Covers for self-dual supercuspidal representations of the Siegel Levi subgroup of classical p-adic groups

David Goldberg^{‡}, Philip Kutzko^{§} and Shaun Stevens^{†}

12 November 2007

Abstract

We study components of the Bernstein decomposition of a p-adic classical group (with p odd) with inertial support a self-dual positive level supercuspidal representation of a Siegel Levi subgroup. More precisely, we use the method of covers to construct a Bushnell-Kutzko type for such a component. A detailed knowledge of the Hecke algebra of the type should have number-theoretic implications.

Introduction

Let F_{0} be a locally compact nonarchimedean local field, and G the F_{0}-points of a connected reductive quasi-split algebraic group G defined over F_{0}. The smooth representation theory of G plays a key role in the modern theory of automorphic forms, and is therefore of deep interest to number theorists. A theorem of Bernstein shows that, in the terminology of [7], it is enough to understand the subcategories ℜ^{𝔰}(G) of smooth representations with supercuspidal support in a given inertial class 𝔰. The class 𝔰 = [L, π]_{G} is determined by a Levi subgroup L ⊂ G and an irreducible supercuspidal representation π of L. The theory of types and covers [7] gives one a method for trying to understand the subcategories ℜ^{𝔰}(G). In particular, if one knows supercuspidal types for all Levi subgroups of G, and how to construct G-covers of all such types, then one has a complete set of types for G. Here, we study the case where F_{0} has odd residual characteristic, G is a classical group, and 𝔰 = [L, π]_{G} with L a Siegel Levi subgroup and π an irreducible supercuspidal representation fixed by the non-trivial Weyl group element. While this work gives a complete set of G-covers (and hence types) in this case, the theory cannot be considered complete without knowledge of the associated Hecke algebras. As shown in [16], a precise understanding of such Hecke algebras gives information on reducibility of induced representations and therefore determines poles of certain Langlands L-functions. These considerations will be the subject of future work.

Let G be any of the groups Sp_{2M}, SO_{2M}, SO_{2M+1}, U_{n} or SO_{n}^{∗}, i.e., a symplectic, quasi-split special orthogonal group, or quasi-split unitary group. (Here SO_{n}^{∗} stands for a quasi-split but nonsplit special orthogonal group.) Let B = T N be a Borel subgroup with unipotent radical N and maximal torus T. There is a choice of standard maximal parabolic subgroup P = LU containing B whose Levi component L is isomorphic to either GL_{M} × G_{0}, or Res_{F/F_{0}}(GL_{M}) × G_{0}; here G_{0} is an anisotropic group and F/F_{0} is the quadratic extension defining the unitary group. We study the theory of types for the inertial class 𝔰 = [L, π]_{G}, where π is fixed by the non-trivial element w_{0} of the Weyl group N_{G}(L)/L. The work of Bushnell and Kutzko [6] gives one a type (J_{L}, λ_{L}) for the inertial class 𝔰_{L} = [L, π]_{L}. The purpose of this paper is to construct a cover (J, λ) of (J_{L}, λ_{L}) for each group G. (This generalizes the work of Blondel [3] when G = Sp_{2M}, though her proofs are somewhat different.) Therefore, by [7], the pair (J, λ) is a type for 𝔰. One then knows that the category ℜ^{𝔰}(G) is isomorphic to the category H(G, λ)-Mod of unital (left) modules over the λ-spherical Hecke algebra, and we also give the structure of this Hecke algebra.

‡The first author was supported by NSF grant DMS0400958.

§The second author was supported by NSF grant DMS0101451.

†The third author was supported by grant NAL/00779/G from the Nuffield Foundation.

This isomorphism of categories has implications both for the location of poles of certain L-functions, and for the classification of local galois representations. One example of this may be seen in recent work of Kutzko and Morris [16], where G is one of the groups Sp_{2M}, SO_{2M} or SO_{2M+1}, and π is a level zero self-dual supercuspidal representation. Specific information about the Hecke algebra H(G, λ) is employed to give a purely local proof of Shahidi’s theorem on the reducibility of parabolic induction [21], which can also be described in terms of the poles of L-functions. On the other hand, recent work of Henniart [15], building on work of Harris and Taylor [12] and Henniart [14], relates this to the classification of local galois representations by their images (e.g. symplectic, orthogonal). In the case of the level zero representations in [16], the classification of galois representations obtained via the Hecke algebra isomorphism was known already. However, in the general situation of this paper, the corresponding classification of galois representations is not fully understood and has been sought for some time. We hope to address this problem in a sequel.

We now give a summary of the contents of the paper. In §1.1, we review the theory of types for self-dual representations, via results of Blondel, which is crucial to the construction (see [3, Proposition 2.2]). (Note that our use of the word self-dual is somewhat non-standard: it does not mean self-contragredient.) In §1.3, we fix a self-dual representation π of L, and use these results to define a well-adapted inner product on an F -vector space V, which defines the group G. In §1.4, we show that there is a choice of simple stratum (see [6]), defining the type in π, which is particularly well suited. In particular, it gives rise to a skew semisimple stratum (see [24]) in End_{F} (V), which underpins the construction of G-cover.

We construct the G-cover (J, λ) in §2, following the recipe of [6, §7]. The first stages, when we are working with pro-p groups, are performed first in Aut_{F} (V) (see §2.1), and then transferred to G using the Glauberman correspondence (see §2.2). Indeed, the principle of the construction is the same as that of Blondel for symplectic groups [3], though the details are complicated by the presence of an anisotropic part. We diverge from [3] in the verification that (J, λ) is indeed a Gcover, where we construct an invertible element f in the Hecke algebra H(G, λ), which is supported on a strongly (P, J)-positive element in the centre of L (see [7]). To do this, we use an argument which harks back to the work of Borel [4]: we find two invertible elements of the Hecke algebra H(G, λ), which each have support in a compact subgroup of G, and whose convolution is supported on a strongly positive element of L (see Lemmas 2.10, 2.11); a suitable power of this is the required element f.

Finally, in §2.3, we prove a result on the Hecke algebra H(G, λ), following the techniques and philosophy of [6, §5] (see also [19]). We show that the subalgebra of elements with support in a fixed maximal compact subgroup is isomorphic to an algebra of the form H(G′, ρ′), where G′ is a (possibly disconnected) classical group over a finite field, and ρ′ is an irreducible cuspidal representation of the Siegel Levi subgroup of G′. In many cases, the calculations in [16] will give the parameters of these Hecke algebras; in general, the calculations will be similar. This will simplify the computation of the structure of the Hecke algebra H(G, λ); these specific calculations, and their implications to the classification of galois representations, are left to future work.

Acknowledgements We would like to thank Kazutoshi Kariyama, for pointing out some mistakes, and, especially, the anonymous referees, whose careful reading, apposite comments and numerous corrections and suggestions have led to a much clearer paper.

1 Preliminaries

Let F be a locally compact nonarchimedean local field. Let μ be an automorphism of F with μ^{2} = 1; we allow the possibility that μ is trivial. We set F_{0} = F^{μ} to be the fixed points of μ. Let 𝔬_{F} be the ring of integers in F, and 𝔭_{F} the maximal ideal in 𝔬_{F}. Denote by k_{F} the residual field 𝔬_{F} /𝔭_{F}, and let q_{F} = |k_{F} |. We adopt similar notation for F_{0} and for any extension of F_{0}.

Let p denote the characteristic of k_{F}. We assume that p is not 2 throughout.

For r a real number, we write ⌊r⌋ for the greatest integer less than or equal to r, and ⌈r⌉ for the least integer greater than or equal to r.

1.1 Self-dual representations of GL_{M}(F)

We begin by looking at the self-dual representations of GL_{M}(F), following Blondel [3, §2.2]. Let W be an M-dimensional F -vector space with basis ℬ = {w_{1}, …, w_{M}}, equipped with the nondegenerate hermitian form ⟨ , ⟩_{W} given by ⟨w_{i}, w_{j}⟩_{W} = δ_{i+j,M+1}. Denote by ε̃_{W} the induced adjoint involution on A_{W} = End_{F} (W) and by ε_{W} the involution g ↦ ε̃_{W} (g^{−1}) of G_{W} = Aut_{F} (W) ≅ GL_{M}(F). Then, writing a ∈ A_{W} with respect to the basis ℬ, we have

ε̃_{W} (a) = ^{†}a^{μ},

where a^{μ} ∈ A_{W} is obtained by applying μ to the coefficients of a, and ^{†} denotes transpose with respect to the off-diagonal. For ρ a representation of a subgroup J of G_{W}, we write ρ^{ε_{W}} for the representation of ε_{W} (J) given by ρ^{ε_{W}} (ε_{W} (j)) = ρ(j), for j ∈ J.

Let π be an irreducible supercuspidal representation of G_{W} such that π ≃ π^{ε_{W}}. Note that, if F = F_{0}, then, by Gel′fand and Kazhdan [11], this is equivalent to π being self-contragredient. Let (J_{W} , λ_{W}) be a maximal simple type in G_{W} corresponding to the inertial equivalence class [G_{W} , π]_{G_{W}}. We will need to use the construction of λ_{W} quite explicitly so we recall it briefly here (see [6] for more details).

It begins with a simple stratum [𝔄_{W} , n_{W} , 0, β], where 𝔄_{W} is a principal hereditary 𝔬_{F} -order in A_{W}, with Jacobson radical 𝔓_{W}, and n_{W} ∈ ℕ is such that β ∈ 𝔓_{W}^{−n_{W}} \ 𝔓_{W}^{1−n_{W}}. Further, E = F [β] is a field extension of F and E^{×} normalizes 𝔄_{W}. We set B_{W} to be the A_{W} -centralizer of E and put 𝔅_{W} = 𝔄_{W} ∩ B_{W}, a maximal hereditary 𝔬_{E}-order in B_{W}. We also fix a uniformizer ϖ_{E} of E.

From the stratum are defined certain subgroups H_{W}^{k} = H^{k}(β, 𝔄_{W}) and J_{W}^{k} = J^{k}(β, 𝔄_{W}), for k ≥ 0, along with some sets C(𝔄_{W} , k, β) of characters of H_{W}^{k+1} called simple characters (see [6, §3]). The construction of the type continues with a simple character θ_{W} ∈ C(𝔄_{W} , 0, β). There is then a unique irreducible representation η_{W} of J_{W}^{1} which contains θ_{W}.

Now we take κ_{W} to be a β-extension of η_{W}, that is, one of a certain family of representations of J_{W} = J_{W}^{0} which restrict to η_{W}. We also recall the construction of κ_{W} here, from [6, §5.1–2]. Let 𝔅_{W}^{m} be a minimal hereditary 𝔬_{E}-order in B_{W} contained in 𝔅_{W} and let 𝔄_{W}^{m} ⊂ 𝔄_{W} be the unique hereditary 𝔬_{F} -order in A_{W} which is normalized by E^{×} and such that 𝔄_{W}^{m} ∩ B_{W} = 𝔅_{W}^{m} . Then [𝔄_{W}^{m} , n_{W}^{m} , 0, β] is also a simple stratum, for some integer n_{W}^{m} , and we can define simple characters associated to this stratum also. Moreover, there is a canonical bijection τ_{𝔄_{W} ,𝔄_{W}^{m} ,β} : C(𝔄_{W} , 0, β) → C(𝔄_{W}^{m} , 0, β) (see [6, §3.6]). Let θ_{W}^{m} be the transfer of θ_{W} under this bijection. There is a unique irreducible representation η_{W}^{m} of J^{1}(β, 𝔄_{W}^{m} ) which contains θ_{W}^{m} .

Now we form the group J̃_{W}^{1} = U^{1}(𝔅_{W}^{m} )J_{W}^{1} . There is a unique representation η̃_{W} of J̃_{W}^{1} such that η̃_{W} |_{J_{W}^{1}} = η_{W} and η̃_{W} , η_{W}^{m} induce equivalent irreducible representations of U^{1}(𝔄_{W}^{m} ). Then κ_{W} is any representation of J_{W} such that κ_{W} |_{J̃1_{W}} = η̃_{W}. Finally, J_{W} /J_{W}^{1} ≅ U(𝔅_{W})/U^{1}(𝔅_{W}) is isomorphic to GL_{r}(k_{E}), where r = M/[E : F]. Then there is a cuspidal representation ρ_{W} of J_{W} /J_{W}^{1} such that

λ_{W} = κ_{W} ⊗ ρ_{W} .

By conjugating (J_{W} , λ_{W}) if necessary, we may and do assume 𝔄_{W} is standard; that is, with respect to our chosen basis ℬ, it consists of matrices with entries in 𝔬_{F} which are upper block triangular modulo 𝔭_{F}. Note that this means that ε̃_{W} (𝔄_{W}) = 𝔄_{W}.

Proposition 1.1 (cf. [3, 2.2 Proposition]). (i) There exists σ ∈ U(𝔄_{W}) such that J_{W} is stable under σ̃ : g → σ ε_{W} (g)σ^{−1} and λ_{W} is equivalent to λ_{W} ∘ σ̃.

(ii) Such an element σ is unique up to left multiplication by J_{W}. It satisfies:

(a) σ ε_{W} (σ) ∈ J_{W} and ϖ_{E}^{−1}σ ε_{W} (ϖ_{E}^{−1}σ) ∈ J_{W}.

(b) The map σ̃ stabilizes H_{W}^{1} and J_{W}^{1} and we have θ_{W} = θ_{W} ∘ σ̃.

(c) The lattices 𝔍_{W}, ℌ_{W}^{1} are stable under X ↦ σε̃_{W} (X)σ^{−1}.

The proof is identical to that of [3, 2.2 Proposition]. Note that we will actually have to be somewhat more careful in our choices of β and σ, as in [3, 2.3 Corollary] – see Remark 1.4 for more details.

1.2 Simple characters

We record here the following useful lemma from [3, 4.3 Lemma]. Note that this uses very strongly the condition that p ≠ 2.

Lemma 1.2 ([3, 4.3 Lemma 1]). Suppose V′′ is any F -vector space and [𝔄′′, n′′, 0, β′′] is a simple stratum in A′′ = End_{F} (V′′), where 𝔄′′ is a hereditary 𝔬_{F} -order in V′′.

(i) [𝔄′′, n′′, 0, \frac{1}{2}β′′] is a simple stratum in A′′, with H^{k}(\frac{1}{2}β′′, 𝔄′′) = H^{k}(β′′, 𝔄′′), for each k ≥ 0, and similarly for J^{k}.

(ii) For each m ≥ 0, the map θ ↦ θ^{2} is a bijection from C(𝔄′′, m, \frac{1}{2}β′′) onto C(𝔄′′, m, β′′) which is compatible with the canonical bijections τ of [6, §3.6]. We denote the inverse bijection by θ ↦ θ^{1/2}.

We will write θ_{L} = θ_{W}^{1/2} ∈ C(𝔄_{W} , 0, \frac{1}{2}β) and θ_{L}^{m} = (θ_{W}^{m} )^{1/2} ∈ C(𝔄_{W}^{m} , 0, \frac{1}{2}β). We also let η_{L} be the unique irreducible representation of J^{1}(β, 𝔄_{W}) which contains θ_{L}, and likewise η_{L}^{m}.

Note also that we have (θ_{L} ∘σ̃)^{2} = θ_{L}^{2} ∘σ̃ = θ_{W}. In particular, since the squaring map is a bijection, we see that θ_{L} ∘ σ̃ = θ_{L}.

1.3 Classical groups

Let V_{0} be an F -vector space equipped with a nondegenerate anisotropic ν-hermitian form ⟨ , ⟩_{0}, with ν = ±1. Thus ⟨v_{0}, w_{0}⟩_{0} = ν ⟨w_{0}, v_{0}⟩_{0}^{μ}, for all v_{0}, w_{0} ∈ V_{0}. We write G_{0} = Aut_{F} (V_{0}) and denote by \overline{G}_{0}^{+} the (anisotropic) group corresponding to the form ⟨ , ⟩_{0}:

\overline{G}_{0}^{+} := \{g_{0} ∈ G_{0} : ⟨g_{0}v_{0}, g_{0}w_{0}⟩_{0} = ⟨v_{0}, w_{0}⟩_{0} for all v_{0}, w_{0} ∈ V_{0}\}.

We put \overline{G}_{0} := \{g_{0} ∈ \overline{G}_{0}^{+} : N_{F/F_{0}} ∘ det_{G_{0}/F} (g_{0}) = 1\}. We allow the possibility that V_{0} = \{0\}.

Set V = W ⊕ V_{0} ⊕ W and define a form ⟨ , ⟩ on V by

⟨(v_{1}, v_{0}, v_{2}), (w_{1}, w_{0}, w_{2})⟩ = ⟨v_{1}, σ^{−1}w_{2}⟩_{W} + ⟨v_{0}, w_{0}⟩_{0} + ν ⟨σ^{−1}v_{2}, w_{1}⟩_{W} ,

for v_{1}, v_{2}, w_{1}, w_{2} ∈ W and v_{0}, w_{0} ∈ V_{0}, where σ is the element given by Proposition 1.1. Note that ⟨ , ⟩ is now a nondegenerate ν-hermitian form in which the two copies of W are (dual) maximal isotropic subspaces, and V_{0} is the maximal anisotropic subspace. Let N = dim_{F} V; then N = 2M + D, where D = dim_{F} V_{0}.

We put A = End_{F} (V) ≅ 𝕄(N, F). We set G = A^{×} = Aut_{F} (V) ≅ GL_{N}(F) and

\overline{G}^{+} = \{g ∈ G : ⟨gv, gw⟩ = ⟨v, w⟩ for all v, w ∈ V \},

a unitary, symplectic or orthogonal group over F_{0}. We also put

\overline{G} := \{g ∈ \overline{G}^{+} : N_{F/F_{0}} ∘ det_{G/F} (g) = 1\}.

More generally, for H a subgroup of G, we will write \overline{H}^{+} for the intersection \overline{H}^{+} = H ∩ \overline{G}^{+}, and \overline{H} for the intersection \overline{H} = H ∩ \overline{G}.

We denote by ε̃ the adjoint involution on A determined by the form; that is, for a ∈ A, ε̃(a) is the unique element of A such that

⟨av, w⟩ = ⟨v, ε̃(a)w⟩ , for all v, w, ∈ V,

We have an involution ε on G given by ε(g) = ε̃(g^{−1}), for g ∈ G, so that \overline{G}^{+} = G^{ε}.

Denote by ϕ_{1} (respectively ϕ_{2}) the embedding of A_{W} into A via the embedding of W into the first (respectively last) factor of V = W ⊕ V_{0} ⊕ W, so that ϕ_{1}(a)(v_{1}, v_{0}, v_{2}) = (av_{1}, v_{0}, v_{2}). The involution ε̃ interchanges the images of these two embeddings, and a straightforward calculation shows that ε̃(ϕ_{1}(a)) = ϕ_{2}(σε_{W} (a)σ^{−1}) = ϕ_{2}(σ̃(a)), using the notation of Proposition 1.1.

Given a representation ρ of a subgroup J of G, we denote by ρ^{ε} the representation of ε(J) given by ρ^{ε}(ε(j)) = ρ(j), for j ∈ J.

For L an 𝔬_{F} -lattice in V, we define its dual lattice to be

L^{#} = {v ∈ V : ⟨v, L⟩ ⊂ 𝔭_{F} }.

We note that, since \overline{G}_{0}^{+} stabilizes lattices L_{0} ⊃ L_{0}^{#} ⊃ 𝔭_{F} L_{0} in V_{0} (see [17, §1.8]), it is contained in a maximal ε-stable compact open subgroup of G_{0} (namely, the G_{0}-stabilizer of these lattices).

Let P be the parabolic subgroup of G stabilizing the self-dual flag

{0} ⊊ W ⊕ {0} ⊕ {0} ⊆ (W ⊕ {0} ⊕ {0})^{⊥} = W ⊕ V_{0} ⊕ {0} ⊊ V,

with unipotent radical U. Let L denote the Levi component of P which stabilizes each copy of W along with V_{0}. So L ≅ GL_{M}(F) × GL_{D}(F) × GL_{M}(F). Let P_{−} denote the opposite parabolic subgroup, P_{−} = LU_{−}. We also put A_{L} = A_{W} ⊕ A_{0} ⊕ A_{W} = Lie(L).

We set \overline{P} = P ∩ \overline{G}, the Siegel parabolic subgroup of \overline{G}, with unipotent radical \overline{U} = U ∩ \overline{G}, and \overline{L} = L ∩ \overline{G}, a Levi component of \overline{P}. Then \overline{L} ≅ GL_{M}(F) × \overline{G}_{0}; in block matrix form, we identify G_{W} × \overline{G}_{0} with \overline{L} via the isomorphism

i(g, g_{0}) = (\begin{matrix}g & 0 & 0 \\ 0 & g_{0} & 0 \\ 0 & 0 & σ̃(g) \end{matrix}) , for g ∈ G_{W} , g_{0} ∈ \overline{G}_{0}.

Note here that we are writing matrices with respect to the non-Witt basis ℬ ∪ ℬ_{0} ∪ ℬ of V = W ⊕ V_{0} ⊕ W, for ℬ_{0} any basis of V_{0}. We choose to write all matrices with respect to this basis, rather than with respect to the Witt basis ℬ ∪ ℬ_{0} ∪ σℬ.

If ρ is a representation of a subgroup J_{W} of G_{W}, and \overline{J}_{0} is a subgroup of \overline{G}_{0}, then we denote by i(ρ) the representation of i(J_{W} × \overline{J}_{0}) given by i(ρ)(i(j, j_{0})) = ρ(j), for j ∈ J_{W}, j_{0} ∈ \overline{J}_{0}. If the group \overline{J}_{0} is not specified then we take it to be the whole of \overline{G}_{0}.

More generally, if \overline{J} is a subgroup of \overline{G} such that \overline{J} ∩ \overline{L} = i(J_{W} × \overline{J}_{0}) and \overline{J} has an Iwahori decomposition with respect to (\overline{L}, \overline{P}), then we denote by ĩ(ρ) the representation of \overline{J} given by

ĩ(ρ)(u_{−}lu) = i(ρ)(l), for u_{−} ∈ \overline{J} ∩ \overline{U}_{−}, l ∈ \overline{J} ∩ \overline{L}, u ∈ \overline{J} ∩ \overline{U},

whenever this defines a representation.

It will also be useful, later, to put V′ = W ⊕ {0} ⊕ W ⊂ V, equipped with the restriction of the form ⟨ , ⟩. We put A′ = End_{F} (V′), G′ = Aut_{F} (V′), ε the involution of G′ associated to the form, and \overline{G}′^{+} = (G′)^{ε}. We also let P′ be the maximal parabolic subgroup of G which stabilizes the flag

{0} ⊊ V′ ⊊ V,

with unipotent radical U′. Let L′ denote the Levi component of P′ which stabilizes the decomposition V = V′ ⊕ V_{0}, so that L′ ≅ G′ × G_{0}, and let P′^{−} denote the opposite parabolic subgroup, with unipotent radical U′^{−}. We note that, while L′ is stable under the involution ε, \overline{L}′ is not a Levi subgroup of \overline{G}.

We consider the inertial class 𝔰_{\overline{L}} = [\overline{L}, i(π)]_{\overline{L}} and a type (J_{\overline{L}}, λ_{\overline{L}}) for it, where

J_{\overline{L}} := i(J_{W}) = i(J_{W} × \overline{G}_{0}) and λ_{\overline{L}} := i(λ_{W}).

We are going to construct a \overline{G}-cover of this type, which will give a \overline{G}-type for the inertial class 𝔰 = [\overline{L}, i(π)]_{\overline{G}}.

1.4 Semisimple characters

We continue with the notation of the previous sections. Let L_{W} = {L_{k}^{W} : k ∈ ℤ} be the 𝔬_{E}-lattice chain in W corresponding to 𝔄_{W}, so that

𝔄_{W} = {a ∈ A_{W} : aL_{k}^{W} ⊂ L_{k}^{W} for all k ∈ ℤ},

normalized so that L_{0}^{W} = 𝔬_{F} w_{1} ⊕ ⋯ ⊕ 𝔬_{F} w_{M}. Let e_{W} denote the 𝔬_{F} -period of 𝔄_{W}. We define L′ to be the 𝔬_{E}-lattice chain of 𝔬_{F} -period 2e_{W} in V′ given by

⋯ ⊃ L_{k}^{W} ⊕ L_{k}^{W} ⊃ L_{k}^{W} ⊕ L_{k+1}^{W} ⊃ L_{k+1}^{W} ⊕ L_{k+1}^{W} ⊃ ⋯

It is straightforward to check, since σ ∈ U(𝔄_{W}), we have

(L_{k}^{W} ⊕ L_{k}^{W} )^{#} = L_{e_{W} −k}^{W} ⊕ L_{e_{W} −k}^{W} and (L_{k}^{W} ⊕ L_{k+1}^{W})^{#} = L_{e_{W} −k−1}^{W} ⊕ L_{e_{W} −k}^{W}

so that L′ is a self-dual lattice chain. We write L′ = {L′_{k} : k ∈ ℤ}, where we number the lattices so that L′_{0} = L_{⌊ \frac{e_{W}}{2} ⌋}^{W} ⊕ L_{⌈ \frac{e_{W}}{2} ⌉}^{W} = (L′_{0})^{#}. Let 𝔄′ = 𝔞_{0}(L′) be the 𝔬_{F} -order associated with L′.

Now let Λ′ be the 𝔬_{E}-lattice sequence of period 4e_{W} in V′ = W ⊕ W given by

Λ′(k) = L′_{⌊ \frac{k}{2} ⌋}

so that every lattice of L′ occurs twice in the sequence and Λ′(k)^{#} = Λ′(1 − k), for k ∈ ℤ.

We consider the element \frac{1}{2}β ⊕ \frac{1}{2}β ∈ A_{W} ⊕ A_{W}, which we will also call \frac{1}{2}β. Then [L′, 2n_{W} , 0, \frac{1}{2}β] and [Λ′, 4n_{W} , 0, \frac{1}{2}β] are both simple strata in A′ so we can define the orders ℌ, 𝔍 and the groups H^{k}, J^{k} for them, and also simple characters (see [8]). These are in fact the same, up to a scaling of the index (loc. cit.); so, for example,

H^{k}(\frac{1}{2}β, Λ′) = H^{⌈ \frac{k}{2} ⌉}(\frac{1}{2}β, L′).

In particular, H^{1}(\frac{1}{2}β, Λ′) = H^{1}(\frac{1}{2}β, L′) and we shall denote this group H′^{1}. Similarly, we put J′^{1} = J^{1}(\frac{1}{2}β, Λ′) = J^{1}(\frac{1}{2}β, L′) and J′ = J(\frac{1}{2}β, Λ′) = J(\frac{1}{2}β, L′). Likewise, the simple characters of H′^{1} are the same: C(Λ′, 0, \frac{1}{2}β) = C(L′, 0, \frac{1}{2}β).

Moreover, the groups H′^{1}, J′^{1} and J′ are described in [3, 2.2 Lemma]. So, for example, with respect to the basis ℬ ∪ ℬ of V′ (which, we recall, is not a Witt basis),

H′^{1} = (\begin{matrix}H_{W}^{1} & 𝔍_{W} \\ ϖ_{E}𝔍_{W} & H_{W}^{1} \end{matrix} )  and  J′^{1} = (\begin{matrix}J_{W}^{1} & ϖ_{E}^{−1}ℌ_{W} \\ ℌ_{W} & J_{W}^{1} \end{matrix} ) .

Finally, we have the bijection τ_{𝔄_{W} ,𝔄′, \frac{1}{2} β} : C(𝔄_{W} , 0, \frac{1}{2}β) → C(𝔄′, 0, \frac{1}{2}β) from [6, §3.6]. Let θ′ be the

image of θ_{L} under this map. Note that, by [6, §7.1–2], θ′ is trivial on H′^{1} ∩ U and H′^{1} ∩ U_{−}, while

θ′|_{H′^{1}∩L} = θ_{L} ⊗ θ_{L}.

It is straightforward to check that Proposition 1.1, together with our definition of the form ⟨ , ⟩, implies that θ′ is fixed by the involution ε (cf. [3, 2.3 Corollary]).

Now let L_{0} be the unique self-dual 𝔬_{F} -lattice chain in V_{0}

⋯ ⊇ L_{0} ⊇ L_{0}^{#} ⊇ 𝔭_{F} L_{0} ⊇ 𝔭_{F} L_{0}^{#} ⊇ · · · .

Note that we may have L_{0} = L_{0}^{#} or L_{0}^{#} = 𝔭_{F} L_{0} so this lattice chain has 𝔬_{F} -period e_{0} = 1 or 2. Let Λ_{0} be the self-dual 𝔬_{F} -lattice sequence of period 4e_{W} given by

Λ_{0}(k) = \begin{cases}𝔭_{F}^{j} L_{0} & if ⌈\frac{k}{2e_{W}} ⌉ = 2j, \\ 𝔭_{F}^{j} L_{0}^{#} & if ⌈\frac{k}{2e_{W}} ⌉ = 2j + 1, \end{cases}

so that every lattice of L_{0} appears with equal multiplicity 4e_{W} /e_{0}. Note also that Λ_{0}(k)^{#} = Λ_{0}(1−k), for k ∈ ℤ. Moreover, the filtration of A_{0} determined by Λ_{0} is the same as that determined by L_{0} up to a scaling of the index. In particular, 𝔞_{0}(Λ_{0}) = 𝔄(L_{0}) and 𝔞_{1}(Λ_{0}) = 𝔓(L_{0}).

Finally, we define Λ to be the 𝔬_{F} -lattice sequence in V given by

Λ(k) = Λ′(k) ⊕ Λ_{0}(k), for k ∈ ℤ.

Then, by construction, Λ is self-dual and of 𝔬_{F} -period e = 4e_{W}. We consider the element \frac{1}{2}β ⊕ 0 ∈ A′ ⊕A_{0} (in fact, in A_{L}); by abuse of notation, we will still call this element \frac{1}{2}β. Then [Λ, n, 0, \frac{1}{2}β] is a semisimple stratum in A, where n = 4n_{W}. (See [24, §3.1] for the definition of semisimple stratum, which is more general than the definition in [22, §3.3]; in particular, null strata are thought of as simple strata so, alternatively, the definition in [22] could be used with “simple” replaced by “simple or null” everywhere. The results of [22] all remain valid in this situation – the proofs are the same and they are also proved in [24].)

We put J = J(\frac{1}{2}β, Λ) (see [22] or [24]) and similarly for J^{1}, H^{1}, 𝔍, ℌ etc. In matrix form we have

𝔍 = (\begin{matrix}𝔍_{W} & 𝔞_{⌈n/2⌉}(Λ) & ϖ_{E}^{−1}ℌ_{W}^{1} \\ 𝔞_{⌈n/2⌉}(Λ) & 𝔄(L_{0}) & 𝔞_{⌈n/2⌉}(Λ) \\ ℌ_{W}^{1} & 𝔞_{⌈n/2⌉}(Λ) & 𝔍_{W} \end{matrix}) and ℌ = (\begin{matrix}ℌ_{W} & 𝔞_{⌊n/2⌋}(Λ) & 𝔍_{W}^{1} \\ 𝔞_{⌊n/2⌋}(Λ) & 𝔄(L_{0}) & 𝔞_{⌊n/2⌋}(Λ) \\ ϖ_{E}𝔍_{W}^{1} & 𝔞_{⌊n/2⌋}(Λ) & ℌ_{W} \end{matrix}) ,

and there are similar decompositions for 𝔍^{k} and ℌ^{k}, k ≥ 1. Since J′ is stable under the involution ε and Λ is self-dual, we see that J is stable under the involution ε, and likewise for J^{1} and H^{1}.

Let θ be the unique semisimple character (see [22, §3.3] or [24, §3.2]) of H^{1} such that

θ|_{H′}_{1} = θ′.

Now θ is trivial on H^{1} ∩U′ and H^{1} ∩U′^{−} by definition so, since θ′ is trivial on H′^{1} ∩U and H′^{1} ∩U_{−}, we see that θ is in fact trivial on H^{1} ∩ U and H^{1} ∩ U_{−}. Since θ|_{H′}_{1} is fixed by ε and θ|_{U^{1}(L_{0})} is trivial (by definition), we see that θ is fixed by ε. Moreover, since θ_{L}^{2} = θ_{W}, we have

θ|_{\overline{H}_{1}} = { \atop i(θ_{W})}.

Proposition 1.3 (cf. [3, 2.3 Theorem]). There exists a semisimple stratum [Λ, n, 0, α] in A with α ∈ A_{L} and α = −ε̃(α) such that θ ∈ C(Λ, 0, α).

Proof Let φ : A′ → A′ be the involution given by conjugation by

h = (\begin{matrix}I_{M} & 0 \\ 0 & −I_{M} \end{matrix}) .

Then (θ′)^{φ} = θ′, since it is trivial on the unipotent parts, and (θ′)^{ε} = θ′. Thus, θ′ is invariant under the subgroup Ω of Aut(G′) generated by ε and φ. As εφ = φε, we see Ω ≅ ℤ_{2} × ℤ_{2}, and thus by [22, Theorem 6.3] (see the Remarks at the bottom of page 139 there also), there is a choice of α, fixed by φ and with α = −ε̃(α), such that θ′ ∈ C(Λ′, 0, α). Then [Λ, n, 0, α] is as required. ■

Remark 1.4. Writing α = diag(α_{W} , 0, −σ̃(α_{W})) for the element of Proposition 1.3, we see that our original type (J_{W} , λ_{W}) could also have been defined using 2α_{W} in place of β. Replacing β by 2α_{W}, the discussion following [3, 2.3 Theorem] applies in this situation (see especially [3, 2.3 Corollary] and the Remarks following). In particular, we may assume that σ is such that σ̃ induces a non-trivial galois involution on E = F [β] which extends the involution μ on F and sends β to −β. From now on, we assume that σ is chosen in this more careful way; this implies that σε_{W} (σ) ∈ B_{W}. Moreover, when μ is trivial, the degree [E : F] is even so that M = dim_{F} (W) is even also.

As in the construction of the type (J_{W} , λ_{W}), we will also need the transfer of θ to a minimal order, which we give now. Let B′ denote the centralizer in A′ of E and put 𝔅′ = 𝔄′ ∩ B′, an 𝔬_{E}-order of period 2 with radical 𝔔′. Recall that we had, in §1.1, a minimal 𝔬_{E}-order 𝔅_{W}^{m} ⊂ 𝔅_{W}. Let 𝔅′^{m} ⊂ 𝔅′ be the ε̃-stable minimal 𝔬_{E}-order in B′ given by

𝔅′^{m} = (𝔅_{W}^{m} ⊕ ε̃(𝔅_{W}^{m} )) + 𝔔′,

(cf. [3, 4.3]). Let L′^{m} be the corresponding self-dual 𝔬_{E}-lattice chain in V′, of 𝔬_{E}-period 2r, where r = M/[E : F]. Let Λ′^{m} be the self-dual 𝔬_{E}-lattice sequence in V′ in which every lattice of L′^{m} occurs twice and with the indexing chosen such that

Λ′^{m}(k)^{#} = Λ′^{m}(1 − k), for all k ∈ ℤ.

Note that [L′^{m}, 2n_{W}^{m} , 0, \frac{1}{2}β] and [Λ′^{m}, 4n_{W}^{m} , 0, \frac{1}{2}β] are simple strata in V′ whose associated groups and characters are the same up to a scaling of index.

Now let Λ^{m} be the 𝔬_{F} -lattice sequence in V defined by

Λ^{m}(k) = Λ′^{m}(k) ⊕ Λ_{0} (⌈\frac{k}{r}⌉) .

It is a self-dual lattice sequence of period re such that 𝔞_{0}(Λ^{m}) ⊂ 𝔞_{0}(Λ). Then [Λ^{m}, n^{m}, 0, \frac{1}{2}β] is a semisimple stratum in A, where n^{m} = 4n_{W}^{m} = nr.

We put θ′^{m} = τ_{Λ′,Λ′^{m}, \frac{1}{2} β}(θ′), a simple character of H^{1}(\frac{1}{2}β, Λ′^{m}); then also θ′^{m} = τ_{𝔄_{W}^{m} ,Λ′^{m}, \frac{1}{2} β}(θ_{L}^{m}). Finally, let θ^{m} be the unique semisimple character of H_{m}^{1} = H^{1}(\frac{1}{2}β, Λ^{m}) such that

θ^{m}|_{H^{1}( \frac{1}{2} β,Λ′^{m})} = θ′^{m}.

In the language of [24, §3.5], we have θ^{m} = τ_{Λ,Λ^{m}, \frac{1}{2} β}(θ).

2 Covers and Hecke Algebras

In this chapter, we construct our cover and find the structure of its Hecke algebra. The method of construction of the cover is (apart from the added complication of the anisotropic part) the same as that in [3] (the reader may find it helpful to consult the diagram on page 552 of [3], especially the left side). However, here we obtain full information on the intertwining of the cover (rather than just a bound), and it is this which allows us, in §2.3, to identify the Hecke algebra. In order to obtain this information on the intertwining, we need to review carefully the construction and keep track of the intertwining at each stage.

We continue with the notation above, so we have [Λ, n, 0, \frac{1}{2}β] a skew semisimple stratum in A, with β ∈ A_{L}, and θ ∈ C(Λ, 0, \frac{1}{2}β) such that

θ|_{\overline{H}_{1}} = { \atop i(θ_{W})}.

We also have E = F [β] and B the A-centralizer of E; note here that, with notation as above, B^{×} = (B′)^{×} × G_{0}. From Remark 1.4, the galois involution μ extends to E (as σ̃ when we consider E embedded in A_{W}, or as ε̃ when we consider E embedded in A), and we write E_{0} for the fixed subfield, which has index 2. We fix a uniformizer ϖ_{E} of E such that ϖ_{E}^{μ} = ±ϖ_{E}. Note that β ∈ A_{W} embeds in A as diag(β, 0, −σ̃(β)) = diag(β, 0, β).

2.1 G-Covers

We begin by doing some work in G, before using Glauberman’s correspondence to transfer this information to \overline{G}.

Recall that, given ρ a representation of a subgroup H of G and g ∈ G, the intertwining space I_{g}(ρ|H) is

I_{g}(ρ|H) = Hom_{H∩^{g}H}(ρ,^{g}ρ),

where ^{g}H = gHg^{−1} and ^{g}ρ is the representation of ^{g}H given by ^{g}ρ(ghg^{−1}) = ρ(h), for h ∈ H. The G-intertwining I_{G}(ρ|H) of ρ is then defined to be

I_{G}(ρ) = I_{G}(ρ|H) = {g ∈ G : I_{g}(ρ|H) ≠ 0}.

In our situation, we recall the following, from [22, Theorem 3.14, Corollary 4.2 and Proposition 4.3]:

Lemma 2.1. (i) I_{G}(θ) = J^{1}B^{×}J^{1}.

(ii) There exists a unique irreducible representation η of J^{1} which contains θ. Moreover, dim η = (J^{1} : H^{1})^{\frac{1}{2}} and, for g ∈ G,

dim I_{g}(η|J^{1}) = \begin{cases}1 & if g ∈ J^{1}B^{×}J^{1}; \\ 0 & otherwise. \end{cases}

From [6, §7.1] (together with [22, §3.3]), H^{1} has an Iwahori decomposition with respect to (L, P) and

H^{1} ∩ L = H^{1}(\frac{1}{2}β, 𝔄_{W}) × U^{1}(Λ_{0}) × H^{1}(\frac{1}{2}β, 𝔄_{W}).

There are also similar decompositions for J^{1} and for J. Moreover, θ is trivial on H^{1} ∩ U and H^{1} ∩ U_{−}, and the restriction of θ to H^{1} ∩ L takes the form

θ|_{H^{1}∩L} = θ_{L} ⊗ 1 ⊗ θ_{L}.

We recall ([22, Proposition 4.1] – see also [6, §3.4]) that the pairing

(j, j′) ↦ θ[j, j′], for j, j′ ∈ J^{1}

induces a nondegenerate alternating bilinear form k_{θ} on J^{1}/H^{1}; likewise, we have a nondegenerate alternating bilinear form k_{θ_{L}} on J^{1}(\frac{1}{2}β, 𝔄_{W})/H^{1}(\frac{1}{2}β, 𝔄_{W}). Then, exactly as in [6, Proposition 7.2.3], we get:

Lemma 2.2 (cf. [6, Proposition 7.2.3]). (i) The subspaces J^{1}∩U_{−}/H^{1}∩U_{−} and J^{1}∩U/H^{1}∩ U of J^{1}/H^{1} are both totally isotropic for the form k_{θ} and orthogonal to the subspace J^{1} ∩ L/H^{1} ∩ L.

(ii) The restriction of k_{θ} to the group

J^{1} ∩ L/H^{1} ∩ L = J^{1}(\frac{1}{2}β, 𝔄_{W})/H^{1}(\frac{1}{2}β, 𝔄_{W}) × J^{1}(\frac{1}{2}β, 𝔄_{W})/H^{1}(\frac{1}{2}β, 𝔄_{W})

is the orthogonal sum of the pairings k_{θ_{L}}, k_{θ_{L}}.

(iii) We have an orthogonal sum decomposition

\frac{J^{1}}{H^{1}} = \frac{J^{1} ∩ L}{H^{1} ∩ L} ⊥ (\frac{J^{1} ∩ U_{−}}{H^{1} ∩ U_{−}} × \frac{J^{1} ∩ U}{H^{1} ∩ U} ) .

We define the groups

H_{P}^{1} = H^{1}(J^{1} ∩ U),  J_{P}^{1} = H^{1}(J^{1} ∩ P),  J_{P} = H^{1}(J ∩ P).

Since J^{1} normalizes θ and θ|_{H^{1}∩U} is trivial, we can define the character θ_{P} of H_{P}^{1} by

θ_{P} (hu) = θ(h), for h ∈ H^{1}, u ∈ J^{1} ∩ U.

As in [6, §7.2], we immediately get:

Corollary 2.3 (cf. [6, Propositions 7.2.4, 7.2.9]). There is a unique irreducible representation η_{P} of J_{P}^{1} such that η_{P} |_{H_{P}^{1}} contains θ_{P}. Moreover, η ≃ Ind_{J_{P}^{1}}^{J^{1}} η_{P} and, for each b ∈ B^{×}, there is a unique (J_{P}^{1} , J_{P}^{1} )-double coset in J^{1}bJ^{1} which intertwines η_{P}.

We note also that we certainly have η_{P} |_{J_{P}^{1} ∩L} ≃ η_{L} ⊗ 1 ⊗ η_{L}.

Proposition 2.4. We have I_{G}(θ_{P}) = J_{P}^{1} B^{×}J_{P}^{1} and hence I_{G}(η_{P}) = J_{P}^{1} B^{×}J_{P}^{1} .

Proof This is a special case of [25, Lemma 5.8]. ■

We will also need similar results for our character θ^{m} of H_{m}^{1} = H^{1}(\frac{1}{2}β, Λ^{m}). Let η^{m} be the unique irreducible representation of J_{m}^{1} = J^{1}(\frac{1}{2}β, Λ^{m}) which contains θ^{m}. As above, H_{m}^{1} and J_{m}^{1} have Iwahori decompositions with respect to (L, P) and we can define the character θ_{P}^{m} of H_{m,P}^{1} = H_{m}^{1}(J_{m}^{1} ∩ U) by trivial extension of θ^{m}. The same proofs (indeed, they are somewhat simpler) as those of Lemma 2.2, Corollary 2.3 and Proposition 2.4 show that

I_{G}(θ_{P}^{m}) = J_{m,P}^{1} B^{×}J_{m,P}^{1} ,

where J_{m,P}^{1} = H_{m}^{1}(J_{m}^{1} ∩ P), that there is a unique irreducible representation η_{P}^{m} of J_{m,P}^{1} which

contains θ_{P}^{m}, and that η^{m} = Ind_{J_{m,P}^{1}}^{J_{m}^{1}} η_{P}^{m}.

Let θ_{L}^{m} denote the transfer to H^{1}(\frac{1}{2}β, 𝔄_{W}^{m} ) of θ_{L} (this is just the restriction of θ^{m} to one of the copies of G_{W} ⊂ L). Then, if η_{L}^{m} denotes the unique irreducible representation of J^{1}(\frac{1}{2}β, 𝔄_{W}^{m} ) which contains θ_{L}^{m}, we have

η_{P}^{m}|_{H_{m}^{1}∩L} ≃ η_{L}^{m} ⊗ 1 ⊗ η_{L}^{m}.

2.2 \overline{G}-Covers

Now we will transfer the information obtained in the last section to \overline{G}, using Glauberman’s correspondence (see [10], or [22] for the situation here). Let Ω denote a 2-group of automorphisms of G. Recall that if H is a pro-p subgroup of G and H^{Ω} is the group of Ω-fixed points, then there is a bijection, denoted ρ ↔ g_{Ω}(ρ) between (equivalence classes of) irreducible representations of H with ρ^{ω} ≃ ρ, for all ω ∈ Ω, and (equivalence classes of) irreducible representations of H^{Ω}. Further, this correspondence commutes with irreducible restriction and irreducible induction. Recall also that the representation g_{Ω}(ρ) is characterized as the unique component of ρ|_{H^{Ω}} appearing with odd multiplicity.

We will usually apply this correspondence with Ω the group of automorphisms of G consisting of ε and the identity, in which case we will just write g for the correspondence. Note also that, for ε-stable pro-p subgroups H of G, we have H^{ε} = \overline{H}^{+} = \overline{H}.

We write \overline{θ} = g(θ) = θ|_{\overline{H}_{1}}, a skew semisimple character (see [22, §3.4]). We also set

\overline{G}_{E}^{+} = B ∩ \overline{G}^{+} and \overline{G}_{E} = B ∩ \overline{G}

so that \overline{G}_{E}^{+} (respectively \overline{G}_{E}) is the direct product of a unitary group, for the quadratic extension E/E_{0}, and the anisotropic group \overline{G}_{0}^{+} (respectively \overline{G}_{0}).

We recall first the following, from [22, Theorem 3.16] (see also the Remarks following op. cit. Corollary 4.2), and [24, Proposition 3.3.1]:

Lemma 2.5. (i) I_{\overline{G}_{+}}(\overline{θ}) = \overline{J}^{1}\overline{G}_{E}^{+}\overline{J}^{1} and I_{\overline{G}}(\overline{θ}) = \overline{J}^{1}\overline{G}_{E}\overline{J}^{1}.

(ii) There exists a unique irreducible representation \overline{η} of \overline{J}^{1} which contains \overline{θ}. Moreover, \overline{η} = g(η), dim \overline{η} = (\overline{J}^{1} : \overline{H}^{1})^{\frac{1}{2}} and, for g ∈ \overline{G}^{+},

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

We also put \overline{θ}_{P} = g(θ_{P}) = θ_{P} |_{\overline{H}1_{P}}. Note that we have

\overline{θ}_{P} = { \atop i(θ_{W})}.

Remark 2.6. (i) Note that, in the language of [25, Definition 5.1], the decomposition V = W ⊕ V_{0} ⊕ W (of which the Levi subgroup L is the stabilizer) is properly subordinate to the stratum [Λ, n, 0, β]. Then [25, Lemma 5.3] implies that U(Λ) ∩ B ∩ U = U^{1}(Λ) ∩ B ∩ U. In particular, \overline{J} ∩ \overline{U} = \overline{J}^{1} ∩ \overline{U}.

(ii) From (i), we have \overline{J}_{P} ∩ \overline{U} = \overline{H}_{P}^{1} ∩ \overline{U}, and the same is true for \overline{U}_{−}. Then the property \overline{θ}_{P} = { \atop i(θ_{W})} implies (see [2, Lemma 1(ii)]) that

(\overline{J}_{P} ∩ \overline{U})(\overline{J}_{P} ∩ \overline{U}_{−}) ⊂ (\overline{J}_{P} ∩ \overline{U}_{−}) Ker(i(θ_{W}))(\overline{J}_{P} ∩ \overline{U}).

In particular (by loc. cit.), if ρ is a representation of a subgroup K of J_{W} × \overline{G}_{0} which restricts to a multiple of θ_{W}, then ĩ(ρ) is a well-defined representation of (\overline{J}_{P} ∩ \overline{U}_{−})(i(K))(\overline{J}_{P} ∩ \overline{U}). We will use this several times, for example, in the definitions of η̃_{P} and \overline{κ}_{P} below.

From Proposition 2.4, together with [22, Corollary 2.5] and[23, Theorem 2.3], we get:

Proposition 2.7. I_{\overline{G}_{+}}(\overline{θ}_{P}) = \overline{J}_{P}^{1} \overline{G}_{E}^{+}\overline{J}_{P}^{1} and I_{\overline{G}}(\overline{θ}_{P}) = \overline{J}_{P}^{1} \overline{G}_{E}\overline{J}_{P}^{1} .

Let \overline{θ}^{m} = g(θ^{m}) = θ^{m}|_{\overline{H}1_{m}}. Let \overline{η} be the unique irreducible representation of \overline{J}^{1} which contains \overline{θ}, as in Lemma 2.5, and let \overline{η}^{m} be the unique irreducible representation of \overline{J}_{m}^{1} which contains \overline{θ}^{m}. Note that \overline{θ}_{P} = g(θ_{P}) is the trivial extension of \overline{θ} to \overline{H}_{P}^{1} and, similarly, \overline{θ}_{P}^{m} = g(θ_{P}^{m}) the trivial extension of \overline{θ}^{m} to \overline{H}_{m,P}^{1} .

We also put \overline{η}_{P} = g(η_{P}) and \overline{η}_{P}^{m} = g(η_{P}^{m}). By taking ε-fixed points in J^{1}/H^{1}, we can imitate Lemma 2.2 and Corollary 2.3 to show that there is a unique irreducible representation of \overline{J}_{P}^{1} which contains \overline{θ}_{P} and, since η_{P} |_{\overline{H}1_{P}} is a multiple of \overline{θ}_{P}, we see that this must be \overline{η}_{P}. Likewise, \overline{η}_{P}^{m} is the unique irreducible representation of \overline{J}_{m,P}^{1} containing \overline{θ}_{P}^{m}. We also have

\overline{η}_{P} = { \atop i(η_{W})} and \overline{η}_{P}^{m} = { \atop i(η_{W}^{m} )},

since the representations on the right restrict to multiples of { \atop i(θ_{W})} = \overline{θ}_{P} and { \atop i(θ_{W}^{m} )} = \overline{θ}_{P}^{m} respectively.

The group U(Λ) ∩ B has Iwahori decomposition with respect to (L, P), and normalizes J^{1}(\frac{1}{2}β, Λ) and H^{1}(\frac{1}{2}β, Λ). Hence U(Λ) ∩ B normalizes J_{P}^{1} and, since U(Λ^{m}) is contained in U(Λ), we can form the group J̃_{P}^{1} = (\overline{U}^{1}(Λ^{m}) ∩ \overline{G}_{E})\overline{J}_{P}^{1} . We also recall that we have the representation η̃_{W} of J̃_{W}^{1} = U^{1}(𝔅_{W}^{m} )J_{W}^{1} (see §1.1) and observe that J̃_{P}^{1} ∩ \overline{L} = i(J̃_{W}^{1} × \overline{U}^{1}(Λ_{0})).

Proposition 2.8 (cf. [6, Propositions 5.1.15, 5.1.19]). There is a unique representation η̃_{P} of J̃_{P}^{1} such that

(i) η̃_{P} |_{\overline{J}1_{P}} = \overline{η}_{P};

(ii) η̃_{P} and \overline{η}_{P}^{m} induce equivalent irreducible representations of \overline{U}^{1}(Λ^{m}).

Moreover, η̃_{P} = { \atop i(η̃_{W})} and

dim I_{g}(η̃_{P}) = \begin{cases}1 & if g ∈ J̃_{P}^{1} \overline{G}_{E}^{+}J̃_{P}^{1} ; \\ 0 & otherwise. \end{cases}

Proof Let Ω_{1} be the group of automorphisms of \overline{G}^{+} generated by conjugation by

h_{1} = (\begin{matrix}−I_{M} \\ & I_{D} \\ & & −I_{M} \end{matrix} ) .

Then (\overline{G}^{+})^{Ω_{1}} = (\overline{G}′)^{+}×\overline{G}_{0}^{+}. We will denote the Glauberman correspondence g_{Ω_{1}} on representations of pro-p subgroups of \overline{G} by g_{1}. We note that, since all our representations are trivial on U′ and U′^{−}, they are fixed by Ω_{1} and, moreover, g_{1} is just restriction for these representations (since their restrictions are irreducible).

Let Ω_{2} be the group of automorphisms of (\overline{G}^{+})^{Ω_{1}} generated by conjugation by

h_{2} = (\begin{matrix}I_{M} \\ & I_{D} \\ & & −I_{M} \end{matrix} ) .

Then ((\overline{G}^{+})^{Ω_{1}})^{Ω_{2}} = \overline{L}. We will denote the Glauberman correspondence g_{Ω_{2}} on representations of pro-p subgroups of (\overline{G}^{+})^{Ω_{1}} by g_{2}. As above, since all our representations are trivial on U and U^{−}, their transfers to (\overline{G}^{+})^{Ω_{1}} are fixed by Ω_{2} and, moreover, g_{2} is again just restriction for these representations (since their restrictions are still irreducible).

We write g_{Ω} for the composition g_{2} ∘ g_{1}. Then we have

g_{Ω}(\overline{η}_{P}) = i(η_{W}) and g_{Ω}(\overline{η}_{P}^{m}) = i(η_{W}^{m} ).

We define η̃_{P} to be { \atop i(η̃_{W})}. Then we clearly have g_{Ω}(η̃_{P}) = i(η̃_{W}) and η̃_{P} |_{\overline{J}1_{P}} = \overline{η}_{P}. Also, since the intertwining of η̃_{P} is contained in that of \overline{η}_{P},

I_{\overline{G}_{+}}(η̃_{P}) ∩ \overline{U}^{1}(Λ^{m}) ⊂ \overline{J}_{P}^{1} \overline{G}_{E}^{+}\overline{J}_{P}^{1} ∩ \overline{U}^{1}(Λ^{m}) = (\overline{U}^{1}(Λ^{m}) ∩ \overline{G}_{E})\overline{J}_{P}^{1} = J̃_{P}^{1} ,

so the induced representation Ind\overline{U}^{1}(Λ^{m}) ; J̃_{P}^{1} η̃_{P} is irreducible. Likewise, Ind_{\overline{J}_{m,P}^{1}}^{\overline{U}^{1}(Λ^{m})} \overline{η}_{P}^{m} is irreducible.

Then, since Glauberman’s correspondence commutes with irreducible induction, we have

g_{Ω} (Ind\overline{U}^{1}(Λ^{m}) ; J̃_{P}^{1} η̃_{P} ) = Ind_{i(J̃_{W}^{1} ×\overline{U}^{1}(Λ_{0}))}^{i(U^{1}(Λ_{W}^{m} )×\overline{U}^{1}(Λ_{0}))} i(η̃_{W})
≃ Ind_{i(J^{1}(β,𝔄_{W}^{m} )×\overline{U}^{1}(Λ_{0}))}^{i(U^{1}(Λ_{W}^{m} )×\overline{U}^{1}(Λ_{0}))} i(η_{W}^{m} ) = g_{Ω} (Ind_{\overline{J}_{m,P}^{1}}^{\overline{U}^{1}(Λ^{m})} \overline{η}_{P}^{m}) .

Condition (ii) now follows as g_{Ω} is injective.

Now we show that these two conditions determine η̃_{P} uniquely. For this, we need only show that \overline{η}_{P} occurs in Ind_{\overline{J}_{m,P}^{1}}^{\overline{U}^{1}(Λ^{m})} \overline{η}_{P}^{m} with multiplicity one. We use the Mackey formula to compute the restriction

Res_{\overline{J}_{P}^{1}}^{\overline{U}^{1}(Λ^{m})} Ind\overline{U}^{1}(Λ^{m}) ; J̃_{P}^{1} η̃_{P} .

If x ∈ \overline{U}^{1}(Λ^{m}) intertwines η̃_{P} with \overline{η}_{P}, then it intertwines \overline{η}_{P} with itself so lies in J̃_{P}^{1} , as above. Thus the multiplicity of \overline{η}_{P} in Ind_{\overline{J}_{m,P}^{1}}^{\overline{U}^{1}(Λ^{m})} \overline{η}_{P}^{m} is equal to its multiplicity in η̃_{P}, which we know to be one.

Finally, we must show that all of \overline{G}_{E}^{+} intertwines η̃_{P}. So suppose b ∈ \overline{G}_{E}^{+}. Since b intertwines \overline{η}_{P}^{m}, it certainly intertwines

Ind\overline{U}^{1}(Λ^{m}) ; J̃_{P}^{1} η̃_{P} ≃ Ind_{\overline{J}_{m,P}^{1}}^{\overline{U}^{1}(Λ^{m})} \overline{η}_{P}^{m}.

We deduce that there exist u, v ∈ \overline{U}^{1}(Λ^{m}) such that ubv intertwines η̃_{P}. In particular, ubv intertwines \overline{η}_{P} so there exist j_{1}, j_{2} ∈ \overline{J}_{P}^{1} such that j_{1}ubvj_{2} ∈ \overline{G}_{E}^{+}. Note that this element also still intertwines η̃_{P}.

Now

U^{1}(Λ^{m})bU^{1}(Λ^{m}) ∩ B^{×} = U^{1}(Λ^{m})bU^{1}(Λ^{m}) ∩ L′ ∩ B^{×}
= (U^{1}(Λ^{m}) ∩ L′)b(U^{1}(Λ^{m}) ∩ L′) ∩ B^{×},

by [23, Lemma 2.1] applied with Γ = Ω_{1}. Further,

(U^{1}(Λ^{m}) ∩ L′)b(U^{1}(Λ^{m}) ∩ L′) ∩ B^{×} = (U^{1}(Λ^{m}) ∩ B^{×})b(U^{1}(Λ^{m}) ∩ B^{×}),

by [6, Theorem 1.6.1] (which we may apply since, in A′ we have a simple stratum). Then, applying [23, Lemma 2.1] again, with Γ = {1, ε}, we get

\overline{U}^{1}(Λ^{m})b\overline{U}^{1}(Λ^{m}) ∩ \overline{G}_{E}^{+} = (\overline{U}^{1}(Λ^{m}) ∩ \overline{G}_{E})b(\overline{U}^{1}(Λ^{m}) ∩ \overline{G}_{E}).

So there exist u′, v′ ∈ \overline{U}^{1}(Λ^{m}) ∩ \overline{G}_{E} such that u′bv′ = j_{1}ubvj_{2} and, since u′, v′ ∈ J̃_{P}^{1} , we see that b intertwines η̃_{P}.

Finally, the assertion that the intertwining spaces are all 1-dimensional follows immediately from Lemma 2.5(ii), since η̃_{P} extends \overline{η}. ■

Now we define \overline{κ}_{P} = { \atop i(κ_{W})}, a representation of \overline{J}_{P}, so that \overline{κ}_{P} |J̃_{P}^{1} ≃ η̃_{P}. We also put \overline{ρ}_{P} = { \atop i(ρ_{W})}, a representation of \overline{J}_{P} trivial on \overline{J}_{P}^{1} , and

\overline{λ}_{P} = { \atop i(λ_{W})} = \overline{κ}_{P} ⊗ \overline{ρ}_{P} .

Writing matrices with respect to the (non-Witt) basis ℬ ∪ ℬ_{0} ∪ ℬ of V = W ⊕ V_{0} ⊕ W, we set

w_{1} = (\begin{matrix}& & I_{M} \\ & I_{D} \\ νσε_{W} (σ) \end{matrix}) ,

where I_{k} denotes the k × k identity matrix. We also put

\overline{W} = \{i(ϖ_{E}^{a}, 1_{V_{0}}) : a ∈ ℤ\} ∪ \{i(ϖ_{E}^{a}, 1_{V_{0}})w_{1} : a ∈ ℤ\} ,

where ϖ_{E} is our fixed uniformizer of E ⊂ A_{W} and 1_{V_{0}} is the identity map on V_{0}.

Proposition 2.9. I_{\overline{G}}(\overline{λ}_{P}) = \overline{J}_{P} \overline{W} \overline{J}_{P}.

Proof Since \overline{J}_{P} ∩ \overline{L} = i(J_{W}), while ϖ_{E} and σε_{W} (σ) normalize J_{W}, the elements of \overline{W} normalize \overline{J}_{P} ∩ \overline{L}. Since ϖ_{E} normalizes λ_{W} and λ_{W} ≃ λ_{W} ∘ σ̃ (see Proposition 1.1), we see that the elements of \overline{W} normalize i(λ_{W}). On the other hand, they either preserve \overline{U} and \overline{U}_{−} or interchange them so we see that every element of \overline{W} intertwines \overline{λ}_{P} = { \atop i(λ_{W})}. Hence we have I_{\overline{G}}(\overline{λ}_{P}) ⊃ \overline{J}_{P} \overline{W} \overline{J}_{P}.

The proof of the opposite containment, which is a variant of the proof of [6, Proposition 5.3.2], is inspired by [3, page 551]; in place of [3, 4.2 Lemma], we use [25, Proposition 1.1], which is a slight generalization of [18, Proposition 4.13]. It is almost identical to the proof of [25, Proposition 6.14], except that the definition of \overline{κ}_{P} there is a priori slightly different.

Suppose g ∈ \overline{G} intertwines \overline{λ}_{P} = \overline{κ}_{P} ⊗ \overline{ρ}_{P}, so that g ∈ I_{\overline{G}}(\overline{η}_{P} |\overline{J}_{P}^{1} ) = \overline{J}_{P} \overline{G}_{E}\overline{J}_{P}, as \overline{ρ}_{P} is trivial on \overline{J}_{P}^{1} . Thus, we may assume g lies in \overline{G}_{E}. Moreover \overline{J}_{P} ∩ \overline{G}_{E} = \overline{U}(Λ) ∩ \overline{G}_{E} is a parahoric subgroup of \overline{G}_{E} containing the Iwahori subgroup \overline{U}(Λ^{m}) ∩ \overline{G}_{E}. Therefore, we may further assume g is a distinguished double coset representative for \overline{U}(Λ) ∩ \overline{G}_{E}\\overline{G}_{E}/\overline{U}(Λ) ∩ \overline{G}_{E} (see [18, §3] or [25, §1] for this notion).

Since dim I_{g}(\overline{η}_{P} , \overline{J}_{P}^{1} ) = 1, we can imitate the proof of [6, Proposition 5.3.2] to get that any non-zero intertwining operator in I_{g}(\overline{λ}_{P} , \overline{J}_{P}) has the form S⊗T, with S ∈ I_{g}(\overline{η}_{P} , \overline{J}_{P}^{1} ) and T an endomorphism of the space of \overline{ρ}_{P}. Now the operator S also intertwines the restriction κ_{P} |J̃_{P}^{1} = η̃_{P} so, again as in [6,

Proposition 5.3.2], it follows that T belongs to I_{g}(\overline{ρ}_{P} |J̃_{P}^{1} ). In particular, g intertwines \overline{ρ}_{P} |_{J̃_{P}^{1} ∩\overline{G}_{E}}.

But J̃_{P}^{1} ∩ \overline{G}_{E} = \overline{U}^{1}(Λ^{m}) ∩ \overline{G}_{E} is the radical of the Iwahori subgroup \overline{U}(Λ^{m}) ∩ \overline{G}_{E} of \overline{G}_{E} contained in \overline{U}(Λ) ∩ \overline{G}_{E}. By [25, Proposition 1.1] and the Remarks that follow it, we conclude that we can assume that g normalizes \overline{U}(Λ) ∩ \overline{G}_{E} ∩ \overline{L}.

Finally, we notice (cf. [3, 4.1]) that N_{\overline{G}_{E}}(\overline{U}(Λ) ∩ \overline{G}_{E} ∩ \overline{L}) = (\overline{U}(Λ) ∩ \overline{G}_{E} ∩ \overline{L})\overline{W}(\overline{U}(Λ) ∩ \overline{G}_{E} ∩ \overline{L}). Since \overline{U}(Λ) ∩ \overline{G}_{E} ∩ \overline{L} ⊂ \overline{J}_{P}, we are done. ■

Recall that we write J_{\overline{L}} = i(J_{W}) and λ_{\overline{L}} = i(λ_{W}). In order to prove that (\overline{J}_{P} , \overline{λ}_{P}) is a cover of (J_{\overline{L}}, λ_{\overline{L}}), the only thing remaining is to find a strongly (\overline{P} , \overline{J}_{P})-positive element in the centre of \overline{L} which supports an invertible element of the spherical Hecke algebra H(\overline{G}, \overline{λ}_{P}). To achieve this, we look at the \overline{λ}_{P} -spherical Hecke algebras of two parahoric subgroups whose intersections with \overline{G}_{E} are non-conjugate maximal compact open subgroups of \overline{G}_{E}. Note that we could here have used instead the methods of [3] – indeed, the crucial ingredients (for example, the elements w_{1}, w_{2} below) are the same.

Let L′^{(1)} be the self-dual 𝔬_{E}-lattice chain of 𝔬_{F} -period e_{W} in V′ = W ⊕ W given by

⋯ ⊃ L_{k}^{W} ⊕ L_{k}^{W} ⊃ L_{k+1}^{W} ⊕ L_{k+1}^{W} ⊃ ⋯,

so that L′^{(1)} consists of every second lattice of L′. Let Λ′^{(1)} be the self-dual 𝔬_{E}-lattice sequence in V′ in which every lattice of L′^{(1)} occurs twice and with the indexing chosen such that

Λ′^{(1)}(k)^{#} = Λ′^{(1)}(1 − k), for all k ∈ ℤ.

Let Λ^{(1)} be the 𝔬_{F} -lattice sequence in V defined by

Λ^{(1)}(k) = Λ′^{(1)}(k) ⊕ Λ_{0}(2k).

It is a self-dual lattice sequence of period 2e_{W} = e/2 such that 𝔞_{0}(Λ^{(1)}) ⊃ 𝔞_{0}(Λ). Put \overline{K}_{1} = \overline{U}(Λ^{(1)}).

We define \overline{K}_{2} = \overline{U}(Λ^{(2)}) by the same process, starting from the self-dual 𝔬_{E}-lattice chain L′^{(2)} in V′ given by

⋯ ⊃ L_{k}^{W} ⊕ L_{k+1}^{W} ⊃ L_{k+1}^{W} ⊕ L_{k+2}^{W} ⊃ ⋯

Then \overline{U}(Λ) ⊂ \overline{K}_{1} ∩ \overline{K}_{2} so, in particular, \overline{J}_{P} ⊂ \overline{K}_{1} ∩ \overline{K}_{2}.

Note that the element w_{1} lies in \overline{W} ∩ \overline{K}_{1}. We also set w_{2} = i(ϖ_{E}^{−1}, 1_{V_{0}})w_{1} ∈ \overline{W} ∩ \overline{K}_{2}.

Lemma 2.10. (i) H(\overline{K}_{1}, \overline{λ}_{P}) = ⟨f_{1}, f_{w_{1}}⟩ where f_{1} is supported on \overline{J}_{P} and f_{w_{1}} is supported on \overline{J}_{P} w_{1}\overline{J}_{P}.

(ii) H(\overline{K}_{2}, \overline{λ}_{P}) = ⟨f_{1}, f_{w_{2}}⟩, with f_{1} as in (i) and f_{w_{2}} supported on \overline{J}_{P} w_{2}\overline{J}_{P}.

Proof Both parts follow from the following consideration. For i = 1, 2 the \overline{K}_{i}-intertwining of \overline{λ}_{P} is given by

I_{\overline{K}_{i}}(\overline{λ}_{P}) = \overline{J}_{P} \overline{W} \overline{J}_{P} ∩ \overline{K}_{i} = \overline{J}_{P} (\overline{W} ∩ \overline{K}_{i}) \overline{J}_{P} .

But \overline{W} ∩ \overline{K}_{i} = \{1, w_{i}\}. Moreover, the restriction of \overline{λ}_{P} to \overline{J}_{P} ∩ \overline{L} is irreducible and w_{i} normalizes \overline{J}_{P} ∩ L so the intertwining space I_{w_{i}}(\overline{λ}_{P}) is 1-dimensional. ■

Lemma 2.11. Consider f_{w_{1}}, f_{w_{2}} as elements of H(\overline{G}, \overline{λ}_{P}). Then the convolution f_{w_{1}} ∗ f_{w_{2}} is supported on \overline{J}_{P} w_{1}w_{2}\overline{J}_{P}.

Proof We know f_{w_{1}} ∗ f_{w_{2}} is supported on

\overline{J}_{P} w_{1}\overline{J}_{P} w_{2}\overline{J}_{P}
= \overline{J}_{P} (w_{1}(\overline{J}_{P} ∩ \overline{U}_{−})w_{1}^{−1}) (w_{1}(\overline{J}_{P} ∩ \overline{L})w_{1}^{−1}) w_{1}w_{2} (w_{2}^{−1}(\overline{J}_{P} ∩ \overline{U})w_{2}) \overline{J}_{P} .

From the explicit matrix descriptions in §1.4, \overline{J}_{P} contains w_{1}(\overline{J}_{P} ∩ \overline{U}_{−})w_{1}^{−1} = w_{1}(\overline{H}^{1} ∩ \overline{U}_{−})w_{1}^{−1}, w_{1}(\overline{J}_{P} ∩ \overline{L})w_{1}^{−1} and w_{2}^{−1}(\overline{J}_{P} ∩ \overline{U})w_{2} = w_{2}^{−1}(\overline{J}^{1} ∩ \overline{U})w_{2}, and the lemma follows. ■

We will prove that f_{ζ} := f_{w_{1}} ∗ f_{w_{2}} is invertible. To accomplish this we prove that f_{w_{1}} and f_{w_{2}} are each invertible. In each case we know

f_{w_{i}} ∗ f_{w_{i}} = c_{1i}f_{1} + d_{i}f_{w_{i}}  (2.12)

by Lemma 2.10. Thus, we only need to show c_{1i} ≠ 0 for each i.

Lemma 2.13. In equation (2.12) c_{1i} ≠ 0 for i = 1, 2.

Proof We treat only the case i = 1, since the other case is identical. We need to check that f_{w_{1}} ∗ f_{w_{1}}(1) ≠ 0. Since

f_{w_{1}}(x) = \begin{cases}0 & if x ∉ \overline{J}_{P} w_{1}\overline{J}_{P} \\ \overline{λ}_{P}^{∨}(j_{1})f_{w_{1}}(w_{1})\overline{λ}_{P}^{∨}(j_{2}) & if x = j_{1}w_{1}j_{2}, with j_{1}, j_{2} ∈ \overline{J}_{P}, \end{cases}

we can write

f_{w_{1}} ∗ f_{w_{1}}(1) = ∫_{\overline{K}_{1}} f_{w_{1}}(y)f_{w_{1}}(y^{−1})dy
= \frac{1}{|\overline{J}_{P} ∩^{w_{1}}\overline{J}_{P} |} ∫_{\overline{J}_{P} ×\overline{J}_{P}} f_{w_{1}}(j_{1}w_{1}j_{2})f_{w_{1}}(j_{2}^{−1}w_{1}^{−1}j_{1}^{−1})dj_{1}dj_{2}

= \frac{|\overline{J}_{P} |}{|\overline{J}_{P} ∩^{w_{1}}\overline{J}_{P} |} ∫_{\overline{J}_{P}} \overline{λ}_{P}^{∨}(j_{1})f_{w_{1}}(w_{1})f_{w_{1}}(w_{1}^{−1})\overline{λ}_{P}^{∨}(j_{1}^{−1}))dj_{1}.

Now w_{1} intertwines \overline{λ}_{P}^{∨} = { \atop i(λ_{W}^{∨})} and normalizes \overline{J}_{P} ∩ \overline{L} = i(J_{W}) so f_{w_{1}}(w_{1}) is an equivalence w_{1}i(λ_{W}^{∨}) ≃ i(λ_{W}^{∨}). But w_{1}^{−1} = h_{1}^{−1}w_{1}, where h_{1} = i(νσε_{W} (σ), 1_{V_{0}}) and, by Proposition 1.1, νσε_{W} (σ) ∈ J_{W}; hence w_{1}^{−1} ∈ J_{P} w_{1} and f_{w_{1}}(w_{1}^{−1}) is an equivalence i(λ_{W}^{∨}) ≃ ^{w_{1}}i(λ_{W}^{∨}). Thus f_{w_{1}}(w_{1})f_{w_{1}}(w_{1}^{−1}) is an equivalence of i(λ_{W}^{∨}) and hence a scalar c ≠ 0. Thus

f_{w_{1}} ∗ f_{w_{1}}(1) = c\frac{|\overline{J}_{P} |^{2}}{|\overline{J}_{P} ∩^{w_{1}}\overline{J}_{P} |} ≠ 0.

Therefore, f_{w_{1}} is invertible, as required. ■

Put ζ = i(ϖ_{E}, 1_{V_{0}}); since σε_{W} (σ) ∈ J_{W} the function f_{ζ} is supported on \overline{J}_{P} ζ\overline{J}_{P}.

Lemma 2.14. For each k ∈ N, the k-fold convolution f_{ζ}^{k} is supported on \overline{J}_{P} ζ^{k}\overline{J}_{P}.

Proof This is by simple induction on k, since

\overline{J}_{p}ζ\overline{J}_{P} ζ^{k}\overline{J}_{P}
= \overline{J}_{P} (ζ(\overline{J}_{P} ∩ \overline{U})ζ^{−1}) (ζ(\overline{J}_{P} ∩ \overline{L})ζ^{−1}) ζ^{k+1} (ζ^{−k}(\overline{J}_{P} ∩ \overline{U}_{−})ζ^{k}) \overline{J}_{P} ,

while \overline{J}_{P} contains ζ(\overline{J}_{P} ∩ \overline{U})ζ^{−1}, ζ(\overline{J}_{P} ∩ \overline{L})ζ^{−1} and ζ^{−k}(\overline{J}_{P} ∩ \overline{U}_{−})ζ^{k} (cf. Lemma 2.11). ■

In particular, since ϖ_{E}^{e(E/F)} and ϖ_{F} differ only by a unit of E, the function f_{ζ}^{e(E/F)} is an invertible element of H(\overline{G}, \overline{λ}_{P}) which is supported on the double coset \overline{J}_{P} ζ^{e(E/F)}\overline{J}_{P} = \overline{J}_{P} ζ_{F} \overline{J}_{P}, where

ζ_{F} = i(ϖ_{F} 1_{W} , 1_{V_{0}})

is a strongly (\overline{P} , \overline{J}_{P})-positive element of the centre of \overline{L}. We conclude:

Theorem 2.15. Let π_{W} be an irreducible supercuspidal representation of G_{W} ≅ GL_{M}(F), with π_{W}^{ε_{W}} ≃ π_{W}. Using the notation above, the pair (\overline{J}_{P} , \overline{λ}_{P}) is a \overline{G}-cover of (J_{\overline{L}}, λ_{\overline{L}}). In particular, it is an 𝔰-type, with 𝔰 = [\overline{L}, i(π_{W})]_{\overline{G}}.

Proof By construction, \overline{J}_{P} is decomposed with respect to \overline{P} so (i) and (ii) of [7, Definition 8.1] are satisfied for \overline{P}, and likewise for \overline{P}_{−}. By Lemma 2.13 the strongly (\overline{P} , \overline{J}_{P})-positive element ζ_{F} supports an invertible element of H(\overline{G}, \overline{λ}_{P}). Thus (\overline{J}_{P} , \overline{λ}_{P}) also satisfies (iii) of [7, Definition 8.1] for \overline{P}, and is therefore a \overline{G}-cover of (J_{\overline{L}}, λ_{\overline{L}}). Then [7, Theorem 8.3] implies (\overline{J}_{P} , \overline{λ}_{P}) is an 𝔰-type.

■

2.3 Hecke Algebras

In this section we derive results analogous to those of Chapter 5 of [6]. In particular, we show that the Hecke algebra of our type, H(\overline{G}, \overline{λ}_{P}) can be computed by using analogous computations in a case where the representation π of \overline{L} is of level zero. For many of these situations, [16] will give us the parameters of the Hecke algebra.

We fix i = 1 or 2 and, in the notation of the previous section, we put Λ^{M} = Λ^{(i)}. Then [Λ^{M}, n_{M}, 0, \frac{1}{2}β] is a skew semisimple stratum and we set \overline{J}_{M} = \overline{J}(\frac{1}{2}β, Λ^{M}), and similarly \overline{J}_{M}^{1} and \overline{H}_{M}^{1}. Let \overline{θ}_{M} denote the transfer of \overline{θ} to \overline{H}_{M}^{1}. By [22, Corollary 4.2], there is a unique irreducible representation \overline{η}_{M} of \overline{J}_{M}^{1} which extends \overline{θ}_{M}, and by [25, Theorem 4.1], we may choose a β-extension, \overline{κ}_{M} of \overline{η}_{M} to \overline{J}_{M} – we recall here what we mean by β-extension:

Recall the lattice sequence Λ^{m} from §1.4 such that \overline{U}(Λ^{m})∩\overline{G}_{E} is an Iwahori subgroup of \overline{U}(Λ)∩\overline{G}_{E}; then we have \overline{θ}^{m} the transfer of \overline{θ} and \overline{η}^{m} the unique irreducible representation of \overline{J}^{1}(\frac{1}{2}β, Λ^{m}) containing \overline{θ}^{m}. We abbreviate \overline{U}(Λ) ∩ \overline{G}_{E} = \overline{U}(Λ_{𝔬_{E}}) (and similarly for other lattice sequences) and define Ĵ_{M} = \overline{U}(Λ_{𝔬_{E}})\overline{J}_{M}^{1} and Ĵ_{M}^{1} = \overline{U}^{1}(Λ_{𝔬_{E}}^{m})\overline{J}_{M}^{1}. Thus, \overline{J}_{M} ⊃ Ĵ_{M} ⊃ Ĵ_{M}^{1} ⊃ \overline{J}_{M}^{1} and Ĵ_{M}^{1} is a pro-p Sylow subgroup of \overline{J}_{M}. By [25, Proposition 3.7], there is a unique irreducible representation, η̂_{M} of Ĵ_{M}^{1} which extends \overline{η}_{M} and such that η̂_{M} and \overline{η}^{m} induce equivalent irreducible representations of \overline{U}^{1}(Λ_{m}). Then a β-extension \overline{κ}_{M} is an extension to \overline{J}_{M} of η̂_{M}.

Similarly (as in [6, Proposition 5.2.5] – see [25, Lemma 4.2]), there is a unique irreducible representation, μ̂_{M} of Ĵ_{M} which extends \overline{η}_{M} and such that

Ind\overline{U}(Λ_{𝔬_{E}})\overline{U}^{1}(Λ) ; Ĵ_{M} μ̂_{M} ≃ Ind_{\overline{J}_{P}}^{\overline{U}(Λ_{𝔬_{E}})\overline{U}^{1}(Λ)} \overline{κ}_{P} .  (2.16)

Moreover, as in [6, Proposition 5.2.6], we have μ̂_{M}|Ĵ_{M}^{1} = η̂_{M}. The following Lemma will be crucial:

Lemma 2.17. Let \overline{τ}_{1}, \overline{τ}_{2} be irreducible representations of \overline{U}(Λ_{𝔬_{E}})/\overline{U}^{1}(Λ_{𝔬_{E}}), which we regard as representations of \overline{J}_{P} or Ĵ_{M}. Then there is a Hecke algebra isomorphism

H(\overline{G}, \overline{κ}_{P} ⊗ \overline{τ}_{1}, \overline{κ}_{P} ⊗ \overline{τ}_{2}) ⟶ H(\overline{G}, μ̂_{M} ⊗ \overline{τ}_{1}, μ̂_{M} ⊗ \overline{τ}_{2})
\overline{f} ⟼ f̂,

which preserves support: if \overline{f} has support \overline{J}_{P} g\overline{J}_{P}, for some g ∈ \overline{G}_{E}, then f̂ has support Ĵ_{M}gĴ_{M}.

Proof Note that we can also think of \overline{τ}_{j} as a representation of \overline{U}(Λ_{𝔬_{E}})\overline{U}^{1}(Λ), and we get

Ind_{\overline{J}_{P}}^{\overline{U}(Λ_{𝔬_{E}})\overline{U}^{1}(Λ)}(\overline{κ}_{P} ⊗ \overline{τ}_{j}) = (Ind_{\overline{J}_{P}}^{\overline{U}(Λ_{𝔬_{E}})\overline{U}^{1}(Λ)} \overline{κ}_{P} ) ⊗ \overline{τ}_{j},

and similarly for μ̂_{M}. The first assertion is then immediate from (2.16), by Lemma A.1 in the Appendix. The assertion concerning supports is a mild generalization of, for example, [25, Lemma 6.1 and Proposition 7.1] (see also [6, Proposition 5.5.13]), using Lemma A.1 in place of [6, Corollary 4.1.5]. More precisely, put \overline{κ} = Ind\overline{J} ; \overline{J}_{P} \overline{κ}_{P} (which is irreducible, since I_{\overline{G}}(\overline{κ}_{P}) ⊂ I_{\overline{G}}(\overline{η}_{P}) = \overline{J}_{P} \overline{G}_{E}\overline{J}_{P}), and consider \overline{τ}_{j} also as representations of \overline{J}_{M}; then the results cited are analogous to showing that there are support-preserving isomorphisms from each Hecke algebra to H(\overline{G}, \overline{κ} ⊗ \overline{τ}_{1}, \overline{κ} ⊗ \overline{τ}_{2}), and our isomorphism is the composition of these. ■

Using this, we obtain:

Lemma 2.18. There is a choice of κ_{W} for which κ_{W} ∘ σ̃ ≃ κ_{W}.

Proof Recall that we have fixed i = 1 or 2; in the notation of the previous section, we put w = w_{i}. Note that, by Proposition 2.8, w intertwines the representation \overline{η}_{P} = { \atop i(η_{W})} of \overline{J}_{P}.

We fix some choice of κ_{W}. Now κ_{W} ∘ σ̃ is also a β-extension of η_{W} so κ_{W} ∘ σ̃ ≃ κ_{W} ⊗ χ_{W}, for some character χ_{W} of U(𝔅_{W})/U^{1}(𝔅_{W}) ≅ J_{W} /J_{W}^{1} which factors through the determinant det_{B_{W} /E}. Then

wi(κ_{W}) ≃ i(κ_{W} ∘ σ̃) = i(κ_{W}) ⊗ i(χ_{W})

and w intertwines \overline{κ}_{P} with \overline{κ}_{P} ⊗ \overline{χ}_{P} (where \overline{χ}_{P} = { \atop i(χ_{W})} on \overline{J}_{P}), since both are trivial on the unipotent parts of \overline{J}_{P}. We also put χ̂_{M} = { \atop i(χ_{W})} on Ĵ_{M}. Then Lemma 2.17 (applied with \overline{τ}_{1} the trivial representation and \overline{τ}_{2} = { \atop i(χ_{W})}) implies that w intertwines μ̂_{M} with μ̂_{M} ⊗ χ̂_{M}.

Recall that we have \overline{κ}_{M}|Ĵ_{M}^{1} = η̂_{M} = μ̂_{M}|Ĵ_{M}^{1}, by construction of β-extensions in [25, Theorem 4.1]. Thus, if we let κ̂_{M} = \overline{κ}_{M}|Ĵ_{M}, we must have κ̂_{M} ≃ μ̂_{M} ⊗ \overline{ψ}_{M} for a character \overline{ψ}_{M} of \overline{U}(Λ_{𝔬_{E}})/\overline{U}^{1}(Λ_{𝔬_{E}}^{m}), that is, a character of the Siegel Levi subgroup of \overline{U}(Λ_{𝔬_{E}}^{M})/\overline{U}^{1}(Λ_{𝔬_{E}}^{M}) which is trivial on the maximal unipotent subgroup \overline{U}^{1}(Λ_{𝔬_{E}}^{m})/\overline{U}^{1}(Λ_{𝔬_{E}}^{M}). Then \overline{ψ}_{M} factorizes through the determinant on the Levi

subgroup and we can write \overline{ψ}_{M} = { \atop i(ψ_{W})}, for some character ψ_{W} of J_{W} /J_{W}^{1} .

Now w ∈ \overline{J}_{M} so w certainly intertwines κ̂_{M} with itself. Hence w intertwines μ̂_{M} ⊗ \overline{ψ}_{M} with μ̂_{M} ⊗ χ̂_{M} ⊗^{w}\overline{ψ}_{M}. Chasing back through the isomorphism of Lemma 2.17, we see that w intertwines \overline{κ}_{P} ⊗ \overline{ψ}_{M} with \overline{κ}_{P} ⊗ χ̂_{M} ⊗^{w}\overline{ψ}_{M}. Since w normalizes \overline{L}, this implies that conjugation by w gives an equivalence κ_{W} ⊗ψ_{W} ≃ κ_{W} ⊗χ_{W} (ψ_{W} ∘σ̃). By [6, Theorem 5.2.2], we deduce that ψ_{W} = χ_{W} (ψ_{W} ∘σ̃) and, in particular, κ_{W} ⊗ ψ_{W} is a β-extension with the required property. ■

We now choose κ_{W} as in Lemma 2.18 and make the same constructions as before: \overline{κ}_{P}, \overline{κ} and μ̂_{M}. Comparing μ̂_{M} with κ̂_{M}, we again get a character \overline{ψ}_{M} but now with the property that ^{w}\overline{ψ}_{M} = \overline{ψ}_{M}. If E/E_{0} is unramified then this implies that \overline{ψ}_{M} = ^{w}\overline{ψ}_{0}\overline{ψ}_{0}, for some character \overline{ψ}_{0}. Then \overline{ψ}_{0} extends to a character of J_{M} and, replacing \overline{κ}_{M} by \overline{κ}_{M} ⊗ \overline{ψ}_{0}^{−1}, we may assume \overline{ψ}_{M} = 1. In the ramified case, the condition on \overline{ψ}_{M} is that \overline{ψ}_{M}^{2} = 1 but it is (at least in principle) possible that \overline{ψ}_{M} ≠ 1.

We write \overline{ψ}_{M} = { \atop i(ψ_{W})}; then we may assume ψ_{W}^{2} = 1.

Proposition 2.19. With the notation as above, there is an algebra isomorphism

H(\overline{J}_{M}, \overline{λ}_{P}) ⟶_{ι}^{∼} H(\overline{U}(Λ_{𝔬_{E}}^{M})/\overline{U}^{1}(Λ_{𝔬_{E}}^{M}), { \atop i(ρ_{W} ⊗ ψ_{W})})

which preserves support: if \overline{f} ∈ H(\overline{J}_{M}, \overline{λ}_{P}) has support \overline{J}_{P} b \overline{J}_{P}, for some b ∈ \overline{J}_{M} ∩ \overline{G}_{E} = \overline{U}(Λ_{𝔬_{E}}^{M}), then ι(\overline{f}) has support (\overline{U}(Λ_{𝔬_{E}})/\overline{U}^{1}(Λ_{𝔬_{E}}^{M})) b (\overline{U}(Λ_{𝔬_{E}})/\overline{U}^{1}(Λ_{𝔬_{E}}^{M})).

Proof From Lemma 2.17, we get a support-preserving isomorphism

H(\overline{G}, \overline{λ}_{P}) ≃ H(\overline{G}, μ̂_{M} ⊗ { \atop i(ρ_{W})}) = H(\overline{G}, (μ̂_{M} ⊗ \overline{ψ}_{M}) ⊗ ({ \atop i(ρ_{W} ⊗ ψ_{W})}),

where we are using the fact that ψ_{W}^{2} = 1. On the other hand, we have support-preserving isomorphisms

H(\overline{J}_{M}, (μ̂_{M} ⊗ \overline{ψ}_{M}) ⊗ ({ \atop i(ρ_{W} ⊗ ψ_{W})}) ≃ H(\overline{J}_{M}, { \atop i(ρ_{W} ⊗ ψ_{W})})
≃ H(\overline{U}(Λ_{𝔬_{E}}^{M})/\overline{U}^{1}(Λ_{𝔬_{E}}^{M}), { \atop i(ρ_{W} ⊗ ψ_{W})}),

where the first isomorphism follows from the fact that μ̂_{M} ⊗ \overline{ψ}_{M} = κ̂_{M} extends to a representation \overline{κ}_{M} of \overline{J}_{M} (cf. [6, Lemma 5.6.3]), and the second by reduction modulo \overline{J}_{M}^{1}, since \overline{J}_{M}/\overline{J}_{M}^{1} ≅ \overline{U}(Λ_{𝔬_{E}}^{M})/\overline{U}^{1}(Λ_{𝔬_{E}}^{M}). Putting these isomorphisms together gives the isomorphism of the Proposition.

■

Remarks 2.20. (i) Note that, writing 𝔅′_{M} for the self-dual 𝔬_{E}-order 𝔞_{0}(Λ^{M}) ∩ B′, we have

\overline{U}(Λ_{𝔬_{E}}^{M})/\overline{U}^{1}(Λ_{𝔬_{E}}^{M}) ≅ \overline{U}(𝔅′_{M})/\overline{U}^{1}(𝔅′_{M}) × \overline{G}_{0}/\overline{G}_{0}^{1} and
\overline{U}(Λ_{𝔬_{E}})/\overline{U}^{1}(Λ_{𝔬_{E}}^{M}) ≅ \overline{U}(𝔅′)/\overline{U}^{1}(𝔅′_{M}) × \overline{G}_{0}/\overline{G}_{0}^{1},

where \overline{G}_{0}^{1} is the pro-p radical of the anisotropic group \overline{G}_{0}. Then we have an isomorphism

H(\overline{U}(Λ_{𝔬_{E}}^{M})/\overline{U}^{1}(Λ_{𝔬_{E}}^{M}), { \atop i(ρ_{W} ⊗ ψ_{W})}) ≃ H(\overline{U}(𝔅′_{M})/\overline{U}^{1}(𝔅′_{M}), { \atop i(ρ_{W} ⊗ ψ_{W})}).

The quotient \overline{U}(𝔅′_{M})/\overline{U}^{1}(𝔅′_{M}) is a unitary (if E/E_{0} is unramified), symplectic or orthogonal group over k_{E_{0}} and the Hecke algebra on the right is described in [13]. Alternatively, reduction modulo \overline{U}^{1}(𝔅′_{M}) gives a support-preserving isomorphism

H(\overline{U}(𝔅′_{M})/\overline{U}^{1}(𝔅′_{M}), { \atop i(ρ_{W} ⊗ ψ_{W})}) ≃ H(\overline{U}(𝔅′_{M}), { \atop i(ρ_{W} ⊗ ψ_{W})}),

and the latter is described in [18]. This will allow us, in a future work, to compute explicitly the parameters for the Hecke algebra.

(ii) Since (in the case where E/E_{0} is ramified) we have ψ_{W}^{2} = 1, we may replace our choice of κ_{W} by κ_{W} ⊗ ψ_{W} (which has the same property of being fixed by σ̃); this replaces ρ_{W} by ρ_{W} ⊗ ψ_{W}, another self-dual cuspidal representation of U(𝔅_{W})/U^{1}(𝔅_{W}), and we lose the character ψ_{W} from the RHS of the isomorphism in Proposition 2.19. However, we cannot do this independently for the two choices Λ^{(1)}, Λ^{(2)} for Λ^{M}. In particular, if we choose (as we always can) to dispose with the character ψ_{W} in one case, then it may a priori still be non-trivial in the other.

(iii) Since (\overline{J}_{P} , \overline{λ}_{P}) is a cover of (J_{\overline{L}}, λ_{\overline{L}}), by [7, Corollary 7.12] we have a canonical embedding of Hecke algebras t_{P} :H(\overline{L}, λ_{\overline{L}}) ↪ H(\overline{G}, \overline{λ}_{P}) and we identify H(\overline{L}, λ_{\overline{L}}) (which is just the algebra of Laurent polynomials in a single variable) with its image B_{P}. We also put K = H(\overline{J}_{M}, \overline{λ}_{P}). Then [9, Theorem 1.5] implies that the map

B_{P} ⊗_{ℂ} K → H(\overline{G}, \overline{λ}_{P})
f ⊗ ϕ ↦ f ∗ ϕ

is an isomorphism of (B_{P} , K)-bimodules.

We conclude with a description of the Hecke algebra as a convolution algebra. We put S = {w_{1}, w_{2}} and recall that we have elements f_{w_{i}} ∈ H(\overline{G}, \overline{λ}_{P}) with support \overline{J}_{P} w_{i}\overline{J}_{P}.

Theorem 2.21 (cf. [1, (1.11) Théorème]). The algebra H(\overline{G}, \overline{λ}_{P}) is a convolution algebra on (\overline{W}, S). As a vector space, it has basis f_{w} = f_{s_{1}} ∗ ⋯ ∗ f_{s_{r}}, where w = s_{1} ⋯ s_{r} is a reduced decomposition of w in \overline{W}. It is generated as an algebra by f_{w_{1}} and f_{w_{2}}, with relations

f_{w_{i}} ∗ f_{w_{i}} = c_{1i}f_{w_{i}} + d_{i}f_{w_{i}}, for i = 1, 2.

Proof The proof is the same as that of [1, (1.11) Théorème]. ■

A Intertwining and Hecke algebras

In this short appendix we give the statement of a result, which is a mild generalization of [6, Proposition 4.1.3, Corollary 4.1.5]. The notation here is independent of that in the rest of this paper.

Let G be a unimodular locally compact totally disconnected topological group with a countable base of open sets, and fix a Haar measure on G. Let K be a compact open subgroup of G and (ρ_{i}, W_{i}) a continuous finite-dimensional representation of K, for i = 1, 2.

Let H(G, ρ_{1}, ρ_{2}) be the complex vector space of compactly supported functions ϕ : G → Hom_{ℂ}(W_{2}^{∨}, W_{1}^{∨}) satisfying

ϕ(k_{1}gk_{2}) = ρ_{1}^{∨}(k_{1}) ∘ ϕ(g) ∘ ρ_{2}^{∨}(k_{2}), for all k_{1}, k_{2} ∈ K, g ∈ G.

Then H(G, ρ_{1}, ρ_{2}) is an (H(G, ρ_{1}), H(G, ρ_{2}))-bimodule, under convolution of functions. If g ∈ G then we write H(G, ρ_{1}, ρ_{2})_{g} for the subspace of functions which are null outside KgK.

Recall that the G-intertwining of ρ_{1} with ρ_{2} is given by

I_{G}(ρ_{1}, ρ_{2}) = {g ∈ G : Hom_{K∩^{g}K}(^{g}ρ_{1}, ρ_{2}) ≠ 0}.

The result we need is the following:

Lemma A.1 (cf. [6, 4.1.1–5]). In the above situation, let g ∈ G.

(i) There is a function ϕ ∈ H(G, ρ_{1}, ρ_{2}) with ϕ(g) ≠ 0 if and only if g^{−1} ∈ I_{G}(ρ_{1}, ρ_{2}). Moreover, if g satisfies these conditions then there is a canonical isomorphism of vector spaces between Hom_{K∩K^{g}}(W_{1}, W_{2}^{g}) and H(G, ρ_{1}, ρ_{2})_{g}.

(ii) Let H be a compact open subgroup containing K. Then there is a canonical isomorphism

H(G, ρ_{1}, ρ_{2}) ≃ H(G, Ind_{K}^{H} ρ_{1}, Ind_{K}^{H} ρ_{2})

of (H(G, ρ_{1}), H(G, ρ_{2}))-bimodules. This restricts to an isomorphism

H(G, Ind_{K}^{H} ρ_{1}, Ind_{K}^{H} ρ_{2})_{g} ≃ ∐_{\begin{matrix}g′∈K\G/K \\ Hg′H=HgH \end{matrix}} H(G, ρ_{1}, ρ_{2})_{g′}.

The proof is very similar to those of [6, 4.1.1–5]; one point to note is that we are using the canonical isomorphisms of [6, Proposition 4.1.3] to identify H(G, ρ_{i}) with H(G, Ind_{K}^{H} ρ_{i}), for i = 1, 2, so that H(G, Ind_{K}^{H} ρ_{1}, Ind_{K}^{H} ρ_{2}) is indeed an (H(G, ρ_{1}), H(G, ρ_{2}))-bimodule.

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  arXiv:math.RT/0607622

David Goldberg, Department of Mathematics, Purdue University, West Lafayette, IN 47907-1395, USA

goldberg@math.purdue.edu Philip Kutzko, Department of Mathematics, University of Iowa, Iowa City, IA 52242-1419, USA

philip-kutzko@uiowa.edu Shaun Stevens, School of Mathematics, University of East Anglia, Norwich, NR4 7TJ, United Kingdom

Shaun.Stevens@uea.ac.uk