Some Results on Bessel Functionals for GSp(4)

Brooks Roberts and Ralf Schmidt^{1}

Received: January 5, 2015 Revised: December 12, 2015

Communicated by Don Blasius

Abstract. We prove that every irreducible, admissible representation π of GSp(4, F), where F is a non-archimedean local field of characteristic zero, admits a Bessel functional, provided π is not onedimensional. If π is not supercuspidal, we explicitly determine the set of all Bessel functionals admitted by π, and prove that Bessel functionals of a fixed type are unique. If π is supercuspidal, we do the same for all split Bessel functionals.

2010 Mathematics Subject Classification: Primary 11F70 and 22E50

Contents

Introduction  468
1 Some definitions  471
1.1 Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471
1.2 Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . 472
2 Generalities on Bessel functionals  475
2.1 Quadratic extensions . . . . . . . . . . . . . . . . . . . . . . . . 475
2.2 2 × 2 symmetric matrices . . . . . . . . . . . . . . . . . . . . . 475
2.3 Another F -algebra . . . . . . . . . . . . . . . . . . . . . . . . . 476
2.4 Bessel functionals . . . . . . . . . . . . . . . . . . . . . . . . . . 477
2.5 Action on Bessel functionals . . . . . . . . . . . . . . . . . . . . 479
2.6 Galois conjugation of Bessel functionals . . . . . . . . . . . . . 480
2.7 Waldspurger functionals . . . . . . . . . . . . . . . . . . . . . . 481

1Supported by NSF grant DMS-1100541.

Documenta Mathematica 21 (2016) 467–553

3 Split Bessel functionals  482
3.1 The group P_{3} and its representations . . . . . . . . . . . . . . . 482
3.2 P_{3}-theory for arbitrary central character . . . . . . . . . . . . . 484
3.3 Generic representations and zeta integrals . . . . . . . . . . . . 485
3.4 Generic representations admit split Bessel functionals . . . . . 486
3.5 Split Bessel functionals for non-generic representations . . . . . 488
4 Theta correspondences  489
4.1 The spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489
4.2 Embeddings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 491
4.3 Example embeddings . . . . . . . . . . . . . . . . . . . . . . . . 496
4.4 Theta correspondences and Bessel functionals . . . . . . . . . . 498
4.5 Representations of GO(X) . . . . . . . . . . . . . . . . . . . . . 504
4.6 GO(X) and GSp(4, F) . . . . . . . . . . . . . . . . . . . . . . . 506
4.7 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 511
5 Twisted Jacquet modules of induced representations  514
5.1 Two useful lemmas . . . . . . . . . . . . . . . . . . . . . . . . . 514
5.2 Siegel induced representations . . . . . . . . . . . . . . . . . . . 515
5.3 Klingen induced representations . . . . . . . . . . . . . . . . . . 523
6 The main results  534
6.1 Existence of Bessel functionals . . . . . . . . . . . . . . . . . . 534
6.2 The table of Bessel functionals . . . . . . . . . . . . . . . . . . 536
6.3 Some cases of uniqueness . . . . . . . . . . . . . . . . . . . . . 542
7 Some applications  545
7.1 Characterizations of non-generic representations . . . . . . . . . 545
7.2 Invariant vectors . . . . . . . . . . . . . . . . . . . . . . . . . . 547
References  550

Introduction

The uniqueness and existence of appropriate models for irreducible, admissible representations of a linear reductive group over a local field has long played an important role in local and global representation theory. Best known are perhaps the Whittaker models for general linear groups, which are instrumental in proving multiplicity one theorems and the analytic properties of automorphic L-functions. Generic representations, i.e., those admitting a Whittaker model, have an important place in the representation theory of GSp(4) as well, the group under consideration in this paper; see [16] for an early example of their use. For GSp(4) they play a less comprehensive role, however, since there are many important non-generic automorphic representations, for example those generated by holomorphic Siegel modular forms.

The use of Bessel models, or equivalently Bessel functionals, as a substitute for the often missing Whittaker models for GSp(4) has been pioneered by Novodvorsky and Piatetski-Shapiro. Similar to the generic case, Bessel models consist of functions on the group with a simple transformation property under a certain subgroup; see below for precise definitions. The papers [17] and [15] are concerned with the uniqueness of Bessel functionals in the case of trivial central character; the first paper treats the case of so-called special Bessel functionals. For the use of Bessel models in the study of analytic properties and special values of L-functions for non-generic representations, see [19], [36], [6], [21]. In this paper, we further investigate the existence and uniqueness of Bessel functionals for irreducible, admissible representations of GSp(4, F), where F is a non-archimedean local field of characteristic zero. To explain our results, we have to introduce some notation. Let F be a non-archimedean local field of characteristic zero, and let ψ be a non-trivial character of F. Let GSp(4, F) be the subgroup of g in GL(4, F) satisfying ^{t}gJg = λ(g)J for some scalar λ(g) in F^{×}, where

J = [ \begin{matrix}& & & 1 \\ & & 1 \\ & −1 \\ −1 \end{matrix} ] .

The Siegel parabolic subgroup P of GSp(4, F) is the subgroup consisting of all matrices whose lower left 2 × 2 block is zero. Let N be the unipotent radical of P. The characters θ of N are in one-to-one correspondence with symmetric 2 × 2 matrices S over F via the formula

θ(\left[{1 X \atop 1}\right]) = ψ(tr(S[ \begin{matrix}& 1 \\ 1 \end{matrix} ]X)).

We say that θ is non-degenerate if the matrix S is invertible, and we say that θ is split if disc(S) = 1; here disc(S) is the class of − det(S) in F^{×}/F^{×2}. For a fixed S, we define

T = [ \begin{matrix}& 1 \\ 1 \end{matrix} ]\{g ∈ GL(2, F) : ^{t}gSg = det(g)S\}[ \begin{matrix}& 1 \\ 1 \end{matrix} ].  (1)

We embed T into GSp(4, F) via the map

t ↦ [ \begin{matrix}t \\ & det(t)t′ \end{matrix} ],

where for a 2×2-matrix g we write g′ = [ \begin{matrix}& 1 \\ 1 \end{matrix} ] ^{t}g^{−1} [ \begin{matrix}& 1 \\ 1 \end{matrix} ]. The group T normalizes N, so that we can define the semidirect product D = T N. This will be referred to as the Bessel subgroup corresponding to S. For t in T and n in N, we have θ(tnt^{−1}) = θ(n). Thus, if Λ is a character of T, we can define a character Λ ⊗ θ of D by (Λ ⊗ θ)(tn) = Λ(t)θ(n). Whenever we regard ℂ as a one-dimensional representation of D via this character, we denote it by ℂ_{Λ⊗θ}. Let (π, V) be an irreducible, admissible representation of GSp(4, F). A non-zero element of the space Hom_{D}(π, ℂ_{Λ⊗θ}) is called a (Λ, θ)-Bessel functional for π. We say that π admits a (Λ, θ)-Bessel functional if Hom_{D}(π, ℂ_{Λ⊗θ}) is non-zero, and that π admits a unique (Λ, θ)-Bessel functional if Hom_{D}(π, ℂ_{Λ⊗θ}) is one-dimensional.

In this paper we prove three main results about irreducible, admissible representations π of GSp(4, F):

• If π is not one-dimensional, we prove that π admits some (Λ, θ)-Bessel functional; see Theorem 6.1.4.

• If θ is split, we determine the set of Λ for which π admits a (Λ, θ)- Bessel functional, and prove that such functionals are unique; see Proposition 3.4.2, Theorem 6.1.4, Theorem 6.2.2 and Theorem 6.3.2.

• If π is non-supercuspidal, or is in an L-packet with a non-super-cuspidal representation, we determine the set of (Λ, θ) for which π admits a (Λ, θ)-Bessel functional, and prove that such functionals are unique; see Theorem 6.2.2 and Theorem 6.3.2.

We point out that all our results hold independently of the residual characteristic of F. To investigate (Λ, θ)-Bessel functionals for (π, V) we use the P_{3}-module V_{Z^{J}}, the G^{J}-module V_{Z^{J} ,ψ}, and the twisted Jacquet module V_{N,θ}. Here,

P_{3} = GL(3, F) ∩ [ \begin{matrix}∗ & ∗ & ∗ \\ ∗ & ∗ & ∗ \\ & & 1 \end{matrix} ] , Z^{J} = GSp(4, F) ∩ [ \begin{matrix}1 & & & ∗ \\ & 1 \\ & & 1 \\ & & & 1 \end{matrix} ] ,

and G^{J} = GSp(4, F) ∩ [ \begin{matrix}1 & ∗ & ∗ & ∗ \\ & ∗ & ∗ & ∗ \\ & ∗ & ∗ & ∗ \\ & & & 1 \end{matrix} ] .

The P_{3}-module V_{Z^{J}} was computed for all π with trivial central character in [28]; in this paper, we note that these results extend to the general case. The G^{J}-module V_{Z^{J} ,ψ} is closely related to representations of the metaplectic group S̃L(2, F). The twisted Jacquet module V_{N,θ} is especially relevant for nonsupercuspidal representations. Indeed, we completely calculate twisted Jacquet modules of representations parabolically induced from the Klingen or Siegel parabolic subgroups. These methods suffice to treat most representations; for the few remaining families of representations we use theta lifts. As a by-product of our investigations we obtain a characterization of non-generic representations. Namely, the following conditions are equivalent: π is non-generic; the twisted Jacquet module V_{N,θ} is finite-dimensional for all non-degenerate θ; the twisted Jacquet module V_{N,θ} is finite-dimensional for all split θ; the G^{J}-module V_{Z^{J} ,ψ} is of finite length. See Theorem 7.1.4. If an irreducible, admissible representation π admits a (Λ, θ)-Bessel functional, then π has an associated Bessel model. For unramified π admitting a (Λ, θ)- Bessel functional, the works [36] and [4] contain explicit formulas for the spherical vector in such a Bessel model. Other explicit formulas in certain cases of Iwahori-spherical representations appear in [32], [20] and [22]. We note that these works show that all the values of a certain vector in the given Bessel model can be expressed in terms of data depending only on the representation and Λ and θ; in this situation it follows that the Bessel functional is unique. As far as we know, a detailed proof of uniqueness of Bessel functionals in all cases has not yet appeared in the literature.

In the case of odd residual characteristic, and when π appears in a generic L-packet, the main local theorem of [23] gives an ε-factor criterion for the existence of a (Λ, θ)-Bessel functional. There is some overlap between the methods of [23] and the present work. However, the goal of this work is to give a complete and ready account of Bessel functionals for all non-supercuspidal representations. We hope these results will be useful for applications where such specific knowledge is needed.

1 Some definitions

Throughout this work let F be a non-archimedean local field of characteristic zero. Let F̄ be a fixed algebraic closure of F. We fix a non-trivial character ψ : F → ℂ^{×}. The symbol 𝔬 denotes the ring of integers of F, and 𝔭 is the maximal ideal of 𝔬. We let ϖ be a fixed generator of 𝔭. We denote by | · | the normalized absolute value on F, and by ν its restriction to F^{×}. The Hilbert symbol of F will be denoted by (·, ·)_{F}. If Λ is a character of a group, we denote by ℂ_{Λ} the space of the one-dimensional representation whose action is given by Λ. If x = [ \begin{matrix}a & b \\ c & d \end{matrix} ] is a 2 × 2 matrix, then we set x^{∗} = [ \begin{matrix}d & −b \\ −c & a \end{matrix} ]. If X is an l-space, as in 1.1 of [3], and V is a complex vector space, then S(X, V) is the space of locally constant functions X → V with compact support. Let G be an l-group, as in [3], and let H be a closed subgroup. If ρ is a smooth representation of H, we define the compactly induced representation (unnormalized) c-Ind_{H}^{G}(ρ) as in 2.22 of [3]. If (π, V) is a smooth representation of G, and if θ is a character of H, we define the twisted Jacquet module V_{H,θ} as the quotient V/V (H, θ), where V (H, θ) is the span of all vectors π(h)v − θ(h)v for all h in H and v in V.

1.1 Groups

Let

GSp(4, F) = \{g ∈ GL(4, F) : ^{t}gJg = λ(g)J, λ(g) ∈ F^{×}\}, J = [ \begin{matrix}& & & 1 \\ & & 1 \\ & −1 \\ −1 \end{matrix} ] .

The scalar λ(g) is called the multiplier or similitude factor of the matrix g. The Siegel parabolic subgroup P of GSp(4, F) consists of all matrices whose lower left 2 × 2 block is zero. For a matrix A ∈ GL(2, F) set A′ = [ \begin{matrix}& 1 \\ 1 \end{matrix} ]^{t}A^{−1}[ \begin{matrix}& 1 \\ 1 \end{matrix} ]. Then the Levi decomposition of P is P = MN, where

M = \{[ \begin{matrix}A \\ & λA′ \end{matrix} ] : A ∈ GL(2, F), λ ∈ F^{×}\},  (2)

and

N = \{[ \begin{matrix}1 & & y & z \\ & 1 & x & y \\ & & 1 \\ & & & 1 \end{matrix} ] : x, y, z ∈ F \}.  (3)

Let Q be the Klingen parabolic subgroup, i.e.,

Q = GSp(4, F) ∩ [ \begin{matrix}∗ & ∗ & ∗ & ∗ \\ & ∗ & ∗ & ∗ \\ & ∗ & ∗ & ∗ \\ & & & ∗ \end{matrix} ] .  (4)

The Levi decomposition for Q is Q = M_{Q}N_{Q}, where

M_{Q} = \{[ \begin{matrix}t \\ & A \\ & & t^{−1} det(A) \end{matrix} ] : A ∈ GL(2, F), t ∈ F^{×}\},  (5)

and N_{Q} is the Heisenberg group

N_{Q} = \{[ \begin{matrix}1 & x & y & z \\ & 1 & & y \\ & & 1 & −x \\ & & & 1 \end{matrix} ] : x, y, z ∈ F \}.  (6)

The subgroup of Q consisting of all elements with t = 1 and det(A) = 1 is called

the Jacobi group and is denoted by G^{J}. The center of G^{J} is Z^{J} = [ \begin{matrix}1 & & & ∗ \\ & 1 \\ & & 1 \\ & & & 1 \end{matrix} ].

The standard Borel subgroup of GSp(4, F) consists of all upper triangular matrices in GSp(4, F). We let

U = GSp(4, F) ∩ [ \begin{matrix}1 & ∗ & ∗ & ∗ \\ & 1 & ∗ & ∗ \\ & & 1 & ∗ \\ & & & 1 \end{matrix} ]

be its unipotent radical. The following elements of GSp(4, F) represent generators for the eight-element Weyl group,

s_{1} = [ \begin{matrix}& 1 \\ 1 \\ & & & 1 \\ & & 1 \end{matrix} ]  and  s_{2} = [ \begin{matrix}1 \\ & & 1 \\ & −1 \\ & & & 1 \end{matrix} ] .  (7)

1.2 Representations

For a smooth representation π of GSp(4, F) or GL(2, F), we denote by π^{∨} its smooth contragredient. For c_{1}, c_{2} in F^{×}, let ψ_{c_{1},c_{2}} be the character of U defined by

ψ_{c_{1},c_{2}}([ \begin{matrix}1 & x & ∗ & ∗ \\ & 1 & y & ∗ \\ & & 1 & −x \\ & & & 1 \end{matrix} ]) = ψ(c_{1}x + c_{2}y).  (8)

An irreducible, admissible representation (π, V) of GSp(4, F) is called generic if the space Hom_{U}(V, ψ_{c_{1},c_{2}}) is non-zero. This definition is independent of the choice of c_{1}, c_{2}. It is known by [30] that, if non-zero, the space Hom_{U}(V, ψ_{c_{1},c_{2}}) is one-dimensional. Hence, π can be realized in a unique way as a space of functions W : GSp(4, F) → ℂ with the transformation property

W (ug) = ψ_{c_{1},c_{2}}(u)W (g), u ∈ U, g ∈ GSp(4, F),

on which π acts by right translations. We denote this model of π by W(π, ψ_{c_{1},c_{2}}), and call it the Whittaker model of π with respect to c_{1}, c_{2}. We will employ the notation of [35] for parabolically induced representations of GSp(4, F) (all parabolic induction is normalized). For details we refer to the summary given in Sect. 2.2 of [28]. Let χ_{1}, χ_{2} and σ be characters of F^{×}. Then χ_{1} ×χ_{2} ⋊σ denotes the representation of GSp(4, F) parabolically induced from the character of the Borel subgroup which is trivial on U and is given by

diag(a, b, cb^{−1}, ca^{−1}) ⟼ χ_{1}(a)χ_{2}(b)σ(c), a, b, c ∈ F^{×},

on diagonal elements. Let σ be a character of F^{×} and π be an admissible representation of GL(2, F). Then π⋊σ denotes the representation of GSp(4, F) parabolically induced from the representation

\left[{A ∗ \atop cA′}\right] ⟼ σ(c)π(A),  A ∈ GL(2, F), c ∈ F^{×},  (9)

of the Siegel parabolic subgroup P. Let χ be a character of F^{×} and π an admissible representation of GSp(2, F) ≅ GL(2, F). Then χ ⋊ π denotes the representation of GSp(4, F) parabolically induced from the representation

[ \begin{matrix}t & ∗ & ∗ \\ & g & ∗ \\ & & det(g)t^{−1} \end{matrix} ] ⟼ χ(t)π(g),  t ∈ F^{×}, g ∈ GL(2, F),  (10)

of the Klingen parabolic subgroup Q. For a character ξ of F^{×} and a representation (π, V) of GSp(4, F), the twist ξπ is the representation of GSp(4, F) on the same space V given by (ξπ)(g) = ξ(λ(g))π(g) for g in GSp(4, F), where λ is the multiplier homomorphism defined above. A similar definition applies to representations π of GL(2, F); in this case, the multiplier is replaced by the determinant. The behavior of parabolically induced representations under twisting is as follows,

ξ(χ_{1} × χ_{2} ⋊ σ) = χ_{1} × χ_{2} ⋊ ξσ,
ξ(π ⋊ σ) = π ⋊ ξσ,
ξ(χ ⋊ π) = χ ⋊ ξπ.

The irreducible constituents of all parabolically induced representations of GSp(4, F) have been determined in [35]. The following table, which is essentially a reproduction of Table A.1 of [28], provides a summary of these irreducible constituents. In the table, χ, χ_{1}, χ_{2}, ξ and σ stand for characters of F^{×}; the symbol ν denotes the normalized absolute value; π stands for an irreducible, admissible, supercuspidal representation of GL(2, F), and ω_{π} denotes the central character of π. The trivial character of F^{×} is denoted by 1_{F^{×}}, the trivial representation of GL(2, F) by 1_{GL(2)} or 1_{GSp(2)}, depending on the context, the trivial representation of GSp(4, F) by 1_{GSp(4)}, the Steinberg representation of GL(2, F) by St_{GL(2)} or St_{GSp(2)}, depending on the context, and the Steinberg representation of GSp(4, F) by St_{GSp(4)}. The names of the representations given in the “representation” column are taken from [35]. The “tempered” column indicates the condition on the inducing data under which a representation is tempered. The “L^{2}” column indicates which representations are square integrable after an appropriate twist. Finally, the “g” column indicates which representations are generic.

474 Brooks Roberts and Ralf Schmidt

constituents of  representation  tempered L^{2} g
I  χ_{1} × χ_{2} ⋊ σ (irreducible)  χ_{i}, σ unitary  •
II  ν^{1/2}χ × ν^{−1/2}χ ⋊ σ a  χSt_{GL(2)} ⋊ σ  χ, σ unitary  •
(χ^{2} ≠ ν^{±1}, χ ≠ ν^{±3/2}) b  χ1_{GL(2)} ⋊ σ
III  χ × ν ⋊ ν^{−1/2}σ  a  χ ⋊ σSt_{GSp(2)}  χ, σ unitary  •
(χ ∉ {1, ν^{±2}})  b  χ ⋊ σ1_{GSp(2)}
IV  ν^{2} × ν ⋊ ν^{−3/2}σ  a  σSt_{GSp(4)}  σ unitary • •
b  L(ν^{2}, ν^{−1}σSt_{GSp(2)})
c L(ν^{3/2}St_{GL(2)}, ν^{−3/2}σ)
d  σ1_{GSp(4)}
V  νξ × ξ ⋊ ν^{−1/2}σ  a  δ([ξ, νξ], ν^{−1/2}σ)  σ unitary • •
(ξ^{2} = 1, ξ ≠ 1)  b L(ν^{1/2}ξSt_{GL(2)}, ν^{−1/2}σ)
c L(ν^{1/2}ξSt_{GL(2)}, ξν^{−1/2}σ)
d  L(νξ, ξ ⋊ ν^{−1/2}σ)
VI  ν × 1_{F^{×}} ⋊ ν^{−1/2}σ a  τ(S, ν^{−1/2}σ)  σ unitary  •
b  τ(T, ν^{−1/2}σ)  σ unitary
c L(ν^{1/2}St_{GL(2)}, ν^{−1/2}σ)
d  L(ν, 1_{F^{×}} ⋊ ν^{−1/2}σ)
VII  χ ⋊ π (irreducible)  χ, π unitary  •
VIII  1_{F^{×}} ⋊ π  a  τ(S, π)  π unitary  •
b  τ(T, π)  π unitary
IX  νξ ⋊ ν^{−1/2}π  a  δ(νξ, ν^{−1/2}π)  π unitary • •
(ξ ≠ 1, ξπ = π)  b  L(νξ, ν^{−1/2}π)
X  π ⋊ σ (irreducible)  π, σ unitary  •
XI  ν^{1/2}π ⋊ ν^{−1/2}σ  a  δ(ν^{1/2}π, ν^{−1/2}σ)  π, σ unitary • •
(ω_{π} = 1)  b  L(ν^{1/2}π, ν^{−1/2}σ)
Va^{∗}  (supercuspidal)  δ^{∗}([ξ, νξ], ν^{−1/2}σ)  σ unitary •
XIa^{∗}  (supercuspidal)  δ^{∗}(ν^{1/2}π, ν^{−1/2}σ)  π, σ unitary •

In addition to all irreducible, admissible, non-supercuspidal representations, the table also includes two classes of supercuspidal representations denoted by Va^{∗} and XIa^{∗}. The reason that these supercuspidal representations are included in the table is that they are in L-packets with some non-supercuspidal representations. Namely, the Va representation δ([ξ, νξ], ν^{−1/2}σ) and the Va^{∗} representation δ^{∗}([ξ, νξ], ν^{−1/2}σ) form an L-packet, and the XIa representation δ(ν^{1/2}π, ν^{−1/2}σ) and the XIa^{∗} representation δ^{∗}(ν^{1/2}π, ν^{−1/2}σ) form an L-packet; see the paper [8]. Incidentally, the other non-singleton L-packets involving non-supercuspidal representations are the two-element packets {τ(S, ν^{−1/2}σ), τ(T, ν^{−1/2}σ)} (type VIa and VIb), as well as {τ(S, π), τ(T, π)} (type VIIIa and VIIIb).

2 Generalities on Bessel functionals

In this section we gather some definitions, notation, and basic results about Bessel functionals.

2.1 Quadratic extensions

Let D ∈ F^{×}. If D ∉ F^{×2}, then let Δ = \sqrt{D} be a square root of D in F̄, and L = F (Δ). If D ∈ F^{×2}, then let \sqrt{D} be a square root of D in F^{×}, L = F × F, and Δ = (−\sqrt{D}, \sqrt{D}) ∈ L. In both cases L is a two-dimensional F -algebra containing F, L = F +F Δ, and Δ^{2} = D. We will abuse terminology slightly, and refer to L as the quadratic extension associated to D. We define a map γ : L → L called Galois conjugation by γ(x + yΔ) = x − yΔ. Then γ(xy) = γ(x)γ(y) and γ(x + y) = γ(x) + γ(y) for x, y ∈ L, and the fixed points of γ are the elements of F. The group Gal(L/F) of F -automorphisms α : L → L is {1, γ}. We define norm and trace functions N_{L/F} : L → F and T_{L/F} : L → F by N_{L/F} (x) = xγ(x) and T_{L/F} (x) = x + γ(x) for x ∈ L. We let χ_{L/F} be the quadratic character associated to L/F, so that χ_{L/F} (x) = (x, D)_{F} for x ∈ F^{×}.

2.2 2 × 2 symmetric matrices

Let a, b, c ∈ F and set

S = [ \begin{matrix}a & b/2 \\ b/2 & c \end{matrix} ].  (11)

Let D = b^{2}/4 − ac = − det(S). Assume that D ≠ 0. The discriminant disc(S) of S is the class in F^{×}/F^{×2} determined by D. It is known that there exists g ∈ GL(2, F) such that ^{t}gSg is of the form [ \begin{matrix}a_{1} \\ & a_{2} \end{matrix} ] and that (a_{1}, a_{2})_{F} is independent of the choice of g such that ^{t}gSg is diagonal; we define the Hasse invariant ε(S) ∈ {±1} by ε(S) = (a_{1}, a_{2})_{F}. In fact, one has:

\begin{matrix}S & g & ^{t}gSg & disc(S) & ε(S) \\ a ≠ 0, c ≠ 0 & \left[{1 \frac{−b}{2a} \atop 1}\right] & [ \begin{matrix}a \\ & c− \frac{b^{2}}{4a} \end{matrix} ] & ( \frac{b^{2}}{4} − ac)F^{×2} & (a, \frac{b^{2}}{4} − ac)_{F} = (c, \frac{b^{2}}{4} − ac)_{F} \\ a ≠ 0, c = 0 & \left[{1 \frac{−b}{2a} \atop 1}\right] & [ \begin{matrix}a \\ & − \frac{b^{2}}{4a} \end{matrix} ] & F^{×2} & 1 \\ a = 0, c ≠ 0 & \left[{1 \atop 1 − \frac{b}{2c}}\right] & [ \begin{matrix}c \\ & − \frac{b^{2}}{4c} \end{matrix} ] & F^{×2} & 1 \\ a = 0, c = 0 & [ \begin{matrix}1 & 1 \\ 1 & −1 \end{matrix} ] & [ \begin{matrix}b \\ & −b \end{matrix} ] & F^{×2} & 1 \end{matrix}

If disc(S) = F^{×2}, then we say that S is split. If S is split, then for any λ ∈ F^{×} there exists g ∈ GL(2, F) such that ^{t}gSg = [ \begin{matrix}& λ \\ λ \end{matrix} ].

2.3 Another F -algebra

Let S be as in (11) with disc(S) ≠ 0. Set D = b^{2}/4 − ac. We define

A = A_{S} = \{[ \begin{matrix}x−yb/2 & −ya \\ yc & x+yb/2 \end{matrix} ] : x, y ∈ F \}.  (12)

Then, with respect to matrix addition and multiplication, A is a twodimensional F -algebra naturally containing F. One can verify that

A = [ \begin{matrix}& 1 \\ 1 \end{matrix} ]\{g ∈ M_{2}(F) : ^{t}gSg = det(g)S\}[ \begin{matrix}& 1 \\ 1 \end{matrix} ].  (13)

We define T = T_{S} = A^{×}. Let L be the quadratic extension associated to D; we also say that L is the quadratic extension associated to S. We define an isomorphism of F -algebras,

A ⟶^{∼} L,  [ \begin{matrix}x−yb/2 & −ya \\ yc & x+yb/2 \end{matrix} ] ⟼ x + yΔ.  (14)

The restriction of this isomorphism to T is an isomorphism T ⟶^{∼} L^{×}, and we identify characters of T and characters of L^{×} via this isomorphism. The automorphism of A corresponding to the automorphism γ of L will also be denoted by γ. It has the effect of replacing y by −y in the matrix (12). We have det(t) = N_{L/F} (t) for t ∈ A, where we identify elements of A and L via (14).

2.3.1 Lemma. Let T be as above, and assume that L is a field. Let B_{2} be the group of upper triangular matrices in GL(2, F). Then T B_{2} = GL(2, F).

Proof. This can easily be verified using the explicit form of the matrices in T and the assumption D ∉ F^{×2}.

2.4 Bessel functionals

Let a, b and c be in F. Define S as in (11), and define a character θ = θ_{a,b,c} = θ_{S} of N by

θ([ \begin{matrix}1 & & y & z \\ & 1 & x & y \\ & & 1 \\ & & & 1 \end{matrix} ]) = ψ(ax + by + cz) = ψ(tr(S[ \begin{matrix}& 1 \\ 1 \end{matrix} ][ \begin{matrix}y & z \\ x & y \end{matrix}]))  (15)

for x, y, z ∈ F. Every character of N is of this form for uniquely determined a, b, c in F, or, alternatively, for a uniquely determined symmetric 2 × 2 matrix S. We say that θ is non-degenerate if det(S) ≠ 0. Given S with det(S) ≠ 0, let A be as in (12), and let T = A^{×}. We embed T into GSp(4, F) via the map defined by

t ⟼ [ \begin{matrix}t \\ & det(t)t′ \end{matrix} ],  t ∈ T.  (16)

The image of T in GSp(4, F) will also be denoted by T; the usage should be clear from the context. For t ∈ T we have λ(t) = det(t) = N_{L/F} (t). It is easily verified that

θ(tnt^{−1}) = θ(n) for n ∈ N and t ∈ T .

We refer to the semidirect product

D = T N  (17)

as the Bessel subgroup defined by character θ (or, the matrix S). Given a character Λ of T (identified with a character of L^{×} as explained above), we can define a character Λ ⊗ θ of D by

(Λ ⊗ θ)(tn) = Λ(t)θ(n) for n ∈ N and t ∈ T .

Every character of D whose restriction to N coincides with θ is of this form for an appropriate Λ. Now let (π, V) be an admissible representation of GSp(4, F). Let θ be a nondegenerate character of N, and let Λ be a character of the associated group T. We say that π admits a (Λ, θ)-Bessel functional if Hom_{D}(V, ℂ_{Λ⊗θ}) ≠ 0. A non-zero element β of Hom_{D}(V, ℂ_{Λ⊗θ}) is called a (Λ, θ)-Bessel functional for π. If such a β exists, then π admits a model consisting of functions B : GSp(4, F) → ℂ with the Bessel transformation property

B(tng) = Λ(t)θ(n)B(g) for t ∈ T, n ∈ N and g ∈ GSp(4, F),

by associating to each v in V the function B_{v} that is defined by B_{v}(g) = β(π(g)v) for g ∈ GSp(4, F). We note that if π admits a central character ω_{π} and a (Λ, θ)-Bessel functional, then Λ|_{F^{×}} = ω_{π}. For a character σ of F^{×}, it is easy to verify that

Hom_{D}(π, ℂ_{Λ⊗θ}) = Hom_{D}(σπ, ℂ_{(σ∘N_{L/F})Λ⊗θ}).  (18)

If π is irreducible, then, using that π^{∨} ≅ ω_{π}^{−1}π (Proposition 2.3 of [37]), one can also verify that

Hom_{D}(π, ℂ_{Λ⊗θ}) ≅ Hom_{D}(π^{∨}, ℂ_{(Λ∘γ)^{−1}⊗θ}).  (19)

The twisted Jacquet module of V with respect to N and θ is the quotient V_{N,θ} = V/V (N, θ), where V (N, θ) is the subspace spanned by all vectors π(n)v −θ(n)v for v in V and n in N. This Jacquet module carries an action of T induced by the representation π. Evidently, there is a natural isomorphism

Hom_{D}(V, ℂ_{Λ⊗θ}) ≅ Hom_{T} (V_{N,θ}, ℂ_{Λ}).  (20)

Hence, when calculating the possible Bessel functionals on a representation (π, V), a first step often consists in calculating the Jacquet modules V_{N,θ}. We will use this method to calculate the possible Bessel functionals for most of the non-supercuspidal, irreducible, admissible representations of GSp(4, F). The few representations that are inaccessible with this method will be treated using the theta correspondence. In this paper we do not assume that (Λ, θ)-Bessel functionals are unique up to scalars. See Sect. 6.3 for some remarks on uniqueness. Finally, instead of GSp(4, F) as defined in this paper, in the literature it is common to work with the group G′ of g ∈ GL(4, F) such that ^{t}g [ \begin{matrix}& 1_{2} \\ −1_{2} \end{matrix} ] g = λ(g) [ \begin{matrix}& 1_{2} \\ −1_{2} \end{matrix} ] for some λ(g) ∈ F^{×}. For the convenience of the reader, we will explain how to translate statements about Bessel functionals from this paper into statements using G′. The groups GSp(4, F) and G′ are isomorphic via the map i : GSp(4, F) ⟶ G′ defined by i(g) = LgL for g ∈ GSp(4, F), where

L = [ \begin{matrix}& 1 \\ 1 \\ & & 1 \\ & & & 1 \end{matrix} ] . We note that ^{t}L = L = L^{−1}, L^{2} = 1, and the inverse of i is

given by i^{−1}(g′) = Lg′L for g′ ∈ G′. If H is a subgroup of GSp(4, F), then we define H′ = i(H), and refer to H′ as the subgroup of G′ corresponding to H. For example, the subgroup N′ of G′ corresponding to N is

N′ = \{[ \begin{matrix}1 & & x & y \\ & 1 & y & z \\ & & 1 \\ & & & 1 \end{matrix} ] : x, y, z ∈ F \}.

If π is a smooth representation of a subgroup H of GSp(4, F) on a complex vector space V, then we define the representation π′ of H′ on V corresponding to π by the formula π′(g′) = π(i^{−1}(g′)) for g′ ∈ H′. Now let S = [ \begin{matrix}a & b/2 \\ b/2 & c \end{matrix} ] be as above, with det(S) ≠ 0. The character θ′ = θ′_{S} of N′ corresponding to the character θ_{S} of N is given by the formula

θ′([ \begin{matrix}1 & & x & y \\ & 1 & y & z \\ & & 1 \\ & & & 1 \end{matrix} ]) = ψ(ax + by + cz) = ψ(tr(S [ \begin{matrix}x & y \\ y & z \end{matrix}]))

for x, y, z ∈ F. The subgroup T′ = T′_{S} of G′ corresponding to T = T_{S} is

T′ = \{[ \begin{matrix}t \\ & det(t)·^{t}t^{−1} \end{matrix} ] : t ∈ GL(2, F) : ^{t}tSt = det(t)S\}.

More explicitly, the group of t ∈ GL(2, F) such that ^{t}tSt = det(t)S consists of the matrices

t = [ \begin{matrix}x+yb/2 & yc \\ −ya & x−yb/2 \end{matrix} ]  (21)

where x, y ∈ F, x^{2}−y^{2}(b^{2}/4−ac) ≠ 0, with b^{2}/4−ac = − det(S), as usual. With L as above, there is an isomorphism T′ ⟶^{∼} L^{×} given by [ \begin{matrix}t \\ & det(t)·^{t}t^{−1} \end{matrix} ] ↦ x+yΔ for t as in (21). Suppose that Λ is a character of L^{×}; identify Λ with a character of T as explained above. The corresponding character Λ′ of T′ is given by the formula

Λ′([ \begin{matrix}t \\ & det(t)·^{t}t^{−1} \end{matrix} ]) = Λ(x + yΔ)

for t as in (21). Finally, suppose that (π, V) is an admissible representation of GSp(4, F), and let π′ be the representation of G′ on V corresponding to π. There is an equality

Hom_{D′}(V, ℂ_{Λ′⊗θ′}) = Hom_{D}(V, ℂ_{Λ⊗θ}),

with D′ = T′N′. The non-zero elements of Hom_{D′}(V, ℂ_{Λ′⊗θ′}) are called (Λ′, θ′)- Bessel functionals for π′, and the last equality asserts that the set of (Λ′, θ′)- Bessel functionals for π′ is the same as the set of (Λ, θ)-Bessel functionals for π.

2.5 Action on Bessel functionals

There is an action of M, defined in (2), on the set of Bessel functionals. Let (π, V) be an irreducible, admissible representation of GSp(4, F), and let β : V → ℂ be a (Λ, θ)-Bessel functional for π. Let a, b, c ∈ F be such that (15) holds. Let m ∈ M, with m = [ \begin{matrix}g \\ & λg′ \end{matrix} ], where λ ∈ F^{×} and g ∈ GL(2, F). Define m · β : V → ℂ by (m · β)(v) = β(π(m^{−1})v) for v ∈ V. Calculations show that m · β is a (Λ′, θ′)-Bessel functional with θ′ defined by

θ′([ \begin{matrix}1 & & y & z \\ & 1 & x & y \\ & & 1 \\ & & & 1 \end{matrix} ]) = ψ(a′x + b′y + c′z) = ψ(tr(S′[ \begin{matrix}& 1 \\ 1 \end{matrix} ][ \begin{matrix}y & z \\ x & y \end{matrix}])), x, y, z ∈ F,

where

S′ = [ \begin{matrix}a′ & b′/2 \\ b′/2 & c′ \end{matrix} ] = λ^{t}hSh with h = [ \begin{matrix}& 1 \\ 1 \end{matrix} ]g^{−1}[ \begin{matrix}& 1 \\ 1 \end{matrix} ].

Since disc(S′) = disc(S), the quadratic extension L′ associated to S′ is the same as the quadratic extension L associated to S. There is an isomorphism of F -algebras

A′ = A_{S′} ⟶^{∼} A = A_{S}, a ↦ g^{−1}ag.

Let T′ = A′^{×}. Finally, Λ′ : T′ → ℂ^{×} is given by Λ′(t′) = Λ(g^{−1}t′g) for t′ ∈ T′. For example, assume that β′ is a split Bessel functional, i.e., a Bessel functional for which the discriminant of the associated symmetric matrix S′ is the class F^{×2}. By Sect. 2.2 there exists m as above such that β′ = m · β, where the symmetric matrix S associated to the (Λ, θ)-Bessel functional β is

S = [ \begin{matrix}& 1/2 \\ 1/2 \end{matrix} ],  (22)

and

θ([ \begin{matrix}1 & & y & z \\ & 1 & x & y \\ & & 1 \\ & & & 1 \end{matrix} ]) = ψ(y).  (23)

In this case

T = T_{S} = \{[ \begin{matrix}a \\ & b \\ & & a \\ & & & b \end{matrix} ] : a, b ∈ F^{×}\}.  (24)

Sometimes when working with split Bessel functionals it is more convenient to work with the conjugate group

N_{alt} = s_{2}^{−1}Ns_{2} = [ \begin{matrix}1 & ∗ & & ∗ \\ & 1 \\ & ∗ & 1 & ∗ \\ & & & 1 \end{matrix} ]  (25)

and the conjugate character

θ_{alt}([ \begin{matrix}1 & −y & & z \\ & 1 \\ & x & 1 & y \\ & & & 1 \end{matrix} ]) = ψ(y).  (26)

In this case the stabilizer of θ_{alt} is

T_{alt} = \{[ \begin{matrix}a \\ & a \\ & & b \\ & & & b \end{matrix} ] : a, b ∈ F^{×}\}.  (27)

2.6 Galois conjugation of Bessel functionals

The action of M can be used to define the Galois conjugate of a Bessel functional. Let S be as in (11), and let A = A_{S} and T = T_{S}. Define

h_{γ} = \{ \begin{matrix}\left[{1 b/a \atop −1}\right] & if a ≠ 0, \\ \left[{1 \atop −b/c −1}\right] & if a = 0 and c ≠ 0, \\ [ \begin{matrix}& 1 \\ 1 \end{matrix} ] & if a = c = 0. \end{matrix}  (28)

Then h_{γ} ∈ GL(2, F), h_{γ}^{2} = 1, S = ^{t}h_{γ}Sh_{γ} and det(h_{γ}) = −1. Set

g_{γ} = [ \begin{matrix}& 1 \\ 1 \end{matrix} ]h_{γ}^{−1}[ \begin{matrix}& 1 \\ 1 \end{matrix} ] = [ \begin{matrix}& 1 \\ 1 \end{matrix} ]h_{γ}[ \begin{matrix}& 1 \\ 1 \end{matrix} ] ∈ GL(2, F), m_{γ} = [ \begin{matrix}g_{γ} \\ & g′_{γ} \end{matrix} ] ∈ M.

We have g_{γ}T g_{γ}^{−1} = T, and the diagrams

A ⟶^{∼} L
conjugation by g_{γ} ↓  ↓^{γ}
A ⟶^{∼} L

T ⟶^{∼} L^{×}
conjugation by g_{γ} ↓  ↓^{γ}
T ⟶^{∼} L^{×}

commute. Let (π, V) be an irreducible, admissible representation of GSp(4, F), and let β be a (Λ, θ)-Bessel functional for π. We refer to m_{γ} · β as the Galois conjugate of β. We note that m_{γ} · β is a (Λ ∘ γ, θ)-Bessel functional for π. Hence,

Hom_{D}(π, ℂ_{Λ⊗θ}) ≅ Hom_{D}(π, ℂ_{(Λ∘γ)⊗θ}).  (29)

In combination with (19), we get

Hom_{D}(π, ℂ_{Λ⊗θ}) ≅ Hom_{D}(π^{∨}, ℂ_{Λ^{−1}⊗θ}).  (30)

2.7 Waldspurger functionals

Our analysis of Bessel functionals will often involve a similar type of functional on representations of GL(2, F). Let θ and S be as in (15), and let T ≅ L^{×} be the associated subgroup of GL(2, F). Let Λ be a character of T. Let (π, V) be an irreducible, admissible representation of GL(2, F). A (Λ, θ)-Waldspurger functional on π is a non-zero linear map β : V → ℂ such that

β(π(g)v) = Λ(g)β(v) for all v ∈ V and g ∈ T.

For trivial Λ, such functionals were the subject of Proposition 9 of [39] and Proposition 8 of [40]. For general Λ see [38], [34] and Lemme 8 of [40]. The (Λ, θ)-Waldspurger functionals are the non-zero elements of the space Hom_{T} (π, ℂ_{Λ}), and it is known that this space is at most one-dimensional. An obvious necessary condition for Hom_{T} (π, ℂ_{Λ}) ≠ 0 is that Λ|_{F^{×}} equals ω_{π}, the central character of π. By Sect. 2.6, Galois conjugation on T is given by conjugation by an element of GL(2, F). Hence,

Hom_{T} (π, ℂ_{Λ}) ≅ Hom_{T} (π, ℂ_{Λ∘γ}).  (31)

Using π^{∨} ≅ ω_{π}^{−1}π, one verifies that

Hom_{T} (π, ℂ_{Λ}) ≅ Hom_{T} (π^{∨}, ℂ_{(Λ∘γ)^{−1}}).  (32)

In combination with (31), we also have

Hom_{T} (π, ℂ_{Λ}) ≅ Hom_{T} (π^{∨}, ℂ_{Λ^{−1}}).  (33)

Let π^{JL} denote the Jacquet-Langlands lifting of π in the case that π is a discrete series representation, and 0 otherwise. Then, by the discussion on p. 1297 of [38],

dim Hom_{T} (π, ℂ_{Λ}) + dim Hom_{T} (π^{JL}, ℂ_{Λ}) = 1.  (34)

It is easy to see that, for any character σ of F^{×},

Hom_{T} (π, ℂ_{Λ}) = Hom_{T} (σπ, ℂ_{(σ∘N_{L/F})Λ}).  (35)

For Λ such that Λ|_{F^{×}} = σ^{2}, it is known that

dim(Hom_{T} (σSt_{GL(2)}, ℂ_{Λ})) = \{ \begin{matrix}0 & if L is a field and Λ = σ ∘ N_{L/F} , \\ 1 & otherwise; \end{matrix}

(36)

see Proposition 1.7 and Theorem 2.4 of [38]. As in the case of Bessel functionals, we call a Waldspurger functional split if the discriminant of the associated matrix S lies in F^{×2}. By Lemme 8 of [40], an irreducible, admissible, infinitedimensional representation of GL(2, F) admits a split (Λ, θ)-Waldspurger functional with respect to any character Λ of T that satisfies Λ|_{F^{×}} = ω_{π} (this can also be proved in a way analogous to the proof of Proposition 3.4.2 below, utilizing the standard zeta integrals for GL(2)).

3 Split Bessel functionals

Irreducible, admissible, generic representations of GSp(4, F) admit a theory of zeta integrals, and every zeta integral gives rise to a split Bessel functional. As a consequence, generic representations admit all possible split Bessel functionals; see Proposition 3.4.2 below for a precise formulation. To put the theory of zeta integrals on a solid foundation, we will use P_{3}-theory. The group P_{3}, defined below, plays a role in the representation theory of GSp(4) similar to the “mirabolic” subgroup in the theory for GL(n). Some of what follows is a generalization of Sects. 2.5 and 2.6 of [28], where P_{3}-theory was developed under the assumption of trivial central character. The general case requires only minimal modifications. While every generic representation admits split Bessel functionals, we will see that the converse is not true. P_{3}-theory can also be used to identify the nongeneric representations that admit a split Bessel functional. This is explained in Sect. 3.5 below.

3.1 The group P_{3} and its representations

Let P_{3} be the subgroup of GL(3, F) defined as the intersection

P_{3} = GL(3, F) ∩ [ \begin{matrix}∗ & ∗ & ∗ \\ ∗ & ∗ & ∗ \\ & & 1 \end{matrix} ] .

We recall some facts about this group, following [3]. Let

U_{3} = P_{3} ∩ [ \begin{matrix}1 & ∗ & ∗ \\ & 1 & ∗ \\ & & 1 \end{matrix} ] , N_{3} = P_{3} ∩ [ \begin{matrix}1 & & ∗ \\ & 1 & ∗ \\ & & 1 \end{matrix} ] .

We define characters Θ and Θ′ of U_{3} by

Θ([ \begin{matrix}1 & u_{12} & ∗ \\ & 1 & u_{23} \\ & & 1 \end{matrix} ]) = ψ(u_{12} + u_{23}), Θ′([ \begin{matrix}1 & u_{12} & ∗ \\ & 1 & u_{23} \\ & & 1 \end{matrix} ]) = ψ(u_{23}).

If (π, V) is a smooth representation of P_{3}, we may consider the twisted Jacquet modules

V_{U_{3},Θ} = V/V (U_{3}, Θ), V_{U_{3},Θ′} = V/V (U_{3}, Θ′)

where V (U_{3}, Θ) (resp. V (U_{3}, Θ′)) is spanned by all elements of the form π(u)v− Θ(u)v (resp. π(u)v−Θ′(u)v) for v in V and u in U_{3}. Note that V_{U_{3},Θ′} carries an action of the subgroup [ \begin{matrix}∗ \\ & 1 \\ & & 1 \end{matrix} ] ≅ F^{×} of P_{3}. We may also consider the Jacquet module V_{N_{3}} = V/V (N_{3}), where V (N_{3}) is the space spanned by all vectors of the form π(u)v − v for v in V and u in N_{3}. Note that V_{N_{3}} carries an action of the subgroup [ \begin{matrix}∗ & ∗ \\ ∗ & ∗ \\ & & 1 \end{matrix} ] ≅ GL(2, F) of P_{3}. Next we define three classes of smooth representations of P_{3}, associated with the groups GL(0), GL(1) and GL(2). Let

τ_{GL(0)}^{P_{3}}(1) := c-Ind_{U_{3}}^{P_{3}}(Θ),  (37)

where c-Ind denotes compact induction. Then τ_{GL(0)}^{P_{3}}(1) is a smooth, irreducible representation of P_{3}. Next, let χ be a smooth representation of GL(1, F) ≅ F^{×}. Define a representation χ ⊗ Θ′ of the subgroup [ \begin{matrix}∗ & ∗ & ∗ \\ & 1 & ∗ \\ & & 1 \end{matrix} ] of P_{3} by

(χ ⊗ Θ′)([ \begin{matrix}a & ∗ & ∗ \\ & 1 & y \\ & & 1 \end{matrix} ]) = χ(a)ψ(y).

Then

τ_{GL(1)}^{P_{3}}(χ) := c-Ind_{[ \begin{matrix}∗ & ∗ & ∗ \\ & 1 & ∗ \\ & & 1 \end{matrix} ]}^{P_{3}}(χ ⊗ Θ′)

is a smooth representation of P_{3}. It is irreducible if and only if χ is onedimensional. Finally, let ρ be a smooth representation of GL(2, F). We define the representation τ_{GL(2)}^{P_{3}}(ρ) of P_{3} to have the same space as ρ, and action given by

τ_{GL(2)}^{P_{3}}(ρ)([ \begin{matrix}a & b & ∗ \\ c & d & ∗ \\ & & 1 \end{matrix} ]) = ρ([ \begin{matrix}a & b \\ c & d \end{matrix} ]).  (38)

Evidently, τ_{GL(2)}^{P_{3}}(ρ) is irreducible if and only if ρ is irreducible.

3.1.1 Proposition. Let notations be as above.

i) Every irreducible, smooth representation of P_{3} is isomorphic to exactly

one of

τ_{GL(0)}^{P_{3}}(1),  τ_{GL(1)}^{P_{3}}(χ),  τ_{GL(2)}^{P_{3}}(ρ),

where χ is a character of F^{×} and ρ is an irreducible, admissible representation of GL(2, F). Moreover, the equivalence classes of χ and ρ are uniquely determined.

ii) Let (π, V) be a smooth representation of P_{3} of finite length. Then there exists a chain of P_{3} subspaces

0 ⊂ V_{2} ⊂ V_{1} ⊂ V_{0} = V

with the following properties,

V_{2} ≅ dim(V_{U_{3},Θ}) · τ_{GL(0)}^{P_{3}}(1),
V_{1}/V_{2} ≅ τ_{GL(1)}^{P_{3}}(V_{U_{3},Θ′}),
V_{0}/V_{1} ≅ τ_{GL(2)}^{P_{3}}(V_{N_{3}}).

Proof. See 5.1 – 5.15 of [3].

3.2 P_{3}-theory for arbitrary central character

It is easy to verify that any element of the Klingen parabolic subgroup Q can be written in a unique way as

[ \begin{matrix}ad−bc \\ & a & b \\ & c & d \\ & & & 1 \end{matrix} ] [ \begin{matrix}1 & −y & x & z \\ & 1 & & x \\ & & 1 & y \\ & & & 1 \end{matrix} ] [ \begin{matrix}u \\ & u \\ & & u \\ & & & u \end{matrix} ]  (39)

with [ \begin{matrix}a & b \\ c & d \end{matrix} ] ∈ GL(2, F), x, y, z ∈ F, and u ∈ F^{×}. Let Z^{J} be the center of the Jacobi group, consisting of all elements of GSp(4) of the form

[ \begin{matrix}1 & & & ∗ \\ & 1 \\ & & 1 \\ & & & 1 \end{matrix} ] .  (40)

Evidently, Z^{J} is a normal subgroup of Q with Z^{J} ≅ F. Let (π, V) be a smooth representation of GSp(4, F). Let V (Z^{J}) be the span of all vectors v − π(z)v, where v runs through V and z runs through Z^{J}. Then V (Z^{J}) is preserved by the action of Q. Hence Q acts on the quotient V_{Z^{J}} := V/V (Z^{J}). Let Q̄ be the subgroup of Q consisting of all elements of the form (39) with u = 1, i.e.,

Q̄ = GSp(4) ∩ [ \begin{matrix}∗ & ∗ & ∗ & ∗ \\ & ∗ & ∗ & ∗ \\ & ∗ & ∗ & ∗ \\ & & & 1 \end{matrix} ] .

The map

i([ \begin{matrix}ad−bc \\ & a & b \\ & c & d \\ & & & 1 \end{matrix} ] [ \begin{matrix}1 & −y & x & z \\ & 1 & & x \\ & & 1 & y \\ & & & 1 \end{matrix} ]) = [ \begin{matrix}a & b \\ c & d \\ & & 1 \end{matrix} ] [ \begin{matrix}1 & & x \\ & 1 & y \\ & & 1 \end{matrix} ]  (41)

establishes an isomorphism Q̄/Z^{J} ≅ P_{3}. Recall the character ψ_{c_{1},c_{2}} of U defined in (8). Note that U maps onto U_{3} under the map (41), and that the diagrams

\xymatrix{U \ar[r]^{i} \ar[dr]_{ψ_{−1,1}} & U_{3} \ar[d]^{Θ} & U \ar[r]^{i} \ar[dr]_{ψ_{−1,0}} & U_{3} \ar[d]^{Θ′} \\  & ℂ^{×} &  & ℂ^{×}}

are commutative. The radical N_{Q} (see (6)) maps onto N_{3} under the map (41). The following theorem is exactly like Theorem 2.5.3 of [28], except that the hypothesis of trivial central character is removed.

3.2.1 Theorem. Let (π, V) be an irreducible, admissible representation of GSp(4, F). The quotient V_{Z^{J}} = V/V (Z^{J}) is a smooth representation of Q̄/Z^{J}, and hence, via the map (41), defines a smooth representation of P_{3}. As a representation of P_{3}, V_{Z^{J}} has finite length. Hence, V_{Z^{J}} has a finite filtration by P_{3} subspaces such that the successive quotients are irreducible and of the form τ_{GL(0)}^{P_{3}}(1), τ_{GL(1)}^{P_{3}}(χ) or τ_{GL(2)}^{P_{3}}(ρ) for some character χ of F^{×}, or some irreducible, admissible representation ρ of GL(2, F). Moreover, the following statements hold: i) There exists a chain of P_{3} subspaces

0 ⊂ V_{2} ⊂ V_{1} ⊂ V_{0} = V_{Z^{J}}

such that

V_{2} ≅ dim Hom_{U}(V, ψ_{−1,1}) · τ_{GL(0)}^{P_{3}}(1),
V_{1}/V_{2} ≅ τ_{GL(1)}^{P_{3}}(V_{U,ψ_{−1,0}}),
V_{0}/V_{1} ≅ τ_{GL(2)}^{P_{3}}(V_{N_{Q}}).

Here, the vector space V_{U,ψ_{−1,0}} admits a smooth action of GL(1, F) ≅ F^{×} induced by the operators

π([ \begin{matrix}a \\ & a \\ & & 1 \\ & & & 1 \end{matrix} ]), a ∈ F^{×},

and V_{N_{Q}} admits a smooth action of GL(2, F) induced by the operators

π([ \begin{matrix}det g \\ & g \\ & & 1 \end{matrix} ]), g ∈ GL(2, F).

ii) The representation π is generic if and only if V_{2} ≠ 0, and if π is generic, then V_{2} ≅ τ_{GL(0)}^{P_{3}}(1).

iii) We have V_{2} = V_{Z^{J}} if and only if π is supercuspidal. If π is supercuspidal and generic, then V_{Z^{J}} = V_{2} ≅ τ_{GL(0)}^{P_{3}}(1) is non-zero and irreducible. If π is supercuspidal and non-generic, then V_{Z^{J}} = V_{2} = 0.

Proof. This is an application of Proposition 3.1.1. See Theorem 2.5.3 of [28] for the details of the proof.

Given an irreducible, admissible representation (π, V) of GSp(4, F), one can calculate the semisimplifications of the quotients V_{0}/V_{1} and V_{1}/V_{2} in the P_{3}- filtration from the Jacquet modules of π with respect to the Siegel and Klingen parabolic subgroups. The results are exactly the same as in Appendix A.4 of [28] (where it was assumed that π has trivial central character). Note that there is a typo in Table A.5 of [28]: The entry for Vd in the “s.s.(V_{0}/V_{1})” column should be τ_{GL(2)}^{P_{3}}(ν(ν^{−1/2}σ × ν^{−1/2}ξσ)).

3.3 Generic representations and zeta integrals

Let π be an irreducible, admissible, generic representation of GSp(4, F). Recall from Sect. 1.2 that W(π, ψ_{c_{1},c_{2}}) denotes the Whittaker model of π with respect to the character ψ_{c_{1},c_{2}} of U. For W in W(π, ψ_{c_{1},c_{2}}) and s ∈ ℂ, we define the zeta integral Z(s, W) by

Z(s, W) = ∫_{F^{×}} ∫_{F} W ([ \begin{matrix}a \\ & a \\ & x & 1 \\ & & & 1 \end{matrix} ])|a|^{s−3/2} dx d^{×}a.  (42)

It was proved in Proposition 2.6.3 of [28] that there exists a real number s_{0}, independent of W, such that Z(s, W) converges for ℜ(s) > s_{0} to an element of ℂ(q^{−s}). In particular, all zeta integrals have meromorphic continuation to all of ℂ. Let I(π) be the ℂ-vector subspace of ℂ(q^{−s}) spanned by all Z(s, W) for W in W(π, ψ_{c_{1},c_{2}}). It is easy to see that I(π) is independent of the choice of ψ and c_{1}, c_{2} in F^{×}.

3.3.1 Proposition. Let π be a generic, irreducible, admissible representation of GSp(4, F). Then I(π) is a non-zero ℂ[q^{−s}, q^{s}]-module containing ℂ, and there exists R(X) ∈ ℂ[X] such that R(q^{−s})I(π) ⊂ ℂ[q^{−s}, q^{s}], so that I(π) is a fractional ideal of the principal ideal domain ℂ[q^{−s}, q^{s}] whose quotient field is ℂ(q^{−s}). The fractional ideal I(π) admits a generator of the form 1/Q(q^{−s}) with Q(0) = 1, where Q(X) ∈ ℂ[X].

Proof. The proof is almost word for word the same as that of Proposition 2.6.4 of [28]. The only difference is that, in the calculation starting at the bottom of p. 79 of [28], the element q is taken from Q̄ instead of Q.

The quotient 1/Q(q^{−s}) in this proposition is called the L-factor of π, and denoted by L(s, π). If π is supercuspidal, then L(s, π) = 1. The L-factors for all irreducible, admissible, generic, non-supercuspidal representations are listed in Table A.8 of [28]. By definition,

\frac{Z(s, W)}{L(s, π)} ∈ ℂ[q^{s}, q^{−s}]  (43)

for all W in W(π, ψ_{c_{1},c_{2}}).

3.4 Generic representations admit split Bessel functionals

In this section we will prove that an irreducible, admissible, generic representation of GSp(4, F) admits split Bessel functionals with respect to all characters Λ of T. This is a characteristic feature of generic representations, which will follow from Proposition 3.5.1 in the next section.

3.4.1 Lemma. Let (π, V) be an irreducible, admissible, generic representation of GSp(4, F). Let σ be a unitary character of F^{×}, and let s ∈ ℂ be arbitrary. Then there exists a non-zero functional f_{s,σ} : V → ℂ with the following properties.

i) For all x, y, z ∈ F and v ∈ V,

f_{s,σ}(π([ \begin{matrix}1 & & y & z \\ & 1 & x & y \\ & & 1 \\ & & & 1 \end{matrix} ])v) = ψ(y)f_{s,σ}(v).  (44)

ii) For all a ∈ F^{×} and v ∈ V,

f_{s,σ}(π([ \begin{matrix}a \\ & 1 \\ & & a \\ & & & 1 \end{matrix} ])v) = σ(a)^{−1}|a|^{−s+1/2}f_{s,σ}(v).  (45)

Proof. We may assume that V = W(π, ψ_{c_{1},c_{2}}) with c_{1} = 1. Let s_{0} ∈ ℝ be such that Z(s, W) is absolutely convergent for ℜ(s) > s_{0}. Then the integral

Z_{σ}(s, W) = ∫_{F^{×}} ∫_{F} W ([ \begin{matrix}a \\ & a \\ & x & 1 \\ & & & 1 \end{matrix} ])|a|^{s−3/2}σ(a) dx d^{×}a  (46)

is also absolutely convergent for ℜ(s) > s_{0}, since σ is unitary. Note that these are the zeta integrals for the twisted representation σπ. Therefore, by (43), the quotient Z_{σ}(s, W)/L(s, σπ) is in ℂ[q^{−s}, q^{s}] for all W ∈ W(π, ψ_{c_{1},c_{2}}). We may therefore define, for any complex s,

f_{s,σ}(W) = \frac{Z_{σ}(s, π(s_{2})W)}{L(s, σπ)},  (47)

where s_{2} is as in (7). Straightforward calculations using the definition (46) show that (44) and (45) are satisfied for ℜ(s) > s_{0}. Since both sides depend holomorphically on s, these identities hold on all of ℂ.

3.4.2 Proposition. Let (π, V) be an irreducible, admissible and generic representation of GSp(4, F). Let ω_{π} be the central character of π. Then π admits a split (Λ, θ)-Bessel functional with respect to any character Λ of T that satisfies Λ|_{F^{×}} = ω_{π}.

Proof. Let θ be as in (23) with T as in (24). Let s ∈ ℂ and σ be a unitary character of F^{×} such that

Λ([ \begin{matrix}a \\ & 1 \\ & & a \\ & & & 1 \end{matrix} ]) = σ(a)^{−1}|a|^{−s+1/2} for all a ∈ F^{×}.

Let f_{s,σ} be as in Lemma 3.4.1. By (45),

f_{s,σ}(π([ \begin{matrix}a \\ & 1 \\ & & a \\ & & & 1 \end{matrix} ])v) = Λ(a)f_{s,σ}(v)  for all a ∈ F^{×}.  (48)

Since Λ|_{F^{×}} = ω_{π} we have in fact f_{s,σ}(π(t)v) = Λ(t)f_{s,σ}(v) for all t ∈ T. Hence f_{s,σ} is a Bessel functional as desired.

3.5 Split Bessel functionals for non-generic representations

The converse of Proposition 3.4.2 is not true: There exist irreducible, admissible, non-generic representations of GSp(4, F) which admit split Bessel functionals. This follows from the following proposition. In fact, using this result and the P_{3}-filtrations listed in Table A.6 of [28], one can precisely identify which non-generic representations admit split Bessel functionals. Other than in the generic case, the possible characters Λ of T are restricted to a finite number.

3.5.1 Proposition. Let (π, V) be an irreducible, admissible and non-generic representation of GSp(4, F). Let the semisimplification of the quotient V_{1} = V_{1}/V_{2} in the P_{3}-filtration of π be given by ∑_{i=1}^{n} τ_{GL(1)}^{P_{3}}(χ_{i}) with characters χ_{i} of F^{×}.

i) π admits a split Bessel functional if and only if the quotient V_{1} in the P_{3}-filtration of π is non-zero.

ii) Let β be a non-zero (Λ, θ)-Bessel functional, with θ as in (23), and a character Λ of the group T explicitly given in (24). Then there exists an i for which

Λ([ \begin{matrix}a \\ & 1 \\ & & a \\ & & & 1 \end{matrix} ]) = |a|^{−1}χ_{i}(a)  for all a ∈ F^{×}.  (49)

iii) If V_{1} is non-zero, then there exists an i such that π admits a split (Λ, θ)- Bessel functional with respect to a character Λ of T satisfying (49).

iv) The space of split (Λ, θ)-Bessel functionals is zero or one-dimensional.

v) The representation π does not admit any split Bessel functionals if and only if π is of type IVd, Vd, VIb, VIIIb, IXb, or is supercuspidal.

Proof. Let N_{alt} be as in (25) and θ_{alt} be as in (26). We use the fact that any (Λ, θ_{alt})-Bessel functional factors through the twisted Jacquet module V_{N_{alt},θ_{alt}}. To calculate this Jacquet module, we use the P_{3}-filtration of Theorem 3.2.1. Since π is non-generic, the P_{3}-filtration simplifies to

0 ⊂ V_{1} ⊂ V_{0} = V_{Z^{J}} ,

with V_{1} of type τ_{GL(1)}^{P_{3}} and V_{0}/V_{1} of type τ_{GL(2)}^{P_{3}}. Taking further twisted Jacquet modules and observing Lemma 2.5.6 of [28], it follows that

V_{N_{alt},θ_{alt}} = (V_{1})_{[ \begin{matrix}1 \\ ∗ & 1 & ∗ \\ & & 1 \end{matrix} ],ψ}, where ψ([ \begin{matrix}1 \\ x & 1 & y \\ & & 1 \end{matrix} ]) = ψ(y).

By Lemma 2.5.5 of [28], after suitable renaming,

0 = J_{n} ⊂ … ⊂ J_{1} ⊂ J_{0} = (V_{1})_{[ \begin{matrix}1 \\ ∗ & 1 & ∗ \\ & & 1 \end{matrix} ],ψ},

where J_{i}/J_{i+1} is one-dimensional, and diag(a, 1, 1) acts on J_{i}/J_{i+1} by |a|^{−1}χ_{i}(a). Table A.6 of [28] shows that all the χ_{i} are pairwise distinct. This proves i), ii), iii) and iv).

v) If π is one of the representations mentioned in v), then V_{1}/V_{2} = 0 by Theorem 3.2.1 (in the supercuspidal case), or by Table A.6 in [28] (in the nonsupercuspidal case). By part i), π does not admit a split Bessel functional. For any representation not mentioned in v), the quotient V_{1}/V_{2} is non-zero, so that a split Bessel functional exists by iii).

4 Theta correspondences

Let S be as in (11), and let θ = θ_{S} be as in (15). Let (π, V) be an irreducible, admissible representation of GSp(4, F), and let (σ, W) be an irreducible, admissible representation of GO(X), where X is an even-dimensional, symmetric, bilinear space. Let ω be the Weil representation of the group R, consisting of the pairs (g, h) ∈ GSp(4, F) × GO(X) with the same similitude factors, on the Schwartz space S(X^{2}). Assume that the pair (π, σ) occurs in the theta correspondence defined by ω, i.e., Hom_{R}(ω, π ⊗ σ) ≠ 0. It is a theme in the theory of the theta correspondence to relate the twisted Jacquet module V_{N,θ} of π to invariant functionals on σ; a necessary condition for the non-vanishing of V_{N,θ} is that X represents S. See for example the remarks in Sect. 6 of [26]. Applications to (Λ, θ_{S})-Bessel functionals also require the involvement of T. The idea is roughly as follows. The group T is contained in M. Moreover, ω(m, h) for (m, h) in R ∩ (M × GO(X)) is given by an action of such pairs on X^{2}. The study of this action leads to the definition of certain compatible embeddings of T into GO(X). Using these embeddings allows us to show that if π has a (Λ, θ)-Bessel functional, then σ admits a non-zero functional transforming according to Λ^{−1}. After setting up notations and studying the embeddings of T mentioned above, we obtain the main result of this section, Theorem 4.4.6. Section 4.7 contains the applications to Bessel functionals.

4.1 The spaces

In this section we will consider non-degenerate symmetric bilinear spaces (X, ⟨·, ·⟩) over F such that

dim X = 2, or dim X = 4 and disc(X) = 1.  (50)

We begin by recalling the constructions of the isomorphism classes of these spaces, and the characterization of their similitude groups. Let m ∈ F^{×}, A = A_{[ \begin{matrix}1 \\ & −m \end{matrix} ]} and T = A^{×} be as in Sect. 2.3, so that

A = \{[ \begin{matrix}x & −y \\ −ym & x \end{matrix} ] : x, y ∈ F \}, T = A^{×} = \{[ \begin{matrix}x & −y \\ −ym & x \end{matrix} ] : x, y ∈ F, x^{2} − y^{2}m ≠ 0\}.

(51)

Let λ ∈ F^{×}. Define a non-degenerate two-dimensional symmetric bilinear space (X_{m,λ}, ⟨·, ·⟩_{m,λ}) by

X_{m,λ} = A_{[ \begin{matrix}1 \\ & −m \end{matrix} ]},  ⟨x_{1}, x_{2}⟩_{m,λ} = λ tr(x_{1}x_{2}^{∗})/2, x_{1}, x_{2} ∈ X_{m,λ}. (52)

Here, ∗ is the canonical involution of 2×2 matrices, given by [ \begin{matrix}a & b \\ c & d \end{matrix} ]^{∗} = [ \begin{matrix}d & −b \\ −c & a \end{matrix} ]. Define a homomorphism ρ : T → GSO(X_{m,λ}) by ρ(t)x = tx for x ∈ X_{m,λ}. We also recall the Galois conjugation map γ : A → A from Sect. 2.3; it is given by γ(x) = x^{∗} for x ∈ A. The map γ can be regarded as an F linear endomorphism

γ : X_{m,λ} ⟶ X_{m,λ},  (53)

and as such is contained in O(X_{m,λ}) but not in SO(X_{m,λ}).

4.1.1 Lemma. If (X_{m,λ}, ⟨·, ·⟩_{m,λ}) is as in (52), then disc(X_{m,λ}) = mF^{×2}, ε(X_{m,λ}) = (λ, m), and the homomorphism ρ is an isomorphism, so that

ρ : T ⟶^{∼} GSO(X_{m,λ}).  (54)

The image ρ(T) and the map γ generate GO(X_{m,λ}). If (X_{m,λ}, ⟨·, ·⟩_{m,λ}) and (X_{m′,λ′}, ⟨·, ·⟩_{m′,λ′}) are as in (52), then (X_{m,λ}, ⟨·, ·⟩_{m,λ}) ≅ (X_{m′,λ′}, ⟨·, ·⟩_{m′,λ′}) if and only if mF^{×2} = m′F^{×2} and (λ, m)_{F} = (λ′, m′)_{F}. Every two-dimensional, non-degenerate symmetric bilinear space over F is isomorphic (X_{m,λ}, ⟨·, ·⟩_{m,λ}) for some m and λ.

Proof. Let m, λ ∈ F^{×}. In X_{m,λ} let x_{1} = [ \begin{matrix}1 \\ & 1 \end{matrix} ] and x_{2} = [ \begin{matrix}& 1 \\ m \end{matrix}]. Then x_{1}, x_{2} is a basis for X_{m,λ}, and in this basis the matrix for X_{m,λ} is λ [ \begin{matrix}1 \\ & −m \end{matrix} ]. Calculations using this matrix show that disc(X_{m,λ}) = mF^{×2} and ε(X_{m,λ}) = (λ, m)_{F}. The map ρ is clearly injective. To see that ρ is surjective, let h ∈ GSO(X_{m,λ}). Write h in the ordered basis x_{1}, x_{2} so that h = [ \begin{matrix}h_{1} & h_{2} \\ h_{3} & h_{4} \end{matrix} ]. By the definition of GSO(X_{m,λ}), we have ^{t}hλ [ \begin{matrix}1 \\ & −m \end{matrix} ] h = det(h)λ [ \begin{matrix}1 \\ & −m \end{matrix} ]. By the definition of T, this implies that t = [ \begin{matrix}& 1 \\ 1 \end{matrix} ] h [ \begin{matrix}& 1 \\ 1 \end{matrix} ] ∈ T. Hence, h = [ \begin{matrix}h_{1} & h_{3}m \\ h_{3} & h_{1} \end{matrix} ] for some h_{1}, h_{3} ∈ F. Calculations now show that ρ(t)x_{1} = h(x_{1}) and ρ(t)x_{2} = h(x_{2}), so that ρ(t) = h. This proves the first assertion. The second assertion follows from the fact that two non-degenerate symmetric bilinear spaces over F with the same finite dimension are isomorphic if and only if they have the same discriminant and Hasse invariant. For the final assertion, let (X, ⟨·, ·⟩) be a two-dimensional, non-degenerate symmetric bilinear space over F. There exists a basis for X with respect to which the matrix for X is of the form [ \begin{matrix}α_{1} \\ & α_{2} \end{matrix} ] for some α_{1}, α_{2} ∈ F^{×}. Then disc(X) = −α_{1}α_{2}F^{×2} and ε(X) = (α_{1}, α_{2})_{F}. An argument shows that there exists λ ∈ F^{×} such that (λ, disc(X))_{F} = ε(X). We now have (X, ⟨·, ·⟩) ≅ (X_{m,λ}, ⟨·, ·⟩_{m,λ}) with m = disc(X) because both spaces have the same discriminant and Hasse invariant.

Next, define a four-dimensional non-degenerate symmetric bilinear space over F by setting

X_{M_{2}} = M_{2}(F),  ⟨x_{1}, x_{2}⟩_{M_{2}} = tr(x_{1}x_{2}^{∗})/2, x_{1}, x_{2} ∈ X_{M_{2}}.  (55)

Here, ∗ is the canonical involution of 2×2 matrices, given by [ \begin{matrix}a & b \\ c & d \end{matrix} ]^{∗} = [ \begin{matrix}d & −b \\ −c & a \end{matrix} ]. Define ρ : GL(2, F)×GL(2, F) → GSO(X_{M_{2}}) by ρ(g_{1}, g_{2})x = g_{1}xg_{2}^{∗} for g_{1}, g_{2} ∈ GL(2, F) and x ∈ X_{M_{2}}. The map ∗ : X_{M_{2}} → X_{M_{2}} is contained in O(X_{M_{2}}) but not in SO(X_{M_{2}}). Finally, let H be the division quaternion algebra over F. Let 1, i, j, k be a quaternion algebra basis for H, i.e.,

H = F + F i + F j + F k, i^{2} ∈ F^{×}, j^{2} ∈ F^{×}, k = ij, ij = −ji.  (56)

Let ∗ be the canonical involution on H so that (a + b · i + c · j + d · k)^{∗} = a − b · i − c · j − d · k, and define the norm and trace functions N, T : H → F by N(x) = xx^{∗} and T(x) = x + x^{∗} for x ∈ H. Define another four-dimensional non-degenerate symmetric bilinear space over F by setting

X_{H} = H,  ⟨x_{1}, x_{2}⟩_{H} = T(x_{1}x_{2}^{∗})/2, x_{1}, x_{2} ∈ X_{H}.  (57)

Define ρ : H^{×} × H^{×} → GSO(X_{H}) by ρ(h_{1}, h_{2})x = h_{1}xh_{2}^{∗} for h_{1}, h_{2} ∈ H^{×} and x ∈ X_{H}. The map ∗ : X_{H} → X_{H} is contained in O(X_{H}) but not in SO(X_{H}).

4.1.2 Lemma. The symmetric bilinear space (X_{M_{2}}, ⟨·, ·⟩_{M_{2}}) is non-degenerate, has dimension four, discriminant disc(X_{M_{2}}) = 1, and Hasse invariant ε(X_{M_{2}}) = (−1, −1)_{F}. The (X_{H}, ⟨·, ·⟩_{H}) symmetric bilinear space is non-degenerate, has dimension four, discriminant disc(X_{H}) = 1, and Hasse invariant ε(X_{H}) = −(−1, −1)_{F}. The sequences

1 ⟶ F^{×} ⟶ GL(2, F) × GL(2, F) ⟶^{ρ} GSO(X_{M_{2}}) ⟶ 1,  (58)
1 ⟶ F^{×} ⟶ H^{×} × H^{×} ⟶^{ρ} GSO(X_{H}) ⟶ 1  (59)

are exact; here, the second maps send a to (a, a^{−1}) for a ∈ F^{×}. The image ρ(GL(2, F) × GL(2, F)) and the map ∗ generate GO(X_{M_{2}}), and the image ρ(H^{×} × H^{×}) and the map ∗ generate GO(X_{H}). Every four-dimensional, nondegenerate symmetric linear space over F of discriminant 1 is isomorphic to (X_{M_{2}}, ⟨·, ·⟩_{M_{2}}) or (X_{H}, ⟨·, ·⟩_{H}).

Proof. See, for example, Sect. 2 of [27].

4.2 Embeddings

Suppose that (X, ⟨·, ·⟩) satisfies (50). We define an action of the group GL(2, F) × GO(X) on the set X^{2} by

(g, h) · (x_{1}, x_{2}) = (hx_{1}, hx_{2})g^{−1} = (g′_{1}hx_{1} + g′_{3}hx_{2}, g′_{2}hx_{1} + g′_{4}hx_{2}) (60)

for (x_{1}, x_{2}) ∈ X^{2}, h ∈ GO(X) and g ∈ GL(2, F) with g^{−1} = [ \begin{matrix}g′_{1} & g′_{2} \\ g′_{3} & g′_{4} \end{matrix} ]. For S as

in (11) with det(S) ≠ 0, we define

Ω = Ω_{S} = Ω_{S,(X,⟨·,·⟩)} = \{(x_{1}, x_{2}) ∈ X^{2} : [ \begin{matrix}⟨x_{1},x_{1}⟩ & ⟨x_{1},x_{2}⟩ \\ ⟨x_{1},x_{2}⟩ & ⟨x_{2},x_{2}⟩ \end{matrix} ] = S\}.  (61)

We say that (X, ⟨·, ·⟩) represents S if the set Ω is non-empty.

4.2.1 Lemma. Let (X, ⟨·, ·⟩) be a non-degenerate symmetric bilinear space over F satisfying (50), and let S be as in (11) with det(S) ≠ 0. The subgroup

B = B_{S} = {(g, h) ∈ GL(2, F) × GO(X) : ^{t}gSg = det(g)S and det(g) = λ(h)}

(62)

maps Ω = Ω_{S} to itself under the action of GL(2, F) × GO(X) on X^{2}.

Proof. Let (g, h) ∈ B, and let g = [ \begin{matrix}g_{1} & g_{2} \\ g_{3} & g_{4} \end{matrix} ]. To start, we note that the assumption ^{t}gSg = det(g)S is equivalent to ^{t}g^{−1}Sg^{−1} = det(g)^{−1}S, which is in turn equivalent to

ag_{4}^{2} − bg_{3}g_{4} + cg_{3}^{2} = det(g)a,
−ag_{4}g_{2} + b(g_{1}g_{4} + g_{2}g_{3})/2 − cg_{3}g_{1} = det(g)b/2,
ag_{2}^{2} − bg_{2}g_{1} + cg_{1}^{2} = det(g)c.

Let (x_{1}, x_{2}) ∈ Ω and set (y_{1}, y_{2}) = (g, h_{1}(t)) · (x_{1}, x_{2}). By the definition of the action and Ω, and using det(g) = λ(h), we have

⟨y_{1}, y_{1}⟩ = det(g)^{−1}(g_{4}^{2}⟨x_{1}, x_{1}⟩ − 2g_{3}g_{4}⟨x_{1}, x_{2}⟩ + g_{3}^{2}⟨x_{2}, x_{2}⟩)
= det(g)^{−1}(g_{4}^{2}a − g_{3}g_{4}b + g_{3}^{2}c)
= a.

Similarly, ⟨y_{1}, y_{2}⟩ = b/2 and ⟨y_{2}, y_{2}⟩ = c. It follows that (y_{1}, y_{2}) = (g, h_{1}(t))(x_{1}, x_{2}) ∈ Ω.

Let (X, ⟨·, ·⟩) be a non-degenerate symmetric bilinear space over F satisfying (50), and let S be as in (11) with det(S) ≠ 0. Assume that Ω is non-empty, and let T = T_{S}, as in Sect. 2.3. The goal of this section is to define, for each z ∈ Ω, a set

E(z) = E_{(X,⟨·,·⟩),S}(z)  (63)

of embeddings τ : T → GSO(X) such that:

τ(t) = t for t ∈ F^{×} ⊂ T;  (64)
λ(τ(t)) = det(t) for t ∈ T, so that (t, τ(t)) ∈ B for t ∈ T;  (65)
([ \begin{matrix}& 1 \\ 1 \end{matrix} ] t [ \begin{matrix}& 1 \\ 1 \end{matrix} ] , τ(t)) · z = z for t ∈ T.  (66)

We begin by noting some properties of Ω. The set Ω is closed in X^{2}. The subgroup O(X) ≅ 1 × O(X) ⊂ B ⊂ GL(2, F) × GO(X) preserves Ω, i.e., if h ∈ O(X) and (x_{1}, x_{2}) ∈ Ω, then (hx_{1}, hx_{2}) ∈ Ω. Since det(S) ≠ 0, the group O(X) acts transitively on Ω. If dim X = 4, then SO(X) acts transitively on Ω. If dim X = 2, then the action of SO(X) on Ω has two orbits.

4.2.2 Lemma. Let (X, ⟨·, ·⟩) be a non-degenerate symmetric bilinear space over F satisfying (50), and let S be as in (11) with det(S) ≠ 0. Assume that dim X = 2 and Ω is non-empty. Let z = (z_{1}, z_{2}) ∈ Ω. For

t = [ \begin{matrix}& 1 \\ 1 \end{matrix} ]g[ \begin{matrix}& 1 \\ 1 \end{matrix} ] = [ \begin{matrix}& 1 \\ 1 \end{matrix} ][ \begin{matrix}g_{1} & g_{2} \\ g_{3} & g_{4} \end{matrix} ][ \begin{matrix}& 1 \\ 1 \end{matrix} ] ∈ T

let τ_{z}(t) : X → X be the linear map that has g as matrix in the ordered basis z_{1}, z_{2} for X, so that

τ_{z}(t)(z_{1}) = g_{1}z_{1} + g_{3}z_{2},
τ_{z}(t)(z_{2}) = g_{2}z_{1} + g_{4}z_{2}.

i) For t ∈ T, the map τ_{z}(t) is contained in GSO(X) and λ(τ_{z}(t)) = det(t).

ii) If z′ lies in the SO(X) orbit of z, and t ∈ T, then τ_{z}(t) = τ_{z′}(t).

iii) The map sending t to τ_{z}(t) defines an isomorphism τ_{z} : T ⟶^{∼} GSO(X).

iv) Let h_{0} ∈ O(X) with det(h_{0}) = −1. Let z′ ∈ Ω not be in the SO(X) orbit of z. Then τ_{z′}(t) = h_{0}τ_{z}(t)h_{0}^{−1} for t ∈ T.

v) Let t ∈ T. The element ([ \begin{matrix}& 1 \\ 1 \end{matrix} ] t [ \begin{matrix}& 1 \\ 1 \end{matrix} ] , τ_{z}(t)) ∈ B acts by the identity on the SO(X) orbit of z, and maps the other SO(X) orbit of Ω to itself.

Proof. i) A computation verifies that τ_{z}(t) ∈ GO(X), with similitude factor λ(τ_{z}(t)) = det(g) = det(t), and the equality det(τ_{z}(t)) = λ(τ_{z}(t)) implies that τ_{z}(t) ∈ GSO(X) by the definition of GSO(X). ii) Suppose that z′ = (z′_{1}, z′_{2}) lies in the SO(X) orbit of z, and let c ∈ SO(X) be such that c(z_{1}) = z′_{1} and c(z_{2}) = z′_{2}. Then τ_{z′}(t) = cτ_{z}(t)c^{−1}. But the group GSO(X) is abelian, so that τ_{z′}(t) = cτ_{z}(t)c^{−1} = τ_{z}(t). iii) Calculations prove that τ_{z} : T → GSO(X) is an isomorphism. iv) Let z′′ = h_{0}(z). A calculation shows that τ_{z′′}(t) = h_{0}τ_{z}(t)h_{0}^{−1} for t ∈ T. By, ii), τ_{z′′}(t) = τ_{z′}(t) for t ∈ T.

v) Write g = [ \begin{matrix}& 1 \\ 1 \end{matrix} ] t [ \begin{matrix}& 1 \\ 1 \end{matrix} ], so that ^{t}gSg = det(g)S. Let g = [ \begin{matrix}g_{1} & g_{2} \\ g_{3} & g_{4} \end{matrix} ]. By the definition of τ_{z}(t), we have

(g, τ_{z}(t)) · z = (det(g)^{−1}g_{4}(g_{1}z_{1} + g_{3}z_{2}) − det(g)^{−1}g_{3}(g_{2}z_{1} + g_{4}z_{2}),

det(g)^{−1}(−g_{2})(g_{1}z_{1} + g_{3}z_{2}) + det(g)^{−1}g_{1}(g_{2}z_{1} + g_{4}z_{2}))

= z.

By ii), it follows that (g, τ_{z}(t)) acts by the identity on all of the SO(X) orbit of z. Next, let z′ ∈ Ω with z′ ∉ SO(X)z. Assume that (g, τ_{z}(t)) · z′ ∈ SO(X)z; we will obtain a contradiction. Since (g, τ_{z}(t)) · z′ ∈ SO(X)z and since we have already proved that (g, τ_{z}(t)) acts by the identity on SO(X)z, we have:

(g, τ_{z}(t)) · ((g, τ_{z}(t)) · z′) = (g, τ_{z}(t)) · z′
(g, τ_{z}(t)) · z′ = z′.

This is a contradiction since z′ ∉ SO(X)z and (g, τ_{z}(t)) · z′ ∈ SO(X)z.

Let (X, ⟨·, ·⟩) be a non-degenerate symmetric bilinear space over F satisfying (50), and let S be as in (11) with det(S) ≠ 0. Assume that dim X = 2 and Ω is non-empty. For z ∈ Ω, we define

E(z) = E_{(X,⟨·,·⟩),S}(z) = {τ_{z}},  (67)

with τ_{z} as defined in Lemma 4.2.2. It is evident from Lemma 4.2.2 that the element of E(z) has the properties (64), (65), and (66).

4.2.3 Lemma. Let (X, ⟨·, ·⟩) be a non-degenerate symmetric bilinear space over F satisfying (50), and let S be as in (11) with det(S) ≠ 0. Assume that dim X = 2. Let λ, λ′ ∈ F^{×}, and set Ω = Ω_{λS} and Ω′ = Ω_{λ′S}. Assume that Ω and Ω′ are non-empty. Then

⋃_{z∈Ω} E(z) = ⋃_{z′∈Ω′} E(z′).  (68)

Proof. Let Ω_{1} and Ω_{2} be the two SO(X) orbits of the action of SO(X) on Ω so that Ω = Ω_{1} ⊔ Ω_{2}, and analogously define and write Ω′ = Ω′_{1} ⊔ Ω′_{2}. Let z = (z_{1}, z_{2}) ∈ Ω_{1} and z′ = (z′_{1}, z′_{2}) ∈ Ω′_{1}. Define a linear map h : X → X by setting h(z_{1}) = z′_{1} and h(z_{2}) = z′_{2}. We have ⟨h(x), h(y)⟩ = (λ′/λ)⟨x, y⟩ for x, y ∈ X, so that h ∈ GO(X). Assume that h ∉ GSO(X). Let z′′ = (z′′_{1}, z′′_{2}) ∈ Ω′_{2}, and let h′ : X → X be the linear map defined by h′(z′_{1}) = z′′_{1} and h′(z′_{2}) = z′′_{2}. Then h′ ∈ O(X) with det(h′) = −1, so that h′h ∈ GSO(X) and (h′h)(z_{1}) = z′′_{1} and (h′h)(z_{2}) = z′′_{2}. Therefore, by renumbering if necessary, we may assume that h ∈ GSO(X). Next, a calculation shows that hτ_{z}(t)h^{−1} = τ_{z′}(t) for t ∈ T. Since GSO(X) is abelian, this means that τ_{z} = τ_{z′}. The claim (68) follows now from ii) and iv) of Lemma 4.2.2.

4.2.4 Lemma. Let (X, ⟨·, ·⟩) be a non-degenerate symmetric bilinear space over F satisfying (50), and let S be as in (11) with det(S) ≠ 0. Assume that dim X = 4 and Ω is non-empty. Let z = (z_{1}, z_{2}) ∈ Ω, and set U = F z_{1} + F z_{2}, so that X = U ⊕ U^{⊥} with dim U = dim U^{⊥} = 2. There exists λ ∈ F^{×} such that (U^{⊥}, ⟨·, ·⟩) represents λS.

Proof. Let M_{4,1}(F) be the F vector space of 4 × 1 matrices with entries from F. Let D = − det(S). Let λ ∈ F^{×}, and define a four-dimensional symmetric bilinear space X_{λ} by letting X_{λ} = M_{4,1}(F) with symmetric bilinear form b given by b(x, y) = ^{t}xMy, where

M = [ \begin{matrix}S \\ & λS \end{matrix} ] .

Evidently, disc(X_{λ}) = 1, and the Hasse invariant of X_{λ} is ε(X_{λ}) = (−1, −1)_{F} (−λ, D)_{F}. Now assume that X is isotropic. Then the Hasse invariant of X is (−1, −1)_{F}. It follows that if λ = −1, then ε(X_{λ}) = ε(X), so that X_{λ} ≅ X. By the Witt cancellation theorem, (U^{⊥}, ⟨·, ·⟩) represents λS. Next, assume that X is anisotropic, so that ε(X) = −(−1, −1)_{F}. By hypothesis, (X, ⟨·, ·⟩) represents S; since X is anisotropic, this implies that D ∉ F^{×2}. Since, D ∉ F^{×2}, there exists λ ∈ F^{×} such that −1 = (−λ, D)_{F}. It follows that ε(X_{λ}) = ε(X), so that X_{λ} ≅ X; again the Witt cancellation theorem implies that (U^{⊥}, ⟨·, ·⟩) represents λS.

Let (X, ⟨·, ·⟩) be a non-degenerate symmetric bilinear space over F satisfying (50), and let S be as in (11) with det(S) ≠ 0. Assume that dim X = 4, Ω = Ω_{S} is non-empty, and let T = T_{S}, as in Sect. 2.3. Let z = (z_{1}, z_{2}) ∈ Ω, and as in Lemma 4.2.4, let U = F z_{1} +F z_{2}, so that X = U ⊕U^{⊥} with dim U = dim U^{⊥} = 2. By Lemma 4.2.4 there exists λ ∈ F^{×} such that (U^{⊥}, ⟨·, ·⟩) represents λS. Let τ_{z} : T → GSO(U) be the isomorphism from Lemma 4.2.2 that is associated to z. Also, let τ_{z′}, τ_{z′′} : T → GSO(U^{⊥}) be the isomorphisms from Lemma 4.2.2, where z′ and z′′ are representatives for the two SO(U^{⊥}) orbits of SO(U^{⊥}) acting on Ω_{λS,(U^{⊥},⟨·,·⟩)}; by Lemma 4.2.3, {τ_{z′}, τ_{z′′}} does not depend on the choice of λ. We now define

E(z) = E_{(X,⟨·,·⟩),S}(z) = {τ_{1}, τ_{2}},  (69)

where τ_{1}, τ_{2} : T → GSO(X) are defined by

τ_{1}(t) = [ \begin{matrix}τ_{z}(t) \\ & τ_{z′} (t) \end{matrix} ] , τ_{2}(t) = [ \begin{matrix}τ_{z}(t) \\ & τ_{z′′} (t) \end{matrix} ]

with respect to the decomposition Z = U ⊕ U^{⊥}, for t ∈ T. For t ∈ T, the similitude factor of τ_{i}(t) is det(t). It is evident that the elements of E(z) satisfy (64), (65), and (66).

4.2.5 Lemma. Let (X, ⟨·, ·⟩) be a non-degenerate symmetric bilinear space over F satisfying (50), and let S be as in (11) with det(S) ≠ 0. Assume that Ω = Ω_{S} is non-empty, and let A = A_{S} and T = T_{S}, as in Sect. 2.3. If dim X = 4, assume that A is a field. Let z ∈ Ω and τ ∈ E(z). Let C be a compact, open subset of Ω containing z. There exists a compact, open subset C_{0} of Ω such that

z ∈ C_{0} ⊂ C and ([ \begin{matrix}& 1 \\ 1 \end{matrix} ] t [ \begin{matrix}& 1 \\ 1 \end{matrix} ] , τ(t)) · C_{0} = C_{0} for t ∈ T.

Proof. Assume dim X = 2. Let C_{0} be the intersection of C with the SO(X) orbit of z in Ω. Then C_{0} is a compact, open subset of Ω because the SO(X) orbit of z in Ω is closed and open in Ω, and C is compact and open. We have ([ \begin{matrix}& 1 \\ 1 \end{matrix} ] t [ \begin{matrix}& 1 \\ 1 \end{matrix} ] , τ(t)) · C_{0} = C_{0} for t ∈ T by v) of Lemma 4.2.2. Assume dim X = 4. The group of pairs ([ \begin{matrix}& 1 \\ 1 \end{matrix} ] t [ \begin{matrix}& 1 \\ 1 \end{matrix} ] , τ(t)) for t ∈ T acts on X^{2} and can be regarded as a subgroup of GL(X^{2}). The group T contains F^{×}, and the pairs with t ∈ F^{×} act by the identity on X^{2}. The assumption that A is a field implies that T/F^{×} is compact, and hence the image K in GL(X^{2}) of this group of pairs is compact. There exists a lattice L of X^{2} such that k · L = L for k ∈ K. Also, by (66) we have that k · z = z for k ∈ K. Let n be sufficiently large so that (z + ϖ^{n}L) ∩ Ω ⊂ C. Then C_{0} = (z + ϖ^{n}L) ∩ Ω is the desired set.

4.3 Example embeddings

In this section we provide explicit formulas for the embeddings of the previous section.

4.3.1 Lemma. Let S be as in (11). Let m, λ ∈ F^{×}, and define (X_{m,λ}, ⟨·, ·⟩_{m,λ}) as in (52). The set Ω = Ω_{S} is non-empty if and only if disc(S) = mF^{×2} and ε(S) = (λ, m)_{F}. Assume that the set Ω is non-empty. Set D = b^{2}/4 − ac so that disc(S) = DF^{×2}, and define Δ and the quadratic extension L = F + F Δ of F (which need not be a field) associated to D as in Sect. 2.1. Similarly, define Δ_{m} with respect to m; the quadratic extension associated m is also L and L = F + F Δ_{m}. The set of compositions

L^{×} ⟶^{∼} T_{S} ⟶^{τ} GSO(X_{m,λ})

for z ∈ Ω and τ ∈ E(z) is the same as the set consisting of the two compositions

L^{×} ⟶^{∼} T_{[ \begin{matrix}1 \\ & −m \end{matrix} ]} ⟶^{∼} GSO(X_{m,λ}), L^{×} ⟶^{γ} L^{×} ⟶^{∼} T_{[ \begin{matrix}1 \\ & −m \end{matrix} ]} ⟶^{∼} GSO(X_{m,λ}).

Here, the maps L^{×} → T_{[ \begin{matrix}1 \\ & −m \end{matrix} ]} are as in (14), and the isomorphism ρ of

T_{[ \begin{matrix}1 \\ & −m \end{matrix} ]} with GSO(X_{m,λ}) is as in (54).

Proof. By definition, Ω is non-empty if and only if there exist x_{1}, x_{2} ∈ X_{m,λ} such that S = [ \begin{matrix}⟨x_{1},x_{1}⟩ & ⟨x_{1},x_{2}⟩ \\ ⟨x_{1},x_{2}⟩ & ⟨x_{2},x_{2}⟩ \end{matrix} ]. Since X_{m,λ} is two-dimensional, this means that Ω is non-empty if and only if (X_{m,λ}, ⟨·, ·⟩_{m,λ}) is equivalent to the symmetric bilinear space over F defined by S. From Lemma 4.1.1, we have disc(X_{m,λ}) = mF^{×2} and ε(X_{m,λ}) = (λ, m)_{F}. Since a finite-dimensional nondegenerate symmetric bilinear space over F is determined by its dimension, discriminant and Hasse invariant, it follows that Ω is non-empty if and only if disc(S) = mF^{×2} and ε(S) = (λ, m)_{F}. Assume that Ω is non-empty, so that disc(S) = mF^{×2} and ε(S) = (λ, m)_{F}. Let e ∈ F^{×} be such that Δ = eΔ_{m}; then b^{2}/4 − ac = D = e^{2}m. Assume first that a ≠ 0. By Sect. 2.2, ε(S) = (a, m)_{F}. Therefore, (a, m)_{F} = (λ, m)_{F}. It follows that there exist g, h ∈ F^{×} such that g^{2} − mh^{2} = λ^{−1}a. Set

z_{1} = [ \begin{matrix}g & h \\ −h(−m) & g \end{matrix} ] , z_{2} = a^{−1} [ \begin{matrix}ehm+gb/2 & eg+hb/2 \\ −(eg+hb/2)(−m) & ehm+gb/2 \end{matrix} ] .

Then z_{1}, z_{2} ∈ X_{m,λ}, and a calculation shows that

[ \begin{matrix}⟨z_{1},z_{1}⟩_{m,λ} & ⟨z_{1},z_{2}⟩_{m,λ} \\ ⟨z_{1},z_{2}⟩_{m,λ} & ⟨z_{2},z_{2}⟩_{m,λ} \end{matrix} ] = S.

It follows that z = (z_{1}, z_{2}) ∈ Ω. Let u ∈ L^{×}. Write u = x + yΔ for some x, y ∈ F^{×}. By (14), u corresponds to t = [ \begin{matrix}x−yb/2 & −ya \\ yc & x+yb/2 \end{matrix} ] ∈ T_{S}. Using the definition of τ_{z}(t), we find that

τ_{z}(t)(z_{1}) = [ \begin{matrix}gx−ehym & hx−egy \\ −(hx−egy)(−m) & gx−ehym \end{matrix} ] ,  (70)

τ_{z}(t)(z_{2}) = \frac{1}{2a} [ \begin{matrix}(2ehm+bg)x−(behm+2e^{2}mg)y & (2eg+bh)x−(beg+2e^{2}mh)y \\ −((2eg+bh)x−(beg+2e^{2}mh)y)(−m) & (2ehm+bg)x−(behm+2e^{2}mg)y \end{matrix} ] . (71)

On the other hand, we also have that u = x + yeΔ_{m}, and u corresponds to the element t′ = [ \begin{matrix}x & −ye \\ (ye)(−m) & x \end{matrix} ] in T_{[ \begin{matrix}1 \\ & −m \end{matrix} ]}. Moreover, calculations show that

ρ(t′)(z_{1}) = t′ · z_{1} and ρ(t′)(z_{2}) = t′ · z_{2} are as in (70) and (71), respectively, proving that the two compositions

L^{×} ⟶^{∼} T_{S} ⟶^{τ_{z}} GSO(X_{m,λ}), L^{×} ⟶^{∼} T_{[ \begin{matrix}1 \\ & −m \end{matrix} ]} ⟶^{ρ} GSO(X_{m,λ})

are the same map. Next, let z′ = (γ(z_{1}), γ(z_{2})). Then z′ ∈ Ω, and calculations as above show that the two compositions

L^{×} ⟶^{∼} T_{S} ⟶^{τ_{z′}} GSO(X_{m,λ}), L^{×} ⟶^{γ} L^{×} ⟶^{∼} T_{[ \begin{matrix}1 \\ & −m \end{matrix} ]} ⟶^{ρ} GSO(X_{m,λ})

are the same. This completes the proof in this case since z and z′ are representatives for the two SO(X_{m,λ}) orbits of Ω, and by ii) of Lemma 4.2.2, ∪_{w∈Ω}E(w) = {τ_{z}, τ_{z′}}. Now assume that a = 0. Set

z_{1} = λ^{−1} [ \begin{matrix}b/2 & −e \\ e(−m) & b/2 \end{matrix} ] , z_{2} = [ \begin{matrix}(cλ^{−1}+1)/2 & −eb^{−1}(cλ^{−1}−1) \\ eb^{−1}(cλ^{−1}−1)(−m) & (cλ^{−1}+1)/2 \end{matrix} ] .

Again, a calculation shows that z = (z_{1}, z_{2}) ∈ Ω. Let u ∈ L^{×} with u = x + yΔ for some x, y ∈ F^{×}. Then u corresponds to t = \left[{x−yb/2 \atop yc x+yb/2}\right] ∈ T_{S}, and

u corresponds to t′ = [ \begin{matrix}x & −ye \\ ye(−m) & x \end{matrix} ] ∈ T_{[ \begin{matrix}1 \\ & −m \end{matrix} ]}. Computations show that

τ_{z}(t)(z_{1}) = ρ(t′)(z_{1}) and τ_{z}(t)(z_{2}) = ρ(t′)(z_{2}), proving that the compositions

L^{×} ⟶^{∼} T_{S} ⟶^{τ_{z′}} GSO(X_{m,λ}), L^{×} ⟶^{∼} T_{[ \begin{matrix}1 \\ & −m \end{matrix} ]} ⟶^{ρ} GSO(X_{m,λ})

are the same. As in the previous case, if z′ = (γ(z_{1}), γ(z_{2})), then z′ ∈ Ω, and the two compositions

L^{×} ⟶^{∼} T_{S} ⟶^{τ_{z′}} GSO(X_{m,λ}), L^{×} ⟶^{γ} L^{×} ⟶^{∼} T_{[ \begin{matrix}1 \\ & −m \end{matrix} ]} ⟶^{ρ} GSO(X_{m,λ})

are the same. As above, this completes the proof.

Let c ∈ F^{×}, and set

S = [ \begin{matrix}1 \\ & c \end{matrix}] .  (72)

Let (X_{M_{2}}, ⟨·, ·⟩_{M_{2}}) be as in (55). Let A = A_{S} and T = T_{S} be as in Sect. 2.3. We embed A in M_{2}(F) via the inclusion map. Set

z_{1} = [ \begin{matrix}1 \\ & 1 \end{matrix} ] , z_{2} = [ \begin{matrix}& 1 \\ −c \end{matrix} ] , z′_{1} = [ \begin{matrix}1 \\ & −1 \end{matrix} ] , z′_{2} = [ \begin{matrix}& 1 \\ c \end{matrix}] .

The vectors z_{1}, z_{2}, z′_{1}, z′_{2} form an orthogonal ordered basis for X_{M_{2}}, and in this basis the matrix for X_{M_{2}} is

[ \begin{matrix}S \\ & −S \end{matrix} ] .

As in Lemma 4.2.4, set U = F z_{1} + F z_{2}. Then U^{⊥} = F z′_{1} + F z′_{2}, and the λ of Lemma 4.2.4 is −1. Calculations show that the set E(z) = E_{X_{M_{2}}} (z) of (69) is

E_{X_{M_{2}}} (z) = {τ_{1}, τ_{2}},  τ_{1}(t) = ρ(t, 1), τ_{2}(t) = ρ(1, γ(t)), t ∈ T^{×}. (73)

Finally, let S be as in (72) with −c ∉ F^{×2}, and let (X_{H}, ⟨·, ·⟩_{H}) be as in (57). Let A = A_{S} and T = T_{S} be as in Sect. 2.3. Let L be the quadratic extension associated to −c as in Sect. 2.1; L is a field. Let e be a representative for the non-trivial coset of F^{×}/N_{L/F} (L^{×}), so that (e, −c)_{F} = −1. We realize the division quaternion algebra H over F as

H = F + F i + F j + F k, i^{2} = −c, j^{2} = e, k = ij, ij = −ji. (74)

We embed A into H via the map defined by

[ \begin{matrix}x & −y \\ cy & x \end{matrix} ] ↦ x − yi

for x, y ∈ F. Let

z_{1} = 1, z_{2} = i, z′_{1} = j, z′_{2} = k.  (75)

The vectors z_{1}, z_{2}, z′_{1}, z′_{2} form an orthogonal ordered basis for X_{H}, and in this basis the matrix for X_{H} is

[ \begin{matrix}S \\ & −eS \end{matrix} ] .

As in Lemma 4.2.4, set U = F z_{1} + F z_{2}. Then U^{⊥} = F z′_{1} + F z′_{2}, and the λ of Lemma 4.2.4 is −e. Calculations again show that the set E(z) = E_{X_{H}} (z) of (69) is

E_{X_{H}} (z) = {τ_{1}, τ_{2}},  τ_{1}(t) = ρ(t, 1), τ_{2}(t) = ρ(1, γ(t)), t ∈ T^{×}. (76)

To close this subsection, we note that (X_{H}, ⟨·, ·⟩_{H}) does not represent S if S is as in (72) but −c ∈ F^{×2}. To see this, assume that −c ∈ F^{×2} and (X_{H}, ⟨·, ·⟩_{H}) represents S; we will obtain a contradiction. Write −c = t^{2} for some t ∈ F^{×}. Since X_{H} represents S, there exist x_{1}, x_{2} ∈ H such that ⟨x_{1}, x_{1}⟩_{H} = N(x_{1}) = 1, ⟨x_{2}, x_{2}⟩_{H} = N(x_{2}) = c = −t^{2} and ⟨x_{1}, x_{2}⟩_{H} = T(x_{1}x_{2}^{∗})/2 = 0. A calculation shows that N(tx_{1} + x_{2}) = 0. Since H is a division algebra, this means that tx_{1} = −x_{2}. Hence, t^{2} = N(tx_{1}) = ⟨tx_{1}, tx_{1}⟩_{H} = ⟨tx_{1}, −x_{2}⟩_{H} = −t⟨x_{1}, x_{2}⟩_{H} = 0, a contradiction.

4.4 Theta correspondences and Bessel functionals

In this section we make the connection between Bessel functionals for GSp(4, F) and equivariant functionals on representations of GO(X). The main result is Theorem 4.4.6 below.

Let (X, ⟨·, ·⟩) be a non-degenerate symmetric bilinear space over F satisfying (50). We define the subgroup GSp(4, F)^{+} of GSp(4, F) by

GSp(4, F)^{+} = {g ∈ GSp(4, F) : λ(g) ∈ λ(GO(X))}.  (77)

The following lemma follows from (54) and the exact sequences (58) and (59), which facilitate the computation of λ(GSO(X)). Note that N(H^{×}) = F^{×}.

4.4.1 Lemma. Let (X, ⟨·, ·⟩) be a non-degenerate symmetric bilinear space over F satisfying (50). Then

[GSp(4, F) : GSp(4, F)^{+}] =\begin{cases}1 & if dim X = 4, or dim X = 2 and disc(X) = 1, \\ 2 & if dim X = 2 and disc(X) ≠ 1. \end{cases}

(78)

4.4.2 Lemma. Let (X, ⟨·, ·⟩) be a non-degenerate symmetric bilinear space over F satisfying (50), and let S be as in (11) with det(S) ≠ 0. Let Ω = Ω_{S} be as in (61), and assume that Ω is non-empty. Let T = T_{S} be as in Sect. 2.3. Embed T as a subgroup of GSp(4, F), as in (16). Then T is contained in GSp(4, F)^{+}.

Proof. By (78) we may assume that dim X = 2 and disc(X) ≠ 1. Since Ω is non-empty and dim X = 2, we make take S to be the matrix of the symmetric bilinear form ⟨·, ·⟩ on X. By definition, GO(X) is then the set of h ∈ GL(2, F) such that ^{t}hSh = λ(h)S for some λ(h) ∈ F^{×}. From (13), we have that ^{t}hSh = det(h)S for h = [ \begin{matrix}& 1 \\ 1 \end{matrix} ] t [ \begin{matrix}& 1 \\ 1 \end{matrix} ] with t ∈ T. It follows that det(T) is contained in λ(GO(X)). This implies that T is contained in GSp(4, F)^{+}.

Let (X, ⟨·, ·⟩) be a non-degenerate symmetric bilinear space over F satisfying (50). Define

R = {(g, h) ∈ GSp(4, F) × GO(X) : λ(g) = λ(h)}.

We consider the Weil representation ω of R on the space S(X^{2}) defined with respect to ψ^{2}, where ψ^{2}(x) = ψ(2x) for x ∈ F. If φ ∈ S(X^{2}), g ∈ GL(2, F) and h ∈ GO(X) with det(g) = λ(h), and x_{1}, x_{2} ∈ X, then

(ω([ \begin{matrix}1 & & y & z \\ & 1 & x & y \\ & & 1 \\ & & & 1 \end{matrix} ] , 1)φ)(x_{1}, x_{2}) = ψ(⟨x_{1}, x_{1}⟩x + 2⟨x_{1}, x_{2}⟩y + ⟨x_{2}, x_{2}⟩z)φ(x_{1}, x_{2}), (79)

(ω([ \begin{matrix}g \\ & det(g)g′ \end{matrix} ] , h)φ)(x_{1}, x_{2}) = (det(g), disc(X))_{F} φ(([ \begin{matrix}& 1 \\ 1 \end{matrix} ] g [ \begin{matrix}& 1 \\ 1 \end{matrix} ] , h)^{−1} · (x_{1}, x_{2})).

(80)

For these formulas, see Sect. 1 of [27]; note that the additive character we are using is ψ^{2}. Also, in (80) we are using the action of GL(2, F) × GO(X) defined in (60). We will also use the Weil representation ω_{1} of

R_{1} = {(g, h) ∈ GL(2, F) × GO(X) : det(g) = λ(h)}

on S(X) defined with respect to ψ^{2}. For formulas, again see Sect. 1 of [27]. The two Weil representations ω and ω_{1} are related as follows.

4.4.3 Lemma. The map

T : S(X) ⊗ S(X) ⟶ S(X^{2}),  (81)

determined by the formula

T (φ_{1} ⊗ φ_{2})(x_{1}, x_{2}) = φ_{1}(x_{1})φ_{2}(x_{2})  (82)

for φ_{1} and φ_{2} in S(X) and x_{1} and x_{2} in X, is a well-defined complex linear isomorphism such that

T ∘ (ω_{1}( [ \begin{matrix}a_{2} & b_{2} \\ c_{2} & d_{2} \end{matrix} ] , h) ⊗ ω_{1}([ \begin{matrix}a_{1} & b_{1} \\ c_{1} & d_{1} \end{matrix} ] , h))

= ω([ \begin{matrix}a_{1} & & & b_{1} \\ & a_{2} & b_{2} \\ & c_{2} & d_{2} \\ c_{1} & & & d_{1} \end{matrix} ] , h) ∘ T  (83)

for g_{1} = [ \begin{matrix}a_{1} & b_{1} \\ c_{1} & d_{1} \end{matrix} ] and g_{2} = [ \begin{matrix}a_{2} & b_{2} \\ c_{2} & d_{2} \end{matrix} ] in GL(2, F) and h in GO(X) such that

det(g_{1}) = det(g_{2}) = λ(h).

This lemma can be verified by a direct calculation using standard generators for SL(2, F). Let θ = θ_{S} be the character of N defined in (15) with respect to a matrix S as in (11). Let S(X^{2})(N, θ) be the subspace of S(X^{2}) spanned by all vectors ω(n)φ − θ(n)φ, where n runs through N and φ runs through S(X^{2}), and set S(X^{2})_{N,θ} = S(X^{2})/S(X^{2})(N, θ).

4.4.4 Lemma. (Rallis) Let (X, ⟨·, ·⟩) be a non-degenerate symmetric bilinear space over F satisfying (50), and let S be as in (11) with det(S) ≠ 0. If (X, ⟨·, ·⟩) does not represent S, then the twisted Jacquet module S(X^{2})_{N,θ} is zero. Assume that (X, ⟨·, ·⟩) represents S. The map S(X^{2}) → S(Ω) defined by φ ↦ φ|_{Ω} induces an isomorphism

S(X^{2})_{N,θ} ⟶^{∼} S(Ω).

Equivalently, S(X^{2})(N, θ) is the space of φ ∈ S(X^{2}) such that φ|_{Ω} = 0.

Proof. See Lemma 2.3 of [13].

Let (X, ⟨·, ·⟩) be a non-degenerate symmetric bilinear space over F satisfying (50), and let S be as in (11) with det(S) ≠ 0. Let Ω = Ω_{S} be as in (61), and assume that Ω is non-empty. In Lemma 4.2.1 we noted that the subgroup B of GL(2, F)×GO(X) acts on Ω. By identifying O(X) with 1×O(X) ⊂ GL(2, F)× GO(X), we obtain an action of O(X) on Ω: this is given by h · (x_{1}, x_{2}) = (hx_{1}, hx_{2}), where h ∈ O(X) and (x_{1}, x_{2}) ∈ Ω. This action is transitive. We obtain an action of O(X) on S(Ω) by defining (h · φ)(x) = φ(h^{−1} · x) for h ∈ O(X), φ ∈ S(Ω) and x ∈ Ω. This action is used in the next lemma.

4.4.5 Lemma. Let (X, ⟨·, ·⟩) be a non-degenerate symmetric bilinear space over F satisfying (50), and let S be as in (11) with det(S) ≠ 0. Let Ω = Ω_{S} be as in (61), and assume that Ω is non-empty. Let (σ_{0}, W_{0}) be an admissible representation of O(X), and let M′ : S(Ω) → W_{0} be a non-zero O(X) map. Let z ∈ Ω. There exists a compact, open subset C of Ω containing z such that if C_{0} is a compact, open subset of Ω such that z ∈ C_{0} ⊂ C, then M′(f_{C_{0}}) ≠ 0. Here, f_{C_{0}} is the characteristic function of C_{0}.

Proof. Let H be the subgroup of h ∈ O(X) such that hz = z. By 1.6 of [3], the map H\O(X) ⟶^{∼} Ω defined by Hh ↦ h^{−1}z is a homeomorphism, so that the map S(Ω) ⟶^{∼} c-Ind_{H}^{O(X)}1_{H} that sends φ to the function f such that f(h) = φ(h^{−1}z) for h ∈ O(X) is an O(X) isomorphism. Via this isomorphism, we may regard M′ as defined on c-Ind_{H}^{O(X)}1_{H}, and it will suffice to prove that that there exists a compact, open neighborhood C of the identity in O(X) such that if C_{0} is a compact, open neighborhood of the identity in O(X) with C_{0} ⊂ C, then M′(f_{HC_{0}}) ≠ 0, where f_{HC_{0}} is the characteristic function of HC_{0}. Since σ_{0} is admissible, by 2.15 of [3] we have (σ_{1})^{∨} ≅ σ_{0} where σ_{1} = σ_{0}^{∨}. Let W_{1} be the space of σ_{1}. We may regard M′ as a non-zero element of Hom_{O(X)}(c-Ind_{H}^{O(X)}1_{H}, σ_{1}^{∨}). Now H and O(X) are unimodular since both are orthogonal groups (H is isomorphic to O(U^{⊥}), where U = F z_{1} +F z_{2}). By 2.29 of [3], there exists an element λ of Hom_{H}(σ_{1}, 1_{H}) such that M′ is given by

M′(f)(v) = ∫_{H\O(X)} f(h)λ(σ_{1}(h)v) dh

for f ∈ c-Ind_{H}^{O(X)}1_{H} and v ∈ W_{1}. Since M′ is non-zero, there exists v ∈ W_{1} such that λ(v) ≠ 0. Let C be a compact, open neighborhood of 1 in O(X) such that σ_{1}(h)v = v for h ∈ C. Let C_{0} be a compact, open neighborhood of 1 in O(X) such that C_{0} ⊂ C. Then

M′(f_{HC_{0}})(v) = ∫_{H\O(X)} f_{HC_{0}}(h)λ(σ_{1}(h)v) dh

= ∫_{H\HC_{0}} λ(σ_{1}(h)v) dh

= vol(H\HC_{0})λ(v),

which is non-zero.

In the following theorem we mention the set E(z) of embeddings of T into GSO(X); see (63), (67) and (69).

4.4.6 Theorem. Let (X, ⟨·, ·⟩) be a non-degenerate symmetric bilinear space over F satisfying (50), and let S be as in (11) with det(S) ≠ 0. Let A = A_{S}, T = T_{S}, and L = L_{S} be as in Sect. 2.3. If dim X = 4, assume that A is a field. Let (π, V) be an irreducible, admissible representation of GSp(4, F)^{+}, and let (σ, W) be an irreducible, admissible representation of GO(X). Assume that there is a non-zero R map M : S(X^{2}) → π ⊗ σ. Let θ = θ_{S} and let Λ be a character of T.

i) If Hom_{N}(π, ℂ_{θ}) ≠ 0, then Ω = Ω_{S} is non-empty and D = T N is contained in GSp(4, F)^{+}.

ii) Assume that Hom_{N}(π, ℂ_{θ}) ≠ 0 so that Ω = Ω_{S} is non-empty, and D = T N ⊂ GSp(4, F)^{+} by i). Assume further that Hom_{D}(π, ℂ_{Λ⊗θ}) ≠ 0. Let z ∈ Ω, and τ ∈ E(z). There exists a non-zero vector w ∈ W such that

σ(τ(t))w = Λ^{−1}(t)w

for t ∈ T.

Proof. i) The assumptions Hom_{R}(S(X^{2}), V ⊗ W) ≠ 0 and Hom_{N}(V, ℂ_{θ}) ≠ 0 imply that Hom_{N}(S(X^{2}), ℂ_{θ}) ≠ 0. This means that S(X^{2})_{N,θ} ≠ 0; by Lemma 4.4.4, we obtain Ω ≠ ∅. Lemma 4.4.2 now also yields that D ⊂ GSp(4, F)^{+}. ii) Let β be a non-zero element of Hom_{D}(π, ℂ_{Λ⊗θ}). We first claim that the composition M′

S(X^{2}) ⟶^{M} V ⊗ W ⟶^{β⊗id} ℂ_{Λ⊗θ} ⊗ W

is non-zero. Let φ ∈ S(X^{2}) be such that M(φ) ≠ 0, and write

M(φ) = ∑_{ℓ=1}^{t} v_{ℓ} ⊗ w_{ℓ}

where v_{1}, …, v_{t} ∈ V and w_{1}, …, w_{t} ∈ W. We may assume that the vectors w_{1}, …, w_{t} are linearly independent and that v_{1} ≠ 0. Since β is non-zero and V is an irreducible representation of GSp(4, F)^{+}, it follows that there exists g ∈ GSp(4, F)^{+} such that β(π(g)v_{1}) ≠ 0. Let h ∈ GO(X) be such that λ(h) = λ(g). Then (g, h) ∈ R. Since M is an R-map, we have

M(ω(g, h)φ) = ∑_{ℓ=1}^{t} π(g)v_{ℓ} ⊗ σ(h)w_{ℓ}.

Applying β ⊗ id to this equation, we get

M′(ω(g, h)φ) = ∑_{ℓ=1}^{t} β(π(g)v_{ℓ}) ⊗ σ(h)w_{ℓ}

in ℂ_{Λ⊗θ} ⊗ W. Since the vectors σ(h)w_{1}, …, σ(h)w_{t} are also linearly independent, and since β(π(g)v_{1}) is non-zero, it follows that the vector M′(ω(g, h)φ) is non-zero; this proves M′ ≠ 0.

Next, the map M′ induces a non-zero map S(X^{2})_{N,θ} → ℂ_{Λ⊗θ} ⊗ W, which we also denote by M′. Lemma 4.4.4 implies that the restriction map yields an isomorphism S(X^{2})_{N,θ} ⟶^{∼} S(Ω). Composing, we thus obtain a non-zero map S(Ω) → ℂ_{Λ⊗θ} ⊗ W, which we again denote by M′. Let z ∈ Ω and τ ∈ E(z). By Lemma 4.2.1, the elements ([ \begin{matrix}& 1 \\ 1 \end{matrix} ] t [ \begin{matrix}& 1 \\ 1 \end{matrix} ] , τ(t)) for t ∈ T act on Ω. We can regard these elements as acting on S(Ω) via the definition (([ \begin{matrix}& 1 \\ 1 \end{matrix} ] t [ \begin{matrix}& 1 \\ 1 \end{matrix} ] , τ(t))·φ)(x) = φ(([ \begin{matrix}& 1 \\ 1 \end{matrix} ] t [ \begin{matrix}& 1 \\ 1 \end{matrix} ] , τ(t))^{−1}·x) for φ ∈ S(Ω) and x ∈ Ω. Moreover, by the definition of M′ and (80), we have

M′(([ \begin{matrix}& 1 \\ 1 \end{matrix} ] t [ \begin{matrix}& 1 \\ 1 \end{matrix} ] , τ(t)) · φ) = (det(t), disc(X))_{F} Λ(t)σ(τ(t))M′(φ)  (84)

for t ∈ T and φ ∈ S(Ω). Let C be the compact, open subset from Lemma 4.4.5 with respect to M′ and z; note that the restriction of σ to O(X) is admissible. By Lemma 4.2.5 there exists a compact, open subset C_{0} of C containing z such that ([ \begin{matrix}& 1 \\ 1 \end{matrix} ] t [ \begin{matrix}& 1 \\ 1 \end{matrix} ] , τ(t)) · C_{0} = C_{0} for t ∈ T. Let φ = f_{C_{0}}. Then ([ \begin{matrix}& 1 \\ 1 \end{matrix} ] t [ \begin{matrix}& 1 \\ 1 \end{matrix} ] , τ(t)) · φ = φ for t ∈ T, and by Lemma 4.4.5, we have M′(φ) ≠ 0. From (84) we have σ(τ(t))M′(φ) = (det(t), disc(X))_{F} Λ(t)^{−1}M′(φ) = χ_{L/F} (N_{L/F} (t))Λ(t)^{−1}M′(φ) = Λ(t)^{−1}M′(φ) for t ∈ T. Since M′(φ) ≠ 0, this proves ii).

Let (X, ⟨·, ·⟩) be a non-degenerate symmetric bilinear space over F satisfying (50), and let S be as in (11) with det(S) ≠ 0. If Ω_{S} is non-empty and z = (z_{1}, z_{2}) ∈ Ω_{S}, then we let O(X)_{z} be the subgroup of h ∈ O(X) such that h(z_{1}) = z_{1} and h(z_{2}) = z_{2}.

4.4.7 Proposition. Let (X, ⟨·, ·⟩) be a non-degenerate symmetric bilinear space over F satisfying (50), and assume that dim X = 4. Let S be as in (11) with det(S) ≠ 0. Assume that Ω_{S} is non-empty, and let z be in Ω_{S}. Let Π and σ be irreducible, admissible, supercuspidal representations of GSp(4, F) and GO(X), respectively. If Hom_{R}(ω, Π ⊗ σ) ≠ 0, then

dim Π_{N,θ_{S}} = dim Hom_{O(X)_{z}}(σ, ℂ_{1}).  (85)

Proof. Assume that Hom_{R}(ω, Π ⊗ σ) ≠ 0. By Proposition 3.3 of [27] the restriction of σ to O(X) is multiplicity-free. By Lemma 4.2 of [24] we have Π|_{Sp(4,F)} = Π_{1} ⊕ ⋯ ⊕ Π_{t}, where Π_{1}, …, Π_{t} are mutually non-isomorphic, irreducible, admissible representations of Sp(4, F), σ|_{O(X)} = σ_{1} ⊕ ⋯ ⊕ σ_{t}, where σ_{1}, …, σ_{t} are mutually non-isomorphic, irreducible, admissible representations of O(X), with Hom_{Sp(4,F)×O(X)}(ω, Π_{i} ⊗ σ_{i}) ≠ 0 for i ∈ {1, …, t}. Let i ∈ {1, …, t}; to prove the proposition, it will suffice to prove that (Π_{i})_{N,θ_{S}} ≅ Hom_{O(X)_{z}}(σ_{i}, ℂ_{1}) as complex vector spaces. By Lemma 6.1 of [26], we have Θ(σ_{i})_{N,θ_{S}} ≅ Hom_{O(X)_{z}}(σ_{i}^{∨}, ℂ_{1}) as complex vector spaces. By 1) a) of the theorem on p. 69 of [14], the representation Θ(σ_{i}) of Sp(4, F) is irreducible. By Theorem 2.1 of [12] we have Π_{i} ≅ Θ(σ_{i}). Therefore, (Π_{i})_{N,θ_{S}} ≅ Hom_{O(X)_{z}}(σ_{i}^{∨}, ℂ_{1}). By the first theorem on p. 91 of [14], σ_{i}^{∨} ≅ σ_{i}. The proposition follows.

4.5 Representations of GO(X)

Let m, λ ∈ F^{×}. By Lemma 4.1.1, the group GSO(X_{m,λ}) is abelian. It follows that the irreducible, admissible representations of GSO(X_{m,λ}) are characters. To describe the representations of GO(X_{m,λ}), let μ : GSO(X_{m,λ}) → ℂ^{×} be a character. We recall that the map γ from (53) is a representative for the non-trivial coset of GSO(X_{m,λ}) in GO(X_{m,λ}). Define μ^{γ} : GSO(X_{m,λ}) → ℂ^{×} by μ^{γ}(x) = μ(γxγ^{−1}). If μ^{γ} ≠ μ, then the representation ind_{GSO(X_{m,λ})}^{GO(X_{m,λ})}μ is irreducible, and we define

μ^{+} = ind_{GSO(X_{m,λ})}^{GO(X_{m,λ})}μ.

Assume that μ = μ^{γ}. Then the induced representation ind_{GSO(X_{m,λ})}^{GO(X_{m,λ})}μ is reducible, and is the direct sum of the two extensions of μ to GO(X_{m,λ}). We let μ^{+} be the extension of μ to GO(X_{m,λ}) such that μ^{+}(γ) = 1 and let μ^{−} be the extension of μ to GO(X_{m,λ}) such that μ^{−}(γ) = −1. Every irreducible, admissible representation of GO(X_{m,λ}) is of the form μ^{+} or μ^{−} for some character μ of GSO(X_{m,λ}). We will sometimes identify characters of GSO(X_{m,λ}) with characters of T_{[ \begin{matrix}1 \\ & −m \end{matrix} ]}, via (54), and in turn identify characters of T_{[ \begin{matrix}1 \\ & −m \end{matrix} ]}

with characters of L^{×}, via (14). Here L is associated to m, as in Sect. 2.1, so that L = F (\sqrt{m}) if m ∉ F^{×2}, and L = F × F if m ∈ F^{×2}. Next, let (X, ⟨·, ·⟩) be either (X_{M_{2}}, ⟨·, ·⟩_{M_{2}}) or (X_{H}, ⟨·, ·⟩_{H}), as in (55) or (57). If X = X_{M_{2}}, set G = GL(2, F), and if X = X_{H}, set G = H^{×}. Let h_{0} be the element of GO(X) that maps x to x^{∗}; then h_{0} represents the nontrivial coset of GSO(X) in GO(X). Let π_{1} and π_{2} be irreducible, admissible representations of G with the same central character. Via the exact sequences (58) and (59), the representations (π_{1}, V_{1}) and (π_{2}, V_{2}) define an irreducible, admissible representation π_{1}⊗π_{2} of GSO(X) which has space V_{1}⊗V_{2} and action given by the formula (π_{1} ⊗ π_{2})(ρ(g_{1}, g_{2})) = π_{1}(g_{1}) ⊗ π_{2}(g_{2}) for g_{1}, g_{2} ∈ G. If π_{1} and π_{2} are not isomorphic, then π_{1} ⊗ π_{2} induces irreducibly to GO(X); we denote this induced representation by (π_{1} ⊗ π_{2})^{+}. Assume that π_{1} and π_{2} are isomorphic. In this case the representation π_{1} ⊗ π_{2} does not induce irreducibly to GO(X), but instead has two extensions σ_{1} and σ_{2} to representations of GO(X). Moreover, the space of linear forms on π_{1}⊗π_{2} that are invariant under the subgroup of GSO(X) of elements ρ(g, g^{∗−1}) for g ∈ G is one-dimensional. Let λ be a non-zero functional in this space. Then λ ∘ σ_{i}(h_{0}) is another such functional, so that λ∘σ_{i}(h_{0}) = ε_{i}λ with {ε_{1}, ε_{2}} = {1, −1}. The representation σ_{i} for which ε_{i} = 1 is denoted by (π_{1} ⊗ π_{2})^{+}, and the representation σ_{j} for which ε_{j} = −1 is denoted by (π_{1} ⊗ π_{2})^{−}. See [26] for details.

4.5.1 Proposition. Let H be as in (56) and let X_{H} be as in (57). Let S be as in (72) with −c ∉ F^{×2}; we may assume that i^{2} = −c, as in (74). Let z = (z_{1}, z_{2}) be as in (75), so that z ∈ Ω_{S}. Set L = F (\sqrt{−c}). We have

dim Hom_{O(X_{H})_{z}}(σ_{0}, ℂ_{1}) = 1

for the following families of irreducible, admissible representations σ_{0} of GO(X_{H}):

i) σ_{0} = (σ1_{H^{×}} ⊗ σχ_{L/F})^{+};

ii) σ_{0} = (σ1_{H^{×}} ⊗ σπ^{JL})^{+}.

Here, σ is a character of F^{×}, and π is a supercuspidal, irreducible, admissible representation of GL(2, F) with trivial central character with Hom_{L^{×}}(π^{JL}, ℂ_{1}) ≠ 0.

Proof. We begin by describing O(X_{H})_{z}. Define g_{1} : X_{H} → X_{H} by

g_{1}(1) = 1, g_{1}(i) = i, g_{1}(j) = j, g_{1}(k) = −k.

Evidently, g_{1} ∈ O(X_{H})_{z}, moreover, det(g_{1}) = −1. It follows that O(X_{H})_{z} = (SO(X_{H}) ∩ O(X_{H})_{z}) ⊔ (SO(X_{H}) ∩ O(X_{H})_{z})g_{1}. Using that z_{1} = 1, z_{2} = i, and the fact that every element of SO(X_{H}) is of the form ρ(h_{1}, h_{2}) for some h_{1}, h_{2} ∈ H^{×}, a calculation shows that SO(X_{H}) ∩ O(X_{H})_{z} is {ρ(h^{∗}^{−1}, h) : h ∈ (F + F i)^{×} = L^{×}}.

i) Since σ_{0}|_{O(X_{H})} = (1_{H^{×}} ⊗ χ_{L/F})^{+}, we may assume that σ = 1. A model for σ_{0} is ℂ ⊕ ℂ, with action defined by

σ_{0}(ρ(h_{1}, h_{2}))(w_{1} ⊕ w_{2}) = χ_{L/F} (N(h_{2}))w_{1} ⊕ χ_{L/F} (N(h_{1}))w_{2},
σ_{0}(∗)(w_{1} ⊕ w_{2}) = w_{2} ⊕ w_{1}

for w_{1}, w_{2} ∈ ℂ and h_{1}, h_{2} ∈ H^{×}; here, ∗ is the canonical involution of H, regarded as an element of O(X_{H}) with determinant −1. Using that g_{1} = ∗ ∘ ρ(k^{∗−1}, k), we find that the restriction of σ_{0} to O(X_{H})_{z} is given by

σ_{0}(ρ(h^{∗−1}, h))(w_{1} ⊕ w_{2}) = w_{1} ⊕ w_{2},
σ_{0}(g_{1})(w_{1} ⊕ w_{2}) = χ_{L/F} (N(k))(w_{2} ⊕ w_{1})

for w_{1}, w_{2} ∈ ℂ and h ∈ (F + F i)^{×} = L^{×}. Therefore, σ_{0}|_{O(X_{H})_{z}} is the direct sum of the trivial character O(X_{H})_{z}, and the non-trivial character of O(X_{H})_{z} that is trivial on SO(X_{H}) ∩ O(X_{H})_{z} and sends g_{1} to −1. This implies that Hom_{O(X_{H})_{z}}(σ_{0}, ℂ_{1}) is one-dimensional. ii) Again, we may assume that σ = 1. Let V be the space of π^{JL}. As a model for σ_{0} we take V ⊕ V with action of GO(X_{H}) defined by

σ_{0}(ρ(h_{1}, h_{2}))(v_{1} ⊕ v_{2}) = π^{JL}(h_{2})v_{1} ⊗ π^{JL}(h_{1})v_{2},
σ_{0}(∗)(v_{1} ⊕ v_{2}) = v_{2} ⊕ v_{1}

for h_{1}, h_{2} ∈ H^{×} and v_{1}, v_{2} ∈ V. By hypothesis, Hom_{L^{×}}(π^{JL}, ℂ_{1}) ≠ 0. This space is one-dimensional; see Sect. 2.7. We have kLk^{−1} = L; in fact, conjugation by k on L is the non-trivial element of Gal(L/F). Since Hom_{L^{×}}(π^{JL}, ℂ_{1}) is one-dimensional, there exists ε ∈ {±1} such that λ ∘ π^{JL}(k) = ελ for λ ∈ Hom_{L^{×}}(π^{JL}, ℂ_{1}). Define a map

Hom_{L^{×}}(π^{JL}, ℂ_{1}) ⟶ Hom_{O(X_{H})_{z}}(σ_{0}, ℂ_{1})

by sending λ to Λ, where Λ is defined by Λ(v_{1} ⊕ v_{2}) = λ(v_{1}) + ελ(v_{2}) for v_{1}, v_{2} ∈ V. A computation using the fact that g_{1} = ∗ ∘ ρ(k^{∗−1}, k) shows that this map is well defined. It is straightforward to verify that this map is injective and surjective, so that

Hom_{L^{×}}(π^{JL}, ℂ_{1}) ≅ Hom_{O(X_{H})_{z}}(σ_{0}, ℂ_{1}).

Hence, Hom_{O(X_{H})_{z}}(σ_{0}, ℂ_{1}) is one-dimensional.

4.6 GO(X) and GSp(4, F)

In this section we will gather together some information about the theta correspondence between GO(X) and GSp(4) when X is as in (50). When dim(X) = 4, we recall in Theorem 4.6.3 some results from [7] and [8]. When dim(X) = 2, we calculate two theta lifts, producing representations of type Vd and IXb, in Proposition 4.6.2. This calculation uses P_{3}-theory. We include this material because, to the best of our knowledge, such a computation is absent from the literature. We let R_{Q} be the group of elements of R of the form ([ \begin{matrix}∗ & ∗ & ∗ & ∗ \\ & ∗ & ∗ & ∗ \\ & ∗ & ∗ & ∗ \\ & & & ∗ \end{matrix} ] , ∗). Let Z^{J} be

the group defined in (40).

4.6.1 Lemma. Let (X, ⟨·, ·⟩) be an even-dimensional symmetric bilinear space satisfying (50); assume additionally that X is anisotropic. There is an isomorphism of complex vector spaces

T_{1} : S(X^{2})_{Z^{J}} ⟶^{∼} S(X)  (86)

that is given by

T_{1}(φ + S(X^{2})(Z^{J}))(x) = φ(x, 0)

for φ in S(X^{2}) and x in X. The subgroup R_{Q} of R acts on the quotient S(X^{2})_{Z^{J}}. Transferring this action to S(X) via T_{1}, the formulas for the resulting action are

([ \begin{matrix}t \\ & a & b \\ & c & d \\ & & & λ(h)t^{−1} \end{matrix} ] , h) · φ = |λ(h)|^{− dim(X)/4} (t, disc(X))_{F} |t|^{dim(X)/2} ω_{1}([ \begin{matrix}a & b \\ c & d \end{matrix}] , h)φ,(87)

([ \begin{matrix}1 & x & y & z \\ & 1 & & y \\ & & 1 & −x \\ & & & 1 \end{matrix} ] , 1) · φ = φ  (88)

for φ in S(X), x, y and z in F, t in F^{×}, and g = [ \begin{matrix}a & b \\ c & d \end{matrix} ] in GL(2, F) and h in GO(X) with λ(h) = det(g).

Proof. We first claim that

S(X^{2})(Z^{J}) = {φ ∈ S(X^{2}) : φ(X × 0) = 0}.  (89)

Let φ be in S(X^{2})(Z^{J}). By the lemma in 2.33 of [3] there exists a positive integer n so that

∫_{𝔭^{−n}} ω([ \begin{matrix}1 & & & b \\ & 1 \\ & & 1 \\ & & & 1 \end{matrix} ] , 1)φ db = 0.  (90)

Evaluating at (x, 0) and using (79) shows that φ(X × 0) = 0. Conversely, assume that φ is contained in the right hand side of (89). For any integer k let

L_{k} = {x ∈ X : ⟨x, x⟩ ∈ 𝔭^{k}}.  (91)

It is known that L_{k} is a lattice, i.e., it is a compact and open 𝔬 submodule of X; see the proof of Theorem 91:1 of [18]. Any lattice is free of rank dim X as an 𝔬 module. Since φ(X × 0) = 0, there exists a positive integer n such that φ(X × L_{n}) = 0. We claim that (90) holds. Let x_{1} and x_{2} be in X. Evaluating (90) at (x_{1}, x_{2}) gives

( ∫_{𝔭^{−n}} ψ(b⟨x_{2}, x_{2}⟩) db)φ(x_{1}, x_{2}).

This is zero if x_{2} is in L_{n} because φ(X × L_{n}) = 0. Assume that x_{2} is not in L_{n}. By the definition of L_{n}, we have ⟨x_{2}, x_{2}⟩ ∉ 𝔭^{n}. This implies that

∫_{𝔭^{−n}} ψ(b⟨x_{2}, x_{2}⟩) db = 0,

proving our claim. This completes the proof of (89). Using (89), it is easy to verify that the map T_{1} is an isomorphism of vector spaces. Equation (87) follows from Lemma 4.4.3, and equation (88) follows from (79) and (80).

4.6.2 Proposition. Let m ∈ F^{×}, and let (X_{m,1}, ⟨·, ·⟩_{m,1}) be as (52). Assume that m ∉ F^{×2}, so that X_{m,1} is anisotropic. Let E = F (\sqrt{m}), and identify characters of GSO(X_{m,1}) and characters of E^{×} via (14) and (54). Let χ_{E/F} be the quadratic character associated to E. Let Π be an irreducible, admissible representation of GSp(4, F), and let σ be an irreducible, admissible representation of GO(X_{m,1}).

i) Assume that σ = μ^{+} with μ = μ ∘ γ, so that μ = α ∘ N_{E/F} for a character α of F^{×}. Then Hom_{R}(ω, Π^{∨} ⊗ σ) ≠ 0 if and only if Π = L(νχ_{E/F} , χ_{E/F} ⋊ ν^{−1/2}α) (type Vd). ii) Assume that σ = μ^{+} = ind_{GSO(X_{m,1})}^{GO(X_{m,1})}(μ) with μ ≠ μ ∘ γ. Then

Hom_{R}(ω, Π^{∨} ⊗ σ) ≠ 0 if and only if Π = L(νχ_{E/F} , ν^{−1/2}π(μ)) (type IXb). Here, π(μ) is the supercuspidal, irreducible, admissible representation of GL(2, F) associated to μ.

Proof. Let (σ, W) be as in i) or ii). In the case of i), set π(μ) = α × αχ_{E/F}. Then Hom_{R_{1}}(ω_{1}, π(μ)^{∨} ⊗σ) ≠ 0, and π(μ) is the unique irreducible, admissible representation of GL(2, F) with this property, by Theorem 4.6 of [11]. Let (Π′, V) be an irreducible, admissible representation of GSp(4, F) such that Hom_{R}(ω, Π′ ⊗ σ) ≠ 0. Let T be a non-zero element of this space. The nonvanishing of T implies that the central characters of Π′ and σ satisfy

ω_{Π′} = ω_{σ}^{−1} = (μ|_{F^{×}})^{−1}.  (92)

We first claim that V is non-supercuspidal. By reasoning as in [9], there exist λ_{1}, …, λ_{t} in F^{×} and an irreducible Sp(4, F) subspace V_{0} of V such that

V = V_{1} ⊕ ⋯ ⊕ V_{t},

where

V_{1} = π([ \begin{matrix}1 \\ & 1 \\ & & λ_{1} \\ & & & λ_{1} \end{matrix} ])V_{0} , …, V_{t} = π([ \begin{matrix}1 \\ & 1 \\ & & λ_{t} \\ & & & λ_{t} \end{matrix} ])V_{0}.  (93)

Similarly, there exist irreducible O(X) subspaces W_{1}, …, W_{r} of W such that

W = W_{1} ⊕ ⋯ ⊕ W_{r}.

There exists an i and a j such that Hom_{Sp(4,F)×O(X)}(ω, V_{i} ⊗W_{j}) ≠ 0. As in the proof of Lemma 4.2 of [24], there is an irreducible constituent U_{1} of π(μ)^{∨} such that Hom_{O(X)}(ω_{1}, U_{1} ⊗ W_{j}) ≠ 0. By Theorem 4.4 of [25], the representation V_{i} is non-supercuspidal, so that V is non-supercuspidal. Since V is non-supercuspidal, we have V_{Z^{J}} ≠ 0 by Tables A.5 and A.6 of [28] (see the comment after Theorem 3.2.1). We claim next that Hom_{R_{Q}}(S(X^{2})_{Z^{J}} , V_{Z^{J}} ⊗ W) ≠ 0. It follows from (93) that (V_{i})_{Z^{J}} ≠ 0. Let p_{i} : V → V_{i} and q_{j} : W → W_{j} be the projections. These maps are Sp(4, F) and O(X) maps, respectively. The composition

S(X^{2}) ⟶^{T} V ⊗ W ⟶^{p_{i}⊗q_{j}} V_{i} ⊗ W_{j} ⟶ (V_{i})_{Z^{J}} ⊗ W_{j}

is non-zero and surjective; note that V_{i} ⊗W_{j} is irreducible. The commutativity of the diagram

S(X^{2}) ⟶^{T} V ⊗ W ⟶^{p_{i}⊗q_{j}} V_{i} ⊗ W_{j}
↓  ↓
V_{Z^{J}} ⊗ W ⟶^{p_{i}⊗q_{j}} (V_{i})_{Z^{J}} ⊗ W_{j}

implies our claim that Hom_{R_{Q}}(S(X^{2})_{Z^{J}} , V_{Z^{J}} ⊗ W) ≠ 0. Let RQ̄ be the subgroup of R_{Q} consisting of the elements of the form

([ \begin{matrix}∗ & ∗ & ∗ & ∗ \\ & ∗ & ∗ & ∗ \\ & ∗ & ∗ & ∗ \\ & & & 1 \end{matrix} ] , ∗). Let R_{P_{3}} be the subgroup of P_{3} × GO(X) consisting of the ele-

ments of the form ([ \begin{matrix}a & b & x \\ c & d & y \\ & & 1 \end{matrix} ] , h), ad−bc = λ(h). There is a homomorphism from RQ̄ to R_{P_{3}} given by

([ \begin{matrix}∗ & ∗ & ∗ & ∗ \\ & a & b & x \\ & c & d & y \\ & & & 1 \end{matrix} ] , h) ↦ ([ \begin{matrix}a & b & x \\ c & d & y \\ & & 1 \end{matrix} ] , h)

for [ \begin{matrix}a & b \\ c & d \end{matrix} ] in GL(2, F), x and y in F, and h in GO(X) with ad − bc = λ(h). We consider Z^{J} a subgroup of RQ̄ via z ↦ (z, 1). The above homomorphism then induces an isomorphism RQ̄/Z^{J} ≅ R_{P_{3}}. We restrict the R_{Q} modules S(X^{2})_{Z^{J}} and V_{Z^{J}} ⊗ W to RQ̄. The subgroup Z^{J} of RQ̄ acts trivially, so that these spaces may be viewed as R_{P_{3}} modules. Let χ be a character of F^{×}. We assert that

Hom_{R_{P_{3}}} (S(X^{2})_{Z^{J}} , τ_{GL(0)}^{P_{3}}(1) ⊗ σ) = 0,  (94)
Hom_{R_{P_{3}}} (S(X^{2})_{Z^{J}} , τ_{GL(1)}^{P_{3}}(χ) ⊗ σ) = 0.  (95)

Let τ be τ_{GL(0)}^{P_{3}}(1) or τ_{GL(1)}^{P_{3}}(χ). Assume that (94) or (95) is non-zero; we will obtain a contradiction. Let S be a non-zero element of (94) or (95). Since S is non-zero, there exists φ in S(X^{2})_{Z^{J}} such that S(φ) is non-zero. Write S(φ) = ∑_{i=1}^{t} f_{i} ⊗ w_{i} for some f_{1}, …, f_{t} in the standard space of τ and w_{1}, …, w_{t} in W. The elements f_{1}, …, f_{t} are functions from P_{3} to ℂ such that

f_{i}([ \begin{matrix}1 & & x \\ & 1 & y \\ & & 1 \end{matrix} ] p) = ψ(y)f_{i}(p)

for x and y in F, p in P_{3}, and i = 1, …, t. We may assume that the vectors w_{1}, …, w_{t} are linearly independent, and that there exists p in P_{3} such that f_{1}(p) is non-zero. Using the transformation properties of S and f_{1}, we may assume that p = [ \begin{matrix}a \\ & 1 \\ & & 1 \end{matrix} ]. Let λ : σ → ℂ be a linear functional such that λ(w_{1}) = 1 and λ(w_{2}) = ⋯ = λ(w_{t}) = 0, and let e : τ → ℂ be the linear functional that sends f to f(p). The composition (e ⊗ λ) ∘ S is non-zero on φ. On the other hand, using (88), for y in F we have

((e ⊗ λ) ∘ S)(([ \begin{matrix}1 \\ & 1 & y \\ & & 1 \end{matrix} ] , 1)φ) = (e ⊗ λ)(([ \begin{matrix}1 \\ & 1 & y \\ & & 1 \end{matrix} ] , 1) · S(φ)),
((e ⊗ λ) ∘ S)(φ) = (e ⊗ λ)(([ \begin{matrix}1 \\ & 1 & y \\ & & 1 \end{matrix} ] , 1) · ∑_{i=1}^{t} f_{i} ⊗ w_{i})

= ∑_{i=1}^{t} f_{i}(p [ \begin{matrix}1 \\ & 1 & y \\ & & 1 \end{matrix} ])λ(w_{i})

= ψ(y)f_{1}(p),

((e ⊗ λ) ∘ S)(φ) = ψ(y)((e ⊗ λ) ∘ S)(φ).

This is a contradiction since ((e ⊗ λ) ∘ S)(φ) is non-zero, and there exist y in F such that ψ(y) ≠ 1. This concludes the proof of (94) and (95).

It follows from (94), (95) and the non-vanishing of Hom_{R_{P_{3}}} (S(X^{2})_{Z^{J}} , V_{Z^{J}} ⊗W) that there exists an irreducible, admissible representation ρ of GL(2, F) that occurs in the P_{3} filtration of V_{Z^{J}} (see Theorem 3.2.1) such that Hom_{R_{P_{3}}} (S(X^{2})_{Z^{J}} , τ_{GL(2)}^{P_{3}}(ρ) ⊗ W) ≠ 0. It follows from (87) that Hom_{R_{1}}(ω_{1}, ν^{−1/2}χ_{E/F} ρ ⊗ σ) ≠ 0. By the uniqueness stated in the first paragraph of this proof, it follows that

ρ = ν^{1/2}χ_{E/F} π(μ)^{∨}.  (96)

As a consequence, ω_{ρ} = ν(μ|_{F^{×}})^{−1}χ_{E/F}. Together with (92), it follows that

ω_{Π′} = χ_{E/F} ν^{−1}ω_{ρ}.  (97)

Going through Table A.5 of [28], we see that only the Π′ = Π^{∨} with Π as asserted in i) and ii) satisfy both (96) and (97). (Observe the remark made after Theorem 3.2.1.)

Conversely, assume that Π is as in i) or ii). Since Hom_{R_{1}}(ω_{1}, π(μ)^{∨}⊗σ) ≠ 0, we have Hom_{O(X)}(S(X^{2}), σ) ≠ 0 by, for example, Remarque b) on p. 67 of [14]. Arguing as in Theorem 4.4 of [24], there exists some irreducible, admissible representation Π′ of GSp(4, F) such that Hom_{R}(ω, Π′ ⊗ σ) ≠ 0. By what we proved above, Π′ = Π^{∨}. This concludes the proof.

4.6.3 Theorem. ([7],[8]) Let (X, ⟨·, ·⟩) be either (X_{M_{2}}, ⟨·, ·⟩_{M_{2}}) or (X_{H}, ⟨·, ·⟩_{H}), as in (55) or (57). If X = X_{M_{2}}, set G = GL(2, F), and if X = X_{H}, set G = H^{×}. Let Π be an irreducible, admissible representation of GSp(4, F), and let π_{1} and π_{2} be irreducible, admissible representations of G with the same central character. We have

Hom_{R}(ω, Π^{∨} ⊗ (π_{1} ⊗ π_{2})^{+}) ≠ 0

for Π, π_{1} and π_{2} as in the following table:

type of Π  Π  π_{1}  π_{2}
I  χ_{1} × χ_{2} ⋊ σ  σχ_{1}χ_{2} × σ  σχ_{1} × σχ_{2}
II a  χSt_{GL(2)} ⋊ σ  σχ^{2} × σ  σχSt_{GL(2)}
b  χ1_{GL(2)} ⋊ σ  σχ^{2} × σ  σχ1_{GL(2)}
III b  χ ⋊ σ1_{GSp(2)}  σχν^{1/2} × σν^{−1/2} σχν^{−1/2} × σν^{1/2}
IV c  L(ν^{3/2}St_{GL(2)}, ν^{−3/2}σ) σν^{3/2} × σν^{−3/2}  σSt_{GL(2)}
d  σ1_{GSp(4)}  σν^{3/2} × σν^{−3/2}  σ1_{GL(2)}
V a  δ([ξ, νξ], ν^{−1/2}σ)  σSt_{GL(2)}  σξSt_{GL(2)}
a^{∗}  δ^{∗}([ξ, νξ], ν^{−1/2}σ)  σ1_{H^{×}}  σξ1_{H^{×}}
b L(ν^{1/2}ξSt_{GL(2)}, ν^{−1/2}σ)  σ1_{GL(2)}  σξSt_{GL(2)}
d  L(νξ ⋊ ν^{−1/2}σ)  σ1_{GL(2)}  σξ1_{GL(2)}
VI a  τ(S, ν^{−1/2}σ)  σSt_{GL(2)}  σSt_{GL(2)}
b  τ(T, ν^{−1/2}σ)  σ1_{H^{×}}  σ1_{H^{×}}
c  L(ν^{1/2}St_{GL(2)}, ν^{−1/2}σ)  σ1_{GL(2)}  σSt_{GL(2)}
d  L(ν, 1_{F^{×}} ⋊ ν^{−1/2}σ)  σ1_{GL(2)}  σ1_{GL(2)}
VIII a  τ(S, π)  π  π
b  τ(T, π)  π^{JL}  π^{JL}
X  π ⋊ σ  σω_{π} × σ  π
XI a  δ(ν^{1/2}π, ν^{−1/2}σ)  σSt_{GL(2)}  σπ
a^{∗}  δ^{∗}(ν^{1/2}π, ν^{−1/2}σ)  σ1_{H^{×}}  σπ^{JL}
b  L(ν^{1/2}π, ν^{−1/2}σ)  σ1_{GL(2)}  σπ

The notation π^{JL} in the table denotes the Jacquet-Langlands lifting of the supercuspidal representation π of GL(2, F) to a representation of H^{×}. See Sect. 4.5 for the definitions of the + representation.

4.7 Applications

We now apply Theorem 4.4.6 along with knowledge of the theta correspondences of the previous section to obtain results about Bessel functionals.

4.7.1 Corollary. Let (X, ⟨·, ·⟩) be either (X_{M_{2}}, ⟨·, ·⟩_{M_{2}}) or (X_{H}, ⟨·, ·⟩_{H}), as in (55) or (57). If X = X_{M_{2}}, set G = GL(2, F), and if X = X_{H}, set G = H^{×}.

Let Π be an irreducible, admissible representation of GSp(4, F), and let π_{1} and π_{2} be irreducible, admissible representations of G with the same central character. Assume that

Hom_{R}(ω, Π^{∨} ⊗ (π_{1} ⊗ π_{2})^{+}) ≠ 0

and that Π has a non-split (Λ, θ)-Bessel functional with θ = θ_{S}. Then

Hom_{T} (π_{1}, ℂ_{Λ}) ≠ 0 and Hom_{T} (π_{2}, ℂ_{Λ}) ≠ 0

where T = T_{S}.

Proof. The assumption that the Bessel functional is non-split means that A = A_{S} is a field. By Sect. 2.2 and Sect. 2.5 we may assume that S has the diagonal form (72). By (19), the contragredient Π^{∨} has a ((Λ ∘ γ)^{−1}, θ)- Bessel functional. The assertion follows now from Theorem 4.4.6, the explicit embeddings in (73) and (76), and the relation (31).

4.7.2 Corollary. Let (Π, V) be an irreducible, admissible representation of GSp(4, F). If Π is one of the representations in the following table, then Π admits a non-zero (Λ, θ)-Bessel functional β if and only if the quadratic extension L associated to β, and Λ, regarded as a character of L^{×}, are as specified in the table.

type of Π  Π  L  Λ
Va^{∗}  δ^{∗}([χ_{E/F} , νχ_{E/F}], ν^{−1/2}α) E  α ∘ N_{E/F}
Vd  L(νχ_{E/F} , χ_{E/F} ⋊ ν^{−1/2}α) E  α ∘ N_{E/F}
IXb  L(νχ_{E/F} , ν^{−1/2}π(μ))  E μ and the Galois conjugate of μ

Proof. First we consider the Va^{∗} case. Let Π = δ^{∗}([χ_{E/F} , νχ_{E/F}], ν^{−1/2}α). By Theorem 4.6.3,

Hom_{R}(ω, Π^{∨} ⊗ (α1_{H^{×}} ⊗ αχ_{E/F} 1_{H^{×}})^{+}) ≠ 0.

First, assume that Π admits a non-zero (Λ, θ)-Bessel functional, and let L be the quadratic extension associated to this Bessel functional; we will prove that E = L and that Λ = α∘N_{E/F}. By v) of Proposition 3.5.1, this Bessel functional is non-split. It follows from Corollary 4.7.1 that

α(N_{L/F} (t)) = Λ(t)  and  (χ_{E/F} α)(N_{L/F} (t)) = Λ(t)

for t in T = L^{×}. It follows that E = L, and that Λ = α ∘ N_{E/F}. Finally, we prove that Va^{∗} admits a Bessel functional as specified in the statement of the corollary. By Theorem 6.1.4 below, Va^{∗} admits some non-zero Bessel functional. The previous paragraph proves that this Bessel functional must be as described in the statement of the corollary. The arguments for the cases Vd and IXb are similar; we will only consider the case of type IXb. Let Π = L(νχ_{E/F} , ν^{−1/2}π(μ)), where E is a quadratic extension of F, χ_{E/F} is the quadratic character associated to E/F, μ is a character of E^{×} that is not Galois invariant, and π(μ) is the supercuspidal, irreducible, admissible representation of GL(2, F) associated to μ. First, assume that Π admits a non-zero (Λ, θ)-Bessel functional, and let L be the quadratic extension associated to this Bessel functional; we will prove that E = L, and that Λ is μ or the Galois conjugate of μ. Let S define θ, as in (15). By (19), Π^{∨} admits a non-zero ((Λ ∘ γ)^{−1}, θ) Bessel functional β. Write E = F (\sqrt{m}) for some m ∈ F^{×}. By Proposition 4.6.2 we have Hom_{R}(ω, Π^{∨} ⊗ μ^{+}) ≠ 0 with μ^{+} as in this proposition. The involved symmetric bilinear space is (X_{m,1}, ⟨·, ·⟩_{m,1}). Let GSp(4, F)^{+} be defined with respect to (X_{m,1}, ⟨·, ·⟩_{m,1}) as in (77). By Lemma 4.4.1 the index of GSp(4, F)^{+} in GSp(4, F) is two. By Lemma 2.1 of [9], the restriction of Π^{∨} to GSp(4, F)^{+} is irreducible or the direct sum of two non-isomorphic irreducible, admissible representations of GSp(4, F)^{+}; the non-vanishing of Hom_{R}(ω, Π^{∨} ⊗ μ^{+}) and Lemma 4.1 of [27] (with m = 2 and n = 2) imply that V^{∨} = V_{1} ⊕ V_{2} with V_{1} and V_{2} irreducible GSp(4, F)^{+} subspaces of V. Moreover, for each i ∈ {1, 2}, there exists λ_{i} ∈ F^{×} such that Π([ \begin{matrix}1 \\ & λ_{i} \end{matrix} ])V_{1} = V_{i}. Since Hom_{R}(ω, V^{∨} ⊗ μ^{+}) is nonzero, we may assume, after possibly renumbering, that Hom_{R}(ω, V_{1} ⊗ μ^{+}) is non-zero. There exists i ∈ {1, 2} such that the restriction of β to V_{i} is non-zero. Let β′ = [ \begin{matrix}1 \\ & λ_{i}^{−1} \end{matrix} ] · β. From Sect. 2.5 it follows that β′ is a ((Λ ∘ γ)^{−1}, θ′) Bessel

functional on Π^{∨} with θ′ defined by S′ = λ_{i}^{−1}S; also, the restriction of β′ to V_{1} is non-zero. We will now apply Theorem 4.4.6, with S′ and V_{1} playing the roles of S and π, respectively. By i) of this theorem we have that Ω_{S′} is non-empty; since S and S′ have the same discriminant, Lemma 4.3.1 implies that L = E. Let z ∈ Ω_{S′} and τ ∈ E(z). By ii) of Theorem 4.4.6, there exists a non-zero vector w in the space of μ^{+} such that μ^{+}(τ(t))w = (Λ ∘ γ)(t)w for t ∈ T_{S′}. By Lemma 4.3.1 again, this implies that μ^{+}(ρ(x))w = Λ(x)w for x ∈ L^{×}, or μ^{+}(ρ(γ(x)))w = Λ(x)w for x ∈ L^{×}. Since w ≠ 0, the definition of μ^{+} now implies that Λ = μ or μ ∘ γ, as desired. Finally, we prove that Π admits Bessel functionals as specified in the statement of the corollary. By Theorem 6.1.4 below, Π admits some non-zero Bessel functional. The previous paragraph proves that this Bessel functional must be as described in the statement of the corollary, and (29) implies that the Π admits both of the asserted Bessel functionals.

The following result will imply uniqueness of Bessel functionals for representations of type Va^{∗} and XIa^{∗}.

4.7.3 Corollary. Let σ be a character of F^{×}. Let c ∈ F^{×} with −c ∉ F^{×2}. Let S be as in (72) and set L = F (\sqrt{−c}). i) If ξ = χ_{L/F}, then dim δ^{∗}([ξ, νξ], ν^{−1/2}σ)_{N,θ_{S}} = 1.

ii) If π is an irreducible, admissible, supercuspidal representation of GL(2, F) with trivial central character such that Hom_{L^{×}}(π^{JL}, ℂ_{1}) ≠ 0, then dim δ^{∗}(ν^{1/2}π, ν^{−1/2}σ)_{N,θ_{S}} = 1.

Proof. This follows from Proposition 4.4.7, Theorem 4.6.3, and Proposition 4.5.1; note that Π^{∨}|_{N} ≅ Π|_{N} for irreducible, admissible representations Π of GSp(4, F) because Π^{∨} ≅ ω_{Π}^{−1}Π.

5 Twisted Jacquet modules of induced representations

Let (π, V) be an irreducible, admissible representation of GSp(4, F). In view of the isomorphism (20), understanding the possible Bessel functionals of π is equivalent to understanding the twisted Jacquet modules V_{N,θ} as T -modules. In this section, we will calculate the twisted Jacquet modules for representations induced from the Siegel and Klingen parabolic subgroup. This information will be used to determine the possible Bessel functionals for many of the nonsupercuspidal representations of GSp(4, F); see Sect. 6.2. The results of this section are similar to Proposition 2.1 and 2.3 of [23]. However, we prefer to redo the arguments, as those in [23] contain some inaccuracies.

5.1 Two useful lemmas

For a positive integer n let S(F^{n}) be the Schwartz space of F^{n}, meaning the space of locally constant, compactly supported functions F^{n} → ℂ. As before, ψ is our fixed non-trivial character of F. Let V be a complex vector space. Let S(F, V) be the space of compactly supported, locally constant functions from F to V. There is a canonical isomorphism S(F, V) ≅ S(F) ⊗ V. The functional on S(F) given by f ↦ ∫_{F} f(x) dx gives rise to a linear map S(F) ⊗ V → V, and hence to a linear map S(F, V) → V. We write this map as an integral

f ⟼ ∫_{F} f(x) dx.

The following lemma will be frequently used when we calculate Jacquet modules in the subsequent sections.

5.1.1 Lemma. Let ρ denote the action of F on S(F, V) by translation, i.e., (ρ(x)f)(y) = f(x + y). Let ρ′ be the action of F on S(F, V) given by (ρ′(x)f)(y) = ψ(xy)f(y).

i) The map f ↦ ∫_{F} f(x) dx induces an isomorphism

S(F, V)/⟨f − ρ(x)f : x ∈ F ⟩ ≅ V.

ii) The map f ↦ ∫_{F} ψ(−x)f(x) dx induces an isomorphism

S(F, V)/⟨ψ(x)f − ρ(x)f : x ∈ F ⟩ ≅ V.

iii) The map f ↦ f(0) induces an isomorphism

S(F, V)/⟨f − ρ′(x)f : x ∈ F ⟩ ≅ V.

Proof. By the Proposition in 1.18 of [3], every translation-invariant functional on S(F) is a multiple of the Haar measure f ↦ ∫_{F} f(x) dx. This proves i) in the case where V = ℂ. The general case follows from this case by tensoring the exact sequence

0 ⟶ ⟨f − ρ(x)f : x ∈ F, f ∈ S(F)⟩ ⟶ S(F) ⟶ ℂ ⟶ 0

by V. Under the isomorphism S(F) ⊗ V ≅ S(F, V), the space ⟨f − ρ(x)f : x ∈ F, f ∈ S(F)⟩ ⊗ V maps onto ⟨f − ρ(x)f : x ∈ F, f ∈ S(F, V)⟩. To prove ii), observe that there is an isomorphism

S(F, V)/⟨f − ρ(x)f : x ∈ F, f ∈ S(F, V)⟩ ⟶ S(F, V)/⟨ψ(x)f − ρ(x)f : x ∈ F, f ∈ S(F, V)⟩

induced by the map f ↦ f′, where f′(x) = ψ(x)f(x). Hence ii) follows from i). Finally, iii) also follows from i), since the Fourier transform f ↦ f̂, where

f̂(y) = ∫_{F} ψ(−uy)f(u) du,

intertwines the actions ρ and ρ′ of F on S(F, V).

5.1.2 Lemma. Let G be an l-group, as in [3], and let H_{1} and H_{2} be closed subgroups of G. Assume that G = H_{1}H_{2}, and that for every compact subset K of G, there exists a compact subset K_{2} of H_{2} such that K ⊂ H_{1}K_{2}. Let (ρ, V) be a smooth representation of H_{1}. The map r : c-Ind_{H_{1}}^{G}ρ → c-Ind_{H_{1}∩H_{2}}^{H_{2}}(ρ|_{H_{1}∩H_{2}}) defined by restriction of functions is a well-defined isomorphism of representations of H_{2}.

Proof. This follows from straightforward verifications.

5.2 Siegel induced representations

Let π be an admissible representation of GL(2, F), let σ be a character of F^{×}, and let π ⋊σ be as defined in Sect. 1.2; see (9). In this section we will calculate the twisted Jacquet modules (π ⋊ σ)_{N,θ} for any non-degenerate character θ of N as a module of T. Lemma 5.2.2 below corrects an inaccuracy in Proposition 2.1 of [23]. Namely, Proposition 2.1 of [23] does not include ii) of our lemma.

5.2.1 Lemma. Let σ be a character of F^{×}, and π an admissible representation of GL(2, F). Let I be the standard space of the Siegel induced representation π ⋊ σ. There is a filtration of P -modules

I^{3} = 0 ⊂ I^{2} ⊂ I^{1} ⊂ I^{0} = I.

with the quotients given as follows.

i) I^{0}/I^{1} = σ_{0}, where

σ_{0}(\left[{A ∗ \atop cA′}\right]) = σ(c) |c^{−1} det(A)|^{3/2} π(A)

for A in GL(2, F) and c in F^{×}.

ii) I^{1}/I^{2} = c-Ind_{[ \begin{matrix}∗ & ∗ & ∗ & ∗ \\ & ∗ & & ∗ \\ & & ∗ & ∗ \\ & & & ∗ \end{matrix} ]}^{P}σ_{1}, where

σ_{1}([ \begin{matrix}t & ∗ & y & ∗ \\ & a & & ∗ \\ & & d & ∗ \\ & & & adt^{−1} \end{matrix} ]) = σ(ad) |a^{−1}t|^{3/2} π(\left[{t y \atop d}\right])

for y in F and a, d, t in F^{×}.

iii) I^{2}/I^{3} = c-Ind_{[ \begin{matrix}∗ & ∗ \\ ∗ & ∗ \\ & & ∗ & ∗ \\ & & ∗ & ∗ \end{matrix} ]}^{P}σ_{2}, where

σ_{2}([ \begin{matrix}A \\ & cA′ \end{matrix} ]) = σ(c) |c det(A)^{−1}|^{3/2} π(cA′)

for A in GL(2, F) and c in F^{×}.

Proof. This follows by going through the procedure of Sections 6.2 and 6.3 of [5].

5.2.2 Lemma. Let σ be a character of F^{×}, and let (π, V) be an admissible representation of GL(2, F). We assume that π admits a central character ω_{π}. Let I be the standard space of the Siegel induced representation π ⋊ σ. Let θ be the character of N defined in (15). Assume that θ is non-degenerate. Let L be the quadratic extension associated to S as in Sect. 2.3.

i) Assume that L is a field. Then I_{N,θ} ≅ V with the action of T given by σω_{π}π′. Here, π′ is the representation of GL(2, F) on V given by π′(g) = π(g′). In particular, if π is irreducible, then the action of T is given by σπ.

ii) Assume that L is not a field; we may arrange that S = [ \begin{matrix}& 1/2 \\ 1/2 \end{matrix} ]. Then

there is a filtration

0 ⊂ J_{2} ⊂ J_{1} = I_{N,θ},

with vector space isomorphisms:

• J_{1}/J_{2} ≅ V_{\left[{1 ∗ \atop 1}\right],ψ} ⊕ V_{\left[{1 ∗ \atop 1}\right],ψ},

• J_{2} ≅ V.

The action of T = T_{S} is given as follows,

diag(a, b, a, b)(v_{1} ⊕ v_{2}) = |\frac{a}{b}|^{1/2}σ(ab)ω_{π}(a)v_{1} ⊕ |\frac{a}{b}|^{−1/2}σ(ab)ω_{π}(b)v_{2},
diag(a, b, a, b)v = σ(ab)π([ \begin{matrix}a \\ & b \end{matrix}])v,

for a, b ∈ F^{×}, v_{1} ⊕ v_{2} ∈ J_{1}/J_{2} ≅ V_{\left[{1 ∗ \atop 1}\right],ψ} ⊕ V_{\left[{1 ∗ \atop 1}\right],ψ}, and v ∈ J_{2}. In particular, if π is one-dimensional, then I_{N,θ} ≅ V, with the action of T given by diag(a, b, a, b)v = σ(ab)π([ \begin{matrix}a \\ & b \end{matrix}])v.

Proof. We may assume that b = 0. Since det(S) ≠ 0 we have a ≠ 0 and c ≠ 0. We use the notation of Lemma 5.2.1. We calculate the twisted Jacquet modules (I^{i}/I^{i+1})_{N,θ} for i ∈ {0, 1, 2}. Since the action of N on I^{0}/I^{1} is trivial and θ is non-trivial, we have (I^{0}/I^{1})_{N,θ} = 0. We consider the quotient I^{1}/I^{2} = c-Ind_{H}^{P}σ_{1}, where

H = [ \begin{matrix}∗ & ∗ & ∗ & ∗ \\ & ∗ & & ∗ \\ & & ∗ & ∗ \\ & & & ∗ \end{matrix} ] ,

and with σ_{1} as in ii) of Lemma 5.2.1. We first show that for each function f in the standard model of this representation, the function f^{∘} : F → V, given by

f^{∘}(w) = f([ \begin{matrix}1 \\ & 1 & w \\ & & 1 \\ & & & 1 \end{matrix} ]),

has compact support. Let K be a compact subset of P such that the support of f is contained in HK. If

[ \begin{matrix}1 \\ & 1 & w \\ & & 1 \\ & & & 1 \end{matrix} ] = [ \begin{matrix}t & ∗ & y & ∗ \\ & a & & ∗ \\ & & d & ∗ \\ & & & adt^{−1} \end{matrix} ] [ \begin{matrix}k_{1} & k_{2} & x_{1} & x_{2} \\ k_{3} & k_{4} & x_{3} & x_{4} \\ & & k_{5} & k_{6} \\ & & k_{7} & k_{8} \end{matrix} ] ,

with the rightmost matrix being in K, then calculations show that k_{3} = k_{7} = 0 and w = k_{4}^{−1}x_{3}. Since k_{4}^{−1} and x_{3} vary in bounded subsets, w is confined to a compact subset of F. This proves our assertion that f^{∘} has compact support. Next, for each function f in the standard model of c-Ind_{H}^{P}σ_{1}, consider the function f̃: F^{2} → V given by

f̃(u, w) = f([ \begin{matrix}1 \\ u & 1 & w \\ & & 1 \\ & & −u & 1 \end{matrix} ])

for u, w in F. Let W be the space of all such functions f̃. Since the map f ↦ f̃ is injective, we get a vector space isomorphism c-Ind_{H}^{P}σ_{1} ≅ W. In this new model, the action of N is given by

([ \begin{matrix}1 & & y & z \\ & 1 & x & y \\ & & 1 \\ & & & 1 \end{matrix} ] f̃)(u, w) = π(\left[{1 y+uz \atop 1}\right])f̃(u, w + x + 2uy + u^{2}z)  (98)

for x, y, z, u, w in F. We claim that W contains S(F^{2}, V). Since W is translation invariant, it is enough to prove that W contains the function

f_{N,v}(u, w) = \begin{cases}v & if u, w ∈ 𝔭^{N}, \\ 0 & if u ∉ 𝔭^{N} or w ∉ 𝔭^{N}, \end{cases}

for any v in V and any positive integer N. Again by translation invariance, we may assume that N is large enough so that

σ_{1}(h)v = v  for h ∈ H ∩ Γ_{N},  (99)

where

Γ_{N} = [ \begin{matrix}1+𝔭^{N} & 𝔭^{N} & 𝔭^{N} & 𝔭^{N} \\ 𝔭^{N} & 1+𝔭^{N} & 𝔭^{N} & 𝔭^{N} \\ & & 1+𝔭^{N} & 𝔭^{N} \\ & & 𝔭^{N} & 1+𝔭^{N} \end{matrix} ] ∩ P.  (100)

Define f : P → V by

f(g) = \begin{cases}σ_{1}(h)v & if g = hk with h ∈ H, k ∈ Γ_{N}, \\ 0 & g ∉ HΓ_{N}. \end{cases}

Then, by (99), f is a well-defined element of c-Ind_{H}^{Q}σ_{1}. It is easy to verify that f̃ = f_{N,v}. This proves our claim that W contains S(F^{2}, V). Now consider the map

W ⟶ S(F, V),  f̃ ⟼ (w ↦ f([ \begin{matrix}1 \\ & 1 & w \\ & & 1 \\ & & & 1 \end{matrix} ] s_{1})),  (101)

where s_{1} is defined in (7). This map is well-defined, since the function on the right is (s_{1}f)^{∘}, which we showed above has compact support. Similar considerations as above show that the map (101) is surjective. We claim that the kernel of (101) is S(F^{2}, V). First suppose that f̃ lies in the kernel; we have to show that f̃ has compact support. Choose N large enough so that f is right invariant under Γ_{N}. Then, for u not in 𝔭^{−N} and w in F,

f̃(u, w) = f([ \begin{matrix}1 \\ u & 1 & w \\ & & 1 \\ & & −u & 1 \end{matrix} ])

= f([ \begin{matrix}1 \\ & 1 & w \\ & & 1 \\ & & & 1 \end{matrix} ][ \begin{matrix}1 & u^{−1} \\ & 1 \\ & & 1 & −u^{−1} \\ & & & 1 \end{matrix} ][ \begin{matrix}−u^{−1} \\ & u \\ & & u^{−1} \\ & & −u \end{matrix} ]s_{1}[ \begin{matrix}1 & u^{−1} \\ & 1 \\ & & 1 & −u^{−1} \\ & & & 1 \end{matrix} ])

= f([ \begin{matrix}1 & u^{−1} \\ & 1 \\ & & 1 & −u^{−1} \\ & & & 1 \end{matrix} ] [ \begin{matrix}1 & & −u^{−1}w & u^{−2}w \\ & 1 & w & −u^{−1}w \\ & & 1 \\ & & & 1 \end{matrix} ] [ \begin{matrix}−u^{−1} \\ & u \\ & & u^{−1} \\ & & −u \end{matrix} ] s_{1})

= π(\left[{1 −u^{−1}w \atop 1}\right])f([ \begin{matrix}1 \\ & 1 & w \\ & & 1 \\ & & & 1 \end{matrix} ] [ \begin{matrix}−u^{−1} \\ & u \\ & & u^{−1} \\ & & −u \end{matrix} ] s_{1})

= π(\left[{1 w^{2}u^{−1} \atop 1}\right])f([ \begin{matrix}1 & −w \\ & 1 \\ & & 1 & w \\ & & & 1 \end{matrix} ] [ \begin{matrix}1 \\ & −u^{−1} \\ & & −u \\ & & & 1 \end{matrix} ] s_{2})

= π(\left[{1 −u^{−1}w \atop 1}\right])f([ \begin{matrix}−u^{−1} \\ & u \\ & & u^{−1} \\ & & −u \end{matrix} ] [ \begin{matrix}1 \\ & 1 & u^{−2}w \\ & & 1 \\ & & & 1 \end{matrix} ] s_{1})

= |u|^{−3}π(\left[{−u^{−1} −u^{−2}w \atop u^{−1}}\right])f([ \begin{matrix}1 \\ & 1 & u^{−2}w \\ & & 1 \\ & & & 1 \end{matrix} ] s_{1}).

This last expression is zero by assumption. For fixed u in 𝔭^{−N}, the function f̃(u, ·) has compact support; this follows because each f^{∘} has compact support. Combining these facts shows that f̃ has compact support. Conversely, assume f̃is in S(F^{2}, V). Then we can find a large enough N such that if u has valuation −N, the function f̃(u, ·) is zero. Looking at the above calculation, we see that, for fixed such u,

f([ \begin{matrix}1 \\ & 1 & u^{−2}w \\ & & 1 \\ & & & 1 \end{matrix} ] s_{1}) = 0

for all w in F. This shows that f̃ is in the kernel of the map (101), completing the proof of our claim about this kernel. We now have an exact sequence

0 ⟶ S(F^{2}, V) ⟶ W ⟶ S(F, V) ⟶ 0.  (102)

Note that the space S(F^{2}, V) is invariant under the action (98) of N. A calculation shows that the action of N on S(F, V) is given by

([ \begin{matrix}1 & & y & z \\ & 1 & x & y \\ & & 1 \\ & & & 1 \end{matrix} ] f)(w) = π(\left[{1 y \atop 1}\right])f(w + z)  (103)

for x, y, z, w in F and f in S(F, V). Since the action of x is trivial and a ≠ 0, it follows that S(F, V)_{N,θ} = 0. Hence, by (102), we have W_{N,θ} ≅ S(F^{2}, V)_{N,θ}. We will compute the Jacquet module S(F^{2}, V)_{N,θ} in stages. The action of N on S(F^{2}, V) is given by (98). By ii) of Lemma 5.1.1, the map f̃ ↦ f′, where f′ : F → V is given by

f′(u) = ∫_{F} ψ^{a}(−u)f̃(u, w) dw,

defines a vector space isomorphism

W_{[ \begin{matrix}1 \\ & 1 & ∗ \\ & & 1 \\ & & & 1 \end{matrix} ],ψ^{a}} ⟶^{∼} S(F, V).

The transfer of the action of the remaining group [ \begin{matrix}1 & & ∗ & ∗ \\ & 1 & & ∗ \\ & & 1 \\ & & & 1 \end{matrix} ] to S(F, V) is given

by

([ \begin{matrix}1 & & y & z \\ & 1 & & y \\ & & 1 \\ & & & 1 \end{matrix} ] f)(u) = ψ(a(2uy + u^{2}z))π(\left[{1 y+uz \atop 1}\right])f(u)

for u, y, z ∈ F and f ∈ S(F, V). The subspace S(F^{×}, V) of elements f of S(F, V) such that f(0) = 0 is preserved under this action, so that we have an exact sequence

0 ⟶ S(F^{×}, V) ⟶ S(F, V) ⟶ S(F, V)/S(F^{×}, V) ⟶ 0  (104)

of representations of the group [ \begin{matrix}1 & & ∗ & ∗ \\ & 1 & & ∗ \\ & & 1 \\ & & & 1 \end{matrix} ]. There is an isomorphism of vector

spaces S(F, V)/S(F^{×}, V) ⟶^{∼} V that sends f to f(0). The transfer of the

action of the group [ \begin{matrix}1 & & ∗ & ∗ \\ & 1 & & ∗ \\ & & 1 \\ & & & 1 \end{matrix} ] to V is given by

[ \begin{matrix}1 & & y & z \\ & 1 & & y \\ & & 1 \\ & & & 1 \end{matrix} ] v = π(\left[{1 y \atop 1}\right])v  (105)

for y, z ∈ F and v ∈ V. The non-vanishing of c and (105) imply that

(S(F, V)/S(F^{×}, V))_{[ \begin{matrix}1 & & & ∗ \\ & 1 \\ & & 1 \\ & & & 1 \end{matrix} ],ψ^{c}} = 0.

Therefore,

(S(F, V)/S(F^{×}, V))_{[ \begin{matrix}1 & & ∗ & ∗ \\ & 1 & & ∗ \\ & & 1 \\ & & & 1 \end{matrix} ],θ} = 0.

Next, we define a vector space isomorphism of S(F^{×}, V) with itself and then transfer the action. For f in S(F^{×}, V), set f̃(u) = π([ \begin{matrix}u \\ & 1 \end{matrix} ])f(u) for u in F^{×}. The map defined by f ↦ f̃ is an automorphism of vector spaces. The transfer

of the action of [ \begin{matrix}1 & & ∗ & ∗ \\ & 1 & & ∗ \\ & & 1 \\ & & & 1 \end{matrix} ] is given by

([ \begin{matrix}1 & & y & z \\ & 1 & & y \\ & & 1 \\ & & & 1 \end{matrix} ] f)(u) = ψ(a(2uy + u^{2}z))π(\left[{1 uy+u^{2}z \atop 1}\right])f(u)

for f ∈ S(F^{×}, V), y, z ∈ F, and u ∈ F^{×}. Now define a linear map

p : S(F^{×}, V) ⟶ S(F^{×}, V_{\left[{1 ∗ \atop 1}\right],ψ^{−2a}})

by composing the elements of S(F^{×}, V) with the natural projection from V to V_{\left[{1 ∗ \atop 1}\right],ψ^{−2a}} = V/V (\left[{1 ∗ \atop 1}\right] , ψ^{−2a}). The map p is surjective. Let f be in S(F^{×}, V). Since f has compact support and is locally constant, we see that f is in the kernel of p if and only if

there exists l > 0 such that ∫_{𝔭^{−l}} ψ(2ay)π(\left[{1 y \atop 1}\right])f(u) dy = 0 for all u ∈ F^{×}.

(106)

Also, f is in S(F^{×}, V)([ \begin{matrix}1 & & ∗ \\ & 1 & & ∗ \\ & & 1 \\ & & & 1 \end{matrix} ]) if and only if

there exists k > 0 such that ∫_{𝔭^{−k}} ψ(2auy)π(\left[{1 uy \atop 1}\right])f(u) dy = 0 for all u ∈ F^{×}.

(107)

Since f is locally constant and compactly supported the conditions (106) and (107) are equivalent. It follows that p induces an isomorphism of vector spaces:

S(F^{×}, V)_{[ \begin{matrix}1 & & ∗ \\ & 1 & & ∗ \\ & & 1 \\ & & & 1 \end{matrix} ],ψ^{b}} = S(F^{×}, V)_{[ \begin{matrix}1 & & ∗ \\ & 1 & & ∗ \\ & & 1 \\ & & & 1 \end{matrix} ]} ⟶^{∼} S(F^{×}, V_{\left[{1 ∗ \atop 1}\right],ψ^{−2a}}).

Transferring the action of [ \begin{matrix}1 & & & ∗ \\ & 1 \\ & & 1 \\ & & & 1 \end{matrix} ] on the first space to the last space results

in the formula

[ \begin{matrix}1 & & & z \\ & 1 \\ & & 1 \\ & & & 1 \end{matrix} ] f(u) = ψ(au^{2}z)π(\left[{1 u^{2}z \atop 1}\right])f(u) = ψ(−au^{2}z)f(u), z ∈ F, u ∈ F^{×},

for f in S(F^{×}, V_{\left[{1 ∗ \atop 1}\right],ψ^{−2a}}). Assume that L is a field; we will prove that

S(F^{×}, V_{\left[{1 ∗ \atop 1}\right],ψ^{−2a}})_{[ \begin{matrix}1 & & & ∗ \\ & 1 \\ & & 1 \\ & & & 1 \end{matrix} ],ψ^{c}} = 0.  (108)

Let f be in S(F^{×}, V_{\left[{1 ∗ \atop 1}\right],ψ^{−2a}}). Since the support of f is compact, and since

there exists no u in F^{×} such that c + au^{2} = 0 as D = b^{2}/4 − ac = −ac is not in F^{×2}, there exists a positive integer l such that

∫_{𝔭^{−l}} ψ(−(c + au^{2})z) dz = 0  (109)

for u in the support of f. Hence, for u in F^{×},

(∫_{𝔭^{−l}} ψ(−cz) [ \begin{matrix}1 & & & z \\ & 1 \\ & & 1 \\ & & & 1 \end{matrix} ] f dz)(u) = ( ∫_{𝔭^{−l}} ψ(−(c + au^{2})z) dz)f(u) = 0. (110)

This proves (108), and completes the argument that (I^{1}/I^{2})_{N,θ} = 0 in the case L is a field. Now assume that L is not a field. We may further assume that a = 1 and c = −1 while retaining b = 0. The group T = T_{[ \begin{matrix}a & b/2 \\ b/2 & c \end{matrix} ]} = T_{[ \begin{matrix}1 \\ & −1 \end{matrix} ]} consists of

the elements

t = [ \begin{matrix}x & y \\ y & x \\ & & x & −y \\ & & −y & x \end{matrix} ]  (111)

with x, y ∈ F such that x^{2} ≠ y^{2}. Define

S(F^{×}, V_{\left[{1 ∗ \atop 1}\right],ψ^{−2a}}) ⟶ V_{\left[{1 ∗ \atop 1}\right],ψ^{−2a}} ⊕ V_{\left[{1 ∗ \atop 1}\right],ψ^{−2a}}  (112)

by f ↦ f(1) ⊕ f(−1). We assert that the kernel of this linear map is

S(F^{×}, V_{\left[{1 ∗ \atop 1}\right],ψ^{−2a}})([ \begin{matrix}1 & & & ∗ \\ & 1 \\ & & 1 \\ & & & 1 \end{matrix} ] , ψ^{c}).

Evidently, this subspace is contained in the kernel. Conversely, let f ∈ S(F^{×}, V_{\left[{1 ∗ \atop 1}\right],ψ^{−2a}}) be such that f(1) = f(−1) = 0. Then there exists a positive integer l such that (109) holds for u in the support of f, implying that (110) holds. This proves our assertion. The map (112) is clearly surjective, so that we obtain an isomorphism

S(F^{×}, V_{\left[{1 ∗ \atop 1}\right],ψ^{−2a}})_{[ \begin{matrix}1 & & & ∗ \\ & 1 \\ & & 1 \\ & & & 1 \end{matrix} ],ψ^{c}} ⟶^{∼} V_{\left[{1 ∗ \atop 1}\right],ψ^{−2a}} ⊕ V_{\left[{1 ∗ \atop 1}\right],ψ^{−2a}}.

We now have an isomorphism (I^{1}/I^{2})_{N,θ} ⟶^{∼} V_{\left[{1 ∗ \atop 1}\right],ψ^{−2a}} ⊕ V_{\left[{1 ∗ \atop 1}\right],ψ^{−2a}}. A

calculation shows that the transfer of the action of T to V_{\left[{1 ∗ \atop 1}\right],ψ^{−2a}} ⊕V_{\left[{1 ∗ \atop 1}\right],ψ^{−2a}} is given by

t(v_{1} ⊕ v_{2}) = |\frac{x − y}{x + y} |^{1/2}σ((x − y)(x + y))ω_{π}(x − y)v_{1}⊕

⊕|\frac{x − y}{x + y} |^{−1/2}σ((x − y)(x + y))ω_{π}(x + y)v_{2}

for t as in (111) and v_{1}, v_{2} ∈ V_{\left[{1 ∗ \atop 1}\right],ψ^{−2a}}. Finally, the result stated in ii) is

written with respect to S = [ \begin{matrix}& 1/2 \\ 1/2 \end{matrix} ]. To change to this choice note that the map

C : (I^{1}/I^{2})_{N,θ_{[ \begin{matrix}a & b/2 \\ b/2 & c \end{matrix} ]}} ⟶ (I^{1}/I^{2})_{N,θ_{[ \begin{matrix}& 1/2 \\ 1/2 \end{matrix} ]}}

defined by v ↦ [ \begin{matrix}g \\ & g′ \end{matrix} ] v, where g = [ \begin{matrix}−1 & 1 \\ 1 & 1 \end{matrix} ], is a well-defined isomorphism; recall that a = 1, b = 0, c = −1. Moreover, C(tv) = t′C(v) for t as in (111) and

t′ = [ \begin{matrix}x−y \\ & x+y \\ & & x−y \\ & & & x+y \end{matrix} ] ∈ T_{[ \begin{matrix}& 1/2 \\ 1/2 \end{matrix} ]}.

It follows that the group T_{[ \begin{matrix}& 1/2 \\ 1/2 \end{matrix} ]} acts on the isomorphic vector spaces

(I^{1}/I^{2})_{N,θ_{[ \begin{matrix}& 1/2 \\ 1/2 \end{matrix} ]}} ≅ V_{\left[{1 ∗ \atop 1}\right],ψ^{−2a}} ⊕ V_{\left[{1 ∗ \atop 1}\right],ψ^{−2a}} ≅ V_{\left[{1 ∗ \atop 1}\right],ψ} ⊕ V_{\left[{1 ∗ \atop 1}\right],ψ}

via the formula in ii).

Next, we consider the quotient I^{2}/I^{3} = c-Ind_{M}^{P}(σ_{2}) from iii) of Lemma 5.2.1. By Lemma 5.1.2, restriction of functions in the standard model of this representation to N gives an N-isomorphism c-Ind_{M}^{P}(σ_{2}) ≅ S(N, V). An application of i) and ii) of Lemma 5.1.1 shows that S(N, V)_{N,θ} ≅ V via the map defined by

f ⟼ ∫_{N} θ(n)^{−1}f(n) dn.

Transferring the action of T we find that t ∈ T acts by σ_{2}(t) on V. If t = [ \begin{matrix}g \\ & det(g)g′ \end{matrix} ] as in (16), then

σ_{2}(t) = σ(det(g))ω_{π}(det(g))π(g′).

This concludes the proof.

In case of a one-dimensional representation of M, it follows from this lemma that

(χ1_{GL(2)} ⋊ σ)_{N,θ} = ℂ_{(σχ)∘N_{L/F}}  (113)

as T -modules. In case L is a field and π is irreducible, it follows from Lemma 5.2.2 that

Hom_{T} ((π ⋊ σ)_{N,θ}, ℂ_{Λ}) = Hom_{T} (σπ, ℂ_{Λ}).  (114)

Hence, in view of (20), the space of (Λ, θ)-Bessel functionals on π ⋊ σ is isomorphic to the space of (Λ, θ)-Waldspurger functionals on σπ.

5.3 Klingen induced representations

Let π be an admissible representation of GL(2, F), let χ be a character of F^{×}, and let χ⋊π be as defined in Sect. 1.2; see (10). In this section we will calculate the twisted Jacquet modules (χ ⋊ π)_{N,θ} for any non-degenerate character θ of N as a module of T. In the split case our results make several corrections to Proposition 2.3 and Proposition 2.4 of [23].

5.3.1 Lemma. Let χ be a character of F^{×} and π an admissible representation of GL(2, F). Let I be the space of the Klingen induced representation χ ⋊ π. There is a filtration of P -modules

I^{2} = 0 ⊂ I^{1} ⊂ I^{0} = I.

with the quotients given as follows.

i) I^{0}/I^{1} = c-Ind_{B}^{P}σ_{0}, where

σ_{0}([ \begin{matrix}t & ∗ & ∗ & ∗ \\ & a & b & ∗ \\ & & d & ∗ \\ & & & adt^{−1} \end{matrix} ]) = χ(t) |t|^{2} |ad|^{−1} π(\left[{a b \atop d}\right])

for b in F and a, d, t in F^{×}.

ii) I^{1}/I^{2} = c-Ind_{[ \begin{matrix}∗ & ∗ & & ∗ \\ & ∗ \\ & & ∗ & ∗ \\ & & & ∗ \end{matrix} ]}^{P}σ_{1}, where

σ_{1}([ \begin{matrix}t & ∗ & & x \\ & a \\ & & d & ∗ \\ & & & adt^{−1} \end{matrix} ]) = χ(d) |a^{−1}d| π(\left[{t x \atop adt^{−1}}\right])

for x in F and a, d, t in F^{×}.

Proof. This follows by going through the procedure of Sections 6.2 and 6.3 of [5].

5.3.2 Lemma. Let χ be a character of F^{×}, and let (π, V) be an admissible representation of GL(2, F). We assume that π has a central character ω_{π}. Let I be the standard space of the Klingen induced representation χ ⋊ π. Let N be the unipotent radical of the Siegel parabolic subgroup, and let θ be the character of N defined in (15). We assume that the associated quadratic extension L is a field. Then, as T -modules,

I_{N,θ} ≅ ⨁_{Λ|_{F^{×}} =χω_{π}} d · Λ, where d = dim Hom\left[{1 ∗ \atop 1}\right](π, ψ).

In particular, I_{N,θ} = 0 if π is one-dimensional.

Proof. We will first prove that (I_{0}/I_{1})_{N,θ} = 0, where the notations are as in Lemma 5.3.1. We may assume that the element b appearing in the matrix S in (15) is zero. For f in the standard space of the induced representation I_{0}/I_{1} = c-Ind_{B}^{P}σ_{0}, let

f̃(u) = f([ \begin{matrix}1 \\ u & 1 \\ & & 1 \\ & & −u & 1 \end{matrix} ]), u ∈ F.

Let W be the space of all functions F → ℂ of the form f̃, where f runs through c-Ind_{B}^{P}σ_{0}. Since the map f ↦ f̃ is injective, we obtain a vector space isomorphism c-Ind_{B}^{P}σ_{0} ≅ W. The identity

[ \begin{matrix}1 \\ u & 1 \\ & & 1 \\ & & −u & 1 \end{matrix} ] = [ \begin{matrix}1 & u^{−1} \\ & 1 \\ & & 1 & −u^{−1} \\ & & & 1 \end{matrix} ] [ \begin{matrix}−u^{−1} \\ & u \\ & & u^{−1} \\ & & & −u \end{matrix} ] s_{1} [ \begin{matrix}1 & u^{−1} \\ & 1 \\ & & 1 & −u^{−1} \\ & & & 1 \end{matrix} ] ,

where s_{1} is as in (7), shows that f̃ satisfies

f̃(u) = χ(−u^{−1})|u|^{−2}π([ \begin{matrix}u \\ & u^{−1} \end{matrix} ])f(s_{1})  for |u| ≫ 0.  (115)

The space W consists of locally constant functions. Furthermore, W is invariant under translations, i.e., if f′ ∈ W, then the function u ↦ f′(u + x) is also in W, for any x in F.

We claim that W contains S(F, V). Since W is translation invariant, it is enough to prove that W contains the function

f_{N,v}(u) = \begin{cases}v & if u ∈ 𝔭^{N}, \\ 0 & if u ∉ 𝔭^{N}, \end{cases}

for any v in V and any positive integer N. Again by translation invariance, we may assume that N is large enough so that

σ_{0}(b)v = v  for b ∈ B ∩ Γ_{N},  (116)

where Γ_{N} is as in (100). Define f : P → V by

f(g) = \begin{cases}σ_{0}(b)v & if g = bk with b ∈ B, k ∈ Γ_{N}, \\ 0 & g ∉ BΓ_{N}. \end{cases}

Then, by (116), f is a well-defined element of c-Ind_{B}^{P}σ_{0}. It is easy to verify that f̃ = f_{N,v}. This proves our claim that W contains S(F, V). We define a linear map W → V by sending f̃ to the vector f(s_{1}) in (115). Then the kernel of this map is S(F, V). We claim that the map is surjective. To see this, let v be in V. Again choose N large enough so that (116) holds. Then the function f : P → V given by

f(g) = \begin{cases}σ_{0}(b)v & if g = bs_{1}k with b ∈ B, k ∈ Γ_{N}, \\ 0 & g ∉ Bs_{1}Γ_{N}. \end{cases}

is a well-defined element of c-Ind_{B}^{P}σ_{0} with f(s_{1}) = v. This proves our claim that the map W → V is surjective. We therefore get an exact sequence

0 ⟶ S(F, V) ⟶ W ⟶ V ⟶ 0.  (117)

The transfer of the action of N to W is given by

([ \begin{matrix}1 & & y & z \\ & 1 & x & y \\ & & 1 \\ & & & 1 \end{matrix} ] f̃)(u) = π(\left[{1 x+2uy+u^{2}z \atop 1}\right])f̃(u)

for all x, y, z, u in F. Evidently, the subspace S(F, V) is invariant under N. Moreover, the action of N on V is given by

[ \begin{matrix}1 & & y & z \\ & 1 & x & y \\ & & 1 \\ & & & 1 \end{matrix} ] v = π(\left[{1 z \atop 1}\right])v  (118)

for all x, y, z in F and v in V. To prove that (I_{0}/I_{1})_{N,θ} = 0, it suffices to show that S(F, V)_{N,θ} = 0 and V_{N,θ} = 0. Since the element a in the matrix S is non-zero, it follows from (118) that V_{N,θ} = 0.

To prove that S(F, V)_{N,θ} = 0, we define a map p from S(F, V) to

S(F, V_{\left[{1 ∗ \atop 1}\right],ψ^{a}}) = S(F, V/V (\left[{1 ∗ \atop 1}\right], ψ^{a}))

by sending f to f composed with the projection from V to V/V (\left[{1 ∗ \atop 1}\right], ψ^{a}). This map is surjective. It is easy to see that p induces an isomorphism

S(F, V)_{[ \begin{matrix}1 \\ & 1 & ∗ \\ & & 1 \\ & & & 1 \end{matrix} ],ψ^{a}} ≅ S(F, V_{\left[{1 ∗ \atop 1}\right],ψ^{a}}).

For the space on the right we have the action

([ \begin{matrix}1 & & y & z \\ & 1 & & y \\ & & 1 \\ & & & 1 \end{matrix} ] f)(u) = π(\left[{1 2uy+u^{2}z \atop 1}\right])f(u), u ∈ F.

By iii) of Lemma 5.1.1, the map f ↦ f(0) induces an isomorphism

S(F, V_{\left[{1 ∗ \atop 1}\right],ψ^{a}})_{[ \begin{matrix}1 & & ∗ \\ & 1 & & ∗ \\ & & 1 \\ & & & 1 \end{matrix} ]} ≅ V_{\left[{1 ∗ \atop 1}\right],ψ^{a}}.

For the space on the right we have the action [ \begin{matrix}1 & & & z \\ & 1 \\ & & 1 \\ & & & 1 \end{matrix} ] v = v. Taking a twisted

Jacquet module with respect to the character ψ^{c} gives zero, since c ≠ 0. This concludes our proof that (I_{0}/I_{1})_{N,θ} = 0. Next let σ_{1} be as in ii) of Lemma 5.3.1. Let

H_{1} = [ \begin{matrix}∗ & ∗ & & ∗ \\ & ∗ \\ & & ∗ & ∗ \\ & & & ∗ \end{matrix} ]

and H_{2} = T N. By Lemma 2.3.1, we have P = H_{1}H_{2}. To verify the hypotheses of Lemma 5.1.2, let K be a compact subset of P. Write P = MN and let p : P → N be the resulting projection map. Since p is continuous, the set p(K) is compact. There exists a compact subset K_{T} of T such that T = F^{×}K_{T}. Then M ⊂ H_{1}K_{T} by Lemma 2.3.1. Therefore K ⊂ H_{1}K_{2} with K_{2} = K_{T} p(K). By Lemma 5.1.2, restriction of functions gives a T N isomorphism

c-Ind_{[ \begin{matrix}∗ & ∗ & & ∗ \\ & ∗ \\ & & ∗ & ∗ \\ & & & ∗ \end{matrix} ]}^{P}σ_{1} ≅ c-Ind_{F^{×}Z^{J}}^{T N} (σ_{1}|_{F^{×}Z^{J}}).

Note that F^{×} acts via the character χω_{π} on this module. Since T is compact modulo F^{×}, the Jacquet module (c-Ind_{F^{×}Z^{J}}^{T N} (σ_{1}|_{F^{×}Z^{J}}))_{N,θ} is a direct sum over characters of T. Let Λ be a character of T. It is easy to verify that

Hom_{T} ((c-Ind_{F^{×}Z^{J}}^{T N} (σ_{1}|_{F^{×}Z^{J}}))_{N,θ}, Λ) = Hom_{T N}(c-Ind_{F^{×}Z^{J}}^{T N} (σ_{1}|_{F^{×}Z^{J}}), Λ ⊗ θ).

By Frobenius reciprocity, the space on the right is isomorphic to

Hom_{F^{×}Z^{J}} (σ_{1}|_{F^{×}Z^{J}} , (Λ ⊗ θ)|_{F^{×}Z^{J}} ).  (119)

This space is zero unless the restriction of Λ to F^{×} equals χω_{π}. Assume this is the case. Then (119) is equal to

Hom\left[{1 ∗ \atop 1}\right](π, ψ^{c}) ≅ Hom\left[{1 ∗ \atop 1}\right](π, ψ).

This concludes the proof.

5.3.3 Lemma. Let χ be a character of F^{×} and π an admissible representation of GL(2, F). Let I be the space of the Klingen induced representation χ ⋊ π. There is a filtration of Q-modules

I^{3} = 0 ⊂ I^{2} ⊂ I^{1} ⊂ I^{0} = I.

with the quotients given as follows.

i) I^{0}/I^{1} = σ_{0}, where

σ_{0}([ \begin{matrix}t & ∗ & ∗ & ∗ \\ & a & b & ∗ \\ & c & d & ∗ \\ & & & (ad−bc)t^{−1} \end{matrix} ]) = χ(t) |t^{2}(ad − bc)^{−1}| π([ \begin{matrix}a & b \\ c & d \end{matrix} ])

for [ \begin{matrix}a & b \\ c & d \end{matrix} ] in GL(2, F) and t in F^{×}.

ii) I^{1}/I^{2} = c-Ind_{[ \begin{matrix}∗ & & ∗ & ∗ \\ & ∗ & ∗ & ∗ \\ & & ∗ \\ & & & ∗ \end{matrix} ]}^{Q}σ_{1}, where

σ_{1}([ \begin{matrix}t & & ∗ & x \\ & a & b & ∗ \\ & & d \\ & & & adt^{−1} \end{matrix} ]) = χ(a) |ad^{−1}| π(\left[{t x \atop adt^{−1}}\right])

for b, x in F and a, d, t in F^{×}.

iii) I^{2}/I^{3} = I^{2} = c-Ind_{[ \begin{matrix}∗ \\ & ∗ & ∗ \\ & ∗ & ∗ \\ & & & ∗ \end{matrix} ]}^{Q}σ_{2}, where

σ_{2}([ \begin{matrix}t \\ & a & b \\ & c & d \\ & & & (ad−bc)t^{−1} \end{matrix} ]) = χ(t^{−1}(ad − bc)) |t^{−2}(ad − bc)| π([ \begin{matrix}a & b \\ c & d \end{matrix} ])

for [ \begin{matrix}a & b \\ c & d \end{matrix} ] in GL(2, F) and t in F^{×}.

Proof. This follows by going through the procedure of Sections 6.2 and 6.3 of [5].

5.3.4 Lemma. Let χ be a character of F^{×}, and let (π, V) be an admissible representation of GL(2, F). Let I be the standard space of the Klingen induced representation χ ⋊ π. Let N be the unipotent radical of the Siegel parabolic subgroup, and let θ be the character of N defined in (23) (i.e., we consider the split case). Then there is a filtration

0 ⊂ J_{3} ⊂ J_{2} ⊂ J_{1} = I_{N,θ},

with the quotients given as follows.

528 Brooks Roberts and Ralf Schmidt • J_{1}/J_{2} ≅ V

• J_{2}/J_{3} ≅ V\left[{1 \atop ∗ 1}\right].

• J_{3} ≅ S(F^{×}, V_{\left[{1 \atop ∗ 1}\right],ψ}). The action of the stabilizer of θ is given as follows,

diag(a, b, a, b)v = χ(a)π([ \begin{matrix}a \\ & b \end{matrix}])v  for v ∈ J_{1}/J_{2},
diag(a, b, a, b)v = χ(b)π([ \begin{matrix}a \\ & b \end{matrix}])v  for v ∈ J_{2}/J_{3},
(diag(a, b, a, b)f)(u) = χ(b)π([ \begin{matrix}a \\ & a \end{matrix}])f(a^{−1}bu)  for f ∈ J_{3}, u ∈ F^{×},

for all a and b in F^{×}. In particular, we have the following special cases.

i) Assume that π = σ1_{GL(2)}. Then the twisted Jacquet module I_{N,θ} = I/⟨θ(n)v − ρ(n)v : n ∈ N, v ∈ I⟩ is two-dimensional. More precisely, there is a filtration

0 ⊂ J_{2} ⊂ J_{1} = I_{N,θ},

where J_{2} and J_{1}/J_{2} are both one-dimensional, and the action of the stabilizer of θ is given as follows,

diag(a, b, a, b)v = χ(a)σ(ab)v  for v ∈ J_{1}/J_{2},
diag(a, b, a, b)v = χ(b)σ(ab)v  for v ∈ J_{2},

for all a and b in F^{×}. ii) Assume that π is infinite-dimensional and irreducible. Then there is a

filtration

0 ⊂ J_{3} ⊂ J_{2} ⊂ J_{1} = I_{N,θ},

with the quotients given as follows.

• J_{1}/J_{2} ≅ V

• J_{2}/J_{3} ≅ V\left[{1 \atop ∗ 1}\right]. Hence, J_{2}/J_{3} is 2-dimensional if π is a principal series representation, 1-dimensional if π is a twist of the Steinberg representation, and 0 if π is supercuspidal.

• J_{3} ≅ S(F^{×}). The action of the stabilizer of θ is given as follows,

diag(a, b, a, b)v = χ(a)π([ \begin{matrix}a \\ & b \end{matrix}])v  for v ∈ J_{1}/J_{2},
diag(a, b, a, b)v = χ(b)π([ \begin{matrix}a \\ & b \end{matrix}])v  for v ∈ J_{2}/J_{3},
(diag(a, b, a, b)f)(u) = χ(b)ω_{π}(a)f(a^{−1}bu)  for f ∈ J_{3}, u ∈ F^{×},

for all a and b in F^{×}.

Documenta Mathematica 21 (2016) 467–553

Proof. It will be easier to work with the conjugate subgroup N_{alt} and the character θ_{alt} of N_{alt} defined in (26). For the top quotient from i) of Lemma 5.3.3 we have

(I^{0}/I^{1})_{N_{alt},θ_{alt}} = 0,

since the subgroup [ \begin{matrix}1 & ∗ \\ & 1 \\ & & 1 & ∗ \\ & & & 1 \end{matrix} ] acts trivially on I^{0}/I^{1}, but θ_{alt} is not trivial on this

subgroup. We consider the quotient I^{1}/I^{2} = c-Ind_{H}^{Q}σ_{1}, where H = [ \begin{matrix}∗ & & ∗ & ∗ \\ & ∗ & ∗ & ∗ \\ & & ∗ \\ & & & ∗ \end{matrix} ],

and with σ_{1} as in ii) of Lemma 5.3.3. We first show that for each function f in the standard model of this representation, the function f^{∘} : F → V, given by

f^{∘}(w) = f([ \begin{matrix}1 & −w \\ & 1 \\ & & 1 & w \\ & & & 1 \end{matrix} ]),

has compact support. Let K be a compact subset of Q such that the support of f is contained in HK. If

[ \begin{matrix}1 & −w \\ & 1 \\ & & 1 & w \\ & & & 1 \end{matrix} ] = [ \begin{matrix}t & & ∗ & x \\ & a & b & ∗ \\ & & d \\ & & & adt^{−1} \end{matrix} ] [ \begin{matrix}k_{0} & x_{1} & x_{2} & x_{3} \\ & k_{1} & k_{2} & x_{4} \\ & k_{3} & k_{4} & x_{5} \\ & & & k_{5} \end{matrix} ] ,

with the rightmost matrix being in K, then calculations show that k_{3} = 0 and w = k_{4}^{−1}x_{5}. Since k_{4}^{−1} and x_{5} vary in bounded subsets, w is confined to a compact subset of F. This proves our assertion that f^{∘} has compact support. Next, for each function f in the standard model of c-Ind_{H}^{Q}σ_{1}, consider the function f̃: F^{2} → V given by

f̃(u, w) = f([ \begin{matrix}1 & −w \\ & 1 \\ & u & 1 & w \\ & & & 1 \end{matrix} ]).

Let W be the space of all such functions f̃. Since the map f ↦ f̃ is injective, we get a vector space isomorphism c-Ind_{H}^{Q}σ_{1} ≅ W. Evidently, in this new model, the action of N_{alt} is given by

([ \begin{matrix}1 & −y & & z \\ & 1 \\ & x & 1 & y \\ & & & 1 \end{matrix} ] f̃)(u, w) = f̃(u + x, w + y).  (120)

We claim that W contains S(F^{2}, V). Since W is translation invariant, it is enough to prove that W contains the function

f_{N,v}(u, w) = \begin{cases}v & if u, w ∈ 𝔭^{N}, \\ 0 & if u ∉ 𝔭^{N} or w ∉ 𝔭^{N}, \end{cases}

for any v in V and any positive integer N. Again by translation invariance, we may assume that N is large enough so that

σ_{1}(h)v = v  for h ∈ H ∩ Γ_{N},  (121)

where

Γ_{N} = [ \begin{matrix}1+𝔭^{N} & 𝔭^{N} & 𝔭^{N} & 𝔭^{N} \\ & 1+𝔭^{N} & 𝔭^{N} & 𝔭^{N} \\ & 𝔭^{N} & 1+𝔭^{N} & 𝔭^{N} \\ & & & 1+𝔭^{N} \end{matrix} ] ∩ Q.  (122)

Define f : Q → V by

f(g) = \begin{cases}σ_{1}(h)v & if g = hk with h ∈ H, k ∈ Γ_{N}, \\ 0 & g ∉ HΓ_{N}. \end{cases}

Then, by (121), f is a well-defined element of c-Ind_{H}^{Q}σ_{1}. It is easy to verify that f̃ = f_{N,v}. This proves our claim that W contains S(F^{2}, V). Now consider the map

W ⟶ S(F, V),  f̃ ⟼ (w ↦ f([ \begin{matrix}1 & −w \\ & 1 \\ & & 1 & w \\ & & & 1 \end{matrix} ] s_{2})),  (123)

where s_{2} is defined in (7). This map is well-defined, since the function on the right is (s_{2}f)^{∘}, which we showed above has compact support. Similar considerations as above show that the map (123) is surjective. We claim that the kernel of (123) is S(F^{2}, V). First suppose that f̃ lies in the kernel; we have to show that f̃ has compact support. Choose N large enough so that f is right invariant under Γ_{N}. Then, for u not in 𝔭^{−N} and w in F,

f̃(u, w) = f([ \begin{matrix}1 & −w \\ & 1 \\ & u & 1 & w \\ & & & 1 \end{matrix} ])

= f([ \begin{matrix}1 & −w \\ & 1 \\ & & 1 & w \\ & & & 1 \end{matrix} ] [ \begin{matrix}1 \\ & 1 & u^{−1} \\ & & 1 \\ & & & 1 \end{matrix} ] [ \begin{matrix}1 \\ & −u^{−1} \\ & & −u \\ & & & 1 \end{matrix} ] s_{2} [ \begin{matrix}1 \\ & 1 & u^{−1} \\ & & 1 \\ & & & 1 \end{matrix} ])

= f([ \begin{matrix}1 & & −wu^{−1} & w^{2}u^{−1} \\ & 1 & u^{−1} & −wu^{−1} \\ & & 1 \\ & & & 1 \end{matrix} ] [ \begin{matrix}1 & −w \\ & 1 \\ & & 1 & w \\ & & & 1 \end{matrix} ] [ \begin{matrix}1 \\ & −u^{−1} \\ & & −u \\ & & & 1 \end{matrix} ] s_{2})

= π(\left[{1 w^{2}u^{−1} \atop 1}\right])f([ \begin{matrix}1 & −w \\ & 1 \\ & & 1 & w \\ & & & 1 \end{matrix} ] [ \begin{matrix}1 \\ & −u^{−1} \\ & & −u \\ & & & 1 \end{matrix} ] s_{2})

= χ(−u^{−1})|u|^{−2}π(\left[{1 w^{2}u^{−1} \atop 1}\right])f([ \begin{matrix}1 & wu^{−1} \\ & 1 \\ & & 1 & −wu^{−1} \\ & & & 1 \end{matrix} ] s_{2}).

This last expression is zero by assumption. For fixed u in 𝔭^{−N}, the function f̃(u, ·) has compact support; this follows because each f^{∘} has compact support. Combining these facts shows that f̃ has compact support. Conversely, assume f̃is in S(F^{2}, V). Then we can find a large enough N such that if u has valuation −N, the function f̃(u, ·) is zero. Looking at the above calculation, we see that, for fixed such u,

f([ \begin{matrix}1 & wu^{−1} \\ & 1 \\ & & 1 & −wu^{−1} \\ & & & 1 \end{matrix} ] s_{2}) = 0

for all w in F. This shows that f̃ is in the kernel of the map (123), completing the proof of our claim about this kernel. We now have an exact sequence

0 ⟶ S(F^{2}, V) ⟶ W ⟶ S(F, V) ⟶ 0.  (124)

Note that the space S(F^{2}, V) is invariant under the action (120) of N_{alt}. A calculation shows that the action of N_{alt} on S(F, V) is given by

([ \begin{matrix}1 & −y & & z \\ & 1 \\ & x & 1 & y \\ & & & 1 \end{matrix} ] f)(w) = π(\left[{1 z−2wy−w^{2}x \atop 1}\right])f(w)  (125)

for x, y, z, w in F and f in S(F, V). We claim that S(F, V)_{N_{alt},θ_{alt}} = 0. To prove this, we calculate this Jacquet module in stages. We define a map p from S(F, V) to

S(F, V\left[{1 ∗ \atop 1}\right]) = S(F, V/V (\left[{1 ∗ \atop 1}\right]))

by sending f to f composed with the natural projection from V to V/V (\left[{1 ∗ \atop 1}\right]).

This map is surjective and has kernel S(F, V)([ \begin{matrix}1 & & & ∗ \\ & 1 \\ & & 1 \\ & & & 1 \end{matrix} ]). Hence, we obtain an

isomorphism

S(F, V)_{[ \begin{matrix}1 & & & ∗ \\ & 1 \\ & & 1 \\ & & & 1 \end{matrix} ]} ≅ S(F, V\left[{1 ∗ \atop 1}\right]).

The action of the group [ \begin{matrix}1 & ∗ \\ & 1 \\ & ∗ & 1 & ∗ \\ & & & 1 \end{matrix} ] on these spaces is trivial. Since θ_{alt} is not

trivial on this group, this proves our claim that S(F, V)_{N_{alt},θ_{alt}} = 0. By (124), we now have W_{N_{alt},θ_{alt}} ≅ S(F^{2}, V)_{N_{alt},θ_{alt}}. The action of N_{alt} on S(F^{2}, V) is given by (120). Since S(F^{2}, V) = S(F) ⊗ S(F) ⊗ V, Lemma 5.1.1 implies that the map

f ⟼ ∫_{F} ∫_{F} f(u, w)ψ(−w) du dw

induces an isomorphism S(F^{2}, V)_{N_{alt},θ_{alt}} ≅ V. Moreover, a calculation shows that diag(a, a, b, b) acts on S(F^{2}, V)_{N_{alt},θ_{alt}} ≅ V by χ(a)π([ \begin{matrix}a \\ & b \end{matrix}]). Finally, we consider the bottom quotient I^{2}/I^{3} = c-Ind_{[ \begin{matrix}∗ \\ & ∗ & ∗ \\ & ∗ & ∗ \\ & & & ∗ \end{matrix} ]}^{Q}σ_{2} with σ_{2} as

in iii) of Lemma 5.3.3. If we associate with a function f in the standard model of this induced representation the function

f̃(u, v, w) = f([ \begin{matrix}1 & −v & u & w \\ & 1 & & u \\ & & 1 & v \\ & & & 1 \end{matrix} ]),

then, by Lemma 5.1.2, we obtain an isomorphism I^{2}/I^{3} ≅ S(F^{3}, V). A calculation shows that the action of N_{alt} on S(F^{3}, V) is given by

([ \begin{matrix}1 & −y & & z \\ & 1 \\ & x & 1 & y \\ & & & 1 \end{matrix} ] f)(u, v, w) = π(\left[{1 \atop x 1}\right])f(u, v + y − ux, w + z + uy)  (126)

for x, y, z, u, v, w in F and f in S(F^{3}, V). This time we take Jacquet modules step by step, starting with the z-variable. Lemma 5.1.1 shows that the map

f ⟼ ((u, v) ↦ ∫_{F} f(u, v, w) dw)

induces an isomorphism S(F^{3}, V)_{[ \begin{matrix}1 & & & ∗ \\ & 1 \\ & & 1 \\ & & & 1 \end{matrix} ]} ≅ S(F^{2}, V). On S(F^{2}, V) we have

the action

([ \begin{matrix}1 & −y \\ & 1 \\ & x & 1 & y \\ & & & 1 \end{matrix} ] f)(u, v) = π(\left[{1 \atop x 1}\right])f(u, v + y − ux)

for x, y, u, v in F and f in S(F^{2}, V). Part ii) of Lemma 5.1.1 shows that the map

f ⟼ (u ↦ ∫_{F} f(u, v)ψ(−v) dv)

induces an isomorphism S(F^{2}, V)_{[ \begin{matrix}1 & ∗ \\ & 1 \\ & & 1 & ∗ \\ & & & 1 \end{matrix} ],ψ} ≅ S(F, V). A calculation shows

that on S(F, V) we have the actions

([ \begin{matrix}1 \\ & 1 \\ & x & 1 \\ & & & 1 \end{matrix} ] f)(u) = ψ(−ux)π(\left[{1 \atop x 1}\right])f(u)  (127)

for x, u in F, and

([ \begin{matrix}a \\ & a \\ & & b \\ & & & b \end{matrix} ] f)(u) = χ(b)π([ \begin{matrix}a \\ & b \end{matrix}])f(a^{−1}bu)  (128)

for u in F and a, b in F^{×}. The subspace S(F^{×}, V) consisting of functions that vanish at zero is invariant under these actions. We consider the exact sequence

0 ⟶ S(F^{×}, V) ⟶ S(F, V) ⟶ S(F, V)/S(F^{×}, V) ⟶ 0.

The quotient S(F, V)/S(F^{×}, V) is isomorphic to V via the map f ↦ f(0). The actions of the above subgroups on V are given by

[ \begin{matrix}1 \\ & 1 \\ & x & 1 \\ & & & 1 \end{matrix} ] v = π(\left[{1 \atop x 1}\right])v  (129)

and

[ \begin{matrix}a \\ & a \\ & & b \\ & & & b \end{matrix} ] v = χ(b)π([ \begin{matrix}a \\ & b \end{matrix}])v.  (130)

Taking Jacquet modules on the above sequence gives

0 ⟶ S(F^{×}, V)_{[ \begin{matrix}1 \\ & 1 \\ & ∗ & 1 \\ & & & 1 \end{matrix} ]} ⟶ S(F, V)_{[ \begin{matrix}1 \\ & 1 \\ & ∗ & 1 \\ & & & 1 \end{matrix} ]}

⟶ (S(F, V)/S(F^{×}, V))_{[ \begin{matrix}1 \\ & 1 \\ & ∗ & 1 \\ & & & 1 \end{matrix} ]} ⟶ 0.

In view of (129), the Jacquet module on the right is isomorphic to V\left[{1 \atop ∗ 1}\right]. The action of the diagonal subgroup on V\left[{1 \atop ∗ 1}\right] is given by the same formula as in (130). We consider the map from S(F^{×}, V) to itself given by f ↦ (u ↦ π([ \begin{matrix}1 \\ & u \end{matrix}])f(u)). This map is an isomorphism of vector spaces. The actions (127) and (128) turn into

([ \begin{matrix}1 \\ & 1 \\ & x & 1 \\ & & & 1 \end{matrix} ] f)(u) = ψ(−ux)π(\left[{1 \atop ux 1}\right])f(u)  (131)

and

([ \begin{matrix}a \\ & a \\ & & b \\ & & & b \end{matrix} ] f)(u) = χ(b)π([ \begin{matrix}a \\ & a \end{matrix}])f(a^{−1}bu).  (132)

We define a map p from S(F^{×}, V) to

S(F^{×}, V_{\left[{1 \atop ∗ 1}\right],ψ}) = S(F^{×}, V/V (\left[{1 \atop ∗ 1}\right], ψ))

by sending f to f composed with the projection from V to V/V (\left[{1 \atop ∗ 1}\right], ψ). This map is surjective. The kernel of p consists of all f in S(F^{×}, V) for which there exists a positive integer l such that

∫_{𝔭^{−l}} ψ(−x)π(\left[{1 \atop x 1}\right])f(u) dx = 0  for all u ∈ F^{×}.  (133)

Let W be the space of f in S(F^{×}, V) for which there exists a positive integer k such that

∫_{𝔭^{−k}} [ \begin{matrix}1 \\ & 1 \\ & x & 1 \\ & & & 1 \end{matrix} ] f dx = 0,  (134)

so that S(F^{×}, V)/W = S(F^{×}, V)_{[ \begin{matrix}1 \\ & 1 \\ & ∗ & 1 \\ & & & 1 \end{matrix} ]}. Let f be in W. The condition

(134) means that

∫_{𝔭^{−k}} ψ(−ux)π(\left[{1 \atop ux 1}\right])f(u) dx = 0  for all u ∈ F^{×}.  (135)

Since f has compact support in F^{×}, the conditions (133) and (135) are equivalent. It follows that

S(F^{×}, V)_{[ \begin{matrix}1 \\ & 1 \\ & ∗ & 1 \\ & & & 1 \end{matrix} ]} ≅ S(F^{×}, V_{\left[{1 \atop ∗ 1}\right],ψ}).

The diagonal subgroup acts on S(F^{×}, V_{\left[{1 \atop ∗ 1}\right],ψ}) by the same formula as in (132).

6 The main results

Having assembled all the required tools, we are now ready to prove the three main results of this paper mentioned in the introduction.

6.1 Existence of Bessel functionals

In this section we prove that every irreducible, admissible representation (π, V) of GSp(4, F) which is not a twist of the trivial representation admits a Bessel functional. The proof uses the P_{3}-module V_{Z^{J}} and the G^{J}-module V_{Z^{J} ,ψ}. The first module is closely related to the theory of zeta integrals. The second module V_{Z^{J} ,ψ} is the quotient of V by the vector subspace generated by the elements of

the form π([ \begin{matrix}1 & & & z \\ & 1 \\ & & 1 \\ & & & 1 \end{matrix} ])v − ψ(z)v for v ∈ V and z ∈ F. Evidently, V_{Z^{J} ,ψ} is a G^{J}

module. This module is closely related to the theory of representations of the metaplectic group S̃L(2, F).

6.1.1 Lemma. Let (π, V) be a smooth representation of N. Then there exists a character θ of N such that V_{N,θ} ≠ 0.

Proof. This follows immediately from Lemma 1.6 of [29].

Let S̃L(2, F) be the metaplectic group, defined as in Sect. 1 of [29]. Let m be in F^{×}. We will use the Weil representation πm_{W} of S̃L(2, F) on S(F) associated to the quadratic form q(x) = x^{2} and ψ^{m}. This is as defined on pp. 3-4 of [39] and p. 223 of [41]. The only explicit property of πm_{W} we will use is

(πm_{W}(\left[{1 b \atop 1}\right], 1)f)(x) = ψ(mbx^{2})f(x),  (136)

for b in F and f in S(F). We define an action of N_{Q}, introduced in (6), on the Schwartz space S(F) by

πm_{S}([ \begin{matrix}1 & λ & μ & κ \\ & 1 & & μ \\ & & 1 & −λ \\ & & & 1 \end{matrix} ] f)(x) = ψ^{m}(κ + (2x + λ)μ)f(x + λ)  (137)

for f in S(F). This representation of N_{Q} is called the Schrödinger representation. Given a smooth, genuine representation (τ, W) of S̃L(2, F), we define a representation τ^{J} of G^{J} on the space W ⊗ S(F) by the formulas

τ^{J}([ \begin{matrix}1 \\ & a & b \\ & c & d \\ & & & 1 \end{matrix} ])(v ⊗ f) = τ([ \begin{matrix}a & b \\ c & d \end{matrix} ], 1)v ⊗ πm_{W}([ \begin{matrix}a & b \\ c & d \end{matrix} ], 1)f,  (138)

τ^{J}([ \begin{matrix}1 & λ & μ & κ \\ & 1 & & μ \\ & & 1 & −λ \\ & & & 1 \end{matrix} ])(v ⊗ f) = v ⊗ πm_{S}([ \begin{matrix}1 & λ & μ & κ \\ & 1 & & μ \\ & & 1 & −λ \\ & & & 1 \end{matrix} ])f.  (139)

Computations show that τ^{J} is a smooth representation of G^{J}. Moreover, the map that sends τ to τ^{J} is a bijection between the set of equivalence classes of smooth, genuine representations of S̃L(2, F), and smooth representations of G^{J} with central character ψ^{m}. The proof of this fact is based on the Stone-von Neumann Theorem; see Theorem 2.6.2 of [2]. Under this bijection, irreducible τ correspond to irreducible τ^{J}.

6.1.2 Lemma. Let m be in F^{×}. Let (τ^{J}, W^{J}) be a non-zero, irreducible, smooth representation of G^{J} with central character ψ^{m}. Then dim W_{N,θ_{a,0,m}}^{J} ≤ 1 for all a in F^{×} and dim W_{N,θ_{a,0,m}}^{J} = 1 for some a in F^{×}. This dimension depends only on the class of a in F^{×}/F^{×2}.

Proof. By the above discussion, there exists an irreducible, genuine, admissible representation τ of S̃L(2, F) such that τ^{J} ≅ τ ⊗ πm_{SW}. Using (136), (137) and iii) of Lemma 5.1.1, an easy calculation shows that

W_{[ \begin{matrix}1 & & ∗ & ∗ \\ & 1 & ∗ & ∗ \\ & & 1 \\ & & & 1 \end{matrix} ],θ_{a,0,m}}^{J} ≅ W_{\left[{1 ∗ \atop 1}\right],ψ^{a}}.

By Lemme 2 on p. 226 of [41], the space on the right is at most one-dimensional, and is one-dimensional for some a in F^{×}. Moreover, the dimension depends only on the class of a in F^{×}/F^{×2}.

6.1.3 Proposition. Let (π, V) be an irreducible, admissible representation of GSp(4, F). Then the following statements are equivalent.

i) π is not a twist of the trivial representation.

ii) There exists a non-trivial character θ of N such that V_{N,θ} ≠ 0.

iii) There exists a non-degenerate character θ of N such that V_{N,θ} ≠ 0.

Proof. i) ⇒ ii) Assume that V_{N,θ} = 0 for all non-trivial θ. By Lemma 6.1.1, it follows that V_{N,1} ≠ 0. In particular, the P_{3}-module V_{Z^{J}} is non-zero. By using Theorem 3.2.1 and inspecting tables A.5 and A.6 in [28], one can see that V_{Z^{J}} contains an irreducible subquotient τ of the form τ_{GL(0)}^{P_{3}}(1), or τ_{GL(1)}^{P_{3}}(χ) for a character χ of F^{×}, or τ_{GL(2)}^{P_{3}}(ρ) for an irreducible, admissible, infinite-dimensional representation ρ of GL(2, F); it is here that we use the hypothesis that π is not one-dimensional. For a, b in F we define a character of the subgroup [ \begin{matrix}1 & ∗ & ∗ \\ & 1 \\ & & 1 \end{matrix} ] of P_{3} by

θ_{a,b}([ \begin{matrix}1 & x & y \\ & 1 \\ & & 1 \end{matrix} ]) = ψ(ax + by).  (140)

By Lemma 2.5.4 or Lemma 2.5.5 of [28], or the infinite-dimensionality of ρ if τ = τ_{GL(2)}^{P_{3}}(ρ),

τ_{[ \begin{matrix}1 & ∗ & ∗ \\ & 1 \\ & & 1 \end{matrix} ],θ_{a,b}} ≠ 0

for some (a, b) ≠ (0, 0). This implies that V_{N,θ_{a,b,0}} ≠ 0, contradicting our assumption. ii) ⇒ iii) The hypothesis implies that V_{Z^{J} ,ψ^{m}} is non-zero for some m in F^{×}. We observe that V_{Z^{J} ,ψ^{m}} is a smooth G^{J} representation. By Lemma 2.6 of [3], there exists an irreducible subquotient (τ^{J}, W^{J}) of this G^{J} module. By Lemma 6.1.2, we have dim W_{N,θ_{a,0,m}}^{J} = 1 for some a in F^{×}. This implies that V_{N,θ_{a,0,m}} ≠ 0. iii) ⇒ i) is obvious.

6.1.4 Theorem. Let (π, V) be an irreducible, admissible representation of GSp(4, F). Assume that π is not one-dimensional. Then π admits a (Λ, θ)- Bessel functional for some non-degenerate character θ of N and some character Λ of T. If π is non-generic and supercuspidal, then every Bessel functional for π is non-split.

Proof. By Proposition 6.1.3, there exists a non-degenerate θ such that V_{N,θ} ≠ 0. Assume that θ is non-split. Then, since the center F^{×} of GSp(4, F) acts by a character on V_{N,θ} and T/F^{×} is compact, V_{N,θ} decomposes as a direct sum over characters of T. It follows that a (Λ, θ)-Bessel functional exists for some character Λ of T. Now assume that θ is split. We may assume that S is the matrix in (22). Let V_{0}, V_{1}, V_{2} be the modules appearing in the P_{3}-filtration, as in Theorem 3.2.1. Since V_{N,θ} ≠ 0, we must have

(V_{0}/V_{1})_{[ \begin{matrix}1 & ∗ & ∗ \\ & 1 \\ & & 1 \end{matrix} ],θ_{0,1}} ≠ 0, (V_{1}/V_{2})_{[ \begin{matrix}1 & ∗ & ∗ \\ & 1 \\ & & 1 \end{matrix} ],θ_{0,1}} ≠ 0, or (V_{2})_{[ \begin{matrix}1 & ∗ & ∗ \\ & 1 \\ & & 1 \end{matrix} ],θ_{0,1}} ≠ 0,

where we use the notation (140). It is immediate from (38) that the first space is zero. If the second space is non-zero, then π admits a split Bessel functional by iii) of Proposition 3.5.1. If the third space is non-zero, then π is generic by Theorem 3.2.1, and hence, by Proposition 3.4.2, admits a split Bessel functional. For the last statement, assume that π is non-generic and supercuspidal. Then V_{Z^{J}} = 0 by Theorem 3.2.1. Hence, V_{N,θ} = 0 for any split θ. It follows that all Bessel functionals for π are non-split.

6.2 The table of Bessel functionals

In this section, given a non-supercuspidal representation π, or a π that is in an L-packet with a non-supercuspidal representation, we determine the set of (Λ, θ) for which π admits a (Λ, θ)-Bessel functional.

6.2.1 Lemma. Let θ be as in (15), and let T be the corresponding torus. Assume that the associated quadratic extension L is a field. Let V_{1}, V_{2}, V_{3} and W be smooth representations of T. Assume that these four representations all have the same central character. Assume further that there is an exact sequence of T -modules

0 ⟶ V_{1} ⟶ V_{2} ⟶ V_{3} ⟶ 0.

Then the sequence of T -modules

0 ⟶ Hom_{T} (V_{3}, W) ⟶ Hom_{T} (V_{2}, W) ⟶ Hom_{T} (V_{1}, W) ⟶ 0

is exact.

Proof. It is easy to see that the sequence

0 ⟶ Hom_{T} (V_{3}, W) ⟶ Hom_{T} (V_{2}, W) ⟶ Hom_{T} (V_{1}, W)

is exact. We will prove the surjectivity of the last map. Let f be in Hom_{T} (V_{1}, W). We extend f to a linear map f_{1} from V_{2} to W. We define another linear map f_{2} from V_{2} to W by

f_{2}(v) = ∫_{T/F^{×}} t^{−1} · f_{1}(t · v) dt.

This is well-defined by the condition on the central characters, the compactness of T/F^{×}, and the smoothness hypothesis. Evidently, f_{2} is in Hom_{T} (V_{2}, W) and maps to a multiple of f.

6.2.2 Theorem. The following table shows the Bessel functionals admitted by the irreducible, admissible, non-supercuspidal representations of GSp(4, F). The column “L ↔ ξ” indicates that the field L is the quadratic extension of F corresponding to the non-trivial, quadratic character ξ of F^{×}; this is only relevant for representations in groups V and IX. The pairs of characters (χ_{1}, χ_{2}) in the “L = F × F ” column for types IIIb and IVc refer to the characters of T = {diag(a, b, a, b) : a, b ∈ F^{×}} given by diag(a, b, a, b) ↦ χ_{1}(a)χ_{2}(b). In representations of group IX, the symbol μ denotes a non-Galois-invariant character of L^{×}, where L is the quadratic extension corresponding to ξ. The Galois conjugate of μ is denoted by μ′. The irreducible, admissible, supercuspidal representation of GL(2, F) corresponding to μ is denoted by π(μ). Finally, the symbol N in the table stands for the norm map N_{L/F}. In the split case, the character σ ∘ N is the same as (σ, σ). In the table, the phrase “all Λ” means all characters Λ of T whose restriction to F^{×} is the central character of the representation of GSp(4, F).

538 Brooks Roberts and Ralf Schmidt

representation  (Λ, θ)-Bessel functional exists exactly for …
L = F × F  L/F a field extension
L ↔ ξ  L ↮ ξ
I  χ_{1} × χ_{2} ⋊ σ (irred.)  all Λ  all Λ
II a  χSt_{GL(2)} ⋊ σ  all Λ  Λ ≠ (χσ) ∘ N
b  χ1_{GL(2)} ⋊ σ  Λ = (χσ) ∘ N  Λ = (χσ) ∘ N
III a  χ ⋊ σSt_{GSp(2)}  all Λ  all Λ
b  χ ⋊ σ1_{GSp(2)} Λ ∈ {(χσ, σ), (σ, χσ)}  —
IV a  σSt_{GSp(4)}  all Λ  Λ ≠ σ ∘ N
b L(ν^{2}, ν^{−1}σSt_{GSp(2)})  Λ = σ ∘ N  Λ = σ ∘ N
c L(ν^{3/2}St_{GL(2)}, ν^{−3/2}σ) Λ = (ν^{±1}σ, ν^{∓1}σ)  —
d  σ1_{GSp(4)}  —  —
V a  δ([ξ, νξ], ν^{−1/2}σ)  all Λ  Λ ≠ σ ∘ N σ ∘ N ≠ Λ ≠ (ξσ) ∘ N
b L(ν^{1/2}ξSt_{GL(2)}, ν^{−1/2}σ) Λ = σ ∘ N  —  Λ = σ ∘ N
c L(ν^{1/2}ξSt_{GL(2)}, ξν^{−1/2}σ) Λ = (ξσ) ∘ N  —  Λ = (ξσ) ∘ N
d L(νξ, ξ ⋊ ν^{−1/2}σ)  —  Λ = σ ∘ N  —
VI a  τ(S, ν^{−1/2}σ)  all Λ  Λ ≠ σ ∘ N
b  τ(T, ν^{−1/2}σ)  —  Λ = σ ∘ N
c L(ν^{1/2}St_{GL(2)}, ν^{−1/2}σ) Λ = σ ∘ N  —
d L(ν, 1_{F^{×}} ⋊ ν^{−1/2}σ)  Λ = σ ∘ N  —
VII  χ ⋊ π  all Λ  all Λ
VIII a  τ(S, π)  all Λ  Hom_{T} (π, ℂ_{Λ}) ≠ 0
b  τ(T, π)  —  Hom_{T} (π, ℂ_{Λ}) = 0
IX a  δ(νξ, ν^{−1/2}π(μ))  all Λ  μ ≠ Λ ≠ μ′  all Λ
b L(νξ, ν^{−1/2}π(μ))  —  Λ = μ or Λ = μ′  —
X  π ⋊ σ  all Λ  Hom_{T} (σπ, ℂ_{Λ}) ≠ 0
XI a  δ(ν^{1/2}π, ν^{−1/2}σ)  all Λ  Λ ≠ σ ∘ N and Hom_{T} (σπ, ℂ_{Λ}) ≠ 0
b L(ν^{1/2}π, ν^{−1/2}σ)  Λ = σ ∘ N  Λ = σ ∘ N and Hom_{T} (π, ℂ_{1}) ≠ 0
Va^{∗}  δ^{∗}([ξ, νξ], ν^{−1/2}σ)  —  Λ = σ ∘ N  —
XIa^{∗}  δ^{∗}(ν^{1/2}π, ν^{−1/2}σ)  —  Λ = σ ∘ N and Hom_{T} (π^{JL}, ℂ_{1}) ≠ 0

Proof. We will go through all representations in the table and explain how

the statements follow from our preparatory sections. \underline{I}: This follows from Proposition 3.4.2 and Lemma 5.3.2. \underline{IIa}: In the split case this follows from Proposition 3.4.2. In the non-split case it follows from Lemma 5.2.2 together with (36). \underline{IIb}: This follows from Lemma 5.2.2; see (113). \underline{IIIa}: This follows from Proposition 3.4.2 and Lemma 5.3.2. \underline{IIIb}: It follows from Lemma 5.3.2 that IIIb type representations have no nonsplit Bessel functionals. The split case follows from either Proposition 3.5.1 or i) of Lemma 5.3.4. Note that the characters (χσ, σ) and (σ, χσ) are Galois conjugates of each other. \underline{IVd}: It is easy to see that the twisted Jacquet modules of the trivial representation are zero. \underline{IVb}: By (2.9) of [28] there is a short exact sequence

0 ⟶ IVb ⟶ ν^{3/2}1_{GL(2)} ⋊ ν^{−3/2}σ ⟶ σ1_{GSp(4)} ⟶ 0.

Taking twisted Jacquet modules and observing (113), we get

(IVb)_{N,θ} ≅ (ν^{3/2}1_{GL(2)} ⋊ ν^{−3/2}σ)_{N,θ} = ℂ_{σ∘N_{L/F}}

as T -modules. \underline{IVc}: By (2.9) of [28] there is a short exact sequence

0 ⟶ IVc ⟶ ν^{2} ⋊ ν^{−1}σ1_{GSp(2)} ⟶ σ1_{GSp(4)} ⟶ 0.

Taking twisted Jacquet modules gives

(IVc)_{N,θ} ≅ (ν^{2} ⋊ ν^{−1}σ1_{GSp(2)})_{N,θ}.

Hence IVc admits the same Bessel functionals as the full induced representation ν^{2} ⋊ ν^{−1}σ1_{GSp(2)}. By Lemma 5.3.2, any such Bessel functional is necessarily split. Assume that θ is as in (23). Then, using Lemma 5.3.4, it follows that IVc admits the (Λ, θ)-Bessel functional for

Λ([ \begin{matrix}a \\ & b \\ & & a \\ & & & b \end{matrix} ]) = ν(ab^{−1})σ(ab),  (141)

which we write as (νσ, ν^{−1}σ). By (29), IVc also admits a (Λ, θ)-Bessel functional for Λ = (ν^{−1}σ, νσ). Again by Lemma 5.3.4, IVc does not admit a (Λ, θ)-Bessel functional for any other Λ. \underline{IVa}: In the split case this follows from Proposition 3.4.2. Assume θ is non-split. By (2.9) of [28], there is an exact sequence

0 ⟶ σSt_{GSp(4)} ⟶ ν^{2} ⋊ ν^{−1}σSt_{GSp(2)} ⟶ IVb ⟶ 0.

Taking Jacquet modules, we get

0 ⟶ (σSt_{GSp(4)})_{N,θ} ⟶ (ν^{2} ⋊ ν^{−1}σSt_{GSp(2)})_{N,θ} ⟶ (IVb)_{N,θ} ⟶ 0.

Keeping in mind Lemma 6.2.1, the result now follows from Lemma 5.3.2 and the result for IVb. \underline{Vd}: This was proved in Corollary 4.7.2. \underline{Vb and Vc}: Let ξ be a non-trivial quadratic character of F^{×}. By (2.10) of [28], there are exact sequences

0 ⟶ Vb ⟶ ν^{1/2}ξ1_{GL(2)} ⋊ ξν^{−1/2}σ ⟶ Vd ⟶ 0

and

0 ⟶ Vc ⟶ ν^{1/2}ξ1_{GL(2)} ⋊ ν^{−1/2}σ ⟶ Vd ⟶ 0.

Taking Jacquet modules and observing (113), we get

0 ⟶ (Vb)_{N,θ} ⟶ ℂ_{σ∘N_{L/F}} ⟶ (Vd)_{N,θ} ⟶ 0  (142)

and

0 ⟶ (Vc)_{N,θ} ⟶ ℂ_{(ξσ)∘N_{L/F}} ⟶ (Vd)_{N,θ} ⟶ 0.  (143)

Hence the results for Vb and Vc follow from the result for Vd. \underline{Va}: In the split case this follows from Proposition 3.4.2. Assume θ is non-split. Assume first that ξ corresponds to the quadratic extension L/F. As we just saw, (Vb)_{N,θ} = 0 in this case. By (2.10) of [28], there is an exact sequence

0 ⟶ Va ⟶ ν^{1/2}ξSt_{GL(2)} ⋊ ν^{−1/2}σ ⟶ Vb ⟶ 0.

Taking Jacquet modules, it follows that

(Va)_{N,θ} = (ν^{1/2}ξSt_{GL(2)} ⋊ ν^{−1/2}σ)_{N,θ}.

By Lemma 5.2.2, the space of (Λ, θ)-Bessel functionals on the representation Va is isomorphic to Hom_{T} (σξSt_{GL(2)}, ℂ_{Λ}). Using (36), it follows that Va admits a (Λ, θ)-Bessel functional if and only if Λ ≠ (σξ) ∘ N_{L/F} = σ ∘ N_{L/F}. Now assume that ξ does not correspond to the quadratic extension L/F. Then, by what we already proved for Vb, we have an exact sequence

0 ⟶ (Va)_{N,θ} ⟶ (ν^{1/2}ξSt_{GL(2)} ⋊ ν^{−1/2}σ)_{N,θ} ⟶ ℂ_{σ∘N_{L/F}} ⟶ 0. (144)

Using Lemma 6.2.1, it follows that the possible characters Λ for Va are those of (ν^{1/2}ξSt_{GL(2)} ⋊ ν^{−1/2}σ)_{N,θ} with the exception of σ ∘ N_{L/F}. By Lemma 5.2.2 and (36), these are all characters other than σ ∘ N_{L/F} and (ξσ) ∘ N_{L/F}. \underline{VIc and VId}: By (2.11) of [28], there is an exact sequence

0 ⟶ VIc ⟶ 1_{F^{×}} ⋊ σ1_{GSp(2)} ⟶ VId ⟶ 0.

It follows from Lemma 5.3.2 that VIc and VId have no non-split Bessel functionals. The split case follows from Proposition 3.5.1. \underline{VIa}: In the split case this follows from Proposition 3.4.2. Assume that θ is non-split. By (2.11) of [28], there is an exact sequence

0 ⟶ VIa ⟶ ν^{1/2}St_{GL(2)} ⋊ ν^{−1/2}σ ⟶ VIc ⟶ 0.  (145)

Taking Jacquet modules and observing the result for VIc, we get (VIa)_{N,θ} = (ν^{1/2}St_{GL(2)} ⋊ ν^{−1/2}σ)_{N,θ}. Hence the result follows from Lemma 5.2.2 and (36). \underline{VIb}: By (2.11) of [28], there is an exact sequence

0 ⟶ (VIb)_{N,θ} ⟶ (ν^{1/2}1_{GL(2)} ⋊ ν^{−1/2}σ)_{N,θ} ⟶ (VId)_{N,θ} ⟶ 0. (146)

By (113), the middle term equals ℂ_{σ∘N_{L/F}}. One-dimensionality implies that the sequence splits, so that

Hom_{T} (ℂ_{σ∘N_{L/F}} , ℂ_{Λ}) = Hom_{D}(VIb, ℂ_{Λ⊗θ}) ⊕ Hom_{D}(VId, ℂ_{Λ⊗θ})  (147)

(D is the Bessel subgroup defined in (17)). Hence the VIb case follows from the known result for VId. \underline{VII}: This follows from Proposition 3.4.2 and Lemma 5.3.2. \underline{VIIIa and VIIIb}: In the split case this follows from Proposition 3.4.2 and v) of Proposition 3.5.1. Assume that θ is non-split. Since we are in a unitarizable situation, the sequence

0 ⟶ VIIIa ⟶ 1_{F^{×}} ⋊ π ⟶ VIIIb ⟶ 0

splits. It follows that

Hom_{D}(1_{F^{×}} ⋊ π, ℂ_{Λ⊗θ}) = Hom_{D}(VIIIa, ℂ_{Λ⊗θ}) ⊕ Hom_{D}(VIIIb, ℂ_{Λ⊗θ}). (148)

By Lemma 5.3.2, the space on the left is one-dimensional for any Λ. Therefore the Bessel functionals of VIIIb are complementary to those of VIIIa. Assume that VIIIa admits a (Λ, θ)-Bessel functional. Then, by Corollary 4.7.1 and Theorem 4.6.3, we have Hom_{T} (π, ℂ_{Λ}) ≠ 0. Conversely, assume that Hom_{T} (π, ℂ_{Λ}) ≠ 0 and assume that VIIIa does not admit a (Λ, θ)-Bessel functional; we will obtain a contradiction. By (148), we have Hom_{D}(VIIIb, ℂ_{Λ⊗θ}) ≠ 0. By Corollary 4.7.1 and Theorem 4.6.3, we have Hom_{T} (π^{JL}, ℂ_{Λ}) ≠ 0. This contradicts (34). The result for VIIIb now follows from (148). \underline{IXb}: This was proved in Corollary 4.7.2. \underline{IXa}: In the split case this follows from Proposition 3.4.2. Assume that θ is non-split. We have an exact sequence

0 ⟶ IXa ⟶ νξ ⋊ ν^{−1/2}π ⟶ IXb ⟶ 0.

By Lemma 5.3.2, the space Hom_{D}(νξ ⋊ ν^{−1/2}π, ℂ_{Λ⊗θ}) is one-dimensional, for any character Λ of L^{×} satisfying the central character condition. It follows that the possible Bessel functionals of IXa are complementary to those of IXb. \underline{X}: In the split case this follows from Proposition 3.4.2. In the non-split case it follows from Lemma 5.2.2. \underline{XIa and XIb}: In the split case this follows from Proposition 3.4.2 and Proposition 3.5.1; note that the V_{1}/V_{2} quotient of XIb equals τ_{GL(1)}^{P_{3}}(νσ) by Table A.6 of [28]. Assume that L/F is not split, and consider the exact sequence

0 ⟶ (XIa)_{N,θ} ⟶ (ν^{1/2}π ⋊ ν^{−1/2}σ)_{N,θ} ⟶ (XIb)_{N,θ} ⟶ 0.  (149)

It follows from Lemma 6.2.1 that

Hom_{D}(ν^{1/2}π ⋊ ν^{−1/2}σ, ℂ_{Λ⊗θ}) = Hom_{D}(XIa, ℂ_{Λ⊗θ}) ⊕ Hom_{D}(XIb, ℂ_{Λ⊗θ}).

(150)

Observe here that, by Lemma 5.2.2, the left side equals Hom_{T} (σπ, ℂ_{Λ}), which is at most one-dimensional. Assume that the representation XIa admits a (Λ, θ)-Bessel functional. Then Λ ≠ σ ∘ N_{L/F} and Hom_{T} (σπ, ℂ_{Λ}) ≠ 0 by Corollary 4.7.1 and Theorem 4.6.3. Conversely, assume that Λ ≠ σ ∘ N_{L/F} and Hom_{T} (σπ, ℂ_{Λ}) ≠ 0. Assume also that XIa does not admit a (Λ, θ)-Bessel functional; we will obtain a contradiction. By the one-dimensionality of the space on the left hand side of (150), we have Hom_{D}(XIb, ℂ_{Λ⊗θ}) ≠ 0. By Corollary 4.7.1 and Theorem 4.6.3, we conclude Λ = σ ∘ N_{L/F}, contradicting our assumption. Assume that the representation XIb admits a (Λ, θ)-Bessel functional. Then Λ = σ ∘ N_{L/F} and Hom_{T} (π, ℂ_{1}) ≠ 0 by Corollary 4.7.1 and Theorem 4.6.3. Conversely, assume that Λ = σ∘N_{L/F} and Hom_{T} (π, ℂ_{1}) ≠ 0. Assume also that XIb does not admit a (Λ, θ)-Bessel functional; we will obtain a contradiction. By our assumption, the space on the left hand side of (150) is one-dimensional. Hence Hom_{D}(XIa, ℂ_{Λ⊗θ}) ≠ 0. By what we have already proven, this implies Λ ≠ σ ∘ N_{L/F}, a contradiction. \underline{Va^{∗}}: This was proved in Corollary 4.7.2. \underline{XIa^{∗}}: By Proposition 3.5.1, the representation XIa^{∗} has no split Bessel functionals. Assume that θ is non-split. By Corollary 4.7.1 and Theorem 4.6.3, if XIa^{∗} admits a (Λ, θ)-Bessel functional, then Λ = σ ∘N and Hom_{T} (π^{JL}, ℂ_{1}) ≠ 0. Conversely, assume that Λ = σ∘N and Hom_{T} (π^{JL}, ℂ_{1}) ≠ 0. By Corollary 4.7.3, the twisted Jacquet module δ^{∗}(ν^{1/2}π, ν^{−1/2}σ)_{N,θ} is one-dimensional. Therefore, XIa^{∗} does admit a (Λ′, θ)-Bessel functional for some Λ′. By what we already proved, Λ′ = Λ. This concludes the proof.

6.3 Some cases of uniqueness

Let (π, V) be an irreducible, admissible representation of GSp(4, F). Using the notations from Sect. 2.4, consider (Λ, θ)-Bessel functionals for π. We say that such functionals are unique if the dimension of the space Hom_{D}(V, ℂ_{Λ⊗θ}) is at most 1. In this section we will prove the uniqueness of split Bessel functionals for all representations, and the uniqueness of non-split Bessel functionals for all non-supercuspidal representations. As far as we know, a complete proof that Bessel functionals are unique for all (Λ, θ) and all representations π has not yet appeared in the literature. In [17] it is proved that (1, θ)-Bessel functionals are unique if π has trivial central character. The main ingredient for this proof is Theorem 1’ of [10]. In [15] it is proved that (Λ, θ)-Bessel functionals are unique if π has trivial central character. The proof is based on a generalization of Theorem 1’ of [10]. In [31] it is stated, without proof, that (Λ, θ)-Bessel functionals are unique if π is supercuspidal and has trivial central character. In [23] it is remarked that the uniqueness of (Λ, θ)-Bessel functionals in the general case can be proven by extending the arguments of [17] and [15], though a proof of this is not given in [23].

6.3.1 Lemma. Let σ_{1} be a character of F^{×}, and let (π_{1}, V_{1}) be an irreducible, admissible representation of GL(2, F). Let the matrix S be as in (22), and θ be as in (24). The resulting group T is then given by (24). Let (π, V) be an irreducible, admissible representation of GSp(4, F). Assume there is an exact sequence

π_{1} ⋊ σ_{1} ⟶ π ⟶ 0.  (151)

Let Λ be a character of T. If Λ is not equal to one of the characters Λ_{1} or Λ_{2}, given by

Λ_{1}(diag(a, b, a, b)) = ν^{1/2}(a)ν^{−1/2}(b)σ_{1}(ab)ω_{π_{1}}(a),  (152)
Λ_{2}(diag(a, b, a, b)) = ν^{−1/2}(a)ν^{1/2}(b)σ_{1}(ab)ω_{π_{1}}(b),  (153)

then (Λ, θ)-Bessel functionals are unique.

Proof. Since π is a quotient of π_{1} ⋊ σ_{1}, it suffices to prove that Hom_{D}(π_{1} ⋊ σ_{1}, ℂ_{Λ⊗θ}) is at most one-dimensional. Any element β of this space factors through the Jacquet module (π_{1} ⋊ σ_{1})_{N,θ}. These Jacquet modules were calculated in Lemma 5.2.2 ii). Using the notation of this lemma, the assumption about Λ implies that restriction of β to J_{2} establishes an injection

Hom_{D}(π_{1} ⋊ σ_{1}, ℂ_{Λ⊗θ}) ⟶ Hom[ \begin{matrix}∗ \\ & ∗ \end{matrix}](σ_{1}π_{1}, ℂ_{Λ}).

The space on the right is at most one-dimensional; see Sect. 5.2. This proves our statement.

6.3.2 Theorem. Let (π, V) be an irreducible, admissible representation of GSp(4, F).

i) Split Bessel functionals for π are unique.

ii) Non-split Bessel functionals for π are unique, if π is not supercuspidal, or if π is of type Va^{∗} or XIa^{∗}.

Proof. i) By Proposition 3.5.1, we may assume that π is generic. Let the matrix S be as in (22), and θ be as in (23). The resulting group T is then given by (24). Let Λ be a character of T. We use the fact that any (Λ, θ)-Bessel functional β on V factors through the P_{3}-module V_{Z^{J}}. Assume that π is supercuspidal. Then, by Theorem 3.2.1, the associated P_{3}- module V_{Z^{J}} equals τ_{GL(0)}^{P_{3}}(1). Therefore, the space of (Λ, θ)-Bessel functionals on V equals the space of linear functionals considered in Lemma 2.5.4 of [28]. By this lemma, this space is one-dimensional. Now assume that π is non-supercuspidal. As in the proof of Proposition 3.5.1, we write the semisimplification of the quotient V_{1}/V_{2} in the P_{3}-filtration as ∑_{i=1}^{n} τ_{GL(1)}^{P_{3}}(χ_{i}) with characters χ_{i} of F^{×}. Let C(π) be the set of characters χ_{i}. Proposition 2.5.7 of [28] states that if the character a ↦ Λ(diag(a, 1, a, 1)) is not contained in the set ν^{−1}C(π), then the set of (Λ, θ)-Bessel functionals is at most one-dimensional (note that the arguments in the proof of this proposition do not require the hypothesis of trivial central character). The table below lists the sets ν^{−1}C(π) for all generic non-supercuspidal representations. This table implies that (Λ, θ)-Bessel functionals for types VII, VIIIa and IXa are unique. Assume that π is not one these types. Then there exists a sequence as in (151) for some irreducible, admissible representation π_{1} of GL(2, F) and some character σ_{1} of F^{×}. These π_{1} and σ_{1} are listed in the table below. Let Λ_{1}, Λ_{2} be the characters defined in (152) and (153). Note that, since Λ_{1} and Λ_{2} are Galois conjugate, we have

dim Hom_{D}(π, ℂ_{Λ_{1}⊗θ}) = dim Hom_{D}(π, ℂ_{Λ_{2}⊗θ})  (154)

by (29). By Lemma 6.3.1, it suffices to prove that these spaces are onedimensional. Define characters λ_{1}, λ_{2} of F^{×} by

λ_{1}(a) = Λ_{1}(diag(a, 1, a, 1)) = ν^{1/2}(a)σ_{1}(a)ω_{π_{1}}(a),
λ_{2}(a) = Λ_{2}(diag(a, 1, a, 1)) = ν^{−1/2}(a)σ_{1}(a).

The set {λ_{1}, λ_{2}} is listed in the table below for each representation. By the previous paragraph, the spaces (154) are one-dimensional if {λ_{1}, λ_{2}} is not a subset of ν^{−1}C(π). This can easily be verified using the table below.

π  π_{1}  σ_{1}  {λ_{1}, λ_{2}}  ν^{−1}C(π)
I  χ_{1} × χ_{2}  σ  {ν^{1/2}χ_{1}χ_{2}σ, ν^{−1/2}σ} {ν^{1/2}χ_{1}χ_{2}σ, ν^{1/2}χ_{1}σ,
ν^{1/2}χ_{2}σ, ν^{1/2}σ}
IIa  χSt_{GL(2)}  σ  {ν^{1/2}χ^{2}σ, ν^{−1/2}σ} {ν^{1/2}χ^{2}σ, ν^{1/2}σ, νχσ}
IIIa  χ^{−1} × ν^{−1} ν^{1/2}χσ  {σ, χσ}  {νχσ, νσ}
IVa ν^{−3/2}St_{GL(2)} ν^{3/2}σ  {ν^{−1}σ, νσ}  {ν^{2}σ}
Va ν^{−1/2}ξSt_{GL(2)} ν^{1/2}ξσ  {ξσ}  {νσ, νξσ}
VIa ν^{−1/2}St_{GL(2)} ν^{1/2}σ  {σ}  {νσ}
VII  —  —  —  ∅
VIIIa  —  —  —  ∅
IXa  —  —  —  ∅
X  π  σ  {ν^{1/2}ω_{π}σ, ν^{−1/2}σ}  {ν^{1/2}ω_{π}σ, ν^{1/2}σ}
XIa  ν^{−1/2}π  ν^{1/2}σ  {σ}  {νσ}

ii) Assume first that π is not supercuspidal. Then there exist an irreducible, admissible representation π_{1} of GL(2, F) and a character σ of F^{×} such that π is either a quotient of π_{1} ⋊ σ, or a quotient of σ ⋊ π_{1}. The assertion of ii) now follows from i) of Lemma 5.2.2 and Lemma 5.3.2. Now assume that π = δ^{∗}([ξ, νξ], ν^{−1/2}σ) is of type Va^{∗}. Suppose that Hom_{D}(π, ℂ_{Λ⊗θ}) is non-zero for some θ and Λ, with L being a field. By our main result Theorem 6.2.2, the quadratic extension L is the field corresponding to ξ and Λ = σ ∘ N_{L/F}. By Corollary 4.7.3, the Jacquet module π_{N,θ} is one-dimensional. This implies that Hom_{D}(π, ℂ_{Λ⊗θ}) is one-dimensional. Finally, assume that π = δ^{∗}(ν^{1/2}π, ν^{−1/2}σ) is of type XIa^{∗}. Suppose that Hom_{D}(π, ℂ_{Λ⊗θ}) is non-zero for some θ and Λ, with L being a field. By our main result Theorem 6.2.2, we have Λ = σ ∘ N_{L/F} and Hom_{T} (π^{JL}, ℂ_{1}) ≠ 0. By Corollary 4.7.3, the Jacquet module π_{N,θ} is one-dimensional. This implies that Hom_{D}(π, ℂ_{Λ⊗θ}) is one-dimensional.

7 Some applications

We present two applications that result from the methods used in this paper. The first application is a characterization of irreducible, admissible, non-generic representations of GSp(4, F) in terms of their twisted Jacquet modules and their Fourier-Jacobi quotient. The second application concerns the existence of certain vectors with good invariance properties.

7.1 Characterizations of non-generic representations

As before, we fix a non-trivial character ψ of F.

7.1.1 Lemma. Let (π, V) be a non-generic, supercuspidal, irreducible, admissible representation of GSp(4, F). Then dim V_{N,θ} < ∞ for all non-degenerate θ.

Proof. If θ is split, then V_{N,θ} = 0 by Theorem 3.2.1. Assume that θ is not split. Let θ = θ_{S} with S as in (11). We may assume that dim V_{N,θ} ≠ 0. Let X be as in (57). By Theorem 5.6 of [8], there exists an irreducible, admissible representation σ of GO(X) such that Hom_{R}(ω, π ⊗ σ) ≠ 0; here, ω is the Weil representation defined in Sect. 4.4. By i) of Theorem 4.4.6, the set Ω_{S} is non-empty. By Proposition 4.4.7, the dimension of V_{N,θ} is finite.

Let W be a smooth representation of N. We will consider the dimensions of the complex vector spaces W_{N,θ_{a,b,c}}. Fix representatives a_{1}, …, a_{t} for F^{×}/F^{×2}. We define

d(W) = ∑_{i=1}^{t} dim W_{N,θ_{a_{i},0,1}}.

If 0 = W_{0} ⊂ W_{1} ⊂ W_{2} ⊂ ⋯ ⊂ W_{k} = W is a chain of N subspaces, then

d(W) = ∑_{j=1}^{k} d(W_{j}/W_{j−1}).  (155)

If one of the spaces W_{N,θ_{a_{i},0,1}} is infinite-dimensional, then this equality still holds in the sense that both sides are infinite.

7.1.2 Lemma. Let W^{J} be a non-zero, irreducible, smooth representation of G^{J} admitting ψ as a central character. Then 1 ≤ d(W^{J}) ≤ #F^{×}/F^{×2}.

Proof. This follows immediately from Lemma 6.1.2.

7.1.3 Lemma. Let (τ^{J}, W^{J}) be a smooth representation of G^{J}. Then W^{J} has finite length if and only if d(W^{J}) is finite. If it has finite length, then

length(W^{J}) ≤ d(W^{J}) ≤ length(W^{J}) · #F^{×}/F^{×2}.

Proof. Assume that W^{J} has finite length. Let

0 = W_{0} ⊂ W_{1} ⊂ W_{2} ⊂ ⋯ ⊂ W_{k} = W^{J}

be a chain of G^{J} spaces such that W_{j}/W_{j−1} is not zero and irreducible. By (155), we have

d(W^{J}) = ∑_{j=1}^{k} d(W_{j}/W_{j−1}).

By Lemma 7.1.2, 1 ≤ d(W_{j}/W_{j−1}) ≤ #F^{×}/F^{×2} for j = 1, …, k. It follows that d(W^{J}) is finite, and that the asserted inequalities hold. If W^{J} has infinite length, a similar argument shows that d(W) is infinite.

7.1.4 Theorem. Let (π, V) be an irreducible, admissible representation of GSp(4, F). The following statements are equivalent.

i) π is not generic.

ii) dim V_{N,θ} < ∞ for all split θ.

iii) dim V_{N,θ} < ∞ for all non-degenerate θ.

iv) The G^{J}-representation V_{Z^{J} ,ψ} has finite length.

Proof. i) ⇒ iii) Assume that π is not generic. Let θ be a non-degenerate character of N. Assume first that θ is split. Then V_{N,θ} can be calculated from the P_{3}-filtration of π. As in the proof of Proposition 3.5.1 we see that V_{N,θ} is finite-dimensional. Now assume that θ is not split. If π is supercuspidal, then dim V_{N,θ} < ∞ by Lemma 7.1.1. Assume that π is not supercuspidal. Then the table of Bessel functionals shows that π admits (Λ, θ)-Bessel functionals only for finitely many Λ. Since every Λ can occur in V_{N,θ} at most once by the uniqueness of Bessel functionals (Theorem 6.3.2), this implies that V_{N,θ} is finite-dimensional. iii) ⇒ ii) is trivial. ii) ⇒ i) Assume that π is generic. Then the subspace V_{2} of the P_{3}-module V_{Z^{J}} from Theorem 3.2.1 is non-zero. In fact, this subspace is isomorphic to the representation τ_{GL(0)}^{P_{3}}(1) defined in (37). By Lemma 2.5.4 of [28], the space

(V_{2})_{[ \begin{matrix}1 & ∗ & ∗ \\ & 1 \\ & & 1 \end{matrix} ],θ_{0,1}},

where θ_{a,b} is defined in (140), is infinite-dimensional. This implies that V_{N,θ_{0,1,0}} is infinite-dimensional, contradicting the hypothesis in ii). iii) ⇔ iv) Let W^{J} = V_{Z^{J} ,ψ}. Then W_{N,θ_{a,0,1}}^{J} = V_{N,θ_{a,0,1}} for any a in F^{×}, so that d(W^{J}) = d(V). Lemma 7.1.3 therefore implies that iii) and iv) are equivalent.

For more thoughts on V_{Z^{J} ,ψ}, see [1]. Theorem 7.1.4 answers one of the questions mentioned at the end of this paper.

7.2 Invariant vectors

Let (π, V) be an irreducible, admissible representation of GSp(4, F). In this section we will prove the existence of a vector v in V such that diag(1, 1, c, c)v = v for all units c in the ring of integers 𝔬 of F. This result was motivated by a question of Abhishek Saha; see [33]. Our main tool will be the G^{J}-module V_{Z^{J} ,ψ} for a smooth representation (π, V) of GSp(4, F). Throughout this section we will make a convenient assumption about the character ψ of F, namely that ψ has conductor 𝔬. By definition, this means that ψ is trivial on 𝔬, but not on 𝔭^{−1}, where 𝔭 is the maximal ideal of 𝔬. We normalize the Haar measure on F such that 𝔬 has volume 1. Let q be the cardinality of the residue class field 𝔬/𝔭. In this section, we will abbreviate

d(c) = [ \begin{matrix}1 \\ & 1 \\ & & c \\ & & & c \end{matrix} ] , z(x) = [ \begin{matrix}1 & & & x \\ & 1 \\ & & 1 \\ & & & 1 \end{matrix} ]

for c in F^{×} and x in F.

7.2.1 Lemma. Let (π, V) be a smooth representation of GSp(4, F). Let p : V → V_{Z^{J} ,ψ} be the projection map, and let w in V_{Z^{J} ,ψ} be non-zero. Then there exists a positive integer m and a non-zero vector v in V with the following properties.

i) p(v) = w.

ii) π(z(x))v = ψ(x)v for all x ∈ 𝔭^{−m}.

iii) π(d(c))v = v for all c ∈ 1 + 𝔭^{m}.

Proof. Let v_{0} in V be such that p(v_{0}) = w. Let m be a positive integer such

that π(d(c))v_{0} = v_{0} for all c ∈ 1 + 𝔭^{m}. Set v = q^{−m} ∫_{𝔭^{−m}} ψ(−x)π(z(x))v_{0} dx.

Then p(v) = w. In particular, v is not zero. Evidently, v has property ii). Moreover, for c in 1 + 𝔭^{m},

π(d(c))(v) = q^{−m} ∫_{𝔭^{−m}} ψ(−x)π(z(xc^{−1})d(c))v_{0} dx

= q^{−m} ∫_{𝔭^{−m}} ψ(−xc)π(z(x))v_{0} dx

= q^{−m} ∫_{𝔭^{−m}} ψ(−x)π(z(x))v_{0} dx

= v.

This concludes the proof.

7.2.2 Lemma. Let (π, V) be a smooth representation of GSp(4, F). Let p : V → V_{Z^{J} ,ψ} be the projection map. Let m be a positive integer. Assume that v in V is such that π(z(x))v = ψ(x)v for all x ∈ 𝔭^{−m}. If c is in 𝔬^{×} but not in 1 + 𝔭^{m}, then p(π(d(c))v) = 0.

Proof. Let w = π(d(c))v. To show that p(w) = 0 it is enough to show that

∫_{𝔭^{−m}} ψ(−x)π(z(x))w dx = 0

because p(∫_{𝔭^{−m}} ψ(−x)π(z(x))w dx) = ∫_{𝔭^{−m}} ψ(−x)ψ(x)p(w) dx = q^{m}p(w). Indeed,

∫_{𝔭^{−m}} ψ(−x)π(z(x))w dx = ∫_{𝔭^{−m}} ψ(−x)π(z(x)d(c))v dx

= π(d(c)) ∫_{𝔭^{−m}} ψ(−x)π(z(xc))v dx

= π(d(c)) ∫_{𝔭^{−m}} ψ(−x)ψ(xc)v dx

= ( ∫_{𝔭^{−m}} ψ(x(c − 1)) dx)π(d(c))v

= 0,

since c ∉ 1 + 𝔭^{m} and ψ has conductor 𝔬.

7.2.3 Proposition. Let (π, V) be a smooth representation of GSp(4, F). Let p : V → V_{Z^{J} ,ψ} be the projection map. Let w be in V_{Z^{J} ,ψ}. Then there exists a unique vector v in V with the following properties.

i) p(v) = w.

ii) π(z(x))v = v for all x ∈ 𝔬.

iii) ∫_{𝔭^{−1}} π(z(x))v dx = 0.

iv) π(d(c))v = v for all c ∈ 𝔬^{×}.

Proof. For the existence part we may assume that w is non-zero. Let the positive integer m and v in V be as in Lemma 7.2.1. Define v_{1} = q^{m} ∫_{𝔬^{×}} π(d(c))v dc.

Then, by Lemma 7.2.2,

p(v_{1}) = q^{m} ∫_{𝔬^{×}} p(π(d(c))v) dc

= q^{m} ∫_{1+𝔭^{m}} p(π(d(c))v) dc

= q^{m} ∫_{1+𝔭^{m}} p(v) dc

= w.

Evidently, v_{1} has property iv). To see properties ii) and iii), let x be in 𝔭^{−1}. By ii) of Lemma 7.2.1,

π(z(x))v_{1} = q^{m} ∫_{𝔬^{×}} π(d(c)z(xc))v dc = q^{m} ∫_{𝔬^{×}} ψ(xc)π(d(c))v dc.

It follows that v_{1} has property ii). Integrating over x in 𝔭^{−1} shows that v_{1} has property iii) as well.

To prove that v_{1} is unique, let V_{1} be the subspace of V consisting of vectors v satisfying properties ii), iii) and iv). We will prove that the restriction of p to V_{1} is injective (so that p induces an isomorphism V_{1} ≅ V_{Z^{J} ,ψ}). Let v be in V_{1} and assume that p(v) = 0. Then there exists a positive integer m such that

∫_{𝔭^{−m}} ψ(−x)π(z(x))v dx = 0.

Applying d(c) to this equation, where c is in 𝔬^{×}, leads to

∫_{𝔭^{−m}} ψ(−cx)π(z(x))v dx = 0.

Integrating over c in 𝔬^{×}, we obtain

q^{−1} ∫_{𝔭^{−1}} π(z(x))v dx = ∫_{𝔬} π(z(x))v dx.

Using properties ii) and iii) it follows that v = 0. This concludes the proof.

7.2.4 Corollary. Let (π, V) be an irreducible, admissible representation of GSp(4, F) that is not a twist of the trivial representation. Then there exists a vector v in V that is invariant under all elements d(c) with c in 𝔬^{×}.

Proof. By Proposition 7.2.3, it is enough to show that V_{Z^{J} ,ψ} is non-zero. By Proposition 6.1.3, there exists a non-trivial character θ of N such that V_{N,θ} ≠ 0. We may assume that θ is of the form (15) with c = 1. The assertion follows.

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[26] Brooks Roberts. The non-Archimedean theta correspondence for GSp(2) and GO(4). Trans. Amer. Math. Soc., 351(2):781–811, 1999.

[27] Brooks Roberts. Global L-packets for GSp(2) and theta lifts. Doc. Math., 6:247–314 (electronic), 2001.

[28] Brooks Roberts and Ralf Schmidt. Local newforms for GSp(4), volume 1918 of Lecture Notes in Mathematics. Springer, Berlin, 2007.

[29] Brooks Roberts and Ralf Schmidt. On the number of local newforms in a metaplectic representation. In Arithmetic geometry and automorphic forms, volume 19 of Adv. Lect. Math. (ALM), pages 505–530. Int. Press, Somerville, MA, 2011.

[30] François Rodier. Whittaker models for admissible representations of reductive p-adic split groups. In Harmonic analysis on homogeneous spaces (Proc. Sympos. Pure Math., Vol. XXVI, Williams Coll., Williamstown, Mass., 1972), pages 425–430. Amer. Math. Soc., Providence, R.I., 1973.

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[36] Takashi Sugano. On holomorphic cusp forms on quaternion unitary groups of degree 2. J. Fac. Sci. Univ. Tokyo Sect. IA Math., 31(3):521–568, 1984.

[37] Ramin Takloo-Bighash. L-functions for the p-adic group GSp(4). Amer. J. Math., 122(6):1085–1120, 2000.

[38] Jerrold Tunnell. Local ϵ-factors and characters of GL(2). Amer. J. Math., 105(6):1277–1307, 1983.

[39] Jean-Loup Waldspurger. Correspondance de Shimura. J. Math. Pures Appl. (9), 59(1):1–132, 1980.

[40] Jean-Loup Waldspurger. Sur les valeurs de certaines fonctions L automorphes en leur centre de symétrie. Compositio Math., 54(2):173–242, 1985.

[41] Jean-Loup Waldspurger. Correspondances de Shimura et quaternions. Forum Math., 3(3):219–307, 1991.

Brooks Roberts

Department of Mathematics University of Idaho Moscow, ID 83844-1103 USA brooksr@uidaho.edu Ralf Schmidt Department of Mathematics University of Oklahoma Norman, OK 73019-3103 USA rschmidt@math.ou.edu 554

Documenta Mathematica 21 (2016)