̃_{γ} := CÃ(γ)), the latter stratum is still pure. Consequently by [BH](4.3) there exists an element δ̄ of B_{γ} ⊗_{F} L, so that the stratum [𝔅_{γ} ⊗_{𝔬_{F}} 𝔬_{L}, ν, ν − 1, δ̄] is simple and equivalent to [𝔅_{γ} ⊗_{𝔬_{F}} 𝔬_{L}, ν, ν − 1, δ_{0}] and so that the stratum [𝔅̃_{γ}, ν, ν − 1, δ̄] remains simple. Now we know, because of [BG](5.1)(ii) and 1.2 that δ̄ generates a totally ramified extension over F and that e(F [δ̄, γ]/F) | e(E/F). Because E^{×} can be embedded in K_{𝔄} in a sound way, we deduce from [G1](2.2)(iii), that this is also the case for the field F [δ̄, γ].

Since, in the totally ramified case, there is only one type of embedding, we may use [BG](4.1.2) to choose this embedding also over F [γ]. We find a simple stratum [𝔅_{γ}, ν, ν − 1, δ̄] such that the strata [𝔅_{γ} ⊗_{𝔬_{F}} 𝔬_{L}, ν, ν − 1, δ̄ ⊗ 1] and [𝔅_{γ} ⊗_{𝔬_{F}} 𝔬_{L}, ν, ν − 1, δ_{0}] intertwine. Therefore, by [BG](4.1.2) and the assumptions on δ_{0}, the strata [𝔅_{γ} ⊗_{𝔬_{F}} 𝔬_{L}, ν, ν − 1, δ̄ ⊗ 1] and [𝔅_{γ} ⊗_{𝔬_{F}} 𝔬_{L}, ν, ν − 1, s_{γ}(β − γ) ⊗ 1] intertwine in K_{𝔅_{γ}⊗_{𝔬_{F}} 𝔬_{L}}. By [BG](4.3.9) we may replace δ̄ by a K_{𝔅_{γ}} -conjugate element δ such that the strata [𝔅_{γ}, ν, ν − 1, δ] and [𝔅_{γ}, ν, ν − 1, s_{γ}(β − γ)] are equivalent.

The stratum [𝔅_{γ}, ν, ν − 1, δ] has the desired properties.

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Using our canonical splitting Ã we easily obtain:

( 1.4 ) Lemma ([BK1](2.2.1)) Let [𝔄, n, ν, β] be a simple stratum and let [𝔄, n, ν − 1, β + b], [𝔄, n, ν − 1, β + b′] be refinements with b, b′ ∈ 𝔓^{−ν}. Let q = −k_{0}(β, 𝔄) and assume that the strata [𝔅, σ_{𝔄/𝔅}(ν), σ_{𝔄/𝔅}(ν) − 1, s_{β}(b)], [𝔅, σ_{𝔄/𝔅}(ν), σ_{𝔄/𝔅}(ν) − 1, s_{β}(b′)] are equivalent, where 𝔅 := 𝔄 ∩ B, B := C_{A}(β) and s_{β} : A → B is a tame corestriction. Then there exists an y ∈ 𝔓^{q−ν} ∩ 𝔑_{−ν}(β, 𝔄) such that [𝔄, n, ν − 1, (1 + y)^{−1}(β + b)(1 + y)] is equivalent to [𝔄, n, ν − 1, β + b′]. We remark that q − ν ≥ 1.

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( 1.5 ) Proposition (Refinement of strata) Let [𝔄, n, ν, β] be a simple stratum, E = F [β], B := C_{A}(E), s_{β} : A → B be a tame corestriction and b ∈ 𝔓^{−ν} such that there exists a simple stratum [𝔅, σ_{𝔄/𝔅}(ν), σ_{𝔄/𝔅}(ν) − 1, c] that is equivalent to the stratum [𝔅, σ_{𝔄/𝔅}(ν), σ_{𝔄/𝔅}(ν) − 1, s_{β}(b)], then there exists a simple stratum [𝔄, n, ν − 1, β_{1}] that is equivalent to [𝔄, n, ν − 1, β + b] such that, with M := F [β_{1}], F [c, β] := E_{1}, we have:

e(M/F) = e(E_{1}/F), f(M/F) = f(E_{1}/F) and k_{0}(β_{1}, 𝔄) = \{ \begin{matrix}k_{0}(β, 𝔄) & if s_{β}(b) ∈ E \\ −ν & else. \end{matrix}

\underline{Proof:}

As in the proof of 1.3, we want to restrict ourselves to the totally ramified case for the extension E_{1}/F. For this purpose we let K := E_{1}^{ur}, ψ_{K} := ψ_{F} ∘tr_{K/F}, ψ_{K[β]} := ψ_{F [β]} ∘tr_{K[β]/F [β]}, C := C_{A}(K), C_{β} := C_{A}(K[β]), s_{K} : A → C the tame corestriction given by ψ_{F} and ψ_{K}, s_{K[β]} : C_{A}(β) → C_{β} the tame corestriction attached to ψ_{F [β]} and ψ_{K[β]}. Moreover, let s_{β}^{K} : C → C_{β} be the tame corestriction given by ψ_{K} and ψ_{K[β]}. By the definition of a tame corestriction we have a commutative diagram:

A →^{s_{β}} C_{A}(β)
s_{K} ↓  ↓  s_{K[β]}
C →^{s_{β}^{K}} C_{β}.

The stratum [ℭ_{β}, σ_{𝔄/ℭ_{β}}(ν), σ_{𝔄/ℭ_{β}}(ν) − 1, s_{K[β]}(s_{β}(b))] = [ℭ_{β}, σ_{𝔄/ℭ_{β}}(ν), σ_{𝔄/ℭ_{β}}(ν) − 1, s_{β}^{K}(s_{K}(b))] is again equivalent to the simple stratum [ℭ_{β}, σ_{𝔄/ℭ_{β}}(ν), σ_{𝔄/ℭ_{β}}(ν) − 1, c].

Assume for a moment that the stratum [ℭ, σ_{𝔄/ℭ}(n), σ_{𝔄/ℭ}(ν) − 1, β + s_{K}(b)] is equivalent to a simple stratum [ℭ, σ_{𝔄/ℭ}(n), σ_{𝔄/ℭ}(ν) − 1, β̄_{1}]. Then the stratum [𝔄, n, ν − 1, β̄_{1}] is still pure and equivalent to the stratum [𝔄, n, ν − 1, β + s_{K}(b)]. We may replace β̄_{1} by a simple representative β′_{1}, which still commutes with K, and the stratum [ℭ, σ_{𝔄/ℭ}(n), σ_{𝔄/ℭ}(ν) − 1, β_{1}] is simple and equivalent to [ℭ, σ_{𝔄/ℭ}(n), σ_{𝔄/ℭ}(ν) − 1, β + s_{K}(b)].

As c commutes with K, [ℭ_{β}, σ_{𝔄/ℭ_{β}}(ν), σ_{𝔄/ℭ_{β}}(ν) − 1, c] is equivalent to the stratum [ℭ_{β}, σ_{𝔄/ℭ_{β}}(ν), σ_{𝔄/ℭ_{β}}(ν) − 1, s_{K[β]}(s_{β}(b))]. However, s_{K[β]}(s_{β}(b)) = s_{β}^{K}(s_{K}(b)) = s_{β}^{K}(s_{K} ∘ s_{K}(b)) = s_{K[β]}(s_{β}(s_{K}(b))) = s_{β}(s_{K}(b)), where we use the same argument again (see [BK1](1.3.8)(ii)) to show that the vertical maps are projectors.

As, consequently, the strata [𝔅, σ_{𝔄/𝔅}(ν), σ_{𝔄/𝔅}(ν) − 1, s_{β}(b)] and [𝔅, σ_{𝔄/𝔅}(ν), σ_{𝔄/𝔅}(ν) − 1, s_{β}(s_{K}(b))] are equivalent, we may also apply 1.4 to restrict ourselves, up to K_{𝔄}-conjugacy, to the case where [𝔄, n, ν − 1, β + b] is equivalent to a simple stratum [𝔄, n, ν − 1, β_{1}]. To show the relation between invariants, we consider [𝔄̃, n, ν − 1, β + b] and [𝔄̃, n, ν − 1, β_{1}] in a canonical splitting Ã ≃ A ⊗ D^{op}, where 𝔄̃ is a canonical splitting of 𝔄 in Ã (see [G2](3)). Here we notice that s_{β} ⊗ 1 : A ⊗_{F} D^{op} → B ⊗_{F} D^{op} is a suitable tame corestriction. The stratum [𝔅̃, ν, ν − 1, s_{β}(b)] is equivalent to the simple stratum [𝔅̃, ν, ν − 1, c], if s_{β}(b) ∉ E. The latter is simple, because the stratum [𝔅, σ_{𝔄/𝔅}(ν), σ_{𝔄/𝔅}(ν) − 1, c] is minimal (cf. [BK1](1.4.14)(1.4.15),[Br2]). However, then also the the stratum [𝔅̃, ν, ν − 1, c] must be minimal, hence simple. Finaly, we observe that k_{0}(β_{1}, 𝔄) = k_{0}(β_{1}, 𝔄̃), k_{0}(β, 𝔄̃) = k_{0}(β, 𝔄) and k_{0}(c, 𝔅̃) = −ν and obtain the desired relation between invariants from [BK1](2.2.3).

Hence, we have reduced to the totally ramified case for the extension E_{1}/F, where we do not have to prove the relation of invariants. We assume now K = F, A = C, ℭ = 𝔄, and ℭ_{β} = 𝔅, we also denote B = C_{β}.

The rest of the proof is similar to that of 1.3 after the last reduction step and we just write down the arguments for completness.

We choose an unramified splitting field L/F in D of degree d, where D is the division algebra of the same Brauer class as A, so A ≃ M_{m}(D), and we have A ⊗_{F} L ≃ M_{m·d}(L). The stratum [𝔅 ⊗_{𝔬_{F}} 𝔬_{L}, ν, ν − 1, c ⊗ 1] is simple and equivalent to the stratum [𝔅 ⊗_{𝔬_{F}} 𝔬_{L}, ν, ν − 1, s_{β}(b) ⊗ 1]. In the same way we can consider the canonical splitting Ã := A⊗_{F} D^{op}, which contains our first splitting by construction. We also consider the canonical splitting 𝔄̃ of our principal order as in [G2](3) and obtain A ⊗_{F} L = CÃ(L) as well as 𝔄 ⊗_{𝔬_{F}} 𝔬_{L} = 𝔄̃ ∩ A ⊗_{F} L.

By [BK1](2.2.8) there exists a simple stratum [𝔄 ⊗_{𝔬_{F}} 𝔬_{L}, n, ν − 1, β̄_{1}] that is equivalent to [𝔄 ⊗_{𝔬_{F}} 𝔬_{L}, n, ν − 1, (β + b) ⊗ 1]. The stratum [𝔄̃, n, ν − 1, β̄_{1}] is still pure (see the proof of 1.3). By [BH](4.3) there also exists an element β′_{1} of A ⊗_{F} L such that the stratum [𝔄 ⊗_{𝔬_{F}} 𝔬_{L}, n, ν − 1, β′_{1}] is simple and equivalent to [𝔄 ⊗_{𝔬_{F}} 𝔬_{L}, n, ν − 1, β̄_{1}] and, therefore, the stratum [𝔄̃, n, ν − 1, β_{1}] remains simple. By [BG](5.1)(ii) and [BK1](2.2.8) we know that β′_{1} also generates a totally ramified extension over F and that e(M/F) = e(E′_{1}/F), where E′_{1} := F [β′_{1}]. On the other hand, M^{×} can be embedded in K_{𝔄} in a sound way. We deduce by [G1](2.2)(iii) that this is also possible for the field E′_{1}.

Hence, we find a simple stratum [𝔄, n, ν − 1, β̃_{1}] such that the strata [𝔄 ⊗_{𝔬_{F}} 𝔬_{L}, n, ν − 1, β̃_{1} ⊗ 1] and [𝔄 ⊗_{𝔬_{F}} 𝔬_{L}, n, ν − 1, (β + b) ⊗ 1] intertwine. By [BG](4.1.2) and the assumptions on β̃_{1} the strata [𝔄 ⊗_{𝔬_{F}} 𝔬_{L}, n, ν − 1, β̃_{1} ⊗ 1] and [𝔄 ⊗_{𝔬_{F}} 𝔬_{L}, n, ν − 1, (β + b) ⊗ 1] intertwine in K_{𝔄⊗_{𝔬_{F}} 𝔬_{L}}. According to [BG](4.3.9), up to K_{𝔄}-conjugacy, we may replace β̃_{1} by an element β_{1} such that the strata [𝔄, n, ν − 1, β_{1}] and [𝔄, n, ν − 1, β + b] are equivalent.

The stratum [𝔄, n, ν − 1, β_{1}] has all desired properties.

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( 1.6 ) Definition Let [𝔄, n, ν, β] be a simple stratum with ν < [\frac{n}{2}]. Further, let q^{+} be the smallest jump of the stratum [𝔄, n, 0, β] such that ν < [\frac{q^{+}}{2} ] and let q^{−} be the previous jump, then each simple stratum [𝔄, n, q^{−}, δ] that is equivalent to [𝔄, n, q^{−}, β] is called a core approximation of [𝔄, n, ν, β].

With the notation of 1.6 we obtain −k_{0}(δ, 𝔄) = q^{+}, 𝔓^{ν+1} ∩ ℌ^{1}(β, 𝔄) = 𝔓^{ν+2} ∩ ℌ^{1}(β, 𝔄) + 𝔓^{ν+1} ∩ C_{A}(δ) and by [G2](5.14) we obtain C(𝔄, β)_{|U^{ν+1}(𝔄)∩H^{1}(β,𝔄)} = C(𝔄, δ)_{|U^{ν+1}(𝔄)∩H^{1}(β,𝔄)}ψ_{β−δ}^{×}.

( 1.7 ) Proposition Let [𝔄, n, ν, β] be a simple stratum with ν ≥ 0 and b ∈ 𝔓^{−(ν+1)}. We consider ψ_{b}^{×} to be a character of U^{ν+1}(𝔄) ∩ H^{1}(β, 𝔄) that is trivial on U^{ν+2}(𝔄) ∩ H^{1}(β, 𝔄).

(i) Suppose that there exists θ ∈ C(β, 𝔄)_{|U^{ν+1}(𝔄)∩H^{1}(β,𝔄)} such that θ·ψ_{b}^{×} ∈ C(β, 𝔄)_{|U^{ν+1}(𝔄)∩H^{1}(β,𝔄)}. Then we have θψ_{b}^{×} ∈ C(β, 𝔄)_{|U^{ν+1}(𝔄)∩H^{1}(β,𝔄)} for all θ ∈ C(β, 𝔄)_{|U^{ν+1}(𝔄)∩H^{1}(β,𝔄)}.

(ii) The intersection C(β, 𝔄)_{|U^{ν+1}(𝔄)∩H^{1}(β,𝔄)}∩C(β, 𝔄)_{|U^{ν+1}(𝔄)∩H^{1}(β,𝔄)}·ψ_{b}^{×} is non empty if and only if, for any core approximation [𝔄, n, q^{−}, δ] of [𝔄, n, ν, β] and any tame corestriction s_{δ} : A → C_{A}(δ), the image s_{δ}(b) lies in the set F [δ] + 𝔓^{−ν} ∩ C_{A}(δ).

\underline{Proof:}

To prove (i) we follow the proof of [BK1](3.5.2) and use [G2](1.9) instead of [BK1](2.4.11), [G2](5.3) instead of [BK1](3.2.1) and [G2](5.14) instead of [BK1](3.3.18). To prove (ii) we follow the proof of [BK1](3.5.6) and use [G2](1.9); [G2](5.3) instead of [BK1](3.2.5).

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( 1.8 ) Proposition Let [𝔄, n, ν, β] be a simple stratum with 0 ≤ k < [\frac{n}{2}] and let [𝔄, n, q^{−}, δ] be an associated core approximation. Furthermore, let θ_{1}, θ_{2} ∈ C(β, 𝔄)_{|U^{ν+1}(𝔄)∩H^{1}(β,𝔄)}. Then the character θ_{1}/θ_{2} of U^{ν+1}(𝔄) ∩ H^{1}(β, 𝔄) = U^{ν+1}(𝔄) ∩ H^{1}(δ, 𝔄) is intertwined by all of S(δ)C_{A}(δ)^{×}S(δ), where S(δ) := 1 + 𝔓^{q^{+}−q^{−}} ∩ 𝔑_{−q^{−}}(β, 𝔄) + 𝔓^{[ \frac{q^{+}+1}{2} ]} ∩ 𝔍^{1}(β, 𝔄).

\underline{Proof:}

We follow the proof of [BK1](3.5.7) and use [G2](5.12) instead of [BK1](3.3.9).

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( 1.9 ) Proposition (Rigidity of a character set) For i = 1, 2 let [𝔄, n, ν, β_{i}] be simple strata. Suppose that C(β_{1}, 𝔄)_{|U^{ν+1}(𝔄)∩H^{1}(β_{1},𝔄)} ∩ C(β_{2}, 𝔄)_{|U^{ν+1}(𝔄)∩H^{1}(β_{2},𝔄)} ≠ ∅. Then we have C(β_{1}, 𝔄)_{|U^{ν+1}(𝔄)∩H^{1}(β_{1},𝔄)} = C(β_{2}, 𝔄)_{|U^{ν+1}(𝔄)∩H^{1}(β_{2},𝔄)}.

\underline{Proof:}

We follow the proof of [BK1](3.5.8) and use [G2](5.11) instead of [BK1](3.3.2), [G2](1.9) instead of [BK1](2.4.11), 1.8 instead of [BK1](3.5.7), 1.1 instead of [BK1](3.5.4) and, finally, 1.7 instead of [BK1](3.5.6).

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( 1.10 ) Proposition (Refinement of simple characters) For i = 1, 2, let [𝔄, n, ν, β_{i}] be simple strata with ν > 1 and C(β_{1}, 𝔄)_{|U^{ν+1}(𝔄)∩H^{1}(β_{1},𝔄)} = C(β_{2}, 𝔄)_{|U^{ν+1}(𝔄)∩H^{1}(β_{2},𝔄)}. Then we have U^{ν}(𝔄) ∩ H^{1}(β_{1}, 𝔄) = U^{ν}(𝔄) ∩ H^{1}(β_{2}, 𝔄) =: H^{ν} and there exists a simple stratum [𝔄, n, ν, β′_{1}] that is equivalent to [𝔄, n, ν, β_{1}] such that C(β′_{1}, 𝔄)_{|H^{ν}} = C(β_{2}, 𝔄)_{|H^{ν}}.

\underline{Proof:}

We follow the proof of [BK1](3.5.9) step by step and use 1.1 instead of [BK1](3.5.4), 1.11 (see below) instead of [BK1](3.5.10), [G2](1.9) instead of [BK1](2.4.11), 1.5 instead of [BK1](2.2.8), [G2](5.13) instead of [BK1](3.3.20), 1.8 instead of [BK1](3.5.7), [G2](5.10) instead of [BK1](3.3.1), [G2](4.2); [G2](5.3); [G2](5.11); [G2](5.12) instead of [BK1](3.3.17) and [G2](5.12) instead of [BK1](3.3.9). As a corollary of the proof we obtain:

( 1.11 ) Lemma Let [𝔄, n, ν, β] be a simple stratum with ν ≥ 1. Let θ ∈ C(β, 𝔄)_{|U^{ν}(𝔄)∩H^{1}(β,𝔄)} and x ∈ A^{×} be an element of the normalizer θ_{|U^{ν+1}(𝔄)∩H^{1}(β,𝔄)}, then we have θ_{|U^{σ_{𝔄/𝔅}(ν)}(𝔅)}^{x} = θ_{|U^{σ_{𝔄/𝔅}(ν)}(𝔅)}, where 𝔅 = 𝔄 ∩ C_{A}(β).

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( 1.12 ) Proposition (Rigidity of a simple character) Let [𝔄, n, q − 1, β] be a simple stratum in A with −k_{0}(β, 𝔄) = q, n > q ≥ 1, and [𝔄, n, q, γ] be a simple special approximation of [𝔄, n, q, β]. Further, let θ ∈ C(𝔄, β)_{|U^{q}(𝔄)∩H^{1}(β,𝔄)}, ϕ ∈ C(𝔄, γ)_{|U^{q}(𝔄)∩H^{1}(γ,𝔄)} with θ_{|H^{q+1}} = ϕ_{|H^{q+1}}, where H^{q+1} := U^{q+1}(𝔄) ∩ H^{1}(β, 𝔄) = U^{q+1}(𝔄) ∩ H^{1}(γ, 𝔄), then θ and ϕ do not intertwine in A^{×}.

\underline{Proof:}

We follow the proof of Lemma [BK1](3.5.12) step by step and take the ramification of continuations of principal orders into account. Moreover, we use [G2](1.9) instead of [BK1](2.4.11), 1.3 instead of [BK1](2.4.1)(iii), [G2](5.11) instead of [BK1](3.3.2) and 1.11 instead of [BK1](3.5.10). At the end of the proof, using a canonical splitting, we may also apply [BK1](2.6.2).

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We recall the Definition [G2](2.4), which leads to a generalization of the notion of a sound embedding and, of course, we also need the notion of the invariant of such an embedding:

( 1.13 ) Notation An embedding (E, 𝔄) is called sound if it fulfils the equivalent conditions from [G2](2.3). The invariant d_{1} from [G2](2.3) is also called the invariant of the sound embedding.

( 1.14 ) Corollary Let (E, 𝔄) be a sound embedding with the invariant d_{1} and (E′, 𝔄) a sound embedding with E^{′ur} ⊂ E^{ur}. Now if, f := ⟨f(E/F), d⟩ and f′ := ⟨f(E′/F), d⟩, then the invariant d′_{1} of the latter embedding is given by d′_{1} = ⟨f′, d_{1}⟩.

\underline{Proof:}

Let L/F and L′/F be the unique unramified subextensions of degree f and f′ of E/F and E′/F, respectively, then, by assumption, L′ ⊂ L and the invariants d_{1} and d′_{1} are the invariants of the embeddings (L, 𝔄) and (L′, 𝔄) (see [BG] for the notation). If s is the invariant of 𝔄 and if s′ and s′′ are the invariants of 𝔄 ∩ C_{A}(L) and 𝔄 ∩ C_{A}(L′), respectively, then we have d_{1} = \frac{f}{( \frac{s}{s′})} and d′_{1} = \frac{f′}{( \frac{s}{s′′})} by [G2](2.2).

On the other hand, by [G1](1.9) s′′ = ⟨s′ · \frac{f}{f′} , m⟩ and, by [G2](2.2), s′ = \frac{s·d_{1}}{f}. Consequently,

d′_{1} = _{\frac{f′}{(_{\frac{s}{⟨ \frac{sd_{1}}{f′} ,m⟩}})}} = ^{\frac{f′⟨ \frac{sd_{1}}{f′} ,m⟩}{s}} = ⟨d_{1}, f′ · r⟩ = ⟨d_{1}, f′⟩ as desired. We use that r, the period of 𝔄, is prime

to d_{1}.

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( 1.15 ) Proposition ([BK1](3.5.11) “intertwining implies conjugacy”) Let A/F be a tame algebra, i.e., the residual characteristic of F is prime to N. For i = 1, 2, let [𝔄, n, ν, β_{i}] be simple strata with ν ≥ 0, which are all purely embedded with maximally possible embedding invariants for the embeddings (F [β_{i}], 𝔄) in the sense of [BG](3.1),1.13. For i = 1, 2, assume that θ_{i} ∈ C(β_{i}, 𝔄)_{|U^{ν+1}(𝔄)∩H^{1}(β_{i},𝔄)} intertwine in A^{×}, then there exists x ∈ K_{𝔄} such that C(β_{2}, 𝔄)_{|U^{ν+1}(𝔄)∩H^{1}(β_{2},𝔄)} = C(x^{−1}β_{1}x, 𝔄)_{|U^{ν+1}(𝔄)∩H^{1}(β_{1},𝔄)} and θ_{2} = θ_{1}^{x}.

\underline{Proof:}

By 1.9 the first assertion follows immediately from the second assertion. We consider the conjugate θ_{1}^{x} by x ∈ K_{𝔄} and first observe that H^{1}(x^{−1}β_{1}x, 𝔄) = x^{−1}H^{1}(β_{1}, 𝔄)x. For ν ≥ [\frac{n}{2}] the result directly follows from [BG](4.1.2). Indeed, in this case θ_{i} = ψ_{β_{i}}^{×} and these characters correspond to the strata [𝔄, n, ν, β_{i}] with simple special approximations [𝔄, n, ν, γ_{i}] for i = 1, 2. Since the latter approximations intertwine by β_{i} − γ_{i} ∈ 𝔓^{−ν}; i = 1, 2, using a canonical splitting 𝔄̃ and [BK1](2.4.1), we see that the invariants of the fields F [γ_{1}] and F [γ_{2}] are the same. By the assumptions on the invariants of the embeddings and as by 1.14, special approximations also have a maximal embedding invariant, the strata [𝔄, n, ν, γ_{i}] intertwine and share a common type of embedding with i = 1, 2. By [BG](4.1.2) the latter strata are conjugated in K_{𝔄}. The equality θ_{2} = θ_{1}^{x} is equivalent to the equivalence of the strata [𝔄, n, ν, β_{2}] and [𝔄, n, ν, x^{−1}β_{1}x]. If for i = 1, 2 the strata [𝔄, n, ν, γ_{i}] are simple special approximations and we have x ∈ K_{𝔄} so that [𝔄, n, ν, γ_{2}] is equivalent to [𝔄, n, ν, x^{−1}γ_{1}x], then also [𝔄, n, ν, β_{2}] and [𝔄, n, ν, x^{−1}β_{1}x] are equivalent.

We deal with the general case by induction on ν and on the approximation sequence. The case ν ≥ [\frac{n}{2}] has been done. All together for the next step we may assume that U^{ν+2}(𝔄) ∩ H^{1}(β_{1}, 𝔄) = U^{ν+2}(𝔄) ∩ H^{1}(β_{2}, 𝔄) =: H^{ν+2}, θ_{2}_{|H^{ν+2}} = θ_{1}_{|H^{ν+2}}, ν < [\frac{n}{2}] and that [𝔄, n, ν + 1, γ_{i}] is a special approximation of the stratum [𝔄, n, ν + 1, β_{i}] for i = 1, 2. With respect to the minimal case, one could assume that the latter strata are equivalent. On the other hand, if we consider the case of ν ≥ q̂, where q̂ is the second last jump, then the last proper approximations [𝔄, n, q̂, γ_{1,μ−1}] and [𝔄, n, q̂, γ_{2,μ−1}], respectively, intertwine and are conjugate by [BG](4.1.2). Therefore, since, by [G2](5.14), replacement of equivalent approximations does not change the simple character, we could inductively assume, up to conjugacy, and in view of the minimal case that γ_{1,μ−1} = γ_{2,μ−1}. We decide for the medium strong assumption F [γ_{1,μ−1}] = F [γ_{2,μ−1}]. With respect to our induction, we assume that there exists simple strata [𝔄, n, ν + 1, γ_{i}] equivalent to the strata [𝔄, n, ν + 1, β_{i}] for i = 1, 2 such that F [γ_{1}] = F [γ_{2}].

With respect to the final assertion and 1.1 we want to show first that the strata [𝔄, n, ν + 1, β_{i}] for i = 1, 2 are either already both simple, or both strata [𝔄, n, ν, β_{i}] for i = 1, 2 have a common jump ν + 1 = −k_{0}(β_{i}, 𝔄) for i = 1, 2.

If not, let us assume that [𝔄, n, ν + 1, β_{1}] is simple and [𝔄, n, ν, β_{2}] has an entire approximation [𝔄, n, ν + 1, γ_{2}]. In particular C(β_{2}, 𝔄)_{|H^{ν+2}} = C(γ_{2}, 𝔄)_{|H^{ν+2}}. By 1.10 we may reduce to the case of C(β_{1}, 𝔄)_{|H^{ν+1}} = C(γ_{2}, 𝔄)_{|H^{ν+1}}. But this contradicts 1.12.

We let B := C_{A}(γ_{1}), 𝔅 := 𝔄 ∩ B, 𝔔 := 𝔓 ∩ B, ν̄ := σ_{𝔄/𝔅}(ν), s be a tame corestriction from A to B and ϵ := s(1), where, in the tame case, it easily follows as in [BK1](1.3) that ϵ ∈ 𝔬_{F [γ_{1}]}^{×}, because the trace tr_{F [γ_{1}]/F} maps 𝔬_{F [γ_{1}]} surjective to 𝔬_{F} and (T rd_{A/F})_{|C_{A}(γ_{1})} = tr_{F [γ_{1}]/F} ∘ T rd_{C_{A}(γ_{1})/F [γ_{1}]}.

We start with the case where the strata [𝔄, n, ν +1, β_{i}] for i = 1, 2 are both simple. By 1.9 together with 1.10, we may assume that U^{ν+1}(𝔄)∩H^{1}(β_{1}, 𝔄) = U^{ν+1}(𝔄)∩H^{1}(β_{2}, 𝔄) =: H^{ν+1} and have shown the first assertion for the induction step. For i = 1, 2 the equivalent strata [𝔄, n, ν +1, γ_{i}] fulfil F [γ_{1}] = F [γ_{2}] =: M by induction and, in particular, s is a tame corestriction for the centralizers with respect to both elements. Using a canonical splitting 𝔄̃ of 𝔄, we may generalize [BK1](2.4.1)(ii) and find δ_{i} ∈ M such that s(β_{i}−γ_{i})−δ_{i} ∈ 𝔓^{−ν}. In particular, we obtain refinements [𝔄, n, ν, β_{i}] = [𝔄, n, ν, γ_{i}+(β_{i}−γ_{i})] and [𝔄, n, ν, γ_{i} + δ_{i} · ϵ^{−1}] of the strata [𝔄, n, ν + 1, γ_{i}] i = 1, 2, and, by 1.4, they are conjugated in Γ := (1+𝔓^{q−ν−1} ∩𝔑_{−ν−1}(γ_{1}, 𝔄)) = (1+𝔓^{q−ν−1} ∩𝔑_{−ν−1}(γ_{2}, 𝔄)), where q = −k_{0}(γ_{1}, 𝔄) = −k_{0}(γ_{2}, 𝔄) and where, by [G2](4.2)(4.4)(5.11) or [G2](5.12), the character (θ_{1})_{|H^{m+2}} = (θ_{2})_{|H^{m+2}} is invariant under this conjugation. So, up to conjugacy, we may fulfil the third assertion of the induction step: The strata [𝔄, n, ν, β_{i}] are equivalent to simple strata [𝔄, n, ν, β′_{i}] for i = 1, 2 with F [β′_{1}] = F [β′_{2}]. In our special situation the latter field is again M and s is again a corestriction with respect to the centralizers of the latter elements.

By 1.10, we may find a simple stratum [𝔄, n, ν, β′′_{1}] equivalent to [𝔄, n, ν, β′_{1}], such that C(β′′_{1}, 𝔄)_{|H^{ν+1}} = C(β′_{2}, 𝔄)_{|H^{ν+1}}. As also [𝔄, n, ν + 1, β′′_{1}] remains simple, we may again use the generalization of [BK1](2.4.1)(ii) to obtain μ ∈ F [β_{1}] ∩ 𝔓^{−ν−1} such that the strata [𝔅, −σ_{𝔄/𝔅}(−ν − 1), −σ_{𝔄/𝔅}(−ν), μ] and [𝔅, −σ_{𝔄/𝔅}(−ν − 1), −σ_{𝔄/𝔅}(−ν), s(β′′_{1} − β′_{1})] are equivalent. Consequently, by 1.4, the refinements [𝔄, n, ν, β′′_{1}] = [𝔄, n, ν, β′_{1} + (β′′_{1} − β′_{1})] and [𝔄, n, ν, β′_{1} + μ · ϵ^{−1}] of the stratum [𝔄, n, ν + 1, β′_{1}] are conjugated by an element of Γ = (1 + 𝔓^{q−ν−1} ∩ 𝔑_{−ν−1}(β′_{1}, 𝔄)) = (1 + 𝔓^{q−ν−1} ∩ 𝔑_{−ν−1}(β′′_{1}, 𝔄)). Since the group Γ normalizes the character θ_{1}_{|H^{ν+2}} by [G2](4.2)(4.4)(5.11) or [G2](5.12) and preserves the equivalence of the strata [𝔄, n, ν + 1, β′_{1}] and [𝔄, n, ν + 1, β_{1}], we may conjugate θ_{1} and its whole approximation sequence to obtain C(β′_{1} +b, 𝔄)_{|H^{ν+1}} = C(β′_{2}, 𝔄)_{|H^{ν+1}}, where we replace β′_{1} by its conjugate and where b is the corresponding conjugate of μ·ϵ^{−1} ∈ M. By [G2](5.14) we have C(β′_{1}, 𝔄)_{|H^{ν+1}}ψ_{b}^{×} = C(β′_{1} + b, 𝔄)_{|H^{ν+1}}.

Let c ∈ 𝔓^{−ν−1} such that θ_{1}ψ_{b+c}^{×} = θ_{2}. Then θ_{1}ψ_{b}^{×} and θ_{2} = θ_{1}ψ_{b+c}^{×} are both in C(β′_{1} + b, 𝔄). Since F [β′_{1} + b] = F [β′_{1}] = M and the restriction of θ_{1} as well as the restriction of θ_{2} to U^{ν+1}(𝔄) ∩ B factor through the reduced norm Nrd_{B/M}, we conclude from [G2](1.9) that s(b+c) ∈ M ∩𝔓^{−ν−1} +𝔓^{−ν} ∩B. We want to show that, in fact, s(b + c) ∈ 𝔓^{−ν} ∩ B and therefore, assume for the next step that −σ_{𝔄/𝔅}(−ν) ≠ −σ_{𝔄/𝔅}(−ν − 1). We deduce that the stratum [𝔅, ν̄ + 1, ν̄, s(b + c)] is equivalent to a simple stratum, where we also consider the case of the zero-stratum. We show that the latter case is valid because the stratum [𝔅, ν̄ + 1, ν̄, s(b + c)] intertwines with the zero-stratum. For this purpose, we use the arguments of the proof of 1.12, which we repeat here because we have left the translation from the split case to the reader.

If g intertwines the characters θ_{1} and θ_{2}, then, by [G2](5.11) and by induction, we have g = x · t · y with x, y ∈ (1 + 𝔓^{q−ν−1} ∩ 𝔑_{−ν−1}(β′_{1}, 𝔄) + 𝔓^{[ \frac{q+1}{2} ]} ∩ 𝔍^{1}(β′_{1}, 𝔄)), where t ∈ B^{×}. Since t now intertwines θ_{1}^{x} with θ_{2}^{y^{−1}}, by 1.11 we may restrict ourselves to the case g = t. As again by 1.11, these characters coincide with θ_{1} and θ_{2} on U^{ν̄+1}(𝔅), the restriction of θ_{1} to B factors through Nrd_{B/M} and because θ_{2} = θ_{1} · ψ_{b+c}^{×}, g intertwines ψ_{s(b+c)}^{×} and ψ_{0}^{×}.

Since, the stratum [𝔅, ν̄ + 1, ν̄, s(b + c)] intertwines the zero-stratum as required, s(b + c) must lie in 𝔔^{−ν̄} according to [BG](4.1.2). Also θ_{1} and θ_{2} coincide on U^{ν̄+1}(𝔅), too. For ν < [\frac{q}{2}] the induction step is reached. For the rest of the proof of the first case we assume ν ≥ [\frac{q}{2}]. Since b + c modulo 𝔓^{−ν} is in the kernel of s, by [G2](2.10), we find a d ∈ (ℌ^{1}(β_{1}, 𝔄) ∩ 𝔓^{ν+2})^{∗} ∩ a_{β}(A) such that θ_{2} = θ_{1} · ψ_{d}^{×}. By [G2](4.5), we find an x ∈ 𝔓^{q−ν−1} ∩ 𝔑_{−ν−1}(β_{1}, 𝔄) + 𝔍^{1}(β_{1}, 𝔄) ∩ 𝔓^{[ \frac{q+1}{2} ]} with d = a_{β}(x) and by [G2](5.12) it follows that θ_{2} = θ_{1}^{1+x} and that 1 + x normalize θ_{1}_{|H^{ν+2}}. Therefore we have handled the first case for both strata and our assertion is valid especialy in the minimal case.

So let ν < [\frac{n}{2}] and for i = 1, 2 assume that the strata [𝔄, n, ν + 1, β_{i}] have proper special approximations [𝔄, n, ν + 1, γ_{i}]. Since, we are at the jump ν + 1 = k_{0}(β_{i}, 𝔄) (i = 1, 2), by [G2](5.14), we may replace the strata by equivalent ones and assume that F [γ_{1}] = F [γ_{2}] =: M. In the above notion we have B = C_{A}(γ_{1}) = C_{A}(M), 𝔅 = 𝔄 ∩ C_{A}(M), ν̄ = σ_{𝔄/𝔅}(ν) and s is a tame corestriction for M/F. Clearly, we have U^{ν+2}(𝔄) ∩ H^{1}(β_{1}, 𝔄) = U^{ν+2} ∩ H^{1}(β_{2}, 𝔄) = U^{ν+2}(𝔄) ∩ H^{1}(γ_{1}, 𝔄) = U^{ν+2}(𝔄) ∩ H^{1}(γ_{2}, 𝔄) =: H^{ν+2} and (θ_{1})_{H^{ν+2}} = (θ_{2})_{H^{ν+2}}.

In principle the proof runs as in the previous case, i.e., first, we reduce to the case where θ_{1} and θ_{2} coincide on U^{ν̄+1}(𝔅) and, second, we use the above argument including [G2](4.5) to find a conjugating element for the characters.

As in the case above, we show that we may change γ_{1} up to equivalence of strata and preserving the assumption F [γ_{1}] = F [γ_{2}] in such a way that C(γ_{1}, 𝔄)_{|U^{ν+1}(𝔄)∩H^{1}(γ_{1},𝔄)} = C(γ_{2}, 𝔄)_{|U^{ν+1}(𝔄)∩H^{1}(γ_{2},𝔄)} =: C(ν). We now choose ϕ ∈ C(ν) such that ϕ_{|H^{ν+2}} = (θ_{1})_{|H^{ν+2}} = (θ_{2})_{|H^{ν+2}}. For i = 1, 2 by 1.3, the strata [𝔅, ν̄ + 1, ν̄, s(β_{i} − γ_{i})] are equivalent to simple strata [𝔅, ν̄ + 1, ν̄, δ_{i}] with F [δ_{i}, γ_{i}]^{ur} ⊂ F [β_{i}]^{ur} and e(F [δ_{i}, γ_{i}]/F) | e(F [β_{i}]/F). By 1.4, we may reduce to β_{i} = γ_{i} + δ_{i} · ϵ^{−1} without restriction of generality. On the other hand, obviously F [β_{i}] = F [γ_{i} + δ_{i} · ϵ^{−1}] ⊂ F [γ_{i}, δ_{i}] and therefore, by the above use of 1.3, F [γ_{i}, δ_{i}] = F [γ_{i} + δ_{i} · ϵ^{−1}] = F [β_{i}].

By [G2](5.13) we have C(β_{i}, 𝔄)_{|U^{ν+1}(𝔄)∩H^{1}(β_{i},𝔄)} = C(γ_{i}, 𝔄)_{|U^{ν+1}(𝔄)∩H^{1}(β_{i},𝔄)} · ψ_{β_{i}−γ_{i}}^{×}, ϕ · ψ_{β_{i}−γ_{i}}^{×} ∈ C(β_{i}, 𝔄)_{|U^{ν+1}(𝔄)∩H^{1}(β_{i},𝔄)} and θ_{i} = ϕ′ · ψ_{β_{i}−γ_{i}}^{×} with ϕ′ ∈ C(ν). By the already used standard argument, for i = 1, 2 there exists c_{i} ∈ 𝔓^{−ν−1} with s(c_{i}) ∈ M ∩ 𝔓^{−ν−1} + 𝔔^{−ν̄} such that ϕ′ = ϕ·ψ_{c_{i}}^{×} and therefore θ_{i} = ϕ · ψ_{β_{i}−γ_{i}+c_{i}}^{×}.

Since, for i = 1, 2 the strata [𝔅, ν̄ + 1, ν̄, δ_{i}] are minimal (see [BK](1.4.14)(1.4.15) eventual applied to a canonical splitting) and non-scalar, the strata [𝔅, ν̄ + 1, ν̄, s(β_{i} − γ_{i} + c_{i})] are equivalent to the simple strata [𝔅, ν̄ + 1, ν̄, δ_{i} + μ_{i}] with μ_{i} ∈ M ∩ 𝔓^{−ν−1} and F [δ_{i} + μ_{i}, γ_{i}] = F [δ_{i}, γ_{i}].

Now, if g is an element that intertwines θ_{1} and θ_{2}, then g = x · t · y with x, y ∈ (1 + 𝔓^{q−ν−1} ∩ 𝔑_{−ν−1}(γ_{1}, 𝔄) + 𝔓^{[ \frac{ν+1}{2} ]} ∩ 𝔍^{1}(γ_{1}, 𝔄)), where t ∈ B^{×}. Again t intertwines θ_{1}^{x} and θ_{2}^{y^{−1}} and by 1.11 these coincide with θ_{1} and θ_{2} on U^{ν̄+1}(𝔅). We obtain that t intertwines ψ_{s(β_{1}−γ_{1}+μ_{1}·ϵ^{−1})}^{×} = ψ_{δ_{1}+μ_{1}}^{×} and ψ_{s(β_{2}−γ_{2}+μ_{2}ϵ^{−1})}^{×} = ψ_{δ_{2}+μ_{2}}^{×}.

Therefore, t intertwines the strata [𝔅, ν̄ + 1, ν̄, δ_{i} + μ_{i}], i = 1, 2, and in particular the fields F [γ_{1}, δ_{1}] and F [γ_{2}, δ_{2}] must have the same invariants. Again, using the above reduction to F [γ_{i}, δ_{i}] = F [γ_{i} + δ_{i} · ϵ^{−1}] = F [β_{i}] and the assumption on the maximal embedding invariant, it is easy to see that, for i = 1, 2, the embeddings (M[δ_{i}], 𝔄) = (F [γ_{i}, δ_{i}], 𝔄) = (F [β_{i}], 𝔄) have the same embedding invariant. However, by [G2](2.2), we may recover the embedding invariant from the inertial degree of the field and the invariants of orders by taking certain centralizers, in particular, equality of embedding invariants holds share, if we take the centralizer with respect to M. Consequently, the strata [𝔅, ν̄ + 1, ν̄, δ_{i} + μ_{i}], i = 1, 2 share a common type of embedding. Since, they have the same type of embedding and intertwine, by [BG](4.1.2) they are conjugated by an element z ∈ K_{𝔅}. Since z commutes with all the elements ϵ, γ_{i} and μ_{i} (i = 1, 2), the strata [𝔄, n, ν, β_{i} + μ_{i} · ϵ^{−1}], i = 1, 2, are conjugated by z. This leads up to conjugacy to the induction step for the assertion that there exists strata [𝔄, n, ν, β′_{i}] equivalent to [𝔄, n, ν, β_{i}] such that F [β′_{1}] = F [β′_{2}], in particular U^{ν+1}(𝔄) ∩ H^{1}(β_{1}, 𝔄) = U^{ν+1}(𝔄) ∩ H^{1}(β_{2}, 𝔄). Explicitly, we may set β′_{1} := β_{2} + (μ_{2} − μ_{1})ϵ^{−1}, β′_{2} = β_{2} for instance and we obtain F [β′_{1}] = F [β′_{2}] = F [β_{2}], because by construction, μ_{1}, μ_{2} ∈ M ⊂ F [β_{2}]. We may also assume that θ_{1} and θ_{2} agree on H^{ν+2} as well as on U^{ν̄+1}(𝔅).

Now, we attain the induction step as before. By [G2](2.10), we find d ∈ (ℌ^{1}(γ_{1}, 𝔄)∩𝔓^{ν+2})^{∗}∩a_{γ_{1}}(A) (i.e., with s(d) = 0 and ψ_{d}^{×} being trivial on H^{ν+2}) such that θ_{2} = θ_{1} · ψ_{d}^{×}. By [G2](4.5) we find x ∈ 𝔓^{q−ν−1} ∩ 𝔑_{−ν−1}(γ_{1}, 𝔄) + 𝔍^{1}(γ_{1}, 𝔄) ∩ 𝔓^{[ \frac{q+1}{2} ]} with d = a_{β_{1}}(x) and with [G2](5.12) we obtain θ_{2} = θ_{1}^{1+x} and 1 + x normalize θ_{1}_{|H^{ν+2}} as required.

□

( 1.16 ) Conjecture Let A be a tame algebra, then the ps-characters defined in [G2](6.6) preserve intertwining of its simple character realizations (i.e. of its values on triples (A′, ι, 𝔄′)) if restricted to all triples (A′, ι, 𝔄′), where ι is an embedding of an element β from a simple pair [0, β] with maximal possible embedding invariant for (F [ι(β)], 𝔄′) and A′ is an algebra of the same reduced degree N as A.

The evidence of this conjecture is given by the results of [BH] on local tame base change if we assume that these generalize from the split case to the mixed tame case A = M_{m}(D).

2 Mixed groups and some group identities

Here we use the notation of [G2](0). Let 𝔄 be a principal 𝔬_{F} -order in A with Jacobson radical 𝔓, β be an 𝔄-sound element in the sense of Definition [G2](2.4)(iii) and [0, β] be a simple pair in the sense of Definition [G2](3.7). In particular the stratum [𝔄, n, 0, β] is simple in the sense of [G2](3.1)(iii). As in Section [G2](5) we may associate the set C(β, 𝔄) of simple characters with the simple stratum [𝔄, n, 0, β]. With any fixed character θ_{𝔄} ∈ C(β, 𝔄) we may associate a unique pscharacter (θ, β) as in Definition [G2](6.6) characterized by θ(A, id_{A}, 𝔄) = θ_{𝔄}. If we apply θ_{𝔄} to commutators of the group J^{1}(β, 𝔄), by [G2](5.6) and [G2](4.1)(4.2), it induces a non-degenerate form J^{1}(β, 𝔄)/H^{1}(β, 𝔄) × J^{1}(β, 𝔄)/H^{1}(β, 𝔄) → ℂ^{×}. Therefore, θ_{𝔄} is a Heisenberg character in the sense of [Z2](4.2) and determines an irreducible Heisenberg representation η_{𝔄} of J^{1}(β, 𝔄). As θ_{𝔄} is K_{𝔅}J^{1}(β, 𝔄)-stable so is η_{𝔄}, where K_{𝔅} denotes the B^{×}-normalizer of 𝔅.

Let E := F [β], B := C_{A}(E) and 𝔅 := 𝔄∩B, then 𝔅 is a principal order and, to each principal 𝔬_{E}- order 𝔅_{m} ⊂ 𝔅, we may assign a simple stratum [𝔄_{m}, n_{m}, 0, β], where 𝔄_{m} is the unique continuation of 𝔅_{m} to A and n_{m} := −ν_{𝔄_{m}}(β).

We want to extend the representation η_{𝔄} to a representation η_{𝔄,𝔅_{m}} of U^{1}(𝔅_{m})J^{1}(β, 𝔄). If 𝔅_{m} is a minimal principal order, then the latter group is a pro p-Sylow subgroup of J(β, 𝔄) := 𝔅^{×}J^{1}(β, 𝔄). For that purpose we construct mixed characters of mixed groups to generalize the construction of Heisenberg representations with respect to the groups H^{1}(β, 𝔄) and J^{1}(β, 𝔄) to the groups U^{1}(𝔅_{m})H^{1}(β, 𝔄) and U^{1}(𝔅_{m})J^{1}(β, 𝔄). If 𝔅_{m} is a minimal principal order, then, by a standard method for extending a representation to from a p-Sylow subgroup of a group one can extend η_{𝔄,𝔅_{m}} to a representation κ_{𝔄} of J(β, 𝔄). The representation κ_{𝔄} will then be used as a tensor factor of our conjectural simple type.

To simplify notation we set H^{1} := H^{1}(β, 𝔄), J^{1} := J^{1}(β, 𝔄), and H_{m}^{1} := H^{1}(β, 𝔄_{m}) and J_{m}^{1} := J^{1}(β, 𝔄_{m}).

We define mixed groups J_{𝔄,𝔅_{m}}^{1} := U^{1}(𝔅_{m})(J^{1} ∩ J_{m}^{1})H^{1} and H_{𝔄,𝔅_{m}}^{1} := U^{1}(𝔅_{m})(J^{1} ∩ H_{m}^{1})H^{1}. We have to check the following lemma:

( 2.1 ) Lemma The sets J_{𝔄,𝔅_{m}}^{1} and H_{𝔄,𝔅_{m}}^{1} are groups and H_{𝔄,𝔅_{m}}^{1} is a normal subgroup of J_{𝔄,𝔅_{m}}^{1}. The factor group is abelian and isomorphic to J_{𝔄,𝔅_{m}}^{1} ∩ J^{1}/H_{𝔄,𝔅_{m}}^{1} ∩ J^{1}.

\underline{Proof:}

The fact that the sets J_{𝔄,𝔅_{m}}^{1} and H_{𝔄,𝔅_{m}}^{1} are groups follows from the properties of the groups H^{1}, J^{1}, H_{m}^{1} and J_{m}^{1} described in Section [G2](4).

We have H_{𝔄,𝔅_{m}}^{1} = U^{1}(𝔅_{m})(J^{1} ∩ H_{m}^{1})H^{1} and J_{𝔄,𝔅_{m}}^{1} = U^{1}(𝔅_{m})(J^{1} ∩ J_{m}^{1})H^{1}. Clearly, H^{1} normalizes itself and the commutator of (J^{1} ∩ J_{m}^{1}) and H^{1} is contained in H^{1}; hence, H^{1} normalizes J_{𝔄,𝔅_{m}}^{1} ∩ J^{1}. Again (J^{1} ∩ H_{m}^{1}) normalizes H^{1} and the commutator of (J^{1} ∩ J_{m}^{1}) and (J^{1} ∩ H_{m}^{1}) is contained in H^{1}; hence, (J^{1} ∩ J_{m}^{1}) normalizes H_{𝔄,𝔅_{m}}^{1} ∩ J^{1}. Finally, U^{1}(𝔅_{m}) ⊂ (K_{𝔅} ∩ K_{𝔅_{m}}); hence, U^{1}(𝔅_{m}) normalizes H_{𝔄,𝔅_{m}}^{1}. Altogether we have proved that H_{𝔄,𝔅_{m}}^{1} is a normal subgroup of J_{𝔄,𝔅_{m}}^{1}.

Since J^{1}/H^{1} is abelian and J_{𝔄,𝔅_{m}}^{1} = (J_{𝔄,𝔅_{m}}^{1} ∩ J^{1})H_{𝔄,𝔅_{m}}^{1}, we also deduce the above statement. □

( 2.2 ) Lemma Let A = M_{m}(D) be an arbitrary central simple algebra over F and K/F an unramified extension in D. Let 𝔅 be a principal order in B = C_{A}(K) = M_{m}(C_{D}(K)), which is given in the normal form and has period r′, then the unique continuation 𝔄 of 𝔅 has period r = \frac{r′}{⟨[K:F],r′⟩} and there exists an r′×r′-(s′ × s′) block monomial matrix g (i.e., g = P M, where P is an invertible r′ × r′-(s′ × s′) block monomial matrix where the entries are either 0 or the s′ × s′-unit matrix I_{s′} and M = diag(d_{1}I_{s′}, …, d_{r′}I_{s′}), where d_{1}, …, d_{r′} are powers of a fixed prime element π of 𝔬_{D}) such that g𝔄g^{−1} is given in normal form and where s′ = \frac{m}{r′} is the invariant of 𝔅.

\underline{Proof:}

The proof is contained in [G1](3.2, 3.3).

□

( 2.3 ) Lemma Let E/F be an extension in A, B := C_{A}(E) and 𝔅 be a principal order in B with period r′. Moreover, let 𝔄 be the unique continuation of 𝔅 to A, then we have e_{𝔄/𝔅} = \frac{⟨[E:F],d⟩}{⟨⟨[E:F],d⟩,e(E/F)r′⟩} in the notation of ([G2](2.5)). In particular, the continuation of 𝔅 by 𝔄 is unramified if and only if ⟨[E : F], d⟩ | e(E/F)r′.

\underline{Proof:}

This is left to the reader or can be found in [G0](2.1.5).

□

( 2.4 ) Lemma Let E/F be an extension in A, B = C_{A}(E) and 𝔅, 𝔅′ be principal orders in B with periods r′ and r̃′, respectively, such that 𝔅′ ⊂ 𝔅. Moreover, let 𝔄 and 𝔄̃ be the continuation of 𝔅 and 𝔅′, respectively, and assume that ⟨[E : F], d⟩ | e(E/F)r′, i.e., e_{𝔄/𝔅} = 1, then we also have 𝔄̃ ⊂ 𝔄.

\underline{Proof:}

Let K/F be the maximal unramified subextension of E/F such that [K : F] | Ind(A) = d, then we have ⟨f(E/K), d̄⟩ = 1, where d̄ := \frac{d}{[K:F]} is the index of C := C_{A}(K). Therefore, if we choose an unramified splitting field L/K of C, the algebra E ⊗_{K} L is a field. Moreover, if ℭ and ℭ̃ are the unique continuations of 𝔅 and 𝔅′, respectively, then ℭ ⊗_{𝔬_{K}} 𝔬_{L}, ℭ̃ ⊗_{𝔬_{K}} 𝔬_{L} are the unique continuations of 𝔅 ⊗_{𝔬_{K}} 𝔬_{L} and 𝔅′ ⊗_{𝔬_{K}} 𝔬_{L}, respectively, from B ⊗_{K} L to C ⊗_{K} L. As, by [BK1](1.2.1), the lattice chain associated with a hereditary order in a simple B ⊗_{K} L- and C ⊗_{K} L-module, respectively, is invariant under continuation, we deduce from 𝔅′ ⊗_{𝔬_{K}} 𝔬_{L} ⊂ 𝔅 ⊗_{𝔬_{K}} 𝔬_{L} (which is valid by assumption) that ℭ̃ ⊗_{𝔬_{K}} 𝔬_{L} ⊂ ℭ ⊗_{𝔬_{K}} 𝔬_{L}; hence, we have ℭ̃ ⊂ ℭ.

Since [K : F] | d and Skolem-Noether theorem, we may assume up to an inner automorphism of A = M_{m}(D) that K ⊂ D and we get C = M_{m}(Δ) with Δ := C_{D}(K). Moreover, up to conjugacy by an element of C^{×} we may assume that the orders ℭ and ℭ̃ are given in simultaneous normal form. By [G1](2.2) the period of ℭ is r′ \frac{e(E/K)}{⟨e(E/K),d̄⟩}; hence, by assumption the situation of [G1](3.2) and its proof apply with t′ = 1 and for the continuation of ℭ by 𝔄. As the period of ℭ divides the period of ℭ′, the same applies to the continuation of ℭ′ by 𝔄̃. So let {Ỹ_{j}}_{j∈ℤ} be the lattice chain of ℭ̃ in the simple C-module Δ^{m} (column-vectors) and π a prime element of 𝔬_{D} that normalizes K. Moreover, let {X̃_{j}}_{j∈ℤ} be the lattice chain of 𝔄̃ in the simple A-module D^{m} (column-vectors) and set f := [K : F], then, by equation (7) in [G1] and with an appropriate numbering of the chains for all j ∈ ℤ, we have X̃_{j} = ⊕_{k=0}^{f−1}Y_{j− \frac{r̂}{f} k}π^{k}, where r̂ is the period of ℭ̃.

Let h be a uniformizer for ℭ̃ and let μ be the quotient of the period r̂ of ℭ̃ and of the period of ℭ, then h^{μ} is a uniformizer of ℭ and, if we choose the numbering of the chain {Ỹ_{j}}_{j∈ℤ} in such a way that Ỹ_{0} is an indecomposable ℭ-module, then {Ỹ_{jμ}}_{j∈ℤ} is the lattice chain of ℭ in Δ^{m}. Applying [G1](equation (7)) again, we see that 𝔄 has the lattice chain {X̃_{jμ}}_{j∈ℤ}; hence, 𝔄̃ ⊂ 𝔄 as required.

□

( 2.5 ) Lemma Let [𝔄_{1}, n_{1}, 0, β], [𝔄_{2}, n_{2}, 0, β] be simple strata in the sense of [G2](3.1)(iii), B = C_{A}(β), 𝔅_{1} = 𝔄_{1} ∩ B and 𝔅_{2} = 𝔄_{2} ∩ B. Moreover, assume that there exists a minimal principal order 𝔅_{m} contained in 𝔅_{1} and 𝔅_{2}. Let 𝔄_{m} be the unique continuation of 𝔅_{m} to A, then there exist integers q_{1}^{(1)}, …, q_{μ}^{(1)}, q_{1}^{(2)}, …, q_{μ}^{(2)} attached to each special defining sequence [𝔄_{m}, n_{m}, q_{i}^{m}, γ_{i}], i = 1, …, μ, such that [𝔄_{1}, n_{1}, q_{i}^{(1)}, γ_{i}], i = 1, …, μ, [𝔄_{2}, n_{2}, q_{i}^{(2)}, γ_{i}], i = 1, …, μ, are special approximation sequences for [𝔄_{1}, n_{1}, 0, β] and [𝔄_{2}, n_{2}, 0, β], respectively.

\underline{Proof:}

If h_{m}, h_{1}, h_{2} are uniformizers for 𝔅_{m}, 𝔅_{1}, 𝔅_{2}, (i.e., K_{𝔅_{m}} = ⟨h_{m}⟩𝔅_{m}^{×}, K_{𝔅_{1}} = ⟨h_{1}⟩𝔅_{1}^{×}, K_{𝔅_{2}} = ⟨h_{2}⟩𝔅_{2}^{×}), then there exist natural numbers t_{1}, t_{2} such that h_{m}^{t_{1}} = h_{1}b_{1} and h_{m}^{t_{2}} = h_{2}b_{2}, where b_{1} ∈ 𝔅_{1}^{×} and b_{2} ∈ 𝔅_{2}^{×}.

Let ℭ_{m}, ℭ_{1}, ℭ_{2} be the unique continuations of 𝔅_{m}, 𝔅_{1}, 𝔅_{2} to C := C_{A}(K), where K = E^{ur} is the unramified part of the extension E/F and E := F [β]. By [G](2.2) these continuations are unramified; hence, h_{m}, h_{1}, h_{2} are also uniformizers for ℭ_{m}, ℭ_{1}, ℭ_{2}. By the given containments of orders, (2.4) and [BG](5.1)(ii), the strata [ℭ_{1}, −ν_{ℭ_{1}}(β), 0, β], [ℭ_{2}, −ν_{ℭ_{2}}(β), 0, β] and [ℭ_{m}, −ν_{ℭ_{m}}(β), 0, β] are simple.

Note that 𝔄_{m} is the unique continuation of ℭ_{m} to A and let [𝔄_{m}, n_{m}, q_{i}^{m}, γ_{i}], i = 1, …, μ, be a special approximation sequence as in [G2](3.11) for [𝔄_{m}, −ν_{𝔄_{m}}(β), 0, β], then there exist integers k_{1}^{m}, …, k_{μ}^{m} such that

( 2.6 ) γ_{i} = h_{m}^{k_{i}^{m}}c_{i},

where c_{i} ∈ ℭ_{m}^{×} for i = 1, …, μ. Finally, ν_{ℭ_{m}}(β) = ν_{ℭ_{m}}(γ_{i}) for i = 1, …, μ, we actually have k_{1}^{m} = … = k_{μ}^{m} =: k^{m}. Moreover, since the representation (2.6) is unique, ℭ_{m}^{×} ⊂ ℭ_{1}^{×}, ℭ_{2}^{×} and β ∈ K_{ℭ_{1}}, K_{ℭ_{2}}, we conclude that t_{1}, t_{2} | k^{m}; hence, γ_{1}, …, γ_{μ} must be ℭ_{1}- and ℭ_{2}-pure. From the remarks following [G2](3.9) we deduce that, with q_{1}^{(1)} = q_{1}^{(2)} = 0, q_{μ}^{(1)} = −ν_{𝔄_{1}}(β), q_{μ}^{(2)} = −ν_{𝔄_{2}}(β) and q_{i+1}^{(1)} = −k_{0}(γ_{i}, 𝔄_{1}), q_{i+1}^{(2)} = −k_{0}(γ_{i}, 𝔄_{2}) for i = 1, …, μ − 1, we obtain defining sequences [𝔄_{1}, −ν_{𝔄_{1}}(β), q_{i}^{(1)}, γ_{i}], [𝔄_{2}, −ν_{𝔄_{2}}(β), q_{i}^{(2)}, γ_{i}], i = 1, …, μ, for the simple strata [𝔄_{1}, −ν_{𝔄_{1}}(β), 0, β], [𝔄_{2}, −ν_{𝔄_{2}}(β), 0, β] as required, because they are common realizations of the sequence of simple pairs [0, β], [−k_{0}(γ_{i}, 𝔄(F [γ_{i}])), γ_{i}] for i = 1, …, μ − 1, and by straightforward arguments using [BG](5.1)(ii) these approximation sequences are special.

□

( 2.7 ) Lemma Let ν, ν̄, ν̃ be non-negative integers, let [𝔄, n, 0, β] be a simple stratum and 𝔅_{m} a principal order contained in 𝔅 := 𝔄∩C_{A}(β). Moreover, let 𝔓 be the Jacobson radical of 𝔄, 𝔄_{m} be the unique continuation of 𝔅_{m}, 𝔓_{m} be the Jacobson radical of 𝔄_{m}, and let [𝔄, n, q_{i}, γ_{i}], [𝔄_{m}, n′, q′_{i}, γ_{i}] with i = 1, …, μ be simultaneous special approximation sequences for the strata [𝔄, n, 0, β], [𝔄_{m}, n′, 0, β] as in (2.5), where n′ := ν_{𝔄_{m}}(β). Then we have

(i) (𝔓^{ν} + 𝔓ν̄ ; m) ∩ C_{A}(γ_{i}) = (𝔓^{ν} ∩ C_{A}(γ_{i}) + 𝔓ν̄ ; m ∩ C_{A}(γ_{i})) and

(ii) (𝔓^{ν} ∩ 𝔓ν̄ ; m + 𝔓ν̃) ∩ C_{A}(γ_{i}) = 𝔓^{ν} ∩ 𝔓ν̄ ; m ∩ C_{A}(γ_{i}) + 𝔓ν̃ ∩ C_{A}(γ_{i})

for i = 1, …, μ.

\underline{Proof:}

Let K_{k}/F be the maximal unramified subextension of F [γ_{k}]/F with respect to the property that [K_{k} : F] | d for k = 1, …, μ. Let us fix an index k. Set ℭ := 𝔄 ∩ C_{A}(K_{k}), ℭ_{m} := 𝔄_{m} ∩ C_{A}(K_{k}), 𝔅 := 𝔄 ∩ C_{A}(γ_{k}) and 𝔄 ∩ C_{A}(γ_{k}).

The algebra C_{A}(K_{k})/K_{k} has index \frac{d}{[K_{k}:F]} and therefore, by (2.3),(2.4), the continuations of 𝔅, 𝔅_{m} by ℭ,ℭ_{m} are unramified and ℭ ⊃ ℭ_{m}. Moreover, up to conjugacy, we may assume that K_{k} ⊂ D and that ℭ, ℭ_{m} are in simultaneous normal form. By (2.2) we may find monomial matrices with entries powers of the same prime element π of 𝔬_{D} such that 𝔄 and 𝔄_{m} are conjugated by them to their normal forms in A = M_{m}(D). Consequently, we have

𝔓^{ν} = (𝔭_{D}^{ν_{i,j}})_{i,j=1,...,m}, 𝔓ν̄ ; m = (𝔭ν̄_{i,j} ; D)_{i,j=1,...,m} and 𝔓ν̃ = (𝔭ν̃_{i,j} ; D)_{i,j=1,...,m}, where these representa-

tions mean m × m-matrices where the (i, j) entry lies in 𝔭_{D}^{ν_{i,j}}, 𝔭ν̄_{i,j} ; D and 𝔭ν̃_{i,j} ; D, respectively. If

we set D_{k} := C_{D}(K_{k}), we obtain (𝔓^{ν} + 𝔓ν̄ ; m) ∩ C_{A}(K_{k}) = (𝔭_{D_{k}}^{η_{i,j}})_{i,j=1,...,m} = 𝔓^{ν} ∩ C_{A}(K_{k}) +

𝔓ν̄ ; m ∩ C_{A}(K_{k}), where η_{i,j} = σ_{𝔬_{D}/𝔬_{D_{k}}} (max(ν_{i,j}, ν̄_{i,j})) = max(σ_{𝔬_{D}/𝔬_{D_{k}}} (ν_{i,j}), σ_{𝔬_{D}/𝔬_{D_{k}}} (ν̄_{i,j})) and

(𝔓^{ν} ∩ 𝔓ν̄ ; m + 𝔓ν̃) ∩ C_{A}(K_{k}) = (𝔭η̂_{i,j} ; D_{k})_{i,j=1,...,m} = (𝔓^{ν} ∩ 𝔓ν̄ ; m ∩ C_{A}(K_{k}) + 𝔓ν̃ ∩ C_{A}(K_{k})), where

η̂_{i,j} = σ_{𝔬_{D}/𝔬_{D_{k}}} (max(min(ν_{i,j}, ν̄_{i,j}), ν̃_{i,j})) = max(min(σ_{𝔬_{D}/𝔬_{D_{k}}} (ν_{i,j}), σ_{𝔬_{D}/𝔬_{D_{k}}} (ν̄_{i,j})), σ_{𝔬_{D}/𝔬_{D_{k}}} (ν̃_{i,j})).

Therefore, we restrict ourselves to the case of K_{k} = F and where F [γ_{i}] ⊗_{F} L is a field for the unramified extension L/F of degree d which splits A. The continuations of 𝔅 and 𝔅_{m} by 𝔄 and 𝔄_{m} are unramified.

With Ā := A ⊗_{F} L, B̄ := CĀ(γ_{k} ⊗ 1), 𝔄̄ := 𝔄 ⊗_{𝔬_{F}} 𝔬_{L}, 𝔄̄_{m} := 𝔄_{m} ⊗_{𝔬_{F}} 𝔬_{L}, 𝔅̄ := 𝔅 ⊗_{𝔬_{F}} 𝔬_{L} and 𝔅̄_{m} := 𝔅_{m} ⊗_{𝔬_{F}} 𝔬_{L}, according to [BK1](1.2.6)(1.2.8), we obtain isomorphisms Ā ≃ A(F [γ_{k}] ⊗_{L} B̄, 𝔄̄ ≃ 𝔄(F [γ_{k}]) ⊗_{𝔬_{L}} 𝔅̄ and 𝔄̄_{m} ≃ 𝔄(F [γ_{k}]) ⊗ 𝔅̄_{m}, where A(F [γ_{k}) := End_{F} (F [γ_{k}]) and 𝔄(γ_{k}) := End_{𝔬_{F}} ({𝔭_{F [γ_{k}]}^{j}}_{j∈ℤ}).

By [BK1](1.2.10) the powers of the Jacobson radicals 𝔓̄ := 𝔓 ⊗_{𝔬_{F}} 𝔬_{L}, 𝔓̄_{m} := 𝔓_{m} ⊗_{𝔬_{F}} 𝔬_{L} of 𝔄̄ and 𝔄̄_{m} fulfil corresponding isomorphisms and are, by [BK1](1.3.12), F [γ_{k} ⊗ 1]-exact in the sense of Definition [BK1](1.3.11) for the tame corestriction s := s_{γ_{k}} ⊗ 1 : Ā → B̄, defined via a tame corestriction s_{γ_{k}} : A → B.

From [BK1](1.3.12)(1.3.10) we deduce (𝔓̄^{ν} +𝔓̄ν̄ ; m)∩B̄ = s(𝔓̄^{ν} +𝔓̄ν̄ ; m) = s(𝔓̄^{ν})+s(𝔓̄ν̄ ; m) = (𝔓̄^{ν} ∩B̄ +

𝔓̄ν̄ ; m∩B̄) and (𝔓̄^{ν} ∩𝔓̄ν̄ ; m+𝔓̄ν̃)∩B̄ = s(𝔓̄^{ν} ∩𝔓̄ν̄ ; m+𝔓̄ν̃) = s(𝔓̄^{ν} ∩𝔓̄ν̄ ; m)+s(𝔓̄ν̃) = (𝔓̄^{ν} ∩𝔓̄ν̄ ; m∩B̄+𝔓̄ν̃∩B̄).

The assertion of our Lemma now follows by taking, on both sides of the equations, the sets of Galois invariant elements with respect to the action of Gal(L/F) on the second tensor factor of A ⊗_{F} L.

□

( 2.8 ) Lemma Let 𝔄, 𝔄_{m} and 𝔅_{m} be as above. Moreover, choose a sequence of elements γ_{1}, ..., γ_{μ} such that the sequences [𝔄, n, q_{i}, γ_{i}]; [𝔄_{m}, n_{m}, q′_{i}, γ_{i}], i = 1, …, μ, are special approximation sequences for the strata [𝔄, n, 0, β] and [𝔄_{m}, n_{m}, 0, β], respectively, where n_{m} := −ν_{𝔄_{m}}(β), n := −ν_{𝔄}(β) and γ_{1}, …, γ_{μ}; q_{1}, …, q_{μ}; q′_{1}, …, q′_{μ} are suitably chosen according to (2.5).

To simplify the notation set J^{1} =: 1 + J^{1}, J_{m}^{1} =: 1 + J_{m}^{1}, H^{1} =: 1 + H^{1} and H_{m}^{1} =: 1 + H_{m}^{1}. Then we have

𝔓^{[ \frac{q_{i}+1}{2} ]} ∩ J^{1} ∩ 𝔓_{m}^{[ \frac{q′_{i}+1}{2} ]} ∩ J_{m}^{1} = ∑_{j=i}^{μ} 𝔓^{[ \frac{q_{j} +1}{2} ]} ∩ 𝔓_{m}^{[^{\frac{q′_{j} +1}{2}} ]} ∩ C_{A}(γ_{j}),

𝔓^{[ \frac{q_{i}+1}{2} ]} ∩ J^{1} ∩ 𝔓_{m}^{[ \frac{q′_{i}}{2} ]+1} ∩ H_{m}^{1} = ∑_{j=i}^{μ} 𝔓^{[ \frac{q_{j} +1}{2} ]} ∩ 𝔓_{m}^{[^{\frac{q′_{j}}{2}} ]+1} ∩ C_{A}(γ_{j})

and

𝔓^{[ \frac{q_{i}}{2} ]+1} ∩ H^{1} ∩ 𝔓_{m}^{[ \frac{q′_{i}}{2} ]+1} ∩ H_{m}^{1} = ∑_{j=i}^{μ} 𝔓^{[ \frac{q_{j}}{2} ]+1} ∩ 𝔓_{m}^{[^{\frac{q′_{j}}{2}} ]+1} ∩ C_{A}(γ_{j})

for i = 1, …, μ. In particular, for i = 1 we obtain the left-hand sides 𝔍^{1} ∩ 𝔍_{m}^{1}, 𝔍^{1} ∩ ℌ_{m}^{1} and ℌ^{1} ∩ ℌ_{m}^{1}.

\underline{Proof:}

As all these assertions are similar to prove, we restrict ourselves to the first case. With the notation of ([G2](4.1)), we have J^{1} = J^{1}(β, 𝔄) and J_{m}^{1} = J^{1}(β, 𝔄_{m}). Moreover, from ([G2](4.3)), we easily deduce

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

and

𝔓_{m}^{[ \frac{q′_{i}+1}{2} ]} ∩ J_{m}^{1} = 𝔓_{m}^{[ \frac{q′_{i}+1}{2} ]} ∩ J^{1}(γ_{i}, 𝔄_{m}) = ∑_{j=i}^{μ} 𝔓_{m}^{[^{\frac{q′_{j} +1}{2}} ]} ∩ C_{A}(γ_{j}) = 𝔓_{m}^{[ \frac{q′_{i}+1}{2} ]} ∩ C_{A}(γ_{i}) + 𝔓_{m}^{[^{\frac{q{′ \atop i+1}+1}{2}} ]} ∩ J_{m}^{1}. (2)

Using induction over the approximation sequence, it is sufficient to show that

\begin{matrix}(𝔓^{[ \frac{q_{i}+1}{2} ]} ∩ C_{A}(γ_{i}) + 𝔓^{[ \frac{q_{i+1}+1}{2} ]} ∩ J^{1}) ∩ (𝔓_{m}^{[ \frac{q′_{i}+1}{2} ]} ∩ C_{A}(γ_{i}) + 𝔓_{m}^{[^{\frac{q{′ \atop i+1}+1}{2}} ]} ∩ J_{m}^{1}) = \\ 𝔓^{[ \frac{q_{i}+1}{2} ]} ∩ 𝔓_{m}^{[ \frac{q′_{i}+1}{2} ]} ∩ C_{A}(γ_{i}) + 𝔓^{[ \frac{q_{i+1}+1}{2} ]} ∩ J^{1} ∩ 𝔓_{m}^{[^{\frac{q{′ \atop i+1}+1}{2}} ]} ∩ J_{m}^{1}. \end{matrix}  (3)

The proof is now the same as that of Lemma [BK1](3.6.10)(p. 140,141) in the split case and we give a more detailed proof for our different situation.

We also consider the adjoint map a_{γ_{i}} ∈ End_{F} (V) of [G2](2) or [BK](1.4) and note that ker(a_{γ_{i}}) = C_{A}(γ_{i}). Let L_{l} be the 𝔬_{F} -lattice on the left-hand side of the equation (3) and L_{r} be the 𝔬_{F} -lattice on the right-hand side of equation (3), then we obviously have L_{l} ⊃ L_{r} and a_{γ_{i}}(L_{l}) ≃ L_{l}/L_{l} ∩ ker(a_{γ_{i}}) ⊃

L_{r}/L_{r} ∩ ker(a_{γ_{i}}) ≃ a_{γ_{i}}(L_{r}). Since L_{l} ∩ ker(a_{γ_{i}}) = L_{l} ∩ C_{A}(γ_{i}) = 𝔓^{[ \frac{q_{i}+1}{2} ]} ∩ 𝔓_{m}^{[ \frac{q′_{i}+1}{2} ]} ∩ C_{A}(γ_{i}) = L_{r} ∩ C_{A}(γ_{i}) = L_{r} ∩ ker(a_{γ_{i}}), it is sufficient to show that a_{γ_{i}}(L_{l}) = a_{γ_{i}}(L_{r}).

We have

a_{γ_{i}}(𝔓^{[ \frac{q_{i+1}+1}{2} ]} ∩ J^{1} ∩ 𝔓_{m}^{[^{\frac{q{′ \atop i+1}+1}{2}} ]} ∩ J_{m}^{1}) = a_{γ_{i}}(L_{r}) ⊆

a_{γ_{i}}(L_{l})⊆a_{γ_{i}}(𝔓^{[ \frac{q_{i}+1}{2} ]} ∩ C_{A}(γ_{i}) + 𝔓^{[ \frac{q_{i+1}+1}{2} ]} ∩ J^{1}) ∩ a_{γ_{i}}(𝔓_{m}^{[ \frac{q′_{i}+1}{2} ]} ∩ C_{A}(γ_{i}) + 𝔓_{m}^{[^{\frac{q{′ \atop i+1}+1}{2}} ]} ∩ J_{m}^{1}) =

a_{γ_{i}}(𝔓^{[ \frac{q_{i+1}+1}{2} ]} ∩ J^{1}) ∩ a_{γ_{i}}(𝔓_{m}^{[^{\frac{q{′ \atop i+1}+1}{2}} ]} ∩ J_{m}^{1}).

Hence, it only remains to prove

a_{γ_{i}}(𝔓^{[ \frac{q_{i+1}+1}{2} ]} ∩ J^{1} ∩ 𝔓_{m}^{[^{\frac{q{′ \atop i+1}+1}{2}} ]} ∩ J_{m}^{1}) = a_{γ_{i}}(𝔓^{[ \frac{q_{i+1}+1}{2} ]} ∩ J^{1}) ∩ a_{γ_{i}}(𝔓_{m}^{[^{\frac{q{′ \atop i+1}+1}{2}} ]} ∩ J_{m}^{1}).

We use the general fact following [BK1](3.6.17), which yields

a_{γ_{i}}(𝔓^{[ \frac{q_{i+1}+1}{2} ]} ∩ J^{1}) ∩ a_{γ_{i}}(𝔓_{m}^{[^{\frac{q{′ \atop i+1}+1}{2}} ]} ∩ J_{m}^{1})/a_{γ_{i}}(𝔓^{[ \frac{q_{i+1}+1}{2} ]} ∩ J^{1} ∩ 𝔓_{m}^{[^{\frac{q{′ \atop i+1}+1}{2}} ]} ∩ J_{m}^{1}) ≃

(𝔓^{[ \frac{q_{i+1}+1}{2} ]} ∩ J^{1} + 𝔓_{m}^{[^{\frac{q{′ \atop i+1}+1}{2}} ]} ∩ J_{m}^{1}) ∩ ker(a_{γ_{i}})/(𝔓^{[ \frac{q_{i+1}+1}{2} ]} ∩ J^{1} ∩ ker(a_{γ_{i}}) + 𝔓_{m}^{[^{\frac{q{′ \atop i+1}+1}{2}} ]} ∩ J_{m}^{1} ∩ ker(a_{γ_{i}}))

and we are reduced to showing that

(𝔓^{[ \frac{q_{i+1}+1}{2} ]} ∩ J^{1} + 𝔓_{m}^{[^{\frac{q{′ \atop i+1}+1}{2}} ]} ∩ J_{m}^{1}) ∩ C_{A}(γ_{i}) = 𝔓^{[ \frac{q_{i+1}+1}{2} ]} ∩ J^{1} ∩ C_{A}(γ_{i}) + 𝔓_{m}^{[^{\frac{q{′ \atop i+1}+1}{2}} ]} ∩ J_{m}^{1} ∩ C_{A}(γ_{i}).

On the other hand, we have

𝔓^{[ \frac{q_{i+1}+1}{2} ]} ∩ C_{A}(γ_{i}) + 𝔓_{m}^{[^{\frac{q{′ \atop i+1}+1}{2}} ]} ∩ C_{A}(γ_{i}) ⊆
𝔓^{[ \frac{q_{i+1}+1}{2} ]} ∩ J^{1} ∩ C_{A}(γ_{i}) + 𝔓_{m}^{[^{\frac{q{′ \atop i+1}+1}{2}} ]} ∩ J_{m}^{1} ∩ C_{A}(γ_{i}) ⊆
(𝔓^{[ \frac{q_{i+1}+1}{2} ]} ∩ J^{1} + 𝔓_{m}^{[^{\frac{q{′ \atop i+1}+1}{2}} ]} ∩ J_{m}^{1}) ∩ C_{A}(γ_{i}) ⊆
(𝔓^{[ \frac{q_{i+1}+1}{2} ]} + 𝔓_{m}^{[^{\frac{q{′ \atop i+1}+1}{2}} ]}) ∩ C_{A}(γ_{i}),

where the first and the last term are equal by (2.7)(i), and we are done.

□ ( 2.9 ) Lemma With the assumptions and notations of (2.8) and i ∈ {1, …, μ} we have

J^{1} ∩ J_{m}^{1} = (∏_{ν=1}^{i−1}(1 + 𝔓^{[ \frac{qν +1}{2} ]} ∩ 𝔓_{m}^{[ \frac{q′_{ν} +1}{2} ]} ∩ C_{A}(γ_{ν})))(U^{[ \frac{q_{i}+1}{2} ]}(𝔄) ∩ J^{1} ∩ U^{[ \frac{q′_{i}+1}{2} ]}(𝔄_{m}) ∩ J_{m}^{1}),

H^{1} ∩ H_{m}^{1} = (∏_{ν=1}^{i−1}(1 + 𝔓^{[ \frac{qν}{2} ]+1} ∩ 𝔓_{m}^{[ \frac{q′_{ν}}{2} ]+1} ∩ C_{A}(γ_{ν})))(U^{[ \frac{q_{i}}{2} ]+1}(𝔄) ∩ H^{1} ∩ U^{[ \frac{q′_{i}}{2} ]+1}(𝔄_{m}) ∩ H_{m}^{1})

and

J^{1} ∩ H_{m}^{1} = (∏_{ν=1}^{i−1}(1 + 𝔓^{[ \frac{qν +1}{2} ]} ∩ 𝔓_{m}^{[ \frac{q′_{ν}}{2} ]+1} ∩ C_{A}(γ_{ν})))(U^{[ \frac{q_{i}+1}{2} ]}(𝔄) ∩ J^{1} ∩ U^{[ \frac{q′_{i}}{2} ]+1}(𝔄_{m}) ∩ H_{m}^{1}).

\underline{Proof:}

The proofs of the first and of the other assertions are essentially the same, so that we restrict ourselves to the first one. For i = 1 there is nothing to prove. Assuming the assertion for a fixed i ∈ {1, …, μ − 1}, we have to show for the proof of the assertion for i + 1 that U^{[ \frac{q_{i}+1}{2} ]}(𝔄) ∩ J^{1} ∩ U^{[ \frac{q′_{i}+1}{2} ]}(𝔄_{m}) ∩ J_{m}^{1} =

(1 + 𝔓^{[ \frac{q_{i}+1}{2} ]} ∩ 𝔓_{m}^{[ \frac{q′_{i}+1}{2} ]} ∩ C_{A}(γ_{i})))(U^{[ \frac{q_{i+1}+1}{2} ]}(𝔄) ∩ J^{1} ∩ U^{[^{\frac{q{′ \atop i+1}+1}{2}} ]}(𝔄_{m}) ∩ J_{m}^{1}).

Using  (2.8)  and  [G2](4.3),  we  have  U^{[ \frac{q_{i}+1}{2} ]}(𝔄) ∩ J^{1} ∩ U^{[ \frac{q′_{i}+1}{2} ]}(𝔄_{m}) ∩ J_{m}^{1} =
U^{[ \frac{q_{i}+1}{2} ]}(𝔄) ∩ J^{1}(γ_{i}, 𝔄) ∩ U^{[ \frac{q′_{i}+1}{2} ]}(𝔄_{m}) ∩ J^{1}(γ_{i}, 𝔄_{m}) =  1 + ∑_{ν=i}^{μ} 𝔓^{[ \frac{qν +1}{2} ]} ∩ 𝔓_{m}^{[ \frac{q′_{ν} +1}{2} ]} ∩ C_{A}(γ_{ν}) =

1 + 𝔓^{[ \frac{q_{i}+1}{2} ]} ∩ 𝔓_{m}^{[ \frac{q′_{i}+1}{2} ]} ∩ C_{A}(γ_{i}) + 𝔓^{[ \frac{q_{i+1}+1}{2} ]} ∩ J^{1} ∩ 𝔓_{m}^{[^{\frac{q{′ \atop i+1}+1}{2}} ]} ∩ J_{m}^{1}.

Therefore, it remains to show that 1 + 𝔓^{[ \frac{q_{i}+1}{2} ]} ∩ 𝔓_{m}^{[ \frac{q′_{i}+1}{2} ]} ∩ C_{A}(γ_{i}) + 𝔓^{[ \frac{q_{i+1}+1}{2} ]} ∩ J^{1} ∩ 𝔓_{m}^{[^{\frac{q{′ \atop i+1}+1}{2}} ]} ∩ J_{m}^{1} =

(1 + 𝔓^{[ \frac{q_{i}+1}{2} ]} ∩ 𝔓_{m}^{[ \frac{q′_{i}+1}{2} ]} ∩ C_{A}(γ_{i}))(1 + 𝔓^{[ \frac{q_{i+1}+1}{2} ]} ∩ J^{1} ∩ 𝔓_{m}^{[^{\frac{q{′ \atop i+1}+1}{2}} ]} ∩ J_{m}^{1}).  By  [G2](4.3),

𝔓^{[ \frac{q_{i+1}+1}{2} ]} ∩ J^{1} = 𝔓^{[ \frac{q_{i+1}+1}{2} ]} ∩ J^{1}(γ_{i}, 𝔄) and 𝔓_{m}^{[ \frac{q′_{i}+1}{2} ]} ∩ J_{m}^{1} = 𝔓_{m}^{[ \frac{q′_{i}+1}{2} ]} ∩ J^{1}(γ_{i}, 𝔄_{m}). We deduce that

𝔓^{[ \frac{q_{i+1}+1}{2} ]} ∩ J^{1} ∩ 𝔓_{m}^{[^{\frac{q{′ \atop i+1}+1}{2}} ]} ∩ J_{m}^{1} is a (𝔓^{[ \frac{q_{i}+1}{2} ]} ∩ 𝔓_{m}^{[ \frac{q′_{i}+1}{2} ]} ∩ C_{A}(γ_{i}), 𝔓^{[ \frac{q_{i}+1}{2} ]} ∩ 𝔓_{m}^{[ \frac{q′_{i}+1}{2} ]} ∩ C_{A}(γ_{i}))-bimodule; hence, the containment ⊇ is clear.

For  the  reverse  containment,  let  x ∈ 𝔓^{[ \frac{q_{i}+1}{2} ]} ∩ 𝔓_{m}^{[ \frac{q′_{i}+1}{2} ]} ∩ C_{A}(γ_{i})  and
y ∈ 𝔓^{[ \frac{q_{i+1}+1}{2} ]} ∩ J^{1} ∩ 𝔓_{m}^{[^{\frac{q{′ \atop i+1}+1}{2}} ]} ∩ J_{m}^{1}.  Then  1 + x + y =  (1 + x)(1 + (1 + x)^{−1}y) =

(1 + x)(1 + ∑_{ν=0}^{∞}(−x)^{ν}y) ∈ (1 + 𝔓^{[ \frac{q_{i}+1}{2} ]} ∩ 𝔓_{m}^{[ \frac{q′_{i}+1}{2} ]} ∩ C_{A}(γ_{i}))(1 + 𝔓^{[ \frac{q_{i+1}+1}{2} ]} ∩ J^{1} ∩ 𝔓_{m}^{[^{\frac{q{′ \atop i+1}+1}{2}} ]} ∩ J_{m}^{1}) as required.

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3 A property of matching in a special situation

( 3.1 ) Lemma Let [𝔄, n, 0, β] be a simple stratum and θ_{𝔄} ∈ C(β, 𝔄) be a simple character. Then, for each defining sequence {(γ_{ν}, λ_{γ_{ν}})}_{ν=1,...,μ}, with underlying special approximation sequence, we obtain an explicit formula for θ_{𝔄} in the following sense:

If 1 + x ∈ H^{1}(β, 𝔄), i ∈ {1, …, μ} and 1 + x = (∏_{ν=1}^{i−1} 1 + x_{ν})(1 + x′_{i}), where x_{ν} ∈ 𝔓^{[ \frac{qν}{2} ]+1} ∩ C_{A}(γ_{ν}) for ν = 1, …, i − 1 and x′_{i} ∈ 𝔓^{[ \frac{q_{i}}{2} ]+1}∩ℌ^{1}(γ_{i}, 𝔄), then we have

θ_{𝔄}(1 + x) = (∏_{ν=1}^{i−1} λ_{γ_{ν}} ∘ Nrd_{C_{A}(γ_{ν})/F [γ_{ν}]}(1 + x_{ν})ψ_{β−γ_{ν}}^{+} (x_{ν}))ψ_{β−γ_{i}}^{+}(x′_{i})θ_{γ_{i}}(1 + x′_{i}),

where θ_{γ_{i}} ∈ C(γ_{i}, 𝔄) is the unique simple character defined by the sequence {(γ_{ν}, λ_{γ_{ν}})}_{ν=i,...,μ} (in particular, we set θ_{γ_{μ}} ≡ 1).

\underline{Proof:}

Our assertion is trivial for i = 1. So let us assume that the assertion holds for μ > i ≥ 1 and write (1 + x′_{i}) = (1 + x_{i})(1 + x′_{i+1}) according to a product representation of H^{1}(β, 𝔄), which we may easily deduce as in (2.9), where x_{i} ∈ 𝔓^{[ \frac{q_{i}}{2} ]+1} ∩ C_{A}(γ_{i}) and x′_{i+1} ∈ 𝔓^{[ \frac{q_{i+1}}{2} ]+1}∩ℌ^{1}(γ_{i+1}, 𝔄).

Since ([\frac{q_{i}}{2} ] + 1) + ([\frac{q_{i+1}}{2} ] + 1) ≥ q_{i} + 1 and β − γ_{i} ∈ 𝔓^{−q_{i}}, we obtain that ψ_{β−γ_{i}}^{+}(x′_{i}) = ψ_{β−γ_{i}}^{+}(x_{i} + x_{i}x′_{i+1} + x′_{i+1}) = ψ_{β−γ_{i}}^{+}(x_{i} + x′_{i+1}). By definition of simple characters, we also know that θ_{γ_{i}}(1 + x′_{i}) = λ_{γ_{i}} ∘ Nrd_{C_{A}(γ_{i})/F [γ_{i}]}(1 + x_{i})ψ_{γ_{i}−γ_{i+1}}^{+}(x′_{i+1})θ_{𝔄}(1 + x′_{i+1}) and the induction step follows from ψ_{β−γ_{i}}^{+}(x′_{i+1})ψ_{γ_{i}−γ_{i+1}}^{+}(x′_{i+1}) = ψ_{β−γ_{i+1}}^{+}(x′_{i+1}).

□

The property of matching as introduced in [G2](6.6) is a generalization of the existence part of [BK1](3.6.1) in a special case:

( 3.2 ) Corollary With the assumptions and notations of (2.5) let θ_{1} ∈ C(β, 𝔄_{1}) and θ_{2} ∈ C(β, 𝔄_{2}) such that θ_{1} = θ(A, id_{A}, 𝔄_{1}) and θ_{2} = θ(A, id_{A}, 𝔄_{2}), where (θ, β) is a ps-character. Then (θ_{1})_{|H^{1}(β,𝔄_{1})∩H^{1}(β,𝔄_{2})} = (θ_{2})_{|H^{1}(β,𝔄_{1})∩H^{1}(β,𝔄_{2})}.

\underline{Proof:}

Choose simultaneous special approximation sequences for [𝔄(E), −ν_{E}(β), 0, β] and [𝔄_{m}, n_{m}, 0, β] that fulfil the assumptions of Theorem ([G2](6.6)). By 2.5 we may find common special approximation sequences for [𝔄_{1}, −ν_{𝔄_{1}}(β), 0, β] and [𝔄_{2}, −ν_{𝔄_{2}}(β), 0, β] that fulfil the assumptions of Theorem ([G2](6.6)) with respect to the same special approximation sequence of [𝔄(E), −ν_{E}(β), 0, β]. Applying Corollary ([G2](6.7)) twice to θ(A(E), id_{A(E)}, 𝔄(E)) it turns out that θ_{1} and θ_{2} have a common defining sequence {(γ_{i}, λ_{γ_{i}})}_{i=0,...,μ}. Thus, our assertion follows directly from expanding H^{1}(β, 𝔄_{1})∩H^{1}(β, 𝔄_{2}) as in (2.9) and (3.1) applied to the case of i = μ.

□

4 Simple characters and their extensions by transferring between orders

( 4.1 ) Proposition Let [𝔄, n, 0, β] be a simple stratum and use the notations of Section (2). Let 𝔅_{m} ⊂ 𝔅 and H_{𝔄,𝔅_{m}}^{1} be the mixed group as in Lemma (2.1). Then there exists a unique character θ_{𝔄,𝔅_{m}} of H_{𝔄,𝔅_{m}}^{1} that concides with θ_{𝔄} on H^{1} and with θ_{𝔄_{m}} := θ(A, id, 𝔄_{m}) on H_{𝔄,𝔅_{m}}^{1} ∩ H_{m}^{1}, where θ is the ps-character characterized by θ(A, id, 𝔄) = θ_{𝔄}. Moreover, the group J_{𝔄,𝔅_{m}}^{1} normalizes the character θ_{𝔄,𝔅_{m}}.

\underline{Proof:}

The uniqueness of the character is clear. That the set H_{𝔄,𝔅_{m}}^{1} is a group follows from (2.1). As, by ([G2](5.11)), K_{𝔅}J^{1} normalizes θ_{𝔄} and K_{𝔅_{m}}J_{m}^{1} normalizes θ_{𝔄_{m}} one only needs the equality (θ_{𝔄})_{|H^{1}∩H_{m}^{1}} = (θ_{𝔄_{m}})_{|H^{1}∩H_{m}^{1}} for the existence of the character; hence, it follows from (3.2).

Moreover, we have to note that (J^{1} ∩J_{m}^{1}) normalizes θ_{𝔄} and θ_{𝔄_{m}}, hence θ_{𝔄,𝔅_{m}}, and since J_{𝔄,𝔅_{m}}^{1} = (J^{1} ∩ J_{m}^{1})H_{𝔄,𝔅_{m}}^{1}, we are also done with the last statement.

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For the construction of our ‘extended Heisenberg representations’ we need the following notion:

( 4.2 ) Definition Let [𝔄, n, 0, β] be a simple stratum and 𝔅_{m} ⊂ 𝔅 = 𝔄 ∩ C_{A}(β). Moreover, keep the notation as above, then the pair ([𝔄, n, 0, β], 𝔅_{m}) is called non-degenerate if

(J^{1} : J_{𝔄,𝔅_{m}}^{1} ∩ J^{1}) = (H_{𝔄,𝔅_{m}}^{1} ∩ J^{1} : H^{1}).

5 Reimann’s method to extend to a pro-p-Sylow subgroup (cf. [HR])

( 5.1 ) Lemma Let [𝔄, n, 0, β] be a simple stratum in A and θ_{𝔄} ∈ C(β, 𝔄). Then there is a unique irreducible Heisenberg representation η_{𝔄} of J^{1}(β, 𝔄) = J^{1} which is, up to isomorphism, determined by the property that it contains θ_{𝔄}. Moreover, (η_{𝔄})_{|H^{1}(β,𝔄)} is a multiple of θ_{𝔄} and dim(η_{𝔄}) = (J^{1} : H^{1})^{\frac{1}{2}} . The A^{×}-intertwining of η_{𝔄} is J^{1}B^{×}J^{1}.

\underline{Proof:}

The proof is the same as that of [BK1](5.1.1).

□

( 5.2 ) Proposition Under the assumption that the underlying pair ([𝔄, n, 0, β], 𝔅_{m}) is nondegenerate, there exists a unique extension η_{𝔄,𝔅_{m}} of η_{𝔄} to U^{1}(𝔅_{m})J^{1} such that

Ind_{H_{𝔄,𝔅m}^{1}}^{U^{1}(𝔅_{m})J^{1}}(θ_{𝔄,𝔅_{m}}) ≃ (J_{𝔄,𝔅_{m}}^{1} : H_{𝔄,𝔅_{m}}^{1})^{\frac{1}{2}} η_{𝔄,𝔅_{m}}.

\underline{Proof:}

The uniqueness of η_{𝔄,𝔅_{m}} is clear. For the existence let us consider the non-degenerate form χ_{θ_{𝔄}} : J^{1}/H^{1} × J^{1}/H^{1} → ℂ^{×}, which is induced by the application of θ_{𝔄} to commutators of J^{1}. Since (θ_{𝔄})_{|H^{1}∩H_{m}^{1}} = (θ_{𝔄_{m}})_{|H^{1}∩H_{m}^{1}}, it is easy to see that the subquotient (J^{1} ∩ H_{m}^{1})H^{1}/H^{1} is isotropic with respect to χ_{θ_{𝔄}}. By assumption (4.2) J_{𝔄,𝔅_{m}}^{1} ∩ J^{1}/H^{1} is the orthogonal of H_{𝔄,𝔅_{m}}^{1} ∩ J^{1}/H^{1} = (J^{1} ∩ H_{m}^{1})H^{1}/H^{1} with respect to χ_{θ_{𝔄}}.

Clearly, θ_{𝔄} extends to a character θ_{𝔄}^{⋆} of H_{𝔄,𝔅_{m}}^{1} ∩ J^{1} and, similarly, we obtain a non-degenerate form χ_{θ_{𝔄}^{⋆}} : J_{𝔄,𝔅_{m}}^{1} ∩ J^{1}/H_{𝔄,𝔅_{m}}^{1} ∩ J^{1} × J_{𝔄,𝔅_{m}}^{1} ∩ J^{1}/H_{𝔄,𝔅_{m}}^{1} ∩ J^{1} → ℂ^{×}. This leads to a Heisenberg representation η̂_{𝔄} of J_{𝔄,𝔅_{m}}^{1} ∩ J^{1}. Due to (2.1) and (4.1) we may identify the form χ_{θ_{𝔄}^{⋆}} with the form

χ_{θ_{𝔄,𝔅m}} : J_{𝔄,𝔅_{m}}^{1}/H_{𝔄,𝔅_{m}}^{1} × J_{𝔄,𝔅_{m}}^{1}/H_{𝔄,𝔅_{m}}^{1} → ℂ^{×},

which comes from θ_{𝔄,𝔅_{m}}. Clearly, this form is non-degenerate.

So we have shown that θ_{𝔄,𝔅_{m}} is a Heisenberg character in the sense of [Z2](4.2) with respect to J_{𝔄,𝔅_{m}}^{1} and that it leads to an irreducible Heisenberg representation η̂_{𝔄,𝔅_{m}} of J_{𝔄,𝔅_{m}}^{1} with

Ind_{H_{𝔄,𝔅m}^{1}}^{J_{𝔄,𝔅m}^{1}} (θ_{𝔄,𝔅_{m}}) ≃ (J_{𝔄,𝔅_{m}}^{1} : H_{𝔄_{,}𝔅_{m}}^{1})^{\frac{1}{2}} η̂_{𝔄,𝔅_{m}}. By construction and Mackey’s formula we have (η̂_{𝔄,𝔅_{m}})_{|J_{𝔄,𝔅m}^{1} ∩J^{1}} ≃ η̂_{𝔄}.

Again by Mackey’s formula and by construction, we have η_{𝔄} ≃ Ind_{J_{𝔄,𝔅m}^{1} ∩J^{1}}^{J^{1}}(η̂_{𝔄}) ≃

(Ind_{J_{𝔄,𝔅m}^{1}}^{U^{1}(𝔅_{m})J^{1}}(η̂_{𝔄,𝔅_{m}}))_{|J^{1}} so that η_{𝔄,𝔅_{m}} := Ind_{J_{𝔄,𝔅m}^{1}}^{U^{1}(𝔅_{m})J^{1}}(η̂_{𝔄,𝔅_{m}}) is irreducible and a continuation

of η_{𝔄}. Finally, by transitivity of induction, we have Ind_{H_{𝔄,𝔅m}^{1}}^{U^{1}(𝔅_{m})J^{1}}(θ_{𝔄,𝔅_{m}}) ≃ (J_{𝔄,𝔅_{m}}^{1} : H_{𝔄,𝔅_{m}}^{1})^{\frac{1}{2}} η_{𝔄,𝔅_{m}}, as required.

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6 A simplification of the problem of non-degeneracy of pairs

( 6.1 ) Lemma Let M be an abstract abelian group and {M_{i}}_{i=1,...,n}, {N_{i}}_{i=1,...,n} be two families of subgroups such that M_{i} ⊃ N_{i} for i = 1, …, n. Then, under the assumption that all group indices involved are finite, we have

(∑_{i=1}^{n} M_{i} : ∑_{i=1}^{n} N_{i}) = \frac{∏_{i=1}^{n}(M_{i} : N_{i})}{∏_{i=1}^{n−1}(M_{i} ∩ (∑_{j=i+1}^{n} M_{j}) : N_{i} ∩ (∑_{j=i+1}^{n} N_{j}))}.

\underline{Proof:}

Lets us prove our lemma by induction on n. The case of n = 1 is trivial. So let us assume that the lemma holds true for n ≥ 1.

Let M′_{2} := ∑_{i=2}^{n+1} M_{i}, N′_{2} := ∑_{i=2}^{n+1} N_{i} and assume that the lemma holds true for the case where n = 2. Then (M_{1} + M′_{2} : N_{1} + N′_{2}) = \frac{(M_{1}:N_{1})(M′_{2}:N′_{2})}{(M_{1}∩M′_{2}:N_{1}∩N′_{2})} and we obtain the desired assertion by including the definitions of M′_{2}, N′_{2} and applying the induction hypothesis to the index (M′_{2} : N′_{2}).

Thus it remains to deal with the case that n = 2. Here we have (M_{1} + M_{2} : N_{1} + N_{2}) = (M_{1} + M_{2} : N_{1} + M_{2})(N_{1} + M_{2} : N_{1} + N_{2}) and our assertions follows from the indentities (M_{1} +M_{2} : N_{1}+M_{2}) = \frac{(M_{1}:N_{1})}{(M_{1}∩M_{2}:N_{1}∩M_{2})}, (N_{1}+M_{2} : N_{1}+N_{2}) = \frac{(M_{2}:N_{2})}{(N_{1}∩M_{2}:N_{1}∩N_{2})} and (M_{1}∩M_{2} : N_{1}∩M_{2})(N_{1}∩M_{2} : N_{1} ∩ N_{2}) = (M_{1} ∩ M_{2} : N_{1} ∩ N_{2}).

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( 6.2 ) Proposition Let [𝔄, n, 0, β] be a simple stratum and 𝔅_{m} be a principal order contained in 𝔅 := 𝔄 ∩ C_{A}(β). Let 𝔄_{m} be the unique continuation of 𝔅_{m} to A and γ_{1}, ..., γ_{μ} be elements of A as in (2.5) such that the sequences [𝔄, n, q_{i}, γ_{i}], [𝔄_{m}, n_{m}, q′_{i}, γ_{i}], i = 1, …, μ, are approximation sequences for the strata [𝔄, n, 0, β] and [𝔄_{m}, n_{m}, 0, β], respectively, where n_{m} := −ν_{𝔄_{m}}(β).

With the notation of Section (2) above we have:

(J^{1} : J_{𝔄,𝔅_{m}}^{1} ∩ J^{1}) = \frac{∏_{i=1}^{μ}(𝔓^{[ \frac{q_{i}+1}{2} ]} ∩ C_{A}(γ_{i}) : 𝔓^{[ \frac{q_{i}+1}{2} ]} ∩ 𝔓_{m}^{[ \frac{q′_{i}+1}{2} ]} ∩ C_{A}(γ_{i}) + 𝔓^{[ \frac{q_{i}}{2} ]+1} ∩ C_{A}(γ_{i}))}{∏_{i=1}^{μ−1}(𝔓^{[ \frac{q_{i+1}+1}{2} ]} ∩ C_{A}(γ_{i}) : 𝔓^{[ \frac{q_{i+1}+1}{2} ]} ∩ 𝔓_{m}^{[^{\frac{q{′ \atop i+1}+1}{2}} ]} ∩ C_{A}(γ_{i}) + 𝔓^{[ \frac{q_{i+1}}{2} ]+1} ∩ C_{A}(γ_{i}))}

and

(H_{𝔄,𝔅_{m}}^{1}∩J^{1} : H^{1}) = \frac{∏_{i=1}^{μ}(𝔓^{[ \frac{q_{i}+1}{2} ]} ∩ 𝔓_{m}^{[ \frac{q′_{i}}{2} ]+1} ∩ C_{A}(γ_{i}) + 𝔓^{[ \frac{q_{i}}{2} ]+1} ∩ C_{A}(γ_{i}) : 𝔓^{[ \frac{q_{i}}{2} ]+1} ∩ C_{A}(γ_{i}))}{∏_{i=1}^{μ−1}(𝔓^{[ \frac{q_{i+1}+1}{2} ]} ∩ 𝔓_{m}^{[^{\frac{q{′ \atop i+1}}{2}} ]+1} ∩ C_{A}(γ_{i}) + 𝔓^{[ \frac{q_{i+1}}{2} ]+1} ∩ C_{A}(γ_{i}) : 𝔓^{[ \frac{q_{i+1}}{2} ]+1} ∩ C_{A}(γ_{i}))}.

\underline{Proof:}

Again we only prove the first identity. It is easy to see that (J^{1} : J_{𝔄,𝔅_{m}}^{1} ∩ J^{1}) = (J^{1} : J^{1} ∩ J_{m}^{1} + H^{1}). By (6.1) and (2.8) the last Lemma (6.1) applies

with M_{i} = 𝔓^{[ \frac{q_{i}+1}{2} ]} ∩ C_{A}(γ_{i}) and N_{i} = 𝔓^{[ \frac{q_{i}+1}{2} ]} ∩ 𝔓_{m}^{[ \frac{q′_{i}+1}{2} ]} ∩ C_{A}(γ_{i}) + 𝔓^{[ \frac{q_{i}}{2} ]+1} ∩ C_{A}(γ_{i}) for i = 1, …, μ. Then it is sufficient to show that, for i = 1, ..., μ − 1, we have

𝔓^{[ \frac{q_{i}+1}{2} ]} ∩ C_{A}(γ_{i}) ∩ (∑_{ν=i+1}^{μ} 𝔓^{[ \frac{qν +1}{2} ]} ∩ C_{A}(γ_{ν})) = 𝔓^{[ \frac{q_{i+1}+1}{2} ]} ∩ C_{A}(γ_{i}) and  (4)

\begin{matrix}(𝔓^{[ \frac{q_{i}+1}{2} ]} ∩ 𝔓_{m}^{[ \frac{q′_{i}+1}{2} ]} ∩ C_{A}(γ_{i}) + 𝔓^{[ \frac{q_{i}}{2} ]+1} ∩ C_{A}(γ_{i}))∩ \\ (∑_{ν=i+1}^{μ} 𝔓^{[ \frac{qν +1}{2} ]} ∩ 𝔓_{m}^{[ \frac{q′_{ν} +1}{2} ]} ∩ C_{A}(γ_{ν}) + 𝔓^{[ \frac{qν}{2} ]+1} ∩ C_{A}(γ_{ν})) = \\ 𝔓^{[ \frac{q_{i+1}+1}{2} ]} ∩ 𝔓_{m}^{[^{\frac{q{′ \atop i+1}+1}{2}} ]} ∩ C_{A}(γ_{i}) + 𝔓^{[ \frac{q_{i+1}}{2} ]+1} ∩ C_{A}(γ_{i}). \end{matrix}  (5)

As (4) is a special case of (5) in some sense, we can restrict ourselves to prove (5).

By (2.8) we have

𝔓^{[ \frac{q_{i+1}+1}{2} ]} ∩ 𝔓_{m}^{[^{\frac{q{′ \atop i+1}+1}{2}} ]} ∩ C_{A}(γ_{i}) + 𝔓^{[ \frac{q_{i+1}}{2} ]+1} ∩ C_{A}(γ_{i}) ⊆

(𝔓^{[ \frac{q_{i}+1}{2} ]} ∩ 𝔓_{m}^{[ \frac{q′_{i}+1}{2} ]} ∩ C_{A}(γ_{i}) + 𝔓^{[ \frac{q_{i}}{2} ]+1} ∩ C_{A}(γ_{i}))∩

(𝔓^{[ \frac{q_{i+1}+1}{2} ]} ∩ J^{1} ∩ 𝔓_{m}^{[^{\frac{q{′ \atop i+1}+1}{2}} ]} ∩ J_{m}^{1} + 𝔓^{[ \frac{q_{i+1}}{2} ]+1} ∩ H^{1}) =

(𝔓^{[ \frac{q_{i}+1}{2} ]} ∩ 𝔓_{m}^{[ \frac{q′_{i}+1}{2} ]} ∩ C_{A}(γ_{i}) + 𝔓^{[ \frac{q_{i}}{2} ]+1} ∩ C_{A}(γ_{i}))∩

( ∑_{ν=i+1}^{μ} 𝔓^{[ \frac{qν +1}{2} ]} ∩ 𝔓_{m}^{[ \frac{q′_{ν} +1}{2} ]} ∩ C_{A}(γ_{ν}) + 𝔓^{[ \frac{qν}{2} ]+1} ∩ C_{A}(γ_{ν})).

On the other hand, (2.7)(ii) leads to

(𝔓^{[ \frac{q_{i}+1}{2} ]} ∩ 𝔓_{m}^{[ \frac{q′_{i}+1}{2} ]} ∩ C_{A}(γ_{i}) + 𝔓^{[ \frac{q_{i}}{2} ]+1} ∩ C_{A}(γ_{i}))∩
(𝔓^{[ \frac{q_{i+1}+1}{2} ]} ∩ J^{1} ∩ 𝔓_{m}^{[^{\frac{q{′ \atop i+1}+1}{2}} ]} ∩ J_{m}^{1} + 𝔓^{[ \frac{q_{i+1}}{2} ]+1} ∩ H^{1}) =
(𝔓^{[ \frac{q_{i}+1}{2} ]} ∩ 𝔓_{m}^{[ \frac{q′_{i}+1}{2} ]} + 𝔓^{[ \frac{q_{i}}{2} ]+1})∩
C_{A}(γ_{i}) ∩ (𝔓^{[ \frac{q_{i+1}+1}{2} ]} ∩ J^{1} ∩ 𝔓_{m}^{[^{\frac{q{′ \atop i+1}+1}{2}} ]} ∩ J_{m}^{1} + 𝔓^{[ \frac{q_{i+1}}{2} ]+1} ∩ H^{1}) =
C_{A}(γ_{i}) ∩ (𝔓^{[ \frac{q_{i+1}+1}{2} ]} ∩ J^{1} ∩ 𝔓_{m}^{[^{\frac{q{′ \atop i+1}+1}{2}} ]} ∩ J_{m}^{1} + 𝔓^{[ \frac{q_{i+1}}{2} ]+1} ∩ H^{1}) ⊆
C_{A}(γ_{i}) ∩ (𝔓^{[ \frac{q_{i+1}+1}{2} ]} ∩ 𝔓_{m}^{[^{\frac{q{′ \atop i+1}+1}{2}} ]} + 𝔓^{[ \frac{q_{i+1}}{2} ]+1}) =
𝔓^{[ \frac{q_{i+1}+1}{2} ]} ∩ 𝔓_{m}^{[^{\frac{q{′ \atop i+1}+1}{2}} ]} ∩ C_{A}(γ_{i}) + 𝔓^{[ \frac{q_{i+1}}{2} ]+1} ∩ C_{A}(γ_{i}).

□

( 6.3 ) Proposition Let [𝔄, n, 0, β] be a simple stratum and let 𝔅_{m} be a principal order contained in 𝔅 := 𝔄 ∩ C_{A}(β). Choose simultaneous special approximation sequences [𝔄, n, q_{i}, γ_{i}], [𝔄_{m}, n_{m}, q′_{i}, γ_{i}], i = 1, …, μ, for [𝔄, n, 0, β] and [𝔄_{m}, n_{m}, 0, β], respectively, where 𝔄_{m} is the unique continuation of 𝔅_{m} to A. Let E := F [β], K/F be the maximal unramified part of the extension E/F and set A_{i} := C_{A}(γ_{i}), C_{i} := C_{A}(K[γ_{i}]), for i = 1, …, μ. Then, in order to show the non-degeneracy of the pair ([𝔄, n, 0, β], 𝔅_{m}), it is now sufficient to show, for all i ∈ {1, …, μ} such that 2e_{𝔄/𝔄∩A_{i}} | q_{i} but 2e_{𝔄/𝔄∩C_{i}} ∤ q_{i}, as well as for the i ∈ {1, …, μ − 1} such that 2e_{𝔄/𝔄∩A_{i}} | q_{i+1} but 2e_{𝔄/𝔄∩C_{i}} ∤ q_{i+1}, that we have

\begin{matrix}(𝔓i^{\frac{−e_{i}}{2}} : 𝔓i^{\frac{−e_{i}}{2}} ∩ 𝔓_{i,m}^{− \frac{r̄m}{r̄} \frac{e_{i}}{2}} + 𝔓_{i}^{\frac{−e_{i}}{2} +1}) = \\ (𝔓i^{\frac{−e_{i}}{2}} ∩ 𝔓_{i,m}^{− \frac{r̄m}{r̄} \frac{e_{i}}{2} +1} + 𝔓_{i}^{\frac{−e_{i}}{2} +1} : 𝔓_{i}^{\frac{−e_{i}}{2} +1}), \end{matrix} where

𝔓_{i}, 𝔓_{i,m} are the Jacobson radical of 𝔄 ∩ A_{i} and 𝔄_{m} ∩ A_{i}, respectively, and r̄, r̄_{m} are the period of 𝔄 ∩ A_{i} and 𝔄_{m} ∩ A_{i} respectively, e_{i} := e_{𝔄∩A_{i}/𝔄∩C_{i}}.

\underline{Proof:}

By (6.2) it is sufficient to show that \begin{matrix}(𝔓^{[ \frac{q_{i}+1}{2} ]} ∩ A_{i} : 𝔓^{[ \frac{q_{i}+1}{2} ]} ∩ 𝔓_{m}^{[ \frac{q′_{i}+1}{2} ]} ∩ A_{i} + 𝔓^{[ \frac{q_{i}}{2} ]+1} ∩ A_{i}) = \\ (𝔓^{[ \frac{q_{i}+1}{2} ]} ∩ 𝔓_{m}^{[ \frac{q′_{i}}{2} ]+1} ∩ A_{i} + 𝔓^{[ \frac{q_{i}}{2} ]+1} ∩ A_{i} : 𝔓^{[ \frac{q_{i}}{2} ]+1} ∩ A_{i}) \end{matrix}

for all i = 1, …, μ, and \begin{matrix}(𝔓^{[ \frac{q_{i+1}+1}{2} ]} ∩ A_{i} : 𝔓^{[ \frac{q_{i+1}+1}{2} ]} ∩ 𝔓_{m}^{[^{\frac{q{′ \atop i+1}+1}{2}} ]} ∩ A_{i} + 𝔓^{[ \frac{q_{i+1}}{2} ]+1} ∩ A_{i}) = \\ (𝔓^{[ \frac{q_{i+1}+1}{2} ]} ∩ 𝔓_{m}^{[^{\frac{q{′ \atop i+1}}{2}} ]+1} ∩ A_{i} + 𝔓^{[ \frac{q_{i+1}}{2} ]+1} ∩ A_{i} : 𝔓^{[ \frac{q_{i+1}}{2} ]+1} ∩ A_{i}) \end{matrix}

for all i = 1, …, μ − 1. If 2e_{𝔄/𝔄∩A_{i}} ∤ q_{i}, the first equation we have to show is trivial, because then we have 𝔓^{[ \frac{q_{i}+1}{2} ]} ∩ A_{i} = 𝔓^{[ \frac{q_{i}}{2} ]+1} ∩ A_{i}. Similarly, for i < μ, the case of 2e_{𝔄/𝔄∩A_{i}} ∤ q_{i+1} leads trivially to the second equation we have to show. Therefore, we exclude these cases and restrict ourselves to show that

\begin{matrix}(𝔓i^{\frac{q_{i}}{2e_{𝔄/𝔄∩A_{i}}}} : 𝔓i^{\frac{q_{i}}{2e_{𝔄/𝔄∩A_{i}}}} ∩ 𝔓i,m^{\frac{q′_{i}}{2e_{𝔄/𝔄∩A_{i}}}} + 𝔓_{i}^{_{\frac{q_{i}}{2e_{𝔄/𝔄∩A_{i}}}} +1}) = \\ (𝔓i^{\frac{q_{i}}{2e_{𝔄/𝔄∩A_{i}}}} ∩ 𝔓_{i,m}^{\frac{q′_{i}}{2e_{𝔄m/𝔄m∩A_{i}}} +1} + 𝔓_{i}^{_{\frac{q_{i}}{2e_{𝔄/𝔄∩A_{i}}}} +1} : 𝔓_{i}^{_{\frac{q_{i}}{2e_{𝔄/𝔄∩A_{i}}}} +1}) \end{matrix}  (6)

for i ∈ {1, …, μ} such that 2e_{𝔄/𝔄∩A_{i}} | q_{i}, and

\begin{matrix}(𝔓i^{\frac{q_{i+1}}{2e_{𝔄/𝔄∩A_{i}}}} : 𝔓i^{\frac{q_{i+1}}{2e_{𝔄/𝔄∩A_{i}}}} ∩ 𝔓i,m^{\frac{q{′ \atop i+1}}{2e_{𝔄/𝔄∩A_{i}}}} + 𝔓_{i}^{\frac{q_{i+1}}{2e_{𝔄/𝔄∩A_{i}}} +1}) = \\ (𝔓i^{\frac{q_{i+1}}{2e_{𝔄/𝔄∩A_{i}}}} ∩ 𝔓_{i,m}^{\frac{q{′ \atop i+1}}{2e_{𝔄m/𝔄m∩A_{i}}} +1} + 𝔓_{i}^{\frac{q_{i+1}}{2e_{𝔄/𝔄∩A_{i}}} +1} : 𝔓_{i}^{\frac{q_{i+1}}{2e_{𝔄/𝔄∩A_{i}}} +1}) \end{matrix}  (7)

for i ∈ {1, …, μ − 1} such that 2e_{𝔄/𝔄∩A_{i}} | q_{i+1}. Our aim is to reduce the latter assertions to the assertion of our proposition.

For that purpose let us fix an arbitrary but fixed element i ∈ {1, …, μ}. Then 𝔄 ∩ A_{i} and 𝔄_{m} ∩ A_{i} are continuations of 𝔄 ∩ C_{i} and 𝔄_{m} ∩ C_{i}, respectively.

Since the extension E/K is totally ramified, 𝔄 ∩ C is an unramified continuation of 𝔅 to C := C_{A}(K), and 𝔄_{m} ∩ C is an unramified continuation of 𝔅_{m} to C. From (2.4) we conclude that 𝔄 ∩ C ⊃ 𝔄_{m} ∩ C. Moreover, we obtain 𝔄 ∩ C_{i} = 𝔄 ∩ C ∩ C_{i} ⊃ 𝔄_{m} ∩ C ∩ C_{i} = 𝔄_{m} ∩ C_{i}.

Let M/F [γ_{i}] be the minimal subextension of K[γ_{i}]/F [γ_{i}] such that the continuation 𝔇 of 𝔄 ∩ C_{i} to C_{A}(M) is unramified. As the period of 𝔄 ∩ C_{i} is a divisor of the period of 𝔄_{m} ∩ C_{i}, we conclude from (2.3) that the unique continuation 𝔇_{m} of 𝔄_{m} ∩ C_{i} to C_{A}(M) is unramified. Consequently we have e_{𝔄∩A_{i}/𝔄∩C_{i}} = e_{𝔄∩A_{i}/𝔇} and e_{𝔄_{m}∩A_{i}/𝔄_{m}∩C_{i}} = e_{𝔄_{m}∩A_{i}/𝔇_{m}}. Moreover, e_{𝔄/𝔇} = e_{𝔄/𝔄∩A_{i}}e_{𝔄∩A_{i}/𝔇} and e_{𝔄_{m}/𝔇_{m}} = e_{𝔄_{m}/𝔄_{m}∩A_{i}}e_{𝔄_{m}∩A_{i}/𝔇_{m}}.

Now let r̄, r̄_{m}, r̂ and r̂_{m} be the periods of 𝔄 ∩ A_{i}, 𝔄_{m} ∩ A_{i}, 𝔇 and 𝔇_{m}, respectively. Let h_{m} be a uniformizer of 𝔇_{m}. Then h := hm^{\frac{r̂m}{r̂}} is a uniformizer for 𝔇 and ν_{𝔄∩A_{i}}(h) = e_{𝔄∩A_{i}/𝔇}.

As the extension M/F [γ_{i}] is unramified by ([G2](2.6))(iii), \frac{r̂_{m}}{r̂} e_{𝔄_{m}/𝔄_{m}∩A_{i}} = \frac{r̂_{m}}{r̂} \begin{matrix}r̄_{m} & d̄ \\ r̂_{m} & d̂ \end{matrix} = \frac{r̄_{m}}{r̄} \begin{matrix}r̄ & d̄ \\ r̂ & d̂ \end{matrix} = \frac{r̄_{m}}{r̄} e_{𝔄/𝔄∩A_{i}}, where d̄is the index of A_{i} and d̂is the index of C_{A}(M). Moreover, we have \frac{r̄_{m}}{r̄} \frac{q_{i}}{2e_{𝔄/𝔄∩A_{i}}} =

\frac{q′_{i}}{2e_{𝔄m/𝔄m∩A_{i}}}.

First assume that 2e_{𝔄/𝔄∩C_{i}} ∤ q_{i}. Then, we may write \frac{q_{i}}{e_{𝔄/𝔄∩C_{i}}} = 2ν + 1 for a non-negative integer ν since e_{𝔄/𝔄∩C_{i}} | q_{i} and, thus \frac{q_{i}}{2e_{𝔄/𝔄∩A_{i}}} = νe_{𝔄∩A_{i}/𝔇} + \frac{e_{𝔄∩A_{i}/𝔇}}{2} .

Multiplying equation (6) by h^{−ν−1} and using the equality ν_{𝔄_{m}∩A_{i}}(h) = \frac{r̂_{m}}{r̂} ν_{𝔄_{m}∩A_{i}}(h_{m}) = \frac{r̂_{m}}{r̂} e_{𝔄_{m}/𝔄_{m}∩A_{i}} = \frac{r̄_{m}}{r̄} e_{𝔄/𝔄∩A_{i}}, we reduce (6) to the equation

\begin{matrix}(𝔓i^{\frac{−e_{𝔄∩A_{i}/𝔄∩C_{i}}}{2}} : 𝔓i^{\frac{−e_{𝔄∩A_{i}/𝔄∩C_{i}}}{2}} ∩ 𝔓_{i,m}^{− \frac{r̄m}{r̄} \frac{e_{𝔄∩A_{i}/𝔄∩C_{i}}}{2}} + 𝔓^{\frac{−e_{𝔄∩A_{i}/𝔄∩C_{i}}}{2} +1}) = \\ (𝔓i^{\frac{−e_{𝔄∩A_{i}/𝔄∩C_{i}}}{2}} ∩ 𝔓_{i,m}^{− \frac{r̄m}{r̄} \frac{e_{𝔄∩A_{i}/𝔄∩C_{i}}}{2} +1} + 𝔓_{i}^{\frac{−e_{𝔄∩A_{i}/𝔄∩C_{i}}}{2} +1} : 𝔓_{i}^{\frac{−e_{𝔄∩A_{i}/𝔄∩C_{i}}}{2} +1}), \end{matrix} as

required. Similarly for i < μ, we may reduce the second equation (7) to the same statement if we assume that 2e_{𝔄∩A_{i}/𝔄∩C_{i}} ∤ q_{i+1}.

Thus, we only have to handle the cases 2e_{𝔄/𝔄∩C_{i}} | q_{i} and 2e_{𝔄/𝔄∩C_{i}} | q_{i+1} for i < μ. However

in this case, we may multiply our equations by h^{−_{\frac{q_{i}}{2e_{𝔄/𝔄∩C_{i}}}}} and h^{− \frac{q_{i+1}}{2e_{𝔄/𝔄∩C_{i}}}} , respectively, and reduce both equations to

(𝔄 ∩ A_{i} : 𝔄 ∩ 𝔄_{m} ∩ A_{i} + 𝔓 ∩ A_{i}) = (𝔄 ∩ 𝔓_{m} ∩ A_{i} + 𝔓 ∩ A_{i} : 𝔓 ∩ A_{i}).  (8)

To show this, let us fix an isomorphism A_{i} ≃ M_{t}(D̄), where D̄ is a central division algebra over F [γ_{i}] of index d̄. Then up to conjugacy in A_{i}^{×}, we may assume that M ⊂ D̄ (note that [M : F [γ_{i}]] = e_{𝔄∩A_{i}/𝔄∩C_{i}} | d̄). Doing so, we identify C_{A}(M) with M_{t}(D̂), where D̂ = CD̄(M). Up to conjugacy in C_{A}(M)^{×} we may assume that 𝔇 and 𝔇_{m} are given in simultaneous normal form. Moreover, we note that, as the continuation of 𝔇 by 𝔄 ∩ A_{i} is totally ramified, we have s̄ = ŝ | ŝ_{m} | s̄_{m}, where s̄, s̄_{m}, ŝ and ŝ_{m} are the invariants of 𝔄 ∩ A_{i}, 𝔄_{m} ∩ A_{i}, 𝔇 and 𝔇_{m}, respectively. By (2.2) 𝔄 ∩ A_{i} is conjugated by an ŝ× ŝ-monomial matrix g with its normal form in M_{t}(D̂) and 𝔄_{m} ∩ A_{i} is conjugated by an ŝ_{m} × ŝ_{m}-monomial matrix with its normal form in the same algebra.

From 𝔄_{m} ∩ A_{i} ∩ C_{A}(M) = 𝔇_{m}, we deduce that the ŝ_{m} × ŝ_{m}-blocks above the block diagonal of 𝔄_{m} ∩ A_{i} must be in a power of 𝔭D̄ of the exponent ≥ 1, the diagonal blocks are in 𝔬D̄ and the ŝ_{m} × ŝ_{m}-blocks below the diagonal blocks are in a power of 𝔭D̄ of the exponent ≤ 0. The same holds true if we restrict this consideration to the diagonal blocks of an ŝ × ŝ-block representation 𝔄_{m} ∩ A_{i} = (A_{i,j}^{m})_{i,j=1,...,r̂} and to the diagonal blocks of an ŝ × ŝ-block representation g𝔄_{m} ∩ A_{i}g^{−1} = (B_{i,j}^{m})_{i,j=1,...,r̂_{ν}}, as conjugation by the ŝ×ŝ-block monomial matrix g just permutes the diagonal blocks and conjugates them by a power of a prime element π̄ of 𝔬D̄.

Further, consider an ŝ × ŝ-block representation 𝔄 ∩ A_{i} = (A_{i,j})_{i,j=1,...,r̂}, and for an ŝ × ŝ-block representation g𝔄 ∩ A_{i}g^{−1} = (B_{i,j})_{i,j=1,...,r̂_{ν}}, the B_{i,i} = Mŝ(𝔬D̄) are again a permutation of the A_{i,i} for i = 1, …, μ, B_{i,j} = Mŝ(𝔬D̄) for i > j and B_{i,j} = Mŝ(𝔭D̄) for i < j.

Similarly, we have 𝔓 ∩ A_{i} = (C_{i,j})_{i,j=1,...,r̂} and g𝔓 ∩ A_{i}g^{−1} = (D_{i,j})_{i,j=1,...,r̂}, where the D_{i,i} are permutations of the blocks C_{i,i} for i = 1, …, r̂, and the blocks represent sets of ŝ × ŝ-matrices with entries from powers of 𝔭D̄, where the exponent depends only on ν, i, j. For the blocks D_{i,j} we just have D_{i,j} = Mŝ(𝔬D̄) for i > j and B_{i,j} = Mŝ(𝔭D̄) for i ≤ j.

Altogether we obtain g𝔄 ∩ 𝔄_{m} ∩ A_{i}g^{−1} + g𝔓 ∩ A_{i}g^{−1} = (B_{i,j} ∩ B_{i,j}^{m} + D_{i,j})_{i,j=1,...,r̂}. In particular, for i > j we have (B_{i,j} ∩ B_{i,j}^{m} + D_{i,j})_{i,j=1,...,r̂} = (Ms̄(𝔬D̄) ∩ B_{i,j}^{m} + Mŝ(𝔬D̄)) = Mŝ(𝔬D̂) and for i < j we have (B_{i,j} ∩ B_{i,j}^{m} + D_{i,j}) = (Mŝ(𝔭D̄) ∩ B_{i,j}^{m} + Mŝ(𝔭D̄)) = Mŝ(𝔭D̄). As 𝔄_{m} ∩ A_{i} is conjugated by an ŝ_{m} × ŝ_{m}-block-monomial matrix with its normal form, the blocks A_{i,i}^{m}, resp. B_{i,i}^{m} have an ŝ_{m} × ŝ_{m}-block description such that the diagonal blocks have entries in 𝔬D̄. Since (as shown above) the ŝ_{m} × ŝ_{m}-blocks below the block diagonal have entries from powers of 𝔭D̄ of the the exponent ≤ 0 and the ŝ_{m} × ŝ_{m}-blocks above the block diagonal have entries in a power of 𝔭D̄ of the exponent ≥ 1, the blocks below the block diagonal in B_{i,i}^{m} are intersected down to 𝔬D̄ in Mŝ(𝔬D̄) ∩ B_{i,i}^{m} and the the blocks above the block diagonal in Mŝ(𝔬D̄) ∩ B_{i,i}^{m} are filled up by 𝔭D̄ in Mŝ(𝔬D̄) ∩ B_{i,i}^{m} + Mŝ(𝔭D̄) for i = 1, …, r̂; hence, we deduce that (B_{i,i} ∩ B_{i,i}^{m} + D_{i,i}) = (Mŝ(𝔬D̄) ∩ B_{i,i}^{m} + Mŝ(𝔭D̄)) is the principal order given in normal form in Mŝ(D̄) with invariant ŝ_{m} and period \frac{ŝ_{m}}{ŝ}. Summarizing we conclude that g𝔄∩𝔄_{m} ∩A_{i}g^{−1} +g𝔓∩A_{i}g^{−1} is the principal order given in normal form in M_{t}(D̄) with invariant ŝ_{m} and period r̂_{m}.

Similar, we may show that g𝔄 ∩ 𝔓_{m} ∩ A_{i}g^{−1} + g𝔓 ∩ A_{i}g^{−1} is the Jacobson radical of g𝔄 ∩ 𝔄_{m} ∩ A_{i}g^{−1} + g𝔓 ∩ A_{i}g^{−1}; hence, by the block-descriptions of the normal forms, we deduce that

(g𝔄 ∩ A_{i}g^{−1} : g𝔄 ∩ 𝔄_{m} ∩ A_{i}g^{−1} + g𝔓 ∩ A_{i}g^{−1}) = (g𝔄 ∩ 𝔓_{m} ∩ A_{i}g^{−1} + g𝔓 ∩ A_{i}g^{−1} : g𝔓 ∩ A_{i}g^{−1}).

Conjugating the latter indices on both sides of the latter equation by g^{−1}, we obtain equation (8), as required.

□

7 Some special cases where all pairs are non-degenerate

( 7.1 ) Corollary All pairs ([𝔄, n, 0, β], 𝔅_{m}) of simple strata [𝔄, n, 0, β] and principal orders 𝔅_{m} ⊂ 𝔅 := 𝔄 ∩ C_{A}(β) are sufficiently non-degenerate if

(i) A = D is a central divison algebra over F.

(ii) A = M_{l}(D), where D is a central division algebra over F and l is a prime number.

(iii) The index d of the algebra A is odd.

\underline{Proof:}

(i) In a central divison algebra D = A over F of finite dimension, there is a unique principal order 𝔄 = 𝔬_{D} and the same holds true for the centralizer B := C_{A}(β); hence, in this case we always have 𝔅_{m} = 𝔅 and our index equation always reduces to the trivial equation 1 = 1.

(ii) If A = M_{l}(D), where l is a prime number and D is a central division algebra over F, we have [E : F] | d = Ind(A) (where E := F [β]) or B := C_{A}(β) is a divison algebra. In the latter case we have 𝔅_{m} = 𝔅 again and the non-degeneracy of all pairs involving β is clear.

If [E : F] | d, then we have B = M_{l}(D′), where D′ is a central division algebra over E of index \frac{d}{[E:F]} and the case 𝔅_{m} = 𝔅 is trivial again. If 𝔅_{m} ≠ 𝔅, then 𝔅_{m} must be the minimal principal order in B, which has period l and invariant 1, and 𝔅 must be the maximal order with period 1 and invariant l. By [G1](1.9,1.8(iii)) we also conlude that 𝔄 is again a maximal order.

Now we use the arguments of the proof of (6.3) in a modified way. We use the notation and assumptions of the proof of (6.3) for γ_{0}, …, γ_{μ}, A_{i}, C_{i}, M/F [γ_{i}], 𝔇 and 𝔇_{𝔪}, where we fix an i such that 2e_{𝔄/𝔄∩A_{i}} | q_{i} and 2e_{𝔄/𝔄∩C_{i}} ∤ q_{i} or 2e_{𝔄/𝔄∩A_{i}} | q_{i+1} and 2e_{𝔄/𝔄∩C_{i}} ∤ q_{i+1} if i < μ. Since 𝔇 as well as 𝔄 ∩ A_{i} are maximal orders of invariant l, by the arguments in the proof of (6.3) we may restrict ourselves to the case where 𝔄 ∩ A_{i} is conjugated by an l × l-block monomial matrix with its normal form; hence, it has to be given itself in normal form in the sense that, if C_{A}(M) = M_{l}(D̂), A_{i} = M_{l}(D̄) and 𝔇 = M_{l}(𝔬D̂), then we have 𝔄 ∩ A_{i} = M_{l}(𝔬D̄). Let us fix a prime element π̄ of 𝔬D̄. We also assume 𝔄_{m} ∩ A_{i} to be conjugated with its normal form by an ŝ_{m} × ŝ_{m}-block monomial matrix in certain powers of π̄ (where ŝ_{m} is the invariant of 𝔇). We deduce that π̄I_{l} is a uniformizer of 𝔄 ∩ A_{i} that normalizes 𝔄_{m} ∩ A_{i}. We may multiply the terms in the index equation

\begin{matrix}(𝔓i^{\frac{−e_{𝔄∩A_{i}/𝔄∩C_{i}}}{2}} : 𝔓i^{\frac{−e_{𝔄∩A_{i}/𝔄∩C_{i}}}{2}} ∩ 𝔓_{i,m}^{− \frac{r̄m}{r̄} \frac{e_{𝔄∩A_{i}/𝔄∩C_{i}}}{2}} + 𝔓_{i}^{\frac{−e_{𝔄∩A_{i}/𝔄∩C_{i}}}{2} +1}) = \\ (𝔓i^{\frac{−e_{𝔄∩A_{i}/𝔄∩C_{i}}}{2}} ∩ 𝔓_{i,m}^{− \frac{r̄m}{r̄} \frac{e_{𝔄∩A_{i}/𝔄∩C_{i}}}{2} +1} + 𝔓_{i}^{\frac{−e_{𝔄∩A_{i}/𝔄∩C_{i}}}{2} +1} : 𝔓_{i}^{\frac{−e_{𝔄∩A_{i}/𝔄∩C_{i}}}{2} +1}) \end{matrix} by

π̄^{\frac{e_{𝔄∩A_{i}/𝔄∩C_{i}}}{2}} and restrict ourselves to show equation (8), which can be proved in the same way as in the proof of (6.3). Therefore, with these modified arguments of the proof of (6.3) we may show that (6.3) applies and that all pairs ([𝔄, n, 0, β], 𝔅_{m}) where 𝔄 is a maximal order, 𝔅_{m} is minimal and [E : F] | d, are non-degenerate, as required.

(iii) Again we want to apply (6.3) and we remark that our conditions from (6.3) will become empty if we may show that

e_{𝔄/𝔄∩C_{i}} | d, q_{i}, q_{i+1},  (9)

as, by assumption on d, e_{𝔄/𝔄∩C_{i}} must be odd and, therefore, we may deduce from 2e_{𝔄/𝔄∩A_{i}} | q_{i}, 2e_{𝔄/𝔄∩A_{i}} | q_{i+1} that 2e_{𝔄/𝔄∩C_{i}} | q_{i}, 2e_{𝔄/𝔄∩C_{i}} | q_{i+1}.

By [G1](1.9)(b) e_{𝔄/𝔄∩C_{i}} is a divisor of f(K[γ_{i}]/F) = [K : F]. If now K_{d}/F is the unique subextension of K/F with [K_{d} : F] =< [K : F], d >, then C_{A}(K_{d}) has index \frac{d}{[K_{d}:F]} (see for example [Z1](Proposition 1)) and by [G1](2.2)(b) the continuation of 𝔄 ∩ C_{i} by 𝔄 ∩ C_{A}(K_{d}) is unramified. Thus, we have e_{𝔄/𝔄∩C_{i}} = e_{𝔄/𝔄∩C_{A}(K_{d})} | [K_{d} : F] | d as well.

By ([G2](3.9))(C) we have \frac{rd}{e(F [γ_{i−1}]/F)} | q_{i} and \frac{rd}{e(F [γ_{i}]/F)} | q_{i+1}. By ([G2](2.6)) we have e_{𝔄/𝔄∩C_{i}} = \frac{rd}{r̄d̄e(K[γ_{i}]/F)}, where r̄ is the period of 𝔄 ∩ C_{i} and d̄ is the index of C_{i}. It follows immediately that e_{𝔄/𝔄∩C_{i}} | q_{i+1}. Now, if we denote the index of C_{A}(K) by d′ and the period of 𝔄 ∩ C_{A}(K) by r′, then [G2](2.2)(i)(a) implies r̄ = \frac{r′<e(K[γ_{i}]/K),d′>}{e(K[γ_{i}]/K)} and, again by [Z1](Proposition 1),

we have d̄ = \frac{d′}{<e(K[γ_{i}]/K),d′>}. Finally, we obtain \frac{\frac{rd}{e(F [γ_{i−1}]/F)}}{e_{𝔄/𝔄∩C_{i}}} = \frac{r̄d̄e(K[γ_{i}]/F)}{e(F [γ_{i−1}]/F)} = \frac{r′d′}{e(K[γ_{i−1}]/K)}. However, by [G1](1.9)(iii), the last number is integral, since γ_{i−1} is 𝔄 ∩ C_{A}(K)-sound. Hence, we also obtain the relation e_{𝔄/𝔄∩C_{i}} | q_{i+1}.

Altogether we have shown (9) and deduced (iii) as described above.

□

( 7.2 ) Corollary All pairs ([𝔄, n, 0, β], 𝔅_{m}) of simple strata [𝔄, n, 0, β] and principal orders 𝔅_{m} ⊂ 𝔅 := 𝔄 ∩ C_{A}(β) with ⟨[F [β] : F], d⟩ | e(F [β]/F)r′, where r′ is the period of the principal order 𝔅 := 𝔄 ∩ C_{A}(β), are non-degenerate.

\underline{Proof:}

By (2.3) and in the notation of (6.3) we have e_{𝔄/𝔄∩A_{i}} = e_{𝔄/𝔄∩C_{i}} = 1 for i = 0, …, μ; hence, the non-degeneracy conditions of (6.3) are empty.

□

To compare our construction in Section (5) with the construction of [BK1] we use the following lemma:

( 7.3 ) Lemma Let [𝔄, n, 0, β] be a simple stratum and 𝔅_{m} ⊂ 𝔅 := 𝔄∩C_{A}(β) a principal order. Let 𝔄_{m} be the unique continuation of 𝔅_{m} to A, let θ_{𝔄}, θ_{𝔄_{m}} be realizations of the same pscharacters for the groups H^{1}, H_{m}^{1}, and let η_{𝔄}, η_{𝔄_{m}} be the associated Heisenberg representations. Moreover, use the assumptions of 7.2 (the unramified case) and let θ_{𝔄,𝔅_{m}}^{⋆} be a character living on a subgroup H_{𝔄,𝔅_{m}}^{1⋆} of U^{1}(𝔅_{m})J^{1} such that (θ_{𝔄,𝔅_{m}}^{⋆})_{|H_{𝔄,𝔅m}^{1⋆} ∩H_{m}^{1}} = (θ_{𝔄_{m}})_{|H_{𝔄,𝔅m}^{1⋆} ∩H_{m}^{1}}. Finally, assume

that Ind_{H_{𝔄,𝔅m}^{1⋆}}^{U^{1}(𝔅_{m})J^{1}}(θ_{𝔄,𝔅_{m}}^{⋆}) is a multiple of an irreducible extension η̃_{𝔄,𝔅_{m}} of η_{𝔄} to U^{1}(𝔅_{m})J^{1}. Then η̃_{𝔄,𝔅_{m}} and η_{𝔄_{m}} induce the same irreducible representation of U^{1}(𝔄_{m}).

\underline{Proof:}

First we note that, in the unramified case, by 2.4, we have U^{1}(𝔄_{m}) ⊃ U^{1}(𝔄) and U^{1}(𝔄) ⊃ J^{1}. Therefore, we also have U^{1}(𝔄_{m}) ⊃ U^{1}(𝔅_{m})J^{1}.

The A^{×}-intertwinig of η̃_{𝔄,𝔅_{m}} is clearly contained in the A^{×}-intertwining of η_{𝔄}, which is J^{1}B^{×}J^{1}. As J^{1}B^{×}J^{1} ∩ U^{1}(𝔄_{m}) = U^{1}(𝔅_{m})J^{1} Mackey’s Theorem applies and the induction of η̃_{𝔄,𝔅_{m}} to

U^{1}(𝔄_{m}) is irreducible. Since the intertwining of η_{𝔄_{m}} is J_{m}^{1}B^{×}J_{m}^{1} (see (5.1)) and J_{m}^{1}B^{×}J_{m}^{1}∩U^{1}(𝔄_{m}) = J_{m}^{1}, the induction of η_{𝔄_{m}} to U^{1}(𝔄_{m}) is also irreducible. One has to show that these induced representations are equivalent. As Ind_{H^{1}}^{J^{1}}(θ_{𝔄}) is a multiple of η_{𝔄} and Ind_{H_{𝔄,𝔅m}^{1⋆}}^{U^{1}(𝔅_{m})J^{1}}(θ_{𝔄,𝔅_{m}}^{⋆}) is a multiple of η̃_{𝔄,𝔅_{m}}, it suffices to show that

∅ ≠ Hom_{U^{1}(𝔄_{m})}(Ind_{H_{m}^{1}}^{U^{1}(𝔄_{m})}(θ_{𝔄_{m}}), Ind_{H_{𝔄,𝔅m}^{1⋆}}^{U^{1}(𝔄_{m})}(θ_{𝔄,𝔅_{m}}^{⋆})) ≃
Hom_{H_{m}^{1}}(θ_{𝔄_{m}}, Res_{H_{m}^{1}}^{U^{1}(𝔄_{m})} ∘ Ind_{H_{𝔄,𝔅m}^{1⋆}}^{U^{1}(𝔄_{m})}(θ_{𝔄,𝔅_{m}}^{⋆})) ≃

Hom_{H_{m}^{1}}(θ_{𝔄_{m}}, ⨁_{x∈H_{𝔄,𝔅m}^{1⋆} \U^{1}(𝔄_{m})/H_{𝔄,𝔅m}^{1⋆}} Ind_{(H_{𝔄,𝔅m}^{1⋆})^{x}∩H_{m}^{1}}^{H_{m}^{1}}((θ_{𝔄,𝔅_{m}}^{⋆})^{x})) ≃

⨁_{x∈H_{𝔄,𝔅m}^{1⋆} \U^{1}(𝔄_{m})/H_{𝔄,𝔅m}^{1⋆}} Hom_{H_{m}^{1}}(θ_{𝔄_{m}}, Ind_{(H_{𝔄,𝔅m}^{1⋆})^{x}∩H_{m}^{1}}^{H_{m}^{1}}((θ_{𝔄,𝔅_{m}}^{⋆})^{x})).

Taking the component x = 1 we see that, by assumption, Hom_{H_{m}^{1}}(θ_{𝔄_{m}}, Ind_{H_{𝔄,𝔅m}^{1⋆} ∩H_{m}^{1}}^{H_{m}^{1}}(θ_{𝔄,𝔅_{m}}^{⋆})) ≃

Hom_{H_{𝔄,𝔅m}^{1⋆} ∩H_{m}^{1}}(Res_{H_{𝔄,𝔅m}^{1⋆} ∩H_{m}^{1}}^{H_{m}^{1}}(θ_{𝔄_{m}}), Res_{H_{𝔄,𝔅m}^{1⋆} ∩H_{m}^{1}}^{H_{𝔄,𝔅m}^{1}}(θ_{𝔄,𝔅_{m}}^{⋆})) = ℂ as required.

□

Since, in a split algebra, we are always in the unramified case (2.3), the combination of (7.1), (5.2), (4.1) and (7.3) provides:

( 7.4 ) Corollary Let A ≃ M_{N}(F) be a split algebra, let [𝔄, n, 0, β] be a simple stratum and 𝔅_{m} ⊂ 𝔅 := 𝔄∩B ⊂ 𝔅_{M}, where 𝔅_{m} is a minimal principal order and 𝔅_{M} is a maximal order in B := C_{A}(β). Moreover, let 𝔄_{m} and 𝔄_{M} be the unique continuations of 𝔅_{m} and 𝔅_{M}, respectively, and let θ_{𝔄}, θ_{𝔄_{m}}, θ_{𝔄_{M}} be the corresponding realizations of a fixed ps-character θ for the simple strata [𝔄, n, 0, β], [𝔄_{m}, −ν_{𝔄_{m}}(β), 0, β], [𝔄_{M}, −ν_{𝔄_{M}} (β), 0, β]. Then there exist representations η_{𝔄,𝔅_{m}}, η_{𝔄_{M} ,𝔅}, η_{𝔄_{M} ,𝔅_{m}} of U^{1}(𝔅_{m})J^{1}, U^{1}(𝔅)J_{M}^{1} and U^{1}(𝔅_{m})J_{M}^{1}, respectively, as in 5.2, and we have:

(i) η_{𝔄,𝔅_{m}} is equivalent to the representation η̃ of [BK1](5.1.15),

(ii) η_{𝔄_{M} ,𝔅} is equivalent to the representation η̂_{M} of [BK1](5.1.16) and

(iii) η_{𝔄_{M} ,𝔅_{m}} is equivalent to the representation η̃_{M} of [BK1](5.1.14).

In particular, in the split case all of B^{×} intertwines the representations η_{𝔄,𝔅_{m}}, η_{𝔄_{M} ,𝔅} and η_{𝔄_{M} ,𝔅_{m}} with themselves.

□

8 Extending to level 0

The main result of this section is obtained by a combination of ideas of [HR](2.4) and [BK1](5.2.4).

( 8.1 ) Proposition In the following, let all groups be subgroups of a locally profinite group G. Let J̄ be a compact group with pro-p-Sylow subgroup J̄_{p} and η̄ be an irreducible representation of J̄^{1} with representation space V so that its isomorphy class is stable under the adjoint action of J̄. Here J̄^{1} ⊂ J̄_{p} is a normal subgroup of J̄ and η̄ has an extension to J̄_{p}. We also assume that dim_{ℂ}(V) is a power of p and J̄/J̄^{1} ≃ J̄∩H̄/J̄^{1} ∩H̄ is isomorphic to a parabolic subgroup of Gl_{n_{1}}(𝔨) × … × Gl_{n_{l}}(𝔨), where 𝔨 is a finite field and 𝔨 ≠ 𝔽_{2}. In the case of 𝔨 = 𝔽_{2}, we do not allow proper parabolic subgroups and assume that n_{i} ≠ 2 for i = 1, … l. Finally, let H̄ be a compact subgroup such that there lives a linear character θ^{⋆} with (J̄∩ H̄)J̄_{p} = J̄_{p}(J̄∩ H̄) = J̄. We assume that η̄_{|J̄^{1}∩H̄} is a multiple of θ_{|J̄^{1}∩H̄}^{⋆}.

Under these assumptions there exists a unique extension κ of η̄ to J̄ such that κ_{|J̄∩H̄} = θ_{|J̄∩H̄}^{⋆} ⊗ W and W is a representation of J̄ ∩ H̄/J̄^{1} ∩ H̄ with det_{V} (W) ≡ 1.

In the extreme case where 𝔨 = 𝔽_{2} and n_{1} = … = n_{l} = 2 for any extension η′ of η̄ to J̄_{p}, there exists a unique extension κ of η′ to J̄.

\underline{Proof:}

Since η̄ is, up to isomorphy, stable under the adjoint action of J̄ and is irreducible, there exists a mapping ζ : J̄ → Aut_{ℂ}(V) such that we have ζ(x)^{−1}η̄(y)ζ(x) = η̄(x^{−1}yx) for all x ∈ J̄ and all y ∈ J̄^{1} and by Schurs Lemma the operators ζ(x) (for fixed x but varying y) are uniquely determined up to a scalar. Since η̄ has an extension η′ to J̄_{p}, we may set ζ(x) := η′(x) for all x ∈ J̄_{p}.

We may find a system of representatives R of the double cosets J̄_{p}\J̄/J̄_{p} contained in J̄ ∩ H̄ and such that x ∈ R and x ∈ J̄_{p} implies x = 1. With this choice we may arange for all j_{1}, j_{2} ∈ J̄_{p} and x ∈ R that ζ(j_{1}xj_{2}) = η′(j_{1})ζ(x)η′(j_{2}). Since replacing ζ(x) by non-zero scalars preserves the assumptions, we may assume that the equation ζ(x) = θ^{⋆}(x) ⊗ W′(x) holds for x ∈ R ⊂ J̄∩ H̄, where

det_{V} (W′(x)) = 1.  (10)

For all x ∈ J̄_{p}, we set W′(x) := θ^{⋆}(x)^{−1}η′(x) and so W′ extents to a function W′ : J̄∩ H̄ → Aut_{ℂ}(V), which is the identity on J̄^{1} ∩ H̄. Since W′_{|J̄_{p}∩H̄} is a representation of J̄_{p} ∩ H̄/J̄^{1} ∩ H̄, we deduce that

det_{V} (W′)_{|J̄_{p}∩H̄} is a linear character with order dividing (J̄_{p} ∩ H̄ : J̄^{1} ∩ H̄).  (11)

For all x_{1}, x_{2} ∈ J̄, we have ζ(x_{1})ζ(x_{2}) = α(x_{1}, x_{2})ζ(x_{1}x_{2}) with a 2-cocycle α of J̄ with values in ℂ^{×}. For all x_{1}, x_{2} ∈ J̄ ∩ H̄ we also have W′(x_{1})W′(x_{2}) = α(x_{1}, x_{2})W′(x_{1}x_{2}). By construction, we have

α_{|J̄_{p}×J̄_{p}} ≡ 1.  (12)

Moreover, we may find elements j, j′ ∈ J̄_{p} and h, h′ ∈ J̄ ∩ H̄ for x_{1}, x_{2} ∈ J̄ such that x_{1} = jh and x_{2} = h′j′. We have

α(x_{1}, x_{2})^{dim_{ℂ}(V)(J̄_{p}∩H̄:J̄^{1}∩H̄)} = det_{V} (ζ(x_{1})ζ(x_{2})ζ(x_{1}x_{2})^{−1})^{(J̄_{p}∩H̄:J̄^{1}∩H̄)} =
det_{V} (η′(j)ζ(h)ζ(h′)η′(j′)(η′(j)ζ(hh′)η′(j′))^{−1})^{(J̄_{p}∩H̄:J̄^{1}∩H̄)} =
det_{V} (ζ(h)ζ(h′)ζ(hh′)^{−1})^{(J̄_{p}∩H̄:J̄^{1}∩H̄)} =
det_{V} (θ^{⋆}(h)W′(h)θ^{⋆}(h′)W′(h′)θ^{⋆}(hh′)^{−1}W′(hh′)^{−1})^{(J̄_{p}∩H̄:J̄^{1}∩H̄)} =
det_{V} (W′(h)W′(h′)W′(hh′)^{−1})^{(J̄_{p}∩H̄:J̄^{1}∩H̄)}.

The above and the properties (10)(11) imply

α^{dim_{ℂ}(V)(J̄_{p}∩H̄:J̄^{1}∩H̄)} ≡ 1.  (13)

As J̄_{p} is a pro-p-Sylow subgroup of J̄, we know from the cohomology theory of finite groups that the mapping H^{2}(J̄/J̄^{1}, ℂ^{×})_{p} →^{res} H^{2}(J̄_{p}/J̄^{1}, ℂ^{×}) is injective, where the subscript p means the subgroup of all cohomology classes of p-power order. By (13) and the assumtions, the class of α is from the first group and by (12) the restriction of α is trivial. Therefore, α must be a 2-coboundary and we find a function χ : J̄/J̄^{1} → ℂ^{×} such that we have α(x_{1}, x_{2}) = χ(x_{1})χ(x_{1}x_{2})^{−1}χ(x_{2}) for all x_{1}, x_{2} ∈ J̄. Of course, κ′ := χζ is a representation of J̄ and an extension of η̄.

Considering the function χ as a complex function of J̄∩ H̄/J̄^{1} ∩ H̄, equation (12) says that χ_{|J̄_{p}∩H̄} is a linear character of J̄_{p} ∩ H̄/J̄^{1} ∩ H̄ and equation (13) indicates that χ^{dim_{ℂ}(V)(J̄_{p}∩H̄:J̄^{1}∩H̄)} is a linear character of J̄ ∩ H̄/J̄^{1} ∩ H̄. Of course, the character χ_{|J̄_{p}∩H̄} has order dividing (J̄_{p} ∩ H̄ : J̄^{1} ∩ H̄) and therefore χ^{dim_{ℂ}(V)(J̄_{p}∩H̄:J̄^{1}∩H̄)} is a character of J̄ ∩ H̄ that is trivial on J̄_{p} ∩ H̄. Let N := ker(χ^{dim_{ℂ}(V)(J̄_{p}∩H̄:J̄^{1}∩H̄)}). Then we have #(J̄_{p}∩H̄/J̄^{1}∩H̄) | #(N/J̄^{1}∩H̄) and therefore #(J̄∩H̄/N) | #(J̄ ∩ H̄/J̄_{p} ∩ H̄), where the last number is prime to p.

We deduce that χ^{dim_{ℂ}(V)(J̄_{p}∩H̄:J̄^{1}∩H̄)} has roots of unity as values with order prime to p. But since dim_{ℂ}(V)(J̄_{p} ∩ H̄ : J̄^{1} ∩ H̄) is a power of p, the k-th power of the inverse of this character, where k is the multiplicative inverse of the this p power mod the order of this character, is a character χ′ of J̄ ∩ H̄/J̄^{1} ∩ H̄ such that (χ′)^{dim_{ℂ}(V)(J̄_{p}∩H̄:J̄^{1}∩H̄)} = χ^{−dim_{ℂ}(V)(J̄_{p}∩H̄:J̄^{1}∩H̄)}. We conclude that

det(W′)(χχ′)^{dim_{ℂ}(V)} is a character of p-power order.  (14)

Let us assume first that 𝔨 ≠ 𝔽_{2} and consider the parabolic subgroup P ≃ J̄ ∩ H̄/J̄^{1} ∩ H̄ from Gl_{n_{1}}(𝔨) × … × Gl_{n_{l}}(𝔨). Without restriction of generality, we may assume that P is a direct product of standard parabolic subgroups and has a decomposition P = LU, where U is the unipotent radical consisting of upper triangular unipotent block matrices and L ≃ Gl_{k_{1}}(𝔨) × … Gl_{k_{m}}(𝔨) is the corresponding standard Levi subgroup consisting of block diagonal matrices. The partition k_{1}, …, k_{m} is a refinement of the partition n_{1}, …, n_{l}. By [HR](2.9) a linear character of P of p-power order would be trivial on L and on the commutator subgroup of U, but, therefore, it has to be supported on the upper beside diagonal elements of matrices from U and should be invariant under conjugation by diagonal matrices from L there. Since 𝔨 has more than two elements and L contains all diagonal matrices, any character trivial on L must be trivial on this upper beside diagonal elements as well. Hence, the character from (14) must also be trivial if 𝔨 ≠ 𝔽_{2}. Assuming that J̄∩ H̄/J̄^{1} ∩ H̄ ≃ Gl_{n_{1}}(𝔨) × … × Gl_{n_{l}}(𝔨) has no factor isomorphic to Gl_{2}(𝔽_{2}), the triviality of the character (14) again follows from [HR](2.9).

In these cases we may set κ := χ′κ′, where χ′ is considered to be a character of J̄/J̄^{1} ≃ J̄ ∩ H̄/J̄^{1} ∩ H̄ and we deduce that κ is again a continuation of η̄. We obtain (κ)_{|J̄∩H̄} = θ^{⋆} ⊗ χχ′W′ and, for W := χχ′W′, we obtain that det_{V} (W) ≡ 1 by (14). The uniqueness of such an extension κ follows again from the same arguments and the fact that another extension η̄ fulfilling (i) must be a twist of κ by a linear character of J̄/J̄^{1} of order dim_{ℂ}(V). Hence, as the order is a power of p, such a character must be trivial again.

Finally, we want to consider the extreme case where J̄/J̄^{1} is isomorphic to ×^{l}Gl_{2}(𝔽_{2}). In this case, each character of J̄_{p}/J̄^{1} must be a product of l characters of a subgroup of Gl_{2}(𝔽_{2}), isomorphic to 𝔽_{2}^{+} and, in fact, all these characters are restrictions of characters of Gl_{2}(𝔽_{2}) ≃ S_{3}, namely, the trivial character or the signum of a permutation of three letters. Any two extensions of η̄ to J̄_{p} must differ by such a character, we may twist κ′ in such a way that it is an extension of η′. The uniqueness follows from the facts that #(J̄/J̄_{p}) = 3^{l}, each other extension of η′ must be a twist by a linear character of J̄/J̄_{p} and such that S_{3} has no factor group of order 3.

□

Using this more general result, we obtain extensions of our Heisenberg representations.

( 8.2 ) Corollary Let ([𝔄, n, 0, β], 𝔅_{m}) be a non-degenerate pair in the sense of (4.2), where 𝔅_{m} is a minimal principal order in B := C_{A}(β) and let θ_{𝔄}^{⋆} be an extension of a simple character θ_{𝔄} ∈ C(β, 𝔄) to H := 𝔅^{×}H^{1} factorizing on 𝔅^{×} through the reduced norm Nrd_{B|F [β]} (i.e., θ_{𝔄}^{⋆} is given in terms of defining sequences by an extension of λ_{β} from U^{1}(𝔬_{F [β]}) to 𝔬_{F [β]}^{×}). Then there exists a unique extension κ_{𝔄} of the Heisenberg representation η_{𝔄} of θ_{𝔄} to J := 𝔅^{×}J^{1} fulfilling one of the following properties:

(i) If V is the representation space of κ_{𝔄} and 𝔅^{×}/U^{1}(𝔅) has no factor isomorphic to Gl_{2}(𝔽_{2}), then (κ_{𝔄})_{|H} = θ_{𝔄}^{⋆} ⊗ W, where W is a representation of H/H^{1} with det_{V} (W) ≡ 1.

(ii) If 𝔅^{×}/U^{1}(𝔅) has a factor isomorphic to Gl_{2}(𝔽_{2}) (in particular 𝔅 ≠ 𝔅_{m}), then κ_{𝔄} is an extension of η_{𝔄,𝔅_{m}}.

\underline{Proof:}

First, note that in the case where dim_{ℂ}(η_{𝔄}) = (J^{1} : H^{1})^{\frac{1}{2}} = 1, we clearly have κ_{𝔄} = θ_{𝔄}^{⋆}; hence, we assume in the following that dim_{ℂ}(V) > 1. In the case where 𝔅 = 𝔅_{m}, we clearly have η_{𝔄} = η_{𝔄,𝔅_{m}} and U^{1}(𝔅_{m})J^{1} = J^{1}; hence our following arguments also work in that case.

We will apply Proposition (8.1) with J̄ = J, J̄^{1} = J^{1}, η̄ = η_{𝔄}, J̄_{p} = U^{1}(𝔅_{m})J^{1}, H̄ = H and θ^{⋆} = θ_{𝔄}^{⋆}. Indeed, η_{𝔄} is stable under the action of J and it is irreducible, η_{𝔄,𝔅_{m}} is an extension of η_{𝔄} to U^{1}(𝔅_{m})J^{1}, the dimension of η_{𝔄} is (J^{1} : H^{1})^{\frac{1}{2}} and therefore a power of p. We have J ⊃ H, J^{1} ∩ H = H^{1}, J = HJ^{1} = J^{1}H, the restriction of η_{𝔄} to H^{1} is a multiple of (θ_{𝔄}^{⋆})_{|H^{1}}, J/J^{1} ≃ H/H^{1} ≃ (Gl_{s′}(𝔨_{D′}))^{r′}, where s′, r′ are the invariants of 𝔅 and D′ is a division algebra of the same Brauer class as B.

Consequently, all assumptions of 8.1 hold and the resulting representation κ_{𝔄} = κ obviously fulfils the conditions (i), (ii), which are equivalent to the conditions (i), (ii) from (8.1) under our assuptions.

□

\underline{Remark:} The diagrams of containments of groups

J  ⊃  H
∪  ∪
U^{1}(𝔅_{m})J^{1} ⊃  J_{𝔄,𝔅_{m}}^{1}  ⊃  H_{𝔄,𝔅_{m}}^{1}  ⊃ U^{1}(𝔅_{m})H^{1}
∪  ∪  ∪  ∪
J^{1}  ⊃ J_{𝔄,𝔅_{m}}^{1} ∩ J^{1} ⊃ H_{𝔄,𝔅_{m}}^{1} ∩ J^{1} ⊃  H^{1}

and of associated representations

κ_{𝔄}  ⊃  θ_{𝔄}^{⋆}
∪  ∪
η_{𝔄,𝔅_{m}} ⊃ η̂_{𝔄,𝔅_{m}} ⊃  θ_{𝔄,𝔅_{m}}  ⊃ (θ_{𝔄,𝔅_{m}})_{|U^{1}(𝔅_{m})H^{1}}
∪  ∪  ∪  ∪
η_{𝔄} ⊃ η̂_{𝔄} ⊃ (θ_{𝔄,𝔅_{m}})_{|H_{𝔄,𝔅m}^{1} ∩J^{1}} ⊃  θ_{𝔄}

show the strategy for constructing κ_{𝔄}.

We have J = HJ^{1}, H^{1} = J^{1} ∩ H, H^{1} is normal in J^{1} and J^{1}/H^{1} is a finite abelian pgroup. The character θ_{𝔄} applied to the commutators of J^{1} induces a nondegenerate alternating form J^{1}/H^{1} × J^{1}/H^{1} → ℂ^{×}. The characters θ_{𝔄} and θ_{𝔄,𝔅_{m}} concide on H^{1} ⊂ H ∩ H_{𝔄,𝔅_{m}}^{1}. Moreover, U^{1}(𝔅_{m})H^{1}/H^{1} = H ∩ H_{𝔄,𝔅_{m}}^{1}/H^{1} is a p-Sylow group in H/H^{1}. Also J_{𝔄,𝔅_{m}}^{1}∩J^{1} is a subgroup of J^{1} containing H_{𝔄,𝔅_{m}}^{1} ∩ J^{1} and normalizing the pair (H_{𝔄,𝔅_{m}}^{1}, θ_{𝔄,𝔅_{m}}). Finally, if the stratum [𝔄, n, 0, β] is non-degenerate and J/J^{1} has no direct factor isomorphic to Gl_{2}(𝔽_{2}), then (J^{1} : J_{𝔄,𝔅_{m}}^{1} ∩J^{1}) = (H_{𝔄,𝔅_{m}}^{1} ∩J^{1} : H^{1}) and H^{1}(J/J^{1}, 𝔽_{p}) = 0, where p is the residual characteristic of F. Hence, [HR](2.4) applies in this case and, in fact, in the case of (8.2(i)) κ_{𝔄} is just the τ constructed in [HR](2.4).

9 Another remark on the split case

( 9.1 ) Lemma Use the previous notations for θ_{𝔄} and η_{𝔄}. Let μ_{𝔄,𝔅_{m}} be an extension of η_{𝔄} to U^{1}(𝔅_{m})J^{1} that is intertwined by the whole of 𝔅^{×}. Then μ_{𝔄,𝔅_{m}} is uniquely determined, except for the case where 𝔅^{×}/U^{1}(𝔅) has a direct factor that is isomorphic to Gl_{2}(𝔽_{2}). In the latter case, 𝔅 must have the invariant 2 and we have 𝔅^{×}/U^{1}(𝔅) ≃ (Gl_{2}(𝔽_{2}))^{r′} ≃ S_{3}^{r′}, where r′ is the period of 𝔅. All other extensions fulfilling the properties of μ_{𝔄,𝔅_{m}} differ from μ_{𝔄,𝔅_{m}} by a twist by a character of U^{1}(𝔅_{m})J^{1}/J^{1} ≃ U^{1}(𝔅_{m})/U^{1}(𝔅) ↪ 𝔅^{×}/U^{1}(𝔅) ≃ (S_{3})^{r′} that has values in {+1, −1} and extends to (S_{3})^{r′}.

\underline{Proof:}

We have 𝔅^{×}/U^{1}(𝔅) ≃ (Gl_{s′}(𝔨_{D′}))^{r′}, where s′ is the invariant of 𝔅, r′ is the period of 𝔅 and D′ is a division algebra of the same Brauer class as B. Moreover, U^{1}(𝔅_{m})/U^{1}(𝔅) ≃ N^{r′}, where N is the set of lower triangular unipotent matrices in Gl_{s′}(𝔨_{D′}). If μ′_{𝔄,𝔅_{m}} is an extension of η_{𝔄} that is intertwined by the whole of 𝔅^{×}, then it differs from μ_{𝔄,𝔅_{m}} by a character ϕ of U^{1}(𝔅_{m})J^{1}/J^{1} ≃ U^{1}(𝔅_{m})/U^{1}(𝔅) ≃ N^{r′} that is intertwined by the whole of 𝔅^{×}; ϕ is the inflation of ϕ_{1} ⊗ … ⊗ ϕ_{r′}, where ϕ_{ν} is a character of N for ν = 1, …, r′.

Fix an arbitrary ν ∈ {1, …, r′}. The commutator group of N consists of all matrices n = (n_{l,m}) ∈ N such that all coefficients are zero except the lower beside diagonal elements n_{l+1,l} for 1 ≤ l ≤ s′ − 1. Thus ϕ_{ν} is of the form n ↦ ψ(∑_{l=1}^{s′−1} a_{l}n_{l+1,l}), where ψ is an additive character of 𝔨_{D′}. Suppose first that 𝔨_{D′} has at least three elements. The character ϕ_{ν} is stable under conjugation by any diagonal matrix of Gl_{s′}(𝔨_{D′}), and this implies that all a_{i} are zero. In this case ϕ_{ν} is trivial. Now assume that 𝔨_{D′} = 𝔽_{2}. The case of s′ = 1 is trivial and we assume first that s′ > 2. If we take into account that ϕ_{ν} is intertwined by the permutation matrix that corresponds to the cycle (1, …, s′), we deduce that, in this case, we must have a_{1} = a_{2} = … = a_{s′−1}. On the other hand, the fact that the permutation matrix corresponding to the transposition (1, 2) intertwines ϕ_{ν} implies a_{1} = 0; hence, ϕ_{ν} is trivial again. As already remarked above, Gl_{2}(𝔽_{2}) ≃ S_{3} (the permutation group of three elements) and N ≃ 𝔽_{2}^{+}, too. Hence, N has only the trivial representation or the character that has the value −1 on the preimage of 1. The first lifts to the trivial representation of S_{3} and the latter to the signum character of a permutation. This proves the last assertion and the case of s′ = 2.

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( 9.2 ) Corollary If A = M_{N}(F) is a split algebra and using the notations, assumptions and conclusions of (8.2), the representation κ_{𝔄} is an extension of η_{𝔄,𝔅_{m}}. Further, assuming that 𝔅 is a maximal order in C_{A}(β), κ_{𝔄} is a β-extension of η_{𝔄} in the sense of [BK1](5.2.1).

\underline{Proof:}

By (7.4) and [BK1](5.1.9) (resp. [BK1](5.1.1) for 𝔅 = 𝔅_{m}) η_{𝔄,𝔅_{m}} (where we set η_{𝔄_{m},𝔅_{m}} := η_{𝔄_{m}}) is intertwined by the whole of B^{×}; hence, (κ_{𝔄})_{|U^{1}(𝔅_{m})J^{1}} and η_{𝔄,𝔅_{m}} are extensions of η_{𝔄} that are intertwined by the whole of 𝔅^{×}, and (9.1) applies. We deduce that even if 𝔅^{×}/U^{1}(𝔅) does not contain any factor isomorphic to Gl_{2}(𝔽_{2}), κ_{𝔄} is an extension of η_{𝔄,𝔅_{m}} and, thus, by Corollary (7.4) and the remark before [BK1](5.2.7), it is one of the extensions obtained as in [BK1](5.2.5). Following the arguments of [BK1](Chapter 5) we obtain the desired reference to [BK1](5.2.1).

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10 A hypothetical list of simple types and the “level-0-case”

In section (1) we have seen that for suitable embedded strata, from the intertwining of two simple characters, follows their conjugacy by an element of the normalizer of the associated principal order. Now, just one simple character should be chosen from each conjugacy class, because we wish to have exactly one conjugacy class of simple characters in each essentially square integrable representation. We follow the plan sketched by [Z4].

For each possible period r of a principal order in A, we fix such an order 𝔄 in the corresponding conjugacy class. For a fixed order 𝔄 let us take the union Ĉ(𝔄) of all character sets C(β, 𝔄), where we vary over all simple strata [𝔄, n, 0, β] with r = \frac{e(F [β]/F)}{⟨d,e(F [β]/F)⟩} and maximally possible invariants d_{1} in the sense of 1.13 for the sound embeddings (F [β], 𝔄). Let C(𝔄) be the set of all possible extensions of the characters θ_{𝔄} with underlying stratum [𝔄, n, 0, β] to H = 𝔅^{×}H^{1}(β, 𝔄) of a representation system of the Orbits in Ĉ(𝔄) under the adjoint action of the normalizer of 𝔄, obtained by extending the associated defining character λ_{β} from U^{1}(𝔬_{F [β]}) to 𝔬_{F [β]}^{×}. For the latter set, we also have to fix a field F [β] for each representative θ_{𝔄} such that we obtain a unique leading term λ_{β} for its defining sequence with respect to an approximation sequence starting with [𝔄, n, 0, β]. The choice is arbitrary but can be taken, for example, from a fundamental domain as in [Z2][Z7] for the division algebra case, which can be transferred to any local central simple algebra of the same reduced degree by [BG]. We call the elements of C(𝔄) for varying 𝔄 fundamental characters for short.

Let A be a central simple algebra of index d and reduced degree N = md and D be a central simple division algebra of index N with a ring of integers O. To a fundamental character θ_{𝔄} with underlying stratum [𝔄, n, 0, β], field F [β] and defining character λ_{β}, we may fix an embedding β_{D} of β into the algebra D. Fixing this embedding, we may find a unique character θ_{O} such that θ_{𝔄} and θ_{O} are realizations of a common “PS-character” in the sense of [G2](6.6). Of course, we may extend all such realizations θ_{O} in the same way by extending λ_{β_{D}} = λ_{β} and obtain a fundamental character set C(O). The character set contains all extended realizations from all character sets C(𝔄) for varying 𝔄.

To each fundamental character θ ∈ C(𝔄), we may assign a hypothetical unique extension κ_{𝔄} of the associated Heisenberg representation η_{𝔄} to J_{A} := J(β, 𝔄).

( 10.1 ) Proposition Let E/F be a field extension in A and let 𝔄 be a principal order with period \frac{e(E/F)}{⟨d,e(E/F)⟩}. Then there exists a sound embedding (E, 𝔄). Moreover, assume that the latter embedding (E, 𝔄) has maximally possible embedding invariants d_{1} in the sense of 1.13. Then 𝔄 ∩ C_{A}(E) is a maximal order.

\underline{Proof:}

The first assertion follows from [G1](1.9)(iii)(b) for example. If d^{⋆} is the largest divisor of d that is prime to the period r of 𝔄, then, by the above choice, \frac{d}{⟨e(E/F),d⟩} | d^{⋆}. The maximally possible embedding invariant for E/F is d_{1} = ⟨f(E/F), d^{⋆}⟩ and, by [G2](2.2), the period of 𝔄 ∩ C_{A}(L), where L/F is unramified in E/F of degree f := ⟨f(E/F), d⟩, is \frac{r·f}{d_{1}} .

Consequently, C_{A}(L) has index \frac{d}{f} and ⟨f(E/L), \frac{d}{f} ⟩ = 1, [G1](2.2) applies, and 𝔄 ∩ C_{A}(E) has

period \frac{r·f·⟨e(E/F), \frac{d}{f} ⟩}{d_{1}·e(E/F)} = \frac{e(E/F)·⟨e(E/F)f,d⟩}{e(E/F)·d_{1}·⟨e(E/F),d⟩} = \frac{⟨f, \frac{d}{⟨e(E/F),d⟩} ⟩}{d_{1}} | \frac{⟨f,d^{⋆}⟩}{d_{1}} = \frac{d_{1}}{d_{1}} = 1, i.e., 𝔄 ∩ C_{A}(E) must be maximal.

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By the choice of the period of 𝔄 and the embedding invariant for (F [β], 𝔄), 10.1 applies and 𝔅_{M} := 𝔄∩C_{A}(β) is a maximal order. If now κ_{O} is the corresponding representation of J_{D} := J(β_{D}, O), we obtain from [Z2], [Z7] or [Br3] that each representation λ_{D} := κ_{O}⊗χ, where χ is a linear character of the cyclic group M := J_{D}/J^{1}(β_{D}, O), is contained in exactly one unramified twist class of irreducible representations of D^{×}. Conversely, we will show below that each irreducible representation of D^{×} can be twisted (by a general twist) in such a way that the twisted representation is contained in an unramified twist class of representations of a minimal Artin conductor in its twist class for arbitrary twist and each such representation of a minimal Artin conductor contains such a representation λ_{D}.

If χ is given, then M is the multiplicative group of a finite extension field 𝔨 of 𝔨_{F [β]} and the Galois group Gal(𝔨/𝔨_{F [β]}) acts on χ. If f is the length of this Galois orbit, m′ the invariant of 𝔅_{M}, d′ the index B := C_{A}(β), f′ := ⟨f, d′⟩ and e′ := ⟨\frac{N}{f} , m′⟩, then, by [SZ](1.1), there exists a supercuspidal representation σ_{0} of the finite group Gl_{f′}(𝔨′), where 𝔅_{M}/𝔔_{M} ≃ M_{m′}(𝔨′) (𝔨′/𝔨_{F [β]} is a finite extension of degree d′ and 𝔔_{M} the Jacobson radical of 𝔅_{M}) and where χ is the so-called Green parameter of the representation. Finally, we choose a principal order 𝔅 ⊃ 𝔅_{M} with period e′ such that 𝔅/𝔔_{M} is isomorphic to the set of lower f′ × f′-block triangular matrices in M_{m′}(𝔨′). We may inflate σ_{0}^{e′} to a representation τ of J := 𝔅^{×}J^{1}(β, 𝔄). We obtain a representation λ := (κ_{𝔄})_{|J} ⊗ τ of J. We choose from each Galois orbit only one χ and this choice also fixes a Galois representative σ_{0} in its orbit under the action of Gal(𝔨′/𝔨_{F [β]}). As in [BK1](5.5.10), we define a list of so-called simple types:

( 10.2 ) Definition A simple type is either

(i) a twisted representation Λ := λ ⊗ μ with μ = λ_{α} ∘ Nrd_{A/F} and λ_{α} a character of 𝔬_{F}^{×}, where λ is constructed as above from a fundamental character and from a character χ of the corresponding finite cyclic group M or

(ii) an irreducible representation of 𝔄^{×} that is in fact inflated from the quotient 𝔄^{×}/(1 + 𝔓) ≃ Gl_{s}(k_{D})^{r} and is given there by a tensor power σ_{0}^{⊗r} of a supercuspidal representation σ_{0} of Gl_{s}(k_{D}), where we choose only one σ_{0} from each orbit under the action of Gal(𝔨_{D}/𝔨_{F}).

The case (ii) can be considered to be the special case κ_{𝔄} ≡ 1 and it is the situation of the level-0-case of the work of [SZ]. In [GSZ](5.3) it is shown that the types defined in (10.2)(ii) are indeed simple types of a single Bernstein component of the category of smooth representations of A^{×} in the sense of [BK4](4.1)(4.2). We conjecture that, in general, the above definition is a list of simple types of a corresponding single Bernstein component containing a single unramified twist class of essentially square integrable irreducible representations.

( 10.3 ) Conjecture (i) All pairs ([𝔄, n, 0, β], 𝔅_{m}) are non-degenerate.

(ii) All extensions κ_{𝔄} are intertwined by the whole of B^{×}.

(iii) For any type of representation Λ, with irreducible factor τ, we have an isomorphism of Hecke algebras H(A^{×}, Λ) ≃ H(B^{×}, τ_{|𝔅^{×}}).

(iv) For any representation Λ with 𝔅 = 𝔅_{M} maximal and τ being inflated from a single factor σ_{0}, there exists an extension to the stabelizer of Λ that induces irreducibly to A^{×}.

(v) Each representation Λ for 𝔅 ≠ 𝔅_{M} is a so-called covering type in the sense of [BK4](8) of a Levi component with e′ factors isomorphic to Gl_{\frac{m}{e′}} (D).

In the split case of A = M_{N}(F), we have already seen in (9.2) that the whole of B^{×} intertwines all extensions κ_{𝔄} with themselves. Our definition differs from the one of [BK1](5.5.10) in two points. First we use types λ′ as defined in [BK1](5.5.10) and used later to calculate the Hecke algebra and, second, we first construct the types for the irreducible essentially square integrable representations of a minimal Artin conductor in its (general) twist class and obtain the others by the fact that induction commutes with twist. By a thesis of Vincent Sécherre [VS] the properties (i) and (ii) can be replaced by another construction that is less close to the ideas of Reimann [HR], but more close to those of Bushnell and Kutzko [BK1].

( 10.4 ) Definition In the following we call a type with maximal order 𝔅 = 𝔅_{M} as in [BK1](6) a maximal simple type.

We have shown the property (i) in case of m = l, l an arbitrary prime number, and would like to show (ii) later, independently of the method of [VS]. Finally, we want to show that the list is also complete in the sense that each single Bernstein component containing an irreducible essentially square integrable representation is parametrized by such a type. Moreover, this will imply that all supercuspidal representations of Gl_{l}(D) are induced by a “compact modulo center subgroup”.

In the following, we need two lemmas, which are generalizations of similar lemmas from [BK1].

( 10.5 ) Lemma Let [𝔄, n, 0, β] be a simple stratum in A and θ_{𝔄} ∈ C(β, 𝔄). Let η_{𝔄} be the Heisenberg representation of J^{1}(β, 𝔄) = J^{1} associated with θ_{𝔄} as in (5.1). Then, for x ∈ A^{×}, we have the equality

dim_{ℂ}(I_{x}(η_{𝔄})) = \{ \begin{matrix}1 & if x ∈ J^{1}B^{×}J^{1} \\ 0 & else, \end{matrix}

where we set I_{x}(ρ) := Hom_{K^{x}∩K}(ρ, ρ^{x}) as in [BK1](5.1.7) if ρ is a representation of an open compact subgroup K of A^{×}.

\underline{Proof:}

The proof is the same as in [BK1](5.1.8)(5.1.9) if we may prove the following generalization of [BK1](5.1.10):

( 10.6 ) Lemma For y ∈ B^{×} we have (J^{1} : J^{1} ∩ (J^{1})^{y}) = (H^{1} : H^{1} ∩ (H^{1})^{y}).

\underline{Proof:}

We follow the proof of [BK1](5.1.10) and have to take into account the following changes. As above we write H^{1} = 1 + ℌ^{1}, J^{1} = 1 + 𝔍^{1} and choose Haar maesures μ^{×} for A^{×} and μ for A^{+} such that μ^{×}(H^{1}) = μ(ℌ^{1}) and μ^{×}(J^{1}) = μ(𝔍^{1}). Since ker(a_{β}) = B and by ([G2](4.5)), there is an exact sequence

0 → 𝔔 → 𝔍^{1} →^{a_{β}} (ℌ^{1})^{∗} →^{𝔰_{β}} 𝔅 → 0.

This sequence remains exact if we conjugate it by y, and we assert that the sequence

0 → 𝔔 + 𝔔^{y} → 𝔍^{1} + (𝔍^{1})^{y} →^{a_{β}} (ℌ^{1})^{∗} + ((ℌ^{1})^{∗})^{y} →^{𝔰_{β}} 𝔅 + 𝔅^{y} → 0

remains exact. By ([G2](4.6)) we have (ℌ^{1})^{∗} = a_{β}(𝔍^{1}) + 𝔄 and thus the last arrow induced by 𝔰_{β} is surjective. Furthermore,

(ℌ^{1})^{∗} + ((ℌ^{1})^{∗})^{y} = a_{β}(𝔍^{1} + (𝔍^{1})^{y}) + 𝔄 + 𝔄^{y}.  (15)

Next we want to examine the exactness at (ℌ^{1})^{∗} + ((ℌ^{1})^{∗})^{y}.

First we may embed into the canonical splitting Ã of A and, as in ([G2](3.3)), we may consider the splitting 𝔄̃ of 𝔄. We set B̃ := CÃ(E) and 𝔅̃ := 𝔄̃ ∩ B̃. Applying [BK](1.4.16) for h_{1} = h_{2} = y, a sufficiently small i, sufficiently big n and j = m = 0, we deduce that, for k = k_{0}(β, 𝔄), we have the equality ya_{β}(𝔔̃^{−k}𝔑_{k}(β, 𝔄̃)) + a_{β}(𝔔̃^{−k}𝔑_{k}(β, 𝔄̃))y = a_{β}(y𝔔̃^{−k}𝔑_{k}(β, 𝔄̃) + 𝔔̃^{−k}𝔑_{k}(β, 𝔄̃)y) = (y𝔄̃ + 𝔄̃y) ∩ ker(𝔰_{β}) = (y𝔄̃ + 𝔄̃y) ∩ a_{β}(Ã). From [BK](1.4.10) we also obtain a_{β}(𝔔̃^{−k}𝔑_{k}(β, 𝔄̃)) = 𝔄̃ ∩ a_{β}(Ã), which implies (y𝔄̃ + 𝔄̃y) ∩ a_{β}(Ã) = y(𝔄̃ ∩ a_{β}(Ã)) + (𝔄̃ ∩ a_{β}(Ã))y.

Taking the intersection with A on both sides and including this into equation (15), we obtain ((ℌ^{1})^{∗} + ((ℌ^{1})^{∗})^{y}) ∩ ker(𝔰_{β}) = a_{β}(𝔍^{1} + (𝔍^{1})^{y}) + 𝔄 ∩ a_{β}(A) + (𝔄 ∩ a_{β}(A))^{y}. Since, by [BK1](3.1.10) the equalities 𝔄 ∩ a_{β}(A) = (𝔄̃ ∩ a_{β}(Ã)) ∩ A = a_{β}(𝔔̃^{−k}𝔑_{k}(β, 𝔄̃)) ∩ A ⊂ a_{β}(U^{[ \frac{k+1}{2} ]}(𝔄̃)𝔍^{1}(β, 𝔄̃)) ∩ A = a_{β}(𝔍^{1}) are true, where k = −k_{0}(β, 𝔄̃), we deduce that the sequence is exact at (ℌ^{1})^{∗} + ((ℌ^{1})^{∗})^{y}. By [BK1](1.3.16) we also have 𝔔 + 𝔔^{y} ⊂ (𝔍^{1} + (𝔍^{1})^{y}) ∩ ker(a_{β}) ⊂ (𝔓 + 𝔓^{y}) ∩ ker(a_{β}) = ((𝔓̃ + 𝔓̃^{y}) ∩ ker(a_{β})) ∩ A = (𝔔̃ + 𝔔̃^{y}) ∩ A = 𝔔 + 𝔔^{y} and, therefore, also exactness at the other positions.

The rest of the proof is exactly the same as for [BK1](5.1.10), where we have to note that [BK1](5.1.5) holds in our more general case of an arbitrary local central simple algebra.

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( 10.7 ) Lemma Let ([𝔄, n, 0, β], 𝔅_{m}) be a non-degenerate pair and κ_{𝔄} be a representation as in Theorem (8.2). Let ξ be an irreducible representation of J/J^{1} = 𝔅^{×}/U^{1}(𝔅). Assuming that the whole of B^{×} intertwines κ_{𝔄} with itself, we have:

(i) I_{A^{×}}(κ_{𝔄} ⊗ ξ) = JI_{B^{×}}(ξ_{|𝔅^{×}})J,

(ii) for all y ∈ B^{×}, the spaces I_{y}(κ_{𝔄} ⊗ ξ) and I_{y}(ξ) have the same dimension and

(iii) the representation κ_{𝔄} ⊗ ξ is irreducible.

\underline{Proof:}

The proof is the same as in [BK1](5.3.2) if we use Lemma (5.1) instead of [BK1](5.1.1), Lemma (10.5) instead of [BK1](5.1.8) and our assumption on the intertwining of κ_{𝔄} instead of [BK1](5.2.7).

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11 Maximal simple types

( 11.1 ) Proposition Let λ = κ_{𝔄} ⊗ σ be a maximal simple type associated with a simple stratum [𝔄, n, 0, β] and assume that the whole of B^{×} intertwines κ_{𝔄} with itself, then we have

(i) λ is a simple type for a single Bernstein component of the category of smooth representations, where the irreducible elements of this component just vary over a single inertial class of supercuspidal representations of A^{×}.

(ii) λ and λ′ = κ_{𝔄} ⊗ σ′ are types for the same Bernstein component iff the supercuspidal representations σ_{0} and σ′_{0} of Gl_{s′}(𝔨_{D′}) belong to the same Γ-orbit, where Γ := Gal(𝔨_{D′}/𝔨_{F [β]}) acts on the entries of a matrix of Gl_{s′}(𝔨_{D′}), hence, on the group and on the representations of the group.

\underline{Proof:}

(i): By assumption on κ_{𝔄}, Lemma (10.7)(iii) applies and we deduce that λ is irreducible. Further, the combination of (10.7)(i), [GSZ](2.1) and the assumption that 𝔅 is a maximal order, lead to the equality I_{A^{×}}(λ) = JI_{B^{×}}(σ_{|𝔅^{×}})J =< h^{l} > J =: J̃, where h ∈ K_{𝔅} is an arbitrary but fixed uniformizer of 𝔅 and l is the length of the Galois orbit of σ_{|𝔅^{×}} or σ_{0}, as in [GSZ](Section 2). As h^{l} normalizes the group J, we may extend the representation λ to a representation λ̃ of J̃.

By [Ca](1.5) the compactly induced representation Π := c-IndA^{×} ; J̃(λ̃) is irreducible and supercuspidal.

Using arguments as in the proof of [GSZ](5.1) or [BK4](5.4) we conclude that λ is a type for a single Bernstein component of the category of smooth representations of A^{×}, where the irreducible elements all belong to the inertial class of Π.

(ii) First we use arguments analogous to those in the beginning of the proof of [GSZ](5.2) to show that λ and λ′ parametrize the same Bernstein component if σ and σ′ resp. σ_{0} and σ′_{0} are conjugated under the action of Γ. For this purpose we consider an extension λ̃ of λ to J̃ as in the proof of (i). As the adjoint action of h (as above) induces the action of a suitable generator of Γ, J̃ is a normal subgroup of index l in < h > J = K_{𝔅}J and the induction of λ̃ to K_{𝔅}J^{1} is irreducible, all h-conjugates must appear in its restriction to J. Therefore Π = c-IndA^{×} ; J̃(λ̃) also contains λ′ and hence, by [BK4](5.2) λ and λ′ parametrize the same Bernstein component.

Conversely, assume that λ and λ′ parametrize the same Bernstein component. Then, by [BK4](5.2), they appear in the same irreducible representation; hence, by [BK5](6.5) they must intertwine. Now we use arguments as in the proof of [BK4](5.3.2) and (10.7), respectively.

Assume that y ∈ A^{×} intertwines λ and λ′, then it also intertwines λ_{|J^{1}} and λ′_{|J^{1}}, which are both multiples of η_{𝔄}. By (5.1) we must have y ∈ J^{1}B^{×}J^{1} and, as J^{1} normalizes λ as well as λ′, we may reduce to the case where y ∈ B^{×}.

Let X be the representation space of κ_{𝔄}, Y be the representation space of σ and Y′ be the representation space of σ′. Then an intertwining operator ϕ ∈ Hom_{J^{1}∩(J^{1})^{y}}(λ′, λ^{y}) associated with y can be considered as an element of End_{ℂ}(X) ⊗ Hom_{ℂ}(Y′, Y) ≃ Hom_{ℂ}(X ⊗ Y′, X ⊗ Y). So let us write ϕ = ∑_{j} S_{j} ⊗ T_{j} with S_{j} ∈ End_{ℂ}(X) such that {T_{j}} is a basis of Hom_{ℂ}(Y′, Y). Then ∑_{j}(κ_{𝔄}(h) ∘ S_{j} − S_{j} ∘ κ_{𝔄}^{y}(h)) ⊗ T_{j} = 0 for all h ∈ J^{1} ∩ (J^{1})^{y} by the triviality of σ and σ′ on arguments from J^{1}. By the linear independence of the T_{j} we conclude that all S_{j} lie in I_{y}(η_{𝔄}). As, by assumption, κ_{𝔄} is intertwined by the whole of B^{×}, all spaces I_{y}(η_{𝔄}) are one-dimensional by (10.5), and κ_{𝔄} is an extension of η_{𝔄}, we conclude that all S_{j} lie in fact in I_{y}(κ_{𝔄}). Again, as the spaces I_{y}(κ_{𝔄}) are onedimensional, we have ϕ = S ⊗ T for suitable S ∈ I_{y}(κ_{𝔄}) and T ∈ Hom_{ℂ}(Y′, Y).

Now choose h ∈ J ∩ J^{y}. Then the intertwining condition takes the form S ∘ κ_{𝔄}^{y} ⊗ T ∘ σ^{y} = κ_{𝔄} ∘ S ⊗ T ∘ σ^{y} = κ_{𝔄} ∘ S ⊗ σ′ ∘ T and, therefore, T ∈ Hom_{J∩J^{y}}(σ′, σ^{y}). We deduce that σ_{|𝔅^{×}} and σ′_{|𝔅^{×}} intertwine in B^{×} and we have reduced to the level-0-case in the algebra B. Finally, we may follow the second part of the proof of [GSZ](5.2) to show that σ and σ′ resp. σ_{0} and σ′_{0} belong to the same orbit under the action of Γ.

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12 The case of tame Gl_{l}(D) with l prime

We first remark that in the case of G_{l}(D) all pairs ([𝔄, n, 0, β], 𝔅_{m}) are non-degenerate by (7.1)(ii).

( 12.1 ) Corollary Let [𝔄, n, 0, β] be a simple stratum such that B = C_{A}(β) is a division algebra. Then the pair ([𝔄, n, 0, β], 𝔅_{m}) is non-degenerate, the associated representation κ_{𝔄} from (8.2) is intertwined by the whole of B^{×} and the types from (10.2) are maximal types that fulfil the conclusions of (11.1).

\underline{Proof:}

If B is a division algebra, then we have 𝔅 = 𝔬_{B} = 𝔅_{m} and each such pair ([𝔄, n, 0, β], 𝔅_{m}) is non-degenerate. The group K_{𝔅} normalizes the group J and each representation η_{𝔄} so that each representation κ_{𝔄} is uniquely determined by properties that are stable under the action of K_{𝔅}. Therefore, the equivalence class of κ_{𝔄} is normalized by K_{𝔅} and, in particular, κ_{𝔄} is intertwined by the whole of B^{×} = K_{𝔅}. As 𝔅 = 𝔬_{B} is a maximal order in B, (11.1) can be applied.

□

( 12.2 ) Corollary Let A = M_{l}(D), where l is prime and [𝔄, n, 0, β] is a simple stratum such that 𝔅 = 𝔅_{M} := 𝔄 ∩ B is a maximal order in B := C_{A}(β) = M_{l}(D′). If 𝔨_{D′} ≠ 𝔽_{2}, then all associated representations κ_{𝔄} from (8.2) are intertwined by the whole of B^{×} and the types from (10.2) are maximal types, satisfying the conclusions of (11.1).

\underline{Proof:}

As above let d be the index of D and E := F [β]. Of course, [E : F] must be a divisor of d, because otherwise B is a division algebra and we are in the case (12.1).

Therefore, we may assume without restriction of generality that E ⊂ D and, thus, B = M_{l}(D′), where D′ := C_{D}(β). Moreover, let 𝔅 be given in normal form, i.e., 𝔅 = M_{l}(𝔬_{D′}) and 𝔄 = M_{l}(𝔬_{D}).

Let J_{0}^{1} := J^{1}(β, 𝔬_{D}) and H_{0}^{1} := H^{1}(β, 𝔬_{D}), then we see easily that J^{1} = I_{l} + M_{l}(J_{0}^{1}) and H^{1} = I_{l} + M_{l}(H_{0}^{1}).

For the proof of the proposition, we have to show that κ := κ_{𝔄} is intertwined by the whole of B^{×}. Since J = 𝔅^{×}J^{1}, we just need to consider representives y of the double cosets 𝔅^{×}\B^{×}/𝔅^{×} and assume that y = diag(π^{′ν_{1}}I_{n_{1}}, …, π^{′ν_{nr}} I_{r}) with ν_{1} > … > ν_{r}, where π′ is a prime element of 𝔬_{D′} and n = (n_{1}, …, n_{r}) is a partition of l.

Let M := diag(M_{n_{1}}(D) × … × M_{n_{r}}(D)) and let U and U^{−} be the groups of upper and lower unipotent blockmatrices, where we work with the block decomposition A = (M_{n_{i}×n_{j}}(D))_{i,j=1,...,r}. To the partition n := (n_{1}, …, n_{r}) we find a hereditary principal standard order 𝔅_{n} ⊂ 𝔅 with invariants n_{1}, …, n_{r}. If we consider the orders in the transposed normal form ([G2](0.3)), i.e., in upper block triangular form modulo π′M_{l}(𝔬_{D′}) = π′𝔅 =: 𝔔, then the following facts, similar to [BK](5.2.9), hold:

( 12.3 ) [BK1](5.2.9)

(i) y normalizes 𝔅_{n} ∩ M and

(ii) 𝔅 ∩ y^{−1}𝔅y ⊂ π′𝔅 + 𝔅_{n} ∩ y^{−1}𝔅_{n}y.

We have (𝔅_{n}^{×}J^{1}) ∩ (𝔅_{n}^{×}J^{1})^{y} = (𝔅_{n} ∩ M)((U^{1}(𝔅_{n})) ∩ (U^{1}(𝔅_{n}))^{y})(J^{1} ∩ (J^{1})^{y}) and (J^{1} ∩ (J^{1})^{y}) = ((U^{1}(𝔅)) ∩ (U^{1}(𝔅))^{y})(J^{1} ∩ (J^{1})^{y}).

First we show the

( 12.4 ) intermediate assertion that κ_{|𝔅_{n}^{×}}J^{1} is intertwined by y.

Proof of the intermediate assertion:

Let (Θ, β) be the PS-character with θ := θ_{𝔄} = Θ(A, id_{A}, 𝔄). We set θ_{i} := Θ(M_{n_{i}}(D), p_{i}, β), J_{i}^{1} := p_{i}(J^{1} ∩ M) and H_{i}^{1} := p_{i}(H^{1} ∩ M) for i = 1, …, r, where p_{i} is the projector to the i-th diagonal component of M. Then due to the product decomposition (2.9) and (3.1), we have the relations H^{1} ∩ M = ∏_{i=1}^{r} H_{i}^{1} and θ_{|H^{1}∩M} = ⨂_{i=1}^{r} θ_{i}. It is easy to show that for the groups J^{1} ∩ U, J^{1} ∩ U^{−}, H^{1} ∩ U and H^{1} ∩ U^{−} the corresponding product decompositions (2.9) hold and, by (3.1), θ vanishes on H^{1} ∩ U and H^{1} ∩ U^{−}.

As in the split case we obtain:

( 12.5 ) [BK1](7.1.19), [BK1](7.2.3) The form χ_{θ} of J^{1}/H^{1} which is defined by application of θ to the commutators of J^{1}, fulfils the following conditions:

(i) The restriction of χ_{θ} to J^{1} ∩ M/H^{1} ∩ M is non-degenerate and is the direct orthogonal sum of the corresponding forms χ_{θ_{i}} of J_{i}^{1}/H_{i}^{1} for i = 1, …, r;

(ii) the spaces J^{1}∩U/H^{1}∩U and J^{1}∩U^{−}/H^{1}∩U^{−} are isotropic and orthogonal to J^{1}∩M/H^{1}∩M with respect to χ_{θ};

(iii) with respect to χ_{θ} we have an orthogonal decomposition J^{1}/H^{1} = J^{1} ∩ M/H^{1} ∩ M ⊥ J^{1} ∩ U/H^{1} ∩ U × J^{1} ∩ U^{−}/H^{1} ∩ U^{−}.

As in the split case we prove:

( 12.6 ) [BK1](7.2.4) Let η = η_{𝔄} (η_{i} respectively) be the Heisenberg representation of J^{1} (J_{i}^{1} respectively) attached to the Heisenberg character θ (resp. θ_{i}) and Φ be a character of the abelian group J^{1} ∩ U/H^{1} ∩ U. Then there exists a unique representation η_{Φ} of J_{MU}^{1} := (J^{1} ∩ M)(J^{1} ∩ U)H^{1} up to isomorphism such that (η_{Φ})_{|J^{1}∩M} ≃ η_{1} ⊗ … ⊗ η_{r}, (η_{Φ})_{|H^{1}} is a multiple of θ and (η_{Φ})_{|J^{1}∩U} is a multiple of Φ.

With the notations as above we also obtain

(i) η_{|J_{MU}^{1}} ≃ ⨁_{Φ∈{ \atop J^{1}∩U/H^{1}∩U}} η_{Φ};

(ii) ∀Φ ∈ { \atop J^{1} ∩ U/H^{1} ∩ U} : Ind_{J_{MU}^{1}}^{J^{1}} (η_{Φ}) ≃ η;

(iii) ∀Φ_{1}, Φ_{2} ∈ { \atop J^{1} ∩ U/H^{1} ∩ U}∃x ∈ J^{1} ∩ U^{−}/H^{1} ∩ U^{−} : η_{Φ_{2}} ≃ η_{Φ_{1}}^{x}.

Let us consider the representations η_{P} := η_{1} and η_{U} := (η_{P})_{|J^{1}∩M} ≃ η_{1} ⊗ … ⊗ η_{r}, where η_{P} can be considered as a representation of J_{MU}^{1} on the space of J^{1} ∩ U fixed vectors in η, and η_{U} as a representation of (J^{1} ∩ M)(J^{1} ∩ U)/(J^{1} ∩ U).

The representation κ_{|𝔅_{n}^{×}J^{1}} fulfils the conditions of proposition (8.1) with respect to the characters θ_{|𝔅_{n}^{×}H^{1}}^{⋆} with J̄ := 𝔅_{n}^{×}J^{1}, H̄ := 𝔅_{n}^{×}H^{1}, J̄^{1} := J^{1}, J̄_{p} := U^{1}(𝔅_{m})J^{1}, η̄ = η and η′ = η_{𝔄,𝔅_{m}}, where 𝔅_{m} is a minimal order contained in 𝔅_{n} and 𝔅_{n}^{×}J^{1}/J^{1} ≃ P_{n}(𝔨_{D′}) ⊂ Gl_{l}(𝔨_{D′}) the parabolic standard subgroup attached to the partition n.

Let us consider the representation η_{U} ≃ η_{1} ⊗…⊗η_{r} of J^{1} ∩M. We want to apply proposition (8.2) to the character θ_{U}^{⋆} = θ_{1}^{⋆} ⊗ … ⊗ θ_{r}^{⋆} and to η_{U}, where θ_{i}^{⋆} is an extension of θ_{i}. For each i we obtain this extension from the same extension of the corresponding character λ_{β} from the defining sequence of θ. We obtain an extension κ_{U} of η_{U} from J^{1} ∩ M to (𝔅_{n}^{×}J^{1}) ∩ M, which is uniquely determined by the properties (κ_{U})_{|J^{1}∩M} = η_{U} and (κ_{U})_{|𝔅_{n}^{×}H^{1}∩M} ≃ θ_{|𝔅_{n}^{×}H^{1}∩M}^{⋆} ⊗ W′, where the determinant character of W′ is trivial and θ_{|𝔅_{n}^{×}H^{1}∩M}^{⋆} = θ_{U}^{⋆}.

Now we extend κ_{U} trivially on U^{1}(𝔅_{n})J^{1} ∩ U and H^{1} ∩ U^{−}. We obtain an extension κ_{P} of η_{P} to (𝔅_{n}^{×} ∩ M)(U^{1}(𝔅_{n}) ∩ U)J_{MU}^{1}, which is intertwind by y due to the choice of the partition n, because y normalizes the group (𝔅_{n}^{×}J^{1}) ∩ M, the representation η_{U} and the character θ_{U}^{⋆} and does not violate the properties that characterize κ_{U}.

Now, we consider κ′ ≃ Ind_{(𝔅_{n}^{×}∩M)(U^{1}(𝔅_{n})∩U)J_{MU}^{1}}^{𝔅_{n}^{×}J^{1}} (κ_{P}), 𝔅_{n}^{×}J^{1} ∩ J^{1} = J^{1} and

(𝔅_{n}^{×} ∩ M)(U^{1}(𝔅_{n}) ∩ U)J_{MU}^{1} ∩ J^{1} = J_{MU}^{1}. We obtain κ′_{|J^{1}} ≃ Ind_{J_{MU}^{1}}^{J^{1}} ((κ_{P})_{|J_{MU}^{1}} ) ≃ Ind_{J_{MU}^{1}}^{J^{1}} (η_{P}) ≃ η. Since 𝔅_{n}^{×}J^{1} ⊃ (𝔅_{n}^{×} ∩ M)(U^{1}(𝔅_{n}) ∩ U)J_{MU}^{1} and since 𝔅_{n}^{×}J^{1} normalize the group 𝔅_{n}^{×}H^{1} and the character θ^{⋆}, we obtain κ′_{|𝔅_{n}^{×}H^{1}} ≃ ⨁_{x∈𝔅_{n}^{×}J^{1}/𝔅_{n}^{×}H^{1}} θ_{|𝔅_{n}^{×}H^{1}}^{⋆} ⊗ W′^{x}, where we consider W′ to be a representation of 𝔅_{n}^{×}H^{1}/H^{1}.

Of course, the representation ⨁_{x∈𝔅_{n}^{×}J^{1}/𝔅_{n}^{×}H^{1}} W′^{x} has a trivial determinant character and therefore, by (8.2), we have κ_{|𝔅_{n}^{×}J^{1}} ≃ κ′. However, since y intertwines κ_{P} and κ′ is induced by κ_{P}, [BK1](4.1.3) leads to the proof of the intermediate assertion.

□

On account of the above properties (12.3)(i)(ii) of 𝔅_{n} we have J ∩ J^{y} = (𝔅^{×} ∩ (𝔅^{×})^{y})(J^{1} ∩ (J^{1})^{y}) = (U^{1}(𝔅) ∩ (𝔅^{×})^{y})(U^{1}(𝔅_{n})J^{1} ∩ (U^{1}(𝔅_{n})J^{1})^{y}). As in the split case, (U^{1}(𝔅) ∩ (𝔅^{×})^{y}) ⊂ (U^{1}(𝔅) ∩ (U^{1}(𝔅))^{y})(𝔅_{n}^{′×} ∩ (𝔅_{n}^{′×})^{y}), where 𝔅′_{n} is the corresponding order in opposite normal form to the partition of n, i.e., 𝔅′_{n} is of upper block traingular form modulo 𝔔.

Now κ on U^{1}(𝔅) is given by θ = θ_{𝔄} and intertwined by y. Also by the above argument, y intertwines the restriction to U^{1}(𝔅_{n})J^{1}. Replacing y by y^{−1}, 𝔅_{n} by 𝔅′_{n}, U by U^{−} and so on, and observing that the intertwining is invariant under taking inverse elements, we conclude that y intertwines the restriction of κ to 𝔅_{n}^{′×}J^{1}. Thus, the assertion follows from the relation (𝔅^{×} ∩ (𝔅^{×})^{y})(J^{1} ∩ (J^{1})^{y}) ⊂ (U^{1}(𝔅) ∩ (U^{1}(𝔅))^{y})(𝔅_{n}^{′×} ∩ (𝔅_{n}^{′×})^{y})(U^{1}(𝔅_{n})J^{1} ∩ (U^{1}(𝔅_{n})J^{1})^{y}) and because, by (10.5), all involved representations can be assumed to be intertwined by the same intertwining operator that finally turns out to be an intertwining operator for κ.

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( 12.7 ) Proposition Let A = M_{l}(D) (l a prime number) and Λ be a type representation with irreducible factor τ as in Definition 10.2. Then we have an isomorphism of Hecke algebras H(A^{×}, Λ) ≃ H(B^{×}, τ_{|𝔅^{×}}).

\underline{Proof:}

First, by Lemma 10.7, it is clear that the supports of the Hecke algebras H(A^{×}, Λ) and that, H(B^{×}, τ_{|𝔅^{×}}) have the same intersection with B^{×} and to any element x of the intersection, there exists a constant multiple of exactly one entire function f, which lives on a double coset JxJ and 𝔅^{×}x𝔅^{×}, respectively. We obtain an isomorphism of ℂ-vector spaces between the algebras.

If Λ = (κ_{𝔄})_{|J} ⊗ μ ⊗ τ and τ is a lift of an entire supercuspidal representation σ_{0} of 𝔅_{M}^{×}/(1 + 𝔔_{M}) ≃ Gl_{l}(k) (also 𝔅 = 𝔅_{M}), then the double cosets J\B^{×}/J, 𝔅^{×}\B^{×}/𝔅^{×} are represented by scalar matrices in D^{×} and there is a scalar matrix h ∈ B^{×} ∩ D^{×}I_{l}, such that the cyclic group ⟨h⟩ forms a system of represetatives of the subsystems of double cosets that form the supports of the Hecke algebras. If f^{A} is then a function given by a non-trivial intertwining operator Φ ∈ I_{h}(Λ) (see [BK1](4.1.1)), then we map f^{A} to f^{B} by shifting the intertwining operator via the isomorphism of 10.7(ii). The functions f^{A} and f^{B} are invertible and the inverses are determined up to a multiple by any function supported on an entire double coset represented by h^{−1}. Indeed: Since h and h^{−1} normalize the groups J and 𝔅^{×}, the products of the respective inverse elements are supported on J and 𝔅^{×} respectively. Moreover, the product has non-trivial value on 1, hence, a non-zero value, which can be normalized to be the identity. As (f^{A})^{m} and (f^{B})^{m} for m ∈ ℤ are functions with support on a single double coset represented by h^{m}, it easily follows that the ℂ-vector space isomorphism is a Hecke algebra isomorphism with respect to the convolutional product of functions.

In our special case of A = M_{l}(D) we just have the other extreme case that 𝔅 ≠ 𝔅_{M}, where 𝔅 is a minimal principal order of period l. Then τ is the lifting of a tensor power ⊗σ_{0}^{l}, where σ_{0} is the character of the multiplicative group of a finite field k, which is the residual field of the centralizer D′ of D with respect to a pure element β, which can be embedded to D without restriction of generality.

Using the fact that the factor (κ_{𝔄})_{|J} of Λ extends to 𝔅_{M}^{×}J, there is a canonical isomorphism H(𝔅_{M}^{×}, τ_{|𝔅^{×}}) ≃ H(𝔅_{M}^{×}J, Λ), given by tensoring a function by κ_{𝔄} (also see [BK](4)(5.6) and [GSZ](4.1)).

By [SZ] Lemma (4.4) we also have an isomorphy H(B^{×}, τ_{|𝔅^{×}}) ≃ H(l, q^{d′}) that extends the above isomorphism of the right-hand side and just depends on the choice of a function f^{B}, which lives on the double coset 𝔅^{×}h𝔅^{×}, where

h := ( \begin{matrix}0 & I_{l−1} \\ π^{′f} & 0 \end{matrix} ) .

Here q is the cardinality of the residual field 𝔨_{E}, where E = F [β] and d′ is the index of D′. We assume that 𝔅 is in standard normal form as in [G2](0.3). Let π′ be a prime element of D′ and f be the length of the galois orbit of χ under the action of Gal(k/k_{E}). To see the isomorphism on the left-hand side, we have to discuss the function f^{A} corrsponding to f^{B} by 10.7(ii) with support JhJ. Since the argumentation is the same as in the proof of [GSZ](4.4) and [BK1](5.6.6), respectively, we may reduce this question to an equality of supports. In detail, we have to show for arbitrary x from the support that the intersections of the triple classes JxJhJ and JxJh^{−1}J with the support of the Hecke algebra H(A^{×}, Λ) are just JxhJ and Jxh^{−1}J, respectively.

If we choose orders 𝔅, 𝔅_{M}, 𝔄 in simultaneous normal form, then for the corresponding minimal order 𝔄_{m} in normal form we have 𝔄_{m}^{×} ⊃ J. Note that 𝔄_{m} is not assumed to be a continuation of 𝔅. Now it is obvious that it is sufficient to show that there is exactly one double coset from 𝔄_{m}^{×}\B^{×}/𝔄_{m}^{×} that appears in the Bruhat decomposition of the triple class 𝔄_{m}^{×}x𝔄_{m}^{×}h𝔄_{m}^{×} resp. 𝔄_{m}^{×}x𝔄_{m}^{×}h^{−1}𝔄_{m}^{×}, which has a non-empty intersection with the support of the Hecke algebra, namely, 𝔄_{m}^{×}xh𝔄_{m}^{×} and 𝔄_{m}^{×}xh^{−1}𝔄_{m}^{×}, respectively.

Indeed: Let ν_{D}(π′) =: k; then, if we multiply the support of the Hecke algebra from both sides by 𝔄_{m}^{×}, the resulting double cosets are represented by W̃_{A_{l,f·k}} in the notation of [GSZ] and we may apply Lemma [GSZ](2.7) with m = r = l. It is easy to see that no representative can get lost, because the triple classes become larger and the system of representatives is respected.

Therefore, we obtain the desired result by the special structure of the possible choices of simultaneous normal forms for 𝔅, 𝔅_{M} and 𝔄 in case of A = M_{l}(D) and the result can be reduced to the level-0-case.

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( 12.8 ) Corollary Let A = M_{l}(D), l a prime number and Λ be a simple type in the sense of Definition 10.2. If 𝔅 ≠ 𝔅_{M}, then τ is lifted from a tensor power ⊗σ_{0}^{l}, where σ_{0} is the character of the multiplicative group of a finite field. Then Λ is a covering type in the sense of [BK4] of a single Bernstein component with representations with supercuspidal support on a Levi group of the form T (D) := ×D^{l}.

Let us be more precise. Let Λ = (κ_{𝔄})_{|J} ⊗ μ ⊗ τ and the fundamental character θ that generates κ_{𝔄}, extended from C(β, 𝔄), where β is without restriction of the generality embedded diagonal into the group T (D) and the orders 𝔅, 𝔅_{M} and 𝔄 are given in simultaneous normal form. Let J_{D}^{1} := J^{1}(β, 𝔬_{D}) and J_{D} := C_{𝔬_{D}}(β) + J^{1}(β, 𝔬_{D}), then J admit a decomposition J = N_{−}(J_{D}^{1})T (J_{D}^{×})N(J_{D}), where N_{−} are the lower unipotent triangular matrices, N are the upper unipotent triangular matrices and T is the above Levi in the stated domain of coefficients. Furthermore, Λ_{|T (J_{D}^{×})} is a tensor power of a maximal simple type Λ_{D} = κ_{𝔬_{D}} ⊗ μ ⊗ σ_{0}, where we restrict μ to a diagonal component by letting the others trivial. The factor κ_{𝔬_{D}} corresponds to a fundamental character θ_{D}, which is, together with θ, a common extended value of a P S-character (see [G2](6.6)). The supercuspidal support of each irreducible subquotient of each representation generated by its Λ-isotypic component is determined up to an unramified twist by a tensor power ⊗Λ_{D}^{l} (see [BK4]).

\underline{Proof:}

The assertion of the corollary already contains some hints for the proof. According to [BK4](6)(8) we just have to find a totally (T (D)N(D), J)-positive element and an invertible function of the Hecke algebra H(A^{×}, Λ) supported on the corresponding double coset.

Indeed, the element t := diag(π_{F}^{l} , π_{F}^{l−1}, …, π_{F}) is totally (T (D)N(D)), J)-positive in the sense of [BK4](6.16) and because t is in B^{×} := C_{A}(β) and intertwines the representation τ, which is inflated from the diagonal, t is in the support of the Hecke algebra H(A^{×}, Λ). There exists a non-trivial function f_{t} with support JtJ. Finally, by reduction to the level-0-case and using 12.7, the statement [GSZ](4.6) applies also in our case and f_{t} must be invertible. This completes the proof.

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( 12.9 ) Corollary Let A = M_{l}(D) (l a prime number) and [𝔄, n, 0, β] a simple stratum such that 𝔅 = 𝔅_{M} := 𝔄 ∩ B is a maximal order in B := C_{A}(β) = M_{l}(D′), where D′ is a division algebra of the same Brauer type as B. If 𝔨_{D′} = 𝔽_{2}, then the associated representation κ_{𝔄} from (8.2) is intertwined by the whole of B^{×} and the type representations from (10.2) are maximal types fulfilling the conclusions of (11.1).

\underline{Proof:}

In our special case we immediatly see that 𝔨_{D′} = 𝔨_{E} = 𝔨_{F} = 𝔽_{2} is the field with two elements and, in particular, F [β] = E/F must be totally ramified and B must be split. Consequently the continuations of 𝔅 by 𝔄 and those of 𝔅_{m} by 𝔄_{m} are unramified and 7.3 applies. By the same argument, we may deduce in the manner of the proof of [BK1](5.1.19) that η_{𝔄,𝔅_{m}} must be intertwined by the whole of B^{×}.

If l ≠ 2, by 9.1 each extension of η_{𝔄} to J must be also an extension of η_{𝔄,𝔅_{m}}. Moreover, another extension of η_{𝔄} to J as κ_{𝔄} must be a twist by a character of J trivial on U^{1}(𝔅_{m})J^{1}. Using the arguments of the proof of [BK1](5.2.4), we may show that the latter character factorizes through the determinant map of the finite linear group 𝔅^{×}/(1 + 𝔔) and is lifted by the isomorphy 𝔅^{×}/(1 + 𝔔) ≃ J/J^{1}(β, 𝔄) to J. But, as the elements of 𝔅^{×}/(1 + 𝔔) ≃ Gl_{l}(𝔽_{2}) have all determinant 1, there is no proper twist of κ_{𝔄}.

If l = 2 and J(β, 𝔄)/J^{1}(β, 𝔄) ≃ Gl_{2}(𝔽_{2}) ≃ S_{3}, by 9.1 we have exactly two extensions of η_{𝔄} to J(β, 𝔄), which are determined uniquely by their restrictions to U^{1}(𝔅_{m})J^{1}(β, 𝔄). By definition our extension κ_{𝔄} is now determined by η_{𝔄,𝔅_{m}} on that subgroup. The other extension is given by a twist with the sign character of S_{3}. However, as the restriction of the sign character to a transposition group corresponding to a group generated by an upper triangular unipotent matrix in Gl_{2}(𝔽_{2}) is intertwined by a matrix of the form diag(π_{E}^{−1}, 1) with the trivial character, again, such a twist cannot lead to an extension as desired.

Therefore, the only possible extension of η_{𝔄} to J that is intertwined by the whole of B^{×} must be κ_{𝔄} and is an extension of η_{𝔄,𝔅_{m}}. In our case this easily implies that the “Weil representation” W must be the identity and we are left to show that θ_{𝔄}^{⋆} is intertwined by the whole of B^{×}. We leave this last fact to the reader or refer to [VS], where it is proved that a “β-extension” always exists.

□

( 12.10 ) Corollary In case A = M_{l}(D), where l is a prime number, all hypothetical simple types are also types in the sense of [BK4](4.1) with Bernstein components containing irreducible essentially square integrable representations.

\underline{Proof:}

The Corollaries 12.1, 12.2, 12.9 and 12.8 cover all cases.

□

We may generalize Lemma [BK1](6.2.5) without a big change if we replace the determinant mapping by the reduced norm mapping and reformulate the modifications of invariants in the calculation within the proof. We obtain:

( 12.11 ) Lemma Let A be an arbitrary central simple algebra over F and Π be an irreducible supercuspidal representation containing the type Λ and fulfilling the assumptions of Proposition 11.1. If the tensor factor κ_{𝔄} is constructed from a simple character attached to a simple stratum [𝔄, n, 0, β], then we may determine the ramification index e(F [β]/F) from the set of unramified characters χ satisfying Π ≃ Π ⊗ χ ∘ Nrd_{A/F}. More precisely, these characters are exactly those of order dividing \frac{N}{e(F [β]/F)}, where N is the reduced degree of A.

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( 12.12 ) Proposition Let A = M_{l}(D), where l is a prime number, be a tame algebra (see Introduction or 1.15). Let Π be an irreducible essentially square integrable representation of Gl_{l}(D) of a minimal Artin conductor in its twist class, containing the types Λ = λ ⊗ μ and Λ′ = λ′ ⊗ μ′, then, without restriction of generality, μ ≡ μ′ ≡ 1, λ = κ_{𝔄} ⊗ τ and λ′ = κ_{𝔄} ⊗ τ′ have a common factor that comes from the same fundamental character, and finally the respective second factors τ and τ′ are liftings of the tensor powers ⊗σ_{0}^{k} and ⊗(σ′_{0})^{k}, respectively, with k = 1 or k = l, where σ_{0} and σ′_{0} are Galois conjugated.

\underline{Proof:}

Indeed: A maximal type can only appear in a supercuspidal representation. But if Π is a discrete series representation, then it has a supercuspidal support and the associated type is the covering type for a supercuspidal representation of the Levi subgroup T (D) as in 12.8. Therefore, we may apply Lemma 12.11 either to the type Λ or to the type Λ_{D} of a diagonal component as in 12.8.

Hence, we can recover the invariant e(F [β]/F) and, by the choice of the period above (10.2), the invariant of the principal order 𝔄 from the representation Π. Therefore, the simple characters θ and θ′ corresponding to the tensor factors of Λ and Λ′, respectively, must be from a common fundamental character set C(𝔄). Also the underlying simple character must be from underlying simple strata [𝔄, n, 0, β] and [𝔄, n, 0, β′], where n can be calculated from the Artin conductor of Π as calculated in [Z6] and [BF] as s(𝔄)n (s(𝔄) the invariant of 𝔄). As the characters θ_{|H^{1}(β,𝔄)} and θ′_{|H^{1}(β′,𝔄)} appear in the same irreducible representation Π, they must intertwine. Consequently, by 1.15, they are conjugated in the normalizer of 𝔄.

We reduce to θ_{|H^{1}} = θ′_{|H^{1}}, where H^{1}(β, 𝔄) = H^{1}(β′, 𝔄) =: H^{1}, and J_{A} = 𝔅_{M}^{×}J^{1}(β, 𝔄) = 𝔅_{M}^{×}J^{1}(β′, 𝔄). Consequently, an intertwining element must be from J_{A}B^{×}J_{A} and J_{A} normalizes θ_{|H^{1}}, we may choose it as a representative of a coset in J_{A}\B^{×}/J_{A}. Fixing an isomorphism B^{×} ≃ Gl_{m′}(D′), we may choose the intertwining element as a monomial matrix in D^{′×}. As it normalizes the set of diagonal matrices ×(𝔬_{D′}^{×})^{m′}, the defining characters λ_{β} and λ_{β′} of θ and θ′, respectively, must coincide because the reduced norm mapping from D′ to F [β] maps 𝔬_{D′}^{×} surjective to 𝔬_{F [β]}^{×} = 𝔬_{F [β′]}^{×}, because we assume that each representative of orbits in Ĉ(𝔄) is extended from the same field F [β] and also by the argument in [BF](1.4.5). We obtain θ = θ′.

The types λ and λ′ have the same factor (κ_{𝔄})_{|J} and again, we may decide from the fact that the representation Π being supercuspidal or not whether the order 𝔅 is maximal or minimal. Applying 10.7, we deduce that the factors τ and τ′ must intertwine in B^{×}. The assertion on σ_{0} and σ′_{0} can be deduced as in the prove of 11.1(ii).

Concerning the assertion on the minimal conductor, we must follow a part of the argument of the proof of 12.14 below. This is possible, because we avoid a cyclic argumentation. We use the abstract matching theorem [BDKV] and the fact that we already have greater knowledge on the side of the division algebra D.

We first show in the general case that the representations containing the types λ (i.e., a type Λ with μ ≡ 1) must have a minimal Artin conductor in its twist class with arbitrary twist. So let Π be chosen in such a way that they contain a λ and assume that Π ⊗ μ̄ has a minimal Artin conductor in its twist class. If [𝔄, n, 0, β] is the stratum corresponding to κ_{𝔄}, then Π⊗μ̄ contains the character (ψ_{β}^{×}μ̄)_{|U^{n}(𝔄)}. If μ̄ is trivial on U^{n+1}(𝔄), then there exists α ∈ F such that μ̄_{|U^{n}(𝔄)} = ψ_{α}^{×}. The representation Π ⊗ μ̄ contains the pure stratum [𝔄, n, n − 1, β + α]. Since the stratum [𝔄, n, n − 1, β] is equivalent to a simple stratum [𝔄, n, n−1, γ] with F [γ] ≠ F, by [BK1](1.4.14)(1.4.15)(eventual applied to a canonical splitting) also the stratum [𝔄, n, n − 1, γ + α] must be simple and equivalent to [𝔄, n, n − 1, β + α]. Then in particular, ψ_{β}^{×}μ̄ is on U^{n}(𝔄) a non-trivial character, which is trivial on U^{n+1}(𝔄) and given by ψ_{γ+α}^{×}. The latter character must be contained in an irreducible representation of K_{𝔄}, which, due to γ + α ∈ K_{𝔄}, is non-degenerate in the sense of [BF](2.2.7)(2.2.8). By [BF](3.3.8)(iv) and the remark [BF](3.4)(C) or [Z6] the Artin conductor of Π ⊗ μ̄ is equal to the Artin conductor of Π, which has the value s(𝔄) · n, where s(𝔄) is the invariant of 𝔄. If μ̄ is now non-trivial on U^{n+1}(𝔄), it is possible to show in the same way that Π ⊗ μ̄ contains a minimal stratum [𝔄, n′, n′ − 1, α] with α ∈ F and n′ > n. However, then the Artin conductor s(𝔄)n′ of Π ⊗ μ̄ is greater than the Artin conductor of Π. As, the latter case cannot appear by assumption, Π was already minimal in its twist class.

As far as the division algebra case is concerned, also the statement with converse implication is valid. By [Z2] resp. [Z7] each irreducible representation of D^{×} can be induced by an extension of a type representation with underlying minimal stratum [𝔬_{D}, n, n − 1, γ]. Here, type representations like ours appear, i.e., F [γ] ≠ F or γ ∈ F. In the latter case the kernel of the representation can be enlarged by a character twist and because the induction commutes with character twist, we obtain the twisted representation by induction. But now the Artin conductor is given according to [Z6] resp.

[BF] by the formula −N \frac{ν_{F} (β)}{e(F [β]/F)} = n, i.e., the bigger the kernel of the type the smaller is the Artin conductor. If we now consider the corresponding types, then, we may assume inductively that they are from the first kind, where all twists make the Artin conductor smaller. Therefore, we obtain all representations of minimal conductor on the division algebra side, if we assume the first case, i.e., of type representations λ_{D} as constructed in this work. We will show later, within the proof of theorem 12.14, that the mapping at the beginning of 10 is, on the set of representations of minimal conductor, a permutation of the abstract matching theorem [BDKV], which is consistent with the character twist and in particular preserves Artin conductor and the central character. If we now assume Π to be of a minimal conductor in its twist class, then we may assume μ ≡ μ′ ≡ 1.

We can use Lemma 10.7 to reduce the final assertion to the level-0-case in the algebra B := C_{A}(β). The last assertion now follows from [GSZ]((5.2) resp. Theorem 1).

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( 12.13 ) Proposition Let Λ be a type representation for A := M_{l}(D), where l is a prime number. Let ωΛ be an extension of Λ to the group F^{×}J. Then there exists a finite number of essentially square integrable representations with a central character ω containing ωΛ. If Λ = λ is a type representation corresponding to a type representation λ for a division algebra D of the same reduced degree N as A by the mapping described before 10.2 and supposed that the parameter χ has a certain Galois length f, then the last finite number is exactly \frac{N}{f(F [β]/F)f}, where β is a pure element that underlies the tensor factor of λ = (κ_{𝔄})_{|J} ⊗ χ as described above 10.2.

\underline{Proof:}

In the level-0-case we set e(F [β]/F) = 1 and refer to [SZ]. So let Λ be a type representation with non-trivial tensor factor κ and assume that Λ is a maximal type first.

If the centralizer B = C_{A}(β) is a division algebra, then χ can also be considered to be a character of J and being of the same Galois length f. If π is a prime element of B, then the stabilizer of Λ is given by ⟨π^{f}⟩J and each extension to this stabilizer induces an irreducible supercuspidal representation of A^{×}. We obtain (⟨π^{f}⟩J : F^{×}J) non-isomorphic representations containing ωΛ. As π^{\frac{N}{[F [β]:F]}} is up to a unit the prime element of F [β], the assertion follows in this case.

If the centralizer is B = C_{A}(β) = M_{l}(D′), then D′ has index d′ = \frac{N}{l[F [β]:F]}. The representation σ_{0} of 𝔅^{×} corresponding to χ has Galois length f̄:= ⟨f, d′⟩ by [SZ] and if π is a prime element of D′, the assertion follows as in the first case by calculating the group index of F^{×}J in the stabilizer ⟨πf̄I_{l}⟩J of Λ. As π^{d′} is up to a unit a prime element of F [β], we obtain the result by transforming the expression \frac{d′}{f̄} e(F [β]/F). Here we note that l is prime to \frac{N}{[F [β]:F]f} by our assumption.

Indeed: If l is a divisor of \frac{N}{[F [β]:F]f}, then Λ is a covering type and f is a divisor of the index d′ of D′. The character χ factors via the norm through a field 𝔨 of degree f over the residual field 𝔨_{F [β]}. The factor σ_{0} is now the uniquely determined character factoring through 𝔨, where 𝔨 is a subfield of 𝔨_{D′} by assumption. If τ is the tensor factor of Λ, inflated from a tensor power ⊗σ_{0}^{l} from T (D) ∩ J/T (D) ∩ J^{1}(β_{A}, 𝔄) to J and Ωτ_{|𝔅^{×}} is an extension of τ_{|𝔅^{×}} to F [β]^{×}𝔅^{×}, then, by [SZ], there are exactly \frac{N}{[F [β]:F]f} irreducible representations of the discrete series of B^{×} containing Ωτ_{|𝔅^{×}} and, thus, the above number contains ωτ_{|𝔅^{×}}. We use the Hecke algebra isomorphism 12.7, to show the corresponding assertion for our types Λ.

Since the arguments of proposition [BK1](7.7.4) and of Corollary [BK1](7.7.5) are founded on the fact that a representation Π containing a simple type Λ is generated by its Λ-isotypic component and uses a Morita equivalence, which was shown more generally in [BK4](2.12), the results of 12.8 and [BK4](4.2)(4.3) can be easily generalized to our case. If we fix a J invariant scalar product ⟨…, …⟩_{W} on the representation space of Λ, then, for ϕ, ψ ∈ H(A^{×}, ωΛ), a scalar product is given by h_{ωΛ}(ϕ, ψ) := tr_{W} (ϕ ⋆ ψ̄(1)), where ψ̄(g) is as usual the adjoint element of ψ(g^{−1}) with respect to the form ⟨…, …⟩_{W}. By completion with respect to this scalar product on the algebra, we obtain a completed algebra H^{2}(A^{×}, ωΛ) that inherits the convolutional product and the unit element. The consequences of the considerations in [BK1] are now that a representation Π containing Λ is square integrable if and only if it is Morita equivalent to a H(A^{×}, ωΛ)-module that can be embedded into H^{2}(A^{×}, ωΛ). Within this proof we have to reduce up to unramified character twist to the case that the central character ω of Π is unitary.

Similarly, we define the product h_{ωτ} on the Hecke algebra H(B^{×}, ωτ_{|𝔅^{×}}), complete it to H^{2}(B^{×}, ωτ_{|𝔅^{×}}) and find a corresponding assertion also for the irreducible level-0-representations of B^{×} containing ωτ. We want to show that the number of square integrable irreducible representations of A^{×} containing ωΛ is as large as the number of irreducible square integrable representations of B^{×} containing τ. By the above arguments it is enough to show that the algebras H^{2}(A^{×}, ωΛ) and H^{2}(B^{×}, ωτ) are isometrically isomorphic, i.e., that there is a Hecke algebra isomorphism that respects the forms h_{ωΛ} and h_{ωτ}.

Now the algebras H(F^{×}𝔅_{M}^{×}, ωτ_{|𝔅^{×}}) ≃ H(F^{×}𝔅_{M}^{×}J, ωλ) are isomorphic by the same argument as in 12.7, using the fact that the factor ω(κ_{𝔄})_{|J} of ωΛ extends to F^{×}𝔅_{M}^{×}J. The isomorphism preserves support of functions and works in the same manner as 12.7, where we use that ω is trivial on π_{F}, i.e., Λ, τ_{|𝔅^{×}} are extended trivially by ω to the infinite cyclic group generated by π_{F}. Therefore, the isomorphy only depends on the functions f^{A} and f^{B}, respectively, or more exactly on their images under a suitable projector. Indeed, the Hecke algebras of the extended representation by the central character ω are quotients of the Hecke algebras in 12.7. For the corresponding projections f_{ω}^{A} and f_{ω}^{B} we have the same relations e_{ωΛ} − (f_{ω}^{A})^{r(𝔅)d′e(F [β]/F)} = 0, e_{ωτ} − (f_{ω}^{B})^{r(𝔅)d′e(F [β]/F)} = 0 respectively, where r(𝔅) is the period of 𝔅, d′ is the index of B and e_{ωΛ} and e_{ωτ} are the unit elements of H(A^{×}, ωΛ) and H(B^{×}, τ_{|𝔅^{×}}) respectively.

The functions f_{ω}^{A} and f_{ω}^{B} up to a scalar multiple can be described explicitly by f_{ω}^{A}(g) ∼ ∑_{n=−∞}^{+∞} f(π_{F}^{n} · g) resp. f_{ω}^{B}(b) ∼ ∑_{n=−∞}^{+∞} f(π_{F}^{n} · b) and it is clear that the mapping that sends f_{ω}^{A} to f_{ω}^{B} extends the above Hecke algebra isomorphism to an isomorphy H(A^{×}, ωΛ) ≃ H(B^{×}, ωτ_{|𝔅^{×}}) preserving support of functions (see 12.7). On the other hand, the forms h_{ωΛ} and h_{ωτ} are defined by evaluating a product of functions on the unit element and then taking the trace. Therefore, the two forms differ just by a constant factor depending on the dimensions of representations and the choice of measures if applied to functions on the left-hand side and images of isomorphism on the right-hand side (see [BK1] page 260). As the isomorphism preserve forms up to a harmless factor, we may extend it to the completions H^{2}(A^{×}, ωΛ) ≃ H^{2}(B^{×}, ωτ_{|𝔅^{×}}).

Since we know that the number of irreducible essentially square integrable representations of B^{×} containing ωτ_{|𝔅^{×}} is given by the formula of our proposition, the same formula is valid for the number of irreducible square integrable representations of A^{×} containing ωΛ as required.

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( 12.14 ) Theorem Let A = M_{l}(D) be a tame algebra and l a suitable prime number, then, assuming that Conjecture 1.16 holds true, the list 10.2 is a complete list of types for those Bernstein components containing an irreducible essentially square integrable representation. In particular, all irreducible supercuspidal representations of Gl_{l}(D) are induced from a compact modulo center subgroup.

\underline{Proof:}

Again, we want to return to the construction in the beginning of Section 10. Let D be a division algebra of the same index N as A. To a type λ_{D} = κ ⊗ χ for an unramified twist class of irreducible representations of D^{×} of a minimal conductor in its twist class with arbitrary twist, can be assigned via the mapping described at the beginning of Section 10 a type λ for a Bernstein component containing an unramified twistclass of essentially square integrable representations, which is of a minimal conductor in its twist class with arbitrary twist, too. By [BF] and [Z6], respectively, this mapping preserves Artin conductor and commutes with the character twist. Due to 12.12 the essentially square integrable representations containing λ have a minimal Artin conductor in there twist classes. Now we may extend λ_{D} and λ by a central character ω to F^{×}J_{D} and F^{×}J, respectively. By [G2](6.4) 8.2 and [SZ] the restrictions to 𝔬_{F}^{×} lead to the same central linear characters of λ_{D} and λ.

Indeed: As 𝔬_{F}^{×}/U^{1}(𝔬_{F}) is of finite order prime to p and the Weil representation has the determinant character 1 and p-power dimension, the Weil representation must be trivial on this quotient and for the factors κ_{O} and κ_{𝔄} it is enough to compare the fundamental characters. Linkwise the case of J/J^{1}(β, 𝔄) ≃ Gl_{2}(𝔽_{2}) does not cause any problem, because the quotient 𝔬_{F}^{×}/U^{1}(𝔬_{F}) has only one element in this case. The level-0-factor can be compared successfully using [SZ]. By 12.13 the extensions ωλ_{D} and ωλ parametrize the same finite number of essentially square integrable representations containing the same extended simple type.

Finally, from [Z2] and [Z7], we know that these finite packets for varying λ_{D} and ω exhaust, on the division algebra side, all irreducible smooth representations with minimal conductor. The packets are also pairwise disjoint and the mapping explained above leads, assuming 1.16 and using 12.12, to an injection of the irreducible representations of D^{×} of a minimal conductor into the set of all irreducible essentially square integrable representations of a minimal conductor of A^{×}. This injection preserves the central character and the Artin conductor of the parametrized representations. By [Kr], there are only a finite number of field extensions F [β]/F of bounded finite degree and, hence, to a fixed n up to conjugacy, just a finite number of simple strata [𝔄, n, 0, β]. There are only a finite number of fundamental characters to such a fixed stratum; hence, the number of corresponding extensions κ_{𝔄} is finite. Finally, there are only a finite number of level-0-components χ on the division algebra side and, therefore, a finite number of corresponding types ωλ for fixed ω and fixed Artin conductor.

Indeed: As by [Z6] the Artin conductor for representations containing λ are given by the expression s(𝔄)n = −N \frac{ν_{F [β]}(β)}{e(F [β]/F)}, there are only a finite number of possiblities to choose strata [𝔄, n, 0, β] to get Bernstein components containing irreducible essentially square integrable representations with these Artin conductor and if we fix the central character ω, we take just a finite number of representations. At least the corresponding assertion for the division algebra side is clear, but the mapping of the abstract matching theorem [BDKV] is surjective and also preserves the Artin conductor and the central character; hence, maps finite packets to finite packets. Therefore, the mapping of packets given by the above correspondence of extended types must be surjective, too.

The matching map [BDKV] is consistent with the character twist and on the division algebra part we obtain all unramified twist classes of irreducible representations by twist with varying arbitrary μ. Hence, also the types Λ parametrize all Bernstein components containing essentially square integrable representations of A^{×}.

In fact, we have just handled the case of 10.2(i), but from [SZ] we know that the abstract matching theorem preserves the level-0-case 10.2(ii), which, for this case, leads to another proof of completeness of construction independent of [GSZ].

Now the Proposition 11.1 and its proof lead to the last assertion.

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