INTERTWINING AND SUPERCUSPIDAL

TYPES FOR P-ADIC CLASSICAL GROUPS^{†}

S.STEVENS

Introduction

Let F be a non-archimedean local field of residual characteristic p ≠ 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 ϵ-hermitian form h and let also denote the adjoint involution on A = End_{F} V induced by h. Let σ be the involution on G̃ = Aut_{F} V ≃ GL(N, F) given by x ↦ \overline{x}^{−1}, for x ∈ G. We put Σ = \{1, σ\} ⊂ AutG̃ and set G = G̃^{Σ}, a unitary group defined over F_{0} (symplectic or orthogonal if F = F_{0}).

Let π be an irreducible supercuspidal representation of G. Following the strategy of Bushnell and Kutzko [4] for the classification of the representations of G, we would like to construct a [G, π]_{G}-type, that is a pair (J, λ) consisting of a compact subgroup J of G and an irreducible representation λ of J such that π is the unique irreducible representation (upto equivalence) of G which contains λ by restriction. We construct such types starting from types for G̃ which are fixed by Σ and the main tool for transferring from G̃ to G is Glauberman’s correspondence ([7]). This gives us a correspondence g between the equivalence classes of irreducible representations ρ of a pro-p subgroup K of G̃ which are fixed by Σ and the equivalence classes of irreducible representations of K^{Σ}. The lemma (2.4) which allows us to make use of this correspondence relates to the intertwining: for g ∈ G, the intertwining space Hom_{^{g}K∩K}(^{g}ρ, ρ) has dimension coprime to p if and only if the intertwining space Hom_{^{g}K^{Σ}∩K^{Σ}}(^{g}g(ρ), g(ρ)) has dimension coprime to p. In particular, if ρ = χ is a character then the intertwining of g(χ) is I_{G}(g(χ)) = IG̃(χ) ∩ G.

However, we cannot use Glauberman’s correspondence directly for types (J, λ) for G̃, since J is not a pro-p subgroup, so we must delve into the construction process for such types ([3, §§2,3,5]). The first step is a simple stratum and the transfer for these (including their intertwining) is described in [15]. In particular, there is a field extension E/F associated to such a stratum.

The crucial step is the construction of simple characters ([3, §3.2]). These are certain arithmetically defined characters θ of a pro-p subgroup H^{1} of G̃. If the simple stratum used in the construction of θ is skew (“contained” in Lie G) then H^{1} is fixed by Σ and we may consider the (non-empty) set of simple characters fixed by Σ. Then we define the simple characters for G to be the Glauberman transfers of these characters (in fact, just their restrictions). Using the intertwining lemma and a theorem on the decomposition of double cosets ([15, (1.1)]), we may calculate the intertwining of these characters (3.7).

†

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

1991 Mathematics Subject Classification: 22E50

There is another pro-p group J^{1} ⊃ H^{1} and the next step is to show that there is a unique irreducible representation η of J^{1} containing θ. Moreover, if θ is fixed by Σ then so is η so we may consider the Glauberman transfer g(η). In fact this is the unique irreducible representation of J^{1} ∩ G containing g(θ). Lemma (2.4) applies again and thus we calculate the intertwining of g(η).

The final stage in the construction is only considered in a special case, which we describe below.

While in G̃ the simple strata are sufficient to give all supercuspidal representations, this is certainly no longer the case for the group G. Indeed, there are compact tori in G of the form N_{1}(E_{1}) × N_{1}(E_{2}), where, for i = 1, 2, E_{i} is a field extension of F with a galois involution which restricts to on F and has fixed field E_{i,0}, and N_{1} denotes the elements e of norm N_{E_{i}/E_{i,0}}e equal to 1.

To take account of this, in [15, §3.2] the notion of a semisimple stratum is introduced – this is a sum of simple strata which are “sufficiently different” from each other. Here we extend this to define semisimple characters for G̃, which are in bijection with products of simple characters for smaller GL(N_{i}, F), ∑ N_{i} = N. The process described above for simple characters works equally well in this semisimple case and we obtain a representation g(η) of J^{1} ∩ G, together with its intertwining.

Finally, we suppose that the field extension E_{i} associated to each simple stratum from which our semisimple stratum is built is maximal – that is, ∑[E_{i} : F] = N. In this case the intertwining of η_{−} = g(η) is contained in J_{−} = J ∩ G and, since J_{−}/J^{1} ∩ G is a product of cyclic groups, η_{−} admits a finite number of extensions to a representation κ_{−} of J_{−}. Then the induced representation π = Ind _{J_{−}}^{G}κ_{−} is irreducible supercuspidal and (J_{−}, κ_{−}) is a [G, π]_{G}-type.

We now give a brief summary of the contents of each chapter. In §1 we introduce the notations and give some definitions. The intertwining lemma for Glauberman’s correspondence is given in §2. In §3 we define simple and semisimple characters for G and calculate their intertwining and in §4 we construct the representation η and its transfer to G. In §5 we construct types and supercuspidal representations of G in the case of a maximal compact torus. We finish with a few remarks concerning possible further work. Finally, in §6, we show that any simple character fixed by Σ does in fact come from a skew simple stratum.

Although the methods used here are somewhat different, I was partly inspired by the work of L.Morris (see [13], [12], for example). Supercuspidal representations have also been constructed in the tame case in [1], [10], [16], and in the general case in [9]. This paper generalizes the constructions of [9].

The results here for simple characters formed 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. The use of Glauberman’s correspondence, in particular with regard to intertwining, was suggested to me by Guy Henniart.

1. 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_{0}}; 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) ≃ 𝕄(N, F) so that G̃ = Aut_{F} (V) may be identified with GL(N, F). 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; 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_{−} = A^{Σ} = \{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 [5, (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. An 𝔬_{F} -lattice sequence Λ is called strict if Λ(n) ≠ Λ(n + 1), 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). 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 nondegeneracy 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 [13, (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}.

(1.1) Lemma. 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 [5, (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:

(1.2) Lemma. 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}.

We now recall some definitions from [3], keeping the language of [5]

(1.3) Definition ([3, (1.5)], [5, (3.1)]). A stratum in A is a 4-tuple [Λ, n, r, b] consisting of a lattice sequence Λ in V, n ∈ ℤ, r ∈ ℝ with r < n, and an element b ∈ 𝔞_{−n}(Λ). We say that two strata [Λ, n, r, b_{i}], i = 1, 2, are equivalent if b_{1} ≡ b_{2} (mod 𝔞_{−r}(Λ)).

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

(1.4)  n > r ≥ [\frac{n}{2}] ≥ 0,

where [x] is the greatest integer less than or equal to x. By (1.2)(i), an equivalence class of strata [Λ, n, r, b] corresponds to the character ψ_{b} of U_{r+1}(Λ)/U_{n+1}(Λ).

(1.5) Definition ([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) = −∞. If F [β] ≠ F then k_{0}(β, Λ) is an integer greater than or equal to −n, with equality if and only if β is minimal (see [3, (1.4.14)]).

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

We now consider the situation for our group G.

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

Again, if(1.4) is satisfied then, by(1.2)(ii), an equivalence class of skew strata [Λ, n, r, b] corresponds to the character ψ_{b}^{−} of P_{r+1}(Λ)/P_{n+1}(Λ).

We finish this section with some technical results concerning simple strata.

(1.8) Lemma. Let Λ be an 𝔬_{F} -lattice sequence in V and let {γ_{t} : t ∈ ℤ} be a sequence of elements of 𝔎(Λ) which converges to a non-zero element γ in A. Then γ ∈ 𝔎(Λ).

Proof. Let ν = ν_{Λ}(γ) < +∞ and let T ∈ ℤ be such that γ_{T} ≡ γ (mod 𝔞_{ν+1}). Let x = γ − γ_{T} ∈ 𝔞_{ν+1}; then γ = γ_{T} (1 + γ_{T}^{−1}x) ∈ 𝔎(Λ)U_{1}(Λ) = 𝔎(Λ). □

(1.9) Proposition. Let {[Λ, n, r, γ_{t}] : t ∈ ℤ} be a sequence of equivalent simple strata such that γ_{t} converges to some γ ∈ A. Then [Λ, n, r, γ] is a simple stratum.

Proof. Let Φ_{t}(X) ∈ F [X] be the characteristic polynomial of γ_{t} and let P_{t}(X) be its minimal polynomial, which is irreducible. Let Φ(X) ∈ F [X] be the characteristic polynomial of γ; then we certainly have lim_{t→∞} Φ_{t}(X) = Φ(X). Let \overline{F} be an algebraic closure of F. Then the set of roots of Φ_{t}(X) in \overline{F}, for all t ∈ ℤ, is bounded. Hence the set of coefficients (in F) of P_{t}(X), t ∈ ℤ, is also bounded and {P_{t}(X) : t ∈ ℤ} has a convergent subsequence. Let P (X) ∈ F [X] be the limit of this subsequence; in particular we have P (γ) = 0. If P (X) factorizes into coprime factors then, by Hensel’s Lemma, P_{t}(X) factorizes for P_{t}(X) sufficiently close to P (X), which is absurd. Hence P (X) = Π(X)^{m}, for some Π(X) ∈ F [X] irreducible, m ∈ ℕ. Now we show Π(γ) = 0. For suppose not, then Π(γ) is nilpotent. But Π(γ_{t}) ∈ 𝔎(Λ) converges to Π(γ) so, by(1.8), Π(γ) is a nilpotent element of 𝔎(Λ), which is absurd. In particular, F [γ] is a field, whose non-zero elements normalize Λ by (1.8) so [Λ, n, r, γ] is a pure stratum. Then, since F [γ] is of degree less than or equal to that of F [γ_{1}], [Λ, n, r, γ] is in fact simple, by [3, (2.4.1)(i)]. □

We now give the skew analogue of [3, (2.4.1)]. In particular, this will allow us to conclude that, for a skew simple stratum, the groups determined by the stratum are invariant under Σ.

(1.10) Proposition. Let [Λ, n, r, β] be a pure stratum with Λ self-dual and β + \overline{β} ∈ 𝔞_{−r}. Then there exists a skew simple stratum [Λ, n, r, γ] equivalent to [Λ, n, r, β].

Proof. By [3, (2.4.1)(i)], there exists a simple stratum [Λ, n, r, γ_{0}] equivalent to [Λ, n, r, β]. Then γ_{0}+\overline{γ}_{0} ∈ 𝔞_{−r} and [Λ, n, r, −\overline{γ}_{0}] is also a simple stratum equivalent to [Λ, n, r, β]. We find, by induction, simple strata [Λ, n, r, γ_{t}] equivalent to [Λ, n, r, β] such that γ_{t} + \overline{γ}_{t} ∈ 𝔞_{t−r} and γ_{t} − γ_{t+1} ∈ 𝔞_{t−r}. Granting this, we let γ be the limit of the γ_{t}, as t → ∞, γ ∈ β + 𝔞_{−r}. Then [Λ, n, r, γ] is simple, by (1.9), equivalent to [Λ, n, r, β] and skew, as required. We have found γ_{0} so assume we have γ_{t} as required, for some t ≥ 0. Let E = F [γ_{t}], B = C_{A}(E), 𝔟_{n} = 𝔞_{n} ∩ B, for n ∈ ℤ, and let s be a tame corestriction relative to E/F (see [3, (1.3.3)]). We will also write Λ_{𝔬_{E}} when we think of Λ as an 𝔬_{E}-lattice sequence. The simple strata [Λ, n, r − t, γ_{t}] and [Λ, n, r − t, −\overline{γ}_{t}] are equivalent so, by [3, (2.4.1)(ii)] and since 2 ∈ 𝔬_{F}^{×}, there exists δ_{t} ∈ E such that s(γ_{t} + \overline{γ}_{t}) ≡ −2δ_{t} (mod 𝔞_{1−r+t}). We put b_{t} = −\frac{1}{2}(γ_{t} +\overline{γ}_{t}); then the stratum [Λ_{𝔬_{E}}, r −t, r −t−1, s(b_{t})] in B is equivalent to the stratum [Λ_{𝔬_{E}}, r − t, r − t − 1, δ_{t}], which is either simple or equivalent to the null stratum [Λ_{𝔬_{E}}, r − t, r − t − 1, 0]. In the latter case, by [3, (2.2.1)], there exists u ∈ U_{1}(Λ) such that the skew stratum [Λ, n, r − t − 1, γ_{t} + b_{t}] is equivalent to the simple stratum [Λ, n, r − t − 1, uγ_{t}u^{−1}] so we put γ_{t+1} = uγ_{t}u^{−1}, with k_{0}(γ_{t+1}, Λ) = k_{0}(γ_{t}, Λ). In the former case, by [3, (2.2.8)], the skew stratum [Λ, n, r − t − 1, γ_{t} + b_{t}] is equivalent to a simple stratum [Λ, n, r − t − 1, γ_{t+1}]; moreover, k_{0}(γ_{t+1}, Λ) = k_{0}(γ_{t}, Λ), since k_{0}(δ_{t}, Λ_{𝔬_{E}}) = −∞. In both cases we have that [Λ, n, r, γ_{t+1}] is simple, as required. □

(1.11) Corollary (cf. [3, (2.2.8)]). Let [Λ, n, r, β] be a skew simple stratum in A. Let B be the A-centralizer of E = F [β] and 𝔟_{n} = 𝔞_{n} ∩ B.

Let b ∈ 𝔞_{−r}^{−} and let s be a tame corestriction on A relative to E/F. Suppose that the stratum [Λ_{𝔬_{E}}, r, r−1, s(b)] is equivalent to a simple stratum in B. Then [Λ, n, r − 1, β + b] is equivalent to a skew simple stratum [Λ, n, r − 1, β_{1}] and, moreover, k_{0}(β_{1}, Λ) = max {k_{0}(β, Λ), k_{0}(c, Λ_{𝔬_{E}})}.

Proof. By [3, (2.2.8)], the skew stratum [Λ, n, r−1, β+b] is equivalent to a simple stratum [Λ, n, r−1, β_{0}]. Hence β_{1}+\overline{β}_{1} ∈ 𝔞_{1−r} and(1.10) implies that [Λ, n, r−1, β_{0}] is equivalent to a skew simple stratum [Λ, n, r−1, β_{1}]. The final assertion follows from [3, (2.2.8)]. □

We remark that (1.10) (and hence also (1.11)) is easily generalizable to the situation of a group Γ of automorphisms of G̃ of order coprime to p and G = G̃^{Γ}.

2. The principal lemma

We now take a digression and consider Glauberman’s correspondence of characters (see [7]). The notation in the first part of this section is independent of that in the remainder of this paper. For the exposition of Glauberman’s results, we follow [2]. Let H be a finite group and Γ a soluble subgroup of AutH such that |H|, |Γ| are relatively prime. We can thus form ΓH = Γ semi-direct product G. We denote the centralizer of Γ in H by H^{Γ}. We write Irr (H) for the set of equivalence classes of irreducible representations of H and use a similar notation for other groups. The group Γ acts on Irr (H); we denote the set of fixed points by Irr (H)^{Γ}. We have the following result of Glauberman, whose formulation is taken from [11]:

(2.1) Theorem. There is a uniquely determined bijection

g = g_{Γ,H} : Irr (H)^{Γ} ⟶^{≃} Irr (H^{Γ})

with the following properties:

(i) if Γ is an l-group, for some prime number l, and ρ ∈ Irr (H)^{Γ} then g(ρ) occurs in ρ|_{H^{Γ}} with multiplicity incongruent to 0 modulo l; (ii) if Δ is a normal subgroup of Γ then

g_{Γ,H} = g_{Γ/Δ,H^{Δ}} ∘ g_{Δ,H}.

In fact, in case (i), one has that g(ρ) occurs in ρ|_{H^{Γ}} with multiplicity congruent to ±1 (mod l), by [7, Corollary 6]. In this situation we also have the following results, from [8]:

(2.2) Theorem ([8, Theorem A]). Let K be a Γ-stable subgroup of H. Let ρ ∈ Irr (H)^{Γ}, σ ∈ Irr (K)^{Γ}.

(i) If ρ ≃ Ind _{K}^{H}σ then g(ρ) ≃ Ind _{K^{Γ}}^{H^{Γ}}g(σ).

(ii) If ρ|_{K} ≃ σ then g(ρ)|_{K^{Γ}} ≃ g(σ).

(2.3) Lemma ([8, (2.3)]). Suppose Γ is an l-group, for l a prime number. Let K be a Γ-stable subgroup of H. Let ρ ∈ Irr (H)^{Γ}, σ ∈ Irr (K)^{Γ}. Then σ occurs in ρ|_{K} with multiplicity incongruent to 0 (mod l) if and only if g(σ) occurs in g(ρ)|_{K^{Γ}} with multiplicity incongruent to 0 (mod l).

In the situation of(2.3), suppose that g(ρ) occurs in ρ with multiplicity congruent to ϵ (mod l) and g(σ) occurs in σ with multiplicity congruent to η (mod l). Then, if σ occurs in ρ|_{K} with multiplicity r (mod l), the proof of [8, (2.3)] in fact shows that g(σ) occurs in g(ρ)|_{K^{Γ}} with multiplicity ϵηr (mod l). We now return to the situation of §1 and apply the results concerning Glauberman’s correspondence (in particular(2.3)) to the intertwining of representations of pro-p subgroups of G̃ = GL(N, F). Firstly though, observe that Glauberman’s correspondence can be applied to representations of pro-p subgroups of G̃ by taking the quotient by a (small enough) normal compact open subgroup. We will apply all the above results to this situation without further comment. For i = 1, 2, let ρ_{i} be a representation of a subgroup H_{i} of G̃. For g ∈ G̃, the intertwining space I_{g}(ρ_{1}, ρ_{2}) is defined to be

I_{g}(ρ_{1}, ρ_{2}) = Hom_{^{g}H_{1}∩H_{2}}(^{g}ρ_{1}, ρ_{2}),

where ^{g}H_{1} = gH_{1}g^{−1} and ^{g}ρ_{1} is the representation x ↦ ρ_{1}(g^{−1}xg) of gH_{1}. We put IG̃(ρ_{1}, ρ_{2}) = {g ∈ G̃ : I_{g}(ρ_{1}, ρ_{2}) ≠ 0} and say that g intertwines ρ_{1} with ρ_{2} if g ∈ IG̃(ρ_{1}, ρ_{2}).

(2.4) Principal lemma. Let Γ be a finite soluble subgroup of AutG̃. Suppose also that Γ is an l-group, l ≠ p a prime number. For i = 1, 2, let H_{i} be Γ-stable pro-p subgroups of G̃ and let ρ_{i} ∈ Irr (H_{i})^{Γ}. Let g ∈ G := G̃^{Γ}. Then

dim_{ℂ}(I_{g}(ρ_{1}, ρ_{2})) ≢ 0 (mod l)
⟺ dim_{ℂ}(I_{g}(g(ρ_{1}), g(ρ_{2}))) ≢ 0 (mod l).

Proof. Let S be the set of triples (σ, m_{1}, m_{2}) consisting of an (equivalence class of) irreducible representation σ of ^{g}H_{1} ∩ H_{2} and the multiplicity m_{1} (respectively m_{2}) with which σ occurs in (the restriction of) ^{g}ρ_{1} (respectively ρ_{2}). The contribution of the triple (σ, m_{1}, m_{2}) to the intertwining space I_{g}(ρ_{1}, ρ_{2}) has dimension m_{1}m_{2}. Let S_{l} be the subset of S consisting of those triples with m_{1}m_{2} ≢ 0 (mod l). The group Γ acts on S_{l} since ^{g}H_{1} ∩ H_{2}, ^{g}ρ_{1} and ρ_{2} are each fixed by Γ. Let O be an orbit for this action; if O is not just one triple then it has the form {(σ^{γ}, m_{1}, m_{2}) : γ ∈ Γ/Stab_{Γ}(σ)}, where Stab_{Γ}(σ) = {γ ∈ Γ : σ ≃ σ^{γ}}. In particular, card O = l^{r}, for some r ∈ ℕ, and the contribution of O to the intertwining space has dimension l^{r}m_{1}m_{2} ≡ 0 (mod l). Let S_{l}^{Γ} = {(σ_{j}, m_{1}^{j}, m_{2}^{j})} be the fixed points of S_{l} for the action of Γ. Then we have

dim_{ℂ}(I_{g}(ρ_{1}, ρ_{2})) ≡ ∑_{j} m_{1}^{j}m_{2}^{j} (mod l).

Similarly, let T be the set of triples (τ, n_{1}, n_{2}) consisting of an (equivalence class of) irreducible representation τ of ^{g}H_{1}^{Γ} ∩ H_{2}^{Γ} and the multiplicity n_{1} (respectively n_{2}) with which τ occurs in (the restriction of) ^{g}g(ρ_{1}) (respectively g(ρ_{2})). Let T_{l} = {(τ_{k}, n_{1}^{k}, n_{2}^{k})} be the subset consisting of those triples with n_{1}n_{2} ≢ 0 (mod l). Then, as above, we have

dim_{ℂ}(I_{g}(g(ρ_{1}), g(ρ_{2}))) ≡ ∑_{k} n_{1}^{k}n_{2}^{k} (mod l).

By (2.3), Glauberman’s correspondence gives a bijection S_{l}^{Γ} ↔ T_{l} and, using the remark following (2.3), we can examine how this bijection affects the multiplicities. For i = 1, 2, let ϵ_{i} = ±1 be such that g(ρ_{i}) occurs in ρ_{i} with multiplicity congruent to ϵ_{i} (mod l). For (σ_{j}, m_{1}^{j}, m_{2}^{j}) ∈ S_{l}^{Γ}, let η_{j} = ±1 be such that g(σ_{j}) occurs in σ_{j} with multiplicity congruent to η_{j} (mod l). Then, if (g(σ_{j}), n_{1}^{j}, n_{2}^{j}) ∈ T_{l}, we have n_{i}^{j} ≡ ϵ_{i}η_{j}m_{i}^{j} (mod l), for i = 1, 2. In particular,

∑_{j} n_{1}^{j}n_{2}^{j} ≡ ∑_{j} ϵ_{1}η_{j}m_{1}^{j}ϵ_{2}η_{j}m_{2}^{j} ≡ ϵ_{1}ϵ_{2} ∑_{j} m_{1}^{j}m_{2}^{j} (mod l),

and the lemma follows. □

(2.5) Corollary. With notation as in(2.4), suppose ρ_{i} = χ_{i} are characters, i = 1, 2. Then I_{g}(χ_{1}, χ_{2}) ≠ 0 if and only if I_{g}(g(χ_{1}), g(χ_{2})) ≠ 0.

We will, in particular, apply(2.4) and(2.5) to the case Γ = Σ, where Σ is as in §2.

3. Characters and intertwining

3.1. Simple characters in G̃

We begin by recalling the definitions of the orders ℌ, 𝔍 from [3, (3.1)]. Throughout this section [Λ, n, 0, β] will be a simple stratum in A. We also assume that Λ is a strict lattice sequence; this restriction is due to the fact that the results of [3, §3] are only available in this case, although it seems likely that they will remain valid in the general case. We set r = −k_{0}(β, Λ) and let {𝔞_{t} : t ∈ ℤ} be the filtration on A associated to Λ. We write B_{β} for the centralizer of β in A, dropping the subscript when the meaning is clear, and put 𝔟_{t} = 𝔟_{β,t} = 𝔞_{t} ∩ B_{β}.

We define ℌ and 𝔍 inductively on r. If β is minimal over F we put

(3.1)  \{ \begin{matrix}ℌ(β) = ℌ(β, Λ) = 𝔟_{β} + 𝔞_{[ \frac{n}{2} ]+1}; \\ 𝔍(β) = 𝔍(β, Λ) = 𝔟_{β} + 𝔞_{[ \frac{n+1}{2} ]}. \end{matrix}

Otherwise, suppose that r < n and let [Λ, n, r, γ] be a simple stratum equivalent to [Λ, n, r, β], which exists by [3, (2.4.1)]; then we put

(3.2)  \{ \begin{matrix}ℌ(β) = ℌ(β, Λ) = 𝔟_{β} + ℌ(γ, Λ) ∩ 𝔞_{[ \frac{r}{2} ]+1}; \\ 𝔍(β) = 𝔍(β, Λ) = 𝔟_{β} + 𝔍(γ, Λ) ∩ 𝔞_{[ \frac{r+1}{2} ]}. \end{matrix}

Note that this inductive definition is independent of the choice of γ such that [Λ, n, r, γ] is a simple stratum equivalent to [Λ, n, r, β]. We also have filtrations on ℌ(β) and 𝔍(β) given by

\begin{matrix}ℌ^{t}(β) = ℌ(β) ∩ 𝔞_{t} \\ 𝔍^{t}(β) = 𝔍(β) ∩ 𝔞_{t} \end{matrix} \} for t ≥ 0.

We define two families of compact open pro-p subgroups of G̃ by

\begin{matrix}H^{t}(β) = H^{t}(β, Λ) = ℌ(β, Λ) ∩ U_{t}(Λ) \\ J^{t}(β) = J^{t}(β, Λ) = 𝔍(β, Λ) ∩ U_{t}(Λ) \end{matrix} \} for t ≥ 0.

We may now recall the definition of the sets C(Λ, m, β) of simple characters associated to the simple stratum [Λ, n, 0, β].

(3.3) Definition ([3, (3.2.1)]). Let β be minimal over F, E = F [β]. For 0 ≤ m ≤ n − 1, let C(Λ, m, β) denote the set of characters θ of H^{m+1}(β) such that

(i) θ|_{H^{m+1}(β)∩U_{[ \frac{n}{2} ]+1}(Λ)} = ψ_{β};

(ii) θ|_{H^{m+1}(β)∩B_{β}^{×}} factors through det_{B_{β}} : B_{β}^{×} → E^{×}.

(3.4) Definition ([3, (3.2.3)]). Suppose r < n and let [Λ, n, r, γ] be a simple stratum equivalent to the pure stratum [Λ, n, r, β]. Then, for 0 ≤ m ≤ r −1, let C(Λ, m, β) denote the set of characters θ of H^{m+1}(β) such that

(i) θ|_{H^{m+1}(β)∩B_{β}^{×}} factors through det_{B_{β}};

(ii) θ is normalized by 𝔎(Λ) ∩ B_{β}^{×}; (iii) if m′ = max \{m, [\frac{r}{2}]\}, the restriction θ|_{H^{m′+1}(β)} is of the form θ_{0}ψ_{c} for some θ_{0} ∈ C(Λ, m′, γ), c = β − γ.

In the latter case, for m ≥ r we set C(Λ, m, β) = C(Λ, m, γ).

For θ ∈ C(Λ, m, β), we write IG̃(θ) for the intertwining set IG̃(θ, θ). For 0 ≤ m ≤ r − 1, we also set 𝔪_{m} = 𝔪_{m}(β, Λ) = 𝔞_{r−m} ∩ 𝔫_{−m}(β, Λ) + 𝔍^{[ \frac{r+1}{2} ]}(β).

(3.5) Theorem ([3, (3.3.2)]). Let [Λ, n, 0, β] be a simple stratum in A, r = −k_{0}(β, Λ). Let 0 ≤ m ≤ r − 1 and θ ∈ C(Λ, m, β). Then

IG̃(θ) = (1 + 𝔪_{m})B_{β}^{×}(1 + 𝔪_{m}).

3.2. Simple characters in G

Recall that σ is the involution of G̃ given by x ↦ \overline{x}^{−1} and Σ = \{1, σ\} ⊂ AutG̃. We now look at the situation in our unitary group G = G̃^{Σ}. We consider again a simple stratum [Λ, n, 0, β] with Λ a strict lattice sequence and continue with the notation of the previous section.

(3.6) Lemma. Let [Λ, n, 0, β] be a skew simple stratum in A. Then the groups H^{t}(β, Λ) and J^{t}(β, Λ), t ≥ 0, are fixed by Σ.

Proof. This follows by induction along r = −k_{0}(β, Λ), since in the definitions of ℌ(β, Λ), 𝔍(β, Λ) ( (3.1), (3.2)) we may choose the simple stratum [Λ, n, r, γ] to be skew, by (1.10). □

From now on, we suppose that the stratum [Λ, n, 0, β] is skew. Hence Σ acts on the set of equivalence classes of irreducible representations of H^{t}(β, Λ). For 0 ≤ m ≤ n − 1, we put C^{Σ}(Λ, m, β) = {θ ∈ C(Λ, m, β) : θ^{σ} = θ}. Note that this set is non-empty since card C(Λ, m, β) is a power of p, by [3, (3.3.21)].

We define two families of compact open subgroups of G by

\begin{matrix}H_{−}^{t}(β, Λ) = H^{t}(β, Λ) ∩ G \\ J_{−}^{t}(β, Λ) = J^{t}(β, Λ) ∩ G \end{matrix} \} for t ≥ 0.

Then, since H_{−}^{t}(β, Λ) = H^{t}(β, Λ)^{Σ}, H^{t}(β, Λ) is a pro-p subgroup of G̃ and p ≠ 2 = card Σ, we have Glauberman’s correspondence g (see(2.1)) between the set of equivalence classes of irreducible representations of H^{t}(β, Λ) fixed by Σ and the set of equivalence classes of irreducible representations of H_{−}^{t}(β, Λ). We put C_{−}(Λ, m, β) = {g(θ) : θ ∈ C^{Σ}(Λ, m, β)} and call an element of C_{−}(Λ, m, β) a simple character for G. Note here that, since θ is a character, g(θ) is just the restriction of θ. Remark. We could also have defined the simple characters for G directly, analogously to the definitions (3.3), (3.4) for G̃. These two definitions coincide (see [14, §6.2]).

For 0 ≤ m ≤ r−1, let Q_{m} = Q_{m}(β, Λ) denote the group (1+𝔪_{m}(β, Λ))∩ G, where 𝔪_{m} = 𝔪_{m}(β, Λ) = 𝔞_{r−m} ∩ 𝔫_{−m} + 𝔍^{[ \frac{r+1}{2} ]}(β) as in (3.5).

(3.7) Theorem. Let [Λ, n, 0, β] be a skew simple stratum in A, 0 ≤ m ≤ r − 1 and θ_{−} ∈ C_{−}(Λ, m, β). Then

I_{G}(θ_{−}) = Q_{m} · B_{β} ∩ G · Q_{m}.

Proof. Let θ ∈ C^{Σ}(Λ, m, β) be such that θ_{−} = g(θ). Then, by (2.5), I_{G}(θ_{−}) = IG̃(θ) ∩ G = (1 + 𝔪_{m})B_{β}^{×}(1 + 𝔪_{m}) ∩ G, by(3.5), and this will decompose as required if we can show that (1 + 𝔪_{m}), B_{β}^{×}, Σ satisfy the conditions of [15, (1.3)]. The only condition which is not automatic is that, for b ∈ B_{β}^{×}, we have

(1 + 𝔪_{m})b(1 + 𝔪_{m}) ∩ B_{β}^{×} = ((1 + 𝔪_{m}) ∩ B_{β}^{×})b((1 + 𝔪_{m}) ∩ B_{β}^{×}).

However, we have 𝔟_{β,m} ⊂ 𝔪_{m} ⊂ 𝔞_{m} so, for b ∈ B_{β}^{×}, we have

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

by [3, (1.6.1)]. □

3.3. Semisimple characters in G̃

We now extend the results of the previous sections to semisimple strata, whose definition we now recall (see [15, §3.2]). 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 y_{β} = π_{F}^{n/g}β^{e/g}. We define the characteristic polynomial φ_{β}(X) of the stratum to be the reduction modulo 𝔭_{F} of the characteristic polynomial of y_{β} ∈ A (which lies in 𝔬_{F} [X]). Let V_{i}, i = 1, 2, be subspaces of V such that V = V_{1} ⊕V_{2}. Let 1^{i} denote the projection V → V^{i} with kernel V^{j}, j ≠ i, and put A^{ij} = 1^{i}·A·1^{j}, for i, j = 1, 2. We identify A^{ij} = Hom_{F} (V^{j}, V^{i}) and abbreviate A^{ii} = A^{i}. We use the notation

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

If S is an 𝔬_{F} -lattice in A, we set S^{ij} = L ∩ A^{ij}. We also put M = A^{1} ⊕ A^{2}, M = M^{×}, a Levi subgroup of G̃, N_{u} = 1 + A^{12}, N_{l} = 1 + A^{21} and P_{u} = MN_{u}, P_{l} = MN_{l}. For i = 1, 2, let Λ^{i} be a lattice sequence in V_{i} and put Λ = Λ^{1} ⊕ Λ^{2}, a lattice sequence in V of period e = lcm (e_{1}, e_{2}). Let β_{i} ∈ A^{i} and put n_{i} = −ν_{Λ^{i}}(β_{i}). Then we put β = β_{1} ⊕β_{2} and n = e·max {n_{1}/e_{1}, n_{2}/e_{2}} so that ν_{Λ}(β) = −n. Thus we obtain a stratum [Λ, n, r, β] in A, for any 0 ≤ r ≤ n − 1.

(3.8) Definition. A 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.

(3.9) Definition. (Inductive on the dimension of V.) A stratum [Λ, n, r, β] is called semisimple if either it is a simple stratum and Λ is strict or it is split as above and [Λ^{i}, n_{i}, r_{i}, β_{i}] is a semisimple stratum, for i = 1, 2, where r_{i} = [re_{i}/e].

Let [Λ, n, 0, β] be a semisimple stratum as above and let {𝔞_{i} : i ∈ ℤ} be the filtration on A associated to Λ. We define the 𝔬_{F} -orders ℌ = ℌ(β, Λ) and 𝔍 = 𝔍(β, Λ) inductively on the dimension of V. If [Λ, n, 0, β] is a simple stratum then ℌ and 𝔍 are as defined in (3.1), (3.2). Otherwise, we put

ℌ(β, Λ) = ( \begin{matrix}ℌ(β_{1}, Λ^{1}) & 𝔞_{[ \frac{n}{2} ]+1}^{12} \\ 𝔞_{[ \frac{n}{2} ]+1}^{21} & ℌ(β_{2}, Λ^{2}) \end{matrix} ) ,

𝔍(β, Λ) = ( \begin{matrix}𝔍(β_{1}, Λ^{1}) & 𝔞_{[ \frac{n+1}{2} ]}^{12} \\ 𝔞_{[ \frac{n+1}{2} ]}^{21} & 𝔍(β_{2}, Λ^{2}) \end{matrix} ) .

For t ≥ 0, we put ℌ^{t}(β, Λ) = ℌ(β, Λ) ∩ 𝔞_{t} and 𝔍^{t}(β, Λ) = 𝔍(β, Λ) ∩ 𝔞_{t} and also define the groups H^{t}(β, Λ) = ℌ(β, Λ) ∩ U_{t}(Λ), J^{t}(β, Λ) = 𝔍(β, Λ) ∩ U_{t}(Λ). We often write J(β, Λ) in place of J^{0}(β, Λ). Note that, since 𝔞_{n+1}(Λ) ∩ A^{i} = 𝔞_{[ne_{i}/e]+1}(Λ^{i}), we have ℌ^{t+1}(β, Λ) ∩ A^{i} = ℌ^{[ \frac{te_{i}}{e}]+1}, for i = 1, 2, and likewise for 𝔍, so that H^{t+1}(β, Λ) ∩ M = H^{[ \frac{te_{1}}{e}]+1}(β_{1}, Λ^{1}) × H^{[ \frac{te_{2}}{e}]+1}(β_{2}, Λ^{2}).

(3.10) Lemma. For [\frac{n}{2}] ≤ t ≤ n−1, J(β, Λ) normalizes ψ_{β} on U_{t+1}(Λ).

Proof. The case of a simple stratum is a special case of [3, (3.3.1)] and the general case follows by induction since, for x ∈ 𝔞_{t+1} and putting x = ∑_{i,j=1}^{2} x_{ij} with x_{ij} = 1^{i}x1^{j} ∈ A^{ij}, we have ψ_{β}(1 + x) = ψ_{β_{1}}(1 + x_{11})ψ_{β_{2}}(1 + x_{22}). □

For 0 ≤ m ≤ n − 1, we would now like to define (inductively) a set of semisimple characters C(Λ, m, β) of the group H^{m+1}(β, Λ). If [Λ, n, 0, β] is a simple stratum then this is just the set of simple characters ((3.3), (3.4)). Otherwise, we put m_{i} = [\frac{me_{i}}{e}] and assume we have defined the sets C(Λ^{i}, m_{i}, β_{i}), for i = 1, 2.

(3.11) Definition. Let [Λ, n, 0, β] be a semisimple stratum as above. For 0 ≤ m ≤ n − 1, let C(Λ, m, β) denote the set of characters θ of H = H^{m+1}(β, Λ) such that

(i) θ|_{H∩U_{[ \frac{n}{2} ]+1}(Λ)} = ψ_{β}; (ii) Writing θ|_{H∩M} = θ_{1} ⊗ θ_{2}, we have θ_{i} ∈ C(Λ^{i}, [\frac{me_{i}}{e}], β_{i}), for i = 1, 2.

(3.12) Lemma. Let [Λ, n, 0, β] be a semisimple stratum as above and fix m ∈ ℤ, 0 ≤ m ≤ n − 1. Let θ_{i} ∈ C(Λ^{i}, [\frac{me_{i}}{e}], β_{i}), for i = 1, 2, and put m′ = max \{m, [\frac{n}{2}]\}. Then there exists a unique character θ of H = H^{m+1}(β, Λ) such that θ|_{H∩M} = θ_{1} ⊗ θ_{2} and θ|_{U_{m′+1}(Λ)} = ψ_{β}. Moreover, θ is normalized by J(β, Λ).

Proof. We proceed by induction on the dimension of V. In the case of a simple stratum, the first statement is empty while the second is [3, (3.3.1)]. So assume [Λ, n, 0, β] is split semisimple; in particular, we assume that θ_{i} is normalized by J(β_{i}, Λ^{i}), for i = 1, 2. As H = H ∩ M · U_{m′+1}(Λ), uniqueness is clear so we need only prove existence of such a character θ. Put θ_{M} = θ_{1} ⊗ θ_{2}, a character of H ∩ M. Now H ∩ M ⊂ J(β, Λ), so H ∩ M normalizes the character ψ_{β} of U_{m′+1}(Λ), by (3.10). Moreover, θ_{M} and ψ_{β} agree on U_{m′+1} ∩ M so we define θ by

θ(uh) = ψ_{β}(u)θ_{M}(h), for u ∈ U_{m′+1}(Λ), h ∈ H ∩ M.

For j ∈ J(β, Λ), h ∈ U_{m′+1}, we have θ(jhj^{−1}) = θ(h) by (3.10). Similarly, we have that H normalizes ψ_{β} on J^{[ \frac{n+1}{2} ]}(β, Λ) so that, for h ∈ H, j ∈ J^{[ \frac{n+1}{2} ]}(β, Λ), we have θ(jhj^{−1}) = ψ_{β}(jhj^{−1}h^{−1})θ(h) = θ(h). Hence, to prove that J(β, Λ) normalizes θ, we need only show that J(β, Λ) ∩ M normalizes θ|_{H∩M} = θ_{M}, which is true by induction. □ Note that we also have

θ(h_{l}h_{M}h_{u}) = θ_{M}(h_{M}), for h_{l} ∈ H∩N_{l}, h_{M} ∈ H∩M, h_{u} ∈ H∩N_{u},

since H ∩ N_{l} ⊂ ker ψ_{β} and likewise for H ∩ N_{u}.

Lemma (3.12) says that we have a bijection

(3.13)  C(Λ, m, β) ⟷ C(Λ^{1}, [\frac{me_{1}}{e}], β_{1}) × C(Λ^{2}, [\frac{me_{2}}{e}], β_{2}).

(3.14) Theorem. Let [Λ, n, 0, β] be a semisimple stratum in A and let θ ∈ C(Λ, 0, β). Then

IG̃(θ) = J^{1}B^{×}J^{1},

where B ⊂ M is the centralizer of β in A.

Proof. We proceed by induction on the dimension of V, the simple case being (3.5). We have IG̃(θ) ⊂ IG̃(ψ_{β}|U_{[ \frac{n}{2} ]+1}(Λ)) ⊂ U_{[ \frac{n+1}{2} ]}MU_{[ \frac{n+1}{2} ]}, by [15, (3.9)]. Now, by(3.12), J^{1} normalizes θ and U_{[ \frac{n+1}{2} ]} ⊂ J^{1} so we have IG̃(θ) = J^{1}I_{M}(θ)J^{1}. But I_{M}(θ) ⊂ J^{1}B^{×}J^{1}, by induction, so the result follows. □

3.4. Semisimple characters in G

We return once more to the group G = G̃^{Σ}, with the notation of the previous sections.

(3.15) Definition. A semisimple stratum [Λ, n, r, β] is called skew if (inductive definition, on the dimension) either [Λ, n, r, β] is a skew simple stratum in the sense of(1.7) or V = V_{1} ⊥ V_{2} and each [Λ^{i}, n_{i}, r_{i}, β_{i}] is a skew semisimple stratum, i = 1, 2, where r_{i} = [re_{i}/e].

From now on, we suppose [Λ, n, 0, β] a skew semisimple stratum. The orders ℌ(β, Λ), 𝔍(β, Λ) are fixed by the involution , since this is true in the skew simple case and \overline{𝔞_{t}^{12}} = 𝔞_{t}^{21}, for t ∈ ℤ. Hence the groups H^{t}(β, Λ) and J^{t}(β, Λ), t ≥ 0, are fixed by Σ and, for 0 ≤ m ≤ n − 1, we put C^{Σ}(Λ, m, β) = {θ ∈ C(Λ, m, β) : θ^{σ} = θ}. From(3.13) we have a bijection

C^{Σ}(Λ, m, β) ⟷ C^{Σ}(Λ^{1}, [\frac{me_{1}}{e}], β_{1}) × C^{Σ}(Λ^{2}, [\frac{me_{2}}{e}], β_{2}).

We define compact open subgroups of G by H_{−}^{t}(β, Λ) = H^{t}(β, Λ) ∩ G, J_{−}^{t}(β, Λ) = J^{t}(β, Λ) ∩ G, for t ≥ 0. As before, we have Glauberman’s correspondence g between the set of equivalence classes of irreducible representations of H^{t}(β, Λ) fixed by Σ and the set of equivalence classes of irreducible representations of H_{−}^{t}(β, Λ). We put C_{−}(Λ, m, β) = {g(θ) : θ ∈ C^{Σ}(Λ, m, β)}. Then we have a bijection

C_{−}(Λ, m, β) ⟷ C_{−}(Λ^{1}, [\frac{me_{1}}{e}], β_{1}) × C_{−}(Λ^{2}, [\frac{me_{2}}{e}], β_{2}).

As in the simple case, we also have C_{−}(Λ, m, β) = {θ|_{H_{−}^{m+1}} : θ ∈ C(Λ, m, β)}.

(3.16) Theorem. Let [Λ, n, 0, β] be a skew semisimple stratum in A and let θ_{−} ∈ C_{−}(Λ, 0, β). Then

I_{G}(θ_{−}) = J_{−}^{1} · B ∩ G · J_{−}^{1},

where B ⊂ M is the centralizer of β in A.

Proof. Let θ ∈ C^{Σ}(Λ, 0, β) be such that θ_{−} = g(θ). We proceed by induction on the dimension of V, the simple case being (3.7). We have I_{G}(θ_{−}) ⊂ I_{G}(ψ_{β}^{−}|P_{[ \frac{n}{2} ]+1}(Λ)) ⊂ P_{[ \frac{n+1}{2} ]}(M ∩ G)P_{[ \frac{n+1}{2} ]}, by [15, (3.15)]. Now, by (3.12), J^{1} normalizes θ so that J_{−}^{1} normalizes θ_{−}. Then P_{[ \frac{n+1}{2} ]} ⊂ J_{−}^{1} so we have I_{G}(θ) = J_{−}^{1}I_{M∩G}(θ_{−})J_{−}^{1}. But I_{M∩G}(θ_{−}) ⊂ J_{−}^{1} · B ∩ G · J_{−}^{1}, by induction, so the result follows. □

4. Heisenberg representations

Let [Λ, n, 0, β] be a skew semisimple stratum and let θ_{−} ∈ C_{−}(Λ, 0, β). We would now like to obtain all irreducible representations of J_{−}^{1}(β, Λ) containing the character θ_{−} of H_{−}^{1}(β, Λ). Since J^{1}(β, Λ) is a pro-p subgroup of G̃, we are able to proceed as above via Glauberman’s correspondence. We remark the the results for G̃ hold regardless of the skewness of the stratum.

(4.1) Proposition. Let θ ∈ C(Λ, 0, β). Then the pairing

k_{θ} : (j, j′) ↦ θ([j, j′]), j, j′ ∈ J^{1}(β, Λ),

induces a nondegenerate alternating bilinear form J^{1}/H^{1}.

Proof. We proceed by induction on dim V, the simple case being given by [3, (3.4.1)]. As in the proof of [3, (3.4.1)], we need only show that, for x ∈ 𝔍^{1},

θ([1 + x, 1 + y]) = 1 ∀y ∈ 𝔍^{1} ⟺ x ∈ ℌ^{1},

the implication ⇐ being clear, since J^{m+1} normalizes θ. We certainly have the implication ⇒ for x ∈ 𝔍^{1} ∩ M, by the induction hypothesis, so we suppose that x ∈ 𝔍^{1} ∩ A^{21} = 𝔞_{[ \frac{n+1}{2} ]}^{21} satisfies the

right hand side (the case x ∈ 𝔍^{1} ∩ A^{12} will follow symmetrically). For y ∈ 𝔍^{1} ∩ A^{12}, we have [1 + x, 1 + y] ⊂ U_{n}(Λ) so

1 = θ([1 + x, 1 + y]) = ψ_{β}(1 + xy − yx) = ψ_{F} ∘ tr_{A/F} (βxy − βyx)

= ψ_{F} ∘ tr_{A/F} (x(yβ − βy)).

Now the map y ↦ yβ − βy = yβ_{2} − β_{1}y sends 𝔞_{[ \frac{n+1}{2} ]}^{12} onto 𝔞_{−[ \frac{n}{2} ]}^{12}, by

[5, (3.7) Lemma 1], so we have ψ_{A}(x𝔞_{−[ \frac{n}{2} ]}^{12}) ≡ 1, that is x ∈ (𝔞_{−[ \frac{n}{2} ]}^{12})^{∗} = 𝔞_{[ \frac{n}{2} ]+1}^{21} = ℌ^{1} ∩ A^{21}. □

As usual, we write B for the centralizer of β in A. Note that we have B ⊂ M.

(4.2) Corollary (cf. [3, (5.1.1)]). Let θ ∈ C(Λ, 0, β). Then there exists a unique irreducible representation η of J^{1}(β, Λ) which contains θ. Moreover, η|_{H^{1}(β,Λ)} is a multiple of θ, dim η = (J^{1}(β, Λ) : H^{1}(β, Λ))^{\frac{1}{2}} and IG̃(η) = J^{1}B^{×}J^{1}.

Now let θ ∈ C^{Σ}(Λ, 0, β) and let η be the unique irreducible representation of J^{1}(β, Λ) containing θ. Then η^{σ} is an irreducible representation of J^{1}(β, Λ) containing θ^{σ} = θ so η^{σ} ≃ η by uniqueness. Let η_{−} = g(η) be the irreducible representation of J_{−}^{1}(β, Λ) corresponding to η via Glauberman’s correspondence. It contains θ_{−} = g(θ) by (2.3), since η contains θ with multiplicity dim η, which is odd. Moreover, η_{−}|_{H_{−}^{1}} is a multiple of θ_{−}, since J_{−}^{1}(β, Λ) ⊂ I_{G}(θ_{−}). Remark. For J^{1}/H^{1} considered as a k_{F} -vector space, σ is a linear map which preserves the bilinear form k_{θ}. The space J^{1}/H^{1} then decomposes into orthogonal eigenspaces and the +1-eigenspace is precisely J_{−}^{1}/H_{−}^{1}. In particular, the restriction of k_{θ} to J_{−}^{1}/H_{−}^{1} is nondegenerate and we deduce that η_{−} is the unique irreducible representation of J_{−}^{1} containing θ_{−}.

In order to obtain information on the intertwining of the representation η_{−}, we need more precise information on the intertwining spaces for η. Recall that for ρ a representation of a subgroup K of G̃, the intertwining space I_{g}(ρ|K) = I_{g}(ρ, ρ) is defined to be

I_{g}(ρ|K) = Hom_{^{g}K∩K}(^{g}ρ, ρ),

with the notations of §2.

(4.3) Proposition (cf. [3, (5.1.8)]). Let θ ∈ C(Λ, 0, β) and let η be the representation of J^{1}(β, Λ) given by (4.2). Then, for g ∈ G̃, we have

dim_{ℂ}(I_{g}(η|J^{1})) = \{ \begin{matrix}1 & if g ∈ J^{1}B^{×}J^{1}, \\ 0 & otherwise. \end{matrix}

Proof. As in the proof of [3, (5.1.8)], and since IG̃(θ) = J^{1}B^{×}J^{1}, the proposition will follow if we can show that, for y ∈ B^{×}, J^{1}yJ^{1} is the union of (J^{1} : H^{1}) distinct (H^{1}, H^{1}) double cosets (where (J^{1} : H^{1}) is the group index). This will, in turn, follow from:

(4.4) Lemma (cf. [3, (5.1.10)]). For y ∈ B^{×} we have (J^{1} : J^{1}∩(J^{1})^{y}) = (H^{1} : H^{1} ∩ (H^{1})^{y}).

Proof. We write J^{1} = 1 + 𝔧, H^{1} = 1 + 𝔥. In the simple case, by [3, (3.1.16)], we have an exact sequence

0 ⟶ 𝔟_{1} ⟶ 𝔧 ⟶^{a_{β}} 𝔥^{∗} ⟶^{s} 𝔟_{0} ⟶ 0,

where 𝔟_{i} = 𝔞_{i} ∩ B, for i = 0, 1, a_{β} is the map A → A given by x ↦ βx − xβ and s is a tame corestriction on A (see [3, (1.3)]). Suppose now we are in the semisimple case. Then we have

𝔥^{∗} = ( \begin{matrix}ℌ(β_{1}, Λ^{1})^{∗} & 𝔞_{−[ \frac{n}{2} ]}^{12} \\ 𝔞_{−[ \frac{n}{2} ]}^{21} & ℌ(β_{2}, Λ^{2})^{∗} \end{matrix} ) .

Moreover, by [5, (3.7) Lemma 1] a_{β} induces a bijection 𝔞_{[ \frac{n+1}{2} ]}^{12} → 𝔞_{−[ \frac{n}{2} ]}^{12},

that is between 𝔧^{12} and 𝔥^{∗}^{12} (and likewise for 𝔞^{21}). Hence, by induction, we have an exact sequence

0 ⟶ 𝔟_{1} ⟶ 𝔧 ⟶^{a_{β}} 𝔥^{∗} ⟶^{s} 𝔟_{0} ⟶ 0,

where s is a tame corestriction on A given by

s ( \begin{matrix}a_{11} & a_{12} \\ a_{21} & a_{22} \end{matrix} ) = ( \begin{matrix}s_{1}(a_{11}) & 0 \\ 0 & s_{2}(a_{22}) \end{matrix} ) ,

for s_{1}, s_{2} tame corestrictions on A^{1}, A^{2} respectively. For y ∈ B^{×}, the map a_{β} also induces a bijection between 𝔧^{12} ∩ (𝔧^{12})^{y} and 𝔥^{∗}^{12} ∩ (𝔥^{∗}^{12})^{y}. Hence, as above (and as in the proof of [3, (5.1.10)] in the simple case) we have an exact sequence

0 ⟶ 𝔟_{1} + 𝔟_{1}^{y} ⟶ 𝔧 + 𝔧^{y} ⟶^{a_{β}} 𝔥^{∗} + (𝔥^{∗})^{y} ⟶^{s} 𝔟_{0} + 𝔟_{0}^{y} ⟶ 0.

The lemma now follows exactly as in [3, (5.1.10)]. □

This also completes the proof of (4.3). □

We may now apply (2.4) to obtain the intertwining of η_{−}

(4.5) Proposition. With notation as above, we have I_{G}(η_{−}) = J_{−}^{1} · B ∩ G · J_{−}^{1}.

Proof. Let g ∈ J^{1}B^{×}J^{1}∩G. Since dim_{ℂ}(I_{g}(η|J^{1})) = 1, by(2.4) we have g ∈ I_{G}(η_{−}) so I_{G}(η_{−}) ⊃ J^{1}B^{×}J^{1}∩G ⊃ J_{−}^{1}·B∩G·J_{−}^{1}. But, since η_{−}|_{H_{−}^{1}} is a multiple of θ_{−}, we also have I_{G}(η_{−}) ⊂ I_{G}(θ_{−}) = J_{−}^{1} · B ∩ G · J_{−}^{1}, by (3.7), so there is equality, as required. □

5. Maximal tori

We now consider the extensions of the representation η_{−} of J_{−}^{1} to a representation κ_{−} of J_{−}. For any such extension, the pair (J_{−}, κ_{−}) will be a type and Ind _{J_{−}}^{G}κ_{−} will be an irreducible supercuspidal representation (see (5.2)). Observe that we cannot use Glauberman’s correspondence for this stage since J is not a pro-p group (more precisely, because 2 divides (J : J^{1})).

We continue with the notation of the previous section; in particular, [Λ, n, 0, β] is a skew semisimple stratum.

(5.1) Definition. A simple stratum [Λ, n, r, β] is called maximal if F [β] is a maximal subfield of A. We extend this definition to semisimple strata inductively, as usual.

From now on we suppose that [Λ, n, 0, β] is a maximal skew semisimple stratum. This implies that the centralizer B_{β} ∩G of β in G is a compact maximal torus in G: if we write B_{β}^{×} = ∏_{i} E_{i}^{×} with E_{i} an extension of F, then each E_{i} is stable under the involution , with fixed field E_{i,0} ≠ E_{i}, and we have B_{β} ∩ G = ∏_{i} N_{1}(E_{i}) where N_{1}(E_{i}) = \{e ∈ E_{i}^{×} : e\overline{e} = 1\} is the norm-1 group of the extension E_{i}/E_{i,0}. We write k_{i}, k_{i,0} for the residue fields of E_{i}, E_{i,0} respectively. We have J_{−} = (B_{β} ∩G)J_{−}^{1} so that J_{−}/J_{−}^{1} = ∏_{i} N_{1}(k_{i}), where N_{1}(k_{i}) is the norm-1 group of the extension k_{i}/k_{i,0}. Each group N_{1}(k_{i}) is cyclic, of order 2 if E_{i}/E_{i,0} is ramified, order card k_{i,0} + 1 otherwise.

Let θ_{−} ∈ C_{−}(Λ, 0, β) and let η_{−} be the (unique) irreducible representation of J_{−}^{1} containing θ_{−}, given by (4.2). The group J_{−} normalizes η_{−} and J_{−}/J_{−}^{1} is a product of cyclic groups so there exists an extension of η_{−} to a representation κ_{−} of J_{−}. Moreover, every extension of η_{−} is of the form κ_{−} ⊗ χ, for χ a character of J_{−} obtained by inflation from J_{−}/J_{−}^{1}.

(5.2) Theorem. Let [Λ, n, 0, β] be a maximal skew semisimple stratum in A, θ_{−} ∈ C_{−}(Λ, 0, β) and η_{−} the unique irreducible representation of J_{−}^{1} containing θ_{−}. Let κ_{−} be any extension of η_{−} to a representation of J_{−}^{1}. Then π = Ind _{J_{−}}^{G}κ_{−} is an irreducible supercuspidal representation of G and (J_{−}, κ_{−}) is a [G, π]_{G}-type.

Proof. We have I_{G}(κ_{−}) ⊂ I_{G}(η_{−}) = J_{−}^{1} · B_{β} ∩ G · J_{−}^{1} = J_{−}. The first assertion now follows from e.g. [6, (1.5)], while the second is a direct consequence, by [4, (5.4)]. □

Remarks. 1. The method described above to construct supercuspidal representations essentially starts from a sequence of simple strata [Λ_{i}, n_{i}, 0, β_{i}] of decreasing level. It should be straightforward to allow the final strata to be of level 0, though we do not pursue the matter here.

2. We have constructed supercuspidal representations starting from split strata. In [5, §6], there is also the notion of “relatively split” strata, where a derived stratum [𝔅, n, n − 1, s(b)] is split. It should be possible to use the results there to repeat the above constructions in this case also.

3. It is possible to do much of the above construction in the more general setting of §2, that is Γ ⊂ AutG̃ is an l-group, l ≠ p, and G = G̃^{Γ}. Indeed, the essential tools (2.4) and [15, (1.3)] are given in this generality (see also the remark following (1.11)).

6. Skew simple characters in G̃

In this technical section we examine the simple characters θ in G̃ which are fixed by the involution σ and show that we may choose the stratum defining θ to be skew. In particular, this implies that we have considered all such characters in §3. The techniques used here rely heavily on the results from [3, §3.5]. Let [Λ, n, m, β] be a simple stratum, with Λ a strict lattice sequence, and let θ ∈ C(Λ, m, β). Let E = F [β], B = C_{A}(E) and 𝔟_{n} = 𝔞_{n}(Λ) ∩ B, as usual, and write Λ_{𝔬_{E}} when we are thinking of Λ as an 𝔬_{E}-lattice sequence. By [3, (3.3.17)], the G̃-normalizer of θ is

NG̃(θ) = 𝔎(Λ_{𝔬_{E}})(1 + 𝔪_{m}),

where 𝔪_{m} = 𝔞_{r−m}∩𝔫_{−m}+𝔍^{[ \frac{r+1}{2} ]}(β). In particular, the unique maximal compact subgroup of NG̃(θ) is ^{0}NG̃(θ) = U(Λ_{𝔬_{E}})(1 + 𝔪_{m}). Let L ⊂ V be an 𝔬_{F} -lattice stabilized by ^{0}NG̃(θ). Then L is an 𝔬_{E}- lattice since 𝔬_{E}^{×} ⊂ ^{0}NG̃(θ). Consider the strict 𝔬_{E}-lattice sequence Λ_{L}(n) = 𝔭_{E}^{n}L and put 𝔞_{L} = 𝔞_{0}(Λ_{L}). Then 𝔞_{L} ∩ B is a maximal 𝔬_{E}- order in B and 𝔟_{0} ⊂ 𝔞_{L}∩B since U(Λ_{𝔬_{E}}) normalizes Λ_{L}. Hence 𝔞_{0} ⊂ 𝔞_{L}. In particular, 𝔞_{0} ⊂ Stab L. This is true for all lattices stabilized by ^{0}NG̃(θ) so we have

𝔞_{0} = ⋂_{L} Stab L,

where the intersection is taken over all lattices stabilized by ^{0}NG̃(θ). In particular, 𝔞_{0} is determined by ^{0}NG̃(θ), hence by θ. Then, since Λ is strict, Λ is also determined by θ. Moreover, the integers n, m are also determined by θ, since n = min {k ∈ ℕ : U_{k}(Λ) ⊂ ker θ} and m = max {k ∈ ℤ, k ≥ 0 : U_{k+1}(Λ) ⊃ H}, where H is the subgroup on which θ is defined.

(6.1) Lemma. Let [Λ, n, m, β] be a simple stratum, with Λ a strict lattice sequence, and let θ ∈ C(Λ, m, β). Suppose that θ is fixed by σ. Then Λ is self-dual.

Proof. We also have that [Λ^{#}, n, m, −\overline{β}] is a simple stratum and θ = θ^{σ} ∈ C(Λ^{#}, m, −\overline{β}). But θ determines Λ so we have Λ = Λ^{#}, as required. □

We require one more preliminary lemma.

(6.2) Lemma. Let [Λ, n, m, β] be a skew simple stratum in A and let θ be a simple character in C^{Σ}(Λ, m, β). Then there exists a simple character θ_{0} ∈ C^{Σ}(Λ, m − 1, β) which restricts to θ on H^{m+1}(β, Λ).

Proof. Consider the set of θ′ ∈ C(Λ, m − 1, β) which restricts to θ. By [3, (3.3.21)], this set has cardinality a power of p. But Σ acts on it and, since p ≠ 2, there is a fixed point. □

We may now state the main result of this section.

(6.3) Theorem. Let [Λ, n, m, β] be a simple stratum, with Λ a strict lattice sequence, and let θ ∈ C(Λ, m, β). Suppose that θ is fixed by σ. Then there exists a skew simple stratum [Λ, n, m, γ] such that θ ∈ C(Λ, m, γ).

Proof. By (6.1), we already have that Λ is self-dual. We proceed by induction along k_{0}(β, Λ) so we assume first that β is minimal. If m ≥ [\frac{n}{2}] then ψ_{β} = θ = θ^{σ} = ψ_{−\overline{β}} so we have β + \overline{β} ∈ 𝔞_{−m}. Then, by (1.10), there exists a skew stratum [Λ, n, m, γ] equivalent to [Λ, n, m, β] and θ = ψ_{γ} ∈ C(Λ, m, γ) as required. Assume now m < [\frac{n}{2}]. We have H^{[ \frac{n}{2} ]+1}(β, Λ) = U_{[ \frac{n}{2} ]+1}(Λ); the restriction of θ to this group is ψ_{β}, while the restriction of θ^{σ} is ψ_{−\overline{β}}. Then, since θ is fixed by σ, we have β+\overline{β} ∈ 𝔞_{−[ \frac{n}{2} ]}. By(1.10), there exists a skew simple stratum [Λ, n, [\frac{n}{2}], γ] equivalent to [Λ, n, [\frac{n}{2}], β] and γ is minimal by [3, (2.1.4)]. By [3, (3.1.9)], we have H^{m+1}(γ, Λ) = H^{m+1}(β, Λ) and, furthermore, we have that θ|U_{[ \frac{n}{2} ]+1}(Λ) = ψ_{β} = ψ_{γ}.

To conclude that θ ∈ C(Λ, m, γ) we need only show that θ|U^{m+1}(Λ_{𝔬_{F [γ]}}) factors through the determinant, or, equivalently by [3, (2.4.11)], that it is intertwined by all of B_{γ}^{×}. But the intertwining of the equivalent simple strata [Λ, n, [\frac{n}{2}], γ] and [Λ, n, [\frac{n}{2}], β] is

U_{[ \frac{n+1}{2} ]}(λ)B_{γ}^{×}U_{[ \frac{n+1}{2} ]}(λ) = U_{[ \frac{n+1}{2} ]}(β)B_{γ}^{×}U_{[ \frac{n+1}{2} ]}(β),

by [3, (1.5.8)]. But this is precisely the intertwining of θ, by [3, (3.3.2)], so θ|U^{m+1}(Λ_{𝔬_{F [γ]}}) is indeed intertwined by all of B_{γ}^{×}, as required.

We now assume k_{0}(β, Λ) = −r > −n. We will first reduce to the case m = r − 1 so we suppose m < r − 1 and that we have the result for m′ > m In particular we have that [Λ, n, m + 1, β] is a simple stratum. The restriction θ̃ = θ|_{H^{m+2}(β)} is a simple character in C(Λ, m+1, β) fixed by Σ so, by induction, we have a skew simple stratum [Λ, n, m + 1, γ′] with θ̃ ∈ C(Λ, m+1, γ′). By [3, (3.5.9)], we have H^{m+1}(β) = H^{m+1}(γ′) Let θ_{1} ∈ C^{Σ}(Λ, m, γ′) be such that θ_{1}|_{H^{m+2}} = θ|_{H^{m+2}}. In particular, we have θ = θ_{1}ψ_{b} for some b ∈ 𝔞_{−1−m} and, since θ, θ_{1} are both fixed by Σ, ψ_{b} = ψ_{−\overline{b}} on H^{m+1}. Hence ψ_{\frac{b−\overline{b}}{2}} = ψ_{b} on H^{m+1} and we may

assume b ∈ A_{−}. Let s be a tame corestriction on A relaive to F [γ′]/F; then, as in the proof of [3, (3.5.9)], we have s(b) ∈ F [γ′] + 𝔟_{γ′,−m}. Hence [Λ_{𝔬_{F [γ′]}}, m + 1, m, s(b)] is simple. Then by (1.11), there exists a skew simple stratum [Λ, n, m, γ] equivalent to [Λ, n, m, γ′ + b]. By [3, (3.3.20)], we have θ = θ_{1}ψ_{b} ∈ C(Λ, m, γ) as required.

Thus we assume m = r − 1. Let [Λ, n, m + 1, ξ] be a simple stratum equivalent to [Λ, n, m + 1, β]. Then θ̃ = θ|_{H^{m+2}} ∈ C^{Σ}(Λ, m + 1, ξ) so, by induction, there exists a skew simple stratum [Λ, n, r, ξ′] with θ̃ ∈ C(Λ, m + 1, ξ′). By [3, (3.5.9)], we have H^{m+1}(β) = H^{m+1}(ξ) = H^{m+1}(ξ′) and we may alter ξ to assume that C(Λ, m, ξ) = C(Λ, m, ξ′).

Let ϕ ∈ C^{Σ}(Λ, m, ξ′) be such that ϕ|_{H^{m+2}} = θ̃. Let b ∈ 𝔞_{−m−1} be such that θ = ϕψ_{b}; as in the previous case, since θ, ϕ are both fixed by Σ, we may assume b ∈ A_{−}. Let s, s′ be tame corestrictions on A relative to F [ξ]/F, F [ξ′]/F respectively. By [3, (3.5.13)], [Λ_{F [ξ]}, m + 1, m, s(b)] is equivalent to a simple stratum. Let I be the G-intertwining of θ and put R = (I ∩ 𝔞_{0}) + 𝔞_{1}/𝔞_{1}. Then, as in [3, (3.5.14)], we have

R = {x ∈ 𝔟_{ξ,0}/𝔟_{ξ,1} : xs(b) ≡ s(b)x (mod 𝔟_{ξ,−m})}
= {x ∈ 𝔟_{ξ′,0}/𝔟_{ξ′,1} : xs′(b) ≡ s′(b)x (mod 𝔟_{ξ′,−m})}.

Since [Λ_{F [ξ]}, m + 1, m, s(b)] is equivalent to a simple stratum, R is a semisimple k_{F} -algebra, by [3, (2.4.13)], and hence [Λ_{F [ξ′]}, m+1, m, s′(b)] is also equivalent to a simple stratum. Then, by (1.11), [Λ, n, m, ξ′ + b] is equivalent to a skew simple stratum [Λ, n, m, γ] and θ ∈ C(Λ, m, γ) by [3, (3.3.20)]. □

Since here we rely only on the results of [3, §3.5], together with (1.10) and (1.11), the above results will remain valid in the situation of an l-group Γ acting on G̃, G = G̃^{Γ} (see the remark following (1.11)).

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