Global L-Packets for GSp(2) and Theta Lifts

Brooks Roberts

Received: February 19, 2001 Revised: September 21, 2001

Communicated by Don Blasius

Abstract. Let F be a totally real number field. We define global L-packets for GSp(2) over F which should correspond to the elliptic tempered admissible homomorphisms from the conjectural Langlands group of F to the L-group of GSp(2) which are reducible, or irreducible and induced from a totally real quadratic extension of F. We prove that the elements of these global L-packets occur in the space of cusp forms on GSp(2) over F as predicted by Arthur’s conjecture. This can be regarded as the GSp(2) analogue of the dihedral case of the Langlands-Tunnell theorem. To obtain these results we prove a nonvanishing theorem for global theta lifts from the similitude group of a general four dimensional quadratic space over F to GSp(2) over F.

2000 Mathematics Subject Classification: Primary 11F70; Secondary 11F27, 11R39. Keywords and Phrases: L-packets, Arthur’s conjecture, GSp(2), theta lifts

Introduction

Let F be a number field with adeles 𝔸 and Weil group W_{F}, and let φ : W_{F} → GL(n, ℂ) be an irreducible continuous representation. There is a unique real number t such that the twist of φ by the canonical norm function on W_{F} raised to the t-th power has bounded image, so assume φ(W_{F}) is bounded; if φ factors through Gal(\overline{F} /F) this is automatic. For all places v of F, let π_{v} be the tempered irreducible admissible representation of GL(n, F_{v}) corresponding to the restriction φ_{v} under the local Langlands correspondence; then conjecturally ⊗_{v}π_{v} is an irreducible unitary cuspidal automorphic representation (hereafter, cuspidal automorphic representation) of GL(n, 𝔸). This conjecture is known in some cases. For example, if n = 2 and the image of φ in PGL(2, ℂ) is not the icosahedral group A_{5}, then the Langlands-Tunnell theorem asserts π is cuspidal automorphic. Inspired by this, one can ask for a complete parameterization of the tempered cuspidal automorphic representations of GL(n) and of other groups ([Ko], Section 12; [LL]). Since there are tempered cuspidal automorphic representations of GL(2, 𝔸_{ℚ}) which do not correspond to any φ : W_{ℚ} → GL(2, ℂ), the Weil group is inadequate. Conjecturally, there exists a locally compact group L_{F}, called the Langlands group of F, which is an extension of W_{F} by a compact group and is formally similar to the Weil group; locally, if v is an infinite place of F, then L_{F_{v}} = W_{F_{v}} and if v is finite, then L_{F_{v}} = W_{F_{v}} × SU(2, ℝ). Moreover, the tempered cuspidal automorphic representations of GL(n, 𝔸) should be in bijection with the n dimensional irreducible continuous complex representations of L_{F} with bounded image. For other connected reductive linear algebraic groups G over F the conjecture is more intricate, and involves L-packets attached to appropriate L-parameters L_{F} → ^{L}G. In this paper we prove results about local and global theta lifts which yield parameterizations of some tempered cuspidal automorphic representations of GSp(2, 𝔸) in agreement with this conjecture. To motivate the results we recall the conjecture, taking into account simplifications for GSp(2). Assume L_{F} exists. Then for GSp(2) one considers elliptic tempered admissible homomorphisms from L_{F} to ^{L} GSp(2) = { \atop GSp(2)} ⋊ W_{F}. Concretely, since GSp(2) is split and one can fix an isomorphism between the dual group { \atop GSp(2)} and GSp(2, ℂ), such homomorphisms amount to continuous homomorphisms φ : L_{F} → GSp(2, ℂ) such that φ(x) is semi-simple for all x ∈ L_{F} and φ(L_{F}) is bounded and not contained in the Levi subgroup of a proper parabolic subgroup of GSp(2, ℂ). Since L_{F} should be an extension of W_{F} a basic example is a continuous homomorphism Gal(\overline{F} /F) → GSp(2, ℂ) which is irreducible as a four dimensional complex representation. Fix such a φ : L_{F} → GSp(2, ℂ). The conjecture first asserts that for each place v of F one can associate to the restriction φ_{v} : L_{F_{v}} → GSp(2, ℂ) a finite set Π(φ_{v}) of irreducible admissible representations of GSp(2, F_{v}), the L-packet of φ_{v}. These packets should have a number of properties [B], but minimally we require that Π(φ_{v}) consists of tempered representations, and if v is finite and φ_{v} is unramified, then Π(φ_{v}) consists of a single representation unramified with respect to GSp(2, 𝔒_{F_{v}}) with Satake parameter φ_{v}(Frob_{v}) where Frob_{v} is a Frobenius element at v; also, the common central character of the elements of Π(φ_{v}) should correspond to λ ∘ φ_{v}, where λ : GSp(2, ℂ) → ℂ^{×} is the similitude quasi-character. Define

This work was partially supported by a NSA Young Investigators Grant.

Π(φ) = {Π = ⊗_{v}Π_{v} ∈ Irr_{admiss}(GSp(2, 𝔸)) : Π_{v} ∈ Π(φ_{v}) for all v}
= ⊗_{v}Π(φ_{v}).

Global L-Packets for GSp(2) and Theta Lifts 249

Arthur’s conjecture ([LL], [Ko], [A1], [A2]) now asserts that if Π ∈ Π(φ) then Π occurs with multiplicity

m(Π) = \frac{1}{|𝕊(φ)|} ∑_{s∈𝕊(φ)}⟨s, Π⟩

in the space of cusp forms on GSp(2, 𝔸) with central character λ∘φ. Here, 𝕊(φ) is the connected component group π_{0}(S(φ)/ℂ^{×}), where S(φ) is the centralizer of the image of φ, and ⟨·, ·⟩ : 𝕊(φ) × Π(φ) → ℂ is defined by

⟨s, Π⟩ = ∏_{v}⟨s_{v}, Π_{v}⟩_{v},

where s_{v} is the image of s under the natural map 𝕊(φ) → 𝕊(φ_{v}) and Π = ⊗_{v}Π_{v}; the ⟨·, ·⟩_{v} : 𝕊(φ_{v}) × Π(φ_{v}) → ℂ should be functions such that ⟨·, Π_{v}⟩_{v} is the character of a finite dimensional complex representation of 𝕊(φ_{v}) which is identically 1 if Π_{v} is unramified. By looking at cases we can be more specific. Elliptic tempered admissible homomorphisms φ : L_{F} → GSp(2, ℂ) can be divided into three types: (a) those which are irreducible and induced as a representation; (b) those which are reducible as a representation; and (c) those which are irreducible and primitive as a representation, i.e., not induced. Our result is motivated by what the conjecture predicts for φ of the first two types. Suppose φ is of type (a). Then one can show that φ is equivalent to φ(η, ρ) for some η and ρ, where φ(η, ρ) = Ind\begin{matrix}L & F \\ L & E \end{matrix} ρ, E is a quadratic extension of F, ρ : L_{E} → GL(2, ℂ) is an irreducible continuous representation with bounded image such that ρ is not Galois invariant but det ρ is, and η : L_{F} → ℂ^{×} extends det ρ; the symplectic form on φ(η, ρ) (regarded as ρ ⊕ ρ) is ⟨v_{1} ⊕ v_{2}, v′_{1} ⊕ v′_{2}⟩ = η(h)⟨v_{1}, v′_{1}⟩ + ⟨v_{2}, v′_{2}⟩ where ⟨·, ·⟩ is any fixed nondegenerate symplectic form on ℂ^{2} (up to multiplication by nonzero scalars there is only one) and h is a representative for the nontrivial coset of L_{E} \ L_{F}. Evidently,

λ ∘ φ(η, ρ) = η, 𝕊(φ(η, ρ)) = 1.

The conjecture thus predicts that every element Π of Π(φ) = Π(φ(η, ρ)) should be cuspidal automorphic with m(Π) = 1; that is, Π(φ) should be a stable global L-packet. Type (b) parameters, however, will in general give unstable L-packets. Suppose φ is of type (b). Then φ ≅ φ(ρ_{1}, ρ_{2}), where φ(ρ_{1}, ρ_{2}) = ρ_{1} ⊕ ρ_{2}, ρ_{1}, ρ_{2} : L_{F} → GL(2, ℂ) are inequivalent irreducible continuous representations with bounded image and the same determinant, and the symplectic form on φ(ρ_{1}, ρ_{2}) is ⟨v_{1} ⊕ v_{2}, v′_{1} ⊕ v′_{2}⟩ = ⟨v_{1}, v′_{1}⟩ + ⟨v_{2}, v′_{2}⟩. We see that

λ ∘ φ(ρ_{1}, ρ_{2}) = det ρ_{1} = det ρ_{2}, S_{φ} = \{[ \begin{matrix}a · I_{2} & 0 \\ 0 & ±a · I_{2} \end{matrix} ] : a ∈ ℂ^{×}\}.

Thus,

𝕊(φ(ρ_{1}, ρ_{2})) ≅ Z_{2}.

Let s ∈ 𝕊(φ(ρ_{1}, ρ_{2})) be nontrivial. If Π ∈ Π(φ) = Π(φ(ρ_{1}, ρ_{2})), the conjecture predicts

m(Π) = \frac{1}{2}(1 + ∏_{v}⟨s_{v}, Π_{v}⟩_{v}).

Now for each v, 𝕊(φ_{v}) = 1 or Z_{2}; and if 𝕊(φ_{v}) = Z_{2}, then s_{v} is a nontrivial element of 𝕊(φ_{v}). Thus, if M is the number of times 𝕊(φ_{v}) = Z_{2} and Π_{v} induces the nontrivial character of 𝕊(φ_{v}), then

m(Π) = \frac{1}{2}(1 + (−1)^{M}).

By the conjecture, Π is cuspidal automorphic if and only if M is even; if so, m(Π) = 1. The conjecture thus provides exact predictions for φ of types (a) and (b). But as precise as they are, these predictions concern conjectural objects. Globally, the hypothetical Langlands group underlies Arthur’s conjecture; locally, the existence of L-packets is required. There are at least two approaches to the avoiding L_{F} and testing the conjecture. One natural alternative is to consider only L-parameters that factor through the Weil group or the Galois group. Another approach is to move matters, when possible, entirely to the automorphic side of the picture and render Arthur’s conjecture into a statement involving only automorphic data. Base change and automorphic induction for GL(n) are important examples of such a shift. There is also a translation for parameters of type (a) and (b). The reason is that for φ of type (a), η corresponds to a Hecke character χ of 𝔸^{×} by Abelian class field theory and ρ should correspond to a non-Galois invariant tempered cuspidal automorphic representation τ of GL(2, 𝔸_{E}) whose central character factors through N_{F}^{E} via χ; for φ of type (b), ρ_{1} and ρ_{2} should correspond to a pair of inequivalent tempered cuspidal automorphic representations τ_{1} and τ_{2} of GL(2, 𝔸) with the same central character χ. Our first main result proves the automorphic version of Arthur’s conjecture for φ of types (a) and (b). To explain this automorphic analogue, suppose we are given, without reference to the global Langlands group, (A) a quadratic extension E of F and a non-Galois invariant tempered cuspidal automorphic representation τ of GL(2, 𝔸_{E}) whose central character factors through N_{F}^{E} via a character χ, or (B) a pair of inequivalent tempered cuspidal automorphic representations τ_{1} and τ_{2} of GL(2, 𝔸) with common central character χ. Then we have a corresponding conjectural ρ or ρ_{1} and ρ_{2}, a corresponding φ of type (a) or (b), and using φ, the statement of Arthur’s conjecture. However, φ can be avoided entirely in arriving at a formulation of Arthur’s conjecture starting from (A) or (B). This is due to two observations: first, the local L-parameters φ_{v} are defined via the local Langlands correspondence for GL(2) independent of the existence of φ; and second, the predictions of Arthur’s conjecture for parameters of type (a) and (b) only involve local data. To be specific, let v be a place of F, E_{v} = F_{v} ⊗_{F} E, and let τ_{v} be the irreducible admissible representation ⊗_{w|v}τ_{w} of GL(2, E_{v}), where w runs over the places of E lying over v (in case (B), E_{v} = F_{v} × F_{v}, and τ_{v} = τ_{1,v} ⊗ τ_{2,v}). Then, as mentioned and using no conjecture, we can associate to χ_{v} and τ_{v} a canonical local L-parameter φ(χ_{v}, τ_{v}) : L_{F_{v}} → GSp(2, ℂ). The automorphic version of Arthur’s conjecture now presumes that we can further associate to χ_{v} and τ_{v} a local L-packet, satisfying certain basic requirements connected with φ(χ_{v}, τ_{v}), and in the unstable case (B) a local pairing. This we do in Section 8: if F′ is a local field of characteristic zero, E′ is a quadratic extension of F′ or E′ = F′ × F′, and τ′ is an infinite dimensional irreducible admissible representation of GL(2, E′) with central character factoring through N_{F′}^{E′} via a quasi-character χ′ (if F′ is nonarchimedean of even residual characteristic we do also assume τ′ is tempered; if F′ is archimedean we assume F′ = ℝ and E′ = ℝ × ℝ), then we define a finite set Π(χ′, τ′) of irreducible admissible representations of GSp(2, F′). We show that this local L-packet has the desired essential properties: the common central character of the elements of Π(χ′, τ′) is χ′, the character corresponding to λ ∘ φ(χ′, τ′); if τ′ is tempered, then φ(χ′, τ′) and the elements of Π(χ′, τ′) are tempered; and if τ′ is unitary and E′/F′ and τ′ are unramified, then Π(χ′, τ′) is a singleton whose Satake parameter is φ(χ′, τ′)(Frob_{F′}) (if E′ = F′ × F′, then we say that E′/F′ is unramified). We also show |Π(χ′, τ′)| = 1 or 2 and |𝕊(φ(χ′, τ′))| = |Π(χ′, τ′)| at least if F′ is not of even residual characteristic and E′ is a field. Additionally, when E′ = F′ × F′ we define a function ⟨·, ·⟩_{F′} : 𝕊(φ(χ′, τ′)) × Π(χ′, τ′) → ℂ, and show that for all Π ∈ Π(χ′, τ′), ⟨·, Π⟩_{F′} is a character of 𝕊(φ(χ′, τ′)), and if |𝕊(φ(χ′, τ′))| = |Π(χ′, τ′)| = 2 then both characters of 𝕊(φ(χ′, τ′)) arise in this way. The following theorem is now the automorphic version of Arthur’s conjecture for parameters of type (a) and (b).

8.6 Theorem. Let F be a totally real number field and let E be a totally real quadratic extension of F or E = F × F. Let τ be a non-Galois invariant tempered cuspidal automorphic representation of GL(2, 𝔸_{E}) whose central character factors through the norm N_{F}^{E} via a Hecke character χ of 𝔸^{×}. Thus, if E = F ×F, then τ is a pair τ_{1}, τ_{2} of inequivalent tempered cuspidal automorphic representations of GL(2, 𝔸) sharing the same central character χ. Define the global L-packet:

Π(χ, τ) = {Π = ⊗_{v}Π_{v} ∈ Irr_{admiss}(GSp(2, 𝔸)) : Π_{v} ∈ Π(χ_{v}, τ_{v}) for all v}

= ⊗_{v}Π(χ_{v}, τ_{v}).

(1) If E is a field, then every element of Π(χ, τ) occurs with multiplicity one in the space of cusp forms on GSp(2, 𝔸) with central character χ.

(2) Suppose E = F × F. Let Π ∈ Π(χ, τ), and let T_{Π} be the set of places v such that 𝕊(φ(χ_{v}, τ_{v})) = Z_{2} and ⟨·, Π_{v}⟩_{v} is the nontrivial character of 𝕊(φ(χ_{v}, τ_{v})). If |T_{Π}| is even, then Π occurs with multiplicity one in the space of cusp forms on GSp(2, 𝔸) with central character χ. Conversely, if Π occurs in the space of cusp forms on GSp(2, 𝔸) with central character χ, then |T_{Π}| is even.

We hope this result will be of some use to investigators of rank four motives, four dimensional symplectic Galois representations, Siegel modular forms or varieties of degree two, or Abelian surfaces. One way to think of this theorem is as an analogue for GSp(2) of the dihedral case of the Langlands-Tunnell theorem. Of course, some applications might require more information about the local L-packets of Theorem 8.6. For example, we still need to prove the sole dependence of the Π(χ_{v}, τ_{v}) on the φ(χ_{v}, τ_{v}), i.e., if φ(χ_{v}, τ_{v}) ≅ φ(χ′_{v}, τ′_{v}), then Π(χ_{v}, τ_{v}) = Π(χ′_{v}, τ′_{v}). Also, detailed knowledge about the local L-packets at the ramified places and at infinity would be useful. We will return to these local concerns in a later work. The intended emphasis of this paper is, as much as possible, global. As remarked, the proof of Theorem 8.6 uses theta lifts. Locally, χ and τ give irreducible admissible representations of GO(X, F_{v}) for various four dimensional quadratic spaces X over F_{v}; theta lifts of these define the local L-packets. Globally, χ and τ induce cuspidal automorphic representations of GO(X, 𝔸) for various four dimensional quadratic spaces X over F. The automorphicity asserted in Theorem 8.6 is a consequence of our second main result, which gives a fairly complete characterization of global theta lifts from GO(X, 𝔸) to GSp(2, 𝔸) for four dimensional quadratic spaces X over F. In particular, it shows that the nonvanishing of the global theta lift to GSp(2, 𝔸) of a tempered cuspidal automorphic representation of GO(X, 𝔸) is equivalent to the nonvanishing of all the involved local theta lifts; in turn, these local nonvanishings are equivalent to conditions involving distinguished representations.

8.3 Theorem. Let F be a totally real number field, and let X be a four dimensional quadratic space over F. Let d ∈ F^{×}/F^{×2} be the discriminant of X(F), and assume that the discriminant algebra E of X(F) is totally real, i.e., either d = 1 or d ≠ 1 and E = F (\sqrt{d}) is totally real. Let σ ≅ ⊗_{v}σ_{v} be a tempered cuspidal automorphic representation of GO(X, 𝔸) with central character ω_{σ}. Let V_{σ} be the unique realization of σ in the space of cusp forms on GO(X, 𝔸) of central character ω_{σ} (Section 7). Then the following are equivalent:

(1) The global theta lift Θ_{2}(V_{σ}) of V_{σ} to GSp(2, 𝔸) is nonzero.

(2) For all places v of F, σ_{v} occurs in the theta correspondence with GSp(2, F_{v}).

(3) For all places v of F, σ_{v} is not of the form π_{v}^{−} for some distinguished

π_{v} ∈ Irr(GSO(X, F_{v})) (Section 3). Let σ lie over the cuspidal automorphic representation π of GSO(X, 𝔸) (Section 7), and let s ∈ O(X, F) be the element of determinant −1 from Lemma 6.1. If s · π ≇ π and one of (1), (2) or (3) holds, then Θ_{2}(V_{σ}) ≠ 0, Θ_{2}(V_{σ}) is an irreducible unitary cuspidal automorphic representation of GSp(2, 𝔸) with central character ω_{σ}, and

Θ_{2}(V_{σ}) ≅ ⊗_{v}θ_{2}(σ_{v}^{∨}) = ⊗_{v}θ_{2}(σ_{v})^{∨},

where θ_{2}(σ_{v}) is the local theta lift of σ_{v}. For all v, θ_{2}(σ_{v}) is tempered.

In this theorem we make no assumptions about Howe duality at the even places: we prove a version of Howe duality for the case at hand in Section 1. As mentioned, in the proof of Theorem 8.6 we use Theorem 8.3 to show that those elements of a global L-packet which should occur in the space of cusp forms really do. Theorem 8.6 also asserts that such elements occur with multiplicity one, and in the case of an unstable global L-packet, if Π is a cuspidal automorphic element, then |T_{Π}| is even. To prove these two remaining claims the key step is to show that if Π is an element of a global L-packet and V is a subspace of the space of cusp forms on GSp(2, 𝔸) with V ≅ Π, then V has a nonzero theta lift to GO(X, 𝔸) for some four dimensional quadratic space X with disc X = d, where E = F (\sqrt{d}) with d = 1 if E = F × F. This step is a consequence of a theorem of Kudla, Rallis and Soudry, which implies that if Π_{1} is a cuspidal automorphic representation of Sp(2, 𝔸), V_{1} is a realization of Π_{1} in the space of cusp forms, and some twisted standard partial L-function L^{S}(s, Π_{1}, χ) has a pole at s = 1, then V_{1} has a nonzero theta lift to O(X, 𝔸) for some four dimensional quadratic space X with (·, disc X)_{F} = χ. Using this key result, multiplicity one follows from the Rallis multiplicity preservation principle and multiplicity one for GO(X, 𝔸) for four dimensional quadratic spaces X; our understanding of the involved local theta lifts and especially the relevant theta dichotomy also plays an important role. The proof of the evenness of |T_{Π}| also uses the key step, local theory, and finally the fact that a quaternion algebra over F must be ramified at an even number of places. Theorems 8.3 and 8.6 depend on many previous works. Locally, we use the papers [R1], [R2] and [R3] which dealt with the local nonarchimedean theta correspondence for similitudes, the nonarchimedean theta correspondence between GO(X, F) and GSp(2, F) for dim_{F} X = 4, and tempered representations and the nonarchimedean theta correspondence, respectively. Globally, the critical nonvanishing results for theta lifts of this paper depend on the main result of [R4]. In turn, the essential idea of [R4] is based on an ingenious insight of [BSP]; [R4] also uses some strong results and ideas from [KR1] and [KR2]. The multiplicity one part of Theorem 8.6 uses one of the main results of [KRS], along with the multiplicity preservation principle of [Ra]. We use nonvanishing results for L-functions at s = 1 from [Sh] to satisfy the hypothesis of Corollary 1.2 of [R4]. Various results and ideas from [HST] are used in this paper. We would also like to mention as inspiration the papers of H. Yoshida [Y1] and [Y2] which first looked at theta lifts of automorphic forms on GSO(X, 𝔸_{ℚ}) for dim_{ℚ} X = 4 to GSp(2, 𝔸_{ℚ}). Using results from [HPS], the paper [V] also defined local discrete series L-packets for GSp(2) using theta lifts in the case of odd residual characteristic. This paper is organized as follows. In Section 1 we consider the local theta correspondence for similitudes. The first main goal of this section is to extend the results of [R1] to the even residual characteristic and real cases. This requires that we prove a version of Howe duality in the even residual characteristic case: we do this for tempered representations when the underlying quadratic and symplectic bilinear spaces have the same dimension. The second main goal is to prove a case of S.S. Kudla’s theta dichotomy conjecture, which is required for a complete theta lifting theory for similitudes for the relevant case. In Section 2 we review the basic theory of four dimensional quadratic spaces and their similitude groups. In particular, we define the four dimensional quadratic spaces X_{D,d} of discriminant d over a field F not of characteristic two; here, D is a quaternion algebra over F and d ∈ F^{×}/F^{×2}. Up to similitude, every four dimensional quadratic space over F is of the form X_{D,d}. The characterization of what irreducible representations of GO(X, F), dim_{F} X = 4, occur in the theta correspondence with GSp(2, F) when F is a local field is given in Section 3. The case when F is nonarchimedean of odd residual characteristic was worked out in [R2] and the remaining cases are similar, but require additional argument. In Section 4 we define the local L-parameters and Lpackets of Theorem 8.6; in fact, the L-parameters and L-packets are associated to irreducible admissible representations of GSO(X_{M_{2×2},d}, F) where X_{M_{2×2},d} is the four dimensional quadratic space from Section 2 over the local field F. The information is summarized in three tables which appear in the Appendix. Section 5 reviews the theory of global theta lifts for similitudes. Sections 6 and 7 explain the transition from cuspidal automorphic representations of a quaternion algebra over a quadratic extension to those of similitude groups of four dimensional quadratic spaces. Finally, in Section 8 we prove the main theorems. I would like to thank N. Nygaard for suggesting this line of research, and S.S. Kudla, S. Rallis and J. Arthur for their interest and encouragement.

Notation. Let F be a field not of characteristic two. A quadratic space over F is a finite dimensional vector space X over F equipped with a nondegenerate symmetric bilinear form (·, ·). Let X be a quadratic space over F. In this and the next two paragraphs, also denote the F points of X by X; the same convention holds when we are considering quadratic spaces solely over a local field, as in Sections 1, 3 and 4. The discriminant disc X ∈ F^{×}/F^{×2} of X is (−1)^{k} det X where dim X = 2k or 2k + 1. If (X′, (·, ·)′) is another quadratic space over F then a similitude from X to X′ is an F linear map t : X → X′ such that for some λ ∈ F^{×}, (tx, tx′) = λ(x, x′) for x, x′ ∈ X; λ is uniquely determined, and we write λ(t) = λ. The group GO(X, F) is the set of h ∈ GL_{F} (X) which are similitudes from X to X. The group O(X, F) is the kernel of λ : GO(X, F) → F^{×}, and SO(X, F) is the subgroup of h ∈ O(X, F) with det h = 1. Assume dim X is even. Then GSO(X, F) is the kernel of sign : GO(X, F) → {±1} defined by h ↦ det(h)/λ(h)^{dim X/2}; SO(X, F) = GSO(X, F) ∩ O(X, F). Let n be a positive integer. Then GSp(n, F) is the group of g ∈ GL(2n, F) such that for some λ ∈ F^{×}

tg [ \begin{matrix}0 & 1_{n} \\ −1_{n} & 0 \end{matrix} ] g = λ [ \begin{matrix}0 & 1_{n} \\ −1_{n} & 0 \end{matrix} ];

λ is uniquely determined, and we write λ(h) = λ. The group Sp(n, F) is the kernel of λ : GSp(n, F) → F^{×}. M_{2×2} = M_{2×2}(F) is the quaternion algebra of 2 × 2 matrices over F with canonical involution ∗.

Suppose F is a nonarchimedean field of characteristic zero with integers 𝔒_{F}, prime ideal 𝔭_{F} = π_{F} 𝔒_{F} ⊂ 𝔒_{F}, Hilbert symbol (·, ·)_{F}, and valuation | · | = | · |_{F} such that if μ is an additive Haar measure on F, then μ(xA) = |x|μ(A) for x ∈ F and A ⊂ F. Let G be a group of td-type, as in [C]. Then Irr(G) is the set of equivalence classes of smooth admissible irreducible representations of G. If π ∈ Irr(G), then π^{∨} ∈ Irr(G) is the contragredient of π and ω_{π} is the central character of π. The trivial representation of G is 1 = 1_{G}. If H is a closed normal subgroup of G, π ∈ Irr(H) and g ∈ G, then g · π ∈ Irr(H) has the same space as π and action defined by (g · π)(h) = π(g^{−1}hg). If G is the F -points of a connected reductive algebraic group defined over F, then π ∈ Irr(G) is tempered (square integrable) if and only if ω_{π} is unitary and every matrix coefficient of π lies in L^{2+ϵ}(G/Z(G)) for all ϵ > 0 (lies in L^{2}(G/Z)). Let X be a quadratic space over F. The quadratic character χ_{X} : F^{×} → {±1} associated to X is (·, disc X)_{F}. We say σ ∈ Irr(O(X, F)) (σ ∈ Irr(GO(X, F)) is tempered if all the irreducible components of σ|_{SO(X,F)} (σ|_{GSO(X,F)})) are tempered. A self-dual lattice L in X is a free 𝔒_{F} submodule of rank dim X such that L = {x ∈ X : (x, y) ∈ 𝔒_{F} for all y ∈ L}. D_{ram} is the division quaternion algebra over F with canonical involution ∗. If E/F is a quadratic extension the quadratic character of F^{×} associated to E/F is ω_{E/F}. Suppose F = ℝ. Let | · | = | · |_{ℝ} be the usual absolute value on ℝ, and let (·, ·)_{ℝ} be the Hilbert symbol of ℝ. If X is quadratic space over ℝ, then the quadratic character χ_{X} : ℝ^{×} → {±1} associated to X is (·, disc X)_{ℝ}. Let G be a real reductive group as in [Wal]. Let K be a maximal compact subgroup of G, and let 𝔤 be the Lie algebra G. Let Irr(G) be the set of equivalence classes of irreducible (𝔤, K) modules. The trivial (𝔤, K) module will be denoted by 1 = 1_{G}. If K_{1} is a closed normal subgroup of K, π is a (𝔤, K_{1}) module and s ∈ K, then s · π is the (𝔤, K_{1}) module with the same space as π and action defined by (s · π)(k) = π(s^{−1}ks) for k ∈ K_{1} and (s · π)(X) = π(Ad(s)X) for X ∈ 𝔤. When G satisfies G^{o} = ^{o}(G^{o}) ([Wal], p. 48-9) the concepts of tempered and square integrable (𝔤, K) modules are defined in [Wal], 5.5.1; this includes G = Sp(n, ℝ), O(p, q, ℝ) and SO(p, q, ℝ) for p and q not both 1. When G^{o} = ^{o}(G^{o}), then π ∈ Irr(G) is tempered (square integrable) if and only if π is equivalent to the underlying (𝔤, K) module of an irreducible unitary representation Π of G such that g ↦ ⟨Π(g)v, w⟩ lies in L^{2+ϵ}(G) for all v, w ∈ π and ϵ > 0 (lies in L^{2}(G) for all v, w ∈ π). When G = GSp(n, ℝ), GO(p, q, ℝ) or GSO(p, q, ℝ) with p and q not both 1, then we say that π ∈ Irr(G) is tempered (square integrable) if π is equivalent to the underlying (𝔤, K) module of an irreducible unitary representation Π of G such that g ↦ ⟨Π(g)v, w⟩ lies in L^{2}(ℝ^{×}\G) for all v, w ∈ π and ϵ > 0 (lies in L^{2}(ℝ^{×}\G) for all v, w ∈ π); this is equivalent to the irreducible constituents of π|_{(𝔤_{1},K_{1})} being tempered (square integrable), where 𝔤_{1} is the Lie algebra and K_{1} ⊂ K is the maximal compact subgroup of Sp(n, ℝ), O(p, q, ℝ) or SO(p, q, ℝ), respectively. D_{ram} is the division quaternion algebra over ℝ with canonical involution ∗. Suppose F is a number field with adeles 𝔸 and finite adeles 𝔸_{f}; set F_{∞} = F ⊗_{ℚ} ℝ. The Hilbert symbol of F is (·, ·)_{F}. If X is quadratic space over F, then the quadratic Hecke character χ_{X} : 𝔸^{×}/F^{×} → {±1} associated to X is (·, disc X(F))_{F}. Let G be a reductive linear algebraic group defined over F, let 𝔤 be the Lie algebra of G(F_{∞}), and let K be a maximal compact subgroup of G(F_{∞}). Then Irr_{admiss}(G(𝔸)) is the set of equivalence classes of irreducible admissible G(𝔸_{f}) × (𝔤, K) modules. If π ∈ Irr_{admiss}(G(𝔸)) then the central character of π is ω_{π} and π = ⊗_{v}π_{v} is tempered if π_{v} is tempered for all places v of F. A cuspidal automorphic representation of G(𝔸) is a π ∈ Irr_{admiss}(G(𝔸)) which is isomorphic to an irreducible submodule of the G(𝔸_{f}) × (𝔤, K) module of cuspidal automorphic forms on G(𝔸) of central character ω_{π}; such a π is unitary.

1. The local theta correspondence for similitudes

In this section we recall and prove results about the local theta correspondence for similitudes. The paper [R1] dealt with the nonarchimedean odd residual characteristic case. Here we do the even residual characteristic and real cases and prove a very special, but adequate, case of S.S. Kudla’s theta dichotomy conjecture. We also show that the theta correspondence for similitudes is independent of the additive character, compatible with contragredients, and respects unramified representations. Fix the following notation. Let F be a local field of characteristic zero, with F = ℝ if F is archimedean. Let n be a positive integer, and let X be a quadratic space of nonzero even dimension m over F. To simplify notation, denote the F points of X by X. Let d = disc X. Fix a nontrivial unitary character ψ of F. The Weil representation ω = ω_{X} = ω_{n} = ω_{X,n} of Sp(n, F) × O(X, F) defined with respect to ψ is the unitary representation on L^{2}(X^{n}) given by

ω(1, h)φ(x) = φ(h^{−1}x),
ω([ \begin{matrix}a & 0 \\ 0 & ^{t}a^{−1} \end{matrix} ] , 1)φ(x) = χ_{X}(det a)| det a|^{m/2}φ(xa),
ω([ \begin{matrix}1 & b \\ 0 & 1 \end{matrix} ] , 1)φ(x) = ψ(\frac{1}{2} tr(bx, x))φ(x),
ω([ \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ] , 1)φ(x) = γφ̂(x).

Here, φ̂ is the Fourier transform defined by

φ̂(x) = ∫_{X^{n}} φ(x′)ψ(tr(x, x′)) dx′

with dx such that ˆ φ̂(x) = φ(−x) for φ ∈ L^{2}(X^{n}) and x ∈ X^{n}, and γ is a certain fourth root of unity depending only on the anisotropic component of X, n and ψ. If h ∈ O(X, F), a ∈ GL(n, F), b ∈ M_{n}(F) with ^{t}b = b and x = (x_{1}, …, x_{n}), x′ = (x′_{1}, …, x′_{n}) ∈ X^{n}, we write h^{−1}x = (h^{−1}x_{1}, …, h^{−1}x_{n}), xa = (x_{1}, …, x_{n})(a_{ij}), (x, x′) = ((x_{i}, x′_{j})), bx = b^{t}(x_{1}, …, x_{n}). Also, χ_{X} is the quadratic character of F^{×} defined by χ_{X}(t) = (t, d)_{F} ; χ_{X} depends only on the anisotropic component of X. Suppose F is nonarchimedean. We will work with smooth representations of groups of td-type such as Sp(n, F) and O(X, F). We thus consider the restriction of ω to a smaller subspace. Let 𝒮(X^{n}) be the space of locally constant, compactly supported functions on X^{n}. Then ω preserves 𝒮(X^{n}). By ω we will usually mean ω acting on 𝒮(X^{n}); context will give the meaning. Let ℛ_{n}(O(X, F)) be the set of elements of Irr(O(X, F)) which are nonzero quotients of ω, and define ℛ_{X}(Sp(n, F)) similarly. Suppose F = ℝ. In the analogy to the last case, we will work with Harish-Chandra modules of real reductive groups. This requires definitions. Fix K_{1} = Sp(n, ℝ) ∩ O(2n, ℝ) as a maximal compact subgroup of Sp(n, ℝ). The Lie algebra of Sp(n, ℝ) is 𝔤_{1} = 𝔰𝔭(n, ℝ). Let X have signature (p, q). We parameterize the maximal compact subgroups of O(X, ℝ) as follows. Let X^{+} and X^{−} be positive and negative definite subspaces of X, respectively, such that X = X^{+} ⊥ X^{−}. Then the maximal compact subgroup J_{1} = J_{1}(X^{+}, X^{−}) associated to (X^{+}, X^{−}) is the set of k ∈ O(X, ℝ) such that k(X^{+}) = X^{+} and k(X^{−}) = X^{−}. Of course, J_{1} = O(X^{+}, ℝ)×O(X^{−}, ℝ) ≅ O(p, ℝ)×O(q, ℝ). Fix one such J_{1} = J_{1}(X^{+}, X^{−}). The Lie algebra of O(X, ℝ) is 𝔥_{1} = 𝔬(X, ℝ). Let 𝒮(X^{n}) = 𝒮_{ψ}(X^{n}) be the subspace of L^{2}(X^{n}) of functions

p(x) exp[−\frac{1}{2}|c|(tr(x^{+}, x^{+}) − tr(x^{−}, x^{−}))].

Here, p : X^{n} → ℂ is a polynomial function on X^{n}, and (x^{+}, x^{+}) and (x^{−}, x^{−}) are the n × n matrices with (i, j)-th entries (x_{i}^{+}, x_{j}^{+}) and (x_{i}^{−}, x_{j}^{−}) respectively, where x_{i} = x_{i}^{+} + x_{i}^{−}, with x_{i}^{+} ∈ X^{+} and x_{i}^{−} ∈ X^{−} for 1 ≤ i ≤ n; c ∈ ℝ^{×} is such that ψ(t) = exp(ict) for t ∈ ℝ. Then 𝒮(X^{n}) is a (𝔤_{1} × 𝔥_{1}, K_{1} × J_{1}) module under the action of ω. By ω we will usually mean the (𝔤_{1} ×𝔥_{1}, K_{1} ×J_{1}) module 𝒮(X^{n}). Let ℛ_{n}(O(X, ℝ)) be the set of irreducible (𝔤_{1}, J_{1}) modules which are nonzero quotients of ω, and define ℛ_{X}(Sp(n, ℝ)) similarly. For uniformity, write Hom_{Sp(n,F)×O(X,F)}(ω, π ⊗ σ) for Hom_{(𝔤_{1}×𝔥_{1},K_{1}×J_{1})}(ω, π ⊗ σ). We have the following foundational result on the theta correspondence for isometries.

1.1 Theorem ([H], [W1]). Suppose F is real or nonarchimedean of odd residual characteristic. The set

{(π, σ) ∈ ℛ_{X}(Sp(n, F)) × ℛ_{n}(O(X, F)) : Hom_{Sp(n,F)×O(X,F)}(ω, π ⊗ σ) ≠ 0}

is the graph of a bijection between ℛ_{X}(Sp(n, F)) and ℛ_{n}(O(X, F)), and

dim_{ℂ} Hom_{Sp(n,F)×O(X,F)}(ω, π ⊗ σ) ≤ 1

for π ∈ ℛ_{X}(Sp(n, F)) and σ ∈ ℛ_{n}(O(X, F)).

When F is nonarchimedean of even residual characteristic partial results are known. For us the following unconditional result suffices. If F is nonarchimedean and σ ∈ Irr(O(X, F)) we say that σ is tempered if all the irreducible constituents of σ|_{SO(X,F)} are tempered.

1.2 Theorem. Suppose F is nonarchimedean of even residual characteristic and m = 2n. Let ℛ_{X}(Sp(n, F))_{temp} and ℛ_{n}(O(X, F))_{temp} be the subsets of ℛ_{n}(O(X, F)) and ℛ_{X}(Sp(n, F)) of tempered elements, respectively. Then the statement of Theorem 1.1 holds with ℛ_{X}(Sp(n, F))_{temp} and ℛ_{n}(O(X, F))_{temp} replacing the sets ℛ_{X}(Sp(n, F)) and ℛ_{n}(O(X, F)), respectively.

Proof. Let σ ∈ ℛ_{n}(O(X, F))_{temp}. By 2) a), p. 69 of [MVW], there exists π ∈ ℛ_{X}(Sp(n, F)) such that the homomorphism space of Theorem 1.1 is nonzero; π is tempered by (1) of Theorem 4.2 of [R3], and is unique by Theorem 4.4 of [R3]. To prove the map from ℛ_{n}(O(X, F))_{temp} to ℛ_{X}(Sp(n, F))_{temp} is injective and the homomorphism space has dimension at most one, let π ∈ Irr(Sp(n, F))_{temp}. By putting together the proofs of Proposition II.3.1 of [Ra] and Theorem 4.4 of [R3] one can show that there is a ℂ linear injection

⊕_{σ∈Irr(O(X,F)) σ unitary} Hom_{Sp(n,F)×O(X,F)}(ω, π ⊗ σ)

↪ Hom_{Sp(n,F)×Sp(n,F)}(𝒮(Sp(n, F)), π ⊗ π^{∨}),

where 𝒮(Sp(n, F)) is the space of locally constant compactly supported functions on Sp(n, F) and the action of Sp(n, F)×Sp(n, F) on 𝒮(Sp(n, F)) is defined by ((g, g′) · ϕ)(x) = ϕ(g^{−1}xg′). The last space is one dimensional as

π ⊗ π^{∨} ≅ 𝒮(Sp(n, F))/ ∩_{f∈Hom_{Sp(n,F)}(𝒮(Sp(n,F)),π⊗U), U a ℂ vector space} ker(f);

see the lemma on p. 59 of [MVW]. This proves the claims about injectivity and dimension. For surjectivity, let π ∈ ℛ_{X}(Sp(n, F))_{temp}. As above, there exists σ ∈ ℛ_{n}(O(X, F)) such that the homomorphism space is nonzero. An argument as in the proof of (1) of Theorem 4.2 of [R3] shows that σ must be tempered. See also [Mu]. □

It is worth noting the following from the proof of Theorem 1.2: Let m = 2n, π ∈ Irr(Sp(n, F)) and σ ∈ Irr(O(X, F)). If Hom_{Sp(n,F)×O(X,F)}(ω, π ⊗ σ) ≠ 0, then π is tempered if and only if σ is tempered. When F is nonarchimedean of odd residual characteristic or F = ℝ, then the bijection from Theorem 1.1 and its inverse are denoted by

θ : ℛ_{X}(Sp(n, F)) ⟶^{∼} ℛ_{n}(O(X, F)), θ : ℛ_{n}(O(X, F)) ⟶^{∼} ℛ_{X}(Sp(n, F));

if F is nonarchimedean of even residual characteristic we use the same notation for the bijections between ℛ_{X}(Sp(n, F))_{temp} and ℛ_{n}(O(X, F))_{temp} from Theorem 1.2. Next we recall and prove a special case of a conjecture of S.S. Kudla on the theta correspondence for isometries. This conjecture has important implications for the theta correspondence for similitudes.

1.3 Theta dichotomy conjecture (S.S. Kudla). Assume F is nonarchimedean. Let m be a positive even integer, let d ∈ F^{×}/F^{×2}, and let n be a positive integer such that m ≤ 2n. There exist at most two quadratic spaces Y and Y′ over F of dimension m and discriminant d; assume both exist. Then ℛ_{Y} (Sp(n, F)) ∩ ℛ_{Y′}(Sp(n, F)) = ∅.

We can prove the conjecture when m is small in comparison to 2n:

1.4 Lemma. Suppose that the notation is as in Conjecture 1.3, and assume the two quadratic spaces Y and Y′ exist. If m ≤ n + 2, then the theta dichotomy conjecture holds for m and n.

Proof. Let Z be the quadratic space over F of dimension 2m with four dimensional anisotropic component. To prove the theta dichotomy conjecture for m and n it suffices to show that 1_{Sp(n)} ∉ ℛ_{Z}(Sp(n, F)). This reduction is well known, but we recall the proof for the convenience of the reader. Assume 1_{Sp(n)} ∉ ℛ_{Z}(Sp(n, F)), and suppose π ∈ ℛ_{Y} (Sp(n, F)) ∩ ℛ_{Y′}(Sp(n, F)). To get a contradiction, let g_{0} ∈ GSp(n, F) be such that λ(g_{0}) = −1. By the first theorem on p. 91 of [MVW], g_{0} · π ≅ π^{∨}. Thus, there is a nonzero Sp(n, F) × Sp(n, F) map from ω_{Y,n} ⊗ g_{0} · ω_{Y′,n} to π ⊗ π^{∨}. By Lemma 1.6 below, g_{0} · ω_{Y′,n} ≅ ω_{−Y′,n}, where −Y′ has the same space as Y′ and form multiplied by −1. Also, (ω_{Y,n} ⊗ ω_{−Y′,n})|_{Δ Sp(n,F)} ≅ ω_{Y ⊥−Y′,n}|_{Sp(n,F)}. Clearly, Y ⊥ −Y′ ≅ Z. Since Hom_{Sp(n,F)}((π ⊗ π^{∨})|_{Δ Sp(n,F)}, 1_{Sp(n,F)}) ≠ 0, there now is a nonzero Sp(n, F) map from ω_{Z,n} to 1_{Sp(n,F)}, i.e., 1_{Sp(n,F)} ∈ ℛ_{Z}(Sp(n, F)). We now show 1_{Sp(n)} ∉ ℛ_{Z}(Sp(n, F)) for m ≤ n + 2. Let i be the Witt index of Z, i.e., i = m − 2, and write V_{i} = Z to indicate that Z is the orthogonal direct sum of the four dimensional anisotropic quadratic space over F with i hyperbolic planes. We must show 1_{Sp(n)} ∉ ℛ_{V_{i}}(Sp(n, F)) for 0 ≤ i ≤ n; we do this by induction on n. The case n = 1 follows from Lemma 7.3 of [R2] (see its proof, which is residual characteristic independent). Let n > 1, and assume the claim for n − 1. Let i ≤ n, and assume 1_{Sp(n,F)} ∈ ℛ_{V_{i}}(Sp(n, F)). To find a contradiction we reduce dimensions using Jacquet functors and Kudla’s filtration of the Jacquet module of the Weil representation. We use the notation of [R3]: write ω_{i,n} = ω_{V_{i},n}. Since 1_{Sp(n,F)} ∈ ℛ_{V_{i}}(Sp(n, F)), by 2) a) of the Theorem on p. 69 of [MVW], there exists σ ∈ Irr(O(V_{i}, F)) and a nonzero Sp(n, F) × O(V_{i}, F) map

ω_{i,n} → 1_{Sp(n,F)} ⊗ σ.

Let N′_{1} be the unipotent radical of the standard maximal parabolic of Sp(n, F) with Levi factor isomorphic to GL(1, F) × Sp(n − 1, F). Applying the normalized Jacquet functor with respect to N′_{1}, which is exact, we obtain a nonzero GL(1, F) × Sp(n − 1, F) × O(V_{i}, F) map

R_{N′_{1}}(ω_{i,n}) → R_{N′_{1}}(1_{Sp(n)}) ⊗ σ = | · |^{−n} ⊗ 1_{Sp(n−1,F)} ⊗ σ.

Suppose first i = 0, so that V_{0} is four dimensional and anisotropic. By Kudla’s computation of the Jacquet functors of ω_{0,n}, (see [R3] for a statement in our notation), R_{N′_{1}}(ω_{0,n})|_{Sp(n−1,F)} ≅ ω_{0,n−1}|_{Sp(n−1,F)}. Thus, 1_{Sp(n−1,F)} ∈ ℛ_{V_{i}}(Sp(n − 1, F)). Since i = 0 ≤ n − 1, by the induction hypothesis this is a contradiction. Suppose i > 0. By Kudla’s filtration (two step, in this case) of R_{N′_{1}}(ω_{i,n}) either there exists a nonzero GL(1, F) × Sp(n − 1, F) × O(V_{i}, F) map

| · |^{dim V_{i}/2−n} ⊗ ω_{i,n−1} → | · |^{−n} ⊗ 1_{Sp(n−1,F)} ⊗ σ,

or there exists a nonzero GL(1, F) × GL(1, F) × Sp(n − 1, F) × O(V_{i−1}, F) map

ξ_{1}ξ′_{1}σ_{1} ⊗ ω_{i−1,n−1} → | · |^{−n} ⊗ 1_{Sp(n−1,F)} ⊗ \overline{R}_{N_{1}}(σ);

here, ξ_{1} and ξ′_{1} are quasi-characters of GL(1, F) and σ_{1} is a representation of GL(1, F) × GL(1, F) whose precise definitions we will not need, | · |^{−n} is regarded as a quasi-character of GL(1, F), N_{1} is the unipotent radical of the standard parabolic of O(V_{i}, F) with Levi factor isomorphic to GL(1, F) × O(V_{i−1}, F), and \overline{R}_{N_{1}}(σ) = R_{N_{1}}(σ^{∨})^{∨}. The first case is ruled out since | · |^{dim V_{i}/2−n} ≠ | · |^{−n}. Since the second case must therefore hold, we get Hom_{Sp(n−1,F)}(ω_{i−1,n−1}, 1_{Sp(n−1,F)}) ≠ 0, i.e., 1_{Sp(n−1,F)} ∈ ℛ_{V_{i−1}}(Sp(n−1, F)). This contradicts the induction hypothesis since i − 1 ≤ n − 1. □

The real analogue of the theta dichotomy conjecture is known. The assumption of the evenness of p and q in the following lemma is a consequence of the same assumptions in [M].

1.5 Lemma. Suppose F = ℝ. Let m be a positive even integer and let n be a positive integer such that m ≤ 2n. Then the sets ℛ_{Y} (Sp(n, ℝ)) as Y runs over the isometry classes quadratic spaces over ℝ of dimension m and signature of the form (p, q) with p and q even are mutually disjoint.

Proof. We argue as in the second paragraph of the proof of Lemma 1.8 of [AB]. Suppose Y and Y′ are quadratic spaces of dimension m with signatures (p, q) and (p′, q′) with p, q, p′ and q′ even. Assume π ∈ ℛ_{Y} (Sp(n, ℝ)) ∩ ℛ_{Y′}(Sp(n, ℝ)). We must show that Y ≅ Y′, i.e., p = p′ and q = q′. We have Hom_{(𝔤_{1},K_{1})}(ω_{Y,n}, π) ≠ 0 and Hom_{(𝔤_{1},K_{1})}(ω_{Y′,n}, π) ≠ 0; as in the proof of Lemma 1.4 this implies Hom_{(𝔤_{1},K_{1})}(ω_{Z,n}, (π ⊗ π^{∨})|_{Δ(𝔤_{1},K_{1})}) ≠ 0, where Z = Y ⊥ −Y′, and −Y′ is the quadratic space with same space as Y′ but with form multiplied by −1; Z has signature (p + q′, p′ + q). Hence, 1_{Sp(n,ℝ)} ∈ ℛ_{Z}(Sp(n, ℝ)). We now use [M] to complete the proof. The representation 1_{Sp(n,ℝ)} has only one K_{1}-type, namely the trivial representation of K_{1}. As 1_{Sp(n,ℝ)} ∈ ℛ_{Z}(Sp(n, ℝ)), the trivial representation of K_{1} appears in the joint harmonics H(K_{1}, O(p + q′, ℝ) × O(p′ + q, ℝ)) for this theta correspondence (see I.1 and the second paragraph of II.1 of [M]). By Corollaire I.4 of [M], which computes the representations of K_{1} occurring in the joint harmonics, p + q′ = p′ + q; since p + q = p′ + q′, we have p = p′ and q = q′. □

We now recall the extended Weil representation which will be used to define the theta correspondence for similitudes; see [R1] for references. Define

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

The Weil representation ω of Sp(n, F) × O(X, F) on L^{2}(X^{n}) extends to a unitary representation of R via

ω(g, h)φ = |λ(h)|^{− \frac{mn}{4}} ω(g_{1}, 1)(φ ∘ h^{−1}),

where

g_{1} = g [ \begin{matrix}1 & 0 \\ 0 & λ(g) \end{matrix} ]^{−1} ∈ Sp(n, F).

Evidently, the group of elements (t, t) = (t·1, t·1) for t ∈ F^{×} is contained in the center of R, and we have ω(t, t)φ = χ_{X}(t)^{n}φ for φ ∈ L^{2}(X^{n}) and t ∈ F^{×}. If F is nonarchimedean, then the extended Weil representation preserves 𝒮(X^{n}); when F is nonarchimedean, by ω we shall often mean ω acting on 𝒮(X^{n}). Suppose F = ℝ; then ω extended to R also preserves 𝒮(X^{n}), but only at the level of Harish-Chandra modules. We need definitions. As a standard maximal compact subgroup K of GSp(n, ℝ) take the group generated by K_{1} and the order two element

k_{0} = [ \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ] .

The Lie algebra 𝔤 = 𝔤𝔰𝔭(n, ℝ) of GSp(n, ℝ) is the direct sum of its center ℝ and 𝔤_{1} = 𝔰𝔭(n, ℝ). If p ≠ q, then any maximal compact subgroup of GO(X, ℝ) is a maximal compact subgroup of O(X, ℝ), and we let J denote the subgroup J_{1} from above. Suppose p = q. Then every maximal compact subgroup of GO(X, ℝ) contains a unique maximal compact subgroup of O(X, ℝ) as a subgroup of index two, and any maximal compact subgroup of O(X, ℝ) is contained in a unique maximal compact subgroup of GO(X, ℝ) as a subgroup of index two. As a maximal compact subgroup for GO(X, ℝ) we take the maximal compact subgroup J = J(X^{+}, X^{−}) containing J_{1} = J_{1}(X^{+}, X^{−}). To get a coset representative j_{0} for the nontrivial coset of J_{1} in J, let i : X^{+} → X^{−} be an isomorphism of ℝ vector spaces such that (i(x^{+}), i(x^{+})) = −(x^{+}, x^{+}) for x^{+} ∈ X^{+} and using X = X^{+} ⊥ X^{−} set

j_{0} = [ \begin{matrix}0 & i^{−1} \\ i & 0 \end{matrix} ] .

The Lie algebra 𝔥 = 𝔤𝔬(n, ℝ) of GO(X, ℝ) is the direct sum of its center ℝ and 𝔥_{1} = 𝔬(n, ℝ). The group R is a real reductive group containing (Sp(n, ℝ) × O(X, ℝ)){(t, t) : t ∈ ℝ^{×}} as an open subgroup of index one if p ≠ q, and index two if p = q. As a maximal compact subgroup L of R we take L = K_{1} × J_{1} if p ≠ q; if p = q, then we take L to be generated by K_{1} × J_{1} and (k_{0}, j_{0}). The Lie algebra 𝔯 of R is the set of pairs (x, y) ∈ 𝔤 × 𝔥 such that x = z + x_{1} and y = z + y_{1} for some z ∈ ℝ, x_{1} ∈ 𝔤_{1} and y_{1} ∈ 𝔥_{1}. The space 𝒮(X^{n}) is evidently closed under the action of ω restricted to L and 𝔯. The (𝔤_{1} × 𝔥_{1}, K_{1} × J_{1}) module 𝒮(X^{n}) thus extends to an (𝔯, L) module, which we will also denote by ω.

Before discussing the theta correspondence for similitudes it will be useful to describe the relationship between the extended Weil representations for similar quadratic spaces, and what happens to the extended Weil representation when the additive character is changed. For λ ∈ F^{×} and g ∈ GSp(n, F) write

g^{[λ]} = [ \begin{matrix}1 & 0 \\ 0 & λ \end{matrix} ] g [ \begin{matrix}1 & 0 \\ 0 & λ \end{matrix} ]^{−1} .

1.6 Lemma. Let X′ another quadratic space over F, and suppose t : X → X′ is a similitude with similitude factor λ. Let ω′ be the Weil representation of R_{X′,n} on L^{2}(X′^{n}). Then

ω(g, h)(φ′ ∘ t) = [ω′(g^{[λ]}, tht^{−1})φ′)] ∘ t

for (g, h) ∈ R_{X,n} and φ′ ∈ L^{2}(X′^{n}).

Proof. By the formulas for the Weil representation the statement holds for g ∈ Sp(n, F) and h = 1, up to a factor α(g) in the fourth roots of unity μ_{4}. The function α : Sp(n, F) → μ_{4} is a character. The only normal subgroups of Sp(n, F) are {±1} and Sp(n, F); α must be trivial. It is now easy to check that the formula holds for all (g, h) ∈ R. □

In the next result the dependence of ω on ψ is indicated by a subscript. Its proof is similar to that of Lemma 1.6.

1.7 Lemma. Let ψ′ be another nontrivial unitary character of F. Let a ∈ F^{×} be such that ψ′(t) = ψ(at) for t ∈ F. Then there is an isomorphism

(ω_{ψ′}, L^{2}(X^{n})) ⟶^{∼} (([ \begin{matrix}1 & 0 \\ 0 & ϵ \end{matrix} ] , 1) · ω_{ψ}, L^{2}(X^{n}))

of representations of R, where ϵ = a if F is nonarchimedean and ϵ = sign(a) if F = ℝ. If F is nonarchimedean, the isomorphism is the identity map; if F = ℝ, the isomorphism sends φ′ to φ, where φ(x) = φ′(\sqrt{|a|}^{−1}x). This isomorphism maps 𝒮_{ψ′}(X^{n}) onto 𝒮_{ψ}(X^{n}) (the subscripts ψ and ψ′ are relevant when F = ℝ).

With this preparation, we recall the theta correspondence for similitudes from [R1]. In analogy to the case of isometries, we ask when does Hom_{R}(ω, π⊗σ) ≠ 0 for π ∈ Irr(GSp(n, F)) and σ ∈ Irr(GO(X, F)) define the graph of a bijection between appropriate subsets of Irr(GSp(n, F)) and Irr(GO(X, F))? In considering this, two initial observations come to mind. First, R only involves GSp(n, F)^{+}, the subgroup of GSp(n, F) (of at most index two) of g ∈ GSp(n, F)^{+} with λ(g) ∈ λ(GO(X, F)); thus, at first it might be better to look at representations of GSp(n, F)^{+} instead of GSp(n, F). Second, there should be a close relationship between Hom_{R}(ω, π ⊗ σ) ≠ 0 and Hom_{Sp(n,F)×O(X,F)}(ω, π_{1} ⊗ σ_{1}) ≠ 0 for π_{1} and σ_{1} irreducible constituents of π|_{Sp(n,F)} and σ|_{O(X,F)}, respectively. The basic result that builds on these remarks is Lemma 4.2 of [R1]. It asserts that if π ∈ Irr(GSp(n, F)^{+}), σ ∈ Irr(GO(X, F)) and Hom_{R}(ω, π ⊗ σ) ≠ 0, then

π|_{Sp(n,F)} = m · π_{1} ⊕ ⋯ ⊕ m · π_{M}, σ|_{O(X,F)} = m′ · σ_{1} ⊕ ⋯ ⊕ m′ · σ_{M}

with θ(π_{i}) = σ_{i} for 1 ≤ i ≤ M and m = 1 if and only if m′ = 1. Here the π_{i} ∈ Irr(Sp(n, F)) and σ_{i} ∈ Irr(O(X, F)) are mutually nonisomorphic. Actually, [R1] considers the nonarchimedean case of odd residual characteristic, but the same proof works if F has even residual characteristic, dim X = 2n and π and σ are tempered, so that Theorem 1.2 applies, or if F = ℝ; in this case m = m′ = 1, as [GSp(n, ℝ)^{+} : ℝ^{×} Sp(n, ℝ)], [GO(X, ℝ) : ℝ^{×} O(X, ℝ)] ≤ 2 (see Table 1 in the appendix for data on GSp(n, ℝ)^{+}). With this in place, [R1] shows that the condition Hom_{R}(ω, π ⊗ σ) ≠ 0 defines the graph of a bijection between ℛ_{X}(GSp(n, F)^{+}) and ℛ_{n}(GO(X, F)), where ℛ_{X}(GSp(n, F)^{+}) is the set of π ∈ Irr(GSp(n, F)^{+}) such that π|_{Sp(n,F)} is multiplicity free and has an irreducible constituent in ℛ_{X}(Sp(n, F)) and ℛ_{n}(GO(X, F)) is similarly defined (again, this also holds if F has even residual characteristic or F = ℝ). Finally, when GSp(n, F)^{+} is proper in GSp(n, F), [R1] shows Hom_{R}(ω, π⊗σ) ≠ 0 defines the graph of a bijection between suitable subsets of Irr(GSp(n, F)) and Irr(GO(X, F)) provided m ≤ 2n and the relevant case of Conjecture 1.3 holds. The idea is that if [GSp(n, F) : GSp(n, F)^{+}] = 2, then X has a certain companion nonisometric quadratic space X′ with the same dimension and discriminant (this determines X′ if F is nonarchimedean; if F = ℝ, then X′ is the quadratic space of signature (q, p)). When it holds and m ≤ 2n, Conjecture 1.3 implies that together the two theta correspondences between GSp(n, F)^{+} and GO(X, F) and between GSp(n, F)^{+} and GO(X′, F) give one theta correspondence between GSp(n, F) and GO(X, F) (which is the same as that between GSp(n, F) and GO(X′, F), using GO(X, F) = GO(X′, F)). Since [R1] explains this in somewhat different language, we recall the argument in the proof of the summary theorem below. For the statement of the theorem we need some notation. If F is nonarchimedean, define ℛ_{n}(GO(X, F)) and ℛ_{X}(GSp(n, F)^{+}) as above, and let ℛ_{X}(GSp(n, F)) be the set of π ∈ Irr(GSp(n, F)) such that some irreducible constituent of π|_{GSp(n,F)^{+}} is contained in ℛ_{X}(GSp(n, F)^{+}). If F = ℝ, let ℛ_{n}(GO(X, ℝ)) be the set of σ ∈ Irr(GO(X, ℝ)) such that σ|_{O(X,ℝ)} has an irreducible constituent in ℛ_{n}(O(X, ℝ)), and let ℛ_{X}(GSp(n, ℝ)) be the set of π ∈ Irr(GSp(n, ℝ)) such that π|_{Sp(n,ℝ)} has an irreducible constituent in ℛ_{X}(Sp(n, ℝ)). Here σ|_{O(X,ℝ)} and π|_{Sp(n,ℝ)} mean σ|_{(𝔥_{1},J_{1})} and π|_{(𝔤_{1},K_{1})}, respectively. If σ ∈ Irr(GO(X, F)) and F is nonarchimedean we say that σ is tempered if all the irreducible constituents of σ|_{GSO(X,F)} are tempered; evidently, σ is tempered if and only if the irreducible constituents of σ|_{O(X,F)} are tempered and σ has unitary central character, and this happens if and only if the irreducible constituents of σ|_{SO(X,F)} are tempered and σ has unitary central character.

1.8 Theorem. Suppose first F is real or nonarchimedean of odd residual characteristic. Then

{(π, σ) ∈ ℛ_{X}(GSp(n, F)) × ℛ_{n}(GO(X, F)) : Hom_{R}(ω, π ⊗ σ) ≠ 0}

is the graph of a bijection between ℛ_{X}(GSp(n, F)) and ℛ_{n}(GO(X, F)), and

dim_{ℂ} Hom_{R}(ω, π ⊗ σ) ≤ 1

for π ∈ ℛ_{X}(GSp(n, F)) and σ ∈ ℛ_{n}(GO(X, F)), in the following cases: (1) F is nonarchimedean and d = 1;

(2) F is nonarchimedean, d ≠ 1, and m ≤ n + 2;

(3) F = ℝ and p = q;

(4) F = ℝ, p ≠ q, p and q are even, and p + q ≤ 2n. Now assume F is nonarchimedean of even residual characteristic and m = 2n. As in Theorem 1.2, let the subscript temp denote the subset of tempered elements. Then the above statement holds with ℛ_{X}(GSp(n, F))_{temp} and ℛ_{n}(GO(X, F))_{temp} in place of ℛ_{X}(GSp(n, F)) and ℛ_{n}(GO(X, F)), respectively, in the following cases:

(5) d = 1 and m = 2n;

(6) d ≠ 1 and m = 2n = n + 2 = 4.

Proof. (1). Since d = 1, GSp(n, F)^{+} = GSp(n, F), and the statement follows from Theorem 4.4 of [R1]. (2). This is dealt with in [R1], but we shall briefly recall the argument for the purposes of explanation. In this case we have [GSp(n, F) : GSp(n, F)^{+}] = 2. Let g ∈ GSp(n, F) be a representative for the nontrivial coset of GSp(n, F)^{+} in GSp(n, F). As mentioned above, by Theorem 4.4 of [R1] the condition Hom_{R}(ω, π′ ⊗ σ) ≠ 0 defines a bijection between ℛ_{X}(GSp(n, F)^{+}) and ℛ_{n}(GO(X, F)), and dim_{ℂ} Hom_{R}(ω, π′ ⊗ σ) ≤ 1 for π′ ∈ ℛ_{X}(GSp(n, F)^{+}) and σ ∈ ℛ_{n}(GO(X, F)). To prove the theorem in this case, we first claim that if π′ ∈ ℛ_{X}(GSp(n, F)^{+}) and σ ∈ ℛ_{n}(GO(X, F)) are such that Hom_{R}(ω, π′ ⊗ σ) ≠ 0, then g · π′ ≇ π′ (so that π = Ind_{GSp(n,F)^{+}}^{GSp(n,F)} π′ is irreducible), and Hom_{R}(ω, π ⊗ σ) ≅ Hom_{R}(ω, π′ ⊗ σ); also, if π ∈ ℛ_{X}(GSp(n, F)) then π|_{GSp(n,F)^{+}} has two irreducible components. Let X′ be the other quadratic space of dimension m and discriminant d nonisometric to X. We may assume that X′ is obtained from X by multiplying the form on X by λ(g); then GO(X′, F) = GO(X, F) and R_{X′,n} = R = R_{X,n}. Let ω′ = ω_{X′}; by Lemma 1.6, g · ω ≅ ω′. Now since Hom_{R}(ω, π′ ⊗ σ) ≠ 0 we have Hom_{R}(g · ω, g · π′ ⊗ σ) ≠ 0, and so Hom_{R}(ω′, g · π′ ⊗ σ) ≠ 0. This gives g · π′ ∈ ℛ_{X′}(GSp(n, F)^{+}). If now π′ ≅ g · π′, then ℛ_{X}(Sp(n, F)) ∩ ℛ_{X′}(Sp(n, F)) ≠ ∅ (see Lemma 4.2 of [R1]), contradicting Lemma 1.4. Thus, g · π′ ≇ π′. Composing with the projection π → π′ gives a map Hom_{R}(ω, π⊗σ) → Hom_{R}(ω, π′⊗σ); by arguments similar to those just given, this map is a ℂ linear isomorphism. Let π ∈ ℛ_{X}(GSp(n, F)), and suppose π|_{GSp(n,F)^{+}} = π′ is irreducible. Then π′ ∈ ℛ_{X}(GSp(n, F)^{+}), and so g · π′ ≇ π′, a contradiction. This completes the proof of our claim. Using the claim, it is straightforward to prove the theorem in this case via analogous arguments.

(3) and (4). The arguments are similar to the nonarchimedean case in [R1]. In fact, they are easier since the indices of the various relevant subgroups are at most two. Thus, the analogues of the lemmas about induction and restriction from [GK] used in [R1] take on a simple form. For the convenience of the reader wishing to look closely at the arguments we present a table of data (See Table 1 in the Appendix). The p = q and p ≠ q cases should be regarded as being analogous to the d = 1 and d ≠ 1 nonarchimedean cases, respectively. In the table K^{+} is a maximal compact subgroup of GSp(n, ℝ)^{+}.

(5) and (6). The arguments are similar to those for (1) and (2) as we have the inputs Theorem 1.2 and Lemma 1.4. The proofs of section 4 of [R1] are made in an abstracted context and thus residual characteristic independent; these arguments also go through with the restriction to tempered representations. The arguments in (2) for the case [GSp(n, F) : GSp(n, F)^{+}] = 2 also work with the restriction to tempered representations. The reader wishing to go through the details should note the remark after Theorem 1.2. □

The proof of Theorem 1.8 only used Lemmas 1.4 and 1.5 when F is nonarchimedean and d ≠ 1 and F = ℝ and p ≠ q, respectively. However, Lemmas 1.4 and 1.5 have important applications when F is nonarchimedean and d = 1, and F = ℝ and p = q: see Lemma 8.4 and the proof of Proposition 4.1. We note that if π ∈ Irr(GSp(n, F)), σ ∈ Irr(GO(X, F)) and Hom_{R}(ω, π⊗σ) ≠ 0 then χ_{X}^{n} = ω_{π}ω_{σ} where ω_{π} and ω_{σ} are the central characters of π and σ, respectively. Here, if F = ℝ then the central character of π ∈ Irr(GSp(n, ℝ)) is defined by ω_{π}(e^{z}) = exp(π(z)) for z ∈ ℝ ⊂ 𝔤, and ω_{π}(−1) = π(−1), where −1 ∈ K; ω_{σ} is defined similarly. The theta correspondence for similitudes from Theorem 1.8 is independent of the choice of character ψ.

1.9 Proposition. Let ψ′ be another nontrivial unitary character of F, and let ω_{ψ′} be the Weil representation of R on 𝒮_{ψ′}(X^{n}) corresponding to ψ′ (the subscript ψ′ in 𝒮_{ψ′}(X^{n}) is relevant when F = ℝ). Let σ ∈ Irr(GO(X, F)) and π ∈ Irr(GSp(n, F)). Then Hom_{R}(ω_{ψ}, π ⊗ σ) ≠ 0 if and only if Hom_{R}(ω_{ψ′}, π ⊗ σ) ≠ 0.

Proof. This follows from Lemma 1.7. □

Assume we are in one of the cases of Theorem 1.8. We then denote the bijection between ℛ_{X}(GSp(n, F)) and ℛ_{n}(GO(X, F)) by θ:

θ : ℛ_{X}(GSp(n, F)) ⟶^{∼} ℛ_{n}(GO(X, F)),

θ : ℛ_{n}(GO(X, F)) ⟶^{∼} ℛ_{X}(GSp(n, F)).

If π ∈ ℛ_{X}(GSp(n, F)) and σ ∈ ℛ_{n}(GO(X, F)) then Hom_{R}(ω, π ⊗ σ) ≠ 0 if and only if θ(π) = σ and θ(σ) = π; if F has even residual characteristic we use ℛ_{X}(GSp(n, F))_{temp} and ℛ_{n}(GO(X, F))_{temp}. If σ ∈ ℛ_{n}(GO(X, F)) we say that σ occurs in the theta correspondence with GSp(n, F); similarly, if π ∈ ℛ_{X}(GSp(n, F)) we say that π occurs in the theta correspondence with GO(X, F). The above definition of θ is not quite compatible with the global definition; a contragredient must be introduced. If π is a cuspidal automorphic representation of GSp(X, 𝔸), the global theta lift Θ(π) is nonzero and cuspidal, and Theorem 1.8 applies at every place, then π_{v}^{∨} ∈ ℛ_{X}(GSp(n, F)) for all places v of F, and Θ(π) = ⊗_{v}θ(π_{v}^{∨}) (See Section 5). However, we have the following proposition. It guarantees that if σ_{v} = θ(π_{v}^{∨}) then θ(σ_{v}^{∨}) = π_{v}.

1.10 Proposition. Let π ∈ Irr(GSp(n, F)) and σ ∈ Irr(GO(X, F)) be unitary. Then Hom_{R}(ω, π ⊗ σ) ≠ 0 if and only if Hom_{R}(ω, π^{∨} ⊗ σ^{∨}) ≠ 0. Suppose one of (1)–(6) of Theorem 1.8 holds. Then π ∈ ℛ_{X}(GSp(n, F)) if and only if π^{∨} ∈ ℛ_{X}(GSp(n, F)) and if π ∈ ℛ_{X}(GSp(n, F)), then θ(π^{∨}) = θ(π)^{∨}. Similarly, σ ∈ ℛ_{n}(GO(X, F)) if and only if σ^{∨} ∈ ℛ_{n}(GO(X, F)), and if σ ∈ ℛ_{n}(GO(X, F)) then θ(σ^{∨}) = θ(σ)^{∨}. (If F has even residual characteristic, replace ℛ_{X}(GSp(n, F)) and ℛ_{n}(GO(X, F)) by ℛ_{X}(GSp(n, F))_{temp} and ℛ_{n}(GO(X, F))_{temp}, respectively, in these statements.)

Proof. Since π and σ are unitary, there exist ℂ antilinear isomorphisms π ⟶^{∼} π^{∨} and σ ⟶^{∼} σ^{∨} intertwining the actions of GSp(n, F) and GO(X, F), respectively. It follows that there is a ℂ antilinear isomorphism π⊗σ ⟶^{∼} π^{∨}⊗σ^{∨} intertwining the action of GSp(n, F)×GO(X, F). Let \overline{ω} be the representation of R on 𝒮(X^{n}) defined by \overline{ω}(r)φ = \overline{ω(r)\overline{φ}} for r ∈ R and φ ∈ 𝒮(X^{n}). Let t : ω → π ⊗ σ be a nonzero R map; then sending φ to t(\overline{φ}) gives a nonzero ℂ antilinear R map \overline{ω} → π ⊗ σ. Composing, we get a nonzero R map \overline{ω} → π^{∨} ⊗ σ^{∨}. On the other hand, there is an R isomorphism

\overline{ω} ≅ ([ \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ] , 1) · ω.

This implies that there is a nonzero R map ω → π^{∨} ⊗ σ^{∨}. The remaining claims of the proposition follow. □

Finally, using [H], we consider how the theta correspondence for similitudes treats unramified representations. This requires some definitions. Assume F is nonarchimedean, and let ℍ be the hyperbolic plane over F. Then X ≅ ℍ ⊥ ⋯ ⊥ ℍ ⊥ X_{0}, where X_{0} is an anisotropic quadratic space over F of dimension 0, 2 or 4. In particular, if dim_{F} X_{0} = 2, then d ≠ 1 and X_{0} ≅ (E, δ N_{F}^{E}), for a quadratic extension E/F, where δ = 1 or is a representative for the nontrivial coset of F^{×}/ N_{F}^{E}(E^{×}). We say that X is unramified if either dim X_{0} = 0 or dim X_{0} = 2, E/F is unramified and δ = 1. If X is unramified, then there exists a lattice L ⊂ X which is self-dual, and if L′ is any other self-dual lattice in X, then there exists h ∈ SO(X, F) such that h(L) = L′. If dim_{F} X_{0} = 0 or dim_{F} X_{0} = 2 and E/F is unramified, we define a maximal compact subgroup J of GO(X, F) in the following way. First, if X is unramified, we let J be the stabilizer in GO(X, F) of a fixed self-dual lattice L, i.e., J is the set of k ∈ GO(X, F) such that k(L) = L. Next assume dim_{F} X_{0} = 2, E/F is unramified but δ ≠ 1. Then there exists a similitude h from X to the unramified quadratic space of the same dimension and discriminant with anisotropic component (E, N_{F}^{E}); we let J be the set of k ∈ GO(X, F) of the form h^{−1}k′h were k′ is in the maximal compact subgroup of GO(X′, F) we have already defined. The definition of J depends on choices, but any two subgroups defined by different choices are conjugate. Let K = GSp(n, 𝔒_{F}).

1.11 Proposition. Suppose F is nonarchimedean of odd residual characteristic and X is such that J is defined. Let σ ∈ ℛ_{n}(GO(X, F)) and π ∈ ℛ_{X}(GSp(n, F)) and assume Hom_{R}(ω, π ⊗ σ) ≠ 0. Then π is unramified with respect to K if and only if σ is unramified with respect to J.

Proof. By Proposition 1.9 we may assume ψ(𝔒_{F}) = 1 but ψ(π_{F}^{−1}𝔒_{F}) ≠ 1; by Lemma 1.6 we may assume X is unramified. We have K = 𝔒_{F}^{×}K_{1} ∪ k_{0}𝔒_{F}^{×}K_{1} and J = 𝔒_{F}^{×}J_{1} ∪ j_{o}𝔒_{F}^{×}J_{1}, where K_{1} = K ∩ Sp(n, F), J_{1} = J ∩ O(X, F), λ(k_{0}) = λ(j_{0}) = μ, and μ is a representative for the nontrivial coset of 𝔒_{F}^{×}/𝔒_{F}^{×2}. For some irreducible component π′ of π|_{GSp(n,F)^{+}}, we have Hom_{R}(ω, π′⊗σ) ≠ 0. As K ⊂ GSp(n, F)^{+}, it will suffice to show that σ is unramified with respect to J if and only if π′ is unramified with respect to K. By the proof of Lemma 4.2 of [R1] we can write

π′|_{Sp(n,F)} = π_{1} ⊕ ⋯ ⊕ π_{M}, σ|_{O(X,F)} = σ_{1} ⊕ ⋯ ⊕ σ_{M}

where the π_{i} ∈ Irr(Sp(n, F)) and the σ_{i} ∈ Irr(O(X, F)) are mutually nonisomorphic and σ_{i} = θ(π_{i}). Let V_{i} and W_{i} be the spaces of π_{i} and σ_{i}, respectively. Assume σ is unramified with respect to J. Let w_{1} ∈ σ be nonzero and fixed by J. Since σ|_{O(X,F)} has exactly one irreducible constituent unramified with respect to J_{1}, we may assume, say, w_{1} ∈ W_{1}. Evidently, σ(j_{0})W_{1} = W_{1}. By (b) of Theorem 7.1 of [H], π_{1} = θ(σ_{1}) is unramified with respect to K_{1}. Let v_{1} ∈ V_{1} be nonzero and fixed by K_{1}. We will show that v_{1} is in fact fixed by K, i.e., π′(k_{0})v_{1} = v_{1}. As π′|_{Sp(n,F)} has exactly one irreducible constituent unramified with respect to K_{1} we have π′(k_{0})V_{1} = V_{1}. Since V_{1}^{K_{1}} is one dimensional, π′(k_{0})v_{1} = ϵv_{1} for some ϵ ∈ {±1}. We must show ϵ = 1. Let T : ω → π′ ⊗ σ be a nonzero R map, and let p : π′ ⊗ σ → π_{1} ⊗ σ_{1} be projection. Let T_{1} = p ∘ T : ω → π_{1} ⊗ σ_{1}; this is a nonzero Sp(n, F) × O(X, F) map. Let φ ∈ ω be such that T_{1}(φ) = v_{1} ⊗ w_{1}; we may assume φ is fixed by K_{1} × J_{1}. By the top of p. 107 of [MVW], there exists a locally constant compactly supported K_{1} bi-invariant function f : Sp(n, F) → ℂ such that

φ = ∫_{Sp(n,F)} f(g)ω(g, 1)φ_{0} dg;

here φ_{0} ∈ ω is a certain element fixed by K_{1} × J_{1}. One can check that φ_{0} is also fixed by (k_{0}, j_{0}). We have T_{1}(ω(k_{0}, j_{0})φ) = ϵ(v_{1} ⊗ w_{1}). However,

ω(k_{0}, j_{0})φ = ∫_{Sp(n,F)} f(k_{0}^{−1}gk_{0})ω(g, 1)φ_{0} dg.

Let g ∈ Sp(n, F). We claim f(k_{0}^{−1}gk_{0}) = f(g). Write g = kak′ with k, k′ ∈ K_{1} and a a diagonal matrix. Since we may assume

k_{0} = [ \begin{matrix}1 & 0 \\ 0 & μ \end{matrix} ] ,

we have k_{0}^{−1}gk_{0} = k_{0}^{−1}kk_{0}ak_{0}^{−1}k′k_{0}; hence, f(k_{0}^{−1}gk_{0}) = f(a) = f(g). Thus, ω(k_{0}, j_{0})φ = φ. This implies v_{1} ⊗ w_{1} = T_{1}(φ) = ϵ(v_{1} ⊗ w_{1}), so that ϵ = 1. The implication in the other direction has a similar argument. □

2. Four dimensional quadratic spaces

In this section we recall background on four dimensional quadratic spaces X over a base field F and their similitude groups. We begin by characterizing the special similitude group GSO(X, F) of X via its even Clifford algebra. We also obtain canonical coset representatives for the nontrivial coset of GSO(X, F) in GO(X, F); these correspond to quaternion algebras over F contained in the even Clifford algebra over F, which in turn are in bijection with Galois actions on the even Clifford algebra. This leads to the concept of a quadratic quaternion algebra over F, an abstraction of the even Clifford algebra of a four dimensional quadratic space. We construct examples of four dimensional quadratic spaces from a given quadratic quaternion algebra over F and quaternion algebras over F contained in the quadratic quaternion algebra, or equivalently, Galois actions on the quadratic quaternion algebra. We prove that any four dimensional quadratic space over F is, up to similitude, one of these examples. We also describe the relationship between the examples that arise from a given quadratic quaternion algebra. To close the section, we consider four dimensional quadratic spaces over local and number fields. The material in this section is essentially well known. As some basic references we use [E], [Sch] and [Kn]. To begin, let F be a field not of characteristic two, and let (X, (·, ·)) be a four dimensional quadratic space over F. For simplicity denote the F points of X by X. Set d = disc X. Let x_{1}, x_{2}, x_{3}, x_{4} be an orthogonal basis for X. Let C be the Clifford algebra of X, let B = B(X) be the even Clifford algebra of X in C, let E = E(X) be the center of B, and let C_{1} = C_{1}(X) be the subspace of C of odd elements. Then C, B, E and C_{1} are 16, 8, 2 and 8 dimensional over F, respectively. The F algebra E is called the discriminant algebra of X and is reduced, i.e., has no nonzero nilpotent elements. Hence, E is either a field or is isomorphic to F × F; these happen when d ≠ 1 and d = 1, respectively. Let Gal(E/F) = {1, α}. Let N_{F}^{E} and T_{F}^{E} be the norm and trace from E to F defined by N_{F}^{E}(z) = zα(z) and T_{F}^{E}(z) = z + α(z), respectively. Let ∗ be the involution of C which takes a product of the x_{i} to the product of the same x_{i} in the reverse order. Clearly, ∗ preserves B and C_{1}. If x ∈ B, then x ∈ E if and only if x^{∗} = x. For x ∈ C, define N(x) = x^{∗}x. Then N(x) ∈ E for x ∈ B. We may regard X as contained in C_{1}. Evidently, X is the set of x ∈ C_{1} such that x^{∗} = x. For x ∈ X, (x, x) = N(x). Also, for z ∈ E and x ∈ C_{1} we have xz = α(z)x. To further describe the structure of B and E, suppose B is an arbitrary F algebra with center E and involution ∗ which is the identity on E. Then we say that B is a quadratic quaternion algebra over F if E is two dimensional over F and reduced, and there exists a quaternion algebra D over F contained in B such that the natural map E ⊗_{F} D → B given by z ⊗ x ↦ zx is an isomorphism of E algebras and ∗ induces the canonical involution on D. Let B be a quadratic quaternion algebra over F with center E and involution ∗. We define the norm N : B → E and trace T : B → E by N(x) = xx^{∗} = x^{∗}x and T(x) = x + x^{∗} respectively. We also define a symmetric E-bilinear form (·, ·) : B × B → E by (x, y) = T(xy^{∗})/2. This form is nondegenerate, i.e., if x ∈ B is nonzero, there exists y ∈ B such that (x, y) ≠ 0. The definition of a quadratic quaternion algebra B includes a particular quaternion algebra over F in B, but the next straightforward result shows that all the quaternion algebras over F in B have equal status.

2.1 Proposition. Let B be a quadratic quaternion algebra over F with center E and involution ∗. Let D be any quaternion algebra over F in B. The natural map E⊗_{F} D → B is an isomorphism of E algebras, and ∗ induces the canonical involution on D.

Given a quadratic quaternion algebra B as above, in general there may be infinitely many nonisomorphic quaternion algebras D over F in B. However, if E ≅ F × F, then B ≅ D × D, and any quaternion algebra over F in B is isomorphic to D.

2.2 Proposition. Let X be a four dimensional quadratic space over X. The F algebra B(X) is a quadratic quaternion algebra over F.

We characterize GSO(X, F). Write B = B(X). Define a left action of F^{×}×B^{×} on C_{1} by ρ(t, g)x = t^{−1}gxg^{∗}. This action preserves X, and a computation shows that if x ∈ X and (t, g) ∈ F^{×} × B^{×}, then N(ρ(t, g)x) = t^{−2} N_{F}^{E}(N(g)) N(x); thus, ρ(t, g) ∈ GO(X, F), with similitude factor t^{−2} N_{F}^{E}(N(g)). In fact, if (t, g) ∈ F^{×} × B^{×}, then ρ(t, g) ∈ GSO(X, F). For the following see for example V (4.6.1) of [Kn], p. 273.

2.3 Theorem. Let X be a four dimensional quadratic space over X, and write B = B(X) and E = E(X). Define an inclusion of E^{×} into F^{×} × B^{×} by a ↦ (N_{F}^{E}(a), a). Then the following sequence is exact:

1 → E^{×} → F^{×} × B^{×} ⟶^{ρ} GSO(X, F) → 1.

This theorem determines GSO(X, F). We also need to understand GO(X, F), and we now explain how to describe certain canonical coset representatives for the nontrivial coset of GSO(X, F) in GO(X, F). These coset representatives will correspond to choices of quaternion algebras over F in B. The following lemma is the key structural result for the construction of the coset representatives. It is an elaboration of a general result about Clifford algebras of even dimensional quadratic spaces (Chapter 9, Theorem 2.10 of [Sch], p. 332).

2.4 Lemma. Let X be a four dimensional quadratic space over X, and write B = B(X) and E = E(X). Let D be a quaternion algebra over F contained in B. Let D′ be the F algebra of elements of C which commute with all the elements of D. Then D′ is a quaternion algebra over F and X ∩ D′ is one dimensional and spanned by an anisotropic vector y, so that D′ = E +Ey. The map x′ ⊗ x ↦ x′x determines an isomorphism D′ ⊗_{F} D ⟶^{∼} C of F algebras. Conversely, if y ∈ X is an anisotropic vector, then the set D of elements of B commuting with y is a quaternion algebra over F in B.

The maps from the previous lemma are evidently inverses of each other; that is, there is a bijection

Quaternion algebras over F in B ⟷ Anisotropic lines in X. For the description of the nontrivial coset representatives of GSO(X, F) in GO(X, F) we also need the following. Suppose B is any quadratic quaternion algebra over F with center E with Gal(E/F) = {1, α}. Then a Galois action on B is an F -automorphism a : B → B such that a^{2} = 1 and a(zx) = α(z)a(x) for z ∈ E and x ∈ B. If a is a Galois action on B, then the fixed points of a are a quaternion algebra over F contained in B; conversely, if D is a quaternion algebra over F contained in B, and a : B → B is defined by a(z⊗x) = α(z)⊗x, then a is a Galois action on B. These two maps are inverses of each other, and establish a bijection:

Quaternion algebras over F in B ⟷ Galois actions on B. Direct computation gives the following:

2.5 Proposition. Let X be a four dimensional quadratic space over X, and write B = B(X) and E = E(X). Let D be a quaternion algebra over F contained in B, and let D′ be as in Lemma 2.4. Let # be the involution of C obtained via the isomorphism D′ ⊗ D ≅ C from the tensor product of the canonical involutions on D′ and D. Then X^{#} = X; define s : X → X by s(x) = −x^{#}. Then s ∈ O(X, F), s^{2} = 1, and det s = −1. Moreover, the following diagram commutes:

1 ⟶ E^{×} ⟶ F^{×} × B^{×} ⟶^{ρ} GSO(X, F) ⟶ 1 ^{α}↓  ^{1×a}↓  ^{conj. by s}↓ 1 ⟶ E^{×} ⟶ F^{×} × B^{×} ⟶^{ρ} GSO(X, F) ⟶ 1.

Here, a is the Galois action on B determined by D.

When F is a local field we shall deal with representations of GSO(X, F) distinguished with respect to subgroups SO(Y, F), where Y is a three dimensional subspace of X. The above development leads to a compatible characterization of such subgroups. For the exactness of the first sequence in the next result see for example [Kn], p. 264.

2.6 Proposition. Let X be a four dimensional quadratic space over X, and write B = B(X) and E = E(X). Let y ∈ X be anisotropic, and set Y = (F ·y)^{⊥} in X. Let D be the quaternion algebra over F in B corresponding to y. For g ∈ D^{×} and x ∈ Y, define ρ(g)x = gxg^{−1}. Then ρ(g) ∈ SO(Y, F) for g ∈ D^{×}, the sequence

1 → F^{×} → D^{×} ⟶^{ρ} SO(Y, F) → 1

is exact, there is a commutative diagram

1 ⟶ F^{×} ⟶ D^{×}  ⟶^{ρ} SO(Y, F) ⟶ 1
↓  ↓  ↓
1 ⟶ E^{×} ⟶ F^{×} × B^{×} ⟶^{ρ} GSO(X, F) ⟶ 1

where the inclusion of D^{×} in F^{×} ×B^{×} is given by g ↦ (N(g), g), and SO(Y, F) is included in GSO(X, F) by regarding SO(Y, F) as the stabilizer of y in GSO(X, F). Moreover, the element s from Proposition 2.5 corresponding to D is such that s(y) = y and s is multiplication by −1 on Y, so that s|_{Y} ∈ O(Y, F), with det s|_{Y} = −1.

It will be important to have some explicitly constructed four dimensional quadratic spaces, and we now reverse matters and construct such examples from a given quadratic quaternion algebra over F equipped with a Galois action. Let B be a quadratic quaternion algebra over F with center E with Gal(E/F) = {1, α}, involution ∗, and let a : B → B be a Galois action on B. Let D be the quaternion algebra over F in B corresponding to a, i.e., the fixed points of a. We let X_{a} be the set of x ∈ B such that a(x) = x^{∗}. Then X_{a} is a four dimensional vector space over F, and equipped with the symmetric bilinear form induced by the norm of B, X_{a} is a quadratic space over F. Define an explicit action of F^{×} × B^{×} on X_{a} by ρ_{a}(t, g)x = t^{−1}gxa(g)^{∗}. Then ρ_{a}(t, g) ∈ GSO(X_{a}, F) for (t, g) ∈ F^{×} × B^{×}. The relationship between the previous characterization of GSO(X_{a}, F) and the homomorphism ρ_{a} is given by the following proposition.

2.7 Proposition. Let B be a quadratic quaternion algebra over F with center E with Gal(E/F) = {1, α}, involution ∗, and let a : B → B be a Galois action on B. Then the sequence

1 → E^{×} → F^{×} × B^{×} ⟶^{ρ_{a}} GSO(X_{a}, F) → 1,

is exact, where the inclusion of E^{×} is defined by z ↦ (N_{F}^{E}(z), z). There exists a unique F algebra isomorphism B(X_{a}) ⟶^{∼} B sending E(X_{a}) onto E so that the diagram

1 ⟶ E(X_{a})^{×} ⟶ F^{×} × B(X_{a})^{×} ⟶^{ρ} GSO(X_{a}, F) → 1
≀↓  ^{≀}↓  ^{id}↓
1 ⟶ E^{×} ⟶ F^{×} × B^{×}  ⟶^{ρ_{a}} GSO(X_{a}, F) → 1

commutes. The map defined by x ↦ a(x) = x^{∗} maps X_{a} onto X_{a}, and is the element s ∈ O(X_{a}, F) from Proposition 2.5 associated to the quaternion algebra over F in B(X_{a}) corresponding to a.

Explicit quadratic quaternion algebras equipped with Galois actions may constructed as follows. Let E be a two dimensional reduced F algebra, so that E is either a quadratic extension of F, or E ≅ F × F. Let Gal(E/F) = {1, α}, and let D be a quaternion algebra over F with canonical involution ∗. Set B_{D,E} = E ⊗_{F} D, endow B_{D,E} with the involution defined by (z ⊗ x)^{∗} = z ⊗ x^{∗}, and define a = a(D, E) : B_{D,E} → B_{D,E} by a(z ⊗ x) = α(z) ⊗ x. Clearly B_{D,E} is a quadratic quaternion algebra over F, and a is a Galois action on B_{D,E}; we will write X_{D,E} = X_{a}. To be even more concrete, let d ∈ F^{×}/F^{×2}. If d ≠ 1, let E_{d} = F (\sqrt{d}); if d = 1, let E_{d} = F × F. We write B_{D,d} = B_{D,E_{d}} and X_{D,d} = X_{D,E_{d}}. Evidently disc X_{D,d} = d. Assume further d = 1. Then there is a canonical isomorphism D×D ⟶^{∼} B_{D,1} of F algebras. With respect to this isomorphism, a is given by a(x, x′) = (x′, x), and ∗ is given by (x, x′)^{∗} = (x^{∗}, x′^{∗}). Thus, X_{D,1} is the set of pairs (x, x^{∗}) for x ∈ D, which can be identified with D. With respect to these identifications, ρ_{a}(t, (g, g′))x = t^{−1}gxg′^{∗} for t ∈ F^{×}, x ∈ D, and g, g′ ∈ D^{×}. Before turning to specific fields we address two natural questions. First, if X is an arbitrary four dimensional quadratic space over F, when can X be related to an X_{D,E}?

2.8 Proposition. Let X be a four dimensional quadratic space over F and write B = B(X) and E = E(X). There exists a quaternion algebra D over F in B and a similitude T : X → X_{D,E} so that

1 ⟶ E^{×} ⟶ F^{×} × B^{×} ⟶^{ρ} GSO(X, F) ⟶ 1
id↓  ^{≀}↓  ↓_{T ·T^{−1}}
1 ⟶ E^{×} ⟶ F^{×} × B_{D,E}^{×} ⟶^{ρ_{a(D,E)}} GSO(X_{D,E}, F) ⟶ 1

commutes, and the element s ∈ O(X, F) corresponding to D from Proposition 2.5 is mapped to the element of O(X_{D,E}, F) defined by x ↦ a(x) = x^{∗}, where a = a(D, E). If X represents 1, then we may further choose T to be an isometry. Conversely, if X is isometric to X_{D,E} for some D, then X represents 1.

Given a quadratic quaternion algebra over F, what is the relationship between the X_{a} for different Galois actions a on the quadratic quaternion algebra? The main ingredient for the following is the Skolem-Noether theorem.

2.9 Proposition. Let B be a quadratic quaternion algebra over F with center E, let Gal(E/F) = {1, α}, and let a and a′ be Galois actions on B. There exists u ∈ B^{×}, uniquely determined up to multiplication by elements of E^{×}, such that a′(x) = u^{−1}a(x)u for x ∈ B. We have ua(u) = ua′(u) ∈ F^{×}. Let μ = ua(u) = ua′(u). Then u can be chosen so that N(u) = μ; choose such a u. The map T : X_{a} → X_{a′} given by T (x) = xu is a well-defined similitude with similitude factor λ(T) = μ. The diagram

1 ⟶ E^{×} ⟶ F^{×} × B^{×} ⟶^{ρ_{a}} GSO(X_{a}, F) ⟶ 1
id↓  ^{id}↓  ↓_{T ·T^{−1}}
1 ⟶ E^{×} ⟶ F^{×} × B^{×} ⟶^{ρ_{a′}} GSO(X_{a′}, F) ⟶ 1

commutes.

To close this section we consider choices of F. Suppose F is nonarchimedean of characteristic zero. Let d ∈ F^{×}/F^{×2}. Up to isometry, there are two four dimensional quadratic spaces of discriminant d; these are distinguished by their Hasse invariant. Both spaces represent 1. One space is isometric to X_{M_{2×2},d}, where M_{2×2} = M_{2×2}(F) is the quaternion algebra of 2 × 2 matrices over F; the other is isometric to X_{D_{ram},d}, where D_{ram} is the division quaternion algebra over F. These spaces have Hasse invariant ϵ(d) and −ϵ(d), respectively, where ϵ(d) = (−1, −d)_{F}. If d = 1, then X_{M_{2×2},1} is isometric to M_{2×2}(F) equipped with the determinant, and X_{D_{ram},1} is isometric to D_{ram} equipped with the norm; see the remarks before Proposition 2.8. Suppose d ≠ 1. Then X_{M_{2×2},d} and X_{D_{ram},d} are both isotropic. Also, B_{M_{2×2},d} and B_{D_{ram},d} are both isomorphic to M_{2×2}(E_{d}). Explicitly, let δ be a representative for the nontrivial coset of F^{×}/ N_{F}^{E_{d}}(E_{d}^{×}). Then we can take

D_{ram} = \{[ \begin{matrix}e & fδ \\ α(f) & α(e) \end{matrix} ] : e, f ∈ E_{d}\} ⊂ M_{2×2}(E_{d}).

The Galois actions a = a(M_{2×2}, E_{d}) and a′ = a(D_{ram}, E_{d}) on M_{2×2}(E_{d}) corresponding to M_{2×2}(F) and D_{ram} are given by

(2.1) a([ \begin{matrix}e & f \\ g & h \end{matrix} ]) = [ \begin{matrix}α(e) & α(f) \\ α(g) & α(h) \end{matrix} ] and a′([ \begin{matrix}e & f \\ g & h \end{matrix} ]) = [ \begin{matrix}α(h) & δα(g) \\ α(f)/δ & α(e) \end{matrix} ] ,

respectively, and X_{M_{2×2},d} and X_{D_{ram},d} are the set of elements in M_{2×2}(E_{d})

[ \begin{matrix}e & f\sqrt{d} \\ g\sqrt{d} & α(e) \end{matrix} ] and [ \begin{matrix}f & −δe \\ α(e) & g \end{matrix} ] ,

respectively, for e ∈ E_{d} and f, g ∈ F. The element u from Proposition 2.9 can be taken to be

\sqrt{d} [ \begin{matrix}0 & δ \\ 1 & 0 \end{matrix} ] .

Evidently, if the residual characteristic of F is odd and E_{d}/F is unramified, then X_{M_{2×2},d} is unramified but X_{D_{ram},d} is not. The quadratic spaces X_{M_{2×2},d} and X_{D_{ram},d} have isomorphic similitude groups, and from the point of view of the theta correspondence for similitudes, they are grouped together. The two quadratic spaces with discriminant 1, however, do not have isomorphic similitude groups and and are distinct from the point of view of the theta correspondence for similitudes. See the remarks before Theorem 1.8 and the proof of Theorem 1.8. Suppose F = ℝ. Let d ∈ ℝ^{×}/ℝ^{×2}. If d = 1, then up to isometry there are three four dimensional quadratic spaces of discriminant 1, with signatures (4, 0), (2, 2) or (0, 4). The quadratic space with signature (4, 0) is X_{D_{ram},1}; the ramified quaternion algebra D_{ram} over ℝ is the Hamilton quaternion algebra. The quadratic space with signature (2, 2) is X_{M_{2×2},1} where M_{2×2} = M_{2×2}(ℝ). Finally, the quadratic space with signature (0, 4) is not of the form X_{D,1}. However, as predicted by Proposition 2.8, there is an intertwining similitude with the space X_{D_{ram},1}: the quadratic space with signature (0, 4) can be taken to be X_{D_{ram},1} with form multiplied by −1. Then the intertwining similitude is just the identity function. If d = −1, then up to isometry there are two quadratics spaces of discriminant −1, with signatures (1, 3) or (3, 1). The quadratic space with signature (3, 1) is X_{M_{2×2},−1}, while the quadratic space with signature (1, 3) is X_{D_{ram},−1}. From the point of view of the theta correspondence for similitudes, the spaces with signature (4, 0) and (0, 4) are grouped together, the spaces with signature (3, 1) and (1, 3) are grouped together, and the space of signature (2, 2) is not grouped with another four dimensional quadratic space. When F = ℝ there are further exact sequences. Let X be a four dimensional quadratic space over ℝ, with even Clifford algebra B; let E be the center of B. We regard F = ℝ, E and B as the Lie algebras of F^{×} = ℝ^{×}, E^{×} and B^{×}, respectively. We take the Lie algebra 𝔤𝔰𝔬(X, ℝ) of GSO(X, ℝ) to be the subalgebra of h ∈ End_{ℝ} X for which there exists a λ ∈ ℝ such that (hx, x′) + (x, hx′) = λ(x, x′) for x, x′ ∈ X; then λ = tr(h)/2. Define an action of ℝ × B on X by ρ(r, h)x = −rx + hx + xh^{∗}, and an inclusion of E into ℝ × B by b ↦ (T_{F}^{E}(b), b). By Theorem 2.3,

(2.2)  0 → E → ℝ × B ⟶^{ρ} 𝔤𝔰𝔬(X, ℝ) → 0

is an exact sequence of Lie algebras. Any two maximal compact subgroups of GSO(X, ℝ) are conjugate. Let J_{0} be a maximal compact subgroup of GSO(X, ℝ). Then there exists a unique maximal compact subgroup K_{B} of B^{×} such that ρ({±1} × K_{B}) = J_{0}. The normalizer of J_{0} is ℝ^{×}J_{0}, and J_{0} is contained in a unique maximal compact subgroup of GO(X, ℝ). There is an exact sequence

(2.3)  1 → K_{B} ∩ E^{×} → {±1} × K_{B} ⟶^{ρ} J_{0} → 1.

Suppose that y ∈ X is anisotropic and Y and D are as in Proposition 2.6. We take the Lie algebra of 𝔰𝔬(Y, ℝ) of SO(Y, ℝ) to be the subalgebra of h ∈ End_{ℝ} Y such that (hx, x′) + (x, hx′) = 0 for x, x′ ∈ Y. We regard D as the Lie algebra of D^{×}, and define an action of D on X by ρ(h)x = hx − xh. By Proposition 2.6 there is an exact sequence

0 → ℝ → D → 𝔰𝔬(Y, ℝ) → 0,

and a commutative diagram

0 ⟶ ℝ ⟶ D ⟶^{ρ} 𝔰𝔬(Y, ℝ) ⟶ 0
↓  ↓  ↓
0 ⟶ E ⟶ ℝ × B ⟶^{ρ} 𝔤𝔰𝔬(X, ℝ) ⟶ 0

where D is included in ℝ × B via h ↦ (T(h), h) and 𝔰𝔬(Y, ℝ) is included in 𝔤𝔰𝔬(X, ℝ) by setting the elements of 𝔰𝔬(Y, ℝ) to be 0 on ℝ·y. Any two maximal compact subgroups of SO(Y, ℝ) are conjugate. Let J_{Y} be a maximal compact subgroup of SO(Y, ℝ). Then there exists a unique maximal compact subgroup K_{D} of D^{×} such that J_{Y} = ρ(K_{D}), and J_{Y} is contained in a unique maximal compact subgroup J_{0} = ρ({±1} × K_{B}) of GSO(X, ℝ). Also, K_{D} ⊂ K_{B}, the diagram

1 ⟶ {±1} ⟶  K_{D}  ⟶^{ρ} J_{Y} ⟶ 1
↓  ↓  ↓
1 ⟶ K_{B} ∩ E^{×} ⟶ {±1} × K_{B} ⟶^{ρ} J_{0} ⟶ 1

commutes, and the element s ∈ O(X, ℝ) from Proposition 2.6 normalizes J_{Y} and J_{0}. Conversely, if J_{0} = ρ({±1} × K_{B}) is a maximal compact subgroup of GSO(X, ℝ) there exists an anisotropic y ∈ X and a maximal compact subgroup J_{Y} = ρ(K_{D}) ⊂ SO(Y, ℝ) such that J_{Y} ⊂ J_{0}; in particular, the unique maximal compact subgroup of GSO(X, ℝ) which contains J_{0} is generated by J_{0} and s. Finally, suppose F is a number field with adeles 𝔸, X is a four dimensional quadratic space over F, B is the even Clifford algebra of X, and E is the center of B. Using Theorem 2.3 one can show that the sequence

1 → 𝔸_{E}^{×} → 𝔸^{×} × B^{×}(𝔸) ⟶^{ρ} GSO(X, 𝔸) → 1

is exact; we identify E^{×}(𝔸) and 𝔸_{E}^{×}. Similarly, if B is a quadratic quaternion algebra over F with center E, and a is a Galois action on B, then the sequence

1 → 𝔸_{E}^{×} → 𝔸^{×} × B^{×}(𝔸) ⟶^{ρ_{a}} GSO(X_{a}, 𝔸) → 1

is exact. In addition, we have the following useful observation. Suppose D and D′ are quaternion algebras over F, and E is a two dimensional reduced algebra over F. Let S_{D,E} be the set of places v of F such that D_{v} is ramified and v splits in E; if E ≅ F × F, we will say that every place of F splits in E. Define S_{D′,E} similarly. Evidently, if S_{D,E} = S_{D′,E}, then B_{D,E} ≅ B_{D′,E} as E algebras. Thus, if S_{D,E} = S_{D′,E}, then by Proposition 2.9 there exists an intertwining similitude from X_{D,E} to X_{D′,E}.

3. Local theta lifts for dim X = 2n = 4

In this section we describe what irreducible representations of GO(X, F) occur in the theta correspondence with GSp(2, F) for X a four dimensional quadratic space over a local field F. This is needed to define local L-packets for GSp(2, F) in the next section. The description below involves distinguished representations, and was given in [R2] when F is a local field of characteristic zero with odd residual characteristic; we also do the even residual characteristic and real cases. Fix the following notation. Let F be a local field of characteristic zero, with F = ℝ if F is archimedean. Let X be a four dimensional quadratic space over F; write X for the F points of X. Let d = disc X. As in Section 2, let B be the even Clifford algebra of X, and let E be the center of B. Let s ∈ O(X, F) be an element as in Proposition 2.5, so that s^{2} = 1, det s = −1, and s is a representative for the nontrivial coset of GSO(X, F) in GO(X, F). Suppose that F = ℝ. Fix a maximal compact subgroup K_{B} of B^{×}, and let J_{0} = ρ({±1} × K_{B}), a maximal compact subgroup of GSO(X, ℝ). As explained in the penultimate paragraph of Section 2, we may assume that s normalizes J_{0}, so that the subgroup J generated by J_{0} and s is a maximal compact subgroup of GO(X, ℝ). As usual, by Irr(B^{×}) we mean the set of equivalence classes of irreducible (B, K_{B}) modules, where B is regarded as the Lie algebra of B^{×}. If τ ∈ Irr(B^{×}), the central character ω_{τ} : E^{×} → ℂ^{×} of τ is defined by ω_{τ} (e^{z}) = exp(τ(z)) for z ∈ E ⊂ B = Lie(B^{×}), and ω_{τ} (ϵ) = τ(ϵ) for ϵ ∈ E^{×} ∩ K_{B}. Using the exact sequences of Section 2, we can describe representations of GSO(X, F) in terms of representations of B^{×}. Let Irr_{f}(F^{×} × B^{×}) be the set of pairs (χ, τ), where τ ∈ Irr(B^{×}) is such that ω_{τ} is Galois invariant, and χ is a quasi-character of F^{×} such that ω_{τ} = χ ∘ N_{F}^{E}. The exact sequences from Theorem 2.3, (2.2) and (2.3) give a bijection

Irr_{f}(F^{×} × B^{×}) ⟶^{∼} Irr(GSO(X, F)), (χ, τ) ↦ π(χ, τ).

If F is nonarchimedean, π(χ, τ) has the same space as τ, and is defined by π(χ, τ)(ρ(t, g)) = χ(t)^{−1}τ(g). Suppose F = ℝ, and let (χ, τ) ∈ Irr_{f}(ℝ^{×} × B^{×}). Since ω_{τ} is Galois invariant, it follows that there exists a unique ℝ linear map l_{τ} : ℝ → ℂ such that τ(z) = l_{τ} (T_{ℝ}^{E}(z)) for z ∈ E ⊂ Lie(B^{×}). We have χ(e^{x}) = exp l_{τ} (x) for x ∈ ℝ. Then π(χ, τ) has the same space as τ, and π(χ, τ) is defined by π(χ, τ)(ρ(ϵ, k)) = χ(ϵ)^{−1}τ(k) for ρ(ϵ, k) ∈ J_{0}, and by π(χ, τ)(ρ(r, h)) = −l_{τ} (r) + τ(h) for ρ(r, h) ∈ 𝔤𝔰𝔬(X, ℝ). The central character of π(χ, τ) is χ. In addition, if X is of the form X_{a} for some Galois action a on a quadratic quaternion algebra B (see Section 2), then it may be convenient to write π = π(χ, τ) with respect to the first exact sequence from Proposition 2.7. By Proposition 2.7, the difference between using the exact sequences from Proposition 2.7 and Theorem 2.3 is inessential.

We describe representations of GO(X, F) via representations of GSO(X, F). Let π ∈ Irr(GSO(X, F)). If the induced representation of π to GSO(X, F) is irreducible, we say that π is regular, and write π^{+} for the induced representation. Here, if F = ℝ, Ind_{GSO(X,ℝ)}^{GO(X,ℝ)} π is the (𝔤𝔬(X, ℝ), J) = (𝔤𝔰𝔬(X, ℝ), J) module with space π ⊕ π and action

π^{+}(k)(w ⊕ w′) = π(k)w ⊕ π(sks)w′, k ∈ J_{0}, π^{+}(s)(w ⊕ w′) = w′ ⊕ w,

and Lie algebra action

π^{+}(X)(w ⊕ w′) = π(X)w ⊕ π(Ad(s)X)w′, X ∈ 𝔤𝔰𝔬(X, ℝ).

If π is not regular, we say that π is invariant. If π is invariant, then s · π ≅ π and π extends to exactly two representations of GO(X, F); if F = ℝ, by s · π we mean the (𝔤𝔰𝔬(X, ℝ), J_{0}) module with same space as π and action defined by (s · π)(k)w = π(sks)w for k ∈ J_{0} and w ∈ π and (s · π)(X)w = π(Ad(s)X)w for X ∈ 𝔤𝔰𝔬(X, ℝ) and w ∈ π. Before we can describe what representations of GO(X, F) occur in the theta correspondence with GSp(2, F) we must be able to adequately tell apart the two extensions of an invariant representation to GO(X, F). To do so we use distinguished representations. Let π ∈ Irr(GSO(X, F)) be invariant. We say that π is distinguished if there exists an anisotropic vector y ∈ X such that Hom_{SO(Y,F)}(π, 1) ≠ 0, and if d ≠ 1, then Y is isotropic. Here, Y = (F · y)^{⊥}, as in Proposition 2.6, and 1 is the trivial representation of SO(Y, F), i.e., the representation with space ℂ and trivial action. In the case F = ℝ more comments are required. Let y ∈ X be anisotropic, and let Y = (F · y)^{⊥}. Let J_{Y} be a maximal compact subgroup of SO(Y, ℝ). Then as mentioned in Section 2, J_{Y} is contained in a unique maximal compact subgroup J′_{0} of GSO(X, ℝ). Since J′_{0} is conjugate to J_{0}, we may regard the (𝔤𝔰𝔬(X, ℝ), J_{0}) module π as a (𝔤𝔰𝔬(X, ℝ), J′_{0}) module, and by restriction, as an (𝔰𝔬(Y, ℝ), J_{Y}) module. Then we say that π is distinguished if for some y, Hom_{(𝔰𝔬(Y,ℝ),J_{Y} )}(π, 1) ≠ 0, and if d ≠ 1, then Y is isotropic. It is easy to verify that the nonvanishing of this homomorphism space does not depend on the choice of maximal compact subgroup of SO(Y, ℝ) or element of GSO(X, ℝ) used to conjugate J′_{0} into J_{0} (use that the normalizer of J_{0} is ℝ^{×}J_{0}). Also, π is distinguished with respect to all anisotropic y if and only if it is distinguished with respect to one anisotropic y. If F is nonarchimedean, then this was pointed out in [R2]; if F = ℝ it follows by a similar argument.

3.1 Proposition. If F = ℝ assume d = 1. Let π ∈ Irr(GSO(X, F)). Assume π is invariant. Then for all anisotropic y ∈ X such that Y = (F · y)^{⊥} is isotropic if d ≠ 1, dim_{ℂ} Hom_{SO(Y,F)}(π, 1) ≤ 1.

Proof. This was proven in Proposition 4.1 of [R2] if F is nonarchimedean. Suppose F = ℝ. Since the homomorphism spaces for different anisotropic y are all isomorphic, it suffices to show this for one y. As d = 1, we have B ≅ D ×D for some quaternion algebra D over ℝ. Identify B with D ×D, and let y ∈ X be an anisotropic vector such that the line ℝ · y corresponds to ΔD, where ΔD consists of the (x, x) ∈ B with x ∈ D (see Section 2). Let K_{D} be a maximal compact subgroup of D^{×}; then ΔK_{D} is a maximal compact subgroup of ΔD^{×}, ΔK_{D} ⊂ K_{D} × K_{D}, and J_{Y} = ρ(K_{D}) ⊂ ρ({±1} × K_{D} × K_{D}). Write π = π(χ, τ), with τ ≅ τ_{1} ⊗ τ_{2}, τ_{1}, τ_{2} ∈ Irr(D^{×}), ω_{τ_{1}} = ω_{τ_{2}} = χ. Since π is invariant, τ_{1} ≅ τ_{2}. We have Hom_{SO(Y,ℝ)}(π, 1) ≅ Hom_{(ΔD,ΔK_{D})}(τ, χ ∘ N) ≅ Hom_{(ΔD,ΔK_{D})}(τ_{1} ⊗ τ_{1}^{∨}, 1). This space is one dimensional. □

As we shall see in the theorem below, we can now tell the two extensions of invariant representations apart to an extent sufficient for our purposes. Suppose π ∈ Irr(GSO(X, F)) is distinguished with respect to an anisotropic y ∈ X, with d = 1 if F = ℝ. Since dim_{ℂ} Hom_{SO(Y,F)}(π, 1) = 1 by Proposition 3.1, it follows that for exactly one extension π^{+} of π to GO(X, F) we have Hom_{O(Y,F)}(π^{+}, 1) ≠ 0. Denote the other extension of π to GO(X, F) by π^{−}. The definitions of π^{+} and π^{−} do not depend on the choice of y. Before characterizing ℛ_{2}(GO(X, F)) we require require two more results.

3.2 Lemma. Let F = ℝ; assume d = 1. Then Hom_{SO(X,ℝ)}(ω, π_{1}) ≠ 0 for π_{1} ∈ Irr(SO(X, ℝ)).

Proof. If the signature of X is (2, 2), this follows from (3.6.10) of [P]. If the signature of X is (4, 0) or (0, 4) this follows by (6.12) of [KV]. □

3.3 Proposition. The elements of Irr(GO(X, F)) have multiplicity free restrictions to O(X, F).

Proof. If F = ℝ then the restriction of any element of Irr(GO(X, ℝ)) is multiplicity free as [GO(X, ℝ) : ℝ^{×} O(X, ℝ)] ≤ 2. If d = 1 and F is nonarchimedean then this is Lemma 7.2 of [HPS]. The case d ≠ 1 and F nonarchimedean remains. If F is of odd residual characteristic then [GO(X, F) : F^{×} O(X, F)] = [N_{F}^{E}(E^{×}) : F^{×2}] = 2 so the proposition follows from Lemma 2.1 of [GK]. We now give an argument for both the even and odd residual characteristic cases. There are two four dimensional quadratic spaces over F of discriminant d. By Proposition 2.9 there is a similitude between them; thus, it suffices to prove the result for one of them. We take X = X_{M_{2×2},d}. Using Proposition 3.2 of [R2] it is easy to verify that the finite dimensional, i.e., one or two dimensional, elements of Irr(GO(X, F)) have multiplicity free restrictions to O(X, F). To complete the proof it will suffice to show that for infinite dimensional π ∈ Irr(GSO(X, F)), the representation σ = Ind_{GSO(X,F)}^{GO(X,F)} π (which may be reducible) has a multiplicity free restriction to O(X, F). Let π ∈ Irr(GSO(X, F)) and using the first exact sequence from Proposition 2.7 write π = π(χ, τ) where τ ∈ Irr(GL(2, E)) and χ is a quasi-character of F^{×} such that χ∘N_{F}^{E} = ω_{τ}; here and below E = E_{d}. We make take s to be given by s(x) = a(x), where a is the usual Galois action on M_{2×2}(E) as given in (2.1). Let V be the space of τ, i.e., the space of π. As a model for σ use V ⊕ V with

(3.1) σ(h)(v⊕v′) = π(h)v⊕π(shs)v′, h ∈ GSO(X, F), σ(s)(v⊕v′) = v′⊕v.

We begin with some remarks about the restriction of π to subgroups. Let ψ_{E} : E → ℂ^{×} be a nontrivial quasi-character of E; we may assume ψ_{E} is Gal(E/F) invariant. Let N be the subgroup of GSO(X, F) of elements

n = ρ_{a}(1, [ \begin{matrix}1 & x \\ 0 & 1 \end{matrix} ])

for x ∈ E. Since the space of Whittaker functionals on τ with respect to ψ_{E} is one dimensional, it follows that dim Hom_{N}(π, ψ_{E}) = 1, where ψ_{E} is the character of N defined by ψ_{E}(n) = ψ_{E}(x). This fact allows us to prove the following statements just as in the proof of Theorem 4.3 of [R2]. Let H_{0} be a closed normal subgroup of GSO(X, F) such that F^{×}H_{0} is open, GSO(X, F)/F^{×}H_{0} is finite and Abelian, and N ⊂ H_{0}. Then the restriction π|_{H_{0}} is multiplicity free: π|_{H_{0}} = V_{1} ⊕ ⋯ ⊕ V_{M}, where V_{i}, 1 ≤ i ≤ M are mutually nonisomorphic irreducible H_{0} subspaces of π (see [GK] for general results about restrictions), and, say, dim_{ℂ} Hom_{N}(V_{1}, ψ_{E}) = 1 and dim_{ℂ} Hom_{N}(V_{i}, ψ_{E}) = 0 for 2 ≤ i ≤ M. Suppose additionally s · π ≅ π, and let π̂ be an extension of π to GO(X, F). Then π̂(s)V_{1} = V_{1}. Now we show σ|_{O(X,F)} is multiplicity free. Suppose first there is no quasicharacter β of E^{×} such that β|_{F^{×}} = 1 and β ⊗ τ ≅ τ ∘ a. Let W be a nonzero irreducible O(X, F) subspace of σ. Then either there is an irreducible SO(X, F) subspace U of (π, V) such that W = U ⊕U, or there is an irreducible SO(X, F) subspace U of (π, V) and i : U → U such that i^{2} = 1, i(π(h)u) = π(shs)i(u) for h ∈ SO(X, F) and W = {u ⊕ i(u) : u ∈ U}. We assert the second case is impossible; suppose it holds. Then π|_{SO(X,F)} and (s · π)|_{SO(X,F)} share an irreducible component. Since π|_{SO(X,F)} is multiplicity free by the last paragraph, by Lemma 2.4 of [GK] there is a quasi-character γ : GSO(X, F) → ℂ^{×} trivial on F^{×} SO(X, F) such that s · π ≅ γ ⊗ π. Since

1 → F^{×} SO(X, F) ⟶^{inc} GSO(X, F) ⟶^{λ} N_{F}^{E}(E^{×})/F^{×2} → 1

is exact, γ = η ∘λ for some quasi-character η : N_{F}^{E}(E^{×}) → ℂ^{×} with η^{2} = 1. Let T : (η ∘ λ) ⊗ π → s · π be a GSO(X, F) isomorphism. Then for g ∈ GL(2, E) and v ∈ V,

T (η(λ(ρ_{a}(1, g))) · π(ρ_{a}(1, g)v)) = π(sρ_{a}(1, g)s)T (v)
(η ∘ N_{F}^{E})(det g)T (τ(g)v) = τ(a(g))T (v).

This implies (η ∘ N_{F}^{E}) ⊗ τ ≅ τ ∘ a, contradicting our assumption; note (η ∘ N_{F}^{E})|_{F^{×}} = 1. Thus, W = U ⊕ U. Let W′ be another nonzero irreducible O(X, F) subspace of σ and assume W′ ≅ W as O(X, F) representations; to show σ|_{O(X,F)} is multiplicity free it will suffice to show W and W′ are identical, i.e., W = W′. Write W′ = U′ ⊕ U′, with U′ an irreducible SO(X, F) subspace of (π, V). Consider the composition U → W ⟶^{∼} W′ → U′ where the first map sends u to u ⊕ 0, the second is our fixed isomorphism W ≅ W′, and the last map sends u ⊕ u′ to u. This is an SO(X, F) map from (U, π|_{SO(X,F)}) to (U′, π|_{SO(X,F)}). We claim it is nonzero; suppose not. Then the same composition with the last map replaced by the map sending u ⊕ u′ to u′ gives a nonzero SO(X, F) map from (U, π|_{SO(X,F)}) to (U′, (s · π)|_{SO(X,F)}). However, we just saw that π|_{SO(X,F)} and (s·π)|_{SO(X,F)} have no common irreducible constituents. Thus, the first composition is nonzero, and U and U′ are isomorphic irreducible subspaces of π|_{SO(X,F)}. Since π|_{SO(X,F)} is multiplicity free, U = U′ and so W = W′. Now suppose there is a quasi-character β of E^{×} such that β|_{F^{×}} = 1 and β⊗τ ≅ τ ∘ a. Let H_{0} be the subgroup of SO(X, F) of ρ_{a}(1, g) for g ∈ Sl(2, E), and let H ⊂ O(X, F) be generated by H_{0} and s. To prove σ|_{O(X,F)} is multiplicity free it will suffice to prove σ|_{H} is multiplicity free. For this, we replace σ with a more tractable representation via twisting. Since β|_{F^{×}} = 1, there is a quasicharacter μ of E^{×} such that β(x) = μ(x/a(x)) for x ∈ E^{×}. Letting τ′ = μ ⊗ τ, we have τ′ ∘ a ≅ τ′. Since ω_{τ} is Galois invariant, β^{2} = 1, which implies μ^{2} is Galois invariant. Let ν be a quasi-character of F^{×} such that μ^{2} = ν ∘ N_{F}^{E}, and set χ′ = νχ; then ω_{τ′} = χ′ ∘ N_{F}^{E}. Set π′ = π(χ′, τ′). Since τ′ ∘ a ≅ τ′ we have s·π′ ≅ π′. Let σ′ = Ind_{GSO(X,F)}^{GO(X,F)} π′, and use the same model for σ′ as above, so the underlying space of σ′ is V ⊕ V. Now σ′ may not be isomorphic to σ, but it is easy to see that the identity map between the models for σ and σ′ gives an isomorphism σ|_{H} ≅ σ′|_{H}. We are reduced to showing σ′|_{H} is multiplicity free. As s · π′ ≅ π′, we have σ′ ≅ π′_{1} ⊕ π′_{2}, where π′_{1} and π′_{2} are the two extensions of π′ to GO(X, F). Since the restrictions π′_{1}|_{H_{0}} = π′_{2}|_{H_{0}} = π|_{H_{0}} are multiplicity free as N ⊂ H_{0}, it follows that π′_{1}|_{H} and π′_{2}|_{H} are multiplicity free. It will now suffice to show π′_{1}|_{H} and π′_{2}|_{H} do not share an irreducible component; suppose they do. By Lemma 2.4 of [GK], π′_{1}|_{H} ≅ π′_{2}|_{H}. Let R : π′_{1}|_{H} → π′_{2}|_{H} be an H isomorphism. As indicated above, there is an irreducible H_{0} subspace V_{1} ⊂ V such that π′_{1}(s)V_{1} = π′_{2}(s)V_{1} = V_{1}, i.e., V_{1} is also an irreducible H subspace for π′_{1}|_{H} and π′_{2}|_{H}. Since π′_{1}|_{H} and π′_{2}|_{H} are multiplicity free, we must have R(V_{1}) = V_{1}. Applying Schur’s lemma to R : V_{1} → V_{1}, with V_{1} regarded as an irreducible H_{0} representation, there exists a nonzero scalar c such that R(v) = cv for v ∈ V_{1}. This implies π′_{1}(s)v = π′_{2}(s)v for v ∈ V_{1}. However, π′_{2}(s)v = −π′_{1}(s)v for v ∈ V, a contradiction. □

3.4 Theorem. Let σ ∈ Irr(GO(X, F)). If F is nonarchimedean and d ≠ 1, assume σ is infinite dimensional; if F = ℝ, assume d = 1. Then σ ∈ ℛ_{2}(GO(X, F)) if and only if σ is not of the form π^{−} for some distinguished π ∈ Irr(GSO(X, F)).

Proof. Suppose first F is nonarchimedean. Then this theorem was proven in [R2] in the case F has odd residual characteristic. To verify the theorem if F has even residual characteristic we proceed as follows. We note first that the background results of [R2] are valid in any residual characteristic; that is, the results of sections 2, 3 and 4 hold, and Lemma 6.1, Corollary 6,2, Lemma 6.3 and Lemma 6.4 also hold with the same proofs. We need to show that Lemmas 6.6 and 6.7 of [R2] also hold if F has even residual characteristic. Consider first the proof of Lemma 6.6. The first paragraph of the proof of Lemma 6.6 is independent of the residual characteristic. In the second paragraph, we used a result from [T] proven only in the case of odd residual characteristic; by [Sa], this also holds in the case of even residual characteristic. The third paragraph of the proof is also valid in even residual characteristic (in spite of the unnecessary mention there of odd residual characteristic). Next, we consider all remaining paragraphs but the last paragraph: these cover the case d = 1 and ϵ = −ϵ(1), in the notation of [R2]. Letting D_{ram} be the ramified quaternion algebra over F, we are given τ, τ′ ∈ Irr(D_{ram}^{×}) with ω_{τ} = ω_{τ′}, and we must show that there exists a quadratic extension E ⊂ D_{ram} of F of such that Hom_{SO(Z)}(π, 1) ≠ 0, where SO(Z) is the subgroup {ρ(x, x^{∗}^{−1}) : x ∈ E^{×}} and π = π(τ, τ′). Embed D_{ram}^{×} into D_{ram}^{×} × D_{ram}^{×} via x ↦ (x, x^{∗}^{−1}), and consider the restriction of τ ⊗ τ′ to D_{ram}^{×}. Let τ′′ be an irreducible component of (τ ⊗ τ′)|_{D_{ram}^{×}}; then ω_{τ′′} = 1. By Proposition 18 of [W2], there exists a quadratic extension E of F in D_{ram} and a nonzero vector v ∈ τ′′ such that τ′′(x)v = v for x ∈ E^{×}. This implies that π(h)v = v for h ∈ SO(Z), proving the required claim. The last paragraph of the proof of Lemma 6.6 is also valid in the case of even residual characteristic, thus completing the verification of Lemma 6.6 in this case. Next, we consider the proof of Lemma 6.7 of [R2]. To make the proof of Lemma 6.7 go through in the case of even residual characteristic it suffices to show that if K is a quadratic extension of F, τ ∈ Irr(GL(2, K)) is Galois invariant with ω_{τ} = χ ∘ N_{F}^{K} and Hom_{GL(2,F)}(τ, χ ∘ det) = 0, then there exist quasicharacters ζ and ζ′ of K^{×} extending χ such that ϵ(τ ⊗ ζ^{−1}, 1/2, ψ_{K}) = χ(−1) and ϵ(τ ⊗ζ′^{−1}, 1/2, ψ_{K}) = −χ(−1). To show the existence of ζ, pick ζ extending χ such that ζ is very ramified (this can be done); then by 3 of Lemma 14 of [HST], ϵ(τ ⊗ζ^{−1}, 1/2, ψ_{K}) = χ(−1). On the other hand, since Hom_{GL(2,F)}(τ, χ∘ det) = 0, by the equivalence of 1 and 2 of Theorem 5.3 of [R2], there exists a quasi-character ζ′ of K^{×} extending χ such that ϵ(τ ⊗ζ′^{−1}, 1/2, ψ_{K}) = −χ(−1). (Note that the proof of the equivalence of 1 and 2 of Theorem 5.3 of [R2] works in any residual characteristic; the use of odd residual characteristic in the proof of Lemma 5.2 is easily seen to be unnecessary.) Now suppose F = ℝ and d = 1. Suppose σ ∈ ℛ_{2}(GO(X, ℝ)). Then an argument as in Theorem 4.3 of [R2] shows that σ cannot be of the form π^{−} for some distinguished π. Conversely, suppose σ is not of the form π^{−} for some distinguished π. Then σ ≅ π^{+} for some regular π or distinguished π. Using Lemma 3.2, an argument as in Theorem 4.4 of [R2] shows that σ ∈ ℛ_{2}(GO(X, ℝ)). □

4. Definition of the local L-packets and parameters

Let F be a local field of characteristic zero, with F = ℝ if F is archimedean. Let d ∈ F^{×}/F^{×2}; assume d = 1 if F = ℝ. Let X_{M_{2×2},d} be the four dimensional quadratic space over F defined after Proposition 2.7 and discussed after Proposition 2.9. We will parameterize Irr(GSO(X_{M_{2×2},d}, F)) as explained at the beginning of Section 3. However, since we are dealing with the concrete quadratic spaces X_{M_{2×2},d} we will use the first exact sequence from Proposition 2.7; by Proposition 2.7, the difference is trivial. We let s ∈ O(X_{M_{2×2},d}, F), det s = −1, be defined by s(x) = a(x), where a is the Galois action on M_{2×2}(E_{d}) defining X_{M_{2×2},d}; see (2.1). Using the results of the last section, we will associate to every element [π] of ⟨s⟩\ Irr(GSO(X_{M_{2×2},d}, F)) a packet Π([π]) of elements of Irr(GSp(2, F)) and a GSp(2) L-parameter φ([π]) : L_{F} → GSp(2, ℂ) over F, where L_{F} is the Langlands group of F (i.e., W_{F} × SU(2, ℝ) if F is nonarchimedean and the Weil group W_{F} if F = ℝ). We expect that Π([π]) is the L-packet associated to φ([π]) under the conjectural Langlands correspondence. Some evidence is provided by Propositions 4.1, 4.2 and 4.3 below which give some basic properties of the Π([π]) and φ([π]). More work on this issue remains to be done: for example, are the packets Π([π]) disjoint, and if φ([π]) and φ([π′]) are equivalent, does it follow that Π([π]) = Π([π′])? We will return to this topic in a subsequent work; the thrust of this paper is global results. To define the L-packets, we begin by noting that there is a surjective map

Irr(GO(X_{M_{2×2},d}, F)) → ⟨s⟩\ Irr(GSO(X_{M_{2×2},d}, F))

which sends σ to the components of σ restricted to GSO(X_{M_{2×2},d}, F). We will define the L-packet of elements of Irr(GSp(2, F)) associated to a point of ⟨s⟩\ Irr(GSO(X_{M_{2×2},d}, F)) by considering the fiber over such a point, and applying the results of Section 3. For π ∈ GSO(X_{M_{2×2},d}, F) denote the element of ⟨s⟩\ Irr(GSO(X_{M_{2×2},d}, F)) determined by π by [π] = {π, s · π}. Let [π] ∈ ⟨s⟩\ Irr(GSO(X_{M_{2×2},d}, F)). We assume that π is infinite dimensional; if F is nonarchimedean of even residual characteristic, we assume additionally that π is tempered. Then how [π] gives rise to irreducible representations of GSp(2, F) is described in Tables 2 and 3 of the Appendix. In the first step, using the results of Section 3, π gives rise to representations of various orthogonal similitude groups. This is summarized in the tables, but certain aspects deserve comment. If d ≠ 1, then it may happen that π is invariant but not distinguished. Then the two extensions of π to GO(X_{M_{2×2},d}, F) are denoted by π_{1} and π_{2}. When d = 1, then π is either regular or invariant and distinguished; in the first case π induces to give π^{+}, and in the second case π extends to give π^{+} and π^{−}. Additionally, if d = 1 and π is essentially square integrable, then π gives an element π^{JL} ∈ Irr(GSO(X_{D_{ram},1}, F)) via the Jacquet-Langlands correspondence, and then analogously elements of Irr(GO(X_{D_{ram},1}, F)). Here, D_{ram} is the ramified quaternion algebra over F, and π is essentially square integrable if and only if π = (α ∘ λ) ⊗ π′ for some quasi-character α : F^{×} → ℂ^{×} and square integrable π′ ∈ Irr(GSO(X_{M_{2×2},1}, F)). To apply the Jacquet-Langlands correspondence, we write as in Section 3, π = π(χ, τ) for τ = τ_{1} ⊗ τ_{2} ∈ Irr(GL(2, F) × GL(2, F)); recall that the exact sequence from Proposition 2.7 is in this case

1 → F^{×} × F^{×} → F^{×} × GL(2, F) × GL(2, F) → GSO(X_{M_{2×2},1}, F) → 1.

We define π^{JL} = π(χ, τ^{JL}) ∈ Irr(GSO(X_{D_{ram},1}, F)), where τ^{JL} is the irreducible representation of D_{ram}^{×} ×D_{ram}^{×} corresponding to τ under the Jacquet-Langlands correspondence (π being essentially square integrable means exactly τ_{1} and τ_{2} are essentially square integrable); the exact sequence from Proposition 2.7 for this is

1 → F^{×} × F^{×} → F^{×} × D_{ram}^{×} × D_{ram}^{×} → GSO(X_{D_{ram},1}, F) → 1.

Next, using Theorem 3.4, the thus constructed representations of orthogonal similitude groups give representations of GSp(2, F) via theta correspondences; note that each theta correspondence used is covered by Theorem 1.8. In Tables 2 and 3 of the Appendix we indicate the appropriate theta correspondences with a subscript. We also indicate when representations do not have theta lifts. Finally, in the Table 4 of the Appendix the packets of representations associated to [π] are defined using the representations constructed in Tables 2 and 3 of the Appendix. Note the introduction of the contragredient. The next proposition describes a few basic properties of the L-packets Π([π]).

4.1 Proposition. Let π ∈ Irr(GSO(X_{M_{2×2},d}, F)). Assume π is infinite dimensional; if F is nonarchimedean of even residual characteristic, assume π is tempered. Then

(1) The common central character of the elements of Π([π]) is ω_{π}.

(2) If d = 1 then |Π([π])| = 1 unless π is essentially square integrable; in this case |Π([π])| = 2. If d ≠ 1, then |Π([π])| = 1 unless π is invariant but not distinguished; in this case |Π([π])| = 2.

(3) If π is tempered, then all the elements of Π([π]) are tempered.

Proof. (1) This follows from the remark on central characters after Theorem 1.8.

(2) Evidently, |Π([π])| = 1 except possibly if d = 1 and π is essentially square integrable, or d ≠ 1 and π is invariant but not distinguished. If d = 1 and π is essentially square integrable, then |Π([π])| = 2 by Lemma 8.4 below. If d ≠ 1 and π is invariant but not distinguished, then |Π([π])| = 2 because θ_{M_{2×2},d} is a bijection.

(3) If F is nonarchimedean, this follows from (1) of Theorem 4.2 of [R3]. If F = ℝ, this follows from IV.3, p. 70 and III.2, p. 49 of [M]. □

Next, we associate to each [π] ∈ ⟨s⟩\ Irr(GSO(X_{M_{2×2},d}, F)) an L-parameter φ([π]) : L_{F} → ^{L} GSp(2). Here L_{F} denotes the Langlands group of F, i.e., L_{F} = W_{F} × SU(2, ℝ) if F is nonarchimedean, and L_{F} = W_{F} if F is archimedean ([Ko], Section 12); W_{F} is the Weil group of F. As is well known, the dual group { \atop GSp(2)} of GSp(2, \overline{F}) ( ^{L} GSp(2)^{0} in the notation of [B]) is isomorphic to GSp(2, ℂ), and we shall use such an isomorphism. But since GSp(2, ℂ) has a non-inner automorphism, we need to be specific (the same issue arises for other groups, but for, say, GL(2) the choice is established). To do so, we will specify an isomorphism from the based root datum of ^{L} GSp(2) to the based root datum of GSp(2, ℂ). As a maximal split torus in GSp(2, \overline{F}) we take the group T of elements t = t(a, b, c) = diag(a, b, a^{−1}c, b^{−1}c). The group X^{∗} of characters of T is the free Abelian group with generators e_{1}, e_{2} and e_{3} defined by e_{1}(t) = a, e_{2}(t) = b and e_{3}(t) = c. The group X_{∗} of cocharacters of T is the free Abelian group with generators f_{1}, f_{2} and f_{3} defined by f_{1}(x) = t(x, 1, 1), f_{2}(x) = t(1, x, 1) and f_{3}(x) = t(1, 1, x). The roots of GSp(2, \overline{F}) with respect to T are \{α_{1} = e_{1} − e_{2}, α_{2} = 2e_{2} − e_{3}, α_{1} + α_{2}, 2α_{1} +α_{2}, −α_{1}, −α_{2}, −(α_{1} +α_{2}), −(2α_{1} +α_{2})}. The coroots are {α_{1}^{∨} = γ_{1} = f_{1} − f_{2}, α_{2}^{∨} = γ_{2} = f_{2}, (α_{1} + α_{2})^{∨} = γ_{1} + 2α_{2}, (2α_{1} + α_{2})^{∨} = γ_{1} + γ_{2}, (−α_{1})^{∨} = −γ_{1}, (−α_{2})^{∨} = −γ_{2}, (−(α_{1}+α_{2}))^{∨} = −(γ_{1}+2γ_{2}), (−(2α_{1}+α_{2}))^{∨} = −(γ_{1}+γ_{2})}. As simple roots we take Δ^{∗} = {α_{1}, α_{2}}; then Δ_{∗} = Δ^{∗∨} = {γ_{1}, γ_{2}}. We have similar notation for GSp(2, ℂ), which we will indicate with the addition of a prime. Let Ψ = (X^{∗}, Δ^{∗}, X_{∗}, Δ_{∗}); the dual of Ψ is Ψ^{∨} = (X_{∗}, Δ_{∗}, X^{∗}, Δ^{∗}); let Ψ′ = (X′^{∗}, Δ′^{∗}, X′_{∗}, Δ′_{∗}). Then an isomorphism Ψ^{∨} ⟶^{∼} Ψ′ amounts to an isomorphism f : X_{∗} ⟶^{∼} X′^{∗} of Abelian groups such that f(Δ_{∗}) = Δ′^{∗} and the matrix of f with respect to our bases is symmetric. One can check that there are exactly two such isomorphisms f, with matrices

[ \begin{matrix}1 & 1 & −1 \\ 1 & −1 & 0 \\ −1 & 0 & c \end{matrix} ] , c = 0 or 1.

As is done implicitly in [HST], we shall fix the isomorphism corresponding to the choice c = 1. Our fixed isomorphism of based root data Ψ^{∨} ⟶^{∼} Ψ′ determines a T conjugacy class of isomorphisms { \atop GSp(2)} ⟶^{∼} GSp(2, ℂ) ([Sp], Theorem 9.6.2); we fix one such isomorphism in the conjugacy class. Additionally, since the action of W_{F} on { \atop GSp(2)} is trivial, ^{L} GSp(2) is the direct product { \atop GSp(2)} × W_{F}. Thus, in considering L-parameters we may just as well look at maps into { \atop GSp(2)}, which we identify with GSp(2, ℂ) (always via our fixed isomorphism). We define a GSp(2) L-parameter over F to be a continuous homomorphism φ : L_{F} → GSp(2, ℂ) such that φ(x) is semisimple for x ∈ W_{F}, and if F is nonarchimedean then φ|_{1×SU(2,ℝ)} is a smooth representation. Let φ be a GSp(2) L-parameter over F. The similitude quasi-character of φ is the quasi-character of L_{F} given by λ ∘ φ, where λ : GSp(2, ℂ) → ℂ^{×} is the usual similitude homomorphism. If F is nonarchimedean, we say φ is unramified if φ(SU(2, ℝ)) = 1 and φ is trivial on the inertia subgroup of W_{F}. We say that φ is tempered if φ(L_{F}) is bounded. If φ′ : L_{F} → GSp(2, ℂ) is another GSp(2) L-parameter over F we say that φ and φ′ are equivalent if there exists g ∈ GSp(2, ℂ) such that gφ(x)g^{−1} = φ′(x) for all x ∈ L_{F}. The connected component group of φ is the group 𝕊(φ) = π_{0}(S(φ)/ℂ^{×}), where S(φ) is the group of g ∈ GSp(2, ℂ) such that gφ(x) = φ(x)g for all x ∈ L_{F}. The parameter φ([π]) will be one of two kinds of examples of GSp(2) Lparameters over F. To define the first kind of example, suppose E/F is a quadratic extension and let ρ : L_{E} → GL(2, ℂ) be a GL(2) L-parameter over E such that det ρ is Galois invariant. Let η : L_{F} → ℂ^{×} be a quasi-character extending det ρ; there are two such quasi-characters. Let V = ℂ^{2}, and regard ρ as a representation on V. Define φ(η, ρ) : L_{F} → GSp(W) by letting the space and action of φ(η, ρ) be W = Ind\begin{matrix}L & F \\ L & E \end{matrix} ρ and defining a nondegenerate symplectic form on W by

⟨v_{1} ⊕ v_{2}, v′_{1} ⊕ v′_{2}⟩ = η(y)⟨v_{1}, v′_{1}⟩ + ⟨v_{2}, v′_{2}⟩.

Here, y is a fixed representative for the nontrivial coset of L_{E} in L_{F}, we identify the space of φ(η, ρ) with V ⊕ V via the map f ↦ f(1) ⊕ f(y), and we have fixed a nondegenerate symplectic form on V (note that up to multiplication by elements of ℂ^{×}, there is only one nondegenerate symplectic form on a two dimensional complex vector space). Then φ(η, ρ) is a GSp(2) L-parameter over F, and the similitude quasi-character of φ(η, ρ) is λ∘φ(η, ρ) = η. Suppose next that ρ_{1} : L_{F} → GL(2, ℂ) and ρ_{2} : L_{F} → GL(2, ℂ) are GL(2) L-parameters over F with det ρ_{1} = det ρ_{2}. Regard ρ_{1} and ρ_{2} as two dimensional representations of L_{F} on V_{1} = ℂ^{2} and V_{2} = ℂ^{2}, respectively, and fix nondegenerate symplectic forms ⟨·, ·⟩_{1} and ⟨·, ·⟩_{2} on the spaces of ρ_{1} and ρ_{2}, respectively. Define φ(ρ_{1}, ρ_{2}) : L_{F} → GSp(W) by letting the space and action of φ(ρ_{1}, ρ_{2}) be W = ρ_{1} ⊕ ρ_{2} and defining a nondegenerate symplectic form on the space of φ(ρ_{1}, ρ_{2}) by

⟨v_{1} ⊕ v_{2}, v′_{1} ⊕ v′_{2}⟩ = ⟨v_{1}, v′_{1}⟩_{1} + ⟨v_{2}, v′_{2}⟩_{2}.

Then φ(ρ_{1}, ρ_{2}) is a GSp(2) L-parameter over F, and the similitude quasicharacter of φ(ρ_{1}, ρ_{2}) is λ ∘ φ(ρ_{1}, ρ_{2}) = det ρ_{1} = det ρ_{2}. Now let [π] ∈ ⟨s⟩\ Irr(GSO(X_{M_{2×2},d}, F)). Write π = π(χ, τ), with τ ∈ Irr(GL(2, E_{d})) and χ a quasi-character of F^{×} such that ω_{τ} = χ ∘ N_{F}^{E_{d}}. Suppose first that d ≠ 1. Let ρ : L_{E_{d}} → GL(2, ℂ) be the GL(2) L-parameter over E_{d} corresponding to τ, and let η : L_{F} → ℂ^{×} be the quasi-character of L_{F} corresponding to χ. Then η extends det ρ and the equivalence class of φ(η, ρ) depends only on [π] and not the choice of representative π. We set φ([π]) = φ(η, ρ). Suppose next d = 1. Then GL(2, E_{d}) ≅ GL(2, F) × GL(2, F). Let τ ≅ τ_{1} ⊗ τ_{2}, with τ_{1}, τ_{2} ∈ Irr(GL(2, F)) such that χ = ω_{τ_{1}} = ω_{τ_{2}}. Let ρ_{1}, ρ_{2} : L_{F} → GL(2, ℂ) be the GL(2) L-parameters over F corresponding to ρ_{1} and ρ_{2}, respectively. Then det ρ_{1} = det ρ_{2} and the equivalence class of φ(ρ_{1}, ρ_{2}) depends only on [π] and not the choice of representative π. We set φ([π]) = φ(ρ_{1}, ρ_{2}). The following is an analogue of Proposition 4.1.

4.2 Proposition. Let π ∈ Irr(GSO(X_{M_{2×2},d}, F)). Assume π is infinite dimensional; if F is nonarchimedean of even residual characteristic, assume π is tempered. Then

(1) The similitude quasi-character of φ([π]) corresponds to ω_{π}.

(2) If d = 1 then |𝕊_{φ([π])}| = 1 unless π is a essentially square integrable; in this case 𝕊(φ([π])) = Z_{2}. If d ≠ 1 and F is not nonarchimedean of even residual characteristic, then |𝕊(φ([π]))| = 1 unless π is invariant but not distinguished; in this case 𝕊(φ([π])) = Z_{2}.

(3) If π is tempered, then φ([π]) is tempered.

Proof. (1). This follows from the definitions and above remarks.

(2) This follows by a case by case analysis following Tables 2 and 3 of the Appendix. We note in particular that by Theorem 5.3 of [R2] if d ≠ 1, then π = π(χ, τ) is distinguished if and only if τ is Galois invariant and τ is the base change of a τ_{0} ∈ Irr(GL(2, F)) such that ω_{τ_{0}} = ω_{E_{d}/F} χ.

(3) Assume π is tempered. Then ρ and η in the case d ≠ 1, and ρ_{1} and ρ_{2} in the case d = 1, have bounded image. This implies that φ([π]) has bounded image. □

4.3 Proposition. Suppose F is nonarchimedean, E_{d}/F is unramified (if d = 1 by convention E_{d}/F is unramified) and π = π(χ, τ) ∈ Irr(GSO(X_{M_{2×2},d}, F)) is infinite dimensional with χ and τ unramified. If the residual characteristic of F is even, assume additionally π is tempered. Then

(1) φ([π]) is unramified, |Π([π])| = 1, and if the residual characteristic of F is odd, then the single element Π of Π([π]) is unramified with respect to GSp(2, 𝔒_{F}).

(2) ( [HST]) Let Π([π]) = {Π}. If π and Π are unitary (e.g., as in global applications, or π tempered), then Π is unramified with respect to GSp(2, 𝔒_{F}) and φ([π]) and Π correspond to the same conjugacy class in GSp(2, ℂ).

Proof. (1) Suppose d = 1. Evidently, φ([π]) is unramified. Write π = π(χ, τ). As mentioned in Section 3 and the beginning of this section, instead of using the exact sequence of Theorem 2.3, let us use the more convenient sequence of Proposition 2.7, and let s be the representative for the nontrivial coset of GSO(X_{M_{2×2},1}, F) in GO(X_{M_{2×2},1}, F) from Proposition 2.7. Also, make the identification of X_{M_{2×2},1} with M_{2×2}(F) equipped with the determinant as remarked before Proposition 2.8. Then s is given by s(x) = x^{∗}, with ∗ the canonical involution of matrices, and τ = τ_{1}⊗τ_{2} with τ_{1}, τ_{2} ∈ Irr(GL(2, F)) and ω_{τ_{1}} = ω_{τ_{2}} = χ. The lattice M_{2×2}(𝔒_{F}) ⊂ X_{M_{2×2},1} is self-dual, and the maximal compact subgroups J_{0} and J of GSO(X_{M_{2×2},1}, F) and GO(X_{M_{2×2},1}, F) which are the stabilizers of M_{2×2}(𝔒_{F}) are ρ_{a}(𝔒_{F}^{×} × GL(2, 𝔒_{F}) × GL(2, 𝔒_{F})) and the subgroup generated by ρ_{a}(𝔒_{F}^{×} × GL(2, 𝔒_{F}) × GL(2, 𝔒_{F})) and s, respectively. Since π is not essentially square integrable, Π([π]) = {Π = θ_{M_{2×2},1}(π^{+})^{∨}} so that |Π([π])| = 1. By Proposition 1.11 to show Π is unramified it will suffice to show π^{+} is unramified. Suppose τ_{1} ≇ τ_{2}. Then π^{+} = Ind_{GSO(X,F)}^{GO(X,F)} π. Using the model for π^{+} as in (3.1), we see that if v is an unramified vector with respect to J_{0}, then v ⊕ v is an unramified vector for π^{+}. Suppose τ_{1} ≅ τ_{2}, so that π is distinguished and π^{+} is the extension to GO(X_{M_{2×2},1}, F) of π defined in Section 3. Let τ = τ_{1}. It will suffice to show π^{+} = π(χ, τ ⊗ τ)^{+} is unramified. Define T : π → π by T (v ⊗ w) = w ⊗ v. To show π^{+} is unramified it suffices to show T = π^{+}(s). Let Y = (F · y)^{⊥}, where y ∈ X_{M_{2×2},1} is the 2 × 2 identity matrix. Then SO(Y, F), identified as usual with the stabilizer of y in GSO(X_{M_{2×2},1}, F), is the group of ρ_{a}(1, g, g^{∗−1}) for g ∈ GL(2, F). To show T = π^{+}(s) it will suffice to show T π(shs) = π(h)T for h ∈ GSO(X_{M_{2×2},1}, F), T^{2} = 1, and L∘T = L for any nonzero element L of Hom_{SO(Y,F)}(π, 1). The first two statements follow from sρ_{a}(t, g, g′)s = ρ_{a}(t, g′, g) for g, g′ ∈ GL(2, F). Let V be the space of τ. Fix a GL(2, F) isomorphism R : (ω_{τ}^{−1} ⊗ τ, V) → (τ^{∨}, V^{∨}). Let S = 1 ⊗ R : V ⊗ V → V ⊗ V^{∨}. We have a nonzero GL(2, F) invariant linear form V ⊗ V^{∨} → 1 given by v ⊗ f ↦ f(v). The composition of this with S gives us L ∈ Hom_{SO(Y,F)}(π, 1). Now L ∘ T = ϵL for some ϵ = ±1. We must show ϵ = 1. Let v ∈ V be nonzero and fixed by GL(2, 𝔒_{F}). Then L(v ⊗ v) = L(T (v ⊗ v)) = ϵL(v ⊗ v). To see L(v ⊗ v) ≠ 0 and hence ϵ = 1, let V = ℂv_{0} ⊕ W be a GL(2, 𝔒_{F}) decomposition, and define f ∈ V^{∨} by letting f be zero on W and setting f(v) = 1. Then τ^{∨}(k)f = f for k ∈ GL(2, 𝔒_{F}). Evidently, S(v⊗v) = c(v⊗f) for some c ∈ ℂ^{×} so that L(v⊗v) = cf(v) = c ≠ 0. Now suppose d ≠ 1. Again, it is clear that φ([π]) is unramified. Write π = π(χ, τ) again using Proposition 2.7. We will also use the notation after Proposition 2.9 regarding X_{M_{2×2},d}. The representative s for the nontrivial coset of GSO(X_{M_{2×2},d}, F) in GO(X_{M_{2×2},d}, F) is given by s(x) = x^{∗} = a(x). The lattice X_{M_{2×2},d} ∩ M_{2×2}(𝔒_{E_{d}}) is self dual, and the maximal compact subgroups J_{0} and J of GSO(X_{M_{2×2},d}, F) and GO(X_{M_{2×2},d}, F) which are the stabilizers of X_{M_{2×2},d} ∩ M_{2×2}(𝔒_{E_{d}}) are ρ_{a}(𝔒_{F}^{×} × GL(2, 𝔒_{E_{d}})) and the subgroup generated by ρ_{a}(𝔒_{F}^{×} × GL(2, 𝔒_{E_{d}})) and s, respectively. Since τ is unramified, τ ≅ Ind_{P}^{GL(2,E_{d})}(μ_{1} ⊗ μ_{2}), where B is the usual Borel subgroup of GL(2, E_{d}) and μ_{1} and μ_{2} are unramified and Galois invariant so that τ is Galois invariant. This implies s · π ≅ π. In the proof of Theorem 5.3 of [R2] it was shown that π is distinguished and that L ∈ Hom_{SO(Y,F)}(π, 1) is given by

L(f) = ∫_{T \ GL(2,F)} f(g_{0}^{−1}g)χ(det g)^{−1} dg

where T and g_{0} are as in [R2]; here Y = (F · y)^{⊥}, where y is the 2 × 2 identity matrix in X_{M_{2×2},d} and SO(Y, F) is the group of ρ_{a}(det g, g) for g ∈ GL(2, F). By definition, we have Π([π]) = {Π = θ_{M_{2×2},d}(π^{+})^{∨}}, so that |Π([π])| = 1. Again, by Proposition 1.11 to show that Π is unramified with respect to GSp(2, 𝔒_{F}) when has F has odd residual characteristic it will suffice to show that π^{+} is unramified. We proceed as in the case d = 1. Define T : π → π by T (f) = f ∘ a. As in the d = 1 case, it will suffice to show L ∘ T = L. For f ∈ π,

(L ∘ T)(f) = ∫_{T \ GL(2,F)} f(a(g_{0}^{−1}g))χ(det g)^{−1} dg

= ∫_{T \ GL(2,F)} f([ \begin{matrix}−1 & 0 \\ 0 & 1 \end{matrix} ] g_{0}^{−1}g)χ(det g)^{−1} dg

= μ_{1}(−1) ∫_{T \ GL(2,F)} f(g_{0}^{−1}g)χ(det g)^{−1} dg

= μ_{1}(−1)L(f).

Since μ_{1}(−1) = 1, we have L ∘ T = L, as desired.

(2) This follows from Lemmas 10 and 11 of [HST] after one reconciles the definitions. ([HST] for example uses Whittaker models instead of distinguished representations to define extensions to GO(X_{M_{2×2},d}, F).) Lemma 1.6 and Proposition 2.9 are also useful for the comparison to [HST]. The reader should be aware that in Lemma 7 of [HST] the Langlands parameter should be diag(χ_{3}(v), χ_{1}(v)χ_{3}(v), χ_{1}(v)χ_{2}(v)χ_{3}(v), χ_{2}(v)χ_{3}(v)), and in Lemma 10 of [HST] the L-parameter should be diag(\sqrt{α}, −\sqrt{α}, \sqrt{β}, −\sqrt{β}). □

5. Global theta lifts for similitudes

In this section we review some foundational results on global theta lifts for similitudes ([HK], [HST]). We use the following definitions. Let F be a totally real number field with ring of integers 𝔒, and let X be a even dimensional quadratic space defined over F of positive dimension. For each infinite place v of F fix maximal compact subgroups J_{1,v} and J_{v} of O(X, F_{v}) and GO(X, F_{v}), and let 𝔥_{1,v} and 𝔥_{v} be the Lie algebras of O(X, F_{v}) and GO(X, F_{v}), respectively, as in Section 1. Let J_{1,∞} and J_{∞} be the products of J_{1,v} and J_{v}, respectively, over the infinite places of F, and let 𝔥_{1,∞} and 𝔥_{∞} be the direct sums of the 𝔥_{1,v} and 𝔥_{v}, respectively, over the infinite places of v. Let n be a positive integer. For each infinite place v of F let K_{1,v} and K_{v} be the usual maximal compact subgroups of Sp(n, F_{v}) and GSp(n, F_{v}), and let 𝔤_{1,v} and 𝔤_{v} be the Lie algebras of Sp(n, F_{v}) and GSp(n, F_{v}), respectively, as in Section 1. Define K_{1,∞}, K_{∞}, 𝔤_{1,∞} and 𝔤_{∞} as in the case of O(X) and GO(X). For v a place of F, define R(F_{v}) ⊂ GSp(n, F_{v}) × GO(X, F_{v}) as in Section 1. Let R(F) and R(𝔸) be the set of pairs (g, h) in GSp(n, F) × GO(X, F) and GSp(n, 𝔸) × GO(X, 𝔸), respectively, such that λ(g) = λ(h). For v an infinite place of F, let L_{v} be the maximal compact subgroup of R(F_{v}) as defined in Section 1, and let 𝔯_{v} be Lie algebra of R(F_{v}). Let L_{∞} and 𝔯_{∞} be defined analogously to the last two cases. To define global theta lifts we need a global version of the Weil representation. Fix a nontrivial unitary character ψ of 𝔸/F. For v a place of F, let ω_{v} be the Weil representation of R(F_{v}) on L^{2}(X(F_{v})^{n}) defined with respect to ψ_{v} as in Section 1. Again, if v is a place of F then 𝒮(X(F_{v})^{n}) ⊂ L^{2}(X(F_{v})^{n}) is an R(F_{v}) module if v is finite and an (r_{v}, L_{v}) module if v is infinite. Let x_{1}, …, x_{m} be a vector space basis for X(F) over F. Let (g, h) ∈ R(𝔸). Then for almost all finite v, ω_{v}(g_{v}, h_{v}) fixes the characteristic function of 𝔒_{v}x_{1} + · · · + 𝔒_{v}x_{m}. Let ⊗_{v}𝒮(X(F_{v})^{n}) be the algebraic restricted direct product over all the places of F of the complex vector spaces 𝒮(X(F_{v})^{n}) with respect to the characteristic function of 𝔒_{v}x_{1} + ⋯ + 𝔒_{v}x_{m} for v finite. We will denote the restricted algebraic direct product ⊗_{v}𝒮(X(F_{v})^{n}) by 𝒮(X(𝔸)^{n}); then 𝒮(X(𝔸)^{n}) is an R(𝔸_{f}) × (𝔯_{∞}, L_{∞}) module, where R(𝔸_{f}) has the obvious meaning. Let φ ∈ 𝒮(X(𝔸)^{n}) and (g, h) ∈ R(𝔸); assume φ = ⊗_{v}φ_{v}. The function ω(g, h)φ : X(𝔸)^{n} → ℂ given by (ω(g, h)φ)(x) = ∏_{v}(ω_{v}(g_{v}, h_{v})φ_{v})(x_{v}) is well defined (note that for infinite v, ω_{v}(g_{v}, h_{v})φ_{v} is a smooth function though it may not be in 𝒮(X(F_{v})^{n}), so that it can be evaluated at a point). Using the universal property of the algebraic restricted direct product, this definition extends to all φ ∈ 𝒮(X(𝔸)^{n}): if (g, h) ∈ R(𝔸) and φ ∈ 𝒮(X(𝔸)^{n}), then ω(g, h)φ may be regarded as a function on X(𝔸)^{n}. In particular, the elements of 𝒮(X(𝔸)^{n}) may be regarded as functions on X(𝔸)^{n}. Global theta lifts are now defined as follows. For φ ∈ 𝒮(X(𝔸)^{n}) and (g, h) ∈ R(𝔸), set

θ(g, h; φ) = ∑_{x∈X(F)^{n}} ω(g, h)φ(x).

This series converges absolutely and is left R(F) invariant. Fix a right O(X, 𝔸) invariant quotient measure on O(X, F)\ O(X, 𝔸). Let f be a cusp form on GO(X, 𝔸) of central character χ and φ ∈ 𝒮(X(𝔸)^{n}). Let GSp(n, 𝔸)^{+} be the subgroup of g ∈ GSp(n, 𝔸) such that λ(g) ∈ λ(GO(X, 𝔸)). For g ∈ GSp(n, 𝔸)^{+} define

θ_{n}(f, φ)(g) = ∫_{O(X,F)\ O(X,𝔸)} θ(g, h_{1}h; φ)f(h_{1}h) dh_{1},

where h ∈ GO(X, 𝔸) is any element such that (g, h) ∈ R(𝔸). This integral converges absolutely, does not depend on the choice of h, and the function θ_{n}(f, φ) on GSp(n, 𝔸)^{+} is left GSp(n, F)^{+} invariant. Moreover, θ_{n}(f, φ) extends uniquely to a GSp(n, F) left invariant function on GSp(n, 𝔸) with support in GSp(n, F) GSp(n, 𝔸)^{+}. This extended function, also denoted by θ_{n}(f, φ), is an automorphic form on GSp(n, 𝔸) of central character χχ_{X}^{n} = χ(·, disc X(F))_{F}^{n}. If V is a GO(X, 𝔸_{f}) × (h_{∞}, J_{∞}) subspace of the space of cusp forms on GO(X, 𝔸) of central character χ, then we denote by Θ_{n}(V) the GSp(n, 𝔸_{f}) × (𝔤_{∞}, K_{∞}) subspace of the space of automorphic forms on GSp(n, 𝔸) of central character χχ_{X}^{n} generated by all the θ_{n}(f, φ) for f ∈ V and φ ∈ 𝒮(X(𝔸)^{n}). Similarly, fix a right Sp(n, 𝔸) invariant quotient measure on Sp(n, F)\ Sp(n, 𝔸), let F be a cusp form on GSp(n, 𝔸) of central character χ′ and φ ∈ 𝒮(X(𝔸)^{n}). For h ∈ GO(X, 𝔸) define

θ_{X}(F, φ)(h) = ∫_{Sp(n,F)\ Sp(n,𝔸)} θ(g_{1}g, h; φ)F (g_{1}g) dg

where g ∈ GSp(n, 𝔸) is any element such that (g, h) ∈ R(𝔸). Again, this integral converges absolutely, does not depend on the choice of g, and the function θ_{X}(F, φ) is an automorphic form on GO(X, 𝔸) of central character χ′χ_{X}^{n}. If W is a GSp(n, 𝔸_{f})×(𝔤_{∞}, K_{∞}) subspace of the space of cusp forms on GSp(n, 𝔸) of central character χ′, then we denote by Θ_{X}(W) the GO(X, 𝔸_{f})× (𝔥_{∞}, J_{∞}) subspace of the space of automorphic forms on GO(X, 𝔸) of central character χ′χ_{X}^{n} consisting of the θ_{X}(F, φ) for F ∈ W and φ ∈ 𝒮(X(𝔸)^{n}). We shall also occasionally consider global theta lifts of O(X, 𝔸_{f})×(𝔥_{1,∞}, J_{1,∞}) subspaces of the space of cusp forms on O(X, 𝔸) and of Sp(n, 𝔸_{f})×(𝔤_{1,∞}, K_{1,∞}) subspaces of the space of cusp forms on Sp(n, 𝔸). These have the obvious analogous definitions. We will need to know how Θ_{n}(V) and Θ_{X}(W) behave if X is changed by a similitude. Let X′ be another quadratic space over F, and suppose there is a similitude t : X(F) → X′(F) with similitude factor λ. Then for each place v of F, there is an isomorphism

GO(X, F_{v}) ⟶^{∼} GO(X′, F_{v})

sending h to tht^{−1}. For each infinite v, let J′_{1,v} and J′_{v} be the maximal compact subgroups of O(X′, F_{v}) and GO(X′, F_{v}) which are the images under the above isomorphism of J_{1,v} and J_{v}, respectively. If v is infinite, then t also determines an isomorphism

𝔤𝔬(X, F_{v}) ⟶^{∼} 𝔤𝔬(X′, F_{v})

given by h ↦ tht^{−1}. Via these two isomorphisms and definitions, for each v we obtain a bijection

Irr(GO(X, F_{v})) ⟶^{∼} Irr(GO(X′, F_{v})),

and thus a bijection

Irr_{admiss}(GO(X, 𝔸)) ⟶^{∼} Irr_{admiss}(GO(X′, 𝔸)).

If f is an automorphic form on GO(X, 𝔸), then tf : GO(X′, 𝔸) → ℂ defined by (tf)(h) = f(t^{−1}ht) is an automorphic form on GO(X′, 𝔸). Under this map, cusp forms are mapped to cusp forms. Let the right O(X′, 𝔸) invariant quotient measure on O(X′, F)\ O(X′, 𝔸) be obtained from the fixed right O(X, 𝔸) invariant quotient measure on O(X, F)\ O(X, 𝔸) via the isomorphism h ↦ tht^{−1}.

5.1 Lemma. Let V be a GO(X, 𝔸_{f}) × (𝔥_{∞}, J_{∞}) subspace of the space of cusp forms on GO(X, 𝔸) of central character χ, and let W be a GSp(n, 𝔸_{f}) × (𝔤_{∞}, K_{∞}) subspace of the space of cusp forms on GSp(n, 𝔸) of central character χ′. Then Θ_{n}(V) = Θ_{n}(tV) and tΘ_{X}(W) = Θ_{X′}(W). Moreover, Θ_{n}(V) and Θ_{X}(W) do not depend on the choice of nontrivial unitary character ψ of 𝔸/F.

Proof. To show Θ_{n}(V) ⊂ Θ_{n}(tV) it will suffice to show that if f ∈ V and φ = ⊗_{v}φ_{v}, then θ_{n}^{X}(f, φ) ∈ Θ_{n}(tV); here and below the superscript X will indicate the dependence on X. By Lemma 1.6, if (g, h) ∈ R_{X,n}(𝔸) then

θ_{n}^{X}(g, h; φ) = θ_{n}^{X′}(g^{[λ]}, tht^{−1}; φ ∘ t^{−1}).

Let g = g_{0}g_{1} ∈ GSp(n, F) GSp(n, 𝔸)^{+} with g_{0} ∈ GSp(n, F) and g_{1} ∈ GSp(n, 𝔸)^{+}. A computation shows that

θ_{n}^{X}(f, φ)(g) = θ_{n}^{X}(f, φ)(g_{1}) = θ_{n}^{X′}(tf, φ ∘ t^{−1})(g_{1}^{[λ]}).

Write

g_{1}^{[λ]} = g′ [ \begin{matrix}1 & 0 \\ 0 & |λ|_{∞}^{−1} \end{matrix} ] .

Here |λ|_{∞} is the element of 𝔸^{×} which is 1 at the finite places and |λ|_{v} at the infinite place v. Then g′ ∈ GSp(n, 𝔸)^{+}. Let h′ ∈ GO(X, 𝔸) be such that λ(h′) = λ(g′). We have

θ_{n}^{X′}(tf, φ ∘ t^{−1})(g_{1}^{[λ]}) = θ_{n}^{X′}(tf, φ ∘ t^{−1})(g′ [ \begin{matrix}1 & 0 \\ 0 & |λ|_{∞}^{−1} \end{matrix} ])

= ∫_{O(X′,F)\ O(X′,𝔸)} θ_{n}^{X′}(g′ [ \begin{matrix}1 & 0 \\ 0 & |λ|_{∞}^{−1} \end{matrix} ] , h_{1}h′\sqrt{|λ|_{∞}}^{−1}; φ ∘ t^{−1})

· (tf)(h_{1}h′\sqrt{|λ|_{∞}}^{−1}) dh_{1}

= χ(\sqrt{|λ|_{∞}}^{−1})θ_{X′}(tf, φ′)(g′),

where

φ′ = ∏_{v inf.} |λ|_{v}^{n dim X/4} · (φ_{f} ∘ t^{−1}) ⊗ (φ_{∞} ∘ \sqrt{|λ|_{∞}}t^{−1}).

Then φ′ ∈ 𝒮(X′(𝔸)^{n}), and

θ_{n}^{X}(f, φ)(g) = χ(\sqrt{|λ|_{∞}}^{−1})θ_{n}^{X′}(tf, φ′)(g′)

= χ(\sqrt{|λ|_{∞}}^{−1})[[ \begin{matrix}1 & 0 \\ 0 & λ \end{matrix} ]_{f}^{−1} [ \begin{matrix}1 & 0 \\ 0 & sign(λ)_{∞} \end{matrix} ]^{−1} θ_{n}^{X′}(tf, φ′)](g).

Here, sign(λ)_{∞} is the element of 𝔸^{×} which is 1 at the finite places and at the infinite place v is the sign of λ in F_{v}. If g ∉ GSp(n, F) GSp(n, 𝔸)^{+}, then also

g [ \begin{matrix}1 & 0 \\ 0 & λ \end{matrix} ]_{f}^{−1} [ \begin{matrix}1 & 0 \\ 0 & sign(λ)_{∞} \end{matrix} ]^{−1} ∉ GSp(n, F) GSp(n, 𝔸)^{+},

so that both sides of the last equality are by definition zero, and hence equal. Since

[ \begin{matrix}1 & 0 \\ 0 & sign(λ)_{∞} \end{matrix} ]^{−1} ∈ K_{∞},

it now follows that θ_{n}^{X}(f, φ) ∈ Θ_{n}(tV), so that Θ_{n}(V) ⊂ Θ_{n}(tV). Similarly, Θ_{n}(tV) ⊂ Θ_{n}(V). The proof of tΘ_{X}(W) = Θ_{X′}(W) and the independence of ψ are analogous. □

The next two results are due to Rallis [Ra] in the case of isometries. The first describes when a theta lift is cuspidal. The second result gives the structure of a theta lift of a space of cusp forms in the case the theta lift is cuspidal. The proofs are similar to or use the proofs in [Ra]. Section 1 is also a basic input for the proof of Proposition 5.3.

5.2 Proposition (Rallis). Let n ≥ 1 be an integer. Let f be a cusp form on GO(X, 𝔸). Suppose that θ_{k}(f, φ) = 0 for all 0 ≤ k ≤ n − 1 and φ ∈ 𝒮(X(𝔸)^{k}). Then θ_{n}(f, φ) is cuspidal (though possibly zero) for all φ ∈ 𝒮(X(𝔸)^{n}).

Here, θ_{0}(f, φ) = 0 is taken to mean

(5.1)  0 = ∫_{O(X,F)\ O(X,𝔸)} f(h_{1}h) dh_{1}

for all h ∈ GO(X, 𝔸). An analogous result holds for lifts from GSp(n) to GO(X). In this case, fix an even dimensional quadratic space X over F such that X(F) is anisotropic. For an integer k ≥ 0, let X_{k} be the orthogonal direct sum of X with k copies of the hyperbolic plane over F. Let f be a cusp form on GSp(n, 𝔸). Let l ≥ 0 be an integer. If l = 0 and dim X = 0, so that X_{l} = 0, then θ_{X_{l}}(f, φ) is not defined; if l = 0 and dim X > 0 so that X_{l} = X, then θ_{X_{l}}(f, φ) is cuspidal for all φ ∈ 𝒮(X_{l}(𝔸)^{n}) since the cuspidal condition is vacuous. Suppose l ≥ 1. Suppose θ_{X_{k}}(f, φ) = 0 for all 0 ≤ k ≤ l − 1 and φ ∈ 𝒮(X_{k}(𝔸)^{n}); then θ_{X_{l}}(f, φ) is cuspidal (though possibly zero) for all φ ∈ 𝒮(X_{l}(𝔸)^{n}). Here, if dim X = 0 and k = 0 then the condition θ_{X_{k}}(f, φ) = 0 is taken to be empty.

5.3 Proposition (Rallis; Multiplicity preservation). Let 2n = dim X. Let V be a GO(X, 𝔸_{f}) × (𝔥_{∞}, J_{∞}) nonzero subspace of the space of cusp forms on GO(X, 𝔸) of central character χ. Assume that for each place v of F, X(F_{v}) satisfies one of the conditions of (1)-(6) of Theorem 1.8. Assume that

V = V_{1} ⊕ ⋯ ⊕ V_{M},

where each V_{i}, 1 ≤ i ≤ M, is a GO(X, 𝔸_{f}) × (𝔥_{∞}, J_{∞}) subspace of V, and all the V_{i} are isomorphic to a single nonzero irreducible GO(X, 𝔸_{f}) × (𝔥_{∞}, J_{∞}) representation σ. Let σ ≅ ⊗_{v}σ_{v}, assume σ_{v}|_{O(X,F)} is multiplicity free for all v, and σ_{v} is tempered for v | 2. Suppose that Θ_{n}(V) is contained in the space of cusp forms on GSp(n, 𝔸) (necessarily of central character χχ_{X}^{n} = χ(·, disc X(F))^{n}), and that for any irreducible nonzero GO(X, 𝔸_{f}) × (𝔥_{∞}, J_{∞}) subspace U of V we have Θ_{n}(U) ≠ 0. Then σ_{v} ∈ ℛ_{n}(GO(X, F_{v})) for all v,

Θ_{n}(V) = Θ_{n}(V_{1}) ⊕ ⋯ ⊕ Θ_{n}(V_{M}),

and each Θ_{n}(V_{i}), 1 ≤ i ≤ M, is isomorphic to Π = ⊗_{v}θ(σ_{v}^{∨}). An analogous result holds if the roles of GSp(n) and GO(X) are interchanged.

6. Tempered cuspidal automorphic representations of B(𝔸)^{×} and GSO(X, 𝔸)

Let F be a totally real number field, and let X be a four dimensional quadratic space over F. As in Section 2, let B be the even Clifford algebra of X(F), and let E be the center of B. Let d = disc X(F). In this section we describe the relationship between tempered cuspidal automorphic representations of B^{×}(𝔸) and GSO(X, 𝔸). From Section 2, we have exact sequences

1 → E^{×} → F^{×} × B^{×}(F) ⟶^{ρ} GSO(X, F) → 1

and

1 → 𝔸_{E}^{×} → 𝔸^{×} × B^{×}(𝔸) ⟶^{ρ} GSO(X, 𝔸) → 1.

6.1 Lemma. There exist s ∈ O(X, F), and for each infinite v, a maximal compact subgroup J_{0,v} of GSO(X, F_{v}), such that det s = −1, s^{2} = 1, and sJ_{0,v}s = J_{0,v} for all infinite v.

Proof. Let y ∈ X(F) be anisotropic, and let Y ⊂ X be the three dimensional quadratic space over F such that Y (F) = (F · y)^{⊥}. Let s ∈ O(X, F) be defined with respect to y as in Propositions 2.5 and 2.6. Then det s = −1 and s^{2} = 1. For each infinite v, choose a maximal compact subgroup J_{Y,v} of SO(Y, F_{v}), and let J_{0,v} be the unique maximal compact subgroup of GSO(X, F_{v}) containing J_{Y,v} mentioned in the penultimate paragraph of Section 2. Then s normalizes J_{Y,v} and J_{0,v} for each infinite v. □

For the remainder of this paper we fix the following choices of compact subgroups. Let s and the maximal compact subgroups J_{0,v} of GSO(X, F_{v}) be as in Lemma 6.1. For each infinite place v of F, let K_{B,v} be the unique maximal compact subgroup of B^{×}(F_{v}) such that ρ({±1} × K_{B,v}) = J_{0,v}. Let J_{0,∞} be the product of the J_{0,v} over the infinite places v of F, and let 𝔥_{∞} be the direct sum of the 𝔥_{v} = 𝔤𝔰𝔬(X, F_{v}) = 𝔤𝔬(X, F_{v}) over the infinite places v of F. Let K_{B,∞} be the product of the K_{B,v} over the infinite places v of F and let B_{∞} be the direct sum over the infinite places v of the Lie algebra B(F_{v}) of B^{×}(F_{v}). We consider B^{×}(𝔸_{f}) × (B_{∞}, K_{B,∞}) and GSO(X, 𝔸_{f}) × (𝔥_{∞}, J_{0,∞}) modules. We will use the following facts about the tempered cuspidal automorphic representations of B^{×}(𝔸). Let Irr_{cusp}^{temp}(B^{×}(𝔸)) be the set of tempered cuspidal automorphic representations τ of B^{×}(𝔸). It is well known that B^{×}(𝔸) has the multiplicity one property, i.e., the elements of Irr_{cusp}^{temp}(B^{×}(𝔸)) of a fixed central character occur with multiplicity one in the space of cusp forms on B^{×}(𝔸) of that central character. If τ ∈ Irr_{cusp}^{temp}(B(𝔸)^{×}), then the unique space of cusp forms on B^{×}(𝔸) isomorphic to τ will be denoted by V_{τ}. Also, B^{×}(𝔸) has the strong multiplicity one property: if τ, τ′ ∈ Irr_{cusp}^{temp}(B^{×}(𝔸)) share the same central character and τ_{v} ≅ τ′_{v} for all but finitely many v, then τ ≅ τ′, so that V_{τ} = V_{τ′}. In addition, the Jacquet-Langlands correspondence gives an injection of Irr_{cusp}^{temp}(B^{×}(𝔸)) into Irr_{cusp}^{temp}(GL(2, 𝔸_{E})). This map is constructed as follows. Suppose E is a field. Since B has center E, we may regard B as an algebra over E, and by Section 2, B is a quaternion algebra over E. There is a canonical isomorphism B^{×}(𝔸) ≅ B^{×}(𝔸_{E}), and thus a bijection Irr_{cusp}^{temp}(B^{×}(𝔸)) ⟶^{∼} Irr_{cusp}^{temp}(B^{×}(𝔸_{E})). Composing with the Jacquet-Langlands map from Irr_{cusp}^{temp}(B^{×}(𝔸_{E})) to Irr_{cusp}^{temp}(GL(2, 𝔸_{E})), we get an injection Irr_{cusp}^{temp}(B^{×}(𝔸)) ↪ Irr_{cusp}^{temp}(GL(2, 𝔸_{E})) which we also call the Jacquet-Langlands correspondence, and denote by τ ↦ τ^{JL}. If E ≅ F × F, we get a similar injection, with GL(2, 𝔸_{E}) taken to be GL(2, 𝔸) × GL(2, 𝔸). Tempered cuspidal automorphic representations of GSO(X, 𝔸) and B^{×}(𝔸) may be related as in the local case. Let Irr_{cusp,f}^{temp}(𝔸^{×} × B^{×}(𝔸)) be the set of pairs (χ, τ), where τ ∈ Irr_{cusp}^{temp}(B^{×}(𝔸)) and χ is a Hecke character of 𝔸^{×} such that ω_{τ} = χ ∘ N_{F}^{E}. Let Irr_{cusp}^{temp}(GSO(X, 𝔸)) denote the set of tempered cuspidal automorphic representations of GSO(X, 𝔸). The above exact sequences give a bijection

Irr_{cusp,f}^{temp}(𝔸^{×} × B^{×}(𝔸)) ⟶^{∼} Irr_{cusp}^{temp}(GSO(X, 𝔸)).

If (χ, τ) ∈ Irr_{cusp,f}^{temp}(𝔸^{×} × B^{×}(𝔸)), then π(χ, τ) ∈ Irr_{cusp}^{temp}(GSO(X, 𝔸)) corresponding to (χ, τ) consists of the space of functions F : GSO(X, 𝔸) → ℂ for which there exists f ∈ τ so that F (ρ(t, g)) = χ(t)^{−1}f(g). The central character of π(χ, τ) is χ. If d = 1, so that E ≅ F × F and B^{×}(𝔸) ≅ D^{×}(𝔸) × D^{×}(𝔸) (see Section 2), then every element τ ∈ Irr_{cusp}^{temp}(B^{×}(𝔸)) is of the form τ_{1} ⊗ τ_{2} for some τ_{1}, τ_{2} ∈ Irr_{cusp}^{temp}(D^{×}(𝔸)), and the condition that ω_{τ} factors through N_{F}^{E} amounts to ω_{τ_{1}} = ω_{τ_{2}}. In this case ω_{τ} factors uniquely through N_{F}^{E} via χ = ω_{τ_{1}} = ω_{τ_{2}}. Also, when dealing with a four dimensional quadratic space X_{a} over F defined by a Galois action a on a given quadratic quaternion algebra B over F with center E (Section 2), we will occasionally parameterize Irr_{cusp}^{temp}(GSO(X_{a}, 𝔸)) with respect to the explicit exact sequence

1 → 𝔸_{E}^{×} → 𝔸^{×} × B^{×}(𝔸) ⟶^{ρ_{a}} GSO(X_{a}, 𝔸) → 1

derived from Proposition 2.7; by that proposition, the difference between the two parameterizations is insignificant. Tempered cuspidal automorphic representations of GSO(X, 𝔸) inherit similar properties from those of B^{×}(𝔸). The elements of Irr_{cusp}^{temp}(GSO(X, 𝔸)) have the multiplicity one property and the strong multiplicity one property. If π ∈ Irr_{cusp}^{temp}(GSO(X, 𝔸)) then the unique space of cusp forms on GSO(X, 𝔸) isomorphic to π will be denoted by V_{π}. If π ∈ Irr_{admiss}(GSO(X, 𝔸)), then we denote by s · π the GSO(X, 𝔸_{f}) × (𝔥_{∞}, J_{0,∞}) module with the same space as π, but with twisted action (s · π)(h) = π(shs) for h ∈ GSO(X, 𝔸_{f}) × J_{0,∞} and (s · π)(x) = π(Ad(s)x) for x ∈ 𝔥_{∞}. Let π ∈ Irr_{cusp}^{temp}(GSO(X, 𝔸)). Then we denote by sV_{π} the space of cusp forms sf on GSO(X, 𝔸) defined by (sf)(h) = f(shs) for h ∈ GSO(X, 𝔸) and f ∈ V_{π}. The map f ↦ sf from V_{π} with the twisted action s·π to sV_{π} with the usual action is an isomorphism; by multiplicity one, s · π ≅ π if and only if sV_{π} = V_{π}.

7. From GSO(X, 𝔸) to GO(X, 𝔸)

In this section F is a totally real number field and X is a four dimensional quadratic space over F. Let the notation be as in Section 6; following [HST], we explain how cuspidal automorphic representations of GO(X, 𝔸) are obtained from those of GSO(X, 𝔸). For each infinite place v of F let J_{0,v} be the maximal compact subgroup of GSO(X, F_{v}) defined in Section 6, and let J_{v} denote the maximal compact subgroup of GO(X, F_{v}) generated by J_{0,v} and s, where s is as in Lemma 6.1. Let J_{∞} be the product of the J_{v} over the infinite places of F. We consider GO(X, 𝔸_{f}) × (𝔥_{∞}, J_{∞}) modules. Let Irr_{cusp}^{temp}(GO(X, 𝔸)) be the set of tempered cuspidal automorphic representations of GO(X, 𝔸).

7.1 Theorem ([HST]). The group GO(X, 𝔸) has the multiplicity one property; if σ ∈ Irr_{cusp}^{temp}(GO(X, 𝔸)), denote by V_{σ} the unique space of cusp forms isomorphic to σ. Let σ ∈ Irr_{cusp}^{temp}(GO(X, 𝔸)), and let V_{σ}^{0} be the nonzero space of cusp forms on GSO(X, 𝔸) obtained by restricting the functions in V_{σ} to GSO(X, 𝔸). Either V_{σ}^{0} is irreducible as a GSO(X, 𝔸_{f}) × (𝔥_{∞}, J_{0,∞}) module, or there exists π ∈ Irr_{cusp}^{temp}(GSO(X, 𝔸)) such that s · π ≇ π and V_{σ}^{0} = V_{π} ⊕ sV_{π} (internal direct sum). Thus, there is a map

Irr_{cusp}^{temp}(GO(X, 𝔸)) → ⟨s⟩\ Irr_{cusp}^{temp}(GSO(X, 𝔸)),

and if σ ↦ [π] = {π, s · π}, then

(7.1)  σ_{v} ↪ Ind_{GSO(X,F_{v})}^{GO(X,F_{v})} π_{v}

for all v. The map σ ↦ [π] is surjective. If [π] ∈ ⟨s⟩\ Irr_{cusp}^{temp}(GSO(X, 𝔸)) and s · π ≇ π, then the fiber over [π] is the set of all σ ∈ Irr_{admiss}(GO(X, 𝔸)) such that (7.1) holds for all places v of F.

Proof. See Section 1 of [HST]. □

8. Proofs of the main theorems

Let F be a totally real number field. In this final section we prove the main results Theorems 8.3 and 8.6 presented in the Introduction. Besides the general foundational work of Sections 1, 2 and 5, the main ingredients for Theorem 8.3 are the local results of Section 3 and the general nonvanishing result for global theta lifts from [R4]. Globally, the result from [R4] requires the nonvanishing of a certain L-function at s = 1; in the case at hand, this L-function turns out to be either a partial GL(2) × GL(2) L-function or a partial twisted Asai L-function, so that the nonvanishing at s = 1 follows from [Sh]. To prove Theorem 8.6 we actually first prove a different version, Theorem 8.5. In this version, using Section 4, a global L-packet Π([π]) of tempered irreducible admissible representations of GSp(2, 𝔸) is assigned to every element π of Irr_{cusp}^{temp}(GSO(X_{M_{2×2},d}, 𝔸)). When s · π ≇ π, Theorem 8.5 determines exactly what elements of Π([π]) are cuspidal automorphic and shows that the cuspidal automorphic elements occur with multiplicity one. In addition to an understanding of the local situation, the main tool for showing cuspidality is Theorem 8.3. For multiplicity one, we use the Rallis multiplicity preservation principle in the context of similitudes (Proposition 5.3), along with the nonvanishing result for global theta lifts from Sp(2, 𝔸) from [KRS]. This result shows that if a twisted partial standard L-function of a cuspidal automorphic representation of Sp(2, 𝔸) has a pole at s = 1, then it has a nonzero theta lift to the isometry group of a certain four dimensional quadratic space. Theorem 8.6 follows directly from Theorem 8.5. We begin with a lemma which computes the standard partial L-function of an O(X, 𝔸) component of a cuspidal automorphic representation of GO(X, 𝔸) for a four dimensional quadratic space X over F. In the following lemma, L_{v}(s, τ^{JL}, χ^{−1}, Asai) is the v-th Euler factor of the Asai L-function of τ^{JL} twisted by χ^{−1} ([HLR], p. 64–5); and L_{v}(s, τ_{1}^{JL} × τ_{2}^{JL∨}) is the v-th Euler factor of the usual Rankin–Selberg GL(2) × GL(2) L-function of τ_{1}^{JL} and τ_{2}^{JL∨}; here, the superscript JL indicates the corresponding element under the Jacquet-Langlands correspondence (Section 6). Also, under the assumption that X(F_{v}) is unramified (Section 1), when we say that an irreducible admissible representation of GO(X, F_{v}) (or of O(X, F_{v}) and SO(X, F_{v})) is unramified we mean with respect to the stabilizer in GO(X, F_{v}) ( or in O(X, F_{v}) and SO(X, F_{v}), respectively) of a self-dual lattice in X(F_{v}).

8.1 Lemma. Let X be a four dimensional quadratic space over F, let B be the even Clifford algebra of X(F), and let E be the center of B. Let d = disc X(F). Let σ ∈ Irr_{cusp}^{temp}(GO(X, 𝔸)), and assume that σ lies over [π = π(χ, τ)] (See Sections 6 and 7). Let v be a finite place of F such that X(F_{v}) and σ_{v} are unramified. Let σ_{1,v} be the unramified component of σ_{v}|_{O(X,F_{v})}. Then the standard L-function of σ_{1,v} is

L(s, σ_{1,v}) = \{ \begin{matrix}L_{v}(s, τ^{JL}, χ^{−1}, Asai) if d ≠ 1, \\ L_{v}(s, τ_{1}^{JL} × τ_{2}^{JL}^{∨}) if d = 1 and τ ≅ τ_{1} ⊗ τ_{2}. \end{matrix}

Proof. By definition, L(s, σ_{1,v}) (see Section 2 of [KR1]) is the standard Lfunction of any irreducible unramified component of σ_{1,v}|_{SO(X,F_{v})}. It will thus suffice to show that the standard L-function of any irreducible unramified component of σ_{v}|_{SO(X,F_{v})} has the stated form; and since π_{v} is an irreducible component of σ_{v}|_{GSO(X,F_{v})}, it will be enough to show that the standard L-function of any irreducible unramified component of π_{v}|_{SO(X,F_{v})} or (s · π_{v})|_{SO(X,F_{v})} has the above form (s is as in Lemma 6.1). Since over a local nonarchimedean field a four dimensional quadratic space represents 1, by Proposition 2.8 there exists a quaternion algebra D over F_{v} contained in B(F_{v}) and an isometry T : X(F_{v}) → X_{D,E_{v}} such that

1 ⟶ E_{v}^{×} ⟶ F_{v}^{×} × B^{×}(F_{v}) ⟶^{ρ} GSO(X, F_{v}) ⟶ 1
id↓  ^{≀}↓  ↓_{T ·T^{−1}}
1 ⟶ E_{v}^{×} ⟶ F_{v}^{×} × B_{D,E_{v}}^{×} ⟶^{ρ_{a(D,Ev)}} GSO(X_{D,E_{v}}, F_{v}) ⟶ 1

commutes, where B^{×}(F_{v}) ⟶^{∼} B_{D,E_{v}}^{×} is the isomorphism induced by the natural isomorphism B(F_{v}) ≅ E_{v} ⊗_{F_{v}} D of E_{v} algebras; E_{v} = E(F_{v}) = F_{v} ⊗_{F} E. Since X(F_{v}) is unramified, D is in particular split, i.e., there exists an isomorphism D ⟶^{∼} M_{2×2}(F_{v}) of quaternion algebras over F_{v}. From this, we obtain an isomorphism B_{D,E_{v}} ⟶^{∼} M_{2×2}(E_{v}) of E_{v} algebras and an isometry t : X_{D,E_{v}} ⟶^{∼} X_{a} so that

1 ⟶ E_{v}^{×} ⟶ F_{v}^{×} × B_{D,E_{v}}^{×}  ⟶^{ρ_{a(D,Ev)}} GSO(X_{D,E_{v}}, F_{v}) ⟶ 1
id↓  ^{≀}↓  ↓_{t·t^{−1}}
1 ⟶ E_{v}^{×} ⟶ F_{v}^{×} × GL(2, E_{v}) ⟶^{ρ_{a}} GSO(X_{a}, F_{v}) ⟶ 1

commutes. Here, a is the Galois action on M_{2×2}(E_{v}) defined by the formula (2.1). Composing, we now have an isomorphism i : B(F_{v}) ⟶^{∼} M_{2×2}(E_{v}) of E_{v} algebras and isometry r : X(F_{v}) ⟶^{∼} X_{a} such that

1 ⟶ E_{v}^{×} ⟶ F_{v}^{×} × B^{×}(F_{v}) ⟶^{ρ} GSO(X, F_{v}) ⟶ 1
id↓  ^{≀}↓  ↓_{r·r^{−1}}
1 ⟶ E_{v}^{×} ⟶ F_{v}^{×} × GL(2, E_{v}) ⟶^{ρ_{a}} GSO(X_{a}, F_{v}) ⟶ 1

commutes. Let π′_{v} be the representation of GSO(X_{a}, F_{v}) corresponding to π_{v}. By definition, (τ_{v})^{JL} = τ_{v} ∘ i, and we have π′_{v} = π(χ_{v}, (τ_{v})^{JL}). Since the standard L-function of any unramified irreducible component of π_{v}|_{SO(X,F_{v})} is the same as the standard L-function of any irreducible unramified component of π′_{v}|_{SO(X_{a},F_{v})}, and the same holds for (s · π_{v})|_{SO(X,F_{v})} and (s · π′_{v})|_{SO(X_{a},F_{v})}, it will now suffice to show that the the standard L-function of any irreducible unramified component of π′_{v}|_{SO(X_{a},F_{v})} or (s · π′_{v})|_{SO(X_{a},F_{v})} has the above form. Assume first d ≠ 1 (i.e., E is a field) and v stays prime in E; let w be the place of E lying over v. Then E_{w} = E_{v}. Since π′_{v} is unramified, so are χ_{v} and (τ^{JL})_{w} = (τ_{v})^{JL} ∈ Irr(GL(2, E_{w})). Let (τ^{JL})_{w} = Ind_{P}^{GL(2,E_{w})}(μ_{1} ⊗ μ_{2}), where P is the usual upper triangular Borel subgroup of GL(2, E_{w}), induction is normalized, μ_{1} and μ_{2} are unramified quasi-characters of E_{w}^{×}, and μ_{1} ⊗ μ_{2} is defined by

(μ_{1} ⊗ μ_{2})([ \begin{matrix}a & b \\ 0 & c \end{matrix} ]) = μ_{1}(a)μ_{2}(c).

The space X_{a} was explicitly described in Section 2. With respect to the ordered basis

[ \begin{matrix}0 & \sqrt{d} \\ 0 & 0 \end{matrix} ] , [ \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ] , [ \begin{matrix}\sqrt{d} & 0 \\ 0 & −\sqrt{d} \end{matrix} ] , [ \begin{matrix}0 & 0 \\ −(2/d)\sqrt{d} & 0 \end{matrix} ]

the symmetric bilinear form on X_{a}, which is given by the determinant, has the form

[ \begin{matrix}0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & −d & 0 \\ 1 & 0 & 0 & 0 \end{matrix} ] .

The stabilizer in GSO(X_{a}, F_{v}) of the isotropic subspace spanned by the first basis vector is a Borel subgroup P′ of GSO(X_{a}, F_{v}), and P′ = ρ_{α}(F_{v}^{×} × P). In particular, we have

ρ_{α}(t, [ \begin{matrix}a & ∗ \\ 0 & c \end{matrix} ]) = t^{−1} [ \begin{matrix}N_{F_{v}}^{E_{w}}(a) & ∗ & ∗ \\ 0 & h & ∗ \\ 0 & 0 & N_{F_{v}}^{E_{w}}(c) \end{matrix} ] ,

with

h = [ \begin{matrix}a_{1}c_{1} − a_{2}c_{2}d & (a_{2}c_{1} − a_{1}c_{2})d \\ a_{2}c_{1} − a_{1}c_{2} & a_{1}c_{1} − a_{2}c_{2}d \end{matrix} ] ,

where a = a_{1} + a_{2}\sqrt{d} and c = c_{1} + c_{2}\sqrt{d}. Here, the middle block h corresponds to multiplication by aα(c) on the two dimensional subspace spanned by the two middle basis vectors, using the obvious identification of this subspace with E_{w}. Recalling that χ_{v} ∘ N_{F_{v}}^{E_{w}} = μ_{1}μ_{2}, a computation shows that

π′_{v} = π(χ_{v}, (τ^{JL})_{w}) = Ind_{P′}^{GSO(X_{a},F_{v})} μ,

where induction is normalized, and on the typical element of P′ μ takes the value

μ([ \begin{matrix}a & ∗ & ∗ \\ 0 & h & ∗ \\ 0 & 0 & λa^{−1} \end{matrix} ]) = (μ_{2}/χ_{v})(λa^{−1})μ_{1}(h),

where again we identify the elements of the middle block with E_{w}^{×} and a, λ ∈ F_{v}^{×}. There is an SO(X_{a}, F_{v}) isomorphism

π′_{v}|_{SO(X_{a},F_{v})} ⟶^{∼} Ind_{P′∩SO(X_{a},F_{v})}^{SO(X_{a},F_{v})} μ|_{P′∩SO(X_{a},F_{v})}

given by restriction of functions. We have

μ|_{P′∩SO(X_{a},F_{v})}([ \begin{matrix}a & ∗ & ∗ \\ 0 & h & ∗ \\ 0 & 0 & a^{−1} \end{matrix} ]) = (χ_{v}/μ_{2})(a)

since N_{F_{v}}^{E_{w}}(h) = 1, so that h ∈ 𝔒_{w}^{×}. By definition, the standard L-function of any irreducible unramified component of π′_{v}|_{SO(X_{a},F_{v})} is now

L(s, χ_{v}/μ_{2})L(s, μ_{2}/χ_{v})ζ_{F_{v}}(2s) = det(1 − χ(π_{F_{v}})^{−1}A|π_{F_{v}}|^{s})^{−1}
= L_{v}(s, τ^{JL}, χ^{−1}, Asai),

where

A = [ \begin{matrix}μ_{1}(π_{F_{v}}) & 0 & 0 & 0 \\ 0 & 0 & μ_{1}(π_{F_{v}}) & 0 \\ 0 & μ_{2}(π_{F_{v}}) & 0 & 0 \\ 0 & 0 & 0 & μ_{2}(π_{F_{v}}) \end{matrix} ] .

For the last equality, see p. 64-65 of [HLR]. Since s · π′_{v} = π(χ_{v}, (τ^{JL})_{w} ∘ α), a similar computation shows that the standard L-function of any irreducible unramified component of (s · π′_{v})|_{SO(X_{a},F_{v})} is also L_{v}(s, τ^{JL}, χ^{−1}, Asai). Now suppose E is a field and v splits in E. Then F_{v} contains a square root of d; fix such a square root \sqrt{d} in F_{v}^{×}. Define an embedding of fields i_{1} : E ↪ F_{v} by sending a fixed square root of d in E to \sqrt{d}, and define another embedding i_{2} : E ↪ F_{v} by sending the fixed square root of d in E to −\sqrt{d}. We denote by w_{1} and w_{2} the places of E determined by i_{1} and i_{2}, respectively. Then w_{1} and w_{2} are the two places of E lying over v, and via i_{1} and i_{2} we take F_{v} to be the completions E_{w_{1}} and E_{w_{2}} of E at w_{1} and w_{2}, respectively. We also have an identification of E_{v} = F_{v}⊗_{F} E with E_{w_{1}} ×E_{w_{2}} and hence with F_{v}×F_{v}. Using the identification E_{v} ≅ F_{v} ×F_{v} we may identify M_{2×2}(E_{v}) with M_{2×2}(F_{v})×M_{2×2}(F_{v}), GL(2, E_{v}) with GL(2, F_{v}) × GL(2, F_{v}) and a with the Galois action defined by (x_{1}, x_{2}) ↦ (x_{2}, x_{1}). Further, as explained after Proposition 2.7, we may identify X_{a} with M_{2×2}(F_{v}) and ρ_{a} with ρ_{a}(t, (g_{1}, g_{2}))x = t^{−1}g_{1}xg_{2}^{∗}. Using the canonical isomorphisms B(F_{v}) ≅ B(E_{w_{1}}) × B(E_{w_{2}}) ≅ D × D write τ_{v} ≅ τ_{w_{1}} ⊗ τ_{w_{2}} with τ_{w_{1}}, τ_{w_{2}} ∈ Irr(D^{×}); then (τ_{v})^{JL} ≅ τ_{w_{1}}^{JL} ⊗ τ_{w_{2}}^{JL}. Let τ_{w_{1}}^{JL} = Ind_{P}^{GL(2,F_{v})}(μ_{1} ⊗ μ_{2}) and τ_{w_{2}}^{JL} = Ind_{P}^{GL(2,F_{v})}(μ′_{1} ⊗ μ′_{2}), with the notation analogous to the previous case. Note that χ = μ_{1}μ_{2} = μ′_{1}μ′_{2}. With respect to the ordered basis

[ \begin{matrix}0 & 1 \\ 0 & 0 \end{matrix} ] , [ \begin{matrix}1 & 0 \\ 0 & 0 \end{matrix} ] , [ \begin{matrix}0 & 0 \\ −2 & 0 \end{matrix} ] , [ \begin{matrix}0 & 0 \\ 0 & 2 \end{matrix} ]

the symmetric bilinear form on X_{a} has the matrix

[ \begin{matrix}0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{matrix} ] .

The stabilizer in GSO(X_{a}, F_{v}) of the isotropic flag

F_{v} · [ \begin{matrix}0 & 1 \\ 0 & 0 \end{matrix} ] ⊂ F_{v} · [ \begin{matrix}0 & 1 \\ 0 & 0 \end{matrix} ] + F_{v} · [ \begin{matrix}1 & 0 \\ 0 & 0 \end{matrix} ]

is a Borel subgroup P′, and P′ = ρ_{α}(F_{v}^{×} × P × P). We have

ρ_{α}(t, [ \begin{matrix}a & ∗ \\ 0 & c \end{matrix} ] , [ \begin{matrix}a′ & ∗ \\ 0 & c′ \end{matrix} ]) = t^{−1} [ \begin{matrix}aa′ & ∗ & ∗ & ∗ \\ 0 & ac′ & ∗ & ∗ \\ 0 & 0 & cc′ & ∗ \\ 0 & 0 & 0 & a′c \end{matrix} ] .

Using μ_{1}μ_{2} = μ′_{1}μ′_{2}, a computation shows that

π′_{v} ≅ Ind_{P′}^{GSO(X_{a},F_{v})} μ

with

μ([ \begin{matrix}a & ∗ & ∗ & ∗ \\ 0 & b & ∗ & ∗ \\ 0 & 0 & λa^{−1} & ∗ \\ 0 & 0 & 0 & λb^{−1} \end{matrix} ]) = μ_{2}(λ)(μ′_{1}/μ_{2})(a)(μ′_{2}/μ_{2})(b).

Again there is an SO(X_{a}, F_{v}) isomorphism

π′_{v}|_{SO(X_{a},F_{v})} ⟶^{∼} Ind_{P′∩SO(X_{a},F_{v})}^{SO(X_{a},F_{v})} μ|_{P′∩SO(X_{a},F_{v})}

given by restriction of functions. We have

μ|_{P′∩SO(X_{a},F_{v})}([ \begin{matrix}a & ∗ & ∗ & ∗ \\ 0 & b & ∗ & ∗ \\ 0 & 0 & a^{−1} & ∗ \\ 0 & 0 & 0 & b^{−1} \end{matrix} ]) = (μ′_{1}/μ_{2})(a)(μ′_{2}/μ_{2})(b).

We now have that the standard L-function of any irreducible unramified component of π′_{v}|_{SO(X_{a},F_{v})} is

L(s, μ′_{1}/μ_{2})L(s, μ_{2}/μ′_{1})L(s, μ′_{2}/μ_{2})L(s, μ_{2}/μ′_{2}) = det(1 − χ(π_{F_{v}})^{−1}A|π_{F_{v}}|^{s})^{−1}
= L_{v}(s, τ^{JL}, χ^{−1}, Asai),

where

A = [ \begin{matrix}(μ_{1}μ′_{1})(π_{F_{v}}) & 0 & 0 & 0 \\ 0 & (μ_{2}μ′_{2})(π_{F_{v}}) & 0 & 0 \\ 0 & 0 & (μ_{1}μ′_{2})(π_{F_{v}}) & 0 \\ 0 & 0 & 0 & (μ_{2}μ′_{1})(π_{F_{v}}) \end{matrix} ];

here we have used χ_{v} = μ_{1}μ_{2} = μ′_{1}μ′_{2}. For the last equality, again see p. 64-65 of [HLR]. Since s · π′_{v} = π(χ_{v}, τ_{w_{2}}^{JL} ⊗ τ_{w_{1}}^{JL}), a similar computation shows that the standard L-function of any irreducible unramified component of (s · π′_{v})|_{SO(X_{a},F_{v})} is also L_{v}(s, τ^{JL}, χ^{−1}, Asai). The argument in the case d = 1 is similar to the last case and will be omitted. □

To prove the nonvanishing part of the main result Theorem 8.3 we will use the following theorem, which follows from Corollary 1.2 of [R4]. In the following L^{S}(s, σ_{1}) is the standard partial L-function of σ_{1} (see Section 2 of [KR1]).

8.2 Theorem ([R4]). Let F be a totally real number field, and let X be a four dimensional quadratic space over F. Let d ∈ F^{×}/F^{×2} be the discriminant of X(F), and assume that the discriminant algebra E of X(F) is totally real, i.e., either d = 1 or d ≠ 1 and E = F (\sqrt{d}) is totally real. Let σ_{1} be a tempered cuspidal automorphic representation of O(X, 𝔸) with σ_{1} ≅ ⊗_{v}σ_{1,v}, and let V_{σ_{1}} be a realization of σ_{1} in the space of cusp forms on O(X, 𝔸). Assume σ_{1,v} occurs in the theta correspondence for O(X, F_{v}) and Sp(2, F_{v}) for all places v. If L^{S}(s, σ_{1}) does not vanish at s = 1 then Θ_{2}(V_{σ_{1}}) ≠ 0.

Proof. This follows from Corollary 1.2 of [R4] (see also the following remark below). Note that by the assumption on E, at each infinite place v of F we have d = 1 in F_{v}^{×}/F_{v}^{×2}, so that the signature of X(F_{v}) is (4, 0), (2, 2) or (0, 4) and the signature assumptions from Corollary 1.2 of [R4] are satisfied. □

We take the opportunity here to make a correction to [R4]. Namely, in Theorem 1.1 of [R4] hypothesis (2) should be replaced with the statement: for all places v, σ_{v} is tempered and if σ_{v} first occurs in the theta correspondence with Sp(n′, F_{v}) with 2n′ > dim X, then the first occurrence of σ_{v} is tempered; in Corollary 1.2 of [R4] σ_{v} should also be assumed to be tempered for infinite v; and finally in Lemma 2.1 of [R4] the assumption on σ (in both the nonarchimedean and real cases) should be that σ is tempered, and if σ first occurs in the theta correspondence with Sp(n′, F) with 2n′ > dim X, then the first occurrence of σ is tempered. The corrections thus also introduce temperedness assumptions at the infinite places entirely analogous to those at the finite places (note that in the corrections to Theorem 1.1 and Lemma 2.1 we have actually weakened the nonarchimedean assumption; this was mentioned in [R4], but not explicitly stated as part of Theorem 1.1 and Lemma 2.1). The omission of these temperedness assumptions at infinity was due to a misreading of [M], Corollaire IV.5 (ii). The only place where the result from [M] is used in [R4] is in the proof of Lemma 2.1 of [R4] where it is asserted that, in the terminology of that lemma, θ_{k+1}(σ) = L(χ_{X}| · |^{s_{X}(k+1)} ⊗ δ_{2} ⋯ ⊗δ_{t} ⊗ τ). The argument for this is as follows. Assume σ first occurs in the theta correspondence with Sp(n′, ℝ) with n′ ≤ dim X/2. Then σ occurs in the theta correspondence with Sp(dim X/2, ℝ) (Lemme I.9, p. 14, [M]) and θ_{dim X/2}(σ) = Ψ_{dim X/2}(σ) (Théorème IV.3, p. 70, [M]). Since σ is tempered, by the definition of Ψ_{dim X/2}(σ) (III.2, p. 49, [M]), θ_{dim X/2}(σ) = Ψ_{dim X/2}(σ) is also tempered. The Langlands data for θ_{k+1}(σ) is obtained from the Langlands data of θ_{dim X/2}(σ) by adjoining the quasicharacters of ℝ^{×}: χ_{X}| · |^{s_{X}(k+1)}, …, χ_{X}| · |^{s_{X}(dim X/2)+2}, χ_{X}| · |^{s_{X}(dim X/2)+1} (Corollaire IV.5 (ii), p. 71, [M]). Since θ_{dim X/2}(σ) is tempered, this implies θ_{k+1}(σ) has the claimed form. Next, assume σ first occurs in the theta correspondence with Sp(n′, ℝ) with n′ > dim X/2. Then θ_{n′}(σ) is tempered by assumption. Again, the Langlands data of θ_{k+1}(σ) is obtained from the Langlands data of θ_{n′}(σ) by adjoining the quasi-characters of ℝ^{×}: χ_{X}| · |^{s_{X}(k+1)}, …, χ_{X}| · |^{s_{X}(n′)+2}, χ_{X}| · |^{s_{X}(n′)+1} (Corollaire IV.5 (ii), p. 71, [M]). Again, since θ_{n′}(σ) is tempered, this implies θ_{k+1}(σ) has the claimed form. This completes the corrected argument for the new statement of Lemma 2.1 of [R4]. The corrected statements of Theorem 1.1 and Corollary 1.2 have exactly the same proofs as in [R4].

Proof of Theorem 8.3. (1) ⟹ (2). Suppose Θ_{2}(V_{σ}) ≠ 0. Suppose Θ_{2}(V_{σ}) is contained in the space of cusp forms. Then by Proposition 5.3, (2) holds. Suppose Θ_{2}(V_{σ}) is not contained in the space of cusp forms. Since σ_{v} is infinite dimensional for at least one v (5.1) holds. By Proposition 5.2, Θ_{1}(V_{σ}) is contained in the space cusp forms; also, Θ_{1}(V_{σ}) is nonzero, for otherwise, by Proposition 5.2, Θ_{2}(V_{σ}) would be contained in the space of cusp forms. A standard argument as in the proof of Proposition 5.3 now shows that for all v, σ_{v} ∈ ℛ_{1}(GO(X, F_{v})). This implies (2) (Lemma 4.2 of [R1]; b), p. 67 of [MVW]; Lemme I.9 of [M]).

(2) ⟺ (3). This is Theorem 3.4. Note that if disc X(F_{v}) ≠ 1, then X(F_{v}) is isotropic, and so σ_{v} is infinite dimensional (as σ_{v} is tempered).

(2) ⟹ (1). Suppose (2) holds. Let σ lie over [π] (Section 7). Restrict the functions in V_{σ} to O(X, 𝔸) to obtain the space of functions V_{σ}^{1}. Then V_{σ}^{1} is nonzero and contained in the space of cusp forms on O(X, 𝔸); let W be an irreducible nonzero O(X, 𝔸_{f}) × (𝔥_{1,∞}, J_{1,∞}) component of V_{σ}^{1}, and denote the isomorphism class of W by σ_{1}. To show Θ_{2}(V_{σ}) ≠ 0 it will suffice to show Θ_{2}(W) ≠ 0 (for if θ_{2}(f, φ) ≠ 0 for some f ∈ W and φ ∈ 𝒮(X(𝔸)^{2}), then θ_{2}(F, φ)|_{Sp(2,𝔸)} = θ_{2}(f, φ) ≠ 0 for any F ∈ V_{σ} with F |_{O(X,𝔸)} = f). For this, we will use Theorem 8.2. We need to see that the hypotheses of Theorem 8.2 are satisfied. For all places v of F, σ_{1,v} is an irreducible constituent of σ_{v}|_{O(X,F_{v})}. Since σ_{v} is tempered for all v, σ_{1,v} is tempered for all v. Also, it is a basic consequence of (2) that σ_{1,v} ∈ ℛ_{2}(O(X, F_{v})) for all v (Lemma 4.2 of [R1]; see the discussion before Theorem 1.8). Finally, we need to see that the partial standard L-function L^{S}(s, σ_{1}) of σ_{1} does not vanish at s = 1. Writing π = π(χ, τ), by Lemma 8.1 we have

L^{S}(s, σ_{1}) = \{ \begin{matrix}L^{S}(s, τ^{JL}, χ^{−1}, Asai) if d ≠ 1 \\ L^{S}(s, τ_{1}^{JL} × τ_{2}^{JL}^{∨}) if d = 1 and τ ≅ τ_{1} ⊗ τ_{2}. \end{matrix}

Showing the nonvanishing of L^{S}(s, σ_{1}) at s = 1 is thus reduced to showing the nonvanishing of these two types of L-functions at s = 1. For the nonvanishing of L^{S}(s, τ_{1}^{JL} × τ_{2}^{JL∨}) at s = 1 see Theorem 5.2 of [Sh]. The nonvanishing of L^{S}(s, τ^{JL}, χ^{−1}, Asai) at s = 1 also follows from [Sh]. For an explanation of this, see p. 296–7 of [F]. Note that L^{S}(s, τ^{JL}, χ^{−1}, Asai) is of the form L^{S}(s, τ′, Asai): there exists a Hecke character χ̂ of 𝔸_{E}^{×} extending χ, and for such a χ̂ we have L^{S}(s, τ ⊗ χ̂^{−1}, Asai) = L^{S}(s, τ^{JL}, χ^{−1}, Asai). By Theorem 8.2 we now have Θ_{2}(W) ≠ 0, and so Θ_{2}(V_{σ}) ≠ 0. Now suppose that one of (1), (2) or (3) holds, and s · π ≇ π. By what we have already shown, Θ_{2}(V_{σ}) ≠ 0. We claim that Θ_{2}(V_{σ}) is contained in the space of cusp forms. Suppose not. Then as in the proof of (1) ⟹ (2), Θ_{1}(V_{σ}) is nonzero and contained in the space of cusp forms, and in particular σ_{v} ∈ ℛ_{1}(GO(X, F_{v})) for all v. By Theorem 7.4 of [R2] this implies s · π_{v} ≅ π_{v} at least for all finite v of odd residual characteristic. However, by strong multiplicity one for GSO(X, 𝔸) (Section 6) and s · π ≇ π, we have s · π_{v} ≇ π_{v} for infinitely many v, a contradiction. Thus, Θ_{2}(V_{σ}) is contained in the space of cusp forms. By Proposition 5.3, Θ_{2}(V_{σ}) is a cuspidal automorphic representation of GSp(2, 𝔸) of central character ω_{σ} and Θ_{2}(V_{σ}) = ⊗_{v}θ_{2}(σ_{v}^{∨}); by Proposition 1.10 this is also ⊗_{v}θ_{2}(σ_{v})^{∨}. The proof that θ_{2}(σ_{v}) is tempered for all v is as in the proof of (3) of Proposition 4.1. □

The next lemma was used in the proof of Proposition 4.1 to show that the two elements of an L-packet defined there are in fact distinct. It will also be used in the proof of Theorem 8.5.

8.4 Lemma. Let v be a place of F. Let D_{ram} be the division quaternion algebra over F_{v}, and define the four dimensional quadratic spaces X_{M_{2×2},1} and X_{D_{ram},1} over F_{v} as in Section 2. Then ℛ_{X_{M_{2×2},1}}(GSp(2, F_{v})) ∩ ℛ_{X_{Dram,1}}(GSp(2, F_{v})) = ∅.

Proof. We will use the notation of Section 1. By Lemmas 1.4 and 1.5 it will suffice to show that if

ℛ_{X_{M_{2×2},1}}(GSp(2, F_{v})) ∩ ℛ_{X_{Dram,1}}(GSp(2, F_{v})) ≠ ∅

then

ℛ_{X_{M_{2×2},1}}(Sp(2, F_{v})) ∩ ℛ_{X_{Dram,1}}(Sp(2, F_{v})) ≠ ∅.

Suppose Π ∈ ℛ_{X_{M_{2×2},1}}(GSp(2, F_{v})) ∩ ℛ_{X_{Dram,1}}(GSp(2, F_{v})). Since Π is contained in ℛ_{X_{M_{2×2},1}}(GSp(2, F_{v})), by definition Π|_{Sp(2,F_{v})} is multiplicity free; let Π|_{Sp(2,F_{v})} = W_{1} ⊕ ⋯ ⊕ W_{M} with the W_{i}, 1 ≤ i ≤ M, mutually nonisomorphic irreducible Sp(2, F_{v}) subspaces of Π. Also by definition, some W_{i}, say W_{1}, is in ℛ_{X_{M_{2×2},1}}(Sp(2, F_{v})). We assert that all the W_{i} are contained in ℛ_{X_{M_{2×2},1}}(Sp(2, F_{v})). Let g ∈ GSp(2, F_{v}) be such that π(g)W_{1} = W_{i} (if F_{v} ≅ ℝ then M = 1 or 2 and we may take g = k_{0} with k_{0} as in Section 1). Since W_{1} ∈ ℛ_{X_{M_{2×2},1}}(Sp(2, F_{v})) there exists a nonzero Sp(2, F_{v}) map t : ω_{X_{M_{2×2},1}} → W_{1}. Let h ∈ GO(X_{M_{2×2},1}, F_{v}) be such that (g, h) ∈ R_{X_{M_{2×2},1}} (if F_{v} ≅ ℝ we take h = j_{0} so that (g, h) ∈ L). Consider the composition

ω_{X_{M_{2×2},1}} ⟶^{ω(g,h)^{−1}} ω_{X_{M_{2×2},1}} ⟶^{t} W_{1} ⟶^{π(g)} W_{i}.

This is a nonzero Sp(2, F) map. Thus, W_{i} ∈ ℛ_{X_{M_{2×2},1}}(Sp(2, F_{v})). On the other hand, since Π ∈ ℛ_{X_{Dram,1}}(GSp(2, F_{v})) we have by definition that some irreducible component of Π|_{Sp(2,F_{v})} is contained in ℛ_{X_{Dram,1}}(Sp(2, F_{v})). We now have ℛ_{X_{M_{2×2},1}}(Sp(2, F_{v})) ∩ ℛ_{X_{Dram,1}}(Sp(2, F_{v})) ≠ ∅ as desired. □

We come now to the definition and analysis of global L-packets for GSp(2). We begin by proving Theorem 8.5, a version of the main result Theorem 8.6. In this version, global L-packets for GSp(2) are associated to elements of Irr_{cusp}^{temp}(GSO(X_{M_{2×2},d}, 𝔸)); Theorem 8.6 will follow easily from Theorem 8.5. As in Section 2, let d ∈ F^{×}/F^{×2}, and let E = E_{d} be F (\sqrt{d}) if d ≠ 1 and E = E_{d} = F × F if d = 1. Assume E is totally real, i.e., in the case d ≠ 1 assume E is totally real. Let π ∈ Irr_{cusp}^{temp}(GSO(X_{M_{2×2},d}, 𝔸)). The packet of irreducible admissible representations of GSp(2, 𝔸) corresponding to [π] is defined to be

Π([π]) = {Π = ⊗_{v}Π_{v} ∈ Irr_{admiss}(GSp(2, 𝔸)) : Π_{v} ∈ Π([π_{v}]) for all v}.

Here, Π([π_{v}]) is defined in Section 4. By Proposition 4.3, for almost all nonarchimedean v, Π([π_{v}]) consists of a single representation unramified with respect to GSp(2, 𝔒_{v}). Thus,

Π([π]) = ⊗_{v}Π([π_{v}]).

Also, by (3) of Proposition 4.1, Π([π_{v}]) consists of tempered representations for all v. If S is any finite set of places such that for v ∉ S, v is nonarchimedean and Π([π_{v}]) consists of a single representation unramified with respect to GSp(2, 𝔒_{v}), then the cardinality of Π([π]) is:

|Π([π])| = ∏_{v∈S} |Π([π_{v}])| = 2^{M}, where M = ∑_{v∈S}(|Π([π_{v}])| − 1).

For Π = ⊗_{v}Π_{v} ∈ Π([π]), let T_{Π} be the set of places v of F such that v splits in E (as usual, if d = 1 so that E = F × F we say that every place of F splits in E) and Π_{v} is of the form θ_{D_{ram},1}(π_{v}^{JL+})^{∨} (so necessarily π_{v} is square integrable); see Section 4.

8.5 Theorem. Assume F is totally real, d ∈ F^{×}/F^{×2}, and let E = E_{d} be F (\sqrt{d}) if d ≠ 1 and F × F if d = 1. Assume E is totally real, i.e., in the case d ≠ 1 assume E is totally real. Let π ∈ Irr_{cusp}^{temp}(GSO(X_{M_{2×2},d}, 𝔸)) and assume s · π ≇ π.

(1) If d ≠ 1, then all the elements of Π([π]) occur with multiplicity one in the space of cusp forms on GSp(2, 𝔸) with central character ω_{π}.

(2) Assume d = 1. Let Π ∈ Π([π]). If |T_{Π}| is even, then Π occurs with multiplicity one in the space of cusp forms on GSp(2, 𝔸) with central character ω_{π}. Conversely, if Π occurs in the space of cusp forms on GSp(2, 𝔸) then |T_{Π}| is even.

Proof. Let Π ∈ Π([π]); if d = 1 assume |T_{Π}| is even. We begin by showing that Π occurs in the space of cusp forms on GSp(2, 𝔸) of central character ω_{π}. To prove this we will construct a four dimensional quadratic space X over F and a σ ∈ Irr_{cusp}^{temp}(GO(X, 𝔸)) such that σ_{v} ∈ ℛ_{2}(GO(X, F_{v})) and θ_{2}(σ_{v})^{∨} = Π_{v} for all v; we will then apply Theorem 8.3 to show Π is cuspidal automorphic. To start, let us set up some definitions involving π. As in Section 6, write π = π(χ, τ); however instead of the abstract exact sequence of Theorem 2.3, let us use the concrete exact sequence

1 → 𝔸_{E}^{×} → 𝔸^{×} × GL(2, 𝔸_{E}) ⟶^{ρ_{a(M_{2×2},E)}} GSO(X_{M_{2×2},d}, 𝔸) → 1

of Proposition 2.7; by this proposition there is no real distinction. Thus, π = π(χ, τ), with τ ∈ Irr_{cusp}^{temp}(GL(2, 𝔸_{E})) and χ a Hecke character of 𝔸^{×} such that ω_{τ} = χ ∘ N_{F}^{E}. The X we will use will be of the form X_{D,d}. Specifically, let D be any quaternion algebra over F which is ramified at the places in T_{Π}, and which is unramified at any other of the places of F which split in E; note that if d = 1, we use the evenness of |T_{Π}| for the existence of D (again, our convention is that if d = 1 so that E = F × F then every place of F is split in E). Evidently, if d = 1, then D is uniquely determined, but if d ≠ 1, then there will be infinitely many choices for D. Nevertheless, if we let B = E ⊗_{F} D, then B is uniquely determined (in the case d ≠ 1, regarded as a quaternion algebra over E, B is split at any place of E lying over a nonsplit place of F), and X_{D,d} is uniquely determined up to similitudes by Proposition 2.9. By Lemma 5.1, it thus follows that our construction will realize Π as a cuspidal automorphic representation in exactly one way in spite of the ambiguity in the choice of D when d ≠ 1. To define the σ mentioned above, note that again by Proposition 2.7 we have an exact sequence

1 → 𝔸_{E}^{×} → 𝔸^{×} × B^{×}(𝔸) ⟶^{ρ_{a(D,E)}} GSO(X_{D,d}, 𝔸) → 1.

By the definition of T_{Π}, τ is in the image of the Jacquet-Langlands correspondence from B^{×}(𝔸) discussed in Section 6; let τ^{JL} ∈ Irr_{cusp}^{temp}(B^{×}(𝔸)) correspond to τ. Let π′ = π(χ, τ^{JL}); this is contained in Irr_{cusp}^{temp}(GSO(X_{D,d}, 𝔸)). We claim that for each place v there exists σ_{v} ∈ ℛ_{2}(GO(X_{D,d}, F_{v})) such that

σ_{v} ↪ Ind_{GSO(X_{D,d},F_{v})}^{GO(X_{D,d},F_{v})} π′_{v}

and θ(σ_{v})^{∨} = Π_{v}. This is clear from the definition of Π([π_{v}]) and D if v is not a nonsplit place with D(F_{v}) ramified; assume we are in this last case. Let w be the place of E lying over v. By Proposition 2.9 and the consideration of examples after this proposition, there exists an isomorphism i : B(F_{v}) ⟶^{∼} M_{2×2}(E_{w}) of E_{w} algebras and a similitude T : X_{D,d}(F_{v}) → X_{M_{2×2},d}(F_{v}) such that

1 ⟶ E_{w}^{×} ⟶ F_{v}^{×} × B^{×}(F_{v}) ⟶ GSO(X_{D,d}, F_{v}) ⟶ 1
↓^{id}  ↓^{id×i}  ↓^{T ·T^{−1}=j}
1 ⟶ E_{w}^{×} ⟶ F_{v}^{×} × GL(2, E_{w}) ⟶ GSO(X_{M_{2×2},d}, F_{v}) ⟶ 1

commutes. By the definition of Π([π_{v}]), there exists π̂_{v} ∈ Irr(GO(X_{M_{2×2},d}, F_{v})) such that π_{v} ↪ π̂_{v}|_{GSO(X_{M_{2×2},d},F_{v})} and θ(π̂_{v}) = Π_{v}^{∨}, i.e.,

Hom_{R_{X_{M_{2×2},d}} (F_{v})}(ω_{X_{M_{2×2},d}(F_{v})}, Π_{v}^{∨} ⊗ π̂_{v}) ≠ 0.

By Lemma 1.6, we obtain

Hom_{R_{X_{D,d}} (F_{v})}(ω_{X_{D,d}(F_{v})}, Π_{v}^{∨} ⊗ σ_{v}) ≠ 0, where σ_{v} = π̂_{v} ∘ j, so that θ(σ_{v}) = Π_{v}^{∨}. Since π_{v} ∘ j ↪ σ_{v}|_{GSO(X_{D,d},F_{v})} and π_{v} ∘ j = π′_{v} by the commutativity of the diagram, we get π′_{v} ↪ σ_{v}|_{GSO(X_{D,d},F_{v})} as desired. Now Π_{v} is unramified for almost all finite v, and so by Proposition 1.11 σ_{v} is unramified for almost all finite v. We may form the restricted direct product σ = ⊗_{v}σ_{v} ∈ Irr_{admiss}(GO(X, 𝔸)). Since s · π ≇ π we have s · π′ ≇ π′. By Theorem 7.1 it follows that σ ∈ Irr_{cusp}^{temp}(GO(X, 𝔸)), and σ lies over [π′]. By Theorem 8.3, Θ_{2}(V_{σ}) is cuspidal and Θ_{2}(V_{σ}) ≅ Π. Having shown that Π occurs in the space of cusp forms on GSp(2, 𝔸) of central character ω_{π}, we will now show that the multiplicity with which Π occurs is one. Our strategy will be to use the multiplicity preservation principle of Rallis (Proposition 5.3) along with the fact that for a four dimensional quadratic space X over F, GO(X, 𝔸) has the (weak) multiplicity one property (Theorem 7.1). Let W be the GSp(2, 𝔸_{f}) × (𝔤_{∞}, K_{∞}) subspace of cusp forms on GSp(2, 𝔸) of central character ω_{π} generated by the subspaces isomorphic to Π. Let U be an irreducible nonzero GSp(2, 𝔸_{f}) × (𝔤_{∞}, K_{∞}) subspace of W. Then U ≅ Π. To be in a position to apply Proposition 5.3 we must show that Θ_{X_{D,d}}(U) is nonzero and contained in the space of cusp forms on GO(X, 𝔸) of central character ω_{π}. As a first step, we will prove that Θ_{X_{D′,d}}(U) is nonzero and cuspidal for some quaternion algebra D′ over F. In the following argument showing that Θ_{X_{D′,d}}(U) is nonzero and cuspidal for some D′ we ask the reader to take note that we only use that Π ∈ Π([π]); this will be germane in a subsequent part of the proof. We begin with a reduction to isometries. Restrict the functions in U to Sp(2, 𝔸). This space of restricted functions is nonzero and is an Sp(2, 𝔸_{f})×(𝔤_{1,∞}, K_{1,∞}) subspace of the space of cusp forms on Sp(2, 𝔸); let U_{1} be a nonzero Sp(2, 𝔸_{f})×(𝔤_{1,∞}, K_{1,∞}) irreducible subspace of this space, and let Π_{1} be the isomorphism class of U_{1}. As in the proof of (2) ⟹ (1) of Theorem 8.3, to show Θ_{X_{D′,d}}(U) ≠ 0 for some D′ it will suffice to show Θ_{X_{D′,d}}(U_{1}) ≠ 0 for some D′. To prove this, we will use Theorem 7.1 of [KRS]. This application requires an understanding the behavior of the partial twisted standard L-function L^{S}(s, Π_{1}, χ_{X_{D,d}}) at s = 1; we now compute this L-function. As U ≅ Π, Π_{1,v} is an irreducible component of Π_{v}|_{Sp(n,F_{v})} for all v. Let S be a finite set of places of F such that for v ∉ S, v is finite, X_{M_{2×2},d}(F_{v}) is unramified (i.e., v is odd and v is unramified in E_{d}) and χ_{v} and τ_{w} for w|v are unramified. For v ∉ S, by Proposition 4.3 and its proof, |Π([π_{v}])| = 1, Π_{v} is the single element of Π([π_{v}]), Π_{v} is unramified and Π_{v} = θ_{M_{2×2},d}(σ′_{v})^{∨} = θ_{M_{2×2},d}(σ′_{v}^{∨}), with σ′_{v} = π_{v}^{+} ∈ Irr(GO(X_{M_{2×2},d}, F_{v})) unramified. Let v ∉ S; we assert that there exists an unramified component σ′_{1,v} of σ′_{v}|_{O(X_{M_{2×2},d},F_{v})} such that Π_{1,v} = θ(σ′_{1,v}^{∨}). To see this let, as in Section 1, GSp(2, F_{v})^{+} be the subgroup of g ∈ GSp(2, F_{v}) such that λ(g) ∈ λ(GO(X_{D,d}, F_{v})); again, GSp(2, F_{v})^{+} has index one or two in GSp(2, F_{v}). Let Π_{v}|_{GSp(2,F_{v})^{+}} = Π_{v}^{1} ⊕ ⋯ ⊕ Π_{v}^{M}, where the Π_{v}^{i} ∈ Irr(GSp(2, F_{v})^{+}) are mutually nonisomorphic and M = 1 or 2. We have by construction

Hom_{R_{X_{M_{2×2},d}} (F_{v})}(ω_{X_{M_{2×2},d}(F_{v})}, Π_{v} ⊗ σ′_{v}^{∨}) ≠ 0.

This implies that for some i,

Hom_{R_{X_{M_{2×2},d}} (F_{v})}(ω_{X_{M_{2×2},d}(F_{v})}, Π_{v}^{i} ⊗ σ′_{v}^{∨}) ≠ 0.

By the proof of Proposition 1.11, Π_{v}^{i} is unramified with respect to GSp(2, 𝔒_{v}) (which is contained in GSp(2, F_{v})^{+}). As Π_{v}|_{Sp(2,F_{v})} has only one irreducible component unramified with respect to Sp(2, 𝔒_{v}), namely Π_{1,v}, it follows that Π_{1,v} is an irreducible component of Π_{v}^{i}|_{Sp(2,F_{v})}. By Lemma 4.2 of [R1], there exists an irreducible component σ′_{1,v} of σ′_{v} such that

Hom_{Sp(2,F_{v})×O(X_{M_{2×2},d},F_{v})}(ω_{X_{M_{2×2},d}(F_{v})}, Π_{1,v} ⊗ σ′_{1,v}^{∨}) ≠ 0.

By (b) of Theorem 7.1 of [H], σ′_{1,v} is unramified. This proves our assertion. By Section 7 of [KR2] and Lemma 8.1 (or rather its proof), the twisted partial standard L-function of Π_{1} now is

L^{S}(s, Π_{1}, χ_{X_{D,d}}) = ζ_{F}^{S}(s) ∏_{v∉S} L(s, σ′_{1,v})

= \{ \begin{matrix}ζ_{F}^{S}(s)L^{S}(s, τ, χ^{−1}, Asai) if d ≠ 1 \\ ζ_{F}^{S}(s)L^{S}(s, τ_{1} × τ_{2}^{∨}) if d = 1 and τ ≅ τ_{1} ⊗ τ_{2}, \end{matrix}

where ζ_{F}^{S}(s) is the partial zeta function of F. We noted in the proof of Theorem 8.3 that L-functions of the type L^{S}(s, τ, χ^{−1}, Asai) or L^{S}(s, τ_{1} × τ_{2}^{∨}) do not vanish at s = 1; hence, L^{S}(s, Π_{1}, χ_{X_{D,d}}) has a pole at s = 1 (in fact, by Corollary 7.2.3 of [KR2] the pole must be simple). Now we apply [KRS]. By Lemma 1.1 of [L], for some f ∈ U_{1}, f has a nonzero T -th Fourier coefficient with det T ≠ 0. Here, T ∈ M_{2}(F) is a symmetric matrix. Define a quadratic Hecke character χ of 𝔸^{×} by χ_{X_{D,d}} = χ_{T} χ, where we also write T for the two dimensional quadratic space defined by T. Since L^{S}(s, Π_{1}, χ_{T} χ) = L^{S}(s, Π_{1}, χ_{X_{D,d}}) has a pole at s = 1, by (i) and (ii) of Theorem 7.1 of [KRS], Θ_{X′}(U_{1}) ≠ 0, where X′ = X_{T} ⊥ X′′, with X′′ some two dimensional quadratic space over F such that χ_{X′′} = χ. We have

χ_{X′} = χ_{X_{T}} · χ_{X′′} = χ_{X_{T}} · χ = χ_{X_{T}}^{2} · χ_{X_{D,d}} = χ_{X_{D,d}}

which implies disc X′(F) = disc X_{D,d}(F) = d. By Proposition 2.8 and Lemma 5.1, we now know that Θ_{X_{D′,d}}(U_{1}) ≠ 0 for some quaternion algebra D′ over F. As mentioned, this implies Θ_{X_{D′,d}}(U) ≠ 0. Next, we claim Θ_{X_{D′,d}}(U) is contained in the space of cusp forms on GO(X_{D′,d}, 𝔸) of central character ω_{π}; suppose not. Then by the remark after Proposition 5.2 there exists a two dimensional quadratic space X_{0} over F such that Θ_{X_{0}}(U) is nonzero and is contained in the space of cusp forms of central character ω_{π} on GO(X_{0}, 𝔸). By a standard argument as in the proof of Proposition 5.3, for all but finitely many places v of F, X_{0}(F_{v}) is unramified, Π_{v} is unramified with respect to GSp(2, 𝔒_{v}), there exists a unitary ρ_{v} ∈ Irr(GO(X_{0}, F_{v})) which is unramified with respect to the stabilizer in GO(X_{0}, F_{v}) of a self-dual lattice and

Hom_{R_{X_{0}} (F_{v})}(ω_{X_{0}(F_{v})}, Π_{v}^{∨} ⊗ ρ_{v}) ≠ 0.

Let v be one such place. Let ρ_{0} be an irreducible unramified component of ρ_{v}|_{O(X_{0},F_{v})}. By Lemma 4.2 of [R1], there exists an irreducible component Π_{0} of Π_{v}^{∨}|_{Sp(2,F_{v})} such that

Hom_{Sp(2,F_{v})×O(X_{0},F_{v})}(ω_{X_{0}(F_{v})}, Π_{0} ⊗ ρ_{0}) ≠ 0.

Now SO(X_{0}, F_{v}) is Abelian as dim X_{0} = 2; since ρ_{0} is unitary, ρ_{0} is therefore tempered. (Recall the definition of a tempered representation of O(X_{0}, F_{v}) preceding Theorem 1.2). Also, it is not difficult to show that ρ_{0} ∈ ℛ_{1}(O(X_{0}, F_{v})) (in fact, the only element of Irr(O(X_{0}, F_{v})) not contained in ℛ_{1}(O(X_{0}, F_{v})) is sign). Applying now Theorem 4.4 of [R3], we conclude that Π_{0} is not tempered, contradicting the temperedness of Π_{v} (see (3) of Proposition 4.1). We have shown Θ_{X_{D′,d}}(U) ≠ 0 is nonzero and cuspidal for some D′; as promised, the argument used only that the cuspidal automorphic representation Π is contained in Π([π]). Now we will show that Θ_{X_{D,d}}(U) is nonzero and contained in the space of cusp forms of central character ω_{π}. By Lemma 5.1, it will suffice to show that there is a similitude between X_{D,d}(F) and X_{D′,d}(F). Let B′ = E ⊗_{F} D′. We assert that B ≅ B′ as E algebras. As in the last paragraph of Section 2, let S_{D,E} be the set of places v of F such that v splits in E and D(F_{v}) is ramified; define S_{D′,E} similarly. As observed in Section 2, it will suffice to show that S_{D,E} = S_{D′,E}. Let v be a place of F that splits in E. As v splits in E, d = 1 in F_{v}^{×}/F_{v}^{×2}. By Proposition 5.3, since Θ_{X_{D′,d}}(U) is nonzero and cuspidal, Π_{v} ∈ ℛ_{X_{D′,d}(F_{v})}(GSp(2, F_{v})); by construction, Π_{v} ∈ ℛ_{X_{D,d}(F_{v})}(GSp(2, F_{v})). By Lemma 8.4 we must have X_{D′,d}(F_{v}) ≅ X_{D,d}(F_{v}). This implies D′(F_{v}) ≅ D(F_{v}) so that D is ramified at v if and only if D′ is ramified at v. This proves S_{D,E} = S_{D′,E}. Since B ≅ B′ as E algebras, by Proposition 2.9 there exists a similitude between X_{D,d}(F) and X_{D′,d}(F). We now apply Proposition 5.3 to conclude that the multiplicity of Π in W is the same as the multiplicity of Θ_{X_{D,d}}(U) in the space of cusp forms on GO(X_{D,d}, 𝔸) of central character ω_{π}. By part of Theorem 7.1, this multiplicity is one. To complete the proof we still must show that if d = 1, Π ∈ Π([π]) and Π occurs in the space of cusp forms on GSp(2, 𝔸), then |T_{Π}| is even. Let U be a realization of Π in the space of cusp forms on GSp(2, 𝔸) of central character ω_{π}. An argument just as above (which just used Π ∈ Π([π]) and nothing about the parity of |T_{Π}|) shows that Θ_{X_{D′,1}}(U) is nonzero and cuspidal for some quaternion algebra D′ over F. We claim that T_{Π} is exactly the set of places where D′ is ramified; this will show that |T_{Π}| is even. Let v ∈ T_{Π}. Then by the definition of T_{Π}, Π_{v} ∈ ℛ_{X_{Dram,1}}(GSp(2, F_{v})). On the other hand, since Θ_{X_{D′,1}}(U) is nonzero and cuspidal, Π_{v} ∈ ℛ_{X_{D′(Fv),1}(F_{v})}(GSp(2, F_{v})) (Proposition 5.3). By Lemma 8.4, X_{D_{ram},1}(F_{v}) ≅ X_{D′(F_{v}),1}(F_{v}), which implies D′(F_{v}) is ramified. Suppose next D′(F_{v}) is ramified. Again, Π_{v} ∈ ℛ_{X_{D′(Fv),1}(F_{v})}(GSp(2, F_{v})). By Lemma 8.4 and the definition of Π([π_{v}]), we must have v ∈ T_{Π}. □

Finally, we prove Theorem 8.6. This result is essentially a restatement of Theorem 8.5, and will follow immediately from that theorem after we make some definitions. First we make the definitions mentioned preceding the statement of Theorem 8.6 in the Introduction. Let F′ be a local field of characteristic zero, and let E′ be a quadratic extension of F′ or E′ = F′ × F′; if F′ is archimedean, assume F′ = ℝ and E′ = ℝ×ℝ. If E′ is a field, write E′ = F′(\sqrt{d}); otherwise, let d = 1. Let τ′ ∈ Irr(GL(2, E′)) be infinite dimensional and assume the central character of τ′ factors through N_{F′}^{E′} via χ′; if F′ has even residual characteristic, assume additionally that τ′ is tempered. By Proposition 2.7 the following sequence is exact:

1 → E′^{×} → F′^{×} × GL(2, E′) ⟶^{ρ_{a(M_{2×2},E′)}} GSO(X_{M_{2×2},d}, F′) → 1.

Using this exact sequence, define π′ = π(χ′, τ′) ∈ Irr(GSO(X_{M_{2×2},d}, F′)) as in Section 3. Define φ(χ′, τ′) = φ([π′]) and Π(χ′, τ′) = Π([π′]), where φ([π′]) and Π([π′]) are defined as in Section 4. If E′ = F′ × F′, define

⟨·, ·⟩_{F′} : 𝕊(φ(χ′, τ′)) × Π(χ′, τ′) → ℂ

as follows. If |𝕊(φ(χ′, τ′))| = |Π(χ′, τ′)| = 1 set ⟨·, ·⟩_{F′} to be identically 1; if |𝕊(φ(χ′, τ′))| = |Π(χ′, τ′)| = 2 (see Propositions 4.1 and 4.2) then define ⟨·, θ_{M_{2×2},1}(π′^{+})^{∨}⟩_{F′} = 1 and let ⟨·, θ_{D_{ram},1}(π′^{JL}^{+})^{∨}⟩_{F′} to be the nontrivial character of 𝕊(φ(χ′, τ′)) = Z_{2} (see Table 4). The claims from the Introduction concerning these definitions follow from Propositions 4.1, 4.2 and 4.3. Next, let E, τ and χ be as in the statement of Theorem 8.6. If E is a field, write E = F (\sqrt{d}); otherwise, let d = 1. By Proposition 2.7 the following sequence is exact:

1 → 𝔸_{E}^{×} → 𝔸^{×} × GL(2, 𝔸_{E}) ⟶^{ρ_{a(M_{2×2},E)}} GSO(X_{M_{2×2},d}, 𝔸) → 1.

Using this exact sequence, define π = π(χ, τ) ∈ Irr_{cusp}^{temp}(GSO(X_{M_{2×2},d}, 𝔸)) as in Section 6.

Proof of Theorem 8.6. This follows from the definitions involved and Theorem 8.5. □

Appendix

p = q  p ≠ q
λ(GO(X, ℝ))  ℝ^{×}  ℝ_{>0}^{×}
[GSp(n, ℝ) : GSp(n, ℝ)^{+}]  1  2
GSp(n, ℝ)^{+}  \begin{matrix}GSp(n, ℝ) = \\ ℝ^{×}(Sp(n, ℝ) ∪ Sp(n, ℝ)k_{0}) \end{matrix} Sp(n, ℝ)ℝ^{×}
K^{+}  K = K_{1} ∪ K_{1}k_{0}  K_{1}
GO(X, ℝ)  ℝ^{×}(O(X, ℝ) ∪ O(X, ℝ)j_{0}) O(X, ℝ)ℝ^{×}
J J_{1} ∪ J_{1}j_{0} J_{1}
L  (K_{1} × J_{1}) ∪ (K_{1} × J_{1})(k_{0}, j_{0}) K_{1} × J_{1}

TABLE 1

d≠1
π regular: π → π^{+} → θ_{M2×2,d}(π^{+})

π regular or distinguished

π invariant but not distinguished

\begin{aligned}& & π_{1} & → θ_{M2×2,d}(π_{1}) \\ & ↗ \\ π \\ & ↘ \\ & & π_{2} & → θ_{M2×2,d}(π_{2}) \end{aligned}

TABLE 2

π distinguished: π^{+} → θ_{M2×2,d}(π^{+}) ↗ π ↘ π^{−} does not lift to GSp(2,F)

Documenta Mathematica 6 (2001) 247–314

d=1

π regular: π → π^{+} → θ_{M2×2,1}(π^{+})

π not ; essentially ; square ; integrable

π^{+} → θ_{M2×2,1}(π^{+})
↗

π invariant and hence distinguished:

π
↘
π^{−} does not lift to GSp(2,F)

π regular: π → π^{+} → θ_{M2×2,1}(π^{+}) ↓ π^{JL} → π^{JL+} → θ_{Dram,1}(π^{JL+})

π essentially

square integrable

π invariant and hence distinguished:

π^{+} → θ_{M2×2,1}(π^{+})
↗
π → π^{−} does not lift to GSp(2,F)
↓
π^{JL} → π^{JL+} → θ_{Dram,1}(π^{JL+})
↘
π^{JL−} does not lift to GSp(2,F)

TABLE 3

d  [π]  Π([π])

1 π not essentially square integrable

{θ_{M2×2,1}(π^{+})^{∨}}

1

π essentially square integrable

{θ_{M2×2,1}(π^{+})^{∨},θ_{Dram,1}(π^{JL+})^{∨}}

≠1 π regular or invariant

and distinguished

{θ_{M2×2,d}(π^{+})^{∨}}

≠1 π invariant but not distinguished

{θ_{M2×2,d}(π_{1})^{∨},θ_{M2×2,d}(π_{2})^{∨}}

TABLE 4

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Brooks Roberts

Department of Mathematics PO Box 441103 University of Idaho Moscow ID 83844-1103 USA brooksr@uidaho.edu