ON LOCAL COEFFICIENTS FOR NON-GENERIC

REPRESENTATIONS OF SOME CLASSICAL GROUPS

Solomon Friedberg and David Goldberg

Abstract. This paper is concerned with representations of split orthogonal and quasi-split unitary groups over a nonarchimedean local field which are not generic, but which support a unique model of a different kind, the generalized Bessel model. The properties of the Bessel models under induction are studied, and an analogue of Rodier’s theorem concerning the induction of Whittaker models is proved for Bessel models which are minimal in a suitable sense. The holomorphicity in the induction parameter of the Bessel functional is established. Last, local coefficients are defined for each irreducible supercuspidal representation which carries a Bessel functional and also for a certain component of each representation parabolically induced from such a supercuspidal.

Introduction. L-functions are a central object of study in representation theory and number theory. Over a global field, one has the Langlands Conjectures, which assert in particular the meromorphic continuation and functional equation of a class of Euler products. Over a local field one has additional conjectures due to Langlands, expressing the Plancherel measure arithmetically as the ratio of certain local L-functions and root numbers.

In many cases these conjectures have been established by Shahidi [Shab,Shac, Shad], following a path laid out by Langlands [Lana]. The framework for Shahidi’s work is the study of Eisenstein series or their local analogues, induced representations. One knows the continuation of these Eisenstein series due to Langlands [Lanb]. Langlands also showed that the constant coefficients of the Eisenstein series may be expressed in terms of local intertwining operators which are almost everywhere quotients of certain L-functions. It remains to study these intertwining operators for the finite set of ‘bad’ places. If the inducing data is generic, that is, admits a Whittaker model, then Shahidi has succeeded in relating them to local L-functions. Thus the careful study of the Eisenstein series, both local and global, affords a proof of certain of the Langlands conjectures for these L-functions.

The aim of this work is to suggest that the Langlands-Shahidi method may be extended beyond the generic spectrum by the use of other models. The Whittaker model is unique (an irreducible admissible representation admits at most one such model up to scalars). In this paper we study the properties of local representations

Typeset by AMS-TEX

1991 Mathematics Subject Classification. Primary 22E50, Secondary 22E35.

Friedberg was partially supported by National Security Administration Grant MDA904-95-H-1053. Goldberg was partially supported by National Science Foundation Fellowship DMS-9206246 and National Science Foundation Career Grant DMS-9501868. The research of both authors at MSRI was supported in part by National Science Foundation Grant DMS-9022140.

of split orthogonal groups and quasi-split unitary groups which are not generic, but which support a unique model of a different kind, the generalized Bessel model. These models involve a character of a proper subgroup of the unipotent radical of a Borel subgroup, but transform under a reductive group of some, in general non-zero, rank. The uniqueness of the models has been proved by S. Rallis [Ral] in the orthogonal case, but as the argument has not yet been written out in full detail in the unitary case we make it a hypothesis throughout the paper.

We first study the properties of Bessel models under induction, and prove an analogue of Rodier’s Theorem [Rodb] concerning the induction of Whittaker models. Our analogue, Theorem 2.1, states that if one parabolically induces a representation with a Bessel model of minimal rank, or more generally one which is minimal in the sense of Definition 1.5 below, then the induced representation has a unique Bessel model of the same rank and compatible type. In the case of rank 0, we recover Rodier’s theorem. To carry out the proof we use Bruhat’s extension [Bru] of Mackey theory and investigate precisely which double cosets of the appropriate type may support a functional with the desired equivariance property. We show that there is a unique such double coset by an extensive combinatorial argument.

Next, we establish the holomorphicity of the Bessel functional which arises from one which is minimal by parabolic induction of the underlying representation. Our approach is based on Bernstein’s theorem [Ber], which uses uniqueness to conclude meromorphicity under some regularity hypotheses, and Banks’s extension [Ban], which allows one to prove holomorphicity as well. We show in Theorem 3.6 that there is a non-zero Bessel functional Λ(ν, π), attached to an irreducible admissible representation π of the Levi subgroup M and a parameter ν in the complexified dual of the Lie algebra of the split component of M, which is holomorphic in ν.

If π is supercuspidal and has a Bessel model, or more generally if π is irreducible and carries a Bessel model corresponding to a minimal Bessel model of the supercuspidal from which it is induced, these results allow us to establish the existence of a local coefficient. In the generic case, such a local coefficient was crucial for Shahidi’s study of the intertwining operators and of the relation between Plancherel measures and L–functions; see Shahidi [Shad]. Let A(ν, π, w) denote the standard intertwining operator attached to inducing data ν, π and Weyl group element representative w (see (3.4) below). We shall prove (cf. Theorem 3.8):

Theorem. Let π be an irreducible representation of M which is a component of the representation parabolically induced from an irreducible admissible supercuspidal representation ρ of a parabolic subgroup of M. Suppose that π carries a Bessel model corresponding (in the sense of Theorem 2.1) to a minimal Bessel model of ρ. For each w̃ in the Weyl group, choose a representative w for w̃. Then there is a complex number C(ν, π, w) so that

Λ(ν, π) = C(ν, π, w)Λ(w̃ν, w̃π)A(ν, π, w).

Moreover, the function ν ↦ C(ν, π, w) is meromorphic and depends only on the class of π and the choice of the representative w.

We call C(ν, π, w) the local coefficient attached to π, ν, and w. We then show in Corollary 3.9 that the local coefficients behave as expected with respect to the Langlands decomposition of the intertwining operators. This generalizes a property of the local coefficients introduced by Shahidi [Shaa] in the generic case.

The authors wish to thank D. Ramakrishnan, who suggested we begin our study of using the Bessel functional to extend the Langlands-Shahidi method by seeking to generalize Rodier’s theorem, and F. Shahidi for his interest in this project. The authors also express their appreciation to W. Banks, D. Bump, M. Furusawa, D. Ginzburg, S. Rallis, F. Shahidi, and D. Soudry for helpful discussions of their work. Finally, the authors wish to thank the Mathematical Sciences Research Institute and the organizers of the 1994-95 Special Year in Automorphic Forms. The authors met and began this work while both were in residence there; without their support this project would not have started.

§1 Preliminaries on Bessel Models. In this section we recall the notion of a Bessel model following [Ral] and [GPR], and review some properties of such models. Let F be a nonarchimedean local field of characteristic zero. Let G be one of the classical groups SO_{2r+1}, U_{2r+1}, U_{2r}, or SO_{2r}, defined over F. We assume that the orthogonal groups are split, and that the unitary groups are quasi-split, and split over a quadratic extension E/F. Let r_{0} = 2r if G = U_{2r} or SO_{2r}, and r_{0} = 2r + 1 otherwise. Denote by B = TU the Borel subgroup of G, where T contains the maximal split subtorus of diagonal elements, and U is the subgroup of upper triangular unipotent matrices in G. We use G to denote the F -rational points of G, and use this notational convention for other algebraic groups defined over F.

Denote by Φ(G, T) the root system of G with respect to T. We choose the ordering on the roots corresponding to our choice of Borel subgroup. Let W = W (G, T_{d}) be the Weyl group of G with respect to the maximal split subtorus T_{d} of T. Thus, W = N_{G}(T_{d})/T. Then,

W ≃ \{ \begin{matrix}S_{r} ⋉ ℤ_{2}^{r} & if G ≠ SO_{2r} \\ S_{r} ⋉ ℤ_{2}^{r−1} & if G = SO_{2r}. \end{matrix}

(See [Gola,Golb] for a more explicit description of T and W.) Here we will denote all elements of W as permutations on r_{0} letters. Thus, the permutation (ij) ∈ S_{r} corresponds to the permutation (ij)(r_{0} + 1 − j r_{0} + 1 − i) in S_{r_{0}}. Similarly, the sign change c_{i} which generates that i-th copy of ℤ_{2} corresponds to the permutation (i r_{0} + 1 − i) in S_{r_{0}}.

Fix an ℓ < r and let ℓ_{0} = r_{0} − 2ℓ. Let U_{ℓ} be the subgroup of U consisting of matrices whose middle ℓ_{0} ×ℓ_{0} block is the identity matrix. For 1 ≤ i ≤ ℓ, let ψ_{i} be a non-trivial additive character of F if G is orthogonal, and let ψ_{i} be the composition of such a character with T r_{E/F} if G is unitary. We let a = (a_{1}, a_{2}, …, a_{ℓ_{0}}) ∈ F^{ℓ_{0}} if G is orthogonal, and let a ∈ E^{ℓ_{0}} if G is unitary. Then define ψ_{ℓ,a_{j}} by ψ_{ℓ,a_{j}}(x) = ψ_{ℓ}(a_{j}x). Now define a character of U_{ℓ} by

χ ((u_{ij})) = ∏_{i=1}^{ℓ−1} ψ_{i}(u_{i,i+1}) ∏_{j=1}^{ℓ_{0}} ψ_{ℓ,a_{j}}(u_{ℓ,ℓ+j}).

Let

M_{ℓ} = \{( \begin{matrix}I_{ℓ} \\ & g \\ & & I_{ℓ} \end{matrix} ) ∈ G\} .

Note that M_{ℓ} ⊂ N_{G}(U_{ℓ}). If g ∈ M_{ℓ}, then define χ^{g} by χ^{g}(u) = χ(g^{−1}ug). We let

M_{χ} = {g ∈ M_{ℓ} | χ^{g} = χ} .

Let R_{χ} = M_{χ}U_{ℓ}. Suppose that ω is an irreducible admissible representation of M_{χ}. (We will denote this by ω ∈ E(M_{χ}).) Let ω_{χ} = ω⊗χ be the associated representation on R_{χ}.

Definition 1.1. We say that two characters χ_{1} and χ_{2} of U_{ℓ} defined as above are equivalent if χ_{1} = χ_{2}^{g} for some g ∈ N_{G}(U_{ℓ}).

The following result is a consequence of Witt’s Theorem.

Lemma 1.2. Any character χ of U_{ℓ} which is defined as above, is equivalent to one for which a = (δ, 0, 0 …, 1), for some δ.

From now on we assume for convenience that χ is given as in Lemma 1.2.

We let ℓ_{1} = [\frac{ℓ_{0}}{2} ] = r − ℓ.

Definition 1.3. Suppose that τ is an admissible representation of G. We say that τ has an ω_{χ}-Bessel model (or a Bessel model with respect to ω_{χ}) if Hom_{G}(τ, Ind_{R_{χ}}^{G}(ω_{χ})) ≠ 0. If χ is a character of U_{ℓ}, and ℓ_{1} is defined as above, then we say that τ has a rank ℓ_{1} Bessel model.

Remarks.

(1) By Frobenius reciprocity [BeZb], we have

Hom_{G}(τ, Ind_{R_{χ}}^{G}(ω_{χ})) ≃ Hom_{M_{χ}}(τ_{U_{ℓ,χ}}, ω_{χ}),

where τ_{U_{ℓ,χ}} is the χ-twisted Jacquet module of τ with respect to U_{ℓ} [BeZa]. Thus, the non-vanishing of τ_{U_{ℓ,χ}}, for some ℓ and χ, would imply that τ has a rank ℓ_{1} Bessel model with respect to some ω_{χ}.

(2) A Whittaker model [Roda,Rodb] is a rank zero Bessel model.

(3) One can make these definitions for any choice of Borel subgroup. We choose the standard one for convenience, but we will need to use others in the sequel.

(4) When G = SO_{2r+1}, Rallis has shown that every irreducible admissible representation of G(F) has a Bessel model for some choice of χ and ω.

(5) Suppose that τ is irreducible, ω ∈ E(M_{χ}), and λ : V_{τ} → V_{ω} = V_{ω_{χ}} satisfies λ(τ(x)v) = δ_{R_{χ}}(x)^{1/2}ω_{χ}(x)λ(v), for all x ∈ R_{χ} and v ∈ V_{τ} . (Such a λ is called a Bessel functional.) Let v ∈ V_{τ} and set B_{v}(g) = λ(τ(g)v). Then the map v ↦ B_{v} realizes an intertwining between τ and Ind_{R_{χ}}^{G}(ω_{χ}). Conversely, if there is an embedding T of τ into Ind_{R_{χ}}^{G}(ω_{χ}), then setting λ(v) = [T (v)](e), we get a map λ : V_{τ} → V_{ω} with the property specified above. Thus, τ has an ω_{χ}-Bessel model if and only if a Bessel functional λ exists.

In this paper we shall make use of the following basic uniqueness principle.

Theorem/Conjecture 1.4. Let τ ∈ E(G). Then for a fixed ω and χ, we have dim_{ℂ} Hom_{G}(τ, Ind_{R_{χ}}^{G}(ω_{χ})) ≤ 1. That is, a Bessel model is unique for irreducible representations. □

Uniqueness for Whittaker models is well-known. For rank one Bessel models, Theorem 1.4 was proved, for both orthogonal and unitary groups, by Novodvorsky [Nov]. For Bessel models of arbitrary rank, Theorem 1.4 has been proved when G is an orthogonal group by S. Rallis ( [Ral]). Though the argument in the unitary case should be similar, it has not yet been written down in full detail.

In the remainder of this paper we study those Bessel models for which the uniqueness principle above is valid. Thus we assume that Theorem/Conjecture 1.4 is true henceforth. Our results are therefore complete for split orthogonal groups and for rank one Bessel models on quasi-split unitary groups, while they are contingent upon the truth of Theorem/Conjecture 1.4 for higher rank Bessel models in the unitary case.

To conclude this section we introduce the notion of a minimal Bessel model for an admissible representation τ of G. This will be a key notion in what follows.

Definition 1.5. Suppose that τ has an ω_{χ}-Bessel model which is of rank ℓ_{1}. We say that this model is minimal if τ has no Bessel model of rank ℓ_{1}−1 with respect to a representation ω′_{χ′} obtained as follows: χ′ is a character of U_{ℓ+1} such that χ′ = χ on the simple roots of U_{ℓ} (this implies that M_{χ′} ⊂ M_{χ}), and ω′ is a component of ω|_{M_{χ′}}.

This condition is used in our proof of Proposition 2.4 below; see the discussion following the proof of Lemma 2.10.

If τ has a Bessel model, we denote by B(τ) the smallest non-negative integer ℓ_{1} such that τ has a Bessel model of rank ℓ_{1}. For example, τ is generic if and only if B(τ) = 0. Then any Bessel model for τ of rank B(τ) is clearly a minimal Bessel model in the sense of Definition 1.5. In particular, any representation which has a Bessel model has a minimal Bessel model.

§2 Induction of Bessel models. In this section we study the behavior of minimal Bessel models under induction and prove an analogue of Rodier’s Theorem [Rodb] for such models.

Suppose that P = MN is an arbitrary parabolic subgroup of G. Then

M ≃ GL_{n_{1}} × ⋯ × GL_{n_{t}} × G(m) (2.1)

if G is orthogonal, and

(2.2) M ≃ Res_{F}^{E}(GL_{n_{1}}) × ⋯ × Res_{F}^{E}(GL_{n_{t}}) × G(m)

if G is unitary, where

G(m) = \{ \begin{matrix}SO_{2m+1} & if G = SO_{2r+1}; \\ SO_{2m} & if G = SO_{2r}; \\ U_{2m+1} & if G = U_{2r+1}; \\ U_{2m} & if G = U_{2r}, \end{matrix}

and we take the convention that SO_{1} = {1}. Here r = n_{1} + ⋯ + n_{t} + m. Let π ∈ E(M). Then

(2.3)  π = σ_{1} ⊗ ⋯ ⊗ σ_{t} ⊗ τ,

where σ_{i} ∈ E(GL_{n_{i}}(F)) or E(GL_{n_{i}}(E)), accordingly, and τ ∈ E(G(m)). Suppose that τ has a Bessel model. We let ℓ_{1} be the rank of a minimal Bessel model for τ, ℓ_{0} = 2ℓ_{1} + r_{0} − 2r, and ℓ′ = m − ℓ_{1}. Let B′ = T′U′ = B ∩ G(m), and U′_{ℓ′} be the subgroup of U′ consisting of matrices whose middle ℓ_{0} × ℓ_{0} block is the identity. Choose a character χ_{1} of U′_{ℓ′} and ω ∈ E(M_{χ_{1}}) for which τ has an ω_{χ_{1}}-Bessel model which is minimal. Suppose that each of the representations σ_{i} is generic. Let ℓ = r − ℓ_{1}, and let χ be a character of U_{ℓ} of the form χ = χ_{0} ⊗ χ_{1}, where χ_{0} is a generic character on each GL block corresponding to a fixed non-trivial additive character ψ of F. (We call this the ψ–generic character of the GL component.) Let w̃_{0} be the longest element of W (G, A_{0})/W (M, A_{0}) and fix a representative w_{0} for w̃_{0}. Our first main result is the following.

Theorem 2.1. Let k = F if G is orthogonal and E if G is unitary. Let P = MN be a parabolic subgroup of G, with M as in (2.1) or (2.2). Let π be as in (2.3) with each σ_{i} generic. Further suppose that τ has a Bessel model, and that χ_{1} is a character of U_{ℓ} ∩ G(m) which gives rise to an ω_{χ_{1}}–Bessel model for τ which is minimal. Let χ be a character of U_{ℓ} such that χ|_{U_{ℓ}∩GL_{n_{i}} (k)} is ψ–generic for each i and such that χ|_{U_{ℓ}∩G(m)} = χ_{1}. Then Ind_{P}^{G}(π) has a unique ω_{χ}^{w_{0}}-Bessel model. Conversely, if any of the σ_{i} are non-generic, or if τ has no Bessel model, then Ind_{P}^{G}(π) has no Bessel model.

The remainder of this section will be devoted to the proof of Theorem 2.1. The first step is to reduce the theorem to the case of a maximal proper parabolic subgroup. To do this, suppose Theorem 2.1 holds for maximal proper parabolic subgroups and let P = MN be an arbitrary parabolic. Then M is of the form (2.1) or (2.2). Let P_{1} = M_{1}N_{1} be the standard maximal proper parabolic with M_{1} = GL_{r−m} × G(m) or M_{1} = Res_{F}^{E}(GL_{r−m}) × G(m) which contains M. Let ρ = Ind_{P ∩M_{1}}^{M_{1}}(π). Then ρ = ρ_{1} ⊗ τ, where ρ_{1} is the representation of GL_{r−m}(k) parabolically induced from σ_{1} ⊗ ⋯ ⊗ σ_{t}. Since each σ_{i} is generic, Rodier’s Theorem implies that ρ_{1} has a unique generic constituent. Now for each irreducible constituent π_{1} of ρ_{1}, the representation π_{1} ⊗ τ satisfies the hypothesis of the Theorem. Then, by assumption,

Ind_{P}^{G}(π) = Ind_{P_{1}}^{G}(Ind_{P ∩M_{1}}^{M_{1}}(π) ⊗ 1_{N_{1}})

will have a unique Bessel model of the desired type.

Now suppose that P = MN is a maximal proper parabolic subgroup of G. Then for some n, 1 ≤ n ≤ r, and m = r−n we have M ≃ GL_{n} ×G(m) if G is orthogonal, and M ≃ Res_{E/F} (GL_{n}) × G(m) if G is unitary. Let π = σ ⊗ τ ∈ E(M), where σ ∈ E(GL_{n}(k)) and τ ∈ E(G(m)). Suppose that τ has an ω_{χ_{1}}-Bessel model of rank ℓ_{1} ≥ 0, and it is minimal. Assume that χ_{1} is of the form given in Lemma 1.2. Let ℓ = r − ℓ_{1}. Note that ℓ_{1} ≤ m implies ℓ ≥ n. Let ℓ′ = ℓ − n = m − ℓ_{1}. Then χ_{1} is a character of U′_{ℓ′}, where U′_{ℓ′} = U_{ℓ′} ∩ G(m). Let χ_{0} be the generic character of the upper triangular unipotent subgroup U_{0} of GL_{n} by given by a fixed additive character ψ. Now define the character χ on U_{ℓ} by χ = χ_{0} ⊗ χ_{1} ⊗ 1_{U′}, where U′ is the complement of U_{0} ×U′_{ℓ′} in U_{ℓ}. Note that M_{χ} = M_{χ_{1}}. We will examine the space of ω_{χ}-Bessel functionals for Ind_{P}^{G}(σ ⊗ τ).

In order to carry out our computation, we have to give a description of the R_{χ} − P double cosets in G. The given list of such double cosets is exhaustive, but overdefined.

Let W_{M} = W (M, T_{d}). Then

W_{M} ≃ \{ \begin{matrix}S_{n} × (S_{m} ⋉ ℤ_{2}^{m}) & if G ≠ SO_{2r} \\ S_{n} × (S_{m} ⋉ ℤ_{2}^{m−1}) & otherwise. \end{matrix}

Note that |W/W_{M}| = 2^{n} ( \begin{matrix}r \\ n \end{matrix} ) . Let w_{0} denote the longest element of W/W_{M}. Then

w_{0} = (1 r_{0})(2 r_{0} − 1) … (n r_{0} + 1 − n),

unless G = SO_{2r} and n is odd, in which case

w_{0} = (1 r_{0})(2 r_{0} − 1) … (n r_{0} + 1 − n)(r r + 1).

We now give a list of coset representatives for W/W_{M}. We will say that a permutation s ∈ S_{r_{0}} “appears” in w if w = w′s, for some w′ which is disjoint from s. We will also use the convention that if 1 ≤ i ≤ r, then i′ = r_{0} + 1 − i.

Lemma 2.2. Suppose that w ∈ W.

(a) If G ≠ SO_{2r}, then there is an element w_{1} of W so that w ≡ w_{1} (mod W_{M}) with w_{1} a product of disjoint transpositions in S_{r_{0}}. More precisely, we may choose w_{1} of the form w′_{1}w′′_{1}, with

w′_{1} = ∏_{i=1}^{k}(a_{i} a′_{i}),

for some {a_{i}} ⊂ {1, …, n}, and

w′′_{1} = ∏_{i=1}^{j}(b_{i} c_{i})(c′_{i} b′_{i}),

with {b_{i}} ⊂ {1, …, n}, and {c_{i}} ⊂ {n+1, n+2, …, r_{0}−n}. Furthermore we may assume that the transpositions appearing in w′_{1} and w′′_{1} are all disjoint.

(b) If G = SO_{2r}, then w ≡ w_{1}w_{2}, where w_{1} is of the form given in part (a), and either w_{2} = 1, w_{2} = (d_{0} d′_{0}), for some n + 1 ≤ d_{0} ≤ r, or w_{2} = (i_{0} j′_{0} i′_{0} j_{0}), for some 1 ≤ i_{0} ≤ n < j_{0} ≤ r. In each case w_{1} and w_{2} are disjoint.

Proof. We first write w = cs, with s ∈ S_{r} and c ∈ ℤ_{2}^{r}. Since c acts on the cycles of s independently, we may assume that s is a pair of “companion” cycles, (a_{1} a_{2} … a_{t})(a′_{1} a′_{2} … a′_{t}). If s = 1, or the length of each of the two companion cycles in s is two, then the claim is trivially true, so we assume that the length of each of the cycles is greater than two. Suppose that the claim holds whenever the length of the two cycles in s is less than t. Without loss of generality, we may assume that a_{1} ≤ n. If, for some i, we have a_{i}, a_{i+1} ≤ n, then

w ≡ w(a_{i} a_{i+1})(a′_{i} a′_{i+1}) = c(a_{1} … a_{i−1} a_{i} a_{i+2} … a_{t})(a′_{1} … a′_{i−1} a′_{i} a′_{i+2} … a′_{t}),

and the claim holds by induction. Similarly, we may assume that if a_{i} > n, then a_{i+1} ≤ n. This argument also shows that we may assume that t is even. Now we see that

w ≡ cs · (a_{1} a_{t−1} a_{t−3} … a_{3})(a′_{1} a′_{t−1} a′_{t−3} … a′_{3})
= c(a_{1} a_{t})(a_{3} a_{2}) … (a_{t−1} a_{t−2})(a′_{1} a′_{t})(a′_{3} a′_{2}) … (a′_{t−1} a′_{t−2}).

Now write c = (b_{1} b′_{1})(b_{2} b′_{2}) … (b_{s} b′_{s}), with b_{i} ≠ b_{j}, for i ≠ j.

If, for a fixed even i ≥ 2, {a_{i}, a_{i+1}} ⊂ {b_{j}}_{j=1}^{s}, then the product

(a_{i} a′_{i})(a_{i+1} a′_{i+1})(a_{i+1} a_{i})(a′_{i+1} a′_{i}) = (a_{i} a′_{i+1})(a_{i+1} a′_{i})

appears in the reduced product for w. The same is true if {a_{1}, a_{t}} ⊂ {b_{j}}_{j=1}^{s}, i.e., (a_{1} a′_{t})(a_{t} a′_{1}) appears in w. If i ≥ 2 is even and {a_{i}, a_{i+1}} ∩ {b_{j}}_{j=1}^{s} = ∅, then c commutes with (a_{i} a_{i+1})(a′_{i} a′_{i+1}), and so this product of transpositions appears in w. Similarly, if {a_{1}, a_{t}} ∩ {b_{j}}_{j=1}^{s} = ∅, then (a_{1} a_{t})(a′_{1} a′_{t}) appears in w.

Suppose i ≥ 2 is even and that exactly one element of {a_{i}, a_{i+1}} belongs to {b_{j}}_{j=1}^{s}. Then, if G ≠ SO_{2r}, we can replace w by w(a_{i} a′_{i}), and we see that either (a_{i+1} a_{i})(a′_{i} a′_{i+1}) or (a_{i+1} a′_{i})(a_{i} a′_{i+1}) appears in w(a_{i} a′_{i}), depending on whether a_{i+1} or a_{i} is in {b_{j}}_{j=1}^{s}. If G = SO_{2r} and w either fixes some d_{0} > n, or interchanges some d_{0} and d′_{0}, then we can instead multiply w by (a_{i} a′_{i})(d_{0} d′_{0}), which shows that one of (a_{i+1} a_{i})(a′_{i} a′_{i+1}) or (a_{i+1} a′_{i})(a_{i} a′_{i+1}) appears. We see that the above considerations apply equally well to the pair {a_{1}, a_{t}}. By fixing the element d_{0} before starting the above process, we can guarantee that, when we have concluded, w_{2} = 1 or w_{2} = (d_{0} d′_{0}).

Finally suppose that no such d_{0} exists. Thus, w(d) ≠ d, d′ for all n + 1 ≤ d ≤ r. So we may now assume that d ∈ {a_{i}}, for each d, n + 1 ≤ d ≤ r. Suppose that the number of d for which d = a_{i} ∈ {b_{j}} with a_{i+1} ∉ {b_{j}} is even. (Here we are including {a_{1}, a_{t}} as one possible pair.) Then we see that

w ≡ w ∏(d d′),

where the product is over precisely those d = a_{i} for which a_{i+1} ∉ {b_{j}}, is of the form w_{1} as claimed. Finally if |{n + 1 ≤ d ≤ r|d ∈ {b_{j}}}| is odd, then we fix some such d_{0}. Without loss of generality, assume that d_{0} = a_{t}. Multiplying on the right by the elements (d d′), for the other such d, we see that we have a factor of (a_{t} a′_{t})(a_{1} a_{t})(a′_{1} a′_{t}) remaining to be dealt with. But this product is indeed w_{2} = (a_{1} a′_{t} a′_{1} a_{t}), as claimed. □

If G = SO_{2r}, and w_{2} is of this final form, then there is some flexibility as to the indices appearing in w_{2}. That is, we may choose, for d_{0}, any of the a_{i} > n for which (a_{i} a′_{i}) appears in c, but (a_{i+1} a′_{i+1}) does not. We will need this below.

Recall that ℓ_{0} = r_{0} − 2ℓ. Let s = ℓ_{0} − 2. Suppose x = (x_{1}, …, x_{s}) ∈ F^{s} if G is orthogonal and x ∈ E^{s} if G is unitary. Let

n(x) = ( \begin{matrix}I_{ℓ} \\ & 1 & x_{1} & … & x_{s} & ∗ \\ & & 1 & 0 & … & −x̄_{s} \\ & & & ... & 0 & ⋮ \\ & & & & 1 & −x̄_{1} \\ & & & & & 1 \\ & & & & & & I_{ℓ} \end{matrix} ) ,

where x̄ is the Galois conjugate of x if G is unitary, and is x if G is orthogonal.

The next result is a straightforward consequence of Witt’s Theorem and the Bruhat decomposition. Here and for the rest of this section, we pass between a Weyl group element and its coset representative without changing the notation.

Proposition 2.3.

(a) Let g ∈ G. Then for some w ∈ W and some x ∈ F^{s}, we have R_{χ}gP = R_{χ}n(x)wP. Clearly we can choose w up to W_{M}, i.e., we may assume w = w_{1} is of the form given in Lemma 2.2.

(b) Denote by ||x|| the standard length of x ∈ F^{s} or E^{s} accordingly. If ||x|| = ||x_{1}||, then R_{χ}n(x)wP = R_{χ}n(x_{1})wP, for all w.

If H ⊂ G, we will use ind_{H}^{G}(π) to denote the representation of G compactly induced from π [BeZa,Cas]. Recall that Ind_{P}^{G}(π) = ind_{P}^{G}(π), by the Iwasawa decomposition. If V is a complex vector space, let C^{∞}(G, V) denote the space of locally constant V –valued functions on G, and let C_{c}^{∞}(G, V) denote the subspace of elements of C^{∞}(G, V) with compact support. Let D(G, V) = C_{c}^{∞}(G, V)^{∗} be the space of V –distributions on G.

Let V_{σ} be the space of σ, V_{τ} be the space of τ, and V_{ω} the space of ω (and hence the space of ω_{χ}). We let V_{π} = V_{σ} ⊗ V_{τ} . Denote by V the vector space Ṽ_{ω} ⊗ V_{π}, where Ṽ_{ω} is the space of the smooth contragredient ω̃ of ω.

We wish to analyze the space Hom_{G}(Ind_{P}^{G}(π), Ind_{R_{χ}}^{G}(ω_{χ})). Dualizing, and using Theorem 2.4.2 of [Cas], this is isomorphic to the space

Hom_{G}(ind_{R_{χ}}^{G}(ω̃_{χ}), Ind_{P}^{G}(σ̃ ⊗ τ̃)).

This space, in turn, is isomorphic to the space of intertwining forms on

ind_{R_{χ}}^{G}(ω̃_{χ}) ⨂ Ind_{P}^{G}(π)

[Har, Lemma 4]. Now by Bruhat’s thesis (see [Rodb, Theorem 4]) this is isomorphic to the space of V –distributions T on G satisfying

(2.4)  ε(r) ∗ T ∗ ε(p^{−1}) = δ_{P}^{1/2}(p)T ∘ [ω̃_{χ}(r) ⊗ π(p)],

for all r ∈ R_{χ} and p ∈ P.

The analysis of this space of distributions will make use of the following proposition. Its proof requires a combinatorial argument, and will be given in several steps later in this section.

Proposition 2.4. If there is a non-zero V -distribution T satisfying (2.4) for all r ∈ R_{χ} and p ∈ P which is supported on R_{χ}n(x)wP, then R_{χ}n(x)wP = R_{χ}w_{0}P and σ is generic.

Lemma 2.5. Suppose that T satisfies (2.4). Then T is completely determined by its restriction to R_{χ}w_{0}P.

Proof. First note that a straightforward matrix computation shows that n(x)w_{0} = w_{0}n(x), for any x. Thus

R_{χ}w_{0}P = ⋃_{x} R_{χ}n(x)w_{0}P = P w_{0}P,

is open. Therefore C = G\R_{χ}w_{0}P is closed. Therefore, we have the exact sequence [BeZa, §1.7]

0 ⟶ C_{c}^{∞}(R_{χ}w_{0}P) ⟶ C_{c}^{∞}(G) ⟶ C_{c}^{∞}(C) ⟶ 0.

Then, by tensoring with V, the above exact sequence yields the exact sequence

0 ⟶ C_{c}^{∞}(R_{χ}w_{0}P , V) ⟶ C_{c}^{∞}(G, V) ⟶ C_{c}^{∞}(C, V) ⟶ 0.

Dualizing, we get the exact sequence

0 ⟶ D(C, V) ⟶ D(G, V) ⟶ D(R_{χ}w_{0}P, V) ⟶ 0.

Let D_{R_{χ},P} be the subspace of distributions satisfying (2.4). Then Proposition 2.4 implies that if T ∈ D(G, V)_{R_{χ},P} and f ∈ C_{c}^{∞}(C, V), then T (f) = 0. Thus, the above sequence tells us that D(G, V)_{R_{χ},P} ≃ D(R_{χ}w_{0}P, V)_{R_{χ},P} , which completes the proof of the Lemma. □

Let R_{χ}^{w_{0}} = w_{0}^{−1}R_{χ}w_{0}, and denote by ω_{χ}^{w_{0}} the representation of R_{χ}^{w_{0}} defined by ω_{χ}^{w_{0}}(r) = ω_{χ}(w_{0}rw_{0}^{−1}). Recall that P = MN is the Levi decomposition of P.

Lemma 2.6. There exists an isomorphism between the vector space D(R_{χ}w_{0}P, V)_{R_{χ},P} and the vector space of distributions in D(U_{ℓ}) ⊗ D(P, V) of the form

χ(u) du ⊗ δ_{P}^{−1/2}(m) dQ(m) dn,

where Q ∈ D(M, V) satisfies

(2.5)  ε(r) ∗ Q ∗ ε(m^{−1}) = Q ∘ [ω̃_{χ}^{w_{0}}(r) ⊗ π(m)],

for all r ∈ R_{χ}^{w_{0}} ∩ M, m ∈ M.

Proof. Define a projection

P : C_{c}^{∞}(U_{ℓ}) ⊗ C_{c}^{∞}(P, V) → C_{c}^{∞}(U_{ℓ}w_{0}P, V)

by specifying that for all f_{1} ∈ C_{c}^{∞}(U_{ℓ}) and f_{2} ∈ C_{c}^{∞}(P, V), one has

P(f_{1} ⊗ f_{2})(uw_{0}p) = ∫_{U_{ℓ}∩w_{0}P w_{0}^{−1}} f_{1}(uu_{1}) f_{2}(w_{0}^{−1}u_{1}^{−1}w_{0}p) du_{1}.

Then it follows from [Sil, Lemma 1.2.1] that P is onto. Let T ∈ D(R_{χ}w_{0}P, V)_{R_{χ},P}. For f_{1}, f_{2} as above, define T′ ∈ D(U_{ℓ}) ⊗ D(P, V) by

T′(f_{1} ⊗ f_{2}) = T (P(f_{1} ⊗ f_{2})).

Then one sees easily that (2.4) implies the equality

(2.6)  ε(u) ∗ T′ ∗ ε(p^{−1}) = ω̃_{χ}(u)T ∘ [π(p)]

for all u ∈ U_{ℓ}, p ∈ P (where π acts on the second factor of V). As in [Sil, Section 1.8], this implies that T′ is in fact a pure tensor of the form

(2.7)  χ(u) du ⊗ δ_{P} (m)^{−1/2} dQ(m) dn.

where Q ∈ D(M, V). (Here we are using that π(mn) = π(m).) It is a formal consequence of the definitions that (2.6) implies that

Q ∗ ε(m^{−1}) = Q ∘ [π(m)]

for all m ∈ M. We claim that, more strongly, equation (2.5) holds. To see this, write

dQ(p) = δ_{P} (m)^{−1/2} dQ(m) dn.

Let f_{1} ∈ C_{c}^{∞}(U_{ℓ}), f_{2} ∈ C_{c}^{∞}(P, V) and r ∈ R_{χ} ∩ w_{0}Mw_{0}^{−1}. Then by (2.4) we have

∫_{U_{ℓ}} f_{1}(u)χ(u) du ∫_{P} ω̃_{χ}(r)f_{2}(p) dQ(p) = ∫_{U_{ℓ}×P} f_{1}(u) ω̃_{χ}(r)f_{2}(p) dT′(u, p)

= T (ω̃_{χ}(r)P(f_{1} ⊗ f_{2}))

= ∫_{U_{ℓ}w_{0}P} P(f_{1} ⊗ f_{2})(ruw_{0}p) dT (uw_{0}p).

But ruw_{0}p = (rur^{−1})w_{0}(w_{0}^{−1}rw_{0}p), so this expression is equal to

∫_{U_{ℓ}×P} f_{1}(rur^{−1})f_{2}(w_{0}^{−1}rw_{0}p) dT′(u, p)

= ∫_{U_{ℓ}} f_{1}(rur^{−1})χ(u) du ∫_{P} f_{2}(w_{0}^{−1}rw_{0}p) dQ(p)

= ∫_{U_{ℓ}} f_{1}(u)χ(u) du ∫_{P} f_{2}(p) d(ε(w_{0}^{−1}rw_{0}) ∗ Q)(p),

where in this last equality the defining properties of R_{χ} = M_{χ}U_{ℓ} have been used to simplify the U_{ℓ} integral. Since this holds for all f_{1} ∈ C_{c}^{∞}(U_{ℓ}) one concludes that

ε(w_{0}^{−1}rw_{0}) ∗ Q = Q ∘ ω̃_{χ}(r)

for all r ∈ R_{χ} ∩ w_{0}Mw_{0}^{−1}, as desired.

Conversely, given a distribution Q satisfying equation (2.5), one reverses the above steps to arrive at a distribution T′ ∈ D(U_{ℓ}) ⊗ D(P, V) satisfying (2.6). Since the map P is onto, one may define a distribution T ∈ D(U_{ℓ}w_{0}P, V) by the formula

T (P(f_{1} ⊗ f_{2})) = T′(f_{1} ⊗ f_{2})

provided one shows that if P(∑_{i} f_{1,i} ⊗ f_{2,i}) = 0, then T′(∑_{i} f_{1,i} ⊗ f_{2,i}) = 0. This follows as in [HeR, Theorem 15.24]. Since M_{χ} ⊆ w_{0}^{−1}Mw_{0} and R_{χ} = U_{ℓ}M_{χ}, it follows from (2.5) and (2.7) that the T so-obtained satisfies (2.4).

The maps T ↦ Q, Q ↦ T described above are clearly inverses. This completes the proof of the Lemma. □

We now complete the proof of Theorem 2.1, modulo the proof of Proposition 2.4. Let Q be as in the proof of Lemma 2.6. Then by Bruhat’s thesis once again, Q corresponds to an element of

Hom_{M}(ind_{R_{χ}^{w_{0}} ∩M}^{M}(ω̃_{χ}^{w_{0}}), σ̃ ⊗ τ̃),

which, by duality gives an element of Hom_{M}(π, Ind_{R_{χ}^{w_{0}} ∩M}^{M}(ω_{χ}^{w_{0}})). Since M = GL_{n}(k)× G(m), where G_{1} is either GL_{n}(F) or GL_{n}(E), depending on whether G is orthogonal or unitary, we see that this last space is exactly the space of Whittaker models for σ tensored with the space of ω_{χ}^{w_{0}}–Bessel models for τ. □

Proof of Proposition 2.4. The remainder of the section will consist of a proof of Proposition 2.4. This is carried out in several steps. We begin by showing that, on many double cosets, the compatibility condition π(p) = ω_{χ}(wpw^{−1}) can not be satisfied for some p ∈ P with r = wpw^{−1} ∈ R_{χ}. By [Sil, Theorem 1.9.5], this is sufficient to imply the Proposition.

Let Σ_{P}^{+} denote the set of positive roots in N. Let Δ denote the simple roots of T in G which give rise to our choice of Borel subgroup. If α ∈ Φ(G, T), then we let X_{α} be the corresponding element of a Chevalley basis for the Lie algebra of U or Ū. Let α_{i} denote the root e_{i} − e_{i+1}, and β = e_{ℓ} + e_{ℓ+1}. Let X = {α_{1}, α_{2}, …, α_{ℓ}, β}. Then X is the set of roots where the character χ is non-trivial. For α ∈ X we have χ(I + tX_{α}) = ψ_{α}(t). Also, note that X ∩ Σ_{P}^{+} = {α_{n}}. We list the elements of Σ_{P}^{+}, for future reference. If G = SO_{2r+1}, then

Σ_{P}^{+} = {e_{i} ± e_{j} | 1 ≤ i ≤ n < j ≤ r} ∪
{e_{i} + e_{j} | 1 ≤ i < j ≤ n} ∪ {e_{i}|1 ≤ i ≤ n} .

If G = U_{2r}, then

Σ_{P}^{+} = {e_{i} ± e_{j} | 1 ≤ i ≤ n < j ≤ r} ∪
{e_{i} + e_{j} | 1 ≤ i < j ≤ n} ∪ {2e_{i}|1 ≤ i ≤ n} .

If G = U_{2r+1}, then

Σ_{P}^{+} = {e_{i} ± e_{j} | 1 ≤ i ≤ n < j ≤ r} ∪
{e_{i} + e_{j} | 1 ≤ i < j ≤ n} ∪ {e_{i}, 2e_{i}|1 ≤ i ≤ n} .

Finally, if G = SO_{2r}, then

Σ_{P}^{+} = {e_{i} ± e_{j} | 1 ≤ i ≤ n < j ≤ r} ∪
{e_{i} + e_{j} | 1 ≤ i < j ≤ n} .

We list the various I + X_{α}, which generate the root subgroups U_{α} of U. Let E_{ij} denote the elementary matrix whose only non-zero entry is a 1 in the ij-th entry. We recall the convention that i′ = r_{0} +1−i. Suppose E = F (γ), where γ̄ = −γ, and a ↦ ā is the Galois automorphism of E/F. If α = e_{i}−e_{j}, then I+X_{α} = I+E_{ij}−E_{j′i′} if G is orthogonal, and I + X_{α} = I + γE_{ij} − γE_{j′i′} if G is unitary. If α = e_{i} + e_{j}, then I + X_{α} = I + E_{ij′} − E_{ji′} if G is orthogonal, and I + X_{α} = I + γE_{ij′} − γE_{ji′} if G is unitary. If α = e_{i}, then I + X_{α} = I + E_{i,r+1} − E_{r+1,i′} if G = SO_{2r+1}, and I + X_{α} = I + γE_{i,r+1} − γE_{r+1,i′} if G = U_{2r+1}. Finally, if G is unitary and α = 2e_{i}, then I + X_{α} = I + γE_{ii′}.

Suppose that, for some α ∈ Σ_{P}^{+}, we have α′ = wα ∈ X. Choose some t ∈ F^{×} for which ψ_{α′}(t) ≠ 1. Now set p = I + tX_{α}, which is in P. Then r = wpw^{−1} = I + tX_{α′} ∈ U_{ℓ} ⊂ R_{χ}. Note that π(p) = 1 ≠ ω_{χ}(r) = ψ_{α}(t). Thus, if w has the above property, R_{χ}wP can support no distribution of the desired type.

Lemma 2.7. Let G = SO_{2r}. Suppose that, as in Lemma 2.2, w ∈ W is equivalent mod W_{M} to w_{1}w_{2}, with w_{2} = (i_{0} j′_{0} i′_{0} j_{0}), for some 1 ≤ i_{0} ≤ n < j_{0} ≤ r. Then wΣ_{P}^{+} ∩ X ≠ ∅.

Proof. From the proof of Lemma 2.2, we may assume that for each n+1 ≤ k ≤ r, we have w(k) = i_{k} or i′_{k}, for some 1 ≤ i_{k} ≤ n. First suppose that (i_{n+1} n+1)(i′_{n+1} (n+ 1)′) appears in w. Consider first the case that for all k, n + 1 ≤ k ≤ ℓ, we have a permutation (i_{k} k)(i′_{k} k′) appearing in w. Since ℓ + 1 = w(i_{ℓ+1}) or ℓ + 1 = w(i′_{ℓ+1}), we have w(e_{i_{ℓ}} + e_{i_{ℓ+1}}) = e_{ℓ} ± e_{ℓ+1}, which will be in X. So now we may suppose that either j_{0} ≤ ℓ, or (i_{k} k′)(i′_{k} k) appears in w, for some k with n + 1 ≤ k ≤ ℓ. Since w changes an even number of signs, we see that in the former case there must be some k with n + 1 ≤ k ≤ r, so that (i_{k} k′)(i′_{k} k) appears in w. Now we can multiply on the right by (j_{0} j′_{0})(k k′), to see that, in fact, we may assume that (i_{k_{0}} k′_{0})(i′_{k_{0}} k_{0}) is appearing, for some k_{0}, with n + 1 ≤ k_{0} ≤ ℓ. Choosing the minimal such k_{0}, we know that k_{0} = w(i′_{k_{0}}), while k_{0} − 1 = w(i_{k_{0}−1}). Thus, α_{k_{0}−1} = w(e_{i_{k_{0}−1}} + e_{i_{k_{0}}} ) ∈ wΣ_{P}^{+} ∩ X, and the Lemma holds.

Thus, we may assume that either (i_{n+1} (n+1)′)(i′_{n+1} n+1) appears in w, or that (i_{n+1} (n + 1)′ i′_{n+1} n + 1) does. In the former case, we may multiply on the right by (n+1 (n+1)′)(j_{0} j′_{0}), to get an equivalent w for which the latter is true, i.e, we may assume that i_{0} = i_{n+1}. First suppose w(n) = n. Then w(e_{n} + e_{i_{0}}) = α_{n} ∈ wΣ_{P}^{+} ∩ X and we are done. Suppose instead that w(n) = n′, i.e., that (n n′) appears in w. Let i be the smallest positive integer so that w(n − i) ≠ n − i′. (By our assumption on the form of w, such an i exists.) Then n − i = i_{k}, for some k ≥ n + 1, and n − i = w(k) or w(k′). Therefore, α_{n−i} is equal to either w(e_{n−i+1} − e_{k}) or to w(e_{n−1+1} + e_{k}). In either case, α_{n−i} ∈ wΣ_{P}^{+} ∩ X. Finally, we may suppose that either n = i_{0} or that one of (n k)(n′ k′) or (n k′)(n′ k) appears in w, for some k, n + 1 ≤ k ≤ r. If (n k)(n′ k′) appears in w, then we may multiply on the right by

((n + 1) k)((n + 1)′ k′)(i_{0} n)(i′_{0} n′),

to replace w by an equivalent element with i_{0} = n. Similarly, if (n k′)(n′ k) appears in w, then we may multiply on the right by

(n + 1 k)((n + 1)′ k′)(i_{0} n)(i′_{0} n′)(k k′)(n + 1 (n + 1)′),

to see that we may assume that i_{0} = n. We are thus reduced to the case where (n (n + 1)′ n′ n + 1) appears in w. In this case, w(e_{n} + e_{n+1}) = α_{n} ∈ wΣ_{P}^{+} ∩ X. Thus, in all cases, the Lemma holds. □

Remark. For future use we make note of the following fact. If w is as in Lemma 2.7, and if w^{−1}α_{ℓ} ∈ Σ_{P}^{+}, then the proof of Lemma 2.7 shows that either w^{−1}α_{ℓ} = e_{i} +e_{j}, for some i, j ≤ n, or that wΣ_{P}^{+} ∩ X ≠ {α_{ℓ}}.

We now describe those w which have the property that wΣ_{P}^{+}∩X = ∅. By Lemmas 2.2 and 2.7, we may assume that w is a product of disjoint transpositions.

Lemma 2.8. Suppose that w ∈ W is a representative for a class in W/W_{M}, and w is in the form specified by Lemma 2.2. Further suppose that, for all α ∈ Σ_{P}^{+}, we have wα ∉ X. Then the following hold:

(a) For all k with n + 1 ≤ k ≤ ℓ, we have w(k) > n;

(b) For all i with 1 ≤ i ≤ n, we have w(i) ≠ i.

Proof. (a) First suppose that w(ℓ) ≤ n. If w(ℓ + 1) ≤ n, then w(β) ∈ Σ_{P}^{+}, contradicting our choice of w. If w(ℓ + 1) = ℓ + 1, then again w(β) ∈ Σ_{P}^{+}. Finally, if w(ℓ + 1) ≥ n′, then w(α_{ℓ}) ∈ Σ_{P}^{+}. So we must have w(ℓ) > n.

Now suppose that for some k, n+1 ≤ k ≤ ℓ−1, we have w(k) ≤ n. If w(k +1) = k + 1, or w(k + 1) ≥ n′, then w(α_{k}) ∈ Σ_{P}^{+}, which is a contradiction. Therefore, w(k + 1) ≤ n. However, this implies, by induction, that w(ℓ) ≤ n, which we have already seen is impossible. Therefore, w(k) > n.

(b) Suppose that w(i) = i for some i, 1 ≤ i ≤ n − 1. If w(i + 1) ≠ i + 1, then w(i + 1) > n, and so w(α_{i}) ∈ Σ_{P}^{+}. Since this contradicts our choice of w, we have w(i + 1) = i + 1. We may thus suppose that w fixes n. Now by part (a), we have w(n + 1) ≥ n + 1, and therefore, wα_{n} ∈ Σ_{P}^{+}. This again is a contradiction, so w cannot fix n. Therefore, w fixes none of the integers 1, 2, …, n. □

Lemma 2.9. Suppose that w is as in Lemma 2.8 and assume that w(n) ≠ n′. Then for n + 1 ≤ k ≤ ℓ, we have w(k) = k.

Proof. By Lemma 2.8(a) it is enough to show that it is impossible that w(k) ≥ n′ for any such k. Suppose to the contrary that there is some k, with n+1 ≤ k ≤ ℓ, for which w(k) ≥ n′. Then there some i ≤ n, for which k = w(i′). If w(k − 1) = k − 1, then α_{k−1} = w(e_{i} + e_{k−1}) ∈ wΣ_{P}^{+} ∩ X. Since this contradicts our choice of w, we must have w(k − 1) ≥ n′. Therefore, by (downwards) induction, w(n + 1) ≥ n′. Set w(n + 1) = i′. Since w(n) ≠ n, and, by assumption, w(n) ≠ n′, either w(n) = k or w(n) = k′ for some k, with n + 1 ≤ k ≤ n′ − 1. Therefore, α_{n} = w(e_{i} + e_{k}) or α_{n} = w(e_{i} − e_{k}). Either one of these possibilities contradicts our assumption on w. Thus w(n + 1) < n′, which then implies the result of the Lemma. □

Lemma 2.10. Suppose that w is as in Lemma 2.8. Suppose that there is some i, 2 ≤ i ≤ n, for which w(i) = i′. Then w(i − 1) = (i − 1)′.

Proof. Suppose w(i − 1) ≠ (i − 1)′. By Lemma 2.8(b), we can choose k, with n + 1 ≤ k ≤ r so that w(i − 1) = k or w(i − 1) = k′. Now α_{i−1} = w(e_{i} − e_{k}) or α_{i−1} = w(e_{i} + e_{k}). Since this contradicts our choice of w we conclude w(i − 1) = (i − 1)′. □

Thus, if w is chosen as in Lemma 2.2 with wΣ_{P}^{+} ∩ X = ∅ and w(n) = n′, then w = w_{0}. If wΣ_{P}^{+} ∩ X = ∅, and w(n) ≠ n′, then by Lemma 2.9, w(ℓ) = ℓ. If w(ℓ + 1) ≠ ℓ + 1, then either α_{ℓ} or β would be of the form wα for some α ∈ Σ_{P}^{+}. Consequently, w(ℓ + 1) = ℓ + 1, and therefore w(α) = α for α ∈ {α_{n+1} …, α_{ℓ}, β}. Thus, for some i_{0} < n, and some {a_{i}} ⊂ {ℓ + 2, …, (ℓ + 2)′},

(2.8) w = (1 r_{0})(2 r_{0} − 1) … (i_{0} i′_{0})(i_{0} + 1 a_{i_{0}+1})((i_{0} + 1)′ a′_{i_{0}+1}) … (n a_{n})(n′ a′_{n}).

Let a = a_{n} if a_{n} ≤ r, and a = a′_{n} otherwise. Then wα_{n} = ±e_{a} − e_{n+1}. Let X_{1} = {α_{n+1} …, α_{ℓ}, β, wα_{n}}. Note that

X_{1} = w({α_{n}, α_{n+1} …, α_{ℓ}, β}),

and is thus a linearly independent subset of the root system Φ(G(m), T′), where we recall that T′ = T ∩ G(m). We extend X_{1} \ {β} to a set of simple roots for G(m). Set B′ = T_{1}U′′ to be the corresponding Borel subgroup of G(m), and suppose that U′′_{ℓ′+1} is the subgroup of U′′ which is conjugate to U′_{ℓ′+1} and generated by the elements of X_{1}. (Recall that U′_{ℓ′} is the subgroup supporting the character χ_{1} which gives rise to the model for τ.) Now let χ′ be the character of U′′_{ℓ′+1} so that χ′(I + tX_{α}) = ψ_{α}(t), for α ∈ X_{1} \ {α_{ℓ}}, and χ′(I + tX_{α_{ℓ}}) = ψ_{α_{ℓ}}(δt). Let M_{χ′} be the corresponding normalizer in M_{ℓ′+1}. Note that M_{χ′} ⊂ M_{χ}. Suppose that m′ ∈ M_{χ′}. If the distribution T satisfies (2.4), then ε(m′) ∗ T = T ∘ ω(m′). So for some component ω′ of ω|_{M_{χ′}} , we have

ε(r) ∗ T ∗ ε(h) = T ∘ [ω̃′_{χ′}(r) ⊗ τ(h)],

for all h ∈ G(m), and r ∈ R_{χ′} = M_{χ′}U′′_{ℓ′+1}. If T is non-zero, this now implies that τ has a Bessel model with respect to U′′, χ′ and ω′. However, since U′′_{ℓ+1} is isomorphic to U_{ℓ+1}, this is a rank ℓ_{1} − 1 Bessel model for τ. This contradicts the minimality of the ω_{χ_{1}}-Bessel model for τ. Hence, no such T exists.

Note that this argument shows that Ind_{P}^{G}(π) cannot have any Bessel model of rank less than B(τ) supported on R_{χ}wP.

Finally, suppose that w = w_{0}. Let u ∈ Ū ∩ GL_{n}(F). Set r = w_{0}^{−1}uw_{0}. Then r ∈ U_{ℓ}, and χ(r) = χ_{0}^{w_{0}}(u). Since

ε(r) ∗ T = T ∘ [χ_{0}^{w_{0}}(u)] = T ∗ ε(u) = T ∘ [σ(u)],

we see that σ must be generic if T is non-zero [Rodb]. This completes the proof of Proposition 2.4 for the cosets R_{χ}wP, with w ∈ W/W_{M}.

We now examine the double cosets represented by n(x)w, where x = (x_{1}, …, x_{s}) is a vector. Recall that

n(x) = ( \begin{matrix}I_{ℓ} \\ & 1 & x_{1} & … & x_{s} & ∗ \\ & 0 & 1 & 0 & … & −x̄_{s} \\ & & & ... & 0 \\ & 0 & 0 & … & 1 & −x̄_{1} \\ & 0 & 0 & 0 & … & 1 \\ & & & & & & I_{ℓ} \end{matrix} ) ,

where x̄ is the Galois conjugate of x if G is unitary, and x̄ = x if G is orthogonal.

We assume that w is of the form given in Lemma 2.2. First note that if α ∈ X \{α_{ℓ}}, then n(x)(I +tX_{α})n(x)^{−1} = I +tX_{α}. Suppose that wΣ_{P}^{+} ∩(X \{α_{ℓ}}) ≠ ∅. Choose α′ ∈ Σ_{P}^{+} with wα′ = α ∈ X \ {α_{ℓ}}, and t for which ψ_{α}(t) ≠ 1. Setting p = I + tX_{α′}, we have n(x)wpw^{−1}n(x)^{−1} = I + tX_{α} ∈ R_{χ}. Furthermore ω_{χ}(x) = ψ_{α}(t) ≠ 1, while π(p) = 1. Thus, R_{χ}n(x)wP supports no distributions satisfying (2.4).

Now suppose that wΣ_{P}^{+} ∩ X = {α_{ℓ}}. First suppose that w^{−1}α_{ℓ} = e_{i} + e_{j}, with i, j ≤ n. Without loss of generality, assume that w(i) = ℓ, and w(j) = (ℓ + 1)′. Suppose that ℓ + 2 ≤ k ≤ r. If w(k) = i_{k} ≤ n then w^{−1}(e_{ℓ} + e_{k}) = e_{i} + e_{i_{k}} ∈ Σ_{P}^{+}. If instead w(k) = i′_{k} for some i_{k} ≤ n, then w^{−1}(e_{ℓ}−e_{k}) = e_{i}+e_{i_{k}}. Finally, if w(k) = k, then w^{−1}(e_{ℓ} ± e_{k}) = e_{i} ± e_{k} ∈ Σ_{P}^{+}. Choose s_{0} ≤ s for which x_{s_{0}} ≠ 0. Let y = x_{s_{0}}. Choose k_{0} with the property that either w^{−1}(e_{ℓ}+e_{k_{0}}) or w^{−1}(e_{ℓ}−e_{k_{0}}) is an element of Σ_{P}^{+}. Denote the root e_{ℓ} ± e_{k_{0}} as α_{0}, with ± chosen so that w^{−1}α_{0} ∈ Σ_{P}^{+}. We may also assume that X_{α_{0}} has −1 as its (r + ℓ_{1}, ℓ + s_{0}) entry (see Lemma 2.3). Now note that

n(x)(I + tX_{α_{0}})n(x)^{−1} = (I + tX_{α_{0}})(I + ytX_{β}).

Thus, if ψ_{α_{0}}(yt) ≠ 0, and p = I + tX_{w^{−1}α_{0}} ∈ N, then π(p) = 1, while

ω_{χ}(n(x)wpw^{−1}n(x)^{−1}) = ψ_{α_{0}}(yt) ≠ 1.

Consequently, R_{χ}n(x)wP cannot support a V -distribution of the desired form.

We are left with the cases wΣ_{P}^{+} ∩ X = {α_{ℓ}}, but w^{−1}α_{ℓ} ≠ e_{i} + e_{j} for all i, j ≤ n, or wΣ_{P}^{+} ∩ X = ∅. For the second of these two cases, the form of w is given by (2.8). In order to complete the proof we will determine the form of w in the first case. To do so we need a few lemmas.

Lemma 2.11. Suppose that wΣ_{P}^{+} ∩X = {α_{ℓ}}, but w^{−1}α_{ℓ} ≠ e_{i} +e_{j}, for all i, j ≤ n. Then w(ℓ) = ℓ.

Proof. If w^{−1}(ℓ) = j′ for some j ≤ n, then w^{−1}α_{ℓ} ∉ Σ_{P}^{+}, which is a contradiction. Suppose w^{−1}(ℓ) = j ≤ n. If w(ℓ + 1) = ℓ + 1, then w^{−1}(β) = e_{j} + e_{ℓ+1} ∈ Σ_{P}^{+}, contradicting our choice of w. If w(ℓ + 1) = i ≤ n, then w^{−1}α_{ℓ} ∉ Σ_{P}^{+}, which also contradicts our choice of w. Finally, if w^{−1}(ℓ + 1) = i′ for some i ≤ n, then w^{−1}α_{ℓ} = e_{j} + e_{i}, which is again a contradiction. Thus, w(ℓ) = ℓ. □

Lemma 2.12. If w is as in Lemma 2.11, then for n + 1 ≤ k ≤ ℓ − 1, we have w^{−1}(k) > n.

Proof. Suppose that w^{−1}(k) = j ≤ n. If w(k+1) = k+1, then w^{−1}α_{k} = e_{j} −e_{k+1} ∈ Σ_{P}^{+}. If w(k + 1) = i′ for some i ≤ n, then w^{−1}α_{k} = e_{j} + e_{i}. Either case contradicts our hypotheses. Therefore w^{−1}(k + 1) ≤ n. Now by induction, w^{−1}(ℓ − 1) ≤ n. On the other hand, by Lemma 2.11, w(ℓ) = ℓ. Therefore w^{−1}α_{ℓ−1} ∈ Σ_{P}^{+}, contradicting our choice of w. Consequently, w^{−1}(k) > n. □

Lemma 2.13. Suppose that w is as in Lemma 2.11. Then w(n) ≠ n.

Proof. Suppose that w(n) = n. If w(n + 1) = n + 1, then w fixes α_{n}, which is in the intersection of X and Σ_{P}^{+}. If w^{−1}(n + 1) = j′ for some j ≤ n, then w^{−1}α_{n} = e_{n} + e_{j}. Both of these possibilities contradict our choice of w. By Lemma 2.12, w^{−1}(n + 1) > n, and so these are the only two choices for w(n + 1). Since each leads to a contradiction, w(n) ≠ n. □

Lemma 2.14. Suppose that w is as in Lemma 2.11.

(a) For all i ≤ n we have w(i) ≠ i.

(b) If w(i_{0}) = i′_{0}, for some i_{0} ≤ n then w(i) = i′ for all i ≤ i_{0}.

Proof. (a) Suppose that w(i) = i for some i ≤ n. Choose the maximal such i. By Lemma 2.13, i < n. Suppose that w(i+1) = (i+1)′. Then w^{−1}α_{i} = e_{i} +e_{i+1} ∈ Σ_{P}^{+}. Thus in this case we have a contradiction. If w(i + 1) = k or w(i + 1) = k′ for some n + 1 ≤ k ≤ r, then w^{−1}α_{i} = e_{i} ± e_{k} ∈ Σ_{P}^{+}. This is also a contradiction, and hence no i ≤ n can be fixed by w.

(b) Suppose that w(i) = i′, for some i ≤ n. If w(i − 1) = k or k′ for some n + 1 ≤ k ≤ r, then w^{−1}α_{i−1} = e_{i} ± e_{k} ∈ Σ_{P}^{+}. But by part (a), w(i − 1) ≠ i − 1, so the only remaining possibility is w(i − 1) = (i − 1)′. This gives the claim by induction. □

Corollary 2.15. If w is as in Lemma 2.11, then w(n) = k_{0} or w(n) = k′_{0} for some n + 1 ≤ k_{0} ≤ r. □

Lemma 2.16. Suppose that w is as in Lemma 2.11. Then w(k) = k for all k with n + 1 ≤ k ≤ ℓ − 1.

Proof. Suppose that w^{−1}(n + 1) = j′ for some j ≤ n. Then, by Corollary 2.15, w^{−1}α_{n} = e_{j} ± e_{k_{0}} ∈ Σ_{P}^{+}, contradicting our choice of w. Thus, by Lemma 2.12, w(n + 1) = n + 1.

Now suppose w^{−1}(k) = j′_{k} for some k with n + 2 ≤ k ≤ ℓ − 1, and some j_{k} ≤ n. If w(k − 1) = k − 1, then w^{−1}α_{k−1} ∈ Σ_{P}^{+}, which is a contradiction. Therefore, by Lemma 2.12, w^{−1}(k − 1) = j′_{k−1}, for some j_{k−1} ≤ n. By induction, this gives w(n + 1) ≠ n + 1, while we have just shown that w(n + 1) = n + 1. Therefore, w(k) = k. □

Lemma 2.17. Suppose that w is as in Lemma 2.11.

(a) Suppose that G ≠ SO_{2r}. Then, for some n_{1}, with 0 ≤ n_{1} < n, and some

{k_{j}|1 ≤ j ≤ n − n_{1}} ⊂ {ℓ + 1, ℓ + 2, …, (ℓ + 1)′},

we have

w =(1 r_{0})(2 r_{0} − 1) … (n_{1} n′_{1})(n_{1} + 1 k_{1})(k′_{1} (n_{1} + 1)′) …
(n k_{n_{2}})(k′_{n_{2}} n′).

Here n = n_{1} + n_{2}. Furthermore, k_{j} = (ℓ + 1)′ for some j.

(b) If G = SO_{2r}, and we write w = w_{1}w_{2} as in Lemma 2.2, then w_{2} = 1 or w_{2} = (d d′), for some ℓ + 2 ≤ d ≤ r. Furthermore w_{1} is of the form

w_{1} =(1 r_{0})(2 r_{0} − 1) … (n_{1} n′_{1})(n_{1} + 1 k_{1})(k′_{1} (n_{1} + 1)′) …
(n k_{n_{2}})(k′_{n_{2}} n′),

with n = n_{1} + n_{2}, and the integers k_{j} are as in part (a). Moreover, k_{j} = (ℓ + 1)′ for some j.

Proof. First note that if G = SO_{2r}, and w = w_{1}w_{2}, then Lemma 2.16 and the remark following Lemma 2.7 imply that w_{2} is not of the form (i j′ i′ j), for some 1 ≤ i ≤ n < j ≤ r. Moreover, since w^{−1}α_{ℓ} ∈ Σ_{P}^{+}, Lemmas 2.16 and 2.11 imply that if w_{2} = (d d′), then ℓ + 2 ≤ d ≤ r. If G ≠ SO_{2r}, let w_{2} = 1.

By Lemma 2.14(a), w(i) ≠ i for all i ≤ n. By Lemma 2.11, Corollary 2.15, and Lemma 2.16, w(n) = k or k′, for some ℓ + 1 ≤ k ≤ r. Let n_{1} be the largest nonnegative integer for which n_{1} < n and w(n_{1}) = n′_{1}. If n_{1} > 0, then by Lemma 2.14(b) w = (1 r_{0})(2 r_{0} − 1) … (n_{1} n′_{1})w_{2}w′, where w′(i) = i for all i ≤ n_{1}, and w_{2} and w′ are disjoint. Now w′(i) ≠ i and w′(i) ≠ i′ for n_{1} + 1 ≤ i ≤ n, and therefore n + 1 ≤ w′(i) ≤ n′ − 1. However, by Lemma 2.16, ℓ + 1 ≤ w′(i) ≤ (ℓ + 1)′. Thus,

w′ = (n_{1} + 1 k_{1})(k′_{1} n′_{1} − 1) … (n k_{n_{2}})(k′_{n_{2}} n′),

as claimed. Finally, Lemma 2.11 implies w(ℓ + 1) ≠ ℓ + 1, and so we must have (w′)^{−1}(ℓ + 1) = w^{−1}(ℓ + 1) = j′, for some j ≤ n. □

We now finish the proof of Proposition 2.4. If w = w_{0}, then n(x)w_{0} = w_{0}n(x), and since n(x) ∈ P, we have R_{χ}w_{0}P = R_{χ}n(x)w_{0}P. If wΣ_{P}^{+} ∩ X = {α_{ℓ}}, or wΣ_{P}^{+} ∩ X = ∅, then Lemma 2.16 and equation (2.8) show that w(e_{n} + e_{ℓ}) = e_{ℓ} ± e_{k}, for some k, with ℓ+2 ≤ k ≤ r. Let α = w(e_{n} +e_{ℓ}), and denote I +tX_{e_{n}+e_{ℓ}} by p. As before, choose x_{0} so that R_{χ}n(x_{0})wP = R_{χ}n(x)wP, and such that x_{0} has a nonzero entry y with ω_{χ}(n(x)wpw^{−1}n(x)^{−1}) = ψ_{α}(yt). Note that π(p) = 1. Choosing t for which ψ_{α}(yt) ≠ 1, we see that R_{χ}n(x)wP cannot support a V -distribution of the desired form. □

From the argument above, it is apparent that Ind_{P}^{G}(π) cannot have a Bessel model of rank less than B(τ). Hence we obtain the following Corollary.

Corollary 2.18. Let the notation be as in Theorem 2.1. Suppose that the ω_{χ}- Bessel model for τ is of rank B(τ). Then the ω_{χ}^{w_{0}}-Bessel model for Ind_{P}^{G}(π) is also minimal, and of rank B(τ).

The proof of Theorem 2.1 also gives the following result.

Corollary 2.19. For any σ, τ, ℓ, χ, and ω, the support of the twisted Jacquet functor π_{U_{ℓ},χ} is a finite number of double cosets. □

§3 Holomorphicity and local coefficients.

In this section we prove the holomorphicity of the Bessel functional and the existence of a local coefficient. To do so, we first adapt the argument used by Banks [Ban] to prove the holomorphicity of Whittaker functions for metaplectic covers of GL_{n}. Banks’s result is an extension of Bernstein’s Theorem, which establishes the meromorphicity under uniqueness and regularity hypotheses. We show that the desired regularity holds in the case of Bessel functionals. We then use an argument similar to Harish-Chandra’s and to Shahidi’s in the generic case to establish the existence of the local coefficient under certain conditions (Theorem 3.8). Corollary 3.9 shows that the local coefficient factors in a manner analogous to the generic case.

Let G be as in Section 1. We use the conventions found in [Cas, §1,Shaa] for subsets of simple roots, Weyl groups, and arbitrary parabolic subgroups. Suppose that Δ is the collection of simple roots corresponding to our choice of Borel subgroup. Let θ ⊂ Δ be a collection of simple roots and set P = P_{θ}. Then P has Levi decomposition P = M_{θ}N_{θ}, with

M = M_{θ} ≃ GL_{n_{1}} × ⋯ × GL_{n_{k}} × G(m),

for some n_{i}, m such that r = n_{1} + ⋯ + n_{k} + m. We abbreviate this by writing M ≃ G_{1} × G(m). We also write N = N_{θ}.

Let A = A_{θ} be the split component of M. Denote by 𝔞_{ℂ}^{∗} = (𝔞_{θ})_{ℂ}^{∗} the complexified dual of the real Lie algebra of A, q_{F} the residual characteristic of F, and denote by H_{P} the Harish-Chandra homomorphism [Har,Shaa]. Suppose that σ ∈ E(G_{1}) and τ ∈ E(G(m)), and let π = σ ⊗ τ. For ν ∈ 𝔞_{ℂ}^{∗}, let I(ν, π, θ) denote the induced representation

Ind_{P}^{G}(π ⊗ q_{F}^{<ν,H_{P} ()>})

and let V (ν, π, θ) denote the space of associated functions. We also use Π_{ν} to denote the representation I(ν, π, θ).

Assume that σ is generic and that τ has an ω_{χ′}–Bessel model which is minimal and of rank ℓ_{0}. Let χ be the character of U_{ℓ} whose restriction to U_{ℓ} ∩ G(m) is χ′ and whose restriction to G_{1}(F) ∩ U_{ℓ} is a ψ–generic character χ_{1}. We will construct a non-zero functional Λ_{χ}(ν, π, θ) on X_{ν} = I(ν, π, θ) ⊗ Ṽ_{ω} so that, for a certain character δ of M_{χ},

Λ_{χ}(ν, π, θ)(Π_{ν}(mu)(f_{ν} ⊗ ṽ)) = δ(m)χ(u)^{−1}Λ_{χ}(ν, π, θ)(f_{ν} ⊗ ω̃(m^{−1})ṽ),

for all choices of f_{ν} ⊗ṽ ∈ X_{ν} and mu ∈ R_{χ}. Then we will show in Theorem 3.6 that the function ν ↦ Λ(ν, π, θ)(x_{ν}) is holomorphic, for a holomorphic section ν ↦ x_{ν}.

Let K = G(O_{F}), where O_{F} is the ring of integers in F. Then K is a good maximal compact subgroup of G [Cas]. Let K_{m} be the corresponding m-th principal congruence subgroup. Then each K_{m} is normal in K. Let Γ_{m} be a complete set of coset representatives for P ∩ K\K/K_{m}. Note that Γ_{m} is of finite cardinality. Let

Y = {f ∈ C^{∞}(K, V_{π}) | f(pk) = (π) (p)f(k), ∀p ∈ P ∩ K, k ∈ K} .

Then F ↦ F |_{K} is a K-isomorphism from V (ν, π, θ) to Y, by the Iwasawa decomposition of G. We will define a certain functional on Y, and use this realization to define an associated functional on X_{ν}. Let

Y_{m} = {f ∈ Y |f(kk_{1}) = f(k), ∀k ∈ K, k_{1} ∈ K_{m}} .

Thus, Y_{m} is the set of K_{m}–fixed vectors of Y under the action of K. Furthermore, the Iwasawa decomposition allows us to realize I(ν, π, θ) on Y for each ν. Denote by V_{π,m} the subspace of V_{π} consisting of P ∩ K_{m}–fixed vectors. Since π is admissible, V_{π,m} is finite dimensional.

The next three results are standard. We include the proof of the first two for completeness. The third is a straightforward consequence of the Iwasawa decomposition.

Lemma 3.1. Y_{m} has a basis {f_{j}} which satisfies the following properties: (1) If γ ∈ Γ_{m}, then the non-zero vectors among {f_{j}(γ)} are a basis for V_{π,m}.

(2) If f_{j} is fixed, then f_{j}(γ) ≠ 0 for some γ ∈ Γ_{m}.

Proof. Suppose that f ∈ Y_{m}. Then f(pkk_{1}) = π(p)f(k), for all p ∈ P ∩ K, k ∈ K, and k_{1} ∈ K_{m}. Thus, f is completely determined by its values on Γ_{m}. Fix γ ∈ Γ_{m}, and let p ∈ P ∩ K_{m}. Since γ^{−1}K_{m}γ = K_{m}, we have γ^{−1}pγ ∈ K_{m}. Therefore,

f(γ) = f(γγ^{−1}pγ) = f(pγ) = π(p)f(γ).

This says that f(γ) is an element of V_{π,m}. Fix a basis {v_{m,i}} of V_{π,m}. Let f_{γ,i} : K ⟶ V_{π,m} be given by

f_{γ,i}(k) = \{ \begin{matrix}π(p)v_{m,i} & if k = pγk_{1}, for some p ∈ P ∩ K, k_{1} ∈ K_{m}, \\ 0 & otherwise. \end{matrix}

Then it is immediate that f_{γ,i} is a well-defined element of Y_{m}. We claim that {f_{γ,i}} is a basis for Y_{m}.

Suppose f ∈ Y_{m}. If γ′ ∈ Γ_{m}, p ∈ P ∩ K, and k ∈ K_{m}, then

f(pγ′k) = π(p)f(γ′).

Since f(γ′) ∈ V_{π,m},

f(γ′) = ∑_{i} c_{γ′,i}v_{m,i}.

This implies that

f(pγ′k) = ∑_{i} c_{γ′,i}π(p)v_{m,i} = ∑_{i} c_{γ′,i}f_{γ′,i}(pγ′k).

Now, taking the collection {c_{γ,i}} for all γ ∈ Γ_{m}, and noting that f_{γ,i}(pγ′k) = 0 for γ ≠ γ′,

f(pγ′k) = ∑_{γ,i} c_{γ,i}f_{γ,i}(pγ′k),

which says that {f_{γ,i}} spans Y_{m}.

On the other hand, suppose that ∑_{γ,i} c_{γ,i}f_{γ,i} = 0. Then, for any γ′ ∈ Γ_{m}, we have

∑_{γ,i} c_{γ,i}f_{γ,i}(γ′) = 0,

which implies that

∑_{i} c_{γ′,i}f_{γ′,i}(γ′) = ∑_{i} c_{γ′,i}v_{m,i} = 0.

But, since the v_{m,i} are linearly independent, c_{γ′,i} = 0, for each γ′ and i. Thus, {f_{γ,i}} are also linearly independent. The collection f_{γ,i} clearly has properties (1) and (2). □

Denote by X the space Y ⊗ Ṽ_{ω}. For each ν ∈ 𝔞_{ℂ}^{∗} let X_{ν} = V (ν, π, θ) ⊗ Ṽ_{ω}. For f ∈ Y denote by f_{ν} the unique element of V (ν, π, θ) satisfying f_{ν}|_{K} = f. Then {f_{ν} ⊗ṽ|f ∈ Y, ṽ ∈ Ṽ_{ω}} spans X_{ν}. Recall that Π_{ν} can be realized on Y via Π_{ν}(g)f = [Π_{ν}(g)f_{ν}] |_{K}. This gives the context in which we discuss the holomorphicity of the map ν ↦ Π_{ν}(g)f_{ν} for a fixed choice of g and f.

Lemma 3.2. Fix g ∈ G, f ∈ Y and ṽ ∈ Ṽ_{ω}. Then the function ν ↦ Π_{ν}(g)f ⊗ ṽ is a regular function from 𝔞_{ℂ}^{∗} to X.

Proof. Choose m_{0} so that f ∈ Y_{m_{0}}, and choose m > m_{0} satisfying g^{−1}K_{m}g ⊂ K_{m_{0}}. Then f ∈ Y_{m} and, for all ν ∈ 𝔞_{ℂ}^{∗} and k ∈ K_{m},

Π_{ν}(k)(Π_{ν}(g)f_{ν})(x) = f_{ν}(xgg^{−1}kg) = f_{ν}(xg) = Π_{ν}(g)f_{ν}(x),

which says that Π_{ν}(g)f ∈ Y_{m} for all ν. Now, by Lemma 3.1,

Π_{ν}(g)f = ∑_{γ,i} c_{γ,i}(ν)f_{γ,i}

for a unique choice of c_{γ,i}(ν) ∈ ℂ. It suffices to show that c_{γ,i} : 𝔞_{ℂ}^{∗} ⟶ ℂ is holomorphic. Fix γ′ ∈ Γ_{m}. Then γ′g = pγ′′k, for some p ∈ P, γ′′ ∈ Γ_{m}, and k ∈ K_{m}. Then

Π_{ν}(g)f(γ′) = q_{F}^{<ν,H_{P} (p)>}δ_{P}^{1/2}(p)π(p)f(γ′′k)

= q_{F}^{<ν,H_{P} (p)>}δ_{P}^{1/2}(p)π(p) ∑_{γ,i} c_{γ,i}(ν)f_{γ,i}(γ′′)

= q_{F}^{<ν,H_{P} (p)>}δ_{P}^{1/2}(p)π(p) ∑_{i} c_{γ′′,i}(ν)v_{i}.

Set c′_{γ′′,i}(ν) = q_{F}^{<−ν,H_{p}(p)>}δ_{P}^{−1/2}(p)c_{γ′′,i}(ν). Then

π(p)f(γ′′) = ∑_{i} c_{γ′′,i}(ν)v_{i},

for all ν. Since the left hand side in the equation above is independent of ν and the v_{i} are linearly independent, c′_{γ′′,i}(ν) is constant for each i. This implies that c_{γ′′,i}(ν) is holomorphic. □

From now on we need to distinguish between a Weyl group element w̃ ∈ W (G, A), for some torus A, and a representative w ∈ N_{G}(A) for w̃. Let w̃_{θ} = w̃_{l,Δ}w̃_{l,θ}, where w̃_{l,Δ} is the longest element of the Weyl group W (G, T), and w̃_{l,θ} is the longest element of W (G, A_{θ}). Fix a representative w_{θ} for w̃_{θ} with w_{θ} ∈ K. Note that w̃_{θ}(θ) ⊂ Δ. Now let M′ = M_{w̃_{θ}(θ)} = w_{θ}M_{θ}w_{θ}^{−1}. Then M′ is a standard Levi subgroup of G. Let N′ be the standard unipotent subgroup of U so that P′ = M′N′ is a standard parabolic subgroup of G. Since M′ ≃ M, we have U_{ℓ} ⊃ N′.

Lemma 3.3. For each m > 0, we have w_{θ}^{−1}N′ ∩ P w_{θ}^{−1}K_{m} is compact.

For m ∈ M_{χ}, let δ(m) = (δ_{P}^{−1/2}δ_{P′})(m). Let X_{R_{χ},ω,ν,θ} be the subspace spanned by functions of the form

Π_{ν}(mu)f ⊗ ṽ − δ(m)χ(u)f ⊗ ω̃(m^{−1})ṽ,

for m ∈ M_{χ}, u ∈ U_{ℓ}, f ∈ Y, and ṽ ∈ Vω̃. Then a non-zero functional Λ on X is a (δω_{χ})-Bessel functional for Π_{ν} if and only if Λ|_{X_{Rχ,ω,ν,θ}} ≡ 0. By Theorem 2.1 the space of such functionals is one-dimensional. Thus X/X_{R_{χ},ω,ν,θ} is one-dimensional.

The construction of this functional will be obtained by taking a direct limit of functionals given by integrating over compact subsets of N′. We show that such a limit exists and is not identically zero. Moreover, we show that there is a function in X which is a complement to X_{R_{χ},ω,ν,θ} for all ν. This will give the regularity condition necessary to apply Bernstein’s Theorem and to obtain the holomorphicity of the functional.

Now let us fix a Whittaker functional for σ and a Bessel functional for ω. (Actually, for notational convenience, we twist ω by δ_{Rχ}^{−1/2}.) That is, suppose that λ_{χ} : V_{π} ⊗ Ṽ_{ω} ⟶ ℂ satisfies

λ_{χ}((σ(u_{1}) ⊗ τ(mu_{2}))(v_{1} ⊗ v_{2}) ⊗ ṽ) = χ_{1}(u_{1})χ′(u_{2})λ_{χ}(v_{1} ⊗ v_{2} ⊗ ω̃(m^{−1})ṽ)

for all u_{1} ∈ U_{ℓ} ∩ G_{1}, u_{2} ∈ U_{ℓ} ∩ G(m), and m ∈ M_{χ}. Let Ω be a compact subgroup of N′. Define a functional on X by

(3.1)  λ_{π,ν,θ}^{Ω}(f ⊗ ṽ) = ∫_{Ω} λ_{χ}(Π_{ν}(w_{θ}^{−1}u)f_{ν}(e) ⊗ ṽ)χ(u)^{−1} du.

This functional depends on the choice of the representative w_{θ} for w̃_{θ}.

Since N′ is exhausted by compact subgroups, the compact subgroups of N′ form a directed set. The following Lemma was suggested to the authors by Prof. Steve Rallis.

Lemma 3.4. For every f ⊗ ṽ ∈ X, the limit lim_{Ω} λ_{π,ν,θ}^{Ω}(f ⊗ ṽ) exists, where the

limit is the direct limit taken over all compact subgroups of N′.

Proof. Fix f ⊗ ṽ ∈ X. Let Ω_{r} be the subset of N′ of elements with all entries of absolute value at most q_{F}^{r}. It suffices to show that there is an r sufficiently large such that if Ω ⊃ Ω_{r} then λ_{π,ν,θ}^{Ω}(f ⊗ ṽ) = λ_{π,ν,θ}^{Ω_{r}}(f ⊗ ṽ). This follows since, for r sufficiently large, if γ ∈ N′ \ Ω_{r} then

∫_{Ω_{r}γ} λ_{χ}(Π_{ν}(w_{θ}^{−1}u)f_{ν}(e) ⊗ ṽ)χ(u)^{−1} du = 0.

Indeed, there is a subgroup K ⊂ Ω_{r} such that f_{ν}(w_{θ}^{−1}ku) = f_{ν}(w_{θ}^{−1}u) for k ∈ K, u ∈ Ω_{r}γ, but such that χ is not identically 1 on K. For one may choose K such that the relevant minors of w_{θ}^{−1}u and w_{θ}^{−1}ku are highly congruent, and use the Iwasawa decomposition. □

Define a functional on X by

(3.2)  Λ_{χ}(ν, π, θ)(f ⊗ ṽ) = lim_{Ω} λ_{π,ν,θ}^{Ω}(f ⊗ ṽ).

Again, this functional depends on the choice of w_{θ}.

Proposition 3.5. Let Λ_{χ}(ν, π, θ) be defined as in (3.2), and extend Λ_{χ} to X_{ν} by the section f ⊗ ṽ ↦ f_{ν} ⊗ ṽ. Then Λ_{χ}(ν, π, θ) defines a non-zero δω_{χ}–Bessel functional for Π_{ν}. □

Proof. Suppose that u_{1} ∈ U_{ℓ}. Since U_{ℓ} ⊂ P′, we can write u_{1} = m_{1}n_{1}, with m_{1} ∈ M′ ∩U_{ℓ}, and n_{1} ∈ N′. Suppose first that u_{1} = n_{1} ∈ N′. Since N′ is exhausted by compact subgroups, we can choose Ω_{0} compact with n_{1} ∈ Ω_{0}. If Ω_{0} ⊂ Ω, then

λ_{π,ν,θ}^{Ω}(Π_{ν}(n_{1})f ⊗ ṽ) = ∫_{Ω} λ_{χ}(f_{ν}(w_{θ}^{−1}un_{1}) ⊗ ṽ)χ^{−1}(u) du

= ∫_{Ω} λ_{χ}(f_{ν}(w_{θ}^{−1}u) ⊗ ṽ)χ^{−1}(un_{1}^{−1}) du

= χ(n_{1})λ_{π,ν,θ}^{Ω}(f ⊗ ṽ).

Therefore,

Λ_{χ}(ν, π, θ)(Π_{ν}(n_{1})f ⊗ ṽ) = χ(n_{1})Λ_{χ}(ν, π, θ)(f ⊗ ṽ).

If u = m_{1} ∈ U_{ℓ} ∩ M′, then since χ|_{G_{1}∩U_{ℓ}} is ψ–generic, χ^{w_{θ}}(m_{1}) = χ(m_{1}). Thus,

λ_{π,ν,θ}^{Ω}(Π_{ν}(m_{1})f ⊗ ṽ) = ∫_{Ω} λ_{χ}(f_{ν}(w_{θ}^{−1}um_{1}) ⊗ ṽ)χ^{−1}(u) du

= ∫_{Ω} λ_{χ}(f_{ν}(w_{θ}^{−1}m_{1}w_{θ}w_{θ}^{−1}m_{1}^{−1}um_{1}) ⊗ ṽ)χ^{−1}(u) du

= ∫_{Ω} λ_{χ}(π(w_{θ}^{−1}m_{1}w_{θ})f_{ν}(w_{θ}^{−1}m_{1}^{−1}um_{1}) ⊗ ṽ)χ^{−1}(u) du

= χ(m_{1}) ∫_{m_{1}^{−1}Ωm_{1}} λ_{χ}(f_{ν}(w_{θ}^{−1}u) ⊗ ṽ)χ^{−1}(u) du

= χ(m_{1})λ_{π,ν,θ}^{m_{1}^{−1}Ωm_{1}}(f ⊗ ṽ).

Therefore,

Λ_{χ}(ν, π, θ)(Π_{ν}(m_{1})f ⊗ ṽ) = χ(m_{1})Λ_{χ}(ν, π, θ)(f ⊗ ṽ).

Similarly, if m ∈ M_{χ} ⊂ M′, then

λ_{π,ν,θ}^{Ω}(Π_{ν}(m)f ⊗ ṽ) = ∫_{Ω} λ_{χ}(f_{ν}(w_{θ}^{−1}um) ⊗ ṽ)χ^{−1}(u) du

= ∫_{Ω} λ_{χ}(π(w_{θ}^{−1}mw_{θ})δ_{P}^{1/2}(w_{θ}^{−1}mw_{θ})f_{ν}(w_{θ}^{−1}m^{−1}um) ⊗ ṽ)χ^{−1}(u) du

= δ_{P}^{1/2}(w_{θ}^{−1}mw_{θ}) ∫_{m^{−1}Ωm} λ_{χ}(f_{ν}(w_{θ}^{−1}m^{−1}um) ⊗ ω̃(m^{−1})ṽ)χ^{−1}(u) du

= δ_{P}^{−1/2}δ_{P′}(m) ∫_{m^{−1}Ωm} λ_{χ}(f_{ν}(w_{θ}^{−1}u) ⊗ ω̃(m^{−1})ṽ)χ^{−1}(mum^{−1}) du

= δ(m)λ_{π,ν,θ}^{m^{−1}Ωm}(f ⊗ ω̃(m^{−1})ṽ).

Taking the limit on Ω on the right and left sides of the above equation completes the proof that Λ_{χ}(ν, π, θ) is a Bessel functional for Π_{ν} with respect to the representation δ(m)ω_{χ}.

It remains to show that Λ_{χ}(ν, π, θ) is not identically zero. Let P̄′ be the parabolic opposite to P′. Then P̄′ = w_{θ}Pw_{θ}^{−1}. By Lemma 3.4, P̄′K_{m} is compact, and if pw_{θ}^{−1}k ∈ P w_{θ}^{−1}K_{m} ∩ N′, then in fact p ∈ P ∩ K_{m}. Choose a v ∈ V_{π} and ṽ ∈ Ṽ_{ω} such that λ_{χ}(v ⊗ ṽ) ≠ 0. Choose m >> 0 such that v ∈ V_{π,m} and such that χ|N′ ∩ P̄′K_{m} ≡ 1. Consider the function in Y defined by

(3.3)  f_{0}(k) = \{ \begin{matrix}π(p)v & if k = pw_{θ}^{−1}k_{1}, p ∈ P ∩ K, k_{1} ∈ K_{m} \\ 0 & otherwise. \end{matrix}

Then

Λ_{χ}(ν, π, θ)(f_{0} ⊗ ṽ) = ∫_{N′∩P̄′K_{m}} λ_{χ}(f_{ν}(w_{θ}^{−1}u) ⊗ ṽ) du

= λ_{χ}(v ⊗ ṽ) |N′ ∩ P̄′K_{m}| ≠ 0.

Thus, Λ_{χ}(ν, π, θ) is non-zero, and f_{0} is a complement to X_{R_{χ},ω,ν,θ} for all ν. □

Suppose r = mu ∈ R_{χ}, f ∈ Y, and ṽ ∈ Ṽ_{ω}. Define an X–valued function on 𝔞_{ℂ}^{∗} by

x_{r,f,ṽ,θ}(ν) = Π_{ν}(r)(f) ⊗ ṽ − δ(m)χ(u)(f ⊗ ω̃(m^{−1})ṽ).

Theorem 3.6. The function ν ↦ Λ_{χ}(ν, π, θ)(x) is holomorphic for each x ∈ X.

Proof. We will apply Banks’s extension of Bernstein’s Theorem. Let

R = {(r, f ⊗ ṽ)|r ∈ R_{χ}, f ∈ Y, ṽ ∈ Vω̃} ∪ {∗}.

For α = (r, f ⊗ ṽ) ∈ R, we let x_{α}(ν) = x_{r,f,ṽ,θ}(ν) in X and let c_{α}(ν) = 0. Fix m, ṽ, v, and f_{0} as in (3.3). For α = ∗, we set x_{∗}(ν) = f_{0} ⊗ ṽ and c_{∗}(ν) = |N′ ∩ P̄′K_{m} | λ_{χ}(v ⊗ ṽ). Now for every ν ∈ 𝔞_{ℂ}^{∗}, we consider the systems of equations in X × ℂ given by:

Ξ(ν) = {(x_{α}(ν), c_{α}(ν))| α ∈ R} .

By Lemma 3.2, the function ν ↦ x_{α}(ν) is holomorphic for each α of the form (r, f ⊗ ṽ). For α = ∗, the function x_{α}(ν) = f_{0} ⊗ ṽ is constant on 𝔞_{ℂ}^{∗}. Note that each c_{α} is constant, hence holomorphic as well.

Now, for each ν the functional Λ_{χ}(ν, π, θ) is a solution to the system Ξ(ν). Moreover, such a solution is unique by the results of Section 2. Thus, Banks’s extension of Bernstein’s theorem [Ban] implies that ν ↦ Λ(ν, π, θ)(f ⊗ ṽ) is holomorphic for all choices of f and ṽ. □

We turn to the question of local coefficients. Let w̃ ∈ W, and fix a representative w for w̃ with w ∈ K. We recall that the intertwining operator

A(ν, π, w) : V (ν, π, θ) → V (w̃(ν), w̃π, w̃(θ))

is defined for ν >> 0 by

(3.4)  A(ν, π, w)f(g) = ∫_{Nw̃} f(w^{−1}ng) dn,

where Nw̃ = U ∩ wN̄w^{−1}, and N̄ is the unipotent radical opposite to N. Then A(ν, π, w) is defined on all of 𝔞_{ℂ}^{∗} by analytic continuation. Note that the intertwining operator depends on the choice of w representing w̃.

We also recall the Langlands decomposition of the intertwining operator, described in Lemma 2.1.2 of [Shaa]. For the convenience of the reader, let us restate this here. For two associate subsets θ and θ′ of Δ, we let

W (θ, θ′) = {w̃ ∈ W | w̃θ = θ′} .

Lemma 3.7 (Langlands (see [Shaa, Lemma 2.1.2])). Suppose that θ, θ′ ⊂ Δ are associate. Let w̃ ∈ W (θ, θ′). Then there exists a family θ_{1}, θ_{2}, …, θ_{n} ⊂ Δ so that

(1) θ_{1} = θ and θ_{n} = θ′;

(2) For each 1 ≤ i ≤ n there is a root α_{i} ∈ Δ \ θ_{i} so that θ_{i+1} is the conjugate of θ_{i} in Δ_{i} = θ_{i} ∪ {α_{i}};

(3) For each 1 ≤ i ≤ n − 1, we let w̃_{i} = w̃_{ℓ,Δ_{i}} w̃_{ℓ,θ_{i}} in W (θ_{i}, θ_{i+1}). Then w̃ = w̃_{n−1} … w̃_{1};

(4) Set w̃′_{1} = w̃, and w̃′_{i+1} = w̃′_{i}w̃_{i}^{−1} for 1 ≤ i ≤ n − 1. Then w̃′_{n} = 1 and

𝔫_{}_{w̃′_{i}} = 𝔫w̃_{i} ⊕ Ad(w_{i}^{−1})𝔫_{}_{w̃′_{i+1}}.

Here 𝔫 is the Lie algebra of N.

Let θ_{∗} ⊂ θ and let ρ be an irreducible supercuspidal representation of M_{θ_{∗}}. If ρ is generic, then Rodier’s Theorem implies that there is a unique constituent π of Ind_{P_{θ∗}}^{M} (ρ) which is generic with compatible character. For this constituent, Shahidi proved that there is a complex number C_{χ}(ν, π, θ, w) which satisfies

Λ_{χ}(ν, π, θ) = C_{χ}(ν, π, θ, w)Λ_{χ}(w̃ν, w̃π, w̃θ)A(ν, π, w),

where Λ_{χ} is the Whittaker functional. Moreover, the function ν ↦ C_{χ}(ν, π, θ, w) is a meromorphic function on (𝔞_{θ})_{ℂ}^{∗}. The value of the local coefficient depends on the choice of representative w for w̃.

Now suppose that ρ is any irreducible supercuspidal which has a minimal Bessel model of a particular type. Then we prove a similar result for the constituent π of Ind_{P_{θ∗}}^{M} (ρ) which has a Bessel model of compatible type; such a constituent exists and is unique by Theorem 2.1.

Theorem 3.8. Let θ and θ′ be associate subsets of Δ. Let θ_{∗} ⊂ θ and let ρ be an irreducible supercuspidal representation of M_{θ_{∗}}. Suppose that ρ has an ω_{χ}–Bessel model which is minimal. Let π be the constituent of Ind_{P_{θ∗}}^{M} (ρ) such that π has an ω_{χ}^{w_{0}}-Bessel model, as in Theorem 2.1. For each w̃ ∈ W (θ, θ′) fix a representative w for w̃. Then there is a complex number C_{χ}(ν, π, θ, w) so that

(3.5)  Λ_{χ}(ν, π, θ) = C_{χ}(ν, π, θ, w)Λ_{χ}(w̃ν, w̃π, w̃θ)A(ν, π, w).

Moreover, the function ν ↦ C_{χ}(ν, π, θ, w) is meromorphic on 𝔞_{ℂ}^{∗}, and depends only on the class of π and the choice of w.

Proof. We first show how to define C_{χ}(ν̃, π, θ_{∗}, w) for ν̃ ∈ (𝔞_{θ_{∗}})_{ℂ}^{∗}. By [Sil, Theorem 5.4.3.7] the representation I(ν̃, ρ, θ_{∗}) is irreducible unless the Plancherel measure μ(ν̃, ρ) = 0 and (ν̃, ρ) is fixed by a nontrivial element of the Weyl group W_{θ_{∗}} (i.e., is singular). Thus, on an open dense subset of (𝔞_{θ_{∗}})_{ℂ}^{∗} the representation I(ν̃, ρ, θ_{∗}) is irreducible, and so Λ_{χ}(w̃ν̃, w̃ρ, w̃θ_{∗})A(ν̃, ρ, w) defines a non-zero Bessel functional on V (ν̃, ρ, θ_{∗}) ⊗ Ṽ_{ω}. By the uniqueness of such a functional (Theorem/Conjecture 1.4), we get the existence of C(ν̃, ρ, θ_{∗}, w) satisfying

Λ_{χ}(ν̃, ρ, θ_{∗}) = C_{χ}(ν̃, ρ, θ_{∗}, w)Λ_{χ}(w̃ν̃, w̃ρ, w̃θ_{∗})A(ν̃, ρ, w)

on the open dense subset. Moreover, it is holomorphic there since both

Λ_{χ}(w̃ν̃, w̃ρ, w̃θ_{∗})

and A(ν̃, ρ, w) are holomorphic there. Thus, C(ν̃, ρ, θ_{∗}, w) extends to a meromorphic function on (𝔞_{θ_{∗}})_{ℂ}^{∗}. Now, write w̃ = w̃_{n−1} … w̃_{1} as in Lemma 3.7. Since C_{χ}(ν̃, ρ, θ_{∗}, w) is now defined, it admits a factorization compatible with the decomposition of the intertwining operators given in Lemma 3.7. (See corollary 3.9.) This implies that on an open dense subset of ν ∈ (𝔞_{θ})_{ℂ}^{∗}, the local coefficient C_{χ}(ν, ρ, θ_{∗}, w) may be defined by the equation C_{χ}(ν, ρ, θ_{∗}, w) = C_{χ}(ν̃, ρ, θ_{∗}, w), where ν̃ is the restriction of ν to (𝔞_{θ_{∗}})_{ℂ}. Suppose that, for some ν in this open dense subset, Λ_{χ}(w̃ν, w̃π, w̃θ)A(ν, π, w) was the zero functional. Then, by inducing in stages and using the discussion preceding this Theorem, we would conclude that Λ_{χ}(w̃ν̃, w̃ρ, w̃θ_{∗})A(ν̃, ρ, w) is also zero. However, since C_{χ}(ν̃, ρ, θ_{∗}, w) is defined there, this would be a contradiction. Thus we may define C(ν, π, θ, w) by the relation (3.5) on this open dense subset, and we have C(ν, π, θ, w) = C(ν̃, ρ, θ_{∗}, w). Since A(ν, π, w) has a meromorphic continuation to (𝔞_{θ})_{ℂ}^{∗}, and Λ_{χ}(w̃ν, w̃π, w̃θ) is holomorphic on (𝔞_{θ})_{ℂ}^{∗}, the function ν ↦ C_{χ}(ν, π, θ, w) must have a meromorphic continuation. □

Corollary 3.9. Let the notation be as in Lemma 3.7 and Theorem 3.8. Let π_{1} = π, and ν_{1} = ν. For each i, 2 ≤ i ≤ n − 1, set π_{i} = w̃_{i}π_{i−1}, ν_{i} = w̃ν_{i−1}. Then the local coefficient factors as

C_{χ}(ν, π, θ, w) = ∏_{i=1}^{n−1} C_{χ}(π_{i}, θ_{i}, w_{i}).

Proof. Let f_{1} = f ∈ V (ν, π, θ) and for 2 ≤ i ≤ n−1, let f_{i} = A(ν_{i−1}, π_{i−1}, w_{i−1})f_{i−1}. Then

Λ_{χ}(ν_{i}, π_{i}, θ_{i})f_{i} =C_{χ}(ν_{i}, π_{i}, θ_{i}, w_{i})Λ_{χ}(ν_{i+1}, π_{i+1}, θ_{i+1})
· A(ν_{i}, π_{i}, w_{i})f_{i},

for each 1 ≤ i ≤ n − 1. The corollary now follows immediately from Lemma 3.7 and iteration of the above equality. □

References

[Ban] W. BANKS, Exceptional representations on the metaplectic group, preprint.

[BeZa] I.N. BERNSTEIN and A.V. ZELEVINSKY, Representations of the group GL(n, F) where F is a local non-archimedean field, Russian Math. Surveys 33 (1976), 1–68.

[BeZb] , Induced representations of reductive p–adic groups. I, Ann. Sci. École Norm. Sup. (4) 10 (1977), 441–472.

[Ber] J. BERNSTEIN, Letter to I.I. Piatetski-Shapiro, Fall 1985, to appear in a book by J. Cogdell and I.I. Piatetski-Shapiro.

[Bru] F. BRUHAT, Sur les représentations induites des groupes de Lie, Bull. Soc. Math. France 84 (1956), 97–205.

[Cas] W. CASSELMAN, Introduction to the theory of admissible representations of p–adic reductive groups, preprint.

[GPR] D. GINZBURG, I. PIATETSKII-SHAPIRO and S. RALLIS, L–functions for the orthogonal group, preprint.

[Gola] D. GOLDBERG, Reducibility of induced representations for Sp(2n) and SO(n), Amer. J. Math. 116 (1994), 1101–1151.

[Golb] , R–groups and elliptic representations for unitary groups, Proc. Amer. Math. Soc. 123 (1995), 1267–1276.

[Har] HARISH-CHANDRA, Harmonic analysis on reductive p–adic groups, Proc. Sympos. Pure Math. 26 (1973), 167–192.

[HeR] E. HEWITT and K. ROSS, Abstract Harmonic Analysis I, Springer-Verlag, New York– Heidelberg–Berlin, 1979.

[Lana] R.P. LANGLANDS, Euler Products, 1971, Yale University.

[Lanb] , On the functional equation satisfied by Eisenstein series, Lecture Notes in Mathematics , no. 544, Springer-Verlag, New York–Heidelberg–Berlin, 1976.

[Nov] M. NOVODVORSKY, New unique models of representations of unitary groups, Compositio Math. 33 (1976), 289–295.

[Ral] S. RALLIS, On certain Gelfand Graev models which are Gelfand pairs, preprint.

[Roda] F. RODIER, Modèle de Whittaker et caractères de représentations, in Non Commutative Harmonic Analysis, Lecture Notes in Math., no. 466, Springer-Verlag, New York–Heidelberg– Berlin, 151–171.

[Rodb] , Whittaker models for admissible representations of reductive p–adic split groups, Proc. Sympos. Pure Math. 26 (1973), 425–430.

[Shaa] F. SHAHIDI, On certain L–functions, Amer. J. Math. 103 (1981), 297–355.

[Shab] , Local coefficients as Artin factors for real groups, Duke Math. J. 52 (1985), 973–1007.

[Shac] , On the Ramanujan conjecture and finiteness of poles for certain L–functions, Ann. of Math. (2) 127 (1988), 547–584.

[Shad] , A proof of Langlands conjecture for Plancherel measures; complementary series for p–adic groups, Ann. of Math. (2) 132 (1990), 273–330.

[Sil] A.J. SILBERGER, Introduction to Harmonic Analysis on Reductive p–adic Groups, Mathematical Notes, no. 23, Princeton University Press, Princeton, NJ, 1979.

Solomon Friedberg, Department of Mathematics, University of California Santa Cruz, Santa Cruz, CA 95064

E-mail address: friedbe@cats.ucsc.edu

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

E-mail address: goldberg@math.purdue.edu