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On a Correspondence Between Cuspidal JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY Volume 12, Number 3, Pages 849{907 S 0894-0347(99)00300-8 Article electronically published on April 26, 1999 ON A CORRESPONDENCE BETWEEN CUSPIDAL REPRESENTATIONS OF GL2n AND Sp2n f DAVID GINZBURG, STEPHEN RALLIS, AND DAVID SOUDRY Introduction Let K be a number field and A its ring of adeles. Let η be an irreducible, automorphic, cuspidal representation of GLm(A). Assume that η is self-dual. By Langlands conjectures, we expect that η is the functorial lift of an irreducible, automorphic, cuspidal representation σ of G(A), where G is an appropriate classical group defined over K. We know by [J.S.1] that the partial (as well as the complete) L-function LS(η η, s) has a (simple) pole at s = 1. Thus, since LS(η η, s)= LS(η, Sym2,s)LS⊗(η, Λ2,s), either LS(η, Λ2,s)orLS(η, Sym2,s) has a pole⊗ at s =1. If m =2n, then one expects that in the first case G =SO2n+1, and in the second S 2 case G =SO2n.Ifm=2n+ 1, we know that L (η, Λ ,s)isentire(see[J.S.2]), S 2 and hence L (η, Sym ,s) has a pole at s = 1. Here, one expects that G =Sp2n. In [G.R.S.1], we started a program to prove the existence of σ as above. See also [G.R.S.2], [G.R.S.3]. We proposed to prove this existence by explicit construc- tion. We constructed a space Vσ(η) of automorphic forms on G(A), invariant under right translations. The elements of Vσ(η) are Bessel coefficients, or Fourier-Jacobi coefficients, restricted to G(A), of residues at s = 1 of certain Eisenstein series on an appropriate (“larger”) group H(A), induced from η.ThespaceVσ(η)has an extra property. It contains all irreducible, automorphic, cuspidal, generic (with respect to a given nondegenerate character) representations σ of G(A) which lift weakly to η (i.e. the unramified parameters of σ, at almost all places, are deter- L mined by those of η through the L-embedding G, GLm(C)). This is a first step → towards the well-known conjecture that every tempered L-packet on G(A) contains a generic representation. Since we believe in the existence of functorial lifting from G to GLm, we expect that Vσ(η) is nontrivial. This remained as a conjecture in [G.R.S.1]. From now on we restrict ourselves to the following special case. Let m =2n, and assume that LS(η, Λ2,s) has a pole at s =1.Now,weaddtheas- sumption that LS(η, 1 ) = 0. (It can be checked that LS(η, 1 ) =0,iffL(η, 1 ) =0, 2 6 2 6 2 6 for a unitary η.) Fix a nontrivial character ψ of K A. One expects that the “back- \ ward” lift of η to SO2n+1(A) is lifted from the metaplectic group Sp2n(A), via the theta correspondence associated to ψ (see [F]). Indeed, in this case we constructed in [G.R.S.1] a space Vσ(η), as above, of automorphic forms on Sp2nf(A). We proved in [G.R.S.1, Theorem 13] that the elements of Vσ(η) are cuspidal in the sense that f Received by the editors July 22, 1998 and, in revised form, March 1, 1999. 1991 Mathematics Subject Classification. Primary 11F27, 11F70, 11F85. The first and third authors' research was supported by The Israel Science Foundation founded by the Israel Academy of Sciences and Humanities. c 1999 American Mathematical Society 849 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 850 DAVID GINZBURG, STEPHEN RALLIS, AND DAVID SOUDRY all their constant terms along radicals of parabolic subgroups are identically zero. The main (global) result of this paper is Main Theorem (global). Let η be an irreducible, automorphic, cuspidal, self-dual S 2 representation of GL2n(A). Assume that L (η, Λ ,s) has a pole at s =1and that S 1 L (η, 2 ) =0. Then the space of automorphic cusp forms Vσ(η) on Sp2n(A) is nontrivial.6 f The space Vσ(η) affords an automorphic, cuspidal, genuine representation σ(η) of Sp2n(A). The construction of σ(η) and the results of [G.R.S.1] show Theorem 1. The cuspidal representation σ(η) contains all irreducible, automor- f 1 phic, cuspidal, genuine representations σ of Sp2n(A), which have a nontrivial ψ− - 1 S Whittaker coefficients (i.e. ψ− -generic) and for which Lψ(σ η, s) has a pole at 1 ⊗ s =1. Each irreducible summand of σ(η) is fψ− -generic. As explained in [G.R.S.4, Sec. 3.1], there is no canonical definition of the (stan- dard) L-function for σν ην on Sp (Kν ) GL n(Kν ) for almost all places ν.It ⊗ 2n × 2 depends on a choice of a nontrivial character of Kν. It turns out that f (0.1) Lψ (σν ην ,s)=L(θψ (σν) ην,s) ν ⊗ ν ⊗ whenever all data is unramified. Here θψν (σν ) is the unramified representation of SO2n+1(Kν ), obtained from σν , under the local theta correspondence θψν , with respect to ψν , from Sp2n(Kν )toSO2n+1(Kν ). The r.h.s. of (0.1) is the standard L-function for SO n GL n. This also suggests that the notion of an L-group 2 +1 × 2 can be extended to fSp2n(F ). Its L-group should be the L-group of SO2n+1(F ), i.e. Sp2n(C). (See [Sa] where this is justified in terms of Hecke algebras.) Note again that the choicef of a nontrivial character of Kν enables us to associate an unramified representation of Sp2n(Kν ) to a semisimple conjugacy class in Sp2n(C). Thus, the standard embedding Sp2n(C) GL2n(C) should yield a (weak) lift from ⊂ genuine automorphic representationsf σ of Sp2n(A) to automorphic representations π of GL2n(A), only after we fixed ψ. In light of (0.1), π should be the weak lift of θψ(σ), the representation of SO2n+1f(A), obtained from σ under the theta correspondence with respect to ψ.Notethatifσon Sp2n(A)hasaψ-weak lift to S η on GL2n(A), then, of course, L (σ η, s) has a pole at s = 1. (The converse is ψ ⊗ not trivial. In a sequel to this paper, we will show, usingf some of the new results of this paper, that σ(η) is exactly the direct sum of all irreducible, automorphic, 1 genuine, cuspidal and ψ− -generic representations σ of Sp2n(A) which, under the ψ-weak lift to GL2n(A), lift to η. This will show, using Theorem 1, that if σ is 1 S ψ− -generic on Sp2n(A), and L (σ η, s) has a pole at sf=1,thenηis the ψ-weak ψ ⊗ lift of σ.) The global theoryf sketched so far has a beautiful local counterpart. The con- structions and theorems are motivated by the global results. However, the passage is not trivial. This local theory occupies the majority of the contents of this pa- per. Let F be a local nonarchimedean field of characteristic zero. Fix a nontrivial character of F still denoted by ψ. We construct an explicit map, which associates an irreducible, supercuspidal representation σ(τ) of the metaplectic group Sp2n(F ) to an irreducible self-dual, supercuspidal representation τ of GL2n(F ), such that L(τ,Λ2,s) has a pole at s = 0. (See [Sh1] for the definition of L(τ,Λ2,sf).) The License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use CUSPIDAL REPRESENTATIONS OF GL2n AND Sp2n 851 representation σ(τ) has a Whittaker model, with respect tof the standard nonde- 1 generate character determined by ψ− . The defining property of σ(τ), and this is in essence the main local result of this work, is Main Theorem (local). The representation σ(τ) is the unique irreducible rep- 1 resentation σ of Sp2n(F ), which is supercuspidal, ψ− -generic and such that the gamma factor γ(σ τ,s,ψ) has a pole at s =1. × f The gamma factor just mentioned is the local gamma factor which appears in the corresponding local theory of global integrals of Shimura type, which represent the standard (partial) L-function for generic cusp forms on Sp2n(A) GL2n(A). (See [G.R.S.4]. In general, the local gamma factor γ(σ τ,s,ψ) may be× defined similarly 1 ⊗ for any irreducible ψ− -generic representation of Sp2k(Ff) GLm(F ). We expect that it is equal, at least up to a simple exponential, to the× local gamma factor associated (say, by Shahidi, or as in [So]) to θψ(σ)f τ on SO k (F ) GLm(F ), ⊗ 2 +1 × where θψ(σ) is the local ψ-theta lift of σ to SO2k+1(F ).) Here we get the added property that σ(τ) is irreducible, which we do not have yet in the global case, but, of course, conjecture to be true. Due to our lack of knowledge of the local representation theory of Sp2n(F ), we do not yet have a good definition of the local L-factor Lψ(σ η, s)andthelocalε-factor ε(σ τ,s,ψ), such ⊗ ⊗ that f ε(σ τ,s,ψ)Lψ(σ τ,1 s) (0.2) γ(σ τ,s,ψ)= ⊗ ⊗ − . ⊗ Lψ(σ τ,s) ⊗ b However, the gamma factor on the l.h.s.
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