On explicit lifts of cusp forms from GL m to classical groups

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1 Annals of Mathematics, 50 (999), On explicit lifts of cusp forms from GL m to classical groups By David Ginzburg, Stephen Rallis, and David Soudry* Introduction In this paper, we begin the study of poles of partial L-functions L S (σ τ,s), where σ τ is an irreducible, automorphic, cuspidal, generic (i.e. with nontrivial Whittaker coefficient) representation of GA GL m (A). G is a split classical group and A is the adele ring of a number field F. We also consider Sp 2n (A) GL m (A), where denotes the metaplectic cover. Examining L S (σ τ,s) through the corresponding Rankin-Selberg, or Shimura-type integrals ([G-PS-R], [G-R-S3], [G], [So]), we find that the global integral contains, in its integrand, a certain normalized Eisenstein series which is responsible for the poles. For example, if G =SO 2k+, then the Eisenstein series is on the adele points of split SO 2m, induced from the Siegel parabolic subgroup and τ det s /2. If G = Sp 2k (this is a convenient abuse of notation), then the Eisenstein series is on Sp 2m (A), induced from the Siegel parabolic subgroup and τ det s /2. The constant term (along the Siegel radical ) of such a normalized Eisenstein series involves one of the L-functions L S (τ,λ 2, 2s ) or L S (τ,sym 2, 2s ). So, up to problems of normalization of intertwining operators, the only pole we expect, for Re(s) > /2, is at s = and then τ should be self-dual. (See [J-S], [B-G].) Thus, let us assume that τ is self-dual. By [J-S], we know that L S (τ τ,s) has a simple pole at s =. Since (0.) L S (τ τ,s)=l S (τ,λ 2,s)L S (τ,sym 2,s) and since each factor is nonzero at s =, it follows that exactly one of the L-functions L S (τ,λ 2,s) and L S (τ,sym 2,s) has a simple pole at s =. We recall that if m is odd, then L S (τ,λ 2,s) is entire ([J-S]). Assume, for example, that m =2n and L S (τ,λ 2,s) has a pole at s =. Then the above-mentioned Eisenstein series (denote it, for this introduction, by E τ,s (h)) on SO 4n (A) has a simple pole at s =. (See Proposition and Remark 2.) Let G =SO 2k+ and σ be as above. Langlands conjectures, predict *This research was supported by the Basic Research Foundation administered by the Israel Academy of Sciences and Humanities.

2 808 DAVID GINZBURG, STEPHEN RALLIS, AND DAVID SOUDRY the existence of a functorial lift of σ, denote it π, ongl 2k (A). π is an irreducible automorphic representation of GL 2k (A). In particular, L S (σ τ,s)= L S (π τ,s). Thus, if k<n, L S (σ τ,s) should be holomorphic at s =. Similarly, if L S (τ,λ 2,s) has a pole at s =,(m =2n), and L S (τ, st, /2) 0, (partial standard L-function) then the above-mentioned Eisenstein series (denote it by E τ,s (h)) on Sp 4n (A) has a simple pole at s = (see Proposition ). Let G = Sp 2k and σ as above. Again, one expects the existence of a functorial lift of σ, denote it π, ongl 2k (A) (such a lift is not canonic, it depends on a choice of a nontrivial character ψ of F \A). One of our main results (Theorem 4) states Theorem A. Let σ τ be an irreducible, automorphic, cuspidal, generic representation of Sp 2k (A) GL 2n (A), such that k<n. Assume that (0.2) L S (τ, st, /2) 0 and L S (τ,λ 2,s) has a pole at s =. Then L S ψ (σ τ,s) is holomorphic at s =. The definition of the partial standard L-function of σ τ depends on a choice of ψ; see [G-R-S3]. The proof of the theorem involves new ideas and results which we regard as the main contribution of this work. In general terms, we actually construct a candidate σ = σ(τ) =σ ψ (τ), on Sp 2n (A), which should lift functorially to τ. Moreover, σ(τ) fits into a fascinating tower of automorphic cuspidal modules σ k (τ) of Sp 2k (A), 0 k 2n. (Define Sp 0 (A) ={}.) The tower property is the same as in the theory of theta series liftings of dual pairs [R]. Thus, for the first index l n, such that σ ln (τ) is nontrivial, σ ln (τ) is cuspidal, and for higher indices, σ k (τ) is noncuspidal. Let us give some more details. We first consider the example where σ is a generic representation of SO 2k+ (A) and τ is on GL 2n (A), such that L S (τ,λ 2,s) has a pole at s =. The Rankin-Selberg integral for L S (σ τ,s), when k<2n, has the form (0.3) ϕ(g)e ψ k τ,s(g)dg SO 2k+ (F )\SO 2k+ (A) where E ψ k τ,s is a certain Fourier coefficient of the Eisenstein series (on SO 4n (A)) along some unipotent subgroup N k of SO 4n, and with respect to a character ψ k of N k (F )\N k (A), stabilized by SO 2k+ (A), for some corresponding embedding of SO 2k+ inside SO 4n. ϕ is a cusp form in the space of σ. The integral (0.3),

3 EXPLICIT LIFTS OF CUSP FORMS 809 in case k = n, suggests that if L S (σ τ,s) has a pole at s =, then σ pairs with the representation σ(τ), of SO 2n+ (A), on the space spanned by [ ] ψn Res s= E τ,s (h). SO 2n+ (A) Now, when we try to compute the various constant terms of σ(τ) along the unipotent radicals of parabolic subgroups of SO 2n+, we find out that these are expressed in terms of [ ] ψl Res s= E τ,s (h) SO 2l+ (A) for l<n. This immediately reveals the tower property. Indeed, we define for any 0 k<2n, the representation σ k (τ) ofso 2k+ (A), acting by right translations on the space of [ ] ψk Res s= E τ,s (h). SO 2k+ (A) This leads us to another main result of this paper. Theorem B. Let τ be an irreducible, automorphic, cuspidal representation of GL 2n (A). Assume that L S (τ,λ 2,s) has a pole at s =. Then the representations {σ k (τ)} 2n k=0 have the tower property, i.e. for the first index l n, such that σ ln (τ) 0,σ ln (τ) is cuspidal and for k>l n, σ k (τ) is noncuspidal. It is easy to see that l n 2n and that σ ln (τ) is generic. Our main conjecture for this case is Conjecture. ) l n = n. 2) σ n (τ) is irreducible and lifts functorially from SO 2n+ (A) to τ. Remark. The conjecture implies that σ n (τ) is a generic member of the L-packet which lifts to τ. We define similar towers, prove Theorem B and state the above conjecture in case conditions (0.2) hold, and also in case L S (τ,sym 2,s) has a pole at s = (so here τ is on GL 2n (A) orongl 2n+ (A)). In each case, we use a corresponding global integral. For example, if conditions (0.2) hold, then the global integral is of Shimura type, and we construct the representations σ k (τ) of Sp 2k (A) using a sequence of Fourier-Jacobi coefficients of Res s= E τ,s (h) (the Eisenstein series is on Sp 4n (A)). In this case, we make one step towards the conjecture and prove

4 80 DAVID GINZBURG, STEPHEN RALLIS, AND DAVID SOUDRY Theorem C. Let τ be an irreducible, automorphic, cuspidal representation of GL 2n (A), such that L S (τ,λ 2,s) has a pole at s =and L S (τ, st, /2) 0. Then σ k (τ) =0, for k<n; that is l n n. The proof of Theorem C is based on the following two key observations. The first is Theorem D. Under the assumptions of Theorem C, Res s= E τ,s (h) has a nontrivial period along the subgroup Sp 2n (A) Sp 2n (A) (direct sum embedding in Sp 4n (A)). The second observation is that for an irreducible, admissible representation of Sp 4n (F ), where F is p-adic, the existence of (nontrivial) Sp 2n (F ) Sp 2n (F ) invariant functionals negates the existence of the Fourier-Jacobi models which enter the definition of σ k (τ), provided k<n. Note added in proof. Since the time of writing this paper, we have proved large parts of the above conjecture. We can now prove that l n = n in all four cases dealt with here. Thus, for τ as above, σ n (τ) is a nonzero cuspidal module on GA (G = Sp 2n,SO 2n+, SO 2n, Sp 2n respectively). The proof for G = Sp 2n appears in [G-R-S5], where we also give the analogous local theory, and construct for an irreducible supercuspidal, self-dual representation τ of GL 2n (k), k ap-adic field, such that its local exterior square L-function has a pole at s = 0, an irreducible, supercuspidal, generic (with respect to a prechosen character ψ) representation σ = σ ψ (τ), such that γ(σ τ, s, ψ) has a pole at s =. γ(σ τ, s, ψ) is the local gamma factor attached to σ τ, by the theory of Shimura type integrals considered here. In case G = Sp 2n,weprove in [G-R-S6] that each irreducible summand of σ n (τ) has indeed the unramified parameters determined by τ, at almost all places (once we fix ψ). Moreover, σ n (τ) is the direct sum of all irreducible, cuspidal, ψ-generic representations of Sp 2n (A), which ψ-lift to τ. Thus, if we have the irreducibility of σ n (τ), which is part of the above conjecture, then σ n (τ) will be the unique, ψ-generic member of the ψ-l-packet determined by τ. The proof of the unramified correspondence and of the fact that l n = n in the remaining cases will appear in a paper which is now in preparation. The results of this paper and those just mentioned were announced in [G-R-S2]; see also [G-R-S].

5 EXPLICIT LIFTS OF CUSP FORMS 8 Finally, this paper is organized as follows. In Section, we prove Theorem D. In Section 2 we define the towers of representations σ k (τ) and prove Theorem B for all cases (SO 2k+, Sp 2k, SO 2k, Sp 2k ). In Section 3 we prove Theorem C. Theorem A then follows as a corollary. Acknowledgement. We thank Jim Cogdell for helpful discussions and for his interest.. Sp 2n Sp 2n Periods of residues of Eisenstein series. The Eisenstein series of study and its pole at s =. Let G =Sp 4n, considered as an algebraic group defined over a number field F. Let P = MU be the Siegel parabolic subgroup of G. Consider an irreducible, automorphic, cuspidal representation τ of GL 2n (A). Assume it is self-dual. Regard τ as a representation of MA as well. Let φ be an element of Ind G A P τ; i.e. φ is a A smooth function on GA, taking values in the space of τ, such that φ(mug; r) =δ /2 P (m)φ(g; rm), for m MA, u UA, g GA, r GL 2n (A). We realize φ as a complex function on GA GL 2n (A), such that r φ(g; r) is a cusp form in the space of τ. Assume that φ is right K-finite, where K is the standard maximal compact subgroup of GA. Let, for s C, ϕ φ τ,s(g; m) =H(g) s /2 φ(g; m), f φ τ,s(g) = ϕφ τ,s(g;), g GA,m GL 2n (A) where if the Iwasawa decomposition of g is âuk, a GL 2n (A), u UA,k K, then H(g) = det a. Now consider the corresponding Eisenstein series E ( g, fτ,s φ ) = γ P F \G F f φ τ,s(γg). The series converges absolutely for Re(s) >n+, and admits a meromorphic continuation to the whole plane. It has a finite number of poles in the halfplane Re(s) 2 and they are all simple [M-W, IV..]. Consider the constant term along U, (.) E U (g, fτ,s) φ = E(ug, fτ,s)du φ U F \U A = f φ τ,s(g)+m(s)f φ τ,s(g), where M(s) is the intertwining operator, given, for Re(s) >n+, by the

6 82 DAVID GINZBURG, STEPHEN RALLIS, AND DAVID SOUDRY convergent integral M(s)fτ,s(g) φ = U A f φ τ,s(w ug)du, ( ) I for w = 2n. Later on, it will be convenient to consider the intertwining operator as evaluation at m =of I 2n M(s) ( ϕ φ ) τ,s (g; m) = ϕ φ τ,s(w ug; m)du U F \U A which converges for Re(s) >n+. These operators are decomposable in the following sense. Fix realizations V τν of the local representations τ ν, and fix an isomorphism l : V τν V τ, where V τ is the space of τ. Now write φ(g)(x) instead of φ(g; x), for g GA, x GL 2n (A), so that φ(g) V τ. For each place ν, let φ ν be an element of Ind Gν P ν τ ν, such that for almost all ν, φ ν = φ 0 ν unramified and φ 0 ν() = ξν 0 a prechosen unramified vector in V τν. Assume that φ corresponds to φ ν, that is l( φ ν (g ν )) = φ(g). For such φ, wehave, ( ) (.2) M(s)fτ,s(g) φ =l M ν (s)ϕ φν τ ν,s(g ν ) (), where ϕ φ τ ν,s(g ν )=H(g ν ) s /2 φ ν (g ν ) and M ν (s) is the vector-valued intertwining operator that is the meromorphic continuation of the absolutely convergent integral given, for Re(s) >n+, by M ν (s)ϕ φν τ ν,s(g ν )= ϕ φν τ ν,s(w ug ν )du. It is well known that for φ ν = φ 0 ν U ν (.3) M ν (s)ϕ φ0 ν τ ν,s = L(τ ν, st, s 2 )L(τ ν, Λ 2, 2s ) L(τ ν, st, s + 2 )L(τ ϕ φ0 νλ 2 ν τ, 2s) ν, s. (Recall that τ = τ.) The factors in (.3) are respectively the standard and the exterior square local L-functions attached to τ ν. Let S be a finite set of places outside which φ ν = φ 0 ν and g ν K ν -the standard maximal compact subgroup of G ν. Then in (.2) (.4) M(s)fτ,s(g) φ = LS (τ, st, s 2 )LS (τ,λ 2, 2s ) L S (τ, st, s + l 2 )LS (τ,λ 2, 2s) where (ξ 0 ) S = ν S ξν. 0 ( ν S M ν (s)ϕ φν τ ν,s(g ν ) (ξ 0 ) S) (),

7 EXPLICIT LIFTS OF CUSP FORMS 83 Proposition. Let S be a finite set of places, including those at infinity, outside which τ ν is unramified. Assume that L S (τ, st, 2 ) 0 and L S (τ,λ 2,s) has a pole (simple) at s =. Then the Eisenstein series E(g, fτ,s) φ has a pole (simple) at s =. Proof. It is enough to show that the constant term E U (g, fτ,s) φ has a pole at s =, and, by (.), it is enough to show this for M(s)fτ,s(g). φ The L-functions L S (τ, st, s + 2 ) and L(τ,Λ2, 2s), are holomorphic at s =. (Actually L S (τ, st, s + 2 ) and LS (τ,λ 2, 2s) are nonzero at s =. Indeed, form [J-S2, Th. 5.3], L S (τ, st, z) is nonzero for Re(z) >. This is also true, by the same theorem, for L S (τ τ,z)=l S (τ,λ 2,z)L S (τ,sym 2,z). From [J-S] and [B-G], L S (τ,λ 2,z) and L S (τ,sym 2,z) are holomorphic at z = 2, and since their product is nonzero at z = 2, it follows that each of these L-function is holomorphic and nonzero at z = 2.) Thus, (.4) will produce ( a pole of M(s)fτ,s(g) φ ats =, provided we can choose data such that l ν S M ν (s)ϕ φν τ ν,s(g ν ) (ξ 0 ) )() S is nonzero at s =. The last expression is holomorphic at a neighbourhood of s =, since otherwise (.4) will produce a high-order pole for M(s)fτ,s(g), φ and this implies that E(g, fτ,s) φ has a high-order pole at s =, contradicting the simplicity of the poles of E(g, fτ,s), φ for Re(s) > /2. It is enough to consider g =. Let ν S be finite. Choose φ ν to have compact support, modulo P ν,in the open cell P ν w U ν, and such that the function on U ν, u φ ν (w u)isthe characteristic function of a small neighbourhood in U ν of. Choose ξ ν V τν, so that φ ν (w )=ξ ν. Then M ν (s)ϕ φν τ ν,s() = c ν ξ ν, where c ν is the measure (in U ν ) of the neighbourhood above. Let ν be archimedean. Consider, as above, φ ν which is compactly supported, modulo P ν, in the open cell P ν w U ν, and such that the function u φ ν (w u) has the form b ν (u)ξ ν, where b ν is a Schwarz-Bruhat function on U ν and ξ ν V τν. ϕ φν τ ν,s is a smooth section. Let a(s) = Hν s /2 (w u)b ν (u)du. ν archimedean U ν Clearly a(s) is holomorphic, and functions b ν can be chosen so that a() 0. For this data, M(s)fτ,s() φ = a(s) LS (τ, st, s 2 )LS (τ,λ 2, 2s ) L S (τ, st, s + l( ξ 2 )LS (τ,λ 2 ν )() ;, 2s) hence M(s)fτ,s() φ has a pole (simple) at s =. Note that fτ,s φ is a smooth section. Let S be the set of archimedean places. Then ( (φ ν ) ν S l ν S M ν ()ϕ φν,() ( ν/ S ξ ν ))() is a continuous nontrivial linear functional on ν S Ind Gν P ν τ ν, and hence it is nontrivial on the dense subspace of K ν -finite vectors. Note, again, that ν S

8 84 DAVID GINZBURG, STEPHEN RALLIS, AND DAVID SOUDRY the intertwining operator is holomorphic at s = at each place (otherwise, the Eisenstein series has a high-order pole at s =, which is impossible.) This provides a pole at s = for M(s)f φ τ,s() with φ being K-finite. Remark. In the last proposition the choice of S is immaterial. If there is a set S 0 such that L S 0 (τ, st, 2 ) 0, then LS (τ, st, 2 ) 0 for any set S (as above), and similarly, if L S 0 (τ,λ 2,s) has a pole at s =, then L S (τ,λ 2,s) has a pole at s =, for any set S (as above). The reason is that locally, at a finite place ν, where τ ν is unramified L(τ ν, st, 2 ) 0 and L(τ ν, Λ 2, ) 0. Indeed, since τ ν is unitary the eigenvalues of its corresponding semisimple conjugacy class (in GL 2n (C)) are all strictly less, in absolute value, then qν /2 [J-S3] and so L(τ ν, st, s) (resp. L(τ ν, Λ 2,s)) is holomorphic and nonzero at s = 2 (resp. at s = ). Similar reasoning in the archimedean case implies that our assumption on the standard L-function of τ is equivalent to L(τ, st, 2 ) 0 (full L-function). Remark 2. Let τ be a self-dual, irreducible, automorphic, cuspidal representation of either GL 2n (A) orgl 2n+ (A). As before, we can construct an Eisenstein series, induced from the Siegel-type parabolic subgroup of G, where G is one of the following (split) groups Sp 4n, SO 4n, SO 4n+, Sp 4n+2 (the last group is the metaplectic cover of Sp 4n+2 ). The analogs of the quotient of (products of) L-functions in (.3) are summarized in the following table (which includes the previous case of Sp 4n ) (.5) Sp 4n L S (τ,st,s /2)L S (τ,λ 2,2s ) L S (τ,st,s+ 2 )LS (τ,λ 2,2s) SO 4n SO 4n+ L S (τ,λ 2,2s ) L S (τ,λ 2,2s) L S (τ,sym 2,2s ) L S (τ,sym 2,2s). Sp 4n+2 L S (τ,sym 2,2s ) L S (τ,sym 2,2s) Note that, in the last case, τ is a representation of GL 2n+ (A), and in the remaining cases, it is a representation of GL 2n (A). In all cases, except the last one, the functions φ and fτ,s φ are defined in exactly the same way. In case Sp 4n+2, we require that φ is a smooth function on Sp 4n+2 (A), taking values in the space of τ, such that (.6) φ(( mu, )g; r) =γ ψ (det m)(det m, det m)δ /2 P ( m)φ(g; rm), for m GL 2n+ (A), u UA, g Sp 4n+2 (A), r GL 2n+ (A). Here γ ψ (x) is the (global) Weil factor associated to a nontrivial additive character ψ

9 EXPLICIT LIFTS OF CUSP FORMS 85 of F \A. γ ψ is a character of the two-fold cover of A. It satisfies γ ψ (x x 2 )= γ ψ (x )γ ψ (x 2 )(x,x 2 ), for x,x 2 in A. (, ) is the (global) Hilbert symbol. fτ,s(g) φ is constructed by multiplying φ(g) byh(g) s /2. We can now repeat the proof of the last proposition (word-for-word) and conclude that the corresponding Eisenstein series has a pole (simple) at s =,ifl S (τ,λ 2,s) has a pole at s = in case SO 4n,orL S (τ,sym 2,s) has a pole at s = in cases SO 4n+ or Sp 4n The Sp 2n Sp 2n -period of Res s= E(g, fτ,s). φ We go back to the case G = Sp 4n, τ self-dual, irreducible, automorphic, cuspidal representation. We assume that L(τ, st, 2 ) 0 and that there exists a finite set of places S, including those at infinity, outside which τ is unramified, such that L S (τ,λ 2,s) has a pole at s =. (See Remark in the last section.) Note that this implies, in particular, that τ has a trivial central character [J-S]. By Proposition, Res s= E(g, fτ,s) φ is nontrivial (s = is a simple pole.) Consider the subgroup H =Sp 2n Sp 2n, embedded in Sp 4n by ( ( ) ( ) ) A B A B (.7) A2 B, 2 i A 2 B 2. C D C 2, D 2 C 2 D 2 C D Each letter represents an n n block. We sometimes identify, to our convenience, h and i(h) insp 4n, for h H. Denote The main result of this chapter is E (g, φ) = Res s= E(g, f φ τ,s). Theorem 2. Under the above assumption, E is integrable over H F \HA and (for a suitable choice of measures) ( ) a (.8) E (h, φ)dh = φ(k; )d(a, b)dk. b H F \H A K H C 2n (A)GL n(f ) 2 \GL n(a) 2 Here K H = K H and C 2n is the center of GL 2n. This theorem and its proof are very similar to Theorem in [J-R]. Note that the inner GL 2 n-integration, on the right-hand side of (.8), is the integral (32) in [F-J.] (with s = 2,χ = η = ). By Theorem 4. in [F-J], a necessary condition, for this integral to be nonzero, is that L S (τ,λ 2,s) has a pole at s =, and, in this case, the integral is of the form α(k; φ)l(τ, st, 2 ), where α(k; φ) is a nontrivial linear form, for each k K H. Under our assumptions on τ, the right-hand side of (.8) is not identically zero. The proof for this is entirely similar to that of Proposition 2 in [J-R]. All the requirements there

10 86 DAVID GINZBURG, STEPHEN RALLIS, AND DAVID SOUDRY are supplied by [F-J.]. Thus α(φ) = α(k, φ)dk is nontrivial and hence the K H period of E along H is nontrivial. Note that, exactly as remarked on the bottom of p. 78 in [J-R], formula (.8) supplies another proof for the fact that if L(τ, st, 2 ) 0 and LS (τ,λ 2,s) has a pole at s =, then E(g, fτ,s) φ has a pole at s =. Moreover, the last remarks, together with (.8) prove Corollary 3. Let τ be a self -dual, irreducible, automorphic, cuspidal representation of GL 2n (A). Then E(g, fτ,s) φ has a pole at s =and E (,φ) admits a nontrivial period along H, if and only if L(τ, st, 2 ) 0and LS (τ,λ 2,s) has a pole at s =. The rest of this chapter is devoted to the proof of Theorem Truncation. As in [J-R], we consider the truncation operator applied to E(g, f φ τ,s). Denote, for a real number c, by χ c the characteristic function of all real numbers larger than c. The only nontrivial constant terms of E(g, f φ τ,s) along unipotent radicals of standard parabolic subgroups, are those taken along U or {}. Thus, the truncation operator Λ c, c>, applied to E(g, f φ τ,s) is (.9) Λ c E(g, f φ τ,s) =E(g, f φ τ,s) γ P F \G F E U (γg,f φ τ,s) χ c(h(γg)). Since E(g, f φ τ,s) is smooth and of moderate growth, Λ c E(g, f φ τ,s) is rapidly decreasing. Also, the sum on the right-hand side of (.9) has finitely many terms (depending on g and c). In particular, Λ c E(g, f φ τ,s) is meromorphic in s. (See [A], [A2], for more details.) Similar remarks are valid for Λ c E (g, φ). By (.) and (.9), we have (.0) Λ c E(g, fτ,s) φ =E(g, fτ,s) φ (fτ,s(γg)+m(s)f φ τ,s(γg)) φ χ c(h(γg)) γ P F \G F = fτ,s(γg) φ χc (H(γg)) M(s)fτ,s(γg) φ χ c(h(γg)). γ P F \G F γ P F \G F This last equality is for Re(s) >n+. χc is the characteristic function of the interval (0,c]. Denote the first sum, in the last expression of (.0), by θ c (g, fφ τ,s), and the second by θ c 2 (g, fφ τ,s), so that (.) Λ c E(g, f φ τ,s) =θ c (g, f φ τ,s) θ c 2(g, f φ τ,s). Note again that the sum defining θ c 2 (g, fφ τ,s) has finitely many terms (depending on g and on c). In particular θ c 2 (g, fφ τ,s) is meromorphic in s, and so θ c (g, fφ τ,s) defines a meromorphic function in the whole plane. (.) is now valid as an equality of meromorphic functions.

11 EXPLICIT LIFTS OF CUSP FORMS 87 Denote Res s= M(s) =M. Since f φ τ,s is holomorphic, then an application of Λ c to E yields similarly, (.2) Λ c E (g, φ) =E (g, φ) θ c 3(g, φ), where (.3) θ3(g, c φ) = γ P F \G F M (f φ τ, (γg)χ c(h(γg)). Since Λ c E(g, f φ τ,s) and Λ c E (g, φ) are rapidly decreasing, they are bounded and hence integrable on H F \HA. We will prove Proposition 4. θ c(,fφ τ,s) is integrable on H F \HA, if Re(s) is sufficiently large, θ2 c(,fφ τ,s) is integrable on H F \HA, if Re(s) > 0, (and M(s) exists) and θ3 c(,φ) is integrable on H F \HA. The following formulae are valid, with a certain choice of measures: (.4) H F \H A θ(h, c fτ,s)dh φ = cs s K H C 2n (A)GL n(f ) 2 \GL n(a) 2 ( a φ(k; b ) d(a, b)dk, (.5) θ c 2(h, f φ τ,s)dh H F \H A (.6) = c s s θ c 3(h, φ)dh K H C 2n (A)GL n(f ) 2 \GL n(a) 2 ( φ a M(s)(ϕτ,s)(k; b ) )d(a, b)dk, H F \H A = c K H ( ) φ a M (ϕτ, (k; d(a, b)dk. b C 2n (A)GL n(f ) 2 \GL n(a) 2 This proposition will finish the proof of Theorem 2, exactly as in [J-R, p. 8]. Integrating (.) along H F \HA, first for Re(s) 0, and using (.4) and (.5), we get (.7) H F \H A Λ c E(h, f φ τ,s)dh = cs s c s s ( a φ(k; b ) d(a, b)dk ( M(s)(ϕ φ a τ,s)(k; b ) )d(a, b)dk.

12 88 DAVID GINZBURG, STEPHEN RALLIS, AND DAVID SOUDRY By analytic continuation, this makes sense in the whole plane. Taking residues in (.7) at s =, (.8) H F \H A Λ c E (h, φ)dh = ( a φ(k; b ) d(a, b)dk ( ) c M (ϕ φ a τ, )(k; )d(a, b)dk b ( ) a = φ(k; )d(a, b)dk θ c b 3(h, φ)dh. H F \H A We used (.6). Comparing (.8) with (.2), we conclude that E is integrable on H F \HA and that (.8) is satisfied. Each of the functions θj c has the form (.9) θ j (g) = ξ j (γg). γ P F \G F where fτ,s(g) φ χc (H(g)), j = (.20) ξ j (g) = M(s)(fτ,s)(g) φ χ c(h(g)), j =2. M (f φ τ, )(g)χ c(h(g)), j =3 The sum (.9), in case j =, converges only for Re(s) 0. Before proving the integrability of θ j on H F \HA, and then compute its period on H, we proceed formally, (.2) θ j (h)dh = ξ j (γh)dh γ P F \G F H F \H A = H F \H A γ P F \G F /H F γ P F γ H F \H A ξ j (γh)dh. The set P F \G F /H F is finite and will be described soon. Our task will be to show the integrability of each ξ j (γh) (in h)) on γ P F γ H F \HA, and then compute the integral. 4. The set P F \G F /H F. The description of this set is known. It appears as an ingredient in the doubling method. See [PS-R, Lemma 2.], for the description of the action of H F on P F \G F, realized as the variety of maximal isotropic subspaces of the symplectic space of dimension 4n. Using this description and passing to a different basis of F 4n, so that H =Sp 2n Sp 2n is

13 EXPLICIT LIFTS OF CUSP FORMS 89 embedded in G by (.7), we arrive at the following set of representatives for P F \G F /H F. (.22) γ d = I d γ d I d, 0 d n, where (.23) I n d I d γ d = 0 I n d I n d I n d 0 I d I n d I n d. Note that I n d I d γ d = 0 I n d I n d 0 I d I n d I d I n d I n d I n d I n d I d I n d. I n d A computation of the stabilizer of γ d shows that (.24) H F γ d P F γ d {( a x y = c x a, b u z c u b ) } a, b GL d (F ) H F. c Sp 2(n d) (F ) Write this as the semidirect product M d n V d, where {( a b ) } M d = c, c a, b GL d (F ) c Sp 2(n d) (F ) a {( I d x y V d = I 2(n d) x I d, b I d u z I 2(n d) u I d ) } H F.

14 820 DAVID GINZBURG, STEPHEN RALLIS, AND DAVID SOUDRY We now collect several conjugation formulae which will be used later. For r = i( a I 2(n d) a b, I 2(n d) b ), (i is the embedding (.7)) (.25) γ d rγ d = For r = i( I d c I d a I n d I d, c b I d I n d I n d b I n d ), where c = ( c c 2 c 3 c 4 a ). is the (n d) (n d) block description of c Sp 2(n d) (F ), (.26) γ d rγ d = I d 0 c c 2 c 2 I d 0 c 3 c 4 c 3 c c 2 I d c 3 c 4 I d. For v = i( I d x e y I n d 0 e I n d x I d, I d x 2 e 2 y 2 I n d 0 e 2 I n d x I d ) V d,

15 EXPLICIT LIFTS OF CUSP FORMS 82 I d x 0 e x 0 0 y I n d 0 0 e x 2 I d e 2 y 2 e 2 0 (.27) γ d vγ I n d x d = I n d e 2 0 e. I d 0 0 x 2 I n d x For 3m = i( I d e ez 0 e I d ),I 2n, I d (.28) γ d m γ d = where I d I n d I d e I n d 0 ez e 0 I d e I n d 0 I d I d I n d I d I n d k 0 =. I n d 0 0 I n d 0 I d I n d I n d 0 I d k 0 5. A formal proof of Proposition 4. We first prove Proposition 4 formally, without paying attention to convergence issues. Denote Q d = γ d Pγ d H, {( a x y b u z ) } Q d = c x, e u H a, b GL d. a b c, e Sp 2(n d)

16 822 DAVID GINZBURG, STEPHEN RALLIS, AND DAVID SOUDRY Q d is a parabolic subgroup. Write its Levi decomposition as M d n V d. Note that M d M d. Following (.2), we have to compute the integrals I j,d = ξ j (γ d h) for 0 d n and j =, 2, 3. Q d (F )\H A Let 0 <d<n. Write the Iwasawa decomposition in HA, h = vm k, m M d (A), v V d(a), k K H, dh = δ (m )dvdm dk, where δ is the modular function corresponding to the parabolic subgroup Q d. Then (.29) I j,d = ξ j (γ d vγ d γ d m k)δ (m )dvdm dk. K H M d (F )\M d (A) V d (F )\V d (A) By (.27), the projection to GL 2n of γ d vγ d,asv varies in V d is a unipotent radical N d in GL 2n, (since 0 <d<n). Then H(γ d vγ d ) =, and from (.20) we see that (.29) involves an inner integration of a cusp form in τ along N d (F )\N d (A), and hence (.30) I j,d =0, for 0 <d<n, j=, 2, 3. In case d =0,Q 0 = {(c, c) c Sp 2n }. We have ( ) (.3) I j,0 = ξ j γ 0 i(c, c)γ0 γ 0i(e, ) dcde. Sp 2n (A) Sp 2n (F )\Sp 2n (A) By (.26), the projection of γ 0 i(b, b)γ0 on GL 2n,asbvaries in Sp 2n is Sp 2n (embedded as a subgroup of GL 2n ). Since H(γ 0 i(c, c)γ0 ) =, we obtain in (.3) an inner integration of a cusp form in τ along Sp 2n (F )\Sp 2n (A), which equals zero by [J-R, Prop. ]. Thus, (.32) I j,0 =0, for j =, 2, 3. The only remaining case is d = n. Here γ n = I 4n and { ( ) ( ) } a y b z Q n = 0 a, 0 b H a, b, GL n = Q n. Using (.25) and (.27), we have

17 EXPLICIT LIFTS OF CUSP FORMS 823 (.33) I j,n = K H = K H GL n(f ) 2 \GL n(a) 2 ξ j GL n(f ) 2 \(GL n(a) 2 ) 0 ( a F \A ξ j b b ( ta a ) k det(ab) (n+) d(a, b)dk tb t b t a ) k t 2n(n+) det(ab) d td(a, b)dk. Here (GL n (A 2 )) 0 = {(a, b) GL n (A) 2 det(ab) =}. Of course, in (.33), we made a choice of measures. d t will soon be specified. Thus, by (.20) (recall also that, by our assumptions, τ has a trivial central character) (.34) I,n = K H GL n(f ) \ (GL n(a) 2 ) 0 ( a φ(k; b ) )d(a, b)dk t F \A t 2n c c t 2n(s ) d t. Choose d t such that the corresponding d t-integral equals ts dt t = cs s, for 0 Re(s) >. The d(a, b)-integration yields one on C 2n (A)GL n (F ) 2 \GL n (A) 2. This proves (.4). In a similar fashion, we get, for Re(s) > 0, (.35) and I 2,n = K H = c s s GL n(f ) 2 \(GL n(a) 2 ) 0 K H (.36) I 3,n = c K H C 2n (A)GL n(f ) 2 \GL n(a) 2 ( φ a M(s)(ϕτ,s)(k; b ( φ a M(s)(ϕτ,s)(k; ) )d(a, b)dk b c ( s) dt t t ) )d(a, b)dk, ( ) φ a M (ϕτ, )(k; )d(a, b)dk. b C 2n (A)GL n(f ) 2 \GL n(a) 2 In the next section, we justify our formal calculations.

18 824 DAVID GINZBURG, STEPHEN RALLIS, AND DAVID SOUDRY 6. Justification of the formal proof of Proposition 4. By (.2), we have to examine the absolute convergence of the integrals I j,d. The case d = n is, actually, rigorously proved in the end of the last section (.34) (.36): I,n (resp. I 2,n ) converges absolutely for Re(s) > (resp. Re(s) > 0) and I 3,n converges absolutely. Proposition 5. Let 0 d<n. Then:. I,d converges absolutely, for Re(s) large enough. 2. I 2,d converges absolutely, for all s where M(s) is defined. 3. I 3,d converges absolutely. In all three cases, I j,d =0. Proof. We start with formula (.29). We will prove the absolute convergence (at the indicated domains in s) of (.37) K H M d (A)\M d (A) M d (F )\M d (A) V d (F )\V d (A) ξ j (γ d vγ d γ d rγ d We realize M d (A)\M d (A) as {( I d x γ d m k)δ (r)dvdrdm dk. I d ) } x,i 2n Sp2(n d) (A) and then use the Iwasawa decomposition, following the Siegel parabolic subgroup R = L n U,inSp 2(n d). x = ( ) e ez e k dx = dedzdk. ( ) Here e GL n d (A), z U A {z = In d z I n d standard maximal compact subgroup of Sp 2(n d) (A). Let ω d = I d 0 I d I n d 0 I n d. } U A, k K the

19 EXPLICIT LIFTS OF CUSP FORMS 825 Then, by (.25) (.28), the integral (.37) equals ( ) a φ j (k e,z k 0 γ d k I2d y (.38) k; b ω I d r e,z ) ab s j r e,z s j 2(n d) c χj,c( ab r e,z )dyd(a, b)dcdedzdk dk. The integration is on y M 2d 2(n d) (F )\M 2d 2(n d) (A), (a, b) GL d (F ) 2 \GL d (A) 2, c Sp 2(n d) (F )\Sp 2(n d) (A), e GL n d (A), z U A, k K (embedded naturally in Sp 4n (A))and k K H. k e,z is the compact part of the Iwasawa decomposition in Sp 4n (A) of the matrix, multiplying k 0, on the righthand side of (.28), and r e,z is the projection to GL 2(n d) (A) of its Siegel parabolic part. φ j is defined as follows (see (.20)). For g GA and m GL 2n (A), ϕ φ τ,s(g; m), j = φ j (g; m) = M(s)(ϕ φ τ,s)(g; m), j =2, M (ϕ φ τ, )(g; m), j =3 and for t R χ j,c (t) = { χ c (t) j = χ c (t) j =2, 3. Finally, s n + d, j = s j = s n + d, j =2 n + d, j =3 s + n, j= s j = s + n, j=2. n, j=3, In (.38), we denote, for a matrix x, x = det x. Examining (.28), we see that (.39) ω d r e,z = m e,zω d, where (.40) m e,z = ( In+d e )( I2d m e,z ),

20 826 DAVID GINZBURG, STEPHEN RALLIS, AND DAVID SOUDRY and m e,z is the projection to GL 2(n d) (A) of the Siegel parabolic part in the Iwasawa decomposition in Sp 2(n d) (A) of I n d u e,z = I n d e z I n d. 0 e I n d We want to establish the absolute convergence of (.38). For this, we use the majorization of cusp forms on GL 2n (A), given in Proposition 6 of [J-R]. Since φ j is K-finite, we find that, for each integer N>, there is a constant c, such that (.4) φ j( k, ( ) m y 0 r ) c max {, m 2(n d) r 2d} N, for all k K, m GL 2d (A), r GL 2(n d) (A). Now, it suffices to replace the d(a, b)-integration in (.38) by an integration on GL 2d (F )\GL 2d (A). By (.40) and (.4), it remains to consider (.42) I j,d = max{, g 2(n d) ( e m e,z ) 2d } N g s j ( e m e,z ) s j χj,c( g e m e,z )dgdedz, where g is integrated over GL 2d (F )\GL 2d (A), e-over GL n d (A) and z over U A. Before proceeding to each case, let us prove two lemmas. Lemma 6. For all e GL n d (A), z U A, (.43) e m e,z. Proof. Write the Iwasawa decomposition of u e,z, at each place ν, following the Borel subgroup of Sp 2(n d) (F ν ), Then t... * t u ν = u eν,zν = 2(n d) t k ν. 2(n d)... t (.44) m eν,z ν = t 2(n d) t = ε 2(n d) u ν ε u ν,

21 EXPLICIT LIFTS OF CUSP FORMS 827 where {ε,...,ε 2(n d) ; ε 2(n d),...,ε 2,ε } is the standard basis of row vectors in Fν 4(n d). For a row vector v =(x,...,x l ) Fν,welet l max i l x i, ν < ( l v = x 2 ) /2 i, Fν = R i= l x i 2, F ν = C. i= (If ν is archimedean, the norm on 2(n d) Fν 4(n d) is defined through the orthonormal basis obtained from the standard basis {ε,...,ε } above.) From (.44), we get (with self-explanatory notation) m eν,z ν (ε (n d) E n d ) (ε E ). where E n d,...e are the rows of the matrix e ν; e ν = (See definition of u e,z.) In particular, E n d. E. m eν,z ν E n d E = e ν = e ν. This implies (.43) at all places ν; hence (.43) is satisfied. This proves the lemma. Lemma 7. Let t be a real number. Then the integral A = ( e m e,z ) t dedz U A GL n d (A) converges, provided t is large enough. Proof. We first consider the local analog of A, at each place ν, A ν = ( e m e,z ) t dedz. U ν GL n d (F ν) Split A ν into two summands A () ν +A (2) ν, where in A () ν, we integrate over e and in A (2) ν over e >. We have A () ν = ( e m e,z )) t dedz = ( e m e,z ) t dedz. e e

22 828 DAVID GINZBURG, STEPHEN RALLIS, AND DAVID SOUDRY We changed variable e e and z z and m e,z is the projection to GL n d (A) of the Siegel parabolic part in the Iwasawa decomposition in Sp 2(n d) (A) of I n d ũ e,z = I n d e z I n d e e I n d. Let d a (e) be the additive measure on M n d (F ν ). Then de = da(e) (.45) A () ν As in (.44), (.46) where Hence e ( e m e,z ) t d a (e)dz. e n d and m e,z = (ε 2(n d) E n d z n d ) (ε (n d) E z ) e = (ε (n d) E n d ) (ε E ), E n d. E,z= z n d (.47) m e,z (ε 2(n d) + E n d + z n d ). z,e = E n d. E. (ε (n d) + E + z ) E n d E = (ε 2(n d) +E n d +z n d ) (ε (n d) +E + z ) ε n d+ ε 2(n d) e = (ε 2(n d) +E n d ) (ε (n d) +E ) ε n d+ ε 2(n d) e α(e) e. From (.46), we also obtain (.48) m e,z (ε 2(n d) + E n d + z n d ) (ε (n d) + E + z ) α(e, z). From (.47) and (.48) hence, for t 0, m e,z α(e) /2 e /2 α(e, z) /2 ; (.49) ( e m e,z ) t ( e α(e) α(e, z) ) t/2.

23 EXPLICIT LIFTS OF CUSP FORMS 829 Let, for a matrix b, max{, b }, ν < λ ν (b) = ( + b 2 ) /2, F ν = R + b 2, F ν = C Then, by the definitions of α(e) and α(e, z), hence Using this in (.49), we get α(e) e and α(e) λ ν (e) α(e, z) e and α(e, z) λ ν (z) ; α(e)α(e, z) e λ ν (e) /2 λ ν (z) /2. ( e m e,z ) t λ ν (e) t/4 λ ν (z) t/4. Thus the integrability of A () ν follows from that of λ ν (e) t/4 d a (e) λ ν (z) t/4 dz e which is valid, for t 0, by Proposition 7 in [J-R]. Similarly, A (2) ν = ( e m e,z ) t dedz = ( e m e,z ) t d a e e e > e < for t>n d. As before hence A (2) ν < All in all, e < U ν e < e t n+d m e,z t d a edz < e <. n d dz m e,z t d a edz, m e,z λ ν (e) and m e,z λ ν (z) ; λ ν (e) t/2 λ ν (z) t/2 d a (e)dz A ν = A () ν + A (2) ν < U ν GL n d (F ν) e < λ ν (e) t/4 λ ν (z) t/4 d a (e)dz. λ ν (e) t/4 λ ν (z) t/4 d a (e)dz r ν (t) and the last integral converges absolutely for t 0, by Proposition 7 in [J-R]. By the same proposition, ν r ν (t) converges for t 0. This proves the absolute convergence of A and Lemma 7.

24 830 DAVID GINZBURG, STEPHEN RALLIS, AND DAVID SOUDRY We go back to the convergence of I j,d in (.4.2). Case j =, 0 <d<n. Here (in (.42)) (.50) g c( e m e,z ). Integral (.42) splits into two summands I () integrate along,d + I(2),d (.5) g ( e m e,z ) d/n d. We have ( e m e,z ) d/n d c( e m e,z ). In the first summand, we due to Lemma 6 and the fact that c>. Thus (.50) is redundant in I (),d.we have (.52) I (),d = g s n+d ( e m e,z ) s+n dgdedz. g ( e m e,z ) d/n d g GL 2d (F )\GL 2d (A) e GL n d (A),z U A The dg-integration in (.52) is proportional to ( e m e,z ) d/n d 0 s n+d dt t t = s n + d ( e m e,z ) d n d (s n+d ) for Re(s) >n d +. I (),d is then proportional to (.53) ( e m e,z ) d n d (s n+d )+s+n dedz (that is, it is enough now to get the absolute convergence of (.53)). By Lemma 7, since Re(s) 0, (.53) converges absolutely. Now consider I (2),d. (.54) I (2),d = ( e m e,z ) d/n d < g <c( e m e,z ) g GL 2d (F )\GL 2d (A) e GL n d (A), z U A g 2(n d)n+s n+d ( e m e,z ) 2dN+s+n dgdedz. The dg-integration is proportional to (c( e m e,z ) ) (n d)(2n+)+s (n d)(2n +)+s ( e m e,z ) d n d ( (n d)(2n+)+s ) d n d ( (n d)(2n +)+s ).

25 EXPLICIT LIFTS OF CUSP FORMS 83 Thus (.54) is a difference of two integrals. One is proportional to J = ( e m e,z ) n(2n+)+n d+ dedz and the other is proportional to J 2 = ( e m e,z ) n n d s+n d+ d n d dedz. The integral J converges absolutely, since we may choose N as large as we want, and then apply Lemma 7 and similarly, in J 2, we assume that Re(s) 0, and get the convergence again. Case j =2, 0 <d<n. Here (in (.42)) (.55) g >c( e m e,z ). Again, split (.42) into two summands I () integrate along (.5). Thus, by (.5) and (.55), 2,d + I(2) 2,d c( e m e,z ) < g ( e m e,z ) d/n d. In the first summand, and, in particular, ( e m e,z ) n/n d > c. Since c >, this contradicts Lemma 6. Hence the domain of integration of I () 2,d is empty, and we need only consider I (2) 2,d. The integration in I(2) 2,d is over g satisfying (.55) and g > ( e m e,z ) d/n d. By Lemma 6, this last condition is redundant (in the presence of (.55)). Thus, (.56) I c,d = I(2) 2,d = g s (n d)(2n+) ( e m e,z )dgdedz. g >c( e m e,z ) g GL 2d (F )\GL 2d (A) e GL n d (A), z U A The dg-integral in (.56) is absolutely convergent, once we take N large enough (relative to s, n, d), and is proportional to c( e m e,z ) s (n d)(2n+) t dt t = (c( e m e,z ) ) s (n d)(2n+) s (n d)(2n +) Thus (.56) is proportional to ( e m e,z ) 2n(N+) d+ dedz, e GL n d (A) z U A which converges absolutely, provided we choose N (Lemma 7). Case j =3, 0 <d<n. This case follows exactly as Case j =2..

26 832 DAVID GINZBURG, STEPHEN RALLIS, AND DAVID SOUDRY Case j =, d = 0. By (.38) (.40), we have to consider integrals of the form ( ) In ϕ(b m e,z )( e m e,z ) s+n dbdedz z U A e GL n(a) e m e,z <c b Sp 2n (F )\Sp 2n (A) for ϕ in the space of τ. Note that, by Lemma 6 (and c>), e m e,z <c always holds. Since ϕ is bounded, (and Sp 2n (F )\Sp 2n (A) has finite volume), it is enough to consider the convergence of ( e m e,z ) s+n dedz. This integral converges absolutely for Re(s) large enough (Lemma 7). Case j =2, 3; d = 0. Similar to case j =, we have to consider ( ) In ϕ(b m e,z )( e m e,z ) s+n dbdedz. z U A e GLn(A) e me,z >c b Sp 2n (F )\Sp 2n (A) The domain of integration is empty here, by Lemma 6 and the fact that c>. We proved assertions () (3) of Proposition 5. They provide the justification to the formal proof in Section 4, which shows that I j,d = 0 for j =, 2, 3 and 0 d<n. This concludes the proof of Theorem 2. e e 2. A tower of correspondences for GL m. Definition of the tower. Let τ be an irreducible, automorphic, cuspidal, representation of GL 2n (A). Assume that τ is self-dual. As recalled in the introduction, exactly one of the partial L-functions L S (τ,λ 2,s)orL S (τ,sym 2,s) has a pole (simple) at s =. Consider the Eisenstein series on SO 4n (A), in the first case, and on SO 4n+ (A), in the second case, induced from the Siegel-type parabolic subgroup, the representation τ and the same complex parameter s, as in Chapter Denote it by E(g, f φ τ,s), keeping the notation of Chapter. In subsection., we have seen that it has a simple pole at s =. Let E (g, φ) = Res s= E(g, fφ τ,s). By taking a sequence of certain Fourier coefficients of E along certain unipotent subgroups of SO 4n, in the first case, and of SO 4n+, in the second case, and restricting to subgroups of the form SO 2k+ (A) (resp. SO 2k (A)), we obtain

27 EXPLICIT LIFTS OF CUSP FORMS 833 a tower of spaces of automorphic representations σ k (τ) ofso 2k+ (A) (resp. of SO 2k (A)) which are generic (2.) SO 4n (A). τ on GL 2n (A) SO 2n+ (A). SO 3 (A) SO (A), (2.2) SO 4n (A). τ on GL 2n (A) SO 2n (A).. SO 2 (A) {I} We will prove that the last step in the tower is always nontrivial and that in the first step k, where σ k (τ) is nonzero, σ k (τ) is also cuspidal. In this sense, the towers resemble Rallis original symplectic-orthogonal towers. In case (2.), we can be more precise and distinguish two cases, according to whether L S (τ, st, 2 ) 0 or not. If LS (τ, st, 2 ) 0, consider the Eisenstein series E(g, fτ,s)onsp φ 4n (A) (as in Chapter ). Again, by., E (g, φ) = Res E(g, s= fφ τ,s) is nontrivial, and by taking a sequence of certain Fourier-Jacobi coefficients of E (g, φ), we construct another tower of liftings to automorphic, generic representations of Sp 2k (A) Sp 4n 2 (A) (2.3). τ on GL 2n (A) Sp2n (A).. Sp 2 (A) = SL 2 (A) {I} Towers (2.) and (2.2). Let G be one of the groups SO 4n or SO 4n+, acting from the left on the column space W = F 4n (resp. W = F 4n+ ) equipped with the quadratic form defined by.... Let {ε,...,ε 2n,ε 2n,...,ε }

28 834 DAVID GINZBURG, STEPHEN RALLIS, AND DAVID SOUDRY (resp. {ε,...,ε 2n,ε 0,ε 2n,...,ε }) be the standard basis of W. Let V + l = Span{ε,...,ε l } ( l 2n). This is an l-dimensional isotropic subspace of W. Its dual in W is V l = Span{ε,...,ε l }. Consider the parabolic subgroup P k of G, which preserves E k = V + 2n k (resp. E k = V + 2n k ). Its Levi decomposition is P k = M k n U k. U k is abelian, in case dim E k =, or in case dim E k =2n, when G =SO 4n, and otherwise, it is a two-step nilpotent group. The Siegel parabolic subgroup P corresponds to k = (resp. k = 0) Let E k be the dual of E k inside W, and let W k be the orthocomplement of E k + E k in W. Then M k = GL(Ek ) SO(W k ), and Uk ab is isomorphic to the space of matrices of size dim E k dim W k. Let α k = dim E k, β k = dim W k. In matrix form, an element of M k has the form a m = b, where a GL(E k ) and b SO(W k ), and an element of U ab k a has the form (2.4) u = I αk Y I βk Y Conjugation by m gives I αk ay b (2.5) mum = I βk by (a ). I αk If we realize Uk ab as E k Wk, where W k is the dual of W k, written in the form of rows in β k coordinates, then conjugation (2.5) by m of u, which corresponds to e v, gives ae v b. Define { t ε v 0 = 0, dim W =4n + t ε 2n + t ε 2n, dim W =4n. Denote (again) the symmetric bilinear form on Wk by (,), where (t v, t v 2 )= (v,v 2 ), for v and v 2 in W k. Denote (2.6) y k = ε αk v 0. Fix a nontrivial character ψ of F \A and define the following character ψ k of U k (F )\U k (A). Let u in Uk ab(a) correspond to y E k(a) Wk (A). Then (2.7) ψ k (u) =ψ ( (y, y k ) ), (where (,)isextended naturally to (E k + E k )(A) W k (A)). In matrix form (u as in (2.4)) { ψ(y2n k,k+ ), dim W =4n + (2.8) ψ k (u) =. ψ(y 2n k,k+ + Y 2n k,k+2 ), dim W =4n I αk.

29 EXPLICIT LIFTS OF CUSP FORMS 835 The stabilizer of ψ k in SO(W k )ish k = SO(W k ), where W k is the orthocomplement of t v 0 in W k. Note that { (2.9) H k SO2k, dim W =4n + =. SO 2k+, dim W =4n The stabilizer of ψ k in GL(E k ) is the mirabolic subgroup (of type (α k, )); in particular, it contains Z αk the standard maximal unipotent subgroup of GL(E k ). Extend ψ k to Z αk (A) by the standard nondegenerate character of Z αk (A), defined by ψ, i.e. it takes z in Z αk (A) toψ( α k z l,l+ ). This defines a character ψ k of N k (A), where N k = Z αk U k. Consider the Eisenstein series E(g, fτ,s) φ of Chapter (mentioned in Remark 2, following Proposition ). Recall that τ is an irreducible, automorphic cuspidal representation of GL 2n (A), which is self-dual. φ is a smooth K-finite function in Ind G(A) P (A) τ. P = M n U is the Siegel-type parabolic subgroup of G = SO(W ). Finally l= (2.0) ϕ φ τ,s(g; m) =H(g) s /2 φ(g; m), g GA, m GL 2n (A) fτ,s(g) φ =ϕ φ τ,s(g;), and if g is written in the Iwasawa decomposition âuk, for a GL 2n (A), u U(A), k K, then H(g) = det a. Assume that E(g, fτ,s) φ has a pole at s =. Let σ k (τ) be the representation, by right translations, of H k (A) on the space spanned by the ψ k -Fourier coefficients of E (g, φ), along N k, restricted to H k (A). That is, the space of σ k (τ) is the space of automorphic functions on H k (A), of the form (2.) (2.) p k (h) =p k (h, φ) = E (vh, φ)ψ k (v)dv. N k (F )\N k (A) The tower (2.3). Here G =Sp 4n. Let W = F 4n (column space), with the... symplectic form (,)defined by J 4n =. Let... {ε,...,ε 2n,ε 2n,...,ε } be the standard basis of W. Define V ± l, l 2n, as in the previous case, and E k = V + 2n k. Let P k be the parabolic subgroup of G, which preserves E k. Its Levi decomposition is P k = M k n U k. U k is abelian in case dim E k =2n (i.e. k = 0), and otherwise it is a two step nilpotent

30 836 DAVID GINZBURG, STEPHEN RALLIS, AND DAVID SOUDRY group. Let E k and W k be as in the previous case. Then M k = GL(Ek ) Sp(W k )=GL 2n k Sp 2k. The Siegel parabolic subgroup P corresponds to k = 0. Let N k = Z 2n k U k (Z 2n k is the standard maximal unipotent subgroup of GL(E k )=GL 2n k ). Define a character ψ k on N k (A) to be trivial on U k (A) and the standard nondegenerate character (corresponding to ψ) onz 2n k (A). Note that ψ k is fixed by Sp(W k )A. The group N k /N k+ is isomorphic to the Heisenberg group H k in 2k + variables. In matrix form, v in N k looks like this z e y v = I 2k e, z Z 2n (k+). z ( Ik Then the isomorphism takes v modulo N k+ to (ẽ; y) where ẽ = e ) 2I k. Denote the composition of this isomorphism with the quotient map N k N k /N k+ by j k. Consider the Weil representation of Sp 2k (A) the global metaplectic cover. It comes with a choice of a nontrivial additive character of F \A. We choose ψ. Denote the Weil representation by ω ψ,k, and extend it to H k (A) Sp 2k (A) by the Stone von-neumann representation (of central characters ψ) onh k (A). We still denote this representation by ω ψ,k. It acts on the space S(A k ) of Schwartz-Bruhat functions on A k. Consider its automorphic realization on the space of theta series (2.2) θ ξ ψ,k (h) = ω ψ,k (h)ξ(x); h HA Sp 2k (A), ξ S(A k ). x F k Let τ be an irreducible, automorphic, cuspidal representation of GL 2n (A). Assume that L S (τ,λ 2,s) has a pole at s = and that L(τ, st, 2 ) 0. Then the Eisenstein series E(g, fτ,s) φ has a (simple) pole at s = (subsection.). Let σ k (τ) be the representation, by right translations, of Sp 2k (A), on the space spanned by coefficients of Fourier-Jacobi type on E (g, φ), namely the functions p k (h), where (2.3) p k (h) =p k (h, φ, ξ) = E (vh, φ)θ ξ ψ,k (j k(v)h)ψ k (v)dv. N k (F )\N k (A) Here h Sp 2k (A), φ- as in subsection. and ξ S(A k ). The appearance of h in E (vh, φ) factors through Sp 2k (A). Denote, for this tower H k = Sp 2k. The case of GL m, where m is odd. Assume that m =2n +. Let τ be an irreducible, automorphic, cuspidal representation of GL 2n+ (A). Assume that τ is self-dual. In this case, it is known that L S (τ,λ 2,s) is holomorphic,

31 EXPLICIT LIFTS OF CUSP FORMS 837 and hence L S (τ,sym 2,s) has a pole at s =. Consider the Eisenstein series E(g, fτ,s) φ onga = Sp 4n+2 (A). It has a pole at s =. As in the last case, consider the groups U k and N k (only now dim E k =2n k + ). ψ k is the character of N k (A) =Z 2n k+ (A)U k (A), which is trivial on U k (A) and is the nondegenerate character on Z 2n k+ (A). As in the tower (2.3) and (2.3), let σ k (τ) be the representation, by right translations, of Sp 2k (A) on the space spanned by coefficients of Fourier-Jacobi type on E (g, φ) namely the functions p k (h), where (2.4) p k (h) =p k (h, φ, ξ) = E (vh, φ)θ ξ ψ,k (j k(v)h)ψ k (v)dv. N k (F )\N k (A) Here h Sp 2k (A), φ as in subsection. and ξ S(A k ). Note that in (2.4), we may take any preimage of h in Sp 2k (A). This defines a tower for τ on GL 2n+ (A) Sp 4n (A). (2.5) τ on GL 2n+ (A) Sp 2n (A).. Sp 2 (A) {I} Denote, for this tower, H k =Sp 2k. The definition of σ k (τ) through formulae (2.), (2.3) and (2.4) is not accidental. It is prescribed by Rankin-Selberg integrals which represent the standard L-function for H k GL m (see [G-PS], [G],[So], [GRS2]). More precisely, let σ be an irreducible, automorphic, cuspidal representation of H k (A) (H k is considered in each of the cases G =SO 4n, SO 4n+, Sp 4n, Sp 4n+2 ). Let ϕ be a cusp form of σ, then the integral (2.6) ϕ(h)p k (h)dh H k (F )\H k (A) is the residue at s = of the Rankin-Selberg integrals mentioned above (it is obtained from (2.6), by replacing E (h, φ) with E(h, f φ τ,s) in the definition of p k (h).) We remark that similar towers of unitary groups can be constructed with the aid of similar global integrals (the theory should resemble the previous cases; see [T], [Wt].) 2. The tower of correspondences. Let G be one of the groups SO 4n, SO 4n+, Sp 4n, Sp 4n+2, and let τ be an irreducible, automorphic, cuspidal and self-dual representation of GL m (A), where m =2n or m =2n +, such that the corresponding Eisenstein series E(g, f φ τ,s) has a pole at s =. Consider

32 838 DAVID GINZBURG, STEPHEN RALLIS, AND DAVID SOUDRY the representation σ k (τ) ofh k (A), acting by right translations in the space spanned by the functions p k (h) ((2.), (2.3), (24)). Our main theorem in this chapter is: Theorem 8. Assume that σ k (τ) =0,for k l. Then σ l (τ) is either zero or cuspidal on H l (A). This is the well-known property of the symplectic-orthogonal towers of theta liftings [R]. (See also [GRS3], for another tower for G 2, within exceptional groups.) The towers we present here should start at level k = n. We cannot prove this at this stage ( ), but at the third chapter, we will prove that, in case G =Sp 4n, σ k (τ) = 0, for k<n. Once we prove that σ n (τ) is nontrivial, then σ n (τ) is a cuspidal representation of H n (A), associated to τ (such that E(g, f φ τ,s) has a pole at s = ). The correspondence τ σ n (τ) should be the explicit backward map to the functorial lift from GL m to H n (m =2n, 2n + ). We first prove (recalling that P = M n U denotes the Siegel parabolic subgroup of G), Proposition 9.. The constant term of E (g, φ) along any unipotent radical of a standard parabolic subgroup, different from P, is zero. 2. The Whittaker coefficient of E (g, φ), along the maximal unipotent subgroup of G, is zero. 3. Let Z m be the standard maximal unipotent subgroup of GL m, embedded naturally in M the Levi part of P. Consider the standard nondegenerate character of Z m (A), defined by ψ, and continue to denote it by ψ. Then the following coefficient is nontrivial, E (zug)ψ(z)dzdu. Z m(f )\Z m(a) U F \U A Proof. Part follows, since E(g, f φ τ,s) is concentrated on P. Part 2 follows since the Whittaker coefficient of E(g, f φ τ,s) is holomorphic at s =. (The proof uses the formula on p. 352 of [Sh], for the Whittaker coefficient of E(g, f φ τ,s). The product of L-functions, which appears there, is that given by the denominator in each case of (.5). As explained (in parenthesis) in the proof of Proposition, each such denominator is nonzero (and holomorphic) at s =. To complete the argument, use Proposition 3. in [Sh]. The proof in case E(g, f φ τ,s) ison Sp 4n is exactly the same.) As for part 3, consider the ( ) Addedinproof. As we mentioned in the introduction, we now can prove that σ n (τ) 0. The case G =Sp 4n appears in [G-R-S5]. The remaining cases will appear in a work under preparation. The fact that the unramified parameters of τ and σ n (τ) correspond, for G =Sp 4n appears in [G-R-S6].