LOCAL FACTORS FOR METAPLECTIC GROUPS: AN ADDENDUM TO LAPID-RALLIS

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1 LOCAL FACTORS FOR METAPLECTIC GROUPS: AN ADDENDUM TO LAPID-RALLIS WEE TECK GAN 1. Introduction In 4.3, we have used the doubling zeta integral and a normalized intertwining operator to define the standard γ-factor associated to an irreducible genuine representation σ of Mp(W). We noted there that this γ-factor satisfies a number of expected properties, as in [LR Thm. 4]. The purpose of this appendix is to verify these expected properties Doubling zeta integrals. Let us recall the basic definitions briefly. Throughout, W will denote a symplectic vector space of dimension 2n. We set W W +( W), so that there is a natural map Mp(W) µ2 Mp(W) Mp(W). Consider the diagonally embedded W which is a maximal isotropic subspace of W. The maximal parabolic subgroup of Sp(W) stabilizing W is P(W ) GL( W). Going to covering groups, one has P(W ) GL(W ) Mp(W). Fix a non-trivial additive character ψ of F. Then ψ determines a genuine character χ ψ of GL(W ). For any character χ of F, one can define the degenerate principal series representation I P(W,ψ(s,χ) : IndMp(W) P(W χ ψ (χ det) det s. Now let σ be an irreducible genuine representation of Mp(W) with contragredient σ. One has a natural Mp(W)-invariant pairing, on σ σ. For v σ, v σ and a holomorphic section f s I P(W,ψ(s,χ), the doubling zeta integral is defined by Z W,ψ (s,χ)(f,v,v ) f s (g,1) gv,v dg Mp(W) for a fixed Haar measure dg on Mp(W). Thus, Z W,ψ (s,χ) defines a Mp(W) Mp(W)-invariant linear form Z W,ψ (s,χ) : I P(W,ψ (s,χ) σ σ C. The analytic properties of this zeta integral, as a function of s, are as summarized in Thm

2 2 WEE TECK GAN 1.2. Normalization of intertwining operator. One has an intertwining operator defined by M(χ,s) : I P(W,ψ (s,χ) I P(W,ψ ( s,χ 1 ) M(χ,s)(f)(g) f(wng)dn where w (1, 1) Mp(W) µ2 Mp(W). This integral converges for Re(s) sufficiently large and has a meromorphic continuation to all of C. Note that this intertwining operator depends on the choice of the Haar measure dn on. In [Sw] and [Z], a normalization of M(χ,s) is given as in [LR], and we recall its definition briefly. We fix a nondegenerate character on the abelian group Sym 2 W as follows. To every homomorphism A : Sym 2 W F, or equivalently a symmetric bilinear form on W, one obtains a character ψ A ψ A of. Now note that A Sym 2 W Hom(W,W ). If one identifies W W via (w, w) 2w and W W via (w,w) w (following [LR]), then A : W W, and so one may consider deta. Henceforth, we assume that A corresponds to the split quadratic form on the even dimensional space W, in which case deta ( 1) n F. Now one can show (cf. [Z]) that, for all s, dimhom (I P(W,ψ (s,χ),ψ A) 1. Moreover, a nonzero element of this 1-dimensional space can be given as a generalized Jacquet integral: W ψ,a (χ,s)(f) f(wn) ψ A (n) 1 dn This integral converges for Re(s) sufficiently large and has an analytic continuation to C. Note that W ψ,a (χ,s) depends on the choice of the Haar measure dn. By the multiplicity one result above, there is a meromorphic function c ψ,a (χ,s) such that W ψ,a (χ 1, s) M(χ,s) c ψ,a (χ,s) W ψ,a (χ,s). The explicit determination of the function c ψ,a (χ,s) has been given in [Sw] (see also [Z]). We shall recall the answer in the unramified case below. In general, the following consequence of Sweet s results is sufficient for our purpose: Proposition 1.1. We have c ψ,a (s,χχ a ) χ a ( 1) n c ψ,a (s,χ) for any quadratic character χ a (associated to A F ), and c ψa,a(s,χ) c ψ (s,χ) χ(a) 2n a 2ns χ a ( 1) n, c ψ,a A (s,χ) c ψ,a (s,χ) χ(a) 2n a 2ns.

3 AN ADDENDUM TO LAPID-RALLIS 3 Indeed, we shall give an independent proof of this proposition later on. Now one may normalize the intertwining operator M(χ,s) by setting so that M ψ,a (χ,s) c ψ,a(χ,s) 1 M(χ,s), W ψ,a (χ 1, s) M ψ,a (χ,s) W ψ,a(χ,s). Then this normalized operator is independent of the choice of the Haar measure dn (we are of course using the same dn in the defintion of M(χ,s) and W ψ,a (χ,s)). Moreover, it satisfies The above proposition implies: Mψ,A (χ 1, s) Mψ,A (χ,s) 1. Corollary 1.2. As linear maps from I P(W,ψ (s,χ) to I P(W ),ψ ( s,χ 1 ), one has: M ψ,a A (χ,s) χ(a)2n a 2ns M ψ,a (χ,s) Definition of local factors. Using the normalized intertwining operator, we define the standard γ-factor associated to the representation σ χ of Mp(W) GL 1 by the functional equation Z W,ψ,A ( s,χ 1 ) (M ψ,a (χ,s) 1 σ 1 σ ) z ψ (σ) χ(det A) det(a) s γ(s+ 1 2,σ χ,ψ) Z W,ψ(s,χ). It follows from the previous Corollary 1.2 that γ(s,σ χ,ψ) is independent of the choice of A. We may then define the local L-factor L(s,σ χ,ψ) and the local epsilon factor ǫ(s,σ χ,ψ) by the procedure of 4.3, i,e, following Shahidi. It follows from definition that γ(s,σ χ t,ψ) γ(s + t,σ χ,ψ). The rest of this appendix is devoted to verifying other less obvious properties of γ(s,σ χ,ψ). The first order of business is to understand the unramified case thoroughly Unramified case. Assume that the residue characteristic p of F is odd and all the data involved (i.e. σ, χ, ψ and A) are unramified. Then one has σ I B,ψ (χ 1,...,χ n ) for some unramified characters χ i of F. In this case, one would like to check that n L(s,σ χ,ψ) L (s,σ χ,ψ) : L(s,χ i χ) L(s,χ 1 i χ). However, this is not immediate from the definition. One can show this desired equality by an explicit calculation of the various quantities used in the definition of the γ-factor. The results are summarized in the following proposition. i1

4 4 WEE TECK GAN Proposition 1.3. Suppose that p is odd and σ, χ, ψ and A are unramified. Let f 0, v 0 and v 0 be the spherical vectors in I P(W ),ψ (s,χ), σ and σ respectively. (i) One has: where Z W,ψ (s,χ)(f 0,v 0,v 0 ) L (s,σ χ,ψ) d(χ, s) d(χ,s) n L(2s,+2k,χ 2 ). k1 (ii) One has (iii) One has (iv) One has W ψ,a (χ,s)(f 0 ) 1 d(χ,s). M(χ,s)(f 0 ) c ψ,a (χ,s) n j1 L(2s 2j + 1,χ 2 ) L(2s + 2j,χ 2. ) n γ(2s (2j 1),χ 2,ψ) 1, where the γ-factors on the right are those of Tate. In particular, one has j1 M ψ,a (χ,s)(f 0) d(χ 1, s) d(χ, s) Proof. The statement (i) was when by J. S Li [Li, Prop. 4.6]. (ii) can be shown in the same way as in [LR, Prop. 4]. (iii) was shown by Zorn [Z, 4]. Finally, (iv) was shown by Sweet [Sw] though it also follows from (ii) and (iii). Corollary 1.4. Assume the same hypotheses as the above proposition. Then one has In particular, one has f 0. γ(s,σ χ,ψ) L (1 s,σ χ 1,ψ) L. (s,σ χ,ψ) L(s,σ χ,ψ) L (s,σ χ,ψ). Proof. The first assertion follows from the above proposition and the definition of γ(s,σ χ,ψ), on noting that z ψ (σ) 1. For the second assertion, suppose that σ I B,ψ (χ 1,...,χ n ). If σ is tempered and χ is unitary, then χ i is unitary for each i and hence there can be no cancellation of factors in L (1 s,σ χ 1,ψ) with those of L (s,σ χ,ψ). This implies the desired equality in the tempered case. The non-tempered case follows by definition and the fact that every non-tempered unramified σ is the unique Langlands quotient of an unramified standard module.

5 AN ADDENDUM TO LAPID-RALLIS Doubling zeta integral for GL(X). In order to establish the crucial property of multiplicativity of the γ-factors defined above, it is necessary to consider the doubling zeta integral associated to a representation of a Levi subgroup M of Mp(W). If X W is an isotropic subspace, with W X + W 0 + X and, P(X) M(X) N(X) is the associated maximal parabolic, then M(X) GL(X) µ2 Mp(W 0 ). Thus, we need to consider the doubling zeta integral associated to a genuine irreducible representation of GL(X). The nonlinear group GL(X) and its genuine representations were described in 2.3 and 2.4. Let X be a vector space and set X X X. Let so that X {(x,x) x X} X and X {(x, x) : x X}, X X X. Let P(X ) be the parabolic subgroup of GL(X) stabilizing X. The Levi subgroup of P(X ) is M(X ) GL(X ) GL(X ), and its unipotent radical is N(X ) Hom(X,X ). For ease of notation, we shall simply write det for the rational character det X det 1 M(X ). Then one has the degenerate principal series representation of GL(X), with χψ defined as in 2.4. I P(X ),ψ (χ,s) : Ind P(X ) χ ψ (χ det) det s We may now repeat the construction of the doubling zeta integral. Let τ be an irreducible representation of GL(X), and set τ ψ : χ ψ τ so that τ ψ is a genuine representation of GL(X). Then one sets Z X,ψ (s,χ)( f,v,v ) f s (g,1) τψ (g)v,v dg GL(X) for f s I P(X ),ψ (χ,s), v τ ψ and v τψ. This is the doubling zeta integral for the representation τ ψ χ of GL(X) GL1, and it defines a GL(X) GL(X)- invariant form Z X,ψ s,χ) : I P(X ),ψ (χ,s) τ ψ τ ψ C. X of As before, one has a standard intertwining operator M(χ,s) : I P(X ),ψ (χ,s) I P(X ),ψ (χ 1, s), and one may define an normalized intertwining operator M ψ (χ,s) as above (cf. also [LR]). Namely, one fixes the nondegenerate character B : N(X ) F corresponding to an isomorphism X X. Then ψ B ψ B is a unitary character of N(X ), and one has the

6 6 WEE TECK GAN corresponding generalized Jacquet integral Wψ,B (χ,s) on I P(X ),ψ(χ,s). By a multiplicity one result, one has W ψ,b (χ 1, s) M(χ,s) c ψ,b (χ,s) Wψ,B (χ,s), for some meromorphic function c ψ,b (χ,s). Then the normalized intertwining operator is defined by: M ψ,b (χ,s) c ψ,b(χ,s) 1 M(χ,s). Finally, one defines a standard γ-factor γ(s,τ χ,ψ) by the local functional equation Z X ( s,χ 1, M ψ (χ,s)( f),v,v ) γ(s + 1/2,τ χ,ψ) (τ χ)( 1 X ) χ(det B) detb 2s ZX (s,χ, f,v,v ), where det B is defined in the same way as in the symplectic or metaplectic case. As before, one has the analog of Corollary 1.2 which ensures that the γ-factor here is independent of the scaling of B Relation to linear case. Now one has the linear version of the construction of the previous subsection, as detailed in [LR]. Thus, for the given representation τ of GL(X), one has the zeta integral Z X (s,χ)(f,,v,v ), the normalized intertwining operator Mψ,B (χ,s) and the γ-factor γ X (s,τ χ,ψ) defined by Z X ( s,χ 1 )(M ψ,b (χ,s)f,v,v ) γ X (s + 1/2,τ χ,ψ) (τ χ)( 1 X ) χ(det B) detb 2s Z X (s,χ,f,v,v ). We note here a typo in [LR, equation (25)], where on the right hand side of the above equation, they had τ( 1) instead of (τ χ)( 1). Moreover, it was shown in [LR] that γ X (s,τ χ,ψ) γ GJ (s,τ χ,ψ) γ GJ (s,τ χ,ψ), where γ GJ refers to the γ-factors of Godement-Jacquet. If one examines the proof of this identity in [LR], one sees that it is the definition using (τ χ)( 1) which leads to this desired relation. The goal of this subsection is to relate the nonlinear doubling zeta integral ZX and the γ-factor γ X (s,τ χ,ψ) defined in the previous subsection to its linear versions. We first note that there is a natural GL(X)-equivariant isomorphism ι(χ,s) : I P(X )(χ,s) χ ψ I P(X ),ψ (χ,s) given by f f f χ ψ. The following lemma is easy to check. Lemma 1.5. One has: and W ψ,b (χ,s) ι(χ,s) χ ψ ( 1 X ) W ψ,b (χ,s) M(χ,s) ι(χ,s) χ ψ ( 1 X ) ι(χ 1, s) M(χ,s)

7 AN ADDENDUM TO LAPID-RALLIS 7 Thus, so that Moreover, c ψ,b (χ,s) χ ψ ( 1 X ) c ψ,b (χ,s), M ψ,b (χ,s) ι(χ,s) ι(χ 1, s) M ψ,b (χ,s). Z X (s,χ,,ι(χ,s)(f),v,v ) 2 Z X (s,χ,f,v,v ). The lemma immediately implies: Proposition 1.6. One has γ X (s,τ χ,ψ) γ X (s,τ χ,ψ). In particular, γ X (s,τ χ,ψ) is independent of the choice of the isomorphism B, and not just indpendent of the scaling of B The main theorem. Now we come to the main theorem of this appendix. Theorem 1.7. One has: (i) (Multiplicativity) Suppose that σ Ind Mp(W) P(X),ψ τ σ 0, where P(X) is a maximal parabolic subgroup with Levi factor GL(X) µ2 Mp(W 0 ), τ is an irreducible representation of GL(X) and σ 0 is an irreducible genuine representation of Mp(W 0 ), then γ W (s,σ χ,ψ) γ X (s,τ χ,ψ) γ W0 (s,σ 0 χ,ψ). (ii) (Variation of ψ) If a F and ψ a (x) ψ(ax), then γ W (s,σ χ,ψ a ) χ(a) 2n a 2n(s 1/2) γ W (s,σ χ a χ,ψ). (iii) (Outer automorphism) Let σ a denote the twist of σ by an outer automorphism of Mp(W) given by conjugation by an element in the similitude group GSp(W) of similitude a. Then one has γ W (s,σ a χ,ψ) γ W (s,σ χ a χ,ψ). (iv) (Duality) One has γ W (s,σ χ,ψ) γ W (s,σ χ,ψ 1 ) γ W (s,σ χ 1 χ,ψ). (v) (Functional equation) One has γ W (s,σ χ,ψ) γ W (1 s,σ χ 1,ψ 1 ) 1.

8 8 WEE TECK GAN (vi) (Archimedean case) Suppose that F is archimedean so that (by the analog of the subrepresentation theorem), σ is contained in a principal series representation IB,ψ (χ 1,...,χ n ), then n γ W (s,σ χ,ψ) γ(s,χ i χ,ψ) γ(s,χ 1 i χ,ψ), i1 where the γ-factors on the right are those of Tate. (vii) (Global property) Suppose that k is a number field with ring of adeles A and W is a symplectic space over k. Let σ be a cuspidal representation of Mp(W) A, χ a Hecke character of A and ψ v ψ v a nontrivial additive charcater of k\a. Let S be a finite set of places outside of which all data involved are non-archimedean and unramified, then one has the global functional equation L S (s,σ χ,ψ) v S γ W (s,σ v χ v,ψ v ) L S (1 s,σ χ 1,ψ), where L S (s,σ χ,ψ) v/ S L(s,σ v χ v,ψ v ) (when Re(s) is sufficiently large) is the partial L-function of σ χ with respect to ψ. Moreover, the functions γ W (s,σ,χ,ψ) are characterized by properties (i), (ii) and (vii) Proof of the theorem. The rest of the appendix is devoted to the proof of the theorem Multiplicativity. The proof of multiplicativity is by the same argument as that in [LR]. Namely, one may apply [LR, Prop 2 and Lemma 9] and relate the doubling zeta integral for σ on Mp(W) to the doubling zeta integral for σ 0 on Mp(W 0 ) and the doubling zeta integral for τ ψ χ ψ τ on GL(X). In particular, by [LR, Prop. 2], one has: (1.8) Z W,ψ (s,χ)(f,α,α ) Z W0 X,ψ(s,χ) ( Ψ(χ,s)(f)(g),α(g),α (g) ) dg. P(X) P(X)\Mp(W) Mp(W) Here, we have set Z W0 X,ψ(s,χ) Z W0,ψ(s,χ) Z X,ψ (s,χ), and we have regarded α σ Ind Mp(W) P(X),ψ τ σ 0 as a function on Mp(W) taking values in τ σ 0. Moreover, Ψ(χ,s) is an intertwining operator as defined in [LR]. Ψ(χ,s) : I P(W ),ψ (χ,s) I P(W 0 ),ψ (χ,s) I P(X ),ψ (χ,s)

9 AN ADDENDUM TO LAPID-RALLIS 9 Furthermore, in the normalization of the intertwining operator for Mp(W), let us choose the element A Hom(W,W ) so that A(X ) X, in which case we also have A(W 0 ) W 0. Thus the restriction of A to X determines an element B Hom(X,X ) Hom(N(X ),F). Similarly, the restriction of A to W 0 determines an element A 0 : N(W 0 ) F. By [LR, Lemma 9], one has Hence, we have: Ψ(χ 1, s) M W,ψ,A ( (χ,s) IndMp(W) P(X) M W0 X,ψ,A 0 B (χ,s)) Ψ(χ,s). (1.9) Z W,ψ ( s,χ 1 )(MW,ψ,A (χ,s)(f),α,α ) Z W0 X(s,χ) ( M P(X) P(X)\Mp(W) Mp(W) W 0 X,ψ,A 0 B (χ,s)(ψ(χ,s)(f)(g)),α(g),α (g) ) dg. Now apply the local functional equations of the two zeta integrals in equation (1.9) and compare with equation (1.8). Noting that and one deduces that z ψ (σ) z ψ (σ 0 ) τ( 1), deta deta 0 detb det( B) ( 1) dim X det(a 0 ) (detb) 2. γ W (s,σ χ,ψ) γ W0 (s,σ 0 χ,ψ) γ X (s,τ χ,ψ). The result then follows by Proposition Archimedean case. We note that statement (vi) of the main theorem follows immediately from statement (i), as a consequence of the subrepresentation theorem Variation of ψ. Now we are ready to prove statement (ii) of the main theorem. We note that Thus, I P(W ),ψ a (χ,s) I P(W ),ψ (χχ a,s), Z W,ψa (s,χ) Z W,ψ (s,χχ a ) M ψa (χ,s) M ψ (χχ a,s) z ψa (σ) z ψ (σ) χ a ( 1) n. M ψ a,a (χ,s) c ψ,a(s,χχ a ) c ψa,a(s,χ) M ψ,a (χχ a,s).

10 10 WEE TECK GAN On comparing the local functional equations for Z W,ψa (s,χ) and Z W,ψ (s,χχ a ), one concludes that γ W (s + 1/2,σ χ,ψ a )/γ W (s + 1/2,σ χ a χ,ψ) c ψ,a(s,χχ a ) c ψa,a(s,χ). Thus statement (ii) holds (for any given σ) if and only if: c ψa,a(s,χ) a 2ns χ(a) 2n c ψ,a (s,χχ a ). This is the first and second identities in Proposition 1.1. However, we can prove this identity without resorting to Proposition 1.1 (which was taken as a blackbox from the unpublished work of Sweet [Sw]). Indeed, it follows from the above that if the identity in (ii) holds for one particular σ, then it holds for arbitrary σ! Now, by multiplicativity (i.e. statement (i)), it is easy to see that the identity in (ii) holds for principal series representations induced from the Borel subgroup B of Mp(W). Hence we are done with (ii) Outer automorphism of Mp(W). To address statement (iii) of the main theorem, we need to recall some facts about outer automorphisms of Mp(W). We follow the treatment of Szpruch [Sz]. Consider the element g a of GSp(W) which acts as identity on W and by the scalar a F on W. Then g a has similitude a. Conjugation by g a defines an (outer) automorphism of Sp(W), which preserves the maximal parabolic P(W ). It is a basic fact that this outer automorphsim has a unique lift to Mp(W); we shall denote this lifted automorphism of Mp(W) by ν a. Thus, ν a (h,ǫ) (g a hg 1 a,ǫ a(g) ǫ), for some ǫ a (g) ±1. The following lemma records some facts about the function ǫ a. Lemma (i) If h, then ǫ a (h) 1. (ii) If h M(W ) GL(W ), then ǫ a (h) χ a (det W h). (iii) If h (1, 1) Sp(W) Sp( W) Sp(W), then ǫ a (h) χ a ( 1) n. It follows from the lemma that the automorphism ν a induces a natural map ν a : I P(W ),ψ (χ,s) I P(W ),ψ (χ a χ,s) given by ν a (f)(h) f(ν a (h)) Some computations. Now we can compute the effect of the change of ψ on various objects used in the definition of the γ-factors. Recall that I P(W ),ψ a (χ,s) I P(W ),ψ (χ χ a,s). The following lemma relates the generalized Jacquet integrals W ψa,a(χ,s) and W ψ,a (χ,s). Lemma As linear functionals on I P(W ),ψ a (χ,s), one has W ψa,a(χ,s) χ 1 (a) n χ(a) 2n a 2ns (W ψ,a (χ,s) ν a )

11 Proof. For f I P(W ),ψ a (χ,s), with Re(s) >> 0, one has W ψa,a(χ,s)(f) f(wn) ψ a A (n) 1 dn Now we note that Hence W ψa,a(χ,s)(f) AN ADDENDUM TO LAPID-RALLIS 11 f(wn) ψ A (ν a 1(n)) 1 dn f(wν a (n)) ψ A (n) 1 a n (2n+1) dn f(ν a (ν a 1(w) n)) ψ A (n) 1 a n (2n+1) dn ν a 1(w) (a 1 W,χ a ( 1) n ) GL(W ) Mp(W). f(wn) ψ a A (n) 1 dn χ a ( 1) n a 2ns χ(a) 2n χ ψ (det a 1 W ) f(ν a (w n)) ψ A (n) 1 dn χ 1 (a) n χ(a) 2n a 2ns (W ψ,a (χ,s) ν a ), since χ ψ (det a 1 W ) 1. The following lemma establishes the relation between ν a and the standard intertwining operator M(χ,s). Lemma As operators from I P(W ),ψ (χ,s) to I P(W ),ψ (χ aχ 1, s), one has ν a M ψ (χ,s) χ 1 (a) n χ(a) 2n a 2ns M ψa (χ,s) ν a. Proof. This is given by a similar computation as in the proof of the previous lemma: ν a (M ψ (χ,s)f)(h) M(χ,s)(f)(ν a (h)) f(wn ν a (h))dn f(ν a (ν a 1(wn) h))dn χ 1 (a) n χ(a) 2n a 2ns f(ν a (wnh))dn χ 1 (a) n χ(a) 2n a 2ns M ψa (χ,s) ν a.

12 12 WEE TECK GAN Corollary One has and c ψa,a(χ,s) χ 1 (a) n χ(a) 2n a 2ns c ψ,a (χ,s) ν a M ψ a,a (χ,s) M ψ,a (χ,s) ν a. In particular, we have given an independent proof of the second identity in Proposition 1.1. Together with the results of 1.11, we have thus proven the first two identities of Proposition 1.1, independently of [Sw]. The third identity in Proposition 1.1 can be checked by computations analogous to those in the two lemmas above Outer automorphism. Let us write W X X with X maximal isotropic. Then one has an outer automorphism of Sp(W) given by conjugation by the element which acts as 1 on X and a on X. The lifting of this to Mp(W) is denoted by µ a. Moreover, the outer automorphism µ a µ a on Mp(W) µ2 Mp(W) is the restriction of the outer automorphism of Mp(W) given by the conjugation of the element of GSp(W) which acts as 1 on X X + X and as a on X (X ) + (X ). We denote this outer automorphism of Mp(W) by µ a as well. For any irreducible representation σ of Mp(W), the twisted representation σ a is realized on the same space as σ, but the action of Mp(W) is defined by: σ a (g) σ(µ 1 a (g)). Now consider the local zeta integral for σ a : Z W,ψ (s,χ,σ a )(f,v,v ) f(g,1) (σ a ) (g)v,v dg. Mp(W) Mp(W) Mp(W) Mp(W) f(g,1) σ (µ 1 a (g))(v ),v dg f(µ a (g,1)) σ (g)(v ),v dg f(µ a ν 1 a ν a (g,1))) σ (g)(v ),v dg. Now it is easy to see that µ a νa 1 is the inner automorphism given by the element m a in the Levi subgroup GL(W ) M(W ) of P(W ) which acts as 1 on X and as a on (X ). Thus we have Z W,ψ (s,χ,σ a )(f,v,v ) f(m a ν a (g,1))m 1 a ) σ (g)(v ),v dg Mp(W) δ P(W )(m a ) 1/2 χ ψ (m a ) a ns χ(a) n Z W,ψa (s,χ,σ)(ν a (m 1 a f),v,v ). We have thus shown that

13 AN ADDENDUM TO LAPID-RALLIS 13 Lemma Z W,ψ (s,χ,σ a ) δ P(W )(m a ) 1/2 χ ψ (m a ) a ns χ(a) n Z W,ψa (s,χ,σ) (ν a m 1 a 1 σ 1 σ ). Similarly, one has Z W,ψ ( s,χ 1,σ a ) M ψ,a (s,χ) δ P(W )(m a ) 1/2 χ ψ (m a ) a ns χ(a) n Z W,ψa ( s,χ 1,σ) (ν a m 1 a M ψ,a (s,χ)) δ P(W )(m a ) 1/2 χ ψ (m a ) a ns χ(a) n Z W,ψa ( s,χ 1,σ) (M ψ a,a (s,χ) ν a m 1 a ) where the last equality follows by Corollary By dividing this identity by that in the above lemma, and noting that we deduce that z ψ (σ a ) z ψa (σ), γ W (s,σ a χ,ψ) a 2ns χ(a) 2n γ W (s,σ χ,ψ a ), which gives the identity in (iii) in view of (ii) Duality. Statement (iv) of the main theorem follows from statement (iii), on noting that the contragredient of σ is simply the twist of σ by an outer automorphism associated to an element of GSp(W) of similitude 1 (see [MVW] and [Sun]) Global property. We shall now deal with the global functional equation, and we use the notation in statement (vi). For this, we need to introduce the global zeta integral associated to a cuspidal representation σ v σ v of Mp(W) A. Consider the global degenerate principal series representation I P(W ),ψ (χ,s) of Mp(W) A and its associated Eisenstein series map E ψ (s,χ) : I P(W ),ψ(χ,s) A(Mp(W)), which is a meromorphic function of s. Here A(Mp(W)) is the space of automorphic forms on Mp(W) A. Then the global zeta integral is defined by: Z W,ψ (s,χ) : I P(W ),ψ (χ,s) σ σ C, (Mp(W) k Mp(W) k )\(Mp(W) A Mp(W) A ) Z W,ψ (s,χ)(f,ϕ,ϕ ) E ψ (s,χ)(f)(g 1,g 2 ) ϕ (g 1 ) ϕ(g 2 )dg 1 dg 2. The following proposition summarizes the key properties of this global zeta integral: Proposition (i) For Re(s) >> 0, one has a factorization: Z W,ψ (s,χ) v Z W,ψv (s,χ v ).

14 14 WEE TECK GAN (ii) If S is a sufficiently large finite set of places of k, then one has the equality of meromorphic functions of s: ( ) Z W,ψ (s,χ)(f,ϕ,ϕ ) Z W,ψv (s,χ v )(f v,ϕ v,ϕ v ) LS (s,σ χ,ψ) d S (s,χ) v S where d S (s,χ) is the product of partial Hecke L-functions as defined in Prop (iii) One has the global functional equation: where Z W,ψ ( s,χ 1 ) (M ψ (s,χ) 1 σ 1 σ ) Z W,ψ (s,χ) is the standard global intertwining operator. M ψ (s,χ) : I P(W ),ψ (χ,s) I P(W ),ψ (χ 1, s) Proof. Statement (i) is due to Piatetski-Shapiro and Rallis [GPSR]. Statement (ii) is a direct consequence of Proposition 1.3. Statement (iii) is a consequence of the functional equation of Eisenstein series [MW]. In addition, we need to relate the global intertwining operator M ψ (s,χ) with the normalized local intertwining operators Mψ v (s,χ v ). We fix a rational homomorphism A : k and thus obtain the automorphic character ψ A ψ A of (A). We consider the associated Fourier coefficient map on I P(W ),ψ(χ,s) defined by: W ψ (s,χ)(f) E ψ (s,χ)(f)(n) ψ A (n) 1 dn. (k)\(a) The following proposition summarizes the properties of W ψa (s,χ). Proposition (i) One has: (ii) When Re(s) >> 0, one has W ψ,a (s,χ)(m ψ (s,χ)f) W ψ,a (s,χ)(f). W ψ,a (s,χ)(f) v W ψv (s,χ v )(f v ). Proof. Statement (i) is a consequence of the functional equation of Eisenstein series [MW]. Statement (ii) follows by unfolding the Eisenstein series when Re(s) >> 0. Now for each place v of k, the element A allows one to define the normalized intertwining operator M ψ v,a (s,χ v) c ψv,a(s,χ v ) 1 M ψv (s,χ v ), such that W ψv,a( s,χ 1 v ) M ψ v,a (s,χ v) W ψv,a(s,χ v ).

15 Morally speaking, the above proposition says that AN ADDENDUM TO LAPID-RALLIS 15 v c ψv,a(s,χ v ) 1, so that (M ψ (s,χ) v M ψ v,a (s,χ v). However, the Euler products above do not converge anywhere, and has to be suitably interpreted. This is given by the following proposition, after taking into account the results of Proposition 1.3: Corollary For a sufficiently large finite set S of places of k, one has n L S (2j 2s,,χ 2 ) c ψv,a(s,χ v ) L S (2s 2j + 1,χ 2 ). v S Proof. Consider the identity j1 W ψ,a (s,χ)(m ψ (s,χ)f) W ψ,a (s,χ)(f), with f v f v such that for all v / S, f v fv 0 is the spherical vector. Then Proposition 1.3 implies that ( ) n W ψv,a( s,χ 1 j1 v )(M(s,χ v )(f v )) LS (2s 2j + 1,χ 2 ) d S (s,χ) d S ( s,χ 1 ) v Thus, we have ( ) c ψv,a(s,χ v ) W ψv,a(s,χ v )(f v ) v Hence, ( ) 1 W ψv,a(s,χ v )(f v ) d S (s,χ). v c ψv,a(s,χ v ) v n j1 LS (2s 2j + 1,χ 2 ) d S ( s,χ 1 ) v n j1 L S (2j 2s,,χ 2 ) L S (2s 2j + 1,χ 2 ). W ψv,a(s,χ ) v(f v ). Combining this with the global functional equation in Proposition 1.15(iii), the results of the above corollary and Proposition 1.3, and the local functional equation of the local zeta integral in 1.3, one deduces the desired result as in [LR] Uniqueness. The uniqueness of the γ-factors considered in the main theorem is proved in the same way as [LR].

16 16 WEE TECK GAN References [GPSR] S. Gelbart, I. Piatetski-Shapiro, and S. Rallis, Explicit constructions of automorphic L-functions, volume 1254 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, [Li] J.-S. Li, Nonvanishing theorems for the cohomology of certain arithmetic quotients, J. Reine Angew. Math. 428 (1992), [LR] E. Lapid and S. Rallis, On the local factors of representations of classical groups, in Automorphic representations, L-functions and applications: progress and prospects, , Ohio State Univ. Math. Res. Inst. Publ., 11, de Gruyter, Berlin, [Mu] G. Muić, A geometric construction of intertwining operators for reductive p-adic groups, Manuscripta Math. 125 (2) (2008), [MVW] C. Moeglin, M.-F. Vigneras and J.-L. Waldspurger, Correspondances de Howe sur un corps p-adique, Lecture Notes in Mathematics, Springer-Verlag, Berlin, [MW] C. Moeglin and J. L. Waldspurger, Spectral decomposition and Eisenstein series, Cambridge Tracts in Mathemtics 113, Cambridge University Press (1995). [PSR] I.Piatetski-Shapiro and S. Rallis, ǫ factor of representations of classical groups, Proc. Nat. Acad. Sci. U.S.A. 83 (1986), no. 13, [Sun] B. Y. Sun, Multiplicity one theorems for Fourier-Jacobi models, available at arxiv: v2. [Sw] J. Sweet, Functional equations of p-adic zeta integrals and representations of the metaplectic group, preprint (1995). [Sz] D. Szpruch, The Langlands-Shahidi method for the metaplectic group and applications, thesis (Tel Aviv University), available at arxiv: v1. [Z] C. Zorn, Theta dichotomy and doubling epsilon factors for Mp(n, F), to appear in American J. of Math.