AUTOMORPHIC FORMS ON PGSp(2)

Yuval Z. Flicker

Abstract. Development of the theory of lifting of automorphic and admissible representations in a new case of great classical interest: Siegel automorphic forms, is reported. The self-contragredient representations of PGL(4) are determined as lifts of representations of rank two either symplectic PGSp(2) or orthogonal SO(4) groups. The method uses the global trace formula and the local fundamental lemma. The lifting is stated in terms of character relations. This permits us to introduce a definition of packets and quasi-packets of representations of PGSp(2). We deal with all representations, not only generic ones. Globally we obtain multiplicity one theorem for the discrete spectrum of PGSp(2), rigidity theorem for packets and quasi-packets, determine all counterexamples to the naive Ramanujan conjecture, and show that each packet which is not such a counterexample has precisely one generic member. The lifting from SO(4) to PGL(4) amounts to establishing a product of two representations of GL(2) with central characters whose product is 1. The rigidity theorem for SO(4) amounts to a strong rigidity statement for a pair of representations of GL(2, 𝔸).

This report summarizes results of [F1]. These consist of a definition of local and global packets and quasi-packets of admissible and automorphic representations of the projective group PGSp(2) of symplectic similitudes, and a description of the lifting from this group to PGL(4), as well as the complementary lifting from SO(4) to PGL(4), and the lifting to PGSp(2) from its elliptic endoscopic group PGL(2)×PGL(2). The packets are defined using the liftings. The liftings are predicted by homomorphisms of dual groups via Langlands’ principle of functoriality, which we describe first. The classification of the automorphic representations of PGSp(2) has applications to the decomposition of the étale cohomology with compact supports and twisted coefficients of the Shimura varieties associated with GSp(2), but this will be discussed elsewhere.

1. Homomorphisms of Dual Groups

Let G be the projective general linear group PGL(4)=PSL(4) over a number field F. Our initial purpose is to determine the automorphic representations π of G(𝔸), 𝔸 is the ring of adeles of F, which are self-contragredient: π ≃ π̌, equivalently ([BZ1]), θ-invariant: π ≃ ^{θ}π. Here θ, θ(g) = J^{−1}^{t}g^{−1}J, is the involution defined by J = ( \begin{matrix}0 & w \\ −w & 0 \end{matrix} ) , w = ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ),

where ^{t}g denotes the transpose of g ∈ G, and ^{θ}π(g) = π(θ(g)). According to the principle of functoriality ([Bo], [A]) these automorphic representations are essentially described by representations of the Weil group W_{F} of F into the dual group Ĝ = SL(4, ℂ) of G which are

Typeset by AMS-TEX

Department of Mathematics, The Ohio State University, 231 W. 18th Ave., Columbus, OH 43210-1174; flicker@math.ohio-state.edu. 2000 MSC: 10D40, 10D30, 12A67, 12A85, 14G10, 22E55, 11F27, 11R42, 11S40.

θ̂-invariant, namely representations of W_{F} into centralizers ZĜ(ŝθ̂) of Int(ŝ)θ̂ in Ĝ. Here θ̂ is the dual involution θ̂(ĝ) = J^{−1}^{t}ĝ^{−1}J, and ŝ is a semisimple element in Ĝ. These centralizers are called twisted (by ŝθ̂) endoscopic groups ([KS]).

A twisted endoscopic group is called elliptic if it is not contained in a proper Levi subgroup of Ĝ. Representations of non elliptic endoscopic groups can be reduced by parabolic induction to known ones of smaller rank groups. For our Ĝ, up to conjugacy the elliptic twisted endoscopic groups are the symplectic group Ĥ = ZĜ(θ̂) = Sp(2, ℂ) and the special

orthogonal group Ĉ = ZĜ(ŝθ̂) = “ SO(4, ℂ)” = \{g ∈ SL(4, ℂ); gŝJ^{t}g = ŝJ = ( \begin{matrix}0 & ω \\ ω^{−1} & 0 \end{matrix} )\} ,

of all A ⊗ B = ( \begin{matrix}aB & bB \\ cB & dB \end{matrix} ); (A = ( \begin{matrix}a & b \\ c & d \end{matrix} ) , B) ∈ (GL(2, ℂ) × GL(2, ℂ))/ℂ^{×} which satisfy

det A · det B = 1. Here z ∈ ℂ^{×} embeds as (z, z^{−1}), ŝ = diag(−1, 1, −1, 1) and ω = ( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ).

The group Ĥ is the dual group of the simple F -group H = PSp(2) = PGSp(2), the projective group of symplectic similitudes. It is the quotient of the group GSp(2) of (g,λ λ λ) ∈ GL(4) × 𝔾_{m} with ^{t}gJg = λ λ λJ, by its center {(λ λ λ,λ λ λ^{2})} ≃ 𝔾_{m}. Since λ λ λ is uniquely determined by g (we write λ λ λ = λ λ λ(g)), we view GSp(2) as a subgroup of GL(4) and PGSp(2) of PGL(4).

The group Ĉ is the dual group of the special orthogonal group C =“SO(4)” of pairs (g_{1}, g_{2}) ∈ (GL(2) × GL(2))/𝔾_{m} with det g_{1} = det g_{2}. Here z ∈ 𝔾_{m} embeds as the central element (z, z). Also we write ((GL(2) × GL(2))/ GL(1))′ for C, where the prime indicates that the two factors in GL(2) have equal determinants, and (…)′′ if their product is 1.

The principle of functoriality suggests that automorphic discrete series representations of H(𝔸) and C(𝔸) parametrize (or lift to) the θ-invariant automorphic discrete series representations of the group G(𝔸) of 𝔸-valued points of G. Our main purpose is to describe this parametrization, in particular define tensor products of two automorphic forms of GL(2, 𝔸) the product of whose central characters is 1, and especially describe the automorphic representations of the projective symplectic group of similitudes of rank two, PGSp(2, 𝔸), in terms of θ-invariant representations of PGL(4, 𝔸).

2. Unramified Lifting

We proceed to explain how the liftings are defined, first for unramified representations. An irreducible admissible representation π of an adele group G(𝔸) is the restricted tensor

product ⊗π_{v} of irreducible admissible ([BZ1]) representations π_{v} of the groups G(F_{v}) of F_{v}-points of G, where F_{v} is the completion of F at the place v of F. Almost all the local components π_{v} are unramified, that is contain a (unique up to a scalar multiple) nonzero K_{v}-fixed vector. Here K_{v} is the standard maximal compact subgroup of G(F_{v}), namely the group G(R_{v}) of R_{v}-points, R_{v} being the ring of integers of the non archimedean local field F_{v}; G is defined over R_{v} at almost all v. For such v, an irreducible unramified G(F_{v})-module π_{v} is the unique unramified irreducible constituent in an unramified principle series representation I(η_{v}), normalizedly induced ([BZ2]) from an unramified character η_{v} of the maximal torus T(F_{v}) of a Borel subgroup B(F_{v}) of G(F_{v}) (extended trivially to the unipotent radical N(F_{v}) of B(F_{v})). The space of I(η_{v}) consists of the smooth functions ϕ : G(F_{v}) → ℂ with ϕ(ank) = (δ_{v}^{1/2}η_{v})(a)ϕ(k), k ∈ K_{v}, n ∈ N(F_{v}), a ∈ T(F_{v}), δ_{v}(a) = det[Ad(a)| Lie N(F_{v})], and the G(F_{v})-action is (g · ϕ)(h) = ϕ(hg), g, h ∈ G(F_{v}).

The character η_{v} is unramified, thus it factorizes as η_{v} : T(F_{v})/T(R_{v}) → ℂ^{×}. As T(F_{v})/T(R_{v}) ≃ X_{∗}(T) = Hom(𝔾_{m}, T), η_{v} lies in Hom(X_{∗}(T), ℂ^{×}) = Hom(X^{∗}(T̂), ℂ^{×}), where T̂ is the maximal torus in the Borel subgroup B̂ of Ĝ, both fixed in the definition of the (complex) dual group Ĝ ([Bo], [Ko]). Now Hom(X^{∗}(T̂), ℂ^{×}) = X_{∗}(T̂), and X_{∗}(T̂) ⊗ ℂ^{×} = T̂ ⊂ Ĝ, thus the unramified irreducible G(F_{v})-module π_{v} determines a conjugacy class t(π_{v}) = t(I(η_{v})) (the “Langlands parameter”) in Ĝ, represented by the image of η_{v} in T̂.

3. The Lifting λ_{1} from SO(4) to PGL(4)

We next describe our results on our secondary lifting λ_{1}, to G = PGL(4) from C = SO(4) = ((GL(2) × GL(2))/ GL(1))′. An unramified (irreducible) GL(2, F_{v})-module π_{1v} is parametrized by a conjugacy class t(π_{1v}) in GL(2, ℂ) (the Langlands parameter; its eigenvalues are called the Hecke eigenvalues of the representation). An unramified irreducible C(F_{v})- module π_{1v}×π_{2v} is parametrized by a class t(π_{1v})×t(π_{2v}) in ((GL(2, ℂ)×GL(2, ℂ))/ℂ^{×})′′ = Ĉ ⊂ Ĝ. If π_{iv} is the unramified constituent of I(η_{iv}), t(π_{iv}) = diag(η_{i1}, η_{i2}), η_{11}η_{12}η_{21}η_{22} = 1, η_{ij} = η_{ijv}(π π π_{v}), we define the “lift” π_{1v}⊠π_{2v} = λ_{1}(π_{1v}×π_{2v}) of π_{1v}×π_{2v} with respect to the dual group homomorphism λ_{1} : Ĉ = SO(4, ℂ) ↪ Ĝ = SL(4, ℂ) (the natural embedding) to be the unramified irreducible constituent π_{v} of the PGL(4, F_{v})-module I(η_{v}) parametrized by the class t(π_{v}) = diag(η_{11}η_{21}, η_{11}η_{22}, η_{12}η_{21}, η_{12}η_{22}) in Ĝ = SL(4, ℂ). In different notations, λ_{1}(I(a_{1}, a_{2}) × I(b_{1}, b_{2})) = I(a_{1}b_{1}, a_{1}b_{2}, a_{2}b_{1}, a_{2}b_{2}) where a_{i}, b_{i} ∈ ℂ^{×}, provided that a_{1}a_{2}b_{1}b_{2} = 1. Note that the inverse image under λ_{1} of I(a_{1}b_{1}, a_{1}b_{2}, b_{1}a_{2}, a_{2}b_{2}) consists only of χI(a_{1}, a_{2}) × χ^{−1}I(b_{1}, b_{2}) and χI(b_{1}, b_{2}) × χ^{−1}I(a_{1}, a_{2}) where χ is any character of F_{v}^{×}. Thus, λ_{1} is two-to-one unless π_{1v} = π̌_{2v} (the contragredient of π_{2v}), where λ_{1} is injective on the set of orbits of multiplication by χ in Hom(F_{v}^{×}, ℂ^{×}).

The rigidity theorem for discrete series automorphic representations of GL(n, 𝔸) asserts that discrete series automorphic representations π_{1} = ⊗π_{1v} and π_{2} = ⊗π_{2v} which have π_{1v} ≃ π_{2v} for almost all places v of F are equivalent ([JS], [MW]). Moreover they are even equal, by the multiplicity one theorem for GL(n) ([Shal]). Representations of PGL(n, 𝔸) (or PGL(n, F_{v})) are simply representations of GL(n, 𝔸) (or GL(n, F_{v})) with trivial central character (since H^{1}(F, 𝔾_{m}) = {0}), and the rigidity theorem applies then to PGL(n). Both multiplicity one theorem, and the rigidity theorem for packets (the latter asserts that π = ⊗π_{v} and π′ = ⊗π′_{v} must lie in the same packet if π_{v} ≃ π′_{v} for almost all v) hold for SL(2) ([F3]) and fail for SL(n), n ≥ 3 ([Bla]).

The rigidity theorem holds for C = SO(4); this is the content of the assertion that the lifting λ_{1} is injective, made in the second paragraph of the following theorem. The first paragraph asserts that the lifting exists. By an elliptic representation we mean one whose character ([H]) is non zero on the set of elliptic elements.

Theorem (SO(4) to PGL(4)). For every two discrete series automorphic representations π_{1} = ⊗π_{1v} ≠ π_{2} = ⊗π_{2v} of GL(2, 𝔸), whose central characters ω_{1}, ω_{2} are equal, and whose components at three places v_{1}, v_{2}, v_{3} are elliptic, there exists a (unique) discrete series automorphic representation π = λ_{1}(π_{1} × π̌_{2}) of PGL(4, 𝔸) with π_{v} = λ_{1}(π_{1v} × π̌_{2v}) for almost all v. Moreover, λ_{1}(π_{1} × π̌_{1}) is the PGL(4, 𝔸)-module normalizedly induced from the PGL(3, 𝔸)-module Sym^{2}(π_{1}) on the Levi factor of the parabolic subgroup of type (3, 1).

Further, the global map λ_{1} is injective on the set of pairs π_{1} × π̌_{2} with ω_{1} = ω_{2} up to the equivalence π_{1} × π̌_{2} ≃ χπ_{1} × χ^{−1}π̌_{2}, χ a character of 𝔸^{×}/F^{×}, and π_{1} × π̌_{2} ≃ π̌_{2} × π_{1}.

The injectivity means that if π_{1}, π_{2}, π_{1}^{0}, π_{2}^{0} are discrete series automorphic representations of GL(2, 𝔸) with central characters ω_{1}, ω_{2}, ω_{1}^{0}, ω_{2}^{0} satisfying ω_{1}ω_{2} = 1 = ω_{1}^{0}ω_{2}^{0}, each of which has elliptic components at least at the three places v_{1}, v_{2}, v_{3}, and if for each v outside a fixed finite set of places of F there is a character χ_{v} of F_{v}^{×} such that the set {π_{1v}χ_{v}, π_{2v}χ_{v}^{−1}} is equal to the set {π_{1v}^{0}, π_{2v}^{0}} (up to equivalence of representations), then there is a character χ of 𝔸^{×}/F^{×} such that the set {π_{1}χ, π_{2}χ^{−1}} is equal to the set {π_{1}^{0}, π_{2}^{0}}. In particular, starting with a pair π_{1}, π_{2} of automorphic discrete series representations of GL(2, 𝔸) with ω_{1}ω_{2} = 1, we cannot get another such pair by interchanging a set of their components π_{1v}, π_{2v} and multiplying π_{1v} by a local character and π_{2v} by its inverse, unless we interchange π_{1}, π_{2} and multiply π_{1} by a global character and π_{2} by its inverse.

Our global results are complemented and strengthened by very precise local results. We define λ_{1}-lifting locally by means of character relations of the form: λ_{1}(π_{1} × π̌_{2}) = π if tr π(f × θ) = tr(π_{1} × π̌_{2})(f_{C}) for all matching functions f, f_{C}. This definition is compatible with the one given above for purely induced π_{1} and π_{2} and unramified representations, and we have λ_{1}(I_{2}(μ, μ′) × π̌_{2}) = I_{4}(μπ̌_{2}, μ′π̌_{2}) (the central character of the GL(2, F)-module π_{2} is μμ′). The local and global results are closely analogous.

4. Special Cases of The Lifting λ_{1}

Let us describe some special cases of the lifting λ_{1}. When π_{2} = π̌_{1} is the contragredient of π_{1}, λ_{1}(π_{1}×π̌_{1}) is the PGL(4, 𝔸)-module normalizedly induced from the maximal parabolic of type (3,1) and the PGL(3, 𝔸)-module Sym^{2}(π_{1}) on the GL(3)-factor of the Levi subgroup (extended trivially to the GL(1)-factor of the Levi, and to the unipotent radical). Here Sym^{2}(π_{1}) is the symmetric square lifting from GL(2) to PGL(3) ([F3]). Indeed, if the local component π_{1v} of π_{1} at v is unramified then t(π_{1v}) = diag(a, b) (thus π_{1v} is a constituent of I_{2}(a, b)), π_{v} = λ_{1}(π_{1v} × π̌_{1v}) has t(π_{v}) = diag(a/b, 1, 1, b/a) (thus π_{v} is a constituent of I_{4}(I_{3}(a/b, 1, b/a), 1), and I_{3}(a/b, 1, b/a) is the symmetric square lifting of I_{2}(a, b)). We write I_{n} to emphasize that the representation is of the group GL(n), and e.g. I_{(3,1)}(π_{3}, π_{1}) to indicate the representation of GL(4) induced from its maximal parabolic subgroup of type (3,1). However, the results of [F3] are stronger, in lifting representations of SL(2, 𝔸) to PGL(3, 𝔸) and consequently providing new results such as multiplicity one for SL(2).

Although we do not obtain a new proof of the existence of the symmetric square lift of discrete series representations of PGL(2, 𝔸), we do obtain new character relations, of the θ-twisted character of I_{(3,1)}(Sym^{2} π_{2}, 1) with that of π_{2} × π̌_{2}. Clearly in this case the lift λ_{1} is injective: if λ_{1}(π_{1} × π̌_{2}) = λ_{1}(π_{0} × π̌_{0})(= I_{(3,1)}(Sym^{2}(π_{0}), 1)) then π_{1} = π_{2} = π_{0}χ for some character χ of 𝔸^{×}/F^{×}.

In particular, if π_{1} is the one dimensional representation g ↦ χ(det g) of GL(2, 𝔸), then λ_{1}(π_{1} × π̌_{1}) = I_{(3,1)}(1_{3}, 1) is the representation of PGL(4, 𝔸) normalizedly induced from the trivial representation of the maximal parabolic subgroup of type (3,1). An alternative purely local computation of this twisted character is developed in [FZ].

Let π_{1} = π(μ) be a cuspidal monomial representation of GL(2, 𝔸) associated with a character μ of 𝔸_{E}^{×}/E^{×} where E is a quadratic extension of F (denote by σ the non trivial element of Gal(E/F)). Then Sym^{2} π_{1} = I_{(2,1)}(π(μ/^{σ}μ), χ_{E/F}), where χ_{E/F} is the quadratic character of 𝔸^{×}/F^{×}N_{E/F} 𝔸_{E}^{×} (N_{E/F} is the norm map from E to F). Moreover, λ_{1}(π(μ) × π̌(μ)) is the representation I_{(2,1,1)}(π(μ/^{σ}μ), χ_{E/F} , 1) induced from the parabolic subgroup of type (2,1,1) of PGL(4). The central character of the GL(2, 𝔸)-module π(μ) is χ_{E/F} ·μ|𝔸^{×}, for any character μ of 𝔸_{E}^{×}/E^{×}. If π(μ) is a PGL(2, 𝔸)-module we have that the restriction of μ to 𝔸_{F}^{×}/F^{×} is χ_{E/F}, non trivial but trivial on F^{×}N_{E/F} 𝔸_{E}^{×}.

If π_{1} = π_{E}(μ_{1}), π_{2} = π_{E}(μ_{2}), cuspidal monomial representations of GL(2, 𝔸) associated with characters μ_{1}, μ_{2} of 𝔸_{E}^{×}/E^{×} where E is a quadratic extension of F such that the restriction of μ_{1}μ_{2} to 𝔸^{×} is 1, then λ_{1}(π_{E}(μ_{1}) × π_{E}(μ_{2})) = I_{(2,2)}(π_{E}(μ_{1}\overline{μ}_{2}), π_{E}(μ_{1}μ_{2})). When μ_{1} = μ_{2}^{−1} we have π(μ_{1}\overline{μ}_{2}) = π(μ_{1}/\overline{μ}_{1}) and π(μ_{1}μ_{2}) = I(χ_{E/F} , 1), thus

λ_{1}(π(μ_{1}) × π̌(μ_{1})) = I_{(2,1,1)}(π(μ_{1}/\overline{μ}_{1}), χ_{E/F} , 1) = I_{(3,1)}(Sym^{2}(π(μ_{1})), 1).

Another interesting special case is when π_{1} is taken to be the trivial representation 1_{2} of PGL(2, 𝔸) while π_{2} is a cuspidal representation of PGL(2, 𝔸). Then λ_{1}(1_{2} × π_{2}) is the discrete series non cuspidal PGL(4, 𝔸)-module J(ν^{1/2}π_{2}, ν^{−1/2}π_{2}), the quotient of the normalizedly induced I(ν^{1/2}π_{2}, ν^{−1/2}π_{2}) from the parabolic of type (2,2) of PGL(4). Here ν(x) = |x|. Indeed, 1_{2} is the quotient of the induced I(ν^{1/2}, ν^{−1/2}). Thus t(λ_{1}(1_{2v} ×π_{2v})) is (t(ν_{v}^{1/2}π_{2v}), t(ν_{v}^{−1/2}π_{2v})) and so λ_{1}(1_{2v} ×π_{2v}) is the quotient J(ν_{v}^{1/2}π_{2v}, ν_{v}^{−1/2}π_{2v}) of the induced I(ν_{v}^{1/2}π_{2v}, ν_{v}^{−1/2}π_{2v}) for all v where π_{2v} is unramified. Hence it is J(ν^{1/2}π_{2}, ν^{−1/2}π_{2}) globally by the rigidity theorem for this non cuspidal discrete series spectrum ([MW]). On the set of pairs π_{1} × π_{2} such that at least one of π_{1} or π_{2} is one dimensional, the lifting λ_{1} is injective. Indeed, a discrete series representation of GL(2, 𝔸) with a one-dimensional component is necessarily one-dimensional. If π_{2} is not cuspidal but rather trivial, then the quotient J(ν^{1/2}1_{2}, ν^{−1/2}1_{2}) of I_{4}(ν^{1/2}1_{2}, ν^{−1/2}1_{2}) is not discrete series, but the induced I_{4}(1_{3}) from the trivial representation of the (3,1)-parabolic; this is λ_{1}(1_{2} × 1_{2}).

5. The Lifting λ from PGSp(2) to PGL(4)

We now turn to the study of our main lifting λ, and of the automorphic representations of the F -group H = PGSp(2) = GSp(2)/𝔾_{m}. The center 𝔾_{m} of GSp(2) = {g ∈ GL(4); ^{t}gJg = λ λ λJ, ∃λ λ λ = λ λ λ(g) ∈ 𝔾_{m}} consists of the scalar matrices. Its dual group is Ĥ = Sp(2, ℂ) = ZĜ(θ̂) ⊂ Ĝ = SL(4, ℂ), where θ̂(g) = J^{−1}^{t}g^{−1}J. It has a single elliptic endoscopic group C_{0} different than H itself. Thus

Ĉ_{0} = ZĤ(ŝ_{0}) = \{( \begin{matrix}a & 0 & 0 & b \\ 0 & α & β & 0 \\ 0 & γ & δ & 0 \\ c & 0 & 0 & d \end{matrix} ) ∈ Ĥ\} ≃ SL(2, ℂ) × SL(2, ℂ),

where ŝ_{0} = diag(−1, 1, 1, −1), and C_{0} = PGL(2) × PGL(2). Write λ_{0} for the embedding Ĉ_{0} ↪ Ĥ, and λ for the embedding Ĥ ↪ Ĝ.

The embedding λ_{0} : Ĉ_{0} = SL(2, ℂ) × SL(2, ℂ) ↪ Ĥ = Sp(2, ℂ) defines the “endoscopic” lifting λ_{0} : π_{2}(μ_{1}, μ_{1}^{−1}) × π_{2}(μ_{2}, μ_{2}^{−1}) ↦ π_{PGSp(2)}(μ_{1}, μ_{2}). Here π_{2}(μ_{i}, μ_{i}^{−1}) is the unramified irreducible constituent of the normalizedly induced PGL(2, F_{v})-module I(μ_{i}, μ_{i}^{−1}) (μ_{i} are unramified characters of F_{v}^{×}, i = 1, 2); π_{PGSp(2)}(μ_{1}, μ_{2}) is the unramified irreducible constituent of the PGSp(2, F_{v})-module I_{PGSp(2)}(μ_{1}, μ_{2}) normalizedly induced from the character n·diag(α, β, γ, δ) ↦ μ_{1}(α/γ)μ_{2}(α/β) of the upper triangular subgroup of PGSp(2, F_{v}) (n is in the unipotent radical, αδ = βγ).

The embedding λ : Ĥ = Sp(2, ℂ) ↪ SL(4, ℂ) = Ĝ defines the lifting λ which maps the unramified irreducible PGSp(2, F_{v})-module π_{PGSp(2)}(μ_{1}, μ_{2}) to the unramified irreducible PGL(4, F_{v})-module π_{4}(μ_{1}, μ_{2}, μ_{2}^{−1}, μ_{1}^{−1}).

The composition λ ∘ λ_{0} : Ĉ_{0} = SL(2, ℂ) × SL(2, ℂ) → Ĝ = SL(4, ℂ) takes π_{2}(μ_{1}, μ_{1}^{−1}) × π_{2}(μ_{2}, μ_{2}^{−1}) to π_{4}(μ_{1}, μ_{2}, μ_{2}^{−1}, μ_{1}^{−1}) = π_{4}(μ_{1}, μ_{1}^{−1}, μ_{2}, μ_{2}^{−1}), namely the unramified irreducible PGL(2, F_{v}) × PGL(2, F_{v})-module π_{2} × π′_{2} to the unramified irreducible constituent π_{4}(π_{2}, π′_{2}) of the PGL(4, F_{v})-module I_{4}(π_{2}, π′_{2}) normalizedly induced from the representation π_{2} ⊗ π′_{2} of the parabolic of type (2,2) of PGL(4, F_{v}) (extended trivially on the unipotent radical). For example λ ∘ λ_{0} takes the trivial PGL(2, F_{v}) × PGL(2, F_{v})-module 1_{2} × 1_{2} to the unramified irreducible constituent π_{4}(1_{2}, 1_{2}) of I_{4}(1_{2}, 1_{2}), and 1_{2} × π_{2} to π_{4}(1_{2}, π_{2}) = π_{4}(ν^{1/2}π_{2}, ν^{−1/2}π_{2}). This last π_{4} is traditionally denoted by J.

The definition of lifting is extended from the case of unramified representations to that of any admissible representations. For this purpose we defined in [F5] norm maps from the θ-stable θ-regular conjugacy classes in G = G(F) to stable conjugacy classes in H = H(F), and from these to conjugacy classes in C_{0}(F), extending the norm maps on the split tori in these groups which are dual to the dual groups homomorphisms λ and λ_{0}. This is used to define a relation of matching functions f, f_{H} and f_{C_{0}} (they have suitably defined matching orbital integrals) and a dual relation of liftings of representations.

To express the lifting results we use the following notations for induced representations of H = PGSp(2, F). For characters μ_{1}, μ_{2}, σ of F^{×} with μ_{1}μ_{2}σ^{2} = 1 we write μ_{1} × μ_{2} ⋊ σ for the H-module normalizedly induced from the character mu ↦ μ_{1}(a)μ_{2}(b)σ(λ λ λ), u ∈ U, m = diag(a, b,λ λ λ/b,λ λ λ/a), a, b,λ λ λ ∈ F^{×}, of the upper triangular minimal parabolic of H.

For a GL(2, F)-module π_{2} and character μ we write π_{2} ⋊ μ for the PGSp(2, F)-module normalizedly induced from the representation p = mu ↦ π_{2}(g)μ(λ λ λ), m = diag(g,λ λ λw^{t}g^{−1}w), u ∈ U_{(2)}, λ λ λ ∈ F^{×} (here the product of the central character ω of π_{2} with μ^{2} is 1) of the Siegel parabolic subgroup (whose unipotent radical U_{(2)} is abelian).

We write μ ⋊ π_{2}, if ωμ = 1, for the representation of PGSp(2, F) normalizedly induced from the representation p = mu ↦ μ(a)π_{2}(g), m = diag(a, g,λ λ λ(g)/a), u ∈ U_{(1)}, λ λ λ(g) = det g, of the Heisenberg parabolic subgroup (whose unipotent radical U_{(1)} is a Heisenberg group).

These inductions are normalized by multiplying the inducing representation by the character p ↦ | det(Ad(p))| Lie U|^{1/2}, as usual. For example, I_{H}(μ_{1}, μ_{2}) = μ_{1}μ_{2} × μ_{1}/μ_{2} ⋊ μ_{1}^{−1}. Note that π ⋊ σ ≃ π̌ ⋊ ωσ and μ(π ⋊ σ) = π ⋊ μσ. Complete results describing reducibility of these induced representations, are recorded in [ST] – whose results and notations we use – following earlier work of [Ro2], [Sh2,3] and [W1]. Our lifting results explain the results recorded in [ST].

For properly induced representations, defining λ- and λ_{0}-liftings by character relations (λ(π_{H}) = π_{4} if tr π_{4}(f × θ) = tr π_{H}(f_{H}) for all matching f, f_{H}, and λ_{0}(π_{1} × π_{2}) = π_{H} if tr π_{H}(f_{H}) = tr(π_{1} × π_{2})(f_{C_{0}}) for all matching f_{H}, f_{C_{0}}), our preliminary results (obtained by local character evaluations), are that ω^{−1} ⋊ π_{2} λ-lifts to π_{4} = I_{G}(π_{2}, π̌_{2}), that μπ_{2} ⋊ μ^{−1} (here ω = 1) λ-lifts to π_{4} = I_{G}(μ, π_{2}, μ^{−1}), and that I_{2}(μ, μ^{−1}) × π_{2} λ_{0}-lifts from C_{0} to μπ_{2} ⋊ μ^{−1} on H = PGSp(2, F).

Let χ be a character of F^{×}/F^{×2}. It defines a one-dimensional representation χ_{H}(h) = χ(λ λ λ(h)) of H, which λ-lifts to the one-dimensional representation χ(g) = χ(det g) of G (if h = Ng then λ λ λ(h) = det g; on diagonal matrices N(diag(a, b, c, d)) = diag(ab, ac, db, dc)). The Steinberg representation of H λ-lifts to the Steinberg representation of G, and for any character χ of F^{×} with χ^{2} = 1 we have λ(χ_{H} St_{H}) = χ St_{G}.

6. Elliptic Representations

Our local liftings, λ, λ_{0} and λ_{1}, are relations of representations, defined by means of character relations. Denote by K(H) the Grothendieck group of H (integral linear combinations of equivalence classes of irreducible representations of H). Similarly we have K(C), K(C_{0}), K(G). Denote by K^{θ}(G) the free abelian group of (equivalence classes of) representations of G ⋊ ⟨θ⟩. It is the ℤ-span of the irreducible θ-invariant representations of G, extended to G ⋊ ⟨θ⟩. Denote by I : K(C_{0}) → K^{θ}(G) the functor of normalized induction. Thus I(π_{0}) denotes the representation of G normalizedly induced from π_{0} (π_{0} ∈ K(C_{0}) is viewed as a representation of the parabolic subgroup of type (2, 2) of G), extended to the semidirect product G ⋊ ⟨θ⟩ (there are two extensions, differing by a sign, but there is a canonical choice, defined first for generic representations).

We define two homomorphisms λ_{0}^{+}, λ_{0}^{−} : K(C_{0}) → K(H) as follows. Note that K(C_{0}) = K_{I}(C_{0})⊕K_{□}(C_{0}), where K_{□}(C_{0}) is the span of the square integrable representations, while K_{I}(C_{0}) is spanned by the π_{1} × π_{2} where at least one of π_{1}, π_{2} is induced and the other is induced or square integrable. We require that λ_{0}^{−} = 0 on K_{I}(C_{0}) and λ_{0}^{+}(K_{I}(C_{0})) lies in K_{I}(H), the span of the induced representations of H, and that for each π_{0} in K_{I}(C_{0}) or K_{□}(C_{0}) we have

tr π_{0}(f_{C_{0}}) = tr λ_{0}^{+}(π_{0})(f_{H}) − tr λ_{0}^{−}(π_{0})(f_{H}),
tr I(π_{0}; f × θ) = tr λ_{0}^{+}(π_{0})(f_{H}) + tr λ_{0}^{−}(π_{0})(f_{H}),

for every triple (f, f_{H}, f_{C_{0}}) of matching functions on G, H, C_{0}.

We define a homomorphism λ : K(H) → K^{θ}(G) and a subgroup K_{1}(H) of K(H), as follows. On λ_{0}^{+}(K_{□}(C_{0})) + λ_{0}^{−}(K_{□}(C_{0})) define λ by λ ∘ λ_{0}^{+} = I, λ ∘ λ_{0}^{−} = I on K_{□}(C_{0}). We require that K_{1}(H) satisfies K_{1}(H) ⊕ [λ_{0}^{+}(K_{□}(C_{0})) + λ_{0}^{−}(K_{□}(C_{0}))] = K(H) and that for any π_{H} ∈ K_{1}(H) we have tr λ(π_{H})(f × θ) = tr π_{H}(f_{H}) for all matching (f, f_{H}). Further we define a homomorphism λ_{1} : K(C) → K^{θ}(G) by λ_{1}(π_{C}) = π if for every matching pair (f, f_{C}) we have tr π(f × θ) = tr π_{C}(f_{C}). Our local results assert the existence of the maps λ_{0}^{+}, λ_{0}^{−}, λ, λ_{1}, and the subgroup K_{1}(H), and describe them and their properties in detail. For example, Im λ_{0}^{+} ∩ Im λ_{0}^{−} = {0}.

Thus our finer local lifting results concern elliptic representations (whose characters are non zero on the set of elliptic elements). They follow on using global techniques. Elliptic representations include the cuspidal ones (terminology of [BZ]. These are called “supercuspidal” by Harish-Chandra, who used the word “cuspidal” for what is currently named “discrete series” or “square integrable” representations).

Local Theorem (PGSp(2) to PGL(4)). For any (unordered) pair π_{1}, π_{2} of square integrable (irreducible) representations of PGL(2, F) there exists a (unique) pair π_{H}^{+}, π_{H}^{−} of tempered (square integrable if π_{1} ≠ π_{2}, cuspidal if π_{1} ≠ π_{2} are cuspidal) representations of H with

tr(π_{1} × π_{2})(f_{C_{0}}) = tr π_{H}^{+}(f_{H}) − tr π_{H}^{−}(f_{H}),
tr I_{G}(π_{1}, π_{2}; f × θ) = tr π_{H}^{+}(f_{H}) + tr π_{H}^{−}(f_{H})

for all triples (f, f_{H}, f_{C_{0}}) of matching functions.

If π_{1} = π_{2} is cuspidal, π_{H}^{+} and π_{H}^{−} are the two inequivalent constituents of 1 ⋊ π_{1}.

If π_{1} = π_{2} = σ sp_{2} where σ is a character of F^{×} with σ^{2} = 1, then π_{H}^{+} and π_{H}^{−} are the two tempered inequivalent constituents τ(ν^{1/2} sp_{2}, σν^{−1/2}) and τ(ν^{1/2}1_{2}, σν^{−1/2}) of 1 ⋊ σ sp_{2}.

If π_{1} = σ sp_{2}, σ^{2} = 1, and π_{2} is cuspidal, then π_{H}^{+} is the square integrable constituent δ(σν^{1/2}π_{2}, σν^{−1/2}) of the induced σν^{1/2}π_{2} ⋊ σν^{−1/2}; π_{H}^{−} is cuspidal, denoted here by δ^{−}(σν^{1/2}π_{2}, σν^{−1/2}).

If π_{1} = σ sp_{2} and π_{2} = ξσ sp_{2}, ξ (≠ 1 = ξ^{2}) and σ (σ^{2} = 1) are characters of F^{×}, then π_{H}^{+} is the square integrable constituent δ(ξν^{1/2} sp_{2}, σν^{−1/2}) of the induced ξν^{1/2} sp_{2} ⋊σν^{−1/2}; π_{H}^{−} is cuspidal, denoted here by δ^{−}(ξν^{1/2} sp_{2}, σν^{−1/2}).

For every character σ of F^{×}/F^{×2} and square integrable π_{2} there exists a non tempered representation π_{H}^{×} of H such that for all matching f, f_{H}, f_{C_{0}}

tr(σ1_{2} × π_{2})(f_{C_{0}}) = tr π_{H}^{×}(f_{H}) + tr π_{H}^{−}(f_{H}),
tr I_{G}(σ1_{2}, π_{2}; f × θ) = tr π_{H}^{×}(f_{H}) − tr π_{H}^{−}(f_{H}).

Here π_{H}^{−} = π_{H}^{−}(σ sp_{2} ×π_{2}) and π_{H}^{×} = L(σν^{1/2}π_{2}, σν^{−1/2}).

For any characters ξ, σ of F^{×}/F^{×2} and matching f, f_{H}, f_{C_{0}} we have

tr(σξ1_{2} × σ1_{2})(f_{C_{0}}) = tr L(νξ, ξ ⋊ σν^{−1/2})(f_{H}) − tr X(ξν^{1/2} sp_{2}, ξσν^{−1/2})(f_{H}),
tr I_{G}(σξ1_{2}, σ1_{2}; f × θ) = tr L(νξ, ξ ⋊ σν^{−1/2})(f_{H}) + tr X(ξν^{1/2} sp_{2}, ξσν^{−1/2})(f_{H}).

Here X = δ^{−} if ξ ≠ 1 and X = L if ξ = 1.

Any θ-invariant irreducible square integrable representation π of G which is not a λ_{1}-lift is a λ-lift of an irreducible square integrable representation π_{H} of H, thus tr π(f × θ) = tr π_{H}(f_{H}) for all matching f, f_{H}. In particular, the square integrable (resp. non tempered) constituent δ(ξν, ν^{−1/2}π_{2}) (resp. L(ξν, ν^{−1/2}π_{2})) of the induced representation ξν ⋊ν^{−1/2}π_{2} of H, where π_{2} is a cuspidal (irreducible) representation of GL(2, F) with central character ξ ≠ 1 = ξ^{2} and ξπ_{2} = π_{2}, λ-lifts to the square integrable (resp. non tempered) constituent S(ν^{1/2}π_{2}, ν^{−1/2}π_{2}) (resp. J(ν^{1/2}π_{2}, ν^{−1/2}π_{2})) of the induced representation I_{G}(ν^{1/2}π_{2}, ν^{−1/2}π_{2}) of G = PGL(4, F). □

These character relations permit us to introduce the notion of a packet of a(n irreducible) representation, and of a quasi-packet, over a local field. Thus we say that the packet of a representation π_{H} of H consists of π_{H} alone unless it is tempered of the form π_{H}^{+} or π_{H}^{−} for some pair π_{1}, π_{2} of (irreducible) square integrable representations of PGL(2, F), in which case the packet {π_{H}} is defined to be {π_{H}^{+}, π_{H}^{−}}, and we write λ_{0}(π_{1} × π_{2}) = {π_{H}^{+}, π_{H}^{−}} and λ({π_{H}^{+}, π_{H}^{−}}) = I_{G}(π_{1}, π_{2}). Further, we define a quasi-packet only for the non tempered (irreducible) representations π_{H}^{×} and L = L(νξ, ξ ⋊ σν^{−1/2}), to consist of {π_{H}^{×}, π_{H}^{−}} and {L, X}, X = X(ξν^{1/2} sp_{2}, ξσν^{−1/2}). We say that σ1_{2} × π_{2} λ_{0}-lifts to the quasi-packet λ_{0}(σ1_{2} × π_{2}) = {π_{H}^{×}, π_{H}^{−}}, which in turns λ-lifts to I_{G}(σ1_{2}, π_{2}), and similarly, σξ1_{2} × σ1_{2} λ_{0}-lifts to λ_{0}(σξ1_{2} × σ1_{2}) = {L, X} which λ-lifts to I_{G}(σξ1_{2}, σ1_{2}).

7. Automorphic Representations

With these local definitions we can state our global results. These global results are partial, since we work with test functions whose components are elliptic at least at three places, and consequently we cannot detect automorphic representations which do not have at least three components whose (θ-) characters are non zero on the (θ-) elliptic set. Thus we fix three places {v_{1}, v_{2}, v_{3}} and discuss only π_{1} × π_{2}, π_{H} and π = π_{G} whose components there are (θ-) elliptic. This constraint will be removed once the trace formulae identity is established for general test functions.

With this reservation, emphasized by a ∗-superscript in the following Global Theorem, the discrete series representations of H(𝔸) = PGSp(2, 𝔸) can now be described by means of the liftings. They consist of two types, stable and unstable, and global packets and quasipackets define a partition of the spectrum. To define a (global) [quasi-] packet {π_{H}}, fix a local [quasi-] packet {π_{Hv}} at each place v of F, such that {π_{Hv}} contains an unramified member π_{Hv}^{0} (and then {π_{Hv}} consists only of π_{Hv}^{0} in case it is a packet) for almost all v. The [quasi-] packet {π_{H}} is then defined to consist of all products ⊗_{v}π′_{Hv} with π′_{Hv} in {π_{Hv}} for all v, and π′_{Hv} = π_{Hv}^{0} for almost all v. The [quasi-] packet {π_{H}} of an automorphic representation π_{H} is defined by the local [quasi-] packets {π_{Hv}} of the components π_{Hv} of π_{H} at almost all places.

The discrete spectrum of PGSp(2, 𝔸) will be described by means of the λ_{0}- and λ-liftings. We say that discrete series π_{1} × π_{2} λ_{0}-lifts to a packet {π_{H}} (or to a member thereof) if {π_{Hv}} = λ_{0}(π_{1v} × π_{2v}) for almost all v, and that a packet {π_{H}} (or a member of it) λ-lifts to an irreducible self-contragredient automorphic representation π if λ({π_{Hv}}) = π_{v} for almost all v. The unstable spectrum of PGSp(2, 𝔸) is defined to be the set of discrete series representations which are λ_{0}-lifts; its complement is the stable spectrum.

Global Theorem^{∗} (PGSp(2) to PGL(4)). The packets and quasi-packets partition the discrete spectrum of the group PGSp(2, 𝔸), namely they satisfy the rigidity theorem: if π_{H} and π′_{H} are discrete series representations locally equivalent at almost all places then their packets or quasi packets are equal.

The λ-lifting is a bijection between the set of packets (resp. quasi-packets) of discrete series representations in the stable spectrum (of PGSp(2, 𝔸)) and the set of self contragredient discrete series representations of PGL(4, 𝔸) which are one dimensional, or cuspidal and not a λ_{1}-lift from C(𝔸) (resp. residual J(ν^{1/2}π_{2}, ν^{−1/2}π_{2}) where π_{2} is a cuspidal representation of GL(2, 𝔸) with central character ξ ≠ 1 = ξ^{2} and ξπ_{2} = π_{2}).

The λ_{0}-lifting is a bijection between the set of pairs of discrete series representations {π_{1} × π_{2}, π_{2} × π_{1}; π_{1} ≠ π_{2}} of PGL(2, 𝔸) × PGL(2, 𝔸), and the set of packets and quasipackets in the unstable spectrum of PGSp(2, 𝔸). The λ-lifting is a bijection from this last set to the set of automorphic representations I_{G}(π_{1}, π_{2}) of PGL(4, 𝔸), normalizedly induced from discrete series π_{1} × π_{2} (π_{1} ≠ π_{2}) on the parabolic subgroup with Levi factor of type (2, 2). If π_{1} × π_{2} is cuspidal, its λ_{0}-lift is a packet, otherwise: quasi-packet.

Each member of a stable packet occurs in the discrete spectrum of PGSp(2, 𝔸) with multiplicity one. The multiplicity m(π_{H}) of a member π_{H} = ⊗π_{Hv} of an unstable [quasi-]packet λ_{0}(π_{1} × π_{2}) (π_{1} ≠ π_{2}) is not (“stable”, or) constant over the [quasi-]packet. If π_{1} × π_{2} is cuspidal, it is m(π_{H}) = \frac{1}{2}(1 + (−1)^{n(π_{H})}) (∈ \{0, 1\}), where n(π_{H}) is the number of components π_{Hv}^{−} of π_{H} and each π_{H} with m(π_{H}) = 1 is cuspidal (n(π_{H}) is bounded by the number of places v where both π_{1v} and π_{2v} are square integrable).

The multiplicity m(π_{H}) (in the discrete spectrum of PGSp(2, 𝔸)) of π_{H} = ⊗π_{Hv} of a quasi-packet λ_{0}(σ1_{2} × π_{2}), where π_{2} is a cuspidal representation of PGL(2, 𝔸) and σ is a character of 𝔸^{×}/F^{×}𝔸^{×2}, is \frac{1}{2}(1 + ε(σπ_{2}, \frac{1}{2})(−1)^{n(π_{H})}) (= 0 or 1), where n(π_{H}) is the number of components π_{Hv}^{−} of π_{H}, and ε(σπ_{2}, s) is the usual ε-factor which appears in the functional equation of the L-function L(π_{2}, s). In particular π_{H}^{×} = ⊗π_{Hv}^{×} (n(π_{H}) = 0) is discrete series if and only if ε(σπ_{2}, \frac{1}{2}) = 1.

Finally we have m(π_{H}) = \frac{1}{2}(1 + (−1)^{n(π_{H})}) for π_{H} = ⊗π_{Hv} in λ_{0}(σξ1_{2} × σ1_{2}) with n(π_{H}) components π_{Hv} = X_{v}. Here π_{H} = ⊗L_{v} (n(π_{H}) = 0) is residual. □

8. Unstable Spectrum

Note that the quasi-packet λ_{0}(σξ1 × σ1_{2}) is defined by the local quasi-packets {L_{v} = L(ν_{v}ξ_{v}, ξ_{v} ⋊ σ_{v}ν_{v}^{−1/2}), X_{v} = X(ξ_{v}ν_{v}^{1/2} sp_{2v}, ξ_{v}σ_{v}ν_{v}^{−1/2})} for every v, where ξ (≠ 1), σ are characters of 𝔸^{×}/F^{×} with ξ^{2} = 1 = σ^{2} and ξ_{v}, σ_{v} are their components. When ξ_{v}, σ_{v} are unramified, this quasi-packet contains the unramified representation π_{Hv}^{0} = L_{v}. Members of this quasi-packet have been studied by means of the theta correspondence by Howe and Piatetski-Shapiro, see, e.g., [PS1], Theorem 2.5. They attracted interest since they violate the naive generalization of the Ramanujan conjecture, which expects the components of a cuspidal representation to be tempered. (The form of the Ramanujan conjecture which is expected to be true asserts that the components of a cuspidal representation of PGSp(2,𝔸) which λ-lifts to a cuspidal representation of PGL(4,𝔸) are tempered.) Members of this quasi-packet are equivalent at almost all places to the quotient of the properly induced representation νξ × ξ ⋊ σν^{−1/2}.

Let π_{2} be a cuspidal representation of PGL(2, 𝔸) and σ a character of 𝔸^{×}/F^{×}𝔸^{×2}. The packet λ_{0}(σ1_{2} × π_{2}) contains the constituent π_{H}^{×} = ⊗_{v}π_{Hv}^{×} of the representation σν^{1/2}π_{2} ⋊ σν^{−1/2} ≃ σν^{−1/2}π_{2} ⋊ σν^{1/2} properly induced from an automorphic representation, hence it is automorphic by [L]. It is known that π_{H}^{×} is residual precisely when L(σπ_{2}, \frac{1}{2}) ≠ 0; hence ε(σπ_{2}, \frac{1}{2}) = 1 in this case.

Let n(π_{2}) denote the number of square integrable components of π_{2}. The quasi-packet λ_{0}(σ1_{2} × π_{2}) thus consists of 2^{n(π_{2})} (irreducible) representations. If n(π_{2}) ≥ 1, half of them are discrete series, all cuspidal if L(σπ_{2}, \frac{1}{2}) = 0, all but one: π_{H}^{×} = ⊗_{v}π_{Hv}^{×}, are cuspidal if L(σπ_{2}, \frac{1}{2}) ≠ 0. If n(π_{2}) ≥ 1 and L(σπ_{2}, \frac{1}{2}) = 0, the automorphic non residual π_{H}^{×} is cuspidal when ε(σπ_{2}, \frac{1}{2}) = 1.

If π_{2} has no square integrable components (n(π_{2}) = 0), the packet λ_{0}(σ1_{2} × π_{2}) consists only of π_{H}^{×}. This π_{H}^{×} is residual if L(σπ_{2}, \frac{1}{2}) ≠ 0; cuspidal (by [PS1], Theorem 2.6 and [PS2], Theorem A.2) if L(σπ_{2}, \frac{1}{2}) = 0 and ε(σπ_{2}, \frac{1}{2}) = 1; or (automorphic but) not discrete series otherwise: L(σπ_{2}, \frac{1}{2}) = 0 and ε(σπ_{2}, \frac{1}{2}) = −1. In this last case the λ_{0}-lift of σ1_{2} × π_{2} is not a discrete series, and there is no discrete series λ-lifting to I_{G}(σ1_{2}, π_{2}).

At a place v where π_{2v} is induced I(μ_{v}, μ_{v}^{−1}), the packet π_{Hv} = λ_{0}(σ_{v}1_{2} × π_{2v}) is the irreducible induced μ_{v}σ_{v}1_{2} ⋊μ_{v}^{−1}, which λ-lifts to I_{G}(μ_{v}, σ_{v}1_{2}, μ_{v}^{−1}), and not the irreducible induced

σ_{v}μ_{v}ν_{v}^{1/2} × σ_{v}μ_{v}^{−1}ν_{v}^{1/2} ⋊ σ_{v}ν_{v}^{−1/2} = σ_{v}μ_{v}ν_{v}^{1/2} ⋊ I(μ_{v}^{−1}, σ_{v}ν_{v}^{−1/2}),

which λ-lifts to the reducible induced I_{G}(μ_{v}, σ_{v}I(ν_{v}^{1/2}, ν_{v}^{−1/2}), μ_{v}^{−1}), which has the constituent I_{G}(μ_{v}, σ_{v}1_{2}, μ_{v}^{−1}).

Members of the quasi-packet λ_{0}(σ1_{2} × π_{2}) were studied numerically by H. Saito and N. Kurokawa, and using the theta correspondence by Piatetski-Shapiro and others, see [PS1], Theorem 2.6. They attracted interest since they violate the naive generalization of the Ramanujan conjecture. They are equivalent at almost all places to the quotient of the properly induced representation σν^{1/2}π_{2} ⋊ σν^{−1/2}.

A discrete series representation π_{H} with component L(ν_{v}ξ_{v}, ν_{v}^{−1/2}π_{2v}) (whose packet consists of itself), where π_{2v} is a cuspidal representation with central character ξ_{v} ≠ 1 = ξ_{v}^{2} and ξ_{v}π_{2v} = π_{2v}, is in the packet of L(νξ, ν^{−1/2}π_{2}). Here π_{2} is cuspidal with central character ξ ≠ 1 = ξ^{2}, hence ξπ_{2} = π_{2}, whose components at v are π_{2v} and ξ_{v}. It λlifts to J_{G}(ν^{1/2}π_{2}, ν^{−1/2}π_{2}). At v with ξ_{v} = 1 the component π_{2v} is induced. If π_{2v} = I(μ_{v}, μ_{v}ξ_{v}), ξ_{v}^{2} = 1 and μ_{v}^{2} = 1 (in particular whenever ξ_{v} ≠ 1 and π_{2v} is not cuspidal), then L(ν_{v}ξ_{v}, ν_{v}^{−1/2}π_{2v}) is L_{v} = L(ν_{v}ξ_{v}, ξ_{v} ⋊ μ_{v}ν_{v}^{−1/2}), which λ-lifts to I_{G}(μ_{v}1_{2}, μ_{v}ξ_{v}1_{2}), and its packet contains also X_{v} = X(ν_{v}^{1/2}ξ_{v} sp_{2v}, ξ_{v}μ_{v}ν_{v}^{−1/2}). Thus the packet of π_{H} is determined by {L_{v}, X_{v}} at all v where π_{2v} = I(μ_{v}, μ_{v}ξ_{v}), μ_{v}^{2} = 1 = ξ_{v}^{2}, and by the singleton {L_{v} = L(ν_{v}ξ_{v}, ν_{v}^{−1/2}π_{2v})} at all other v, where π_{2v} is cuspidal, or ξ_{v} = 1 and π_{2v} = I(μ_{v}, μ_{v}^{−1}), μ_{v}^{2} ≠ 1. Each member of this infinite packet occurs in the discrete series with multiplicity one, and is cuspidal, with the exception of L(νξ, ν^{−1/2}π_{2}) = ⊗_{v}L(ν_{v}ξ_{v}, ν_{v}^{−1/2}π_{2v}), which is residual ([Kim], Theorem 7.2). Members of the packet L(νξ, ν^{−1/2}π_{2}) are considered in the Appendix of [PS1] and its corrigendum.

If π_{1} and π_{2} are cuspidal but there is no place v where both are square integrable, λ_{0}(π_{1} × π_{2}) consists of a single irreducible cuspidal representation. This instance of the lifting λ_{0} – where π_{i} are cuspidal – can also be studied using the theta correspondence for suitable dual reductive pairs (SO(4), PGSp(2)) for the isotropic and anisotropic forms of the orthogonal group, to describe further properties of the packets, such as their periods.

9. Generic Representations

Our proof of the existence of the lifting λ uses only the trace formula, orbital integrals and character relations. To compute the multiplicities in the discrete spectrum we use two independent techniques. The first is purely local. It uses only character relations. It is based on a twisted analogue of Rodier’s result [Ro1] that the number of Whittaker models is captured as a coefficient of a regular nilpotent orbit in the germ expansion of the character. Such a technique was first used in [F4;II], Proof of Proposition 3.5, p. 47, to prove multiplicity one for the discrete spectrum of U(3). For further details see [F4;IV]. Our second proof is global. It relies on results of [GRS], [KRS], and [Sh1] from the theory of generic representations.

This second proof in our case of PGSp(2) is similar to the second proof of [F4;II], Proposition 3.5, p. 48, in the case of U(3). The 2nd proof of [F4;II] is also based on the theory of generic representations. But it is not complete. Indeed, the claim in Proposition 2.4(i) in reference [GP] to [F4;II], that “L_{0,1}^{2} has multiplicity 1”, is interpreted in [F4;II] as asserting that generic representations of U(3) occur in the discrete spectrum with multiplicity one. But it should be interpreted as asserting that irreducible π in L_{0,1}^{2} have multiplicity one only in the subspace L_{0,1}^{2} of the discrete spectrum. This assertion does not exclude the possibility of having a cuspidal π′ perpendicular and equivalent to π ⊂ L_{0,1}^{2}. Multiplicity one for the generic spectrum of U(3) could be deduced from Proposition 2.4(i) of [GP] of [F4;II] using the statement that a locally generic representation is globally generic. (In fact this statement is equivalent to multiplicity one). In our case of PGSp(2) this statement follows from [KRS], [GRS], [Sh1]. A proof for U(3) (independent of the local proof of [F4;II], p.

47) still needs to be written down.

There is some overlap between our results on the existence of the λ-lifting and the work of [GRS]. However, the methods of [GRS] apply only to generic representations, while our methods apply to all representations of PGSp(2). Thus we can define packets, describe their structure, establish multiplicity one theorem and rigidity theorem for packets of PGSp(2), conclude that in a packet which λ-lifts to a generic self-contragredient representation of PGL(4, 𝔸) there is precisely one generic member, and we can also λ-lift the non generic non tempered (at almost all places) packets to residual self-contragredient representations of PGL(4, 𝔸). Our liftings are proven in terms of all places, not only almost all places. In addition we establish the lifting λ_{1} from SO(4) to PGL(4), determine its fibers (that is, prove multiplicity one theorem for SO(4) and rigidity in the sense explained above), and show that any self-contragredient discrete series representation of PGL(4, 𝔸) which is not a λ-lift from PGSp(2, 𝔸) is a λ_{1}-lift from SO(4, 𝔸).

Our work is an analogue for (SO(4), PGSp(2), PGL(4)) of [F3], which dealt with the symmetric square lifting, thus with (PGL(2), SL(2), PGL(3)), and of [F4], which dealt with quadratic base change for the unitary group U(3,E/F), thus with (U(2,E/F), U(3,E/F), GL(3,E)). These works use the twisted – by transpose-inverse (and the Galois action in the unitary groups case) – trace formulae on PGL(4), PGL(3), GL(3,E). They are based on the fundamental lemma: [F5] in our case, [F3;V] and [F4;I] in the other cases. The technique employed in these last works benefited from work of Weissauer [W] and Kazhdan [K]. The present work, which deals with the applications of the fundamental lemma and the trace formula to character relations, liftings and the definition of packets, is analogous to [F3;II] and [F4;IV]. The trace formula identity is proven in [F3;III] and [F4;III] for all test functions. Here we deal only with test functions which have at least three elliptic components. The trace formulae identity for a general test function has not yet been proven in our case. There is no doubt that the method of [AC] could be used for that, as it has been applied in a general rank case of GL(n). It would be interesting to pursue the elementary techniques of [F3;III] and [F4;III], and [F2], which establishes the trace formulae identity for base change for GL(2) by elementary means, based on the usage of regular, Iwahori test functions. In particular our work does not develop the trace formula. It only uses a form of it.

Our approach to the lifting uses the trace formula, developed by Arthur, as envisaged by Langlands e.g. in his work on base change for GL(2). It is compatible with the conjectures of Arthur [A]. Of course Siegel modular forms have been extensively studied by many authors (e.g. Siegel, Maass, Shimura, Andrianov,...) over a long period of time.

An important representation theoretic approach alternative to the trace formula, based on the theta correspondence, Weil representation, Howe’s dual reductive pairs, L-functions and converse theorems, has been fruitfully developed in our context of the symplectic group by many authors, see, e.g., [PS], [KRS], [GRS].

Another – purely local – approach is developed in [FZ].

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