REMARKS ON METAPLECTIC TENSOR PRODUCTS FOR COVERS OF GL(r)

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1 REMARKS ON METAPLECTC TENSOR PRODUCTS FOR COVERS OF GL(r) SHUCHRO TAKEDA Abstract. n our previous paper, we constructed a metaplectic tensor product of automorphic representations of covers of GL(r). To be more precise, let M = GL r1 GL rk GL r be a Levi subgroup of GL r, where r = r r k, and M its metaplectic preimage in the n-fold metaplectic cover GL r of GL r. For automorphic representations π 1,..., π k of GL r1 (A),..., GL rk (A), we constructed (under certain technical assumptions, which is always satisfied when n = 2) an automorphic representation π of M(A) which can be considered as the tensor product of the representations π 1,..., π k. n the present paper, we will significantly simplify and generalize our previous construction without the technical assumptions mentioned above. 1. ntroduction Let F be a number field and A be the ring of adeles. For a partition r = r r k of r, one has the Levi subgroup M(A) := GL r1 (A) GL rk (A) GL r (A) of the (r 1,..., r k )-parabolic. Let π 1,..., π k be automorphic representations of GL r1 (A),..., GL rk (A), respectively. t is a trivial construction to obtain the automorphic representation π 1 π k of the Levi M(A) simply by taking the usual tensor product. Though highly trivial, this construction is of great importance in the theory of automorphic forms, especially when one would like to formulate Eisenstein series. Now if one considers the metaplectic n-fold cover GL r (A) constructed by Kazhdan and Patterson in [KP], the analogous construction turns out to be far from trivial. Namely for the metaplectic preimage M(A) of M(A) in GL r (A) and automorphic representations π 1,..., π k of the metaplectic n-fold covers GL r1 (A),..., GL rk (A), one cannot construct a representation of M(A) simply by taking the tensor product π 1 π k. This is simply because M(A) is not the direct product of GL r1 (A),..., GL rk (A), namely M(A) GL r1 (A) GL rk (A), and even worse there is no natural map between them. For the local case, P. Mezo ([Me]), whose work, we believe, is based on the work by Kable ([K]), carried out a construction of an irreducible admissible representation of the Levi M starting with representations π 1,..., π k of GL r1,..., GL rk, which can be called the metaplectic tensor product of π 1,..., π k, and characterized it uniquely up to certain character twists. n our previous paper [T3], we carried out an analogous construction for the global case and defined the global metaplectic tensor product. Further, we showed that the global metaplectic tensor product satisfies various expected properties. We, however, needed to impose certain technical assumptions for the group M, most notably Hypothesis ( ) in [T3, p.202]. n this paper, we will modify the construction of [T3] so that the metaplectic tensor product can be defined without those technical assumptions and show that the new version also satisfies all the expected properties. ndeed, it seems 1

2 2 SHUCHRO TAKEDA our construction in [T3] was unnecessarily complicated, and here we will give a simpler construction. To be more precise, the main theorem of the paper is the following. Main Theorem. Let M = GL r1 GL rk be a Levi subgroup of GL r, and let π 1,..., π k be automorphic subrepresentations of GL r1 (A),..., GL rk (A). Then there exists an automorphic subrepresentation π of M(A) such that π = vπ v, where each π v is the local metaplectic tensor product of Mezo. Moreover, if π 1,..., π k are cuspidal (resp. square-integrable modulo center), then π is cuspidal (resp. square-integrable modulo center). Further the metaplectic tensor product satisfies various expected properties. n the above theorem, v indicates the metaplectic restricted tensor product, the meaning of which will be explained later in the paper. Also we require π i be an automorphic subrepresentation, so that it is realized in a subspace of automorphic forms and hence each element in π i is indeed an automorphic form. (Note that in general an automorphic representation is a subquotient.) As we will see, strictly speaking the metaplectic tensor product of π 1,..., π k might not be unique even up to equivalence but is dependent on a character ω on the center Z GLr of GL r. Hence we write π ω := (π 1 π k ) ω for the metaplectic tensor product to emphasize the dependence on ω. Notations Throughout the paper, F is a number field and A is the ring of adeles of F. For each place v, F v is the corresponding local field and O Fv is the ring of integers of F v. For each algebraic group G over a global F, and g G(A), by g v we mean the v-th component of g, and so g v G(F v ). For any group G, we denote its center by Z G. For a positive integer r, we denote by r the r r identity matrix. Throughout we fix an integer n 2, and we let µ n be the group of n-th roots of unity in the algebraic closure of the prime field. We always assume that µ n F. We fix an ordered partition r r k = r of r, and we let M = GL r1 GL rk GL r and assume it is embedded diagonally as usual. f π is a representation of a group G, we denote the space of π by V π, though we often confuse π with V π when there is no danger of confusion. We say π is unitary if V π is equipped with a Hermitian structure invariant under the action of G, but we do not necessarily assume that the space V π is complete. Now assume that the space V π is a space of functions or maps on the group G and π is the representation of G on V π defined by right translation. (This is the case, for example, if π is an automorphic subrepresentation.) Let H G be a subgroup. We define π H to be the representation of H realized in the space V π H := {f H : f V π } of restrictions of f V π to H, on which H acts by right translation. Namely π H is the representation obtained by restricting the functions in V π. Occasionally, we confuse π H with its space when there is no danger of confusion. Note that there is an H-intertwining surjection π H π H, where π H is the (usual) restriction of π to H. Also for any subset X G and any f V π, we denote by π(x)f the vector space generated by π(x)f for all x X. f X is a subgroup, this gives rise to a representation of X, which is a subrepresentation of π X.

3 REMARKS ON METAPLECTC TENSOR PRODUCTS 3 Acknowledgements This paper grew out of a question asked by Solomon Friedberg on the construction of the metaplectic tensor product of [T3], when both he and the author were attending the conference in honor of Professor Roger Howe on the occasion of his 70th birthday. The author would like to thank him for asking the question and the organizers of the conference (James Cogdell, Ju-Lee Kim, Jian-Shu Li, David Manderscheid, Gregory Margulis, Cheng-Bo Zhu and Gregg Zuckerman) for creating a good opportunity for the interaction. Also a part of the paper was completed when the author was visiting the National University of Singapore in the spring of 2016, and he would greatly appreciate their hospitality. Finally, he would like to thank the anonymous referee for his/her careful reading and letting him know about the independence of the multiplicities in Proposition 3.5. The author is partially supported by Simons Foundations Collaboration Grants # The metaplectic cover GL r of GL r 2.1. The groups. n this subsection, we set up our notations for the metaplectic n-fold cover GL r of GL r for both local and global cases. Most of the time, we work both locally and globally at the same time. Hence we let { F v in the local case; R = A in the global case. By the metaplectic n-fold cover GL r (R) of GL r (R) with a fixed parameter c {0,..., n 1}, we mean the central extension of GL r (R) by µ n as constructed by Kazhdan and Patterson in [KP]. More concretely, as a set, whereas the multiplication is defined by GL r (R) = GL r (R) µ n = {(g, ξ) : g GL r (R), ξ µ n } (g, ξ) (g, ξ ) = (gg, τ r (g, g )ξξ ), where τ r is a certain 2-cocycle. (See [T3, Sect. 2 and 3] more about various issues on cocycles.) f P is a parabolic subgroup of GL r whose Levi part is M = GL r1 GL rk, we often write for the metaplectic preimage of M(R). Next let and M(R) = GL r1 (R) GL rk (R) GL r (R) = {g GL r (R) : det g R n }, GL r (R) its metaplectic preimage. Also we define and often denote its preimage by M (R) = {(g 1,..., g k ) M(R) : det g i R n } M (R) = GL r 1 (R) GL r k (R). The groups M (R) and M (R) are normal subgroups of M(R) and M(R), respectively. ndeed, if we define (2.1) Det M : M(R) = GL r1 (R) GL rk (R) R R }{{} k times to be the map given by determinant on each factor GL ri, then M (R) is the kernel of the composition of Det M with projection to R n \R R n \R. Hence for the local case (R = F v ), the groups M (R) and M (R) are of finite index.

4 4 SHUCHRO TAKEDA Let us mention the following important fact. Let Z GLr (R) GL r (R) be the center of GL r (R). Then its metaplectic preimage Z GLr(R) is not the center of GL r (R) in general. (t might not be even commutative for n > 2.) The center, which we denote by Z GLr(R), is (2.2) Z GLr(R) = {(a r, ξ) : a r 1+2rc R n, ξ µ n } = {(a r, ξ) : a R n d, ξ µn }, where d = gcd(r 1 + 2rc, n). (The second equality is proven in [CO, Lemma 1].) Also the center of Z M(R) M(R) is described as 1 r1 = { Z M(R) a... a k rk : a r 1+2cr i R n and a 1 a r mod R n }. See Proposition 3.10 of [T3]. Let us mention that the above descriptions of Z GLr(R) and Z M(R) give (2.3) Z GLr(R) M (R) = Z M(R) M (R). Let π be a representation of a subgroup H GL r (R) containing µ n. We say π is genuine if each element (1, ξ) H acts as multiplication by ξ, where we view ξ as an element of C in the natural way. n [T3], we considered how the metaplectic tensor product behaves when it is restricted to a smaller Levi. n this paper, we will also consider the same question. For this purpose, let us here set up our notations. Let = {i 1,..., i l } {1,..., k} be a non-empty subset with i 1 < < i l. We set (2.4) M (R) = GL ri1 (R) GL ril (R) which is embedded into M(R) in the obvious way and hence viewed as a subgroup of M(R). Let M (R) be the metaplectic preimage of M (R), so we have Also set M (R) M(R). M (R) := M (R) M (R) The global metaplectic cover GL r (A). n this subsection we only consider the global case, i.e. R = A. First let us mention that both the F rational points GL r (F ) and the unipotent radical N B (A) of the Borel subgroup B split in GL r (A) via a certain partial map s : GL r (A) GL r (A). Via this splitting we identify GL r (F ) with a subgroup of GL r (A). Let us mention, however, that this partial map is not given by the map g (g, 1) for our choice of cocycle τ r. But rather the map g (g, 1) splits some compact subgroup. For our purpose here, we have only to mention the following. Let S be a finite set of places containing all archimedean places and those v with v n. Then we have a group homomorphism (2.5) GL r (O Fv ) GL r (A) under the map g (g, 1). v / S

5 REMARKS ON METAPLECTC TENSOR PRODUCTS 5 We can also describe GL r (A) as a quotient of a restricted direct product of the groups GL r (F v ) as follows. Consider the restricted direct product GL v r (F v ) with respect to the groups K v for all v with v n and v. f we denote each element in this restricted direct product by Π v(g v, ξ v ) so that g v K v and ξ v = 1 for almost all v, we have the surjection (2.6) ρ : v GLr (F v ) GL r (A), Π v(g v, ξ v ) (Π vg v, Π v ξ v ), where the product Π v ξ v is literary the product inside µ n. This is indeed a group homomorphism and GLr (F v )/ ker ρ = GL r (A), v where ker ρ consists of the elements of the form (1, ξ) with ξ v µ n and Π v ξ v = 1. We set GL r (F v ) := GLr (F v )/ ker ρ v v and call it the metaplectic restricted direct product. Let us note that each GL r (F v ) naturally embeds into v GLr (F v ). By composing it with ρ, we have the natural inclusion (2.7) GLr (F v ) GL r (A), which allows us to view GL r (F v ) as a subgroup of GL r (A). Let us mention that all the discussions above on GL r (A) can be generalized to M(A), though there is a subtle issue on cocycles for M(A), which is discussed in detail in [T3, Sect. 3]. This issue will not play any role in this paper. We have the notion of automorphic representations as well as automorphic forms on GL r (A) or M(A). n this paper, by an automorphic form, we mean a smooth automorphic form instead of a K-finite one, namely an automorphic form is K f -finite, Z-finite and of uniformly moderate growth. (See [C, p.17].) Hence if π is an automorphic representation of GLr (A) (or M(A)), the full group GL r (A) (or M(A)) acts on π. An automorphic form f on GL r (A) (or M(A)) is said to be genuine if f(g, ξ) = ξf(g, 1) for all (g, ξ) GL r (A) (or M(A)). n particular every automorphic form in the space of a genuine automorphic representation is genuine. We denote the space of genuine automorphic forms on GL r (A) (resp. M(A)) by A( GL r ) (resp. A( M)). Suppose we are given a collection of irreducible admissible representations π v of GL r (F v ) such that π v is K v -spherical for almost all v. Then we can form an irreducible admissible representation of GL v r (F v ) by taking a restricted tensor product vπ v as usual. Suppose further that ker ρ acts trivially on vπ v, which is always the case if each π v is genuine. Then it descends to an irreducible admissible representation of GL r (A), which we denote by vπ v, and call it the metaplectic restricted tensor product. Let us emphasize that the space for vπ v is the same as that for vπ v. Conversely, if π is an irreducible genuine admissible representation of GL r (A), it is written as vπ v where π v is an irreducible genuine admissible representation of GL r (F v ), and for almost all v, π v is K v -spherical. (To see it, view π as a representation of the restricted direct product GL v r (F v ) by pulling it back by ρ in (2.6) and apply the usual tensor product theorem for the restricted direct product. This gives the restricted tensor product vπ v, where each π v is genuine, and hence it descends to vπ v.)

6 6 SHUCHRO TAKEDA n what follows, we will list some important properties of various groups we consider. Lemma 2.8. Let S be a finite set of places containing all the archimedean ones, and set Then the set is finite. Proof. This is [T3, Lemma 14]. This implies Lemma 2.9. The group is compact. O S := v / S O F v. F A n \A /O S F A n \A Proof. Let S be any finite set of places containing all the archimedean ones. By the above lemma, we know F A n \A is a finite union of sets of the form F A n ao S for a A. But this set, which is the image of the compact set ao S under the quotient map A F A n \A, is compact in the topology of F A n \A. Hence the lemma follows. This in turn implies Lemma The group M(F ) M (A) is a closed normal subgroup of M (A) whose quotient M(F ) M (A)\ M(A) is a compact abelian group. ndeed, we have an isomorphism of topological groups. M(F ) M (A)\ M(A) = F A n \A F A n \A }{{} k times Proof. That it is closed is [T3, Proposition A.4]. To show it is normal, one can check that the group M(F ) M (A) is indeed the kernel of the composite M(A) A A F A n \A F A n \A where the first map is the determinant map Det M as in (2.1). By the previous lemma, the last group on the right hand side is compact. We should also mention the following. Lemma Let S be a finite set of places containing all the archimedean ones and those v with v n. Define (2.12) K S := v / S M(O Fv ), which can be viewed as a subgroup of M(A) as in (2.5). Then the set is finite. M(F ) M (A)\ M(A)/K S Proof. This is immediate from Lemma 2.8, because M(F ) M (A)\ M(A)/K S is a product of k copies of F A n \A /O S

7 REMARKS ON METAPLECTC TENSOR PRODUCTS 7 Next let us mention the following lemma on general topology. Lemma Let A be a Hausdorff compact abelian group, and m 1,..., m k be positive integers. Define H := {(a m1,..., a m k ) : a A} = A m1 A m k A A. }{{} k times Then H is a closed subgroup of A A. Proof. Note that for each i {1,..., k}, the m i -th power map A A mi A is continuous, and hence the image A mi of the compact A is compact. Recall that in a Hausdorff topological group, every compact subgroup is closed by, say, [D-E, Lemma 1.1.4]. So each A mi is closed. Hence H is closed. This implies the following important fact. Proposition We have M(F )Z M(A) M (A) = M(F )Z GLr(A) M (A) and this group is a closed (hence locally compact) subgroup of M(A). Proof. The equality is immediate from (2.3). To prove this group is closed, it suffices to show that the image of in the quotient M(F ) M Z M(A) (A)\ M(A) is closed. But one can see that the image of Z GLr(A) under the isomorphism is the subgroup of the form M(F ) M (A)\ M(A) = F A n \A F A n \A {(a nr 1 nr k d,..., a d ) : a F A n \A }, where d = gcd(r 1 + 2rc, n). By Lemma 2.9, we know that F A n \A is compact, and hence by the previous lemma, this is closed. 3. The metaplectic tensor product n this section, after reviewing the local metaplectic tensor product of Mezo [Me] with the modification made by the author in [T3], we will construct the global metaplectic tensor product Mezo s local metaplectic tensor product. n this subsection, all the groups are over the local field F v, and accordingly we simply write GL r, M, etc, instead of GL r (F v ), M(F v ), etc. Let π 1,, π k be irreducible admissible genuine representations of GLr1,..., GL rk, respectively. For each i = 1,..., k, let σ i := π i GL r i. Note that σ i, as a representation of GL r i, is completely reducible, and the multiplicities of all the irreducible constituents are all equal. Namely, we have (3.1) σ i = m i τi, where τ i is an irreducible representation of GL r i such that τ i τ j for i j, and m i is a positive multiplicity which is independent of τ i. For the non-archimedean case, this is precisely [GK, Lemma 2.1], and the archimedean case can be proven in the same way as the non-archimedean case because

8 8 SHUCHRO TAKEDA the index of GL r i in GL ri is at most 2. n [Me], Mezo first picks up an irreducible constituent τ i of σ i and consider the (usual) tensor product V τ1 V τk, which, of course, gives a representation of the direct product GL r 1 GL r k. The genuineness of the representations τ 1,..., τ k implies that this tensor product representation descends to a representation of the group M := GL r 1 GL r k, i.e. the representation factors through the natural surjection GL r 1 We denote this representation of M by GL r k τ := τ 1 τ k. GL r 1 GL r k. Let us emphasize that the space V τ of τ is the usual tensor product V τ1 V τk n this paper, however, we will take a different approach. nstead of picking up a τ i, we will consider all of σ i at the same time and define the representation σ := σ 1 σ k of M in the same way as τ. Let us again emphasize that the space V σ of σ is the usual tensor product V σ1 V σk. Further note that because of (3.1), we have (3.2) σ = m τ where m = m 1 m k and the sum is over all possible equivalence classes of representations of the form τ = τ 1 τ k. Then we define Π = Π(π 1,..., π k ) := nd M M σ. By (3.2), we have Π = m τ where the sum is over all the equivalence classes of irreducible subrepresentations τ of σ. Note that since σ is completely reducible and the index of M in M is finite, one can see that the representation Π is completely reducible. Certainly, it is highly unlikely that Π is irreducible. Rather it contains all the metaplectic tensor products constructed by Mezo. To see it, we need to take the action of the center Z GLr into account. For this purpose, let us first define τ nd M M τ, (3.3) Ω = Ω(π 1,..., π k ) := {ω : ω is a character on Z GLr which appears in σ}, where we say ω appears in σ if there is an irreducible constituent τ σ that agrees with ω on the overlap, namely ω Z GLr M = τ Z GLr M. Now Mezo s construction can be summarized as follows. Let τ σ be an irreducible representation, so τ = τ 1 τ k for some τ i s, and let ω Ω be such that it agrees with τ on the overlap. Then we can extend τ to the representation τ ω := ωτ of Z GLr M by letting Z GLr act by ω. Then if we induce it to the group M, it is isotypic (though possibly reducible), and we denote this isomorphism class by (π 1 π k ) ω

9 REMARKS ON METAPLECTC TENSOR PRODUCTS 9 and call it the metaplectic tensor product of π 1,..., π k with respect to ω. With this notation, we have nd M Z GLr M τ ω = m (π 1 π k ) ω for some finite multiplicity m, which will be seen to be independent of τ and ω but only dependent on the representations π 1, π k. Clearly we have the inclusions (3.4) nd M τ M Z GLr ω nd M M τ nd M M σ = Π(π 1,..., π k ) = m τ nd M M τ, because τ σ. Further we have Proposition 3.5. For each fixed τ, let Then we have nd M M τ = Ω(τ) := {ω Ω : ω Z GLr M = τ Z GLr M }. ω Ω(τ) nd M τ M Z GLr ω = m ω Ω(τ) (π 1 π k ) ω, where m is the positive multiplicity of (π 1 π k ) ω in nd M τ M Z GLr ω, which is independent of τ and ω but is only dependent on π 1,..., π k. Proof. The proof is an elementary exercise in representation theory. But we will give a brief explanation for each equality. First note that by inducing in stages we have nd M M τ = nd M Z GLr Then similarly to [Me, Lemma 4.1], one can see nd Z GLr M τ = M nd Z GLr M τ. M M ω Ω(τ) τ ω, because the quotient M \Z GLr M = Z GLr M \Z GLr precise, for a fixed ω Ω(τ) we can write has the same size as Ω(τ). To be more Ω(τ) = {ωχ : χ is in the dual of Z GLr M \Z GLr }. To show the next equality, the only non-trivial part is to show that the multiplicity m is independent of τ and ω. The independence from ω follows from the fact that the restrictions (nd M τ M ω ) M Z GLr and (nd M τ M Z GLr ω ) M have the same number of constituents and the latter is independent of ω, and further the restrictions (π 1 π k ) ω and (π M Z GLr 1 π k ) ω M have the same number of constituents and again the latter is independent of ω. To show it is independent of τ, let us note that m is indeed equal to [ H : Z GLr M ], where H is a maximal subgroup of M such that τ ω can be extended to H so that Mackey s irreducible criterion is satisfied as constructed in [Me, p.89-90]. (This can be proven in the same way as in [T3, Proposition 4.7].) From the construction of H, one can see that H is independent of the choice of τ 1,..., τ k but only dependent on π 1,..., π k. (Also see [Ca, Section 3.4] for this issue.) This proposition immediately implies the main theorem for local metaplectic tensor product. Z GLr

10 10 SHUCHRO TAKEDA Theorem 3.6. Keeping the above notation, we have Π = Π(π 1,..., π k ) = ω Ω m(ω)(π 1 π k ) ω, where Ω is as in (3.3) and m(ω) is the positive multiplicity of (π 1 π k ) ω. Proof. By the previous proposition, we have which implies the theorem. Π = mm τ ω Ω(τ) (π 1 π k ) ω, Remark 3.7. n the above theorem, one may wonder if m(ω) = mm. This is certainly the case if the Ω(τ) s are all distinct for distinct τ. But it may be the case that Ω(τ) Ω(τ ) even when τ τ. Also it should be mentioned that if Ω(τ) Ω(τ ), then necessarily Ω(τ) = Ω(τ ) With this theorem, one can tell that the presentation Π contains all the metaplectic tensor products, and one can call each irreducible constituent of Π a metaplectic tensor product. Next we consider the behavior of metaplectic tensor products upon restriction to a smaller Levi. So let, M, etc, be as in the end of Section 2.1. Proposition 3.8. Let π Π(π 1,..., π k ) be a metaplectic tensor product. Then the restriction π M is completely reducible (with most likely infinite multiplicity). Further each constituent of is of τ M the form (π i1... π il ) ω for some ω Ω(π i1,..., π il ). Proof. Note that π nd M M τ for some irreducible representation τ of M. Hence it suffices to show the restriction (nd M M is completely irreducible. But since the group M \ M is finite, τ) M one has the following Mackey type theorem M (nd M M = τ) M M g M g 1(τ g ), g M \ M/ M nd where, as usual, τ g is the representation of τ twisted by g viewed as a representation of M g M g 1 by restriction. But note that M \ M/ M = M \M/M and each element in this double coset is represented by an element in M which has the identity on all the components for the GL ri factors for i. Hence M g M g 1 = M, and τ g = τ as a representation of M. Hence we have M (nd M M = ). τ) M (τ M g M \ M/ M nd M But note that the space V τ of τ is of the form V τ1 V τk for some irreducible representations τ 1,..., τ k of GL r 1,..., GL r k, which are irreducible constituents of the restrictions π 1 GL,..., π k r GL. 1 r k Hence when it is restricted to M, it is completely reducible. (Yet note that the multiplicity is infinite unless all the σ i for i / are one dimensional.) ndeed τ M is isotypic with all the irreducible constituents equivalent to τ i1 τ il. Hence first of all, is completely reducible. Second of π M all, each irreducible constituent in is contained in the induced representation π M nd M τ M i1 τ il.

11 REMARKS ON METAPLECTC TENSOR PRODUCTS 11 This, together with Theorem 3.6 applied to the group M, implies that each constituent of π M the form (π i1... π il ) ω. is of 3.2. The global metaplectic tensor product. By essentially following the local metaplectic tensor product, the global metaplectic tensor product was constructed in [T3] with some technical assumptions, most notably Hypothesis ( ) in [T3, p.202]. Here we will simplify the construction of [T3] and remove the technical assumptions imposed there. Throughout this subsection, let π 1,..., π k be automorphic subrepresentations of the groups GL r1 (A),..., GL rk (A) realized in the spaces of automorphic forms. Namely we assume V πi A( GL ri ). Also let H i := GL ri (F ) GL r i (A). Note that by Lemma 2.10, H i is a closed normal subgroup of GL ri (A) whose quotient is a compact abelian group. First let σ i := π i Hi, where recall the notation from the notation section. Each element ϕ in the space of V σi is a restriction to H i of an automorphic form on GL ri (A), and hence we may view it as a function on H i with the property that ϕ(γg) = ϕ(g) for all γ GL ri (F ) and g GL r i (A). Namely the representation σ i is a representation of the group H i realized in a space of automorphic forms on H i. We should mention Proposition 3.9. Let π be an irreducible smooth representation of GLr (A). Then the restriction π GLr(F ) GL is completely reducible, and hence π is a subrepresentation of r (A) GL r(f ) GL r (A) π GLr(F ) GL. r (A) Proof. n this proof, let us write H = GL r (F ) GL r (A). We will prove the proposition by modifying the proof of [GK, Lemma 2.1]. First we will show that the restriction π H has an irreducible subrepresentation. For this, consider the contragredient ˆπ of π. Since π is irreducible, so is ˆπ. Let ϕ ˆπ be nonzero. Then ˆπ is generated by ϕ as a representation of GL r (A). One of the key points is that the restriction ˆπ H is also finitely generated as a representation of H. To see it, let S be a sufficiently large finite set of places such that ϕ is fixed by K S := v / S GL r(o Fv ). We know that the set H\ GL(A)/K S is finite by Lemma Let {g 1,..., g l } be a complete set of representatives of this finite set. Then one can see that the vectors ˆπ(g i )ϕ generate ˆπ H, i.e. ˆπ H is finitely generated. Hence ˆπ H has an irreducible quotient. (t is an elementary exercise of Zorn s lemma to show that every finitely generated representation of any group has an irreducible quotient.) Let W be the kernel of the surjection from Vˆπ to this irreducible quotient. Let Ann(W ) := {f V π : f, ϕ = 0 for all ϕ W } be the annihilator of W, which gives a representation of H. Then one can see Ann(W ) is an irreducible subrepresentation of π H as follows. Let X Ann(W ) be any nonzero subrepresentation of Ann(W ). Consider the annihilator Ann(X) of X, which is a subrepresentation of Vˆπ. Note that W Ann(X) Vˆπ. But since the pairing V π Vˆπ C is non-degenerate, we have Ann(X) Vˆπ. Hence W = Ann(X) by the irreducibility of Vˆπ /W. Hence the pairing V π Vˆπ C gives rise to a non-degenerate pairing X Vˆπ /W C,

12 12 SHUCHRO TAKEDA which is H invariant. This implies that X is canonically isomorphic to the representation realized in the space { f, : f X}, where f, : Vˆπ /W C is the functional given by ϕ f, ϕ. But this space is independent of X. Hence X = Ann(W ), which shows Ann(W ) is irreducible. Now let V be an irreducible subrepresentation of π H and let f V be a fixed non-zero vector. As above there exists a sufficiently large S, possibly (most likely) different from the above one, such that the group K S fixes f. Again let {g 1,, g l } be a complete set of representatives of H\ GL(A)/K S, which is most likely different from the one above. Then one can see V π = l π(hg i K S )f = i=1 l π(hg i )f. Note that each space π(hg i )f gives rise to a representation of H, which is equivalent to the g i twist of π(h)f. But π(h)f = V because V is a space of an irreducible representation of H, and hence each π(hg i )f is irreducible. Let {g i1,..., g in } be the smallest subset of {g 1,..., g l } such that V π = i=1 N π(hg ij )f. Then this sum is actually a direct sum because for each k {1,..., N}, if the intersection j=1 π(hg ik )f j k π(hg ij )f, which is a representation of H, is nonzero, then it is actually equal to π(hg ik )f by irreducibility, which contradicts to the minimality of the set {g i1,..., g in }. This competes the proof. Next note that each element in H i is of the form (h i, ξ i ) for h i GL ri (F ) GL ri (A) and ξ i µ n. As in [T3, p.215], we have the natural surjection (3.10) H 1 H k M(F ) M (A) given by the map ((h 1, ξ 1 ),..., (h k, ξ k )) (h 1 h k, ξ 1 ξ k ). Then consider the space V σ1 V σk of functions on the direct product H 1 H k, which gives rise to a representation of the direct product H 1 H k. But each element in V σ, which is a function on this direct product, descends to a function on M(F ) M (A), which is automorphic in the sense that it is left invariant on M(F ). (t should be mentioned that this is not as immediate as it looks, especially due to some issues on cocycles. See [T3, Proposition 5.2] for details.) f ϕ i V σi for i = 1,..., k, we denote this function by ϕ 1 ϕ k, and denote the space generated by those functions by V σ. We call each function in V σ an automorphic form on M(F ) M (A). The group M(F ) M (A) acts on V σ by right translation, and denote this representation by σ. We define σ 1 σ k := σ. Proposition With the above notation, σ is completely reducible. Further, if all of π 1,..., π k are unitary, so is σ.

13 REMARKS ON METAPLECTC TENSOR PRODUCTS 13 Proof. Each σ i is completely reducible by Proposition 3.9. Hence one can see σ is completely reducible. f π 1,..., π k are unitary and each σ i is a subrepresentation of π i Hi, the unitary structure on π i descends to σ i. Hence one can define a unitary structure on σ 1 σ k, which descends to σ. Now just as we did for the local case, consider the smooth induced representation (3.12) Π = Π(π 1,..., π k ) := nd M(A) M(F ) M (A) σ. Then we have the obvious map (3.13) nd M(A) M(F ) M (A) σ A( M), where f is defined by f(m) = f(m)(1), f f for m M(A) and A( M) is the space of automorphic forms on M(A). Further this map is one-to-one, and one can identify Π as a subspace of A( M), namely we have Next, let us mention the following. Π A( M). Proposition f all of π 1,..., π k are cuspidal, then Π, viewed as a subspace of A( M), is in the space of cusp forms. f all of π 1,..., π k are realized in the spaces of square integrable automorphic forms, then Π is also in the space of square integrable automorphic forms. Also if all of π 1,..., π k are unitary, so is Π. Proof. The proofs for the first two parts (cuspidality and square-integrability) are simple modifications of the proofs of Theorem 5.12 and 5.13 of [T3]. But let us repeat the key points. For this purpose, note that for f Π and m M(A), we have f(m) V σ, and hence f(m) is a sum of functions of the form ϕ 1 ϕ k, where each ϕ i is a restriction of a function in V πi to H i. Assume that π 1,..., π k are cuspidal, and N := N 1 N k is a unipotent radical of a parabolic of M, where each N i is a unipotent radical of a parabolic of GL ri. We view N(A) as a subgroup of M(A) via the splitting N(A) M(A). Noting that N(A) M(F ) M (A), we have f(nm) dn = f(nm)(1) dn N(F )\N(A) = N(F )\N(A) N(F )\N(A) f(m) dn = N(F )\N(A)(ϕ 1 ϕ k ) dn = N(F )\N(A) ϕ 1 ϕ k dn = ϕ 1 (n 1 ) dn 1 ϕ k (n k ) dn k N 1(F )\N 1(A) N k (F )\N k (A) = 0, where the last equality follows from the cuspidality of ϕ i s. Next let us show the square integrability. By [T3, Lemma 5.17], it suffices to show that f(m) 2 dm < Z M(F )\M(A) M(A)

14 14 SHUCHRO TAKEDA for each f Π, where But Z M(A) = f(m) 2 dm = a n 1 r1... a n k r k : a i A. f(m m) 2 dm dm Z M(F )\M(A) M(A) = = Z M(A) M(F )M (A)\M(A) Z M(A) M(F )M (A)\M(A) Z M(A) M (F )\M (A) Z M(A) M (F )\M (A) f(m m)(1) 2 dm dm f(m)(m ) 2 dm dm. Z M(A) M(F )M (A)\M(A) Z M(A) M (F )\M (A) Note that the outer integral is over a compact set, and hence we only need to show the convergence of the inner integral. But this follows from the square integrability of the function f(m) V σ as an automorphic form on M(F ) M (A). Finally, assume π 1,..., π k are unitary. By Proposition 3.11, we know σ is unitary. But by Lemma 2.11, we know the induction defining Π is a compact induction, which makes Π unitary. Remark n the above proof for square integrability, we implicitly used the fact that the group M(F )Z M(A) M (A) is closed, which can be shown by the same argument as Proposition This justifies the existence of the quotient measure for Z M(A) M(F )M (A)\M(A). The author has to admit that this subtle point was not addressed in the proof of [T3, Theorem 5.13]. Also there the author, for some reason, did not realize that the group Z M(A) M(F )M (A)\M(A) is compact when writing [T3], which made the proof in [T3] unnecessarily long. As in the local case, we would like to have that the representation Π = Π(π 1,..., π k ) is completely reducible. And this is immediate if π 1,..., π k are cuspidal because then Π is in the space of cusp forms. But the author does not know if this is true in general. For our purposes, however, the following is enough. Proposition Let τ σ be an irreducible subspace. Then the space nd M(A) M(F ) M (A) τ has an irreducible subrepresentation. Hence Π has an irreducible subrepresentation. Proof. n this proof, let us write H = M(F ) M (A). First note that since σ is completely reducible by Proposition 3.9, an irreducible τ σ always exists. Let ϕ V τ be nonzero. Since π 1,..., π k are smooth, there exists a finite set of places such that ϕ is fixed by the group H K S, where K S = v / S M(O F v ). Let g 1,..., g l be a complete set of representatives of the double cosets H\ M(A)/ K S, which, we know, is finite by Lemma 2.11, where we assume g 1 = 1. Hence each vector in nd M(A) H τ fixed by K S is completely determined by its values at g 1,..., g l. With this said, let us define an element

15 REMARKS ON METAPLECTC TENSOR PRODUCTS 15 f : M(A) V τ in nd M(A) H τ by setting f(hg i k) = { τ(h)ϕ, if i = 1; 0, otherwise, where h H and k K S. This is well-defined because ϕ is fixed by H K S, and has the property that f(hm) = τ(h)f(m) for all h H and m M(A). To show f is indeed in nd M(A) H τ, we need to show that f is smooth. This can be checked at each v by viewing M(F v ) as a subgroup of M(A) as in (2.7) (or its M(A) analogue). Namely for each v / S, clearly f is fixed by M(O Fv ) and hence f is smooth at v. f v is archimedean, then since the Lie algebra of M(F v ) is the same as that of M (F v ), the smoothness follows from that of ϕ. Finally let v S be a non-archimedean place in S. Then by the smoothness of ϕ, there is an open compact subgroup U of M (F v ). Since M (F v ) is an open subgroup of M(F v ), U is also an open compact subgroup of M(F v ). Then one can see that the intersection of all g 1 i Ug i, which is also an open compact subgroup of M(F v ), fixes f. Hence f is smooth and indeed in nd M(A) H τ. Now consider the space Π( M(A))f generated by f. Then we can write Π( M(A))f = l Π(Hg i )f, where each space Π(Hg i )f is H invariant and hence a subrepresentation of Π( M(A))f H. Now to prove the proposition, it suffices to show that Π(Hg i )f is irreducible, because, then, Π( M(A))f H has only finite length, and hence a fortiori Π( M(A))f is of finite length, which implies that Π( M(A))f has an irreducible subrepresentation. Moreover, one can see that, as abstract representations, each Π(Hg i )f is equivalent to the g i twist of Π(H)f. Hence it suffices to show that Π(H)f is irreducible. To show each Π(H)f is irreducible, consider the evaluation map at 1, namely i=1 Π(H)f τ, f f (1) for f Π(H)f, which is H-intertwining. Since f(1) = ϕ 0, this map is nonzero. But note that each nonzero f is supported on HK S, which implies f (1) 0 for all nonzero f Π(H)f. Therefore Π(H)f = τ, which shows Π(H)f is irreducible. Now as in the local case, Π(π 1,... π k ) essentially contains all the possible metaplectic tensor products. To see it, we need to carry out a construction analogous to the local metaplectic tensor product of Mezo by taking the central character into account. Namely, we now need to consider an irreducible subrepresentation of σ and extend it to a representation of Z GLr(A) M(F ) M (A) by letting the center Z GLr (A) act as a character. First note that since σ is completely reducible by Proposition 3.11, it has an irreducible subrepresentation Fix such τ from now on. We need τ σ. Lemma For each irreducible τ σ, the abelian group acts as a character, which we denote by ω τ. Z GLr(A) M(F ) M (A)

16 16 SHUCHRO TAKEDA Proof. By Proposition 3.16, there exists an irreducible subrepresentation π of nd M(A) τ. Let M(F ) M (A) ω be the central character of π. Now by Frobenius reciprocity we have an M(F ) M (A)-intertwining map π τ, which shows that the group Z GLr(A) M(F ) M (A) acts via the character ω on τ. Remark Of course, if τ is unitary, which is the case if π 1,, π k are, then τ actually has a central character because M(F ) M (A) is locally compact. But the author does not know if τ admits a central character in general. By the automorphy of each element in V τ, we can see that the character ω τ in the above lemma is automorphic in the sense that ω τ (γz) = ω τ (z) for all z Z GLr(A) M(F ) M (A) and γ M(F ) (Z GLr(A) M(F ) M (A)). Then we can find a Hecke character ω on Z GLr(A) by extending ω τ, namely ω is a character on Z GLr(A) such that ω(z) = ω τ (z) for all z Z GLr(A) M(F ) M (A). Such ω always exists because both Z GLr(A) and Z GLr(A) M(F ) M (A) are locally compact abelian groups. Also note that any such ω is indeed a Hecke character in the sense that ω(γz) = ω(z) for all z Z GLr(A) and γ GL r(f ) Z GLr(A), simply because by GL r (F ) Z GLr(A) Z GLr(A) M(F ) M (A). For each f V τ, which is a function on M(F ) M (A), we can extend it to a function f ω (zm) = ω(z)f(m) f ω : Z GLr(A) M(F ) M (A) C for all z Z GLr (A) and m M(F ) M (A). This is well-defined because of our choice of ω, and is considered as an automorphic form on the group Z GLr(A) M(F ) M (A). We define V τω := {f ω : f V τ }. The group Z GLr(A) M(F ) M (A) irreducibly acts on this space, giving rise to an automorphic representation τ ω of Z GLr(A) M(F ) M (A). As abstract representations, we have (3.19) τ ω = ωτ where by ωτ we mean the representation of the group Z GLr(A) M(F ) M (A) extended from τ by letting Z GL(A) act via the character ω. As we did before, let us consider the smooth induced representation Note that we have the obvious inclusion Π(τ ω ) := nd M(A) τ ω. Z GLr (A) M(F ) M (A) nd M(A) Z GLr (A) M(F ) M (A) τ ω nd M(A) M(F ) M (A) σ, which allows us to view Π(τ ω ) as a subrepresentation of Π realized in the space of automorphic forms on M(A), namely Π(τ ω ) Π A( M).

17 REMARKS ON METAPLECTC TENSOR PRODUCTS 17 Then we have Proposition The representation Π(τ ω ) has an irreducible subrepresentation. Proof. This can be proven identically to Proposition Finally, we can define our metaplectic tensor product as follows. Definition Keeping the above notations, let π ω Π(τ ω ) be an irreducible subrepresentation. Then we write π ω = (π 1 π k ) ω and call it a metaplectic tensor product of π 1,..., π k with respect to the character ω. The definition of metaplectic tensor product along with Proposition 3.14 immediately implies the following. Proposition f all of π 1,, π k are cuspidal (resp. square integrable modulo center, resp. unitary), then so is (π 1 π k ) ω. This π ω is precisely the metaplectic tensor product we want. Namely we have Theorem The representation π ω constructed above has the desired local-global compatibility. Namely if we write π ω = v π ω,v, then for each v we have π ω,v = (π 1,v π k,v ) ωv. Thus π ω is unique up to equivalence, and depends only on π 1,... π k and ω. Proof. Note that the uniqueness assertion follows from the corresponding local statement that the local metaplectic tensor product only depends on π 1,v,... π k,v and ω v. Hence we have only to show the local-global compatibility. First, note that, since π ω nd M(A) Z GLr (A) M(F ) M (A) τ ω, we have the natural surjection π ω Z GLr (A) M(F ) M (A) τ ω. Recall that as abstract representations, we have τ ω = ωτ, where τ is an irreducible representation of M(F ) M (A). So by restricting further down to Z GLr(A) M (A), we have π ω Z GLr (A) M (A) ωτ Z GLr (A) M (A). Now by Lemma A.1 in Appendix, π ω Z GLr (A) M (A) is completely reducible. Hence ωτ Z GLr (A) M (A) is completely reducible. Let ωπ τ ω Z GLr (A) M (A) be an irreducible subrepresentation, where π is an irreducible representation of M (A). By complete reducibility, this is also a quotient, and hence we have a surjection π ω Z GLr (A) M (A) ωπ. Recall that τ is realized as a space of automorphic forms on M(F ) M (A) and is written as V τ = V τ1 V τk where each V τi is a space of restrictions of automorphic forms in the space V πi. By the automorphy, one can see that τ ω M Z GLr (A) (A) = τ ω M Z GLr (A) (A).

18 18 SHUCHRO TAKEDA Hence we have where V π i V τi Therefore we can write V π = V π 1 for each i, and indeed we have π i τ i GL r i V π k = τ (A) i π GL (A) i GL ωπ = v ω v (π 1,v π k,v ), r i r i (A). where ω v (π 1,v π k,v ) is the irreducible representation of Z M GLr(F v) (F v ) constructed from π 1,v,..., π k,v as is done for the local metaplectic tensor product. Then if we write π ω = vπ ω,v where π ω,v is an irreducible representation of M(F v ), we have the surjection ( vπ ω,v ) Z GLr (A) M (A) v ω v (π 1,v π k,v ). Hence by Lemma 5.5 of [T3], we conclude that at each place v, the representation ω v (π 1,v π is a quotient of π ω,v M Z GLr (Fv ) (F v). By Frobenius reciprocity, we have k,v ) π ω,v nd M(F v) Z GLr (Fv ) M (F v) ω v(π 1,v π k,v ). Thus by definition of local metaplectic tensor product, we have π ω,v = (π 1,v π k,v ) ωv. Hence we have the desired local-global compatibility. Remark With the theorem, we can say that the notation (π 1 π k ) ω is unambiguous in the sense that it only depends on π 1,..., π k and ω as long as we consider the metaplectic tensor product as an equivalence class of representations, which we usually do. This theorem immediately implies the following. Corollary For fixed ω, all the irreducible subrepresentations of nd M(A) τ Z GLr (A) M(F ) M ω are (A) equivalent. Next we will show that Π(π 1,..., π k ) contains all the possible metaplectic tensor products of π 1,..., π k. For this purpose, let us define Ω = Ω(π 1,... π k ) := {ω : ω is a Hecke character on Z GLr(A) which appears in σ}, where we say ω appears in σ if there exists a nonzero function ϕ σ such that (3.26) ϕ(zm) = ω(z)ϕ(m) for all z Z GLr(A) M(F ) M (A) and m M(F ) M (A). We need Proposition Let ω Ω be as above, i.e. ω appears in σ. Then there exists a metaplectic tensor product π ω = (π 1 π k ) ω such that π ω Π.

19 REMARKS ON METAPLECTC TENSOR PRODUCTS 19 Proof. Since ω appears in σ, there exists ϕ V σ with the property (3.26). Consider the space σ(m(f ) M (A))ϕ generated by ϕ inside V σ. Since each z Z GLr(A) M(F ) M (A) is in the center of M(F ) M (A), one can see that σ(z)ϕ = ω(z)ϕ for all ϕ σ(m(f ) M (A))ϕ. Hence if we pick up an irreducible τ σ(m(f ) M (A))ϕ, we can extend it to τ ω, and an irreducible subrepresentation of nd M(A) τ Z GLr (A) M(F ) M ω is the desired metaplectic tensor product. (A) With this proposition, we can state the global analogue of Proposition 3.5 as follows. Proposition First we have the decomposition Π(π 1,..., π k ) = τ m(τ) nd M(A) M(F ) M (A) τ, where the sum is over all the equivalence classes τ σ and m(τ) is the positive multiplicity of τ in σ. Further for each fixed τ, let Then we have Ω(τ) := {ω Ω : ω Z GLr (A) M(F ) M (A) = τ Z GLr (A) M(F ) M (A) }. nd M(A) M(F ) M (A) τ ω Ω(τ) where m(τ, ω) is the positive multiplicity. nd M(A) Z GLr (A) M(F ) M (A) τ ω ω Ω(τ) m(τ, ω)(π 1 π k ) ω, Proof. The proposition can be proven in the same way as the local case. Yet, we should mention that unlike the local case, we do not seem to know the precise information on the multiplicities. From the proposition, the following is immediate. Theorem We have Π(π 1,..., π k ) ω Ω m(ω)(π 1 π k ) ω, where m(ω) is the multiplicity of (π 1 π k ) ω. Also if all of π 1,..., π k are cuspidal, then the inclusion is actually an equality. Proof. The first part is obvious from the above proposition. The second part follows from Proposition 3.14 because if Π is in the cuspidal spectrum, it is completely reducible. Let us note that if we know the multiplicity one property for the group M(A), we could set m(ω) = 1. Yet, the author does not know if the multiplicity one property holds even for the cuspidal spectrum Restriction to a smaller Levi. As we did for the local case, we will discuss the restriction of our metaplectic tensor products to a smaller Levi M, where M is as in (2.4). Recall σ = σ 1 σ k, whose space V σ is essentially identified with V σ1 V σk, which is a space of functions on the direct product H 1 H k. Hence if we let σ := σ i1 σ il,

20 20 SHUCHRO TAKEDA one can see that if ϕ V σ, we have ϕ M (F ) M (A) V σ. ndeed, we have an M (F ) M (A)- intertwining surjection σ σ, ϕ ϕ M (F ) M (A). n other words, we have σ = σ M (F ) M (A). Now we can prove Theorem For each π ω = (π 1 π k ) ω Π(π,..., π k ), we have π ω M (A) where Ω = Ω(π i1,, π ik ) is defined analogously to Ω. ω Ω m(ω )(π i1 π il ) ω Proof. Let us view π ω as a subspace of the induced space of (3.12). Then we have the commutative diagram A( M) π ω nd M(A) M(F ) M (A) σ π ω M (A) nd M (A) σ M (F ) M (A) A( M ) where all the vertical arrows are restriction of functions and the hooked arrow on the top is the automorphic realization map as in (3.2), and the one on the bottom is its analogue for M. By Lemma A.1, we know that π ω M (A) is completely irreducible, and hence so is π. Note that ω M (A) every irreducible subrepresentation of nd M (A) M (F ) M with respect to some ω. Hence the theorem follows by identifying nd M (A) of A( M ). (A) σ is a metaplectic tensor product of π i1,..., π il M (F ) M (A) σ with a subspace 3.4. Other properties. n [T3], a couple of other properties of metaplectic tensor product are discussed. To be precise, they have the expected behavior under the Weyl group action and the compatibility with parabolic induction, which are, respectively, Theorems 5.19 and 5.22 of [T3]. But both of them follow from the corresponding local statements, and hence they also hold in our new construction. t should be also mentioned that recently it has been shown by W. T. Gan in [G] that the metaplectic tensor product can be interpreted as an instance of Langlands functoriality by using the L-group formalism of covering groups developed by of Weissman. (See [W, GG] for this formalism.) This shows that the construction of the metaplectic tensor product is indeed a natural one Some remarks on past literature. The notion of metaplectic tensor product has been implicitly used in many of the past works on automorphic forms on GL r (A), especially when one would like to construct Eisenstein series on GL r (A). But there are various discrepancies in the past literature in this subject, which, we believe, was due to the lack of a foundation on metaplectic tensor product. n this final subsection, let us briefly discuss some of the previous works and how they can be reconciled with the theory developed in this paper. The first work that considered Eisenstein series on GL r (A) is, of course, the important work of Kazhdan and Patterson [KP]. There they only considered those Eisenstein series which are induced

21 REMARKS ON METAPLECTC TENSOR PRODUCTS 21 from the Borel subgroup B. Namely they only considered the case M = GL 1 GL 1. n this case, one can show that the group Z GLr(A) M(F ) M (A) is a maximal abelian subgroup of M, and accordingly, from the outset they considered a character on Z GLr(A) M(F ) M (A) instead of starting with characters on GL 1 (A). (See [KP, Sec..1].) Yet, one can see that this is the same as constructing some τ ω in our notation. Then Kazhdan-Patterson considered the induced representation nd GL r(a) Z GLr (A) M(F ) M (A)N B τ ω, to construct Eisenstein series. By inducing in stages, nd GL r(a) B(A) Z GLr (A) M(F ) M (A)N B τ ω = nd GL r(a) nd M(A)N B Z GLr (A) M(F ) M (A)N B τ ω, and hence the metaplectic tensor product that is implicitly used in [KP] is our Π(τ ω ) = nd M(A) Z GLr (A) M(F ) M (A) τ ω. Further the fact that Z GLr(A) M(F ) M (A) is maximal abelian implies that Π(τ ω ) is irreducible. (See [KP, Sec. 0.3].) The next important set of works on this subject is probably the one by Bump and Ginzburg ([BG]) on the symmetric square L-function, and the work by Banks [B] on the twisted case for GL 3, both of which dealt with only the case n = 2. There are two main parabolic subgroups considered there: the Borel and the (r 1, 1)-parabolic. For the Borel, they use the same formulation as [KP]. For the (r 1, 1)-parabolic, they first start with a representation of GL r 1 (A) viewed as a subgroup of GL r (A) and they extend it to a representation of Z GL GLr(A) r 1 (A) by letting Z GLr(A) act by an appropriate character. Now if r is odd (and n = 2), this gives a representation of GL r 1 (A) GL 1 (A) because Z GL GLr(A) r 1 (A) = GL r 1 (A) GL 1 (A). But if r is even, we only have Z GL GLr(A) r 1 (A) = (2) (2) GL r 1 (A) GL 1 (A). Then, in [BG], they induced the representation of GLr 1 (A) GL 1 (A) to GL r 1 (A) GL 1 (A). (See the middle of p.159 [BG].) However, it seems to the author that one cannot show the automorphy of this induced representation if it is simply induced from GL (2) r 1 (A) GL 1 (A), and probably this is another technical issue to be addressed in [BG]. At any rate, one can see that at least if r is odd this construction is also obtained as our metaplectic tensor product, say, by first restricting to Z GLr(A) M(F ) M (A) and then inducing one of the constituents to M(A). t should be also mentioned that in [BG, B] various properties of metaplectic tensor product, such as the behavior of metaplectic tensor product upon restriction to a smaller Levi, are implicitly used. Based on [BG, B], the author generalized their work in [T1], in which the parabolic subgroups considered are mainly (2,..., 2) and (r 1, 1) parabolic. For the (2,..., 2)-parabolic, the inducing representation for each GL 2 factor is only the Weil representation, and hence by using the Schrodinger (2) model for GL 2, we explicitly constructed what we called the Weil representation of M P. One can see that this is also an instance of our metaplectic tensor product, because for the case at hand we have Z GLr(A) M (A), which means that the central character does not play any role in the formation of metaplectic tensor product and hence the metaplectic tensor product only depends locally on restrictions to M (F v ). For the (r 1, 1)-parabolic case in [T1], however, depending on the parity of r, we took a different approach. For r odd, we did just as in [BG, B]. For r even, we directly constructed a representation of the Levi GL r 1 GL 1 as residues of Eisenstein series induced from the Borel, instead of starting with representations of GL r 1 and GL 1 separately. One can show that this construction is the same as our metaplectic tensor product by using the compatibility of our metaplectic tensor product with parabolic induction as discussed in the previous subsection. Besides those applications to symmetric square L-functions, the works of Suzuki in [S1, S2] should be mentioned. n [S1], he considered the (r 1, r 2 )-parabolic for r 1 +r 2 = r. To construct an automorphic