EXPLICIT FORMULAS FOR THE

WALDSPURGER AND BESSEL MODELS

Daniel Bump, Solomon Friedberg, and Masaaki Furusawa

Stanford University University of California, Santa Cruz Mathematical Sciences Research Institute

In this paper we will study certain models of irreducible admissible representations of the split special orthogonal group SO(2n + 1) over a nonarchimedean local field. If n = 1, these models were considered by Waldspurger [Wa1,Wa2], and arose in his profound studies of the Shimura correspondence. If n = 2, they were considered by Novodvorsky and Piatetski-Shapiro [NP], who called them Bessel models, and for general n they were studied by Novodvorsky [No]. In the works cited, these authors established the uniqueness of these models; in this paper we establish functional equations and explicit formulas for them. In general, these models arise from a variety of Rankin-Selberg integrals (for example, those of Andrianov [An], Furusawa [Fu1], and Sugano [Su]), and the results of this paper will naturally have applications to the study of Lfunctions. Moreover, these models arise in the study of the theta correspondence between SO(2n + 1) and the double cover of Sp(2n), and they will therefore be of importance in generalizing the work of Waldspurger (see Furusawa [Fu2]).

In the final Section, we present a global application of the explicit formulas: we consider the Eisenstein series (6.1) on SO(2n + 1) formed with a cuspidal automorphic representation π on GL(n), and we show that its Bessel period (6.2) is essentially a product of L-series

L(n(s − 1/2) + 1/2, π) L(n(s − 1/2) + 1/2, π ⊗ η),

where η is a quadratic character. This result generalizes work for n = 2 and base field ℚ of Mizumoto [Mi] and Böcherer [Bö], and puts it in a representation-theoretic context.

*Research supported in part by National Science Foundation grants DMS 9023441 (Bump) and DMS 9123845 (Friedberg), by the AMS Centennial Research Fellowship (Bump), and by the NSF Postdoctoral Research Fellowship DMS 9206242 (Furusawa). Research at MSRI is supported in part by NSF grant DMS 9022140.

11–14–94 Explicit Formulas for the Waldspurger and Bessel Models 2

This global application is closely related to the results of Bump, Friedberg, and Hoffstein [BFH2]. That paper computes the spherical Whittaker functions on the metaplectic double cover of Sp(2n). (Whittaker models on that group are also unique.) The computation in [BFH2] has the following consequence: if one forms the metaplectic Eisenstein series on the double cover of GSp(2n) with a cuspidal automorphic representation π of GL(n) (which is possible because the cover splits over GL(n) ⊂ Sp(2n)), the Whittaker coefficients of this Eisenstein series are quadratic twists of the standard L-function of π. The close relation between these two computations is a reflection of the following result of Furusawa [Fu2], generalizing the case n = 2 in Piatetski-Shapiro and Soudry [PS]: the (special) Bessel coefficient of a cusp form on SO(2n + 1) essentially agrees with the Whittaker coefficient of the theta lift on the double cover of Sp(2n). If instead of a cusp form one considers the Eisenstein series (6.1), the theta correspondent on the metaplectic group is the metaplectic Eisenstein series, and our calculation implies that this result of Furusawa for cusp forms is true for these Eisenstein series also. (Our calculation of the Bessel period is in fact direct and independent of [Fu2].)

These results should have an application to the nonvanishing of L-functions under quadratic twists. Namely, there are Rankin-Selberg integrals on the double cover of GSp(6) ([BG]) and on SO(7) ([Gi]) unfolding to Dirichlet series involving the Whittaker (resp. Bessel) periods described above, that is, to Dirichlet series whose individual coefficients are the quadratic twists of a standard GL(3) L-series. (The two constructions give Dirichlet series whose individual coefficients are Euler products which agree at almost all places.) Arguing as in [BFH1], one should be able to show that an infinite number of these quadratic twists are nonzero. In fact, these integrals are the next members of a series beginning with Siegel’s calculation of the Mellin transform of a metaplectic GL(2) Eisenstein series and including integrals of Hecke type on the double cover of GSp(4) (due in a nonmetaplectic context to Novodvorsky; see [BFH1]) and on SO(5) (due to Maass). The elucidation of this scenario owes much to discussions with Duke, Ginzburg, Goldfeld, and Hoffstein. In particular, the verification that our results could be applied to the evaluation of (6.2) was first worked out in conversation with Ginzburg.

1. Notations and Statement of Results. Let F be a nonarchimedean local field of characteristic different from 2. Let O denote the ring of integers of F, ϖ denote a local uniformizer, q denote the cardinality of the residue field O/ϖO, and | |_{F} denote the absolute value on F, normalized so that |ϖ|_{F} = q^{−1}.

We begin by describing our results on the Waldspurger model. Let G_{2} = GL(2, F), and let T_{2} be a maximal torus in G_{2}. Then T_{2} is the connected component of the identity in a group of orthogonal similitudes of degree two corresponding to some quadratic form. If (π_{2}, V_{π_{2}}) is an irreducible admissible representation of G_{2}, and if σ : T_{2}(F) → ℂ is a character, then there exists at most one linear functional W : V_{π_{2}} → ℂ (up to scalar multiplication) such that

(1.1)  W (π_{2}(t) v) = σ(t) W (v)

for all t ∈ T_{2} and v ∈ V_{π_{2}}. This is proved when σ = 1 in Waldspurger [Wa1], Proposition 9′, and the proof in the general case is identical (as pointed out in [Wa2], Lemme 8). In order for such a functional to exist, since T_{2} contains the center Z_{2} of G_{2}, it is necessary that the restriction of σ to Z_{2} match the central character of π_{2}. We will call a functional satisfying (1.1) a Waldspurger functional. The Waldspurger model for π_{2} will be the space of all functions of the form g ↦ W (π_{2}(g)v) with v ∈ V_{π_{2}}.

First let us consider the case where T_{2} = T_{2}^{a} is nonsplit. We will limit ourselves to the case where T_{2}^{a} has the form

(1.2)  \{( \begin{matrix}x & y \\ ϵy & x \end{matrix} ) |x^{2} − y^{2}ϵ ≠ 0\} ,

where ϵ ∈ O^{×} is a nonsquare. In this case, let T_{2}^{a}(O) = T_{2}^{a} ∩ GL(2, O). We will further assume that σ is trivial on T_{2}^{a}(O). Since T_{2}^{a} is generated by T_{2}^{a}(O) and by the center of G_{2}, on which σ is to agree with the central character of π_{2}, it follows that σ is uniquely determined by these conditions. (We note that if T_{2}^{a} and σ are obtained by localizing global data then these conditions will be satisfied locally almost everywhere at places where the global quadratic form defining T_{2} is nonsplit.)

Suppose that π_{2} is in the unramified principal series. Then the contragredient representation of π_{2} is also spherical, and the GL(2, O)-fixed vector in the contragredient representation is clearly T_{2}^{a}(O)-invariant. Since it also has the correct transformation property with respect to the center of G_{2}, it is thus a Waldspurger functional. Thus if ϕ is the spherical vector in V_{π_{2}}, the function g ↦ W (π_{2}(g)ϕ) is the spherical function for π_{2}, which is given by the Macdonald formula (see [Ca1]).

One of our results will be a formula analogous to the Macdonald formula for the split Waldspurger functional. Thus let T_{2} = T_{2}^{s} be a split torus of G_{2}. Specifically, we may take T_{2}^{s} to be the group of diagonal matrices in G_{2}; also, let B_{2} be the Borel subgroup consisting of upper triangular matrices in G_{2}.

Let us construct a Waldspurger functional for the unramified principal series. Let ξ_{1}, ξ_{2} be unramified quasicharacters of F^{×}, and let ξ be the character of B_{2} given by

(1.3)  ξ (\binom{a b}{d}) = ξ_{1}(a)ξ_{2}(d).

Suppose π_{2} = Ind(ξ) (we also write π_{2} = Ind(ξ_{1}, ξ_{2})) is the representation obtained by normalized induction from the character ξ. Thus V_{π_{2}} consists of the complex-valued locally constant functions f on G_{2} such that

(1.4)  f(bg) = ξ(b) δ_{B_{2}}^{1/2}(b) f(g)

for all b ∈ B_{2}, g ∈ G_{2}, where δ_{B_{2}} is the modular character of B_{2}, and π_{2} is the right regular representation.

Let σ be an unramified quasicharacter of F^{×}. Extend σ to T_{2}^{s} (we use the same letter) by the formula σ (( \begin{matrix}ab \\ & b \end{matrix} )) = σ(a) ξ_{1}ξ_{2}(b). Then a Waldspurger functional on V_{π_{2}} is defined as follows. Suppose f ∈ V_{π_{2}}. Let

(1.5)  \begin{aligned}W (f) & = ∫_{Z_{2}\T_{2}^{s}} f (( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ) t) σ^{−1}(t) d^{×}t \\ & = ∫_{F^{×}} f (( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ) ( \begin{matrix}a & 0 \\ 0 & 1 \end{matrix} )) σ^{−1}(a) d^{×}a. \end{aligned}

The Haar measure on F^{×} is normalized so that the measure of O^{×} is 1. Suppose that ξ_{i}(ϖ) = γ_{i} for i = 1, 2, and that σ(ϖ) = τ. As we shall show in Section 2 below, the integral (1.5) is absolutely convergent in the region

(1.6)  |γ_{1}τ^{−1}| < q^{1/2},  |γ_{2}^{−1}τ| < q^{1/2}.

For these representations, (1.1) holds.

Let ϕ_{ξ} be the K_{2} = GL(2, O)-fixed vector in V_{π_{2}} such that ϕ_{ξ}(I_{2}) = 1. Define a function Wa_{ξ}: G_{2} → ℂ by the formula

Wa_{ξ}(g) = W (π_{2}(g)ϕ_{ξ}).

This function is analogous to the Whittaker function obtained from the standard Whittaker functional.

Our main result on the Waldspurger functional gives the analytic continuation of the function Wa = Wa_{ξ} to the full space of unramified quasicharacters ξ_{1}, ξ_{2}, and σ, and an explicit formula for its value. To describe this, note that

(1.7)  Wa_{ξ}(t_{2}gκ_{2}) = σ(t_{2}) Wa(g)

for all t_{2} ∈ T_{2}^{s}, κ_{2} ∈ K_{2}. Hence it suffices to determine Wa_{ξ} on a set of coset representatives for the double cosets T_{2}^{s}\G_{2}/K_{2}. Using the Iwasawa decomposition, it follows that a set of coset representatives is given by the matrices

η_{k} = ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ) ( \begin{matrix}ϖ^{k} & 0 \\ 0 & 1 \end{matrix} ) ,

with k ≥ 0. In Section 2 we shall prove

Theorem 1.1. Suppose that (1.6) holds. Let

𝒲a_{ξ}(g) = \frac{(1 − γ_{1}τ^{−1}q^{−1/2})(1 − γ_{2}^{−1}τq^{−1/2})}{1 − γ_{1}γ_{2}^{−1}q^{−1}}Wa_{ξ}(g).

Then 𝒲a_{ξ} is given by the formula

(1.8) 𝒲a_{ξ}(η_{k}) = (1 − q^{−1})^{−1}q^{−k/2}×

[γ_{1}^{k} \frac{(1 − γ_{2}τ^{−1}q^{−1/2})(1 − γ_{1}^{−1}τq^{−1/2})}{1 − γ_{1}^{−1}γ_{2}} + γ_{2}^{k} \frac{(1 − γ_{1}τ^{−1}q^{−1/2})(1 − γ_{2}^{−1}τq^{−1/2})}{1 − γ_{1}γ_{2}^{−1}}] .

In particular, the function 𝒲a_{ξ}, originally defined as an integral when the inequalities (1.6) hold, has holomorphic continuation to all γ_{1}, γ_{2}, τ ∈ ℂ^{×}, and is invariant under the interchange of γ_{1} and γ_{2}.

Note that if γ_{1} = γ_{2}, this must be interpreted correctly: both the numerator and the denominator in (1.8) vanish, but their ratio is analytic, so the formula still makes sense.

We also address the analytic continuation of the Waldspurger functionals. Let Λ be the domain of (γ_{1}, γ_{2}, τ) ∈ (ℂ^{×})^{3} such that τγ_{1}^{−1} ≠ q^{1/2} and τγ_{2}^{−1} ≠ q^{−1/2}. We note that for fixed ξ_{1} and ξ_{2}, the space Ind(ξ) may be identified with the space C^{∞}((B_{2} ∩K_{2})\K_{2}) of locally constant functions on K_{2} which are left invariant by B_{2} ∩ K_{2}; indeed, such a function may be uniquely extended to a function on G_{2} satisfying (1.4), and every element of Ind(ξ) arises uniquely from an element of C^{∞}((B_{2} ∩K_{2})\K_{2}) in this way. We shall also prove in Section 2:

Theorem 1.2. Fix an element f ∈ C^{∞}((B_{2} ∩ K_{2})\K_{2}). For (γ_{1}, γ_{2}, τ) satisfying (1.6), f may be extended uniquely to an element of Ind(ξ), and the function W (f) defined by (1.5) may thus be regarded as an analytic function of three variables (γ_{1}, γ_{2}, τ). This function has analytic continuation to all of Λ, so the domain of definition of W : Ind(ξ) → ℂ may be extended to all ξ for which the parameters are in Λ. If (γ_{1}, γ_{2}, τ) ∈ Λ, then W defines a (possibly zero) Waldspurger functional on Ind(ξ).

The analytic continuation assertion here can actually be deduced from Theorem 1.1, since if ξ is in general position then the representation π_{2} is irreducible, and hence every element of Ind(ξ) is a linear combination of right translates of the spherical vector. Thus there is some overlap between these two theorems. In fact, however, we shall give a direct proof of Theorem 1.2, independent from Theorem 1.1.

Next let us describe our results concerning the Bessel models. For r ≥ 2 let SO(r, F) denote the split group of determinant one orthogonal matrices

SO(r, F) = {g ∈ SL(r, F) | (gx, gy) = (x, y) for all x, y ∈ F^{r}},

where ( , ) is the quadratic form

(x, y) = ∑_{i=1}^{r} x_{i} y_{r+1−i}.

Let G = SO(2n + 1, F), and let U be the subgroup of G consisting of upper triangular unipotent matrices whose center 3 × 3 block is the identity. An element of U is of the form u = (u_{ij}) with u_{ij} = 0 for i, j = n, n + 1, n + 2 and i ≠ j. Let ψ be a character of F of conductor O. Given S = (a, b, c) ∈ O such that b^{2} + 2ac ≠ 0, define a character θ_{S} of U by the formula

(1.9) θ_{S}(u) = ψ(u_{12} + u_{23} + ⋯ + u_{n−2,n−1} + au_{n−1,n} + bu_{n−1,n+1} + cu_{n−1,n+2}).

Let T be the subgroup of G consisting of the matrices of the form

( \begin{matrix}I_{n−1} \\ & g \\ & & I_{n−1} \end{matrix} )

which, acting by conjugation, stabilize θ_{S}. This constrains g to lie in a suitable torus in SO(3). Thus T is a torus in G, which may be either split or nonsplit over F, depending on S. Note that T normalizes U, and hence R := T U is again a subgroup of G. Let λ be a character of T, and extend θ_{S} to a character of R by θ_{S}(tu) = λ(t) θ_{S}(u).

Let π: G → End(V_{π}) be an admissible representation of G. Then a Bessel functional on π is a linear functional B: V_{π} → ℂ such that

(1.10)  B(π(tu)v) = θ_{S}(tu) B(v),

for all t ∈ T, u ∈ U, and v ∈ V_{π}. As shown by Novodvorsky [No], if π is irreducible then the dimension of the space of such functionals is at most 1. Note also that, in view of the isomorphism of SO(3, F) and P GL(2, F), a Bessel functional on SO(3, F) may be identified with a Waldspurger functional.

One may similarly define a Bessel functional on the larger group of orthogonal similitudes; however, since this group is the direct product of G with its center, there is no gain in generality by doing so.

Once again we shall consider this notion for the unramified principal series. Let χ_{1}, ⋯, χ_{n} be unramified quasicharacters of F^{×}. We shall consider the principal series representation of G

π = Ind(χ_{1}, ⋯, χ_{n}).

In our notation, this will be the representation on space of locally constant functions Ψ on G which satisfy

(1.11)  Ψ(bg) = δ_{B}^{1/2}(b) (∏_{i=1}^{n} χ_{i}(y_{i})) · Ψ(g)

for all

b = ( \begin{matrix}y_{1} & ∗ & ∗ & ∗ & ∗ & ∗ & ∗ \\ & ... & ∗ & ∗ & ∗ & ∗ & ∗ \\ & & y_{n} & ∗ & ∗ & ∗ & ∗ \\ & & & 1 & ∗ & ∗ & ∗ \\ & & & & y_{n}^{−1} & ∗ & ∗ \\ & & & & & ... & ∗ \\ & & & & & & y_{1}^{−1} \end{matrix} )

in the standard Borel subgroup B of G. Here

δ_{B}(b) = |∏_{i=1}^{n} y_{i}^{2n−2i+1}|_{F}

is the modular character of B_{F}. The group action is by right translation. Let us write α_{i} = χ_{i}(ϖ). If the α_{i} are in general position (α_{i} ≠ α_{j}^{±1} for i ≠ j and α_{i}^{2} ≠ 1), then the isomorphism class of this representation is invariant under permutations of the α_{i}, as well as transformations of the form α_{i} ↦ α_{i}^{±1}. Also, let us write K for the standard maximal compact subgroup SO(2n + 1, O) of G.

We may construct a Bessel functional explicitly as follows. There are two cases: T nonsplit, and T split. In the split case, we shall take λ unramified, i.e. identically one on T ∩ K.

Suppose first that T is a nonsplit torus. We assume that a ∈ O^{×}. Write T (O) for the subgroup T ∩ K. For a permutation s in the symmetric group S_{2n+1}, we shall also use s to denote the corresponding signed permutation matrix in SL(2n+1, F) (the matrix with sgn(s) in the (s(i), i) position and 0 elsewhere). Let w_{1} = (1, 2n+1)(2, 2n) ⋯ (n−1, n+3). Suppose Ψ ∈ V_{π}. Then we define

(1.12)  B(Ψ) = ∫_{T (O)} ∫_{U} Ψ(w_{1}ut) θ_{S}(u)^{−1} du dt,

and this is a Bessel functional on V_{π}. The integral is absolutely convergent if the quasicharacters χ_{i} are in a suitable region. Indeed, since T (O) is compact, comparing with the standard intertwining operator T_{w_{1}} defined in Section 3 below (see Lemma 3.1), one finds that if

(1.13)  |α_{1}| < ⋯ < |α_{n−1}| < min(|α_{n}|, |α_{n}^{−1}|),

then the integral (1.12) is absolutely convergent.

For the second case, suppose that T is instead split. We may suppose that S = (0, 1, 0). Let us introduce the following notation. For x ∈ F, let us write n(x) for the unipotent matrix

n(x) = ( \begin{matrix}I_{n−1} \\ & 1 & x & −x^{2}/2 \\ & & 1 & −x \\ & & & 1 \\ & & & & I_{n−1} \end{matrix} ) ,

and for a ∈ F^{×}, let us write t(a) for the diagonal matrix in T given by

t(a) = ( \begin{matrix}I_{n−1} \\ & a \\ & & 1 \\ & & & a^{−1} \\ & & & & I_{n−1} \end{matrix} ) .

The matrices t(a) give all of T. Also we write λ(a) = λ(t(a)), β = λ(ϖ). Let w_{0} = w_{1}(n, n + 2) be (a representative for) the long element of the Weyl group Ω. Then for π as above, and Ψ ∈ V_{π}, we define

(1.14)  B(Ψ) = ∫_{F^{×}} ∫_{U} Ψ(w_{0} n(1) u t(a)) θ_{S}(u)^{−1} λ^{−1}(a) du d^{×}a.

Then this is a Bessel functional on V_{π}. Once again, the Haar measure on F^{×} is normalized so that the measure of O^{×} is 1. As we shall show in Section 3 below, the integral (1.14) is absolutely convergent if

(1.15)  |α_{1}| < ⋯ < |α_{n−1}| < min(|α_{n}|, |α_{n}^{−1}|),  |α_{n}| < q^{1/2} min(|β|, |β^{−1}|).

Our first pair of results concerns the functional equations satisfied by B. We shall show that, in both cases, the Bessel functional may be extended to all characters χ = (χ_{1}, ⋯, χ_{n}) by a variation of the familiar process whereby the standard intertwining operators are analytically continued, and has a functional equation under certain transformations of these characters χ. More precisely, the Weyl group Ω of G acts on the characters χ = (χ_{1}, ⋯, χ_{n}), or what is the same thing in the unramified case, on the parameters α_{1}, ⋯, α_{n}. In terms of these parameters, Ω is the group of transformations of (α_{1}, ⋯, α_{n}) ∈ (ℂ^{×})^{n} generated by

(α_{1}, …, α_{n−1}, α_{n}) ↦ (α_{1}, …, α_{n−1}, α_{n}^{−1}),

and by the action of the symmetric group S_{n} on (α_{1}, ⋯, α_{n}). The cardinality of Ω is 2^{n}n!.

Let Φ_{χ} ∈ V_{π} be the standard nonramified vector. This is the unique function in V_{π} taking value one on K. We define a function H = H_{χ}: G → ℂ by

H(g) = B(π(g) Φ_{χ}).

This function is once again analogous to the Whittaker function obtained from the standard Whittaker functional.

Theorem 1.3. Suppose that T is nonsplit. Then the function H_{χ}, originally defined as an integral when

|α_{1}| < ⋯ < |α_{n−1}| < min(|α_{n}|, |α_{n}^{−1}|),

has a meromorphic continuation to all nonzero complex α_{1}, …, α_{n}. Moreover the function

ℋ_{χ}(g) = ∏_{1≤i<j≤n}(1 − α_{i}α_{j}q^{−1})^{−1}(1 − α_{i}α_{j}^{−1}q^{−1})^{−1} H_{χ}(g)

is invariant under the action of Ω on the α_{i}, and is holomorphic for all (α_{1}, …, α_{n}) ∈ (ℂ^{×})^{n}.

Theorem 1.4. Suppose that T is split. Then the function H_{χ}, originally defined as an integral when

|α_{1}| < ⋯ < |α_{n−1}| < min(|α_{n}|, |α_{n}^{−1}|), |α_{n}| < q^{1/2} min(|β|, |β^{−1}|),

has a meromorphic continuation to all nonzero complex α_{1}, …, α_{n}, β. Moreover the function

ℋ_{χ}(g) = \frac{∏_{i=1}^{n}(1 − α_{i}βq^{−1/2})(1 − α_{i}β^{−1}q^{−1/2})}{∏_{1≤i<j≤n}(1 − α_{i}α_{j}q^{−1})(1 − α_{i}α_{j}^{−1}q^{−1}) ∏_{i=1}^{n}(1 − α_{i}^{2}q^{−1})} H_{χ}(g)

is invariant under the action of Ω on the α_{i}, and is holomorphic for all (α_{1}, …, α_{n}) ∈ (ℂ^{×})^{n} and β satisfying q^{−1/2} < min(|β|, |β|^{−1}).

The proofs of Theorems 1.3 and 1.4 are given in Section 3 below.

Our next pair of results gives an explicit formula for ℋ_{χ}(g). As in the case of the Waldspurger model (see (1.7)), since

(1.16)  ℋ_{χ}(rgκ) = θ_{S}(r)ℋ_{χ}(g)

for all r ∈ R, κ ∈ K, it suffices to determine ℋ on a set of coset representatives for R\G/K. Let k = (k_{1}, ⋯, k_{n}) be a vector of integers with k_{1} ≥ 0. If T is nonsplit then one finds, using the Iwasawa and Cartan decompositions (compare Sugano [Su], Lemma 2-4), that a set of coset representatives for R\G/K is given by the diagonal matrices of the form

d_{k} = diag(ϖ^{k′_{n}}, ϖ^{k′_{n−1}}, ⋯, ϖ^{k′_{1}}, 1, ϖ^{−k′_{1}}, ⋯, ϖ^{−k′_{n−1}}, ϖ^{−k′_{n}}),

with

k′_{i} = k_{1} + ⋯ + k_{i}.

If T is split, then one finds instead that a set of coset representatives is given by the matrices g_{k} of the form

g_{k} = \{ \begin{matrix}d_{k} & if k_{1} = 0 \\ n(1) d_{k} & if k_{1} > 0. \end{matrix}

For convenience, we write h(k_{1}, ⋯, k_{n}) for the quantity ℋ(d_{k}) (resp. ℋ(g_{k})).

¿From equation (1.16) it follows that H(g) = 0 unless θ_{S} is identically one on R ∩ gKg^{−1}. A short calculation shows that this condition implies that h(k_{1}, ⋯, k_{n}) = 0 unless each k_{i} ≥ 0.

Let A be the alternator ∑_{w∈Ω}(−1)^{length(w)} w in the group algebra ℂ[Ω]. Let Δ = (−1)^{n}A(α_{1}^{n}α_{2}^{n−1} ⋯ α_{n}). According to Weyl’s identity for Sp(2n, ℂ)

(1.17)  Δ = ∏_{i=1}^{n} α_{i}^{−1+i−n}(1 − α_{i}^{2}) ∏_{1≤i<j≤n}(1 − α_{i}α_{j})(1 − α_{i}α_{j}^{−1}).

Also let

e_{k} = −\frac{1}{2} ∑_{i=1}^{n} (n^{2} − (i − 1)^{2}) k_{i}.

Then the evaluation of h(k_{1}, ⋯, k_{n}) is given by

Theorem 1.5. Suppose T is nonsplit and k_{i} ≥ 0 for i = 1 to n. Then

h(k_{1}, ⋯, k_{n}) = q^{e_{k}}(1 + q^{−1})^{−1} Δ^{−1} A( ∏_{i=1}^{n} α_{n+1−i}^{−k′_{i}−i}(1 − α_{i}^{2}q^{−1})) .

In particular, ℋ_{χ} (I_{2n+1}) = 1.

Theorem 1.6. Suppose T is split and k_{i} ≥ 0 for i = 1 to n.

(1) If k_{1} = 0 then

h(0, k_{2}, ⋯, k_{n}) = q^{e_{k}} Δ^{−1}

× A(α_{n}^{−1} ∏_{i=1}^{n−1} α_{i}^{−k′_{n+1−i}−n−1+i}(1 − α_{i}βq^{−1/2})(1 − α_{i}β^{−1}q^{−1/2})) .

(2) If k_{1} > 0 then

h(k_{1}, k_{2}, ⋯, k_{n}) = q^{e_{k}}(1 − q^{−1})^{−1} Δ^{−1}

× A(∏_{i=1}^{n} α_{n+1−i}^{−k′_{i}−i}(1 − α_{i}βq^{−1/2})(1 − α_{i}β^{−1}q^{−1/2})) .

In particular, the function ℋ_{χ} is holomorphic for all (α_{1}, …, α_{n}, β) ∈ (ℂ^{×})^{n+1}, and ℋ_{χ} (I_{2n+1}) = 1.

These theorems are proved in Section 4 below.

Let us finally formulate the meromorphic continuation of the Bessel functional to all values of χ. We will formulate this result only in the split case; the nonsplit case is nearly identical. Suppose that Ψ ∈ C^{∞}((B ∩ K)\K). Given χ, there is a unique extension Ψ_{χ} of Ψ to an element of Ind(χ) satisfying (1.11).

Theorem 1.7. There exists a dense open subset Γ of (ℂ^{×})^{n+1} such that there exists a Bessel functional B on V_{π} for all (α_{1}, ⋯, α_{n}, β) ∈ Γ; if (1.15) is satisfied, this functional agrees with that defined by (1.14); and if Ψ ∈ C^{∞}((B ∩ K)\K), then B(Ψ_{χ}) is a meromorphic function of α_{1}, ⋯, α_{n}, β, whose polar set is contained in the complement of Γ.

As with Theorem 1.2, there is some overlap between this result and our previous Theorems. We will prove Theorem 1.7 by means of a theorem of Bernstein [Be] in Section 5. Actually Bernstein’s theorem implies that the complement of Γ may be taken to be a countable union of hyperplanes.

In view of the isomorphism of SO(5, F) to the projective group of symplectic similitudes P GSp(4, F), the theorems above imply the functional equation and the explicit formula for the Bessel model on G_{4} := GSp(4). The Bessel model on GSp(4, F) was first studied by Novodvorsky and Piatetski-Shapiro [NP]. A formula for the generating series of this model was given by Sugano in [Su], Proposition 2-5. To state the explicit formula, consider the group

G_{4} = \{g ∈ GL(4, F) | ^{t}g ( \begin{matrix}0 & I_{2} \\ −I_{2} & 0 \end{matrix} ) g = ν(g) ( \begin{matrix}0 & I_{2} \\ −I_{2} & 0 \end{matrix} ) , ν(g) ∈ F^{×}\} .

Let S ∈ M_{2}(O) be a nonsingular symmetric matrix S = ( \begin{matrix}c & b/2 \\ b/2 & a \end{matrix} ). Let U_{4} be the unipotent radical of the Siegel parabolic

U_{4} = \{u = ( \begin{matrix}I_{2} & X \\ 0 & I_{2} \end{matrix} ) | X =^{t} X\} ,

and let θ_{S} be the character of U_{4} given by θ_{S}(u) = ψ(tr(SX)). Let T be the torus in G_{4} consisting of those matrices of the form

( \begin{matrix}h & 0 \\ 0 & det(h)^{t}h^{−1} \end{matrix} )

which, acting by conjugation, stabilize θ_{S}; thus h ∈ G_{2} is required to satisfy ^{t}hSh = det(h)S. Then T normalizes U, and hence R := T U is a subgroup of G_{4}. Let λ be a character of T, and once again extend θ_{S} to a character of R by θ_{S}(tu) = λ(t) θ_{S}(u).

If π is an admissible representation of G_{4}, a Bessel functional on π is once again a linear functional B: V_{π} → ℂ such that

B(π(tu)v) = θ_{S}(tu) B(v),

for all t ∈ T, u ∈ U, and v ∈ V_{π}. If π transforms by a central character, then since T contains the center Z_{4} of G_{4}, this notion requires that λ |_{Z_{4}} match this character.

Let π = Ind(χ) be the principal series representation obtained by normalized induction from a character χ of the standard Borel subgroup of G_{4}. For χ in a suitable domain a Bessel functional may be obtained by integration as above. If Φ_{χ} ∈ V_{π} is the standard nonramified vector, then define H = H_{χ}: G_{4} → ℂ by the formula H(g) = B(π(g)Φ_{χ}).

To describe the explicit formula for this function, define parameters α_{1} to α_{4} by

χ ( \begin{matrix}ϖ \\ & ϖ \\ & & 1 \\ & & & 1 \end{matrix} ) = α_{1}  χ ( \begin{matrix}ϖ \\ & 1 \\ & & 1 \\ & & & ϖ \end{matrix} ) = α_{2}

χ ( \begin{matrix}1 \\ & 1 \\ & & ϖ \\ & & & ϖ \end{matrix} ) = α_{3}  χ ( \begin{matrix}1 \\ & ϖ \\ & & ϖ \\ & & & 1 \end{matrix} ) = α_{4}.

The Weyl group Ω_{4} acts on these parameters through all permutations of the α_{i} which preserve the relation α_{1}α_{3} = α_{2}α_{4}. Let A_{4} be the alternator ∑_{w∈Ω_{4}}(−1)^{length(w)} w in the group algebra ℂ[Ω_{4}]. Observe moreover that the function H is completely determined by its values on the elements

a_{k,l} = ( \begin{matrix}ϖ^{k+2l} \\ & ϖ^{k+l} \\ & & 1 \\ & & & ϖ^{l} \end{matrix} )

with k, l ≥ 0 if T is nonsplit, and by its values on the elements

b_{k,l} = \{ \begin{matrix}a_{k,0} & if l = 0 \\ m(1)a_{k,l} & otherwise \end{matrix}

with k, l ≥ 0 if T is split, where we set

m(1) = ( \begin{matrix}1 & 1 \\ & 1 \\ & & 1 \\ & & −1 & 1 \end{matrix} ) .

Then we have

Corollary 1.8. Suppose that T is nonsplit and a ∈ O^{×}.

(1) The function H_{χ} may be defined by continuation for all unramified characters χ. Moreover the function

ℋ_{χ}(g) = (1 − α_{1}α_{2}^{−1}q^{−1})^{−1}(1 − α_{2}α_{3}^{−1}q^{−1})^{−1}H_{χ}(g)

is holomorphic and invariant under the action of Ω.

(2) The function ℋ_{χ} is given by the explicit formula

ℋ_{χ}(a_{k,l}) = (1 + q^{−1})^{−1}q^{−3k/2−2l} \frac{A_{4} (α_{3}^{k+l+2}α_{2}^{l}α_{4}^{−1}(1 − α_{1}α_{3}^{−1}q^{−1})(1 − α_{4}α_{2}^{−1}q^{−1}))}{A_{4} (α_{3}^{2}α_{4}^{−1})}

valid for k, l ≥ 0.

In the split case, let us suppose without loss that a = c = 0, b = 1. Let β_{1}, β_{2} be the parameters

β_{1} = λ (( \begin{matrix}ϖ \\ & 1 \\ & & 1 \\ & & & ϖ \end{matrix} )) , β_{2} = λ (( \begin{matrix}1 \\ & ϖ \\ & & ϖ \\ & & & 1 \end{matrix} )) .

We are concerned with characters χ such that α_{1}α_{3} = β_{1}β_{2}. Then we have

Corollary 1.9. Suppose that T is split.

(1) The function H_{χ} may be defined by continuation for all unramified characters χ, λ such that χ |_{Z_{4}}= λ |_{Z_{4}}. Moreover the function

ℋ_{χ}(g) = \frac{∏_{i,j=1}^{2}(1 − α_{i}β_{j}^{−1}q^{−1/2})}{(1 − α_{1}α_{2}^{−1}q^{−1})(1 − α_{1}α_{3}^{−1}q^{−1})(1 − α_{2}α_{3}^{−1}q^{−1})(1 − α_{2}α_{4}^{−1}q^{−1})} H_{χ}(g)

is holomorphic and invariant under the action of Ω.

(2) The function ℋ_{χ} is given by the explicit formula

ℋ_{χ}(b_{k,0}) =q^{−3k/2} \frac{A_{4} (α_{3}^{k+2}α_{4}^{−1} ∏_{j=1}^{2}(1 − α_{1}β_{j}^{−1}q^{−1/2}))}{A_{4} (α_{3}^{2}α_{4}^{−1})}

valid for k ≥ 0 and by

ℋ_{χ}(b_{k,l}) =(1 − q^{−1})^{−1} q^{−3k/2−2l} \frac{A_{4} (α_{3}^{k+l+2}α_{2}^{l}α_{4}^{−1} ∏_{i=1,4;j=1,2}(1 − α_{i}β_{j}^{−1}q^{−1/2}))}{A_{4} (α_{3}^{2}α_{4}^{−1})}

valid for k ≥ 0, l > 0.

2. The Waldspurger Functional. We begin the study of the split Waldspurger functional with the following Lemma.

Lemma 2.1. Suppose that the inequalities (1.6) hold. Then the integral (1.5) is absolutely convergent.

Proof. Let f ∈ Ind(ξ) be given. Since K_{2} is compact, there is a number C such that |f(κ)| ≤ C for all κ ∈ K_{2}. One has the matrix identities

(2.1)  ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ) ( \begin{matrix}a & 0 \\ 0 & 1 \end{matrix} ) = \{ \begin{matrix}( \begin{matrix}a & 1 \\ 0 & 1 \end{matrix} ) ( \begin{matrix}−1 & 0 \\ a & 1 \end{matrix} ) & if |a|_{F} ≤ 1 \\ ( \begin{matrix}1 & 0 \\ 0 & a \end{matrix} ) ( \begin{matrix}0 & 1 \\ 1 & a^{−1} \end{matrix} ) & if |a|_{F} > 1, \end{matrix}

where the last matrix in each case is in K_{2}. Since f ∈ Ind(ξ), one obtains the inequalities

|f (( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ) ( \begin{matrix}a & 0 \\ 0 & 1 \end{matrix} ))| ≤ \{ \begin{matrix}C|a|_{F}^{1/2}|ξ_{1}(a)| & if |a|_{F} ≤ 1 \\ C|a|_{F}^{−1/2}|ξ_{2}(a)| & if |a|_{F} > 1. \end{matrix}

Thus one sees that the integral (1.5) is majorized by the sum of the integrals

∫_{|a|_{F} ≤1} |ξ_{1}(a)| |σ^{−1}(a)| |a|_{F}^{1/2} d^{×}a

and

∫_{|a|_{F} >1} |ξ_{2}(a)| |σ^{−1}(a)| |a|_{F}^{−1/2} d^{×}a.

But when the inequalities (1.6) hold, each of these integrals is an absolutely convergent geometric series. The Lemma follows.

We turn to the proof of Theorem 1.1. The proof makes essential use of ideas of Casselman and Shalika [CS], and of Banks [Ba].

Let us define certain elements of the representation π = π_{2} = Ind(ξ) as follows. Let ϕ_{ξ} be the normalized K_{2}-fixed vector, and for k ≥ 0, let

(2.2)  F_{k}(g) = ∫_{O} ϕ_{ξ}(g \binom{1 z}{1} ( \begin{matrix}ϖ^{k} \\ & 1 \end{matrix} )) dz,

ζ_{k}(g) = ϕ_{ξ}(g \binom{1 1}{1} ( \begin{matrix}ϖ^{k} \\ & 1 \end{matrix} )) .

It is easy to see that in the integral (2.2), the integrand has constant value when z has constant valuation; and if z is a unit, this constant value is ζ_{k}(g). Thus

(1 − q^{−1})ζ_{k}(g) = F_{k}(g) − ∫_{ϖO} ϕ_{ξ}(g \binom{1 z}{1} ( \begin{matrix}ϖ^{k} \\ & 1 \end{matrix} )) dz.

A simple change of variables shows that the second integral equals

q^{−1} π( \begin{matrix}ϖ \\ & 1 \end{matrix} ) F_{k−1}(g),

and so applying the Waldspurger functional W, we obtain

(2.3)  (1 − q^{−1})W (ζ_{k}) = W (F_{k}) − q^{−1}τW (F_{k−1}).

As in Casselman [Ca1] and Casselman and Shalika [CS], the vectors F_{k} are fixed by the Iwahori subgroup B_{2} of G_{2}. We remind the reader of the Casselman basis of the Iwahori fixed vectors of G_{2}. Assuming that ξ is regular, we define linear functionals T_{w} for w a Weyl group representative of G_{2} by

T_{w}f = ∫_{N∩wNw^{−1}\N} f(w^{−1}n) dn,

where N is the group of upper triangular unipotent matrices in G_{2}. These functionals are linearly independent on the Iwahori fixed vectors, and the Casselman basis f_{w} is defined by T_{w}f_{w′} = δ(w, w′) (Kronecker delta) for w and w′ in the Weyl group. If w_{0} is (a representative of) the long element of the Weyl group, then

(2.4)  f_{w_{0}}(g) = \{ \begin{matrix}ϕ_{ξ}(g) & if g ∈ B_{2}w_{0}B_{2}, \\ 0 & otherwise. \end{matrix}

The element f_{1} is given by a more complicated formula, and we do not need to know it. Since F_{k} is an Iwahori fixed vector, we can write

F_{k} = c(1, ξ) f_{1} + c(w_{0}, ξ) f_{w_{0}},

and by definition of the f_{w}, c(w, ξ) = T_{w}F_{k}. It is easy to see (and proved in Casselman and Shalika [CS]) that

(2.5)  T_{1}F_{k} = q^{−k/2} γ_{1}^{k},  T_{w_{0}}F_{k} = (1 − q^{−1}γ_{1}γ_{2}^{−1})(1 − γ_{1}γ_{2}^{−1})^{−1} q^{−k/2} γ_{2}^{k}.

Thus

W (F_{k}) = W (f_{1}) q^{−k/2} γ_{1}^{k} + (1 − q^{−1}γ_{1}γ_{2}^{−1})(1 − γ_{1}γ_{2}^{−1})^{−1} q^{−k/2} γ_{2}^{k} W (f_{w_{0}}).

Now using (2.3), we find that

(1 − q^{−1})W (ζ_{n}) = (1 − q^{−1/2}τγ_{1}^{−1})W (f_{1}) q^{−k/2} γ_{1}^{k}+

(1 − q^{−1/2}τγ_{2}^{−1})(1 − q^{−1}γ_{1}γ_{2}^{−1})(1 − γ_{1}γ_{2}^{−1})^{−1} q^{−k/2} γ_{2}^{k} W (f_{w_{0}}).

We may compute W (f_{w_{0}}) explicitly. By definition this equals

∫_{F^{×}} f_{w_{0}}(( \begin{matrix}& 1 \\ 1 \end{matrix} ) \binom{1 1}{1} ( \begin{matrix}a \\ & 1 \end{matrix} )) σ(a)^{−1} d^{×}a,

and it follows from (2.4) that the integrand here equals

ϕ_{ξ}(( \begin{matrix}& 1 \\ 1 \end{matrix} ) \binom{1 1}{1} ( \begin{matrix}a \\ & 1 \end{matrix} ))

if |a| ≥ 1, zero otherwise. Thus

W (f_{w_{0}}) = ∫_{|a|≥1} ϕ_{ξ}(( \begin{matrix}& 1 \\ 1 \end{matrix} ) \binom{1 1}{1} ( \begin{matrix}a \\ & 1 \end{matrix} )) σ(a)^{−1} d^{×}a.

If |a| = q^{k}, k ≥ 0, then the integrand is readily evaluated using (2.1), and equals τ^{k} γ_{2}^{−k} q^{−k/2}, so, assuming (1.6), the last integral is absolutely convergent and equals (1 − q^{−1/2}τγ_{2}^{−1})^{−1}. Thus

(1 − q^{−1})W (ζ_{n}) = (1 − q^{−1/2}τγ_{1}^{−1})W (f_{1}) q^{−k/2} γ_{1}^{k}+

(1 − q^{−1}γ_{1}γ_{2}^{−1})(1 − γ_{1}γ_{2}^{−1})^{−1} q^{−k/2} γ_{2}^{k}.

We have not yet proved the Theorem since we have not evaluated W (f_{1}). Note however that the Theorem will follow if we prove that 𝒲a_{ξ} is invariant under the interchange of γ_{1} and γ_{2}; indeed, we may assume without loss of generality that ξ is regular, so γ_{1} ≠ γ_{2}. Then, because the two functions q^{−k/2}γ_{1}^{k} and q^{−k/2}γ_{2}^{k} of k are linearly independent, the unknown value of W (f_{1}) will be determined.

We will show that 𝒲a_{ξ}(1) = 1. This is sufficient: indeed, recalling Waldspurger’s theorem on the uniqueness of the model, there is a unique spherical vector in this unique model which is normalized to equal 1 at g = 1, if 𝒲a_{ξ}(1) = 1 then clearly 𝒲a_{ξ} must be this vector; then, since the isomorphism class of the representation π_{2} = Ind(ξ) is unchanged when we interchange γ_{1} and γ_{2}, 𝒲a_{ξ} is thus invariant under this interchange. By definition

Wa_{ξ}(1) = ∫_{F^{×}} ϕ_{ξ}(( \begin{matrix}& 1 \\ 1 \end{matrix} ) \binom{1 1}{1} ( \begin{matrix}a \\ & 1 \end{matrix} )) σ(a)^{−1} d^{×}a,

and in this integral, the integrand is constant when a has constant valuation. (This is not true for Wa_{ξ}(η_{k}), which is the reason for the somewhat elaborate proof of this Theorem!) Once again applying (2.1), we find that the integrand equals τ^{−k}γ_{1}^{k}q^{−k/2} if |a| = q^{−k}, k ≥ 0, or τ^{−k}γ_{2}^{k}q^{k/2} if k < 0. Assuming (1.6) holds, it is then simple to sum the two geometric series and check that 𝒲a_{ξ}(1) = 1.

This completes the proof of Theorem 1.1

We turn now to the proof of Theorem 1.2. Fix an element f ∈ C^{∞}((B_{2} ∩ K_{2})\K_{2}), and extend f to Ind(ξ). Then, since f is locally constant, it follows from (2.1) that the integrand on the right in (1.5) equals

|a|^{1/2}ξ_{1}(a)σ^{−1}(a) f( \begin{matrix}−1 & 0 \\ 0 & 1 \end{matrix} ) if |a| is sufficiently small;
|a|^{−1/2}ξ_{2}(a)σ^{−1}(a) f( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) if |a| is sufficiently large.

Hence the integral (1.5) is equal to an integral over a compact set plus two integrals giving geometric series, whose values have analytic continuation to the region Λ. This gives the analytic continuation of W, and it only remains to be seen that it represents a Waldspurger functional. Thus we must show that with t ∈ T_{2}^{s}(F), W (π_{2}(t) f − σ(t) f) = 0. It is clear that this is true when (1.6) is satisfied, and that the left side is analytic, so this is true for all (γ_{1}, γ_{2}, τ) ∈ Λ. This completes the proof of Theorem 1.2.

3. Proof of the Analytic Continuation and Functional Equation for the Bessel Model. In this Section we shall prove (most of) Theorems 1.3 and 1.4. The proof of the continuation in the parameters α_{i} and of the functional equation is based on homomorphisms from GL(2, F) into G, similarly to Jacquet’s proof of the analytic continuation and functional equation of the Whittaker functions on Chevalley groups [Ja]. Jacquet’s method suffices to give most of the functional equations, but an extra step is needed (different in the nonsplit and split cases). One then applies Hartog’s theorem. At this point, we will have proved Theorems 1.3 and 1.4 except for one point, namely the meromorphic continuation in the split case outside the region q^{−1/2} < min(|β|, |β|^{−1}). This meromorphic continuation follows from the explicit formula in Theorem 1.6, or from Theorem 1.7; so for this minor point, the proof will be completed in subsequent sections.

Our proof will follow to the extent possible the notation and organization of [BFG], where a similar method was used to study another unique functional.

First, we recall two well-known lemmas. To give the first, let ω ∈ Ω be a Weyl group element represented by the permutation matrix w (we shall frequently abuse the notation and write w ∈ Ω). Let N denote the full subgroup of upper triangular unipotent matrices in G. Given a character χ as above, let ^{w}χ be the character satisfying ^{w}χ(a) = χ(w^{−1}aw) for all diagonal a ∈ G. Define the intertwining operator T_{w}: Ind(χ) → Ind(^{w}χ) by the integral

(3.1)  (T_{w}Ψ)(g) = ∫_{N_{w}\N} Ψ(w^{−1}ng) dn,

where N_{w} = N ∩ wNw^{−1}. Given an unramified character χ as above, let a_{α} be the diagonal matrix in G

a_{α} = (α_{1}, ⋯, α_{n}, 1, α_{n}^{−1}, ⋯, α_{1}^{−1}).

Order the roots of SO(2n + 1, ℂ) so that N corresponds to the positive ones. Then one has ([Ca2], Section 6.4)

Lemma 3.1. The intertwining integral (3.1) is absolutely convergent if |r(a_{α})| < 1 for all positive roots r of SO(2n+1, ℂ) such that w(r) < 0. Moreover, T_{w} varies holomorphically with χ and has meromorphic continuation to the space of all unramified characters.

The second lemma concerns the G_{2} = GL(2, F) Whittaker function. As in Section 1, given two unramified quasicharacters ξ_{1}, ξ_{2} of F^{×}, ξ_{i}(ϖ) = γ_{i}, define a character ξ of the standard Borel subgroup of G_{2} by equation (1.3). Let π_{2} = Ind(ξ) be the normalized induced representation of G_{2}, and ϕ_{ξ} be the K_{2}-fixed vector such that ϕ_{ξ}(I_{2}) = 1. If |γ_{1}| > |γ_{2}|, let Wh_{ξ} be the Whittaker function

(3.2)  Wh_{ξ}(g) = ∫_{F} ϕ_{ξ} (( \begin{matrix}& 1 \\ 1 \end{matrix} ) \binom{1 x}{1} g) ψ(x) dx.

Then one has Lemma 3.2.

(1) The Whittaker function Wh_{ξ}, originally defined by the integral (3.2) when |γ_{1}| > |γ_{2}|, has a meromorphic continuation to all nonzero complex γ_{1}, γ_{2}. Moreover, the function

𝒲h_{ξ}(g) = (1 − γ_{1}γ_{2}^{−1}q^{−1})^{−1}Wh_{ξ}(g)

is holomorphic in (ℂ^{×})^{2} and is invariant under the interchange of γ_{1} and γ_{2}.

(2) Let γ′_{1} = (1 + ϵ) max(|γ_{1}|, |γ_{2}|), γ′_{2} = (1 + ϵ)^{−1} min(|γ_{1}|, |γ_{2}|), where ϵ ≥ 0 is chosen so that γ′_{1} ≠ γ′_{2}. Define corresponding unramified quasicharacters ξ′_{i} for i = 1, 2 by ξ′_{i}(ϖ) = γ′_{i}. Then

|Wh_{ξ}(g)| ≪ |ϕ_{ξ′}(g)|

uniformly in g, where ϕ_{ξ′} is the normalized K_{2}-fixed vector in Ind(ξ′_{1}, ξ′_{2}).

Just as the continuation of the Waldspurger function Wa_{ξ} is deduced from its evaluation, Lemma 3.2 may be deduced from the explicit evaluation of the Whittaker function Wh_{ξ}. Assuming |γ_{1}| > |γ_{2}|, a computation shows that the integral (3.2) is zero if g = ( \begin{matrix}ϖ^{k} & 0 \\ 0 & 1 \end{matrix} ) with k < 0, and is given by

Wh_{ξ} (( \begin{matrix}ϖ^{k} & 0 \\ 0 & 1 \end{matrix} )) = q^{−k/2}(1 − γ_{1}γ_{2}^{−1}q^{−1}) \frac{γ_{1}^{k+1} − γ_{2}^{k+1}}{γ_{1} − γ_{2}}

if k ≥ 0. The Lemma then follows.

We give next a third Lemma, concerning the convergence of the integrals which arise in the consideration of the split case. Let χ be an unramified character as above. Let w ∈ Ω be a given permutation, and factor w^{−1} as w^{−1} = w′^{−1}(n, n + 2). Set U_{w} = U ∩ w^{−1}Uw. Also, define the integer g and the complex number α by the equations

δ_{B} (w′^{−1}t(ϖ)w′) = q^{−g}, χ (w′^{−1}t(ϖ)w′) = α.

Lemma 3.3. For Ψ ∈ Ind(χ), the integral

(3.3)  ∫_{F^{×}} ∫_{U_{w}\U} Ψ(w^{−1}n(1) u t(a)) λ^{−1}(a) du d^{×}a

is absolutely convergent provided

|α| < q^{g/2} min(|β|, |β^{−1}|)

and provided that the intertwining integral T_{w′}(Ψ) converges absolutely.

Proof. We may interchange u and t(a) in the above integral. We have w^{−1}n(1) t(a) u = w′^{−1}(n, n + 2) n(1) t(a) u. A computation similar to (2.1) shows that

(n, n + 2) n(1) t(a) = \{ \begin{matrix}t(a) n(a^{−1}) κ_{1}(a) & if |a|_{F} ≤ 1 \\ t(a^{−1}) κ_{2}(a) & if |a|_{F} > 1, \end{matrix}

with κ_{1}(a), κ_{2}(a) in K of the form

( \begin{matrix}I_{n−1} \\ & ∗ \\ & & I_{n−1} \end{matrix} ) .

Since matrices of this form normalize U, we may move the κ_{i}(a) to the right in the integral (3.3). Using the condition Ψ ∈ Ind(χ) and comparing with the definition (3.1) of the intertwining operator T_{w′}, one sees that the integral (3.3) is absolutely bounded by the sum of the two integrals

∫_{|a|_{F} ≤1} |α|^{ord(a)} |λ^{−1}(a)| |a|_{F}^{g/2} |(T_{w′}Ψ)(κ_{1}(a))| d^{×}a

and

∫_{|a|_{F} >1} |α|^{−ord(a)} |λ^{−1}(a)| |a|_{F}^{−g/2} |(T_{w′}Ψ)(κ_{2}(a))| d^{×}a.

Since (T_{w′}Ψ)(g) is a locally constant function, it is absolutely bounded on K. Hence both integrals are bounded by geometric series, which converge under the hypotheses of the Lemma.

This proof may also be rephrased by using the isomorphism of P GL(2, F) with SO(3, F) to write the integral (3.3) as the Waldspurger integral of an intertwining integral, and then applying Lemmas 2.1 and Lemma 3.1.

Observe that applying Lemma 3.3 with w = w_{0}, so that w′ = w_{1}, g = 1 and α = α_{n}, one finds that the integral (1.14) is absolutely convergent in the region (1.15), as claimed.

To prove Theorems 1.3 and 1.4 we now proceed in two stages. First, using the standard functional equation for the GL(2) p-adic Whittaker function (Lemma 3.2), we shall establish functional equations under the transpositions (j, j + 1) ∈ Ω, 1 ≤ j ≤ n − 1. The proofs of these results also give the analytic continuation to certain unions of Weyl chambers properly larger than the original region of convergence. Then, using Theorem 1.1 in the split case and the invariance of the G_{2}-spherical function in Ind(ξ_{1}, ξ_{2}) under the interchange of ξ_{1} and ξ_{2} in the nonsplit case, we obtain a functional equation for the Weyl group element taking α_{n} to α_{n}^{−1} and fixing the rest. Since these two elements generate Ω, these steps imply that ℋ_{χ} is invariant under Ω.

To obtain the functional equations under the transpositions (j, j + 1), 1 ≤ j ≤ n − 1, let ι_{j} be the embedding of GL(2, F) into G given by

g ↦ ( \begin{matrix}I_{j−1} \\ & det(g)^{−1}g \\ & & I_{2n−2j−1} \\ & & & g^{♯} \\ & & & & I_{j−1} \end{matrix} ) ,

where g^{♯} = ( \begin{matrix}−1 \\ & 1 \end{matrix} ) g ( \begin{matrix}−1 \\ & 1 \end{matrix} ). Let w denote w_{0} if T is split, and w_{1} if T is nonsplit. Let v_{1} = (j, j + 1)(2n − j + 1, 2n − j + 2), and factor w = v_{1}v_{2}. Let B_{2} and U_{2} be, as above, the subgroups of GL(2, F) consisting respectively of upper triangular and of upper triangular unipotent matrices, and let B_{j} and U_{j} be the subgroups of G given by

B_{j} = ι_{j}(B_{2}), U_{j} = v_{2}^{−1} ι_{j}(U_{2}) v_{2}.

For u_{j} ∈ U_{j}, write v_{2} u_{j} = ι_{j}(û_{j}) v_{2}. Let U′_{j} and B′_{j} be the complementary subgroups in U and B to U_{j} and B_{j}, respectively, so that U = U_{j}U′_{j} and B = B_{j}B′_{j} (uniquely). Note that v_{1} and ι_{j}(U_{2}) normalize B′_{j} and fix Φ_{χ}|_{B′_{j}}. Let u ∈ U, t ∈ T, a ∈ G. Factor u = u_{j}u′_{j}, u_{j} ∈ U_{j}, u′_{j} ∈ U′_{j}. Applying the Iwasawa decomposition, for g ∈ G, we may write v_{2}u′_{j}tg = b′_{j}b_{j}κ, with b_{j} ∈ B_{j}, b′_{j} ∈ B′_{j}, κ ∈ K (we suppress the dependence of b′_{j} and b_{j} on u′_{j}, t, and g from the notation).

Suppose first that T is nonsplit. Then with the above notation we have

w_{1}utg = v_{1}v_{2}u_{j}u′_{j}tg
= v_{1}ι_{j}(û_{j}) v_{2}u′_{j}tg
= v_{1}ι_{j}(û_{j}) b′_{j}b_{j}κ.

Since Φ_{χ} is right K-invariant, we may thus express H_{χ}(g) as the iterated integral

(3.4)  H_{χ}(g) = ∫_{T (O)} ∫_{U′_{j}} [ ∫_{U_{j}} Φ_{χ}(v_{1}ι_{j}(û_{j})b_{j}) θ_{S}(u_{j})^{−1} du_{j}] Φ_{χ}(b′_{j}) θ_{S}(u′_{j})^{−1} du′_{j} dt.

However for h ∈ G, it follows from (1.11) that the function ϕ: GL(2, F) → ℂ given by ϕ(a) = Φ_{χ}(ι_{j}(a)h) is in the space Ind(χ_{j+1}^{−1}μ^{j−n}, χ_{j}^{−1}μ^{j−n}), where μ is the quasicharacter of F^{×} given by μ(x) = |x|_{F}. Accordingly the inner integral in (3.4) is a constant multiple of the GL(2) Whittaker function associated to this representation. Applying Lemma 3.2, we obtain the analytic continuation of the function

\frac{1}{1 − α_{j}α_{j+1}^{−1}q^{−1}} H_{χ}(g)

to the region C_{j} of (ℂ^{×})^{n} defined by the inequalities

(3.5) |α_{1}| < ⋯ < |α_{j−1}| < min(|α_{j}|, |α_{j+1}|) ≤ max(|α_{j}|, |α_{j+1}|)

< |α_{j+2}| < ⋯ < |α_{n−1}| < min(|α_{n}|, |α_{n}^{−1}|)

if 1 ≤ j < n − 1, and defined by the inequalities

(3.6)  |α_{1}| < |α_{2}| < ⋯ < |α_{n−2}| < min(|α_{n−1}|, |α_{n}|) ≤ max(|α_{n−1}|, |α_{n}|) < 1

if j = n − 1, and the invariance of this function under (j, j + 1) there. The region C_{j} is obtained by replacing the inner integral by the estimate given in Lemma 3.2, part (2), and comparing with the intertwining operator T_{v_{2}^{−1}}, whose convergence is given in Lemma 3.1. The difference in inequalities is due to this comparison; note that for 1 ≤ j < n − 1, v_{2} is a product of transpositions

v_{2} = (1, 2n+1) ⋯ (j−1, 2n−j+3)(j, 2n−j+1)(j+1, 2n−j+2)(j+2, 2n−j) ⋯ (n−1, n+3),

while for j = n − 1, v_{2} includes a four-cycle

v_{2} = (1, 2n + 1) ⋯ (n − 2, n + 4)(n − 1, n + 2, n + 3, n).

Also, let us remark that if 1 ≤ j < n − 1, then the region C_{j} properly contains the original region of convergence C_{0}, while if j = n − 1, Hartog’s theorem gives at once the analytic continuation of H_{χ} to C_{0} ∪ C_{n−1}.

Suppose now that T is split. Let u, u_{j}, u′_{j}, û_{j} be as above; note that U_{j} = ι_{j}(U_{2}). We have

w_{0}n(1)ut(a) = v_{1}v_{2}u_{j}(u_{j}^{−1}n(1)u_{j})u′_{j}t(a)
= v_{1}ι_{j}(û_{j})v_{2}(u_{j}^{−1}n(1)u_{j})u′_{j}t(a).

Moreover, for u_{j} = u_{j}(x) := ι_{j} (\binom{1 x}{1}), a calculation shows that u_{j}^{−1}n(1)u_{j} = n(1)u′′_{j}

with u′′_{j} ∈ U′_{j} and θ_{S}(u_{j})^{−1}θ_{S}(u′′_{j}) = ψ(−x). Using the Iwasawa decomposition, write v_{2}n(1)u′_{j}t(a)g = b′_{j}b_{j}κ, with b_{j} ∈ B_{j}, b′_{j} ∈ B′_{j}, κ ∈ K (once again we suppress from the notation the dependence of b′_{j} and b_{j} on u′_{j}, t(a), and g). Then the integral (1.14) representing H_{χ}(g) becomes

H_{χ}(g) = ∫_{F^{×}} ∫_{U′} [ ∫_{F} Φ_{χ}(v_{1}u_{j}(−x)b_{j}) ψ(−x) dx] Φ_{χ}(b′_{j}) θ_{S}(u′_{j})^{−1} λ^{−1}(a) du d^{×}a.

The inner integral is once again a GL(2) Whittaker function. Applying Lemma 3.2, we obtain the analytic continuation of the function

\frac{1}{1 − α_{j}α_{j+1}^{−1}q^{−1}} H_{χ}(g),

to the region C′_{j} of (ℂ^{×})^{n} defined by requiring the inequalities (3.5) and in addition the inequality

|α_{n}| < q^{1/2} min(|β|, |β^{−1}|)

if 1 ≤ j < n − 1, and defined by requiring the inequalities (3.6) and in addition the inequality

min(|α_{n−1}|, |α_{n}|) < q^{3/2} min(|β|, |β^{−1}|)

if j = n − 1. We also obtain the invariance of this function under (j, j + 1) there. The region C′_{j} is obtained by replacing the inner integral by the estimate given in Lemma 3.2, part (2), and applying Lemma 3.3. For later use, we denote the original region of convergence given by (1.15) as C′_{0}.

It remains to obtain the functional equation under the interchange α_{n} ↔ α_{n}^{−1}. To obtain this, let ι_{n} denote the homomorphism of GL(2, F) into G given by

(3.7) ι_{n} (( \begin{matrix}a & b \\ c & d \end{matrix} )) =

\frac{1}{ad − bc} ( \begin{matrix}(ad − bc)I_{n−1} \\ & a^{2} & ab & −b^{2}/2 \\ & 2ac & ad + bc & −bd \\ & −2c^{2} & −2cd & d^{2} \\ & & & & (ad − bc)I_{n−1} \end{matrix} ) .

Consider first the case that T is nonsplit. Since w_{1} fixes T, the integral (1.12) representing H_{χ}(g) is expressed as an iterated integral

(3.8)  H_{χ}(g) = ∫_{U} [ ∫_{T (O)} Φ_{χ}(tw_{1}ug) dt] θ_{S}(u)^{−1} du.

Now for h ∈ G, the function ϕ: GL(2, F) → ℂ given by ϕ(a) = Φ_{χ}(ι_{n}(a)h) is in the space Ind(χ_{n}, χ_{n}^{−1}). Recalling that ι_{n} gives the isomorphism between P GL_{2}(F) and SO(3, F), one sees that the inner integral in (3.8) is a multiple of the nonsplit Waldspurger functional on P GL_{2}(F), evaluated on a suitable right-translate of the spherical vector in this induced space. As explained in Section 1, it is thus a multiple of the spherical function in Ind(χ_{n}, χ_{n}^{−1}). By the invariance of this spherical function under the interchange of χ_{n} and χ_{n}^{−1}, we conclude that H_{χ}(g) is invariant under α_{n} ↔ α_{n}^{−1} in the domain (1.13).

Consider instead the case that T is split. Factor w_{0} = w_{1}w_{2} with w_{2} = (n, n + 2). In this case the integral (1.14) is an iterated integral of the form

H_{χ}(g) = ∫_{U} [ ∫_{F^{×}} Φ_{χ}(w_{2} n(1) t(a) w_{1} u g) λ^{−1}(a) d^{×}a] θ_{S}(u)^{−1} du.

Once again using the homomorphism ι_{n}, one sees that the inner integral is a multiple of the split Waldspurger functional computed in Section 2, applied to a function in Ind(χ_{n}, χ_{n}^{−1}). Note that this integral is absolutely convergent provided |α_{n}| < q^{1/2} min(|β|, |β^{−1}|). Applying Theorem 1.1 and arguing as above, we obtain the invariance of the function

\frac{(1 − α_{n}βq^{−1/2})(1 − α_{n}β^{−1}q^{−1/2})}{(1 − α_{n}^{2}q^{−1})} H_{χ}(g),

under α_{n} ↔ α_{n}^{−1} in the domain (1.15).

To complete the proof of Theorem 1.3, let C be the set of α = (α_{1}, ⋯, α_{n}) ∈ (ℂ^{×})^{n} such that at most one of the equalities |α_{i}| = |α_{j}|, 1 ≤ i < j ≤ n, and |α_{i}| = |α_{j}|^{−1}, 1 ≤ i ≤ j ≤ n, is satisfied. It may be deduced from Hartog’s theorem that any analytic function on C can be extended to an analytic function on (ℂ^{×})^{n}, and so it is sufficient to extend the function ℋ_{χ}(a) to the domain C in such a way that the corresponding functional equations are satisfied on C. Now if α ∈ C, then there is an ω ∈ Ω such that ωα is in C_{k} for some k, 0 ≤ k ≤ n − 1. We then define ℋ_{χ}(a) to equal ℋ_{^{ω}χ}(a). This is well defined by the above discussion. It is apparent that this extends ℋ_{χ}(a) to an analytic function of C satisfying the corresponding functional equations, as required. This completes the proof of Theorem 1.3.

The proof of Theorem 1.4 is similar. Suppose first that λ is chosen so that q^{−1/2} < min(|β|, |β|^{−1}). Then using the regions C′_{j} in place of C_{j}, the argument in the paragraph above gives the analytic continuation of ℋ_{χ} to α ∈ (ℂ^{×})^{n}, and the functional equation there. This completes the proof of Theorem 1.4, except, as we have noted, for the point of the meromorphic continuation to all α_{i} and β, and this point is a consequence of the explicit formula in Theorem 1.6 or of Theorem 1.7.

4. Explicit Formulas for the Bessel Model. The evaluation of the Bessel model given in Theorems 1.5 and 1.6 is obtained by applying the method of Casselman and Shalika [Ca1], [CS]. This method was also used in Section 2. A similar evaluation is carried out [BFG]; however, the particulars are different in the case at hand, and the cases T nonsplit and T split are once again different from each other. Throughout the proofs we shall assume that χ is regular, i.e. ^{ω}χ ≠ χ for all nonidentity ω ∈ Ω. The general case then follows from the analytic continuation given in Theorems 1.3 and 1.4.

Let B be the Iwahori subgroup of K, consisting of integral matrices which are upper triangular invertible modϖO. It follows from the Iwasawa decomposition of G and the Bruhat decomposition over O/ϖO that the space of right-Iwahori-fixed vectors Ind(χ)^{B} is |Ω|-dimensional; moreover, a basis is given by the functions ϕ_{w} defined by

ϕ_{w}(bw^{′ −1}b_{1}) = \{ \begin{matrix}Φ_{χ}(b) & if w = w′; \\ 0 & otherwise, \end{matrix}

where b ∈ B, w ∈ Ω, b_{1} ∈ B (note that this differs from Casselman’s notation in [Ca1], but is consistent with [BFG]).

If χ is regular, then it is shown in [Ca1], Section 3, that the linear functionals on Ind(χ)^{B} given by f ↦ (T_{w}f)(I_{2n+1}) are linearly independent. Here T_{w} is the intertwining operator defined in (3.1). Let f_{w}, w ∈ Ω, be the dual basis, characterized by

(T_{w}f_{w′})(I_{2n+1}) = \{ \begin{matrix}1 & if w = w′; \\ 0 & otherwise. \end{matrix}

Suppose first that T is nonsplit. Let d = d_{k} be as given in Section 1, and suppose that all k_{i} ≥ 0. Let F_{d} be the function

F_{d}(g) = ∫_{T (O)} ∫_{U∩K} Φ_{χ}(gutd) du dt.

Clearly F_{d} ∈ Ind(χ). Let us show that F_{d} is right Iwahori invariant; then we may write

(4.1)  F_{d}(g) = ∑_{w∈Ω} R(d, w; χ) f_{w}.

To prove the Iwahori invariance, we first note that by Jacquet’s first lemma (see the proof of Proposition 2.5 of Casselman [Ca1]), F_{d} is (right) invariant under the group Σ of elements of SO(2n + 1, O) whose lower entries are divisible by ϖ, and whose middle 3 × 3 block is the identity; and also, if ι_{n} is as in (3.7), it is clear that F_{d} is invariant by ι_{n}(T_{2}^{a}(O)). Thus

(4.2)  F_{d}(bgσ) = (δ_{B}^{1/2}χ)(b)F_{d}(g)

when b ∈ B(F) and σ ∈ ι_{n}(T_{2}^{a}(O))Σ. We will deduce invariance by the group ι_{n}(K_{2})Σ (which contains the Iwahori subgroup) from the fact that the canonical map

(4.3)  B\G/ι_{n}(T_{2}^{a}(O))Σ → B\G/ι_{n}(K_{2})Σ

is a bijection. To see this, we note that since ι_{n}(K_{2})Σ contains the Iwahori subgroup, every double coset in B\G/ι_{n}(K_{2})Σ contains a Weyl group element w; so what we must show is that Bwι_{n}(T_{2}^{a}(O))Σ = Bwι_{n}(K_{2})Σ. We consider an element bwι_{n}(k)σ of the right side, where b ∈ B, k ∈ K_{2} and σ ∈ Σ. We can write k = β^{+}t^{+} = β^{−}t^{−} where t^{±} are in T_{2}^{a}(O), β^{+} is upper triangular and β^{−} is lower triangular. Then one of wι_{n}(β^{±})w^{−1} is upper triangular, so bwι_{n}(k)σ = b′wι_{n}(t^{±})σ where b′ = bwι_{n}(β^{±})w^{−1}, which shows that this element lies in Bwι_{n}(T_{2}^{a}(O))Σ. Thus (4.3) is a bijection. We may now prove the Iwahori invariance of F_{d}. If g ∈ G and σ ∈ ι_{n}(K_{2})Σ, we write gσ = bgσ′ with b ∈ B and σ′ ∈ ι_{n}(T_{2}^{a}(O))Σ. We note that b is conjugate to σσ′^{−1} ∈ K, and so the eigenvalues of b are units, and χ(b) = 1. It thus follows from (4.2) that F_{d}(gσ) = F_{d}(bgσ′) = F_{d}(g).

Let us compute the coefficient R(d, w; χ). It equals

(T_{w}F_{d})(I_{2n+1}) = ∫_{N_{w}\N} ∫_{T (O)} ∫_{U∩K} Φ_{χ}(w^{−1}nutd) du dt dn

= ∫_{T (O)} T_{w}(Φ_{χ})(td) dt.

However, let Φ^{+} denote the set of positive roots of SO(2n + 1, ℂ), and if r ∈ Φ^{+}, let ι_{r} be the corresponding embedding of SL(2, F) into G, and

a_{r} = ι_{r} ( \begin{matrix}ϖ \\ & ϖ^{−1} \end{matrix} ) .

Then it is shown in [Ca1], Theorem 3.1, that

T_{w}(Φ_{χ}) = c_{w}(χ) Φ_{^{w}χ},

where Φ_{^{w}χ} is the standard nonramified vector in Ind(^{w}χ), and the coefficient c_{w}(χ) is given by

(4.4)  c_{w}(χ) = ∏_{{r∈Φ^{+} \atop w(r)<0}} (\frac{1 − q^{−1} χ(a_{r})}{1 − χ(a_{r})}) .

We find that

R(d, w; χ) = c_{w}(χ) σ_{^{w}χ}(d),

where

σ_{^{w}χ}(d) = ∫_{T (O)} Φ_{^{w}χ}(td) dt.

Let ι_{n} be the homomorphism from GL(2, F) into G given by equation (3.7). Factor d = d′ι_{n}(d′′) (all these matrices depending on the k_{i}) with

d′ = diag(ϖ^{k′_{1}}, ⋯, ϖ^{k′_{2}}, 1, 1, 1, ϖ^{−k′_{2}}, ⋯, ϖ^{−k′_{n}})

and

d′′ = ( \begin{matrix}ϖ^{k_{1}} \\ & 1 \end{matrix} ) .

Then one sees that

σ_{^{w}χ}(d) = ^{w}χδ_{B}^{1/2}(d′) ∫_{T (O)} Φ_{^{w}χ}(t ι_{n}(d′′)) dt.

Now if

w(χ_{1}, ⋯, χ_{n}) = (χ′_{1}, ⋯, χ′_{n}),

then the function g ↦ Φ_{^{w}χ}(ι_{n}(g)) lies in the induced space Ind(χ′_{n}, χ′_{n}^{−1}). Recalling that ι_{n} identifies P GL(2, F) with SO(3, F) and applying the analysis of the nonsplit Waldspurger functional in Section 1, one concludes that this last integral is equal to the spherical function on GL(2, F) in this induced space, evaluated at d′′. This value is given by the Macdonald formula (see [Ca1]). Substituting this formula, we obtain

(4.5) σ_{^{w}χ}(d) = (1 + q^{−1})^{−1} q^{e_{k}} ∏_{i=2}^{n} χ′_{n+1−i}(ϖ^{k′_{i}})

× \frac{(χ′_{n}(ϖ) − χ′_{n}^{−1}(ϖ)q^{−1}) χ′_{n}(ϖ)^{k_{1}} − (χ′_{n}^{−1}(ϖ) − χ′_{n}(ϖ)q^{−1}) χ′_{n}(ϖ)^{−k_{1}}}{χ′_{n}(ϖ) − χ′_{n}^{−1}(ϖ)} .

This completes the evaluation of R(d, w; χ).

If f ∈ Ind(χ) with χ dominant, let

L(f) = ∫_{U} f(w_{1}u) θ_{S}(u)^{−1} du.

This integral converges absolutely by comparison with T_{w_{1}}(f). Then, by (1.12) and (4.1), we have

(4.6)  H_{χ}(d) = L(F_{d}) = ∑_{w∈Ω} R(d, w; χ) L(f_{w}).

It is clear from this formula and our evaluation of R(d, w; χ) that h(k_{1}, ⋯, k_{n}) is a linear combination of functions of the α_{i} which lie in exactly one Ω-orbit. To complete the proof of Theorem 1.5, we shall compute the full contribution to h(k_{1}, ⋯, k_{n}) of the terms on the right of (4.6) with w = w_{0}, w = w_{1}. By the computation of R(d, w; χ) given above, no other w ∈ Ω contributes a rational function of the form

α_{1}^{−k′_{n}} ⋯ α_{n}^{−k′_{1}}

times a function independent of the k_{i}. Hence, by Theorem 1.3, we will have computed one piece of the Ω-orbit, and the final formula is then the Ω-symmetric sum of these terms, taking into account the normalizing factor in the functional equation of Theorem 1.3.

To compute the contribution of the w = w_{0}, w_{1} terms to (4.6), we need the following lemma relating the two bases for the Iwahori fixed vectors, {ϕ_{w}} and {f_{w}}.

Lemma 4.1.

(1) ϕ_{w_{0}} = f_{w_{0}}.

(2) ϕ_{w_{1}} = f_{w_{1}} + \frac{(1−q^{−1})α_{n}^{2}}{1−α_{n}^{2}} f_{w_{0}}.

(3) L(f_{w_{0}}) = 0.

(4) L(f_{w_{1}}) = 1.

Proof. For w′ ∈ Ω we may write

(4.7)  ϕ_{w′}(g) = ∑_{w∈Ω} c_{χ}(w, w′) f_{w}(g).

where the coefficients c_{χ} are given by

c_{χ}(w, w′) = T_{w}(ϕ_{w′})(I_{2n+1})

= ∫_{N_{w}\N} ϕ_{w′}(w^{−1}n) dn.

This integral is zero unless w^{−1}n ∈ Bw′^{−1}B for some n ∈ N_{w}\N. If w′ = w_{0}, it is easy to see that this may happen only for w = w_{0} and n ∈ N ∩ K; part (1) follows. Similarly, if w′ = w_{1}, then c_{χ}(w, w′) = 0 unless w = w_{0} or w = w_{1}, and c_{χ}(w_{1}, w_{1}) = 1. Finally, to determine c_{χ}(w_{1}, w_{0}), it suffices to take g in (4.7) to be of the form ι_{n}(g′), with g′ ∈ GL(2, F). In that case the ϕ_{w_{i}} and f_{w_{i}} necessarily match the analogously defined GL(2) functions in Ind(χ_{n}, χ_{n}^{−1}), and the determination of this coefficient follows from the analogous statement on GL(2, F), which is equivalent to (2.5).

To prove the remaining parts of the Lemma, it suffices to show that

(4.8)  L(ϕ_{w_{0}}) = 0,  L(ϕ_{w_{1}}) = 1.

However, for w ∈ Ω,

L(ϕ_{w}) = ∫_{U} ϕ_{w}(w_{1}u) θ_{S}(u)^{−1} du.

A matrix calculation shows that w_{1}u ∈ Bw^{−1}B is impossible if w = w_{0}, and holds when w = w_{1} if and only if u ∈ U ∩ K. Thus (4.8) holds.

Combining (4.6), Lemma 4.1, and the evaluation of R(d, w; χ), one finds that

(4.9)  H_{χ}(d) = c_{w_{1}}(χ) σ_{^{w_{1}}χ}(d) + ∑_{w∈Ω;w≠w_{0},w_{1}} R(d, w; χ) L(f_{w}).

The final value for ℋ_{χ}(d) may now be obtained by taking the first term on the right hand side of (4.9), multiplying by the normalizing factor in the functional equation of Theorem 1.3, and symmetrizing with respect to the action of Ω on the parameters α_{i}. To do this, observe that (4.4) gives

(4.10)  c_{w_{1}}(χ) = ∏_{1≤i<j≤n} \frac{(1 − α_{i}α_{j}q^{−1})(1 − α_{i}α_{j}^{−1}q^{−1})}{(1 − α_{i}α_{j})(1 − α_{i}α_{j}^{−1})} ∏_{1≤i<n} \frac{1 − α_{i}^{2}q^{−1}}{1 − α_{i}^{2}} .

Substituting this formula and (4.5) into (4.9) and making use of Weyl’s identity (1.17), the explicit formula follows.

This completes the proof of Theorem 1.5.

We turn to the proof of Theorem 1.6. Suppose that T is split, so that the Bessel functional B is given by equation (1.14). Recall that N denotes the unipotent radical of the standard Borel subgroup of G. Let N(O) = N ∩ K. Given k = (k_{1}, ⋯, k_{n}) with all k_{i} ≥ 0, define

P_{k}(g) = ∫_{N(O)} Φ_{χ}(gnd_{k}) dn.

Then using the bijection (4.3), an argument similar to the one given there demonstrates that P_{k} ∈ Ind(χ)^{B}.

Though we will ultimately use the Casselman-Shalika method, we first establish the following Lemma, which implies that the determination of h(k_{1}, ⋯, k_{n}) follows from the determination of the quantities B(P_{k}). For convenience, let us set

H(k) := H_{χ}(g_{k}) B(k) := B_{χ}(P_{k}).

Then we have Lemma 4.2. Let k = (k_{1}, ⋯, k_{n}) with all k_{i} ≥ 0.

(1) Suppose k_{1} = 0. Then H(k) = B(k).

(2) Suppose k_{1} > 0. Then

H(k) = (1 − q^{−1})^{−1} (B(k) − q^{−1}βB(k_{1} − 1, k_{2} + 1, k_{3}, ⋯, k_{n})).

Proof. Comparing the definitions,

B(k) = ∫_{O} B (π (n(x)d_{k}) Φ_{χ}) dx.

A matrix calculation shows that if x ∈ O, then

n(x)d_{k} = \{ \begin{matrix}d_{k} n(ϖ^{−k_{1}}x) & if x ∈ ϖ^{k_{1}}O \\ t(x) n(1) d_{k_{1}−m,k_{2}+m,k_{3},··· ,k_{m}} t(ϖ^{m}x^{−1}) & if x ∈ ϖ^{m}O^{×}, 0 ≤ m < k_{1}. \end{matrix}

If x ∈ ϖ^{k_{1}}O then n(ϖ^{−k_{1}}x) ∈ K; moreover Φ_{χ} is K-fixed. Factoring

d_{k} = t(ϖ^{k_{1}}) d_{0,k_{1}+k_{2},k_{3},··· ,k_{m}}

and using property (1.10), one thus obtains

∫_{ϖ^{k_{1}} O} B (π (n(x)d_{k}) Φ_{χ}) dx = q^{−k_{1}}β^{k_{1}}H(0, k_{1} + k_{2}, k_{3}, ⋯, k_{n}).

Similarly if x ∈ ϖ^{m}O^{×}, then t(ϖ^{m}x^{−1}) ∈ K. Arguing similarly, one then finds that

(4.11) B(k) = q^{−k_{1}}β^{k_{1}}H(0, k_{1} + k_{2}, k_{3}, ⋯, k_{n})

+ (1 − q^{−1}) ∑_{m=0}^{k_{1}−1} q^{−m}β^{m}H(k_{1} − m, k_{2} + m, k_{3}, ⋯, k_{n}).

If k_{1} = 0, Lemma 4.2, part (1), follows at once from (4.11). For k_{1} > 0, equation (4.11) also implies that

(4.12) q^{−1}βB(k_{1} − 1, k_{2} + 1, k_{3}, ⋯, k_{n}) = q^{−k_{1}}β^{k_{1}}H(0, k_{1} + k_{2}, k_{3}, ⋯, k_{n})

+ (1 − q^{−1}) ∑_{m=1}^{k_{1}−1} q^{−m}β^{m}H(k_{1} − m, k_{2} + m, k_{3}, ⋯, k_{n}).

Subtracting (4.12) from (4.11), part (2) of the Lemma follows.

Write now P_{k} in terms of the Casselman basis {f_{w}}:

P_{k} = ∑_{w∈Ω} S(k, w; χ) f_{w}.

The coefficients S(k, w; χ) are given by

S(k, w; χ) = ∫_{N_{w}\N} P_{k}(w^{−1}n) dn

= ∫_{N_{w}\N} Φ_{χ}(w^{−1}nd_{k}) dn

= c_{w}(χ) (^{w}χδ_{B}^{1/2})(d_{k}), (4.13)

where c_{w}(χ) is defined by equation (4.4) above.

In this case, unlike the nonsplit case, no two terms for different w contribute the same rational function of the α_{i}. Hence it suffices to determine B(f_{w_{0}}); the value of B(P_{k}) is then obtained by symmetrization, using the functional equation of Theorem 1.4. The value of B(f_{w_{0}}) is given by

Lemma 4.3. Suppose |α_{n}β| < q^{1/2}. Then

B(f_{w_{0}}) = (1 − α_{n}βq^{−1/2})^{−1} .

Proof. Suppose |α_{n}β| < q^{1/2}. Since T normalizes U and fixes θ_{S},

B(f_{w_{0}}) = ∫_{F^{×}} ∫_{U} f_{w_{0}}(w_{0} n(1) t(a) u) θ_{S}(u)^{−1} λ^{−1}(a) du d^{×}a.

Suppose that w_{0}n(1)t(a)u ∈ Bw_{0}B. Then (t(a^{−1})n(1)t(a)) u ∈ w_{0}Bw_{0}B. This relation implies that u ∈ U ∩ K. Thus

B(f_{w_{0}}) = ∫_{F^{×}} f_{w_{0}}(w_{0} n(1) t(a)) λ^{−1}(a) d^{×}a.

Suppose first that a ∈ F^{×} satisfies |a|_{F} < 1. Then one sees by a matrix calculation that in fact w_{0}n(1)t(a) is never is the cell Bw_{0}B. If instead |a|_{F} ≥ 1, then

w_{0} n(1) t(a) = w_{0} t(a) (t(a)^{−1}n(1)t(a)),

and the last factor t(a)^{−1}n(1)t(a) is in B. Since f_{w_{0}} is right B-invariant, this gives

f_{w_{0}}(w_{0}n(1)t(a)) = f_{w_{0}}(w_{0}t(a)w_{0}^{−1} · w_{0})
= χδ_{B}^{1/2} (t(a^{−1})) .

Hence

B(f_{w_{0}}) = ∑_{m=0}^{∞} α_{n}^{m} β^{m} q^{−m/2}

= (1 − α_{n}βq^{−1/2})^{−1} ,

as claimed.

To conclude the proof of Theorem 1.6, observe that

B(k) = B_{χ}(P_{k}) = ∫_{O} H_{χ}(n(x)d_{k}) dx.

By Theorem 1.4, this satisfies a functional equation under the action of Ω on the α_{i}. Similarly to (4.10), (4.4) gives

c_{w_{0}}(χ) = ∏_{1≤i<j≤n} \frac{(1 − α_{i}α_{j}q^{−1})(1 − α_{i}α_{j}^{−1}q^{−1})}{(1 − α_{i}α_{j})(1 − α_{i}α_{j}^{−1})} ∏_{1≤i≤n} \frac{1 − α_{i}^{2}q^{−1}}{1 − α_{i}^{2}} .

Substituting this expression into (4.13) to obtain S(k, w_{0}; χ), and symmetrizing, one obtains the pleasant formula

\frac{∏_{i=1}^{n}(1 − α_{i}βq^{−1/2})(1 − α_{i}β^{−1}q^{−1/2})}{∏_{1≤i<j≤n}(1 − α_{i}α_{j}q^{−1})(1 − α_{i}α_{j}^{−1}q^{−1}) ∏_{i=1}^{n}(1 − α_{i}^{2}q^{−1})}B(k)

= q^{e_{k}}Δ^{−1}×

A((1 − α_{n}β^{−1}q^{−1/2})α_{1}^{−k′_{n}−n} ∏_{i=1}^{n−1} α_{n+1−i}^{−k′_{i}−i}(1 − α_{i}βq^{−1/2})(1 − α_{i}β^{−1}q^{−1/2})) .

Then applying Lemma 4.2, one obtains without difficulty the expression for h(k) given in Theorem 1.6. (Note that if k_{1} = 0 then in fact one obtains the formula

h(0, k_{2}, ⋯, k_{n}) = q^{e_{k}} Δ^{−1}

× A(α_{n}^{−1}(1 − α_{n}β^{−1}q^{−1/2}) ∏_{i=1}^{n−1} α_{i}^{−k′_{n+1−i}−n−1+i}(1 − α_{i}βq^{−1/2})(1 − α_{i}β^{−1}q^{−1/2})) .

However, expanding the second factor, the term

A(α_{n}^{−1}(−α_{n}β^{−1}q^{−1/2}) ∏_{i=1}^{n−1} α_{i}^{−k′_{n+1−i}−n−1+i}(1 − α_{i}βq^{−1/2})(1 − α_{i}β^{−1}q^{−1/2}))

is independent of α_{n}, and hence its alternator is zero. Thus one is in fact left with the expression of Theorem 1.6, part (1).)

This completes the proof of Theorem 1.6.

5. Continuation of the Bessel Functionals: The Application of Bernstein’s Theorem. Bernstein [Be] gave a powerful new method for the meromorphic continuation of functionals satisfying a suitable uniqueness property; in the case at hand, this uniqueness is the uniqueness of the Bessel model, which, as we have already noted, was proved by Novodvorsky [No]. Bernstein’s result will appear as an appendix to a book in progress of Cogdell and Piatetski-Shapiro; in the meantime, a statement (but no proof) may be found in Gelbart and Piatetski-Shapiro [GP]. In this Section we will show that Bernstein’s theorem implies Theorem 1.7. We assume familiarity with either [Be] or its paraphrase in [GP].

Let X = C^{∞}((B ∩ K)\K). Given the α_{i}, we may identify X with V_{π} by extending an element Ψ ∈ X uniquely to an element of Ψ_{χ} ∈ V_{π} satisfying (1.11). Let D = (ℂ^{×})^{n+1}. If (α_{1}, ⋯, α_{n}, β) ∈ D, we consider functionals B which satisfy the following two conditions:

(1) B is a Bessel functional,

(2) B(Φ_{χ}) = 1.

We claim that these conditions may be expressed by a polynomial system of equations (in Bernstein’s sense) in the parameters (α_{1}, ⋯, α_{n}, β). To see this, note that if Ψ ∈ X and (with the notation as in (1.10)) if t ∈ T, u ∈ U, then there exist Ψ_{j} ∈ X and polynomial functions f_{j} (j = 1, ⋯, N) of the α_{i} and β such that π(tu)Ψ_{χ} = ∑ f_{j}Ψ_{j,χ}. Thus (1.10) may be expressed by the polynomial equation

∑_{j=1}^{N} f_{j}B(Ψ_{j}) − θ_{S}(tu)B(Ψ) = 0.

It is also evident that condition (2) above is a polynomial equation. By Novodvorsky’s theorem, the solution, if it exists, is unique, and we have proved that a solution exists on a non-empty open subset of D. Consequently Bernstein’s theorem is applicable, yielding the meromorphic continuation of the functional B in the sense made precise by Theorem 1.7.

6. Bessel Periods of Eisenstein Series. In this Section we present a global application of these formulas. (In fact, the application makes use of only a particular case of them, namely the formula (6.6) for the value of the local Bessel functional at the identity.) Accordingly, we now let F be a global field and 𝔸 be its ring of adeles. Let π = ⊗π_{v} be a cuspidal automorphic representation of GL(n, 𝔸). Let P be the standard maximal parabolic subgroup of G = SO(2n + 1) with Levi factor GL(n). Denoting by δ_{P} the modular character of P, let f_{s} ∈ Ind_{P_{𝔸}}^{G(𝔸)}(π ⊗ δ_{P}^{s−1/2}), and let

(6.1)  E(g, s, f_{s}) = L_{S}(π, 2n(s − 1/2) + 1, ∨^{2}) ∑_{γ∈P_{F} \G_{F}} f_{s}(γg)

be the Eisenstein series attached to f_{s}. Here S is a finite set of places including the archimedean ones and those where π is ramified, and L_{S}(π, s, ∨^{2}) is the partial symmetric square L-function, which is the normalizing factor of the Eisenstein series. Let a, b and c be elements of F such that b^{2} + 2ac is not a square. Let Q be the standard parabolic subgroup of G with Levi factor GL(1) × ⋯ × GL(1) × SO(3), let U be the unipotent radical of Q, and let ψ be a nontrivial character of 𝔸/F. Let θ be the character of U(𝔸) defined by (1.9). Then Q(𝔸) acts on U(𝔸) and hence on its character group by conjugation; let R(𝔸) be the subgroup of the stabilizer consisting of elements whose projection to the Levi factor of Q(𝔸) lies in the embedded SO(3, 𝔸). We may naturally extend θ to a character of R(𝔸). Then R(𝔸) is the group of adelic points for an algebraic group R which is the semidirect product of U with a one-dimensional torus T, which for simplicity we are assuming nonsplit—this is our hypothesis that b^{2} + 2ac is a nonsquare. In this case, there exists a unique quadratic field extension K of F over which T splits. Let η = ⊗η_{v} be the quadratic Hecke character of F attached to K. In this Section we will prove

Theorem 6.1. The integral

(6.2)  ∫_{R(F)\R(𝔸)} E(r, s, f_{s}) θ(r) dr

is an Euler product, whose local factor at a good place v equals

(6.3)  L(n(s − 1/2) + 1/2, π_{v}) L(n(s − 1/2) + 1/2, π_{v} ⊗ η_{v}).

If n = 2 this result is essentially due to Böcherer [Bö] and Mizumoto [Mi] (these authors consider holomorphic Siegel modular forms on P GSp_{4} over ℚ, but the unramified local computation is the same for general base field and infinity type). The precise conditions to make v good are described below.

We turn to the proof of Theorem 6.1. Unfolding the integral, we see that (6.2) equals

L_{S}(π, 2n(s − 1/2) + 1, ∨^{2}) ∑_{γ∈P_{F} \G_{F} /R_{F}} ∫_{R_{𝔸}^{γ}\R_{𝔸}} ∫_{R_{F}^{γ} \R_{𝔸}^{γ}} f_{s}(γur) θ(ur) du dr,

where R^{γ} denotes the algebraic group R ∩ γ^{−1}P γ. Using the fact that π is cuspidal, one may show that only the open orbit in P \G/R contributes, and we may take the representative

γ = ( \begin{matrix}& & I_{n} \\ & (−1)^{n} \\ I_{n} \end{matrix} ) .

Then γR^{γ}γ^{−1} = U is a maximal unipotent subgroup in GL(n), the Levi factor of P, and the character γuγ^{−1} ↦ θ(u) is nondegenerate; indeed, since b^{2} + 2ac is a nonsquare, a ≠ 0. It follows that

W_{s}(g) = ∫_{U_{F} \U_{𝔸}} f_{s}(ug) θ(γ^{−1}uγ) du

lies in Ind_{P_{𝔸}}^{G_{𝔸}}(W_{π} ⊗ δ_{P}^{s−1/2}), where W_{π} is the Whittaker model of π (relative to the appropriate character of its maximal unipotent group). Writing

Ind_{P_{𝔸}}^{G_{𝔸}}(W_{π} ⊗ δ_{P}^{s−1/2}) = ⨂_{v} Ind_{P (F_{v})}^{G(F_{v})}(W_{π,v} ⊗ δ_{P}^{s−1/2})

as a restricted tensor product over all places v of F, where W_{π,v} is the local Whittaker model of π_{v}, there is no loss of generality in assuming that W_{s}(g) is a pure tensor; thus we write W_{s}(g) = ∏_{v} W_{s,v}(g_{v}). The integral (6.2) thus equals

L_{S}(π, 2n(s − 1/2) + 1, ∨^{2}) ∫_{R_{𝔸}^{γ}\R_{𝔸}} W_{s}(γr) θ(r) dr

and hence is factorizable, with local factor

(6.4)  L(π_{v}, 2n(s − 1/2) + 1, ∨^{2}) ∫_{R^{γ}(F_{v})\R(F_{v})} W_{s,v}(γr) θ(r) dr.

We compute this local factor at a good place. More precisely, suppose that the finite place v does not ramify in K. Then we compute (6.4) for any nonramified principal series π_{v} with Satake parameters α_{1}, ⋯, α_{n}, under the assumption that W_{s,v} is the unramified spherical vector in Ind_{P (F_{v})}^{G(F_{v})}(W_{π,v} ⊗δ_{P}^{s−1/2}), normalized so that W_{s,v}(1) = 1. To carry out this computation we first make the further assumption that |α_{i}| < |α_{i+1}|; the general case then follows by analytic continuation. (Note that the assumption |α_{i}| < |α_{i+1}| is unrealistic for a representation π_{v} which is a local component of an automorphic representation π, as this condition violates the Ramanujan conjecture.) According to the results of Casselman and Shalika [CS], we may then write

W_{s,v}(g) = ∏_{1≤i<j≤n} (1 − α_{i}α_{j}^{−1}q_{v}^{−1})^{−1} ∫_{U(F_{v})} Φ_{χ,v} (( \begin{matrix}J_{n} \\ & 1 \\ & & J_{n} \end{matrix} ) ug) θ(γ^{−1}uγ) du,

where q_{v} is the cardinality of the residue field at v, J_{n} is the n×n matrix with ones on the sinister diagonal and zeros elsewhere, and Φ_{χ,v} is the normalized spherical vector in the representation denoted (in the notation of Section 1) Ind(χ_{1}, ⋯, χ_{n}), where χ_{i}(ϖ_{i}) = α_{i} q^{−n(s−1/2)}. Substituting this into (6.4), we obtain precisely the integral (1.12) if the place v is inert in K, that is, if T is a nonsplit torus in F_{v}. If on the other hand the place v splits in K, we obtain the integral (1.14). To see this, one must remember to conjugate the torus so as to make it diagonal; and conjugating w_{1} in this way produces w_{0}n(1) as in (1.14). One sees that, in either case, the local factor (6.4) is equal to

(6.5)  ∏_{1≤i≤j≤n} (1 − α_{i}α_{j} q_{v}^{−1−2n(s−1/2)})^{−1} ∏_{1≤i<j≤n} (1 − α_{i}α_{j}^{−1}q_{v}^{−1})^{−1} H_{χ}(1).

By Theorems 1.5 and 1.6, ℋ_{χ}(1) = 1, so by Theorems 1.3 and 1.4 (with α_{i} q_{v}^{−n(s−1/2)} replacing α_{i}, and β = 1),

(6.6) H_{χ}(1) =

∏_{1≤i<j≤n} (1 − α_{i}α_{j} q_{v}^{−1−2n(s−1/2)}) (1 − α_{i}α_{j}^{−1}q_{v}^{−1}) ∏_{i=1}^{n} \frac{1 + η_{v}(ϖ_{v})α_{i} q_{v}^{−n(s−1/2)−1/2}}{1 − α_{i} q_{v}^{−n(s−1/2)−1/2}} ,

where the quadratic character η_{v}(ϖ_{v}) equals 1 if v splits in K, and −1 if v is inert. Substituting this formula into (6.5) and simplifying, we obtain the local factor (6.3) for v.

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Department of Mathematics, Stanford University, Stanford, CA 94305-2125 E-mail address: bump@gauss.stanford.edu

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

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

Mathematical Sciences Research Institute, 1000 Centennial Drive, Berkeley, CA 94720-5070

E-mail address: furusawa@msri.org