ELECTRONIC RESEARCH ANNOUNCEMENTS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 10, Pages 39–50 (April 23, 2004) S 1079-6762(04)00128-3

AUTOMORPHIC FORMS ON PGSp(2)

YUVAL Z. FLICKER

(Communicated by David Kazhdan)

Abstract. The theory of lifting of automorphic and admissible representations is developed in a new case of great classical interest: Siegel automorphic forms. The self-contragredient representations of PGL(4) are determined as lifts of representations of either symplectic PGSp(2) or orthogonal SO(4) rank two split groups. Our approach to the lifting uses the global tool of the trace formula together with local results such as the 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 the projective symplectic group of similitudes PGSp(2), and analyse the structure of all packets. All representations, not only generic or tempered ones, are studied. Globally we obtain a multiplicity one theorem for the discrete spectrum of the projective symplectic group PGSp(2), a rigidity theorem for packets and quasi-packets, determine all counterexamples to the naive Ramanujan conjecture, and compute the multiplicity of each member in a packet or quasi-packet in the discrete spectrum. 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).

According to the “principle of functoriality”, “Galois” representations ρ : L_{F} → LG of the hypothetical Langlands group L_{F} of a global field F into the complex dual group ^{L}G of a reductive group G over F should parametrize “packets” of automorphic representations of the adèle group G(𝔸). Thus a homomorphism λ : ^{L}H → ^{L}G of complex dual groups should give rise to lifting of automorphic representations π_{H} of H(𝔸) to those π of G(𝔸).

We report here on the work of [F1]. It proves the existence of the expected lifting of automorphic representations of the projective symplectic group of similitudes H = PGSp(2) to those on G = PGL(4). The image is the set of the self-contragredient representations of PGL(4) which are not lifts of representations of the rank two split orthogonal group SO(4).

The global lifting is defined by means of local lifting. We define the local lifting in terms of character relations. This permits us to introduce a definition of packets and quasi-packets of representations of PGSp(2) as the sets of representations that

Received by the editors March 4, 2004. 2000 Mathematics Subject Classification. Primary 11F70; Secondary 22E50, 22E55, 22E45. Key words and phrases. Automorphic representations, symplectic group, liftings, twisted en-

doscopy, packets, quasi-packets, multiplicity one, rigidity, functoriality, twisted trace formula, character relations.

Partially supported by a Lady Davis Visiting Professorship at the Hebrew University, 2004, and Max-Planck-Institut für Mathematik, Bonn, 2003.

\begin{matrix}c \\ ◯ \end{matrix}2004 American Mathematical Society

occur in these relations. Our main local result is that packets exist and partition the set of tempered representations. We give a detailed description of the structure of packets (in Sect. 4).

Our global results include a detailed description of the structure of the global packets and quasi-packets (the latter are almost everywhere nontempered). We obtain a multiplicity one theorem for the discrete spectrum of PGSp(2), a rigidity theorem for packets and quasi-packets, determine all counterexamples to the naive Ramanujan conjecture, and compute the multiplicity of each member in a packet or quasi-packet in the discrete spectrum, in Sect. 5.

Interesting phenomena of instability are described in Sect. 6. Sect. 7 discusses the special case of generic representations. It serves as another introduction. The first three sections are preparatory. They describe the relevant parts of abstract functoriality in our case.

We also prove the lifting from SO(4) to PGL(4). This amounts to establishing a product of two representations of GL(2) with central characters whose product is 1. Our rigidity theorem for SO(4) amounts to a strong rigidity statement for a pair of representations of GL(2, 𝔸); see [F6]. Our method uses the global tool of the trace formula and the local tool of the fundamental lemma. We deal with all, not only generic or tempered, representations.

For applications to decomposition of cohomology of Shimura varieties see [F7].

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 being the ring of adèles 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 θπ(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 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 the duals of the twisted (by ŝθ̂) endoscopic groups ([KS]).

A twisted endoscopic group is called elliptic if its dual is not contained in a proper parabolic subgroup of Ĝ. Representations of nonelliptic 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 have as duals 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 spectrum representations of H(𝔸) and C(𝔸) parametrize (or lift to) the θ-invariant automorphic discrete spectrum 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 adèle 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 nonarchimedean 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 principal 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) ⊗ 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.

For lack of space here we shall describe elsewhere (see [F6]) our results on our secondary lifting λ_{1}, to G = PGL(4) from C = SO(4) = ((GL(2)×GL(2))/ GL(1))′.

3. 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 from 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 to 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). These norm maps extend the norm maps on the split tori in these groups. The latter maps are dual to the homomorphisms λ and λ_{0} of the dual groups. 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],

[Sh3] and [W]. Our lifting results explain the results of [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} λlifts 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}.

4. Elliptic representations

Our local liftings, λ, λ_{0} and λ_{1}, are relations of representations, defined by means of character relations. Thus our finer local lifting results concern elliptic representations (whose characters are nonzero on the set of elliptic elements). They follow on using global techniques. Elliptic representations include the cuspidal ones (terminology of [BZ1], [BZ2]; 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 nontempered 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. nontempered) 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}, λlifts to the square integrable (resp. nontempered) 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 an 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 nontempered (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 turn 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}).

5. 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 two places. This constraint will be removed once the trace formulae identity is established for general test functions. With further work, known techniques can reduce the constraint to a single place.

With this reservation, emphasized by a ∗-superscript in the following Global Theorem, the discrete spectrum representations of H(𝔸) = PGSp(2, 𝔸) can now be described by means of the liftings. They consist of two types, stable and unstable. Global packets and quasi-packets 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 the discrete spectrum representation 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 selfcontragredient 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 spectrum 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 spectrum 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 spectrum representations in the stable spectrum (of PGSp(2, 𝔸)) and the set of self-contragredient discrete spectrum 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 spectrum 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 quasi-packets 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 the discrete spectrum representation 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, a 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 in the discrete spectrum 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. □

6. 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 quasipacket λ_{0}(σ1_{2} × π_{2}) thus consists of 2^{n(π_{2})} (irreducible) representations. If n(π_{2}) ≥ 1, half of them are in the discrete spectrum, all cuspidal if L(σπ_{2}, \frac{1}{2}) = 0, and 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 nonresidual 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 in the discrete spectrum 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 in the discrete spectrum, and there is no discrete spectrum representation 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 spectrum 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 λ-lifts 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 λlifts to I_{G}(μ_{v}1_{2}, μ_{v}ξ_{v}1_{2}). 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 spectrum 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 ([Rb]) 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.

7. 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 a global proof. It relies on results of [GRS], [KRS], and [Sh1] from the theory of generic representations.

This global 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) in [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) a similar 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 given.

The first proof of Proposition 3.5 in [F4, II], p. 47, namely that the multiplicity one theorem holds for the discrete spectrum of U(3), 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 (0 or 1) is encoded in the behavior of the character near the origin. For further details see [F4, IV]. Such a technique was first used in [FK], to prove the multiplicity one theorem for the metaplectic group of GL(n).

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 the multiplicity one theorem and rigidity theorem for packets of PGSp(2), specify which member in a packet or a quasi-packet is in the discrete spectrum, and we can also λ-lift the nongeneric nontempered (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 the multiplicity one theorem for SO(4) and rigidity in the sense explained in [F6] and [F1]), and show that any self-contragredient discrete spectrum 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 papers benefited from the work of Weissauer [We] 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, IV] and [F4, II]. The trace formulae identity is proven in [F3, VI] and [F4, III] for all test functions. Here we deal only with test functions which have at least two elliptic components. The trace formulae identity for a general test function has not yet been proven in our case. Arthur is working on this. It would be interesting to pursue the elementary techniques of [F3, VI] 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, and others) 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., [PS1], [PS2], [KRS], [GRS].

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

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Department of Mathematics, The Ohio State University, 231 W. 18th Ave., Columbus, OH 43210-1174

   E-mail address: flicker@math.ohio-state.edu