Essential Whittaker Functions for GL(n)

Nadir Matringe

Received: July 6, 2012 Revised: July 19, 2013

Communicated by Patrick Delorme

Abstract. We give a constructive proof of the existence of the essential Whittaker function of a generic representation of GL(n, F), for F a non-archimedean local field, using mirabolic restriction techniques.

2010 Mathematics Subject Classification: 22E50

Introduction

Let F be nonarchimedean local field, we denote by 𝔒 its ring of integers, and by 𝔓 = ϖ𝔒 the maximal ideal of this ring, where ϖ is a uniformiser of F. We denote by q the cardinality of 𝔒/𝔓 and by |.| the absolute value on F normalised such that |ϖ| is equal to q^{−1}. For n ≥ 1, we denote the group GL(n, F) by G_{n}, the group GL(n, 𝔒) by G_{n}(𝔒), and we set G_{0} = {1}. We denote by A_{n} the torus of diagonal matrices in G_{n}, and by N_{n} the unipotent radical of the Borel subgroup of G_{n} given by upper triangular matrices. For m ≥ 1, we denote by K_{n}(m) the subgroup

of G_{n}, given by matrices (\begin{matrix}g & v \\ l & t \end{matrix}), for g in G_{n−1}(𝔒), v in 𝔒^{n−1}, l with every

coefficient in 𝔓^{m}, and t in 1 + 𝔓^{m}. We set K_{n}(0) = G_{n}(𝔒).

If π is a generic representation of G_{2}, the essential vector of π was first considered in [C], for G_{n} with n ≥ 2, it was studied in [J-P-S]. Here is one of its main properties: if one calls d the conductor (the power of q^{−s} in the ϵ factor with respect to an unramified additive character of F) of the representation π, the complex vector space π^{K_{n}(d)} of vectors in π fixed under K_{n}(d), is generated by the essential vector of π, and π^{K_{n}(k)} becomes the null space for k < d. However, to prove its existence, one has to study properties of the Rankin-Selberg integrals associated to the pairs (π, π′), where π′ varies through the set of unramified generic representations of G_{n−1}.

We set a few more notations before explaining this. We denote by ν the positive character |.| ∘ det of G_{n}. We use the product notation for normalised parabolic induction (see Section 1.2). For any sequence of complex numbers s_{1}, …, s_{n}, the representation |.|^{s_{1}} × ⋯ × |.|^{s_{n}} of G_{n} is unramified, and its subspace of G_{n}(𝔒)-invariant vectors is of dimension 1 (see Section 1.5). We choose a character θ of (F, +) trivial on 𝔒 but not on 𝔓^{−1}, and use it to define a non degenerate character, still denoted θ, of the standard unipotent subgroup N_{n} of G_{n}, by θ(n) = θ(∑_{i=1}^{n−1} n_{i,i+1}).

For n ≥ 2, let π and π′ be representations of Whittaker type (see Section 1.4) of G_{n} and G_{n−1} respectively, and denote by W (π, θ) and W (π′, θ^{−1}) their respective Whittaker models (which are quotients of π and π′) with respect to θ and θ^{−1}. If W and W′ belong respectively to W (π, θ) and W (π′, θ^{−1}), we denote I(W, W′, s) the associated Rankin-Selberg integral (see Section 1.4). For example, for a sequence of complex numbers a_{1}, …, a_{m}, the induced representation |.|^{a_{1}} × ⋯ × |.|^{a_{m}} of G_{m}, is of Whittaker type. If moreover Re(a_{1}) ≥ ⋯ ≥ Re(a_{m}), the representation |.|^{a_{1}} × ⋯ × |.|^{a_{m}} is of Langlands’ type, and its Whittaker model contains a unique normalised spherical Whittaker function W (q^{−a_{1}}, …, q^{−a_{m}}). It is the unique Whittaker function on G_{m}, fixed by G_{m}(𝔒), which equals 1 on G_{m}(𝔒), and associated to the Satake parameter {q^{−a_{1}}, …, q^{−a_{m}}} (see [S]). For fixed g in G_{m}, the function W (q^{−s_{1}}, …, q^{−s_{m}})(g) is an element of the ring ℂ[q^{±s_{1}}, …, q^{±s_{m}}]^{S_{m}} of invariant Laurent polynomials. To define the essential vector of π, one needs to show as in [J-P-S], the following theorem (see [G-J] for the definition of the L function of an irreducible representation of G_{n}):

Theorem. Let π be a generic representation of G_{n} with Whittaker model W (π, θ), then there exists in W (π, θ) a unique G_{n−1}(𝔒)-invariant function W_{π}^{ess}, such that for every sequence of complex numbers s_{1}, …, s_{n−1}, one has the equality I(W_{π}^{ess}, W (q^{−s_{1}}, …, q^{−s_{n−1}}), s) = ∏_{i=1}^{n−1} L(π, s + s_{i}).

Hence, the statement of the theorem is equivalent to say that for any unramified representation π′ of Langlands’ type of G_{n−1} with normalised spherical Whittaker function W_{π′}^{0} in W (π′, θ), one has the equality I(W_{π}^{ess}, W_{π′}^{0}, s) = L(π, π′, s) (see Section 1.4 for the definition of L(π, π′, s) and the equality L(π, π′, s) = ∏_{i=1}^{n−1} L(π, s + s_{i}) when π′ = |.|^{s_{1}} × ⋯ × |.|^{s_{m}}). Using this theorem, it is then shown in [J-P-S], using the functional equation of L(π, π′, s), that the space W (π, θ)^{K_{n}(d)} is a complex line spanned by W_{π}^{ess}, and that W (π, θ)^{K_{n}(k)} is zero for k < d.

In this paper, we will show the following result, using the interpretation in terms of restriction of Whittaker functions of the Bernstein-Zelevinsky derivatives.

Essential Whittaker Functions for GL(n) 1193

Let π be a ramified generic representation of G_{n}, and π_{u} be the unramified component of the first nonzero spherical Bernstein-Zelevinsky derivative π^{(n−r)} of π (see Definition 1.3 for the precise definition). The representation π_{u} is an unramified representation of Langlands’ type of G_{r} when r ≥ 1. In this situation, we show in Corollary 3.2, that there is a unique Whittaker function W_{π}^{ess} in W (π, θ), which is right G_{n−1}(𝔒)-invariant, and which satisfies, for a = diag(a_{1}, …, a_{n−1}) ∈ A_{n−1} and a′ = diag(a_{1}, …, a_{r}) ∈ A_{r}, the equality:

W (diag(a, 1)) = W_{π_{u}}^{0}(a′)ν(a′)^{(n−r)/2}1_{𝔒}(a_{r}) ∏_{r<i<n} 1_{𝔒^{∗}}(a_{i}),  (1)

when r ≥ 1, and

W (diag(a, 1)) = ∏_{0<i<n} 1_{𝔒^{∗}}(a_{i})  (2)

when r = 0. Computing the integral I(W_{π}^{ess}, W_{π′}^{0}, s) for an unramified representation π′ of Langlands’ type of G_{n−1}, we will obtain in Corollary 3.3 the statement (more precisely a slightly more general statement) of the theorem stated above.

For GL(2, F), a detailed account about newforms can be found in [Sc], the author obtains Formula (1) (see Section 2.4 of [loc. cit.]) up to normalisation by an ϵ-factor. For GL(n, F), Miyauchi ([Mi]) recently obtained Formula (1), assuming the existence of the essential vector, by using Hecke algebras, i.e. generalising Shintani’s method for spherical representations.

Remark. The reason why we got interested in reproving the existence of such a vector is the following. In [J-P-S], the uniqueness of such a vector is proved. The proof of the existence is valid only for generic representations π appearing as subquotients of representations parabolically induced by ramified characters of GL(1, F) and cuspidal representations of GL(r, F) for r ≥ 2, i.e. generic representations whith L-function equal to one. Before we explain this, let us mention that Jacquet (see [J]) found a simple fix for the proof of [J-P-S], so that the motivation of writing our note is really to give a constructive proof of the existence of this vector, which provides a nice application of the techniques developed in [C-P].

In [J-P-S], the following is shown: for fixed W in W (π, θ), the function

P (W, q^{−s_{1}}, …, q^{−s_{n−1}}) = I(W, W (q^{−s_{1}}, …, q^{−s_{n−1}}), 0)/ ∏_{i=1}^{n−1} L(π, s_{i})

belongs to the ring ℂ[q^{±s_{1}}, …, q^{±s_{n−1}}]^{S_{n−1}} of symmetric Laurent polynomials in the variables q^{−s_{i}}. It is also shown that the existence of the essential vector is equivalent to the fact that the vector space

I(π) = {P (W, q^{−s_{1}}, …, q^{−s_{n−1}}), W ∈ W (π, θ)}, which is actually an ideal, is equal to the the full ring

ℂ[q^{±s_{1}}, …, q^{±s_{n−1}}]^{S_{n−1}}.

The argument used to prove it goes like this: For W well chosen, P (W, q^{s_{1}}, …, q^{−s_{n−1}}) is equal to

∏_{i=1}^{n−1} 1/L(π, s_{i}).

We denote by Q the element 1/L(π, s) of ℂ[q^{−s}], so that

P (W, q^{−s_{1}}, …, q^{−s_{n−1}}) = ∏_{i=1}^{n−1} Q(q^{−s_{i}}).

Because of the functional equation of the L-function L(π, |.|^{s_{1}} × ⋯ × |.|^{s_{n−1}}, s), denoting π^{∨} the smooth contragredient of π, one shows that I(π) also contains the product ∏_{i=1}^{n−1} Q′(q^{−1}q^{s_{i}}), where

Q′(q^{−s}) = 1/L(π^{∨}, s).

Proposition 2.1. of the paper then shows that Q′(q^{−1}q^{s}) and Q(q^{−s}) are prime to one another in ℂ[q^{±s}], and they deduce from this that no maximal ideal

I_{q^{−a_{1}} ,...,q^{−a_{n−1}}} = {R ∈ ℂ[q^{±a_{1}}, …, q^{±a_{n−1}}], R(q^{±a_{1}}, …, q^{±a_{n−1}}) = 0}

for (a_{1}, …, a_{n−1}) in ℂ^{n−1}, contains ∏_{i=1}^{n−1} Q′(q^{−1}q^{s_{i}}) and ∏_{i=1}^{n−1} Q(q^{−s_{i}}) together, which implies the result. This last step is false as soon as n ≥ 3, and there are a and b in ℂ^{∗} such that Q(a) = Q′(q^{−1}b^{−1}) = 0, because then both products belong to any ideal I_{a,b,...,x_{n−1}}. This is the case as soon as the degree d^{∘}(Q) of Q satisfies d^{∘}(Q) ≥ 1. However, using the functional equation of L(π, |.|^{z_{1}} × ⋯ × |.|^{z_{n−1}}), and the cyclicity of W (q^{−z_{1}}, …, q^{−z_{n−1}}) in W (|.|^{z_{1}} × ⋯ × |.|^{z_{n−1}}, θ) when Re(z_{i}) ≥ Re(z_{i+1}), Jacquet noticed (see [J]) that one can find for every (a_{1}, …, a_{n−1}) in ℂ^{n−1}, a polynomial in I(π), taking the value 1 when evaluated at (q^{±a_{1}}, …, q^{±a_{n−1}}), so I(π) is indeed equal to ℂ[q^{±s_{1}}, …, q^{±s_{n−1}}]^{S_{n−1}}.

1 Preliminaries

In this section, we first recall basic facts about smooth representations of locally profinite groups. We then focus on G_{n}, recall results from [B-Z] about derivatives, then introduce the L-function of a pair of representations of Whittaker type, we discuss espacially the unramified case.

1.1 Smooth representations, restriction and induction

When G is an l-group (locally compact totally disconnected group), we denote by Alg(G) the category of smooth complex G-modules. We denote by Ĝ the group of smooth characters (smooth representations of dimension 1) of G. If (π, V) belongs to Alg(G), H is a closed subgroup of G, and χ is a character of H, we denote by V (H, χ) the subspace of V generated by vectors of the form π(h)v − χ(h)v for h in H and v in V. This space is stable under the action of the subgroup N_{G}(χ) of the normalizer N_{G}(H) of H in G, which fixes χ. We denote by δ_{G} the positive character of G such that if μ is a right Haar measure on G, and int is the action of G on smooth functions f with compact support in G, given by (int(g)f)(x) = f(g^{−1}xg), then μ ∘ int(g) = δ_{G}(g)μ for g in G. The space V (H, χ) is N_{G}(χ)-stable. Thus, if L is a closed-subgroup of N_{G}(χ), and δ′ is a (smooth) character of L (which will be a normalising character dual to that of normalised induction later), the quotient V_{H,χ} = V/V (H, χ) (that we simply denote by V_{H} when χ is trivial) becomes a smooth L-module for the (normalised) action l.(v + V (H, χ)) = δ′(l)π(l)v + V (H, χ) of L on V_{H,χ}. We denote by V^{H} the subspace of vectors of V fixed by H; for H compact and open, the functor V ↦ V^{H} from Alg(G) to Alg(G_{0}) is exact ([B-H], 2.3., Corollary 1). We say that (π, V) in Alg(G) is admissible if for any compact open subgroup H of G, the vector space V^{H} is finite dimensional. If H is a closed subgroup of an l-group G, and (ρ, W) belongs to Alg(H), we define the objects (ind_{H}^{G}(ρ), V_{c} = ind_{H}^{G}(W)) and (Ind_{H}^{G}(ρ), V = Ind_{H}^{G}(W)) of Alg(G) as follows. The space V is the space of smooth functions from G to W, fixed under right translation by the elements of a compact open subgroup U_{f} of G, and satisfying f(hg) = ρ(h)f(g) for all h in H and g in G. The space V_{c} is the subspace of V, consisting of functions with support compact mod H, in both cases, the action of G is by right translation on the functions. We recall that by Frobenius reciprocity law ([B-H], 2.4.), the spaces Hom_{G}(π, Ind_{H}^{G}(ρ)) and Hom_{H}(π_{|H}, ρ) are isomorphic when π (resp. ρ) belongs to Alg(G) (resp. Alg(H)). If the group G is exhausted by compact subsets (which is the case of closed subgroups of G_{n}), and (π, V) is irreducible, it is known ([B-H], 2.6., Corollary 1) that the center Z of G acts on V by the so-called central character of π which we will denote c_{π}. When G = G_{n}, then Z identifies with F^{∗}. By definition, the real part Re(χ) of a character χ of F^{∗} is the real number r such that |χ(t)|_{ℂ} = |t|^{r}, where |z|_{ℂ} = \sqrt{zz̄} for z in ℂ.

1.2 Parabolic induction and segments for GL(n)

Now we focus on the case G = G_{n}, we will only consider smooth representations of its closed subgroups. It is known that irreducible representations of G_{n} are admissible (see [C2]).

If n ≥ 1, let n̄ = (n_{1}, …, n_{t}) be a partition of n of length t (i.e. an ordrered set of t positive integers whose sum is n), we denote by Mn̄ to be the Levi subgroup of G_{n}, of matrices diag(g_{1}, …, g_{t}), with each g_{i} in G_{n_{i}}, by Nn̄ the

unipotent subgroup of matrices (\begin{matrix}I_{n_{1}} & ⋆ & ⋆ \\ & ... & ⋆ \\ & & I_{n_{t}} \end{matrix} ), and by Pn̄ the standard

parabolic subgroup Mn̄Nn̄ (where Mn̄ normalises Nn̄). Note that M_{(1,...,1)} is equal to A_{n}, and N_{(1,...,1)} = N_{n}. For each i, let π_{i} be a smooth representation of G_{n_{i}}, then the tensor product π_{1} ⊗ ⋯ ⊗ π_{t} is a representation of Mn̄, which can be considered as a representation of Pn̄ trivial on Nn̄. We will use the product notation

π_{1} × ⋯ × π_{t} = Ind_{Pn̄}^{G_{n}}(δ_{Pn̄}^{1/2}π_{1} ⊗ ⋯ ⊗ π_{t})

for the normalised parabolic induction. Parabolic induction preserves finite length and admissibility (see [B-Z] or [C2]). We say that an irreducible representation (ρ, V) of G_{n} is cuspidal, if the Jacquet module V_{Nn̄} is zero whenever n̄ is a proper partition of n (i.e. we exclude n̄ = (n)). Suppose that n̄ = (m, …, m) is a partition of n of length l, and that ρ is a cuspidal representation of G_{m}. Then Theorem 9.3. of [Z] implies that the G_{n}- module ν^{−(l−1)}ρ × ν^{−(l−2)}ρ × ⋯ × ν^{−1}ρ × ρ has a unique irreducible quotient which we denote [ν^{−(l−1)}ρ, ν^{−(l−2)}ρ, …, ν^{−1}ρ, ρ]. We will call such a representation a segment, it is known that segments are the quasi square integrable representations of G_{n}, but we won’t need this result. We end this paragraph with a word about induced representations of Langlands’ type:

Definition 1.1. Let Δ_{1}, …, Δ_{t} be segments of respectively G_{n_{1}}, …, G_{n_{t}}, and suppose that Re(c_{Δ_{i}}) ≥ Re(c_{Δ_{i+1}}). Let n = n_{1}+⋯+n_{t}, then the representation Δ_{1} × ⋯ × Δ_{t} of G_{n} is said to be induced of Langlands’ type.

These representations enjoy many remarkable properties, some of which we will recall later, here is a first one (which is the main result of [Sil]).

Proposition 1.1. Let π be induced of Langlands’ type, then π has a unique irreducible quotient Q(π). Moreover, considering that isomorphic representations are equal, the map π ↦ Q(π) gives a bijection between the set of induced representations of Langlands’ type of G_{n}, and the set of irreducible representations of G_{n}.

1.3 Berstein-Zelevinsky derivatives

For n ≥ 2 we denote by U_{n} the group of matrices of the form \binom{I_{n−1} v}{1}.

For n > k ≥ 1, the group G_{k} embeds naturally in G_{n}, and is given by matrices of the form diag(g, I_{n−k}). We denote by P_{n} the mirabolic subgroup G_{n−1}U_{n} of G_{n} for n ≥ 2, and P_{1} = {1_{G_{1}}}. If one sees P_{n−1} as a subgroup of G_{n−1} itself embedded in G_{n}, then P_{n−1} is the normaliser of θ_{|U_{n}} in G_{n−1} (i.e. if g ∈ G_{n−1}, then θ(g^{−1}ug) = θ(u) for all u ∈ U_{n} if and only if g ∈ P_{n−1}). We define the following functors:

• The functor Φ^{−} from Alg(P_{k}) to Alg(P_{k−1}) such that, if (π, V) is a smooth P_{k}-module, Φ^{−}V = V_{U_{k},θ}, and P_{k−1} acts on Φ^{−}(V) by Φ^{−}π(p)(v + V (U_{k}, θ)) = δ_{P_{k}}(p)^{−1/2}π(p)(v + V (U_{k}, θ)).

• The functor Φ^{+} from Alg(P_{k−1}) to Alg(P_{k}) such that, for π in Alg(P_{k−1}), one has Φ^{+}π = ind_{P_{k−1}U_{k}}^{P_{k}}(δ_{P_{k}}^{1/2}π ⊗ θ).

• The functor Φ̂^{+} from Alg(P_{k−1}) to Alg(P_{k}) such that, for π in Alg(P_{k−1}), one has Φ̂^{+}π = Ind_{P_{k−1}U_{k}}^{P_{k}}(δ_{P_{k}}^{1/2}π ⊗ θ).

• The functor Ψ^{−} from Alg(P_{k}) to Alg(G_{k−1}), such that if (π, V) is a smooth P_{k}-module, Ψ^{−}V = V_{U_{k},1}, and G_{k−1} acts on Ψ^{−}(V) by Ψ^{−}π(g)(v + V (U_{k}, 1)) = δ_{P_{k}}(g)^{−1/2}π(g)(v + V (U_{k}, 1)).

• The functor Ψ^{+} from Alg(G_{k−1}) to Alg(P_{k}), such that for π in Alg(G_{k−1}), one has Ψ^{+}π = ind_{G_{k−1}U_{k}}^{P_{k}}(δ_{P_{k}}^{1/2}π ⊗ 1) = δ_{P_{k}}^{1/2}π ⊗ 1.

These functors have the following properties which can be found in [B-Z]:

Proposition 1.2. a) The functors Φ^{−}, Φ^{+}, Ψ^{−}, and Ψ^{+} are exact.

b) Ψ^{−} is left adjoint to Ψ^{+}. b’) Φ^{−} is left adjoint to Φ̂^{+}.

c) Φ^{−}Ψ^{+} = 0 and Ψ^{−}Φ^{+} = 0.

d) Ψ^{−}Ψ^{+} ≃ Id and Φ^{−}Φ^{+} ≃ Id.

e) One has the exact sequence 0 → Φ^{+}Φ^{−} → Id → Ψ^{+}Ψ^{−} → 0.

Following [C-P], if τ belongs to Alg(P_{n}), we will denote (Φ^{−})^{k}τ by τ_{(k)}, and as usual, τ^{(k)} will be defined as Ψ^{−}τ_{(k−1)}.

Because of e), τ has a natural filtration of P_{n}-modules 0 ⊂ τ_{n} ⊂ ⋯ ⊂ τ_{1} = τ, where τ_{k} = Φ^{+}^{k−1}Φ^{−}^{k−1}τ. We will use the notation τ_{(k),i} for (τ_{(k)})_{i}. The following observation is just a restatement of the definitions:

Lemma 1.1. If τ belongs to Alg(P_{n}), then τ_{k} = Φ^{+}(τ_{(1),k−1}) for k ≥ 1.

1.4 Representations of Whittaker type and their L-functions

We recall that we fixed a character θ of conductor 𝔒 in the introduction.

Definition 1.2. Let π be an admissible representation of G_{n}, we say that π is of Whittaker type π if Hom(π, Ind_{N_{n}}^{G_{n}}(θ)) is of dimension 1, or equivalently, according to Frobenius reciprocity law, if the space Hom_{N_{n}}(π, θ) is of dimension 1. We denote by W (π, θ) the image of π in Ind_{N_{n}}^{G_{n}}(θ), it is called the Whittaker model of π (with respect to θ), it is a quotient of π.

Being of Whittaker type does not depend on the character θ of (F, +), as another non trivial character θ′ of (F, +) will give birth to a character θ′ of N_{n}, conjugate to θ by A_{n}.

In terms of derivatives, as the representation Ind_{N_{n}}^{P_{n}}(θ) is isomorphic to (Φ̂^{+})^{n−1}Ψ^{+}(1), where 1 is the trivial representation of G_{0}, applying b) and b′) of Proposition 1.2, we obtain that Hom_{N_{n}}(π, θ) ≃ ℂ if and only if π^{(n)} = 1. Applying this to product of segments, and using the rules of “derivation“ given in Lemma 3.5 of [B-Z] and Proposition 9.6. of [Z], we obtain that if Δ_{1}, …, Δ_{t} are segments of G_{n_{1}}, …, G_{n_{t}} respectively, the representation π = Δ_{1} ×⋯×Δ_{t} of G_{n} (for n = n_{1} + ⋯ + n_{t}) is of Whittaker type. If the segments Δ_{i} are ordered so that π is of Langlands’ type, we can say more according to the main result of [J-S 3].

Proposition 1.3. For n ≥ 1, let π be a representation of G_{n}, which is induced of Langlands’ type, then it has an injective Whittaker model, i.e. π ≃ W (π, θ) (equivalently π embeds in Ind_{N_{n}}^{G_{n}}(θ)).

If π is irreducible and embeds in Ind_{N_{n}}^{G_{n}}(θ), it is a well-known theorem of Gelfand and Kazhdan ([G-K]) that the multiplicity of π in Ind_{N_{n}}^{G_{n}}(θ) is 1, we then say that π is generic. We recall (Theorem 9.7 of [Z]), that every generic representation π of GL(n, F) can be written uniquely, up to permutation of the terms in the product, as a commutative product of unlinked (see 4.1. of [Z]) segments

[ν^{−(k_{1}(π)−1)}ρ_{1}(π), …, ρ_{1}(π)] × ⋯ × [ν^{−(k_{t}(π)−1)}ρ_{t}(π), …, ρ_{t}(π)].

In particular, generic representations are the representations of Langlands’ type which are irreducible.

We now recall from [J-P-S 2], some results about the L-function of a pair of representations of Whittaker type. Let π be a representation of G_{n} of Whittaker type, and π′ be a representation of Whittaker type of G_{m}, with respective Whittaker models W (π, θ) and W (π′, θ^{−1}), for n ≥ m ≥ 1. When n > m, and W and W′ are respectively in W (π, θ) and W (π′, θ^{−1}), we write

I(W, W′, s) = ∫_{N_{m}\G_{m}} W (\begin{matrix}g \\ & I_{n−m} \end{matrix}) W′(g)ν(g)^{s−(n−m)/2}dg.

When n = m, and W and W′ are respectively in W (π, θ) and W (π′, θ^{−1}), ϕ is in C_{c}^{∞}(F^{n}), and η is the row vector (0, …, 0, 1) in the space M(1, n, F) of row matrices 1 by n with entries in F, we write

I(W, W′, ϕ, s) = ∫_{N_{n}\G_{n}} W (g)W′(g)ϕ(ηg)ν(g)^{s}dg.

It is shown in [J-P-S 2] that these integrals converge absolutely for Re(s) large, and define elements of ℂ(q^{−s}). If n > m, the integrals I(W, W′, s) (which we shall also write I(W′, W, s) when convenient) span, when (W, W′) varies in W (π, θ) × W (π′, θ^{−1}), a fractional ideal of ℂ[q^{s}, q^{−s}], which is generated by a unique Euler factor L(π, π′, s). If n = m, the integrals I(W, W′, ϕ, s) span, when (W, W′, ϕ) varies in W (π, θ) × W (π′, θ^{−1}) × C_{c}^{∞}(F^{n}), a fractional ideal of ℂ[q^{s}, q^{−s}], which is generated by a unique Euler factor L(π, π′, s). If n < m, we define L(π, π′, s) to be L(π′, π, s). We recall Proposition 9.4 of [J-P-S 2].

Proposition 1.4. For n ≥ m ≥ 1, if π = Δ_{1} × ⋯ × Δ_{t} is a representation of G_{n} induced of Langlands’ type, and π′ = Δ′_{1} × ⋯ × Δ′_{u} is a representation of G_{m} induced of Langlands’ type, then L(π, π′, s) = ∏_{i,j} L(Δ_{i}, Δ′_{j}, s).

Finally, we recall that it is proved in Section 5 of [J-P-S 2], that if π is a generic representation of G_{n}, and χ is a character of G_{1}, one has the equality L(π, χ, s) = L(χ⊗π, s) between the Rankin-Selberg L-function on the left, and the Godement-Jacquet L-function on the right.

1.5 Unramified representations

We say that a representation (π, V) of G_{n} is unramified (or spherical), if it admits a nonzero G_{n}(𝔒)-fixed vector in its space. If it is the case, we recall that the Hecke convolution algebra H_{n} (whose elements are the functions with compact support on G_{n}, which are left and right invariant under G_{n}(𝔒)), acts on V^{G_{n}(𝔒)} (when V^{G_{n}(𝔒)} is of dimension 1, the action is necessarily by a character). The Hecke algebra H_{n} is commutative and isomorphic by the Satake isomorphism to the algebra ℂ[X_{1}^{±1}, …, X_{n}^{±1}]^{S_{n}} according to [Sat]. Hence, a character of H_{n} is associated to a unique set of nonzero complex numbers {z_{1}, …, z_{n}}, corresponding to the evaluation P ↦ P (z_{1}^{±1}, …, z_{n}^{±1}) from ℂ[X_{1}^{±1}, …, X_{n}^{±1}]^{S_{n}} to ℂ. It is known that when π is unramified and irreducible (see Section 4.6 in [Bu] for example), then V^{G_{n}(𝔒)} is one dimensional, and that the corresponding character of H_{n} determines π, in which case the associated set of nonzero complex numbers is called the Satake parameter of π. We recall with proofs, some classical facts about parabolically induced spherical representations.

Proposition 1.5. Let π_{i} be a representation of G_{n_{i}} for i between 1 and t, n = n_{1} + ⋯ + n_{t}, Mn̄ be the standard Levi subgroup of n corresponding to the partition (n_{1}, …, n_{t}), and π = π_{1} × ⋯ × π_{t}. The vector space Hom_{G_{n}(𝔒)}(1, π) is isomorphic to Hom_{Mn̄∩G_{n}(𝔒)}(1, π_{1} ⊗ ⋯ ⊗ π_{t}).

Proof. From Mackey’s theory ([B-Z], Theorem 5.2.), as G_{n} is equal to the double-class Pn̄G_{n}(𝔒), the restriction of π to G_{n}(𝔒) is equal to Ind_{Mn̄∩G_{n}(𝔒)}^{G_{n}(𝔒)}(π_{1} ⊗⋯⊗π_{t}). The result then follows from Frobenius reciprocity law.

We have the following corollaries to this.

Corollary 1.1. Let χ_{1}, …, χ_{n} be unramified characters of F^{∗}, then the representation π = χ_{1}×⋯×χ_{n} of G_{n} is unramified, and π^{G_{n}(𝔒)} is one dimensional. In particular, by the exactness of the functor V ↦ V^{G_{n}(𝔒)} from Alg(G_{n}) to Alg(G_{0}), the representation π has only one irreducible spherical subquotient.

When π = |.|^{s_{1}} × ⋯ × |.|^{s_{n}} is an unramified representation of G_{n} induced from the Borel subgroup, we just saw that π^{G_{n}(𝔒)} is of dimension 1. The character of H_{n}, given by its action on π^{G_{n}(𝔒)} corresponds to the set {q^{−s_{1}}, …, q^{−s_{n}}}. If π is moreover of Langlands’ type, it is the Satake parameter of Q(π) thanks to the next corollary.

Corollary 1.2. If π = χ_{1} × ⋯ × χ_{n} is as above, but moreover of Langlands’ type, then it is the irreducible quotient of π which is spherical. In particular, π is spanned by π^{G_{n}(𝔒)}.

Proof. One checks by induction on n the following assertion: the representation π can be written π_{1} × ⋯ × π_{t}, where for each i, the representation π_{i} is equal to ν^{(l_{i}−1)/2}μ_{i} × ν^{(l_{i}−3)/2}μ_{i} ⋯ × ν^{(1−l_{i})/2}μ_{i} for a positive integer l_{i} and an unramified character μ_{i} of F^{∗}, and such that the segments [π′_{i}] = [ν^{(1−l_{i})/2}μ_{i}, …, ν^{(l_{i}−1)/2}μ_{i}] are unlinked, the character ν^{(1−l_{t})/2}μ_{t} is equal to χ_{n}, and the segment [π′_{t}] contains any other [π′_{i}] in which χ_{n} occurs (i.e. if χ_{n} = ν^{r}μ_{i} for r ∈ \frac{1}{2}ℤ between (1−l_{i})/2 and (l_{i}−1)/2, then r = (1−l_{i})/2 and l_{i} ≤ l_{t}). From [Z], Section 2, the irreducible quotient of π_{i} is the irreducible submodule of ν^{(1−l_{i})/2}μ_{i} × ⋯ × ν^{(l_{i}−1)/2}μ_{i}, i.e. the character μ̃_{i} = μ_{i} ∘ det of G_{l_{i}}. The representation μ̃_{1} ×⋯×μ̃_{l} is thus a quotient of π, which is irreducible by Theorem 4.2 of [Z], hence it is its irreducible quotient. It is spherical from Proposition 1.5, as the representations μ̃_{i} are spherical.

Corollary 1.3. A segment Δ of G_{n}, for n ≥ 2, is always ramified.

Proof. If Δ was unramified, as it is irreducible, its Satake parameter would be equal to a set {q^{−s_{1}}, …, q^{−s_{n}}}, hence the same as that of Q(π), for π = |.|^{s_{1}} × ⋯ × |.|^{s_{n}} (if we order the s_{i}’s in a correct way). In particular, Δ would be equal to Q(π), which is absurd according to Proposition 1.1.

For unramified representations of Langlands’ type, normalised spherical Whittaker functions are test functions for L factors. If π is an unramified representation of Langlands’ type, we denote by W_{π}^{0} the spherical function in W (π, θ) which is equal to 1 on G_{n}(𝔒), and call it the normalised spherical Whittaker function of π. In [S], Shintani gave an explicit formula for W_{π}^{0} in terms of the Satake parameter of π.

Using this formula, Jacquet and Shalika found (Proposition 2.3 of [J-S], and equality (3) in Section 1 of [J-S 2]), for π and π′ two unramified representations of Langlands’ type (see the discussion after Equations (3) and (4) below) of G_{n} and G_{m} respectively, and for correct normalisations of Haar measures, the equalities:

L(π, π′, s) = I(W_{π}^{0}, W_{π′}^{0}, 1_{𝔒^{n}}, s)
= ∫_{A_{n}} W_{π}^{0}(a)W_{π′}^{0}(a)1_{𝔒}(a_{n})δ_{B_{n}}(a)^{−1}ν(a)^{s}d^{∗}a,  (3)

when n = m, and

L(π, π′, s) = I(W_{π}^{0}, W_{π′}^{0}, s)
= ∫_{A_{m}} W_{π}^{0} (\begin{matrix}a \\ & I_{n−m} \end{matrix}) W_{π′}^{0}(a)δ_{B_{m}}(a)^{−1}ν(a)^{s−(n−m)/2}d^{∗}a  (4)

when n > m. In the aforementioned papers, Jacquet and Shalika work with generic representations, however, their proofs extend verbatim to unramified representations of Langlands’ type. Indeed, the formula for W_{π}^{0} in terms of Satake parameters is still valid, and thanks to Proposition 1.4, the factor L(π, π′, s) is still equal to (using notations of [J-S] and [J-S 2]) the Artin factor 1/det(1 − q^{−s}A ⊗ A′) for A and A′ diagonal matrices corresponding to the Satake parameters of π and π′ respectively. In particular, when π is generic, W_{π}^{0} is the essential vector of π.

Now let π = Δ_{1} × ⋯ × Δ_{t} be a generic representation of G_{n}, written as a unique product of the unlinked segments Δ_{i} = [ν^{−(k_{i}(π)−1)}ρ_{i}(π), …, ρ_{i}(π)], and π′ = μ_{1} × ⋯ × μ_{m} be an unramified representation of G_{m} of Langlands’ type, for 1 ≤ m ≤ n. One has, according to Proposition 1.4 and Theorem 8.2. of [J-P-S 2] (whose proof is independant of [J-P-S]), the equality of Rankin-Selberg L-functions L(π, π′, s) = ∏_{i,j} L(ρ_{i}(π), μ_{j}, s). We notice that L(ρ_{i}(π), μ_{j}, s) is equal to 1 unless ρ_{i}(π) is an unramified character of G_{1}. Hence, one has the equality

L(π, π′, s) = ∏_{\{i,ρ_{i}(π)∈{ \atop F^{∗}/𝔒^{∗}},j\}} L(ρ_{i}(π), μ_{j}, s).

This incites us to introduce the following representation.

Definition 1.3. Let π = Δ_{1} × ⋯ × Δ_{t} be a generic representation of G_{n}, with Δ_{i} = [ν^{−(k_{i}(π)−1)}ρ_{i}(π), …, ρ_{i}(π)]. Let r be the cardinality of the set \{ρ_{j}(π), ρ_{j}(π) ∈ { \atop F^{∗}/𝔒^{∗}}\}. When this set is non empty, denote by χ_{1}, …, χ_{r} its elements ordered such that Re(χ_{i}) ≥ Re(χ_{i+1}) for 1 ≤ i ≤ r − 1. We define π_{u} as the trivial representation of G_{0} when r = 0, and as the unramified representation of Langlands type χ_{1} × ⋯ × χ_{r} of G_{r} when r ≥ 1.

Let π a generic representation of G_{n}, and π′ be an unramified representation of G_{m} of Langlands’ type, with 1 ≤ m ≤ n. If we set L(π_{u}, π′, s) = 1 when π_{u} is the trivial representation of G_{0}, we have, according to Proposition 1.4:

L(π, π′, s) = L(π_{u}, π′, s).  (5)

From now on, we will order the segments Δ_{i} in the generic representation π, such that ρ_{i}(π) is an unramified character χ_{i} of G_{1} for 1 ≤ i ≤ r, is not such a character for i ≥ r + 1, and Re(χ_{i}) ≥ Re(χ_{i+1}) for 1 ≤ i ≤ r − 1.

2 Mirabolic restriction, sphericity, and restriction of Whittaker functions

In this section, we first give results on the derivative functors and how they act on subspaces fixed by compact subgroups, then we recall some results from [C-P] about their interpretation in terms of restriction of Whittaker functions. We introduce a few more notations, in order to get a handy parametrisation of the diagonal torus of G_{n}, in terms of simple roots. For k ≤ n, let Z_{k} be the center of G_{k} naturally embedded in G_{n}; we parametrise it by F^{∗} using the morphism β_{k} : z_{k} ↦ diag(z_{k}I_{k}, I_{n−k}). Hence the maximal torus A_{n} of G_{n} is the direct product Z_{1}.Z_{2} … Z_{n−1}.Z_{n}. We will sometimes (but not always) omit the β_{k}’s in this parametrisation and write (z_{1}, …, z_{n}) for the element β_{1}(z_{1}) … β_{n}(z_{n}) of A_{n}. Notice that the i-th simple root α_{i} has the property that α_{i}(z_{1}, …, z_{n}) = z_{i}.

2.1 Mirabolic restriction and sphericity

We first give a corollary of Proposition 1.2, about a concrete interpretation of Φ^{−}, when restricted to Φ^{+}(Alg(P_{n})). Property d) of the aforementioned proposition says that Φ^{−} sends Φ^{+}τ surjectively onto a P_{n}-module isomorphic to τ. Writing Φ^{+}τ as Ind_{P_{n}U_{n+1}}^{P_{n+1}}(δ_{P_{n+1}}^{1/2}τ ⊗ θ), we want to make the map Φ^{−}

explicit between Ind_{P_{n}U_{n+1}}^{P_{n+1}}(δ_{P_{n+1}}^{1/2}τ ⊗ θ) and τ.

Proposition 2.1. For n ≥ 1, if τ belongs to Alg(P_{n}), then Φ^{−} identifies with the map f ↦ f(I_{n+1}) from Φ^{+}τ to τ.

Proof. Call E_{I} the map f ↦ f(I_{n+1}) from Φ^{+}τ to τ. Let’s first show that E_{I} is surjective. If v belongs to the space V of τ, let U = I_{n} + M(n, 𝔓^{l}) be a congruence subgroup of G_{n}, with l large enough for v to be fixed by U′ = U ∩ P_{n}. Call f the function from P_{n} to V, defined by

f \binom{pu x}{1} = δ_{P_{n+1}}^{1/2}(p)θ(x_{n})τ(p)v,

for p ∈ P_{n}, u ∈ U, x ∈ F^{n} with bottom coordinate x_{n}, and f(p′) = 0 when p′ is not in P_{n}U_{n+1}U. The map f is well defined, because v is fixed by U′. It is smooth as it is right invariant under U_{n+1}(𝔒)U (as θ is trivial on 𝔒). One checks (see the proof of next Propositon for a detailed similar computation), that f satsifies the requested left invariance under P_{n}U_{n+1}, so that f ∈ Φ^{+}(V). Finally f(I_{n+1}) = v, thus f is a preimage of v via E_{I}. An easy adaptation of the Proposition 1.1. of [C-P] then shows that the P_{n}- submodule Φ^{+}τ(U_{n+1}, θ) of Φ^{+}(τ) is equal to Ker(E_{I}). As a consequence, the map E_{I} induces an isomorphism \overline{E_{I}} between Φ^{−}Φ^{+}(τ) and τ, which is P_{n}- equivariant. Hence, the following diagram, with the right isomorphism equal to \overline{E_{I}}, commutes:

Φ^{+}(τ) ↠^{Φ^{−}} Φ^{−}Φ^{+}(τ)
=  ≃
Φ^{+}(τ) ↠^{E_{I}}  τ

We recall that for n ≥ 1, as a consequence of the Iwasawa decomposition, any element g of G_{n} can be written in the form zpk with z in F^{∗}, p in P_{n}, and k in G_{n}(𝔒). We now notice that the restriction of Φ^{−} to (Φ^{+}τ)^{P_{n+1}(𝔒)} is surjective onto τ^{P_{n}(𝔒)}.

Proposition 2.2. For n ≥ 1, the map f ↦ f(I_{n+1}) from (Φ^{+}τ)^{P_{n+1}(𝔒)} to τ^{P_{n}(𝔒)} is surjective.

Proof. Let v_{0} be a vector in the space of τ which is P_{n}(𝔒)-invariant, we claim that the function f defined by

f \binom{zpk x}{1} = δ_{P_{n+1}}^{1/2}(p)θ(x_{n})1_{𝔒^{∗}}(z)τ(p)v_{0},

for z in F^{∗}, p in P_{n}, k in G_{n}(𝔒), and x in F^{n} (with x the transpose of (x_{1}, …, x_{n})), is a preimage of v_{0} in (Φ^{+}τ)^{P_{n+1}(𝔒)}. First we check that f is well-defined: if zpk = z′p′k′, this implies that z′ is equal to z mod 𝔒^{∗}, and p′ is equal to p mod P_{n}(𝔒). Hence δ_{P_{n+1}}^{1/2}(p) = δ_{P_{n+1}}^{1/2}(p′), 1_{𝔒^{∗}}(z′) = 1_{𝔒^{∗}}(z), and τ(p′)v_{0} = τ(p)v_{0} as v_{0} is P_{n}(𝔒)-invariant. Then we check that f indeed belongs to Φ^{+}(τ). Let p_{0} belong to P_{n} embedded in P_{n+1} by p ↦ diag(p, 1), let u_{0} belong to U_{n+1}, and p_{1} belong to P_{n+1}, we need to check the relations

f(p_{0}p_{1}) = δ_{P_{n+1}}^{1/2}(p_{0})τ(p_{0})f(p_{1})

and

f(u_{0}p_{1}) = θ(u_{0})f(p_{1}).

Write p_{1} = \binom{zpk x}{1}, we have f(p_{0}p_{1}) = f \binom{zp_{0}pk p_{0}x}{1}. As θ((p_{0}x)_{n}) =

θ(x_{n}), we have

f(p_{0}p_{1}) = δ_{P_{n+1}}^{1/2}(p_{0}p)θ(x_{n})1_{𝔒^{∗}}(z)τ(p_{0}p)v_{0} = δ_{P_{n+1}}^{1/2}(p_{0})τ(p_{0})f(p_{1})

as wanted. Write u_{0} = \binom{I_{n} x_{0}}{1}, we also have

f(u_{0}p_{1}) = f \binom{zpk x_{0} + x}{1}

= δ_{P_{n+1}}^{1/2}(p)θ((x_{0} + x)_{n})1_{𝔒^{∗}}(z)τ(p)v_{0} = θ(u_{0})f(p_{1})

as wanted. If \binom{k_{0} x_{0}}{1} belongs to P_{n+1}(𝔒) (i.e. k_{0} ∈ G_{n}(𝔒) and x_{0} ∈ 𝔒^{n}), as

θ((zpkx_{0})_{n}) = 1 when z belongs to 𝔒^{∗}, we have

f(\binom{zpk x}{1} \binom{k_{0} x_{0}}{1}) = f \binom{zpkk_{0} zpkx_{0} + x}{1} = f \binom{zpk x}{1} ,

and f is right P_{n+1}(𝔒)-invariant. Finally, it is obvious that f(I_{n+1}) = v_{0}.

Now we are able to prove the following property of Φ^{−}, that we will be of great use later.

Proposition 2.3. For n ≥ 1, if τ belongs to Alg(P_{n}), then Φ^{−} maps τ^{P_{n}(𝔒)} surjectively onto τ_{(1)}^{P_{n−1}(𝔒)}, and Ψ^{−} maps τ^{G_{n−1}(𝔒)} surjectively onto

τ^{(1)}^{G_{n−1}(𝔒)}.

Proof. For the first part, we use the filtrations 0 ⊂ τ_{n} ⊂ ⋯ ⊂ τ_{1} = τ of τ, and 0 ⊂ τ_{(1),n−1} ⊂ ⋯ ⊂ τ_{(1),1} = τ_{(1)} of τ_{(1)}. But τ_{i} equals Φ^{+}(τ_{(1),i−1}) because of Lemma 1.1, so that Φ^{−} maps τ_{i}^{P_{n}(𝔒)} onto τ_{(1),i−1}^{P_{n−1}(𝔒)} surjectively

according to Proposition 2.2. In particular, Φ^{−} maps τ_{2}^{P_{n}(𝔒)}, hence τ^{P_{n}(𝔒)} (as τ_{2}^{P_{n}(𝔒)} ⊂ τ^{P_{n}(𝔒)}), onto τ_{(1),1}^{P_{n−1}(𝔒)} = τ_{(1)}^{P_{n−1}(𝔒)} surjectively.

Ψ^{−} maps τ^{G_{n−1}(𝔒)} surjectively onto τ^{(1)}^{G_{n−1}(𝔒)}, because Ψ^{−} is surjective from τ to τ^{(1)}, and the functor V ↦ V^{G_{n−1}(𝔒)} is exact from Alg(G_{n−1}) to Alg(G_{0}) as G_{n−1}(𝔒) is compact open in G_{n−1} (τ is a G_{n−1}-module by restriction).

2.2 Mirabolic restriction for Whittaker functions

We start by recalling Proposition 1.1 of [C-P], which gives an interpretation of Φ^{−} in terms of restriction of Whittaker functions.

Proposition 2.4. For k ≥ 2, and any submodule τ of (ρ, C^{∞}(N_{k}\P_{k}, θ)) (where ρ denotes the action of P_{k} by right translation), the map R : W ↦ δ_{P_{k}}^{−1/2}W_{|P_{k−1}} is P_{k−1}-equivariant from (ρ, C^{∞}(N_{k}\P_{k}, θ)) to (ρ, C^{∞}(N_{k−1}\P_{k−1}, θ)), with kernel τ(U_{k}, θ). Hence it inducues a P_{k−1}- modules isomorphism between Φ^{−}τ and Im(R) ⊂ C^{∞}(N_{k−1}\P_{k−1}, θ), so that (ρ, Im(R)) is a model for Φ^{−}τ.

Notice that for k ≥ 2, if g ∈ G_{k−1} equals zpk with z ∈ F^{∗}, p ∈ P_{k−1}, and k ∈ G_{n−1}(𝔒), then the absolute value of z depends only on g, so we can write it |z(g)|. We now state a proposition that follows from the proofs of Proposition 1.6. of [C-P], about the interpretation of Ψ^{−} in terms of Whittaker functions.

Proposition 2.5. Let τ be a P_{k}-submodule of C^{∞}(N_{k}\P_{k}, θ), and suppose that τ^{(1)} is a G_{k−1}-module with central character c. Then, for any W in τ, for any g in G_{k−1}, the quantity c^{−1}(z)|z|^{−(k−1)/2}W (diag(zg, 1)) is constant whenever z is in a punctured neighbourhood of zero (maybe depending on g) in F^{∗}.

Remark 2.1. Notice that in the proof of the Proposition 1.6. of [C-P], τ is of a particular form, and τ^{(1)} is supposed to be irreducible. The only fact that is actually needed is that τ^{(1)} has a central character.

This has the following consequence.

Corollary 2.1. For k ≥ 2, let τ be a P_{k}-submodule of (ρ, C^{∞}(N_{k}\P_{k}, θ)), and suppose that τ^{(1)} is a G_{k−1}-module with central character c, then Ψ^{−} identifies with the map

S : W ↦ [g ↦ lim_{z→0} c^{−1}(z)|z|^{(1−k)/2}W (diag(zg, 1))δ_{P_{k}}^{−1/2}(g)]

from τ to to C^{∞}(N_{k−1}\G_{k−1}, θ). To be more precise, S has kernel τ(U_{k}, 1), and it induces a G_{k−1}-modules isomorhism \overline{S} between τ^{(1)} and S(τ) ⊂ C^{∞}(N_{k−1}\G_{k−1}, θ).

Proof. For W in τ, call \overline{W} its image in τ^{(1)}, and call S(W) the function

[g ↦ lim_{z→0} c^{−1}(z)|z|^{(1−k)/2}W (diag(zg, 1))δ_{P_{k}}^{−1/2}(g)]

in C^{∞}(N_{k−1}\G_{k−1}, θ), which is well defined according to Proposition 2.5.

If u(x) = \binom{I_{k−1} x}{1} belongs to U_{k}, then ρ(u(x))W (diag(zg, 1)) =

θ(z(gx)_{k−1})W (diag(zg, 1)). As θ(z(gx)_{k−1}) = 1 for z small enough, we deduce that S(ρ(u(x))W) = S(W), hence the kernel of S contains τ(U_{k}, 1). Conversely, if S(W) = 0, the smoothness of W and the Iwasawa decomposition imply that W (g) is null for |z(g)| in a punctured neighbourhood of zero depending only on W. According to Proposition 2.3. of [M] (which is a restatement of Proposition 1.3. of [C-P]), this means that W belongs to τ(U_{k}, 1). The ℂ-linear map S : W ↦ S(W) induces a ℂ-linear isomorphism \overline{S} : \overline{W} ↦ S(W) between τ^{(1)} and its image in (ρ, C^{∞}(N_{k−1}\G_{k−1}, θ)). Moreover, it is a G_{k−1}-equivariant because for g_{0} ∈ G_{k−1}, one has that \overline{S}(τ^{(1)}(g_{0})\overline{W}) equals

\overline{S}(δ_{P_{k}}^{−1/2}(g_{0})\overline{ρ(g_{0})W}) = δ_{P_{k}}^{−1/2}(g_{0})S(ρ(g_{0})W) = ρ(g_{0})S(W) = ρ(g_{0})\overline{S}(\overline{W}).

We end this section by stating two technical lemmas about Whittaker functions fixed under a maximal compact subgroup, the first is inspired from Lemma 9.2 of [J-P-S 2].

Lemma 2.1. For n ≥ 2, let τ be a P_{n}-submodule of C^{∞}(N_{n}\P_{n}, θ), and let W belong to τ^{P_{n}(𝔒)}, then there exists W′ in τ^{P_{n}(𝔒)}, such that W′(pβ_{n−1}(z)) = W (p)1_{𝔒^{∗}}(z) for p in P_{n−1} and z in F^{∗}.

Proof. For l in ℤ, we denote by ϕ_{l} the characteristic function 1_{𝔓^{l}} of 𝔓^{l}. We fix

a Haar measure dt on F. The Fourier transform ϕ̂_{l}^{θ} with respect to θ and dt is equal to λ_{l}ϕ_{−l} for λ_{l} = dt(𝔓^{l}) > 0. We denote by Φ_{l} the function ⊗_{i=1}^{n−1}ϕ_{l}, which is the characteristic of the lattice ϖ^{l}𝔒^{n−1} in F^{n−1}. We denote by u the natural isomorphism between F^{n−1} and U_{n}. We also recall that any element of τ is determined by its restriction to G_{n−1}. We set W^{l}(p) = ∫_{x∈F^{n−1}} W (pu(x))Φ_{l}(x)dx for p in P_{n} and dx = dt_{1} ⊗ ⋯ ⊗ dt_{n−1}, hence W^{l} belongs to τ. Moreover if k belongs to G_{n−1}(𝔒), and g belongs to G_{n−1}, then W^{l}(gk) is equal to ∫_{x∈F^{n−1}} W (gku(x))Φ_{l}(x)dx = ∫_{x∈F^{n−1}} W (gu(kx))Φ_{l}(x)dx because W is P_{n}(𝔒)-invariant, and this last integral is equal to ∫_{x∈F^{n−1}} W (gu(x))Φ_{l}(k^{−1}x)dx = ∫_{x∈F^{n−1}} W (gu(x))Φ_{l}(x)dx = W^{l}(g) because of the invariance of dx and Φ_{l} under G_{n−1}(𝔒). It is also clear that W^{l} is invariant U_{n}(𝔒) because W is, hence W^{l} belongs to τ^{P_{n}(𝔒)}. Now, for p in P_{n−1} ⊂ P_{n}, and z in F^{∗}, we obtain:

W^{l}(pβ_{n−1}(z)) = ∫_{x∈F^{n−1}} W (pβ_{n−1}(z)u(x))Φ_{l}(x)dx
= ∫_{x∈F^{n−1}} W (u(zpx)pβ_{n−1}(z))Φ_{l}(x)dx
= ∫_{x∈F^{n−1}} θ((zpx)_{n−1})W (pβ_{n−1}(z))Φ_{l}(x)dx
= W (pβ_{n−1}(z)) ∫_{x∈F^{n−1}} Φ_{l}(x)θ(zx_{n−1})dx
= W (pβ_{n−1}(z)) ∫_{y∈ϖ^{l}𝔒^{n−2}} dy ∫_{t∈F} ϕ_{l}(t)θ(zt)dt
= λ_{l}^{n−2}W (pβ_{n−1}(z))ϕ̂_{l}^{θ}(z) = λ_{l}^{n−1}W (pβ_{n−1}(z))ϕ_{−l}(z)

The function W′ = W^{0}/λ_{0}^{n−1} − W^{−1}/λ_{−1}^{n−1} thus satisfies

W′(pβ_{n−1}(z))=W (pβ_{n−1}(z))(ϕ_{0} −ϕ_{1})(z)=W (pβ_{n−1}(z))1_{𝔒^{∗}}(z)=W (p)1_{𝔒^{∗}}(z)

because W is invariant under β_{n−1}(𝔒^{∗}) ⊂ P_{n}(𝔒).

Lemma 2.2. For n ≥ 2, let τ be a P_{n}-submodule of C^{∞}(N_{n}\P_{n}, θ), and let W belong to τ^{G_{n−1}(𝔒)}, then there exists W′ in τ^{P_{n}(𝔒)}, such that W′(z_{1}, …, z_{n−1}, 1) = W (z_{1}, …, z_{n−1}, 1)1_{𝔒}(z_{n−1}) for z_{i} in F^{∗}.

Proof. Let du be the Haar measure on U_{n}, corresponding to the Haar measure dx = dt_{1} ⊗ ⋯ ⊗ dt_{n−1} on F^{n−1}, normalised by dt_{i}(𝔒) = 1 for i between 1 and n − 1. Now set W′(g) = ∫_{u∈U_{n}(𝔒)} W (gu)du. The vector W′ is a linear combination of right translates of W by elements of U_{n}(𝔒), so it belongs to τ. It is clearly invariant under U_{n}(𝔒), and still invariant under G_{n−1}(𝔒), as G_{n−1}(𝔒) normalises U_{n}(𝔒). The following computation then gives the result:

W′(z_{1}, …, z_{n−1}, 1) = ∫_{x∈𝔒^{n−1}} W (β_{1}(z_{1}) … β_{n−1}(z_{n−1})u(x))dx

= ∫_{x∈𝔒^{n−1}} W (u(β_{1}(z_{1}) … β_{n−1}(z_{n−1})x)β_{1}(z_{1}) … β_{n−1}(z_{n−1}))dx
= ∫_{x∈𝔒^{n−1}} θ(z_{n−1}x_{n−1})W (β_{1}(z_{1}) … β_{n−1}(z_{n−1}))dx
= ˆ1_{𝔒}^{θ}(z_{n−1})W (z_{1}, …, z_{n−1}, 1) = 1_{𝔒}(z_{n−1})W (z_{1}, …, z_{n−1}, 1),

the last equality because of the normalisation of the Haar measure on F.

3 Construction of the essential Whittaker function

We are now going to produce the essential vector of a generic representation π of G_{n}, which will now be fixed untill the end. We recall recall that we associated to π, an integer 0 ≤ r ≤ n, and an unramified representation of Langlands’ type π_{u} of G_{r} in Section 1.5.

We first notice that the subspace of π^{(n−r)} fixed by G_{r}(𝔒) is a complex line.

Proposition 3.1. Let π be generic representation of G_{n}. Then (π^{(n−r)})^{G_{r}(𝔒)} is of dimension 1. If v^{0} is a generator of (π^{(n−r)})^{G_{r}(𝔒)}, then the submodule < G_{r}.v^{0} > of π^{(n−r)} spanned by v^{0} surjects onto π_{u}.

Proof. Write π = Δ_{1}×⋯×Δ_{t} for the ordering of the Δ_{i}’s fixed after Definition 1.3. According to Lemma 3.5. of [B-Z] the representation π^{(n−r)} has a filtration with subquotients Δ_{1}^{(a_{1})} × ⋯ × Δ_{t}^{(a_{t})}, with ∑_{i} a_{i} = n − r. According to Proposition 9.6 of [Z], π_{u} appears as one of these subquotients, and by the choice of r, the other nonzero subquotients amongst them all contain either a segment as a factor (in the product notation) of some G_{k} for k ≥ 2, or a ramified character of G_{1}. According to Proposition 1.3, and Proposition 1.5 1.5, these other subquotients contain no nonzero G_{n}(𝔒)-invariant vector. The result then follows from the exactness of the functor V ↦ V^{G_{n}(𝔒)} from Alg(G_{n}) to Alg(G_{0}).

We also notice the following facts. First, from the theory of Kirillov models (see [B-Z], Theorem 4.9), for n ≥ 2, the map W ∈ W (π, θ) ↦ W_{|P_{n}} is injective, we denote by W (π_{(0)}, θ) its image. We choose this notation because P_{n}-module π_{(0)} = π_{|P_{n}} is isomorphic to the submodule W (π_{(0)}, θ) of (ρ, C^{∞}(N_{n}\P_{n}, θ)). Now if one applies Proposition 2.4 repeatedly to π_{(0)}, then for r ≤ n − 1, the P_{r+1}-module π_{(n−r−1)} is isomorphic to the submodule of (ρ, C^{∞}(N_{r+1}\P_{r+1}, θ)), whose vectors are the functions (∏_{k=r+2}^{n} δ_{P_{k}}^{−1/2})W_{|P_{r+1}} for W ∈ W (π, θ) (where P_{r+1} is embedded in P_{n} via p ↦ diag(p, I_{n−r−1})), we denote by W (π_{(n−r−1)}, θ) this P_{r+1}-module.

Proposition 3.1 has the following corollary.

Corollary 3.1. Under the condition 1 ≤ r ≤ n − 1, there exists W̃_{0} in W (π_{(n−r−1)}, θ)^{G_{r}(𝔒)} such that

W_{π_{u}}^{0}(g) = lim_{z→0} c^{−1}(z)|z|^{−r/2}W̃_{0}(diag(zg, 1))δ_{P_{r+1}}^{−1/2}(g)

for all g in G_{r}. This implies that the representation π_{u} occurs as a submodule of π^{(n−r)}.

Proof. We take Ψ^{−}(W (π_{(n−r−1)}, θ)) as a model for π^{(n−r)}, i.e. π^{(n−r)} = Ψ^{−}(W (π_{(n−r−1)}, θ)). Let W̃_{0} be a preimage (which we shall normalise later), of v^{0} under Ψ^{−}, which we take in W (π_{(n−r−1)}, θ)^{G_{r}(𝔒)} thanks to Proposition 2.3. We denote by < P_{r+1}.W̃_{0} > the P_{r+1}-submodule of W (π_{(n−r−1)}, θ) spanned by W̃_{0}. By definition of Ψ^{−}, we have

Ψ^{−}(< P_{r+1}.W̃_{0} >) = Ψ^{−}(< G_{r}.W̃_{0} >) =< G_{r}.v^{0} > .

Now, Z_{n} acts by a character c on < G_{r}.v^{0} > (which is the central character of π_{u} as well according to Proposition 3.1). Let S be the map defined in Corollary 2.1 from < P_{r+1}.W̃_{0} > to C^{∞}(N_{r}\G_{r}, θ). We know from this corollary, that S factors to give an isomorphism \overline{S} between < G_{r}.v^{0} > and S(< P_{r+1}.W̃_{0} >). Define W^{0} as W^{0} = S(W̃_{0}) in C^{∞}(N_{r}\G_{r}, θ), so that S(< P_{r+1}.W̃_{0} >) is equal to G_{r}.W^{0}. As the G_{r}-module < G_{r}.W^{0} > is isomorphic to < G_{r}.v^{0} >, there is a surjective G_{r}-module morphism from < G_{r}.W^{0} > onto W (π_{u}, θ) according to Proposition 3.1. It sends W^{0} to a nonzero multiple of W_{π_{u}}^{0}. We normalise W̃_{0}, such that the Whittaker function W^{0} = S(W̃_{0}) is equal to 1 on G_{n}(𝔒). The Hecke algebra H_{r} thus multiplies W^{0} and W_{π_{u}}^{0} by the same character, as W_{π_{u}}^{0} is the image of W^{0} via a G_{r}-intertwining operator. Both are normalised spherical Whittaker functions, they are thus equal according to [S]. In particular, we have S(W̃_{0}) = W_{π_{u}}^{0}, which is the first statement of the corollary. Next, this implies the equalities < G_{r}.W^{0} >=< G_{r}.W_{π_{u}}^{0} >= W (π_{u}, θ), so the surjection from < G_{r}.v^{0} > onto W (π_{u}, θ) is actually equal to the isomorphism \overline{S}, hence π_{u} occurs as a submodule of π^{(n−r)}.

The following proposition then holds.

Proposition 3.2. Under the condition 1 ≤ r ≤ n − 1, there exists in W (π_{(n−r−1)}, θ)^{P_{r+1}(𝔒)} an element W_{0}, such that W_{0}(z_{1}, …, z_{r}, 1) = δ_{P_{r+1}}^{1/2}(z_{1}, …, z_{r})W_{π_{u}}^{0}(z_{1}, …, z_{r})1_{𝔒}(z_{r}) for z_{r} in F^{∗}.

Proof. Let W̃_{0} be as in Corollary 3.1. By the claim in the proof of Theorem 2.1 of [M] (the arguments of the claim in Proposition 1.6. of [C-P] are actually sufficient here), there is N in ℕ, such that W̃_{0}(z_{1}, …, z_{r}a, 1) = c_{π_{u}}(a)|a|^{r/2}W̃_{0}(z_{1}, …, z_{r}, 1) (parametrizing A_{r+1} with the β_{i}’s) for |z_{r}| ≤ q^{−N} and |a| ≤ 1. For b in F^{∗}, call W̃_{0,b} the function p ↦ W̃_{0}(pβ_{r}(b))/(c_{π_{u}}(b)|b|^{r/2}) defined on P_{r+1}, then W̃_{0,b} still belongs to W (π_{(n−r−1)}, θ)^{G_{r}(𝔒)}, and W̃_{0,b}(z_{1}, …, z_{r}, 1)/(c_{π_{u}}(z_{r})|z_{r}|^{r/2}) is constant with respect to z_{r} whenever |z_{r}| ≤ q^{−N}/|b|. We choose b in F^{∗} satisfying |b| = q^{−N}, so that the function W̃_{0,b}(z_{1}, …, z_{r}, 1)/(c_{π_{u}}(z_{r})|z_{r}|^{r/2}) is constant with respect to z_{r} for |z_{r}| ≤ 1. Hence, according to Corollary 3.1, for |z_{r}| ≤ 1, we have the equalities

W̃_{0,b}(z_{1}, …, z_{r}, 1)/(c_{π_{u}}(z_{r})|z_{r}|^{r/2}) = W̃_{0}(z_{1}, …, z_{r}b, 1)/(c_{π_{u}}(z_{r}b)|z_{r}b|^{r/2})

= δ_{P_{r+1}}^{1/2}(z_{1}, …, z_{r−1}, b)W_{π_{u}}^{0}(z_{1}, …, z_{r−1}, b)/(c_{π_{u}}(b)|b|^{r/2})
= δ_{P_{r+1}}^{1/2}(z_{1}, …, z_{r−1}, 1)W_{π_{u}}^{0}(z_{1}, …, z_{r−1}, 1).

They imply the equality

W̃_{0,b}(z_{1}, …, z_{r}, 1) = δ_{P_{r+1}}^{1/2}(z_{1}, …, z_{r−1}, z_{r})W_{π_{u}}^{0}(z_{1}, …, z_{r−1}, z_{r})

for |z_{r}| ≤ 1. On the other hand, applying Lemma 2.2, there is W_{0} is in W (π_{(n−r−1)}, θ)^{P_{r+1}(𝔒)}, such that W_{0}(z_{1}, …, z_{r}, 1) is equal to W̃_{0,b}(z_{1}, …, z_{r}, 1)1_{𝔒}(z_{r}), it is then clear that W_{0} has the desired property.

We now prove the main result of this paper.

Theorem 3.1. For n ≥ 2, let π be a ramified generic representation of G_{n} (i.e. r ≤ n − 1). Then one can produce a G_{n−1}(𝔒)-invariant function W_{π}^{ess} in W (π, θ), whose restriction to A_{n−1} (when A_{n−1} is parametrised by its simple roots), is given by formula

W_{π}^{ess}(z_{1}, …, z_{n−1}, 1)

= W_{π_{u}}^{0}(z_{1}, …, z_{r})ν(z_{1}, …, z_{r})^{(n−r)/2}1_{𝔒}(z_{r}) ∏_{j=r+1}^{n−1} 1_{𝔒^{∗}}(z_{j})  (6)

when r ≥ 1, and by

W_{π}^{ess}(z_{1}, …, z_{n−1}, 1) = ∏_{j=1}^{n−1} 1_{𝔒^{∗}}(z_{j})  (7)

when r = 0. A function W_{π}^{ess} with such properties is unique, and has image W_{π_{u}}^{0} in π^{(n−r)}.

Proof. Suppose first that we have r ≥ 1. We already constructed in the previous proposition a vector W_{0} in W (π_{(n−r−1)}, θ)^{P_{r+1}(𝔒)} such that

W_{0}(z_{1}, …, z_{r}, 1) = δ_{P_{r+1}}^{1/2}(z_{1}, …, z_{r})W_{π_{u}}^{0}(z_{1}, …, z_{r})1_{𝔒}(z_{r}).

Then, applying Proposition 2.3 and then Lemma 2.1, we obtain W_{1} in W (π_{(n−r−2)}, θ)^{P_{r+2}(𝔒)}, that satisfies

\begin{matrix}W_{1}(z_{1}, …, z_{r+1}, 1) = δ_{P_{r+2}}^{1/2}(z_{1}, …, z_{r+1})W_{0}(z_{1}, …, z_{r}, 1)1_{𝔒^{∗}}(z_{r+1}) \\ = δ_{P_{r+2}}^{1/2}(z_{1}, …, z_{r}, 1)δ_{P_{r+1}}^{1/2}(z_{1}, …, z_{r})W_{π_{u}}^{0}(z_{1}, …, z_{r})1_{𝔒}(z_{r})1_{𝔒^{∗}}(z_{r+1}). \end{matrix}

Repeating this last step (Proposition 2.3 and then Lemma 2.1), we obtain by induction for all k between 1 and n − r − 1, an element W_{k} in W (π_{(n−r−1−k)}, θ)^{P_{r+k+1}(𝔒)}, that satisfies

W_{k}(z_{1}, …, z_{r+k}, 1) = δ_{P_{r+k+1}}^{1/2}(z_{1}, …, z_{r+k})W_{k−1}(z_{1}, …, z_{r+k−1}, 1)1_{𝔒^{∗}}(z_{r+k})

= W_{π_{u}}^{0}(z_{1}, …, z_{r})1_{𝔒}(z_{r}) ∏_{j=r+1}^{r+k} 1_{𝔒^{∗}}(z_{j}) ∏_{i=r+1}^{r+k+1} δ_{P_{i}}^{1/2}(z_{1}, …, z_{r}, \underbrace{1, …, 1}_{i−(r+1)×}).

We define W_{π}^{ess} to be the element of W (π, θ) which restricts to P_{n} as W_{n−r−1}, it is thus G_{n−1}(𝔒)-invariant and satisfies Equation (6) of the statement of the theorem, because

∏_{i=r+1}^{n} δ_{P_{i}}^{1/2}(z_{1}, …, z_{r}, \underbrace{1, …, 1}_{i−(r+1)×}) = |det(z_{1}, …, z_{r})|^{\frac{n−r}{2}}

as δ_{P_{i}}(z_{1}, …, z_{r}, 1, …, 1) = |det(z_{1}, …, z_{r})| for i > r. If r = 0, we take for W_{0} the constant function on the trivial group P_{1} equal to 1 in W (π_{(n−1)}, θ) = W (π_{(n−1)}, θ)^{P_{1}(𝔒)}. Again, thanks to Proposition 2.3 and Lemma 2.1, there is W_{1} in W (π_{(n−2)}, θ)^{P_{2}(𝔒)} such that W_{1}(z_{1}, 1) = 1_{𝔒^{∗}}(z_{1}) for z_{1} in F^{∗}, and we end as in the case r ≥ 1. The function W_{π}^{ess} is unique by the theory of Kirillov models, and its image in π^{(n−r)} is W_{π_{u}}^{0} by construction.

The expression of the restriction of W_{π}^{ess} to A_{n−1} in the usual coordinates is the same.

Corollary 3.2. Let π be a ramified generic representation of G_{n}, then if a = diag(a_{1}, …, a_{n−1}) belongs to A_{n−1}, and a′ = diag(a_{1}, …, a_{r}) ∈ A_{r}, we obtain Formulas (1) and (2) of the introduction:

W_{π}^{ess}(diag(a, 1)) = W_{π_{u}}^{0}(a′)ν(a′)^{(n−r)/2}1_{𝔒}(a_{r}) ∏_{j=r+1}^{n−1} 1_{𝔒^{∗}}(a_{j})

when r ≥ 1, and

W_{π}^{ess}(diag(a, 1)) = ∏_{j=1}^{n−1} 1_{𝔒^{∗}}(a_{j})

when r = 0.

Proof. We do the case r ≥ 1, the case r = 0 being simpler. If diag(a_{1}, …, a_{n−1}) = (z_{1}, …, z_{n−1}) belongs to A_{n−1}, we have a_{i} = z_{i} … z_{n−1}, hence if a′ = diag(a_{1}, …, a_{r}), we have (z_{1}, …, z_{r}) = (∏_{i=r+1}^{n−1} z_{i})^{−1}a′ in A_{r}. Equation (6) can thus be read:

W_{π}^{ess}(diag(a, 1))

= W_{π_{u}}^{0}(( ∏_{i=r+1}^{n−1} z_{i})^{−1}a′)ν(( ∏_{i=r+1}^{n−1} z_{i})^{−1}a′)^{(n−1)/2}1_{𝔒}(z_{r}) ∏_{i=r+1}^{n−1} 1_{𝔒^{∗}}(z_{i})

= W_{π_{u}}^{0}(a′)ν(a′)^{(n−r)/2}1_{𝔒}(a_{r}) ∏_{i=r+1}^{n−1} 1_{𝔒^{∗}}(a_{i}),

the last equality because if it is not 0 = 0, this means that z_{r+1}, …, z_{n−1} all belong to 𝔒^{∗}, hence the inverse of their product as well, and W_{π_{u}}^{0}, ν, 1_{𝔒^{∗}} and 1_{𝔒} are all invariant under 𝔒^{∗}.

We then have the following corollary.

Corollary 3.3. Let π be a generic representation of G_{n} with Whittaker model W (π, θ). There exists in W (π, θ) a unique G_{n−1}(𝔒)-invariant function W_{π}^{ess} equal to 1 on G_{n−1}(𝔒), such that for every 1 ≤ m ≤ n − 1, and every unramified representation π′ of Langlands’ type of G_{m}, with normalised spherical function W_{π′}^{0} in W (π′, θ^{−1}), the equality I(W_{π}^{ess}, W_{π′}^{0}, s) = L(π, π′, s) holds for an appropriate normalisation of the invariant measure on N_{m}\G_{m}.

Proof. The unicity of a function W_{π}^{ess} with such properties follows from [J-P-S]. If π is unramified (i.e. r = n), we set W_{π}^{ess} = W_{π}^{0} and Equations (3) and (4) show that it is the correct choice. When r ≤ n−1, we again only treat the case r ≥ 1, the case r = 0 being similar, but simpler (using Equation (2) instead of Equation (1)). We show that the function W_{π}^{ess} from the previous corollary satisfies the wanted equalities. Thanks to Iwasawa decomposition, we have

I(W_{π}^{ess}, W_{π′}^{0}, s) = ∫_{A_{m}} W_{π}^{ess}(diag(a, I_{n−m}))W_{π′}^{0}(a)δ_{B_{m}}^{−1}(a)ν(a)^{s− \frac{(n−m)}{2}} d^{∗}a′.

If m > r, using Equations (1) and δ_{B_{m}} (\begin{matrix}a \\ & I_{m−r} \end{matrix}) = δ_{B_{m}}(a)ν(a)^{m−r}, we find

I(W_{π}^{ess}, W_{π′}^{0}, s) =∫_{A_{r}}W_{π_{u}}^{0}(a′)W_{π′}^{0} (\begin{matrix}a′ \\ & I_{m−r} \end{matrix}) δ_{B_{r}}^{−1}(a′)1_{𝔒}(a_{r})ν(a′)^{s− \frac{(m−r)}{2}} d^{∗}a′

= ∫_{A_{r}} W_{π_{u}}^{0}(a′)W_{π′}^{0} (\begin{matrix}a′ \\ & I_{m−r} \end{matrix}) δ_{B_{r}}^{−1}(a′)ν(a′)^{s− \frac{(m−r)}{2}} d^{∗}a′ = I(W_{π′}^{0}, W_{π_{u}}^{0}, s),

as W_{π′}^{0} (\begin{matrix}a′ \\ & I_{m−r} \end{matrix}) vanishes for |a_{r}| > 1. Hence, by Equations (4) and (5), we

obtain

I(W_{π}^{ess}, W_{π′}^{0}, s) = L(π_{u}, π′, s) = L(π, π′, s).

If m = r, using Equation (1), we find

I(W_{π}^{ess}, W_{π′}^{0}, s) = ∫_{A_{r}} W_{π_{u}}^{0}(a′)W_{π′}^{0} (a′) δ_{B_{r}}^{−1}(a′)1_{𝔒}(a_{r})ν(a′)^{s}d^{∗}a′,

but this integral is equal to

I(W_{π_{u}}^{0}, W_{π′}^{0}, 1_{𝔒^{m}}, s) = L(π_{u}, π′, s) = L(π, π′, s)

by Equations (3) and (5). If m < r, Equation (1) gives

I(W_{π}^{ess}, W_{π′}^{0}, s) = ∫_{A_{m}} W_{π_{u}}^{0}(diag(a, I_{r−m}))W_{π′}^{0}(a)δ_{B_{m}}^{−1}(a)ν(a)^{s− \frac{(r−m)}{2}} d^{∗}a,

and this integral is equal to

I(W_{π_{u}}^{0}, W_{π′}^{0}) = L(π_{u}, π′, s) = L(π, π′, s)

by Equations (4) and (5). In all cases, we have

I(W_{π}^{ess}, W_{π′}^{0}, s) = L(π, π′, s).

Acknowledgements. I thank the p-adic workgroup in Poitiers for allowing me to give talks about [J-P-S]. I thank Guy Henniart, Hervé Jacquet, and Michitaka Miyauchi, for useful comments and corrections about this note. Finally and most importantly, I thank the referee for his very careful reading of the paper, which allowed to correct some mistakes, and improve many arguments.

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Nadir Matringe Université de Poitiers Laboratoire de Mathématiques et Applications Téléport 2 - BP 30179 Boulevard Marie et Pierre Curie 86962, Futuroscope Chasseneuil France matringe@math.univ-poitiers.fr