The supercuspidal representations of p-adic classical groups

Shaun Stevens^{∗}

12 November 2007

Abstract

Let G be a unitary, symplectic or special orthogonal group over a locally compact nonarchimedean local field of odd residual characteristic. We construct many new supercuspidal representations of G, and Bushnell-Kutzko types for these representations. Moreover, we prove that every irreducible supercuspidal representation of G arises from our constructions.

Introduction

Let G be a unitary, symplectic or special orthogonal group over a locally compact non-archimedean local field of odd residual characteristic. This article completes a series of papers (in particular [31, 32]) whose major goal was the construction of all irreducible supercuspidal representations of G. More precisely, in this paper, we construct pairs (J, λ), where J is a compact open subgroup of G and λ is an irreducible representation of J, such that the compactly induced representation c-Ind _{J}^{G}λ is irreducible and supercuspidal. Moreover, any irreducible supercuspidal representation π of G contains some such pair, in the sense that λ is a component of the restriction of π to J, and hence π ≃ c-Ind _{J}^{G}λ. In particular, this shows that every irreducible supercuspidal representation of G can be obtained by irreducible compact induction from a compact open subgroup.

The study of the representation theory of p-adic groups by restriction to compact open subgroups goes back to the work of Howe [17] and, for the general linear group in arbitrary (residual) characteristic, was completed by Bushnell and Kutzko in [8]. For classical groups G in odd residual characteristic, we can go back to the work of Moy on U(2, 1) [23] and GSp_{4} [24, 25], and, for a general classical group, of Morris in a series of papers (see, for example, [19, 20]) which laid much of the necessary groundwork. More recently, results have been obtained for arbitrary connected reductive groups. The level-zero representations – that is, those representations which contain the trivial character on restriction to the pro-unipotent radical of some parahoric subgroup – were classified independently by Morris [22] and Moy and Prasad [26]. For positive-level representations in the so-called tame case, the most general constructions of supercuspidal representations are due to Yu [33]. Recently in [18], Kim has proved that, under some restrictive conditions on the ground field (dependent on the group G), Yu’s constructions do give all supercuspidal representations. Although our work here is not for such a general class of group, it has the notable advantage that the only restriction on the field is that the residual characteristic be odd.

∗The research for this paper was supported by EPSRC grant GR/T21714/01.

Our work for classical groups G follows the spirit, and the methods, of Bushnell and Kutzko’s work [8]. So we begin with a non-archimedean local field F of odd residual characteristic with a (possibly trivial) galois involution with fixed field F_{0}, and a finite-dimensional F -vector space V equipped with a nondegenerate ε-hermitian form h. Then G is the group of F_{0}-rational points of the connected component of the linear algebraic group preserving the form. It will also be useful to have the full group G^{+} of automorphisms of V preserving the form. The construction of supercuspidal representations begins with certain local data, namely a skew semisimple stratum [Λ, n, 0, β] (see §2.1). Here we recall only that Λ is a lattice sequence in V and β is an endomorphism of V which is skew for the adjoint involution induced by h and such that E = F [β] is a direct sum of fields each preserved by this involution – that is, β is an elliptic semisimple element of the Lie algebra of G. We write G_{E} for the centralizer of β in G; this is a product of classical groups over extensions of F_{0}. The lattice sequence Λ defines a compact open subgroup P(Λ) in G, together with a filtration, and we write P(Λ_{𝔬_{E}}) for the intersection P(Λ)∩G_{E}.

In [32], we defined the set C_{−}(Λ, 0, β) of skew semisimple characters attached to a skew semisimple stratum, generalizing the notion of simple character from [8]. These are certain arithmetically defined abelian characters θ of a compact open subgroup H^{1}(β, Λ) with nice intertwining properties. Moreover, for fixed β but varying lattice sequence Λ, these semisimple characters exhibit remarkable functorial properties, known as transfer properties – again, these generalize such properties for general linear groups. Attached to our stratum, there are two other compact open subgroups J(β, Λ) ⊇ J^{1}(β, Λ) containing H^{1}(β, Λ) and normalizing each semisimple character θ. Moreover, it is shown in [32] that there is a unique irreducible representation η of J^{1}(β, Λ) (a Heisenberg extension) containing a given semisimple character θ, that the intertwining of η is the same as that of θ and, moreover, that every intertwining space is 1-dimensional.

Our first task in this article is to extend the representation η to the group J(β, Λ) – this is known as the β-extension problem. In fact, finding such an extension is not difficult – given the formalism and transfer properties of semisimple characters, the same arguments as in the case of general linear groups in [8] can be applied. The major difficulty is in knowing whether we have a suitable extension – in general, many of the extensions are not at all appropriate for our purposes. For GL_{n}(F) (or, more generally, general linear groups over a division algebra – see [27]) the crucial property of β-extensions is that their intertwining is the same as that of the representation η which they extend. In our situation for classical groups, we have not been able to prove that any extensions have this property. Instead, we characterize β-extensions by certain compatibility (or transfer) properties, beginning with the case where the group P(Λ_{𝔬_{E}}) is a maximal compact open subgroup of G_{E}, analogously to the techniques in [8]. However, even these compatibilities have subtleties which do not appear for GL_{n}(F). In the case where the order P(Λ_{𝔬_{E}}) is maximal, we actually extend η to the group J^{+}(β, Λ), which is the group analogous to J(β, Λ) in G^{+} – the representation of J(β, Λ) we seek will then be given by restriction. The methods of [8] generalize to our case and we can construct an extension κ of η whose restriction to a pro-p Sylow subgroup of J^{+}(β, Λ) is determined by the transfer properties of semisimple characters. Indeed, in most cases any extension will do – the only exception is in the case where one of the simple components of the quotient J^{+}(β, Λ)/J^{1}(β, Λ), which is a product of reductive groups over finite fields but may be disconnected, is isomorphic to SL_{2}(𝔽_{3}). This is similar to the situation for GL_{n}(F), where the only difficulty is with GL_{2}(𝔽_{2}).

For general Λ, we choose a lattice sequence Λ^{M} such that P(Λ_{𝔬_{E}}^{M}) is a maximal compact subgroup of G_{E} containing P(Λ_{𝔬_{E}}) and let κ_{M} be a β-extension to J^{+}(β, Λ^{M}) corresponding to the semisimple character in C_{−}(Λ^{M}, 0, β) which is the transfer of θ. We observe here that, unlike the case of GL_{n}(F), this does not imply that the corresponding subgroups of G satisfy P(Λ^{M}) ⊇ P(Λ). This complication has already been encountered for general linear groups over division algebras in [27]. In the case where we do have this inclusion, we may follow [8] and define a β-extension κ to J^{+}(β, Λ) to be the (unique) one which satisfies an induction compatibility with respect to κ_{M}, namely

Ind _{J^{+}(β,Λ)}^{P^{+}(Λ^{M})}κ ≃ Ind _{P^{+}(Λ_{𝔬_{E}})J^{1}(β,Λ^{M})}^{P^{+}(Λ^{M})}(κ_{M}|_{P^{+}(Λ_{𝔬_{E}})J^{1}(β,Λ^{M})}),

where P^{+}(Λ) is the compact open subgroup of G^{+} defined by Λ. When we do not have the containment of subgroups, we imitate [27] and proceed in steps from Λ to Λ^{M} by tracing a line through the Bruhat-Tits building of G and cutting it into sufficiently small pieces. Here we are using the description of the building in terms of lattice functions (see [6] or §2.2).

There is an important difference with previous work to note here, namely that the different possible choices for Λ^{M} are not conjugate in G^{+} (and, in general, not in the group of similitudes either). This gives the added complication that the definition of β-extension depends a priori on the choice of Λ^{M}. Indeed, it is easy to see that different choices can give different numbers of β-extensions. It is not clear whether or not there is, in general, an extension of η to J^{+}(β, Λ) which is a β-extension for every choice of Λ^{M} and this is at the heart of the problem of computing intertwining. On the other hand, we do show (Corollary 6.13) that β-extensions for different choices of Λ^{M} cannot differ by too much. It is essentially this same problem which prevents us from adopting the “bottom-up” approach to β-extensions of [27], beginning with the case when P(Λ_{𝔬_{E}}) is an Iwahori subgroup. There is, nonetheless, a “canonical” choice for Λ^{M}. (In fact, there are two choices which could be described as canonical, and we choose one of them – see (4.11).) We call β-extensions defined relative to this canonical choice standard β-extensions. The point here is that P(Λ_{𝔬_{E}}^{M}) is chosen so that the Weyl group of its reductive quotient is as large as possible – so it is “as close as possible” to being special. Using this large Weyl group, and an Iwahori decomposition with respect to a suitable Levi subgroup of G, we are able to compute a lower bound for the intertwining of standard β-extensions (see Proposition 6.14), which is good enough for our purposes.

Now we add a level zero piece. Let J^{o}(β, Λ) denote the inverse image in J(β, Λ) of the connected component of the reductive quotient J(β, Λ)/J^{1}(β, Λ) ≅ P(Λ_{𝔬_{E}})/P^{1}(Λ_{𝔬_{E}}); likewise, we write P^{o}(Λ_{𝔬_{E}}) for the inverse image in P(Λ_{𝔬_{E}}), which is a parahoric subgroup of G_{E}. Let ρ be the inflation to J^{o}(β, Λ) of an irreducible cuspidal representation of the quotient P^{o}(Λ_{𝔬_{E}})/P^{1}(Λ_{𝔬_{E}}) and set λ = κ ⊗ ρ. We can now bound above the G-intertwining of λ (see § 6.4). The key point is that the intertwining of the restriction of κ to a pro-p Sylow subgroup is the same as that of η so we can first bound the intertwining in terms of the intertwining of the restriction of ρ to a pro-p Iwahori subgroup of P^{o}(Λ_{𝔬_{E}}). Now this intertwining can be bounded by a slight weakening of the hypotheses of a result of Morris in [21] (see Proposition 1.1). The idea that knowing the intertwining of the restriction to the pro-p Iwahori can be enough to bound intertwining has been used already in a special case by Blondel (see [3, page 553]). Of course, this only works because the level zero piece ρ is cuspidal.

In the special case where P(Λ_{𝔬_{E}}) is a maximal compact subgroup of G_{E}, the intertwining of λ is found to be just J(β, Λ). In particular, if τ is any irreducible representation of J(β, Λ) whose restriction to J^{o}(β, Λ) contains cuspidal ρ, then the compactly induced representation

π = c-Ind _{J(β,Λ)}^{G}(κ ⊗ τ)

is irreducible and hence supercuspidal. Moreover, the pair (J(β, Λ), κ ⊗ τ) is a [G, π]_{G}-type, in the sense of Bushnell-Kutzko [9].

Finally, it remains to show that we have constructed every supercuspidal representation of G in this way. The ideas are similar to those in [8] (see also [28] for the case of general linear groups over division algebras) though the details are somewhat more complicated. From [32], we know already that any irreducible supercuspidal representation π contains a semisimple character θ. Then, possibly after changing the lattice sequence Λ, it is not hard to see that π must contain a representation ϑ = κ ⊗ ρ of J^{o}(β, Λ), for κ a standard β-extension and ρ a cuspidal representation. The idea then is to show that, if the compact subgroup P(Λ_{𝔬_{E}}) is not maximal then π has a non-zero Jacquet module, contradicting the cuspidality of π. In order to construct such a Jacquet module we use Bushnell-Kutzko’s method of covers. There is a (choice of) Levi subgroup M associated to the stratum such that, for any parabolic subgroup P = MU with Levi factor M, the group J^{o}(β, Λ) has an Iwahori decomposition with respect to (M, P). We form the group

J_{P}^{o} = H^{1}(β, Λ) (J^{o}(β, Λ) ∩ P)

and let ϑ_{P} be the natural representation of J_{P}^{o} on the (J^{o}(β, Λ) ∩ U)-fixed vectors of ϑ. Then π certainly also contains ϑ_{P}. Moreover, the pair (J_{P}^{o} , ϑ_{P}) is a decomposed pair – that is, ϑ_{P} restricts trivially to the unipotent radical of any parabolic subgroup with Levi component M, and the restriction ϑ_{P} |_{J_{P}^{o} ∩M} is irreducible. In order to show that (J_{P}^{o} , ϑ_{P}) is a cover, it remains to show that there is an invertible element of the Hecke algebra H(G, ϑ_{P}) supported on J_{P}^{o} ζJ_{P}^{o} , for some strongly (P, J_{P}^{o} )-positive element ζ of the centre of M. Then the theory of covers shows that π has a non-zero Jacquet module.

We note that, since the compact open subgroup P(Λ_{𝔬_{E}}) is non-maximal, the reductive quotient P^{o}(Λ_{𝔬_{E}})/P^{1}(Λ_{𝔬_{E}}) has a simple component isomorphic to some general linear group (over an extension of the residue field of F_{0}). There is then an involution on these simple components, induced by the Weyl group elements introduced earlier. We will need to divide into two cases:

(i) If the component of ρ on some such general linear group is not fixed by the involution, then the bound on intertwining from above gives the bound

I_{G}(ϑ_{P}) ⊆ J_{P}^{o} M′J_{P}^{o} ,

for some proper Levi subgroup M′ of G containing M. Then it is straightforward that (J_{P}^{o} , ϑ_{P}) is a cover of (J_{P}^{o} ∩ M′, ϑ_{P} |_{J_{P}^{o} ∩M′}) (see §7.2.1).

(ii) Otherwise every component of ρ is fixed by the involution and we use an idea of Kutzko (which was also used in [16]). If Λ^{M} is a lattice sequence such that P(Λ_{𝔬_{E}}^{M}) is a maximal compact subgroup of G_{E} containing P(Λ_{𝔬_{E}}) then, thinking of ρ as a representation of P(Λ_{𝔬_{E}}), we get a support-preserving injection of Hecke algebras (see (7.3))

H(P(Λ_{𝔬_{E}}^{M}), ρ ⊗ χ) ↪ H(G, ϑ_{P}),

for some self-dual character χ (which comes from the problem of incompatibility of β-extensions relative to different maximal compact subgroups of G_{E}). Moreover, the Hecke algebra on the left, which is just in level zero, is understood via the very general results of Morris in [21]. In particular, it contains an invertible function supported on a certain Weyl group element (see §7.2.2 for details). If we do this for two choices of Λ^{M}, we obtain two invertible elements of H(G, ϑ_{P}) whose convolution is supported on a single double coset J_{P}^{o} ξJ_{P}^{o} , with ξ ∈ M. Raising this to a suitable power, we obtain the invertible function we were seeking.

We end with some remarks on questions which are still unanswered. Firstly, although we present here a construction which gives all irreducible supercuspidal representations of G, we do not know when two apparently distinct constructions give rise to equivalent representations. More precisely, we do not have an “intertwining implies conjugacy” result for the types we construct – indeed, we do not even have such a result for the strata underlying the types. Secondly, there still remains the problem of constructing types for the non-supercuspidal components of the Bernstein spectrum. Types have been constructed in certain special cases: see [3] and [16], where types are constructed for self-dual supercuspidal representations of the Siegel Levi subgroup; and [4], where (sometimes putative) covers in smaller symplectic groups are “propagated” to larger ones – note that our work here shows that the hypothesis (H4) of [4] can always be satisfied. There are also some results in this direction in [14], where it is shown that any smooth irreducible representation of G contains some (suitably generalized) semisimple character. We remark that, throughout this work and its predecessor [32], numerous covers are constructed. It seems likely that a more careful examination, especially of the Hecke algebras in level zero used in proving exhaustion, should give types in all cases.

Acknowledgements I would like to thank: David Goldberg, and Purdue University, for their hospitality at an early stage of this work; the University of East Anglia, for granting me study leave, which gave me the chance finally to write up all the details of the arguments; and, in no particular order, Colin Bushnell, Guy Henniart, Corinne Blondel, Phil Kutzko and Vincent Sécherre, for their support, advice, and interest in this work at various times. I would also like to thank Corinne Blondel, Michitaka Miyauchi and the referee for their careful reading, and for pointing out some errors and missing explanations.

Contents

1 A variant of a result of L. Morris  6
2 Notations and preliminaries  7
2.1 Strata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Buildings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3 Semisimple characters and Heisenberg extensions  13
3.1 Semisimple characters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.2 Heisenberg extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.3 Characters of Sylow subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.4 Analogous results for G̃ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
4 Beta extensions  19
4.1 Beta-extensions in the maximal case . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
4.2 Beta-extensions in the general case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
4.3 Analogous results for G̃ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

5 Iwahori factorizations  25
5.1 Iwahori decompositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
5.2 Iwahori factorization of semisimple characters . . . . . . . . . . . . . . . . . . . . . . 28
5.3 Glauberman transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
6 Intertwining and supercuspidal representations  34
6.1 A Hecke algebra isomorphism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
6.2 Some Weyl group elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
6.3 Intertwining of beta extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
6.4 Intertwining and supercuspidal types . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
7 Exhaustion  44
7.1 A Hecke algebra injection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
7.2 Covers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
7.2.1 The case ρ ≄ ρ ∘ σ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
7.2.2 The case ρ ≃ ρ ∘ σ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
7.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

1 A variant of a result of L. Morris

In this section, we give a mild weakening of the hypotheses in a result of Lawrence Morris ([21, Proposition 4.13]). Since this result is valid in great generality, we retain all the notation of op. cit.; in particular, the notation of this section is independent of that in the rest of the paper. We also refer the reader to op. cit. (or to [11], [12]) for details of the situation, which we recall briefly below.

Let G be a connected reductive group over a non-archimedean local field F and let G = G(F) be its group of F -points. Let T be a maximal F -split torus in G and let N be its normalizer; we also write T = T (F) and N = N(F). Let B be an Iwahori subgroup of G, which is the stabilizer of a chamber in the apartment of the enlarged Bruhat-Tits building of G associated to T. Then (G, B, N) is a generalized affine BN-pair. Associated to this affine BN-pair, we have the generalized affine Weyl group W = N/B ∩ N; it is the semi-direct product of a subgroup Ω and a normal subgroup W′ which is the affine Weyl group for some split affine root system Σ. We also have a set S of fundamental reflections in W′ corresponding to a simple system of affine roots. Given any proper subset J ⊆ S, we let W_{J} be the subgroup of W generated by the reflections in J and let N_{J} be the corresponding subgroup of N. Then P_{J} = BN_{J}B is a subgroup of G, which we call a standard parahoric subgroup. For J, K subsets of S, Morris defines in [21, §3.10] (see also loc. cit. Variant 3.22) a (non-unique, in general) set of distinguished double coset representatives for W_{J}\W/W_{K}, which we denote D_{J,K}. We can then choose D′_{J,K} to be a set of inverse images in N of D_{J,K}, and this is a set of distinguished double coset representatives for P_{J}\G/P_{K}. Given P_{J} a standard parahoric subgroup of G, we denote by U_{J} its pro-unipotent radical, and by M_{J} the quotient P_{J}/U_{J}, (the points of) a connected reductive group over a finite field. We write U_{B} for the pro-radical U_{∅} of B = P_{∅}.

Finally, for ρ a representation of a compact subgroup K of G, we write H(ρ|K) = End_{G} (c-Ind _{K}^{G}ρ) for the associated Hecke algebra and I_{G}(ρ|K) for the intertwining of ρ, which is the support of the Hecke algebra.

Proposition 1.1 (cf. [21, Proposition 4.13]). Let J be a proper subset of S, let D′_{J,J} be a set of distinguished double coset representatives for P_{J}\G/P_{J} and let n ∈ D′_{J,J}. Let ρ be the inflation to P_{J} of a cuspidal representation of M_{J}. If n lies in the support of H(ρ|U_{B}), then wJ = J, where w ∈ W is the projection of n.

Remark The only difference between this and [21, Proposition 4.13] is that the hypothesis there is that n lies in the support of H(ρ|P_{J}).

Proof Write P = P_{J} and let P′ denote nP n^{−1}, with radical U′, and U′_{B} = nU_{B}n^{−1}. We note that we have P ∩ U′ ⊆ U_{J∩wJ} ⊆ U_{B}, by [21, Lemma 3.21], and U′ ⊆ U′_{B} so we have P ∩ U′ ⊆ U_{B} ∩ U′_{B}. Then, if we replace every P ∩ P′ in the proof of op. cit. Proposition 4.13 by U_{B} ∩ U′_{B}, the proof is identical. ■

The interest in weakening the hypothesis to one which says only that the restriction of ρ to U_{B} is intertwined by n is suggested by the work of Blondel in [3], in which she is able to compute sufficient information on intertwining without calculating it completely (op. cit. Proposition IV.1 and Lemma IV.2). We will wish to use it in two different cases:

(i) The case when P = P_{J} is a non-maximal parahoric subgroup of G. In this case, let ^{c}Φ_{J} denote the closure of the root system generated by the gradients of the affine roots in J and let 𝔐_{J} = 𝔐_{J}(F) be the group of points of the corresponding reductive subgroup of G (see [21, §3.15]). Then 𝔐_{J} ∩ P_{J} is a parahoric subgroup of 𝔐_{J} with reductive quotient M_{J} again. We also put N_{J} = N ∩𝔐_{J} ∩P_{J}. Then a set of representatives for {n ∈ N : wJ = J}/N_{J} can be taken in the normalizer N_{N}(𝔐_{J} ∩ P_{J}) of 𝔐_{J} ∩ P_{J} in N (see [21, §4.16]).

(ii) The case where P = P_{J} is a maximal parahoric subgroup of G. In this case, the conclusion from the previous lemma is that n normalizes P (see [21, Appendix A.1]). In the classical groups we will be looking at, the G-normalizer of P is compact.

We will also need to use the following stronger result from [21]:

Theorem 1.2 ([21, Theorem 4.15]). Let J be a proper subset of S and let ρ be the inflation to P_{J} of a cuspidal representation of M_{J}. Then I_{G}(ρ) ⊆ P_{J}N_{N}(ρ)P_{J}, where N_{N}(ρ) = {n ∈ N_{N}(𝔐_{J} ∩ P_{J}) : nρ ≃ ρ}.

2 Notations and preliminaries

After setting up the notation for the remainder of this paper, we will recall some of the basic objects needed in the theory. We give quite a rapid overview, and refer the reader to [8] and [32] for more details.

2.1 Strata

Let F be a non-archimedean local field equipped with a galois involution with fixed field F_{0}; we allow the possibility F = F_{0}. Let 𝔬_{F} be the ring of integers of F, 𝔭_{F} its maximal ideal and k_{F} = 𝔬_{F} /𝔭_{F} the residue field; we assume throughout that the residual characteristic p := char k_{F} is not 2. We denote by 𝔬_{0}, 𝔭_{0}, k_{0} the same objects in F_{0}, and will use similar notation for any non-archimedean local field. We fix a uniformizer ϖ_{F} of F such that \overline{ϖ_{F}} = −ϖ_{F} if F/F_{0} is ramified, \overline{ϖ_{F}} = ϖ_{F} otherwise. We put ϖ_{0} = ϖ_{F}^{2} if F/F_{0} is ramified, ϖ_{0} = ϖ_{F} otherwise; so ϖ_{0} is a uniformizer of F_{0}. We will use analogous notation for any field extension E of F to which the involution extends.

Let V be an N-dimensional vector space over F, equipped with a nondegenerate ϵ-hermitian form, with ϵ = ±1. We put A = End_{F} V and denote by the adjoint (anti-)involution on A induced by h. Set also G̃ = Aut_{F} V and let σ be the involution given by g ↦ \overline{g}^{−1}, for g ∈ G̃. We also have an action of σ on the Lie algebra A given by a ↦ −\overline{a}, for a ∈ A. We put Σ = \{1, σ\}, where 1 acts as the identity on both G̃ and A. We put G^{+} = G̃^{Σ} = {g ∈ G̃ : h(gv, gw) = h(v, w) for all v, w ∈ V }, the F_{0}-points of a unitary, symplectic or orthogonal group G^{+} over F_{0}. Let G be the F_{0}-points of the connected component G of G^{+}, so that G = G^{+} except in the orthogonal case. We put A_{−} = A^{Σ} ≃ Lie G. In general, for S a subset of A, we will write S_{−} or S^{−} for S ∩ A_{−}, and, for H̃ a subgroup of G̃, we will write H for H̃ ∩ G.

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}}, a character of the additive group of F with conductor 𝔭_{F}, and ψ_{A} = ψ_{F} ∘ tr_{A/F}.

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) Λ(k) ⊆ Λ(j), for k ⩾ j;

(ii) there exists a positive integer e = e(Λ|𝔬_{F}), called the 𝔬_{F} -period of Λ, such that ϖ_{F} Λ(k) = Λ(k + e), for all k ∈ ℤ.

For L an 𝔬_{F} -lattice in V, we put L^{#} = {v ∈ V : h(v, L) ⊆ 𝔭_{F} }. Then we call an 𝔬_{F} -lattice sequence Λ self-dual if there exists d ∈ ℤ such that Λ(k)^{#} = Λ(d − k), for all k ∈ ℤ. Without changing any of the objects associated to a self-dual 𝔬_{F} -lattice sequence Λ (except for a scale of the indices), we may (and do) normalize all self-dual lattice sequences Λ so that d = 1 and so that all periods are even. There is also a well-defined notion of the direct sum of lattice sequences (see [10, §2]). The orthogonal direct sum of self-dual lattice sequences is itself self-dual, by the assumption d = 1.

Associated to an 𝔬_{F} -lattice sequence Λ in V, we have a decreasing filtration {𝔞_{n}(Λ) : n ∈ ℤ} of A by 𝔬_{F} -lattices; 𝔞_{0} is a hereditary 𝔬_{F} -order in A and 𝔞_{1} is its Jacobson radical. The filtration on A also gives rise to a valuation ν_{Λ} on A, with ν_{Λ}(0) = +∞. If Λ is self-dual, then each 𝔞_{n}(Λ) is fixed by σ and 𝔞_{n}^{−} = 𝔞_{n}^{−}(Λ) = 𝔞_{n}(Λ) ∩ A_{−} gives a filtration of A_{−} by 𝔬_{F} -lattices. Moreover, ν_{Λ} is fixed by σ.

Given an 𝔬_{F} -lattice sequence Λ, we also put P̃ = P̃(Λ) = 𝔞_{0}(Λ)^{×}, a compact open subgroup of G̃, and P̃_{n} = P̃_{n}(Λ) = 1 + 𝔞_{n}(Λ), for n > 0, a filtration of P̃(Λ) by normal subgroups. If Λ is self-dual, then P̃, P̃_{n} are fixed by σ and we put P^{+} = P^{+}(Λ) = P̃ ∩ G^{+}, a compact open subgroup of G^{+}, and P = P(Λ) = P^{+} ∩ G. We have a filtration of P(Λ) by normal subgroups P_{n} = P_{n}(Λ) = P̃_{n}^{Σ} = P̃_{n} ∩ G, for n > 0. We also have, for n > 0, a bijection 𝔞_{n}^{−} → P_{n} given by the Cayley map x ↦ C(x) = (1 + \frac{x}{2})(1 − \frac{x}{2})^{−1}, which is equivariant under conjugation by P. The quotient group G = P/P_{1} is (the group of rational points of) a reductive group over the finite field k_{F_{0}}. However, it is not, in general, connected. We denote by P^{o} = P^{o}(Λ) the inverse image in P of (the group of rational points of) the connected component G^{o} of G; then P^{o} is a parahoric subgroup of G.

Definition 2.1 ([8, §1.5], [10, §3.1]). (i) A stratum in A is a 4-tuple [Λ, n, r, b], where Λ is an 𝔬_{F} -lattice sequence, n ∈ ℤ and r ∈ ℤ with n ⩾ r ⩾ 0 and b ∈ 𝔞_{−n}(Λ).

(ii) Two strata [Λ, n, r, b_{i}], i = 1, 2, are called equivalent if b_{1} − b_{2} ∈ 𝔞_{−r}(Λ).

(iii) A stratum [Λ, n, r, b] is called skew if Λ is self-dual and b ∈ A_{−}.

(iv) A stratum [Λ, n, r, b] is called null if n = r and b = 0.

For n ⩾ r ⩾ \frac{n}{2} > 0, an equivalence class of strata corresponds to a character of P̃_{r+1}(Λ), by

[Λ, n, r, b] ↦ (ψ_{b} : x ↦ ψ_{A}(b(x − 1)), for x ∈ P̃_{r+1}),

while an equivalence class of skew strata corresponds to a character of P_{r+1}(Λ), by

[Λ, n, r, b] ↦ ψ_{b}^{−} = ψ_{b}|_{P_{r+1}(Λ)}.

Definition 2.2 ([8, Definition 1.5.5], [10, Definition 5.1]). A stratum [Λ, n, r, β] in A is called simple if either it is null or the following four conditions are satisfied:

(i) the algebra E = F [β] is a field;

(ii) Λ is an 𝔬_{E}-lattice chain (we usually write Λ_{𝔬_{E}} when we are thinking of it as such);

(iii) ν_{Λ}(β) = −n;

(iv) k_{0}(β, Λ) < −r.

Here, k_{0}(β, Λ) is an integer related to the action of the adjoint map a_{β}(x) = βx−xβ on the filtration associated to Λ. We refer the reader to [8, §1.5] or [29, §5] for the formal definition and a discussion of it.

Now let [Λ, n, r, β] be a stratum in A and suppose we have a decomposition V = ⨁_{i=1}^{l} V^{i} into F -subspaces. Let Λ^{i} be the lattice sequence in V^{i} given by Λ^{i}(k) = Λ(k) ∩ V^{i} and put β_{i} = 1^{i}β1^{i}, where 1^{i} is the projection onto V^{i} with kernel ⨁_{j≠i} V^{j}.

Definition 2.3. (i) We say that V = ⨁_{i=1}^{l} V^{i} is a splitting for the stratum [Λ, n, r, β] if we have Λ(k) = ⨁_{i=1}^{l} Λ^{i}(k), for all k ∈ ℤ, and β = ∑_{i=1}^{l} β_{i}.

(ii) If B = {v_{1}, ..., v_{N}} is a basis for V then we say that B splits the stratum [Λ, n, r, β] if V = ⨁_{i=1}^{N} F v_{i} is a splitting for [Λ, n, r, β].

Definition 2.4 ([32, Definition 3.2]). A stratum [Λ, n, r, β] in A is called semisimple if either it is null or ν_{Λ}(β) = −n and there exists a splitting V = ⨁_{i=1}^{l} V^{i} for the stratum such that

(i) for 1 ⩽ i ⩽ l, [Λ^{i}, q_{i}, r, β_{i}] is a simple or null stratum in A^{ii}, where q_{i} = r if β_{i} = 0, q_{i} = −ν_{Λ^{i}}(β_{i}) otherwise; and

(ii) for 1 ⩽ i, j ⩽ l, i ≠ j, the stratum [Λ^{i} ⊕ Λ^{j}, q, r, β_{i} + β_{j}] is not equivalent to a simple or null stratum, with q = max {q_{i}, q_{j}}.

In this case, the splitting is uniquely determined (upto ordering) by the stratum. We will use the block notation A^{ij} = Hom_{F} (V^{j}, V^{i}) and put M_{β} = ⨁_{i=1}^{l} A^{ii}.

Let [Λ, n, r, β] be a semisimple stratum as above and put E = F [β] = ⨁_{i=1}^{l} E_{i}, where E_{i} = F [β_{i}]. We will sometimes say that Λ is an 𝔬_{E}-lattice sequence to mean that Λ = ⨁_{i=1}^{l} Λ^{i} and each Λ^{i} is an 𝔬_{E_{i}}-lattice sequence in V^{i}. We let B = B_{β} denote the A-centralizer of β, so that B = ⨁_{i=1}^{l} B_{i}, where B_{i} is the centralizer of β_{i} in A^{ii}. We write G̃_{E} = B^{×}, G̃_{i} = Aut_{F} (V^{i}) and G̃_{E_{i}} = B_{i}^{×} = G̃_{i} ∩ G̃_{E}, so that M̃_{β} = M_{β}^{×} is a Levi subgroup of G̃ and G̃_{E} = ∏_{i=1}^{l} G̃_{E_{i}} ⊆ M̃_{β}. Each G̃_{E_{i}} is (the group of F_{0}-points of) the restriction of scalars to F_{0} of a general linear group over E_{i}. We also write 𝔟_{n}(Λ) = 𝔞_{n}(Λ) ∩ B, for n ∈ ℤ, which gives the filtration induced on B by thinking of Λ as an 𝔬_{E}-lattice sequence.

Definition 2.5. We say that a semisimple stratum [Λ, n, r, β] is skew if Λ is self-dual, β ∈ A_{−} and the associated splitting V = ⨁_{i=1}^{l} V^{i} is orthogonal with respect to the form h.

If [Λ, n, r, β] is skew semisimple with splitting V = ⨁_{i=1}^{l} V^{i}, then the strata [Λ^{i}, q_{i}, r, β_{i}] are all skew simple strata, for the form h_{i} induced by h on V^{i}. The involution on F extends to each E_{i}, as the adjoint involution, and we write E_{i,0} for the subfield of fixed points; it is a subfield of index 2 except in the case E_{i} = F = F_{0} (so that β_{i} = 0). For [Λ, n, r, β] a skew semisimple stratum, we put G_{E}^{+} = B ∩ G^{+} and G_{E} = B ∩ G; then G_{E}^{+} is (the group of F_{0}-points of) a reductive group over F_{0} and G_{E} is (the group of F_{0}-points of) its connected component. We write G_{E}^{+} = ∏_{i=1}^{l} G_{E_{i}}^{+} and G_{E} = ∏_{i=1}^{l} G_{E_{i}}; then each G_{E_{i}}^{+} is (the group of F_{0}-points of) the restriction of scalars to F_{0} of a unitary, symplectic or orthogonal group over E_{i,0}; moreover, at most one of the G_{E_{i}}^{+} is symplectic or orthogonal – this happens in the case β_{i} = 0 and F = F_{0}. If G_{E_{i}}^{+} is not orthogonal, then we have G_{E_{i}} = G_{E_{i}}^{+}; if it is orthogonal, then G_{E_{i}} is special orthogonal. Note that, by [6, Lemmas 5.2–5.5], for each i there is a nondegenerate E_{i}/E_{i,0} ϵ-hermitian form f_{i} on V^{i} such that the two notions of lattice duality for 𝔬_{E_{i}}-lattices in V^{i} given by h_{i} and by f_{i} coincide. Then G_{E_{i}}^{+} is the group of fixed points in G̃_{E_{i}} of the involution determined by f_{i}.

We write P̃_{n}(Λ_{𝔬_{E}}) = P̃_{n}(Λ) ∩ G̃_{E} = 1 + 𝔟_{n}(Λ) and P_{n}(Λ_{𝔬_{E}}) = P_{n}(Λ) ∩ G_{E}, for n > 0. Similarly, we write P̃(Λ_{𝔬_{E}}) = P̃(Λ) ∩ G̃_{E} and P^{+}(Λ_{𝔬_{E}}) = P^{+}(Λ) ∩ G_{E}^{+}, P(Λ_{𝔬_{E}}) = P(Λ) ∩ G_{E}. Let P^{o}(Λ_{𝔬_{E}}) denote the inverse image in P(Λ_{𝔬_{E}}) of the connected component of the quotient P(Λ_{𝔬_{E}})/P_{1}(Λ_{𝔬_{E}}); note that this can be smaller than P^{o}(Λ) ∩ G_{E}.

Lemma 2.6 (cf. [8, Theorem 1.6.1]). Let [Λ, n, 0, β] be a skew semisimple stratum. Then

P_{1}(Λ)xP_{1}(Λ) ∩ G_{E}^{+} = P_{1}(Λ_{𝔬_{E}})xP_{1}(Λ_{𝔬_{E}}), for x ∈ G_{E}^{+};
P_{1}(Λ)xP_{1}(Λ) ∩ G_{E} = P_{1}(Λ_{𝔬_{E}})xP_{1}(Λ_{𝔬_{E}}), for x ∈ G_{E}.

Proof We begin by proving the corresponding statement for G̃: for x ∈ G̃_{E},

P̃_{1}(Λ)xP̃_{1}(Λ) ∩ G̃_{E} = P̃_{1}(Λ_{𝔬_{E}})xP̃_{1}(Λ_{𝔬_{E}}).  (2.7)

The simple case for this is given by [8, Theorem 1.6.1]. For the general case, we have G̃_{E} ⊆ M̃_{β} and M̃_{β} is the fixed-point subgroup in G̃ of a 2-group of automorphisms (namely the group of conjugations by ∑_{i=1}^{l} ε_{i}1^{i}, for ε_{i} = ±1). Since p ≠ 2, x ∈ M̃_{β} and P̃_{1}(Λ) is a pro-p group, [29, Lemma 2.1] implies that P̃_{1}(Λ)xP̃_{1}(Λ)∩M̃_{β} = (P̃_{1}(Λ)∩M̃_{β})x(P̃_{1}(Λ)∩M̃_{β}). The result now follows from the simple case, by intersecting with G̃_{E}. Now to prove the first statement of the Lemma, we take the σ-fixed points of (2.7) and apply [29, Lemma 2.1] again. The second statement follows by intersection with G, since P_{1}(Λ) ⊆ G. ■

2.2 Buildings

We will need a few results on inclusions of the hereditary orders given by lattice sequences, which are most easily proved using the Bruhat-Tits affine building. We recall briefly the description of the building in terms of lattice sequences – see [5] and [6] for more details and proofs. There is a (unique upto translation) G̃-set isomorphism between the (extended) affine building I(G̃, F_{0}) and the set of lattice functions on V, of which the lattice sequences are the subset of “rational points” (i.e. barycentres of vertices with rational weights). The involution σ on G̃ induces an involution on I(G̃, F_{0}) which, for a unique choice of the isomorphism above, coincides with the map Λ ↦ Λ^{#}, where Λ^{#}(k) = (Λ(1 − k))^{#}, on lattice sequences. Moreover, we can identify the set of fixed points with the affine building I(G^{+}, F_{0}) of G^{+}. Thus the affine building is identified with the set of self-dual lattice functions on V. In particular, we will identify a self-dual lattice function with the corresponding points in I(G^{+}, F_{0}) and I(G̃, F_{0}); thus we can, for example, talk about the line segment joining two lattice sequences.

Lemma 2.8 (cf. [27, Lemme 1.7]). Given (self-dual) 𝔬_{E}-lattice sequences Λ, Λ′ with 𝔟_{0}(Λ) ⊇ 𝔟_{0}(Λ′), there exists a (self-dual) 𝔬_{E}-lattice sequence Λ′′ such that

𝔟_{0}(Λ′′) = 𝔟_{0}(Λ′)  and  𝔞_{0}(Λ) ⊇ 𝔞_{0}(Λ′′).

Note that we are identifying an 𝔬_{E}-lattice sequence both with a point of I(G̃_{E}, F_{0}) and a point of I(G̃, F_{0}), which gives a continuous affine injection of buildings I(G̃_{E}, F_{0}) ↪ I(G̃, F_{0}) (see [5]).

Proof We use the following facts, translated into the language of lattice sequences from [27, Lemme 1.5]):

(i) There is a neighbourhood of Λ in I(G̃, F_{0}) such that, for all Λ′′ in this neighbourhood, 𝔞_{0}(Λ) ⊇ 𝔞_{0}(Λ′′).

(ii) There is a sequence of 𝔬_{E}-lattice sequences Λ_{j} such that 𝔟_{0}(Λ_{j}) = 𝔟_{0}(Λ′) and the limit of (Λ_{j}) is Λ.

Note that (ii) is given by [27, Lemme 1.5(ii)] only in the simple case; in the general case, we use the facts that I(G̃_{E}, F_{0}) ≃ ∏_{i=1}^{l} I(G̃_{E_{i}}, F_{0}) and I(G̃_{E_{i}}, F_{0}) ≃ I(B_{i}^{×}, E_{i}), so we can apply the simple case in each I(B_{i}^{×}, E_{i}). The result is now immediate: for large enough j, the sequence Λ_{j} of (ii) is inside the neighbourhood given by (i) so Λ′′ = Λ_{j} has the required properties. For the self-dual case, note that we may assume that the sequence in (ii) consists of self-dual lattice sequences, by replacing Λ_{j} by the barycentre of Λ_{j} and Λ_{j}^{#}. The result now follows in the same way. ■

Corollary 2.9. Given any (self-dual) 𝔬_{E}-lattice sequence Λ, there exists a (self-dual) 𝔬_{E}-lattice sequence Λ^{m} such that 𝔟_{0}(Λ^{m}) is a minimal (self-dual) 𝔬_{E}-order in B and 𝔞_{0}(Λ^{m}) ⊆ 𝔞_{0}(Λ).

Remark The same result is not true for maximal (self-dual) orders: given a (self-dual) 𝔬_{E}-lattice sequence Λ, there is not necessarily a (self-dual) 𝔬_{E}-lattice sequence Λ^{M} such that 𝔟_{0}(Λ^{M}) is a maximal (self-dual) 𝔬_{E}-order in B and 𝔞_{0}(Λ^{M}) ⊇ 𝔞_{0}(Λ).

Lemma 2.10. Let Λ, Λ′ be (self-dual) 𝔬_{E}-lattice sequences in A with 𝔟_{0}(Λ) = 𝔟_{0}(Λ′). Then there is a sequence Λ = Λ_{0}, Λ_{1}, ..., Λ_{t} = Λ′ of (self-dual) 𝔬_{E}-lattice sequences on the line segment [Λ, Λ′] such that, for 1 ⩽ i ⩽ t:

(i) 𝔟_{0}(Λ_{i}) = 𝔟_{0}(Λ);

(ii) either 𝔞_{0}(Λ_{i}) ⊆ 𝔞_{0}(Λ_{i−1}) or 𝔞_{0}(Λ_{i}) ⊇ 𝔞_{0}(Λ_{i−1}).

Proof Note that any lattice sequence Λ′′ on the line segment [Λ, Λ′] is certainly a (self-dual) 𝔬_{E}- lattice sequence and we have 𝔟_{0}(Λ′′) = 𝔟_{0}(Λ). On the other hand, the map Λ′′ ↦ 𝔞_{0}(Λ′′) is locally constant on this line segment, except at a discrete (so, in particular, finite) set of points – namely, the points where the line segment meets some wall in the building. This divides the line segment into a finite set of subsegments, on the interior of which Λ′′ ↦ 𝔞_{0}(Λ′′) is constant. Taking one point from the interior of each subsegment, together with the endpoints of the subsegments, gives the required sequence. ■

Remark 2.11. Suppose Λ_{1}, Λ_{2}, Λ are 𝔬_{E}-lattice sequences such that 𝔞_{0}(Λ_{i}) ⊇ 𝔞_{0}(Λ), for i = 1, 2. Then, for any Λ′ on the line segment [Λ_{1}, Λ_{2}], we also have 𝔞_{0}(Λ′) ⊇ 𝔞_{0}(Λ) – this is clear since 𝔞_{0}(Λ′) ⊇ 𝔞_{0}(Λ) if and only if Λ′ is contained in the closure of the facet containing Λ. Likewise, if 𝔞_{0}(Λ_{i}) ⊆ 𝔞_{0}(Λ), for i = 1, 2, then, for any Λ′ on the line segment [Λ_{1}, Λ_{2}], we also have 𝔞_{0}(Λ′) ⊆ 𝔞_{0}(Λ). One can check thus using the explicit description of the building in terms of lattice sequences as follows. We choose an apartment containing both Λ_{1} and Λ_{2}; then also Λ is in the same apartment, since it is in the closure of each Λ_{i}. One can then use the explicit description of 𝔞_{0}(Λ_{i}) from [5, Corollary 4.5]. The details are left to the reader.

If Λ_{1}, …, Λ_{t} are (self-dual) 𝔬_{E}-lattice sequences then we write [Λ_{1}, …, Λ_{t}] for their (closed) convex hull. If this is actually a polygon then we assume that the Λ_{i} are distinct and that we have ordered the lattice sequences so that the boundary is ∪_{i=1}^{t}[Λ_{i}, Λ_{i+1}], with the understanding that Λ_{t+1} = Λ_{0}.

Lemma 2.12. Let Λ, Λ′, Λ′′ be (self-dual) 𝔬_{E}-lattice sequences with 𝔟_{0}(Λ) and 𝔟_{0}(Λ′) both contained in 𝔟_{0}(Λ′′). Then there is a decomposition of the triangle [Λ, Λ′, Λ′′] into a union of triangles [Λ_{i}, Λ′_{i}, Λ′′_{i}], where Λ_{i}, Λ′_{i}, Λ′′_{i} are (self-dual) 𝔬_{E}-lattice sequences with the property that 𝔞_{0}(Λ_{i}) ⊆ 𝔞_{(}Λ′_{i}) ⊆ 𝔞_{0}(Λ′′_{i}).

Note that the hypotheses imply that Λ, Λ′, Λ′′ lie in a common apartment of I(G̃_{E}, F_{0}); since the embedding of buildings I(G̃_{E}, F_{0}) ↪ I(G̃, F_{0}) respects apartments, they also lie in a common apartment of I(G̃, F_{0}), so that the convex hull [Λ, Λ′, Λ′′] is genuinely a triangle. The same is true in the self-dual case.

Proof The idea is essentially the same as that of Lemma 2.10; indeed, it already gives the result if Λ, Λ′, Λ′′ are collinear, so we assume this is not the case. Removing all walls of the building from [Λ, Λ′, Λ′′] splits it into a disjoint union of convex polygons. We split the closure [Γ_{1}, ..., Γ_{t}] of each such polygon into triangles as follows: choose a point Γ_{0} in the interior of the polygon and points Γ′_{i} on each (open) edge (Γ_{i}, Γ_{i+1}), for 1 ⩽ i ⩽ t; then the triangles are

[Γ_{0}, Γ′_{i}, Γ_{i}] and [Γ_{0}, Γ′_{i}, Γ_{i+1}], for 1 ⩽ i ⩽ t.

Taking all these triangles for all the polygons gives a decomposition as required. ■

We note that, in the situation of Lemma 2.12, if we actually have 𝔟_{0}(Λ) = 𝔟_{0}(Λ′) = 𝔟_{0}(Λ′′), then also 𝔟_{0}(Λ_{i}) = 𝔟_{0}(Λ′_{i}) = 𝔟_{0}(Λ′′_{i}) = 𝔟_{0}(Λ).

3 Semisimple characters and Heisenberg extensions

We will now recall results from [32] on the definition and properties of semisimple characters, in particular their “transfer property”. Then we look at the same transfer properties for the Heisenberg extensions which are induced by these characters, as in [8, §5.1] and [27, §2.2], and deduce the existence of certain extensions to some larger pro-p groups. Finally, we characterize these extensions in terms of their intertwining.

3.1 Semisimple characters

We will not repeat in detail the definitions of the objects used here but refer the reader to [8, §3] and [32, §3] for more details, in the simple and semisimple cases respectively. Let [Λ, n, 0, β] be a semisimple stratum with associated splitting V = ⨁_{i=1}^{l} V^{i}. We use the same notation as in §2.1, so Λ = ⨁_{i=1}^{l} Λ^{i}, β = ∑_{i=1}^{l} β_{i}, B = C_{A}(E) = ⨁_{i=1}^{l} B_{i}, G̃_{E} = B^{×}, P̃(Λ_{𝔬_{E}}) = P̃(Λ) ∩ G̃_{E}, etc. In [32, §3.2] we define, inductively along k_{0}(β, Λ), a pair of 𝔬_{F} -orders

ℌ(β, Λ) ⊆ 𝔍(β, Λ) ⊆ 𝔞_{0}(Λ).

This gives us two compact open subgroups H̃(β, Λ) = ℌ(β, Λ)^{×} and J̃(β, Λ) = 𝔍(β, Λ)^{×} of G̃, filtered by H̃^{m+1}(β, Λ) = H̃(β, Λ) ∩ P̃_{m+1}(Λ) and J̃^{m+1}(β, Λ) = J̃(β, Λ) ∩ P̃_{m+1}(Λ), for m ⩾ 0. Moreover, we have J̃(β, Λ) = P̃(Λ_{𝔬_{E}})J̃^{1}(β, Λ). When the stratum [Λ, n, 0, β] is skew, ℌ and 𝔍 are stable under the involution so, as usual, we put J^{+}(β, Λ) = J̃(β, Λ) ∩ G^{+} and J(β, Λ) = J̃(β, Λ) ∩ G etc. We also put J^{o}(β, Λ) = P^{o}(Λ_{𝔬_{E}})J^{1}(β, Λ), the inverse image in J(β, Λ) of the connected component of J(β, Λ)/J^{1}(β, Λ).

Returning to the case of a general semisimple stratum, in [32, Definition 3.13] we define, again inductively along k_{0}(β, Λ), a set C(Λ, 0, β) of abelian characters θ̃ of H̃^{1}(β, Λ), called semisimple characters, by the following two properties:

(i) we have θ̃|_{H̃^{1}∩M̃_{β}} = ⨂_{i=1}^{l} θ̃_{i}, for some simple characters θ̃_{i} ∈ C(Λ^{i}, 0, β_{i}) (in the sense of [8, §3.3]);

(ii) writing r = −k_{0}(β, Λ) there is, by [32, Proposition 3.4], a semisimple stratum [Λ, n, r, γ] equivalent to [Λ, n, r, β], with γ ∈ M_{β}; then θ̃|_{H̃^{[ \frac{r}{2} ]+1}(β,Λ)} = θ̃_{0}|_{H̃^{[ \frac{r}{2} ]+1}(β,Λ)}ψ_{β−γ}, for some

semisimple character θ̃_{0} ∈ C(Λ, 0, γ).

Moreover, by [32, Lemma 3.15], θ̃ is trivial on the unipotent radical Ũof any parabolic subgroup of G̃ containing M̃_{β}. When the stratum is skew, this set C(Λ, 0, β) is stable under the involution and the set C_{−}(Λ, 0, β) of restrictions to H^{1}(β, Λ) of semisimple characters is called the set of skew semisimple characters. This can also be described in slightly different terms via the Glauberman correspondence (see [32, §3.6] and also §5.3).

Recall that, given a representation ρ of a subgroup K of G and g ∈ G^{+}, the g-intertwining space of ρ is

I_{g}(ρ) = I_{g}(ρ|K) = Hom_{K∩^{g}K}(ρ,^{g}ρ),

where ^{g}ρ is the representation of ^{g}K = gKg^{−1} given by ^{g}ρ(gkg^{−1}) = ρ(k). The G-intertwining of ρ is

I_{G}(ρ) = I_{G}(ρ|K) = {g ∈ G : I_{g}(ρ) ≠ {0}},

and, similarly, we have the G^{+}-intertwining I_{G^{+}}(ρ). We recall the following intertwining result:

Proposition 3.1 ([32, Proposition 3.27]). Let [Λ, n, 0, β] be a skew semisimple stratum and θ ∈ C_{−}(Λ, 0, β) a skew semisimple character. Then θ is normalized by J(β, Λ) and

I_{G^{+}}(θ) = J^{1}(β, Λ)G_{E}^{+}J^{1}(β, Λ), I_{G}(θ) = J^{1}(β, Λ)G_{E}J^{1}(β, Λ).

Of equally great importance is the so-called “transfer property” of semisimple characters:

Proposition 3.2 ([32, Propositions 3.26, 3.32]). Let [Λ, n, 0, β] and [Λ′, n′, 0, β] be semisimple strata in A. Then there exists a canonical bijection

τ_{Λ,Λ′,β} : C(Λ, 0, β) → C(Λ′, 0, β)

such that, for θ̃ ∈ C(Λ, 0, β), θ̃′ := τ_{Λ,Λ′,β}(θ̃) is the unique semisimple character in C(Λ′, 0, β) such that G̃_{E} ∩ IG̃(θ̃, θ̃′) ≠ ∅. Moreover G̃_{E} ⊆ IG̃(θ̃, θ̃′). If the strata are skew then the bijection τ_{Λ,Λ′,β} commutes with the involution σ; in particular, τ_{Λ,Λ′,β} restricts to a bijection τ_{Λ,Λ′,β} : C_{−}(Λ, 0, β) → C_{−}(Λ′, 0, β).

Remark 3.3. In [32, §3.5], the transfer is given in a more general situation between C(Λ, m, β) and C(Λ′, m′, β), where C(Λ, m, β) denotes the set of restrictions to H̃^{m+1}(β, Λ) and m, m′ are related by ⌊m/e(Λ|𝔬_{F})⌋ = ⌊m′/e(Λ′|𝔬_{F})⌋. This is not quite the correct condition on the integers m, m′. It should be replaced as follows: put

e = \frac{e(Λ|𝔬_{F})}{lcm_{1⩽i⩽l}\{e(E_{i}/F)\}},

and likewise e′; then we require that

⌊m/e⌋ = ⌊m′/e′⌋.  (3.4)

In particular, this ensures that ⌊m/e(Λ^{i}|𝔬_{E_{i}})⌋ = ⌊m′/e(Λ^{′i}|𝔬_{E_{i}})⌋, for each i, so the transfer from the simple case can be used in each block. Notice that the integers called m_{0}, m′_{0} in [32, §3.5] also satisfy this condition since, by [32, §3.1] we have k_{0}(β, Λ)/e(Λ|𝔬_{F}) = k_{0}(β, Λ′)/e(Λ′|𝔬_{F}) so surely k_{0}(β, Λ)/e = k_{0}(β, Λ′)/e′. In particular, the proof of [32, Proposition 3.26] is valid under the condition (3.4).

3.2 Heisenberg extensions

Let [Λ, n, 0, β] be a skew semisimple stratum and θ ∈ C_{−}(Λ, 0, β) a skew semisimple character.

Proposition 3.5 ([32, Corollary 3.29, Proposition 3.31]). There is a unique irreducible representation η of J^{1} such that η contains θ. Moreover, dim η = (J^{1} : H^{1})^{\frac{1}{2}} and the intertwining of η is given by

dim I_{g}(η) = \begin{cases}1 & if g ∈ J^{1}G_{E}^{+}J^{1}; \\ 0 & otherwise. \end{cases}

We also have a relationship between the representations η for different 𝔬_{E}-lattice sequences Λ.

Lemma 3.6 (cf. [8, Proposition 5.1.2]). Let [Λ^{i}, n_{i}, 0, β] be skew semisimple strata, for i = 1, 2, and let θ_{i} ∈ C(Λ^{i}, 0, β); let η_{i} be the representation of J_{i}^{1} = J^{1}(β, Λ^{i}) given by Proposition 3.5. Then

\frac{dim η_{1}}{dim η_{2}} = \frac{(J_{1}^{1} : J_{2}^{1})}{(P_{1}(Λ_{𝔬_{E}}^{1}) : P_{1}(Λ_{𝔬_{E}}^{2}))}.

Proof From [32, Lemma 3.17], we have exact sequences in A_{−}

0 ⟶ 𝔟_{1}(Λ^{i})_{−} ⟶ 𝔍^{1}(β, Λ^{i})_{−} ⟶^{a_{β}} (ℌ^{1}(β, Λ^{i}))_{−}^{∗} ⟶^{s} 𝔟_{0}(Λ^{i})_{−} ⟶ 0,

where s : A → B is a tame corestriction relative to E/F which commutes with the involution (see [8, §1.3] and [30, Lemma 4.4] for the definition). Since the Cayley map is a bijection 𝔍^{1}(β, Λ^{i})_{−} → J_{i}^{1} (and similarly for H), the result follows from Proposition 3.5 exactly as in the proof of [8, Proposition 5.1.2]. ■

Now suppose we are given:

• skew semisimple strata [Λ, n, 0, β], [Λ^{m}, n_{m}, 0, β] and [Λ^{M}, n_{M}, 0, β] with (necessarily) the same splitting, such that 𝔞_{0}(Λ^{m}) ⊆ 𝔞_{0}(Λ) ⊆ 𝔞_{0}(Λ^{M}); by duality we then also have that 𝔞_{1}(Λ^{m}) ⊇ 𝔞_{1}(Λ) ⊇ 𝔞_{1}(Λ^{M}).

• semisimple characters θ ∈ C_{−}(Λ, 0, β), θ_{m} ∈ C_{−}(Λ^{m}, 0, β) and θ_{M} ∈ C_{−}(Λ^{M}, 0, β), related by

the correspondences:

θ_{m} = τ_{Λ,Λ^{m},β}(θ), θ_{M} = τ_{Λ,Λ^{M},β}(θ)

• η (respectively η_{m}, η_{M}) the unique irreducible representation of J^{1} = J^{1}(β, Λ) (respectively J_{m}^{1} = J^{1}(β, Λ^{m}), J_{M}^{1} = J^{1}(β, Λ^{M})) containing θ (respectively θ_{m}, θ_{M}), given by Proposition 3.5.

We form the group J_{m,M}^{1} = P_{1}(Λ_{𝔬_{E}}^{m})J_{M}^{1}; note that this is indeed a group, since P(Λ_{𝔬_{E}}^{M}) normalizes J_{M}^{1} and contains P_{1}(Λ_{𝔬_{E}}^{m}).

Proposition 3.7 (cf. [8, Propositions 5.1.4, 5.1.19], [27, Proposition 2.12]). There exists a unique irreducible representation η_{m,M} of J_{m,M}^{1} such that

(i) η_{m,M}|_{J_{M}^{1}} = η_{M};

(ii) η_{m,M} and η_{m} induce equivalent irreducible representations of P_{1}(Λ^{m}).

Moreover, the intertwining of η_{m,M} is given by

dim I_{g}(η_{m,M}) = \begin{cases}1 & if g ∈ J_{m,M}^{1}G_{E}^{+}J_{m,M}^{1}; \\ 0 & otherwise. \end{cases}

Proof Given Proposition 3.5, Lemma 3.6 and Lemma 2.6, the proof of the existence and uniqueness of η_{m,M} is the same as that of [8, Proposition 5.1.14] while the proof of the intertwining is the same as that of op. cit. Proposition 5.1.19. ■

Remark The representation η_{m,M} depends only on 𝔟_{0}(Λ^{m}), not on the lattice sequence Λ^{m}. For suppose Λ^{m′} is another 𝔬_{E}-lattice sequence with 𝔟_{0}(Λ^{m}) = 𝔟_{0}(Λ^{m′}) and 𝔞_{0}(Λ^{m′}) ⊆ 𝔞_{0}(Λ^{M}), and let η_{m′,M} be the representation of J_{m,M}^{1} given by Proposition 3.7 applied with Λ^{m′} in place of Λ^{m}.

By Lemma 2.10, there is a sequence Λ^{m} = Λ_{0}, Λ_{1}, ..., Λ_{t} = Λ^{m′} of 𝔬_{E}-lattice sequences on the line segment [Λ^{m}, Λ^{m′}] such that, for 1 ⩽ i ⩽ t:

(i) 𝔟_{0}(Λ_{i}) = 𝔟_{0}(Λ^{m});

(ii) either 𝔞_{0}(Λ_{i}) ⊆ 𝔞_{0}(Λ_{i−1}) or 𝔞_{0}(Λ_{i}) ⊇ 𝔞_{0}(Λ_{i−1}).

Let θ_{i} = τ_{Λ,Λ^{i},β}(θ) and let η_{i} be the unique irreducible representation of J_{i}^{1} = J^{1}(β, Λ_{i}) containing θ_{i}. Since each Λ_{i} is on the line segment [Λ^{m}, Λ^{m′}], we have 𝔞_{0}(Λ_{i}) ⊆ 𝔞_{0}(Λ^{M}) (see Remark 2.11). Let η_{i,M} denote the representation of J_{m,M}^{1} given by Proposition 3.7 applied with Λ_{i} in place of Λ^{m}. Fix i and suppose 𝔞_{0}(Λ_{i}) ⊆ 𝔞_{0}(Λ_{i−1}) (the opposite case is symmetrically similar). Applying Proposition 3.7 with the pair (Λ_{i}, Λ_{i−1}) in place of (Λ^{m}, Λ^{M}), we see that η_{i}, η_{i−1} induce equivalent irreducible representations of P_{1}(Λ_{i}). But then

Ind _{J_{m,M}^{1}}^{P_{1}(Λ_{i})}η_{i−1,M} ≃ Ind _{P_{1}(Λ_{i−1})}^{P_{1}(Λ_{i})}Ind _{J_{m,M}^{1}}^{P_{1}(Λ_{i−1})}η_{i−1,M} ≃ Ind _{P_{1}(Λ_{i−1})}^{P_{1}(Λ_{i})}Ind _{J_{i−1}^{1}}^{P_{1}(Λ_{i−1})}η_{i−1} ≃ Ind _{J_{i}^{1}}^{P_{1}(Λ_{i})}η_{i}.

Hence, by the uniqueness in Proposition 3.7, we have η_{i,M} = η_{i−1,M}. Applying this with all pairs (i, i − 1), we see that η_{m′,M} = η_{m,M}, as required. Let η_{Λ,M} denote the representation of J_{Λ,M}^{1} = P_{1}(Λ_{𝔬_{E}})J_{M}^{1} given by applying Proposition 3.7 with Λ in place of Λ^{m}. As in [8, Proposition 5.1.18] (and with the same proof), we have the following compatibility property:

Proposition 3.8. With notation as above, we have

η_{m,M}|_{J_{Λ,M}^{1}} = η_{Λ,M}.

3.3 Characters of Sylow subgroups

We continue with the notation of the previous section but suppose, in addition, that 𝔟_{0}(Λ^{m}) is a minimal self-dual 𝔬_{E}-order in B. It will be useful, in this situation, to have another characterization of the extension η_{m,M} of η_{M}. For this we examine the characters of the quotient group J_{m,M}^{1}/J_{M}^{1} ≅ P_{1}(Λ_{𝔬_{E}}^{m})/P_{1}(Λ_{𝔬_{E}}^{M}). We will think of this as a subgroup of J_{M}^{+}/J_{M}^{1} ≅ P^{+}(Λ_{𝔬_{E}}^{M})/P_{1}(Λ_{𝔬_{E}}^{M}), where J_{M}^{+} = J^{+}(β, Λ^{M}). Since 𝔟_{0}(Λ^{m}) is a minimal self-dual 𝔬_{E}-order, the group J_{m,M}^{1} is in fact a Sylow pro-p subgroup of J_{M}^{+}. We identify P^{+}(Λ_{𝔬_{E}}^{M})/P_{1}(Λ_{𝔬_{E}}^{M}) with the product of (the points of) reductive groups G_{i} over k_{E_{i,0}}, which are not necessarily connected; the possibilities here are as follows:

• if E_{i}/E_{i,0} is quadratic unramified then G_{i} is the product of (the restriction of scalars to k_{E_{i,0}} of) some general linear groups over k_{E_{i}} and at most two unitary groups over k_{E_{i,0}};

• if E_{i}/E_{i,0} is quadratic ramified then G_{i} is the product of some general linear groups, at most one symplectic group, and at most one orthogonal group, over k_{E_{i}} = k_{E_{i,0}}.

• if E_{i} = F = F_{0} then G_{i} is a product of some general linear groups and at most two symplectic (if ϵ = −1) or orthogonal (if ϵ = +1) groups over k_{F}.

The image of P_{1}(Λ_{𝔬_{E}}^{m}) identifies with U, the product of the unipotent radicals U_{i} of Borel subgroups of G_{i}. We also remark, for future reference, that P(Λ_{𝔬_{E}}^{M}) is a maximal compact subgroup of G_{E} if and only if there are no factors isomorphic to a general linear group in any of the three cases above.

Lemma 3.9. Let ϕ be a character of J_{m,M}^{1}/J_{M}^{1} ≅ U which is intertwined by all of J_{M}^{+}. Then ϕ extends to a character of J_{M}^{+}.

Proof We think of ϕ = ⊗ϕ_{i} as a character of U = ∏_{i=1}^{l} U_{i}, where ϕ_{i} is a character of U_{i} intertwined by all of G_{i}. We claim that the characters ϕ_{i} extend to characters, also denoted ϕ_{i}, of G_{i}; indeed, in all but one case, ϕ_{i} will be trivial. Inflating ϕ = ⊗ϕ_{i} to J_{M} gives the required extension.

We fix i and now omit it from our notation. Let T be a maximal torus in G, and B a Borel subgroup with unipotent radical U containing T . Let Σ be the root system of G relative to T , and let Σ^{+} be the positive roots, and Δ the simple roots, determined by B. For α ∈ Σ, write U_{α} for the corresponding root subgroup, so U = ∏_{α∈Σ^{+}} U_{α}. By [15, page 129], the derived subgroup of U is ∏_{α∈Σ^{+}\Δ} U_{α} (note that the only (quasi-simple) exceptions to this are groups of type B_{2} or F_{4} over the field of 2 elements and G_{2} over the field of 3 elements, none of which can occur in our situation). Hence ϕ must take the form

ϕ = ∏_{α∈Δ} ϕ_{α}, for ϕ_{α} a character of U_{α}.

Now write Δ = ∪Δ_{j}, with each Δ_{j} irreducible. If card Δ_{j} ⩾ 2 then, for all α ∈ Δ_{j}, there is a Weyl-group element w such that w(α) ∈ Σ^{+} \ Δ. If n in the G-normalizer of T reduces to w then, since n intertwines ϕ, we have

ϕ(u_{α}) = ϕ(nu_{α}n^{−1}) = 1, for u_{α} ∈ U_{α},

since nu_{α}n^{−1} ∈ U_{w(α)} ⊆ ker ϕ. Hence ϕ_{α} is trivial. In particular, if card Δ_{j} ⩾ 2 for all j, then ϕ is trivial and extends to the trivial representation of G. This leaves the cases when card Δ_{j} = 1, for some j; in this case the corresponding factor of G is one of:

(i) a general linear group of dimension 2, that is GL_{2}(k_{E});

(ii) a symplectic group of dimension 2, that is SL_{2}(k_{E});

(iii) a unitary group of dimension 2 or 3;

(iv) an orthogonal group of dimension 3 or 4.

These cases need to be treated individually. We omit the subscript j.

(i) The proof in this case is contained in the proof of [8, Proposition 5.2.4] (see bottom of page 168). Indeed, since p ≠ 2, we have |k_{E}| > 2 so the proof there shows that the character ϕ is in fact trivial and extends to the trivial character of G.

(ii) We write the elements of G ≃ SL_{2}(k_{E}) as matrices and U as the group of upper triangular unipotent matrices. The character ϕ is given by

ϕ (\begin{matrix}1 & n \\ 0 & 1 \end{matrix}) = χ(n), for n ∈ k_{E},

where χ is an additive character of k_{E}. Since ϕ is intertwined by all of G, it is normalized by diag(a, a^{−1}), for a ∈ k_{E}^{×}. Hence

χ ((a^{2} − 1)n) = 1, for all a ∈ k_{E}^{×}, n ∈ k_{E}.

Providing |k_{E}| > 3, there is an a ∈ k_{E}^{×} such that a^{2} ≠ 1 and hence χ is trivial and ϕ extends to the trivial representation of G. If |k_{E}| = 3 then the two non-trivial characters of U are also intertwined by all of G = SL_{2}(𝔽_{3}). However, in this case U is a quotient of G (by its 2-Sylow subgroup) so ϕ extends in any case. (In the case of the non-trivial characters of U, it extends to a 1-dimensional cuspidal representation of G.)

(iii) The proof here is almost identical, though there is no exceptional case as in (ii). In the 2dimensional case we conjugate by diag(a, \overline{a}^{−1}) and use the facts that the norm map k_{E} → k_{E,0} is surjective and |k_{E,0}| > 2; in the 3-dimensional case, we conjugate by diag(a, 1, \overline{a}^{−1}).

(iv) Again, the proof here is the same, with no exceptional case. In the 3-dimensional case we conjugate by diag(a, 1, a^{−1}) and in the 4-dimensional case by diag(a, 1, 1, a^{−1}). ■

Lemma 3.10. Let ϕ be a character of J_{m,M}^{1}/J_{M}^{1} ≅ U which is intertwined by all of G_{E}. Then ϕ is trivial.

Proof Using the notation of the proof of Lemma 3.9, we have ϕ = ⊗ϕ_{i}, and each character ϕ_{i} is intertwined by G_{E_{i}}. It is enough to show that each ϕ_{i} is trivial so it is enough to check the simple case. Moreover, we have seen in the proof of Lemma 3.9 that ϕ is trivial except in the case where G has a factor isomorphic to SL(2, 𝔽_{3}). Hence we may restrict to the case (in the notation of the proof of Lemma 3.9) when |Δ| = 1 and G ≃ SL_{2}(𝔽_{3}). We choose a self-dual E-basis which splits the lattice sequence Λ_{𝔬_{E}}^{m} (so also Λ_{𝔬_{E}}^{M}). Then, after scaling and permuting the basis if necessary, this identifies

𝔟_{0}(Λ_{𝔬_{E}}^{m}) with (\begin{matrix}𝔬_{E} & 𝔬_{E} \\ 𝔭_{E} & 𝔬_{E} \end{matrix});  and
𝔟_{0}(Λ_{𝔬_{E}}^{M}) with (\begin{matrix}𝔬_{E} & 𝔬_{E} \\ 𝔬_{E} & 𝔬_{E} \end{matrix}) or (\begin{matrix}𝔬_{E} & 𝔭_{E}^{−1} \\ 𝔭_{E} & 𝔬_{E} \end{matrix}) .

(The second case occurs when E/E_{0} is ramified and the form h is hermitian.) We consider

x = (\begin{matrix}1 & 1 \\ 0 & 1 \end{matrix}) or (\begin{matrix}1 & 0 \\ ϖ_{E} & 1 \end{matrix})

in the two cases, so that xP_{1}(Λ_{𝔬_{E}}^{M}) generates U. We put g = diag(ϖ_{E}, ϖ_{E}^{−1}) or diag(ϖ_{E}^{−1}, ϖ_{E}) in the two cases. Then gxg^{−1} ∈ P_{1}(Λ_{𝔬_{E}}^{M}). Since g ∈ G_{E} intertwines ϕ and ϕ is trivial on P_{1}(Λ_{𝔬_{E}}^{M}), we see that

ϕ(x) = ϕ(gxg^{−1}) = 1.

Since xP_{1}(Λ_{𝔬_{E}}^{M}) generates U, the character ϕ is indeed trivial. ■

Corollary 3.11. If 𝔟_{0}(Λ^{m}) is a minimal self-dual 𝔬_{E}-order contained in 𝔟_{0}(Λ^{M}), then the representation η_{m,M} is the unique extension of η_{M} to J_{m,M}^{1} which is intertwined by all of G_{E}.

Proof Suppose η′ is another such extension. Then η′ = η_{m,M} ⊗ϕ, for some character ϕ of J_{m,M}^{1}/J_{M}^{1} which is intertwined by all of G_{E}. By Lemma 3.10, such a character is trivial so η′ = η_{m,M}. ■

3.4 Analogous results for G̃

The results of the previous two sections all have analogues for G̃. The proofs are almost identical, though a little simpler. Indeed, since so far we have only been looking at representations of pro-p subgroups, we could have obtained the results above by first proving the analogous results for G̃ and then using the Glauberman transfer, as in previous work (see [30], [32]). We state those results we will need for G̃ here, leaving the modifications of the proofs of the previous sections as an exercise for the reader.

Proposition 3.12. Let [Λ^{m}, n_{m}, 0, β] and [Λ^{M}, n_{M}, 0, β] be semisimple strata with 𝔞_{0}(Λ^{m}) ⊆ 𝔞_{0}(Λ^{M}). Let θ̃_{m} ∈ C(Λ^{m}, 0, β) be a semisimple character and θ̃_{M} = τ_{Λ^{m},Λ^{M},β}(θ̃_{m}). Then there is a unique irreducible representation η̃_{m} (respectively η̃_{M}) of J̃^{1}(β, Λ^{m}) (respectively J̃^{1}(β, Λ^{M})) which contains θ̃_{m} (respectively θ̃_{M}). Put J̃_{m,M}^{1} = P̃_{1}(Λ_{𝔬_{E}}^{m})J̃^{1}(β, Λ^{M}). There is a unique irreducible representation η̃_{m,M} of J̃_{m,M}^{1} such that

(i) η̃_{m,M}|_{J̃^{1}(β,Λ^{M})} = η̃_{M};

(ii) η̃_{m,M} and η̃_{m} induce equivalent irreducible representations of P̃_{1}(Λ^{m}).

Moreover, if 𝔟_{0}(Λ^{m}) is a minimal 𝔬_{E}-order in B then η̃_{m,M} is the unique extension of η̃_{M} to J̃_{m,M}^{1} which is intertwined by G̃_{E}. Finally, the intertwining of η̃_{m,M} is given by

dim I_{g}(η̃_{m,M}) = \begin{cases}1 & if g ∈ J̃_{m,M}^{1}G̃_{E}J̃_{m,M}^{1}; \\ 0 & otherwise. \end{cases}

4 Beta extensions

This section is devoted to the study of β-extensions, which are certain rather special extensions of the Heisenberg representations η to J^{+}(β, Λ). In the case of GL_{N}(F) (or GL_{m}(D) for D an F -central division algebra) these extensions are characterized by their intertwining, which should contain the whole centralizer of β (see [8, §5.2], [27, §2.4] respectively). In our situation we have been unable to do this, but instead give a different characterization (which, from the results of [8, §5.2], is also valid for GL_{N}(F)) in terms of the representation η_{m,M} of the previous section and by transfer properties.

4.1 Beta-extensions in the maximal case

Let [Λ^{M}, n, 0, β] be a skew semisimple stratum such that 𝔟_{0}(Λ^{M}) is a maximal self-dual 𝔬_{E}-order in B. Let θ_{M} ∈ C_{−}(Λ^{M}, 0, β) be a skew semisimple character and let η_{M} be the unique irreducible representation of J_{M}^{1} = J^{1}(β, Λ^{M}) containing θ_{M}. By Corollary 2.9, there exists a self-dual 𝔬_{E}-lattice sequence Λ^{m} in V such that 𝔟_{0}(Λ^{m}) is a minimal self-dual 𝔬_{E}-order in B and 𝔞_{0}(Λ^{m}) ⊆ 𝔞_{0}(Λ^{M}). Let η_{m,M} be the representation of J_{m,M}^{1} = P_{1}(Λ_{𝔬_{E}}^{m})J_{M}^{1} given by Proposition 3.7. Recall that this depends only on the group J_{m,M}^{1}, not on the choice of Λ^{m}.

Following the ideas of [8, §5.2], we show that there is a representation of J_{M}^{+} = J_{M}^{+}(β, Λ) which extends η_{m,M}.

Theorem 4.1 (cf. [8, Proposition 5.2.4]). With notation as above, there exists a representation κ_{M} of J_{M}^{+} which extends η_{m,M}. Moreover, if κ′_{M} is another such representation then κ′_{M} = κ_{M} ⊗ χ, for some character χ of P^{+}(Λ_{𝔬_{E}}^{M})/P_{1}(Λ_{𝔬_{E}}^{M}) which is trivial on the subgroup generated by all its unipotent subgroups.

We call such a representation κ_{M} a β-extension of η_{M}. Note that the definition of κ_{M} is independent of the choice of Λ^{m}, since P^{+}(Λ_{𝔬_{E}}^{M}) acts transitively by conjugation on the minimal self-dual 𝔬_{E}- orders contained in 𝔟_{0}(Λ^{M}) and the representation η_{m,M} depends only on 𝔟_{0}(Λ^{m}), not on the lattice sequence Λ^{m}.

Proof The proof is essentially the same as that of [8, Proposition 5.2.4] (see also [2, Lemme 6.4]) so we will omit some of the details and refer the reader to loc. cit.. Since J_{M}^{+} normalizes η_{M}, we can extend η_{M} to a projective representation of J_{M}^{+}. This gives rise to a 2-cocycle α of J_{M} in ℂ^{×}. Since the dimension of η_{M} is a power of p, the order of the cohomology class of this cocycle is a power of p. But η_{M} admits an extension to J_{m,M}^{1} so the restriction of the cocycle to J_{m,M}^{1} is cohomologous to zero. However, J_{m,M}^{1} is a Sylow pro-p subgroup of J_{M}^{+} so the original cohomology class of α is trivial and hence η_{M} extends to a linear representation λ of J_{M}^{+}. Now we compare λ|_{J_{m,M}^{1}} to η_{m,M}; it is of the form η_{m,M} ⊗ ϕ, for some abelian character ϕ of P_{1}(Λ_{𝔬_{E}}^{m})/P_{1}(Λ_{𝔬_{E}}^{M}) which is intertwined by all of P^{+}(Λ_{𝔬_{E}}^{M}). By Lemma 3.9, any such character extends to a character, also denoted ϕ, of P^{+}(Λ_{𝔬_{E}}^{M}). Then κ_{M} := λ⊗ϕ^{−1} extends η_{m,M}, as required. Note also that the final assertion of the Theorem is clear. ■

Remark 4.2. The conclusions of Theorem 4.1 are also true without the assumption that 𝔟_{0}(Λ^{M}) is maximal, as is the case for G̃ (see [8, Remark, p.170]). However, as for G̃, this gives many more extensions than are useful for our purposes – instead, we will define β-extensions in the non-maximal case by a compatibility property with the maximal case.

4.2 Beta-extensions in the general case

Let [Λ, n, 0, β] be any skew semisimple stratum, let θ ∈ C_{−}(Λ, 0, β) be a skew semisimple character and let η be the representation of J^{1} = J^{1}(β, Λ) given by Proposition 3.5. We are going to define βextensions of η in the case where 𝔟_{0}(Λ) is not necessarily a maximal self-dual 𝔬_{E}-order, by transfer from the maximal case of Theorem 4.1. In order to obtain compatibility bijections, we will see that we do not want to extend our representations all the way to the group J^{+}(β, Λ), but rather to the group J^{o} = J^{o}(β, Λ) = P^{o}(Λ_{𝔬_{E}})J^{1}, the inverse image in J(β, Λ) of the connected component of J(β, Λ)/J^{1}(β, Λ) ≅ P(Λ_{𝔬_{E}})/P_{1}(Λ_{𝔬_{E}}). Nevertheless, we initially define β-extensions to the group J^{+}(β, Λ) and then restrict back to J^{o}(β, Λ); this is because we need to be sure that our β-extensions to J^{o} extend to J^{+}(β, Λ).

Lemma 4.3 (cf. [8, Proposition 5.2.5], [27, Lemme 2.23, Proposition 2.9]). Using the notation above, let [Λ^{m}, n_{m}, 0, β] be a skew semisimple stratum such that 𝔟_{0}(Λ) ⊇ 𝔟_{0}(Λ^{m}). Let θ_{m} = τ_{Λ,Λ^{m},β}(θ) and let η_{m} be the unique irreducible representation of J_{m}^{1} = J^{1}(β, Λ^{m}) which contains θ_{m}. Then there is a canonical bijection between the set of extensions κ_{m} of η_{m} to J_{m}^{+} = J^{+}(β, Λ^{m}) and the set of extensions κ of η to J_{m,Λ}^{+} = P^{+}(Λ_{𝔬_{E}}^{m})J^{1}. If 𝔞_{0}(Λ) ⊇ 𝔞_{0}(Λ^{m}) then it is given as follows: given κ_{m} (respectively κ) there is a unique κ (respectively κ_{m}) such that κ_{m}|_{J_{m}^{+}} and κ|_{J_{m,Λ}^{+}} induce equivalent irreducible representations of P^{+}(Λ_{𝔬_{E}}^{m})P_{1}(Λ^{m}). Moreover, if g ∈ G_{E}^{+} then g intertwines κ_{m}|_{J_{m}^{+}} if and only if it intertwines the corresponding κ|_{J_{m,Λ}^{+}}.

Note that, in the special case of Lemma 4.3 when 𝔟_{0}(Λ) = 𝔟_{0}(Λ^{m}), we have J_{m,Λ}^{+} = J^{+}(β, Λ).

Proof \underline{Case 1} We assume first that 𝔞_{0}(Λ) ⊇ 𝔞_{0}(Λ^{m}). The proof of the first statement is identical to that of [8, Proposition 5.2.5]. Given the simple intersection property Lemma 2.6, the proof of the second statement is identical to that of [27, Proposition 2.9]. For the general case, we first need a compatibility condition:

Lemma 4.4 (cf. [8, (5.2.14)]). Suppose [Λ, n, 0, β], [Λ′, n′, 0, β] and [Λ^{M}, n_{M}, 0, β] are semisimple strata such that 𝔞_{0}(Λ) ⊆ 𝔞_{0}(Λ′) ⊆ 𝔞_{0}(Λ^{M}). Let θ ∈ C_{−}(Λ, 0, β) and θ′ = τ_{Λ,Λ′,β}, θ_{M} = τ_{Λ,Λ^{M},β}, and let η (respectively η′, η_{M}) be the irreducible representation of J^{1}(β, Λ) (respectively, J^{1}(β, Λ′), J^{1}(β, Λ^{M})) given by Proposition 3.5. Put J_{Λ,M}^{+} = P^{+}(Λ_{𝔬_{E}})J^{1}(β, Λ^{M}), J_{Λ′,M}^{+} = P^{+}(Λ′_{𝔬_{E}})J^{1}(β, Λ^{M}) and J_{Λ,Λ′}^{+} = P^{+}(Λ_{𝔬_{E}})J^{1}(β, Λ′) so J_{Λ,M}^{+} ⊆ J_{Λ′,M}^{+}. Then the following diagram commutes, where the maps are given by (Case 1 of) Lemma 4.3, or by restriction:

\xymatrix{\{extensions of η_{M} to J_{Λ′,M}^{+}\} \ar[r]^{res} \ar[d] & \{extensions of η_{M} to J_{Λ,M}^{+}\} \ar@{-->}[dr] \\ \{extensions of η′ to J^{+}(β, Λ′)\} \ar[r]^{res} & \{extensions of η′ to J_{Λ,Λ′}^{+}\} \ar[r] & \{extensions of η to J^{+}(β, Λ)\}}

Proof Suppose κ_{M} is an extension of η_{M} to J_{Λ′,M}^{+}, let κ′ be the corresponding extension of η′ to J^{+}(β, Λ′), and let κ be the extension of η to J^{+}(β, Λ) which corresponds to κ′|_{J_{Λ,Λ′}^{+}} . We form the

group P^{+}(Λ_{𝔬_{E}})P_{1}(Λ′) and note that we have

Ind _{J_{Λ,Λ′}^{+}}^{P^{+}(Λ_{𝔬_{E}})P_{1}(Λ′)}Res _{J_{Λ,Λ′}^{+}}^{J^{+}(β,Λ′)}κ′ ≃ Res _{P^{+}(Λ_{𝔬_{E}})P_{1}(Λ′)}^{P^{+}(Λ_{𝔬_{E}})P_{1}(Λ)}Ind _{J^{+}(β,Λ′)}^{P^{+}(Λ_{𝔬_{E}})P_{1}(Λ)}κ′,

since there is only one double coset in the Mackey restriction formula here. Similarly, we have

Ind _{J_{Λ,M}^{+}}^{P^{+}(Λ_{𝔬_{E}})P_{1}(Λ′)}Res _{J_{Λ,M}^{+}}^{J_{Λ′,M}^{+}}κ_{M} ≃ Res _{P^{+}(Λ_{𝔬_{E}})P_{1}(Λ′)}^{P^{+}(Λ′_{𝔬_{E}})P_{1}(Λ′)}Ind _{J_{Λ′,M}^{+}}^{P^{+}(Λ′_{𝔬_{E}})P_{1}(Λ′)}κ_{M}
≃ Res _{P^{+}(Λ_{𝔬_{E}})P_{1}(Λ′)}^{P^{+}(Λ′_{𝔬_{E}})P_{1}(Λ′)}Ind _{J^{+}(β,Λ′)}^{P^{+}(Λ′_{𝔬_{E}})P_{1}(Λ′)}κ′,

where the second equivalence is from the compatibility of κ′ and κ_{M}. Hence

Ind _{J_{Λ,M}^{+}}^{P^{+}(Λ_{𝔬_{E}})P_{1}(Λ)}Res _{J_{Λ,M}^{+}}^{J_{Λ′,M}^{+}}κ_{M} ≃ Ind _{P^{+}(Λ_{𝔬_{E}})P_{1}(Λ′)}^{P^{+}(Λ_{𝔬_{E}})P_{1}(Λ)}Res _{P^{+}(Λ_{𝔬_{E}})P_{1}(Λ′)}^{P^{+}(Λ′_{𝔬_{E}})P_{1}(Λ′)}Ind _{J^{+}(β,Λ′)}^{P^{+}(Λ′_{𝔬_{E}})P_{1}(Λ′)}κ′
≃ Ind _{J_{Λ,Λ′}^{+}}^{P^{+}(Λ_{𝔬_{E}})P_{1}(Λ)}Res _{J_{Λ,Λ′}^{+}}^{J^{+}(β,Λ′)}κ′ ≃ Ind _{J^{+}(β,Λ)}^{P^{+}(Λ_{𝔬_{E}})P_{1}(Λ)}κ.

Hence κ does indeed correspond to κ_{M}|_{J_{Λ,M}^{+}} via (Case 1 of) Lemma 4.3. ■

\underline{Case 2} We return to the proof of Lemma 4.3 and consider now the case where 𝔟_{0}(Λ) = 𝔟_{0}(Λ^{m}). Let Λ = Λ_{0}, Λ_{1}, ..., Λ_{t} = Λ^{m} be a sequence of self-dual 𝔬_{E}-lattice sequences given by Lemma 2.10, so that 𝔟_{0}(Λ_{i}) = 𝔟_{0}(Λ) and either 𝔞_{0}(Λ_{i}) ⊆ 𝔞_{0}(Λ_{i−1}) or 𝔞_{0}(Λ_{i}) ⊇ 𝔞_{0}(Λ_{i−1}), for 1 ⩽ i ⩽ t. For each i, let θ_{i} = τ_{Λ,Λ_{i},β}(θ) and let η_{i} be the representation of J^{1}(β, Λ_{i}) given by Proposition 3.5. Then, using the first case repeatedly (with {Λ_{i}, Λ_{i+1}} in place of {Λ, Λ^{m}}) and composing the bijections it gives, we get the required bijection between the set of extensions κ_{m} of η_{m} to J_{m}^{+} and the set of extensions κ of η to J_{m,Λ}^{+} = J^{+}(β, Λ). The assertion concerning intertwining also follows by repeated use of the first case. To prove that this bijection is independent of the choices of the Λ_{i}, we must use Lemma 4.4. So suppose that Λ = Λ′_{0}, Λ′_{1}, ..., Λ′_{t′} = Λ^{m} is another sequence of self-dual 𝔬_{E}-lattice sequences such that 𝔟_{0}(Λ_{i}) = 𝔟_{0}(Λ) and either 𝔞_{0}(Λ′_{i}) ⊆ 𝔞_{0}(Λ′_{i−1}) or 𝔞_{0}(Λ′_{i}) ⊇ 𝔞_{0}(Λ′_{i−1}), for 1 ⩽ i ⩽ t′. We consider the triangles

[Λ, Λ_{i}, Λ_{i+1}], for 1 ⩽ i ⩽ t − 1, and [Λ′_{i−1}, Λ′_{i}, Λ^{m}], for 1 ⩽ i ⩽ t′ − 1.

Using Lemma 2.12, each can be decomposed into smaller triangles [Γ_{j}, Γ′_{j}, Γ′′_{j}] such that 𝔞_{0}(Γ_{j}) ⊆ 𝔞_{0}(Γ′_{j}) ⊆ 𝔞_{0}(Γ′′_{j}). We write θ_{Γ_{j}}, θ_{Γ′_{j}}, θ_{Γ′′_{j}} for the transfers of θ, and η_{Γ_{j}}, η_{Γ′_{j}}, η_{Γ′′_{j}} for the representations given by Proposition 3.5. Then Lemma 4.4 gives us commutative diagrams

\xymatrix{\{extensions of η_{Γ′′_{j}} to J^{+}(β, Γ′′_{j})\} \ar[d] \ar@{-->}[dr] \\ \{extensions of η_{Γ′_{j}} to J^{+}(β, Γ′_{j})\} \ar[r] & \{extensions of η_{Γ_{j}} to J^{+}(β, Γ_{j})\}}

Putting together all these commutative diagrams, we see that the bijection

{extensions of η to J^{+}(β, Λ)} ⟷ {extensions of η_{m} to J_{m}^{+}}

is indeed independent of the choices. \underline{Case 3} Finally, consider the general case 𝔟_{0}(Λ) ⊇ 𝔟_{0}(Λ^{m}). By Lemma 2.8, there is a self-dual 𝔬_{E}-lattice sequence Λ′ such that 𝔟_{0}(Λ′) = 𝔟_{0}(Λ^{m}) and 𝔞_{0}(Λ) ⊇ 𝔞_{0}(Λ′). Put θ′ = τ_{Λ,Λ′,β} and let η′ be the irreducible representation of J^{1}(β, Λ′) given by Proposition 3.5. Applying Case 2 (with Λ′ in place of Λ), we get a canonical bijection between the set of extensions κ_{m} of η_{m} to J_{m}^{+} and the set of extensions κ′ of η′ to J^{+}(β, Λ′). Now applying Case 1 (with Λ′ in place of Λ^{m}), we get a canonical bijection between the set of extensions κ′ of η′ to J^{+}(β, Λ′) and the set of extensions κ of η to J_{m,Λ}^{+} = P^{+}(Λ′_{𝔬_{E}})J^{1}. The required bijection is the composition of these two and the assertion on intertwining is immediate. It remains to check that the bijection is independent of the choice of Λ′. Suppose Λ′′ is another 𝔬_{E}-lattice sequence such that 𝔟_{0}(Λ′′) = 𝔟_{0}(Λ^{m}) and 𝔞_{0}(Λ) ⊇ 𝔞_{0}(Λ′′), put θ′′ = τ_{Λ,Λ′′,β} and let η′′ be the irreducible representation of J^{1}(β, Λ′′) given by Proposition 3.5. Then we have a diagram in which the arrows are given by Cases 1 and 2 of Lemma 4.3

\xymatrix{\{extensions of η to J_{m,Λ}^{+}\} \ar[r] \ar[d] & \{extensions of η′′ to J^{+}(β, Λ′′)\} \ar[d] \ar[dl] \\ \{extensions of η′ to J^{+}(β, Λ′)\} \ar[r] & \{extensions of η_{m} to J_{m}^{+}\}}

The lower right triangle commutes by Case 2, while we can see that the upper left triangle commutes by splitting [Λ′, Λ′′, Λ] into triangles (using Lemma 2.12) and arguing as in Case 2. Hence the whole diagram commutes, as required. ■

We can now define β-extensions for an arbitrary skew semisimple stratum [Λ, n, 0, β] and character θ ∈ C_{−}(Λ, 0, β).

Definition 4.5. Let [Λ^{M}, n_{M}, 0, β] be a skew semisimple stratum with 𝔟_{0}(Λ^{M}) a maximal selfdual 𝔬_{E}-order in B and 𝔟_{0}(Λ^{M}) ⊇ 𝔟_{0}(Λ). Let θ_{M} = τ_{Λ,Λ^{M},β}, let η_{M} be the unique irreducible representation of J_{M}^{1} = J^{1}(β, Λ^{M}) which contains θ_{M}, and let κ_{M} be a β-extension of η_{M} to J_{M}^{+} = J^{+}(β, Λ^{M}), given by Theorem 4.1. Then the β-extension of η to J^{+} relative to Λ^{M}, compatible with κ_{M} is the unique representation κ of J^{+} which corresponds to κ_{M}|_{P^{+}(Λ_{𝔬_{E}})J_{M}^{1}} under the canonical bijection of Lemma 4.3.

The definition of β-extension, together with Lemma 4.3, immediately gives us a lower bound for their intertwining

Corollary 4.6. Let κ be a β-extension of η relative to Λ^{M}. Then P^{+}(Λ_{𝔬_{E}}^{M}) ⊆ I_{G^{+}}(κ).

Proof Note first that certainly P^{+}(Λ_{𝔬_{E}}^{M}) ⊆ I_{G^{+}}(κ_{M}), since P^{+}(Λ_{𝔬_{E}}^{M}) ⊆ J_{M}^{+}. In particular, P^{+}(Λ_{𝔬_{E}}^{M}) intertwines κ_{M}|_{P^{+}(Λ_{𝔬_{E}})J_{M}^{1}} so, by Lemma 4.3, it also intertwines κ. ■

This is, of course, much less than the situation for simple characters for G̃, where β-extensions are intertwined by the whole G̃-centralizer of β. When P(Λ_{𝔬_{E}}^{M}) is a good maximal compact subgroup of G_{E} it should also be possible to prove such a result here, using the Cartan decomposition and imitating the proofs in [8, §5.2]. Alternatively, if G is an unramified unitary group then the techniques of §5 (see also [27]) may provide a proof of this for any Λ^{M}. In general, the situation is much less clear. Although, in §6.3 we do improve our lower bound for β-extensions, we do not pursue these strong intertwining results here since we will not need them.

It is useful also to note that we can describe the restriction of a β-extension to the pro-p Sylow subgroup of J^{+}(β, Λ) in terms of the representation given by Proposition 3.7. The proof is identical to that of [8, Proposition 5.2.6].

Proposition 4.7 (cf. [8, Proposition 5.2.6]). Let [Λ, n, 0, β], [Λ^{m}, n_{m}, 0, β] be skew semisimple strata with 𝔟_{0}(Λ^{m}) a minimal self-dual 𝔬_{E}-order in B and 𝔞_{0}(Λ^{m}) ⊆ 𝔞_{0}(Λ). Let θ ∈ C_{−}(Λ, 0, β), let θ_{m} = τ_{Λ^{m},Λ,β}(θ), and let η (respectively η_{m}) be the irreducible representation of J^{1}(β, Λ) (respectively, J^{1}(β, Λ^{m})) given by Proposition 3.5. Put J_{m,Λ}^{1} = P_{1}(Λ_{𝔬_{E}}^{m})J^{1}(β, Λ), a pro-p Sylow subgroup of J^{+}(β, Λ), and let η_{m,Λ} be the irreducible representation of J_{m,Λ}^{1} given by Proposition 3.7 applied with Λ in place of Λ^{M}. Then, for any β-extension κ of η to J^{+}(β, Λ),

κ|_{J_{m,Λ}^{1}} = η_{m,Λ}.

If κ is a β-extension of η to J^{+}(β, Λ) then we say that the restriction κ|_{J^{o}(β,Λ)} is a β-extension of η to J^{o}(β, Λ). We have a compatibility property analogous to Lemma 4.3 for extensions to J^{o}(β, Λ). Indeed, the following Proposition shows that this compatibility is a bijection for β-extensions to J^{o}(β, Λ), as in [8, (5.2.14)].

Proposition 4.8 (cf. [8, (5.2.14)]). Let [Λ, n, 0, β] and [Λ′, n′, 0, β] be skew semisimple strata such that there exists [Λ^{M}, n_{M}, 0, β] a skew semisimple stratum with 𝔟_{0}(Λ^{M}) a maximal self-dual 𝔬_{E}- order containing both 𝔟_{0}(Λ) and 𝔟_{0}(Λ′). Let θ ∈ C_{−}(Λ, 0, β) be a skew semisimple character and let θ′ = τ_{Λ,Λ′,β}(θ). Let η, η′ be the representations of J^{1}(β, Λ), J^{1}(β, Λ′) respectively, given by Proposition 3.5. Then there is a canonical bijection between the set of β-extensions κ′ of η′ to J^{o}(β, Λ′) relative to Λ^{M} and the set of β-extensions κ of η to J^{o}(β, Λ) relative to Λ^{M}. If 𝔞_{0}(Λ) ⊆ 𝔞_{0}(Λ′) then it is given as follows: given κ (respectively κ′) there is a unique κ′ (respectively κ) such that κ′|_{P^{o}(Λ_{𝔬_{E}})J^{1}(β,Λ′)} and κ induce equivalent irreducible representations of P^{o}(Λ_{𝔬_{E}})P_{1}(Λ).

We say that κ′ is compatible with κ if they correspond under the bijection of Proposition 4.8.

Proof We observe first that, using Lemma 4.3 (or rather, its analogue for J^{o}), we can reduce to the case where 𝔞_{0}(Λ^{M}) contains both 𝔞_{0}(Λ) and 𝔞_{0}(Λ′): Lemma 2.8 shows that there is a self-dual 𝔬_{E}-lattice sequence Λ′′ with 𝔟_{0}(Λ′′) = 𝔟_{0}(Λ) and 𝔞_{0}(Λ′′) ⊆ 𝔞_{0}(Λ^{M}) and, by Lemma 4.3, there is a canonical bijection between the extensions of η to J^{o}(β, Λ) and the extensions of η′′ to J^{o}(β, Λ′′). Then we replace Λ by Λ′′ (and similarly for Λ′). When 𝔞_{0}(Λ) ⊆ 𝔞_{0}(Λ′), from the commutativity of the diagram in Lemma 4.4, and by restriction, we get a well-defined surjection

{β-extensions of η′ to J^{o}(β, Λ′) relative to Λ^{M}} → {β-extensions of η to J^{o}(β, Λ) relative to Λ^{M}} .

(4.9)

To see that we have a bijection, we need the following Lemma:

Lemma 4.10. Let [Λ^{M}, n_{M}, 0, β] be a skew semisimple stratum with 𝔟_{0}(Λ^{M}) a maximal self-dual 𝔬_{E}- order in B containing 𝔟_{0}(Λ). Let θ_{M} ∈ C_{−}(Λ^{M}, 0, β), let η_{M} be the representation of J_{M}^{1} = J^{1}(β, Λ^{M}) given by Proposition 3.5, and let κ_{M}, κ′_{M} be β-extensions of η_{M} to J_{M}^{+} = J^{+}(β, Λ^{M}), given by Theorem 4.1. Put J_{M}^{o} = J^{o}(β, Λ^{M}) and J_{Λ,M}^{o} = P^{o}(Λ_{𝔬_{E}})J_{M}^{1}. Then

κ_{M}|_{J_{Λ,M}^{o}} ≃ κ′_{M}|_{J_{Λ,M}^{o}} ⟺ κ_{M}|_{J_{M}^{o}} ≃ κ′_{M}|_{J_{M}^{o}}.

Proof By Theorem 4.1, we have κ′_{M} = κ_{M} ⊗ χ, for some character χ of G^{+} = P^{+}(Λ_{𝔬_{E}}^{M})/P_{1}(Λ_{𝔬_{E}}^{M}) which is trivial on the subgroup generated by all its unipotent subgroups. If κ_{M}|_{J_{Λ,M}^{o}} ≃ κ′_{M}|_{J_{Λ,M}^{o}} then χ is also trivial on P^{o} = P^{o}(Λ_{𝔬_{E}})/P_{1}(Λ_{𝔬_{E}}^{M}), which is a parabolic subgroup of G^{o}. Hence χ is trivial on all of G^{o} and κ_{M}|_{J_{M}^{o}} ≃ κ′_{M}|_{J_{M}^{o}}. The converse is trivial. ■

Returning to Proposition 4.8, we see that Lemmas 4.10 and 4.3 (together with the analogous Lemma for J^{o}(β, Λ)) imply that both sides of (4.9) have the same number of elements, which completes the proof when 𝔞_{0}(Λ) ⊆ 𝔞_{0}(Λ′). The general case follows immediately. ■

Remark In the simple case for G̃ = GL_{N}(F), all the maximal 𝔬_{E}-orders in B containing 𝔟_{0}(Λ) are conjugate by G̃_{E} so the set of β-extensions relative to Λ^{M} is independent of the choice of Λ^{M}. This is no longer true in our situation. However, we will see in Corollary 6.13 that there is some sort of compatibility between the β-extensions defined relative to different maximal self-dual 𝔬_{E}-orders in B. On the other hand, it will be useful in many situations to have a “canonical” choice for Λ^{M}, as follows: Given a skew semisimple stratum [Λ, n, 0, β], write Λ = ⊕_{i}Λ^{i}, with Λ^{i} an 𝔬_{E_{i}}-lattice sequence. Define the 𝔬_{E_{i}} lattice sequence 𝔐_{Λ}^{i} by

𝔐_{Λ}^{i}(2r + s) = 𝔭_{E_{i}}^{r}Λ^{i}(s),  for r ∈ ℤ, s = 0, 1,  (4.11)

and put 𝔐_{Λ} = ⊕_{i}𝔐_{Λ}^{i}. Then [𝔐_{Λ}, n_{𝔐}, 0, β] is a skew semisimple stratum, for some integer n_{𝔐}, and 𝔟_{0}(𝔐_{Λ}) is a maximal self-dual 𝔬_{E}-order containing 𝔟_{0}(Λ). Let θ ∈ C_{−}(Λ, 0, β) and let η be the unique irreducible representation of J^{1}(β, Λ) containing θ. We call a β-extension of η relative to 𝔐_{Λ} a standard β-extension.

In particular, if [Λ, n, 0, β] and [Λ′, n′, 0, β] are skew semisimple strata with 𝔐_{Λ} = 𝔐_{Λ′} then, by Proposition 4.8 there is a bijection between the standard β-extensions of η to J^{o}(β, Λ) and those of η′ to J^{o}(β, Λ′), where η′ is the unique irreducible representation of J^{1}(β, Λ′) containing θ′ = τ_{Λ,Λ′,β}(θ).

4.3 Analogous results for G̃

If [Λ, n, 0, β] is a semisimple stratum and θ̃ ∈ C(Λ, 0, β) is a semisimple character then the results of the previous sections can be imitated to construct β-extensions κ̃ of η̃ to J̃(β, Λ). Given Proposition 3.12, the proofs are identical to those for G and are left as an exercise for the reader. In this situation the notion of β-extension is independent of the choice of Λ^{M} such that 𝔟_{0}(Λ^{M}) is a maximal 𝔬_{E}-order containing 𝔟_{0}(Λ), the reductive quotient J̃(β, Λ)/J̃^{1}(β, Λ) is always connected and there is a canonical bijection between β-extensions, as in Proposition 4.8. We will not, in fact, use these results for semisimple characters, but only for simple characters, when they all come from [8, §5.2].

5 Iwahori factorizations

In this section we imitate the constructions of [8, §7] to get Iwahori factorizations of semisimple characters. We do this first for G̃ and then, by Glauberman transfer, for G^{+} and G. We end with some similar results for β-extensions.

5.1 Iwahori decompositions

Let [Λ, n, 0, β] be a semisimple stratum with associated splitting V = ⨁_{i=1}^{l} V^{i} and all the other usual notation. Let V = ⨁_{j=1}^{m} W^{(j)} be a decomposition of V into subspaces such that

(i) W^{(j)} = ⨁_{i=1}^{l} (W^{(j)} ∩ V^{i}), for 1 ⩽ j ⩽ m, so also V^{i} = ⨁_{j=1}^{m} (W^{(j)} ∩ V^{i}), for 1 ⩽ i ⩽ l;

(ii) W^{(j)} ∩ V^{i} is an E_{i}-subspace of V^{i}, for 1 ⩽ j ⩽ m and 1 ⩽ i ⩽ l.

Definition 5.1. We say that the decomposition V = ⨁_{j=1}^{m} W^{(j)} is:

(i) subordinate to [Λ, n, 0.β] if, for all r ∈ ℤ, Λ(r) = ⨁_{j=1}^{m} (Λ(r) ∩ W^{(j)});

(ii) properly subordinate to [Λ, n, 0, β] if it is subordinate to [Λ, n, 0, β] and, for all r ∈ ℤ and for 1 ⩽ i ⩽ l, there is at most one j, 1 ⩽ j ⩽ m, such that

(Λ(r) ∩ W^{(j)} ∩ V^{i}) ⊋ (Λ(r + 1) ∩ W^{(j)} ∩ V^{i}) .

An explanation of how these decompositions arise is in order at this point. We consider first the case when the stratum [Λ, n, 0, β] is simple. Let B be an E-basis for V which splits the lattice sequence Λ. If we decompose B as a disjoint union of subsets B_{j}, 1 ⩽ j ⩽ m, and define W^{(j)} to be the E-linear span of B_{j}, then we obtain a decomposition of V subordinate to the stratum. Moreover, any decomposition subordinate to the stratum arises in this way. For a properly subordinate decomposition, we need to be a little more careful. Writing e = e(Λ|𝔬_{E}) for the 𝔬_{E}-period of Λ, we decompose B into a disjoint union of subsets B_{j}, 1 ⩽ j ⩽ e, by saying that a vector w ∈ B lies in B_{j} if and only if there exists r ∈ ℤ such that w ∈ Λ(j +re)\Λ(j +re+1). Let W^{(j)} be the E-linear space of B_{j}. Then any decomposition of V which coarsens the decomposition V = ⨁_{j=1}^{e} W^{(j)} (that is, each subspace is a sum of some W^{(j)}) is properly subordinate to the stratum. Again, any decomposition properly subordinate to the stratum arises in this way.

For the semisimple case we take, for 1 ⩽ i ⩽ l, a decomposition V^{i} = ⨁_{j=1}^{m} W^{(i,j)} which is (properly) subordinate to the simple stratum [Λ^{i}, n_{i}, 0, β_{i}] (and where we allow some of the subspaces W^{(i,j)} to be trivial). Then we get a decomposition of V which is (properly) subordinate to [Λ, n, 0, β] by setting W^{(j)} = ⨁_{i=1}^{l} W^{(i,j)}.

Proposition 5.2 (cf. [8, Theorem 7.1.14]). Let [Λ, n, 0, β] be a semisimple stratum and let V = ⨁_{j=1}^{m} W^{(j)} be a decomposition as above. Let M̃ be the Levi subgroup of G̃ which is the stabilizer of this decomposition and let P̃ be any parabolic subgroup with Levi component M̃ and unipotent radical Ũ.

(i) If the decomposition is subordinate to [Λ, n, 0, β], then the groups H̃^{1}(β, Λ) and J̃^{1}(β, Λ) have Iwahori decompositions with respect to (M̃, P̃).

(ii) If the decomposition is properly subordinate to [Λ, n, 0, β], then the group J̃(β, Λ) also has an Iwahori decomposition with respect to (M̃, P̃) and, moreover, J̃(β, Λ) ∩ Ũ = J̃^{1}(β, Λ) ∩ Ũ.

Proof (i) For 1 ⩽ j ⩽ m, let e^{(j)} denote the projection onto W^{(j)} with kernel ⨁_{k≠j} W^{(k)}. The definition of subordinate (Definition 5.1(i)) ensures that e^{(j)} ∈ 𝔞_{0}(Λ). Moreover, the property that each W^{(j)} ∩ V^{i} is an E_{i}-subspace of V^{i}, for 1 ⩽ i ⩽ l, ensures that e^{(j)} ∈ B. Hence e^{(j)} ∈ 𝔟_{0}(Λ). Now J̃^{1}(β, Λ) = 1 + 𝔍^{1}(β, Λ) and, by [32, Lemma 3.10], 𝔍^{1}(β, Λ) is a 𝔟_{0}(Λ)-bimodule. In particular, e^{(j)}𝔍^{1}(β, Λ)e^{(k)} ⊆ 𝔍^{1}(β, Λ) so, by [7, Proposition 10.4], J̃^{1}(β, Λ) has an Iwahori decomposition with respect to (M̃, P̃). The same proof works for H̃^{1}(β, Λ). (ii) Suppose now that the decomposition is properly subordinate to [Λ, n, 0, β]. We start with a lemma.

Lemma 5.3 (cf. [8, Lemma 7.1.15]). In this situation we have P̃(Λ_{𝔬_{E}}) ∩ Ũ = P̃_{1}(Λ_{𝔬_{E}}) ∩ Ũ.

Proof We prove the corresponding additive statement, which is e^{(j)}𝔟_{0}(Λ)e^{(k)} ⊆ 𝔟_{1}(Λ), for j ≠ k. So suppose x ∈ e^{(j)}𝔟_{0}(Λ)e^{(k)}. For any r ∈ ℤ and for s ≠ k we have x(Λ(r) ∩ W^{(s)}) = 0. On the other hand, for any r ∈ ℤ and 1 ⩽ i ⩽ l, either Λ(r) ∩ W^{(k)} ∩ V^{i} = Λ(r + 1) ∩ W^{(k)} ∩ V^{i} or Λ(r) ∩ W^{(j)} ∩ V^{i} = Λ(r + 1) ∩ W^{(j)} ∩ V^{i}, since the decomposition is properly subordinate. In either case, we get x(Λ(r) ∩ W^{(k)} ∩ V^{i}) ⊆ Λ(r + 1) ∩ W^{(j)} ∩ V^{i}. Hence

xΛ(r) = x (Λ(r) ∩ W^{(k)}) = ⨁_{i=1}^{l} x (Λ(r) ∩ W^{(k)} ∩ V^{i}) ⊆ Λ(r + 1),

and x ∈ 𝔞_{1}(Λ) as required. ■

Now the same proof as that of [8, Theorem 7.1.14] shows that P̃(Λ_{𝔬_{E}}) has an Iwahori decomposition with respect to (M̃, P̃) and that P̃(Λ_{𝔬_{E}}) = (P̃(Λ_{𝔬_{E}}) ∩ M̃) P̃_{1}(Λ_{𝔬_{E}}). Hence J(β, Λ) =

(P̃(Λ_{𝔬_{E}}) ∩ M̃) J^{1}(β, Λ) has an Iwahori decomposition with respect to (M̃, P̃), and the final assertion follows also. ■

Let [Λ, n, 0, β] be a semisimple stratum and let V = ⨁_{j=1}^{m} W^{(j)} be a decomposition subordinate to the stratum. We write Λ^{(j)} for the lattice sequence in W^{(j)} given by

Λ^{(j)}(r) = Λ(r) ∩ W^{(j)}, for r ∈ ℤ,

and put β^{(j)} = e^{(j)}βe^{(j)}, where e^{(j)} denotes the projection onto W^{(j)} with kernel ⨁_{k≠j} W^{(k)}. Then, since each V^{i} ∩ W^{(j)} is an E_{i}-subspace of V^{i}, the stratum [Λ^{(j)}, n^{(j)}, 0, β^{(j)}] is a semisimple stratum in A^{(j)} = End_{F} (W^{(j)}), for some integer n^{(j)}, with splitting W^{(j)} = ⨁_{i=1}^{l} (V^{i} ∩ W^{(j)}). Note that some of the pieces of this splitting may be trivial so, strictly speaking, the splitting consists only of the non-zero pieces. We also write B^{(j)} for the centralizer of β^{(j)} in A^{(j)}.

Proposition 5.4. Let [Λ, n, 0, β] be a semisimple stratum and let V = ⨁_{j=1}^{m} W^{(j)} be a decomposition subordinate to the stratum. Let M̃ be the Levi subgroup of G̃ which is the stabilizer of this decomposition. Then

H̃^{1}(β, Λ) ∩ M̃ = ∏_{j=1}^{m} H̃^{1}(β^{(j)}, Λ^{(j)}),

and likewise for J̃^{1}(β, Λ). If the decomposition is properly subordinate to the stratum then we also have

J̃(β, Λ) ∩ M̃ = ∏_{j=1}^{m} J̃(β^{(j)}, Λ^{(j)}).

Proof We will prove the additive statement

e^{(j)}ℌ(β, Λ)e^{(j)} = ℌ(β^{(j)}, Λ^{(j)}).

The same proof will give the corresponding statement for 𝔍(β, Λ) and the result follows.

Let M denote the stabilizer in A of the decomposition, so that M̃ = M^{×}. We proceed by induction on k_{0}(β, Λ), noting that the base case is a null stratum, where the result is clear since ℌ(β, Λ) = 𝔞_{0}(Λ). So we turn to the inductive step and put r = −k_{0}(β, Λ). Put W^{(i,j)} = V^{i} ∩ W^{(j)} so that V^{i} = ⨁_{j=1}^{m} W^{(i,j)} is (properly) subordinate to the simple stratum [Λ^{i}, n_{i}, 0, β_{i}]. We choose an E_{i}-basis B_{i,j} for each W^{(i,j)} which splits the lattice sequence Λ^{(i,j)} given by Λ^{(i,j)}(r) = Λ(r)∩W^{(i,j)}, and let Y^{(i,j)} be the F -linear span of B_{i,j}. Putting Y^{i} = ⨁_{j=1}^{m} Y^{(i,j)}, we get a “generalized (W, E)-decomposition” V^{i} = Y^{i} ⊗_{F} E_{i} (see [10, §5.3]), and hence an embedding ι_{Y^{i}} : End_{F} (E_{i}) ↪ A^{ii} = End_{F} (V^{i}) with image in A^{ii} ∩ M. According to [32, Proposition 3.4], we can choose a semisimple stratum [Λ, n, r, γ] equivalent to [Λ, n, r, β] with γ in the image of ∏_{i} ι_{Y^{i}}. In particular, we have γ ∈ M so the decomposition is subordinate to [Λ, n, r, γ] also. Hence, by the inductive hypothesis, we have

e^{(j)}ℌ(γ, Λ)e^{(j)} = ℌ(γ^{(j)}, Λ^{(j)}),

where γ^{(j)} = e^{(j)}γe^{(j)}. The semisimple stratum [Λ^{(j)}, n^{(j)}, r, γ^{(j)}] is equivalent to [Λ^{(j)}, n^{(j)}, r, β^{(j)}] so, since e^{(j)} ∈ 𝔟_{0}(Λ), we get

e^{(j)}ℌ(β, Λ)e^{(j)} = e^{(j)}𝔟_{0}(Λ)e^{(j)} + e^{(j)}ℌ^{[^{\frac{r}{2}}]+1}(γ, Λ)e^{(j)}
= 𝔟_{0}(Λ^{(j)}) + ℌ^{[^{\frac{r}{2}}]+1}(γ^{(j)}, Λ^{(j)})
= ℌ(β^{(j)}, Λ^{(j)}),

as required. ■

Continuing in the same situation, let P̃ be any parabolic subgroup of G̃ with Levi subgroup M̃, and let Ũ denote its unipotent radical. We define the groups

H̃1 ; P̃ = H̃^{1}(β, Λ) (J̃^{1}(β, Λ) ∩ Ũ) ,  and J̃1 ; P̃ = H̃^{1}(β, Λ) (J̃^{1}(β, Λ) ∩ P̃) .

If the decomposition is properly subordinate to the stratum, we also put

J̃P̃ = H̃^{1}(β, Λ) (J̃(β, Λ) ∩ P̃) .

All these groups have Iwahori decompositions with respect to any parabolic subgroup with Levi component M̃.

5.2 Iwahori factorization of semisimple characters

We continue in the same situation, so [Λ, n, 0, β] is a semisimple stratum and V = ⨁_{j=1}^{m} W^{(j)} is a decomposition subordinate to the stratum. Let M̃ be the Levi subgroup of G̃ which is the stabilizer of this decomposition and let P̃ be any parabolic subgroup with Levi component M̃. Write Ũ for the unipotent radical of P̃.

Proposition 5.5 (cf. [8, Proposition 7.1.19]). In this situation, let θ̃ ∈ C(Λ, 0, β). Then the character θ̃|_{H̃^{1}(β,Λ)∩Ũ} is trivial. After identifying H̃^{1}(β, Λ) ∩ M̃ with ∏_{j=1}^{m} H̃^{1}(β^{(j)}, Λ^{(j)}), we have

θ̃|_{H̃^{1}(β,Λ)∩M̃} = ⨂_{j=1}^{m} θ̃^{(j)},

where θ̃^{(j)} ∈ C(Λ^{(j)}, 0, β^{(j)}).

In this situation, we shall write θ̃^{(j)} = τ_{Λ,Λ^{(j)},β}(θ̃). By the definition of simple character in [10, §5], this is consistent with the notion of transfer between different algebras for simple characters. We do not pursue the functorial properties of this transfer map but note that it gives a map τ_{Λ,Λ^{(j)},β} : C(Λ, 0, β) → C(Λ^{(j)}, 0, β^{(j)}) which is not in general injective; this happens if one of the subspaces W^{(j)} ∩ V^{i} = {0} but β_{i} ≠ 0.

Proof We proceed by induction on k_{0}(β, Λ), the base case again being the null case, when the only semisimple character is the trivial one and the result is clear. So we consider the induction step and put r = −k_{0}(β, Λ). We will use again the notation of the proof of Proposition 5.4, so that [Λ, n, r, γ] is the carefully chosen semisimple stratum equivalent to [Λ, n, r, β] from there. Recall that θ̃ has the properties:

(i) we have θ̃|_{H̃^{1}∩G̃_{i}} = θ̃_{i}, for some simple character θ̃_{i} ∈ C(Λ^{i}, 0, β_{i}), where G̃_{i} = Aut_{F} (V^{i});

(ii) θ̃|_{H̃^{[ \frac{r}{2} ]+1}(β,Λ)} = θ̃_{0}|_{H̃^{[ \frac{r}{2} ]+1}(β,Λ)}ψ_{β−γ}, for some semisimple character θ̃_{0} ∈ C(Λ, 0, γ).

Moreover, these properties characterize the semisimple characters in C(Λ, 0, β).

For the first assertion of the Lemma, let M̃_{β} denote the stabilizer of the splitting V = ⨁_{i=1}^{l} V^{i} and let Ũ_{β} be the unipotent radical of some parabolic subgroup P̃_{β} with Levi component M̃_{β}. According to [32, Lemma 3.15(i)], the restriction θ̃|_{H̃^{1}∩Ũ_{β}} is trivial. Since H̃^{1} ∩Ũ has an Iwahori decomposition

with respect to (M̃_{β}, P̃_{β}), we are reduced to showing that θ̃ restricts trivially to H̃^{1} ∩ Ũ ∩ M̃_{β}. But here we may work block-by-block so the result follows from the simple case in [10, Proposition 5.2]. Now we turn to the second assertion. From Proposition 5.4 we have that H̃^{1}(β, Λ) ∩ Aut_{F} (W^{(j)}) = H̃^{1}(β^{(j)}, Λ^{(j)}) so we may consider θ̃^{(j)} = θ̃|_{H̃^{1}(β^{(j)},Λ^{(j)})}. To show that it is a semisimple character in C(Λ^{(j)}, 0, β^{(j)}) we need only check that it satisfies the two properties (i), (ii) above. The splitting of [Λ^{(j)}, n^{(j)}, 0, β^{(j)}] is given by W^{(j)} = ⨁_{i=1}^{l} W^{(i,j)}, where W^{(i,j)} = V^{i} ∩ W^{(j)} and we ignore any trivial pieces of the splitting. We put G̃^{(i,j)} = Aut_{F} (W^{(i,j)}), let 1^{(i,j)} be the projection from W^{(j)} to W^{(i,j)} with kernel ⨁_{k≠i} W^{(k,j)}, and let e^{(i,j)} be the projection from V^{i} to W^{(i,j)} with

kernel ⨁_{k≠j} W^{(i,k)}. We put β_{i}^{(j)} = e^{(i,j)}β_{i}e^{(i,j)} = 1^{(i,j)}β^{(j)}1^{(i,j)}.

(i) We have H̃^{1}(β^{(j)}, Λ^{(j)}) ∩ G̃^{(i,j)} = H̃^{1}(β_{i}^{(j)}, Λ^{(i,j)}) and

θ̃^{(j)}|_{H̃^{1}(β^{(j)},Λ^{(j)})∩G̃^{(i,j)}} = (θ̃|_{H̃^{1}(β,Λ)∩G̃^{i}})|_{H̃^{1}(β_{i}^{(j)},Λ^{(i,j)})} = θ̃_{i}|_{H̃^{1}(β_{i}^{(j)},Λ^{(i,j)})},

which, by [10, §5], is the simple character τ_{Λ^{i},Λ^{(i,j)},β_{i}}(θ̃_{i}) in C(Λ^{(i,j)}, 0, β_{i}^{(j)}).

(ii) The stratum [Λ^{(j)}, n^{(j)}, 0, γ^{(j)}] is a semisimple stratum equivalent to [Λ^{(j)}, n^{(j)}, 0, β^{(j)}] and it is straightforward that ψ_{β−γ}|_{H̃^{[ \frac{r}{2} ]+1}(β^{(j)},Λ^{(j)})} = ψ_{β^{(j)}−γ^{(j)}}. Thus we have

θ̃^{(j)}|_{H̃^{[ \frac{r}{2} ]+1}(β^{(j)},Λ^{(j)})} = (θ̃|_{H̃^{[ \frac{r}{2} ]+1}(β,Λ)})|_{H̃^{[ \frac{r}{2} ]+1}(β^{(j)},Λ^{(j)})}
= (θ̃_{0}|_{H̃^{[ \frac{r}{2} ]+1}(β,Λ)}ψ_{β−γ})|_{H̃^{[ \frac{r}{2} ]+1}(β^{(j)},Λ^{(j)})} = θ̃_{0}^{(j)}|_{H̃^{[ \frac{r}{2} ]+1}(β^{(j)},Λ^{(j)})}ψ_{β^{(j)}−γ^{(j)}},

for some θ̃_{0}^{(j)} ∈ C(Λ^{(j)}, 0, γ^{(j)}), by the inductive hypothesis. ■

We continue with the same situation. From [32, Proposition 3.24], the pairing

kθ̃(x, y) = θ̃[x, y], x, y ∈ J̃^{1}(β, Λ),

induces a nondegenerate alternating form on J̃^{1}(β, Λ)/H̃^{1}(β, Λ). Likewise, for j = 1, ..., m we have a form kθ̃^{(j)} on J̃^{1}(β^{(j)}, Λ^{(j)})/H̃^{1}(β^{(j)}, Λ^{(j)}).

Let Ũ_{l} denote the unipotent radical of the parabolic subgroup opposite P̃. As an immediate consequence of Proposition 5.5 and the Iwahori decompositions for H̃^{1}(β, Λ) and J̃^{1}(β, Λ), we get:

Lemma 5.6 (cf. [8, Proposition 7.2.3]). (i) The subspaces J̃^{1} ∩ Ũ_{l}/H̃^{1} ∩ Ũ_{l} and J̃^{1} ∩ Ũ/H̃^{1} ∩ Ũ of J̃^{1}/H̃^{1} are both totally isotropic for the form kθ̃ and orthogonal to J̃^{1} ∩ M̃/H̃^{1} ∩ M̃.

(ii) The restriction of kθ̃ to the group

J̃^{1} ∩ M̃/H̃^{1} ∩ M̃ = ∏_{j=1}^{m} J̃^{1}(β^{(j)}, Λ^{(j)})/H̃^{1}(β^{(j)}, Λ^{(j)})

is the orthogonal sum of the pairings kθ̃^{(j)}, where θ̃^{(j)} = τ_{Λ,Λ^{(j)},β}(θ̃).

(iii) We have an orthogonal sum decomposition

\frac{J̃^{1}}{H̃^{1}} = \frac{J̃^{1} ∩ M̃}{H̃^{1} ∩ M̃} ⊥ (\frac{J̃^{1} ∩ Ũ_{l}}{H̃^{1} ∩ Ũ_{l}} × \frac{J̃^{1} ∩ Ũ}{H̃^{1} ∩ Ũ} ) .

Recall that we have defined the group H̃1 ; P̃ = H̃^{1}(β, Λ) (J̃^{1}(β, Λ) ∩ Ũ), and, similarly, J̃1 ; P̃. We define

the character θ̃P̃ of H̃1 ; P̃ by

θ̃P̃(hj) = θ̃(h), for h ∈ H̃^{1}(β, Λ), j ∈ J̃^{1}(β, Λ) ∩ Ũ.

This is well-defined by Lemma 5.6. As in [8, §7.2], we immediately get

Corollary 5.7 (cf. [8, Propositions 7.2.4, 7.2.9]). There is a unique irreducible representation η̃P̃

of J̃1 ; P̃ such that η̃P̃|H̃1 ; P̃ contains θ̃P̃. Moreover, η̃ ≃ Ind J̃^{1}(β,Λ) ; J̃1 ; P̃η̃P̃ and, for each y ∈ G̃_{E}, there is a

unique (J̃1 ; P̃, J̃1 ; P̃)-double coset in J̃^{1}(β, Λ)yJ̃^{1}(β, Λ) which intertwines η̃P̃.

Lemma 5.8 (cf. [16, Proposition 2.4]). We have IG̃(θ̃P̃) = IG̃(η̃P̃) = J̃1 ; P̃G̃_{E}J̃1 ; P̃.

Proof We have IG̃(θ̃P̃) = IG̃(η̃P̃) so, by Corollary 5.7, we need only check that all of G̃_{E} intertwines θ̃P̃.

Let B_{i,j} be an E_{i}-basis for W^{(i,j)} = V^{i} ∩ W^{(j)} which splits the lattice sequence Λ^{(i,j)} and consider the decomposition

V = ⨁_{i,j} ⨁_{v∈B_{i,j}} E_{i}v.  (5.9)

This is subordinate to the stratum [Λ, n, 0, β] and refines both the decomposition V = ⨁_{j=1}^{m} W^{(j)} and the splitting V = ⨁_{i=1}^{l} V^{i} of the stratum. Let M̃_{0} denote the stabilizer of the decomposition (5.9) and let P̃_{0} ⊆ P̃ be a parabolic subgroup with Levi component M̃_{0}. We write Ũ_{0} for the unipotent radical of P̃_{0}, so Ũ_{0} ⊇ Ũ. Let P_{0} ⊆ A denote the Lie algebra of P̃_{0}. Then P_{0} ∩ 𝔟_{0}(Λ) + 𝔟_{1}(Λ) is a minimal 𝔬_{E}-order in B, which must be of the form 𝔟_{0}(Λ^{m}), for some 𝔬_{E}-lattice sequence Λ^{m} in V. Then, for a suitable integer n_{m}, the stratum [Λ^{m}, n_{m}, 0, β] is semisimple and the decomposition (5.9) is subordinate to it. Moreover, P̃(Λ_{𝔬_{E}}^{m}) is an Iwahori subgroup of G̃_{E} contained in P̃(Λ_{𝔬_{E}}).

Now, by construction, P̃(Λ_{𝔬_{E}}^{m}) = (P̃(Λ_{𝔬_{E}}) ∩ P̃_{0})P̃^{1}(Λ_{𝔬_{E}}) ⊆ J̃P̃ and, since J̃P̃ normalizes θ̃P̃, we need only show that some set of double coset representatives for P̃(Λ_{𝔬_{E}}^{m})\G̃_{E}/P̃(Λ_{𝔬_{E}}^{m}) intertwines θ̃P̃. By the Bruhat decomposition, we may take these representatives to be in the G̃_{E}-normalizer of the maximal torus M̃_{0} ∩ G̃_{E}.

So suppose y ∈ G̃_{E} normalizes M̃_{0}. Since H̃1 ; P̃ has Iwahori decompositions with respect to both

(M̃_{0}, P̃_{0}) and (M̃_{0}, P̃_{0}^{y}), the group H̃1 ; P̃ ∩^{y}H̃1 ; P̃ has an Iwahori decomposition with respect to (M̃_{0}, P̃_{0}). Moreover, θ̃P̃ is trivial on both H̃1 ; P̃ ∩ Ũ_{0} and H̃1 ; P̃ ∩ Ũ_{0}^{y}. Hence, to check that y intertwines θ̃P̃, we

need only check that it intertwines θ̃P̃|_{H̃1 ; P̃∩M̃_{0}}. But, since M̃_{0} ⊆ M̃ and H̃1 ; P̃ ∩ M̃ = H̃^{1} ∩ M̃, this

restriction is just θ̃|_{H̃^{1}(β,Λ)∩M̃_{0}}, which is surely intertwined by y, by Proposition 3.1. ■

We remark that, in the same way as in [8, §7.2], we can describe η̃P̃ as the natural representation of J̃1 ; P̃ on the space of (J̃^{1}(β, Λ) ∩ Ũ)-fixed vectors in η̃. It is also useful to note that the restriction

of η̃P̃ to J̃1 ; P̃ ∩ M̃ = J̃^{1}(β, Λ) ∩ M̃ is given by

η̃P̃|_{J̃^{1}(β,Λ)∩M̃} ≃ ⨂_{j=1}^{m} η^{(j)},

where η^{(j)} is the unique irreducible representation of J̃^{1}(β^{(j)}, Λ^{(j)}) which contains the character θ̃^{(j)} = τ_{Λ,Λ^{(j)},β}(θ̃).

5.3 Glauberman transfer

In this section, we consider the case when we have a skew semisimple stratum [Λ, n, 0, β]. We say that a decomposition V = ⨁_{j=−m}^{m} W^{(j)} is self-dual if, for −m ⩽ j ⩽ m, the orthogonal complement of W^{(j)} is ⨁_{k≠−j} W^{(k)}; we allow the possibility that W^{(0)} = {0}. Then, immediately from Proposition 5.2 we get:

Corollary 5.10. Let [Λ, n, 0, β] be a skew semisimple stratum and let V = ⨁_{j=−m}^{m} W^{(j)} be a self-dual decomposition as above. Let M̃ be the Levi subgroup of G̃ which is the stabilizer of this decomposition and let P̃ be any σ-stable parabolic subgroup with Levi component M̃. Put M^{+} = M̃ ∩ G^{+}, P^{+} = P̃ ∩ G^{+} and M = M̃ ∩ G, P = P̃ ∩ G.

(i) If the decomposition is subordinate to [Λ, n, 0, β], then the groups H^{1}(β, Λ) and J^{1}(β, Λ) have Iwahori decompositions with respect to (M, P).

(ii) If the decomposition is properly subordinate to [Λ, n, 0, β], then the group J^{+}(β, Λ) also has an Iwahori decomposition with respect to (M^{+}, P^{+}) and the groups J(β, Λ), J^{o}(β, Λ) have Iwahori decompositions with respect to (M, P).

Proof Note that, if g ∈ G^{+} has Iwahori decomposition g = lmu, with respect to some self-dual (M̃, P̃) in G̃, then also g = l^{σ}m^{σ}u^{σ} so, by uniqueness of decomposition, l, m, u ∈ G^{+}. Now, since U ⊆ G, if g ∈ G then l, m, u ∈ G also. ■

Notice that, in this situation, the involution swaps the blocks e^{(j)}Ae^{(k)} and e^{(−k)}Ae^{(−j)} so that the stratum [Λ^{(0)}, n^{(0)}, 0, β^{(0)}] is skew semisimple. From Proposition 5.4 we get:

Corollary 5.11. Let [Λ, n, 0, β] be a skew semisimple stratum and let V = ⨁_{j=−m}^{m} W^{(j)} be a self-dual decomposition subordinate to the stratum. Let M̃ be the Levi subgroup of G̃ which is the stabilizer of this decomposition and put M^{+} = M̃ ∩ G^{+}, M = M̃ ∩ G. Then

H^{1}(β, Λ) ∩ M ≃ H^{1}(β^{(0)}, Λ^{(0)}) × ∏_{j=1}^{m} H̃^{1}(β^{(j)}, Λ^{(j)}),

and there is a similar decomposition for J^{1}(β, Λ). If the decomposition is properly subordinate to the stratum then we also have

J^{o}(β, Λ) ∩ M ≃ J^{o}(β^{(0)}, Λ^{(0)}) × ∏_{j=1}^{m} J̃(β^{(j)}, Λ^{(j)}),

and similar decompositions for J^{+}(β, Λ) ∩ M^{+} and J(β, Λ) ∩ M.

Note here that H̃^{1}(β^{(j)}, Λ^{(j)}) has been identified with

\{(k, h) ∈ H̃^{1}(β^{(−j)}, Λ^{(−j)}) × H̃^{1}(β^{(j)}, Λ^{(j)}) such that h\overline{k} = 1\}

(Recall that the involution swaps the blocks e^{(j)}Ae^{(j)} and e^{(−j)}Ae^{(−j)}.)

We can also form the groups H_{P}^{1} = H̃1 ; P̃ ∩ G = H^{1}(β, Λ) (J^{1}(β, Λ) ∩ U) and, likewise, J_{P}^{1} and (when the decomposition is properly subordinate to the stratum) J_{P}^{+} = H^{1}(β, Λ) (J^{+}(β, Λ) ∩ P), and J_{P}, J_{P}^{o} .

Now, using the Glauberman correspondence, we transfer the results of §5.2 to the case of skew semisimple strata. We recall briefly the nature of the correspondence, for details of which (in our situation) we refer the reader to [30, §2].

Let K̃ be a σ-stable pro-p subgroup of G̃ and K = K̃^{σ} the group of σ-fixed points. Then σ acts on the representations ρ̃ of K̃ by ρ^{σ}(k) = ρ(k^{σ}), for k ∈ K̃. There is a bijection, denoted ρ̃ ↔ ρ = g(ρ̃), between (equivalence classes of) irreducible representations of K̃ with ρ^{σ} ≃ ρ̃ and (equivalence classes of) irreducible representations of K. Further, this correspondence commutes with irreducible restriction and irreducible induction. Recall also that the representation ρ = g(ρ̃) is characterized as the unique component of ρ̃|_{K} appearing with odd multiplicity; in particular, if ρ̃ is a character then ρ = ρ̃|_{K}. Moreover, for g ∈ G^{+}, we have that dim I_{g}(ρ̃) is odd if and only if dim I_{g}(ρ) is odd; in particular, if ρ̃ is a character, or if all the intertwining spaces are one-dimensional (as is often the case for us), then I_{G^{+}}(ρ|K) = IG̃(ρ̃|K̃) ∩ G^{+}.

Let [Λ, n, 0, β] be a skew semisimple stratum with a subordinate self-dual decomposition V = ⨁_{j=−m}^{m} W^{(j)}. Let M̃ be the stabilizer in G̃ of this decomposition and let P̃ be a σ-stable parabolic subgroup with Levi component M̃. Put M = M̃ ∩ G and P = P̃ ∩ G. Let θ ∈ C_{−}(Λ, 0, β) be a skew semisimple character of H^{1}(β, Λ), so that θ = θ̃|_{H^{1}(β,Λ)}, for some σ-stable semisimple character θ̃ ∈ C(Λ, 0, β). From Corollary 5.11 and Proposition 5.5, we have

θ|_{H^{1}(β,Λ)∩M} = θ^{(0)} ⊗ ⨂_{j=1}^{m} \binom{}{θ^{(j)}}^{2} ,

for some θ^{(0)} ∈ C_{−}(Λ^{(0)}, 0, β^{(0)}) and θ̃^{(j)} ∈ C(Λ^{(j)}, 0, β^{(j)}). Notice that we have H̃^{1}(β^{(j)}, Λ^{(j)}) =

H̃^{1}(2β^{(j)}, Λ^{(j)}) and \binom{}{θ^{(j)}}^{2} ∈ C(Λ^{(j)}, 0, 2β^{(j)}), as in [3, §4.3 Lemma 1].

We define the character θ_{P} of H_{P}^{1} = H^{1}(β, Λ) (J^{1}(β, Λ) ∩ U) by

θ_{P} (hj) = θ(h), for h ∈ H^{1}(β, Λ), j ∈ J^{1}(β, Λ) ∩ U.

Then θ_{P} = θ̃P̃|_{H_{P}^{1}} and is the Glauberman transfer of θ̃P̃.

Let η be the unique irreducible representation of J^{1}(β, Λ) containing θ and let η̃ be the unique irreducible representation of J̃^{1}(β, Λ) containing θ̃. Then η = g(η̃). Let η̃P̃ be the unique irreducible representation of J̃1 ; P̃ containing θ̃P̃, given by Corollary 5.7, and define the representation η_{P} of J_{P}^{1} = H^{1}(β, Λ) (J^{1}(β, Λ) ∩ P ) by η_{P} = g(η̃P̃). Then Corollary 5.7 and Lemma 5.8, together with the properties of the Glauberman correspondence, give us

Lemma 5.12. The representation η_{P} is the unique irreducible representation of J_{P}^{1} such that η_{P} |_{H_{P}^{1}}

contains θ_{P}. We have η ≃ Ind _{J_{P}^{1}}^{J^{1}(β,Λ)}η_{P}. Moreover, for each y ∈ G_{E}^{+}, the coset J_{P}^{1} yJ_{P}^{1} is the unique (J_{P}^{1} , J_{P}^{1} )-double coset in J^{1}(β, Λ)yJ^{1}(β, Λ) which intertwines η_{P} and

dim I_{g}(η_{P}) = \begin{cases}1 & if g ∈ J_{P}^{1} G_{E}^{+}J_{P}^{1} ; \\ 0 & otherwise. \end{cases}

Proof The statement about the dimensions of the intertwining spaces comes from the corresponding statement for η, which is Proposition 3.5. ■

As above we can describe η_{P} as being the natural representation of J_{P}^{1} on the space of (J^{1}(β, Λ)∩U)- fixed vectors in η and the restriction of η_{P} to J_{P}^{1} ∩ M = J^{1}(β, Λ) ∩ M is given by

η_{P} |_{J^{1}(β,Λ)∩M} ≃ η^{(0)} ⊗ ⨂_{j=1}^{m} η^{(j)},

where η^{(0)} is the unique irreducible representation of J^{1}(β^{(0)}, Λ^{(0)}) which contains the character θ^{(0)} and, for j non-zero, η^{(j)} is the unique irreducible representation of J̃^{1}(β^{(j)}, Λ^{(j)}) = J̃^{1}(2β^{(j)}, Λ^{(j)})

which contains the character \binom{}{θ^{(j)}}^{2}.

We continue in the same situation and suppose from now on that the self-dual decomposition V = ⨁_{j=−m}^{m} W^{(j)} is properly subordinate to the skew semisimple stratum [Λ, n, 0, β]. Let κ be a β-extension of η to J^{+}(β, Λ) (relative to some Λ^{M}). We can form the natural representation κ_{P} of J_{P}^{+} on the space of (J^{+}(β, Λ) ∩ U) = (J^{1}(β, Λ) ∩ U)-fixed vectors in κ. Then κ_{P} |_{J_{P}^{1}} = η_{P} so, in particular, κ_{P} is irreducible. As in the proof of [8, Proposition 7.2.15], the Mackey restriction formula gives:

Proposition 5.13 (cf. [8, Proposition 7.2.15]). We have Ind _{J_{P}^{+}}^{J^{+}(β,Λ)}κ_{P} ≃ κ.

As for η_{P}, we can describe the restriction of κ_{P} to J_{P}^{+} ∩ M = J^{+}(β, Λ) ∩ M. Since κ_{P} |_{J^{+}(β,Λ)∩M} restricts further to η_{P} |_{J^{1}(β,Λ)∩M}, it is irreducible and

κ_{P} |_{J^{+}(β,Λ)∩M} = κ^{(0)} ⊗ ⨂_{j=1}^{m} κ^{(j)},

where κ^{(0)} is some irreducible representation of J^{+}(β^{(0)}, Λ^{(0)}) which extends η^{(0)} and κ^{(j)} is some irreducible representation of J̃(β^{(j)}, Λ^{(j)}) which extends η^{(j)}. We will see below in Proposition 6.3 that, at least in certain circumstances, κ^{(0)} is a β^{(0)} extension and κ^{(j)} is a 2β^{(j)}-extension for 1 ⩽ j ⩽ m.

We can also make the same constructions for the restriction κ|_{J^{o}(β,Λ)}, and we clearly obtain the restriction κ_{P} |_{J_{P}^{o}} . Moreover, we have

Ind _{J_{P}^{o}}^{J^{o}(β,Λ)}(κ_{P} |_{J_{P}^{o}} ) ≃ κ|_{J^{o}(β,Λ)}  and  Ind _{J_{P}}^{J(β,Λ)}(κ_{P} |_{J_{P}}) ≃ κ|_{J(β,Λ)}.

Indeed, these are special cases of Lemma 6.1 below. We also have

κ_{P} |_{J^{o}(β,Λ)∩M} = κ^{(0)}|_{J^{o}(β^{(0)},Λ^{(0)})} ⊗ ⨂_{j=1}^{m} κ^{(j)}.

6 Intertwining and supercuspidal representations

The aim of this section is two-fold. First we seek to improve the lower bound for the intertwining of β-extensions from Corollary 4.6, in particular for standard β-extensions. In order to do this we need first to define certain Weyl group elements. Secondly, we seek to bound above the intertwining of representations of the form κ|_{J^{o}} ⊗ ρ, where κ is a standard β-extension and ρ is a cuspidal representation of the reductive quotient J^{o}/J^{1}. This will use both the lower bound for the intertwining of κ and, critically, the result of Morris, Lemma 1.1.

6.1 A Hecke algebra isomorphism

Let [Λ, n, 0, β] be a skew semisimple stratum and let θ ∈ C_{−}(Λ, 0, β) be a skew semisimple character. We use our usual notation. Let η be the unique irreducible representation of J^{1}(β, Λ) containing θ and let κ be a β-extension of η (relative to some Λ^{M}). Let V = ⨁_{j=−m}^{m} W^{(j)} be a self-dual decomposition which is properly subordinate to [Λ, n, 0, β]. As in §5, let M^{+} be the Levi subgroup of G^{+} which stabilizes the decomposition V = ⨁_{j=−m}^{m} W^{(j)} and let P^{+} be a parabolic subgroup with Levi component M^{+}, and unipotent radical U. We use all the other related notation from §5. The following Lemma will be useful in several situations.

Lemma 6.1. Let K be a compact open subgroup of G with J^{1}(β, Λ) ⊆ K ⊆ J^{+}(β, Λ), such that K has an Iwahori decomposition with respect to (M^{+}, P^{+}). We write

K ∩ M^{+} = K^{(0)} × ∏_{j=1}^{m} K̃^{(j)}

Let ρ be the inflation to K of an irreducible representation of K/J^{1}(β, Λ), and put λ = κ|_{K} ⊗ ρ. Let λ_{P} denote the natural representation of K_{P} = H^{1}(β, Λ) (K ∩ P) on the space of J^{1}(β, Λ) ∩ U-fixed vectors in λ. Then

(i) λ_{P} is irreducible and λ ≃ Ind _{K_{P}}^{K} λ_{P}.

(ii) We have λ_{P} ≃ κ_{P} ⊗ ρ, considering ρ as a representation of K_{P} /J_{P}^{1} ≅ K/J^{1}(β, Λ).

(iii) λ_{P} |_{K∩M} = λ^{(0)} ⊗ ⨂_{j=1}^{m} λ̃^{(j)}, where λ^{(0)} = κ^{(0)}|_{K^{(0)}} ⊗ ρ^{(0)} is a representation of K^{(0)} and λ̃^{(j)} = κ^{(j)}|K̃^{(j)} ⊗ ρ^{(j)} is a representation of K̃^{(j)}, for 1 ⩽ j ⩽ m.

(iv) There is an algebra isomorphism

H(G^{+}, λ_{P}) ≅ H(G^{+}, λ)

which preserves support: if ϕ ∈ H(G^{+}, λ) has support KyK, for some y ∈ G_{E}^{+}, then the corresponding function ϕ_{P} ∈ H(G^{+}, λ_{P}) has support K_{P} yK_{P}.

Proof (i), (ii) and (iii) are clear. We deduce from (ii) and [8, Proposition 4.1.3] that we have an algebra isomorphism H(G^{+}, λ_{P}) ≅ H(G^{+}, λ). Moreover, if ϕ ∈ H(G^{+}, λ) has support KyK, for some y ∈ G_{E}^{+}, then [8, Corollary 4.1.5] shows that the corresponding function ϕ_{P} ∈ H(G^{+}, λ_{P}) has support on a union of double cosets K_{P} xK_{P} contained in KyK. Using the Iwahori decomposition for K, we may assume further that x ∈ (K ∩ U_{l})y(K ∩ U_{l}) ⊆ J^{1}(β, Λ)yJ^{1}(β, Λ), where U_{l} is the unipotent radical of the parabolic subgroup opposite P^{+}. Since λ_{P} |_{J_{P}^{1}} is a multiple of η_{P}, we also have that x intertwines η_{P}. Meanwhile, by Lemma 5.12, J_{P}^{1} yJ_{P}^{1} is the unique (J_{P}^{1} , J_{P}^{1} )-double coset in J^{1}(β, Λ)yJ^{1}(β, Λ) which intertwines η_{P}. Hence x ∈ J_{P}^{1} yJ_{P}^{1} and, since J_{P}^{1} ⊆ K_{P}, we have K_{P} xK_{P} = K_{P} yK_{P}. Hence ϕ_{P} has support K_{P} yK_{P}, as required. Since

Supp H(G^{+}, λ) ⊆ I_{G^{+}}(λ|J^{1}(β, Λ)) = I_{G^{+}}(η|J^{1}(β, Λ)) = J^{1}(β, Λ)G_{E}^{+}J^{1}(β, Λ) = KG_{E}^{+}K,

we are done. ■

Corollary 6.2. Let κ be a β-extension of η to J^{+}(β, Λ) (relative to Λ^{M}). Then P^{+}(Λ_{𝔬_{E}}^{M}) ⊆ I_{G}(κ_{P} |J_{P}^{+}).

Proof Let y ∈ P^{+}(Λ_{𝔬_{E}}^{M}). By Corollary 4.6, y intertwines κ so the result follows by applying Lemma 6.1 with K = J^{+}(β, Λ) and λ = κ. ■

We write

κ_{P} |_{J^{+}(β,Λ)∩M} = κ^{(0)} ⊗ ⨂_{j=1}^{m} κ^{(j)},

as before.

Proposition 6.3 (cf. [8, Corollary 7.2.16]). (i) Suppose 𝔟_{0}(Λ^{(0)}) is a maximal self-dual 𝔬_{E}-order in B^{(0)}. Then κ^{(0)} is a β^{(0)}-extension of η^{(0)} to J^{+}(β^{(0)}, Λ^{(0)}).

(ii) If 1 ⩽ j ⩽ m, W^{(j)} ⊆ V^{i}, for some i, and 𝔟_{0}(Λ^{(j)}) is a maximal 𝔬_{E_{i}}-order in B^{(j)}, then κ^{(j)} is a 2β^{(j)}-extension of η^{(j)} to J̃(β^{(j)}, Λ^{(j)}).

Note that this Proposition is clear when |k_{F} | > 3, since then any extension of η^{(0)} to J^{+}(β^{(0)}, Λ^{(0)}) is a β^{(0)}-extension in this maximal case and, for j ≠ 0, any extension of η^{(j)} to J̃(β^{(j)}, Λ^{(j)}) is a 2β^{(j)}-extension (the latter only requires |k_{F} | > 2).

Proof For i = 1, ..., l, let Λ_{m}^{i} be an 𝔬_{E_{i}}-lattice sequence in V^{i} such that 𝔟_{0}(Λ_{m}^{i}) is a minimal self-dual 𝔬_{E_{i}}-order contained in 𝔟_{0}(Λ^{i}) and such that the self-dual decomposition

V^{i} = ⨁_{j=−m}^{m} V^{i} ∩ W^{(j)}

splits Λ_{m}^{i}. Put Λ^{m} = ⨁_{i=1}^{l} Λ_{m}^{i}, a self-dual 𝔬_{E}-lattice sequence in V such that P(Λ_{𝔬_{E}}^{m}) is an Iwahori subgroup contained in P(Λ_{𝔬_{E}}) and such that the self-dual decomposition V = ⨁_{j=−m}^{m} W^{(j)} is (properly) subordinate to [Λ^{m}, n_{m}, 0, β].

For j = −m, ..., m, set E^{(j)} = F [β^{(j)}]. We also form the 𝔬_{E^{(j)}}-lattice sequence Λ^{m^{(j)}} in W^{(j)}, given by

Λ^{m^{(j)}}(r) = Λ^{m}(r) ∩ W^{(j)}, for r ∈ ℤ.

Then 𝔟_{0}(Λ^{m^{(j)}}) is a minimal (self-dual, in the case j = 0) 𝔬_{E^{(j)}}-order contained in 𝔟_{0}(Λ^{(j)}). In particular, we can think of the pair (Λ^{m^{(j)}}, Λ^{(j)}) in the same way as the pair (Λ^{m}, Λ), and make the constructions analogous to those below in these cases too.

We form the group J_{m,Λ}^{1} = P_{1}(Λ_{𝔬_{E}}^{m})J^{1}(β, Λ) and let η_{m,Λ} be the unique extension of η to J_{m,Λ}^{1} which is intertwined by G_{E}^{+} (using Corollary 3.11). Recall that, by Proposition 4.7, we have κ|_{J_{m,Λ}^{1}} = η_{m,Λ}.

Notice that, since P_{1}(Λ_{𝔬_{E}}^{m}) ⊆ P(Λ_{𝔬_{E}}) and P(Λ_{𝔬_{E}})∩U = P_{1}(Λ_{𝔬_{E}})∩U, we have J_{m,Λ}^{1}∩U = J^{1}(β, Λ)∩U.

Then we can form the natural representation η_{P}^{m,Λ} of J_{m,Λ,P}^{1} = H^{1}(β, Λ) (J_{m,Λ}^{1} ∩ P ) on the space of J^{1}(β, Λ) ∩ U-fixed vectors in η_{m,Λ}. By applying Lemma 6.1, we have

η_{m,Λ} ≃ Ind _{J_{m,Λ,P}^{1}}^{J_{m,Λ}^{1}} η_{P}^{m,Λ}.

It is irreducible and we can describe its restriction to

J_{m,Λ}^{1} ∩ M = J_{m^{(0)},Λ^{(0)}}^{1} × ∏_{j=1}^{m} J̃_{m^{(j)},Λ^{(j)}}^{1},

where J_{m^{(0)},Λ^{(0)}}^{1} = P_{1}(Λ_{𝔬_{E^{(0)}}}^{m^{(0)}})J^{1}(β^{(0)}, Λ^{(0)}) and J̃_{m^{(j)},Λ^{(j)}}^{1} = P̃_{1}(Λ_{𝔬_{E^{(j)}}}^{m^{(j)}})J̃^{1}(β^{(j)}, Λ^{(j)}), for j = 1, ..., m. We have

η_{P}^{m,Λ}|_{J_{m,Λ}^{1}∩M} = η̂ = η̂^{(0)} ⊗ ⨂_{j=1}^{m} ˆ η̃^{(j)},

where η̂^{(0)} is the restriction of κ^{(0)}, and ˆ η̃^{(j)} the restriction of κ^{(j)}, for j = 1, ..., m. By Lemma 6.1(iv), G_{E}^{+} intertwines η_{P}^{m,Λ} so certainly G_{E}^{+} ∩ M intertwines η̂ = η_{P}^{m,Λ}|_{J_{m,Λ}^{1}∩M}. Then

the uniqueness in Corollary 3.11 shows that η̂^{(0)} is the unique extension η_{m^{(0)},Λ^{(0)}} of η^{(0)} to J_{m^{(0)},Λ^{(0)}}^{1}

which is intertwined by all of G_{E^{(0)}}^{+} = G^{+}∩Aut_{E}(W^{(0)}). But then, if P^{+}(Λ_{𝔬_{E}}^{(0)}) is a maximal compact subgroup of G_{E^{(0)}}^{+}, the construction of β-extensions in Proposition 4.1 shows that κ^{(0)} is a β^{(0)}- extension of η^{(0)}, since it restricts to η_{m^{(0)},Λ^{(0)}}. The same argument works for κ^{(j)}, for j = 1, .., m, using the uniqueness in Proposition 3.12, whenever W^{(j)} ⊆ V^{i} and P̃(Λ_{𝔬_{E_{i}}}^{(j)}) is a maximal compact subgroup of G̃_{E^{(j)}} = Aut_{E_{i}}(W^{(j)}). ■

We also record the following for later use, which follows from the application of Lemma 6.1 in the proof above:

Corollary 6.4. The intertwining of η_{P}^{m,Λ} is given by

dim I_{g}(η_{P}^{m,Λ}) = \begin{cases}1 & if g ∈ J_{m,Λ,P}^{1} G_{E}^{+}J_{m,Λ,P}^{1} ; \\ 0 & otherwise. \end{cases}

Proposition 6.3 motivates the following definition:

Definition 6.5. Let [Λ, n, 0, β] be a skew semisimple stratum. We say that a self-dual decomposition V = ⨁_{j=−m}^{m} W^{(j)} is exactly subordinate to [Λ, n, 0, β] if it is minimal amongst all self-dual decompositions which are properly subordinate to [Λ, n, 0, β] – that is, no refinement of the decomposition is properly subordinate to the stratum.

A properly subordinate decomposition is exactly subordinate to [Λ, n, 0, β] if and only if 𝔞_{0}(Λ^{(0)}) ∩ B^{(0)} is a maximal self-dual 𝔬_{E}-order in B^{(0)} and, for each j ≠ 0, there is an i such that W^{(j)} is contained in V^{i} and 𝔞_{0}(Λ^{(j)}) ∩ B^{(j)} is a maximal 𝔬_{E_{i}}-order in B^{(j)}. In particular, Proposition 6.3 can be applied to all blocks when the decomposition is exactly subordinate.

6.2 Some Weyl group elements

Let [Λ, n, 0, β] be a skew semisimple stratum and let V = ⨁_{j=−m}^{m} W^{(j)} be a self-dual decomposition which is subordinate (but not necessarily exactly subordinate) to the stratum. We also assume, throughout this section, that each W^{(j)} is contained in some V^{i}, for j ≠ 0, and that 𝔞_{0}(Λ^{(j)})∩B^{(j)} is a maximal 𝔬_{E_{i}}-order in B^{(j)}. In particular, this hypothesis is satisfied whenever the decomposition is exactly subordinate. It will be useful to have a fixed ordering for the subspaces in the decomposition – in effect, this is giving a preferred choice of the parabolic subgroup P whose Levi component M is the stabilizer of the decomposition. For each j ≠ 0, write i = i_{j} for the unique integer such that W^{(j)} ⊆ V^{i}. Since 𝔞_{0}(Λ^{(j)}) ∩ B^{(j)} is a maximal 𝔬_{E_{i}}-order, for each j ≠ 0 there is a unique integer q_{j} = q_{j}(Λ) such that

(i) −e_{i}/2 ⩽ q_{j} ⩽ e_{i}/2, where e_{i} = e(Λ^{i}|𝔬_{E_{i}});

(ii) Λ(q_{j}) ∩ W^{(j)} ⊋ Λ(q_{j} + 1) ∩ W^{(j)};

(iii) for j > 0, q_{j} > −e_{i}/2; for j < 0, q_{j} < e_{i}/2.

By duality, we have i_{−j} = i_{j} and q_{−j} = −q_{j}, for j ≠ 0. In particular, by swapping the numbering of W^{(j)} and W^{(−j)} if necessary, we may, and do, assume that q_{j} ⩾ 0 for j > 0. We put

ν(j) = ν(j, Λ) = (i_{j}, q_{j}) ∈ ℤ^{2}.

Then, for j > 0 we order the subspaces W^{(j)} lexicographically according to ν(j) – that is, by reordering if necessary, we assume that, for 0 < j < k ⩽ m, either i_{j} < i_{k} or i_{j} = i_{k} and q_{j} ⩽ q_{k}. (If ν(j) = ν(k) then we may order them either way.) The order for j < 0 is then determined by duality. We remark also that this ordering depends on the ordering of the pieces in the splitting V = ⨁_{i=1}^{l} V^{i}, which may be put in any convenient order.

We now need to define certain elements of G_{E}^{+}, which are essentially Weyl group elements. We use the block decomposition for A given by the decomposition V = ⨁_{j=−m}^{m} W^{(j)}, ordering the blocks according to ν(j). For each j > 0 we choose an ordered 𝔬_{E_{i}}-basis B^{(j)} = {v_{j,1}, ..., v_{j,d_{j}}} for the lattice Λ^{(j)}(0) which splits the lattice sequence Λ^{(j)}, where i = i_{j}. (Indeed, the second condition follows from the first since 𝔟_{0}(Λ^{(j)}) is a maximal 𝔬_{E_{i}}-order.) Note that then B^{(j)} ⊆ Λ^{(j)}(q_{j}) \ Λ^{(j)}(q_{j} + 1). Recall from §2.1 that we have a nondegenerate ϵ-hermitian E_{i}/E_{i,0}-form f_{i} on V^{i} such that the duality of lattices in V^{i} induced by f_{i} coincides with that induced by the restriction of the form h.

On W^{(j)} ⊕ W^{(−j)}, we have the nondegenerate ϵ-hermitian form which is the restriction of f_{i}, and we take the ordered basis B^{(−j)} = {v_{−j,1}, ..., v_{−j,d_{j}}} of W^{(j)} for which

f_{i}(v_{j,s}, v_{−j,t}) = \begin{cases}ϖ_{i} & if s = t, \\ 0 & otherwise, \end{cases}

where ϖ_{i} is a (fixed) uniformizer for E_{i} such that \overline{ϖ_{i}} = (−1)^{e(E_{i}/E_{i,0})−1}ϖ_{i}. By duality, this basis is an 𝔬_{E_{i}}-basis for the lattice Λ^{(−j)}(1) and splits the lattice sequence Λ^{(−j)}; indeed B^{(−j)} ⊆ Λ^{(−j)}(e_{i} − q_{j}) \ Λ^{(−j)}(e_{i} − q_{j} + 1). Finally, for each i = 1, …, l, we choose a (possibly empty) self-dual E_{i}-basis B^{(i,0)} for W^{(i,0)} = W^{(0)} ∩ V^{i} which splits Λ^{(i,0)} = e^{(0)}Λ^{i}. Suppose j, k ≠ 0, are such that i_{j} = i_{k} = i and dim_{E_{i}}W^{(j)} = dim_{E_{i}}W^{(k)}. Then we write I_{j,k} for the element of B^{i} which sends the ordered basis B^{(k)} to the ordered basis B^{(j)} and all other basis elements to 0. Note that, in the case j = k, we have I_{j,j} = e^{(j)}, the projection onto W^{(j)}. We also have the following properties, which are immediate from the definitions:

Lemma 6.6. For j, k, s, t ≠ 0 such that I_{j,k} and I_{s,t} are defined, we have

(i) I_{j,k}I_{s,t} = \begin{cases}I_{j,t} & if k = s; \\ 0 & otherwise. \end{cases}

(ii) if j, k > 0 then \overline{I_{j,k}} = I_{−k,−j} and \overline{I_{j,−j}} = ϵ(−1)^{e(E_{i}/E_{i,0})−1}I_{j,−j}.

(iii) I_{j,k} ∈ \begin{cases}𝔟_{q_{j}−q_{k}}(Λ) \ 𝔟_{q_{j}−q_{k}+1}(Λ) & if j, k > 0, \\ 𝔟_{q_{j}−q_{k}−e_{i}}(Λ) \ 𝔟_{q_{j}−q_{k}−e_{i}+1}(Λ) & if j > 0, k < 0. \end{cases}

We remark that, if the decomposition is exactly subordinate then: for j > 0, we have 0 < q_{j} < e_{i}/2: and, for j ≠ k, if i_{j} = i_{k} then q_{j} ≠ q_{k}.

We are now ready to define our Weyl group elements. If j, k > 0 are distinct, such that i_{j} = i_{k} = i and dim_{E_{i}}W^{(j)} = dim_{E_{i}}W^{(k)}, then we put

s_{j,k} = I_{j,k} + I_{k,j} + I_{−k,−j} + I_{−j,−k} + ∑_{t≠±j,±k} I_{t,t}.

Then s_{j,k} exchanges the block e^{(j)}Ae^{(j)} with e^{(k)}Ae^{(k)} and the block e^{(−j)}Ae^{(−j)} with e^{(−k)}Ae^{(−k)}. Note also that Lemma 6.6 implies that \overline{s_{j,k}} = s_{j,k} = s_{j,k}^{−1}, so that s_{j,k} ∈ G_{E}^{+}; indeed, we have s_{j,k} ∈ G_{E} since det_{B/E}(s_{j,k}) = 1. For j > 0, we put

s_{j} = I_{−j,j} + ϵ(−1)^{e(E_{i}/E_{i,0})−1}I_{j,−j} + ∑_{t≠±j} I_{t,t},
s_{j}^{ϖ} = ϖ_{i}^{−1}I_{−j,j} + ϵϖ_{i}I_{j,−j} + ∑_{t≠±j} I_{t,t}.

These elements exchange the blocks e^{(j)}Ae^{(j)} and e^{(−j)}Ae^{(−j)}. Again, Lemma 6.6 implies that s_{j}, s_{j}^{ϖ} ∈ G_{E}^{+}. Indeed, except in the case when E_{i} = F = F_{0}, ϵ = 1 and dim_{F} (W^{(j)}) is odd, they lie in G_{E}.

From Lemma 6.6(iii), we also get:

Lemma 6.7. If j, k > 0 are distinct, such that i_{j} = i_{k} = i and dim_{E_{i}}W^{(j)} = dim_{E_{i}}W^{(k)}, then:

(i) s_{j,k} ∈ P^{+}(Λ_{𝔬_{E}}) if and only if q_{j}(Λ) = q_{k}(Λ);

(ii) \begin{cases}s_{j} ∈ P^{+}(Λ_{𝔬_{E}}) & if and only if q_{j}(Λ) = e_{i}/2; \\ s_{j}^{ϖ} ∈ P^{+}(Λ_{𝔬_{E}}) & if and only if q_{j}(Λ) = 0. \end{cases}

We continue in the same situation. Let Λ^{M} be an 𝔬_{E}-lattice sequence in V such that 𝔟_{0}(Λ^{M}) is a self-dual 𝔬_{E}-order containing 𝔟_{0}(Λ). Then the decomposition V = ⨁_{j=−m}^{m} W^{(j)} is subordinate to Λ^{M} and, for j ≠ 0, 𝔞_{0}(Λ^{M}) ∩ B^{(j)} = 𝔞_{0}(Λ) ∩ B^{(j)} is a maximal 𝔬_{E_{i}}-order in B^{(j)}, where i = i_{j}; hence we can define the integers q_{j}(Λ^{M}), as above. In particular, this has the following consequences:

(i) if j > 0 then Λ^{M}(0) ∩ W^{(j)} = Λ(0) ∩ W^{(j)};

(ii) if j > 0 then q_{j}(Λ^{M}) ⩾ 0;

(iii) if j > k > 0 then q_{j}(Λ^{M}) ⩾ q_{k}(Λ^{M}).

Then (ii)–(iii) imply that the ordering of the terms in the decomposition V = ⨁_{j=−m}^{m} W^{(j)} is also a good ordering for Λ^{M}, and, having fixed such an ordering, (i) implies that the elements s_{j,k}, s_{j} and s_{j}^{ϖ} are the same for Λ and Λ^{M}.

Definition 6.8. We say that two blocks W^{(j)} and W^{(k)}, with j, k ≠ 0, are companion with respect to Λ^{M} if both are contained in the same subspace V^{i}, dim_{E_{i}}W^{(j)} = dim_{E_{i}}W^{(k)} and, for r ∈ ℤ,

Λ^{M}(r) ∩ W^{(j)} = Λ^{M}(r + 1) ∩ W^{(j)} ⟺ Λ^{M}(r) ∩ W^{(k)} = Λ^{M}(r + 1) ∩ W^{(k)}.

So, in the notation above, W^{(j)} and W^{(k)} are companion with respect to Λ^{M} if and only if i_{j} = i_{k} = i, dim_{E_{i}}W^{(j)} = dim_{E_{i}}W^{(k)} and q_{j}(Λ^{M}) ≡ q_{k}(Λ^{M}) (mod e_{i}^{M}), where e_{i}^{M} = e(Λ^{M,i}|𝔬_{E_{i}}).

Lemma 6.7 then says that, for j, k > 0 distinct, W^{(j)} and W^{(k)} are companion with respect to Λ^{M} if and only if s_{j,k} ∈ P^{+}(Λ_{𝔬_{E}}^{M}), while W^{(j)} and W^{(−j)} are companion with respect to Λ^{M} if and only if either s_{j} or s_{j}^{ϖ} lies in P^{+}(Λ_{𝔬_{E}}^{M}).

Finally, suppose that Λ^{M} is such that 𝔟_{0}(Λ^{M,i}) is a maximal self-dual 𝔬_{E_{i}}-order, for some (fixed) i. Then, with our numbering (in particular, Λ^{M}(r)^{#} = Λ^{M}(1 − r)), the lattices Λ^{M}(0) ∩ V^{i} and Λ^{M}(1) ∩ V^{i} determine all the lattices in the image of Λ^{M,i} (they are all E_{i}-multiples of these two lattices). Then we see that

Λ^{M,i}(0) ⊇ Λ^{M,i}(1) = ⋯ = Λ^{M,i}(e_{i}^{M}/2) ⊇ Λ^{M,i}(e_{i}^{M}/2 + 1) = ⋯ = Λ^{M,i}(e_{i}^{M}).

[Note that the converse of this is not quite true: if precisely one of these containments is proper then 𝔟_{0}(Λ^{M,i}) is necessarily maximal, but if both are proper then 𝔟_{0}(Λ^{M,i}) can be the “Siegel” order.] In particular, for j > 0 with i_{j} = i, we have q_{j}(Λ^{M}) ∈ {0, e_{i}^{M}/2} so W^{(j)} and W^{(−j)} are companion with respect to Λ^{M}, and

s_{j}^{ϖ} ∈ P^{+}(Λ_{𝔬_{E}}^{M}) ⟺ q_{j}(Λ^{M}) = 0; s_{j} ∈ P^{+}(Λ_{𝔬_{E}}^{M}) ⟺ q_{j}(Λ^{M}) = e_{i}^{M}/2.

6.3 Intertwining of beta extensions

We continue with the notation of the previous section and, throughout this section and the next, we suppose that the self-dual decomposition V = ⨁_{j=−m}^{m} W^{(j)} is exactly subordinate to the skew semisimple stratum [Λ, n, 0, β], with the numbering as in the previous section. In particular, we are in a special case of the situation of §6.2. Using the Weyl group element s_{j}, we can define an involution σ_{j} on G̃^{(j)} = Aut_{F} (W^{(j)}) as follows. Identifying G̃^{(j)} with the subgroup \{(\overline{g}^{−1}, g) ∈ G̃^{(−j)} × G̃^{(j)}\} ⊆ M ∩ G, we put, for g ∈ G̃^{(j)},

σ_{j}(g) = s_{j}g(s_{j})^{−1}.

Lemma 6.9. (i) If 1 ⩽ j < k ⩽ m and W^{(j)} ≅ W^{(k)} as E_{i}-vector spaces (for some i = i_{j} = i_{k}) then conjugation by s_{j,k} induces an isomorphism J̃(β^{(j)}, Λ^{(j)}) ≅ J̃(β^{(k)}, Λ^{(k)}).

(ii) For 1 ⩽ j ⩽ m, the group J̃(β^{(j)}, Λ^{(j)}) is stable under the involution σ_{j}.

Proof Suppose 1 ⩽ j ⩽ m. We can identify β^{(j)} with β_{i}, for i = i_{j}. Since the decomposition is exactly subordinate, the 𝔬_{E_{i}}-order 𝔟_{0}(Λ^{(j)}) is maximal in End_{E_{i}}(W^{(j)}). Moreover, Λ^{(j)} is the unique (upto translation of index) 𝔬_{E_{i}}-lattice sequence of period e(Λ^{i}|𝔬_{E_{i}}) whose associated 𝔬_{E_{i}}-order is 𝔟_{0}(Λ^{(j)}).

(i) If W^{(j)} ≅ W^{(k)} as E_{i}-vector spaces then conjugation by s_{j,k} identifies an 𝔬_{E_{i}}-basis for Λ^{(j)}(0) with an 𝔬_{E_{i}}-basis for Λ^{(k)}(0), so we get an isomorphism 𝔟_{0}(Λ^{(j)}) ≅ 𝔟_{0}(Λ^{(k)}) of 𝔬_{E_{i}}-orders. By the uniqueness above, this must then identify Λ^{(j)} with a translate of Λ^{(k)}. In particular 𝔞_{r}(Λ^{(j)}) is identified with 𝔞_{r}(Λ^{(k)}), for all r ∈ ℤ, and, since β^{(j)} is identified with β^{(k)} via β_{i}, we have an isomorphism J̃(β^{(j)}, Λ^{(j)}) ≅ J̃(β^{(k)}, Λ^{(k)}). (ii) As in (i), conjugation by s_{j} stabilizes J̃(β^{(j)}, Λ^{(j)}) × J̃(β^{(−j)}, Λ^{(−j)}), swapping the two factors. Since s_{j} ∈ G^{+}, taking intersections with G^{+} gives the result. ■

Let Λ^{M} be an 𝔬_{E}-lattice sequence in V such that 𝔟_{0}(Λ^{M}) is a maximal self-dual 𝔬_{E}-order containing 𝔟_{0}(Λ) and let κ be a β-extension relative to Λ^{M}. Let M be the Levi subgroup of G which is the stabilizer of the decomposition and let P be a parabolic subgroup with Levi component M. We form the representation κ_{P} of J_{P}^{o} and write

κ_{P} |_{J_{P}^{o} ∩M} = κ^{(0)} ⊗_{j=1}^{m} κ^{(j)}.

We recall that, since the decomposition is exactly subordinate to the stratum, κ^{(0)} is a β^{(0)}-extension and κ^{(j)} is a 2β^{(j)}-extension, for j = 1, …, m, by Proposition 6.3.

Corollary 6.10. (i) If 1 ⩽ j < k ⩽ m and W^{(j)} is companion to W^{(k)} with respect to Λ^{M} then conjugation by s_{j,k} induces an equivalence κ^{(j)} ≃ κ^{(k)}.

(ii) For 1 ⩽ j ⩽ m, conjugation by s_{j} induces an equivalence κ^{(j)} ∘ σ_{j} ≃ κ^{(j)}.

Proof (i) In this situation s_{j,k} ∈ P^{+}(Λ_{𝔬_{E}}^{M}) so, by Corollary 6.2, s_{j,k} intertwines κ_{P}. Moreover, it normalizes J_{P}^{o} ∩ M so, since the restriction of κ_{P} to this group is irreducible, it normalizes κ_{P} |_{J_{P}^{o} ∩M} = κ_{0} ⊗ ⨂_{j=1}^{m} κ^{(j)}. The result is then clear, from Lemma 6.9.

(ii) If s_{j} ∈ P^{+}(Λ_{𝔬_{E}}^{M}) then the proof is the same as for (i). Otherwise s_{j}^{ϖ} ∈ P^{+}(Λ_{𝔬_{E}}^{M}) so we get an equivalence κ^{(j)} ∘ σ_{j} ≃ ^{ϖ_{i}}κ^{(j)}. But κ^{(j)} is normalized by ϖ_{i} and the result follows. ■

Corollary 6.11. For 1 ⩽ j ⩽ m, there exists a 2β^{(j)}-extension κ^{(j)} which is σ_{j}-stable.

Remark 6.12. Once we have such a σ_{j}-stable κ^{(j)}, it follows (using [8, Theorem 5.2.2]) that all σ_{j}-stable 2β^{(j)}-extensions are of the form κ^{(j)} ⊗ χ̃ ∘ det^{(j)}, where:

• det^{(j)} : End_{E_{i}}(W^{(j)}) → E_{i} is the determinant; and

• χ̃ is the inflation of a σ_{i}-invariant character of κ_{E_{i}}^{×}, where σ_{i} is the involution x ↦ (\overline{x})^{−1} on κ_{E_{i}}^{×} and is the Galois involution on k_{E_{i}} with fixed field k_{E_{i,0}}.

If E_{i}/E_{i,0} is ramified, then there are only two such characters: the trivial character and the quadratic character. If E_{i}/E_{i,0} is unramified, then there are |k_{E_{i,0}}| + 1 such characters.

Since κ_{P} |_{J_{P}^{o} ∩M} = κ^{(0)} ⊗_{j=1}^{m} κ^{(j)} and κ_{P} is trivial on both J_{P}^{o} ∩ U and J_{P}^{o} ∩ U_{l}, this observation gives us the following useful corollary:

Corollary 6.13. Let Λ^{M}, Λ^{M′} be 𝔬_{E}-lattice sequences in V such that 𝔟_{0}(Λ^{M}) and 𝔟_{0}(Λ^{M′}) are maximal self-dual 𝔬_{E}-orders containing 𝔟_{0}(Λ). Let κ be a β-extension of η relative to Λ^{M}, and let κ′ be a β-extension of η relative to Λ^{M′}. There are σ_{i}-invariant characters χ^{(j)} of κ_{E_{i}}^{×} and a character χ^{(0)} of J^{o}(β^{(0)}, Λ^{(0)})/J^{1}(β^{(0)}, Λ^{(0)}) such that, writing χ = χ^{(0)} ⊗ ⨂_{j=1}^{m} χ^{(j)} ∘ det^{(j)}, we have

κ′ = Ind _{J_{P}^{o}}^{J^{o}(β,Λ)}(κ_{P} ⊗ χ).

Moreover, if W^{(j)} is companion to W^{(k)} with respect to both Λ^{M} and Λ^{M′}, then χ^{(j)} = χ^{(k)}.

Corollary 6.10 can also be applied to get more information about the intertwining of κ_{P} (and hence of κ). Suppose that κ is a standard β-extension, so Λ^{M} = 𝔐_{Λ}. If j, k ≠ 0 and W^{(j)} ≃ W^{(k)} as E_{i}-spaces, for some i, then W^{(j)} and W^{(k)} are companion with respect to 𝔐_{Λ}. Moreover, by Lemma 6.7(ii), we have s_{j} ∈ P(𝔐_{Λ,𝔬_{E}}).

For each i, write B^{i} = B^{(i,0)} ∪⋃_{j:i_{j}=i} B^{(j)}, where B^{(j)} is the E_{i}-basis for W^{(j)} chosen in the previous section and B^{(i,0)} is the basis for V^{i} ∩ W^{(0)}. Then B^{i} is an E_{i}-basis for V^{i} which splits the lattice sequence Λ^{i}. We write T̃_{E_{i}} for the maximal split torus in G̃_{E_{i}} which corresponds to the basis B^{i} – that is, T̃_{E_{i}} is the stabilizer in G̃_{E_{i}} of the decomposition V^{i} = ⨁_{v∈B^{i}} E_{i}v.

We write T̃_{E} = ∏_{i=1}^{l} T̃_{E_{i}} and put T_{E}^{+} = T̃_{E} ∩ G_{E}^{+} and T_{E} = T̃_{E} ∩ G_{E}. We also put N^{+} = N_{G_{E}^{+}}(T_{E}^{+}) and N = N_{G_{E}}(T_{E}) = N^{+} ∩ G.

Proposition 6.14. Let w ∈ N^{+} be such that w normalizes P^{o}(Λ_{𝔬_{E}}) ∩ M. Then w normalizes κ_{P} |_{J^{o}(β,Λ)∩M} and intertwines κ_{P} |_{J_{P}^{o}} .

Proof Suppose w ∈ N normalizes P^{o}(Λ_{𝔬_{E}}) ∩ M. In particular, it then permutes the blocks of the decomposition V = ⨁_{j=−m}^{m} W^{(j)}. Moreover, since w is fixed by the involution σ, we have that wW^{(j)} = W^{(k)} if and only if wW^{(−j)} = W^{(−k)}. In particular, this implies that w stabilizes W^{(0)} and its orthogonal complement ⨁_{j≠0} W^{(j)}. But then the 0-component w^{(0)} = e^{(0)}we^{(0)} of w normalizes

P^{o}(Λ_{𝔬_{E}}^{(0)}) so, since 𝔟_{0}(Λ^{(0)}) is a maximal self-dual 𝔬_{E}-order in B^{(0)}, we have w^{(0)} ∈ P(Λ_{𝔬_{E}}^{(0)}), and w^{(0)} certainly normalizes κ^{(0)}. The remaining part of w can be written as the product of:

• a block diagonal element z with z^{(j)} = e^{(j)}ze^{(j)} a non-zero scalar in E_{i_{j}} for 1 ⩽ j ⩽ m; and • a block permutation matrix s which, in turn, can be expressed as a product of the matrices

s_{j,k} and s_{j}, for 1 ⩽ j < k ⩽ m.

Since κ is a standard β-extension, Corollary 6.10 implies that s normalizes κ_{P} |_{J^{o}(β,Λ)∩M} = κ^{(0)} ⊗ ⨂_{j=1}^{m} κ^{(j)}. On the other hand, z^{(j)} certainly normalizes κ^{(j)}, since it is a 2β̃^{(j)}-extension. Hence w normalizes κ_{P} |_{J^{o}(β,Λ)∩M}, as required. To show that w also intertwines κ_{P} |_{J_{P}^{o}} , we imitate [27, Théorème 2.19]. Since J^{o}(β, Λ) has an Iwahori decomposition with respect to both (M, P) and (M, P^{w}), the group J_{P}^{o} ∩^{w}J_{P}^{o} also has an Iwahori decomposition with respect to (M, P). Now let Φ be a non-zero intertwining operator of κ_{P} |_{J^{o}(β,Λ)∩M} with ^{w}κ_{P} |_{J^{o}(β,Λ)∩M}. For g ∈ J_{P}^{o} ∩^{w}J_{P}^{o} we write g = lmu, with l ∈ U_{l}, m ∈ M and u ∈ U. Then

Φ ∘ κ_{P} (g) = Φ ∘ κ_{P} (m) = ^{w}κ_{P} (m) ∘ Φ.

It remains only to check that l, u ∈ ker (^{w}κ_{P}) or, equivalently, that l^{w}, u^{w} ∈ ker (κ_{P}). By elementary row and column operations (cf. [27, Lemme 2.20]) we have Ũ^{w} = (Ũ^{w} ∩ Ũ)(Ũ^{w} ∩ Ũ_{l}) so, by uniqueness of Iwahori decomposition, U^{w} = (U^{w} ∩ U)(U^{w} ∩ U_{l}) and u^{w} ⊆ UU_{l}. Moreover, since u^{w} ∈ J_{P}^{o} , the uniqueness of the Iwahori decomposition again implies u^{w} ⊆ (J_{P}^{o} ∩U)(J_{P}^{o} ∩U_{l}) ⊆ ker (κ_{P}). The same argument applies for l^{w}. ■

Write N_{Λ} = {w ∈ N : w normalizes P^{o}(Λ_{𝔬_{E}}) ∩ M}. It will be useful to understand this group in terms of the notation of Morris in §1. If P^{o}(Λ_{𝔬_{E}}) is a maximal parahoric subgroup of G_{E} (so the decomposition is just V = W^{(0)} and M = P = G) then we have N_{Λ} ⊆ P(Λ_{𝔬_{E}}), since this is the normalizer of P^{o}(Λ_{𝔬_{E}}) in G_{E}. Otherwise, the group called 𝔐_{J} in §1 is just M ∩ G_{E} (because the decomposition is exactly subordinate) and the group 𝔐_{J} ∩ P_{J} is just P^{o}(Λ_{𝔬_{E}}) ∩ M. In particular, our group N_{Λ} is just the group written N_{N}(𝔐_{J} ∩ P_{J}).

6.4 Intertwining and supercuspidal types

We continue with the notation of the previous section so that we have: a self-dual decomposition V = ⨁_{j=−m}^{m} W^{(j)} exactly subordinate to a skew semisimple stratum [Λ, n, 0, β], numbered as usual; a semisimple character θ ∈ C_{−}(Λ, 0, β); a standard β-extension κ to J^{o}(β, Λ) of the unique irreducible representation η of J^{1}(β, Λ) containing θ; and a parabolic subgroup P of G whose Levi component M is the stabilizer of the decomposition. The following proposition bounds the intertwining of a pair (J_{P}^{o} , λ_{P}), where λ_{P} = κ_{P} ⊗ ρ and ρ is (the inflation of) a cuspidal irreducible representation of J_{P}^{o} /J_{P}^{1} ≅ P^{o}(Λ_{𝔬_{E}})/P^{1}(Λ_{𝔬_{E}}).

Proposition 6.15. We have I_{G}(λ_{P} |J_{P}^{o} ) ⊆ J_{P}^{o} N_{Λ}J_{P}^{o} .

Remark The proof, which is a variant of the proof of [8, Proposition 5.3.2], is inspired by [3, page 553] (using Proposition 1.1 in place of op. cit. Lemma 4.2).

Proof Suppose g ∈ G intertwines λ_{P} = κ_{P} ⊗ ρ, so that g ∈ I_{G}(η_{P} |J_{P}^{1} ) = J_{P}^{o} G_{E}J_{P}^{o} , as ρ is trivial on J_{P}^{1} . Thus we may assume g lies in G_{E}. Now let Λ^{m} be a self-dual 𝔬_{E}-lattice sequence such that 𝔟_{0}(Λ_{𝔬_{E}}^{m}) is a minimal self-dual 𝔬_{E}-order contained in 𝔟_{0}(Λ_{𝔬_{E}}) and put J_{m,Λ,P}^{o} = P^{o}(Λ_{𝔬_{E}}^{m})J_{P}^{1} and J_{m,Λ,P}^{1} = P_{1}(Λ_{𝔬_{E}}^{m})J_{P}^{1} . Then G_{E} ∩J_{P}^{o} = P^{o}(Λ_{𝔬_{E}}) is a parahoric subgroup of G_{E} containing the Iwahori subgroup P^{o}(Λ_{𝔬_{E}}^{m}). Thus we may further assume that g is a distinguished double coset representative for P^{o}(Λ_{𝔬_{E}})\G_{E}/P^{o}(Λ_{𝔬_{E}}) (see §1).

Since dim I_{g}(η_{P} , J_{P}^{1} ) = 1, we can imitate the proof of [8, Proposition 5.3.2] to show that any nonzero intertwining operator in I_{g}(κ_{P} ⊗ ρ, J_{P}^{o} ) has the form S ⊗ T, with S ∈ I_{g}(η_{P} , J_{P}^{1} ) and T an endomorphism of the space of ρ. Now, by Lemma 5.12 and Corollary 6.4, the operator S also intertwines the restriction κ_{P} |_{J_{m,Λ,P}^{1}} so, again as in the proof of [8, Proposition 5.3.2], it follows that T belongs to I_{g}(ρ|J_{m,Λ,P}^{1} ). In particular, g intertwines the restriction of ρ to J_{m,Λ,P}^{1} ∩ G_{E} = P_{1}(Λ_{𝔬_{E}}^{m}). But P_{1}(Λ_{𝔬_{E}}^{m}) is the radical of an Iwahori subgroup of G_{E} so, by Proposition 1.1 and the remarks following it (see also the end of the previous section for a translation of this into our notations here), g ∈ N_{Λ}, as required. ■

Corollary 6.16. Put N_{Λ}(ρ) = {g ∈ N_{Λ} : ^{g}ρ ≃ ρ}. Then I_{G}(λ_{P} |J_{P}^{o} ) ⊆ J_{P}^{o} N_{Λ}(ρ)J_{P}^{o} .

Proof By Proposition 6.15, we need only show that I_{N_{Λ}}(λ_{P} |J_{P}^{o} ) ⊆ J_{P}^{o} N_{Λ}(ρ)J_{P}^{o} . So suppose g ∈ N_{Λ} intertwines λ_{P}. As in the proof of Proposition 6.15, any non-zero intertwining operator in I_{g}(κ_{P} ⊗ ρ, J_{P}^{o} ) has the form S ⊗ T, with S ∈ I_{g}(η_{P} , J_{P}^{1} ) and T an endomorphism of the space of ρ. Now, by Proposition 6.14, g intertwines κ_{P} so, by Lemma 5.12, the operator S also intertwines κ_{P} and it follows that T belongs to I_{g}(ρ|J_{P}^{o} ). In particular, g intertwines the restriction of ρ to J_{P}^{o} ∩ G_{E} = P^{o}(Λ_{𝔬_{E}}). Then, by Theorem 1.2, g ∈ P^{o}(Λ_{𝔬_{E}})N_{Λ}(ρ)P^{o}(Λ_{𝔬_{E}}), as required. ■

In the case when P(Λ_{𝔬_{E}}^{M}) is a maximal compact subgroup of G_{E} we can now obtain supercuspidal representations from this construction: Let [Λ^{M}, n, 0, β] be a skew semisimple stratum with 𝔟_{0}(Λ^{M}) a maximal self-dual 𝔬_{E}-order in B. Let θ_{M} be a skew semisimple character in C_{−}(Λ^{M}, 0, β), let η_{M} be the unique irreducible representation of J_{M}^{1} = J^{1}(β, Λ^{M}) and let κ_{M} be a β-extension of η_{M} to J_{M} = J(β, Λ^{M}). Put λ_{M} = κ_{M} ⊗ τ, for τ an irreducible cuspidal representation of J_{M}/J_{M}^{1} ≃ P(Λ_{𝔬_{E}}^{M})/P_{1}(Λ_{𝔬_{E}}^{M}). Recall that P(Λ_{𝔬_{E}}^{M})/P_{1}(Λ_{𝔬_{E}}^{M}) is not, in general, connected: by an irreducible cuspidal representation we mean an irreducible representation whose restriction to the connected component contains an irreducible cuspidal representation ρ. Write J_{M}^{o} = P^{o}(Λ_{𝔬_{E}}^{M})J_{M}^{1}.

Definition 6.17. We call a pair (J_{M}, λ_{M}) as above a maximal simple type for G.

Proposition 6.18. Let (J_{M}, λ_{M}) be a maximal simple type for G. Then I_{G}(λ_{M}|J_{M}) = J_{M}.

Proof Suppose g ∈ G intertwines λ_{M} = κ_{M} ⊗ τ, so that g ∈ I_{G}(η_{M}|J_{M}^{1}) = J_{M}G_{E}J_{M}, as τ is trivial on J_{M}^{1}. Thus we may assume g lies in G_{E}. Now κ_{M}|_{J_{M}^{o}} is irreducible and is normalized by J_{M}. If ρ is an irreducible cuspidal component of τ|_{J_{M}^{o}} then, by Clifford Theory, the restriction of λ_{M} to J_{M}^{o} takes the form

λ_{M}|_{J_{M}^{o}} = m ∑_{p} κ_{M} ⊗ τ^{p} ≃ m ∑_{p}(κ_{M} ⊗ ρ)^{p},

where m ∈ ℕ is a multiplicity and the sum is taken over a set of representatives for P(Λ_{𝔬_{E}}^{M})/N_{P(Λ_{𝔬_{E}}^{M} )}(τ). Since g certainly intertwines λ_{M}|_{J_{M}^{o}}, we see that there exist p_{1}, p_{2} ∈ P(Λ_{𝔬_{E}}^{M}) such that p_{1}gp_{2} intertwines κ_{M} ⊗ ρ and, since p_{i} ∈ J_{M}, we may assume that g intertwines κ_{M} ⊗ ρ. But then, since N_{Λ}(ρ) ⊆ N_{G_{E}}(P^{o}(Λ_{𝔬_{E}}^{M})) = P(Λ_{𝔬_{E}}^{M}), Corollary 6.16 implies that g ∈ J_{M}, as required. ■

From Proposition 6.18, together with [13, Proposition 1.5] and [9, Proposition 5.4], we immediately get:

Corollary 6.19. Let (J_{M}, λ_{M}) be a maximal simple type in G. Then π = c-Ind _{J_{M}}^{G}λ_{M} is an irreducible supercuspidal representation of G and (J_{M}, λ_{M}) is a [G, π]_{G}-type.

We will show in the following section that all irreducible supercuspidal representations of G arise from this construction.

7 Exhaustion

In this final section, we show that any irreducible supercuspidal representation of G contains a maximal simple type. In particular, every irreducible supercuspidal representation of G arises from our constructions. We do not, however, address any unicity issues – for example, we do not have an intertwining implies conjugacy result for maximal simple types, so we do not know when two apparently different maximal simple types might induce to equivalent irreducible supercuspidal representations of G.

7.1 A Hecke algebra injection

Let [Λ, n, 0, β] be a skew semisimple stratum and let θ ∈ C_{−}(Λ, 0, β) be a skew semisimple character. We use our usual notation: V = ⨁_{i=1}^{l} V^{i} is the splitting associated to [Λ, n, 0, β], E = F [β] = ⨁_{i=1}^{l} E_{i}, B = End_{E}(V) = ⨁_{i=1}^{l} B^{i}, G_{E} = B ∩ G = ∏_{i=1}^{l} G_{E_{i}}, etc. Let [Λ′, n′, 0, β] be another skew semisimple stratum, with 𝔟_{0}(Λ) ⊆ 𝔟_{0}(Λ′) and let θ′ = τ_{Λ,Λ′,β}(θ). Let κ (respectively κ′) be a β-extension of J^{o}(β, Λ) (respectively J^{o}(β, Λ′)) such that κ, κ′ are compatible. Let ρ be the inflation of some irreducible representation of J^{o}(β, Λ)/J^{1}(β, Λ) and put ϑ = κ ⊗ ρ. Set

G^{o} = J^{o}(β, Λ)/J^{1}(β, Λ) ≅ ∏_{i=1}^{l} P^{o}(Λ_{𝔬_{E_{i}}}^{i})/P_{1}(Λ_{𝔬_{E_{i}}}^{i})

and write ρ = ⊗_{i=1}^{l}ρ_{i}, for ρ_{i} the inflation of an irreducible representation of the finite reductive group G_{i}^{o} = P^{o}(Λ_{𝔬_{E_{i}}}^{i})/P_{1}(Λ_{𝔬_{E_{i}}}^{i}).

We put J_{Λ,Λ′}^{o} = P^{o}(Λ_{𝔬_{E}})J^{1}(β, Λ′); then we can identify J_{Λ,Λ′}^{o}/J^{1}(β, Λ′) with P^{o}(Λ_{𝔬_{E}})/P_{1}(Λ′_{𝔬_{E}}), which has P^{o}(Λ_{𝔬_{E}})/P_{1}(Λ_{𝔬_{E}}) as a quotient. Hence we can view ρ as a representation of J_{Λ,Λ′}^{o}. We define ϑ′ to be the (irreducible) representation of J_{Λ,Λ′}^{o} given by

ϑ′ = (κ′|_{J_{Λ,Λ′}^{o}} ) ⊗ ρ.

Proposition 7.1 (cf. [8, Proposition 5.5.13]). There is a canonical algebra isomorphism

H(G, ϑ) ≅ H(G, ϑ′)

which preserves support: if ϕ ∈ H(G, ϑ) is supported on J^{o}(β, Λ)yJ^{o}(β, Λ), for some y ∈ G_{E}, then the corresponding function ϕ′ ∈ H(G, ϑ′) has support J_{Λ,Λ′}^{o}yJ_{Λ,Λ′}^{o}.

Proof Suppose first that 𝔞_{0}(Λ) ⊆ 𝔞_{0}(Λ′). Given the simple intersection property from Lemma 2.6, the proof in this case is identical to that of [8, Proposition 5.5.13].

In general, by Lemma 2.8, there is an 𝔬_{E}-lattice sequence Λ′′ with 𝔟_{0}(Λ′′) = 𝔟_{0}(Λ) and 𝔞_{0}(Λ′′) ⊆ 𝔞_{0}(Λ′). By Lemma 2.10, we have a sequence Λ_{0} = Λ, Λ_{1}, ..., Λ_{t} = Λ′′ of 𝔬_{E}-lattice sequences such that, for each i, 𝔟_{0}(Λ_{i}) = 𝔟_{0}(Λ) and either 𝔞_{0}(Λ_{i}) ⊆ 𝔞_{0}(Λ_{i−1}) or 𝔞_{0}(Λ_{i}) ⊇ 𝔞_{0}(Λ_{i−1}). For each i, let κ_{i} be the β-extension of J^{o}(β, Λ_{i}) compatible with κ and put ϑ_{i} = κ_{i} ⊗ ρ, where we use the fact that J^{o}(β, Λ_{i})/J^{1}(β, Λ_{i}) ≅ J^{o}(β, Λ)/J^{1}(β, Λ) to view ρ as a representation of J^{o}(β, Λ_{i}). Applying the first case with {Λ_{i−1}, Λ_{i}} in place of {Λ, Λ′}, we see that we have support-preserving Hecke algebra isomorphisms

H(G, ϑ_{i−1}) ≅ H(G, ϑ_{i}),

for 1 ⩽ i ⩽ t. Again applying the first case, with Λ′′ = Λ_{t} in place of Λ, we have a support-preserving Hecke algebra isomorphism

H(G, ϑ_{t}) ≅ H(G, ϑ′).

Putting these together gives the required isomorphism. ■

Proposition 7.2 (cf. [8, (5.6.2), Lemma 5.6.3]). Thinking of ρ as an irreducible representation of P^{o}(Λ_{𝔬_{E}}), there is an algebra isomorphism

H(J(β, Λ′), ϑ′) ≅ H(P(Λ′_{𝔬_{E}}), ρ)

which preserves support: if ϕ ∈ H(J(β, Λ′), ϑ′) is supported on J_{Λ,Λ′}^{o}yJ_{Λ,Λ′}^{o}, for some y ∈ P(Λ′_{𝔬_{E}}), then the corresponding function ϕ_{E} has support P^{o}(Λ_{𝔬_{E}})yP^{o}(Λ_{𝔬_{E}}).

Proof As in [8, Lemma 5.6.3], the irreducibility of κ′|_{P(Λ_{𝔬_{E}})J^{1}(β,Λ′)} implies that we have an algebra isomorphism H(J(β, Λ′), ρ) ≅ H(J(β, Λ′), ϑ′), given as follows: map ϕ ∈ H(J(β, Λ′), ρ) to κ′^{∨} ⊗ ϕ ∈ H(J(β, Λ′), ϑ′), where ^{∨} denotes the contragredient. The isomorphism with H(P(Λ′_{𝔬_{E}}), ρ) follows by reduction to J(β, Λ′)/J^{1}(β, Λ′) ≅ P(Λ′_{𝔬_{E}})/P^{1}(Λ′_{𝔬_{E}}). ■

Let V = ⨁_{j=−m}^{m} W^{(j)} be a self-dual decomposition which is properly subordinate to [Λ, n, 0, β]. As in §5, let M be the Levi subgroup of G which stabilizes the decomposition V = ⨁_{j=−m}^{m} W^{(j)} and let P be a parabolic subgroup with Levi component M, and unipotent radical U. We use all the other related notation from §5. By definition, the representation ρ is trivial on J^{1}(β, Λ) so we can apply Lemma 6.1 with K = J^{o}(β, Λ) and λ = ϑ = κ ⊗ ρ to obtain Iwahori factorization results for ϑ. In particular, we get a support-preserving Hecke algebra isomorphism

H(G, ϑ_{P}) ≅ H(G, ϑ).

Putting all this together, we see that there is an injective algebra map

H(P(Λ′_{𝔬_{E}}), ρ) ≅ H(J(β, Λ′), ϑ′) ↪ H(G, ϑ′) ≅ H(G, ϑ) ≅ H(G, ϑ_{P})  (7.3)

which preserves support: if ϕ_{E} ∈ H(P(Λ′_{𝔬_{E}}), ρ) has support P^{o}(Λ_{𝔬_{E}})yP^{o}(Λ_{𝔬_{E}}), for some y ∈ P(Λ′_{𝔬_{E}}), then the corresponding function ϕ_{P} ∈ H(G, ϑ_{P}) has support J_{P}^{o} yJ_{P}^{o} .

7.2 Covers

Let π be an irreducible representation of G and suppose that there is a pair ([Λ, n, 0, β], θ), consisting of a skew semisimple stratum [Λ, n, 0, β] and a skew semisimple character θ ∈ C_{−}(Λ, 0, β), such that π|_{H^{1}(β,Λ)} contains θ.

Among all such pairs, for fixed β, we choose one for which the order 𝔟_{0}(Λ) = 𝔞_{0}(Λ) ∩ B is minimal. Since there is a unique irreducible representation η of J^{1} = J^{1}(β, Λ) containing θ, π must also contain η and hence some irreducible representation ϑ of J^{o} = J^{o}(β, Λ) containing η. Since η extends to J^{o}, we see that ϑ = κ ⊗ ρ, where κ is some standard β-extension of η and ρ is the inflation of some irreducible representation of J^{o}/J^{1}. As above, we put

G^{o} = J^{o}/J^{1} ≅ ∏_{i=1}^{l} P^{o}(Λ_{𝔬_{E_{i}}}^{i})/P_{1}(Λ_{𝔞_{i}}^{i})

and write ρ = ⊗_{i=1}^{l}ρ_{i}, for ρ_{i} the inflation of an irreducible representation \overline{ρ}_{i} of the finite reductive group G_{i}^{o} = P^{o}(Λ_{𝔬_{E_{i}}}^{i})/P_{1}(Λ_{𝔬_{E_{i}}}^{i}).

Lemma 7.4. In the situation above, all the representations \overline{ρ}_{i} are cuspidal.

Proof Suppose, for contradiction, that \overline{ρ}_{1} is not cuspidal. Then there exists a proper parabolic subgroup P_{1}^{o} of G_{1}^{o}, with unipotent radical U_{1}, such that \overline{ρ}_{1}|_{U_{1}} contains the trivial character. Let Λ′^{1} be a self-dual 𝔬_{1}-lattice sequence in V^{1} such that 𝔟_{0}(Λ′^{1}) ⊆ 𝔟_{0}(Λ^{1}) and the image of P^{o}(Λ′_{𝔬_{1}}^{1}) under the quotient map P^{o}(Λ_{𝔬_{1}}^{1}) → G_{1} is P_{1}^{o}. Such a lattice sequence Λ′^{1} is obtained as a refinement of Λ^{1}: P_{1}^{o} is defined by a self-dual k_{E_{1}}-flag in Λ^{1}(0)/Λ^{1}(1), k_{E_{1}}-flags in Λ^{1}(r)/Λ^{1}(r +1), for 1 ⩽ r < \frac{e}{2}, where e = e(Λ^{1}|𝔬_{1}), and a self-dual k_{E_{1}}-flag in Λ^{1}(⌊\frac{e+1}{2}⌋)/Λ^{1}(⌊\frac{e}{2}⌋+1); then the image of Λ′^{1} consists of all the lattices which are inverse images of these flags, their duals, and their E_{1}-scalar multiples. We put Λ′ = Λ′^{1} ⊕ ⨁_{i=2}^{l} Λ^{i} so that 𝔟_{0}(Λ′) ⊆ 𝔟_{0}(Λ). By Lemma 2.8, we can choose Λ′′ an 𝔬_{E}-lattice sequence so that 𝔟_{0}(Λ′′) = 𝔟_{0}(Λ′) and 𝔞_{0}(Λ′′) ⊆ 𝔞_{0}(Λ). Then [Λ′′, n′′, 0, β] is a skew semisimple stratum in A, for a suitable integer n′′. Let θ′′ = τ_{Λ,Λ′′,β}(θ) be the transfer of θ to H^{1}(β, Λ′′) and let η′′ be the unique irreducible representation of J^{1}(β, Λ′′) which contains θ′′. We show that π contains η′′, and hence also θ′′, contradicting the minimality of 𝔟_{0}(Λ). Since P_{1}(Λ′′_{𝔬_{E}}) projects onto U_{1} under the projection map J^{o} ↦ G, ρ|_{P_{1}(Λ′′_{𝔬_{E}})J^{1}} contains the trivial character so π contains the restriction of κ to P_{1}(Λ′′_{𝔬_{E}})J^{1}. The lemma now follows from:

Lemma 7.5 (cf. [8, Lemma 8.1.6]). The representations of P_{1}(Λ′′) induced by η′′ and by κ|_{P_{1}(Λ′′_{𝔬_{E}})J^{1}} are irreducible and equivalent to each other.

The proof of Lemma 7.5 is identical to that of [8, Lemma 8.1.6]. This also finishes the proof of Lemma 7.4. ■

We continue with the same situation, so that the semisimple character θ contained in π has 𝔟_{0}(Λ) minimal for this property and hence the representations ρ_{i} are all cuspidal. For the remainder of this section, we also make the following hypothesis:

(H) There is no skew semisimple stratum [Λ′, n′, 0, β] with 𝔟_{0}(Λ′) a maximal self-dual 𝔬_{E}-order in B such that π contains a representation of the form κ′ ⊗ρ′, where κ′ is a standard β-extension and ρ′ is a cuspidal representation of P^{o}(Λ′_{𝔬_{E}}).

In particular, hypothesis (H) implies that 𝔟_{0}(Λ) is not a maximal self-dual 𝔬_{E}-order in B. Let [Λ′, n′, 0, β] be another skew semisimple stratum with 𝔐_{Λ′} = 𝔐_{Λ} and 𝔟_{0}(Λ′) ⊇ 𝔟_{0}(Λ). We let θ′ = τ_{Λ,Λ′,β}θ be the transfer of θ to H^{1}(β, Λ′) and let η′ be the unique irreducible representation of J^{1}(β, Λ′) containing θ′. Let κ′ be the unique standard β-extension of η′ compatible with κ and let ρ′ be the inflation to J^{o}(β, Λ′) of an irreducible component of

Ind _{P^{o}(Λ_{𝔬_{E}})/P_{1}(Λ′_{𝔬_{E}})}^{P^{o}(Λ′_{𝔬_{E}})/P_{1}(Λ′_{𝔬_{E}})}\overline{ρ}.

Definition 7.6. With notation as above, we say that a representation of the form ϑ′ = κ′ ⊗ ρ′ lies over ϑ = κ ⊗ ρ.

Note that, thinking of ρ as a representation of J_{Λ,Λ′}^{o} = P^{o}(Λ_{𝔬_{E}})J^{1}(β, Λ′) trivial on J^{1}(β, Λ′), the representations lying over ϑ are precisely the irreducible constituents of

Ind _{J_{Λ,Λ′}^{o}}^{J^{o}(β,Λ′)}(κ′|_{J_{Λ,Λ′}^{o}} ⊗ ρ) = κ′ ⊗ Ind _{J_{Λ,Λ′}^{o}}^{J^{o}(β,Λ′)}ρ.

Lemma 7.7 (cf. [8, Proposition 8.3.5]). Let ϑ and Λ′ be as above and let π be an irreducible smooth representation of G. Then π contains ϑ if and only if it contains some ϑ′ lying over ϑ.

Proof We can think of ρ as a representation of P^{o}(Λ_{𝔬_{E}})P_{1}(Λ) trivial on P_{1}(Λ). Notice also that Ind _{J^{o}(β,Λ)}^{P^{o}(Λ_{𝔬_{E}})P_{1}(Λ)}κ is irreducible and restricts irreducibly to the representation Ind _{J^{1}(β,Λ)}^{P_{1}(Λ)}η of

P_{1}(Λ). Since ρ is irreducible, but trivial on P_{1}(Λ), we deduce that (Ind _{J^{o}(β,Λ)}^{P^{o}(Λ_{𝔬_{E}})P_{1}(Λ))}κ) ⊗ ρ is also irreducible. Suppose first that we also have 𝔞_{0}(Λ′) ⊇ 𝔞_{0}(Λ). Then, by the compatibility of κ and κ′, we have

Ind _{J^{o}(β,Λ)}^{P^{o}(Λ_{𝔬_{E}})P_{1}(Λ)}ϑ ≃ (Ind _{J^{o}(β,Λ)}^{P^{o}(Λ_{𝔬_{E}})P_{1}(Λ)}κ) ⊗ ρ
≃ (Ind _{J_{Λ,Λ′}^{o}}^{P^{o}(Λ_{𝔬_{E}})P_{1}(Λ)}κ′|_{J_{Λ,Λ′}^{o}} ) ⊗ ρ
≃ Ind _{J_{Λ,Λ′}^{o}}^{P^{o}(Λ_{𝔬_{E}})P_{1}(Λ)}(κ′|_{J_{Λ,Λ′}^{o}} ⊗ ρ).

Since these representations are irreducible, π contains ϑ if and only if it contains κ′|_{J_{Λ,Λ′}^{o}} ⊗ ρ. But this is the case if and only if π contains some representation lying over ϑ.

For the general case, by Lemma 2.8, there is an 𝔬_{E}-lattice sequence Λ′′ with 𝔟_{0}(Λ′′) = 𝔟_{0}(Λ) and 𝔞_{0}(Λ′′) ⊆ 𝔞_{0}(Λ′). By Lemma 2.10, we have a sequence Λ_{0} = Λ, Λ_{1}, ..., Λ_{t} = Λ′′ of 𝔬_{E}-lattice sequences such that, for each i, 𝔟_{0}(Λ_{i}) = 𝔟_{0}(Λ) and either 𝔞_{0}(Λ_{i}) ⊆ 𝔞_{0}(Λ_{i−1}) or 𝔞_{0}(Λ_{i}) ⊇ 𝔞_{0}(Λ_{i−1}). For each i, let κ_{i} be the β-extension of J^{o}(β, Λ_{i}) compatible with κ and put ϑ_{i} = κ_{i} ⊗ ρ. Applying the first case with {Λ_{i}, Λ_{i−1}} in place of {Λ, Λ′}, we see that π contains ϑ_{i} if and only if it contains ϑ_{i−1}. (Note that, when 𝔞_{0}(Λ_{i}) ⊆ 𝔞_{0}(Λ_{i−1}), we have J_{Λ_{i},Λ_{i−1}}^{o} = J^{o}(β, Λ_{i−1}) so ϑ_{i−1} is the only representation lying over ϑ_{i}, and vice versa.) Applying the first case again with Λ′′ = Λ_{t} in place of Λ, we see that π contains ϑ_{t} if and only if it contains some representation ϑ′ lying over ϑ_{t}. Since P^{o}(Λ_{𝔬_{E}}) = P^{o}(Λ′′_{𝔬_{E}}), these are precisely the representations lying over ϑ = ϑ_{0}, and the result follows. ■

We continue under the hypothesis (H), so that 𝔟_{0}(Λ) is not a maximal self-dual 𝔬_{E}-order in B. Indeed, we can suppose moreover that P(Λ_{𝔬_{E}}) is not a maximal compact subgroup of G_{E}; for, if it were, then there would exist a self-dual 𝔬_{E}-lattice sequence Λ′ such that 𝔟_{0}(Λ′) is a maximal self-dual 𝔬_{E}-order in B containing 𝔟_{0}(Λ) and P(Λ′_{𝔬_{E}}) = P(Λ_{𝔬_{E}}); then, by Lemma 7.7 and using the notation there, π would also contain the unique representation ϑ′ of J^{o}(β, Λ′) which lies over ϑ, and we have ϑ′ = κ′ ⊗ ρ, contradicting (H). We may also make another assumption from now on, namely that the subspaces V^{i} are numbered so that β_{i} ≠ 0, for i > 1 – that is, if any block of the semisimple stratum [Λ, n, 0, β] is a null stratum then it is the 1^{st} block.

Let V = ⨁_{j=−m}^{m} W^{(j)} be a self-dual decomposition exactly subordinate to the stratum [Λ, n, 0, β]. Since the order 𝔟_{0}(Λ) is not maximal self-dual, this decomposition is non-trivial (that is m ⩾ 1). We use the usual numbering for exactly subordinate decompositions (see §6.2). Then we have

J^{o}(β, Λ)/J^{1}(β, Λ) ≅ P^{o}(Λ_{𝔬_{E}}^{(0)})/P_{1}(Λ_{𝔬_{E}}^{(0)}) × ∏_{j=1}^{m} P̃(Λ_{𝔬_{E}}^{(j)})/P̃_{1}(Λ_{𝔬_{E}}^{(j)})  (7.8)

and we can write

\overline{ρ} = \overline{ρ}^{(0)} ⊗ ⨂_{j=1}^{m} \overline{ρ̃}^{(j)},

where we can interpret ρ^{(j)} as an irreducible representation of P̃(Λ_{𝔬_{E}}^{(j)}) or J̃(β^{(j)}, Λ^{(j)}), as appropriate. Suppose that 0 < j ⩽ m − 1 is such that i_{j} = i_{j+1} = i and \overline{ρ̃}^{(j)} ≄ \overline{ρ̃}^{(j+1)}. Since the decomposition is exactly subordinate, we have 0 < q_{j}(Λ) < q_{j+1}(Λ) (see the remarks following Lemma 6.6). We write e_{i} = e(Λ^{i}|𝔬_{E_{i}}) and define another self-dual 𝔬_{E}-lattice sequence Λ′ in V, of the same 𝔬_{F} -period as Λ, by

Λ′(ke_{i}+r) = \begin{cases}Λ(ke_{i} + r)  if − e_{i}/2 < r ⩽ −q_{j+1}, or − q_{j} < r ⩽ q_{j}, or q_{j+1} < r ⩽ e_{i}/2; \\ ϖ_{i}^{k}Λ^{(j)}(q_{j}) ⊕ ϖ_{i}^{k}Λ^{(j+1)}(q_{j+1}) ⊕ ⨁_{t≠j,j+1} Λ^{(t)}(ke_{i} + r)  if q_{j} < r ⩽ q_{j+1}; \\ ϖ_{i}^{k}Λ^{(−j)}(1 − q_{j}) ⊕ ϖ_{i}^{k}Λ^{(−j−1)}(1 − q_{j+1}) ⊕ ⨁_{t≠−j,−j−1} Λ^{(t)}(ke_{i} + r) \\ if − q_{j+1} < r ⩽ −q_{j}. \end{cases}

Then [Λ′, n, 0, β] is a skew semisimple stratum, the decomposition is exactly subordinate to this stratum, and

q_{t}(Λ′) = q_{t}(Λ) for 0 < t < j or j + 1 < t < m,
q_{j}(Λ′) = q_{j+1}(Λ), q_{j+1}(Λ′) = q_{j}(Λ).

In particular, for the standard numbering relative to Λ′, the j^{th} and (j + 1)^{th} blocks have been exchanged (and likewise for −j and −j −1). Note also that we have 𝔐_{Λ} = 𝔐_{Λ′} so there is a unique standard β-extension κ′ to J^{o}(β, Λ′) compatible with κ. We have

J^{o}(β, Λ′)/J^{1}(β, Λ′) ≅ J^{o}(β, Λ)/J^{1}(β, Λ)

(where the j^{th} and (j + 1)^{th} terms in the decomposition (7.8) have been interchanged) and we denote by ρ′ the inflation to J^{o}(β, Λ′) of \overline{ρ}. We put ϑ′ = κ′ ⊗ ρ′.

Lemma 7.9 (cf. [8, Proposition 8.3.4]). The representation π contains ϑ′.

Proof We define a third self-dual 𝔬_{E}-lattice sequence Λ′′ in V, of the same 𝔬_{F} -period as Λ, by

Λ′′(ke_{i}+r) = \begin{cases}Λ(ke_{i} + r)  if − e_{i}/2 < r ⩽ −q_{j+1}, or − q_{j} < r ⩽ q_{j}, or q_{j+1} < r ⩽ e_{i}/2; \\ ϖ_{i}^{k}Λ^{(j)}(q_{j}) ⊕ ϖ_{i}^{k}Λ^{(j+1)}(q_{j}) ⊕ ⨁_{t≠j,j+1} Λ^{(t)}(ke_{i} + r)  if q_{j} < r ⩽ q_{j+1}; \\ ϖ_{i}^{k}Λ^{(−j)}(1 − q_{j}) ⊕ ϖ_{i}^{k}Λ^{(−j−1)}(1 − q_{j}) ⊕ ⨁_{t≠−j,−j−1} Λ^{(t)}(ke_{i} + r) \\ if − q_{j+1} < r ⩽ −q_{j}. \end{cases}

Then [Λ′′, n, 0, β] is a skew semisimple stratum but the decomposition is no longer exactly (or even properly) subordinate. We have

q_{t}(Λ′′) = q_{t}(Λ) for 0 < t < j, j + 1 < t ⩽ m,
q_{j}(Λ′′) = q_{j+1}(Λ′′) = q_{j+1}(Λ).

Now the self-dual 𝔬_{E}-order 𝔟_{0}(Λ′′) contains both 𝔟_{0}(Λ) and 𝔟_{0}(Λ′). Moreover, P^{o}(Λ_{𝔬_{E}})/P_{1}(Λ′′_{𝔬_{E}}) is a maximal parabolic subgroup in P^{o}(Λ′′_{𝔬_{E}})/P_{1}(Λ′′_{𝔬_{E}}), and P^{o}(Λ′_{𝔬_{E}})/P_{1}(Λ′′_{𝔬_{E}}) is the opposite parabolic.

Since \overline{ρ̃}^{(j)} ≄ \overline{ρ̃}^{(j+1)}, the induced representation

Ind

P^{o}(Λ′′_{𝔬E})/P_{1}(Λ′′_{𝔬E})
P^{o}(Λ_{𝔬E})/P_{1}(Λ′′_{𝔬E})

\overline{ρ}

is irreducible and there is a unique irreducible representation ϑ′′ of J^{o}(β, Λ′′) lying over ϑ. By Lemma 7.7, π therefore contains ϑ′′. But this representation also lies over ϑ′ so, again by Lemma 7.7, π contains ϑ′. ■

In particular, this means that we may repeat this process several times and hence we may (and do) assume that:

(A) if k > j > 0 and i_{j} = i_{k} (so q_{j} < q_{k}) then dim_{E_{i}}(W^{(j)}) ⩽ dim_{E_{i}}(W^{(k)}).

7.2.1 The case ρ ≄ ρ ∘ σ

We suppose first that there is some index k such that the representation ρ^{(k)} of J̃(β^{(k)}, Λ^{(k)}) is not fixed by the involution σ_{k} (see §6.3). For j > 0 we define ρ^{(−j)} = ρ^{(j)} ∘ σ_{j} (a representation of J̃(β^{(j)}, Λ^{(j)})) and set

J = {−m ⩽ j ⩽ m : ρ^{(j)} ≃ ρ^{(k)}}.

Note that, since ρ^{(k)} is not fixed by σ_{k}, if j ∈ J then −j ∉ J. Put −J = {j : −j ∈ J} and J_{0} = {j : ±j ∉ J}; then we set

Y_{1} = ⨁_{j∈J} W^{(j)}, Y_{0} = ⨁_{j∈J_{0}} W^{(j)}, Y_{−1} = ⨁_{j∈−J} W^{(j)}.

The decomposition V = Y_{1} ⊕ Y_{0} ⊕ Y_{−1} is then properly subordinate to the stratum [Λ, n, 0, β]. Let M′ be the Levi subgroup of G which is the stabilizer of this decomposition, and let P′ = M′U′ be a parabolic subgroup with Levi factor M′. Let M be the Levi subgroup of G which stabilizes the decomposition V = ⨁_{j=−m}^{m} W^{(j)} and let P = MU be a parabolic subgroup with Levi factor M such that P ⊆ P′. We can then form the representation ϑ_{P} of J_{P}^{o} , as in Lemma 6.1.

Proposition 7.10. The pair (J_{P}^{o} , ϑ_{P}) is a cover of (J_{P}^{o} ∩M′, ϑ_{P} |_{J_{P}^{o} ∩M′}). In particular, any smooth irreducible representation π of G which contains ϑ is not supercuspidal.

Proof By the choice of the subspaces Y_{1}, Y_{0}, Y_{−1}, we have N_{Λ}(ρ) ⊆ M′, where we recall N_{Λ}(ρ) = {g ∈ N_{Λ} : g normalizes ρ}. By Corollary 6.16, we get I_{G}(ϑ_{P}) ⊆ J_{P}^{o} M′J_{P}^{o} and it follows from [9, Theorem 7.2] that any smooth irreducible representation π of G which contains ϑ_{P} is not supercuspidal. However, π contains ϑ if and only if it contains ϑ_{P}, by Lemma 6.1. ■

7.2.2 The case ρ ≃ ρ ∘ σ

We suppose now that ρ^{(j)} ∘ σ_{j} ≃ ρ^{(j)}, for all 1 ⩽ j ⩽ m. Let M be the Levi subgroup of G which stabilizes the decomposition V = ⨁_{j=−m}^{m} W^{(j)}. We fix, throughout this section, a choice of parabolic subgroup P with Levi component M, namely the stabilizer in G of the self-dual flag

{0} ⊆ W^{(−m)} ⊆ ⋯ ⊆ ⨁_{j=−m}^{k} W^{(j)} ⊆ ⋯ ⊆ ⨁_{j=−m}^{m} W^{(j)} = V.

Let U bet he unipotent radical of P, and U_{l} its opposite relative to M.

According to the numbering of §6.2, i_{m} = l so that W^{(m)} ⊆ V^{l}. For t = 0, 1, we define a self-dual 𝔬_{E_{l}}-lattice sequence 𝔐_{t}^{l} in V^{l} as follows: if t = 1,

𝔐_{1}^{l}(2k + r) = \begin{cases}ϖ_{l}^{k}Λ^{l}(1 − e_{l}/2) & for r = 0, \\ ϖ_{l}^{k}Λ^{l}(e_{l}/2) & for r = 1. \end{cases}

if t = 0 then recall that, again according to the numbering of §6.2, q_{m} ⩾ 1 is the greatest integer such that Λ^{l}(q_{m}) ⊋ Λ^{l}(e_{l}/2) and we put

𝔐_{0}^{l}(2k + r) = \begin{cases}ϖ_{l}^{k}Λ^{l}(1 − q_{m}) & for r = 0, \\ ϖ_{l}^{k}Λ^{l}(q_{m}) & for r = 1; \end{cases}

Then the 𝔟_{0}(𝔐_{t}^{i}) are maximal self-dual 𝔬_{E_{l}}-orders containing 𝔟_{0}(Λ^{i}). For each t, we put 𝔐_{t} = 𝔐_{t}^{i} ⊕ ⨁_{k≠i} Λ^{k}.

We recall the elements s_{m} and s_{m}^{ϖ} of §6.2. Note that s_{m} ∈ P^{+}(𝔐_{0}) and s_{m}^{ϖ} ∈ P^{+}(𝔐_{1}), by Lemma 6.7. We define U^{m} = s_{m}Us_{m}^{−1} and U_{l}^{m} = s_{m}U_{l}s_{m}^{−1}.

Lemma 7.11. We have:

(i) J_{P}^{o} ∩ M, J_{P}^{o} ∩ U ∩ U^{m} and J_{P}^{o} ∩ U_{l} ∩ U_{l}^{m} are stable under conjugation by s_{m} and by s_{m}^{ϖ};

(ii) s_{m}(J_{P}^{o} ∩ U_{l} ∩ U^{m})s_{m}^{−1} ⊆ J_{P}^{o} ∩ U ∩ U_{l}^{m};

(iii) (s_{m}^{ϖ})^{−1} (J_{P}^{o} ∩ U ∩ U_{l}^{m})s_{m}^{ϖ} ⊆ J_{P}^{o} ∩ U_{l} ∩ U^{m};

In particular, J_{P}^{o} s_{m}J_{P}^{o} s_{m}^{ϖ}J_{P}^{o} = J_{P}^{o} s_{m}s_{m}^{ϖ}J_{P}^{o} .

Proof We prove the corresponding statements in G̃ and the Lemma follows by taking fixed points under σ. Recall that, since the decomposition is properly subordinate, 𝔍(β, Λ) can be decomposed as

𝔍(β, Λ) = ⨁_{−m⩽j,k⩽m} e^{(j)}𝔍(β, Λ)e^{(k)},

and similarly for ℌ(β, Λ). We can also write, for example,

Ũ ∩ s_{m}Ũs_{m}^{−1} = 1 + ∑_{−m<j<k<m} e^{(j)}Ae^{(k)}.

Then the properties from Lemma 6.6 imply that the conjugations by s_{m} and by s_{m}^{ϖ} are trivial on Ũ ∩ s_{m}Ũs_{m}^{−1} and (i) follows easily.

Parts (ii) and (iii) are rather similar so we will only address (iii). We have

J̃P̃ ∩ Ũ ∩ s_{m}Ũ_{l}s_{m}^{−1} = 1 + ∑_{−m<k⩽m} e^{(−m)}𝔍(β, Λ)e^{(k)} + ∑_{−m<j<m} e^{(j)}𝔍(β, Λ)e^{(m)}.

For −m < k < m, we have

(s_{m}^{ϖ})^{−1}e^{(−m)}𝔍(β, Λ)e^{(k)}s_{m}^{ϖ} = ϖ_{l}I_{m,−m}e^{(−m)}𝔍(β, Λ)e^{(k)}I_{k,k}
= e^{(m)}ϖ_{l}I_{m,−m}𝔍(β, Λ)e^{(k)}
⊆ e^{(m)}ℌ(β, Λ)e^{(k)},

since, by Lemma 6.6, ϖ_{l}I_{m,−m} ∈ 𝔟_{1}(Λ) and, by [32, Lemma 3.11(ii)], 𝔟_{1}(Λ)𝔍(β, Λ) ⊆ ℌ(β, Λ). But 1 + e^{(m)}ℌ(β, Λ)e^{(k)} ⊆ J̃P̃, as required. The situation for the other blocks is similar.

The final assertion is immediate by the Iwahori decomposition of J_{P}^{o} : we have

s_{m}J_{P}^{o} s_{m}^{ϖ} = (s_{m}(J_{P}^{o} ∩ U_{l})s_{m}^{−1})(s_{m}(J_{P}^{o} ∩ M)s_{m}^{−1})s_{m}s_{m}^{ϖ}((s_{m}^{ϖ})^{−1}(J_{P}^{o} ∩ U)s_{m}^{ϖ}) ⊆ J_{P}^{o} s_{m}s_{m}^{ϖ}J_{P}^{o} ,

by (i)–(iii). ■

We put ζ = s_{m}s_{m}^{ϖ}, so that J_{P}^{o} s_{m}J_{P}^{o} s_{m}^{ϖ}J_{P}^{o} = J_{P}^{o} ζJ_{P}^{o} .

Corollary 7.12. For k ⩾ 0, we have:

(i) ζ^{k}(J_{P}^{o} ∩ U)ζ^{−k} ⊆ J_{P}^{o} ∩ U;

(ii) ζ^{−k}(J_{P}^{o} ∩ U_{l})ζ^{k} ⊆ J_{P}^{o} ∩ U_{l};

(iii) ζ^{k}(J_{P}^{o} ∩ M)ζ^{−k} = J_{P}^{o} ∩ M.

In particular, for k_{1}, k_{2} ⩾ 0,

J_{P}^{o} ζ^{k_{1}}J_{P}^{o} ζ^{k_{2}}J_{P}^{o} = J_{P}^{o} ζ^{k_{1}+k_{2}}J_{P}^{o} .

Proof This is immediate from Lemma 7.11. For example, we have J_{P}^{o} ∩ U = (J_{P}^{o} ∩ U ∩ U^{m})(J_{P}^{o} ∩ U ∩ U_{l}^{m}) and certainly ζ^{k} normalizes the first factor here. But, noting that s_{m}, s_{m}^{ϖ} differ from their inverses only by an element of J_{P}^{o} ∩ M, by Lemma 7.11(iii),(ii) we have

ζ(J_{P}^{o} ∩ U ∩ U_{l}^{m})ζ^{−1} ⊆ s_{m}(J_{P}^{o} ∩ U_{l} ∩ U^{m})s_{m}^{−1} ⊆ (J_{P}^{o} ∩ U ∩ U_{l}^{m}),

as required. ■

Now we will construct certain invertible elements of H(G, ϑ_{P}). We assume for now that s_{m}, s_{m}^{ϖ} are elements of G. We will treat the other case below. For t = 0, 1, let κ_{t} denote a β-extension of η compatible with some standard β-extension of J^{o}(β, 𝔐_{t}). By Corollary 6.13, we have

κ_{t} ≃ Ind _{J_{P}^{o}}^{J^{o}(β,Λ)}κ_{P} ⊗ χ_{t},

for some self-dual character χ_{t} = χ_{t}^{(0)}⊗⨂_{j=1}^{m} χ_{t}^{(j)}∘det^{(j)}. We write ρ_{t} = ρ⊗χ_{t}^{−1} so that ϑ = κ_{t}⊗ρ_{t}.

Since the character χ_{t}^{(m)} is self-dual, we still have that ρ_{t}^{(m)} ≃ ρ_{t}^{(m)} ∘σ_{m} and, since it factors through the determinant det^{(m)}, the representation ρ_{t}^{(m)} is still cuspidal. By (7.3), we have support-preserving injective algebra maps

H(P(𝔐_{t,𝔬_{E}}), ρ_{t}) ↪ H(G, ϑ_{P}).

The former has structure understood by the results of Morris [21, Theorem 7.12]. In particular:

• when t = 0, it has an invertible element with support P^{o}(Λ_{𝔬_{E}})s_{m}P^{o}(Λ_{𝔬_{E}}) and we denote by T the image in H(G, ϑ_{P}) of this, so that T has support J_{P}^{o} s_{m}J_{P}^{o} .

• when t = 1, it has an invertible element with support P^{o}(Λ_{𝔬_{E}})s_{m}^{ϖ}P^{o}(Λ_{𝔬_{E}}) and we denote by T^{ϖ} the image in H(G, ϑ_{P}) of this, so that T^{ϖ} has support J_{P}^{o} s_{m}^{ϖ}J_{P}^{o} ;

Now we define S = T ∗ T^{ϖ}. By Lemma 7.11, it is supported on the single double coset J_{P}^{o} ζJ_{P}^{o} and it is invertible, since both T and T^{ϖ} are. Finally, put S′ = S^{e(E_{l}/F)}. Then S′ is invertible and, by Lemma 7.12, it has support J_{P}^{o} ζ^{e(E_{l}/F)}J_{P}^{o} . We also put Y_{1} = W^{(m)}, Y_{0} = ⨁_{j≠±m} W^{(j)} and Y_{−1} = W^{(−m)}.

Now we turn to the case when s_{m}, s_{m}^{ϖ} are not elements of G. Recall that this means that E_{l} = F = F_{0}, that ϵ = 1, and that dim_{F} W^{(m)} is odd. In particular, this means we have β_{l} = 0 so i = i_{m} = 1 and V^{k} ⊆ W^{(0)}, for k > 1, by our ordering of the blocks V^{i}; in particular, P(Λ_{𝔬_{E_{k}}}^{k}) is a maximal compact subgroup of G_{E_{k}}, for k > 1.

Moreover, J̃(β^{(m)}, Λ^{(m)}) = P(Λ^{(m)}) and κ^{(m)} is just the trivial character (not the quadratic character, since that does not extend to P^{+}(Λ^{(m)})). The involution σ_{m} is transpose-inverse for the basis B_{m} chosen to define s_{m}. In particular, the conditions that ρ^{(m)} be cuspidal and stable under σ_{m} imply that dim_{F} W^{(m)} = 1 (see for example [1, Theorem 7.1]). But then, by our ordering of the subspaces W^{(j)}, (see assumption (A) above) we have dim_{F} (W^{(j)}) = 1, for all j = 1, ..., m. We have two distinct cases here:

(i) V^{1} ∩ W^{(0)} ≠ {0}. In this case there is an element p ∈ P^{+}(Λ_{𝔬_{E}}^{(0)}) \ P(Λ_{𝔬_{E}}^{(0)}) such that p^{2} = 1. (In fact, p ∈ P^{+}(Λ_{𝔬_{E}}^{(l,0)}), in the notation of §5.1–2.) Then we replace s_{m} and s_{m}^{ϖ} by ps_{m} and ps_{m}^{ϖ} respectively, which lie in G. (Note that p commutes with s_{m} and s_{m}^{ϖ}.) The argument is then as above and we obtain an invertible element S′ of H(G, ϑ_{P}) supported on J_{P}^{o} ζJ_{P}^{o} (we have e(E_{1}/F) = 1). We define the subspaces Y_{1} = W^{(m)}, Y_{0} = ⨁_{j≠±m} W^{(j)} and Y_{−1} = W^{(−m)} as above also.

(ii) V^{1} ∩ W^{(0)} = {0}. In this case, the first observation to make is that m > 1; for if m = 1 then G_{E_{1}} is a 2-dimensional special orthogonal group, which is just F^{×}, so P(Λ_{𝔬_{1}}^{1}) is certainly a maximal compact subgroup of G_{E_{1}}. But then, P(Λ_{𝔬_{E}}) is a maximal compact subgroup of G_{E}, contradicting hypothesis (H).

Hence m ⩾ 2 and we may imitate the constructions above but with the elements s_{m}s_{m−1} and s_{m}^{ϖ}s_{m−1}^{ϖ} in place of s_{m} and s_{m}^{ϖ} respectively. (Note that the element ζ does change in this case.) Then the analogue of Lemma 7.11 holds in this situation (with U^{m} replaced by s_{m}s_{m−1}Us_{m−1}^{−1}s_{m}^{−1}) and the argument is again as above. So we obtain an invertible element S′ of H(G, ϑ_{P}) supported on J_{P}^{o} ζJ_{P}^{o} . In this case we put Y_{1} = W^{(m)} ⊕ W^{(m−1)}, Y_{0} = ⨁_{j≠±m,±(m−1)} W^{(j)} and Y_{−1} = W^{(−m)} ⊕ W^{(1−m)}.

Finally, we suppose we are in any of the cases – that is s_{m} ∈ G or not . The decomposition V = Y_{1} ⊕ Y_{0} ⊕ Y_{−1} is properly subordinate to the stratum [Λ, n, 0, β]. Let M′ be the Levi subgroup of G which is the stabilizer of this decomposition, and let P′ = M′P, a parabolic subgroup with Levi factor M′. We again form the representation ϑ_{P} of J_{P}^{o} .

Proposition 7.13. The pair (J_{P}^{o} , ϑ_{P}) is a G-cover of (J_{P}^{o} ∩ M′, ϑ_{P} |_{J_{P}^{o} ∩M′}). In particular, any smooth irreducible representation π of G which contains ϑ is not supercuspidal.

Proof Certainly (J_{P}^{o} , ϑ_{P}) has the Iwahori factorization properties required, since U′ ⊆ U. It remains only to observe S′ ∈ H(G, ϑ_{P}) is invertible with support J_{P}^{o} ζ^{e(E_{l}/F)}J_{P}^{o} = J_{P}^{o} ζ′J_{P}^{o} , for ζ′ a strongly (P, J_{P}^{o} )-positive element of the centre of M′. The result now follows as in Proposition 7.10, by appealing to [9, Theorem 7.2] and Lemma 6.1. ■

7.3 Conclusions

Let π be any irreducible supercuspidal representation of G. We know from [32, Theorem 5.1] that π contains some semisimple character θ ∈ C_{−}(Λ, 0, β), where [Λ, n, 0, β] is a skew semisimple stratum. As in §7.2, it contains some representation ϑ = κ ⊗ ρ of J^{o}(β, Λ), with κ a standard β-extension and ρ cuspidal. If hypothesis (H) were satisfied then either Proposition 7.10 or Proposition 7.13 would give us a contradiction. Hence π contains some such ϑ = κ ⊗ ρ with 𝔟_{0}(Λ) a maximal self-dual 𝔬_{E}-order in B. Then π contains some irreducible component λ of Ind _{J^{o}(β,Λ)}^{J(β,Λ)}ϑ, and we have λ = κ ⊗ τ, for κ a

β-extension and τ some irreducible component of Ind _{J^{o}(β,Λ)}^{J(β,Λ)}ρ. Such a τ is cuspidal so (J(β, Λ), λ) is a maximal simple type and, by Corollary 6.19, c-Ind _{J}^{G}λ is irreducible with π ≃ c-Ind _{J}^{G}λ. We have proved our main result:

Theorem 7.14. Let π be an irreducible supercuspidal representation of G. Then there exists a maximal simple type (J_{M}, λ_{M}), in the sense of Definition 6.17, such that π ≃ c-Ind _{J_{M}}^{G}λ_{M}.

In particular, this verifies, for the group G, the folklore conjecture that all irreducible supercuspidal representations are irreducibly compact-induced from compact (mod-centre) subgroups.

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Shaun Stevens School of Mathematics University of East Anglia Norwich NR4 7TJ United Kingdom

email: Shaun.Stevens@uea.ac.uk