SOME APPLICATIONS OF DEGENERATE EISENSTEIN SERIES ON Sp 2n

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1 SOME APPLICATIONS OF DEGENERATE EISENSTEIN SERIES ON Sp n GORAN MUIĆ Introduction In this paper we generalize usual notion of Siegel Eisenstein series see for example [35]) to give a simple and natural construction of some classes of square integrable automorphic representations. The construction of automorphic representations obtained in this paper is an automorphic version of the local construction of strongly negative unramified representations [4] 1 or of discrete series obtained by Tadić [40] in early 90 s see also later work [31]). Let G = Sp n be a split symplectic group of rank n over a number field k. Let A be the ring of adéles of k. Let W k be the Weil group of k. Let ĜC) = SOn + 1, C) be the complex dual group of G. We look at the following types of Arthur parameters [3]; see Section 4): where we require ϕ : W k SL, C) ĜC), 1) ϕ Wk is continuous, semisimple, and its image is bounded ) ϕ SL,C) is a morphism of algebraic groups 3) The image is discrete, that is, the image is not contained in a proper Levi subgroup of SOn + 1, C) 4) ϕ Wk factors through Wk ab, where W ab k is the maximal abelian Hausdorff) quotient of W k. We remark that under 1) 4), ϕ Wk is a sum of quadratic characters. These parameters are called quadratic unipotent, and the corresponding automorphic representations that is, which belong to corresponding Arthur packets; [3]) are called quadratic unipotent square integrable automorphic representations. They were considered by many authors including Mœglin [7], [8]), Kim [13], [14], [15]), and Kim Shahidi [16]. Those authors follow the standard procedure of Langlands spectral decomposition as described in [34] in order to construct square integrable automorphic representations. On the other hand, recently, Arthur have announced the classification of whole discrete automorphic spectrum for all split classical groups over k using the methods of Arthur Selberg trace formula see [4]). In this approach, we obtain the formula for the multiplicity of irreducible representations in L disc Gk) \ GA)), but it is not clear which representations are cuspidal and which are residual. Thus, it is still important to characterize and construct residual representations among all in the discrete spectrum. A result on the characterization of residual representations is obtained in [8]. Also, in Arthur s approach, the existence of formula for multiplicity of representations in L disc Gk) \ GA)) enables us to concentrate on construction of automorphic representations but not on a difficult procedure of a spectral decomposition [34]. In this paper we adopt Arthur s point of view. We are interested in an explicit construction of certain classes of square integrable automorphic representations. Some of them are already constructed by other means in the papers mentioned above, but the majority of them are not. 1 An unramified representations is strongly negative resp., negative) if its exponents relative to a Borel belong to resp., the closure) in the obtuse Weyl chamber. 1

2 GORAN MUIĆ The present paper gives a construction of spherical representations and certain other automorphic representations of L disc Gk) \ GA)) attached to a spherical unipotent Arthur parameter which by definition is of the form: ϕ : SL, C) ĜC). This is somehow the simplest possible global case. We use our technique of degenerate Eisenstein series sketched in [5]. As opposed to [5], we do not require that archimededan components are spherical unless we consider the spherical component itself). This approach uses some general results on construction and analytic continuation of Eisenstein series [34] as a global input, while the local input is given by the normalization of intertwining operators [17] at ramified places, and local results of [4] on unramified representations. The proofs of [4] are entirely based on results of Bernstein Zelevinsky [8], [9]) and a result of Waldspurger [3] that describes the contragredient of an irreducible representation of a split classical p adic group. In Section 4 we classify Arthur parameters mentioned above. This is more or less well known exercise in Linear algebra, but we were unable to find a convenient reference. The main result of Section 4 is Theorem 4-1. In the same section we recall the classification of negative unramified representations from [4] see Theorem 4-4). According to Arthur [3], [4]), if a quadratic unipotent representation π = v π v of a Hecke algebra of) GA) belongs to L disc Gk) \ GA)), then for almost all places π v is a negative representation of Gk v ). After these preparations, we recall how to match square integrable automorphic representations and Arthur parameter via Hecke operators see Definition 4-6) In Section 5 we present our construction of square integrable automorphic representations attached to a spherical unipotent Arthur parameter. First, we consider the case of a spherical representation in Theorem 5-. Thanks to the Gindikin Karpelevič formula, we avoid the computation of poles of normalized) intertwining operators. Next, we consider the general case in Theorem 5-1. To deal with the constant term of a degenerate Eisenstein series, we need to use some non standard normalization of intertwining operators. The usual normalization procedure [3], [36]) is not applicable here because we are dealing with some induced representations see 5-16)) which are induced from representations for which is difficult to write their Langlands data especially in the archimedean case). Nevertheless, the normalization satisfies some usual properties see 5-18) and 5-19)). The local components of our automorphic representations are Aubert Schneider Sthuler Zelevinsky duals of discrete series discussed in [40]) for finite places, but for archimedean places everything is much more complicated. We give some examples in the last section see Section 6). Needless to say that our construction can be generalized using the local results of [31] but we leave this for some other ocassion since to cover the most general case we need to generalize and adapt some core constructions of [31] allowing multisets instead of sets) of Jordan blocks. I would like to thank J. Schwermer for his invitation to the Erwin Schrödinger Institute in Vienna where the final version of paper was written. I would like to thank N. Grbac, C. Mœglin, W.-T. Gan for some stimulating discussions. 1. Preliminaries Let Z, R, and C be the ring of rational integers, the field of real numbers, and the field of complex numbers, respectively. Let k be an algebraic number field. If A is an k algebra and H an algebraic group over k, we letha) to be the group of A points of H. The symplectic group of rank n over k is defined as follows: Sp n = {g GLn) g 0 Jn J n 0 ) g t = ) 0 Jn }, J n 0

3 where where J n is the n n matrix defined by AUTOMORPHIC FORMS ON Sp n 3 J n = The Langlands dual group of Sp n is SOn + 1, C). We realize this group as a group SOV ) of isommetries of determinant = 1 of the orthogonal space V,, )), where V is a complex vector space of dimension n + 1 and, ) is a non degenerate symmetric form on V. We fix the usual Borel subgroup B n consisting of upper triangular matrices in Sp n. It comes with a Levi decomposition B n = T n U n, where U n is a unipotent radical, and T n is a split torus consisting of the elements of the form: 1-1) t = diagt 1, t,..., t n, t 1 n,..., t 1 t 1 1 ). We let W = W n be the Weyl group of T n in Sp n. Let XT n ) be the group of characters of T n. It is a free Z module XT n ) Zφ 1 Zφ Zφ n, where φ i is defined by φ i t) = t i, 1 i n. The Weyl group W n is isomorphic to S n {±1}) n, where S n is the group of permutations of n letters. This semidirect product, as well as the isomorphism with W n, is described in the standard action of the Weyl group W n on XT n ). More precisely, for p S n and ɛ = ɛ 1,..., ɛ n ) {±1}) n, we have pɛ)φ i ) = φ ɛ i pi). We write Σ for the root system of T n with respect to G n. Further, Σ + and will denote positive roots and simple roots with respect to U n. We have Σ = Σ + Σ + ), and we have: Σ + = {φ i φ j, φ i + φ j ; 1 i < j n} {φ i ; 1 i n}. Let α i = φ i φ i+1, 1 i n 1, and α n = φ n. Then = {α 1, α,, α n }. Now, we recall [38], Lemma 4.4) the following lemma: Lemma 1-. Fix i, 1 i n. The set {w W n ; w \{α i }) > 0} can be described as the disjoint union of the sets 0 j i W j, where W j is the set of all pɛ see the description above) such that the following conditions hold: { 1, 1 k j, i + 1 k n ɛ k = 1, j + 1 k i, and we have the following:. p restricted to {1,..., j} or {i + 1,..., n} is increasing but not necessarily when restricted to their union).. p restricted to {j + 1,..., i} is decreasing. The next corollary follows directly from Lemma 1-. We leave the details to the reader. Corollary 1-3. Fix i, 1 i n. Let w = pɛ W j. This means wφ k ) = φ pk), 1 k j or i + 1 k n, and wφ k ) = φ 1 pk), j k i.) Then the set of all roots α Σ+ such that wα) < 0 is given by the union of the following cases:

4 4 GORAN MUIĆ. {φ k ; j + 1 k i}. {φ k + φ l ; j + 1 k < l i}. {φ k φ l ; j + 1 k i, i + 1 l n}. {φ k + φ l ; 1 k j, j + 1 l i, pl) < pk)}. {φ k φ l ; 1 k j, i + 1 l n, pl) < pk)}. {φ k + φ l ; j + 1 k i, i + 1 l n, pl) > pk)}. We define the homomorphism ϕ i : GL1) T n, 1 i n, given by ϕ i p) = t, where t is the matrix given by 1-1) for which t j = 1, j i, and t i = p. We let X T n ) Zϕ 1 Zϕ Zϕ n. We write the group operation in X T n ) additively. We have the usual pairing, : XT n ) X T n ) Z, defined by a b)p) = p a,b, p GL1). We have φ i ϕ j )p) = 1 i j) and φ i ϕ i )p) = p. Thus φ i, ϕ j = δ ij. Finally, we have the following list of roots and corresponding coroots:. α = φ i ± φ j α = ϕ i ± ϕ j. α = φ i α = ϕ i.. Construction of Eisenstein Series Attached to Principal Series In this section we give the set-up for the computation of poles of some degenerate Eisenstein series given in Section 3. We follow the book of Mœglin and Waldspurger [33] for the theory of automorphic forms, we use the results of Keys and Shahidi [17] on normalization of intertwining operators and we follow [37] for the results of Tate on L functions associated to grössencharacters. We continue using the notation introduced in the previous section. Let k be a global field, A its ring of adeles and A its group of ideles. We write {v} for the set of places of k. For each place v of k, we let k v be its completion at that place. Let v be the normalized absolute value of k v. Put = v v. If v is finite denoted by v < ), then we let O v be the ring of integers of k v and ϖ v a generator of the maximal ideal in O v. Then A v k v is a restricted product over all places {v}. We choose a non trivial additive character ψ : k \ A C and write it as a restricted product of local characters ψ = v ψ v. Note that for all but a finite set of places v we have that ψ v is unramified: ψ v Ov = 1, ψ v ϖ 1 v O v 1.) We fix a Haar measure dx = v dx v such that dx v is self dual with respect to ψ v ; this means for any f Cc k v ) the Fourier transform ˆfx v ) = k v fy v )ψ v x v y v )dy v satisfies fx v ) = k ˆfyv v )ψ v x v y v )dy v. The induced measure on k \ A has total measure equal to one. Let µ : k \ A C be a unitary) grössencharacter. We can write it as a restricted tensor product µ = v µ v of local characters µ v : k v C. In the above set-up, Tate associated local factors Ls, µ v ) and ɛs, µ v, ψ v ) such that the product ɛs, µ) = v ɛs, µ v, ψ v ) is independent of ψ and is a finite product since for all but finite places ψ v and µ v are unramified, and, in that case, ɛs, µ v, ψ v ) = 1. Moreover, we have that the product Ls, µ) = v Ls, µ v) initially converges for Res) > 1 and extends to the whole complex plane as a meromorphic function satisfying the following functional equation: Ls, µ) = ɛs, µ)l1 s, µ 1 ).

5 AUTOMORPHIC FORMS ON Sp n 5 Moreover, if µ 1, then Ls, µ) is holomorphic. Ls, 1) has simple poles at s = 0 and s = 1 and no other poles. We have Sp n A) v Sp nk v ) where the restricted product is taken with respect to K v := Sp n O v ), v < ). Analogous notation is used for the torus T n, unipotent radical U n and the Borel group B n introduced in Section 1. We fix a maximal compact subgroup K = v K v Sp n A) such that for any finite place v, K v is defined as above, and for an archimedean place v we fix some maximal compact subgroup K v. We have the usual Iwasawa decomposition Sp n A) = K B n A). In order to normalize intertwining operators, we need to fix Haar measures on unipotent radicals. We follow [17]. For any root subgroup U n,α U n, we have U n,α A) A and we transfer Haar measure from A to U n,α A). The Haar measure on U n,α A) will be denoted by du n,α. Similarly, for any place v, we have U n,α k v ) k v and we transfer the Haar measure from k v to U n,α k v ). The Haar measure on U n,α k v ) will be denoted by du n,α,v. We have U n,α A) v U n,αk v ) and du n,α = v du n,α,v. Furthermore, this also fixes Haar measures du n and du n,v on both A points and k v points of U n α Σ + U n,α the product map is an isomorphism of affine k varieties). The induced measure on U n k) \ U n A) has total volume equal to 1. Also, in this way we obtain a unique Haar measure on any subgroup of A points or k v points that can be written as a product of root subgroups. Typically, we use this in the following way. Let w W be an element of the Weyl group of Sp n with respect to T n. Then U n wu n w 1 = U n,α, α Σ +, w 1 α)<0 and our remark applies to this subgroup. We also need to fix a representative w of w W properly. We do this as follows. If α, the simple reflection r α has the representative ) 0 1 r α = ϕ α, 1 0 where ϕ α : SL) Sp n is a k morphism of algebraic groups attached to α see [17]). In general, for w W we obtain the representative in the following way. For any reduced expression w = r α1 r α r αl, we let w = r α1 r α r αl. The representative w is independent of the choice of the reduced expression used [17], Section ). We write W for the set of representatives. It is clear W Sp n k). Now, we discuss automorphic forms following [34]. Put s = s 1, s,..., s n ) C n XT n ) Z C. Next, given a sequence of grössencharacters k \ A C : λ = λ 1, λ,..., λ n ), we consider a family of characters λs) : T n k)\t n A) C defined as follows t is given by 1-1)): n -1) λs)t) = λ i t i ) t i s i, i=1 parametrized by s C n. All characters given by -1), have the same restriction to T n A) K: λ 0 = λ 1 λ n ) Tn A) K. We also write Ind K T n A) K)U n A) K) λ 0)

6 6 GORAN MUIĆ for the space of K finite functions spanned by the factorizable functions f = v f v : K C such that ftk) = λ 0 t)fk), for all t T n A) K, k K, and, for all but a finite number of places v, f v is right K v invariant and f v 1) = 1. K acts by right translations there. Now, if we take the character -1) and f Ind K T n A) K)U n A) K) λ 0), we define a function f λs) on Sp n A) using the Iwasawa decomposition: f λs) tuk) := δ 1/ B n t)λs)t)fk), t T n A), u U n A), k K. One checks that it is well defined. In the notation of [34], the space of such functions is denoted by AU n A)T n k) \ Sp n A), λs)). It is a space of K finite automorphic forms. It is a v< Sp nk v ) g, K ) module, where g is a real) Lie algebra of the Lie group v archimedean Sp nk v ) considered as a real Lie group and K = v archimedean K v. Moreover, as an abstract representation of v< Sp nk v ) g, K ), AU n A)T n k) \ Sp n A), λs)) is isomorphic to the global induced representation Ind Sp n A) B na) λs)) = Ind Sp n A) B na) s 1 λ 1 s λ s n λ n ) normalized induction; K finite vectors; the character λs) = s 1 λ 1 s λ s n λ n is extended to P n A) trivially over U n A)) that factors into the restricted tensor product of local representations: Ind Sp n A) B n A) λs)) v Ind Sp n kv) B n k v ) λs) v ), where we write and λs) v = s 1 v λ 1,v s v λ,v sn v λ n,v, λ i = v λ i,v. Let f Ind K T n A) K)U n A) K) λ 0). Then we define the Eisenstein series [34], II.1.5) -) Ef λs), g) = f λs) γ g), g Sp n A). γ B n k)\sp n k) The constant term along B n of -) determines the poles and analytic continuation of the Eisenstein series [34], I.4.10): -3) E 0 f λs), g) = f λs) ug)du = w W f λs) w 1 ug). U n k)\u n A) U n A) wu n A)w 1 Next, the global intertwining operator -4) Mλs), w)fk) = f λs) w 1 uk), k K, U na) wu na)w 1 does not depend on the choice of the representative for w in Sp n k). Thus, the constant term given by -3) does not depend on the set of representatives W. For Res) large enough, the global intertwining operator factors into product of local intertwining operators where Mλs), w)f = v Aλs) v, w)f v, f = v f v Ind K T n A) K)U n A) K) λ 0).

7 Away from poles, Aλs) v, w) intertwines representations: Its normalizing factor is defined by We let rλs) v, w) = AUTOMORPHIC FORMS ON Sp n 7 Ind Sp n k v) B n k v ) λs) v ) Ind Sp n k v) B n k v ) wλs) v )). α Σ +,wα)<0 L1, λs) v α )ɛ0, λs) v α, ψ v ) L0, λs) v α. ) N λs) v, w) = rλs) v, w)aλs) v, w). The following theorem summarizes the basic properties of the normalization [17] iii) for v infinite is a well known Gindikin Karpelevič formula). Theorem -5. Under the above assumptions, we have the following: i) N λs) v, w 1 w ) = N w λs) v ), w 1 )N λs) v, w ). ii) N wλs) v ), w 1 )N λs) v, w) = N λs) v, w)n wλs) v ), w 1 ) = id. iii) The Hermitian contragredient N λs) v, w) of N λs) v, w) is N wλ conjs)) v ), w 1 ), where for s = s 1, s,..., s n ) C n we write conjs) = s 1, s,..., s n ). 3 iv) Assume that f λs)v Ind Sp n k v) B n k v ) λs) v ) is K v invariant, f λs)v 1) = 1. Then N λs) v, w)f λs)v = f wλs)v ). v) Assume that λs) v is in the closure of positive Weyl chamber with respect to Σ +. This means that for any α Σ +, the character λs) v α can be written as a product of a unitary character and a v, for some a 0.) Then, if w is the longest element of the Weyl group W, then N λs) v, w) is holomorphic and non zero. 3. Construction of Degenerate Eisenstein Series We will need the extension of results of Section on Eisenstein series. While the results of Section have little to do with with our particular choice of a symplectic group, the results of the present section depend on that. Motivated by the construction of Piatetski Shapiro and Rallis [35] on a particular case of a Siegel Eisenstein series we perform the following construction. We follow the notation introduced in [34]. Let P = MN be a maximal k parabolic subgroup of Sp n. Assume M GLm) Sp n. Let V ASp n k) \ Sp n A)) be an v< Sp n k v) g, K ) irreducible subspace of the space of automorphic forms. Let us call the corresponding representation Π. Assume that V is concentrated on B n. That is, there is a constant term along B n that does not vanish.) Let V 0 be the space of constant terms along B n of V. The map V V 0 defined by ϕ g ϕu g )du ) U n k)\u n A) is an intertwining operator. In particular, since V is irreducible and concentrated on B n, the map is an isomorphism. It is a subspace of the space of cuspidal) automorphic forms AU n A)T n k) \ Sp n A)) see [34], I..17) for a definition). In this paper w 1 stands for the representative of w 1 and w) 1 stands for the inverse of w in Sp n k). 3 We also have Aλs)v, w) = Awλ conjs)) v), w) 1 ).

8 8 GORAN MUIĆ On the space AU n A)T n k) \ Sp n A)), we have an action of T n A) by left translations: lt)f g ) = F t 1 g ) t T n A), g Sp n A)). The action of v< Sp n k v) g, K ) commutes with that of T n A). It was explained in the proof of [34], Lemma I.3.) that every F AU n A)T n k) \ Sp n A)) is T n A) finite. For t T n A), we let V0 t = lt)v 0. The representation of v< Sp n k v) g, K ) on V0 t is irreducible and isomorphic to V 0 and to V ). Let 0 F t T n A) V 0 t such that 0 < dim C {lt)f ; t T n A)} is minimal possible. We conclude that dimension is equal to one arguing as in [3], Lemma 6). Thus, there exists a character λ : T n A) C, necessarily trivial on T n k), such that Hence, we have the following: F t g ) = δ 1/ B n t )λ t )F g ), t T n A), g Sp n A). 3-1) F t u g ) = δ 1/ B n t )λ t )F g ), t T n A), u U n A), g Sp n A). The same identity holds for all functions in the v< Sp n k v) g, K ) subrepresentation V 0 t T n A) V 0 t generated by F. Clearly, V 0 contains an irreducible submodule4 isomorphic to V ). Therefore, we may assume that V 0 is itself irreducible. Then 3-1) implies the embedding 3-) Π Ind Sp n A) B n A) λ ). Remark 3-3. The type of embedding constructed we usually refer as the one obtained from the computation of the constant term along B n. We have shown that there exists at least one such embedding. Let µ : k \ A C be a unitary) grössencharacter. The representation µ1 GLm,A) is an automorphic representation of GLm, A) on the one dimensional space W AGLm, k) \ GLm, A)). The computation of the constant term W 0 of W along Borel subgroup Bm GL gives an embedding: ) µ1 GLm,A) Ind GL ma) s m 1)/ µ s+m 1)/ µ. Bm GL A) Let us fix a unitary) grössencharacter µ : k \ A C and an embedding 3-) that is obtained from the computation of the constant term V 0 along B n of V. Then we consider the 4 Indeed, we use [0], Chapter XVII, Section ). There the author considers modules over rings with identity which is not directly applicable to our situation. Nevertheless, the chain of implications SS1 = SS = SS3 is true for v< Sp n kv) g, K ) modules. Using this, we may replace a sum in U := t T n A) V 0 t with a direct sum, say U = t T V0 t, where T T n A) is a suitable subset, and, for an arbitrary submodule U U, we can find a submodule U U such that U = U U. Projecting to U V0 t, t T, we see that V 0 has at least one irreducible quotient. Let U be the kernel. Then U = U U. Hence V 0 = U V 0 U, and V 0 U V 0/U is irreducible.

9 AUTOMORPHIC FORMS ON Sp n 9 following global induced representation: 3-4) Ind Sp n A) P A) det s µ1 GLm,A) Π ) s C) normalized induction; K finite vectors; the representation det s µ1 GLm,A) Π is extended to a representation of P A) trivially over NA)). Then in the context of abstract representations, the induction in stages implies Ind Sp n A) P A) det s µ1 GLm,A) Π ) Ind Sp n A) B na) ) s m 1)/ µ s+m 1)/ µ λ 1 λ n. But since we consider the issues of convergence of Eisenstein series we need to be careful in choosing the realization of such representation. First of all, let us emphasize that Π is realized on the space V 0 AU n A)T n k) \ Sp n A)) defined above. Next, the induced representation in 3-4) is realized on the subspace A NA)Mk) \ Sp n A)) s,µ,v 0 of K finite) automorphic forms see Section ) 3-5) AU n A)T n k) \ Sp n A), s m 1)/ µ s+m 1)/ µ λ 1 λ n ) consisting of all functions such that, for k K, the function Ind Sp n A) B na) s m 1)/ µ s+m 1)/ µ λ 1 λ n f s : NA)Mk) \ Sp n A) C, x, y) GLm, A) Sp n A) det x s µdet x) 1 δ 1/ P x)f s x, y)k) is independent of x and of s, and, considered as a function of y, belongs to V 0. We remark that this construction using the realization of Π on V ASp n k) \ Sp n A)) would not work. Let f s A NA)Mk) \ Sp n A)) s,µ,v. Then we define a degenerate Eisenstein series as follows: 0 3-6) Ef s, g) = f s γg). γ P k)\sp n k) Arguing as in [34], II.1.5), we see that this series converges for Res) sufficiently large. Obviously it as an automorphic form in ASp n k) \ Sp n A)). Then, as in [34], II.1.7) we compute its constant term along the Borel B n. To accomplish that, we write α for the simple root in so that P is attached to \ {α}. Now, we have the following: Theorem 3-7. Assume that Π is concentrated on the Borel subgroup that is, in our fixed automorphic realization V some constant term along B n does not vanish). Let us fix the realization V 0 as above. Let µ : k \ A C be a unitary) grössencharacter. Put λs) = s m 1)/ µ s+m 1)/ µ λ 1 λ n. Then the Eisenstein series given by 3-6) is also concentrated on the Borel subgroup, and its constant term along B n is given by 3-8) E 0 f s, g) = Ef s, ug)du = Mλs), w)f s g). U nk)\u na) Proof. This is proved in [5], Section ). w W, w \{α})>0 )

10 10 GORAN MUIĆ We can argue as in [34], IV.4.3) to prove the meromorphic continuation of 3-6). Next, Theorem 3-7 and results recalled in the previous section along with [34], IV.1.11) imply that the constant term E 0 f s, ) admits a meromorphic continuation to whole C in the sense that there exists a non zero entire analytic) function s ds) C such that ds)e 0 f s, ) is entire. Hence, [34], I.4.10) implies that the Eisenstein series 3-6) admits an analytic continuation to whole C such that ds)ef s, ) is entire. The Eisenstein series 3-6) strongly depends on the choice of automorphic realization V 0 of Π in the space of cuspidal forms AU n A)T n k) \ Sp n A)). Later, we will choose carefully that realization in order to compute poles of Eisenstein series given by 3-6). 4. Weil group W k and Arthur parameters Let k a be an algebraic closure of the number field k. We describe the Weil group following [37]. The Weil group W k is a dense subgroup of Galk a /k) equipped with a relative topology. If k is a finite extension of k in k a, then W k is a subgroup of W k since we consider Galk a /k ) Galk a /k), and W k = Galk a /k ) W k by the way, this is an open subgroup). As a consequence W k /W k = Hom k k, k a ) = Galk a /k)/galk a /k ). In particular, if k k is finite Galois extension, then Next we have the following isomorphism: W k /W k = Galk /k). Wk ab A /k, where Wk ab = W k/wk c and W k c is the closure of the commutator subgroup of W k. We look the equivalence classes of Arthur parameters 5 ϕ : W k SL, C) ĜC) = SOV ). such that the following holds: 1) ϕ Wk is continuous, semisimple, and its image is bounded. ) ϕ SL,C) is a morphism of algebraic groups. 3) The image is discrete, that is, the image is not contained in a proper Levi subgroup of SOV ). 4) ϕ Wk factors through Wk ab. Under the requirements 1) and ), we can think of ϕ as a representation ϕ : W k SL, C) GLV ): ϕ ρ,a) Jordϕ) ρ V a. Here Jordϕ) is a finite multiset consisting of the pairs ρ, a) where a Z >0, ρ is an irreducible representation of W k, and V a is the unique irreducible algebraic representation of SL, C) of dimension a. Under 1) and ), it is not in general a set since ϕ may decompose with multiplicity. But the requirement 3) forces that it is a set; we call the set of Jordan blocks of ϕ. Indeed, all Levi subgroups of SOV ) can be described as follows. Let V V be a non zero subspace such that the restriction of, ) is trivial an isotropic subspace). Let V be another isotropic subspace of 5 We do not invoke the conjectural Langlands dual group Lk. Rather we work with its quotient W k L k /L 0 k, where L 0 k is the component of identity. In general, the Arthur parameter is of the form ϕ : L k SL, C) ĜC).

11 AUTOMORPHIC FORMS ON Sp n 11 the same dimension such that the restriction of, ) to V + V is non degenerate. Then the sum V + V is direct. The corresponding Levi factor M consists of all g SOV ) such that gv V, gv V. Our claim is clear. Next, under requirements 1), ), 3), and 4), we may write ϕ µ,a) Jordϕ) µ V a, where Jordϕ) is a set consisting of a finite number of pairs µ, a) where µ is a grössencharacter of k and a Z >0. Next, V a poses a non degenerate SL, C) invariant form. This form is skew symmetric resp., symmetric) if and only if a is even resp., odd). This form is unique up to a non zero scalar multiple; let us fix one of them, a. Then the requirement 3) forces that the restriction of, ) to µ V a is non degenerate symmetric form. This forces µ to be quadratic and a must be odd. This discussion proves the following: Theorem 4-1. The set of equivalence classes of Arthur parameters ϕ satisfying the requirements 1), ), 3), and 4) is in the one to one correspondence with the collection of finite sets Jord consisting of the pairs µ, a) where i) µ : k \ A C is a quadratic grössencharacter of k ii) a Z >0, a is odd. iii) µ,a) Jord a = n + 1 iv) µ,a) Jord µ = 1. Since the determinant gives the trivial character.) The correspondence is given by Jordϕ) = Jord. More precisely, ϕ µ,a) Jord µ V a. We let Jordϕ) µ = {a; µ, a) Jordϕ)}. We would like to construct square integrable automorphic residual representations supported on Borel) attached to the parameter ϕ. We need a local version of above construction for ϕ. For almost all places v, the local components µ v of µ such that Jordϕ) µ is not empty are unramified characters. More precisely, there exists a finite set of places S containing all archimedean places) such that, for v S, we have the following: Jordϕ) µ = µ v {1 k v, sgn v }, where sgn v is the unique unramified character of order two of k v : sgn v ϖ v ) = 1 where ϖ v is a generator of the maximal ideal in the ring of integers of k v. Let v S. Then, for the local Arthur parameter ϕ v : W kv SL, C) SOV ), defined as follows: ϕ v = µ,a) Jord µ v V a, we let Jordϕ v ) = {µ v, a); µ, a) Jord}. In general, it is a multiset. Clearly, we have the following: ϕ v = µv,a) Jordϕ v ) µ v V a. They are related to the unramified representations Irr unr Sp n k v )) as follows see [4]). We remind the reader that an irreducible representation of Sp n k v ) is unramified if its space of invariants under Sp n O v ) is non trivial. Unramified representations are classified using the Satake classification see [10]) as follows:

12 1 GORAN MUIĆ Theorem 4-. i) Assume that χ 1,..., χ n is a sequence of unramified characters of k v. Then the induced representation Ind Sp n kv) B n k v ) χ 1 χ n ) contains the unique unramified irreducible subquotient, say σχ 1,..., χ n ). ii) Assume that χ 1,..., χ n and χ 1,..., χ n are two sequences of unramified characters of k v. Then σχ 1,..., χ n ) σχ 1,..., χ n) if and only if there is w W such that χ 1 χ n = wχ 1 χ n ). In another words, if and only if there is a permutation α of and a sequence ɛ 1,..., ɛ n {±1} such that χ i = χɛ i αi), i = 1,..., n. iii) Assume that σ IrrSp n k v )) is an unramified representation. Then there exists a sequence χ 1,..., χ n of unramified characters of k v such that σ σχ 1,..., χ n ). The local functorial lift σ GLn+1) of σ = σχ 1,..., χ n ) Irr unr Sp n k v )) to GLn + 1, k v ) is always defined and it is an unramified representation described as the unique irreducible unramified subquotient of Ind GL n+1k v ) Bn+1 GL kv) χ 1 χ n 1 k v χ 1 n χ 1 1 ) The defined mapping σ σ GLn+1) from Irr unr Sp n k v )) into Irr unr GLn + 1, k v )) is injective. Moreover, the image consists of all self dual unramified representations with trivial central characters as it is easy to verify directly). Example 4-3. It is well known that 1 Spn k v ) = σ,,..., n ). Then its lift 1 GLn+1) Sp n k v) unique unramified irreducible subquotient of Ind GLn+1,kv) Bn+1 GL k v) which is the trivial representation 1 GLn+1,kv ). n 1 k v 1 n ) is the We say that σ Irr unr Sp n k v )) is negative if all exponents in the normalized Jacquet modules of σ with respect to B n k v ) are of the form: 6 x α α, x α 0. α There are various ways to classify those representations. One way is to note that their duals under Iwahori Matsumoto involution are tempered representations and then use [1]. The other more elementary way is described in [4] using the work of Bernstein and Zelevinsky [8], [9]), and the work of Waldspurger [3] describing the contragredient of an irreducible representation of Sp n k v ) which is also based on the mentioned work of Bernstein and Zelevinsky). It follows from [6], [7]) using local methods or from [5] using global methods that every negative representation is unitary. This is a deep result which we do not use in this paper. We use the following algebraic result [4]; the formulation is taken from [6]) Theorem 4-4. There is a one to one correspondence between the set of equivalence classes of negative unramified representations σ Irr unr Sp n k v )) and the set of equivalence classes of Arthur parameters ψ : W kv SL, C) SOV ) satisfying: ψ Wkv is continuous, semisimple, and its image is bounded. ψ SL,C) is a morphism of algebraic groups. ψ Wk factors through W ab v k v k v k v /O v 6 We remind the reader that an automorphic form F ASpn k) \ Sp n A)) concentrated on B n is square integrable if and only if all of exponents of F are of the form α xαα, xα > 0.

13 AUTOMORPHIC FORMS ON Sp n 13 given by the order of characters is irrelevant; the parabolic subgroup P depends on the taken order) σ GLn+1) Ind GLn+1,F ) ) P χ,l) Jordψ) χ det 1 GLl,kv ), where ψ = χ,l) Jordψ) χ V l. If ψ is an Arthur parameter given by Theorem 4-4, then we write σψ) Irr unr Sp n k v )) for the corresponding negative representation. Example 4-5. The trivial representation 1 Spn k v ) is negative. The corresponding Arthur parameter is 1 kv V n+1 : σ1 kv V n+1 ) = 1 Spn k v ). Let ϕ a,µ) Jord µ V a be an Arthur parameter given by Theorem 4-1. We attach square integrable automorphic representations to ϕ in a usual way []. We match automorphic representations and Arthur parameters via Hecke operators: Definition 4-6. Let X A Sp n k) \ Sp n A)), be an irreducible subrepresentation of the space of square integrable automorphic forms on Sp n A). We say that X is attached to ϕ if for almost all places X v is a negative unramified representation attached to ϕ v : X v σϕ v ). For example, the trivial representation 1 Spn A) is attached to the parameter: ϕ 1 V n+1. The general case is more interesting. The automorphic representation X attached ϕ to might be cuspidal. The classical examples follow from the works of Gelbart and Jacquet [11], 3.7) and Labesse and Langlands []. 5. Spherical Unipotent Arthur Parameters and the Corresponding Representations In this section we prove main results of this paper. The construction of automorphic representations obtained in Theorems 5- and 5-1 is an automorphic version of the local construction of discrete series obtained by Tadić [40] in early 90 s see also later work [31]) or of strongly negative unramified representations [4]. The technical point of dealing with degenerate Eisenstein series instead of considering Eisenstein series attached to the principal series is that the recursive construction described in Theorems 5- and 5-1 gives simple and easy control on the constant term. For example, in the first step of the construction we need just to control two terms to decide non vanishing while in the second step the constant term after clearing the poles has only one term and then the square integrability is obvious form the local theory see Theorem 4-4). Let G = Sp n. We say that an Arthur parameter ϕ : W k SL, C) ĜC) see Theorem 4-1) is spherical unipotent if it is trivial on W k. Thus, it is of the form ϕ : SL, C) ĜC). Using the notation introduced in Section 4, we can find a unique increasing sequence of positive integers m 1,..., m k ), k m i + 1) = n + 1, i=1

14 14 GORAN MUIĆ such that 5-1) ϕ = k i=1v mi +1. We remark that k must be odd. Let λϕ) : T n k) \ T n A) C be defined by: m k mk+1 m k 1 ) m 3 m3+1 m ) m 1 m1+1 1 ). We remark that for k = 1, the trivial representation 1 Spm1 A) has a unique automorphic realization jv m1 +1) : 1 Spm1 A) A Sp m1 k) \ Sp m1 A)). The usual embedding 1 Spm1 A) Ind Sp m 1 A) B m1 A) m 1 1 ) is obtained computing the constant term along B m1 on the space of constant functions which is the same as ImagejV m1 +1)). The first main result of this paper is the following theorem: Theorem 5-. Let k > 0 be an odd integer. Under above assumptions, the unique irreducible K spherical subquotient σϕ) of the globally induced representation Ind Sp n A) B n A) λϕ)) is its subrepresentation, and there is a non zero embedding jϕ) : σϕ) A Sp n k) \ Sp n A)) constructed recursively as follows. Let k 3. Put ϕ = k i=1 V m i +1 and n + 1 = k i=1 m i + 1. Consider the global induced representation 5-3) Ind Sp n +4m k 1 + A) P A) det s 1 GLmk 1+1,A) Imagejϕ )) ), where P is the standard parabolic subgroup of Sp n +4m k 1 + with Levi factor GLm k 1 +1) Sp n. At s = 0 this representation is unitary and therefore semisimple of infinite length).) Then the map obtained from a degenerate Eisenstein series 7 5-4) f s Ef s, ) s=0 is an intertwining operator 1GLmk 1+1,A) Imagejϕ )) ) ASp n +4m k 1+k) \ Sp n +4m k 1+A)), Ind Sp n +4m k 1 + A) P A) which is non trivial on the unique irreducible K spherical subrepresentation of 5-3) for s = 0; let us write X for the image of the K spherical subrepresentation. Taking the constant term of X along Borel B n +m k 1 +1 see Remark 3-3) we obtain the following embedding: 5-5) X Ind Sp n +4m k 1 + A) B n +mk 1 +1 A) m k 1 m k 1+1 m k 1 λϕ ) ) 7 We remark that 1GLmk 1 +1,A) Imagejϕ )) is square integrable modulo centre on GLm k 1 + 1, A) Sp n A). Then the degenerate Eisenstein series defined by Ef s, g) = γ P k)\sp n +4mk 1 + k) fsγg), for Res) large enough, is defined by Langlands. It is holmorphic at s = 0 by a result of Langlands [1], Main theorem).

15 AUTOMORPHIC FORMS ON Sp n 15 which we use to construct degenerate Eisenstein series f s Ef s, g) = γ P k)\sp n k) f sγg) attached to the global induced representation 5-6) Ind Sp n A) P A) det s 1 GLmk m k 1,A) X ), where P is the standard parabolic subgroup of Sp n with Levi factor GLm k m k 1 ) Sp n +4m k 1 +. Then the map K det m k 1 +m k +1 1 GLmk m k 1,A) X) ASp n k) \ Sp n A)) given by Ind Sp n A) P A) 5-7) f m k 1 +m k +1 s m ) k 1 + m k + 1 Ef s, ) s= mk 1+mk+1 is well defined and non trivial. Let E be a v< Sp nk v ) g, K ) subrepresentation of ASp n k)\ Sp n A)) generated by the image of the space of K invariants. Then E is irreducible, contained in the space of square integrable automorphic forms A Sp n k) \ Sp n A)), and it induces the required embedding jϕ) : σϕ) E A Sp n k) \ Sp n A)). Finally, the embedding σϕ) Ind Sp n A) B na) λϕ)) is obtained computing the constant term of E along B n. Proof. We let ξ s = s m k 1 s m k 1+1 s+m k 1 ) m k m k +1 m k 3 ) m 3 m 3+1 m ) m 1 m ). Then the constant term along the Borel of the Eisenstein series 5-4) can be computed as follows see Theorem 3-7): 5-8) E 0 f s, g) = Ef s, ug)du U n +mk 1 +1 k)\u n +m k 1 +1 A) = Mξ s, w)f s g) w W, w \{α})>0 = f s g) + Mξ s, w 0 )f s g)) + w W, w w 0,1, w \{α})>0 Mξ s, w)f s g) where α is the root defining P Sp n +4m k 1 +, and w 0 W is defined as follows: w 0 = w l, w 1 l, \{α}, where w l, is the longest Weyl group element for Σ + of Sp n +4m k 1 +) and w l, \{α} is the longest Weyl group element for Σ + \{α}. The main point is the following observation see [5], Claim 6): 5-9) at s = 0, wξ s ) w 0 ξ s ) = ξ s for all w {1, w 0 }, w \ {α}) > 0 which follows by a direct computation using the explicit description of all w W, w \ {α}) > 0 see Lemma 1-). Using factorizable f s = v f ξs,v, such that for all v the function f ξs,v is K v invariant see Theorem -5 iii)), the constant term 5-8) can be rewritten as follows: 5-10) E 0 f s, g) = f ξs g) + rξ s, w 0 ) 1 f ) w0 ξ s)g) + rξ s, w) 1 f wξs)g). w W, w w 0,1, w \{α})>0

16 16 GORAN MUIĆ Since, the Eisenstein series is holomorphic at s = 0, this computation of the constant term tell us that the map 5-4) is not trivial on the K spherical subrepresentation if we show 5-11) lim s 0 rξ s, w 0 ) = 1. This is accomplished by a direct computation. First, we have ξ s ϕ i = s m k 1+i 1, 1 i m k ξ s ϕ i + ϕ j ) = s m k 1+i+j, 1 i < j m k For other two cases, we have the following we warn the reader that j has now a different meaning): ξ s ϕ i ± ϕ) = s m k 1+i 1± m e+1 +j), 1 i m k 1 + 1, 0 j m e+1 + m e, where e = 0,... k 3)/, and where we let m 0 = 1. Now, we have the following formula: L1, ξ s α )ɛ0, ξ s α ) rξ s, w 0 ) = L0, ξ s α ) α Σ +,w 0 α)<0 Then, we compute the normalization factor we write Ls) = Ls, 1) and ɛs) = ɛs, 1)): Ls m k 1 + i)ɛs m k 1 + i 1) Ls m k 1 + i 1) 1 i m k i<j m k 1 +1 k 3)/ e=0 k 3)/ e=0 Ls m k 1 + i + j 1)ɛs m k 1 + i + j ) Ls m k 1 + i + j ) Ls m k 1 + i + m e+1 + j))ɛs m k 1 + i 1 + m e+1 + j)) Ls m 1 i m k 1 +1, 0 j m e+1 +m k 1 + i 1 + m e+1 + j)) e Ls m k 1 + i m e+1 + j))ɛs m k 1 + i 1 m e+1 + j)) Ls m 1 i m k 1 +1, 0 j m e+1 +m k 1 + i 1 m e+1 + j)) e Using the functional equations Ls) = ɛs)l1 s) changing the indices in the denumerators of the first two products and rearranging the terms) we transform the expression into the following form: 1 i m k 1 +1 k 3)/ e=0 k 3)/ e=0 Ls m k 1 + i) L s m k 1 + i) 1 i m k 1 +1, 0 j m e+1 +m e 1 i m k 1 +1, 0 j m e+1 +m e 1 i<j m k 1 +1 Ls m k 1 + i + j 1) L s m k 1 + i + j 1) Ls m k 1 + i + m e+1 + j)) L s m k 1 + i + m e+1 + j)) Ls m k 1 + i m e+1 + j)) L s m k 1 + i m e+1 + j)). In order to compute the limit s 0, we remark the following elementary fact: { Ls + t) lim s 0 L s + t) = 1; t {0, 1} 1 t {0, 1}.

17 AUTOMORPHIC FORMS ON Sp n 17 which follows from the fact that Ls) has a simple pole at s = 0 and s = 1 and it is holomorphic otherwise. Since k 3 by our assumption, m k 1. This forces that the limit of the first two products is equal to 1. The limit of the expression in the second and the third line is 1) k 1)/. Therefore, we conclude that lim rξ s, w 0 ) = 1. s 0 This clearly completes the first stage of the construction. Next, we let λ s = s m k m k 1 1)/ s m k m k 1 1)/+1 s+m k m k 1 1)/ Clearly m k 1 m k 1+1 m k 1 ) m k m k +1 m k 3 ) m 3 m 3+1 m ) m 1 m ). λ mk 1 +m k +1)/ = λϕ). Now, we use the induced representation 5-6). We compute the constant term of the Eisenstein series f s Ef s, g) = γ P k)\sp n k) f sγg) using Theorem 3-7: 5-1) E 0 f s, g) = = U nk)\u na) w W, w \{α})>0 = Mλ s, w 0 )f s g) + Ef s, ug)du Mλ s, w)f s g) w W, w w 0, w \{α})>0 where α is the root defining P Sp n, and w 0 W is defined as follows: w 0 = w l, w 1 l, \{α}, Mλ s, w)f s g) where w l, is the longest Weyl group element for Σ + of Sp n )and w l, \{α} is the longest Weyl group element for Σ + \{α}. Let f s = v f λs,v be the factorizable function in the space of the induced representation 5-6) such that f λs,v is K v spherical for all v. Then we can write the constant term as follows see Theorem -5 iii)): E 0 f s, ) = Mλ s, w 0 )f s + Mλ s, w)f s 5-13) w W, w w 0, w \{α})>0 = rλ s, w 0 ) 1 v f w0 λ s,v )) + w W, w w 0, w \{α})>0 rλ s, w) 1 v f wλs,v )). We show later that the order of pole of rλ s, w 0 ) 1 at s = m k 1 + m k + 1)/ is two while the order of rλ s, w) 1, w w 0, at s = m k 1 + m k + 1)/ is at most one. Granting this, we put ηw 0 ) = lim s m ) k 1 + m k + 1 rλ s, w 0 ) 1 0. s m k 1 +m k +1)/

18 18 GORAN MUIĆ Hence 5-13) implies 5-14) lim s m ) k 1 + m k + 1 E 0 f s, g) = ηw 0 ) v f s m k 1 +m k +1)/ w0λmk 1+mk+1)/,v)). ) Hence, we see that s m k 1+m k +1 E0 f s, g), g Sp n A), is holomorphic and non zero at s = m k 1 + m k + 1)/. Therefore, the normalized Eisenstein series s m ) k 1 + m k + 1 Ef s, ), is holomorphic and non zero at s = m k 1 + m k + 1)/. Moreover, this computation implies that the space E of automorphic forms generated as a v< Sp nk v ) g, K ) representation by the normalized Eisenstein series is isomorphic using the constant term map along B n ) to the representation v Y v, where Y v is a subrepresentation of 5-15) Ind Sp n k v) B n k v ) w 0 λ mk 1 +m k +1)/,v)) generated by f w0 λ mk 1 +m k +1)/)),v. We remark that w 0 λ mk 1 +m k +1)/) = λ mk 1 +m k +1)/ = λϕ). The results of [4] shows that all spaces Y v v finite) are irreducible and in fact strongly negative). Finally, 5-14) implies that for a non zero F E, the only possible exponent in its constant term along B n comes from the character λϕ) = w 0 λ mk 1 +m k +1)/). Since, at an arbitrary finite place v, the representation Y v is strongly negative and a subrepresentation of the induced representation in 5-15), we see that the exponent of F is of the form α x αα, x α > 0. Therefore, the automorphic form F is square integrable by the usual square integrability criterion see [34], Lemma I.4.11). This implies that E is an unitary representation. Hence every Y v is unitary. But since Y v is generated by a K v invariant vector f w0 λ mk 1 +m k +1)/)),v, it is irreducible this is applied to Archimedean v only). Hence E is irreducible. It remains to prove that the order of pole of rλ s, w 0 ) 1 at s = m k 1 + m k + 1)/ is two while the order of rλ s, w) 1, w w 0, at s = m k 1 + m k + 1)/ is at most one. We recall L1, λ s α )ɛ0, λ s α ) rλ s, w) = L0, λ s α. ) α Σ +,wα)<0 First, we consider the case w = w 0. We list all α such that α Σ +, w 0 α) < 0 as follows 8 : {ϕ i ; 1 i m k m k 1 } {ϕ i + ϕ j ; 1 i < j m k m k 1 } {ϕ i ϕ j ; 1 i m k m k 1, m k m k j n} {ϕ i + ϕ j ; 1 i m k m k 1, m k m k j n} We have λ s ϕ i = s m k m k 1 1)/+i 1, 1 i m k m k 1. λ s ϕ i + ϕ j ) = s m k+m k 1 +i+j 1, 1 i < j m k m k 1. 8 Those are just all α Σ + \ Σ + \{α mk m }. In fact, using the notation Lemma 1-,. we have k 1 the following: w 0φ 1) = φ 1 m k m k 1, w 0φ ) = φ 1 m k m k 1 1,..., w0ϕm k m k 1 ) = φ 1 1, w0ϕm k m k 1 +1) = ϕ mk m k 1 +1,..., w 0ϕ n) = ϕ n.

19 AUTOMORPHIC FORMS ON Sp n 19 For other two cases, we have the following we warn the reader that j has now a different meaning): λ s ϕ i ± ϕ) = s m k m k 1 1)/+i 1± m e+1 +j), 1 i m k m k 1, 0 j m e+1 + m e, where e = 1,... k 3)/, and where we let m 0 = 1, and λ s ϕ i ± ϕ) = s m k m k 1 1)/+i 1± m k 1 +j), 1 i m k m k 1, 0 j m k 1. Now, we compute rλ s, w 0 ) 1 we write Ls) = Ls, 1) and ɛs) = ɛs, 1)) up to exponential functions: Ls m k m k 1 1)/ + i 1) Ls m 1 i m k m k m k 1 1)/ + i) k 1 k 3)/ e=0 k 3)/ e=0 Ls m k + m k 1 + i + j 1) Ls m 1 i<j m k m k + m k 1 + i + j) k 1 Ls m k m k 1 1)/ + i 1 + m e+1 + j)) Ls m 1 i m k m k 1, 0 j m e+1 +m k m k 1 1)/ + i + m e+1 + j)) e Ls m k m k 1 1)/ + i 1 m e+1 + j)) Ls m 1 i m k m k 1, 0 j m e+1 +m k m k 1 1)/ + i m e+1 + j)) e Ls m k m k 1 1)/ + i 1 m k 1 + j)) Ls m 1 i m k m k 1, 0 j m k m k 1 1)/ + i m k 1 + j)) k 1 Ls m k m k 1 1)/ + i 1 + m k 1 + j)) Ls m 1 i m k m k 1, 0 j m k m k 1 1)/ + i + m k 1 + j)) k 1 Now, we use the fact that Ls) is holomorphic for s 0, 1, that it has simple poles at 0 and 1 and it never vanish for s > 1. We see that the pole at s = m k 1 + m k + 1)/ is double. We consider L1, λ s α )ɛ0, λ s α ) rλ s, w) = L0, λ s α, ) α Σ +,wα)<0 where w \ {α mk m k 1 }) > 0. We use Lemma 1- and Corollary 1-3 to determine w. So, let 0 i 1 m k m k 1 such that w = pɛ W i1. This means wφ i ) = φ pi), 1 i i 1 or m k m k i n, and wφ i ) = φ 1 pi), i 1 i m k m k 1. We list all α such that α Σ + and wα) < 0 see Corollary 1-3): 1) {ϕ i ; i i m k m k 1 } ) {ϕ i + ϕ j ; i i < j m k m k 1 } 3) {ϕ i + ϕ j ; 1 i i 1, i j m k m k 1, pj) < pi)} 4) {ϕ i ϕ j ; i i m k m k 1, m k m k j n} 5) {ϕ i ϕ j ; 1 i i 1, m k m k j n, pj) < pi)} 6) {ϕ i + ϕ j ; i i m k m k 1, m k m k j n, pj) > pi)}. For α in the cases 1), ), and 3) the quotient L0, λ s α )/L1, λ s α ) is holomorphic and non zero at s = m k 1 + m k + 1)/. The product α 4) L0, λ s α )/L1, λ s α ) is holomorphic and non zero unless i 1 = 0 in which case it has a simple pole at s = m k 1 + m k + 1)/. The product α 5) L0, λ s α )/L1, λ s α ) is holomorphic and non zero unless i 1 1 in which case it has at most a simple pole at s = m k 1 + m k + 1)/..

20 0 GORAN MUIĆ The product α 6) L0, λ s α )/L1, λ s α ) is holomorphic and non zero unless i 1 = 0, i = 1, j = m k m k 1 + 1, pm k m k 1 + 1) > p1) in which case it has at most a simple pole at s = m k 1 + m k + 1)/. This case see Lemma 1-) we have the following: pm k m k 1 ) < pm k m k 1 1) < < p1) < pm k m k 1 + 1) < < pn). This implies w = w 0. There are two obvious problems to consider:. describe the image of the map given by 5-7) not just its restriction to K invariant vectors). generalize the construction from the previous theorem starting from some other irreducible representation of σ with an automorphic embedding j : σ A Sp n k) \ Sp n A)) such that the constant term map along B n induces the embedding σ Ind Sp n A) B λϕ n A) )). See Remark 3-3.) We remark that we can write the decomposition of σ into the restricted tensor product of local representations σ v σ v. Then the corresponding global induced representation is written as a restricted tensor product of local induced representations as follows compare to 5-3)): 5-16) Ind Sp n +4m k 1 + A) P A) det s 1 GLmk 1+1,A) σ ) v Ind Sp n +4m k 1 + kv) ) P k v ) det s v 1 GLmk 1+1,k v) σ v. Let w 0 has the same meaning as in the related part of the proof of Theorem 5- see 5-8)). We consider the unnormalized) standard intertwining operator Ind Sp n +4m k 1 + k v) ) P k v) det s v 1 GLmk 1+1,k v ) σ v Ind Sp n +4m k 1 + kv) ) P k v ) det s v 1 GLmk 1 +1,k v) σ v defined for Res) large enough by the integral As, σ v, w 0 )f v g v ) = Nk v) f v w 1 0 n vg v )dn v. Still using the notation from the first part of the proof of Theorem 5-, we have the following commutative diagram: 0 Ind Sp n +4m k 1 + k v) ) P k v ) det s i v 1 GLmk 1+1,k v ) σ s Sp v Ind n k v ) B n k v ) ξ s,v ) As,σ v, w 0 ) Aξ s,v, w 0 ) 0 Ind Sp n +4m k 1 + k v) ) P k v ) det s i s v 1 GLmk 1 +1,k v ) σ v Ind Sp n k v) B n k v ) w 0 ξ s,v )). We normalize As, σ v, w 0 ) as follows: N s, σ v, w 0 ) = rξ s,v, w 0 )As, σ v, w 0 ). This might not be a standard normalization provided by the procedure explained in [3] or in [36]. Nevertheless, it satisfies some standard properties as we explain below.

21 AUTOMORPHIC FORMS ON Sp n 1 Above diagram implies the following commutative diagram: Ind Sp n +4m k 1 + k v) ) P k v ) det s i v 1 GLmk 1+1,k v) σ s Sp v Ind n k v ) B n k v ) ξ s,v ) 5-17) N s,σ v, w 0 ) N ξ s,v, w 0 ) Ind Sp n +4m k 1 + k v) ) P k v ) det s i s v 1 GLmk 1 +1,k v) σ v Ind Sp n k v) B n k v ) w 0 ξ s,v )). The assumption j : σ A Sp n k) \ Sp n A)) implies that σ v is unitary for all v. Hence, by general theory, the Hermitian contragredient As, σ v, w 0 ) of As, σ v, w 0 ) is A s, σ v, w 0 ) 1 ). Combining this with Theorem -5 iii), we obtain Next, since w 0 N s, σ v, w 0 ) = N s, σ v, w 1 0 ). = 1, the representative w 1 0 of w 1 0 is equal to w 0. Hence, we have the following: 5-18) N s, σ v, w 0 ) = N s, σ v, w 0 ). Moreover, Theorem -5 ii) implies the following: N w 0 ξ s,v ), w 0 )N ξ s,v, w 0 ) = id. Hence, the commutative diagram 5-17) implies he following: 5-19) N s, σ v, w 0 )N s, σ v, w 0 ) = id. The expressions 5-18) and 5-19) imply that N s, σ v, w 0 ) is holomorphic and unitary operator at s = 0. 9 We define the representations X v ± σ) as follows: X v ± σ) = {f Ind Sp n +4m k 1 + kv) ) P k v ) 1GLmk 1+1,k v) σ v ; N s, σv, w 0 )f = ±f}. Then Ind Sp n +4m k 1 + k v) ) P k 1GLmk 1+1,k v) v ) σ v = X + v σ) Xv σ). Moreover, if σ v is K v spherical, then X v + σ) 0 since, by Theorem -5 iv) and the definition of N s, σ v, w 0 ), it contains Ind Sp n +4m k 1 + kv) ) Kv P k v ) 1GLmk 1+1,k v) σ v 0. The representation X v ± σ) are semisimple. If v is Archimedean, then Xv might be trivial. But if v is non Archimedean and σ v is K v spherical, then representation X v ± σ) are irreducible [4], [6]). Now, we can decompose the global induced representation as follows: 5-0) Ind Sp n +4m k 1 + A) P A) 1GLmk 1+1,A) σ ) S v S X v + σ) v S Xv σ) ), Sp 9 n +4mk 1 + k v) ) Indeed, we realize the family of the induced representations Ind det s v 1 GLmk 1+1,kv ) σ v s C) on K v using the usual restriction of the functions. Let f 1, f ) = K v f 1 k v ), f k v )) σv dk v, where, ) σv is a scalar product on σ v. Then N s, σ v, w 0 )f 1, N s, σ v, w 0 )f ) = N s, σ v, w 1 0 ) N s, σ v, w 0 )f 1, f ) = f 1, f ) shows the claim. P k v )