manuscripta mathematica manuscript No. (will be inserted by the editor)

Shaun Stevens

Double Coset Decompositions and Intertwining

Received: date / Revised version: date

Abstract. Let F be a non-archimedean local field of residual characteristic different from 2. This paper is a first step in the description of the smooth representation theory of a unitary group G ⊂ GL(N, F) via types. We intersect certain double cosets in GL(N, F) with G and hence obtain the intertwining of certain characters ψ_{β}^{−} of open compact subgroups of G, for β ∈ Lie G. In the case when β is elliptic regular “modulo 𝔭_{F} ”, we will then obtain supercuspidal representations of G.

1. Introduction

Let F be a non-archimedean local field of residual characteristic different from 2, with a (possibly trivial) galois involution with fixed field F_{0}. Let V be an N-dimensional F -vector space equipped with a nondegenerate ϵhermitian form h. Put A = End_{F} V and let be the adjoint involution on A associated to h. We then set G̃ = Aut_{F} V ≃ GL(N, F) and let G be the subgroup of G̃ of fixed points of the map σ : g ↦ \overline{g}^{−1}; thus G is a unitary group (symplectic or orthogonal if F = F_{0}) defined over F_{0}.

This paper is a step in the attempt to describe the smooth representation theory of G via the theory of types (see [5]). Such types, in particular those associated to supercuspidal representations, are intrinsically difficult to construct so we seek to “transfer” types from G̃ to G. (Note that the types for G̃ have been constructed in [3], [4].) In particular, we obtain new supercuspidal representations of G, many of which are transferred from types for non-supercuspidal representations of G̃.

The types for G̃ are constructed from pairs (J, θ), where J is a compact open subgroup of G̃ and θ is a character of J. The irreducible representations containing θ are classified by the simple modules over the Hecke algebra H(G̃, θ) and this algebra is “recognizable”: that is, it is isomorphic (in a support-preserving way) to a Hecke algebra of standard type in a smaller reductive group. In particular, if the support of the Hecke algebra (which is just the intertwining of θ) is compact modulo centre then any irreducible representation containing θ is supercuspidal.

Shaun Stevens: Mathematical Institute, St. Giles’ 24-29, Oxford OX1 3LB, United Kingdom. e-mail: ginnyshaun@bigfoot.com Mathematics Subject Classification (1991): 22E50

This philosophy works well for G̃ and also for SL(N) (see [6], [7] and the work of Goldberg and Roche [8], in which a complete set of types for SL(N) is obtained) and this paper is a first step in extending this to the classical group G, in a manner which is compatible with the situation for G̃: that is, we show that the intertwining transfers.

The first result of this paper (2.3) states that, if U is a pro-p subgroup of G̃, H is a subgroup of G̃ such that, for all h ∈ H, we have UhU ∩ H = (U ∩ H)h(U ∩ H), and U, H are both fixed by the involution σ, we have

UHU ∩ G = U ∩ G · H ∩ G · U ∩ G.

We may now apply this to computing intertwining of characters for G.

We consider skew strata, that is quadruples [Λ, n, r, β], where Λ is an 𝔬_{F} -lattice sequence in V fixed by the duality induced by h, n > r ≥ 0 are integers and β ∈ Lie G ∩ 𝔞_{−n}(Λ), where 𝔞_{−n}(Λ) is the level −n filtration lattice of A induced by Λ (see §3). If r ≥ [\frac{n}{2}] then a skew stratum corresponds to a character ψ_{β} of U_{r} = 1 + 𝔞_{r}(Λ) and to the character ψ_{β}^{−} of P_{r} = U_{r} ∩ G obtained by restriction.

Now we consider skew strata which are simple, that is where β satisfies certain arithmetical properties relative to Λ (see §4). When the 𝔬_{F} -lattice sequence Λ is strict, [3] (1.5.8) gives the formal intertwining of a (skew) simple stratum in the form UHU and we generalize this (4.5) to arbitrary lattice sequences. If r ≥ [\frac{n}{2}], this is precisely the intertwining of ψ_{β} and the intertwining of ψ_{β}^{−} is UHU ∩ G. The groups U, H satisfy the conditions of (2.3) so we obtain the intertwining of ψ_{β}^{−} as U ∩ G · H ∩ G · U ∩ G.

We now combine the notion of a simple skew stratum with that of a split stratum from [4] (3.6) to define semisimple strata. These are orthogonal direct sums of simple skew strata which are “sufficiently different” (see (4.9)) and generalize the characters considered in [10] (2.17). Then we may adapt the intertwining result of [4] (3.7) to calculate the formal intertwining of such a semisimple stratum in the same form UHU. Once again, if r ≥ [\frac{n}{2}], the stratum corresponds to a character ψ_{β}^{−} of P_{r} and its intertwining is UHU ∩ G, which decomposes as in the simple case.

Finally, suppose that [Λ, n, r, β] is a semisimple stratum with r ≥ [\frac{n}{2}] and F [β] is maximal in A (of degree N). Then the intertwining of the character ψ_{β}^{−} is compact and contained in P = 𝔞_{0} ∩ G, where 𝔞_{0} := 𝔞_{0}(Λ) (see § 3). In particular, if ρ is an irreducible representation of P containing the character ψ_{β}^{−}, then the (compact) induced representation Ind _{P}^{G}ρ is irreducible and supercuspidal. If r < n − 1 and F [β] is wildly ramified then this is a new supercuspidal representation. The construction of the representation ρ (in the more general setting of simple characters) is described in [14] and, again, is more general than that in [10].

We now give a brief summary of the contents of each chapter. In §2 we describe the double coset decomposition, in the generality of an l-group acting on some group G, l ≠ p. In §3 we introduce the notations for our padic groups and recall the definition of a lattice sequence and its associated objects. In §4 we recall the notion of a simple stratum from [3], [4], define semisimple strata and calculate their intertwining, and see that, in the case described above, we obtain supercuspidal representations of G. Finally, in §5, we give the proof of (4.5).

The main results of this paper are based on part of my PhD thesis. I would like to thank my supervisor, Colin Bushnell, for starting me on the project and for his support and encouragement. I would also like to thank Guy Henniart, for suggesting a generalization of the main result and its application to semisimple strata, and Paul Broussous, for his patient explanations during my doctorate and for his corrections to earlier versions of this paper.

2. Double cosets

This section is written in much more generality than the remainder of this paper. Indeed, we begin with a group G and a group Γ of automorphisms of G; we denote the fixed points G^{Γ}.

Lemma 2.1. Let G be a group and Γ a group of automorphisms of G. Let U be a subgroup of G fixed by Γ and suppose that, for all g ∈ G^{Γ}, we have

H^{1}(Γ, gUg^{−1} ∩ U) = 1.

Then, for g ∈ G^{Γ}, we have

(UgU)^{Γ} = U^{Γ} gU^{Γ} .

Proof. We have a Γ -equivariant exact sequence

1 → gUg^{−1} ∩ U →^{δ} U × U →^{π} UgU → 1,

where δ(u) = (u, g^{−1}ug) and π(u, v) = ugv^{−1}. By [13] Prop. 36, this gives rise to a long exact sequence of pointed sets

1 → (gUg^{−1} ∩ U)^{Γ} → (U × U)^{Γ} →^{π} (UgU)^{Γ}
→ H^{1}(Γ, gUg^{−1} ∩ U) → H^{1}(Γ, U × U).

We have (U × U)^{Γ} = U^{Γ} × U^{Γ} and H^{1}(Γ, gUg^{−1} ∩ U) = 1. Hence the map π is surjective and (UgU)^{Γ} = U^{Γ} gU^{Γ}. ⊓ ⊔

Lemma 2.2. Let G be a group and Γ an l-group of automorphisms of G, l prime. Let U be a pro-p subgroup of G fixed by Γ, p ≠ l prime. Let g ∈ G. Then UgU is fixed by Γ if and only if (UgU)^{Γ} ≠ ∅.

Proof. The implication “if” is clear so suppose UgU is fixed by Γ. We can decompose UgU as a disjoint union of single cosets g_{i}U and Γ permutes the cosets in this decomposition. The number of such single cosets is the index of gUg^{−1} ∩ U in U, which is a power of p. In particular, since l ≠ p, there is a fixed coset for the action of Γ, say kU. For all γ ∈ Γ, we have k^{γ} ∈ kU so γ ↦ k^{γ}k^{−1} defines a 1-cocycle of Γ in U. However, H^{1}(Γ, U) = 1 so there exists u ∈ U such that k^{γ}k^{−1} = u^{γ}u^{−1} for all γ ∈ Γ. Then ku^{−1} ∈ (UgU)^{Γ}. ⊓ ⊔

Theorem 2.3. Let G be a group and Γ an l-group of automorphisms of G, l prime. Let U be a pro-p subgroup of G fixed by Γ, p ≠ l prime. Let H be a subgroup of G fixed by Γ such that, for all h ∈ H,

UhU ∩ H = (U ∩ H)h(U ∩ H).  (2.4)

Then we have

(UHU)^{Γ} = U^{Γ} H^{Γ} U^{Γ} .

We remark that condition (2.4) will be satisfied if H is the fixed point subgroup in G of some group of automorphisms of G satisfying the conditions of Lemma 2.1.

Proof. Suppose that h ∈ H is such that (UhU)^{Γ} ≠ ∅; hence UhU is fixed by Γ. In particular, for all γ ∈ Γ, h^{γ} ∈ UhU ∩ H = (U ∩ H)h(U ∩ H). Thus (U ∩ H)h(U ∩ H) is fixed by Γ so, by (2.2), there exists h′ ∈ ((U ∩ H)h(U ∩ H))^{Γ}. In particular, we have UhU = Uh′U and h′ ∈ H^{Γ}. Hence

(UHU)^{Γ} = (UH^{Γ} U)^{Γ}
= U^{Γ} H^{Γ} U^{Γ}  by (2.1),

since hUh^{−1} ∩ U is a pro-p subgroup for all h ∈ H^{Γ}. ⊓ ⊔

Note that the above results will of course remain valid if Γ is a solvable group of order coprime to p.

3. Preliminaries

Let F be a non-archimedean local field, 𝔬_{F} its ring of integers, 𝔭_{F} the maximal ideal of 𝔬_{F}, k_{F} = 𝔬_{F} /𝔭_{F} the residue class field and q_{F} = p^{f_{F}} = card k_{F}. We assume throughout that the residual characteristic p is not 2.

Suppose that F comes equipped with a galois involution , with fixed field F_{0}; we allow the possibility that F = F_{0}. Then we denote by 𝔬_{0}, 𝔭_{0}, k_{0}, q_{0} = p^{f_{0}} the objects for F_{0} analogous to those above for F. We also fix a uniformizer ϖ_{F} of F such that \overline{ϖ_{F}} = ±ϖ_{F} (the sign depending on whether F/F_{0} is ramified or not).

Let ψ_{0} be a character of the additive group of F_{0}, with conductor 𝔭_{0}. Then we put ψ_{F} = ψ_{0}∘tr_{F/F_{o}}; since p ≠ 2, F/F_{0} is at worst tamely ramified so ψ_{F} is a character of the additive group of F with conductor 𝔭_{F}.

Let V be an N-dimensional F -vector space and put A = End_{F} (V), G̃ = Aut_{F} (V). Let ψ_{A} be the character of A given by ψ_{A} = ψ_{F} ∘ tr_{A/F}. Let h be a nondegenerate ϵ-hermitian form on V and let be the adjoint involution on A associated to h so that h(av, w) = h(v, \overline{a}w) for all v, w ∈ V, a ∈ A; this extends the involution on F (for F embedded diagonally in A). We also denote by σ the involution on G̃ given by x ↦ \overline{x}^{−1} and by Σ the subgroup of Aut G̃ consisting of σ and the identity. Note that the action of σ on Lie G̃ ≃ A, via the differential, is given by x ↦ −\overline{x}.

We put G = G̃^{Σ} = {g ∈ G̃ : h(gv, gw) = h(v, w) for all v, w ∈ V }, a unitary group over F_{0} (possibly symplectic or orthogonal). We also put A_{−} = \{x ∈ A : x+\overline{x} = 0\} ≃ Lie G and A_{+} = \{x ∈ A : x = \overline{x}\}; since F is not of characteristic 2 we have A = A_{−} ⊕A_{+} and, moreover, this decomposition is orthogonal with respect to the pairing induced by tr_{0} = tr_{F/F_{0}} ∘ tr_{A/F} since, for x ∈ A_{−}, y ∈ A_{+}, we have

tr_{0}(xy) = \overline{tr_{0}(xy)} = tr_{0}(\overline{xy}) = tr_{0}(−yx) = −tr_{0}(xy).

For S any subset of A, we write S_{−} (or sometimes S^{−}) for S ∩ A_{−} and S_{+} for S ∩ A_{+}. If S is an 𝔬_{F} -lattice fixed by the involution then we have S = S_{−} ⊕ S_{+}, since the residual characteristic of F is not 2.

Recall from [4] (2.1) that an 𝔬_{F} -lattice sequence in V is a function Λ from ℤ to the set of 𝔬_{F} -lattices in V such that

(i) n ≥ m implies Λ(n) ⊂ Λ(m); (ii) there exists a positive integer e = e(Λ) (the 𝔬_{F} -period of Λ) such that

Λ(n + e) = 𝔭_{F} Λ(n), for all n ∈ ℤ.

A lattice sequence Λ gives rise to a filtration on A by

𝔞_{n} = 𝔞_{n}(Λ) = {x ∈ A : xΛ(m) ⊂ Λ(m + n), m ∈ ℤ}, n ∈ ℤ.

This then gives rise to a “valuation” ν_{Λ} on A by

ν_{Λ}(x) = sup{n ∈ ℤ : x ∈ 𝔞_{n}}, for x ∈ A,

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

From a lattice sequence Λ we obtain a compact open subgroup U = U(Λ) = 𝔞_{0}(Λ)^{×} of G̃, equipped with a filtration

U_{n} = U_{n}(Λ) = 1 + 𝔞_{n}(Λ), n ∈ ℤ, n > 0.

This is also the Moy-Prasad filtration associated to a certain rational point in the building of GL(N, F) (see [12], [2]). We define the normalizer of the filtration to be

𝔎(Λ) = ⋂_{r≥0} NG̃(U_{r}),

where NG̃ denotes normalizer.

For L an 𝔬_{F} -lattice in V, we define the dual lattice L^{#} by

L^{#} = {v ∈ V : h(v, L) ⊂ 𝔭_{F} }.

Then L^{#} can be identified with Hom_{𝔬_{F}} (L, 𝔭_{F}) by the non-degeneracy of h and we have L^{##} = L. For Λ an 𝔬_{F} -lattice sequence, define the dual sequence Λ^{#} by

Λ^{#}(n) = Λ(−n)^{#}, n ∈ ℤ.

We say that Λ is self-dual if there exists d ∈ ℤ such that Λ^{#}(n) = Λ(n + d), for all n ∈ ℤ. In this case, the filtration 𝔞_{n} on A induced by Λ satisfies \overline{𝔞}_{n} = 𝔞_{n}, for n ∈ ℤ. In particular, the groups U, U_{n}, n ≥ 1, are fixed by Σ and we put P = U^{Σ}, a compact open subgroup of G, and P_{n} = U_{n}^{Σ}, for n ≥ 1, a filtration on P. Further, by [11] (2.13)(c) we have a bijection 𝔞_{n}^{−} → P_{n} given by the Cayley map x ↦ (1 + \frac{x}{2})(1 − \frac{x}{2})^{−1}.

Lemma 3.1. Let Λ be an 𝔬_{F} -lattice sequence in V and let m, n ∈ ℤ satisfy 2n ≥ m > n ≥ 1.

(i) The map x ↦ 1 + x induces an isomorphism of abelian groups 𝔞_{n}/𝔞_{m} ˜→U_{n}/U_{m}.

(ii) If Λ is self-dual then the map x ↦ 1 + x induces an isomorphism of abelian groups 𝔞_{n}^{−}/𝔞_{m}^{−} ˜→P_{n}/P_{m}.

Let S be an 𝔬_{F} -lattice in A, hence an 𝔬_{0}-lattice in A. We define the 𝔬_{F} -lattice

S^{∗} = {a ∈ A : tr_{0}(aS) ⊂ 𝔭_{0}}
= {a ∈ A : tr_{A/F} (aS) ⊂ 𝔭_{F} },

since F is at worst tamely ramified over F_{0}. If S is also stable under the involution, we can define

(S_{−})^{∗} = {a ∈ A_{−} : tr_{0}(aS_{−}) ⊂ 𝔭_{0}} = (S^{∗})_{−}

since the direct sum S = S_{−} ⊕ S_{+} is orthogonal with respect to tr_{0}.

We recall from [4] (2.10) that, if Λ is an 𝔬_{F} -lattice sequence in V with associated filtration 𝔞_{n}, then we have 𝔞_{n}^{∗} = 𝔞_{1−n}.

Let “hat” ˆ denote the Pontrjagin dual. Then we have the following:

Lemma 3.2. Let Λ be an 𝔬_{F} -lattice sequence in V and let m, n ∈ ℤ satisfy 2n ≥ m > n ≥ 1.

(i) There is a 𝔎(Λ)-equivariant isomorphism of abelian groups

𝔞_{1−m}/𝔞_{1−n} ˜→ (U_{n}/U_{m})ˆ
b + 𝔞_{1−n} ↦ ψ_{b}

where ψ_{b}(u) = ψ_{F} (tr_{A/F} (b(u − 1))), for u ∈ U_{n}.

(ii) If Λ is self-dual then there is a P -equivariant isomorphism of abelian groups

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

where ψ_{b}^{−}(p) = ψ_{0}(tr_{0}(b(p − 1))), for p ∈ P_{n}. Moreover, for b ∈ (𝔞_{1−m}^{−}), ψ_{b}^{−} is the restriction to P_{n} of ψ_{b}.

4. Strata and supercuspidal representations

We now give two applications of theorem (2.3) to the intertwining of certain characters involved in representation theory. The language used will be that of [3] and [4], as in the previous section.

4.1. Simple strata

Definition 4.1 (cf. [3] (1.5), [4] (3.1)). A stratum in A is a 4-tuple [Λ, n, r, β] consisting of a lattice sequence Λ in V, n, r ∈ ℤ with r < n, and an element β ∈ 𝔞_{−n}(Λ). We say that two strata [Λ, n, r, β_{i}], i = 1, 2, are equivalent if β_{1} ≡ β_{2} (mod 𝔞_{−r}(Λ)).

Let [Λ, n, r, β] be a stratum in A and suppose that the integers r, n satisfy

n > r ≥ [\frac{n}{2}] ≥ 0,  (4.2)

where [x] is the greatest integer less than or equal to x. By (3.2)(i), an equivalence class of strata [Λ, n, r, β] corresponds to the character ψ_{β} of U_{r+1}(Λ)/U_{n+1}(Λ).

Definition 4.3 ([3] (1.5.5)). Let [Λ, n, r, β] be a stratum in A. It is pure if

(i) the algebra E = F [β] is a field; (ii) Λ is an 𝔬_{E}-lattice sequence; (iii) ν_{Λ}(β) = −n.

If [Λ, n, r, β] is a pure stratum, we put, for k ∈ ℤ,

𝔫_{k}(β, Λ) = {x ∈ 𝔞_{0}(Λ) : βx − xβ ∈ 𝔞_{k}},

an 𝔬_{F} -lattice in A. Then we define k_{0}(β, Λ) to be the least integer k such that 𝔫_{k+1}(β, Λ) is contained in B ∩ 𝔞_{0} + 𝔞_{1}, where B is the A-centralizer of β; we understand that if F [β] = F then k_{0}(β, Λ) = ∞. Note that this is the same definition as in [4] (5.1), though this is not clear a priori (see §5 for further discussion). If F [β] ≠ F then k_{0}(β, Λ) is an integer greater than or equal to −n.

Definition 4.4 ([3] (1.5.5)). A pure stratum [Λ, n, r, β] is called simple if r < −k_{0}(β, Λ).

We now define the formal intertwining of a stratum [Λ, n, r, β] to be

IG̃[Λ, n, r, β] = {x ∈ G̃ : x^{−1}(β + 𝔞_{−r})x ∩ (β + 𝔞_{−r}) ≠ ∅} .

If (4.2) is satisfied, then this is nothing other than the intertwining in G̃ of the character ψ_{β}:

IG̃(ψ_{β}) := {x ∈ G̃ : ψ_{β}(u) = ψ_{β}^{x}(u) for any u ∈ U_{n}^{x} ∩ U_{n}} ,

where U_{n}^{x} = x^{−1}U_{n}x and ψ_{β}^{x}(u) = ψ_{β}(xux^{−1}), for u ∈ U_{n}^{x} (see, e.g. [9] pp.484–485). Then we have the following result, which is a generalization of [3] (1.5.8).

Theorem 4.5. Let [Λ, n, r, β] be a simple stratum in A. Let B denote the A-centralizer of β, 𝔟_{n} = 𝔞_{n} ∩ B, for n ∈ ℤ. Write k = k_{0}(β, Λ) and put 𝔪_{i} = 𝔫_{k+i}(β, Λ) ∩ 𝔞_{i}, for i ≥ 0. Then

IG̃[Λ, n, r, β] = (1 + 𝔪_{−(k+r)})B^{×}(1 + 𝔪_{−(k+r)}).

Essentially, the proof is identical to [3] (1.5.8), once we have proved a few preliminary lemmas. We postpone the proof until §5.

We now consider the situation in our group G.

Definition 4.6. A stratum [Λ, n, r, β] in A is called skew if β + \overline{β} = 0 and Λ is self-dual.

Again, if (4.2) is satisfied then, by (3.2)(ii), an equivalence class of skew strata [Λ, n, r, β] corresponds to the character ψ_{β}^{−} of P_{r+1}(Λ)/P_{n+1}(Λ).

We define the formal intertwining in G of a skew stratum [Λ, n, r, β] to be

I_{G}[Λ, n, r, β] = {x ∈ G : x^{−1}(β + 𝔞_{−r}^{−})x ∩ (β + 𝔞_{−r}^{−}) ≠ ∅}.

If (4.2) is satisfied, then this is precisely the intertwining in G of the character ψ_{β}^{−}.

Theorem 4.7. Let [Λ, n, r, β] be a skew simple stratum in A. Let B denote the A-centralizer of β, 𝔟_{n} = 𝔞_{n} ∩ B, for n ∈ ℤ. Write k = k_{0}(β, Λ) and put 𝔪_{i} = 𝔫_{k+i}(β, Λ) ∩ 𝔞_{i}, for i ≥ 0. Then 𝔪_{i} is fixed by the involution and we define the group Q_{i} = (1 + 𝔪_{i}) ∩ G, for i > 0. Then

I_{G}[Λ, n, r, β] = Q_{−(k+r)}(B ∩ G)Q_{−(k+r)}.

Proof. We write I_{G} for I_{G}[Λ, n, r, β] and IG̃ for IG̃[Λ, n, r, β]. We begin with a lemma which will also prove useful later:

Lemma 4.8. I_{G} = IG̃ ∩ G.

Proof. We clearly have I_{G} ⊂ IG̃∩G. Suppose, on the other hand, x ∈ IG̃∩G; then there exist b_{i} ∈ 𝔞_{−r}, i = 1, 2 such that

x(β + b_{1})x^{−1} = β + b_{2}.

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

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

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

x(β + v_{1})x^{−1} = β + v_{2},

i.e. x ∈ I_{G}. ⊓ ⊔

Returning to the proof of the Theorem, we see that I_{G} = IG̃ ∩ G = (1 + 𝔪_{d})B^{×}(1 + 𝔪_{d}) ∩ G by (4.5), where d = −r − k > 0.

Now 1+𝔪_{d} is a pro-p subgroup of G̃ fixed by Σ and, since 𝔟_{d} ⊂ 𝔪_{d} ⊂ 𝔞_{d} we have, for b ∈ B^{×},

(1 + 𝔟_{d})b(1 + 𝔟_{d}) ⊂ (1 + 𝔪_{d})b(1 + 𝔪_{d}) ∩ B
⊂ (1 + 𝔞_{d})b(1 + 𝔞_{d}) ∩ B = (1 + 𝔟_{d})b(1 + 𝔟_{d}),

by [3] (1.6.1). Hence all the conditions of (2.3) are satisfied and the result follows. ⊓ ⊔

4.2. Semisimple strata

In this section we use the techniques and results of [4] (in particular §3) to generalize the above result to the case where the element β does not generate a field extension over F. We first recall some notions, in particular that of the characteristic polynomial of a stratum (see e.g. [3] (2.3)).

Let [Λ, n, r, β] be a stratum in A and put e = e(Λ), the 𝔬_{F} -period of Λ. Set g = (n, e) and consider the element

y_{β} = ϖ_{F}^{n/g}β^{e/g}.

This is an element of 𝔞_{0}(Λ) so its characteristic polynomial as an F -endo-morphism of V lies in 𝔬_{F} [X]. We define φ_{β}(X) to be the reduction modulo 𝔭_{F} of this characteristic polynomial; this depends only on the equivalence class of the stratum.

For i = 1, 2, let V_{i} be subspaces of V such that V = V_{1} ⊥ V_{2}. In particular, h_{i}, the restriction of h to V_{i} × V_{i}, is a nondegenerate ϵ-hermitian form on V_{i}, for i = 1, 2. We put A^{ij} = Hom_{F} (V^{j}, V^{i}), for i, j = 1, 2, and abbreviate A^{ii} = A^{i}. We use the notation

A = (\begin{matrix}A^{11} & A^{12} \\ A^{21} & A^{22} \end{matrix}) .

For i = 1, 2, let Λ^{i} be a self-dual lattice sequence in V_{i}, e_{i} = e(Λ^{i}|𝔬_{F}) the 𝔬_{F} -period of Λ^{i} and let d_{i} be the (unique) integer such that Λ^{i}(k)^{#} = Λ^{i}(d_{i} − k), for all k ∈ ℤ. After renumbering, we may assume that d_{i} = 0 or −1. If d_{i} = 0 then we replace Λ^{i} by the self-dual lattice sequence Λ′^{i} given by

Λ′^{i}(k) = Λ^{i}([\frac{k+1}{2}]) for k ∈ ℤ;

that is, every lattice is taken with double the multiplicity. Note that the filtration on A^{i} determined by Λ′^{i} is the same as that determined by Λ^{i}, upto a factor in the index. We have Λ′^{i}(k)^{#} = Λ′^{i}(−1 − k), for all k ∈ ℤ, so we may assume d_{i} = −1, for i = 1, 2.

We set Λ = Λ^{1} ⊥ Λ^{2}, a self-dual lattice sequence in V of 𝔬_{F} -period e = lcm (e_{1}, e_{2}) (see [4] (2.8) for the definition of direct sum of lattice sequences). Let β_{i} ∈ A^{i} be skew elements and put n_{i} = −ν_{Λ^{i}}(β_{i}). Put β = β_{1} ⊕ β_{2} and n = e · max {n_{1}/e_{1}, n_{2}/e_{2}}. Then we obtain a skew stratum [Λ, n, r, β] in A, with ν_{Λ}(β) = −n, for any 0 ≤ r ≤ n − 1.

Definition 4.9 (cf. [4] (3.6)). A skew stratum [Λ, n, r, β] as above is called split if

(i) β_{1} ∈ 𝔎(Λ^{1}); (ii) either n_{1}/e_{1} > n_{2}/e_{2} or else all the following conditions hold:

(a) n_{1}/e_{1} = n_{2}/e_{2},

(b) β_{2} ∈ 𝔎(Λ^{2}),

(c) gcd (φ_{β_{1}}, φ_{β_{2}}) = 1.

It is called semisimple if (inductive definition, on the dimension)

(iii) either V_{2} = 0 and [Λ, n, r, β] is a simple stratum or it is split and each [Λ^{i}, n_{i}, r_{i}, β_{i}] is a semisimple stratum, i = 1, 2, where r_{i} = [re_{i}/e].

We remark that the characters considered in [10] arise from semisimple strata of the form [Λ, n, n − 1, β] with Λ a strict lattice sequence.

Theorem 4.10 (cf. [4] (3.7)). Let [Λ, n, r, β] as above be split, with r ≥ 0. Put M = Aut_{F} (V^{1}) × Aut_{F} (V^{2}), considered as a Levi subgroup of G̃. Then

IG̃[Λ, n, r, β] ⊂ U_{n−r} M U_{n−r},

where U_{s} = U_{s}(Λ) = 1 + 𝔞_{s}(Λ), for s > 0.

Proof. We proceed as in the proof of [4] Theorem 3.7.

Proposition 4.11. Under the hypotheses of (4.10), let c ∈ 𝔞_{−r} ∩ A^{21} and define a map δ_{c} : A^{12} → A^{12} by

δ_{c}(x) = β_{1}x − xβ_{2} + xcx, x ∈ A^{12}.

Then δ_{c}(𝔞_{n−r} ∩ A^{12}) = 𝔞_{−r} ∩ A^{12}.

Proof. We need some lemmas.

Lemma 4.12. The map δ = δ_{0} maps 𝔞_{s} ∩ A^{12} onto 𝔞_{s−n} ∩ A^{12} for all s.

Proof. Suppose first that n_{1}/e_{1} > n_{2}/e_{2}. The Λ^{1}-invertibility of β_{1} implies that 𝔞_{s−n} ∩ A^{12} = β_{1}(𝔞_{s} ∩ A^{12}), while (𝔞_{s} ∩ A^{12})β_{2} ⊂ 𝔞_{s−n+1} ∩ A^{12}. Hence

𝔞_{s−n} ∩ A^{12} = δ_{0}(𝔞_{s} ∩ A^{12}) + 𝔞_{s−n+1} ∩ A^{12}

and the lemma follows immediately in this case. For the other case, see [4] (3.7) Lemma 1. ⊓ ⊔

Lemma 4.13. We have

δ_{c}(𝔞_{n−r} ∩ A^{12}) + 𝔞_{k−r} ∩ A^{12} = 𝔞_{−r} ∩ A^{12},

for all integers k ≥ 0.

Proof. Since c ∈ 𝔞_{−r} we have, for x ∈ 𝔞_{n−r} ∩ A^{12}, xcx ∈ 𝔞_{2n−3r} ⊂ 𝔞_{−r}, as n − r > 0, so the lemma holds if k = 0. The proof then follows by induction exactly as [4] (3.7) Lemma 2. ⊓ ⊔

The proposition now follows as in [4] (3.7). ⊓ ⊔

Write M for the algebra A^{11} ⊕ A^{22} ⊂ A, so that M = M^{×}. We first need:

Lemma 4.14. Let x = β + y, y ∈ 𝔞_{−r}. Then there exists u ∈ U_{n−r} such that uxu^{−1} ∈ β + 𝔞_{−r} ∩ M.

Proof. This is essentially identical to [4] (3.7) Lemma 3. ⊓ ⊔

We now prove the theorem. Let g ∈ IG̃[Λ, n, r, β]; thus there exist x, y ∈ 𝔞_{−r} such that g^{−1}(β + x)g = β + y. By (4.14), we may replace g by u_{1}gu_{2}, u_{i} ∈ U_{n−r}, and assume that x, y ∈ 𝔞_{−r} ∩ M. We put

β + x = (\begin{matrix}β′_{1} & 0 \\ 0 & β′_{2} \end{matrix}) , β + y = (\begin{matrix}β′′_{1} & 0 \\ 0 & β′′_{2} \end{matrix}) ,

and write out the equation (β + x)g = g(β + y). Comparing (1, 2)-entries, we have β′_{1}g_{12} = g_{12}β′′_{2}. But the map A^{12} → A^{12} given by z ↦ β′_{1}z − zβ′′_{2} is injective, as in [4] (3.7) Lemma 4, so g_{12} = 0. Likewise g_{21} = 0, whence g ∈ M as required. ⊓ ⊔ In fact, we even have

I = IG̃[Λ, n, r, β] = U_{n−r} I_{M}[Λ, n, r, β] U_{n−r},  (4.15)

where I_{M} is the intertwining in M. For suppose x ∈ I, so that

βx ≡ xβ (mod 𝔞_{−r}x + x𝔞_{−r}),

and let y ∈ 𝔞_{n−r}. Then a_{β}(y) ∈ 𝔞_{−r} so (1 + y)β(1 + y)^{−1} = β − a_{β}(y)(1 + y)^{−1} ≡ β (mod 𝔞_{−r}). Then βx ≡ xβ ≡ x(1 + y)β(1 + y)^{−1} (mod 𝔞_{−r}x + x𝔞_{−r}) so x(1 + y) ∈ I, since (1 + y) ∈ 𝔞_{0}^{×}. By symmetry, we have I = U_{n−r} I U_{n−r} and (4.15) follows.

Now suppose [Λ, n, r, β] is a skew stratum in A; by (4.8), we have that the formal intertwining of the stratum in G is I_{G}[Λ, n, r, β] = IG̃[Λ, n, r, β] ∩ G. Hence, applying (2.3) with H = M we have:

Corollary 4.16. Let [Λ, n, r, β] as above be a skew split stratum, with r ≥ 0. Then, with notation as above, we have

I_{G}[Λ, n, r, β] ⊂ P_{n−r}(M ∩ G)P_{n−r},

where P_{s} = U_{s} ∩ G, for s > 0.

Proof. We need only show that M, U_{n−r} satisfy the condition (2.4). But M is the fixed point subgroup in G̃ of the subgroup Γ = {1} × {±1} of M acting by inner automorphisms and U_{n−r} is fixed by Γ so (2.4) is satisfied, by (2.1). ⊓ ⊔

Indeed, as above for (4.15) but using the Cayley map, we have

I_{G}[Λ, n, r, β] = P_{n−r} I_{M∩G}[Λ, n, r, β] P_{n−r}.  (4.17)

For [Λ, n, r, β] a skew semisimple stratum as above, with r ≥ 0, we define a compact open subgroup K = K(β, Λ) of G inductively as follows: if the stratum is simple then K = Q_{−(k+r)} (see (4.7)); otherwise

K = K(β_{1}, Λ^{1}) × K(β_{2}, Λ^{2}) · P_{n−r}.  (4.18)

Then, putting together (4.16) with (4.7), we have the following result.

Theorem 4.19. Let [Λ, n, r, β] as above be a skew semisimple stratum, with r ≥ 0. Then we have

I_{G}[Λ, n, r, β] = KZK,

where K = K(β, Λ) and Z = Z_{G}(β) is the centralizer in G of β.

4.3. Supercuspidal representations

Let [Λ, n, r, β] be a skew semisimple stratum. We will call it maximal if the centralizer B = Z_{A}(β) in A of β has dimension N. In this case, we have that B = ∑_{i=1}^{r} E_{i} is a sum of fields, each of which is stable under the involution but not fixed pointwise. Moreover, the G-centralizer of β is the compact maximal torus Z = ∏_{i=1}^{r} N_{1}(E_{i}), where N_{1}(E_{i}) = \{e ∈ E_{i} : e\overline{e} = 1\} is the group of norm 1 elements of E_{i}.

In particular, we have in this situation that Z ⊂ P; moreover, the group K = K(β, Λ) defined in (4.18) satisfies K ⊂ P_{1}. Hence, (4.19) gives us

I_{G}[Λ, n, r, β] ⊂ P,  (4.20)

and we deduce the following:

Theorem 4.21. Let [Λ, n, r, β] be a maximal skew semisimple stratum such that (4.2) is satisfied. Let ρ be an irreducible representation of P containing the character ψ_{β}^{−}. Then the (compactly) induced representation π = Ind _{P}^{G}ρ is irreducible and supercuspidal.

Proof. Since P normalizes P_{n}, the restriction of ρ to P_{n} is a sum of P - conjugates of ψ_{β}^{−}. Then, if g ∈ G intertwines ρ, there exist p, p′ ∈ P such that g intertwines ^{p}ψ_{β}^{−} with ^{p′}ψ_{β}^{−}. Hence p^{−1}gp′ ∈ I_{G}(ψ_{β}^{−}) and this is contained in P, by (4.20). We have shown that I_{G}(ρ) = P and the result now follows. ⊓ ⊔

5. Proof of Theorem 4.5

Let [Λ, n, r, β] be a pure stratum in A. Let 𝔞_{n} = 𝔞_{n}(Λ) be the associated filtration on A, E = F [β], B = Z_{A}(β), the A-centralizer of β, and 𝔟_{n} = 𝔞_{n} ∩ B, for n ∈ ℤ. Note that 𝔞_{1} is the Jacobson radical of the 𝔬_{F} -hereditary order 𝔞_{0} and, likewise, that 𝔟_{1} is the Jacobson radical of the 𝔬_{E}-hereditary order 𝔟_{0}.

Let a_{β} : A → A be the map x ↦ βx − xβ. Recall from §4.1 that, for k ∈ ℤ, we put 𝔫_{k} = 𝔫_{k}(β, Λ) = {x ∈ 𝔞_{0}(Λ) : a_{β}(x) ∈ 𝔞_{k}}, an 𝔬_{E}-lattice in A. Then, for i ≥ 0, we define

h_{i} = h_{i}(β, Λ) = inf{k ∈ ℤ : 𝔫_{k+e+i}(β, Λ) ⊂ 𝔟_{0} + 𝔭_{E}𝔞_{i}}.

with the understanding that if E = F then h_{i}(β, Λ) = ∞,

Suppose first that Λ is a strict lattice sequence (that is Λ(n) ⊋ Λ(n + 1) for n ∈ ℤ). Then the filtration induced by Λ on A is given by

𝔞_{n} = (𝔞_{1})^{n},

with (𝔞_{1})^{0} := 𝔞_{0} and likewise for the filtration 𝔟_{n}. We now show that h_{i}(β, Λ) coincides with k_{0}(β, Λ) as defined in [3] (1.4.5).

Lemma 5.1. Let [Λ, n, r, β] be a pure stratum in A, with Λ a strict lattice sequence of 𝔬_{E}-period e and let i ≥ 0. Then

h_{i}(β, Λ) = inf{k ∈ ℤ : 𝔫_{k+1}(β, Λ) ⊂ 𝔟_{0} + 𝔞_{1}}.

Proof. Put l = inf{k ∈ ℤ : 𝔫_{k+1}(β, Λ) ⊂ 𝔟_{0} + 𝔞_{1}} and h = h_{i}(β, Λ). By [3] (1.4.8)(i), we have 𝔫_{l+e+i} = 𝔟_{0} +(𝔟_{1})^{e+i}𝔫_{l} ⊂ 𝔟_{0} +(𝔟_{1})^{e+i}𝔞_{0} = 𝔟_{0} +(𝔞_{1})^{e+i} = 𝔟_{0} + 𝔭_{E}𝔞_{i}. Hence h ≤ l.

However, by definition 𝔫_{h+e+i} ⊂ 𝔟_{0} + 𝔭_{E}𝔞_{i} so, by [3] (1.4.8)(i), we have 𝔫_{h+1} = (𝔟_{1})^{1−e−i}𝔫_{h+e+i} ∩𝔞_{0} ⊂ (𝔟_{1})^{1−e−i}(𝔟_{0} +(𝔞_{1})^{e+i})∩𝔞_{0} = 𝔟_{0} +𝔞_{1}. Hence l ≤ h also, so we have equality. ⊓ ⊔

The main ingredients in proving the intertwining theorem 4.5 are the exact sequences analogous to those in [3] (1.4) so we proceed by proving these. We start with some preliminary lemmas, where Λ is no longer assumed to be strict and e is the 𝔬_{E}-period of Λ.

Lemma 5.2 (cf. [3] (1.4.8)).

(i) We have 𝔭_{E}𝔫_{k} = 𝔫_{k+e} ∩ 𝔭_{E}𝔞_{0} and 𝔫_{k} = 𝔭_{E}^{−1}𝔫_{k+e} ∩ 𝔞_{0}, for all k ∈ ℤ. (ii) For k ∈ ℤ, the following are equivalent:

(a) k ≥ h_{i}(β, Λ);

(b) 𝔫_{k+e+i} = 𝔟_{0} + 𝔭_{E}(𝔫_{k+i} ∩ 𝔞_{i}).

Proof. This is identical to [3] (1.4.8), since 𝔭_{E}𝔭_{E}^{−1} = 𝔬_{E}. ⊓ ⊔

In particular, we have that for k ≥ h_{i}(β, Λ),

𝔫_{k+e+i} ∩ 𝔞_{i} = 𝔟_{i} + 𝔭_{E}(𝔫_{k+i} ∩ 𝔞_{i}).

Also observe that the lemma implies that the application i ↦ h_{i} is periodic, with period dividing e.

Recall ([3] (1.3.3)) that a tame corestriction on A relative to E/F is a (B, B)-bimodule homomorphism s : A → B such that s(𝔄) = 𝔄 ∩ B for any hereditary 𝔬_{F} -order 𝔄 in A which is normalized by E^{×}. Such a homomorphism exists and is uniquely determined upto multiplication by an element of 𝔬_{E}^{×}. We also have ker s = im a_{β} and ker a_{β} = im s.

Lemma 5.3 (cf. [3] (1.3.4)(ii)). Let Λ be any 𝔬_{E}-lattice sequence in V and let s be a tame corestriction on A relative to E/F. Then we have s(𝔞_{n}) = 𝔟_{n}, for n ∈ ℤ.

Proof. This is identical to [3] (1.3.4)(ii), having observed that, from [4] (2.10), we have 𝔞_{n}^{∗} = 𝔞_{1−n}. ⊓ ⊔

Now we have all the tools necessary to prove (and the proof is mutatis mutandis that of [3] (1.4.7)):

Proposition 5.4 (cf. [3] (1.4.7), (1.4.10), [1] (5.2)). Let i, k, r ∈ ℤ and suppose i ≥ 0, k ≥ h_{i}(β, Λ), r ≥ 1. Then the following sequences are exact:

0 → 𝔫_{k+i} ∩ 𝔞_{i}/𝔫_{k+r+i} ∩ 𝔞_{i} →^{a_{β}} 𝔞_{k+i}/𝔞_{k+r+i} →^{s} 𝔟_{k+i}/𝔟_{k+r+i} → 0,
0 → 𝔫_{k+i} ∩ 𝔞_{i}/𝔟_{i} →^{a_{β}} 𝔞_{k+i} →^{s} 𝔟_{k+i} → 0.

We now give another characterization of the integers h_{i} (cf. [3] (1.4.11)).

Lemma 5.5. For i ≥ 0 we have h_{i}(β, Λ) = inf{k ∈ ℤ : a_{β}(𝔞_{i}) ⊃ 𝔞_{k+i} ∩ im a_{β}}.

Proof. We have a_{β}(𝔞_{i}) ⊃ a_{β}(𝔫_{h_{i}+i} ∩ 𝔞_{i}) = ker (s|𝔞_{h_{i}+i}) by (5.4) and this is precisely 𝔞_{h_{i}+i}∩im a_{β}. Conversely, suppose k < h_{i} and a_{β}(𝔞_{i}) ⊃ 𝔞_{k+i}∩im a_{β}. Then

a_{β}(𝔭_{E}𝔞_{i}) ⊃ 𝔞_{k+e+i} ∩ im a_{β} ⊃ 𝔞_{h_{i}+e+i} ∩ im a_{β} = a_{β}(𝔫_{h_{i}−1+e+i} ∩ 𝔞_{i})

by (5.4) (and since e − 1 ≥ 0). Hence we have 𝔫_{h_{i}−1+e+i} ∩ 𝔞_{i} ⊂ 𝔟_{0} + 𝔭_{E}𝔞_{i}. Now by (5.2) we have

𝔫_{h_{i}−1+2e+i} = 𝔟_{0} + 𝔭_{E}(𝔫_{h_{i}−1+e+i} ∩ 𝔞_{i}) ⊂ 𝔟_{0} + 𝔭_{E}(𝔟_{0} + 𝔭_{E}𝔞_{i}) = 𝔟_{0} + 𝔭_{E}^{2}𝔞_{i}

so, also by (5.2),

𝔫_{h_{i}−1+e+i} = 𝔭_{E}^{−1}𝔫_{h_{i}−1+2e+i} ∩ 𝔞_{0} ⊂ 𝔭_{E}^{−1}(𝔟_{0} + 𝔭_{E}^{2}𝔞_{i}) ∩ 𝔞_{0} = 𝔟_{0} + 𝔭_{E}𝔞_{i},

contradicting the minimality of h_{i}. ⊓ ⊔

In order to compare the integers h_{i} with the value k_{0}(β, Λ) as defined in [4] (5.1), we must introduce the notion of a (W, E)-decomposition (see [3] (1.2)). Let W be the F -span of an E-basis of V so the canonical map E ⊗_{F} W → V is an isomorphism. This induces an algebra isomorphism A(E) ⊗_{F} End_{F} (W) ≃ A, where A(E) := End_{F} (E), which in turn induces an isomorphism of (A(E), B)-bimodules

A(E) ⊗_{E} B ≃ A.  (5.6)

In the space A(E) there is a unique hereditary 𝔬_{F} -order normalized by E^{×}; we denote it 𝔄(E) and write Λ(E) for the associated (strict) lattice sequence in E. Then, for a suitable choice of W, the isomorphism (5.6) restricts to an isomorphism

𝔄(E) ⊗_{𝔬_{E}} 𝔟_{r} ≃ 𝔞_{r},

for each r ∈ ℤ (see [4] (5.3)).

We may also consider the lattices 𝔫_{k}(β, Λ(E)). Then we also have

𝔫_{ek}(β, Λ) = 𝔫_{k}(β, Λ(E)) ⊗_{𝔬_{E}} 𝔟_{0},

for k ∈ ℤ. This is [3] (1.4.13)(i) in the case of a strict lattice sequence and the general case follows since 𝔞_{ek} = 𝔭_{E}^{k}𝔞_{0} for all k ∈ ℤ.

We put k_{F} (β) = inf{k ∈ ℤ : 𝔫_{k+1}(β, Λ(E)) ⊂ 𝔬_{E} + 𝔓(E)}, where 𝔓(E) = 𝔭_{E}𝔄(E) is the Jacobson radical of 𝔄(E). (Note that k_{F} (β) is denoted k_{0}(β, 𝔄(E)) in [3].) We also have k_{F} (β) = inf{k ∈ ℤ : a_{β}(𝔄(E)) ⊃ 𝔓(E)^{k} ∩ im a_{β}}. Then we define, as in [4] (5.1),

k_{0}(β, Λ) = e · k_{F} (β).

Lemma 5.7 (cf. [3] (1.4.13)(ii)). With notation as above, k_{0}(β, Λ) ≥ h_{i}(β, Λ), for all i ≥ 0, and h_{1}(β, Λ) = k_{0}(β, Λ).

Proof. Writing ⊗ for ⊗_{𝔬_{E}} and k = k_{F} (β), we have

a_{β}(𝔞_{i}) = a_{β}(𝔄(E) ⊗ 𝔟_{i}) ⊃ (𝔓(E)^{k} ∩ im a_{β}) ⊗ 𝔟_{i}
= 𝔄(E)𝔭_{E}^{k} ⊗ 𝔟_{i} ∩ im a_{β} = 𝔞_{ek+i} ∩ im a_{β},

so certainly h_{i}(β, Λ) ≤ ek = k_{0}(β, Λ).

Now suppose ek > h_{1}(β, Λ); then 𝔫_{ek+e} ⊂ 𝔟_{0} + 𝔭_{E}𝔞_{1} so we have 𝔫_{ek} = 𝔭_{E}^{−1}𝔫_{ek+e} ∩ 𝔞_{0} ⊂ 𝔟_{0} + 𝔞_{1}. Then

𝔫_{k}(β, Λ(E)) ⊂ (𝔫_{k}(β, Λ(E)) ⊗ 𝔟_{0}) ∩ A(E)
= 𝔫_{ek} ∩ A(E) ⊂ (𝔟_{0} + 𝔞_{1}) ∩ A(E) ⊂ 𝔬_{E} + 𝔓(E)

contrary to the definition of k = k_{F} (β). ⊓ ⊔

Observe that the proof also shows that k_{0}(β, Λ) = inf{k ∈ ℤ : 𝔫_{k+1} ⊂ 𝔟_{0} + 𝔞_{1}}, which is the definition given in §4.

In order to prove (4.5), we need one more exact sequence (cf. [3] (1.4.16)). Let i, j, m, n ∈ ℕ, b_{1}, b_{2} ∈ B^{×}, k = k_{0}(β, Λ) and put

L = 𝔞_{i+k} ∩ (b_{1}𝔞_{j+k} + 𝔞_{m+k}b_{2} + 𝔞_{n+k}),
M = (𝔞_{i} ∩ 𝔫_{i+k}) ∩ (b_{1}(𝔞_{j} ∩ 𝔫_{j+k}) + (𝔞_{m} ∩ 𝔫_{m+k})b_{2} + (𝔞_{n} ∩ 𝔫_{n+k})).

Then we require that the sequence

M →^{a_{β}} L →^{s} A  (5.8)

be exact. This will follow exactly as in [3] (1.4.16) if we can show that s(L) = L ∩ B. (L is E-exact, in the language of [3] (1.3).) But this holds as in [3] (1.3.16) so (5.8) is indeed an exact sequence.

We now have all the tools necessary to prove that

I = IG̃[Λ, n, r, β] = (1 + 𝔪_{d})B^{×}(1 + 𝔪_{d}),

for [Λ, n, r, β] a simple stratum and where 𝔪_{d} = 𝔞_{−r−k_{0}} ∩ 𝔫_{r}. We have (1 + 𝔪_{d})I(1 + 𝔪_{d}) = I as in [3] (1.5.8) and B^{×} ⊂ I. Furthermore, x ∈ I if and only if ϖ_{F} x ∈ I so we only need show that I ∩𝔞_{0} ⊂ (1+𝔪_{d})B^{×}(1+𝔪_{d}). The remainder of the proof is then mutatis mutandis that of [3] (1.5.8). ⊓ ⊔

Acknowledgements. The research for this paper was partially supported by the EU TMR network “Arithmetical Algebraic Geometry”.

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