260 I.I. PIATETSKI-SHAPIRO and one can associate to f() a Dirichlet series L(f; s) = X T modulo integral equivalence a T jt j s : Hecke's original pro

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1 pacific journal of mathematics Vol. 181, No. 3, 1997 L-FUNCTIONS FOR GSp 4 I.I. Piatetski-Shapiro Dedicated to Olga Taussky-Todd 1. Introduction. To a classical modular cusp form f(z) with Fourier expansion f(z) = one can associate the Dirichlet series 1X n=1 L(f; s) = a n e 2inz 1X n=1 a n n s : Hecke showed that L(f; s) converges absolutely in some right half plane, has an analytic continuation to the whole s-plane, and satises a nice functional equation. In addition, Hecke discovered that if f(z) is an eigenfunction for the Hecke operators, then the associated Dirichlet series has an Euler product expansion L(f; s) = Y p prime 1? p p s?1 1? p p s?1 : Siegel introduced a theory of modular forms on the domains H n = f = X + iy 2 M n (C )j t = ; Y > 0g; called the Siegel upper half plane of genus n. If f() is a Siegel modular cusp form, then f() has a Fourier expansion f() = X T >0 integral, symmetric a T e 2itr(T ) ; 259

2 260 I.I. PIATETSKI-SHAPIRO and one can associate to f() a Dirichlet series L(f; s) = X T modulo integral equivalence a T jt j s : Hecke's original proof of the analytic continuation and functional equation for the one dimensional case goes over with little change to the higher dimensional case, giving the analytic continuation and functional equation for Dirichlet series attached to Siegel modular cusp forms of any genus. However, Hecke's proof of the Euler product expansion for the Dirichlet series attached to eigenfunctions of the Hecke operators on the upper half plane completely fails in the higher dimensional case. It is clear that these Dirichlet series do not have Euler product expansions. For a Siegel modular cusp form f() of genus n = 2, Andrianov suggested the following construction [1]. Dene 1X a mt0 L(f; T 0 ; s) = m ; s where T 0 is a positive denite integral symmetric matrix. He was then able to prove that L(f; T 0 ; s) converges in some half plane, has a meromorphic continuation to the whole s-plane, and satises a nice functional equation. Moreover, if f() is an eigenfunction of the Hecke operators, then the Dirichlet series L(f; T 0 ; s) have Euler product expansions. At the same time, Shimura was studying modular forms of half-integral weight [15]. A modular cusp form of half-integral weight has a Fourier expansion of the usual type f(z) = 1X n=1 m=1 a n e 2inz : Consider the analogue of the classical Dirichlet series attached to f(z) L(f; s) = 1X n=1 a n n s : As in Hecke, L(f; s) has an analytic continuation to the whole s-plane and satises a nice functional equation. But L(f; s) has an Euler product expansion only in very special cases. Then Shimura considered a family of Dirichlet series attached to f(z) L(f; t 0 ; s) = 1X n=1 a t0n 2 n s

3 L-FUNCTIONS FOR GSp with t 0 a positive integer. As in the Siegel modular case, Shimura was able to prove that the L(f; t 0 ; s) converge in some half plane, have meromorphic continuation to the whole s-plane, satisfy a nice functional equation, and all have Euler product expansions. Both Andrianov and Shimura obtained the meromorphic continuation of their Dirichlet series by using the Rankin-Selberg method. In this method, the analytic continuation of a Dirichlet series follows from the analytic continuation of a certain Eisenstein series. Andrianov's construction, like Hecke's, does not provide a denition of the local L- and -factors. In [12] we suggested a reformulation of Andrianov's construction which, like J. Tate's thesis for GL 1 or Jacquet-Langlands for GL 2, allows for the denition of L and factors for GSp 4 over local and global elds. Our construction was based on a generalization of a Whittaker model. The uniqueness of this generalized model was proved by Novodvorsky and the author in [10]. Later on it was considered by Rodier [14]. The complete proof of uniqueness in all cases was given by Novodvorsky. The computation of the L and factors, however, was not done explicitly. This seems to be an interesting problem. One of the aims of this paper is to present the results of these computations in some cases. For instance, let be a representation of GSp 4 (k) induced from a Borel subgroup (k a local non-archimedean eld). Denote by L the representation of GL 4 (k) which corresponds to under the Langlands correspondence associated with the natural embedding of L GSp 4 = GSp 4 (C ) in L GL 4 = GL 4 (C ) [3]. Then we have that our L and facotrs for are equal to the standard L and for L. Let be a mysterious cuspidal nongeneric representation of PGSp 4. (For a nite eld this representation was found by Mrs. Srinivasan and labelled 10 [16].) We prove that if L exists then it should be an induced representation K which will be dened at the end of Section 4. During the discussion of this result with R. Langlands, he pointed out that he had suggested a dierent approach to the computation of L [9]. There is a striking dierence between L-functions for GL n and L-functions for GSp 4. The L-function of a representation of GL n depends only upon embeddings in representations induced from the appropriate parabolic subgroups. The same is true for generic representations of GSp 4. However, for nongeneric representations of GSp 4, embeddings in representations induced from some reductive subgroups give a contribution to the L-function. This implies that cuspidal representations of GSp 4 can have nontrivial L- functions. In the proceedings of the Corvallis conference, there is an article by Novodvorsky where similar questions were studied by him and me [11]. It also contains a study of the L-function corresponding to, where is a

4 262 I.I. PIATETSKI-SHAPIRO representation of GSp 4 as before and a representation of GL 2. We would like to express our gratitude to R. Howe, R. Langlands, A. Weil, and G. uckerman for interesting and stimulating discussions. This paper is based on two lectures given in G. D. Mostow's \Lie Group Seminar" at Yale in the late 1970's. Note taking and nal editing were done by J. Cogdell, to whom we are very grateful. This article has circulated as a preprint for a number of years. We would like to thank Dinakar Ramakrishnan for giving us the opportunity to publish it in this volume dedicated to Olga Taussky- Todd. 2. GSp 4 and its subgroups.? 0 I Let k be any eld of characteristic 6= 2. Let J = 2?I 2 0 be the standard skew symmetric matrix of dimension 4. Then dene GSp 4 = fg 2 GL 4 (k)j t gjg = J for some 2 k g; which is an algebraic group dened over k. We let C denote the center and B denote the standard Borel subgroup of GSp 4. Let S denote the subgroup of GSp 4 comprised of all matrices of the form! I 2 A with t A = A: 0 I 2 Then S is an abelian group and is equal to the unipotent radical of the parabolic subgroup (!) P = g 2 GSp 4 g = : 0 Let P = M S be the Levi decomposition of P, M being the reductive part. Any homomorphism of S into k can be written in the form s 7! tr(s) with 2 M 2 (k) with t =. A linear form s 7! tr(s) on S is called nondegenerate if det() 6= 0. Now x a non degenerate linear form `(s) = tr(s) on S. Let D denote the connected component of the stabilizer of ` in M. Then there is a unique semisimple algebra K over k, with (K : k) = 2, such that D ' K as an algebraic group over k. It is known that either K = k k or K is a quadratic extension of k. The subgroup of GSp 4 which will be most important in our investigation of L-functions will be R = DS. We introduce N = fs 2 S j tr(s) = 0g:

5 L-FUNCTIONS FOR GSp If K is the semisimple algebra over k determined by D, as above, then let V = K 2 and consider the group G = fg 2 GL 2 (K)j det(g) 2 k g: We will write vectors of V in row form and let G act on the right. On V we consider the skew symmetric form (x; y) = Tr K=k (x 1 y 2? x 2 y 1 ); where x = (x 1 ; x 2 ) and y = (y 1 ; y 2 ) are elements of V. Then it is easy to see that G preserves up to a factor in k. (More precisely, the action of g 2 G changes Tr K=k (x 1 y 2? x 2 y 1 ) by det(g) 2 k.) If we then consider V as a 4 dimensional vector space over k, we get a natural embedding G,! GSp = fg 2 GL 4 (k)j(xg; yg) = (x; y)g: If we dene the k-linear transformation on V by : (x 1 ; x 2 ) 7! (x 1 ; x 2 ) then preserves and gives us a well dened element of GSp. (Here x 7! x denotes the non-trivial automorphism of K=k.) Proposition 2.1. There exists an isomorphism GSp ' GSp 4 such that G \ R = DN. We x such an isomorphism and consider G as a subgroup of GSp 4. Let H be the subgroup of G dened by H = f( ) j 2 k g. Lemma. Let K be a eld and let L 1 = f` 2 Hom k (S; k) j ` is a nontrivial linear functional but `j N is trivialg. Then H acts simply transitively on L 1. Remark. All the groups dened above are algebraic groups dened over k and all statements made are true in this context. In particular, these constructions and results are valid for local and global elds and also for the ring of adeles. 3. Local L and factors. In this section we let k denote a local eld. The Jacquet-Langlands construction of local L and factors for GL 2 is based on the denition of a Whittaker model. For a representation of GSp 4 (k) there does not always exist a Whittaker model dened with respect

6 264 I.I. PIATETSKI-SHAPIRO to the maximal unipotent subgroup. For GSp 4 (k) we use a generalized Whittaker model dened with respect to the subgroup R introduced in Section 2. Let be a character on D k and a nondegenerate character on S k, i.e., (s) = 0 (tr(s)) with det() 6= 0, t = and 0 a nontrivial character of k. Then we get a character ; on R k by dening ; (r) = ; (ds) = (d) (s) where r = ds 2 R k, d 2 D k, s 2 S k. We have D k ' K. Theorem 3.1. Let (; V ) be an irreducible smooth admissible preunitary representation of GSp 4 (k) and let ; be a character on R k as above. Then there exists at most one (up to scalar multiple) linear functional ` : V?! C such that `((r)) = ; (r)`() for all r 2 R k, 2 V. We will call such a linear functional an ; -eigenfunctional. (If k is archimedean, we also require that ` be continuous in the C 1 -topology.) Using the well-known Gelfand-Kazhdan method, this theorem was proved by Novodvorsky and the author for the case K a eld [10] and for cuspidal representations by Rodier for arbitrary k [14]. Novodvorsky later proved the theorem for arbitrary K and nonarchimedean local elds using the same method (unpublished). The archimedean case is even simpler to handle by using the fact that in this case any representation is a quotient of a principal series representation. R. Howe has proven results from which it follows that, if k 6= C, then for any innite dimensional representation (; V ) of GSp 4 (k) there exists a nontrivial ; -eigenfunctional on V for an appropriate choice of,, and R [5]. If k = C, the only representations for which this fails will be the Weil representations. One can prove that these representations never occur as components of cuspidal automorphic representations. Let (; V ) be an irreducible smooth representation of GSp 4 (k) and let ` be an ; -eigenfunctional. For a vector 2 V we dene the generalized Whittaker function W (g) on GSp 4 (k) by W (g) = `((g)): If we let W ; denote this space of functions and let GSp 4 (k) act on W ; by right translations, then the representation of GSp 4 (k) on W ; is equivalent to the representation. From the dening property of ` we see that the functions W (g) satisfy W (rg) = ; (r)w (g)

7 L-FUNCTIONS FOR GSp for r 2 R k. We call the space W ; a generalized Whittaker model for. If we let W (g) = W (g), then the space W comprised of the functions W (g) for 2 V is the generalized Whittaker model associated to the character ;, where is the character on D k dened by (d) = ( d). Let V = K 2 be the vector space upon which the group G, dened in Section 2, acts. Let 2 S(V ), the space of Schwartz-Bruhat functions on V, let be the character of K ' D k chosen above, and let be a character of k. Then for s 2 C we dene a function on G by f (g; ; ; s) = (det(g))j det(g)j s+ 1 2 k ((0; t)g)jttj s+ k (tt)(t)d t: K Then f (g; ; ; s) 2 ind G B 0() where B0 is the Borel subgroup of G and is the character on B 0 dened by!!!! x 0 t 0 1 n = (x)jxj 1 2 +s?1 (t): t 0 1 Now let W 2 W ;. Then we dene L(W; ; ; s) = = DNnG NnG W (g)f (g; ; ; s) dg W (g)((0; 1)g)(det g)j det gj s+ 1 2 k In the usual manner, the integral dening L(W; ; ; s) converges in some half plane Re(s) > s 0 and has a meromorphic continuation to the whole s-plane. As in Tate's thesis [17], [8], or [7], the L(W; ; ; s) admit a greatest common denominator for all W 2 W ; and 2 S(V ). This allows us to dene the function L(; ; s) by the property that L(W;;s) is an entire L(;;s) function for all W 2 W ; and 2 S(V ). In our denition, L(; ; s) depends upon the choice of,, and R. It is easy to see that for a xed R, L(; ; s) does not depend on. In the most important cases we can prove that it does not depend on the choice of or R either. See, for example, Theorem 4.4 below and [13]. For simplicity we do not include in our notation for L(; ; s), which is called the local L-factor, or the local Euler factor, associated to the pair (; ). 1 2 dg: Proposition 3.2. There is a (local ) functional equation L(W; ; ; s) (; ; ; s) L(; ; s) = L(W ; ^;?1 ; 1? s) L(^;?1 ; 1? s)

8 266 I.I. PIATETSKI-SHAPIRO where ^ denotes the Fourier transform of with respect to the trace form on V, ^ is the contragredient representation to, and (; ; ; s) is an entire function without zeros. Proof. Denote by (; W ) the representation ind G B0 where is the character on B 0 dened by x 0 0 1! t 0 0 t! 1 n 0 1!! = (x)jxj 1 2 +s?1 (t): Then the map : 7! f (g; ; ; s) denes a homomorphism : S(V )?! W. Let T = ker(). Then L(W; ; ; s) is a bilinear G quasi-invariant functional on V S(V ) which is equal to zero for any 2 T, and hence is actually a bilinear functional on V W. Here, being G quasi-invariant means that for (; w) 2 V W we have ((g); (g)w) =?1 (g)j det gj? 1 2?s (; w): It is clear that L(W ; ^;?1 ; 1? s) can also be considered as a bilinear functional on V S(V ). Using properties of intertwining operators, it is possible to show that L(W ; ^;?1 ; 1? s) is zero for the same 2 T. In fact, the composition map [ 7! ^ 7! f ^(g;?1 ; ; 1? s)] also has kernel T. Hence L(W ; ^;?1 ; 1? s) is also a bilinear G-quasi-invariant form on V W. Then, using the uniqueness of our generalized Whittaker models, it is easy to show that such bilinear G quasi-invariant linear forms on V W are unique up to scalar multiples. Letting s vary, we obtain a meromorphic function (; ; ; s) such that L(W ; ^;?1 ; 1? s) = (; ; ; s)l(w; ; ; s): Then (; ; ; s) is given by L(; ; s) (; ; ; s) = (; ; ; s) L(^;?1 ; 1? s) : 4. Computation of the local L and factors. In this section we restrict our attention to the case where k is a nonarchimedean local eld. Let S 0 (V ) = f 2 S(V )j((0; 0)) = 0g. Then we divide the poles of L(; ; s) into two types. We call a pole of L(; ; s) regular if it is a pole of some L(W; ; ; s) with 2 S 0 (V ). A pole of

9 L-FUNCTIONS FOR GSp L(; ; s) is called exceptional if it is not a pole of any L(W; ; ; s) when is restricted to lie in S 0 (V ). We assume that is in general position in the natural meaning. Consider the collection of all characters of H such that there exists a linear functional ` on V such that () `((hts)) = (h)jhj 3=2 (t)`() where h 2 H; t 2 D, and s 2 S. It is not dicult to prove that the collection of characters for which this linear functional is nontrivial is independent of. Of course, we consider here only the such that for some nondegenerate character of S there exists a nontrivial ; -eigenfunctional. Theorem 4.1. The regular part of L(; ; s) Q is equal to 2 L(s; ), where the L(s; ) are the standard Tate L-functions. Proof. It is easy to see that all regular poles come from the asymptotic behavior of the W (h) when h 2 H is small. Recall that h 2 H has the form h = x x 1 1 for x 2 k. By h small we mean x approaching 0 in k. For small h, we can write W (h) in the form W (h) = i C i () i (h)jhj 3=2 : Then it is easy to see that the C i () are linear functionals satisfying (). For not in general positiion a similar result holds [13]. Theorem 4.2. If s 0 is an exceptional pole of L(; ; s), then there exists a linear functional ` on V such that for all g 2 G; ; 2 V. `((g)) =?1 (det g)j det gj?s0? 1 2 `() Proof. Assume that L(; ; s) has an exceptional pole at s 0, and assume that for some integral L(W; ; ; s) we have L(W; ; ; s) = A(W; ) s? s 0 + : Then it is obvious that A is linear in and satises A(W; ) = 0 if ((0; 0)) = 0. Hence A(W; ) is of the form A(W; ) = ((0; 0))A(W ). It is also clear

10 268 I.I. PIATETSKI-SHAPIRO that A(W ) is a linear function of W. Since L(W; ; ; s) is G quasi-invariant, we see that A(W ) will be a linear functional of the desired type. It is likely that this condition is also sucient for s 0 to be an exceptional pole of L(; ; s), but this has not been proven. If our representation has additional structure we are sometimes able to say more about the form of L(; ; s). We recall that is called generic if has a standard Whittaker model, that is, a Whittaker model dened with respect to a maximal unipotent subgroup U of GSp 4. Theorem 4.3. If is generic, then L(; ; s) has only regular poles. Proof. It suces to show that for any generic representation there cannot exist any linear functional on V which is G quasi-invariant. The proof is based on a description of the double cosets P ngsp 4 =G. This set consists of exactly two double cosets. If g 0 is any representative of the double coset not containing 1, then P \ g 0 Gg?1 0 = GL 2 (k). This fact is true over an arbitrary eld k. In particular, we can take k to be a nite eld. Then if is a generic representation, denote by a standard Whittaker vector and consider '(g) = `((g)) where g 2 GSp 4 and ` is a linear functional on V which is G quasiinvariant. Then it is easy to see that '(g) satises the functional equations '(g 1 g) = (g 1 )'(g) '(gu) = (u)'(g) where g 2 GSp 4 ; g 1 2 G; a character on G; u 2 U, a maximal unipotent subgroup of GSp 4, and a generic character on U. Then using the above double coset decomposition one can prove that any function satisfying such functional equations must be identically 0. Now take k to be a non-archimedean local eld. We can similarly dene a distribution B(') on the space of smooth functions with compact support on GSp 4 (k) satisfying B(L g ') = (g)b(') B(R u ') = (u)b(') with g 2 G; u 2 U, and where L g and R u are operators of left and right translations respectively. Then, using the standard Gelfand-Kazhdan technique and the double coset decomposition above, we can prove that such a distribution must be identically 0.

11 L-FUNCTIONS FOR GSp Now let be any character of a Cartan subgroup of GSp 4 and extend it trivially on U to all of B. For x 2 k, dene characters 1 (x) = 2 (x) = x x 1 1 x 1 1 x 11 C A C A = 3(x)?1 ; 11 C A C A = 4(x)?1 : Theorem 4.4. If = ind GSp 4 B L(; ; s) = 4Y i=1 and is irreducible, then L(s; i ); (; ; ; s) = 4Y i=1 (s; i ; ); where the L(s; i ) and (s; i ; ) are the Tate L and factors, [17]. In particular, L(; ; s) and (; ; ; s) are independent of and R. The computation of L(; ; s) and (; ; ; s) when is a Weil lift from a split GO 4 can be found in [13]. Consider now a representation coming from the dual reductive pair Sp 4 O 2 [4], which corresponds to the one dimensional nontrivial representation of O 2. It is known that if O 2 is anisotropic then the corresponding representation of Sp 4 is cuspidal [2], [6]. The anisotropic O 2 correspond in a 1-1 manner to the quadratic extensions K of k. Since the Weil representation depends on the choice of, and for (x) and (x); 2 N K=k (K ), we get isomorphic representations, we see that to any quadratic extension K of k we can attach two cuspidal representations of Sp 4. If we extend these representations to representations of PGSp 4 (k), we will get two irreducible cuspidal representations. Take to be either of these two representations. It is possible to show that has the same L-factor as the following representation K of GL 4 (k). Denote by 0 the special representation of GL 2 which is the uniquely dened subrepresentation of ind GL2, where B 0 is the standard Borel subgroup of GL 2 and b1 x 0 b 2 = subgroup P 22 = ( g11 g 12 0 g 22! g ij 2 M 2 (k) B 0 1 b1 2 b 2. Denote by P 2;2 the parabolic ) \ GL 4 (k):

12 270 I.I. PIATETSKI-SHAPIRO Then K = ind GL4 P 2;2 ( 0 ( 0 K (det g))) where K is the character of k given by ( 1 if x 2 NK=k (K ) K (x) =?1 if x 62 N K=k (K ) : One can prove that K is the only generic unitary representation of GL 4 (k) with the same L-function as. A very simple proof of this statement was communicated to the author by H. Jacquet. One can also prove, by using other properties of the conjectured Langlands correspondence, that for any other cuspidal representation of PGSp 4 (k) the representation L of GL 4 (k) must be generic and cuspidal. Hence if L exists, it equals K. There is a very interesting discussion of how to nd L in [9]. The approach in [9] is completely dierent from ours. 5. Eisenstein Series and Global L-functions. In this section we let k be a global eld. Let denote a nondegenerate character on S A and a character on D A = IK. Let V A be the adelic points of the vector space on which G A acts. Take 2 S(V A ), the Schwartz-Bruhat functions on V A. Let be a Groencharacter on k (i.e., a character on I k, trivial on the principal ideles). Then to we can associate a function on G A f (g; ; ; s) = (det g)j det gj s+ 1 2 dened by I K ((0; t)g)jttj s+ 1 2 (t t)(t) d t where j j denotes the idele norm on I K. Note that f (g; ; ; s) 2 ind G A B, A 0 where is a character on B 0 dened by A!!!! x 0 t 0 1 n = (x)jxj s+ 2 1?1 (t): t 0 1 We can then form the Eisenstein series E (g; ; ; s) = X 2B 0 k ng k f (g; ; ; s): Theorem 5.1 ([7]). E (g; ; ; s) is a holomorphic function of s, 8s 2 C, except for a nite number of poles, and satises the functional equation E (g; ; ; s) = E ^(g;?1?1 ; ; 1? s) where?1 is the Groencharacter on k obtained by restriction from I K to I k, is the character of I K dened by (a) = (a), and ^ is the Fourier

13 L-FUNCTIONS FOR GSp transform of. Suppose that and are a normalized pair, i.e., (tt)(t) = jttj implies = 0. Then if (tt)(t) 6= 1, E (g; ; ; s) has no poles. If (tt)(t) 1, the poles are only at s =? 1, with residue (0)(det g), and 2 at s = 3, with residue ^(0)?1 (det g)?1 (det g). 2 We now turn to the study of global L-functions. So let (; V ) be an automorphic cuspidal representation of GSp 4 (A ). Then there exists a pair (; ), where is a nondegenerate character on S A and a character on the associated group D A = IK, such that for the character ; on R A = D A S A, there exists a cusp form ' 2 V such that C A R k nr A '(r)?1 ; (r)dr 6= 0: This again is a consequence of the results of Howe referred to above and the fact that B k n B A is dense in GSp 4 (k)n GSp 4 (A ). We x now such a pair (; ) and the associated adelic groups as in Section 2. The nonvanishing of the above integral allows us to dene a global generalized Whittaker model for. If ' 2 V we dene the function W ' on GSp 4 (A ) by These functions satisfy W ' (g) = C A R k nr A '(rg)?1 ; (r)dr: W ' (rg) = ; (r)w ' (g) for r 2 R A. Denote this space of functions by W ;. The representation of GSp 4 (A ) on W ; via right translation is then equivalent to. For any local component P of there exists a nontrivial P ; P -eigenfunctional as in Section 3, where P and P are the local components of the characters and respectively. The global (generalized) Whittaker model will then factor as a restricted tensor product of the local (generalized) Whittaker models associated to the P ; P as in Section 3. The uniqueness of the global model will then follow from the uniqueness of the associated local Whittaker models. Once we have a global (generalized) Whittaker model, we can make constructions analogous to those in Section 3 for local elds. In particular for W ' (g) 2 W ; and 2 S(V A ) we dene L(W ' ; ; ; s) = = D A N A ng A W ' (g)f (g; ; ; s) dg N A ng A W ' (g)((0; 1)g)(det g)j det gj 1 2 +s dg:

14 272 I.I. PIATETSKI-SHAPIRO Theorem 5.2. Assume that is a cuspidal automorphic representation and ' 2 V. Then L(W ' ; ; ; s) converges in some half plane Re(s) > s 0 and has a meromorphic continuation to the whole s-plane. Further, if (tt)(t) is nontrivial ( and a normalized pair); then L(W ' ; ; ; s) is holomorphic everywhere. If (tt)(t) 1, then L(W ' ; ; ; s) has poles at s =? 1, with residue 2 (0) (det g)'(g) dg; C A G k ng A and at s = 3=2 with residue ^(0) C A G k ng A?1 (det g)?1 (det g)'(g) dg: Proof. The convergence for s in some half plane is standard. The meromorphic continuation follows immediately from the identity C A G k ng A '(g)e (g; ; ; s) dg = D A N A ng A W ' (g)f (g; ; ; s) dg; and the meromorphic continuation of E (g; ; ; s). To prove this identity, we let I denote the left hand side above and expand the Eisenstein series. Then X I = '(g) f (g; ; ; s) dg = '(g)f (g; ; ; s) dg; C A G k ng A C A Bk 0 ng A 2B 0 k ng k since '(g) is an automorphic form on GSp 4 (A ) and thus invariant under GSp 4 (k) G k. Next we expand ' in its Fourier expansion '(g) = X 2 Char(S k ns A ) nontrivial where the Fourier coecients are given by ' (g) = ' (g); S k ns A '(sg)?1 (s)ds: If we substitute this Fourier expansion in the above expression for I, we note from the left invariance of f (g; ; ; s) under N A, that the Fourier coecients involving characters which are not trivial on N A will be killed upon integration. If we let denote the set of nontrivial characters on S k ns A which are trivial on N A, then we get I = CAHkDkNAnGA X 2 ' (g)f (g; ; ; s) dg:

15 L-FUNCTIONS FOR GSp Here we have used the decomposition B 0 = HDN. Now, from our lemma at the end of Section 2, it follows that H k acts simply transitively on. Then, since f (g; ; ; s) is invariant under H k by our choice of, we can rewrite the above as I = CAHkDkNAnGA X where is our character on S k n S A I = = = = = = C A D k N A ng A ' (g)f (g; ; ; s) dg DANAnGA DANAnGA D A N A ng A 0 DANAnGA as desired. h2h k ' (hg)f (g; ; ; s) dg xed above. Then C A D k nd A ' (tg)f (tg; ; ; s) dt C A D k nd A ' (tg)?1 (t) dt CADknDA S k ns A C A R k nr A '(rg)?1 ; (r) dr D A N A ng A W ' (g)f (g; ; ; s) dg;!! dg f (g; ; ; s) dg 1 C '(tsg)?1 (s)?1 (t) ds dta f (g; ; ; s) dg! f (g; ; ; s) dg This method of obtaining the meromorphic continuation of L(W; ; ; s) through the continuation of an Eisenstein series is what is commonly called the Rankin-Selberg method [7], [12]. Once we have the meromorphic continuation of L(W; ; ; s) we can dene L(; ; s) as in Section 3, i.e., L(; ; s) is a function dened so that L(W;;;s) L(;;s) is entire for all choices of W 2 W ; and 2 S(V A ). As noted above, to the local factors P ; P, and P of ; and respectively, we can associate the local (generalized) Whittaker model W P ; P. Then we can form the local integrals L(W P ; P ; P ; s) for W P 2 W P ; P and P 2 S(V P ) and thereby dene the local L and -factors L( P ; P ; s) and ( P ; P ; P ; s). Theorem 5.3. L(; ; s) = Q P L( P ; P ; s). Then: 1) The product converges in some half plane Re(s) > s 0.

16 274 I.I. PIATETSKI-SHAPIRO 2) The product has a meromorphic continuation to the whole s-plane. 3) We have the functional equation L(; ; s) = (; ; s) L(^;?1 ; 1? s) where (; ; s) = Q ( P ; P ; P ; s) and, as usual, this product does not P depend on. 4) If for some P 0 ; P0 is generic, then L(; ; s) is holomorphic. Proof. Choose ' 2 V and 2 S(V A ), which are factorizable. Then we can decompose the integral Y W ' (g)f (g; ; ; s) dg = W 'P (g)f P (g; P ; P ; s) dg D A N A ng A P D P N PnG P which gives L(W; ; ; s) = Q P L(W P; P ; P ; s). Then the convergence and meromorphic continuation of this product follows from the previous theorem. Statement 3) follows quickly from this. In order to prove 4) we have to use the following local result: If is generic, then an embedding of into ind GSp 4 G! cannot exist for any one dimensional representation! of G. The main idea of the proof of this statement was given in the proof of Theorem 4.3. References [1] A.N. Andrianov, eta functions and the Siegel modular forms, Proc. Summer School Bolya-Janos Math. Soc. (Budapest, 1970), Halsted, New York, 1975, [2] C. Asmuth, Weil representations of symplectic p-adic groups, Amer. J. Math., 101(4) (1979), [3] A. Borel, Automorphic L-functions, Proc. Sympos. Pure Math., 33(2) (1979), [4] R. Howe, -series and invariant theory, Proc. Sympos, Pure Math., 33(1) (1979), [5], On a notion of rank for unitary representations of the classical groups, C.I.M.E. Summer School on Harmonic Analysis and Group Representations, Cortona, 1980, [6] R. Howe and I.I. Piatetski-Shapiro, A counterexample to the \generalized Ramanujan conjecture" for (quasi-) split groups, Proc. Sympos. Pure Math., 33(1) (1979), [7] H. Jacquet, Automorphic Forms on GL(2) II, Lecture Notes in Math., 278, Springer, New York, [8] H. Jacquet and R. Langlands, Automorphic Forms on GL(2), Lecture Notes in Math., 114, Springer, New York, 1970.

17 L-FUNCTIONS FOR GSp [9] R. Langlands, Letter to Roger Howe, March 24, [10] M.E. Novodvorsky and I.I. Piatetski-Shapiro, Generalized Bessel models for the symplectic group of rank 2, Math. Sb., 90(2) (1973), [11] M.E. Novodvorsky, Automorphic L-functions for the symplectic group GSp(4), Proc. Symp. Pure Math., 33(2) (1979), [12] I.I. Piatetski-Shapiro, Euler subgroups, Proc. Summer School Bolya-Janos Math. Soc. (Budapest, 1970), Halsted, New York, 1975, [13] I.I. Piatetski-Shapiro and D. Soudry, L and " factors for GSp 4, J. Fac. Sci. Univ. Tokyo, Sec. IA, 28 (1981), [14] F. Rodier, Le representations de GSp(4; k) ou k est un corps local, C. R. Acad. Sci. Paris Ser. A, 283 (1976), [15] G. Shimura, On modular forms of half integral weight, Ann. of Math., 97 (1973), [16] B. Srinivasan, The characters of the nite symplectic group Sp(4; q), Trans. AMS, 131 (1968), [17] J. Tate, Fourier Analysis on Number Fields and Hecke's eta Functions, Thesis, Princeton, 1950, in `Algebraic Number Theory', Proceedings of the Brighton Conference, Academic Press, New York, 1968, Yale University New Haven, CT