On the realisation of maximal simple types and epsilon factors of pairs

Vytautas Paskunas and Shaun Stevens

February 2, 2006

Abstract

Let G be the group of rational points of a general linear group over a non-archimedean local field F. We show that certain representations of open, compact-mod-centre subgroups of G, (the maximal simple types of Bushnell and Kutzko) can be realized as concrete spaces. In the level zero case our result is essentially due to Gel′fand. This allows us, for a supercuspidal representation π of G, to compute a distinguished matrix coefficient of π. By integrating, we obtain an explicit Whittaker function for π. We use this to compute the εfactor of pairs, for supercuspidal representations π_{1}, π_{2} of G, when π_{1} and the contragredient of π_{2} differ only at the ‘tame level’ (more precisely, π_{1} and π̌_{2} contain the same simple character). We do this by computing both sides of the functional equation defining the epsilon factor, using the definition of Jacquet, Piatetskii-Shapiro, Shalika. We also investigate the behaviour of the ε-factor under twisting of π_{1} by tamely ramified quasi-characters. Our results generalise the special case π_{1} = π_{2} totally wildly ramified, due to Bushnell and Henniart.

Contents

Introduction 3

1 Notation and Preliminaries 7

1.1 Unipotent subgroups and characters . . . . . . . . . . . . . . . 7

1.2 Induced supercuspidals . . . . . . . . . . . . . . . . . . . . . . 8 2 A note on β-extensions 9

2.1 Iwahori decompositions . . . . . . . . . . . . . . . . . . . . . . 10 2.2 β-extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3 A particular unipotent subgroup 17 3.1 An F -basis of E . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.2 A choice of maximal flag . . . . . . . . . . . . . . . . . . . . . 20

4 Supercuspidal representations 25 4.1 Some decompositions . . . . . . . . . . . . . . . . . . . . . . . 27 4.2 The proof of Theorem 4.4 in a special case . . . . . . . . . . . 29 4.3 The proof of Theorem 4.4 in the general case . . . . . . . . . . 32 4.4 A characterisation of Ind_{J}^{𝔎(𝔄)} Λ . . . . . . . . . . . . . . . . . . 33

5 Realisation of maximal simple types 34 5.1 Bessel functions . . . . . . . . . . . . . . . . . . . . . . . . . . 34 5.2 Explicit Whittaker functions . . . . . . . . . . . . . . . . . . . 37 5.3 Multiplicative property . . . . . . . . . . . . . . . . . . . . . . 38

6 A numerical invariant 40 6.1 Behaviour under tame lifting . . . . . . . . . . . . . . . . . . . 41

7 Application to ε-factors of pairs 43 7.1 Preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 7.2 Computation . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 7.3 The case σ_{1} = σ_{2} . . . . . . . . . . . . . . . . . . . . . . . . . 51 7.4 The case σ_{1} ≠ σ_{2} . . . . . . . . . . . . . . . . . . . . . . . . . 52 7.5 Twists by tamely ramified quasi-characters . . . . . . . . . . . 54

Introduction

Let F be a non-archimedean local field and fix an additive character ψ_{F} of F, with conductor 𝔭_{F} (the maximal ideal of the ring of integers 𝔬_{F} of F). Let V be an N-dimensional F -vector space and let G = Aut_{F} (V). This paper concerns the supercuspidal representations of G, so we adopt the notation of [6], where these are classified in terms of maximal simple types ([6]§6).

A maximal simple type is a pair (J, λ), consisting of a (rather special) compact open subgroup J of G and an irreducible representation λ of J. It is constructed from a simple stratum [𝔄, n, 0, β], where 𝔄 is a principal hereditary 𝔬_{F} -order in A and β ∈ A satisfy various properties (see [6] (1.5.5) for the full definition). The algebra E = F [β] is a field, and E^{×} normalises J. If we set J = E^{×}J and let Λ be a representation of J such that Λ|_{J} ≅ λ, then c-Ind_{J}^{G} Λ is an irreducible supercuspidal representation of G. Conversely, any irreducible supercuspidal representation of G arises this way.

It follows from the fact that c-Ind_{J}^{G} Λ has a Whittaker model, that there exist a maximal unipotent subgroup U of G and a non-degenerate character ψ_{α} of U such that Hom_{U∩J}(ψ_{α}, Λ) ≠ 0. Moreover, the uniqueness of the Whittaker model implies that the pair (U, ψ_{α}) is determined up to conjugation by J (see [3]).

Associated to the maximal simple type, we have two other groups H^{1} ⊂ J^{1} ⊂ J (see [6]§3.1). They are normal subgroups of J and λ|_{H^{1}} is a multiple of a simple character θ. Hence, ψ_{α}|_{U∩H^{1}} = θ|_{U∩H^{1}} and we may define a character Ψ of (J ∩ U)H^{1}, by

Ψ(uh) = ψ_{α}(u)θ(h), ∀u ∈ J ∩ U, ∀h ∈ H^{1}.

Let 0 ⊂ V_{1} ⊂ ⋯ ⊂ V_{N} = V be the maximal flag corresponding to U, and set M = {g ∈ G : (g − 1)V ⊆ V_{N−1}}, so that M is a mirabolic subgroup of G. Our first main result, in §4, is

Theorem A. Let U be a maximal unipotent subgroup of G and let ψ_{α} be a non-degenerate character of U such that Hom_{U∩J}(ψ_{α}, Λ) ≠ 0. Then Λ|_{(M∩J)J^{1}} is irreducible and

Λ|_{(M∩J)J^{1}} ≅ Ind_{(U∩J)H^{1}}^{(M∩J)J^{1}} Ψ.

Moreover, the same result holds if we replace: J by 𝔎(𝔄), the G-normaliser of 𝔄; Λ by ρ = Ind_{J}^{𝔎(𝔄)} Λ; and (M∩J)J^{1} by (M∩𝔎(𝔄))U^{1}(𝔄), where U^{1}(𝔄) is the group of principal units of 𝔄.

We show in §4.4 that this property in fact characterises the representations of the form Ind_{J}^{𝔎(𝔄)} Λ. More precisely, writing M_{𝔄} = (M ∩ 𝔎(𝔄))U^{1}(𝔄) we have

Proposition B. Let τ be a representation of 𝔎(𝔄) such that

τ|_{M_{𝔄}} ≅ Ind_{(J∩U)H^{1}}^{M_{𝔄}} Ψ.

Then

τ ≅ Ind_{J}^{𝔎(𝔄)} Λ′,

for some representation Λ′ of J, such that (J, Λ′|_{J}) is a maximal simple type containing θ.

This is an analogue of Gel′fand’s characterisation of the cuspidal representations of GL_{N}(F_{q}).

Before continuing with the applications of Theorem A, we will say a few words about its proof. The strategy is as follows: We first construct a special pair (U, ψ_{α}), such that Hom_{U∩J}(ψ_{α}, Λ) ≠ 0, by carefully picking a basis of V, and then letting U be the group of upper-triangular matrices with respect to this basis, and ψ_{α} be the ’standard’ nondegenerate character. The choice of basis is made so that we can control the restriction of ψ_{α} to U ∩ B^{×}, where B is the centraliser of β. We then prove Theorem A for this particular pair (U, ψ_{α}). The general case follows from the fact that any other such pair (U′, ψ_{α′}) is conjugate to our particular choice by some g ∈ J.

We remark also that Theorem A should follow easily from [3] Theorem 2.9, but there are problems with the proof of that result: the unipotent group U used in the proof is not (in general) the group required by the statement of the Theorem; moreover, there is a gap in the proof of [3] Lemma 2.10 which, so far as we know, nobody has been able to fix. In the course of the proof of Theorem A, we end up proving an analogue of [3] Theorem 2.9, see Theorem 3.3. We get around the problem of [3] Lemma 2.10 by using the case when E is maximal in A as a ‘black box’. If [3] Lemma 2.10 were true then the basis of V referred to above could be written explicitly in terms of β and some of our proofs would simplify.

The interest in Theorem A is that, since F^{×} Ker Λ is of finite index in J, it allows us to apply a very general Theorem of Alperin and James [1] for finite groups. Following [1], we define the Bessel function J_{Λ} of Λ by

J_{Λ}(g) = Q^{−1} ∑_{u∈(U∩J)H^{1}/U^{n+1}(𝔄)}Ψ(u) tr_{Λ}(gu^{−1}).

where Q = ((U ∩ J)H^{1} : U^{n+1}(𝔄)). Now, [1] implies:

Theorem C. Let S be the space of functions f : (M ∩ J)J^{1} → ℂ, such that

f(ug) = Ψ(u)f(g), ∀u ∈ (U ∩ J)H^{1}, ∀g ∈ (M ∩ J)J^{1},

and, for all g ∈ J, let L(g) ∈ End_{ℂ}(S) be the operator:

[L(g)f](m) = ∑_{m_{1}∈(M∩J)J^{1}/(U∩J)H^{1}} J_{Λ}(mgm_{1})f(m_{1}^{−1}),

Then L defines a representation of J on S, which is isomorphic to Λ.

The analogous result for GL_{N}(F_{q}) (or, equivalently, for level zero supercuspidal representations of G) was first observed by Gel′fand [7].

Now we proceed as in [3] §3 and construct a Whittaker function for π = c-Ind_{J}^{G} Λ. However, the observation that Λ can be realised via Bessel functions results in explicit formulae and extra information about this Whittaker function. Our main result concerning Whittaker functions (see §5.2) is as follows:

Theorem D. Define W ∈ Ind_{U}^{G} ψ_{α} by Supp W ⊆ UJ and

W(ug) = ψ_{α}(u)J_{Λ}(g), ∀u ∈ U, ∀g ∈ J,

then W is a Whittaker function for π = c-Ind_{J}^{G} Λ. Moreover, (Supp W) ∩ M = U(H^{1} ∩ M) and

W(uh) = ψ_{α}(u)θ(h), ∀u ∈ U, ∀h ∈ H^{1} ∩ M.

It is the second part of Theorem D that really requires Theorem A and the explicit realization in terms of Bessel functions. The first part of Theorem D has also been obtained by Roberto Johnson [9], in the special case when π is a Carayol representation.

Finally, in §7 we use our Whittaker functions to compute ε-factors of pairs in the following situation. We fix a simple stratum as above and consider two supercuspidal representations π_{1} = c-Ind_{J}^{G} Λ_{1} and π_{2} = c-Ind_{J}^{G} Λ_{2}, such that Λ_{1}|_{H^{1}} and Λ_{2}|_{H^{1}} are multiples of the same simple character θ. Then, for i = 1, 2, we can write Λ_{i}|_{J} ≅ κ ⊗ σ_{i}, where κ is a β-extension (see [6] (5.2.1)) and σ_{i} is the lift of a cuspidal representation of J/J^{1} ≅ GL_{r}(𝔨_{E}), where 𝔨_{E} is the residue field of E and r = dim_{E}(V).

One may show that, for i = 1, 2, we have Λ_{i} ≅ κ̃ ⊗ Σ_{i}, where κ̃ and Σ_{i} are representations of J which restrict to κ and σ_{i} respectively. Moreover, we may think of Σ_{i} as a representation of 𝔎(𝔄) ∩ B^{×} ≅ E^{×} GL_{r}(𝔬_{E}) (where, we recall, B is the centraliser of E). We set τ_{i} = c-Ind_{𝔎(𝔅)}^{B^{×}} Σ_{i}; then τ_{i} is a supercuspidal level zero representation of B^{×} ≅ GL_{r}(E).

In Theorem 7.1 we relate ε(π × π̌_{2}, s, ψ_{F}) and ε(τ_{1} × τ̌_{2}, s, ψ_{E}), where ψ_{E} is an additive character of E with conductor 𝔭_{E} which extends ψ_{F}. We obtain:

Theorem E.

ε(π_{1} × π̌_{2}, s, ψ_{F}) = ζω_{τ_{1}}(ν^{−r})ω_{τ_{2}}(ν^{r})q^{(s−1/2)rv_{E}(ν)N/e}ε(τ_{1} × τ̌_{2}, s, ψ_{E}),

where: ζ = ω_{τ_{2}}(−1)^{r−1}ω_{π_{2}}(−1)^{N−1}; q = q_{F} is the cardinality of 𝔨_{F}; v_{E} is the additive valuation on E with image ℤ; and ν = ν(θ, ψ_{F} , ψ_{E}) ∈ E^{×}/(1 + 𝔭_{E}) is an invariant which we define in §6.

We prove this by computing both sides of the functional equation for the epsilon factor, using the definition of Jacquet, Piatetskii-Shapiro, Shalika [8], with the Whittaker functions of Theorem D. We are able to do the calculation because the fact that the operator L in Theorem C defines a group action imposes various identities on the Bessel function J_{Λ}.

Theorem E implies:

Corollary F. Let χ : F^{×} → ℂ^{×} be a tamely ramified quasi-character and put χ_{E} = χ ∘ N_{E/F}; then

\frac{ε(π_{1}χ × π̌_{2}, s, ψ_{F})}{ε(π_{1} × π̌_{2}, s, ψ_{F})} = χ(N_{E/F} (ν^{−r^{2}}))\frac{ε(τ_{1}χ_{E} × τ̌_{2}, s, ψ_{E})}{ε(τ_{1} × τ̌_{2}, s, ψ_{E})},

where ν = ν(θ_{F} , ψ_{F} , ψ_{E}).

We recover a result of Bushnell and Henniart [5], concerning the effect on ε(π × π̌, s, ψ_{F}) of twisting π by tamely ramified quasi-characters, when π is totally wildly ramified, as a special case of the above Corollary. Moreover, we show in §6.1 that the invariant ν behaves well under the tame lifting operation for simple characters of Bushnell and Henniart [2], which implies [5] Theorem 7.1.

We end the introduction with a brief summary of the contents of each section. We begin in §1 with notation and some elementary results about nondegenerate characters and induced supercuspidal representations. In §2 we begin the groundwork for the proof of Theorem A, proving a similar result for a β-extension κ. In §3 we define the particular unipotent subgroup, and the basis, used in the proof of Theorem A; this proof appears in §4, along with the proof of Proposition B. In §5, we apply the Theorem of Alperin and James to define Bessel functions, and construct our explicit Whittaker function from Theorem D. In §6, we define the numerical invariant ν which appears in Theorem E. Finally, the proof of Theorem E, and its application to twisting by tamely ramified quasi-characters, appears in §7.

Acknowledgements. This work has grown out of discussions which started at the LMS Symposium “L-functions and Galois representations” in Durham, July 2004. The authors would like to thank the organisers of the symposium, and also Richard Hill for some productive conversations there. Part of this work was done while the first-named author was a CNRS post-doc at Jussieu; the first-named author would like to thank Marie-France Vigneras, the Institut de Mathématiques de Jussieu and the CNRS for providing excellent working conditions in a stimulating environment. The second-named author would like to thank the SFB 701 “Spektrale Strukturen und Topologische Methoden in der Mathematik”, Bielefeld, for their invitation and hospitality at a crucial stage in the writing of the paper.

1 Notation and Preliminaries

Let F be a locally compact non-archimedean local field, with ring of integers 𝔬_{F}, maximal ideal 𝔭_{F}, and residue field 𝔨_{F} = 𝔬_{F} /𝔭_{F} with q_{F} = p^{f} elements, p prime. We fix ϖ_{F} a uniformizing element of F and let v_{F} denote the additive valuation of F, normalised so that v_{F} (ϖ_{F}) = 1. We use similar notation for any field extension of F.

Let V be an N-dimensional F -vector space, A = End_{F} (V) and G = Aut_{F} (V) so, after choosing a basis for V, we have

A ≅ 𝕄_{N}(F), G ≅ GL_{N}(F).

1.1 Unipotent subgroups and characters

The results of this section are stated without proof, since these proofs are straightforward. One way to prove them would be to choose a suitable basis for V with respect to which the unipotent subgroups considered consist of matrices which are upper triangular.

We fix, once and for all, an additive character ψ_{F} : F → ℂ which is trivial on 𝔭_{F}, non-trivial on 𝔬_{F}. For any a ∈ A, we define a function ψ_{a} : A → ℂ by

ψ_{a}(x) = (ψ_{F} ∘ tr_{A/F})(a(x − 1)), for x ∈ A,

where tr_{A/F} denotes the matrix trace. We use the same notation for the restriction of ψ_{a} to various subsets of A.

Let F be an F -flag in V,

F : 0 = V_{0} ⊂ V_{1} ⊂ V_{2} ⊂ ⋯ ⊂ V_{s} = V,

Let P = P_{F} be the G-stabiliser of F, a parabolic subgroup of G, and let U = U_{F} be its unipotent radical. We also put

X_{F} = {x ∈ A : xV_{i} ⊆ V_{i+1}, 0 ≤ i ≤ s − 1},
X_{F}^{+} = {x ∈ X_{F} : xV_{i} ⊄ V_{i}, 0 ≤ i ≤ s − 1},
X_{F}^{−} = {x ∈ X_{F} : xV_{i} ⊆ V_{i}, 0 ≤ i ≤ s − 1}.

Lemma 1.1. Let a ∈ A. The function ψ_{a} defines a linear character of U if and only if a ∈ X_{F}. Moreover, ψ_{a} is trivial on U if and only if a ∈ X_{F}^{−}.

Now suppose that F is a maximal F -flag so that s = N and dim_{F} V_{i} = i, for 0 ≤ i ≤ N. A smooth linear character χ of U is said to be nondegenerate if its G-normaliser is F^{×}U. We can describe this more concretely by choosing a basis v_{1}, ..., v_{N} for V such that V_{i} = ⨁_{j=1}^{i} F v_{j}, for 1 ≤ i ≤ N. Then U is identified with the upper triangular unipotent matrices in GL_{N}(F) and the smooth characters χ of U are given by

χ(u) = ψ_{F} (∑_{i=1}^{N−1} μ_{i}u_{i,i+1}) , for u = (u_{ij}) ∈ U,

where μ_{i} ∈ F, 1 ≤ i ≤ N − 1, are fixed scalars. It is easy to see that χ is nondegenerate if and only if all μ_{i} ≠ 0; or, equivalently, if and only if, for all 1 ≤ j ≤ N − 1, there exists u^{(j)} ∈ U with u_{i,i+1}^{(j)} = 0, for i ≠ j, such that χ(u^{(j)}) ≠ 1.

Lemma 1.2. Let F be a maximal F -flag and a ∈ X_{F}. Then ψ_{a} is nondegenerate if and only if a ∈ X_{F}^{+}.

1.2 Induced supercuspidals

Since the supercuspidal representations of G are all obtained by irreducible induction from compact-mod-centre subgroups, the following Proposition (which is mostly taken from [3] §1) will be useful.

Proposition 1.3. Let 𝔎 be an open, compact-mod-centre subgroup of G, and suppose that ρ is a representation of 𝔎, such that π = c-Ind_{𝔎}^{G} ρ is an irreducible supercuspidal representation of G. Let U be a maximal unipotent subgroup of G, and let χ be a smooth character of U.

(i) If Hom_{U∩𝔎}(ρ, χ) ≠ 0 then χ is non-degenerate.

(ii) If χ is non-degenerate then there exists g ∈ G such that Hom_{U∩𝔎^{g}}(χ, ρ^{g}) ≠ 0.

(iii) If Hom_{U∩𝔎}(χ, ρ) ≠ 0 and Hom_{U∩𝔎^{g}}(χ, ρ^{g}) ≠ 0, for some g ∈ G, then there exists u ∈ U such that 𝔎^{u} = 𝔎^{g} and ρ^{u} ≅ ρ^{g}.

Proof. (i) Let ϕ ∈ Hom_{U∩𝔎}(ρ, χ) ≠ 0 be such that ϕ ≠ 0, and fix a Haar measure du on U. Then this gives a non-zero Φ ∈ Hom_{U}(π, χ), by

Φ(f) = ∫_{U} χ(u^{−1})ϕ(f(u))du, ∀f ∈ c-Ind_{𝔎}^{G} ρ.

If χ is degenerate then there exists a unipotent radical U′ of some proper parabolic subgroup of G, such that the restriction of χ to U′ is trivial, but this implies Hom_{U′}(π, 1) ≠ 0. However this may not happen as π is supercuspidal.

Parts (ii) and (iii) follow from [3] Proposition 1.6 and (1.8).

2 A note on β-extensions

The main result of this section is Theorem 2.6, which asserts that the restriction of a β-extension κ to a certain subgroup of J is isomorphic to a representation induced from a linear character. This result will be used in §4. In section 2.1 we recall some results on Iwahori decompositions. We will use the definitions and notations of [6] with little introduction.

Let [𝔄, n, 0, β] be a principal simple stratum in A (see [6] (1.5.5)). In particular, 𝔄 is a hereditary, principal 𝔬_{F} -order in A, with Jacobson radical 𝔓, and β ∈ 𝔓^{−n} \ 𝔓^{1−n} is such that E = F [β] is a field with E^{×} normalising 𝔄. We denote by B the A-centraliser of E and put 𝔅 = 𝔄 ∩ B.

Let L = {L_{k} : k ∈ ℤ} be the 𝔬_{F} -lattice chain in V associated to 𝔄, see [6] (1.1.2). Since E^{×} normalises 𝔄 we may consider L also as an 𝔬_{E}-lattice chain. Let e = e(𝔅|𝔬_{E}) be the 𝔬_{E}-period of L. We fix an E-basis {w_{1}, …, w_{r}} of V, where r[E : F] = N, such that

L_{0} = 𝔬_{E}w_{1} + 𝔬_{E}w_{2} + ⋯ + 𝔬_{E}w_{r}

L_{i} = 𝔬_{E}w_{1} + ⋯ + 𝔬_{E}w_{\frac{r}{e} (e−i)} + 𝔭_{E}w_{\frac{r}{e} (e−i)+1} + ⋯ + 𝔭_{E}w_{r}

for 0 < i < e. The choice of this basis identifies B with 𝕄_{r}(E) and 𝔅 with a subring of 𝕄_{r}(𝔬_{E}), which consists of block upper-triangular matrices modulo 𝔭_{E}, such that each block on the diagonal is of the size \frac{r}{e} × \frac{r}{e}.

Let F_{0} be the F -flag in V, given by

F_{0} : 0 ⊂ Ew_{1} ⊂ ⋯ ⊂ ⊕_{i=1}^{j}Ew_{i} ⊂ ⋯ ⊂ ⊕_{i=1}^{r}Ew_{i} = V.

Let P_{0} be the G-stabiliser of F_{0}. Moreover, put G_{i} = Aut_{F} (Ew_{i}), for 1 ≤ i ≤ r, and

M_{0} = ∏_{i=1}^{r} G_{i},

a Levi component of the parabolic subgroup P_{0} of G. Let U_{0} be the unipotent radical of P_{0}, so that P_{0} = M_{0}U_{0}. We also denote by U_{0}^{−} the unipotent radical of the parabolic subgroup opposite to P_{0} relative to M_{0}.

We note that F_{0} is also a maximal E-flag in V. This yields:

Lemma 2.1. Let F be an F -flag refining F_{0} and let U be the unipotent radical of the G-stabiliser of F, then

U ∩ B^{×} = U_{0} ∩ B^{×}.

2.1 Iwahori decompositions

Put J = J(β, 𝔄), J^{1} = J^{1}(β, 𝔄) and H^{1} = H^{1}(β, 𝔄) (see [6] §3 for the definitions of these groups). By [2] Example 10.9, J^{1} and H^{1} have Iwahori decompositions with respect to (M_{0}, P_{0}):

J^{1} = (J^{1} ∩ U_{0}^{−})(J^{1} ∩ M_{0})(J^{1} ∩ U_{0})
H^{1} = (H^{1} ∩ U_{0}^{−})(H^{1} ∩ M_{0})(H^{1} ∩ U_{0})

It will also be useful for us to form the group

(J^{1} ∩ P_{0})H^{1} = (H^{1} ∩ U_{0}^{−})(J^{1} ∩ M_{0})(J^{1} ∩ U_{0}).

Now let F be a maximal F -flag refining F_{0} and U the corresponding unipotent subgroup. We want to understand the group (J ∩ U)H^{1}. For 1 ≤ i ≤ r, let F_{i} be the maximal F -flag in Ew_{i} given by intersection of F with Ew_{i}. Let U_{i} be the unipotent radical of the G_{i}-stabiliser of F_{i}; then U_{i} = U ∩ G_{i} and

U ∩ M_{0} = ∏_{i=1}^{r} U_{i}.

For 1 ≤ i ≤ r, we denote by 𝔄_{i} the hereditary 𝔬_{E}-order in A_{i} = End_{F} (Ew_{i}) given by the lattice chain L_{i} = {L_{k} ∩ Ew_{i} : k ∈ ℤ}.

Lemma 2.2. (J^{1}∩U)H^{1} has an Iwahori decomposition with respect (M_{0}, P_{0}) and

((J^{1} ∩ U)H^{1}) ∩ M_{0} = ∏_{i=1}^{r}(J^{1}(β, 𝔄_{i}) ∩ U_{i})H^{1}(β, 𝔄_{i}).

Proof. Since J^{1} and U have Iwahori decompositions with respect to (M_{0}, P_{0}), so does J^{1} ∩ U. Since this group normalises H^{1}, which also has an Iwahori decomposition, we see that (J^{1} ∩ U)H^{1} has an Iwahori decomposition with respect (M_{0}, P_{0}) and, in particular,

((J^{1} ∩ U)H^{1}) ∩ M_{0} = (J^{1} ∩ U ∩ M_{0})(H^{1} ∩ M_{0}).

The lemma now follows from the decompositions of J^{1} ∩ M_{0} and H^{1} ∩ M_{0} (see [2] §10) and the decompositions above.

Since dim_{E}(Ew_{i}) = 1, the algebra E = F [β] is a maximal subfield of A_{i}. We will use the lemma above as a reduction step to the case when E is a maximal subfield of A.

Lemma 2.3. Let U′ be a unipotent radical of some parabolic subgroup of G, then the image of

U′ ∩ J → J/J^{1} ≅ U(𝔅)/U^{1}(𝔅)

is contained in the unipotent radical of some Borel subgroup of U(𝔅)/U^{1}(𝔅).

Corollary 2.4. We have

J ∩ U = (U(𝔅) ∩ U_{0})(J^{1} ∩ U).

Moreover, (J ∩ U)H^{1} has an Iwahori decomposition with respect to (M_{0}, P_{0}).

Proof. The E-basis {w_{1}, …, w_{r}} of V identifies B^{×} with GL_{r}(E) and U(𝔅)∩ U_{0} with a subgroup of unipotent upper-triangular matrices with entries in 𝔬_{E}. This implies that the image of U(𝔅) ∩ U_{0} in U(𝔅)/U^{1}(𝔅) is the unipotent radical of some Borel subgroup of U(𝔅)/U^{1}(𝔅). Lemma 2.3 implies that

(J ∩ U)J^{1} = (U(𝔅) ∩ U_{0})J^{1}.

Intersecting both sides with U gives the first part of the lemma. The second follows immediately from Lemma 2.2

2.2 β-extensions

Let C(𝔄, 0, β) be the set of simple characters of H^{1}, in the sense of [6] (3.2.3). Let θ ∈ C(𝔄, 0, β); then θ is a linear character and, there exists a unique irreducible representation η of J^{1} containing θ, [6] (5.1.1).

Lemma 2.5. Let k_{θ} be the nondegenerate alternating form on J^{1}/H^{1} given by

k_{θ}(x, y) = θ([x, y]), for x, y ∈ J^{1},

introduced in [6] §3.4. Let U be a subgroup of J^{1} containing H^{1} and let \overline{U} be the image of U in J^{1}/H^{1}. Suppose there exists a linear character χ of U such that χ|_{H^{1}} = θ. Then the following are equivalent:

(i) χ occurs in η with multiplicity one;

(ii) \overline{U} is a maximal totally isotropic subspace of J^{1}/H^{1} for the form k_{θ};

(iii) η ≅ Ind_{U}^{J^{1}} χ.

Proof. Since H^{1} is normal in J^{1} and J^{1}/H^{1} is abelian (by [6] (3.1.15)), U is a normal subgroup of J^{1}. Moreover, since χ|_{H^{1}} = θ and J^{1}/H^{1} is abelian, the commutator subgroup of U will lie in the kernel of θ and hence \overline{U} is a totally isotropic subspace of J^{1}/H^{1} for the form k_{θ}.

(i)⇒(ii) Let \overline{U}_{max} be a maximal totally isotropic subspace of J^{1}/H^{1} containing \overline{U} and let U_{max} be its inverse image in J^{1}, so that U_{max}/ Ker(θ) is a maximal abelian subgroup of J^{1}/ Ker(θ). The character χ admits extension to a linear character of U_{max} in exactly (U_{max} : U) ways and every one of these extensions occurs in η. Then χ occurs in η with multiplicity at least this index so that U_{max} = U.

(ii)⇒(iii) Suppose that j ∈ J^{1} intertwines χ with itself. Let U′ be the subgroup of J^{1} generated by j and U and let \overline{U′} be the image of U′ in J^{1}/H^{1}. A typical element of \overline{U′} is a coset j^{a}xH^{1}, where a is an integer and x ∈ U. Since

χ([j^{a}x, j^{b}y]) = χ(j^{a}xj^{−a})χ(j^{a+b}yx^{−1}j^{−a−b})χ(j^{b}y^{−1}j^{−b}) = χ([x, y]) = 1

for all x, y ∈ U, the subspace \overline{U′} is totally isotropic for the form k_{θ}. Hence \overline{U′} = \overline{U} and j ∈ U. In particular, Ind_{U}^{J^{1}} χ is irreducible. Since η is the unique irreducible representation of J^{1} containing θ and χ contains θ, we have η ≅ Ind_{U}^{J^{1}} χ.

(iii)⇒(i) Since η is irreducible, this is just Frobenius reciprocity.

Now let κ be a representation of J, such that κ|_{J^{1}} ≅ η and κ is intertwined by the whole of B_{β}^{×}, that is, a β-extension of η in the sense of [6] (5.2.1).

Theorem 2.6. Let [𝔄, n, 0, β] be a simple stratum. Let θ ∈ C(𝔄, 0, β) and let κ be a β-extension of η as above. Let F be a maximal F -flag in V, let U be the unipotent radical of the G-stabiliser of F and let χ be a smooth character of U, such that

(i) F_{0} ⊆ F,

(ii) θ|_{U∩H^{1}} = χ|_{U∩H^{1}},

(iii) χ is trivial on U_{0}.

Let Θ be the linear character of (J ∩ U)H^{1} defined by

Θ(uh) = χ(u)θ(h), ∀u ∈ U ∩ J, ∀h ∈ H^{1}.

Then

κ|_{(J∩U)J^{1}} ≅ Ind_{(J∩U)H^{1}}^{(J∩U)J^{1}} Θ.

Before proving Theorem 2.6, we remark that it is not clear that there exist a flag F and a character χ satisfying the hypotheses. This would follow from [3] Lemma 2.10, if we could fix the proof of that result. Instead, we will have to wait for Theorem 3.3 to see that there are indeed such a flag and character.

Proof. We begin by proving

Ind_{(J^{1}∩U)H^{1}}^{J^{1}} Ψ ≅ η.  (∗)

Step 1. We will prove (∗) in the special case when E = F [β] is a maximal subfield of A.

Suppose that E is a maximal subfield of A so that B = E and J = 𝔬_{E}^{×}J^{1}. The pair (J, κ) is a simple type in the sense of [6] (5.5.10) and, since 𝔅 = 𝔬_{E}, we have e(𝔅|𝔬_{E}) = 1. Now [6] (6.2.2) and (6.2.3) imply that there exists a representation Λ of E^{×}J such that Λ|_{J} ≅ κ and π = c-Ind_{E^{×}J}^{G} Λ is an irreducible supercuspidal representation of G. If u ∈ U, then det_{A}(u) = 1 and this implies that

(E^{×}J) ∩ U = J ∩ U.

Since dim_{E} V = 1 the unipotent radical U_{0} is trivial and hence Corollary 2.4 implies that

J ∩ U = J^{1} ∩ U.

Hence Λ|_{(E^{×}J)∩U} ≅ η|_{J^{1}∩U}. Since η is the unique irreducible representation containing θ and Θ|_{H^{1}} = θ, we obtain that Θ occurs in η|_{(J^{1}∩U)H^{1}}. Since Θ|_{U∩J} = χ|_{U∩J}, we obtain

1 ≤ dim Hom_{(J^{1}∩U)H^{1}}(Θ, η) ≤ dim Hom_{J^{1}∩U}(χ, η)
= dim Hom_{(E^{×}J^{1})∩U}(χ, Λ) ≤ 1,

where the last inequality follows from [3] Proposition 1.6(iii). Hence

dim Hom_{(J^{1}∩U)H^{1}}(Θ, η) = 1.

The equivalence (∗) now follows immediately from Lemma 2.5 applied to U = (J^{1} ∩ U)H^{1} and χ = Θ.

Step 2. We will prove (∗) in the general case by reducing to Step 1.

For 1 ≤ j ≤ r we have an equality of sets:

{L_{i} ∩ Ew_{j} : L_{i} ∈ L} = {𝔭_{E}^{k}w_{j} : k ∈ ℤ}.

This follows from the explicit description of lattices in L, in terms of the Ebasis {w_{1}, … w_{r}} of V. Hence [2] Example 10.9 implies that the character θ is trivial on its restrictions to H^{1} ∩ U_{0} and H^{1} ∩ U_{0}^{−}. Moreover, by [2] page 167, the restriction of θ to H^{1} ∩ G_{i} = H^{1}(β, 𝔄_{i}) is the simple character θ_{i} in C(𝔄_{i}, 0, β) corresponding to θ under the canonical bijection τ_{𝔄,𝔄_{i},β} of [6] §3.6, where 𝔄_{i} is the hereditary 𝔬_{F} -order corresponding to the lattice chain {𝔭_{E}^{k}w_{i} : k ∈ ℤ} in Ew_{i}. In particular, [2] Example 10.9 implies that the analogue of [6] (7.2.3) holds in our situation:

(i) the subspaces (J^{1} ∩ U_{0})/(H^{1} ∩ U_{0}) and (J^{1} ∩ U_{0}^{−})/(H^{1} ∩ U_{0}^{−}) of J^{1}/H^{1} are both totally isotropic for the form k_{θ}, and orthogonal to the subspace (J^{1} ∩ M_{0})/(H^{1} ∩ M_{0});

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

(J^{1} ∩ M_{0})/(H^{1} ∩ M_{0}) = ∏_{i=1}^{r} J^{1}(β, 𝔄_{i})/H^{1}(β, 𝔄_{i})

is the orthogonal sum of the pairings k_{θ_{i}};

(iii) we have an orthogonal sum decomposition

\frac{J^{1}}{H^{1}} = \frac{J^{1} ∩ M_{0}}{H^{1} ∩ M_{0}} ⊥ (\frac{J^{1} ∩ U_{0}^{−}}{H^{1} ∩ U_{0}^{−}} × \frac{J^{1} ∩ U_{0}}{H^{1} ∩ U_{0}} ) .

In particular, the restriction of k_{θ} to the group (J^{1} ∩ U_{0}^{−})/(H^{1} ∩ U_{0}^{−}) ×

(J^{1} ∩ U_{0})/(H^{1} ∩ U_{0}) is non-degenerate.

Let \overline{((J^{1} ∩ U)H^{1}) ∩ M_{0}} be the image of the natural homomorphism

((J^{1} ∩ U)H^{1}) ∩ M_{0} → (J^{1} ∩ M_{0})/(H^{1} ∩ M_{0}).

Lemma 2.2 and (ii) above imply that

\overline{((J^{1} ∩ U)H^{1}) ∩ M_{0}} = ∏_{i=1}^{r} \overline{J^{1}(β, 𝔄_{i}) ∩ U_{i}}

where \overline{J^{1}(β, 𝔄_{i}) ∩ U_{i}} is the image of the natural homomorphism

J^{1}(β, 𝔄_{i}) ∩ U_{i} → J^{1}(β, 𝔄_{i})/H^{1}(β, 𝔄_{i}).

Since E is a maximal subfield of A_{i} and we have proved (∗) when E is maximal, \overline{J^{1}(β, 𝔄_{i}) ∩ U_{i}} is a maximal isotropic subspace in J^{1}(β, 𝔄_{i})/H^{1}(β, 𝔄_{i}) for the form k_{θ_{i}}.

Now (ii) implies that \overline{((J^{1} ∩ U)H^{1}) ∩ M_{0}} is a maximal isotropic subspace of (J^{1} ∩ M_{0})/(H^{1} ∩ M_{0}) for k_{θ}. Moreover, (i) and (iii) imply that (J^{1} ∩ U_{0})/(H^{1} ∩ U_{0}) is a maximal isotropic subspace of (J^{1} ∩ U_{0}^{−})/(H^{1} ∩ U_{0}^{−}) × (J^{1} ∩ U_{0})/(H^{1} ∩ U_{0}).

It follows from the orthogonal sum decomposition in (iii) that

\overline{((J^{1} ∩ U)H^{1}) ∩ M_{0}} × (J^{1} ∩ U_{0})/(H^{1} ∩ U_{0})

is a maximal isotropic subspace in J^{1}/H^{1} for the form k_{θ}.

Since (J^{1}∩U)H^{1} contains J^{1}∩U_{0}, the image of (J^{1}∩U)H^{1} in J^{1}/H^{1} contains a maximal totally isotropic subspace of J^{1}/H^{1} described above. Since there exists a linear character Θ of (J^{1} ∩ U)H^{1} extending θ, we see that this image is itself isotropic and hence must be a maximal totally isotropic subspace. The equivalence (∗) now follows from Lemma 2.5.

Step 3. Finally, we will deduce Theorem 2.6 from (∗) by examining the construction of κ in [6] §5.

Let L_{m} be the 𝔬_{E}-lattice chain in V given by

L_{m} = {ϖ_{E}^{k}(𝔬_{E}w_{1} + ⋯ + 𝔬_{E}w_{j} + 𝔭_{E}w_{j+1} + ⋯ + 𝔭_{E}w_{r}) : k ∈ ℤ, 1 ≤ j ≤ r}

Let 𝔅_{m} = End_{𝔬_{E}}^{0}(L_{m}) so that 𝔅_{m} is a minimal 𝔬_{E}-order in B. Similarly, let 𝔄 = End_{𝔬_{F}}^{0}(L_{m}) so that 𝔄_{m} is the unique hereditary 𝔬_{F} -order in A normalised by E^{×} such that 𝔄_{m} ∩ B = 𝔅_{m}. Moreover, [𝔄_{m}, nr/e, 0, β], where e = e(𝔅|𝔬_{E}), is a simple stratum in A and the groups

H_{m}^{1} = H^{1}(β, 𝔄_{m}), J_{m}^{1} = J^{1}(β, 𝔄_{m})

have Iwahori decompositions with respect to (M_{0}, P_{0}).

We denote by θ_{m} the simple character in C(𝔄_{m}, 0, β) corresponding to θ via the canonical bijection τ_{𝔄,𝔄_{m},β} of [6] §3.6. Then θ_{m} is trivial on H_{m}^{1} ∩ U_{0} and H_{m}^{1} ∩ U_{0}^{−} (see [2] (10.9)). We also denote by η_{m} the unique irreducible representation of J_{m}^{1} which contains θ_{m}.

Since L ⊆ L_{m} we have 𝔄_{m} ⊆ 𝔄 and 𝔅_{m} ⊆ 𝔅. Moreover,

U(𝔅) ∩ U_{0} = U(𝔅_{m}) ∩ U_{0}

since with respect to the E-basis {w_{1}, …, w_{r}} of V, both groups are identified with unipotent upper-triangular matrices, with entries in 𝔬_{E}.

We define

η̃ := Ind_{(J∩U)H^{1}}^{(J∩U)J^{1}} Θ.

Note first that η̃|_{J^{1}} = Ind_{(J^{1}∩U)H^{1}}^{J^{1}} Θ ≅ η so η̃ is certainly irreducible. Moreover, [6] (5.1.1) implies that the U^{1}(𝔄_{m})-intertwining of η̃ is contained in

I_{U^{1}(𝔄_{m})}(η) = (J^{1}B^{×}J^{1}) ∩ U^{1}(𝔄_{m}) = U^{1}(𝔅_{m})J^{1} = (J ∩ U)J^{1}

where the last equality follows from Corollary 2.4. Hence Ind_{(J∩U)J^{1}}^{U^{1}(𝔄_{m})} η̃ is irreducible.

Now we claim that

Ind_{(J∩U)J^{1}}^{U^{1}(𝔄_{m})} η̃ ≅ Ind_{J_{m}^{1}}^{U^{1}(𝔄_{m})} η_{m}.

It is enough to show that

Hom_{U^{1}(𝔄_{m})} (Ind_{(J∩U)J^{1}}^{U^{1}(𝔄_{m})} η̃, Ind_{H_{m}^{1}}^{U^{1}(𝔄_{m})} θ_{m}) ≠ 0,

since the latter is a multiple of Ind_{J_{m}^{1}}^{U^{1}(𝔄_{m})} η_{m}, which is irreducible by the same argument as above. By Mackey Theory, it is enough to show that

Hom_{((J∩U)H^{1})∩H_{m}^{1}}(Θ, θ_{m}) ≠ 0.

But ((J ∩U)H^{1})∩H_{m}^{1} has an Iwahori decomposition with respect to (M_{0}, P_{0}), since both (J ∩U)H^{1} and H_{m}^{1} do. Moreover, Θ is trivial on ((J ∩U)H^{1})∩U_{0}^{−}, since ((J∩U)H^{1})∩U_{0}^{−} = H^{1}∩U_{0}^{−} and θ is trivial on H^{1}∩U_{0}^{−}. Since χ is trivial on U_{0} we get that Θ is trivial on ((J ∩ U)H^{1}) ∩ U_{0}. Hence both Θ and θ_{m} are trivial on the subgroups ((J ∩ U)H^{1}) ∩ H_{m}^{1} ∩ U_{0}^{−} and ((J ∩ U)H^{1}) ∩ H_{m}^{1} ∩ U_{0}. Finally, by [2] §10,

H_{m}^{1} ∩ M_{0} = ∏_{i=1}^{r} H^{1}(β, 𝔄_{i}) = H^{1} ∩ M_{0}

so

(J ∩ U)H^{1} ∩ H_{m}^{1} ∩ M_{0} = H^{1} ∩ M_{0} ⊂ H^{1} ∩ H_{m}^{1},

where Θ = θ and θ_{m} agree by [6] (3.6.1).

In particular, we have shown that, upto equivalence, η̃ satisfies the conditions of the representation (also denoted η̃) in [6] (5.1.15). Since these conditions uniquely determine η̃, we conclude that our η̃ is, up to equivalence, the same one used in the construction of κ in [6] (5.2.4) and hence

κ|_{(J∩U)J^{1}} ≅ η̃ ≅ Ind_{(J∩U)H^{1}}^{(J∩U)J^{1}} Θ

as required.

3 A particular unipotent subgroup

The main result of this section says that we may choose a maximal F -flag F in V and a character χ of the G-stabiliser U of F which satisfy the hypotheses of Theorem 2.6 and, moreover, such that we can control the restriction of certain characters of U to U ∩ U(𝔅) (see Theorem 3.3 for details). We will continue with the notation of the previous section but, since we will consider only supercuspidal representations in the applications, we will assume that e(𝔅|𝔬_{E}) = 1.

3.1 An F -basis of E

Although we would like to work in a basis-free way, we are forced to choose one, since the zeta function in §7, whose functional equation defines ε-factors of pairs, is defined on matrices. In this subsection we consider the special case when E is maximal, so we may identify V = E. We show that there exists an F -basis B of E with ’nice’ properties. The basis is chosen to ease the pain of calculations in §7, see also Corollary 3.4. The conditions imposed on B imply certain uniqueness result, which will be used to define a numerical invariant in §6.

Let F = {V_{i} : 1 ≤ i ≤ d}, where d = [E : F], be a maximal F -flag in E, let U be the unipotent radical of the Aut_{F} (E)-stabiliser of F, and let χ be a smooth, non-degenerate character of U. Let ψ_{E} be an additive character of E, trivial on 𝔭_{E}, and such that

ψ_{E}(x) = ψ_{F} (x), ∀x ∈ F.

Definition 3.1. An F -basis B = {x_{1}, …, x_{d}} of E is (U, χ, ψ_{E})-balanced if the following hold:

(i) V_{i} = F x_{1} + ⋯ + F x_{i}, 1 ≤ i ≤ d;

(ii) there exists functions a_{i} : ℤ → ℤ, for 1 ≤ i ≤ d such that

𝔭_{E}^{k} = ∑_{i=1}^{d} 𝔭_{F}^{a_{i}(k)}x_{i}, ∀k ∈ ℤ;

(iii) if u ∈ U, and (u_{ij}) is the matrix of u with respect to B, then

χ(u) = ψ_{F} (∑_{i=1}^{d−1} u_{i,i+1});

(iv) if we embed naturally E ↪ End_{F} (E) and, for ξ ∈ E, we let (ξ_{ij}) be the matrix of ξ with respect to B, then

ψ_{E}(ξ) = ψ_{F} (ξ_{dd}), ∀ξ ∈ E.

Proposition 3.2. There exists an (U, χ, ψ_{E})-balanced basis. Moreover, if B = {x_{1}, … x_{d}} is (U, χ, ψ_{E})-balanced and B′ = {y_{1}, …, y_{d}} is a basis of E over F which satisfies Definition 3.1 (i),(iii) and (iv), then x_{d}x_{1}^{−1} = y_{d}y_{1}^{−1}.

Proof. Proposition II-3 in [13] applied to F and {𝔭_{E}^{i} : i ∈ ℤ} implies that there exists an F -basis B = {x_{1}, …, x_{d}} of E which satisfies (i) and (ii). Since χ is non-degenerate, after replacing x_{i} by some λ_{i}x_{i}, where λ_{i} ∈ F^{×}, we may ensure that B satisfies (iii).

For ξ ∈ E, let (ξ_{ij}) be a matrix of ξ with respect to B. Consider the function ϕ_{B} : E → ℂ^{×} given by ϕ_{B}(ξ) = ψ_{F} (ξ_{dd}). It is clear that ϕ_{B} is an additive character. If x ∈ F, then ϕ_{B}(x) = ψ_{F} (x), hence ϕ_{B} is nontrivial on 𝔬_{E}. Since x_{d} ∈ 𝔭_{E}^{v_{E}(x_{d})} and x_{d} ∉ 𝔭_{E}^{v_{E}(x_{d})+1} we have a_{d}(v_{E}(x_{d})) = 0 and a_{d}(v_{E}(x_{d}) + 1) = 1, hence if ξ ∈ 𝔭_{E}, then ξ_{dd} ∈ 𝔭_{F}, and so ϕ_{B} is trivial on 𝔭_{E}. Since ψ_{E} and ϕ_{B} have the same conductor, there exists α ∈ 𝔬_{E}^{×}, such that ψ_{E}(ξ) = ϕ_{B}(αξ), for all ξ ∈ E.

Set A(E) = End_{F} (E), let 𝔄(E) be the hereditary order corresponding to the lattice chain L(E) = {𝔭_{E}^{i} : i ∈ ℤ}, and, for 1 ≤ i, j ≤ d, let 1_{ij} ∈ A(E) be given by 1_{ij}x_{k} = δ_{ik}x_{j}, where δ_{ik} is the Kronecker delta. Since B satisfies (ii) we have 1_{ij} ∈ 𝔄.

Set u = 1 + (α − 1)1_{dd} ∈ 𝔄(E). We have

ψ_{F} (λ) = ψ_{E}(λ) = ϕ_{B}(αλ) = ψ_{F} (α_{dd}λ), ∀λ ∈ F.

Hence α_{dd} = 1 and so u ∈ U ∩ 𝔄(E). So the basis uB = {ux_{1}, …, ux_{d}} also satisfies (i),(ii) and (iii). Now

ϕ_{uB}(ξ) = ϕ_{B}(u^{−1}ξu) = ψ_{F} (∑_{i=1}^{d} ξ_{di}α_{id}) = ψ_{F} ((ξα)_{dd}) = ϕ_{B}(ξα) = ψ_{E}(ξ).

Hence uB is (U, χ, ψ_{E})-balanced.

Suppose that B = {x_{1}, … x_{d}} is (U, χ, ψ_{E})-balanced and that B′ = {y_{1}, …, y_{d}} satisfies Definition 3.1(i),(iii) and (iv). After translating by some λ ∈ F^{×}, we may assume that x_{1} = y_{1}. Parts (i),(iii) imply that there exists u ∈ U such that B′ = uB. Let (u_{ij}) be the matrix of u with respect to B. Then, for all ξ ∈ E, we have

ψ_{E}(ξ) = ϕ_{uB}(ξ) = ψ_{F} ((u^{−1}ξu)_{dd}) = ψ_{F} (∑_{i=1}^{d} ξ_{di}u_{id}) = ψ_{A} (ξ (∑_{i=1}^{d} u_{id}1_{di})).

where ψ_{A} = ψ_{F} ∘ tr_{A(E)/F}. Moreover, 1_{id} = x_{i}x_{d}^{−1}1_{dd} so

ψ_{E}(ξ) = ψ_{A} (ξ (∑_{i=1}^{d} u_{id}x_{i}x_{d}^{−1}) 1_{dd}) .

By [6] (1.3.4) there exists an (E, E)-bimodule homomorphism s : A → E (a tame corestriction), such that

ψ_{A}(ξa) = ϕ_{B}(ξs(a)), ∀ξ ∈ E, ∀a ∈ A.

Since ϕ_{B}(ξ) = ψ_{F} (ξ_{dd}) = ψ_{A}(ξ1_{dd}) = ϕ_{B}(ξs(1_{dd})), for all ξ ∈ E, we obtain that s(1_{dd}) = 1. Hence we get

ψ_{E}(ξ) = ϕ_{B} (ξ (∑_{i=1}^{d} u_{id}x_{i}x_{d}^{−1})) = ψ_{E} (ξ (∑_{i=1}^{d} u_{id}x_{i}x_{d}^{−1})) .

Thus x_{d} = ∑_{i=1}^{d} u_{id}x_{i}, which implies that u_{id} = 0, if i ≠ d. In particular, we get y_{d} = ux_{d} = x_{d}.

3.2 A choice of maximal flag

Let [𝔄, n, 0, β] be a simple stratum, such that e(𝔅|𝔬_{E}) = 1. Let θ ∈ C(𝔄, 0, β) and let F_{0} be a maximal E-stable flag in V, U_{0} the unipotent radical of the Gstabiliser of F_{0}.

Theorem 3.3. There exist a maximal F -flag F in V, a smooth character χ of the unipotent radical U = U_{F} and an element b ∈ X_{F} ∩ 𝔄 , such that the following hold:

(i) F_{0} ⊆ F,

(ii) θ|_{U∩H^{1}} = χ|_{U∩H^{1}},

(iii) χ is trivial on U_{0},

(iv) the character

\overline{ψ}_{b} : (U(𝔅) ∩ U)/(U^{1}(𝔅) ∩ U) → ℂ^{×}, u(U^{1}(𝔅) ∩ U) ↦ ψ_{b}(u)

defines a non-degenerate character of a maximal unipotent subgroup of U(𝔅)/U^{1}(𝔅).

Proof. Let us consider first, as in Step 1 of the proof of Theorem 2.6 the case when E is a maximal subfield of A. We identify V = E and 𝔄 = 𝔄(E) is the hereditary order associated to the lattice chain L(E) = {𝔭_{E}^{k} : k ∈ ℤ}. The parts (i),(iii) and (iv) are empty in this case since F_{0} = {0 ⊆ V }, U_{0} = {1}, and U(𝔅) = 𝔬_{E}^{×} does not contain non-trivial unipotent elements, and so U(𝔅) ∩ U = {1}. Consider the supercuspidal representation π = c-Ind_{E^{×}J}^{G} Λ, as in Step 1 of the proof of Theorem 2.6. According to [3] Proposition 1.6(i), there exist a maximal F -flag F in E, and a smooth character χ of the unipotent radical U of the G-stabiliser of F such that

Hom_{U∩(E^{×}J)}(χ, Λ) ≠ 0.

Since Λ|_{H^{1}} = (dim Λ)θ, we obtain

θ|_{U∩H^{1}} = χ|_{U∩H^{1}}.

Hence the theorem holds for E maximal. Note that, since E^{×} normalises H^{1} and θ = θ^{x}, for all x ∈ E^{×}, we may replace the pair (U, χ) by a conjugate (U^{x}, χ^{x}), where x ∈ E^{×}.

Now let us consider the general case. Put d = [E : F].

Construction of F and χ. According to [13] Proposition II-3 we may choose an E-basis B_{E} = {w_{1}, …, w_{r}} of V such that

F_{0} = {∑_{i=1}^{j} Ew_{i} : 1 ≤ j ≤ r}, and L_{k} = 𝔭_{E}^{k}w_{1} + ⋯ + 𝔭_{E}^{k}w_{r}, ∀k ∈ ℤ.

Let P_{0} be the G-stabiliser of F_{0}. Moreover, put G_{i} = Aut_{F} (Ew_{i}), for 1 ≤ i ≤ r, and

M_{0} = ∏_{i=1}^{r} G_{i},

a Levi component of the parabolic subgroup P_{0} of G. Then U_{0} is the unipotent radical of P_{0}, so that P_{0} = M_{0}U_{0}. We are in the situation considered in Step 2 of the proof of Theorem 2.6. In particular,

H^{1}(β, 𝔄) ∩ M_{0} = ∏_{i=1}^{r} H^{1}(β, 𝔄_{i}) ≅ ∏_{i=1}^{r} H^{1}(β, 𝔄(E)),

where the last isomorphism is induced by identifying Ew_{i} with E. Moreover, according to [2] Corollary 10.16, there exists θ_{F} ∈ C(𝔄(E), 0, β) such that

θ|_{H^{1}(β,𝔄)∩M_{0}} = θ_{F} ⊗ ⋯ ⊗ θ_{F}

via the above identification. Now, by the case when E is maximal considered above, we know that there exist a maximal F -flag F_{1} in E, and a smooth character χ_{1} : U_{F_{1}} → ℂ^{×} such that

χ_{1}(u) = θ_{F} (u), ∀u ∈ U_{F_{1}} ∩ H^{1}(β, 𝔄(E)),

where U_{F_{1}} is the unipotent radical of Aut_{F} (E)-stabiliser of F_{1}. Proposition 1.3 (i) implies that χ_{1} is non-degenerate. Choose an additive character ψ_{E} of E, such that ψ_{E}(x) = ψ_{F} (x), for all x ∈ F, and ψ_{E} trivial on 𝔭_{E}. Proposition 3.2 gives a (U_{F_{1}}, χ_{1}, ψ_{E})-balanced F -basis B_{1} = {x_{11}, … x_{1d}} of E. Set y = x_{1d}x_{11}^{−1} and for 2 ≤ j ≤ r, let B_{j} = {x_{j1}, …, x_{jd}} be the basis of E over F given by

x_{ji} = y^{j−1}x_{1i}, 1 ≤ i ≤ d.

Note that, in particular, x_{j1} = x_{j−1,d}, for 2 ≤ j ≤ r.

Let F_{j} = {∑_{i=1}^{k} F x_{ji} : 1 ≤ k ≤ d}; then F_{j} = yF_{j−1} and hence U_{F_{j}} = U_{F_{j−1}}^{y} so we may define a character χ_{j} : U_{F_{j}} → ℂ^{×} by χ_{j} = χ_{j−1}^{y}. Since y ∈ E^{×} normalises θ_{F}, we obtain that

χ_{j}(u) = θ_{F} (u), ∀u ∈ U_{F_{j}} ∩ H^{1}(β, 𝔄(E)), 1 ≤ j ≤ r.

Let F = {V_{k} : 1 ≤ k ≤ N} be the maximal F -flag in V given by

V_{(i−1)d+j} = F x_{11}w_{1} + ⋯ + F x_{ji}w_{i}, 1 ≤ i ≤ r, 1 ≤ j ≤ d,

and let U_{F} be the unipotent radical of the G-stabiliser of F. Since U_{F}/U_{0} ≅ U_{F} ∩ M_{0} ≅ ∏_{i=1}^{r} U_{i} we may define χ : U_{F} → ℂ^{×} by

χ|_{U_{F} ∩M_{0}} = χ_{1} ⊗ ⋯ ⊗ χ_{r}, χ|_{U_{0}} = 1.

The Iwahori decomposition implies that

\begin{matrix}U_{F} ∩ H^{1}(β, 𝔄) = & (U_{F} ∩ M_{0} ∩ H^{1}(β, 𝔄))(H^{1}(β, 𝔄) ∩ U_{0}) \\ ≅ & (∏_{i=1}^{r} U_{F_{i}} ∩ H^{1}(β, 𝔄(E)))(H^{1}(β, 𝔄) ∩ U_{0}) \end{matrix}

Since θ is trivial on U_{0} ∩ H^{1}(β, 𝔄), it follows that

θ(u) = χ(u), ∀u ∈ U ∩ H^{1}(β, 𝔄).

Construction of b. For μ = (μ_{1}, ..., μ_{r−1}) ∈ 𝔬_{F}^{r−1}, we define b = b(μ) ∈ X_{F} by

b(x_{ji}w_{j}) = \begin{cases}μ_{j}x_{j+1,1}w_{j+1}, & if i = d, 1 ≤ j ≤ r − 1; \\ 0 & otherwise. \end{cases}

We claim that such b also lies in 𝔄. For 1 ≤ j ≤ r we have constructed a (U_{F_{j}}, χ_{j}, ψ_{E})-balanced basis B_{j} = {x_{j1}, …, x_{jd}}, of E over F, so there exist functions a_{ji} : ℤ → ℤ, for 1 ≤ i ≤ d, such that

𝔭_{E}^{k} = ∑_{i=1}^{d} 𝔭_{F}^{a_{ji}(k)}x_{ji}, ∀k ∈ ℤ.

Hence,

L_{k} = ∑_{j=1}^{r} 𝔭_{E}^{k}w_{j} = ∑_{j=1}^{r} ∑_{i=1}^{d} 𝔭_{F}^{a_{ji}(k)}x_{ji}w_{j}, ∀k ∈ ℤ.

Since, by construction, x_{jd} = x_{j+1,1}, for 1 ≤ j ≤ r − 1, we have a_{jd}(k) = a_{j+1,1}(k) for all k ∈ ℤ. Hence

bL_{k} = ∑_{j=1}^{r−1} μ_{j}𝔭_{F}^{a_{jd}(k)}x_{j+1,1}w_{j+1} ⊆ L_{k}, ∀k ∈ ℤ

which implies that b ∈ 𝔄.

Since b ∈ X_{F}, Lemma 1.1 implies that ψ_{b} defines a linear character of U. Since b ∈ 𝔄 we have ψ_{b}(u) = 1 for all u ∈ U^{1}(𝔄). Hence the character \overline{ψ}_{b} is well defined. Moreover, (U(𝔅) ∩ U)/(U^{1}(𝔅) ∩ U) is the unipotent radical of the Aut_{𝔨_{E}}(L_{0}/L_{1})-stabiliser of the flag {∑_{j=1}^{i} 𝔨_{E}(w_{j} + L_{1}) : 0 ≤ i ≤ r}. This follows from Lemma 2.1.

Let μ = (1, ..., 1) and let b = b(μ) as above. (Indeed, any μ ∈ (𝔬_{F}^{×})^{r−1} would do.) We claim that \overline{ψ}_{b} is non-degenerate. For 2 ≤ k ≤ r and ξ ∈ E we define u_{ξ,k} ∈ U ∩ B^{×} = U_{0} ∩ B^{×} by

u_{ξ,k}(w_{j}) = \begin{cases}w_{k} + ξw_{k−1}, & if k = j; \\ w_{j}, & otherwise. \end{cases}

We will compute (χψ_{b})(u_{ξ,k}) = ψ_{b}(u_{ξ,k}). Since x_{ji} ∈ E, it commutes with u_{ξ,k} and hence

(u_{ξ,k} − 1)(x_{ji}w_{j}) = \{ \begin{matrix}ξx_{ki}w_{k−1} & if j = k , \\ 0 & otherwise. \end{matrix}

Since B_{k−1} is an F -basis of E we may write uniquely

ξx_{ki} = ∑_{j=1}^{d} λ_{ji}^{(k)}x_{k−1,j}

where λ_{ji}^{(k)} ∈ F. Then

(b(u_{ξ,k} − 1))(x_{ji}w_{j}) = \{ \begin{matrix}λ_{di}^{(k)}x_{k1}w_{k} & if j = k , \\ 0 & otherwise. \end{matrix}

Hence,

tr_{A/F} (b(u_{k} − 1)) = λ_{d1}^{(k)}.

Since, by construction, x_{k1} = x_{k−1,d}, we have λ_{d1}^{(k)} = ξ_{dd}^{(k)}, where (ξ_{ij}^{(k)}) is the matrix of ξ ∈ End_{F} (E) with respect to B_{k−1}. However, since B_{k} is (U_{F_{k}}, χ_{k}, ψ_{E})-balanced

ψ_{b}(u_{ξ,k}) = ψ_{F} (ξ_{dd}^{(k)}) = ψ_{E}(ξ).

Now, ψ_{E} has conductor 𝔭_{E}, and so \overline{ψ}_{b} is non-degenerate.

We record the following corollary to the proof of the theorem.

Corollary 3.4. Fix an additive character ψ_{E} : E → ℂ^{×} such that ψ_{E}(x) = ψ_{F} (x), for all x ∈ F and ψ_{E} is trivial on 𝔭_{E}. Choose an E-basis B_{E} = {w_{1}, …, w_{r}} of V such that

F_{0} = {∑_{i=1}^{j} Ew_{i} : 1 ≤ j ≤ r}, and L_{k} = 𝔭_{E}^{k}w_{1} + ⋯ + 𝔭_{E}^{k}w_{r}, ∀k ∈ ℤ.

Let F, χ and ψ_{b} be as in Theorem 3.3. There exists a basis {x_{1}, …, x_{d}} of E over F, satisfying Definition 3.1(iv) with respect to ψ_{F} and ψ_{E}, such that x_{1} = 1 and such that, if we putt

v_{d(i−1)+j} = x_{d}^{i−1}x_{j}w_{i}, 1 ≤ i ≤ r, 1 ≤ j ≤ d,

then the set B_{F} = {v_{1}, …, v_{N}} is an F -basis of V with the following properties:

(i) F = {∑_{i=1}^{j} F v_{i} : 1 ≤ j ≤ N};

(ii) there exist functions a_{i} : ℤ → ℤ, for 1 ≤ i ≤ N such that

L_{k} = ∑_{i=1}^{N} 𝔭_{F}^{a_{i}(k)}v_{i}, ∀k ∈ ℤ;

(iii) if u ∈ U has matrix (u_{ij}) with respect to B_{F}, then

(χψ_{b})(u) = ψ_{F} (∑_{i=1}^{N−1} u_{i,i+1});

(iv) if u ∈ U ∩ B^{×} has matrix (u_{ij}) with respect to B_{E}, then

(χψ_{b})(u) = ψ_{E}(∑_{i=1}^{r−1} u_{i,i+1}).

If we put β = 0, so that E = F, 𝔬_{E} = 𝔬_{F} and r = N then the proof of Theorem 3.3 formally goes through with F = F_{0}, B_{j} = {1}, for 1 ≤ j ≤ N, and b = b(μ), for any μ ∈ (𝔬_{F}^{×})^{N−1}. We obtain a ‘level zero version’ of Theorem 3.3:

Corollary 3.5. Let 𝔄 be a maximal 𝔬_{F} -order in A. Then there exist a maximal F -flag F in V and an element b ∈ X_{F} ∩ 𝔄 such that

\overline{ψ}_{b} : (U(𝔄) ∩ U)/(U^{1}(𝔄) ∩ U) → ℂ^{×}, u(U^{1}(𝔄) ∩ U) ↦ ψ_{b}(u)

defines a non-degenerate character of a maximal unipotent subgroup of the group U(𝔄)/U^{1}(𝔄).

4 Supercuspidal representations

Let π be an irreducible supercuspidal representation of G. By [6] §6, π is compactly induced:

π ≅ c-Ind_{J}^{G} Λ,

from a (rather special) open compact-mod-centre subgroup J of G. More precisely, J has a unique maximal compact open subgroup J and if we put λ := Λ|_{J} then (J, λ) is a maximal simple type in the sense of [6] §6.

Definition 4.1. The pair (J, λ) is a maximal simple type if one of the following holds:

(a) J = J(β, 𝔄) is a subgroup associated to a simple stratum [𝔄, n, 0, β], such that if we write E = F [β] and B = End_{E}(V) then 𝔅 = 𝔄 ∩ B a maximal 𝔬_{E}-order in B. Moreover, there exists a simple character θ ∈ C(𝔄, 0, β, ψ_{F}) such that

λ ≅ κ ⊗ σ,

where κ is a β-extension of the unique irreducible representation η of J^{1} = J^{1}(β, 𝔄), which contains θ, and σ is the inflation to J of a cuspidal representation of J/J^{1} ≅ U(𝔅)/U^{1}(𝔅) ≅ GL_{r}(𝔨_{E}).

(b) (J, λ) = (U(𝔄), σ), where 𝔄 is a maximal hereditary 𝔬_{F} -order in A and σ is an inflation of a cuspidal representation of U(𝔄)/U^{1}(𝔄) ≅ GL_{N}(𝔨_{F}).

In case (a), J = E^{×}J, and in case (b), J = F^{×}U(𝔄). In practice we will treat (b) as a special case of (a), with β = 0, E = F, 𝔅 = 𝔄, J^{1} = H^{1} = U^{1}(𝔄) and θ, η, κ all trivial. We will refer to (b) as the level zero case.

We are going to describe the main result of this section. Let F = {V_{i} : 1 ≤ i ≤ N} be any maximal F -flag in V, let U be the unipotent radical of the Gstabiliser of F, and let ψ_{α} : U → ℂ^{×} be any non-degenerate character of U. Let π be a supercuspidal representation of G. We know by [3] Proposition 1.6 that there exists (J, Λ) as above, such that

π ≅ c-Ind_{J}^{G} Λ, Hom_{U∩J}(ψ_{α}, Λ) ≠ 0.

Moreover we know by Proposition 1.3 that the above properties determine such a pair (J, Λ) up to conjugation by u ∈ U. Since Λ|_{H^{1}} = (dim Λ)θ, we obtain that

θ(u) = ψ_{α}(u), ∀u ∈ U ∩ H^{1}.

Since J normalises H^{1} and intertwines θ, we may define:

Definition 4.2. Let Ψ_{α} : (J ∩ U)H^{1} → ℂ^{×} be the character given by

Ψ_{α}(uh) = ψ_{α}(u)θ(h), ∀u ∈ J ∩ U, ∀h ∈ H^{1}.

We also define the following subgroups:

Definition 4.3. Set

M_{F} = {g ∈ G : (g − 1)V ⊆ V_{N−1}}, M_{𝔄} = (M_{F} ∩ U(𝔄))U^{1}(𝔄),

and

G_{F} = {g ∈ G : gv_{1} = v_{1}, ∀v_{1} ∈ V_{1}}, G_{𝔄} = (G_{F} ∩ U(𝔄))U^{1}(𝔄).

Let 𝔎(𝔄) = {g ∈ G : g^{−1}𝔄g = 𝔄} be the G-normaliser of 𝔄 and let ρ be the representation of 𝔎(𝔄) given by

ρ = Ind_{J}^{𝔎(𝔄)} Λ.

The main result of this Section is the following Theorem.

Theorem 4.4. (i) The restriction Λ|_{M_{𝔄}∩J} is an irreducible representation of M_{𝔄} ∩ J. Moreover,

Λ|_{M_{𝔄}∩J} ≅ Ind_{(J∩U)H^{1}}^{M_{𝔄}∩J} Ψ_{α}.

(ii) The restriction ρ|_{M_{𝔄}} is an irreducible representation of M_{𝔄}. Moreover,

ρ|_{M_{𝔄}} ≅ Ind_{(J∩U)H^{1}}^{M_{𝔄}} Ψ_{α}.

Further, both (i) and (ii) hold if we replace M_{𝔄} with G_{𝔄}.

The strategy is to show that Theorem 4.4 holds for a particular choice of U and ψ_{α}, constructed from Theorem 3.3, and then show that the general result may be obtained by conjugating by some g ∈ J. Before proceeding with the proof we note that Ψ_{α} occurs in Λ with multiplicity 1, since ψ_{α} occurs in Λ with multiplicity 1 and the proof of [3] Lemma 3.1 implies:

Corollary 4.5. The character Ψ_{α} occurs in π with multiplicity 1.

We note that the level 0 case can be formally recovered from the general case with β = 0 and θ the trivial character, and is a well known result of Gel′fand [7].

4.1 Some decompositions

We will need some decompositions of G, and also of other general linear groups, so we state the following Theorem for a general field F.

Theorem 4.6. Let F be any field, let V be an N-dimensional F-vector space, set G = Aut_{F}(V). Let K be an extension of F of degree N and suppose that we are given an embedding of algebras ι : K ↪ End_{F}(V). Then the following hold:

(i) Let v ∈ V, v ≠ 0, and put G_{v} = {g ∈ G : gv = v}; then every g ∈ G can be uniquely decomposed as

g = xh, x ∈ ι(K^{×}), h ∈ G_{v};

(ii) Let V′ ⊂ V be an F-subspace of V of dimension N − 1 and set M = {g ∈ G : (g − 1)V ⊆ V′}; then every g ∈ G can be uniquely decomposed as

g = xh, x ∈ ι(K^{×}), h ∈ M.

Proof. We can view V as a K-vector space, via ι. Since, [K : F] = dim_{F} V, we obtain dim_{K} V = 1. Hence K^{×} acts transitively on the set of non-zero vectors in V and that G = ι(K^{×})G_{v}. If x ∈ G_{v} then it has eigenvalue 1, and so ι(K^{×}) ∩ G_{v} = {1}. This establishes Part (i).

Choose a basis B = {v_{1}, …, v_{N}} of V such that V′ = Fv_{1} + ⋯ + Fv_{N−1}. Let δ : G → G be the map g ↦ w(g^{⊤})^{−1}w, where w ∈ G is defined on B by w(v_{i}) = v_{N−i+1}, and g^{⊤} denotes the transpose of g with respect to the basis B. We have δ^{2} = id and δ(M) = G_{v_{1}}. Part (ii) follows from Part (i) with ι replaced with δ ∘ ι and v = v_{1}.

We apply our decomposition theorem to prove several results on the intersection with the groups M_{F} and G_{F}. Analogously, we set

M_{E} = {g ∈ B^{×} : (g − 1)V ⊆ V_{N−d}}, M_{𝔅} = (M_{E} ∩ U(𝔅))U^{1}(𝔅)

and

G_{E} = {g ∈ B^{×} : gw_{1} = w_{1}}, G_{𝔅} = (G_{E} ∩ U(𝔅))U^{1}(𝔅).

Let K be a maximal unramified extension of E, which normalises 𝔄, so that [K : E] = N/d. Theorem 4.6 implies that B^{×} = K^{×}M_{E}. Since V_{N−d} ⊆ V_{N−1}, we obtain that

M_{F} ∩ B^{×} = M_{E}.

Corollary 4.7. The following decompositions hold:

(i) 𝔎(𝔄) = K^{×}(U(𝔄) ∩ M_{F}), U^{1}(𝔄) = (U^{1}(𝔄) ∩ M_{F})(1 + 𝔭_{K});

(ii) J = K^{×}(J ∩ M_{F}), J^{1} = (J^{1} ∩ M_{F})(1 + 𝔭_{K});

(iii) 𝔎(𝔅) = K^{×}(U(𝔅) ∩ M_{E}), U^{1}(𝔅) = (U^{1}(𝔅) ∩ M_{E})(1 + 𝔭_{K}).

Moreover, the statements remain true if we replace M_{F} with G_{F} and M_{E} with G_{E}.

Proof. Note, that K^{×} ⊂ 𝔎(𝔅) ⊂ J ⊂ 𝔎(𝔄). It is enough to prove (i), since (ii) and (iii) are obtained by intersecting with J and 𝔎(𝔅), respectively. According to Theorem 4.6 we may write 𝔎(𝔄) = K^{×}(𝔎(𝔄) ∩ M_{F}).

Let L = {L_{i} : i ∈ ℤ} be the 𝔬_{E}-lattice chain in V associated to 𝔄. According to [13] Theorem II-1 there exists a decomposition of V = ∑_{i=1}^{N} V^{i} into one dimensional subspaces such that V_{N−1} = ∑_{i=1}^{N−1} V^{i} and L_{j} = ∑_{i=1}^{N} L_{j} ∩ V^{i}, for all j ∈ ℤ. If g ∈ 𝔎(𝔄) ∩ M_{F} then by projecting to the V^{N} subspace we obtain L_{j+v_{𝔄}(g)} ∩ V^{N} = L_{j} ∩ V^{N}, for all j ∈ ℤ. This implies that v_{𝔄}(g) = 0, hence g ∈ U(𝔄) ∩ M_{F}. Hence 𝔎(𝔄) ∩ M_{F} = U(𝔄) ∩ M_{F}.

Let g ∈ U^{1}(𝔄). We may write by above g = hx, where h ∈ U(𝔄) ∩ M_{F} and x ∈ 𝔬_{K}^{×}. Let ḡ, h̄ and x̄ be the images of g, h, x in Aut_{𝔨_{F}}(L_{m}/L_{m+1}), where L_{m} ∈ L, such that L_{m+1} + L_{m} ∩ V_{N−1} ≠ L_{m}. Since g ∈ U^{1}(𝔄), we have ḡ = 1. Now

(h̄ − 1)(L_{m}/L_{m+1}) ⊆ (L_{m} ∩ V_{N−1} + L_{m+1})/L_{m+1}.

Our assumption on L_{m}, implies that h̄ has eigenvalue 1. Since h̄x̄ = 1, x̄ also has eigenvalue 1, hence x̄ = 1, which implies that x ∈ 1 + 𝔭_{K}, and so h ∈ U^{1}(𝔄) ∩ M_{F}.

Corollary 4.8. We have

𝔎(𝔄) = M_{𝔄}J, M_{𝔄} ∩ J = (M_{F} ∩ J)J^{1} = M_{𝔅}J^{1}.

Moreover, the statement is true if we replace M_{𝔄} with G_{𝔄}; M_{F} with G_{F} and M_{B} with G_{B}.

We end this section with an observation which will prove useful later.

Lemma 4.9. Suppose that g ∈ 𝔎(𝔄) then

v_{F} (det g) = \frac{Nv_{𝔄}(g)}{e(𝔄|𝔬_{F})}.

In particular, J ∩ U = J ∩ U.

Proof. Let e = e(𝔄|𝔬_{F}) and set h = g^{e}ϖ_{F}^{−v_{𝔄}(g)}, then h ∈ 𝔎(𝔄) and v_{𝔄}(h) = 0. Hence, h ∈ U(𝔄). Since U(𝔄) is compact, this implies that det h ∈ 𝔬_{F}^{×} and this yields the Lemma.

4.2 The proof of Theorem 4.4 in a special case

Let us fix a maximal simple type (J, λ). Let [𝔄, n, 0, β] be an associated simple stratum, and let θ ∈ C(𝔄, 0, β) be such that λ|_{H^{1}} = (dim λ)θ. Let (F, χ, b) be a triple given by Theorem 3.3 and let U be the unipotent radical of the Gstabiliser of F. By construction, a subflag F_{0} := {V_{di} : 1 ≤ i ≤ r} of F is a maximal stable E-flag in V, and let B_{E} = {w_{1}, … w_{r}} denote an E-basis of V chosen as in Corollary 3.4. We choose some a ∈ X_{F} such that χ = ψ_{a}, and we set

α = a + b.

Then we have

ψ_{α}(u) = ψ_{a}(u)ψ_{b}(u) = ψ_{a}(u) = θ(u), ∀u ∈ U ∩ H^{1}.

The first equality is trivial; the second holds, since b ∈ 𝔄 and u ∈ U^{1}(𝔄); the third is Theorem 3.3(ii). Hence we may define Ψ_{α} : (J ∩ U)H^{1} → ℂ^{×} by Ψ_{α}(uh) = ψ_{α}(u)θ(h), for u ∈ J ∩ U and h ∈ H^{1}, as above.

Lemma 4.10. The function ψ_{b} defines a linear character on (J ∩ U)J^{1} and

Ψ_{α}(j) = Θ(j)ψ_{b}(j), ∀j ∈ (J ∩ U)H^{1}

where the character Θ is given by Θ(uh) = ψ_{a}(u)θ(h), for u ∈ (J ∩ U) and h ∈ H, as in Theorem 2.6.

Proof. Since b ∈ X_{F}, by Lemma 1.1, ψ_{b} defines a linear character of U. Since b ∈ 𝔄, we have ψ_{b}(u) = 1, for all u ∈ U^{1}(𝔄). This implies that

ψ_{b}(uj) = ψ_{b}(u), ∀u ∈ U ∩ J, j ∈ J^{1}.

Now J normalises J^{1} and hence ψ_{b} is a character on (J∩U)J^{1}. Since α = a+b, we have an equality of functions ψ_{α} = ψ_{a}ψ_{b} and hence for every u ∈ J ∩ U and h ∈ H^{1} we have

Ψ_{α}(uh) = ψ_{b}(u)ψ_{a}(u)θ(h) = ψ_{b}(uh)Θ(uh).

Let L = {L_{i} : i ∈ ℤ} be the 𝔬_{E}-lattice chain in V associated to 𝔄, and put

M_{𝔨_{E}} = {b ∈ U(𝔅)/U^{1}(𝔅) : (b − 1)(L_{0}/L_{1}) ⊆ ∑_{j=1}^{r−1} 𝔨_{E}(w_{j} + L_{1})} ,

the image of M_{𝔅} in U(𝔅)/U^{1}(𝔅). Similarly, let

G_{𝔨_{E}} = {b ∈ U(𝔅)/U^{1}(𝔅) : b(w_{1} + L_{1}) = w_{1} + L_{1}}

be the image of G_{𝔅} in U(𝔅)/U^{1}(𝔅).

The group M_{𝔨_{E}} is known as a mirabolic subgroup. We note that it contains the image of U(𝔅) ∩ U in U(𝔅)/U^{1}(𝔅). Moreover, with respect to the basis w_{1} + L_{1}, ..., w_{r} + L_{1} of L_{0}/L_{1}, the group M_{𝔨_{E}} is identified with the subgroup of matrices of the form

(\begin{matrix}∗ & ⋯ & ∗ & ∗ \\ ⋮ & ... & ⋮ & ⋮ \\ ∗ & ⋯ & ∗ & ∗ \\ 0 & ⋯ & 0 & 1 \end{matrix}) .

and the image of U(𝔅) ∩ U in U(𝔅)/U^{1}(𝔅) is identified with the subgroup of unipotent upper-triangular matrices.

Lemma 4.11. Let σ be a cuspidal representation of U(𝔅)/U^{1}(𝔅). Let U_{𝔨_{E}} be the image of U(𝔅) ∩ U in U(𝔅)/U^{1}(𝔅), so that

U_{𝔨_{E}} ≅ (U(𝔅) ∩ U)/(U^{1}(𝔅) ∩ U),

and let ψ be any non-degenerate character of U_{𝔨_{E}}. Then

σ|_{M_{𝔨_{E}}} ≅ Ind_{U_{𝔨_{E}}}^{M_{𝔨_{E}}} ψ, σ|_{G_{𝔨_{E}}} ≅ Ind_{U_{𝔨_{E}}}^{G_{𝔨_{E}}} ψ.

Moreover, the representations σ|_{M_{𝔨_{E}}} and σ|_{G_{𝔨_{E}}} are irreducible representations of M_{𝔨_{E}} and G_{𝔨_{E}}, respectively.

Proof. The statement for M_{𝔨_{E}} is [7] Theorem 8. Set w̄_{i} = w_{i}+L_{1}, for 1 ≤ i ≤ r, and B_{𝔨_{E}} = {w̄_{1}, …, w̄_{r}}, a 𝔨_{E}-basis of L_{0}/L_{1}. We identify U(𝔅)/U^{1}(𝔅) with GL_{r}(𝔨_{E}), via B_{𝔨_{E}}. Let δ : GL_{r}(𝔨_{E}) → GL_{r}(𝔨_{E}) be the automorphism given by δ(g) = w(g^{⊤})^{−1}w, where g^{⊤} denotes the transpose of g and w is given by w(w̄_{i}) = w̄_{r−i+1}, for 1 ≤ i ≤ r. Then δ(U_{𝔨_{E}}) = U_{𝔨_{E}}, and ψ^{δ}, given by ψ^{δ}(u) = ψ(δ(u)), for all u ∈ U_{𝔨_{E}}, is a non-degenerate character. The representation of GL_{r}(𝔨_{E}), given by σ^{δ}(g) = σ(δ(g)) is cuspidal. Hence, by the statement for M_{𝔨_{E}},

σ^{δ}|_{M_{𝔨_{E}}} ≅ Ind_{U_{𝔨_{E}}}^{M_{𝔨_{E}}} ψ^{δ}.

Since δ(G_{𝔨_{E}}) = M_{𝔨_{E}} and δ^{2} = id, by twisting by δ we obtain

σ|_{G_{𝔨_{E}}} ≅ Ind_{U_{𝔨_{E}}}^{G_{𝔨_{E}}} ψ.

The irreducibility follows from the irreducibility of σ^{δ}|_{M_{𝔨_{E}}}.

Theorem 4.12. The restriction λ|_{M_{𝔅}J^{1}} is an irreducible representation of M_{𝔅}J^{1} and

λ|_{M_{𝔅}J^{1}} ≅ Ind_{(J∩U)H^{1}}^{M_{𝔅}J^{1}} Ψ_{α}.

The statement remains true if we replace M_{𝔅} with G_{𝔅}.

Proof. Set M = M_{𝔅}. In the level 0 case, M = M_{𝔄} and the Theorem asserts that σ|_{M_{𝔄}} is irreducible and

σ|_{M_{𝔄}} ≅ Ind_{(U∩U(𝔄))U^{1}(𝔄)}^{M_{𝔄}} ψ_{b}.

Since σ is an inflation of a cuspidal representation and F and b were chosen in Corollary 3.5 so that \overline{ψ}_{b} is non-degenerate, the assertion is Lemma 4.11.

Let us consider the general case. Since σ is an inflation of a cuspidal representation of U(𝔅)/U^{1}(𝔅) and F and b were chosen in Theorem 3.3, so that the character \overline{ψ}_{b} is non-degenerate, we may again apply Lemma 4.11 to obtain

λ|_{MJ^{1}} ≅ κ|_{MJ^{1}} ⊗ σ|_{MJ^{1}} ≅ κ|_{MJ^{1}} ⊗ Ind_{(J∩U)J^{1}}^{MJ^{1}} ψ_{b} ≅ Ind_{(J∩U)J^{1}}^{MJ^{1}} κ ⊗ ψ_{b}.

Theorem 2.6 implies that κ|_{(J∩U)J^{1}} ≅ Ind_{(J∩U)H^{1}}^{(J∩U)J^{1}} Θ and hence

λ|_{MJ^{1}} ≅ Ind_{(J∩U)J^{1}}^{MJ^{1}} Ind_{(J∩U)H^{1}}^{(J∩U)J^{1}} Θ ⊗ ψ_{b} ≅ Ind_{(J∩U)H^{1}}^{MJ^{1}} Ψ_{α}

where the last isomorphism is given by Lemma 4.10 and the transitivity of induction. Moreover, since by Lemma 4.11 the restriction σ|_{MJ^{1}} is irreducible, a straightforward modification of [6] (5.3.2) implies that λ|_{MJ^{1}} is an irreducible representation of MJ^{1}. The proof for G_{𝔅} is analogous.

Since, by Corollary 4.8, we have M_{𝔄} ∩ J = M_{𝔅}J^{1}, we have now proved Theorem 4.4(i) in our special case. We also record the following: Since J ∩ U = J ∩ U, we immediately get Corollary 4.13. Hom_{U∩J}(ψ_{α}, Λ) ≠ 0.

Finally, part (ii) of Theorem 4.4 is given by:

Proposition 4.14. The restrictions of ρ to M_{𝔄} and to G_{𝔄} are irreducible representations of M_{𝔄} and G_{𝔄}, respectively. Moreover,

ρ|_{M_{𝔄}} ≅ Ind_{(J∩U)H^{1}}^{M_{𝔄}} Ψ_{α}, ρ|_{G_{𝔄}} ≅ Ind_{(J∩U)H^{1}}^{G_{𝔄}} Ψ_{α}.

Proof. We will prove statement for M_{𝔄}, the proof for G_{𝔄} is analogous. We have

ρ|_{M_{𝔄}} ≅ Ind_{M_{𝔄}∩J}^{M_{𝔄}} Λ|_{M_{𝔄}∩J} ≅ Ind_{M_{𝔅}J^{1}}^{M_{𝔄}} λ|_{M_{𝔅}J^{1}} ≅ Ind_{(J∩U)H^{1}}^{M_{𝔄}} Ψ_{α}

where the last two isomorphisms follow from Corollary 4.8 and Theorem 4.12. Since Ψ_{α}|_{H^{1}} = θ we have

I_{G}(Ψ_{α}) ⊆ I_{G}(θ) = J^{1}B^{×}J^{1}

by [6] (3.3.2). Since λ|_{M_{𝔅}J^{1}} ≅ Ind_{(J∩U)H^{1}}^{M_{𝔅}J^{1}} Ψ_{α}, [6] (4.1.1) and (4.1.5) imply that

I_{G}(λ|_{M_{𝔅}J^{1}}) = M_{𝔅}J^{1}I_{G}(Ψ_{α})M_{𝔅}J^{1} ⊆ J^{1}B^{×}J^{1}.

Hence,

I_{M_{𝔄}}(λ|_{M_{𝔅}J^{1}}) ⊆ M_{𝔄} ∩ (J^{1}B^{×}J^{1}) = (U(𝔅) ∩ M_{𝔄})J^{1} = M_{𝔅}J^{1}

and hence ρ|_{M_{𝔄}} is irreducible. We note that M_{𝔄} contains U^{1}(𝔄), and hence J^{1}; U(𝔄) ∩ B^{×} = U(𝔅) and the last equality above follows from Corollary 4.7.

This completes the proof of Theorem 4.4 for our special choice of F and ψ_{α}.

4.3 The proof of Theorem 4.4 in the general case

We will prove Theorem 4.4 in the general case, by showing that after conjugation by some g ∈ J we end up in the special case, considered above.

Proof. Let F′ = {V′_{i} : 1 ≤ i ≤ N} be any maximal F -flag in V, and let U′ be the unipotent radical of the G-stabiliser of F′, and let ψ_{α′} be any smooth non-degenerate character of U′. Let π be a supercuspidal representation of G, then there exists a pair (J, Λ), such that π ≅ c-Ind_{J}^{G} Λ and Hom_{U′∩J}(ψ_{α′}, Λ) ≠ 0, and Λ|_{J} ≅ λ, where J is the maximal compact open subgroup of J, and (J, λ) is a maximal simple type, with the stratum [𝔄, n, 0, β]. Moreover, λ|_{H^{1}} = (dim λ)θ, where θ ∈ C(𝔄, 0, β). We define M′_{𝔄} as in Definition 4.3 and Ψ_{α′} as in Definition 4.2. Let F = {V_{i} : 1 ≤ i ≤ N}, U, ψ_{α} and M_{𝔄} be as in §4.2. By Corollary 3.4 the character ψ_{α} is non-degenerate, and we know that

Hom_{U′∩J}(ψ_{α′}, Λ) ≠ 0, and Hom_{U∩J}(ψ_{α}, Λ) ≠ 0.

Hence by [3] Proposition 1.6 (ii), there exists g ∈ J, such that U′ = U^{g} and ψ_{α′} = ψ_{α}^{g}. In particular, F′ = gF, and hence V′_{N−1} = gV_{N−1}, which implies that M′_{𝔄} = M_{𝔄}^{g}. Since g ∈ J, we have J = J^{g}, H^{1} = (H^{1})^{g}, θ = θ^{g}. Hence (J ∩ U′)H^{1} = ((J ∩ U)H^{1})^{g} and Ψ_{α′} = Ψ_{α}^{g}. We have proved the Theorem for F and ψ_{α}, now twisting by g, we obtain the result for F′ and ψ_{α′}.

Remark 4.15. It follows from the proof that any F′ and ψ_{α′}, with the property that Hom_{U′∩J}(ψ_{α′}, Λ) ≠ 0, arise from the construction in Theorem 3.3, once we replace β by gβg^{−1}, for some g ∈ J, and so the construction in Theorem 3.3 is a natural one.

4.4 A characterisation of Ind_{J}^{𝔎(𝔄)} Λ

We observe that a result of Gel′fand characterising cuspidal representations of GL_{N}(F_{q}) implies a very similar result, for the representations of 𝔎(𝔄) of the form Ind_{J}^{𝔎(𝔄)} Λ.

Proposition 4.16. Let τ be a representation of 𝔎(𝔄) such that

τ|_{M_{𝔄}} ≅ Ind_{(J∩U)H^{1}}^{M_{𝔄}} Ψ_{α},

then

τ ≅ Ind_{J}^{𝔎(𝔄)} Λ,

for some representation Λ of J, such that (J, Λ|_{J}) is a maximal simple type, as in Definition 4.1.

Proof. Since Ind_{(J∩U)H^{1}}^{M_{𝔄}} Ψ_{α} is irreducible, so is τ. Now,

Ind_{(J∩U)H^{1}}^{J} Ψ_{α} ≅ κ ⊗ Ind_{(J∩U)J^{1}}^{J} ψ_{b} ≅ ∏_{σ} κ ⊗ σ,

where the product runs over all the generic representations σ of J/J^{1} ≅ U(𝔅)/U^{1}(𝔅) ≅ GL_{r}(𝔨_{E}). Hence τ|_{J} will contain a summand of the form κ⊗ σ. It follows from [6] (5.3.2) that the I_{U(𝔄)}(κ ⊗ σ) ⊆ (JB^{×}J) ∩ U(𝔄) = J. Hence Ind_{J}^{U(𝔄)} κ ⊗ σ is irreducible, and so is isomorphic to τ. Restricting to M_{𝔄}, we obtain that σ|_{M_{𝔅}J^{1}} is irreducible, and [7] implies that σ is cuspidal. Hence, τ|_{J} will contain some simple type λ, it follows from [6] §6.2 that τ|_{J} will contain some Λ, and hence τ ≅ Ind_{J}^{𝔎(𝔄)} Λ.

5 Realisation of maximal simple types

We continue with the situation described in the beginning of §4. Let U be a maximal unipotent subgroup of G, and ψ_{α} non-degenerate character of U. Let π ≅ c-Ind_{J}^{G} Λ be a supercuspidal representation, and Hom_{U∩J}(ψ_{α}, Λ) ≠ 0. Theorem 4.4 allows us to use a rather general result of Alperin and James [1], and realize the representation Λ as a concrete space, and describe the action of J on this space in terms of the character of Λ and Ψ_{α}. This concrete realization enables us to compute a certain matrix coefficient of π, and by integrating it we obtain an explicit Whittaker function for π.

5.1 Bessel functions

We will adapt the result of Alperin and James [1] to our setting. Let K be an open, compact-modulo-centre subgroup of G and let τ be an irreducible smooth representation of K.

Assumption 5.1. Suppose that there exists compact open subgroups U ⊆ M ⊆ K, and a linear character Ψ of U, such that the following hold:

(i) τ|_{M} is irreducible representation of M;

(ii) τ|_{M} ≅ Ind_{U}^{M} Ψ.

Let N be an open, normal subgroup of K contained in the Ker τ. Set

e_{Ψ} = (U : N)^{−1} ∑_{h∈U/N} Ψ(h)h^{−1}.

Let χ = χ_{τ} be the (trace) character of τ and let ω = ω_{τ} be the central character of τ, so that

χ(xg) = ω(x)χ(g), ∀x ∈ F^{×}, ∀g ∈ K.

Definition 5.2. The Bessel function J : K → ℂ of τ is defined by:

J (g) = tr_{τ}(e_{Ψ}g) = (U : N)^{−1} ∑_{h∈U/N} Ψ(h^{−1})χ(gh).

Proposition 5.3. The Bessel function J has the following properties:

(i) J (1) = 1;

(ii) J (xg) = J (gx) = ω(x)J (g), ∀x ∈ F^{×}, ∀g ∈ K;

(iii) J (hg) = J (gh) = Ψ(h)J (g), ∀h ∈ U, ∀g ∈ K;

(iv) if J (g) ≠ 0 then g intertwines Ψ; in particular, if m ∈ M then J (m) ≠ 0 if and only if m ∈ U;

(v) for all g_{1}, g_{2} ∈ K we have

∑_{m∈M/U} J (g_{1}m)J (m^{−1}g_{2}) = J (g_{1}g_{2}).

Proof. We observe that it is enough to prove the Proposition for a twist of τ by an unramified character. Twisting by [g ↦ (ω(ϖ_{F}))^{−v_{F} (det(g))/N}] we ensure that ϖ_{F}^{ℤ}N lies in the kernel of τ. Hence, Ker τ is of finite index in K and we may consider τ as a representation of a finite group.

Part (i) is a reformulation of the fact that Ind_{U}^{M} Ψ is irreducible. Since χ is defined by matrix trace, we have χ(gg_{1}) = χ(g_{1}g), for all g, g_{1} ∈ K, and Parts (ii) and (iii) are straightforward consequences of the definition of J.

Part (iv): Part (iii) implies that J is a Ψ̌-spherical function on K, in the sense of [6] (4.1), where Ψ̌ is the dual of Ψ. Hence if J (g) ≠ 0 then according to [6] (4.1.1), g intertwines Ψ.

Since Ind_{U}^{M} Ψ is irreducible, the M-intertwining of Ψ is equal to U. Now Parts (i),(iii) and the argument above finish the proof of Part (iv).

Part (v) is [1] Lemma 2, or [7] Theorem 9.

Theorem 5.4 (cf. [1]). Let S be the space of functions from M to ℂ satisfying the condition

f(hm) = Ψ(h)f(m), ∀h ∈ U, ∀m ∈ M.

For each g ∈ K we define an operator L(g) on S by the formula

[L(g)f](m) = ∑_{m_{1}∈M/U} J (mgm_{1})f(m_{1}^{−1}).

Then L defines a representation of K, which is isomorphic to τ.

Proof. Again, it is enough to prove the statement after twisting by unramified character, and this way we may ensure that K/ Ker τ is finite. The assertion now follows from the main Theorem in [1]. The level 0 case is [7] Theorem 10.

If τ̌ is the dual of τ, then τ̌, M, U and Ψ̌ satisfy Assumption 5.1. Hence Theorem 5.4 holds for τ̌, with Ψ̌ instead of Ψ and with Š the space of functions from M to ℂ satisfying the condition

f(hm) = Ψ̌(h)f(m), ∀h ∈ U, ∀m ∈ M.

Moreover, the Bessel function J̌ = Jτ̌ satisfies

J̌(g) = J (g^{−1}), ∀g ∈ K.

Let ( , ) be a non-degenerate K-invariant pairing on S × Š. Since, τ is irreducible, the pairing is determined up to a scalar multiple. Let φ ∈ S and φ̌ ∈ Š be such that Supp φ = Supp φ̌ = U and φ(1) = φ̌(1) = 1. Since φ and φ̌ span the Ψ- and Ψ̌-isotypical subspaces in S and Š respectively, we may normalise ( , ), so that

(φ, φ̌) = 1.

This determines the pairing uniquely.

Lemma 5.5. We have

(L(g)φ, φ̌) = J (g), ∀g ∈ K.

Proof. It follows from Theorem 5.4 that

L(g)φ = ∑_{m∈M/U} J (m^{−1}g)L(m)φ.

Since the M-intertwining of Ψ is just U, for m ∈ M we have (L(m)φ, φ̌) ≠ 0 if and only if m ∈ U. This implies the Lemma.

We will apply the preceding results in several situations but, for now, we observe that Theorem 4.4 implies (in the notation of §4):

Theorem 5.6. Assumption 5.1 (and hence Proposition 5.3 and Theorem 5.4) holds in the following contexts:

(i) K = 𝔎(𝔄), τ = ρ, M = M_{𝔄} or M = G_{𝔄}, U = (J ∩ U)H^{1}, Ψ = Ψ_{α};

(ii) K = J, τ = Λ, M = M_{𝔄}∩J or M = G_{𝔄}∩J, U = (J ∩U)H^{1}, Ψ = Ψ_{α}.

If π has level 0 then we recover the result of Gel′fand [7].

5.2 Explicit Whittaker functions

Now we argue along the lines of [3] §3. However, not only do we get a uniqueness statement as in [3], but we also obtain explicit formulae in terms of the character of Λ and Ψ_{α}.

Let V be the underlying vector space of π and let (π̌, V̌) be the smooth dual of (π, V). We denote by ⟨, ⟩ the pairing on V × V̌ given by the evaluation. Let v_{α} ∈ V and v̌_{α} ∈ V̌ be non-zero vectors such that π(h)v_{α} = Ψ_{α}(h)v_{α} and π̌(h)v_{α} = Ψ̌_{α}(h)v̌_{α}, for all h ∈ (J ∩ U)H^{1}. Corollary 4.5 implies that such vectors exist and they are unique up to scalar multiple. We rescale so that ⟨v_{α}, v̌_{α}⟩ = 1.

Proposition 5.7. The representation π admits a unique coefficient function f = f_{α,U} with the following properties:

(i) f(1) = 1, and

(ii) f(h_{1}gh_{2}) = Ψ_{α}(h_{1}h_{2})f(g), ∀h_{1}, h_{2} ∈ (J ∩ U)H^{1}, ∀g ∈ G.

Moreover, Supp f ⊆ J and

f(g) = ⟨π(g)v_{α}, v̌_{α}⟩ = J (g), ∀g ∈ J

where J = J_{Λ} is the Bessel function.

Proof. For Parts (i) and (ii) we argue as in the proof of [3] Proposition 3.2. If we set f(g) = ⟨π(g)v_{α}, v̌_{α}⟩, then f satisfies (i) and (ii); the uniqueness is implied by Corollary 4.5.

If ⟨gv_{α}, v̌_{α}⟩ ≠ 0 then e_{Ψ}π(g)v_{α} ≠ 0 and, since v_{α} ∈ V^{Λ}, Corollary 4.5 implies e_{Λ}(π(g)V^{Λ}) ≠ 0. Hence g intertwines Λ and so g ∈ J.

If g ∈ J then Theorem 5.4 and Lemma 5.5 imply that

⟨π(g)v_{α}, v̌_{α}⟩ = (L(g)φ, φ̌) = J (g).

Theorem 5.8. Let du be an invariant Haar measure on U, normalised so that ∫_{U∩J} du = 1. Let Υ : π → Ind_{U}^{G} ψ_{α} be a linear map given by

v ↦ [g ↦ ∫_{U} ψ_{α}(u)⟨π(u^{−1}g)v, v̌_{α}⟩du].

Then Υ is non-zero and G-equivariant. Moreover, Supp Υ(v_{α}) ⊆ UJ and

[Υ(v_{α})](ug) = ψ_{α}(u)J (g), ∀u ∈ U, ∀g ∈ J.

Further, Supp Υ(v_{α}) ∩ M_{F} = U(H^{1} ∩ M_{F}) and

[Υ(v_{α})](uh) = ψ_{α}(u)θ(h), ∀u ∈ U, ∀h ∈ H^{1} ∩ M_{F} .

Proof. The first assertion follow directly from Proposition 5.7. Now

Supp Υ(v_{α}) ∩ M_{F} ⊆ (UJ) ∩ M_{F} = U(J ∩ M_{F}) = U(J ∩ M_{𝔄}) ∩ M_{F} ,

where the last equality follows from Corollary 4.8. The second assertion now follows from Proposition 5.3(iv).

Remark 5.9. The Whittaker function Υ(v_{α}) above, and the bound on the support, can be obtained by integrating the matrix coefficient appearing in [2] (see also [10]) and this is sufficient for the purposes of [5]. However, the fact that we can realize the representation Λ via Bessel functions gives us the precise knowledge of Supp Υ(v_{α}) ∩ M_{F}. We use this in §7 to compute, in some cases, epsilon factors of pairs.

Corollary 5.10. Let J_{ρ} be the Bessel function of ρ. For g ∈ 𝔎(𝔄),

J_{ρ}(g) = \{ \begin{matrix}J (g) & if g ∈ J, \\ 0 & otherwise. \end{matrix}

Proof. This is implied by the uniqueness of the matrix coefficient in Proposition 5.7.

5.3 Multiplicative property

Our maximal simple types are of the form (J, λ), where λ = κ ⊗ σ. In this section, we show that the Bessel function associated to the extension Λ of λ to J can be split as a product of two Bessel function (see Proposition 5.13).

Lemma 5.11. There exists a representation κ̃ of J such that κ̃|_{J} ≅ κ. Moreover, given such a representation κ̃, there exists a unique representation Σ of J, such that Σ|_{J} ≅ σ and Λ ≅ κ̃ ⊗ Σ.

Proof. We may extend the action of J on κ to the action of F^{×}J, by making some uniformiser ϖ_{F} act trivially. By definition of κ, [6] (5.2.1), E^{×}, (and hence J) intertwines κ. Since, F^{×}J is normal in J and the quotient J/(F^{×}J) ≅ E^{×}/F^{×}𝔬_{E}^{×} is a cyclic group, we may extend the action to J.

Now suppose that we are given a representation κ̃ of J such that κ̃|J ≅ κ. By the same argument as above, there exists a representation Σ of J such that Σ|_{J} ≅ σ. Moreover, we may ensure that ωκ̃(ϖ_{F})ω_{Σ}(ϖ_{F}) = ω_{Λ}(ϖ_{F}), where ω denotes the central character. Hence

Λ|_{F^{×}J} ≅ (κ̃ ⊗ Σ)|_{F^{×}J}.

Thus Λ is a direct summand of

Ind_{F^{×}J}^{J} κ̃ ⊗ Σ ≅ κ̃ ⊗ Σ ⊗ Ind_{F^{×}J}^{J} 1 ≅ ⨁_{χ} κ̃ ⊗ Σ ⊗ χ,

where χ runs over characters of J/F^{×}J ≅ E^{×}/F^{×}𝔬_{E}^{×}. Hence, after replacing Σ by some Σ ⊗ χ, we may ensure that Λ ≅ κ̃ ⊗ Σ. Moreover, by [6] §6 we know that Λ ≅ Λ ⊗ χ implies that χ is the trivial character. Hence, such Σ is unique.

Let us now fix some κ̃ as above and let Σ be the unique representation of J, given by Lemma 5.11. Let (U, ψ_{α}) be as in §4.2. In particular, we require that U ∩ B^{×} is a maximal unipotent subgroup of B^{×} and we may write ψ_{α} = ψ_{a}ψ_{b}, such that ψ_{a} is trivial on U ∩ B^{×}; ψ_{b} is a non-degenerate character on U ∩ B^{×}, which descends to a non-degenerate character of a maximal unipotent subgroup of U(𝔅)/U^{1}(𝔅) (see Theorem 3.3(iv)).

Lemma 5.12. Assumption 5.1 (and hence Proposition 5.3 and Theorem 5.4) holds in the following contexts:

(i) K = J, τ = κ̃, M = J^{1}, U = (J^{1} ∩ U)H^{1}, Ψ = Θ, where Θ(uh) = ψ_{α}(u)θ(h), for u ∈ J^{1} ∩ U and h ∈ H^{1}.

(ii) K = J, τ = Σ, M = M_{𝔄} ∩ J or G_{𝔄} ∩ J, U = (J ∩ U)J^{1}, Ψ = ψ_{b}.

Proof. Part (i) is just Theorem 2.6. Since J/J^{1} ≅ 𝔎(𝔅)/U^{1}(𝔅) and J^{1} acts trivially on Σ, part (ii) is given by Theorem 5.6(i) (together with Corollary 4.8) applied in the level zero case.

Theorem 5.6 and Lemma 5.12 imply that we may associate Bessel functions to Λ, κ̃ and Σ, via Definition 5.2.

Proposition 5.13. We have

J_{Λ}(g) = Jκ̃(g)J_{Σ}(g), ∀g ∈ J.

Proof. We have Λ ≅ κ̃ ⊗ Σ, and Ψ_{α} = Θψ_{b}, see Lemma 4.10. Let Vκ̃ and V_{Σ} be the underlying vector spaces of κ̃ and Σ, respectively. We claim that

e_{Ψ_{α}}(v ⊗ w) = (e_{Θ}v) ⊗ (e_{ψ_{b}}w), ∀v ∈ Vκ̃, ∀w ∈ V_{Σ}.  (†)

We choose non-zero vectors v_{Θ} ∈ e_{Θ}Vκ̃ and w_{ψ_{b}} ∈ e_{ψ_{b}}V_{Σ}. It follows from Theorem 2.6 that dim e_{Θ}Vκ̃ = 1 and from Lemma 4.11 that dim e_{ψ_{b}}V_{Σ} = 1. Now (J ∩ U)H^{1} acts on v_{Θ} ⊗ w_{ψ_{b}} via Ψ_{α} and hence Theorem 4.12 implies that the set

{c(v_{Θ} ⊗ w_{ψ_{b}}) : c̄ ∈ M_{𝔅}J^{1}/(J ∩ U)H^{1}},

is a basis of Vκ̃ ⊗ V_{Σ}, where c denotes a coset representative of a coset c̄. It is enough to show the claim (†) holds for the elements of this basis.

If g ∈ M_{𝔅}J^{1} and e_{Ψ_{α}}g(v_{Θ} ⊗ w_{ψ_{b}}) is not equal to zero then g intertwines Ψ_{α} and hence, by Theorem 4.12, we obtain that g ∈ (J ∩ U)H^{1}. Conversely, if g ∈ (J ∩ U)H^{1} then e_{Ψ_{α}}g(v_{Θ} ⊗ w_{ψ_{b}}) = Ψ_{α}(g)v_{Θ} ⊗ w_{ψ_{b}}.

If g ∈ M_{𝔅}J^{1} and (e_{Θ}gv_{Θ}) ⊗ (e_{ψ_{b}}gw_{ψ_{b}}) ≠ 0 then g intertwines ψ_{b} and hence, by Lemma 4.11, g ∈ (J ∩ U)J^{1}. Moreover, g intertwines Θ and so by Theorem 2.6, g ∈ (J ∩ U)H^{1}. If g ∈ (J ∩ U)H^{1} then

(e_{Θ}gv_{Θ}) ⊗ (e_{ψ_{b}}gw_{ψ_{b}}) = Θ(g)ψ_{b}(g)v_{Θ} ⊗ w_{ψ_{b}} = Ψ_{α}(g)v_{Θ} ⊗ w_{ψ_{b}}.

Hence we obtain the claim (†).

Now let Θ′ be the restriction of Θ to (J^{1} ∩ U)H^{1}. According to Theorem 2.6, Θ′ also occurs in κ with multiplicity one and hence e_{Θ′}v = e_{Θ}v, for all v ∈ Vκ̃. Hence,

J_{Λ}(g) = tr_{Λ}(e_{Ψ_{α}}g) = trκ̃(e_{Θ′}g) tr_{Σ}(e_{ψ_{b}}g) = Jκ̃(g)J_{Σ}(g).

6 A numerical invariant

In this section, we will define a certain numerical invariant which appears in our formula for epsilon factors in §7. We continue with the notation of previous sections.

We suppose that E = F [β] is maximal in A and we identify V with E. Let θ ∈ C(𝔄, 0, β). Let F = {V_{i} : 1 ≤ i ≤ d} be a maximal F -flag in E; let U be a unipotent radical of Aut_{F} (E)-stabiliser of F; let χ be a smooth, non-degenerate character of U, such that

χ|_{U∩H^{1}} = θ|_{U∩H^{1}};

and let ψ_{E} be an additive character of E, trivial on 𝔭_{E}, and such that

ψ_{E}(x) = ψ_{F} (x), ∀x ∈ F.

Definition 6.1. Choose a basis B = {x_{1}, …, x_{d}} of E over F, which satisfies Definition 3.1(i),(iii) and (iv) with (U, χ, ψ_{E}). We define ν ∈ E^{×}/(1 + 𝔭_{E}), by

ν = ν(θ, ψ_{F} , ψ_{E}) = x_{d}x_{1}^{−1} (mod 𝔭_{E}).

Proposition 6.2. ν(θ, ψ_{F} , ψ_{E}) depends only on θ, ψ_{F} and ψ_{E}.

Proof. Suppose that we have another triple U′, χ′, B′ which satisfy the conditions above. Then Proposition 1.3 and the first part of the proof of Theorem 3.3 imply that there exists g ∈ J such that U′ = U^{g} and χ′ = χ^{g}. Since E is maximal, we may write g = xh, where x ∈ E^{×} and h ∈ J^{1}.

Let ξ ∈ E be the unique element such that ξhx_{1} = hx_{d}. Then (x_{1}x_{d}^{−1}h^{−1}ξh) ∈ G ∩ J, where G = {g ∈ Aut_{F} (E); gx_{1} = x_{1}}. Now, according to Corollary 4.7, G ∩ J = G ∩ J^{1} and hence the image of (x_{1}x_{d}^{−1}h^{−1}ξh) in J/J^{1} ≅ E^{×}/1 + 𝔭_{E}, is equal to 1. This implies that ξ ≡ x_{d}x_{1}^{−1} (mod 𝔭_{E}) and hence we may assume that U = U′ and χ = χ′. The second part of Proposition 3.2 implies that ν does not depend on the choice of basis B.

Remark 6.3. Suppose E is not necessarily maximal. Let θ ∈ C(𝔄, β, ψ_{F}) and let θ_{F} ∈ C(𝔄(E), β, ψ_{F}) be the simple character corresponding to θ via the correspondence of [6] §3.6, where 𝔄(E) is the hereditary 𝔬_{F} -order in End_{F} (E), corresponding to the lattice chain {𝔭_{E}^{i} : i ∈ ℤ}. Let {x_{1}, ..., x_{d}} be the F - basis of E given by Corollary 3.4, with x_{1} = 1. Then it follows from the construction in the proof of Theorem 3.3, that ν(θ_{F} , ψ_{F} , ψ_{E}) ≡ x_{d} (mod 𝔭_{E}).

6.1 Behaviour under tame lifting

We continue with the assumption that E is maximal in A and let us further assume that E is totally wildly ramified. Let K be a tame extension of F. The algebra L = K ⊗_{F} E is a field, which is the compositum of E and K.

Let θ ∈ C(𝔄, 0, β); then Bushnell and Henniart in [2] define the tame lift of θ, which is a simple character θ^{K} of H_{K}^{1} = H^{1}(β, 𝔄_{K}), where 𝔄_{K} is the hereditary 𝔬_{K}-order in End_{K}(L) corresponding to the 𝔬_{K}-lattice chain {𝔭_{L}^{j} : j ∈ ℤ}. We will investigate how the invariant ν varies with the tame lifting.

Let ψ_{L} and ψ_{K} be the additive characters of L and K, respectively, given by

ψ_{L}(x) = ψ_{E}(tr_{L/E} x), ∀x ∈ L, ψ_{K}(x) = ψ_{F} (tr_{K/F} x), ∀x ∈ K.

Since K is tame over F, ψ_{K} has conductor 𝔭_{K}; likewise, ψ_{L} has conductor 𝔭_{L}.

Let U be a maximal unipotent subgroup of G and let F = {V_{i} : 1 ≤ i ≤ d} be a maximal flag corresponding to U. Let χ be a smooth non-degenerate character of U, and let B = {x_{1}, …, x_{d}} be an F -basis of E, with respect to which U is the group of unipotent upper-triangular matrices and, if u ∈ U and (u_{ij}) ∈ 𝕄_{d}(F) is a matrix of u with respect to B, then

χ(u) = ψ_{F} (∑_{i=1}^{d−1} u_{i,i+1}).

Set F_{K} = {V_{i} ⊗_{F} K : 1 ≤ i ≤ d} and let U_{K} be the unipotent radical of the Aut_{K}(L)-stabiliser of F_{K}. For u ∈ U_{K}, write (u_{ij}) ∈ 𝕄_{d}(K) for the matrix of u with respect to {x_{1}, …, x_{d}}, and let χ^{K} : U_{K} → ℂ^{×} be the character given by

χ^{K}(u) = ψ_{K}(∑_{i=1}^{d−1} u_{i,i+1}).

Proposition 6.4. We have θ|_{U∩H^{1}} = χ|_{U∩H^{1}} if and only if

θ^{K}|_{U_{K}∩H_{K}^{1}} = χ^{K}|_{U_{K}∩H_{K}^{1}}.

Proof. By [2] Corollary 9.13(iii), tame lifting is transitive in the field extension: if K′ is a subfield of K containing F, then

(θ^{K′})^{K} = θ^{K}.

So it is enough to prove the Proposition when K/F is Galois, cyclic, and either unramified or totally tamely ramified, as in [2] (12.2). Let Γ be the Galois group of K/F and fix a generator σ of Γ. For g ∈ Aut_{K}(L), let N_{σ} be the cyclic norm map, given by N_{σ}g = gσ(g) ⋯ σ^{l−1}(g), where l = [K : F]. Define ℌ_{F}^{1} , ℌ_{K}^{1}, 𝔘_{F}, 𝔘_{K} by

H^{1} = 1 + ℌ_{F}^{1} , H_{K}^{1} = 1 + ℌ_{K}^{1}, U = 1 + 𝔘_{F} , U_{K} = 1 + 𝔘_{K}.

We observe that the proof of [2] (12.3) Proposition (including the results required from [2]§11) goes through if we replace ℌ_{F}^{1} with ℌ_{F}^{1} ∩ 𝔘_{F} and ℌ_{K}^{1} with ℌ_{K}^{1} ∩ 𝔘_{K}. We obtain the following:

(i) For x ∈ H_{K}^{1} ∩ U_{K}, there exists u ∈ H_{K}^{1} ∩ U_{K} such that y_{x} = uxσ(u)^{−1} satisfies N_{σ}y_{x} ∈ H^{1} ∩ U.

(ii) The map x ↦ N_{σ}y_{x} induces a bijection between σ-conjugacy classes in H_{K}^{1} ∩ U_{K} and conjugacy classes in H^{1} ∩ U.

Now, [2] (12.6), and the fact that both θ^{K} and χ^{K} are stable under Γ, imply that

θ^{K}(x) = θ^{K}(y_{x}) = θ(N_{σ}y_{x}), χ^{K}(x) = χ^{K}(y_{x}) = χ(N_{σ}y_{x}).

The above coupled with (ii) gives the Proposition.

Remark 6.5. The above Proposition would follow easily from [3] Lemma 2.10, if the gap in its proof were fixed.

Corollary 6.6. We have ν(θ^{K}, ψ_{K}, ψ_{L}) ≡ ν(θ, ψ_{F} , ψ_{E}) (mod 𝔭_{L}).

Proof. Let (U, χ) be such that θ|_{H^{1}∩U} = χ|_{H^{1}∩U}, let B = {x_{1}, …, x_{d}} be an F -basis of E, which satisfies Definition 3.1(i),(iii) and (iv), with respect to U,χ,ψ_{F} and ψ_{E}. Propositions 6.2 and 6.4 imply that it is enough to show the following: If a ∈ L and (a_{ij}) is the matrix of a ∈ End_{K}(L) with respect to B then ψ_{L}(a) = ψ_{K}(a_{dd}).

Since ψ_{L} and ψ_{K} are additive, and L = K ⊗_{F} E, it is enough to prove this for a = c ⊗ b, where c ∈ K and b ∈ E. Let (b_{ij}) be a matrix of b with respect to B then b_{ij} ∈ F and a_{ij} = cb_{ij}. Hence,

ψ_{K}(a_{dd}) = ψ_{F} (b_{dd} tr_{K/F} c) = ψ_{E}(b tr_{K/F} c) = ψ_{E}(tr_{L/E}(bc)) = ψ_{L}(a).

7 Application to ε-factors of pairs

We will use the Whittaker function constructed in Theorem 5.8 to compute εfactor of pairs in the following situation:

As before, let [𝔄, n, 0, β] be a simple stratum in A, such that e(𝔅|𝔬_{E}) = 1. Let θ ∈ C(𝔄, 0, β)be a simple character and let κ and η be representations of J = J(β, 𝔄) and J^{1} = J^{1}(β, 𝔄), as in Definition 4.1. Let σ_{1} and σ_{2} be lifts of cuspidal representations of J/J^{1} ≅ U(𝔅)/U^{1}(𝔅) ≅ GL_{r}(𝔨_{E}) to J. We allow the case σ_{1} ≅ σ_{2}. For i = 1, 2, set λ_{i} = κ ⊗ σ_{i} and let π_{i} be a supercuspidal representation of G such that π_{i}|_{J} contains λ_{i}. According to [6] §6, there exists an irreducible representation Λ_{i} of J = E^{×}J, such that Λ_{i}|_{J} ≅ λ_{i} and

π_{i} ≅ c-Ind_{J}^{G} Λ_{i}.

We fix some extension κ̃ of κ to J, as in Lemma 5.11. For i = 1, 2, let Σ_{i} be the unique representation of J, also given by Lemma 5.11, such that Λ_{i} ≅ κ̃ ⊗ Σ_{i} and Σ_{i}|_{J} ≅ σ_{i}; we view Σ_{i} as a representation of 𝔎(𝔅) = E^{×}U(𝔅) and set

τ_{i} = c-Ind_{𝔎(𝔅)}^{B^{×}} Σ_{i}.

Then τ_{1} and τ_{2} are level zero supercuspidal representations of B^{×} ≅ GL_{r}(E). Let 𝔄(E) be the hereditary order in End_{F} (E), corresponding to the lattice chain {𝔭_{E}^{i} : i ∈ ℤ}. Let θ_{F} ∈ C(𝔄(E), n, 0, β)be the simple character corresponding to θ via the correspondence of [6] §3.6.

Theorem 7.1. Choose an additive, unitary character ψ_{E} : E → ℂ^{×}, such that ψ_{E} is trivial on 𝔭_{E} and ψ_{E}(x) = ψ_{F} (x), for all x ∈ F. Then

ε(π_{1} × π̌_{2}, s, ψ_{F}) = ζω_{τ_{1}}(ν^{−r})ω_{τ_{2}}(ν^{r})q^{(s−1/2)rv_{E}(ν)N/e}ε(τ_{1} × τ̌_{2}, s, ψ_{E}),

where: ν = ν(θ_{F} , ψ_{F} , ψ_{E}) ∈ E^{×}/(1 + 𝔭_{E}) is the invariant defined in Definition 6.1; r = dim_{E}(V); π̌ denotes the contragredient of π; q = q_{F} is the cardinality of 𝔨_{F}; and ζ = ω_{τ_{2}}(−1)^{r−1}ω_{π_{2}}(−1)^{N−1}.

We remark that, although the representation τ_{1} and τ_{2} depend on the choice of β-extension κ, and the choice of κ̃, the ε-factor in Theorem 7.1 does not. For a different choice of κ̃ would twist τ_{1} and τ_{2} by the same tamely ramified character χ and we have

ε(τ_{1}χ × τ̌_{2}χ^{−1}, s, ψ_{E}) = ε(τ_{1} × τ̌_{2}, s, ψ_{E}).

In §7.5, we use our invariant ν to recover (and generalise) certain results in [5] on the behaviour of ε-factors of pairs under twists by tamely ramified characters.

7.1 Preparation

Let U be a maximal unipotent subgroup of G, which is the G-stabiliser of the maximal flag F = {V_{i} : 1 ≤ i ≤ N}, and let ψ_{α} = χψ_{b} be a smooth nondegenerate character of U, as constructed in Theorem 3.3. Let L = {L_{i} : i ∈ ℤ} be the lattice chain in V corresponding to 𝔄. Recall that, in Corollary 3.4, we constructed an F -basis B_{F} = {v_{1}, …, v_{N}} of V, with respect to which U is the group of upper triangular matrices and ψ_{α} is the ‘standard’ character. Moreover, the vectors v_{i} are of the form

v_{d(i−1)+j} = x_{d}^{i−1}x_{j}w_{i}, 1 ≤ i ≤ r, 1 ≤ j ≤ d,

where B_{E} = {w_{1}, …, w_{r}} is an E-basis of V such that L_{0} = ∑_{i=1}^{r} 𝔬_{E}w_{i}, and {x_{1}, …, x_{d}} is an F -basis of E, which depends on ψ_{E} (see Corollary 3.4). Further, x_{1} = 1. Whenever it is required of us we will identify G with GL_{N}(F) via B_{F}. Let w ∈ G be the element defined on the basis B_{F} by

w(v_{i}) = v_{N−i+1}, 1 ≤ i ≤ N.

We also define an involution δ : G → G, by

δ(g) = wg^{⊤−1}w,

where g^{⊤} is the transpose of g with respect to B_{F} and g^{⊤−1} = (g^{⊤})^{−1} = (g^{−1})^{⊤}.

We will briefly recall the definition of ε-factors of pairs, using the the formulation of Jacquet, Piatetskii-Shapiro and Shalika [8], rather than Shahidi [11].

Let W_{1} = W(π, ψ_{α}) and W_{2} = W(π̌_{2}, \overline{ψ}_{α}) be the Whittaker models of π_{1} and π̌_{2} respectively. Let S(F^{N}) be the set of compactly-supported, locally constant functions ϕ : F^{N} → ℂ. We denote by e_{1} = (1, 0, …, 0), …, e_{N} = (0, …, 0, 1) the standard basis of F^{N}. Given W_{1} ∈ W_{1}, W_{2} ∈ W_{2} and Φ ∈ S(F^{N}), we define the zeta function

Z(W_{1}, W_{2}, Φ, s) = ∫_{U\G} W_{1}(g)W_{2}(g)Φ(e_{N}g)| det g|^{s}dg,

where dg is a G-equivariant measure on U\G. Note that, under our identification of G with GL_{N}(F), via B_{F}, the term e_{N}g is the N^{th} row of the matrix of g with respect to B_{F}. The integral converges absolutely for Re(s) sufficiently large, and is a rational function of q^{−s}. This zeta function satisfies a functional equation, [8](2.7):

\frac{Z(W̃_{1}, W̃_{2}, Φ̂, 1 − s)}{L(π̌_{1} × π_{2}, 1 − s)} = ω_{π_{2}}(−1)^{N−1}ε(π_{1} × π̌_{2}, s, ψ_{F})\frac{Z(W_{1}, W_{2}, Φ, s)}{L(π_{1} × π̌_{2}, s)},

where, for i = 1, 2, W̃_{i}(g) = W_{i}(wg^{⊤−1}); Φ̂ is the Fourier transform of Φ, given by

Φ̂(y) = ∫_{F^{N}} Φ(x)ψ_{F} (xy^{⊤})dx, ∀y ∈ F^{N},

where dx is normalised so that ˆ Φ̂(x) = Φ(−x); and L(π_{1} × π̌_{2}, s) is the Lfunction. Since π_{1} and π_{2} are supercuspidal, it is enough for our purposes to know that

L(π_{1} × π̌_{2}, s) = ∏_{χ} L(χ, s),

where the product is taken over all the unramified characters χ : F^{×} → ℂ^{×} such that π_{1} ≅ π_{2}⊗χ∘det, and L(χ, s) is as in Tate’s thesis [12]. In particular, if σ_{1} ≅ σ̸_{2} then the product is taken over an empty set and so L(π_{1}×π̌_{2}, s) = 1. If σ_{1} ≅ σ_{2} then there exists χ as above such that π_{1} ≅ π_{2} ⊗ χ ∘ det. In this case, it follows from [6] (6.2.5) that

L(π_{1} × π̌_{2}, s) = (1 − χ(ϖ_{F})^{−N/e}q^{−sN/e})^{−1},

where e = e(𝔄|𝔬_{F}) = e(E|F).

7.2 Computation

Let W_{1} ∈ W(π_{1}, ψ_{α}) and W_{2} ∈ W(π̌_{2}, \overline{ψ}_{α}) be the Whittaker functions constructed in Theorem 5.8. Then Supp W_{1} ⊆ UJ, Supp W_{2} ⊆ UJ and

W_{1}(ug) = ψ_{α}(u)J_{Λ_{1}}(g), W_{2}(ug) = \overline{ψ}_{α}(u)JΛ̌_{2}(g), ∀u ∈ U, ∀g ∈ J.

Set J_{1} = J_{Λ_{1}} and J_{2} = JΛ̌_{2}, and let Φ ∈ S(F^{N}) be the indicator function on the set e_{N}J^{1}. We are going to compute the zeta functions on both sides of the functional equation for this particular choice of W_{1}, W_{2} and Φ. This will give us Theorem 7.1.

For X a subset of G which is a union of right U-cosets, we write vol_{U}(X) for the volume of U\X with respect to the measure du on U\G.

Proposition 7.2. Let F : G → ℂ be the function given by

F(g) = W_{1}(g)W_{2}(g)Φ(e_{N}g).

Then F is an indicator function on the set UH^{1}. In particular,

Z(W_{1}, W_{2}, Φ, s) = vol_{U}(UH^{1}).

Proof. We have

Supp W_{1} ⊆ UJ, Supp W_{2} ⊆ UJ, Supp[g ↦ Φ(e_{N}g)] = M_{F} J^{1},

where M_{F} = {g ∈ G : (g − 1)V ⊆ V_{N−1}} is the mirabolic subgroup, as in §4. Hence,

Supp F ⊆ UJ ∩ M_{F} J^{1} = U(J ∩ M_{F})J^{1} = UM_{𝔅}J^{1},

where the last equality is given by Corollary 4.8. We have

F(ug) = W_{1}(g)W_{2}(g)Φ(e_{N}g) = J_{1}(g)J_{2}(g), ∀u ∈ U, ∀g ∈ M_{𝔅}J^{1}.

It follows from Proposition 5.3(ii) and (iv) that J_{1}(g)J_{2}(g) = Ψ_{α}(g)\overline{Ψ_{α}}(g) = 1, if g ∈ (J ∩ U)H^{1}, and J_{1}(g)J_{2}(g) = 0, otherwise. Hence F is an indicator function on the set UH^{1}. Since H^{1} is compact and U is unipotent, we obtain that | det g| = 1, for all g ∈ UH^{1}. Hence Z(W_{1}, W_{2}, s, Φ) = vol_{U}(UH^{1}).

We write G_{F} = {g ∈ G : gv_{1} = v_{1}}, as in §4.

Lemma 7.3. For all g_{1} ∈ G, h ∈ G and g_{2} ∈ (J ∩ M_{F})J^{1} we have:

Φ̂(e_{1}(g_{1}hg_{2})^{⊤−1}) = Φ̂(e_{1}h^{⊤−1}).

Proof. Since e_{1}g_{1}^{⊤−1} = e_{1}, we obtain Φ̂(e_{1}(g_{1}h)^{⊤−1}) = Φ̂(e_{1}h^{⊤−1}). Since Φ is an indicator function on the set e_{N}J^{1} = e_{N}(J ∩ M_{F})J^{1}, we have g_{2}Φ = Φ, hence

Φ̂(e_{1}(hg_{2})^{⊤−1}) = ∫_{F^{N}} Φ(x)ψ_{F} (x(g_{2}^{−1}h^{−1}e_{1}^{⊤}))dx
= ∫_{F^{N}} [g_{2}Φ](x)ψ_{F} (x(h^{−1}e_{1}^{⊤}))dx = Φ̂(e_{1}h^{⊤−1}).

For L a lattice in F^{N}, we write vol_{F} (L) for the volume of L with respect to the measure dx on F^{N}.

Lemma 7.4. Let i, j ∈ ℤ; then vol_{F} (e_{N}𝔓^{i}) = q^{(j−i)N/e} vol_{F} (e_{N}𝔓^{j}).

Proof. Since there exists γ ∈ 𝔎(𝔄) such that e_{N} = e_{1}γ, and then e_{N}𝔓^{i} = e_{1}γ𝔓^{i} = e_{1}𝔓^{i+v_{𝔄}(γ)}, it is enough to prove that

vol_{F} (e_{1}𝔓^{i}) = q^{(j−i)N/e} vol_{F} (e_{1}𝔓^{j}).

Since our basis B_{F} splits the lattice chain we may write

𝔓^{k} = ⨁_{1≤i,j≤N} 𝔭_{F}^{c_{ij}(k)}1_{ij},

where 1_{ij} ∈ A are the projections given by 1_{ij}(v_{k}) = δ_{ik}v_{j}, for 1 ≤ k ≤ N, and δ_{ik} is the Kronecker delta. Hence, e_{1}𝔓^{k} = ∑_{j=1}^{N} 𝔭_{F}^{c_{1j}(k)}e_{j}. According to [6] (1.1.4) we have

𝔓^{1−k} = {a ∈ A : ψ_{F} (tr_{A/F} (xa)) = 1, ∀x ∈ 𝔓^{k}}, ∀k ∈ ℤ,

which, since ψ_{F} has conductor 𝔭_{F}, implies that 1 − c_{ij}(k) = c_{ji}(1 − k), for all k ∈ ℤ. Since we have chosen v_{1}, so that v_{1} ∈ L_{0}, v_{1} ∉ L_{1} and the lattice chain is principal, we have 𝔓^{k}v_{1} = L_{k}, for all k ∈ ℤ. Now, 𝔓^{k}v_{1} = ∑_{j=1}^{N} 𝔭_{F}^{c_{j1}(k)}v_{j} = ∑_{j=1}^{N} 𝔭_{F}^{1−c_{1j}(1−k)}v_{j}. Hence

(e_{1}𝔓^{i} : e_{1}𝔓^{j}) = (L_{1−j} : L_{1−i}),

where the brackets denote the generalised index. Since (L_{i} : L_{i+e}) = q^{N} and L is principal, we have (L_{i} : L_{i+1}) = q^{N/e}, for all i ∈ ℤ, and the result follows.

We write q_{𝔄} = q^{N/e}, so Lemma 7.4 says that vol_{F} (e_{N}𝔓^{i}) = q_{𝔄}^{j−i} vol_{F} (e_{N}𝔓^{j}).

Let w_{E} ∈ U(𝔅) be the element defined by its action on the basis B_{E} by

w_{E}(w_{i}) = w_{r−i+1}, 1 ≤ i ≤ r.

From our construction of the bases B_{E} and B_{F}, we have x_{d}^{r}w_{E}v_{1} = x_{d}^{r}w_{r} = v_{N}. In terms of matrices with respect to B_{F} we can rephrase this as

(x_{d}^{r}w_{E})e_{1}^{⊤} = e_{N}^{⊤}.

Lemma 7.5. Let ϕ : A → ℂ^{×} be the function

ϕ : a ↦ ψ_{F} ((e_{N}a)e_{1}^{⊤}) = ψ_{F} (a_{N1}),

where (a_{ij}) is the matrix of a ∈ A with respect to B_{F}. Then

ϕ(ax_{d}^{r}w_{E}) = ψ_{F} (a_{NN}), ∀a ∈ A.

Hence ϕ defines an additive character on A, which is trivial on 𝔓^{1+rv_{E}(x_{d})}, and non-trivial on 𝔓^{rv_{E}(x_{d})}.

Lemma 7.6. Let b ∈ B, let (b_{ij}) be the matrix of b with respect to B_{E}, and define ϕ : A → ℂ^{×} as in Lemma 7.5; then

ϕ(b) = ψ_{E}(x_{d}^{−r}b_{r1}).

Proof. Set a = bx_{d}^{−r}w_{E}, let (a_{ij}^{F}) be the matrix of a with respect to B_{F} and let (a_{ij}^{E}) be the matrix of a with respect to B_{E}. According to Lemma 7.5, we have ϕ(b) = ψ_{F} (a_{NN}^{F}). We have aw_{r}+V_{N−r} = a_{rr}^{E}w_{r}+V_{N−r} and, since x_{d} ∈ E, we obtain a(x_{d}^{r}w_{r})+V_{N−r} = a_{rr}^{E}(x_{d}^{r}w_{r})+V_{N−r}. Since a_{rr}^{E} ∈ E, we may consider it as a_{rr}^{E} ∈ End_{F} (E). Let (α_{ij}) be the matrix of a_{rr}^{E} with respect to the basis {x_{1}, …, x_{d}}. Since v_{d(i−1)+j} = x_{d}^{i−1}x_{j}w_{i}, for 1 ≤ i ≤ r and 1 ≤ j ≤ d, and in particular v_{N} = x_{d}^{r}w_{r}, we obtain that

av_{N} + V_{N−1} = α_{dd}v_{N} + V_{N−1}.

In particular, α_{dd} = a_{NN}. Recall that the basis {x_{1}, … x_{d}} was chosen so that ψ_{E}(a_{rr}^{E}) = ψ_{F} (α_{dd}), see Definition 3.1(iv). Hence, ϕ(b) = ψ_{E}(a_{rr}^{E}). Since x_{d} ∈ E, we obtain that a_{rr}^{E} = x_{d}^{−r}(bw_{E})_{rr} = x_{d}^{−r}b_{r1}.

To ease the notation, we set

c = vol_{F} (e_{N}𝔓^{1+rv_{E}(x_{d})}).

As in §4.1 let K be a maximal unramified extension of E, such that K^{×} normalises 𝔄.

Lemma 7.7. Let h ∈ 𝔎(𝔄) and set j = v_{𝔄}(h) − v_{E}(x_{d}^{−r}); then

Φ̂(e_{1}h^{⊤−1}) = \{ \begin{matrix}0 & if j > 0; \\ c| det h|ϕ(h^{−1}) & if j = 0; \\ c| det h|q_{𝔄}^{j} & if j < 0. \end{matrix}

Proof. Corollary 4.7 implies that e_{N}U^{1}(𝔄) = e_{N}J^{1} = e_{N}(1 + 𝔭_{K}). Hence,

Φ̂(e_{1}h^{⊤−1}) = ∫_{e_{N} U^{1}(𝔄)} ψ_{F} ((xh^{−1})e_{1}^{⊤})dx
= ψ_{F} ((e_{N}h^{−1})e_{1}^{⊤}) ∫_{e_{N} 𝔓} ψ_{F} ((xh^{−1})e_{1}^{⊤})dx
= | det h|ψ_{F} ((e_{N}h^{−1})e_{1}^{⊤}) ∫_{e_{N} 𝔓^{1−v_{𝔄}(h)}} ψ_{F} (xe_{1}^{⊤})dx.

Lemmas 7.4, 7.5 and the orthogonality of characters imply the Lemma.

Lemma 7.8. vol_{U}(Uδ(H^{1})) = vol_{U}(UH^{1}).

Proof. Let K_{m} = {g ∈ GL_{N}(𝔬_{F}) : g ≡ 1 (mod 𝔭_{F}^{m})}. Since K_{m}, for m ≥ 1, form a basis of neighbourhoods of 1 in G, there exists m, such that K_{m} ⊆ H^{1} ∩ δ(H^{1}). Since δ(U) = U, δ(K_{m}) = K_{m} and the measure on U\G is Ginvariant, we obtain that vol_{U}(Uδ(H^{1})) = vol_{U}(UH^{1}).

Let F̃ : G → ℂ be the function given by

F̃(g) = W̃_{1}(g)W̃_{2}(g)Φ̂(e_{N}g).

We have Supp F̃ ⊆ Supp W̃_{1} = δ(Supp W_{1})w ⊆ Uδ(J)w, and

F̃(uδ(g)w) = J_{1}(g)J_{2}(g)Φ̂(e_{1}g^{⊤−1}), ∀g ∈ J, ∀u ∈ U.

For x ∈ E^{×} we define S(x), by

S(x) = ∫_{Uδ(Jx)w} F̃(g)| det g|^{1−s}dg.

Then S(x) depends only on v_{E}(x). For Re(−s) sufficiently large,

Z(W̃_{1}, W̃_{2}, Φ̂, 1 − s) = ∑_{x∈𝔬_{E}^{×}\E^{×}} S(x).

Since H^{1} is normal in J, Proposition 5.3(iii), and Lemma 7.3 imply that

F̃(δ(h)g) = F̃(g), ∀h ∈ H^{1}, ∀g ∈ G.

Hence,

S(x) = vol_{U}(Uδ(H^{1}))| det x|^{s−1} ∑_{h∈(J∩U)H^{1}\J} J_{1}(hx)J_{2}(hx)Φ̂(e_{1}(hx)^{⊤−1}).

We forget the volume term, by using Lemma 7.8 and normalising the measure on U\G, so that vol_{U}(Uδ(H^{1})) = vol_{U}(UH^{1}) = 1.

As in §4.1, we put

G_{E} = {g ∈ B^{×} : gw_{1} = w_{1}}, G_{𝔅} = (G_{E} ∩ U(𝔅))U^{1}(𝔅),

where B_{E} = {w_{1}, ..., w_{r}} is our E-basis of V. Corollary 4.7 implies that we have J = (G_{𝔅}J^{1})𝔬_{K}^{×}. Then, using Lemma 7.3, we obtain:

Lemma 7.9.

S(x) = | det x|^{s−1} ∑_{y∈(1+𝔭_{K})\𝔬_{K}^{×}} Φ̂(e_{1}(yx)^{⊤−1}) ∑_{h∈(J∩U)H^{1}\G_{𝔅}J^{1}} J_{1}(hyx)J_{2}(hyx).

7.3 The case σ_{1} = σ_{2}

Suppose that σ_{1} ≅ σ_{2}; then it follows from [6] (6.2.3) that there exists an unramified quasi-character χ : F^{×} → ℂ^{×}, such that Λ_{1} ≅ Λ_{2} ⊗ χ ∘ det, and hence π_{1} ≅ π_{2} ⊗ χ ∘ det. Then J_{2}(g) = J_{1}(g^{−1})χ(det g), for all g ∈ J. Hence, for all g ∈ J, we have

∑_{h} J_{1}(hg)J_{2}(hg) = χ(det g) ∑_{h∈(J∩U)H^{1}\G_{𝔅}J^{1}} J_{1}(hg)J_{1}(g^{−1}h^{−1}) = χ(det g),

where the last equalities follow from Proposition 5.3(v),(i). It follows from Corollary 4.7 that (J : G_{𝔅}J^{1}) = (𝔬_{K}^{×} : 1 + 𝔭_{K}) = q^{N/e} − 1 = q_{𝔄} − 1. There exists a ∈ ℂ such that χ(x) = |x|_{F}^{a}, for all x ∈ F. If x ∈ E then Lemma 4.9 implies that χ(det x) = q_{𝔄}^{−av_{E}(x)}. Let x ∈ E^{×} and set j = v_{E}(x) − v_{E}(x_{d}^{−r}), Lemmas 4.9 and 7.7 imply that

S(x) = \{ \begin{matrix}0 & if j > 0; \\ cq_{𝔄}^{rv_{E}(x_{d})(s−a)} ∑_{y} ϕ(yx_{d}^{r}) & if j = 0; \\ cq_{𝔄}^{−(s−1−a)j+(s−a)rv_{E}(x_{d})}(q_{𝔄} − 1) & if j < 0. \end{matrix}

where, in the sum, y runs over the cosets 𝔬_{K}^{×}/1 + 𝔭_{K} and ϕ is defined in Lemma 7.5. It follows from 7.5 that ϕ restricted to K defines an additive character, which is trivial on 𝔭_{K}^{rv_{E}(x_{d})+1} and non-trivial on 𝔭_{K}^{rv_{E}(x_{d})}. Hence

∑_{y∈𝔬_{K}^{×}/1+𝔭_{K}} ϕ(yx_{d}^{r}) = −ϕ(0) = −1.

Set Z̃ = Z(W̃_{1}, W̃_{2}, Φ̂, 1 − s); then, for Re(−s) sufficiently large, we obtain that

\begin{aligned}Z̃ & = cq_{𝔄}^{rv_{E}(x_{d})(s−a)}(−1 + (q_{𝔄} − 1) ∑_{k≥1} q_{𝔄}^{(s−1−a)k}) \\ & = cq_{𝔄}^{rv_{E}(x_{d})(s−a)} \frac{q_{𝔄}^{s−a} − 1}{1 − q_{𝔄}^{s−1−a}} = cq_{𝔄}^{(rv_{E}(x_{d})+1)(s−a)} \frac{L(π̌_{1} × π_{2}, 1 − s)}{L(π_{1} × π̌_{2}, s)}. \end{aligned}

It follows from the functional equation and Proposition 7.2 that

ε(π_{1} × π̌_{2}, ψ_{F} , s) = ω_{π_{2}}(−1)^{N−1}cq_{𝔄}^{(s−a)(rv_{E}(x_{d})+1)}.

Following [4], we observe that the symmetry in the functional equation implies that ε(π_{1} × π̌_{1}, ψ_{F} , 1/2)^{2} = 1. Hence,

c = vol_{F} (e_{N}𝔓^{1+rv_{E}(x_{d})}) = q_{𝔄}^{−(rv_{E}(x_{d})+1)/2}.

We will now prove Theorem 7.1 in the case σ_{1} ≅ σ_{2}.

Proof. Note that Λ_{1} ≅ Λ_{2}⊗χ∘det_{A} implies Σ_{1} ≅ Σ_{2}⊗χ_{E} ∘det_{B}, where χ_{E} = χ ∘ N_{E/F} and N_{E/F} denotes the field norm. Moreover, Remark 6.3 implies that v_{E}(x_{d}) = v_{E}(ν), where ν = ν(θ_{F} , ψ_{F} , ψ_{E}). Hence,

q^{av_{E}(x_{d}^{−r})N/e} = χ(det ν^{−r}) = ω_{τ_{1}}(ν^{−r})ω_{τ_{2}}(ν^{r}).

If we compute ε(τ_{1} × τ̌_{2}, s, ψ_{E}) by the same recipe we obtain that

ε(τ_{1} × τ̌_{2}, s, ψ_{E}) = ω_{τ_{2}}(−1)^{r−1}q_{𝔅}^{−1/2}q_{𝔅}^{s−a},

where q_{𝔅} = q_{E}^{r} = q_{F}^{N/e} = q_{𝔄}. Hence,

ε(π_{1} × π̌_{2}, s, ψ_{F}) = ζω_{τ_{1}}(ν^{−r})ω_{τ_{2}}(ν^{r})q^{(s−1/2)rv_{E}(ν)N/e}ε(τ_{1} × τ̌_{2}, s, ψ_{E}),

where ζ = ω_{π_{2}}(−1)^{N−1}ω_{τ_{2}}(−1)^{r−1}.

7.4 The case σ_{1} ≠ σ_{2}

Now let us suppose that σ_{1} ≅ σ̸_{2}.

Lemma 7.10. Let x ∈ E^{×}. If v_{E}(x) ≠ v_{E}(x_{d}^{−r}) then S(x) = 0.

Proof. Set j = v_{E}(x) − v_{E}(x_{d}^{−r}). If j > 0 then Lemma 7.7 gives us S(x) = 0. If j < 0 then Lemma 7.7 implies that

S(x) = c| det x|^{s}q_{𝔄}^{j} ∑_{h∈(J∩U)H^{1}\J} J_{1}(hx)J_{2}(hx).

Recall from §5.1 that we have the idempotent e_{Ψ_{α}} given by

e_{Ψ_{α}} = Q^{−1} ∑_{h∈U^{n+1}(𝔄)\(J∩U)H^{1}} Ψ_{α}(h)h^{−1},

where Q = ((J ∩ U)H^{1} : U^{n+1}(𝔄)), and similarly e_{\overline{Ψ}_{α}}. Now,

∑_{h∈(J∩U)H^{1}\J} J_{1}(hx)J_{2}(hx) = ∑_{h∈(J∩U)H^{1}\J} tr_{Λ_{1}}(xe_{Ψ_{α}}h) trΛ̌_{2}(xe_{\overline{Ψ}_{α}}h)

= Q^{−1} tr_{Λ_{1}⊗Λ̌_{2}}(x(e_{Ψ_{α}} ⊗ e_{\overline{Ψ}_{α}}) ∑_{h∈U^{n+1}(𝔄)\J} h),

Since σ_{1} ≅ σ̸_{2} we have Λ_{1} ≅ Λ̸_{2}, hence Hom_{J}(1, Λ_{1} ⊗ Λ̌_{2}) = 0. This implies that

∑_{h∈U^{n+1}(𝔄)\J} Λ_{1}(h) ⊗ Λ̌_{2}(h) = 0,

and so S(x) = 0.

Lemma 7.11. Let b ∈ 𝔎(𝔅); then

∑_{h∈(J^{1}∩U)H^{1}\J^{1}} J_{1}(hb)J_{2}(hb) = J_{Σ_{1}}(b)JΣ̌_{2}(b).

Proof. According to Proposition 5.13, we have J_{Λ_{i}}(g) = Jκ̃(g)J_{Σ_{i}}(g), for i = 1, 2 and for all g ∈ J. The assertion follows from the fact that J^{1} acts trivially on Σ_{i} and Proposition 5.3(v),(i) applied to κ̃, via Lemma 5.12.

We now prove Theorem 7.1 in the case when σ_{1} ≅ σ̸_{2}.

Proof. Set Z̃ = Z(W̃_{1}, W̃_{2}, Φ̂, 1 − s). It follows from Lemma 7.10 that, for Re(−s) sufficiently large, Z̃ = S(x_{d}^{−r}). Lemmas 7.9, 4.9 and 7.7 imply that

Z̃ = cq_{𝔄}^{−rv_{E}(x_{d})s} ∑_{y∈(1+𝔭_{K})\𝔬_{K}^{×}} ϕ(y^{−1}x_{d}^{r}) ∑_{h∈(J∩U)H^{1}\G_{𝔅}J^{1}} J_{1}(hyx_{d}^{−r})J_{2}(hyx_{d}^{−r}),

where c = q_{𝔄}^{(−rv_{E}(x_{d})−1)/2}. Lemma 7.11 implies that

Z̃ = cq_{𝔄}^{rv_{E}(x_{d})s} ∑_{y∈(1+𝔭_{K})\𝔬_{K}^{×}} ϕ(y^{−1}x_{d}^{r}) ∑_{h∈U^{1}(𝔅_{m})\G_{𝔅}} J_{Σ_{1}}(hyx_{d}^{−r})JΣ̌_{2}(hyx_{d}^{−r}),

where U^{1}(𝔅_{m}) = (U ∩ U(𝔅))U^{1}(𝔅). Now x_{d} ∈ E so we can use Proposition 5.3(ii) and Lemma 7.3 to obtain

Z̃ = cq_{𝔄}^{rv_{E}(x_{d})s}ω_{Σ_{1}}(x_{d}^{−r})ω_{Σ_{2}}(x_{d}^{r}) ∑_{h∈U^{1}(𝔅_{m})\U(𝔅)} ϕ(hx_{d}^{r})J_{Σ_{1}}(h^{−1})JΣ̌_{2}(h^{−1}).

Lemma 7.6 implies that

Z̃ = cq_{𝔄}^{rv_{E}(x_{d})s}ω_{Σ_{1}}(x_{d}^{−r})ω_{Σ_{2}}(x_{d}^{r}) ∑_{h∈U^{1}(𝔅_{m})\U(𝔅)} ψ_{E}(h_{r1})J_{Σ_{1}}(h^{−1})JΣ̌_{2}(h^{−1}),

where h_{r1} is the r1-coefficient of the matrix of h with respect to the basis B_{E}. It now follows from the functional equation and Proposition 7.2 that ε(π_{1} × π̌_{2}, s, ψ_{F}) = ω_{π_{2}}(−1)^{N−1}Z̃. If we compute ε(τ_{1} × τ̌_{2}, s, ψ_{E}) by the same recipe, we obtain that

ε(τ_{1} × τ̌_{2}, s, ψ_{E}) = q_{𝔅}^{−1/2}ω_{τ_{2}}(−1)^{r−1} ∑_{h∈U^{1}(𝔅_{m})\U(𝔅)} ψ_{E}(h_{r1})J_{Σ_{1}}(h^{−1})JΣ̌_{2}(h^{−1}),

where q_{𝔅} = q_{E}^{r} = q_{F}^{N/e} = q_{𝔄}. Hence,

ε(π_{1} × π̌_{2}, s, ψ_{F}) = ζω_{τ_{1}}(x_{d}^{−r})ω_{τ_{2}}(x_{d}^{r})q^{rv_{E}(x_{d})(s−1/2)N/e}ε(τ_{1} × τ̌_{2}, s, ψ_{E}),

where ζ = ω_{τ_{2}}(−1)^{r−1}ω_{π_{2}}(−1)^{N−1}. Now ω_{τ_{1}} and ω_{τ_{2}} are trivial on 1 + 𝔭_{E} and Remark 6.3 finishes the proof, as in the case σ_{1} ≅ σ_{2}.

7.5 Twists by tamely ramified quasi-characters

We continue in the same situation as above. Theorem 7.1 immediately implies the following:

Corollary 7.12. Let χ : F^{×} → ℂ^{×} be a tamely ramified quasi-character and put χ_{E} = χ ∘ N_{E/F}; then

\frac{ε(π_{1}χ × π̌_{2}, s, ψ_{F})}{ε(π_{1} × π̌_{2}, s, ψ_{F})} = χ(N_{E/F} (ν^{−r^{2}}))\frac{ε(τ_{1}χ_{E} × τ̌_{2}, s, ψ_{E})}{ε(τ_{1} × τ̌_{2}, s, ψ_{E})},

where ν = ν(θ_{F} , ψ_{F} , ψ_{E}).

In the case E is maximal, totally ramified over F and π_{1} = π_{2} we recover [5]§6.1 Corollaire 2, with (in the notation of [5]) c(π_{1}, π̌_{1}, ψ_{F}) = N_{E/F} (ν). Moreover, Corollary 6.6 implies [5]§7.1 Théorème, which describes how the constant c(π_{1}, π̌_{1}, ψ_{F}) changes under the tame lifting operation.

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Vytautas Paskunas Shaun Stevens Fakultät für Mathematik School of Mathematics Universität Bielefeld University of East Anglia Postfach 100131 Norwich NR4 7TJ D-33501 Bielefeld United Kingdom Germany

paskunas@math.uni-bielefeld.de ginnyshaun@bigfoot.com