Sp(2N)-covers for self-contragredient supercuspidal representations of GL(N)

Corinne Blondel

November 15, 2003

Let G be the group of F -points of a connected reductive algebraic group defined over F, a local non-archimedean field. The goal of the theory of types is the description of direct summands of the category R(G) of smooth complex representations of G as categories of modules over Hecke algebras.

More precisely, the Bernstein decomposition of this category states that it is the direct sum, over the set of inertial classes in G, of full subcategories R^{[M,π]}(G) attached to each inertial class. Recall that an inertial class in G is the equivalence class [M, π]_{G} of a pair (M, π) made up of a F -Levi subgroup M of G and an irreducible supercuspidal representation π of M; the equivalence relation includes G-conjugacy and twisting of π by an unramified character of M. The subcategory R^{[M,π]}(G) consists of representations each of whose irreducible subquotients is a subquotient of a representation parabolically induced to G from an unramified twist of π.

Finding a type (J, λ) for this subcategory means finding a compact open subgroup J of G and a smooth irreducible representation λ of J such that the subcategory R^{[M,π]}(G) consists of representations generated by their isotypic component of type λ under J. If (J, λ) is a type for [M, π]_{G}, i.e. for R^{[M,π]}(G), this subcategory is then equivalent to the category of non-degenerate modules over the Hecke algebra H(G, J, λ) (for all this see [BK2]).

The problem of finding types in G naturally breaks into two pieces which are very different in nature. One is finding types for the inertial classes of supercuspidal representations of G. The other is finding types for inertial classes [M, π]_{G} where M is a proper Levi subgroup of G. C. J. Bushnell and P. C. Kutzko in [BK2] have developped a method to address this second problem, based on the definition of covers.

We say that (J, λ) is a G-cover of (J_{M}, λ_{M}), an analogous pair in M, if there is an F -parabolic subgroup P of G with unipotent radical U and Levi decomposition P = MU such that :

(i) (J, λ) is a decomposed pair with respect to (M, P), i.e.

– J = (J ∩ U^{−})(J ∩ M)(J ∩ U), where U^{−} is the unipotent radical of the parabolic subgroup P^{−} opposite of P relative to M, and

– λ is trivial on J ∩ U^{−} and J ∩ U; (ii) J ∩ M = J_{M} and λ|J_{M} ≃ λ_{M}; (iii) for any smooth irreducible representation (σ, V) of G, the restriction to V^{λ} of

the Jacquet functor r_{U} is injective.

Typeset by AMS-TEX

Note that the definition in [BK2] requires those properties to hold for any such parabolic subgroup P; nonetheless it follows from [Bu2] that one may restrict the definition to just one parabolic subgroup.

C. J. Bushnell and P. C. Kutzko have shown that:

if (J_{M}, λ_{M}) is a type for [M, π]_{M} in M and if (J, λ) is a G-cover of (J_{M}, λ_{M}), then (J, λ) is a type for [M, π]_{G} in G.

Let now Ḡ be Sp_{2N}(F) where F has odd residual characteristic, let P̄ be the Siegel parabolic subgroup, and let M̄ be the Siegel Levi subgroup, which we identify with GL_{N}(F) (see §I.1). Let π be an irreducible supercuspidal representation of GL_{N}(F) and (JM̄, λM̄) be a Bushnell-Kutzko type for [GL_{N}(F), π]_{GL_{N} (F)} in M̄. Observe that the non-trivial element s in NḠ(M̄)/M̄ stabilizes the inertial class [M̄, π]Ḡ if and only if π and its contragredient representation are equivalent up to twisting by an unramified character of GL_{N}(F) – yet, since any unramified character of GL_{N}(F) is a square, π and its contragredient representation are in the same inertial class if and only if this class actually contains a self-contragredient representation.

If this is not the case, it should follow from [BK2], Theorem 12.1, that the Hecke algebra H(Ḡ, J, λ) of a Ḡ-cover (J, λ) of (JM̄, λM̄) is commutative, isomorphic to H(M̄, JM̄, λM̄), and the corresponding subcategories are equivalent. In any case, a recent result of A. Roche ([R], Theorem 3.1) states, in our present setting, that parabolic induction from R^{[M̄,π]}(M̄) to R^{[M̄,π]}(Ḡ) is an equivalence of categories if and only if s does not stabilize [M̄, π]Ḡ.

Hence, although the question of existence of Ḡ-covers is interesting in itself, the most interesting case is the case when π is self-contragredient. Indeed, given a Ḡcover (J, λ) of (JM̄, λM̄), one expects the description of H(Ḡ, J, λ) to give insight into reducibility problems for parabolically induced representations in R^{[M̄,π]}(Ḡ) (see e.g. [BB2] for details in the case of Sp_{4}(F); although, for this group, the results concerning reducibility were already known). In particular, obtaining Ḡ-covers and their Hecke algebras for all such representations π should lead to an exact knowledge of the real numbers α such that the parabolically induced representation to Ḡ of the twisted representation π ⊗ |det|^{α} is reducible (those numbers are known to belong to {0, ±1/2)} if N > 1, by the work of Shahidi

[Sh1] [Sh2]).

We construct in this paper Ḡ-covers for Bushnell-Kutzko types attached to inertial classes [GL_{N}(F), π]_{GL_{N} (F)} where π is self-contragredient, which is the first step in the above program. We do not compute the corresponding Hecke algebras. The principle of the construction is to start with a well chosen GL_{2N}(F)-cover attached to the inertial class [GL_{N}(F) × GL_{N}(F), π ⊗ π]_{GL_{2N} (F)} and then restrict it to Sp_{2N}(F). In the process we need some strong properties of simple types attached to self-contragredient supercuspidal representations. We prove the following in part II (Corollary II.3):

Sp(2N)-covers for self-contragredient supercuspidals of GL(N) 3

Theorem 1. Let (Γ, γ) be a maximal simple type (in the sense of [BK1]) in GL_{N}(F) such that the corresponding inertial class contains a self-contragredient representation. Then either (Γ, γ) has level zero, or the simple character θ_{0} attached to γ can be attached to a simple stratum [𝔄_{0}, n_{0}, 0, β] in M_{N}(F) with the following properties: 1 –The field F [β] is a quadratic extension of F [β^{2}] (in particular N is even). 2 –Let ν be an element in 𝔄_{0}^{×} realizing the Galois conjugation of F [β] over F [β^{2}].

The character θ_{0} satisfies θ_{0}(νxν^{−1}) = θ_{0}(x^{−1}) (x in H^{1}(β, 𝔄_{0})).

This property of self-contragredient supercuspidal representations was known in the tame case: such a representation is then attached to an admissible character of a maximal field extension contained in GL_{N}(F) and Adler [A] proved that this character is trivial on the group of norms relative to a quadratic subextension (or N = 1 and the character is quadratic). In loc. cit. Adler also gives a full description of level zero self-contragredient supercuspidal representations (which exist only if N is even or N = 1).

Let g ↦ ^{τ}g be the transposition relative to the anti-diagonal. Theorem 1 essentially amounts to saying that, for a suitable order 𝔄 in M_{2N}(F) related to the order 𝔄_{0} above, the stratum Λ = [𝔄, 2n_{0}, 0, ( \begin{matrix}β & 0 \\ 0 & −^{τ}β \end{matrix} )] in M_{2N}(F) is simple. Let G = GL_{2N}(F) and P be the maximal parabolic subgroup in G of upper block-diagonal matrices with Levi subgroup M isomorphic to GL_{N}(F)×GL_{N}(F). The process in [BK1], §7.2, provides us (Corollary II.2) with a G-cover (J_{P} , λ_{P}) of (Γ × ^{τ}Γ, γ ⊗ γ^{∗}), with γ^{∗}(x) = γ(^{τ}x^{−1}), attached to the stratum Λ. It will lead us (Theorem III.1) to the cover we are looking for:

Theorem 2. Let (Γ, γ) be as in Theorem 1 and (J_{P} , λ_{P}) be as above. The unique representation ω of Ω = J_{P} ∩ Ḡ such that (Ω, ω) is a decomposed pair with respect to (M̄, P̄) with Ω ∩ M̄ = Γ and ω|Ω ∩ M̄ = γ is a Ḡ-cover of (Γ, γ).

In the case when (Γ, γ) has level zero, the cover given by Theorem 2 has previously been obtained by L. Morris in [M2]. Also recall that Ju-Lee Kim [Ki] has constructed a set of types in classical groups, under the assumption that the characteristic of F is 0 and the residual characteristic is “big enough”. The types in her work that correspond to our present setting need not be the same as those above, in particular they may not be Ḡ-covers (see [BB1]).

The main goal of this paper is Theorem 2, while Theorem 1 appears as a necessary tool. In part I, we establish notation and explain the basic mechanism allowing one to build decomposed pairs in Sp_{2N}(F) from the restriction of decomposed pairs in GL_{2N}(F). In part II, we detail the structure of the maximal simple type (Γ, γ) and of suitable GL_{2N}(F)-covers of (Γ × ^{τ}Γ, γ ⊗ γ^{∗}). This leads us to a proof of Theorem 1, as a Corollary of Theorem II.3. In part III, we first build a periodic infinite sequence (Ω_{i}, ω_{i}) of decomposed pairs in Ḡ, with Ω_{i} ∩ M̄ = Γ and ω_{i}|Ω_{i} ∩ M̄ = γ, then we show that certain sufficient criteria for this sequence to be a sequence of Ḡ-covers are satisfied. Part IV is devoted to the proof of an intertwining property (Proposition IV.1) that has been assumed in part III.

From paragraph II.2 on, we assume that the order 𝔄_{0} is standard. Any maximal simple type (Γ, γ) in GL_{N}(F) is conjugate to a maximal simple type satisfying this property, hence theorems 1 and 2 hold without this restriction (see the remarks after Corollary II.3 and Theorem III.1).

I would like to thank here Bertrand Lemaire for helpful conversations, David Manderscheid and Shaun Stevens for linguistic comments, Shaun Stevens for his careful reading of an earlier draft of the manuscript and pointing out some mistakes, and Guy Henniart for his constant advice.

I - Framework and basic tool

I.1 - Notations

Let F be a non-archimedean local field of residual characteristic p different from 2, let 𝔬_{F} or 𝔬 be its ring of integers, 𝔭_{F} or 𝔭 the maximal ideal of 𝔬_{F}, ϖ_{F} or ϖ a uniformizing element and k_{F} = 𝔬_{F} /𝔭_{F} the residue class field, of cardinality q_{F}. We will be working with the group G = GL_{2N}(F) and its subgroup Ḡ = Sp_{2N}(F) viewed as the symplectic group of the F -vector space V = F^{2N} equipped with the symplectic form < , > with matrix ( \begin{matrix}0 & −w_{N} \\ w_{N} & 0 \end{matrix} ) in the canon-

ical basis \{e_{1}, …, e_{2N}\}, where w_{N} = ( \begin{matrix}0 & ... & 0 & 1 \\ 0 & ... & 1 & 0 \\ . & . & . & . & . & . \\ 1 & 0 & ... & 0 \end{matrix} ). Most matrices written below

will be 2 × 2 block matrices with N × N blocks. Hence:

Sp_{2N}(F) = \{g ∈ GL_{2N}(F) ; ( \begin{matrix}0 & −w_{N} \\ w_{N} & 0 \end{matrix} ) ^{t}g^{−1} ( \begin{matrix}0 & w_{N} \\ −w_{N} & 0 \end{matrix} ) = g\}.

We will let X ↦ ^{T}X denote the corresponding involution on M_{2N}(F):

TX = ( \begin{matrix}0 & −w_{N} \\ w_{N} & 0 \end{matrix} ) ^{t}X ( \begin{matrix}0 & w_{N} \\ −w_{N} & 0 \end{matrix} ) ; ^{T}( \begin{matrix}A & B \\ C & D \end{matrix} ) = ( \begin{matrix}^{τ}D & −^{τ}B \\ −^{τ}C & ^{τ}A \end{matrix} ) ,

where g ↦ ^{τ}g, g ∈ GL_{i}(F), is the transposition relative to the antidiagonal; in other words: ^{tτ}g = ^{τt}g = w_{i} g w_{i}.

For any subgroup H of G, we put H̄ = H ∩ Ḡ. Let P be the stabilizer of the subspace < e_{1}, …, e_{N} > in F^{2N}, a parabolic subgroup of G. Let U be its unipotent radical and let M be the Levi factor of P consisting of matrices stabilizing < e_{N+1}, …, e_{2N} >. We let P^{−} be the parabolic subgroup of G opposite of P relative to M and we let U^{−} be its unipotent radical. We have

\begin{aligned}M & = \{ ( \begin{matrix}g_{1} & 0 \\ 0 & g_{2} \end{matrix} ) ; g_{1}, g_{2} ∈ GL_{N}(F)\}, U = \{ ( \begin{matrix}I & X \\ 0 & I \end{matrix} ) ; X ∈ M_{N}(F)\}, \\ M̄ & = \{ ( \begin{matrix}g & 0 \\ 0 & ^{τ}g^{−1} \end{matrix} ) ; g ∈ GL_{N}(F)\}, Ū = \{ ( \begin{matrix}I & X \\ 0 & I \end{matrix} ) ; X ∈ M_{N}(F), X = ^{τ}X\}. \end{aligned}

We will accordingly identify M with GL_{N}(F) × GL_{N}(F) and M̄ with GL_{N}(F), the latter through the isomorphism i from GL_{N}(F) to M̄ defined by

i(g) = ( \begin{matrix}g & 0 \\ 0 & ^{τ}g^{−1} \end{matrix} ) , g ∈ GL_{N}(F).

If μ is a representation of a subgroup H of GL_{N}(F), i(μ) will be the representation of i(H) defined by i(μ)(i(g)) = μ(g) (g ∈ H).

Let H be a compact open subgroup of G and ρ a smooth irreducible representation of H. The G-intertwining of ρ is:

I_{G}(ρ) = I_{G}(ρ, H) = {g ∈ G / Hom_{H∩H^{g}} (ρ, ρ^{g}) ≠ {0}}.

For any g in G we define the intertwining space of ρ at g to be

I_{g}(ρ) = I_{g}(ρ, H) = Hom_{H∩H^{g}} (ρ, ρ^{g}).

I.2 - Some decomposed pairs in Sp_{2N}(F)

Let π be a smooth irreducible self-contragredient supercuspidal representation of GL_{N}(F), hence viewed as a representation of M̄; likewise π ⊗ π is viewed as a representation of M. We want to find types in G and Ḡ for the inertial classes attached to these representations, and we want those types to be a G-cover and a Ḡ-cover respectively, of types attached to π ⊗ π in M and to i(π) in M̄. The situation in G has been settled by Bushnell and Kutzko in [BK1], as we will recall in part II. Indeed we will use the types build in loc. cit. to construct the Ḡ-covers we are looking for: the process will involve a suitable conjugation followed by a restriction to Sp_{2N}(F). The basic mechanism is the following:

Proposition. Let Γ be a compact open subgroup of GL_{N}(F) and let γ be a smooth finite-dimensional complex representation of Γ. Let γ^{∗} be the representation of ^{τ} Γ defined by: γ^{∗}(g) = γ(^{τ}g^{−1}), g ∈ ^{τ} Γ.

If (J, λ) is a decomposed pair in G relative to (M, P) such that J ∩M = Γ×^{τ} Γ and λ_{|J∩M} ≃ γ ⊗ γ^{∗}, then J̄∩ M̄ = i(Γ) and there exists a unique representation λ̂ of J̄ such that (J̄, λ̂) is a decomposed pair in Ḡ relative to (M̄, P̄) with

λ̂_{|i(Γ)} = i(γ).

The representation λ̄ = λ_{|J̄} of J̄ is isomorphic to λ̂ ⊗ λ̂.

Proof. We recall the following useful fact: let x ∈ U^{−}, m ∈ M and y ∈ U be such that their product xmy belongs to Ḡ; then x, m and y already belong to Ḡ. Indeed the involution defining Sp_{2N}(F) stabilizes U^{−}, M and U.

Hence taking intersections with Ḡ provides a decomposed pair (J̄, λ̄) in Ḡ relative to (M̄, P̄). We have J̄ ∩ M̄ = \overline{J ∩ M} = \overline{Γ ×^{τ} Γ} = i(Γ) and

λ̄ (( \begin{matrix}g & 0 \\ 0 & ^{τ}g^{−1} \end{matrix} )) ≃ γ(g) ⊗ γ^{∗}(^{τ}g^{−1}) = γ(g) ⊗ γ(g), g ∈ Γ.

In particular: (J̄ ∩ Ū)(J̄ ∩ Ū^{−}) ⊂ (J̄ ∩ Ū^{−}) T (J̄ ∩ Ū), with

T = \{ ( \begin{matrix}g & 0 \\ 0 & ^{τ}g^{−1} \end{matrix} ) ; g ∈ Γ, γ(g) ⊗ γ(g) = I\}.

To go from there to the decomposed pair we are looking for, it is enough ([Bl1], Lemme 1) to prove a similar inclusion with T replaced by

T′ = \{ ( \begin{matrix}g & 0 \\ 0 & ^{τ}g^{−1} \end{matrix} ) ; g ∈ Γ, γ(g) = I\}.

Indeed, the representation λ̂ will then be uniquely defined by the condition λ̂_{|i(Γ)} = i(γ), plus the fact that it is trivial on J̄ ∩ Ū and J̄ ∩ Ū^{−}.

Now the subgroup of J̄ generated by J̄∩Ū and J̄∩Ū^{−} is a pro-p-group ([Bl3]) and so is its intersection with M̄, hence we can replace T in the above inclusion by a suitable pro-p-subgroup of T. All we have to show is :

Lemma. Let γ be a finite-dimensional smooth complex representation of a pro-p-group H, with p odd. If the representation γ ⊗ γ of H is trivial, so is γ.

Indeed γ factors through a finite quotient of H, so it is unitarisable. In particular each operator γ(h), h ∈ H, is diagonalisable, and the triviality of γ ⊗ γ implies that any product of two eigenvalues of γ(h) is equal to 1. Hence γ(h) is a scalar operator, namely ±I. Now −I is impossible for p odd. □

If the pair (Γ, γ) in the proposition is a maximal simple type in GL_{N}(F) and the pair (J, λ) is a G-cover of (Γ ×^{τ} Γ, γ ⊗ γ^{∗}), one would like to know whether or not, under relevant conditions on (J, λ), the associated pair (J̄, λ̂) is a Ḡ-cover of (Γ, γ). We do address this question here in the special case of a pair (Γ, γ) attached to the inertial class of a self-contragredient representation ; the object of part II is to use Bushnell and Kutzko’s simple types to produce in this context a G-cover (J, λ) with suitable properties for that purpose.

II - “Self-contragredient” GL_{2N}(F)-covers

II.1 - Bushnell and Kutzko’s GL_{2N}(F)-covers

All references in this paragraph are to [BK1], any undefined notion or notation comes from [BK1].

Let π be an irreducible supercuspidal representation of GL_{N}(F) and (Γ, γ) a maximal simple type in GL_{N}(F) attached to the inertial class of π. From Definition (5.5.10) – where we do treat (b) as a special case of (a) – and Theorems (6.2.1), (6.2.2), the pair (Γ, γ) = (J(β, 𝔄_{0}), λ(β, 𝔄_{0})) comes equipped with the following data:

(i) A principal 𝔬_{F} -order 𝔄_{0} and a simple stratum [𝔄_{0}, n_{0}, 0, β] in M_{N}(F); in particular E = F [β] is a field extension of F. We let 𝔓_{0} be the radical of 𝔄_{0} and B_{0} be the commutant of E in M_{N}(F). Then 𝔅_{0} = 𝔄_{0} ∩ B_{0} is a maximal 𝔬_{E}-order with radical 𝔔_{0} = 𝔓_{0} ∩ B_{0}.

(ii) A simple character θ_{0} ∈ C(𝔄_{0}, 0, β) and a β-extension κ_{0} to J(β, 𝔄_{0}) of the

unique irreducible representation η_{0} of J^{1}(β, 𝔄_{0}) which contains θ_{0}. (iii) An irreducible cuspidal representation σ_{0} of GL(f, k_{E}) inflated to Γ through:

J(β, 𝔄_{0})/J^{1}(β, 𝔄_{0}) ≃ U(𝔅_{0})/U^{1}(𝔅_{0}) ≃ GL(f, k_{E}), f = N/[E : F].

We now write M_{2N}(F) as 2 × 2 block matrices with entries in M_{N}(F). This amounts to a decomposition of the underlying vector space V = F^{2N}, written as column matrices, as a direct sum V = V^{(1)} ⊕ V^{(2)} with V^{(1)} (resp. V^{(2)}) the subspace of column matrices having their first (resp. last) N entries equal to 0.

Let (Λ_{0,i})_{i∈ℤ} be the lattice chain in F^{N} associated to the order 𝔄_{0}. It determines lattice chains (Λ_{0,i}^{(j)})_{i∈ℤ} in V^{(j)}, j = 1, 2, under the natural identification of V^{(j)} with F^{N}. Let (Λ_{i})_{i∈ℤ} be the lattice chain in V defined by:

Λ_{2i} = Λ_{0,i}^{(1)} ⊕ Λ_{0,i}^{(2)}, Λ_{2i+1} = Λ_{0,i+1}^{(1)} ⊕ Λ_{0,i}^{(2)} (i ∈ ℤ).

The corresponding principal order in M_{2N}(F) is 𝔄 = ( \begin{matrix}𝔄_{0} & 𝔄_{0} \\ 𝔓_{0} & 𝔄_{0} \end{matrix} ).

We identify E with its block-diagonal image in M_{2N}(F), hence we also write β for ( \begin{matrix}β & 0 \\ 0 & β \end{matrix} ). We write B for the commutant of E in M_{2N}(F) and define the 𝔬_{E}-order 𝔅 = 𝔄 ∩ B, with radical 𝔔 = 𝔓 ∩ B (where 𝔓 is the radical of 𝔄).

Since the period of (Λ_{i})_{i∈ℤ} is twice the period of (Λ_{0,i})_{i∈ℤ}, we get (see 1.2.11, 1.4.13, 1.2.4):

Fact. [𝔄, 2n_{0}, 0, β] is a simple stratum in M_{2N}(F) and all assumptions in (7.1.11), (7.2.1) are satisfied, with t = e(𝔅|𝔬_{E}) = 2.

The following proposition can be regarded as obvious: it is a paraphrase of [BK1], §7. We state it to fix notations and make references easy, and give a sketch of proof as a matter of conscientiousness. The groups P, U, M are defined in (7.1.13) or equivalently in §I.1 above.

Proposition. There exists a unique representation λ of J = J(β, 𝔄) which is a simple type with the following property.

Let λ_{P} denote the natural representation of J_{P} = (J ∩ P)H^{1}(β, 𝔄) on the space of J^{1}(β, 𝔄) ∩ U-fixed vectors in λ. The pair (J_{P} , λ_{P}) is a decomposed pair in G relative to (M, P) with :

J_{P} ∩ M = J ∩ M = Γ × Γ and (λ_{P})_{|J∩M} = γ ⊗ γ.

Proof. Indeed this is Theorem (7.2.17) in [BK1], except that we want an actual equality between representations instead of an equivalence.

From (7.1.16): J ∩ M = J(β, 𝔄^{(1)}) × J(β, 𝔄^{(2)}); but we have arranged 𝔄 so that 𝔄^{(1)} = 𝔄^{(2)} = 𝔄_{0}, hence J(β, 𝔄^{(i)}) = Γ.

If β belongs to 𝔬_{F} we just note that J = J_{P} and J ∩ M = U(𝔄_{0}) × U(𝔄_{0}); we take the representation σ_{0} ⊗ σ_{0} there.

We now assume that β ∉ 𝔬_{F}. From (7.1.19), the restriction to H^{1} = H^{1}(β, 𝔄) of any simple type λ is a multiple of a simple character θ ∈ C(𝔄, 0, β) and the pair (H^{1}, θ) is a decomposed pair in G relative to (M, P), satisfying: H^{1} ∩ M = H^{1}(β, 𝔄^{(1)}) × H^{1}(β, 𝔄^{(2)}) and θ_{|H^{1}∩M} = θ^{(1)} ⊗ θ^{(2)}, where θ^{(i)} ∈ C(𝔄^{(i)}, 0, β) is the image of θ under the bijection C(𝔄^{(i)}, 0, β) ⟶^{≈} C(𝔄, 0, β) given by Theorem (3.6.14). Since the family of bijections given by (3.6.14) is unique and 𝔄^{(1)} = 𝔄^{(2)} = 𝔄_{0}, we must have θ^{(1)} = θ^{(2)}, and from (3.6.14) there is a unique θ ∈ C(𝔄, 0, β) such that θ_{|H^{1}∩M} = θ_{0} ⊗ θ_{0}.

With this choice of θ, the next step towards λ is the choice of a β-extension κ. From (7.2.5), (7.2.15), (7.2.16), it has the form κ = Ind_{J_{P}}^{J} κ_{P} where again the pair (J_{P} , κ_{P}) is decomposed and (κ_{P})_{|J∩M} = κ^{(1)} ⊗κ^{(2)}, both being β-extensions

of η_{0}. Since (κ_{P})_{|J∩M} is normalized by ( \begin{matrix}0 & I_{N} \\ ϖ_{E} & 0 \end{matrix} ) (7.2.15) and ϖ_{E} intertwines

κ^{(i)}, we have κ^{(1)} ≃ κ^{(2)} ≃ κ_{0} ⊗ χ ∘ det_{B_{0}} in the notation of (5.2.2). Since κ ⊗ χ^{−1} ∘ det_{B} is another β-extension we may pick κ in the first place so that (κ_{P})_{|J∩M} = κ_{0} ⊗ κ_{0}.

All we have to do now is to tensor κ with σ_{0} ⊗ σ_{0} as before (7.2.17). □

Corollary. The pair (J_{P} , λ_{P}) is a G-cover of the pair (Γ × Γ, γ ⊗ γ) in M. The pair (J_{P} , λ_{P}) is a type in G attached to the inertial class [M, π ⊗ π]_{G}.

The first assertion follows from (7.3.2) and the results in [BK2, §7], the second from Theorem 8.3 in [BK2]. Note that by symmetry – see (7.1.13) – this also holds for the pair (J_{P^{−}}, λ_{P^{−}}), with J_{P^{−}} = (J ∩ P^{−}) H^{1}.

Before turning to the case of self-contragredient supercuspidals in the next paragraphs, let us fix some more notations and write down some properties that will be used later on ; they all derive from [BK1, §3.1].

Let 𝕌 = ( \begin{matrix}0 & M_{N}(F) \\ 0 & 0 \end{matrix} ) and 𝕌^{−} = ( \begin{matrix}0 & 0 \\ M_{N}(F) & 0 \end{matrix} ). We will identify 𝕌^{−} with

M_{N}(F) through ( \begin{matrix}0 & 0 \\ X & 0 \end{matrix} ) ↦ X and 𝕌 with M_{N}(F) through ( \begin{matrix}0 & X \\ 0 & 0 \end{matrix} ) ↦ X. We also use the isomorphisms i^{+} and i^{−} from M_{N}(F) to U = 1 + 𝕌 and U^{−} = 1 + 𝕌^{−} respectively, defined by

i^{+}(X) = ( \begin{matrix}I & X \\ 0 & I \end{matrix} ) , i^{−}(X) = ( \begin{matrix}I & 0 \\ X & I \end{matrix} ) .

Write J^{1} = J^{1}(β, 𝔄), 𝔍^{1} = 𝔍^{1}(β, 𝔄), 𝔍_{−}^{1} = 𝔍^{1} ∩ 𝕌^{−}, 𝔍_{+}^{1} = 𝔍^{1} ∩ 𝕌 (both viewed as lattices in M_{N}(F)), and similarly for H^{1}, ℌ^{1}. From [BK1, §3.1, 7.1] we have :

\begin{aligned}J^{1} & = 1 + 𝔍^{1} = i^{−}(𝔍_{−}^{1}) (J^{1} ∩ M) i^{+}(𝔍_{+}^{1}), \\ H^{1} & = 1 + ℌ^{1} = i^{−}(ℌ_{−}^{1}) (H^{1} ∩ M) i^{+}(ℌ_{+}^{1}). \end{aligned}

The lattices ℌ^{1} and 𝔍^{1} are invariant under conjugation by 𝔎(𝔅), hence by

( \begin{matrix}0 & I_{N} \\ ϖ_{E} & 0 \end{matrix} ). Hence

𝔍_{−}^{1} = ϖ_{E}𝔍_{+}^{1} = 𝔍_{+}^{1}ϖ_{E} ; ℌ_{−}^{1} = ϖ_{E}ℌ_{+}^{1} = ℌ_{+}^{1}ϖ_{E}.

Those lattices in M_{N}(F) also satisfy :

ϖ_{E}𝔍_{−}^{1} ⊂ ℌ_{−}^{1} ⊂ 𝔍_{−}^{1} ⊂ ℌ_{+}^{1} ⊂ 𝔍_{+}^{1} ⊂ ϖ_{E}^{−1}ℌ_{+}^{1}.

II.2 - Intertwining properties in the self-contragredient case

Let (Γ, γ) = (J(β, 𝔄_{0}), λ(β, 𝔄_{0})) be a maximal simple type in GL_{N}(F) as above, attached to the inertial class of π. We want to use the decomposed pair (J_{P} , λ_{P}) given by Proposition II.1 to produce a decomposed pair in Ḡ through the process described in Proposition I.2. This is easy if Γ is equal to ^{τ} Γ and γ^{∗} equivalent to γ, which implies that π and its contragredient representation belong to the same inertial class. We actually want to show that the converse is true up to conjugacy.

Proposition. Let (Γ, γ) be a maximal simple type in GL_{N}(F) such that the corresponding inertial class of irreducible supercuspidal representations of GL_{N}(F) contains a self-contragredient representation π. We keep the notation in §II.1 and assume the order 𝔄_{0} is standard.

(i) There exists σ in U(𝔄_{0}) such that Γ is stable under σ̃ : x ↦ σ^{τ}x^{−1}σ^{−1}, and γ is equivalent to γ ∘ σ̃.

(ii) Such an element σ is unique up to left multiplication by Γ. It satisfies: (a) σ^{τ}σ^{−1} ∈ Γ and ϖ_{E}^{−1}σ^{τ}ϖ_{E} ^{τ}σ^{−1} ∈ Γ.

(b) The map σ̃ stabilizes H^{1}(β, 𝔄_{0}) and J^{1}(β, 𝔄_{0}) and we have: θ_{0} = θ_{0} ∘ σ̃.

(c) The lattices 𝔍_{+}^{1}, 𝔍_{−}^{1}, ℌ_{+}^{1} and ℌ_{−}^{1} in M_{N}(F) defined in II.1 are stable under X ↦ σ^{τ}Xσ^{−1}.

Example. Assume N = 2 and E is a quadratic extension of F generated by an element of matrix form ( \begin{matrix}0 & a \\ b & 0 \end{matrix} ). Then we may take σ = ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ); here conjugation by σ realizes the conjugation of E over F and σ plays an explicit part in the construction of Ḡ-covers in [BB1], [BB2].

Proof. We call here standard an order whose matrix form is given by [BK1], (2.5.1) or [Bu1], (1.9). Note that we can always conjugate (Γ, γ) into a maximal simple type whose associated order is standard. The property we need here is that 𝔄_{0}, being standard and principal, is stable under the map x ↦ ^{τ}x.

(i) By a theorem of Gelfand and Kazhdan ([GK], Theorem 2), the contragredient representation of π is equivalent to the representation π^{∗} defined by π^{∗}(x) = π(^{τ}x^{−1}). Since 𝔄_{0} is stable under τ, the automorphism x ↦ ^{τ}x^{−1} transforms the pair (Γ, γ) = (J(β, 𝔄_{0}), λ(β, 𝔄_{0})) into (Γ^{∗}, γ^{∗}) = (J(−^{τ}β, 𝔄_{0}), λ(−^{τ}β, 𝔄_{0})), a maximal simple type underlaid by the simple stratum [𝔄_{0}, n_{0}, 0, −^{τ}β]. Since π^{∗} is equivalent to π, those two types, (Γ, γ) and (Γ^{∗}, γ^{∗}), intertwine in GL_{N}(F) and we may use [BK1], Theorem (5.7.1), to derive that they are conjugate in GL_{N}(F). Since the two types are associated to the same order 𝔄_{0}, the proof of loc. cit. actually tells us more: indeed it says that there is an element σ in U(𝔄_{0}) that conjugates the simple character θ_{0} ∈ C(𝔄_{0}, 0, β) into θ_{0}^{∗} = θ_{0} ∘ (x ↦ ^{τ}x^{−1}) ∈ C(𝔄_{0}, 0, −^{τ}β), and that, eventually, that same element σ conjugates (Γ, γ) into (Γ^{∗}, γ′) where γ′ is equivalent to γ^{∗} (if γ has level 0, it says that γ and γ^{∗} are equivalent).

(ii) Let σ_{1} be another such element. The automorphism σ̃_{1}∘σ̃ normalizes (Γ, γ) and is a conjugation by σ_{1} ^{τ}σ^{−1} that must therefore belong to E^{×}Γ ∩ U(𝔄_{0}) = Γ ([BK1], (6.2.2)). Then the first part in (a) follows whence σ_{1} ∈ Γ^{τ}σ = Γσ.

For the last part in (a), we use the same argument: the element σ_{1} = ϖ_{E}^{−1}σ also satisfies the conditions in (i) – except that it does not belong to U(𝔄_{0}) – because ϖ_{E} normalizes (Γ, γ), and the determinant of σ_{1}(^{τ}σ_{1})^{−1} is 1, whence the result.

Now, for (b) we only have to note that the element σ produced in the proof of (i) satisfies the required properties.

The first step in the proof of (c) is the description of 𝔍^{1}(β, 𝔄) and ℌ^{1}(β, 𝔄) in terms of the lattices in GL_{N}(F) attached to [𝔄_{0}, n_{0}, 0, β], as in [BK1], (7.1.12). Since the given lattices in M_{2N}(F) are the direct sums of their intersections with the Hom_{F} (V^{(i)}, V^{(j)}) we use the corresponding block-matrix notation.

Lemma. For any non-negative integer k, we have:

\begin{aligned}ℌ^{k}(β, 𝔄) & = ( \begin{matrix}ℌ^{[ \frac{k+1}{2} ]}(β, 𝔄_{0}) & 𝔍^{[ \frac{k}{2} ]}(β, 𝔄_{0}) \\ ϖ_{E}𝔍^{[ \frac{k}{2} ]}(β, 𝔄_{0}) & ℌ^{[ \frac{k+1}{2} ]}(β, 𝔄_{0}) \end{matrix} ) and \\ 𝔍^{k}(β, 𝔄) & = ( \begin{matrix}𝔍^{[ \frac{k+1}{2} ]}(β, 𝔄_{0}) & ϖ_{E}^{−1}ℌ^{[ \frac{k}{2} ]+1}(β, 𝔄_{0}) \\ ℌ^{[ \frac{k}{2} ]+1}(β, 𝔄_{0}) & 𝔍^{[ \frac{k+1}{2} ]}(β, 𝔄_{0}) \end{matrix} ) . \end{aligned}

Proof. The diagonal blocks are already described in [BK1] Proposition (7.1.12), (iii). Since the lattices ℌ^{k}(β, 𝔄), 𝔍^{k}(β, 𝔄) are invariant under conjugation by

( \begin{matrix}0 & I_{N} \\ ϖ_{E} & 0 \end{matrix} ) ∈ 𝔎(𝔅), all we have to show is ℌ^{k}(β, 𝔄)∩ℕ = 𝔍^{[ \frac{k}{2} ]}(β, 𝔄_{0}) and 𝔍^{k}(β, 𝔄)∩

ℕ^{−} = ℌ^{[ \frac{k}{2} ]+1}(β, 𝔄_{0}). The proof of this goes exactly as in loc. cit.: the equality is first checked for β a minimal element, then obtained by induction along β. □

Since 𝔍_{+}^{1} = ϖ_{E}^{−1}𝔍_{−}^{1} and ℌ_{−}^{1} = ϖ_{E}ℌ_{+}^{1} it is easy, using property (a), to check that 𝔍_{+}^{1} and ℌ_{−}^{1} are stable under the map Σ : X ↦ σ^{τ}Xσ^{−1} if 𝔍_{−}^{1} and ℌ_{+}^{1} are. From the lemma we have 𝔍_{−}^{1} = ℌ^{1}(β, 𝔄_{0}) and ℌ_{+}^{1} = 𝔍(β, 𝔄_{0}).

We know from (b) that Σ stabilizes H^{1}(β, 𝔄_{0}) = 1+ℌ^{1}(β, 𝔄_{0}) hence ℌ^{1}(β, 𝔄_{0}). We have 𝔍(β, 𝔄_{0}) = 𝔅_{0} + 𝔍^{1}(β, 𝔄_{0}) ([BK1] (3.1.8)) and 𝔍^{1}(β, 𝔄_{0}) is stable

under Σ by (b), so we have to show that Σ(𝔅_{0}) is contained in 𝔍(β, 𝔄_{0}). We know that Σ(𝔔_{0}) = Σ(𝔅_{0} ∩ 𝔓_{0}) is contained in 𝔍^{1}(β, 𝔄_{0}). Since J(β, 𝔄_{0}) = Γ is invariant under σ̃ we also know that Σ(𝔅_{0}^{×}) is contained in J(β, 𝔄_{0}). Our claim then follows from the fact that 𝔅_{0} is the 𝔬_{E}-linear span of 𝔅_{0}^{×}: 𝔅_{0} = 𝔬_{E}[𝔅_{0}^{×}], as asserted in [Bu1] on page 190 (recall p ⩾ 3). □

Let S = ( \begin{matrix}I & 0 \\ 0 & σ \end{matrix} ); it belongs to U(𝔄). Define J^{S} = S^{−1}JS and λ^{S}(x) =

λ(SxS^{−1}), x ∈ J^{S}, where (J, λ) is the simple type given by Proposition II.1. Recall that the restriction of λ to H^{1} = H^{1}(β, 𝔄) is a multiple of a simple character θ ∈ C(𝔄, 0, β) such that (H^{1}, θ) is a decomposed pair in G relative to (M, P) with H^{1} ∩ M = H^{1}(β, 𝔄_{0}) × H^{1}(β, 𝔄_{0}) and θ_{|H^{1}∩M} = θ_{0} ⊗ θ_{0}.

Corollary. The representation λ^{S} of J^{S} = J(S^{−1}βS, 𝔄) is a simple type with the following properties:

(i) The pair (J_{P}^{S}, λ_{P}^{S}) is a decomposed pair in G relative to (M, P) with :

J_{P}^{S} ∩ M = J^{S} ∩ M = Γ × ^{τ} Γ and (λ_{P}^{S})_{|J^{S}∩M} ≃ γ ⊗ γ^{∗}.

The same holds for the pair (J_{P^{−}}^{S}, λ_{P^{−}}^{S}). (ii) The groups J^{S}, J_{P}^{S} and J_{P^{−}}^{S} are invariant under the involution X ↦ ^{T}X^{−1}. (iii) The group H^{1}(β, 𝔄)^{S} is invariant under the involution X ↦ ^{T}X^{−1} and so is

the simple character θ^{S} . We have θ_{|H^{1}^{S}∩M}^{S} = θ_{0} ⊗ θ_{0}^{∗}.

Indeed, using Iwahori decompositions of those groups, one checks easily that the invariance of their intersections with M, U and U^{−} derives from the properties in the proposition.

II.3 - A block-diagonal skew-simple stratum.

The simple character θ^{S} ∈ C(𝔄, 0, S^{−1}βS) in the above Corollary is fixed under the involution X ↦ ^{T}X^{−1}; it follows from [St1], Theorem 6.3, that this character can be viewed as a simple character attached to a skew simple stratum, namely:

There exists a simple stratum [𝔄, 2n_{0}, 0, δ] in End_{F} (V) satisfying δ = −^{T}δ such that θ^{S} belongs to C(𝔄, 0, δ).

In our situation we want, though, to work with a field extension both stable under the involution and contained in M. We will thus derive a number of properties of the above stratum that will lead us to that goal: we will first study the hermitian structure of V over the field L = F [δ] and show that we can conjugate δ into a block diagonal element g^{−1}δg (lemma 1); then we will use the very strong intertwining properties of simple types (lemma 2) to show that we can pick g in J^{S}.

If δ is equal to 0, then L = F is already contained in M anyway. We thus assume that δ ≠ 0. The involution X ↦ ^{T}X induces on L an automorphism of order 2; let L_{0} be the fixed field of this automorphism. Then L is a separable extension of L_{0} (recall p ≠ 2) with

Tr_{L/L_{0}}(X) = X + ^{T}X (X ∈ L).

Let ϕ be any non-zero F -linear form on L_{0}; then ϕ ∘ Tr_{L/L_{0}} is a non-zero F - linear form on L, invariant under the involution T. We define a non-degenerate L-anti-hermitian form b_{ϕ} on V through :

∀a ∈ L, ∀x ∈ V, ∀y ∈ V, < ax, y >= ϕ ∘ Tr_{L/L_{0}}(a b_{ϕ}(x, y)).

The intersection of End_{L}(V) with Sp(V) is the unitary group U(V, b_{ϕ}) relative to that form.

We first want to find a decomposition of the symplectic F -space V as a direct sum of maximal isotropic subspaces stable under L; it amounts to showing that the anisotropic part of the (anti-)hermitian space (V, b_{ϕ}) is equal to {0}. To show this we will use lattice duality, and it will actually be easier, as Shaun Stevens pointed out to us, to work with an hermitian form. We then fix an element u in L satisfying Tr_{L/L_{0}}(u) = 0 and such that val_{L}u = 0 if L/L_{0} is unramified, val_{L}u = 1 if L/L_{0} is ramified, and we define: d_{ϕ}(x, y) = ub_{ϕ}(x, y) (x, y ∈ V). This is an hermitian form.

Recall ([BK1], (3.5.1)) that the field extensions E = F [β] and L = F [δ] have the same ramification index and residual degree over F (hence [E : F] is even). In particular the self-dual lattice chain (Λ_{i})_{i∈ℤ} attached to 𝔄 (II.1) has period 2 over L (by definition of a stratum, those lattices are 𝔬_{L}-lattices). For Y an 𝔬_{L}-lattice in V we put :

Y^{♯} = {v ∈ V / < v, Y >⊆ 𝔬_{F} } ; Y^{♮} = {v ∈ V / d_{ϕ}(v, Y) ⊆ 𝔬_{L}}.

We fix ϕ such that ϕ(𝔬_{L_{0}}) = 𝔬_{F} and ϕ(𝔭_{L_{0}}^{−1}) = 𝔭_{F}^{−1}; we then have Y^{♮} = Y^{♯} . Since 𝔄_{0} is standard, we can number the lattice chain (Λ_{0,i})_{i∈ℤ} in such a way

that Λ_{0,0} = ( 𝔬_{F} ; ⋮ ; 𝔬_{F} ). We get the sequence Λ_{1}^{♯} = Λ_{−1} ⊃ Λ_{0} = Λ_{0}^{♯} ⊃ Λ_{1} = ϖ_{L}Λ_{−1},

that reads:

Λ_{1}^{♮} = Λ_{−1} ⊃ Λ_{0} = Λ_{0}^{♮} ⊃ Λ_{1} = ϖ_{L}Λ_{−1}.

This is the self-dual slice of the lattice chain in the sense of Morris [M1]. Propositions 1.7, 1.10 in [M1] tell us that we can find a decomposition of the hermitian space (V, d_{ϕ}) into a direct orthogonal sum V = V_{H} ⊕ V_{a}, where V_{a} is anisotropic and the anisotropic part of V_{H} is null, such that, for all i ∈ ℤ : Λ_{i} = Λ_{i} ∩ V_{H} ⊕ Λ_{i} ∩ V_{a}. We now use the following fact:

Let (W, b) be an anisotropic hermitian space over L. Assume there is an 𝔬_{L}- lattice Y in W satisfying:

– Y = Y^{♮} if L/L_{0} is unramified;

– Y = ϖ_{L}Y^{♮} if L/L_{0} is ramified.

Then W has dimension 0 or 1 over L.

Remark : The first case is a remark in [MVW], 5.I.1). Both cases rely on the classification of anisotropic hermitian spaces (see, e.g., [MVW], 1.I.4, or [M1], 1.8); indeed such a configuration cannot occur in two-dimensional spaces.

We can now conclude, since V has even dimension over L as over E, that the anisotropic part of V is null. Again, from Propositions 1.7, 1.10 in [M1], we can find a decomposition V = W_{1} ⊕ W_{2} into a direct sum of maximal b_{ϕ}-isotropic L-subspaces such that:

– for all i ∈ ℤ, Λ_{i} = Λ_{i} ∩ W_{1} ⊕ Λ_{i} ∩ W_{2};

– the induced lattice chains on W_{1} and W_{2} have period 1 over L; – Λ_{1} = Λ_{0} ∩ W_{1} ⊕ ϖ_{L}Λ_{0} ∩ W_{2}.

Let (f_{1}, ⋯, f_{N}) be an 𝔬_{F} -basis for (Λ_{i} ∩ W_{1})_{i∈ℤ} (see definition (1.1.7) in [BK1]) such that Λ_{0} ∩ W_{1} = 𝔬_{F} f_{1} + ⋯ + 𝔬_{F} f_{N} ; one checks easily that the basis (f_{N+1}, ⋯, f_{2N}) of W_{2} defined by < f_{k}, f_{2N−k+1} >= −1 for 1 ⩽ k ⩽ N and < f_{j}, f_{2N−k+1} >= 0 for 1 ⩽ k ⩽ N, 1 ⩽ j ⩽ N and j ≠ k, is an 𝔬_{F} -basis for (Λ_{i} ∩ W_{2})_{i∈ℤ}.

Let g be the element of Sp(V) that sends the canonical basis (e_{1}, ⋯, e_{2N}) on (f_{1}, ⋯, f_{2N}). We started (see II.1) with a decomposition V = V^{(2)} ⊕ V^{(1)} having the same properties with respect to E as the above decomposition with respect to L, and (e_{1}, ⋯, e_{N}) is an 𝔬_{F} -basis for (Λ_{i} ∩ V^{(2)})_{i∈ℤ}. We thus have g(Λ_{i} ∩ V^{(2)}) = Λ_{i} ∩ W_{1} for all i ∈ ℤ, hence, using duals, g(Λ_{i}) = Λ_{i}, so g belongs to 𝔄^{×}. We sum up what we have just proved :

Lemma 1. There exists an element g in Sp_{2N}(F) ∩ 𝔄^{×} such that g^{−1}δg is block diagonal, namely

g^{−1}δg = ( \begin{matrix}δ_{0} & 0 \\ 0 & −^{τ}δ_{0} \end{matrix} ) .

Let us come back now to the simple type (J^{S}, λ^{S}) in Corollary II.2, related to the simple character θ^{S} in C(𝔄, 0, S^{−1}βS) = C(𝔄, 0, δ). Its conjugate (J^{Sg}, λ^{Sg}) is related to the simple character θ^{Sg} in C(g^{−1}𝔄g, 0, g^{−1}δg); note that g^{−1}𝔄g = 𝔄 since g belongs to 𝔄^{×}. Since g^{−1}δg is block diagonal, the machinery of [BK1], §7.1 and 7.2 applies: (J^{Sg}, λ^{Sg}) determines equivalent maximal simple types ρ^{(1)} and ρ^{(2)}, attached respectively to the strata [𝔄_{0}, n_{0}, 0, δ_{0}] and [𝔄_{0}, n_{0}, 0, −^{τ}δ_{0}]; from [BK1], Theorem 7.2.17, we have: (λ^{Sg})_{U} = ρ^{(1)} ⊗ ρ^{(2)} (see also terminology 7.2.18, (iii)).

We now recall [BK1], Corollary 7.3.12. Let π′ be any smooth irreducible representation of G containing λ. Its supercuspidal support consists of unramified twists of an irreducible supercuspidal representation π of GL_{N}(F) containing γ, the maximal simple type we started with in II.2: (Γ, γ) = (J(β, 𝔄_{0}), λ(β, 𝔄_{0})).

But π′ contains λ if and only if it contains λ^{Sg}, hence π also contains the maximal simple type ρ^{(1)}. Since the maximal simple types λ(β, 𝔄_{0}) and ρ^{(1)} intertwine in GL_{N}(F) and are associated to the same order 𝔄_{0}, they are conjugate in 𝔄_{0}^{×} ([BK1], Theorem 5.7.1 and its proof). We sum up :

Lemma 2. Let (J(δ_{0}, 𝔄_{0}), ρ^{(1)}) be the maximal simple type associated to (J^{Sg}, λ^{Sg}). There exists an element a in 𝔄_{0}^{×} such that J(δ_{0}, 𝔄_{0}) = a^{−1}J(β, 𝔄_{0})a and ρ^{(1)} ≃ [λ(β, 𝔄_{0})]^{a}.

Now the element a above is related as follows to the element g in lemma 1:

Proposition. Let c = ( \begin{matrix}a & 0 \\ 0 & ^{τ}a^{−1} \end{matrix} ) be the element of Sp_{2N}(F) ∩ 𝔄^{×} associated

to a. Then g belongs to the coset cJ^{Sg} in 𝔄^{×}.

Proof. To simplify notation here we let H = J^{Sg} = J(g^{−1}δg, 𝔄) and μ = λ^{Sg}. We use [BK1], Theorem 7.2.17 to produce two decomposed pairs (H_{1}, μ_{1}) and (H_{2}, μ_{2}) in H such that μ = Ind_{H_{1}}^{H}μ_{1} = Ind_{H_{2}}^{H}μ_{2}.

For the first one we let μ_{1} be the natural action of H_{1} = (H ∩P)H^{1}(g^{−1}δg, 𝔄) on the space of (H ∩ U)-fixed vectors in μ; indeed H_{1} = (J^{Sg})_{P}, μ_{1} = (λ^{Sg})_{P}. We obtain a decomposed pair (H_{1}, μ_{1}) relative to (M, P) with μ = Ind_{H_{1}}^{H}μ_{1}.

For the second one, we let H_{2} = (J_{P})^{Sg}, μ_{2} = (λ_{P})^{Sg} and obtain a decomposed pair relative to (g^{−1}Mg, g^{−1}P g) with μ = Ind_{H_{2}}^{H}μ_{2}.

We now apply Mackey’s theorem [Ku] to the irreducible representation μ. The intertwining of μ in H is one-dimensional, hence there exists a unique double coset H_{2}zH_{1} in H such that the restrictions of μ_{1} and μ_{2}^{z} to their common domain H_{1} ∩ z^{−1}H_{2}z intertwine.

Let us look at the induced representation Ind_{H}^{𝔄^{×}}μ. It is irreducible – indeed the intertwining of μ in 𝔄^{×} is contained in the intersection with 𝔄^{×} of the intertwining of the simple character θ^{Sg}, hence in (HD^{×}H) ∩ 𝔄^{×} = H(D ∩ 𝔄)^{×}H = H, where D is the commutant algebra of g^{−1}δg. Applying the same theorem to Ind_{H_{1}}^{𝔄^{×}}μ_{1} = Ind_{H_{2}}^{𝔄^{×}}μ_{2} produces a unique double coset H_{2}z′H_{1} in 𝔄^{×} with the previous properties. We must have H_{2}zH_{1} = H_{2}z′H_{1}, hence the proposition will follow from:

Claim. g^{−1}c intertwines μ_{1} and μ_{2}.

First note that the pairs (H_{1}, μ_{1}) and (H_{2}, μ_{2}) are both invariant (up to equivalence of the representations) under the involution x ↦ ^{T}x^{−1}: this follows from Corollary II.2, since g belongs to Sp(V).

We have c^{−1}gH_{2}g^{−1}c = c^{−1}J_{P}^{S}c, so the pairs (H_{1}, μ_{1}) and (H_{2}^{g^{−1}c}, μ_{2}^{g^{−1}c}) are both decomposed with respect to (M, P) and the representations μ_{1} and μ_{2}^{g^{−1}c} intertwine if and only if their restrictions to H_{1} ∩ M and H_{2}^{g^{−1}c} ∩ M intertwine.

Now H_{1} ∩ M = J(g^{−1}δg, 𝔄) ∩ M = J(δ_{0}, 𝔄_{0}) × ^{τ}J(δ_{0}, 𝔄_{0}) and the restriction of μ_{1} there is ρ^{(1)} ⊗ ρ^{(2)}, equivalent to ρ^{(1)} ⊗ (ρ^{(1)})^{∗}. On the other hand:

H_{2}^{g^{−1}c} ∩ M = c^{−1}(J_{P} ∩ M)^{S}c = (a^{−1}J(β, 𝔄_{0})a) × ^{τ}(a^{−1}J(β, 𝔄_{0})a)

and the restriction of μ_{2}^{g^{−1}c} there is isomorphic to [λ(β, 𝔄_{0})]^{a} ⊗ ([λ(β, 𝔄_{0})]^{a})^{∗}. We now conclude with lemma 2. □

Since c^{−1}g belongs to J^{Sg}, then c^{−1} belongs to g^{−1}J^{S} so we can write g = hc with h ∈ J^{S}; note that h belongs to Ḡ since g and c do. Since the elements g^{−1}δg = c^{−1}h^{−1}δhc and c belong to M, so does h^{−1}δh. Furthermore, since h belongs to J^{S}, it stabilizes the simple character θ^{S}.

We have finally proved that, given any simple stratum [𝔄, 2n_{0}, 0, δ] in M_{2N}(F) satisfying δ = −^{T}δ and θ^{S} ∈ C(𝔄, 0, δ), there exists an element h ∈ J^{S} ∩ Ḡ such that h^{−1}δh belongs to M. We conclude:

Theorem. Let θ^{S} be the simple character in Corollary II.2. There exists a simple stratum [𝔄, 2n_{0}, 0, α] in M_{2N}(F), satisfying α ∈ M and α = −^{T}α, such that θ^{S} belongs to C(𝔄, 0, α).

Now write α = ( \begin{matrix}α_{0} & 0 \\ 0 & −^{τ}α_{0} \end{matrix} ) and note that such an element generates a field over F if and only if α_{0} = 0 (case ruled out from the start) or the field K = F [α_{0}] is a quadratic extension of K_{0} = F [α_{0}^{2}].

Recall that the pair (H^{1}(α, 𝔄), θ^{S}) is a decomposed pair above (H^{1}(α_{0}, 𝔄_{0}) × H^{1}(−^{τ}α_{0}, 𝔄_{0}), θ_{0} ⊗ θ_{0}^{∗}) and is T -stable, i.e.:

\{ \begin{matrix}H^{1}(α_{0}, 𝔄_{0}) × H^{1}(−^{τ}α_{0}, 𝔄_{0}) = H^{1}(α_{0}, 𝔄_{0}) × ^{τ}H^{1}(α_{0}, 𝔄_{0}) \\ θ_{S} (( \begin{matrix}g_{1} & 0 \\ 0 & g_{2} \end{matrix} )) = θ_{0}(g_{1})θ_{0}(^{τ}g_{2}^{−1}). \end{matrix}

From [BK1], Proposition 7.1.19, we conclude that the character g ↦ θ_{0}(^{τ}g^{−1}) on H^{1}(−^{τ}α_{0}, 𝔄_{0}) is the image of θ_{0} under the canonical transfer of simple characters from [BK1], Theorem 3.6.14:

C(𝔄_{0}, 0, α_{0}) ⟶^{≈} C(𝔄_{0}, 0, −^{τ}α_{0})

It is difficult here to use the original notations to denote the canonical map; indeed C(𝔄_{0}, 0, −^{τ}α_{0}) is still a set of simple characters attached to α_{0}, but we change the action of K on the underlying vector space by composing it with τ and with the Galois conjugacy over K_{0}, denoted by x ↦ x̄. We now use the following

Fact. Let ψ_{1}, ψ_{2} be two F -embeddings of K into M_{N}(F) such that ψ_{1}(K^{×}) and ψ_{2}(K^{×}) both normalise 𝔄_{0}. There exists u in U(𝔄_{0}) such that, for all x in K, we have ψ_{2}(x) = u^{−1}ψ_{1}(x)u. The canonical transfer map between the set of simple characters C(𝔄_{0}, m, ψ_{1}(α_{0})) and C(𝔄_{0}, m, ψ_{2}(α_{0})) (m in ℕ) is then given by θ ↦ θ^{u}.

(The first assertion above is Lemma 1.6 in [BH1]. The second is so tautological that it is implicit in [BK1]. In any case properties 3.6.13 in [BK1] are easily checked.)

We may then choose an element σ in U(𝔄_{0}) such that σ^{−1}xσ = ^{τ}x̄ for all x in K. The canonical transfer map from C(𝔄_{0}, 0, α_{0}) to C(𝔄_{0}, 0, −^{τ}α_{0}) hence transforms a simple character μ into the simple character x ↦ μ(σxσ^{−1}). We get:

Corollary. Let (Γ, γ) be a maximal simple type in GL_{N}(F) such that the corresponding inertial class of irreducible supercuspidal representations of GL_{N}(F) contains a self-contragredient representation π, and assume the corresponding principal order 𝔄_{0} is standard. Then either (Γ, γ) has level zero, or the simple character θ_{0} underlying it can be attached to a simple stratum [𝔄_{0}, n_{0}, 0, α_{0}] in M_{N}(F) with the following properties. 1 –The field F [α_{0}] is a quadratic extension of F [α_{0}^{2}] – in particular N is even. 2 –Let x ↦ x̄ denote the Galois conjugation of F [α_{0}] over F [α_{0}^{2}]. There is an

element σ in U(𝔄_{0}) such that σ^{−1}xσ = ^{τ}x̄ for all x in F [α_{0}]. The simple character θ_{0} then satisfies:

θ_{0}(σ^{τ}xσ^{−1}) = θ_{0}(x^{−1}) (x ∈ H^{1}(α_{0}, 𝔄_{0})).

Remark 1. The element σ above satisfies all assumptions in Proposition II.2 (see the proof of II.2). It is unique up to left multiplication by U(𝔅_{0}), and σ^{τ}σ^{−1} belongs to U(𝔅_{0}).

Remark 2. We can apply the above fact to the embedding x ↦ ^{τ}x of K into M_{N}(F), and get u in U(𝔄_{0}) such that u^{−1}xu = ^{τ}x for all x in K. The transfer map between C(𝔄_{0}, m, α_{0}) and C(𝔄_{0}, m, ^{τ}α_{0}) is then given by θ ↦ θ^{u}. Since it is also given by θ ↦ θ ∘ τ, any simple character θ in C(𝔄_{0}, m, α_{0}) satifies θ(g) = θ(u^{τ}gu^{−1}). We let ν = σu^{−1} and combine this with the above Corollary: we have ν^{−1}xν = x̄ for x ∈ K and θ_{0}(νxν^{−1}) = θ_{0}(x^{−1}) for x in H^{1}(α_{0}, 𝔄_{0}). This is the formulation given in the introduction; it is conjugacy-invariant, hence the assumption that 𝔄_{0} is standard can be removed there.

Remark 3. One can actually go further along the same lines and show that f = N/[E : F] is either even or equal to 1; hence either [E : F] is equal to N, or N is a multiple of 4.

III - A sequence of Sp_{2N}(F)-covers

III.1 - Construction of the sequence

We do not need in this paragraph the results obtained in §II.3; their use would not help. Henceworth we keep the notations and assumptions in §II.2 – in particular (Γ, γ) is a maximal simple type in GL_{N}(F), attached to the inertial class of a self-contragredient supercuspidal representation π, and 𝔄_{0} is standard – and start with a sequence (J_{i}, λ_{i})_{0⩽i⩽4} of G-covers of the pair (Γ × Γ, γ ⊗ γ) as obtained from Proposition II.1:

(J_{3}, λ_{3}) = (J_{P} , λ_{P}) ; (J_{2}, λ_{2}) = (J_{P^{−}}, λ_{P^{−}}) ;

(J_{1}, λ_{1}) = (J_{2}^{s}, λ_{2}^{s}) ; (J_{0}, λ_{0}) = (J_{3}^{s}, λ_{3}^{s}) with s = ( \begin{matrix}0 & I \\ I & 0 \end{matrix} ) ;

(J_{4}, λ_{4}) = (J_{0}^{c}, λ_{0}^{c}) with c = ( \begin{matrix}ϖ_{E} & 0 \\ 0 & ϖ_{E}^{−1} \end{matrix} ) .

Note that the elements s and c normalize (Γ × Γ, γ ⊗ γ). Let us write down the Iwahori decompositions of the J_{i}’s to visualize them :

J_{0} = i^{−}(𝔍_{+}^{1}) (J ∩ M) i^{+}(ℌ_{−}^{1}) , J_{1} = i^{−}(ℌ_{+}^{1}) (J ∩ M) i^{+}(𝔍_{−}^{1}) ,
J_{2} = i^{−}(𝔍_{−}^{1}) (J ∩ M) i^{+}(ℌ_{+}^{1}) , J_{3} = i^{−}(ℌ_{−}^{1}) (J ∩ M) i^{+}(𝔍_{+}^{1}) ,
J_{4} = i^{−}(𝔍_{−}^{1}ϖ_{E}) (J ∩ M) i^{+}(ϖ_{E}^{−1}ℌ_{+}^{1}).

The process in Proposition I.2, applied to the conjugates J_{i}^{S} of Corollary II.2, provides us with a corresponding sequence of decomposed pairs (\overline{J_{i}^{S}}, λ̂_{i}^{S}) in Ḡ, that we will denote by (Ω_{i}, ω_{i}) to simplify notations; namely:

(Ω_{i}, ω_{i})_{0⩽i⩽4} with Ω_{i} = J_{i}^{S} ∩ Ḡ, Ω_{i} ∩ M̄ = i(Γ) and (ω_{i})_{|Ω_{i}∩M̄} = i(γ).

We have: J_{4}^{S} = ( \begin{matrix}ϖ_{E}^{−1} & 0 \\ 0 & σ^{−1}ϖ_{E}σ \end{matrix} ) J_{0}^{S} ( \begin{matrix}ϖ_{E} & 0 \\ 0 & σ^{−1}ϖ_{E}^{−1}σ \end{matrix} ) with

( \begin{matrix}ϖ_{E}^{−1} & 0 \\ 0 & σ^{−1}ϖ_{E}σ \end{matrix} ) = ( \begin{matrix}ϖ_{E}^{−1} & 0 \\ 0 & ^{τ}ϖ_{E} \end{matrix} ) ( \begin{matrix}I & 0 \\ 0 & ^{τ}ϖ_{E}^{−1}σ^{−1}ϖ_{E}σ \end{matrix} )

and we know from Proposition II.2 that ^{τ}ϖ_{E}^{−1}σ^{−1}ϖ_{E}σ belongs to σ^{−1}Γσ = ^{τ} Γ. Hence Ω_{4} is equal to z^{−1}Ω_{0}z where z = ( \begin{matrix}ϖ_{E} & 0 \\ 0 & ^{τ}ϖ_{E}^{−1} \end{matrix} ) belongs to Ḡ. We can thus derive from the above (Ω_{i}, ω_{i})_{0⩽i⩽4} an infinite sequence of decomposed pairs in Ḡ through :

Ω_{i+4j} = z^{−j}Ω_{i}z^{j}, i ∈ {0, 1, 2, 3}, j ∈ ℕ.

Theorem. The pairs (Ω_{i}, ω_{i}), i ∈ ℕ, are Ḡ-covers of the pair (i(Γ), i(γ)).

Remark. Let (Γ′, γ′) be another maximal simple type attached to the inertial class of π. From [BK1], (6.2.4), we can find a ∈ GL_{N}(F) such that Γ′ = Γ^{a}, γ′ = γ^{a}. The conjugates of the subgroups in Corollary II.2 by the element A = i(a) in Sp_{2N}(F) satisfy analogous properties with respect to (Γ′, γ′). The process in Proposition I.2 then gives us decomposed pairs above (i(Γ′), i(γ′)) which are Ḡ-covers of (i(Γ′), i(γ′)), as A-conjugates of the above. Hence Theorem III.1 actually provides us with a construction of a Ḡ-cover of (i(Γ), i(γ)) whether or not 𝔄_{0} is standard.

The proof of this theorem will occupy the remainder of this paper; it is organised as follows. From the properties recalled in §II.1 we know that the sequences Ω_{i} ∩ Ū and Ω_{i} ∩ Ū^{−}, i ∈ ℕ, are respectively increasing and decreasing, with ⋃_{i∈ℕ} Ω_{i} = Ū. We can then use [BB1], Theorem I.3.4: to show that the sequence of decomposed pairs (Ω_{i}, ω_{i}) is actually a sequence of covers, it is enough to show that each couple of consecutive pairs ((Ω_{i}, ω_{i}), (Ω_{i+1}, ω_{i+1})), i ∈ ℕ, satisfies one of three criteria. In the present paragraph we will prove a convenient periodicity lemma, allowing us to reduce this checking of criteria to the cases i = 0 to 3. For i = 0 or 2, criterion 1 in loc. cit. is satisfied (III.2). For i = 1 or 3, criterion 2 is used, but the proof in III.3 takes for granted an intertwining property, property (⋆). Part IV is then devoted to proving property (⋆), or rather Proposition IV.1 which implies the former; for this we will need Theorem II.3, i.e. the stability of the underlying field extension under the involution T.

Since an appropriate power of ϖ_{E} belongs to ϖ_{F} Γ, the sequence (Ω_{i}, ω_{i}) is periodic in the sense of [Bl2], Lemma 1, with period 4e(E/F). Since we would rather restrict the checking of criteria to the smallest possible number of cases, we have to generalize this lemma to the case of our element z, which does not lie in the center of M̄. Note that although we state the periodicity lemma below in our present context, it actually holds in the more general situation of [Bl2], Lemma 1.

Lemma. Let z be an element of M̄ which normalizes Ω_{i} ∩ M̄ = i(Γ) and such that γ and ^{z}γ : x ↦ γ(z^{−1}xz), are equivalent representations of Γ. Let i, k ∈ ℕ such that Ω_{i+k} = z^{−1}Ω_{i}z and Ω_{i+k+1} = z^{−1}Ω_{i+1}z. Let (τ, W) be a smooth representation of Ḡ and define, for w ∈ W^{ω_{i}} :

𝔑_{i}^{τ} (w) = ∫_{Ω_{i}∩Ū^{−}} τ(y) dy ∫_{Ω_{i+1}∩Ū} τ(n)w dn.

We have 𝔑_{i+k}^{τ} = τ(z^{−1}) 𝔑_{i}^{τ} τ(z), hence 𝔑_{i+k}^{τ} is injective on W^{ω_{i+k}} if and only if 𝔑_{i}^{τ} is injective on W^{ω_{i}}.

Proof. Since z belongs to M̄ we have : Ω_{i+k} ∩ Ū^{−} = z^{−1}(Ω_{i} ∩ Ū^{−})z and Ω_{i+k+1}∩Ū = z^{−1}(Ω_{i+1}∩Ū)z. From a change of variables (y, n) ↦ (z^{−1}yz, z^{−1}nz) in the integral defining 𝔑_{i+k}^{τ}, we get 𝔑_{i+k}^{τ} = τ(z^{−1}) 𝔑_{i}^{τ} τ(z) (indeed the moduli of the action of z on Ū and Ū^{−} are mutually inverse). The consequence on injectivity relies on the equality τ(z)W^{ω_{i+k}} = W^{ω_{i}}, due to the fact that, since γ and ^{z}γ are equivalent, the representations ω_{i} and ^{z}ω_{i+k} of Ω_{i} are equivalent. □

Since our present element z = ( \begin{matrix}ϖ_{E} & 0 \\ 0 & ^{τ}ϖ_{E}^{−1} \end{matrix} ) satisfies the assumptions in the lemma, showing injectivity of the operators 𝔑_{i}^{τ} for i ∈ ℕ amounts to showing it for i = 0 to 3. In other words (see [Bl2], Proposition 1), we only need to check the criteria in [BB1], Theorem I.3.4, for i = 0 to 3.

III.2 - Injectivity of 𝔑_{i}^{τ} for i = 0 or 2

We start with the case i = 2 and will prove that criterion 1 is satisfied, namely: for any y in Ω_{3} ∩ Ū, y ∉ Ω_{2}, there is a closed subgroup X of Ω_{2} ∩ Ū^{−} such

that y^{−1}Xy is contained in Ω_{2} and has no non-zero fixed vectors under ω_{2}.

We use the groups H^{1} = H^{1}(β, 𝔄) and J^{1} = J^{1}(β, 𝔄) from §II. From our definitions of J_{2} and J_{3}, both groups contain H^{1} and the restrictions of λ_{2} and λ_{3} to H^{1} are a multiple of the simple character θ which satisfies ([BK1], (7.2.3)):

Fact. For x in J^{1} ∩ U^{−} and y in J^{1} ∩ U, the commutator [x, y] = xyx^{−1}y^{−1} belongs to H^{1} and the map:

\begin{matrix}(J^{1} ∩ U^{−}/H^{1} ∩ U^{−}) × (J^{1} ∩ U/H^{1} ∩ U) & ⟶ ℂ^{×} \\ (x, y) & ⟼ θ ([x, y]) \end{matrix}

is a perfect duality between those two groups.

We have by definition J_{2} ∩U^{−} = J^{1} ∩U^{−}, J_{3} ∩U^{−} = H^{1} ∩U^{−}, J_{3} ∩U = J^{1} ∩U and J_{2} ∩ U = H^{1} ∩ U. Conjugating by S then gives us a perfect duality:

\begin{matrix}(J_{2}^{S} ∩ U^{−}/J_{3}^{S} ∩ U^{−}) × (J_{3}^{S} ∩ U/J_{2}^{S} ∩ U) & ⟶ ℂ^{×} \\ (x, y) & ⟼ θ^{S} ([x, y]) . \end{matrix}

Corollary II.2 states that the involution X ↦ ^{T}X^{−1} on G preserves J_{2}^{S}, J_{3}^{S} and the above duality given by θ^{S}. From Stevens’s remark in [St1, §4], we conclude that by restriction to Sp_{2N}(F) we still have a perfect duality:

\begin{matrix}(Ω_{2} ∩ Ū^{−}/Ω_{3} ∩ Ū^{−}) × (Ω_{3} ∩ Ū/Ω_{2} ∩ Ū) & ⟶ ℂ^{×} \\ (x, y) & ⟼ θ^{S} ([x, y]) . \end{matrix}

Now for x in Ω_{2} ∩ Ū^{−} and y in Ω_{3} ∩ Ū, the commutator [x, y] belongs to (H^{1} ∩ Ū^{−}) i(g) (H^{1} ∩ Ū) for some g in H^{1}(β, 𝔄_{0}), and we have θ^{S} ([x, y]) = θ_{0}(g)^{2}. On the other hand ω_{2}([x, y]) is a multiple of θ_{0}(g). Since H^{1} is a pgroup with p odd, the perfect duality above implies that, if y ∉ Ω_{2}, the subgroup y^{−1} (Ω_{2} ∩ Ū^{−}) y acts in ω_{2} through a non-trivial character, q.e.d.

The case i = 0 is entirely similar : indeed J_{1} = sJ_{2}s^{−1} and J_{0} = sJ_{3}s^{−1}.

III.3 - Injectivity of 𝔑_{i}^{τ} for i = 1 or 3

Those steps are more involved that the previous ones – indeed, in cases when 𝔍_{±}^{1} = ℌ_{±}^{1}, we have Ω_{2} = Ω_{3}. We start with i = 1 and want to show that criterion 2 is satisfied, i.e.:

there is a compact subgroup K̄ of Ḡ, containing Ω_{1}, such that the Hecke algebra H(K̄, Ω_{1}, ω_{1}) is supported on Ω_{1} ∪ Ω_{1}tΩ_{1} for some t in K̄ satisfying:

t^{−1}i(Γ)t = i(Γ), t^{−1} (Ω_{1} ∩ Ū) t = Ω_{2} ∩ Ū^{−}, t^{−1} (Ω_{1} ∩ Ū^{−}) t = Ω_{2} ∩ Ū.

We certainly have Ω_{1} = t Ω_{2}t^{−1} with

t = ( \begin{matrix}0 & σ \\ −^{τ}σ^{−1} & 0 \end{matrix} ) = S^{−1} ( \begin{matrix}0 & I \\ −σ^{τ}σ^{−1} & 0 \end{matrix} ) S (t ∈ Ḡ).

Note that ( \begin{matrix}0 & I \\ −σ^{τ}σ^{−1} & 0 \end{matrix} ) belongs to the coset s (Γ × Γ) from Proposition II.2(a);

hence t normalizes (Ω_{2} ∩ M̄, i(γ)) and intertwines ω_{2}.

Since J_{1} = sJ_{2}s^{−1} and J_{2}, s and S are contained in K = GL_{2N}(𝔬) (recall 𝔄_{0} is standard), the subgroup generated by Ω_{1} and Ω_{2} is contained in the maximal compact subgroup K̄ = Sp_{2N}(𝔬). Note that working with Ω_{2} or Ω_{1} here amounts to the same since the element t^{2} belongs to i(Γ). The support of H(Ḡ, Ω_{2}, ω_{2}) is the Ḡ-intertwining of ω_{2}, criterion 2 hence amounts to showing :

Proposition. We have : IḠ(ω_{2}) ∩ K̄ ⊂ Ω_{2} ∪ Ω_{2} t Ω_{2}.

Proof. From Proposition I.2 we know that IḠ(ω_{2}) is contained in IḠ (\overline{λ_{2}^{S}}),

itself contained in IḠ (\overline{θ^{S}}), since the restriction of λ_{2}^{S} to H′ = S^{−1}H^{1}S is a

multiple of θ^{S}.

We must now make an essential use of Shaun Stevens’s results in [St1]. Indeed the character θ^{S} of H′ is fixed under the involution x ↦ ^{T}x^{−1}. From [St1], Theorem 6.3, it follows that θ^{S} can be viewed as a simple character attached to a skew simple stratum, hence satisfies the properties shown in [St1], §3. In particular we have by [St1], Theorem 3.7:

IḠ (\overline{θ^{S}}) = I_{G}(θ^{S}) ∩ Ḡ.

We thus have the following information on the support we are looking for :

IḠ(ω_{2}) ∩ K̄ ⊂ I_{G}(θ^{S}) ∩ K̄.

From [BK1], (5.1.1) and (5.5.11), the intertwining of θ is equal to JB^{×}J = JW̃J, where W̃ is the affine Weyl group of B^{×} relative to the basis given in loc. cit. Assume for a moment that [E : F] = N. Since B^{×} is isomorphic to GL_{i}(E) with i[E : F] = 2N, we are considering in this case the affine Weyl group of GL_{2}(E), whose intersection with a maximal compact subgroup has at most two elements, so I_{G}(θ) ∩ K consists of the two double classes J and JsJ and we get:

(⋆)  IḠ(ω_{2}) ∩ K̄ ⊂ (J^{S} ∪ J^{S}tJ^{S}) ∩ K̄.

We now drop our assumption on [E : F] and get on with our proof assuming that (⋆) holds.

Since J = i^{−}(𝔍_{−}^{1}) (J ∩ M) i^{+}(𝔍_{+}^{1}) = (J_{2} ∩ U^{−})(J_{2} ∩ M)(J_{3} ∩ U) = J_{2}J_{3}, the group Ω = J^{S} ∩Ḡ satisfies Ω = (Ω_{2}∩Ū^{−})(Ω_{2}∩M̄)(Ω_{3}∩Ū) = Ω_{2}Ω_{3} = Ω_{2}(Ω_{3}∩Ū). Furthermore we have:

Lemma. J^{S} t J^{S} ∩ Ḡ = Ω t Ω.

Proof. Since t(J^{S} ∩ U^{−})t^{−1} is contained in J^{S} ∩ U, we have

J^{S} t J^{S} = (J^{S} ∩ U) t (J^{S} ∩ M)(J^{S} ∩ U).

We apply [St2], Theorem 2.3, to the automorphism x ↦ ^{T}x^{−1} of G, the pro-p-subgroup J^{S} ∩ U, and the subgroup H = (J^{S} ∩ M) ∪ t (J^{S} ∩ M). Condition (2.1) in [St2] is easily checked, hence

J^{S} t J^{S} ∩ Ḡ = (Ω ∩ Ū) t (Ω ∩ M̄)(Ω ∩ Ū) = Ω t Ω. □

At this point we know: IḠ(ω_{2}) ∩ K̄ ⊂ Ω ∪ Ω t Ω. Now note that the above argument can be applied in exactly the same way to the representation ω_{3} of Ω_{3}; indeed J_{3}^{S} also contains H′. We thus get IḠ(ω_{3}) ∩ K̄ ⊂ Ω ∪ Ω t Ω, with the pleasant feature that Ω t Ω = Ω_{3} t Ω_{3} since Ω ∩ Ū = Ω_{3} ∩ Ū.

We now use [BK1], (4.1.5) : for i = 2, 3, the dimension of the subspace of the Hecke algebra H(Ḡ, Ω_{i}, ω_{i}) supported on Ω (resp. Ω t Ω) is equal to the dimension of the subspace of the Hecke algebra H(Ḡ, Ω, Ind_{Ω_{i}}^{Ω}ω_{i}) supported on Ω (resp. Ω t Ω). But the argument in §III.2 shows that the induced representation of ω_{i} to Ω is irreducible, hence the first dimension – dimension of the subspaces supported on Ω – is equal to 1. Furthermore the two induced representations Ind_{Ω_{i}}^{Ω}ω_{i}, i = 2, 3, are isomorphic (use for instance Mackey restriction formula, plus the irreducibility and the fact that the representations ω_{2} and ω_{3} coincide on Ω_{2} ∩ Ω_{3}), so the second dimension – dimension of the subspaces supported on Ω t Ω – is the same for i = 2 and i = 3. For i = 3 it is equal to 1, because any intertwining between ω_{3} and ω_{3}^{t} must intertwine the irreducible representation i(γ), then it is also equal to 1 for i = 2, and since t does intertwine ω_{2} we get the required property. □

The last case left, i = 3, is dealt with exactly in the same manner, after observing that Ω_{4} = q Ω_{3}q^{−1} with :

q = ( \begin{matrix}0 & −^{τ}σ^{τ}ϖ_{E}^{−1} \\ σ^{−1}ϖ_{E} & 0 \end{matrix} ) = S^{−1} ( \begin{matrix}−^{τ}σ^{τ}ϖ_{E}^{−1}σ^{−1}ϖ_{E} & 0 \\ 0 & I \end{matrix} ) wS, w = ( \begin{matrix}0 & ϖ_{E}^{−1} \\ ϖ_{E} & 0 \end{matrix} ) .

Again we have J_{4} = wJ_{3}w^{−1}. Furthermore, let y = ( \begin{matrix}0 & I \\ ϖ_{E} & 0 \end{matrix} ); then:

yJ_{1}y^{−1} = J_{4}, yJ_{2}y^{−1} = J_{3}, yJ_{3}y^{−1} = J_{2} and ysy^{−1} = w.

So we can repeat the previous argument, and we obtain a formula analogous to (⋆), where we replace K by (K^{y})^{S}, ω_{2} by ω_{3}, t by q. The lemma becomes J^{S} q J^{S} ∩ Ḡ = Ω q Ω, with the same proof except that the roles of U and U^{−} are exchanged. Since Ω is contained in (K^{y})^{S}, the last part of the argument also follows through after exchanging the roles of ω_{2} and ω_{3}. The proof of Theorem III.1 is now complete, provided we prove (⋆) in part IV below.

Remark. Along the lines of the above proof we might get on to show that the Hecke algebra of the pair (Ω_{2}, ω_{2}) for instance, in Ḡ, is the algebra with generators T_{t} and T_{q} (elements with support the double coset respectively of t and q) and relations the quadratic relations satisfied by T_{t} and T_{q} (they belong to a two-dimensional subalgebra). We cannot expect though that these quadratic relations be the same as the relations satisfied by T_{s} and T_{w} in the Hecke algebra of the pair (J_{2}, λ_{2}) in G – and it is not to be expected either ! For instance, for N = 2, the Ḡ-covers constructed in [BB1] are, in the self-contragredient case, instances of the above (Ω_{2}, ω_{2}). The corresponding Hecke algebras are described in [BB2]; one can check there that the quadratic relations are, in a number of cases, different from the ones in GL_{4}(F).

IV - Glauberman’s correspondence and intertwining

We must in this last part complete the proof of Theorem III.1, that is, establish the property (⋆) in III.3, as well as the analogous property needed in the last case (i = 3) in III.3. We will first show that these properties follow from a bound on intertwining, namely proposition IV.1 (compare [BK1], Proposition 5.5.11). To prove this proposition, we will detail in IV.2 properties of the representations involved and use the argument in [BK1], Proposition 5.3.2, to reduce the proof to a very precise intertwining assertion: Proposition IV.3. At last we will establish that assertion using Glauberman’s correspondence together with arguments from [BK1], §5.1 and 5.2.

IV.1 - Intertwining and Weyl group

We must now use the full content of Theorem II.3, so we change notations in this last part, both to simplify them and to stick to the notations in [BK1]. We call (J, λ), θ, and so on, what we previously called (J^{S}, λ^{S}), θ^{S} and so on (Corollary II.2), call β the element in M_{N}(F) previously called α_{0} and call β̃ the element in M_{2N}(F) previously called α in Theorem II.3, i.e. β̃ = ( \begin{matrix}β & 0 \\ 0 & −^{τ}β \end{matrix} ).

We know that (J, λ) = (J(β̃, 𝔄), λ(β̃, 𝔄)) is a simple type attached to the simple stratum [𝔄, 2n_{0}, 0, β̃] and the simple character θ ∈ C(𝔄, 0, β̃). Recall that J, λ and θ are stable under the involution X ↦ ^{T}X^{−1}. We will abbreviate H^{1} = H^{1}(β̃, 𝔄) and the same for, e.g., J^{1}, when there is no risk of confusion.

We let E be the field F [β] in M_{N}(F) and B_{0} be its commutant; we still call E the field embedding F [β̃] in M_{2N}(F) and call B its commutant. The crucial fact is that the embedding F [β̃] of E in M_{2N}(F) is stable under the involution T and B̄^{×} is a unitary group (§II.3).

Let W be the affine Weyl group of B̄^{×} relative to the subgroup of diagonal matrices. We have B̄^{×} = Ū(𝔅)W Ū(𝔅), since Ū(𝔅) contains a standard Iwahori subgroup of B̄^{×}. As in [BK1] §5.5, we let 𝔐(𝔅)^{×} be the intersection with M of 𝔅^{×}.

Let I(λ) be the representation of J̄ defined by I(λ) = IndJ̄ ; J̄_{P} λ̂_{P} (notation defined in I.2). We already know (III.3) that its intertwining is contained in \overline{JB^{×}J} = \overline{J^{1}B^{×}J^{1}} (recall J = U(𝔅)J^{1}). Since B is now stable under T we can use fully [St1], Theorem 3.7, to get: \overline{J^{1}B^{×}J^{1}} = J̄^{1}B̄^{×}J̄^{1}. We will prove in the next paragraphs the following proposition:

Proposition. The intertwining of I(λ) is contained in J̄^{1}NB̄^{×}(𝔐(𝔅)^{×})J̄^{1}, equal to J̄N_{W} (𝔐(𝔅)^{×})J̄.

To derive property (⋆) we need only note that the normalizer of 𝔐(𝔅)^{×} in W is equal to (Ū(𝔅) ∩ W) W_{2} (Ū(𝔅) ∩ W) with

W_{2} = \{( \begin{matrix}ϖ_{E}^{i} & 0 \\ 0 & ^{τ}ϖ_{E}^{−i} \end{matrix} ) , i ∈ ℤ\} ∪ \{( \begin{matrix}0 & ϖ_{E}^{i}σ \\ −^{τ}ϖ_{E}^{−i} ^{τ}σ^{−1} & 0 \end{matrix} ) , i ∈ ℤ\} ,

where σ is an element in U(𝔄_{0}) satisfying the property in Corollary II.3 (or σ = I in the level zero case).

The intersection of W_{2} with any compact subgroup has at most two elements. It follows that the intersection with any compact subgroup of the intertwining of I(λ) contains at most two J̄-double classes, q.e.d.

IV.2 - A one-dimensional intertwining space

To prove the above proposition we have to collect informations on the representation I(λ). We need more notation. We let J_{−} = J ∩ U^{−} = J^{1} ∩ U^{−}, J_{+} = J ∩U = J^{1} ∩U, J_{M} = J ∩M, J_{M}^{1} = J^{1} ∩M, H_{−} = H^{1} ∩U^{−}, H_{+} = H^{1} ∩U, H_{M}^{1} = H^{1} ∩ M, and so on. We define the auxiliary subgroups J_{P}^{1} = H_{−}J_{M}^{1}J_{+} and K = H_{−}H_{M}^{1}J_{+}. We will move around the following diagram (where arrows mean inclusion):

J̄ =J̄_{−}J̄_{M} J̄_{+}  I(κ)
induction ↗ ↖ extension
{ \atop i(κ_{0})}  J̄_{P} = H̄_{−}J̄_{M} J̄_{+}  J̄^{1} = J̄_{−}J̄_{M}^{1} J̄_{+}  I(η)
extension ↖ ↗ induction
J̄_{P}^{1} =H̄_{−}J̄_{M}^{1} J̄_{+}  η̂_{P} = { \atop i(η_{0})}

↑ Heisenberg

K̄ =H̄_{−}H̄_{M}^{1} J̄_{+}  { \atop i(θ_{0})}

↑ extension

H̄^{1} =H̄_{−}H̄_{M}^{1} H̄_{+}  { \atop i(θ_{0})}

Here, for any subgroup N of Ḡ admitting an Iwahori decomposition with respect to (M, P), we denote by ĩ(μ) the representation of N trivial on N ∩ U^{−} and N ∩ U and with restriction i(μ) to N ∩ M, whenever it makes sense. We define I(κ) = IndJ̄ ; J̄_{P} { \atop i(κ_{0})} and I(η) = IndJ̄^{1} ; J̄_{P}^{1} { \atop i(η_{0})}. We let η_{P} be the representation η_{|J_{P}^{1}} of J_{P}^{1} in the space of J̄_{+}-fixed vectors in η, as in [BK1] §7.2. By definition of

η̂_{P} (proposition I.2) we have { \atop i(η_{0})} = η̂_{P}. Furthermore, since the representation { \atop i(σ_{0})} of J̄ is trivial on J̄^{1}, we have:

λ̂_{P} = { \atop i(κ_{0} ⊗ σ_{0})} = { \atop i(κ_{0})} ⊗ { \atop i(σ_{0})}

hence I(λ) = IndJ̄ ; J̄_{P} λ̂_{P} ≃ (IndJ̄ ; J̄_{P} { \atop i(κ_{0})}) ⊗ { \atop i(σ_{0})} = I(κ) ⊗ { \atop i(σ_{0})}.

We need the following properties:

Proposition.

(i) The representations I(κ) and I(η) are irreducible. (ii) The restriction of I(κ) to J̄^{1} is isomorphic to I(η). (iii) The representation I(η) is the Heisenberg representation above (H̄^{1}, ĩ(θ_{0})). (iv) The intertwining of I(η) is equal to J̄^{1}B̄^{×}J̄^{1} and for any g in J̄^{1}B̄^{×}J̄^{1} the

dimension of the intertwining space I_{g}(I(η), J̄_{1}) is equal to 1.

Proof. The irreducibility of I(η) is a consequence of the fact in III.2 and (ii) follows from Frobenius reciprocity. For (iii) the argument is in [St1, §4].

(iv) is more intricate but the proof is entirely in [BK1]. We recall the main points. First of all we already know that the intertwining is contained in J̄^{1}B̄^{×}J̄^{1} so we may assume that g belongs to B̄^{×}. Since I(η) is a Heisenberg representation, the argument in [BK1], 5.1.8, 5.1.9, reduces us to proving that

[J̄^{1} : J̄^{1} ∩ (J̄^{1})^{g}] = [H̄^{1} : H̄^{1} ∩ (H̄^{1})^{g}]

namely lemma 5.1.10 in [BK1], but for Ḡ instead of G. Using the Cayley transform x ↦ (1 + x/2)(1 − x/2)^{−1}, which is defined on (J^{1})^{−} = {X ∈ J^{1}/^{T}X = −X} and establishes bijections between (J^{1})^{−} and J̄^{1}, (H^{1})^{−} and H̄^{1}, and so on (see [St3]), we replace the equality to be proved by

[(J^{1})^{−} : (J^{1})^{−} ∩ ((J^{1})^{−})^{g}] = [(H^{1})^{−} : (H^{1})^{−} ∩ ((H^{1})^{−})^{g}].

(the Cayley transform on 𝔓 is easily seen to preserve subgroup indices).

Now the proof of loc. cit. applies mutatis mutandis: all exact sequences there remain exact after replacing each lattice involved, say Z , by Z^{−} = {X ∈ Z/^{T}X = −X}. Indeed, since g belongs to B^{×} and satisfies g = ^{T}g^{−1}, all the lattices involved are T -invariant; furthermore, the map a_{β} is easily seen to commute with the involution T and from [St3], Lemma 2.1.1, we may (and must here) choose a corestriction s that also commutes with T. □

We are now in a position to work out the intertwining of I(λ). Let g belong to W and intertwine I(λ); we have to show that g normalizes 𝔐(𝔅)^{×}. Since I(λ) is isomorphic to I(κ) ⊗ { \atop i(σ_{0})} and the following two facts hold:

– { \atop i(σ_{0})} is trivial on J̄^{1};

– dim I_{g}(I(η), J̄_{1}) = 1; we can imitate the proof of [BK1], Proposition 5.3.2, to get that any non-zero intertwining operator in I_{g}(I(λ), J̄) has the form S ⊗ R with S ∈ I_{g}(I(η), J̄_{1}) and R an endomorphism in the space of { \atop i(σ_{0})}.

Let us use Proposition IV.3 below: for any T -stable minimal 𝔬_{E}-order 𝔅_{m} contained in 𝔅, the operator S also intertwines the restriction of I(κ) to the subgroup Ū^{1}(𝔅_{m})J̄^{1} (use one-dimensionality for I(η)). Again as in loc. cit., this implies that R belongs to I_{g}({ \atop i(σ_{0})}, Ū^{1}(𝔅_{m})J̄^{1}).

Proposition IV.1 now follows from:

Lemma. Let g ∈ W intertwine the restriction of { \atop i(σ_{0})} to Ū^{1}(𝔅_{m})J̄^{1} for any T -stable minimal 𝔬_{E}-order 𝔅_{m} contained in 𝔅. Then g normalizes the group 𝔐(𝔅)^{×}.

Proof. Indeed we almost recognize [BK1], Proposition 5.5.5, that again we will imitate. The sequence of lemmas there holds unchanged, so we assume g does not normalize 𝔐(𝔅)^{×} and produce an hereditary order 𝔅′_{0}, with radical 𝔔′_{0}, and a parabolic subgroup Q of GL_{N}(F), with unipotent radical N = 1 + 𝔑, such that:

(i) 𝔅′_{0} ⊂ 𝔅_{0}; 𝔅′_{0} ∩ 𝔑 = 𝔔′_{0} ∩ 𝔑 ⊈ 𝔔_{0}; g^{−1}(𝔔′_{0} ∩ 𝔑) g ⊂ 𝔔_{0}; (ii) the image of 1 + 𝔔′_{0} ∩ 𝔑 in U(𝔅_{0})/U^{1}(𝔅_{0}) is the unipotent radical of a

proper parabolic subgroup of U(𝔅_{0})/U^{1}(𝔅_{0}). Indeed, in the notations of loc. cit., 𝔅′_{0} is 𝔅̄, contained in some B^{i}, and one

can decompose V^{i} into a direct sum V^{i} = W^{1} ⊕W^{2} of E- vector spaces such that L̄_{1} ∩ W^{1} = L̄_{0} ∩ W^{1} and L̄_{1} ∩ W^{2} = ϖ_{E}L̄_{0} ∩ W^{2}; then 𝔑 = Hom_{E}(W^{2}, W^{1}) ⊂ End_{E}V^{i} satisfies the assumptions. Furthermore, since g belongs to B̄^{×}, we can as well assume here that B^{i} = B^{1}.

We put 𝔅′ = 𝔅′_{0} ⊕ ^{τ}𝔅′_{0}; hence (𝔅′)^{×} is contained in M and equal to U(𝔅′_{0})× τU(𝔅′_{0}). We put 𝔔′ = 𝔔′_{0} ⊕ ^{τ}𝔔′_{0}. Since g belongs to B̄^{×} we still have

g^{−1} (1 + [𝔔′_{0} ∩ 𝔑 ⊕ ^{τ}(𝔔′_{0} ∩ 𝔑)]) g ⊂ 1 + 𝔔 and 1 + 𝔔′ ⊂ U(𝔅).

We now pick a T -stable minimal 𝔬_{E}-order 𝔅_{m} contained in 𝔅′ + 𝔔. Then 1 + 𝔔′ ⊂ 1 + 𝔔_{m} = U^{1}(𝔅_{m}).

Assume then that g does intertwine the restriction of { \atop i(σ_{0})} to Ū^{1}(𝔅_{m})J̄^{1}; then there is a non-zero operator R in the space of the representation such that:

∀x ∈ (Ū^{1}(𝔅_{m})J̄^{1}) ∩ (Ū^{1}(𝔅_{m})J̄^{1})^{g} R ∘ { \atop i(σ_{0})}(x) = { \atop i(σ_{0})}(gxg^{−1}) ∘ R.

This relation holds in particular for x = g^{−1}i(y)g with y ∈ 1 + 𝔔′_{0} ∩ 𝔑, because x belongs to J̄^{1} and i(y) belongs to Ū^{1}(𝔅_{m}). We get R = σ_{0}(y) ∘ R, which, with (ii), contradicts the cuspidality of σ_{0}. □

Remark. The above lemma itself is the full proof of Proposition IV.1 in the case of level 0 representations.

IV.3 - Glauberman’s correspondence

This last paragraph will be devoted to the proof of the proposition below.

Proposition. Let 𝔅_{m} be a T -stable minimal 𝔬_{E}-order in B contained in 𝔅. The restriction of I(κ) to Ū^{1}(𝔅_{m})J̄^{1} has the same intertwining as I(η).

The property we want to prove is invariant under conjugation by 𝔅̄^{×}; hence we may assume (see [BK1], 1.1.9 and 7.1.15) that 𝔅_{m} = (𝔅_{0,m} ⊕ ^{τ}𝔅_{0,m}) + 𝔔 where 𝔅_{0,m} is a minimal 𝔬_{E}-order in B_{0} contained in 𝔅_{0}. We then have: Ū^{1}(𝔅_{m}) = i (U^{1}(𝔅_{0,m})) Ū^{1}(𝔅) and

Ū^{1}(𝔅_{m})J̄^{1} = i (U^{1}(𝔅_{0,m})) J̄^{1} = J̄_{−} [i (U^{1}(𝔅_{0,m})) J̄_{M}^{1}] J̄^{+}.

Looking at the diagram and proposition in IV.2, we find that

I(κ)_{|Ū^{1}(𝔅_{m})J̄^{1}} ≃ Ind_{i(U^{1}(𝔅_{0,m}))J̄_{P}^{1}}^{i(U^{1}(𝔅_{0,m}))J̄^{1}} { \atop i(κ′_{0})} where κ′_{0} = κ_{0}_{|U^{1}(𝔅_{0,m})J^{1}(β,𝔄_{0})}.

It is enough to show that the representations { \atop i(κ′_{0})} and { \atop i(η_{0})} have the same intertwining. Indeed, by [BK1], 4.1.5, we have: IḠ (I(κ)_{|Ū^{1}(𝔅_{m})J̄^{1}}) = J̄^{1}IḠ \binom{}{i(κ′_{0})} J̄^{1}

and IḠ (I(η)) = J̄^{1}IḠ \binom{}{i(η_{0})} J̄^{1}.

Glauberman’s correspondence is the tool we need here. We recall briefly what it is in our setting; more general and precise statements can be found in [St1], §2, or [BH2], §A2, as well as the original references.

Let ϵ be the involution x ↦ ^{T}x^{−1} on G, with fixed points Ḡ. For any open compact pro-p-subgroup H of G which is ϵ-stable, Glauberman’s correspondence gives us a unique bijection 𝔤: ρ ↦ 𝔤(ρ), between the set Irr (H)^{ϵ} of ϵ-stable equivalence classes of smooth irreducible representations of H and the set Irr (H̄) of equivalence classes of smooth irreducible representations of H̄, characterized by the property that 𝔤(ρ) occurs in ρ_{|H̄} with odd multiplicity.

Let K be a subgroup of H satisfying the same assumptions as H, let σ ∈ Irr (K)^{ϵ} and ρ ∈ Irr (H)^{ϵ}. Then ρ ≃ Ind_{K}^{H}σ implies 𝔤(ρ) ≃ IndH̄ ; K̄𝔤(σ), and ρ_{|K} ≃ σ implies 𝔤(ρ)_{|K̄} ≃ 𝔤(σ).

A crucial property of this correspondence is the following:

Fact. [Stevens [St1], Lemma 2.4] Let ρ ∈ Irr (H)^{ϵ}, g ∈ Ḡ. The dimension of I_{g}(ρ, H) is odd if and only if the dimension of I_{g}(𝔤(ρ), H̄) is odd.

We now proceed to find the inverse images under Glauberman’s correspondence of the representations { \atop i(κ′_{0})} and { \atop i(η_{0})}. We start a series of lemmas. The first one is valid for any simple stratum in M_{N}(F).

Lemma 1. (i) Let [𝔄, n, 0, β] be a simple stratum in M_{N}(F) and let α ∈ 𝔬_{F}^{×}. Then, for any defining sequence [𝔄, n, r_{i}, γ_{i}] for [𝔄, n, 0, β] (see [BK1], 2.4.2), the sequence [𝔄, n, r_{i}, αγ_{i}] is a defining sequence for [𝔄, n, 0, αβ]. In particular we have J^{k}(β, 𝔄) = J^{k}(αβ, 𝔄) and H^{k}(β, 𝔄) = H^{k}(αβ, 𝔄).

(ii) The map θ ↦ θ^{2} is a bijection from C(𝔄, m, \frac{1}{2}β) onto C(𝔄, m, β) (m ∈ ℕ), which is compatible with the canonical bijections of [BK1], §3.6. We will denote the inverse bijection by θ ↦ θ^{1/2}.

Proof. (i) is simple checking. Since the groups H^{m+1}(β, 𝔄) are p-groups with p odd, (ii) is easily checked by induction along a defining sequence for β. □

Let again θ be the simple character that underlies our simple type (J, λ) and let θ^{1/2} in C(𝔄, m, \frac{1}{2} β̃) be its inverse image under the square map. To the simple character θ^{1/2} we attach representations η^{1/2}, κ^{1/2}, and η_{P}^{1/2}, κ_{P}^{1/2} in the usual way of [BK1], §5 and 7. For instance, κ^{1/2} is a representation of J(\frac{1}{2} β̃, 𝔄) = J(β̃, 𝔄) which is a beta-extension of η^{1/2}; note that we have to choose one here, while η^{1/2} is completely determined by θ^{1/2}.

Lemma 2. We still write θ^{1/2} for the extension of θ^{1/2} to K trivial on J^{+}.

We have 𝔤(θ^{1/2}) ≃ ĩ(θ_{0}) (on K, K̄) and 𝔤(η_{P}^{1/2}) ≃ { \atop i(η_{0})} (on J_{P}^{1} , J̄_{P}^{1} ).

Proof. The character θ^{1/2} on K is trivial on J^{+} and H^{−}; on H_{M}^{1} it is given

by θ^{1/2} (( \begin{matrix}g_{1} & 0 \\ 0 & g_{2} \end{matrix} )) = θ_{0}^{1/2}(g_{1})θ_{0}^{1/2}(^{τ}g_{2}^{−1}) for g_{1}, ^{τ}g_{2} in H^{1}(β, 𝔄_{0}) ([BK1], 7.1.19,

and Corollary II.2). We thus have: θ^{1/2}(i(g)) = θ_{0}^{1/2}(g)θ_{0}^{1/2}(g) = θ_{0}(g), for g ∈ H^{1}(β, 𝔄_{0}), which proves the first assertion. The restriction of η_{P}^{1/2} to K is a multiple of θ^{1/2} so its restriction to K̄ is a multiple of { \atop i(θ_{0})}; unicity of Heisenberg representations says that the restriction of η_{P}^{1/2} to J̄_{P}^{1} is then a multiple of { \atop i(η_{0})}. □

Of course Glauberman’s correspondence does not apply to J(β̃, 𝔄) which is not a p-group, nor to J_{P}. But it does apply to the following group, intermediate between J_{P}^{1} and J_{P}:

\begin{aligned}L & = [U^{1}(𝔅_{0,m}) × ^{τ}U^{1}(𝔅_{0,m})] J_{P}^{1} \\ & = H_{−} [ (U^{1}(𝔅_{0,m})J^{1}(β, 𝔄_{0})) × ^{τ}(U^{1}(𝔅_{0,m})J^{1}(β, 𝔄_{0}))] J_{+}. \end{aligned}

The subgroup L is certainly stable under x ↦ ^{T}x^{−1}; let us check that the restriction to L of the representation κ_{P}^{1/2} is also stable by this involution, up to isomorphism. From [BK1], §7.2, κ_{P}^{1/2} is trivial on L_{−} and L_{+} and its restriction

to L ∩ M has the form κ_{P}^{1/2} (( \begin{matrix}g_{1} & 0 \\ 0 & g_{2} \end{matrix} )) ≃ κ_{1}(g_{1}) ⊗ κ_{2}(^{τ}g_{2}^{−1}) where κ_{1} and κ_{2} are

both beta-extensions of the Heisenberg representation η_{0}^{1/2} of J^{1}(β, 𝔄_{0}) attached to θ_{0}^{1/2}. Hence κ_{1} and κ_{2} differ from a character χ ∘ det_{B_{0}} where χ is a character of 𝔬_{E}^{×}/1 + 𝔭_{E} ([BK1], 5.2.2). This implies that κ_{1} and κ_{2} agree on U^{1}(𝔅_{0,m})J^{1} whence the stability of (κ_{P}^{1/2})_{|L} under ϵ.

Lemma 3. We have 𝔤((κ_{P}^{1/2})_{|L}) ≃ { \atop i(κ′_{0})} (on L, L̄).

Proof. Both representations have trivial restrictions to L̄_{−} and L̄_{+} and irreducible restrictions to L̄∩M. So what we have to show is: 𝔤((κ_{P}^{1/2})_{|L∩M}) ≃ i(κ′_{0}).

Let κ_{M}^{1/2} = (κ_{P}^{1/2})_{|L∩M}; this is a representation of

L ∩ M = (U^{1}(𝔅_{0,m})J^{1}(β, 𝔄_{0})) × ^{τ}(U^{1}(𝔅_{0,m})J^{1}(β, 𝔄_{0})) .

Denote by 𝔄_{0,m} the unique hereditary 𝔬_{F} -order in 𝔄_{0} stable under conjugation by E^{×} such that 𝔄_{0,m} ∩ B_{0} = 𝔅_{0,m}. Let θ_{0,m}^{1/2} be the image of θ_{0}^{1/2} under the canonical tranfer map: C(𝔄_{0}, 0, \frac{1}{2}β) → C(𝔄_{0,m} , 0, \frac{1}{2}β), and let η_{0,m}^{1/2} be the unique irreducible representation of J^{1}(β, 𝔄_{0,m}) containing θ_{0,m}^{1/2}. Let μ_{0}^{1/2} be the unique extension of η_{0}^{1/2} to L_{0} = U^{1}(𝔅_{0,m})J^{1}(β, 𝔄_{0}) satisfying

Ind_{L_{0}}^{U^{1}(𝔄_{0,m})}μ_{0}^{1/2} ≃ Ind_{J^{1}(β,𝔄_{0,m})}^{U^{1}(𝔄_{0,m})}η_{0,m}^{1/2}.

From [BK1], 5.2.6 and 5.1.15 (where μ_{0}^{1/2} is denoted by η̃, or in our context η̃_{0}^{1/2}), we have κ_{M}^{1/2} ≃ μ_{0}^{1/2} ⊗ μ_{0}^{1/2}^{∗}. The induced representations above are irreducible and loc. cit. implies:

Ind_{L∩M}^{U^{1}(𝔄_{0,m})×^{τ}U^{1}(𝔄_{0,m})}κ_{M}^{1/2} ≃ Ind_{L_{0}}^{U^{1}(𝔄_{0,m})}μ_{0}^{1/2} ⊗ [Ind_{L_{0}}^{U^{1}(𝔄_{0,m})}μ_{0}^{1/2}]^{∗}
≃ Ind_{J^{1}(β,𝔄_{0,m})}^{U^{1}(𝔄_{0,m})}η_{0,m}^{1/2} ⊗ [Ind_{J^{1}(β,𝔄_{0,m})}^{U^{1}(𝔄_{0,m})}η_{0,m}^{1/2}]^{∗}
≃ Ind_{J^{1}(β,𝔄_{0,m})×^{τ}J^{1}(β,𝔄_{0,m})}^{U^{1}(𝔄_{0,m})×^{τ}U^{1}(𝔄_{0,m})} η_{0,m}^{1/2} ⊗ [η_{0,m}^{1/2}]^{∗}

Since the representations involved are irreducible, one gets through Glauberman’s correspondence an isomorphism:

Ind_{i(L_{0})}^{i(U^{1}(𝔄_{0,m}))} 𝔤 (κ_{M}^{1/2}) ≃ Ind_{i(J^{1}(β,𝔄_{0,m}))}^{i(U^{1}(𝔄_{0,m}))} 𝔤 (η_{0,m}^{1/2} ⊗ [η_{0,m}^{1/2}]^{∗}) .

We already know that 𝔤(η_{P}^{1/2}) is isomorphic to { \atop i(η_{0})} and that i(κ′_{0}) extends i(η_{0}). Again from loc. cit., the representation i(κ′_{0}) is the unique irreducible representation of i(L_{0}) extending i(η_{0}) and satisfying:

Ind_{i(L_{0})}^{i(U^{1}(𝔄_{0,m}))} i(κ′_{0}) ≃ Ind_{i(J^{1}(β,𝔄_{0,m}))}^{i(U^{1}(𝔄_{0,m}))} i(η_{0,m}).

Hence it is enough to show that 𝔤 (η_{0,m}^{1/2} ⊗ [η_{0,m}^{1/2}]^{∗}) ≃ i(η_{0,m}). But the restriction

of η_{0,m}^{1/2} ⊗ [η_{0,m}^{1/2}]^{∗} to i (H^{1}(β, 𝔄_{0,m})) is a multiple of

i(g) ↦ θ_{0,m}^{1/2}(g) ⊗ [θ_{0,m}^{1/2}]^{∗}(^{τ}g^{−1}) = [θ_{0,m}^{1/2}(g)]^{2} = θ_{0,m}(g),

so its image is the Heisenberg representation above i (θ_{0,m}) namely i(η_{0,m}). □

End of proof of Proposition IV.3. The intertwining of { \atop i(η_{0})} contains the intertwining of { \atop i(κ′_{0})} since the second representation restricts to the first. Let us now take g in IḠ({ \atop i(η_{0})} and show that g belongs to IḠ({ \atop i(κ′_{0})}. The above fact about intertwining spaces and Glauberman’s correspondence, combined with the lemmas, gives us:

– g intertwines η_{P}^{1/2} (Fact and Lemma 2). Indeed dim I_{g}({ \atop i(η_{0})}) is equal to 1, from Proposition IV.2 and [BK1], 4.1.5.

– g intertwines κ_{P}^{1/2}. Indeed, from [BK1], §7.2, g intertwines η^{1/2} (induced from η_{P}^{1/2}), hence g intertwines κ^{1/2} (that has the same intertwining as η^{1/2}); furthermore J_{P}^{1} gJ_{P}^{1} is the unique J_{P}^{1} -double coset in J^{1}gJ^{1} that intertwines η_{P}^{1/2} ([BK1], 4.1.5 and 5.1.8). Similarly, in JgJ there is a unique J_{P} -double coset J_{P} bJ_{P} that intertwines κ_{P}^{1/2}. Since J = J_{P} J_{−} we may assume that b belongs to J_{−} g J_{−}, hence to J^{1}gJ^{1}. But then, since b also intertwines η_{P}^{1/2}, we must have b ∈ J_{P}^{1} gJ_{P}^{1} ; so b ∈ J_{P} gJ_{P} whence the result.

– the dimension of I_{g}(η_{P}^{1/2}) is equal to 1, so is the dimension of I_{g}((κ_{P}^{1/2})_{|L}) (again the second representation restricts to the first, by [BK1], §7.2). Hence g intertwines the image of this representation by the Glauberman correspondence, namely { \atop i(κ′_{0})} (Fact and Lemma 3). □

Remark. Of course proposition IV.3 says something about the restriction of I(κ) to a suitable subgroup being a β-extension of I(η). To prove Theorem III.1, we actually do not need to know whether or not I(κ) itself is a β-extension of I(η). It should follow from the study of the Hecke algebra of the Ḡ-cover that the bound on intertwining given by Proposition IV.1 is actually an equality, i.e. the intertwining of I(λ) is equal to J̄N_{W} (𝔐(𝔅)^{×})J̄ – see the remark at the end of III.3.

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C.N.R.S. - Théorie des Groupes - Case 7012, Institut de Mathématiques de Jussieu, Université Paris 7, F-75251 PARIS Cedex 05. blondel@math.jussieu.fr