METAPLECTIC TENSOR PRODUCTS FOR AUTOMORPHIC REPRESENTATIONS OF GL(r)

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1 METAPLECTIC TENSOR PRODUCTS FOR AUTOMORPHIC REPRESENTATIONS OF GL(r SHUICHIRO TAKEDA Abstract. Let M = GL r 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,..., π k of GL r (A,..., GL rk (A, we construct (under a certain technical assumption, 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,..., π k. This is the global analogue of the metaplectic tensor product defined by P. Mezo in the sense that locally at each place v, π v is equivalent to the local metaplectic tensor product of π,v,..., π k,v defined by Mezo. Then we show that if all of π i are cuspidal (resp. square-integrable modulo center, then the metaplectic tensor product is cuspidal (resp. square-integrable modulo center. We also show that (both locally and globally the metaplectic tensor product behaves in the expected way under the action of a Weyl group element, and show the compatibility with parabolic inductions.. Introduction Let F be either a local field of characteristic 0 or a number field, and R be F if F is local and the ring of adeles A if F is global. Consider the group GL r (R. For a partition r = r + + r k of r, one has the Levi subgroup M(R := GL r (R GL rk (R GL r (R. Let π,..., π k be irreducible admissible (resp. automorphic representations of GL r (R,..., GL rk (R where F is local (resp. F is global. Then it is a trivial construction to obtain the representation π π k, which is an irreducible admissible (resp. automorphic representation of the Levi M(R. Though highly trivial, this construction is of great importance in the representation theory of GL r (R. Now if one considers the metaplectic n-fold cover GL r (R constructed by Kazhdan and Patterson in [KP], the analogous construction turns out to be far from trivial. Namely for the metaplectic preimage M(R of M(R in GL r (R and representations π,..., π k of the metaplectic n-fold covers GL r (R,..., GL rk (R, one cannot construct a representation of M(R simply by taking the tensor product π π k. This is simply because M(R is not the direct product of GL r (R,..., GL rk (R, namely M(R GL r (R GL rk (R, and even worse there is no natural map between them. When F is a local field, for irreducible admissible representations π,..., π k of GL r (F,..., GL rk (F, P. Mezo ([Me], whose work, we believe, is based on the work by Kable [K2], constructed an irreducible admissible representation of the Levi M(F, which can be called the metaplectic tensor product of π,..., π k, and characterized it uniquely up to certain character twists. (His construction will be reviewed and expanded further in Section 4. The theme of the paper is to carry out a construction analogous to Mezo s when F is a number field, and our main theorem is

2 2 SHUICHIRO TAKEDA Main Theorem. Let M = GL r GL rk be a Levi subgroup of GL r, and let π,..., π k be unitary automorphic subrepresentations of GLr (A,..., GL rk (A. Assume that M and n are such that Hypothesis ( (see section 3.4 is satisfied, which is always the case if n = 2. Then there exists an automorphic representation π of M(A such that π = vπ v, where each π v is the local metaplectic tensor product of Mezo. Moreover, if π,..., π k are cuspidal (resp. square-integrable modulo center, then π is cuspidal (resp. square-integrable modulo center. In the above theorem, v indicates the metaplectic restricted tensor product, the meaning of which will be explained later in the paper. The existence and the local-global compatibility in the main theorem are proven in Theorem 5.3, and the cuspidality and square-integrability are proven in Theorem 5.6 and Theorem 5.8, respectively. Let us note that by unitary, we mean that π i is equipped with a Hermitian structure invariant under the action of the group. 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 usually an automorphic representation is a subquotient. We need those two conditions for technical reasons, and they are satisfied if π i is in the discrete spectrum, namely either cuspidal or residual. Also we should emphasize that if n > 2, we do not know if our construction works unless we impose a technical assumption as in Hypothesis (. We will show in Appendix A that this assumption is always satisfied if n = 2, and if n > 2 it is satisfied, for example, if gcd(n, r + 2cr =, where c is the parameter to be explained. We hope that even for n > 2 it is always satisfied, though at this moment we do not know how to prove it. As we will see, strictly speaking the metaplectic tensor product of π,..., 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 π k ω for the metaplectic tensor product to emphasize the dependence on ω. Also we will establish a couple of important properties of the metaplectic tensor product both locally and globally. The first one is that the metaplectic tensor product behaves in the expected way under the action of the Weyl group. Namely Theorem 4.9 and Let w W M be a Weyl group element of GL r that only permutes the GL ri -factors of M. Namely for each (g,..., g k GL r GL rk, we have w(g,..., g k w = (g σ(,..., g σ(k for a permutation σ S k of k letters. Then both locally and globally, we have w (π π k ω = (πσ( π σ(k ω, where the left hand side is the twist of (π π k ω by w. The second important property we establish is the compatibility of the metaplectic tensor product with parabolic inductions. Namely Theorem 4.9 and Both locally and globally, let P = MN GL r be the standard parabolic subgroup whose Levi part is M = GL r GL rk. Further for each i =,..., k let P i = M i N i GL ri be the standard parabolic of GL ri whose Levi part is M i = GL ri, GL ri,li. For each i, we are given a representation σ i := (τ i, τ i,li ωi

3 METAPLECTIC TENSOR PRODUCTS 3 of M i, which is given as the metaplectic tensor product of the representations τ i,,..., τ i,li of GL ri,,..., GL ri,li. σ i. Then the meta- Assume that π i is an irreducible constituent of the induced representation Ind GL ri P i plectic tensor product π ω := (π π k ω is an irreducible constituent of the induced representation Ind M Q (τ, τ,l τ k, τ k,lk ω, where Q is the standard parabolic subgroup of M whose Levi part is M M k. In the above two theorems, it is implicitly assumed that if n > 2 and F is global, the metaplectic tensor products in the theorems exist in the sense that Hypothesis ( is satisfied for the relevant Levi subgroups. Finally at the end, we will discuss the behavior of the global metaplectic tensor product when restricted to a smaller Levi. Namely for each automorphic form ϕ (π π k ω in the metaplectic tensor product, we would like to know which space the restriction belongs to, where M ϕ M2 2 = {I r } GL r2 GL rk M, viewed as a subgroup of M, is the Levi for the smaller group GL r r. Somehow similarly to the non-metaplectic case, the restriction belongs to the metaplectic tensor ϕ M2 product of π 2,..., π k. But the precise statement is a bit more subtle. Indeed, we will prove Theorem Assume Hypothesis ( (see section 5.6 is satisfied, which is always the case if n = 2 or gcd(n, r + 2cr = gcd(n, r r + 2c(r r =. Then there exists a realization of the metaplectic tensor product π ω = (π π k ω such that, if we let π ω M2(A = { ϕ M2(A : ϕ π ω}, then π ω M2(A δ m δ (π 2 π k ωδ, as a representation of M 2 (A, where (π 2 π k ωδ is the metaplectic tensor product of π 2,..., π k, ω δ is a certain character twisted by δ which runs through a finite subset of GL r (F and m δ Z 0 is a multiplicity. The precise meanings of the notations will be explained in Section 5.6. Even though the theory of metaplectic groups is an important subject in representation theory and automorphic forms, and used in various important literatures such as [B, F, BBL, BFH, BH, S], and most importantly for the purpose of this paper, [BG] which concerns the symmetric square L-function on GL(r, it has an unfortunate history of numerous technical errors and as a result published literatures in this area are often marred by those errors which compromise their reliability. As is pointed out in [BLS], this is probably due to the deep and subtle nature of the subject. At any rate, this has made people who work in the area particularly wary of inaccuracies in new works. For this reason, especially considering the foundational nature of this paper, we tried to provide detailed proofs for most of our assertions at the expense of the length of the paper. Furthermore, for large part, we rely only on the two fundamental works, namely the work on the metaplectic cocycle by Banks, Levy and Sepanski ([BLS] and the local metaplectic tensor product by Mezo ([Me], both of which are written carefully enough to be reliable.

4 4 SHUICHIRO TAKEDA Finally, let us mention that the result of this paper will be used in our forthcoming [T2], which will improve the main result of [T]. Notations Throughout the paper, F is a local field of characteristic zero or a number field. If F is a number field, we denote the ring of adeles by A. As we did in the introduction we often use the notation { F if F is local R = A if F is global. The symbol R has the usual meaning and we set R n = {a n : a R }. Both locally and globally, we denote by O F the ring of integers of F. 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 a positive integer r, we denote by I 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, where F is either local or global. So in particular if n 3, for archimedean F, we have F = C, and for global F, F is totally complex. The symbol (, F denotes the n th order Hilbert symbol of F if F is local, which is a bilinear map (, F : F F µ n. If F is global, we let (, A := v (, F v, where the product is finite. We sometimes write simply (, for (, R when there is no danger of confusion. Let us recall that both locally and globally the Hilbert symbol has the following properties: (a, b = (b, a (a n, b = (a, b n = (a, a = for a, b R. Also for the global Hilbert symbol, we have the product formula (a, b A = for all a, b F. We fix a partition r + + r k = r of r, and we let M = GL r GL rk GL r and assume it is embedded diagonally as usual. We often denote each element m M by m = g g k, or m = diag(g,..., g k or sometimes simply m = (g,..., g k, where g i GL ri. For GL r, we let B = T N B be the Borel subgroup with the unipotent radical N B and the maximal torus T. If 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

5 METAPLECTIC TENSOR PRODUCTS 5 is an automorphic subrepresentation. Let H G be a subgroup. Then 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. For any group G and elements g, h G, we define g h = ghg. For a subgroup H G and a representation π of H, we define g π to be the representation of ghg defined by g π(h = π(g h g for h ghg. We let W be the set of all r r permutation matrices, so for each element w W each row and each column has exactly one and all the other entries are 0. The Weyl group of GL r is identified with W. Also for our Levi M, we let W M be the subset of W that only permutes the GL ri -blocks of M. Namely W M is the collection of block matrices W M := {(δ σ(i,j I rj W : σ S k }, where S k is the permutation group of k letters. Though W M is not a group in general, it is in bijection with S k. Note that if w W M corresponds to σ S k, we have w diag(g,..., g k = w diag(g,..., g k w = diag(g σ (,..., g σ (k. In addition to W, in order to use various results from [BLS], which gives a detailed description of the 2-cocycle σ r defining our metaplectic group GL r, one sometimes needs to use another set of representatives of the Weyl group elements, which we as well as [BLS] denote by M. The set M is chosen to be such that for each element η M we have det(η =. To be more precise, each η with length l is written as where w αi η = w α w αl is a simple root reflection corresponding to a simple root α i and is the matrix of the form w αi =. Though the set M is not a group, it has the advantage that we can compute the cocycle σ r in a systematic way as one can see in [BLS]. For each w W, we denote by η w the corresponding element in M. If w W M, one can see that η w is of the form (ε j δ σ(i,j I rj for ε j {±}. Namely η w is a k k block matrix in which the non-zero entries are either I rj or I rj. Acknowledgements The author would like to thanks Paul Mezo for reading an early draft and giving him helpful comments, and Jeff Adams for sending him the preprint [Ad] and explaining the construction of the metaplectic tensor product for the real case. The author is partially supported by NSF grant DMS Also part of this research was done when he was visiting the I.H.E.S. in the summer of 202 and he would like to thank their hospitality.

6 6 SHUICHIRO TAKEDA 2. The metaplectic cover GL r of GL r In this section, we review the theory of the metaplectic n-fold cover GL r of GL r for both local and global cases, which was originally constructed by Kazhdan and Patterson in [KP]. 2.. The local metaplectic cover GL r (F. Let F be a (not necessarily non-archimedean local field of characteristic 0 which contains all the n th roots of unity. In this paper, by the metaplectic n-fold cover GL r (F of GL r (F with a fixed parameter c {0,..., n }, we mean the central extension of GL r (F by µ n as constructed by Kazhdan and Patterson in [KP]. To be more specific, let us first recall that the n-fold cover SL r+ (F of SL r+ (F was constructed by Matsumoto in [Mat], and there is an embedding ( det(g (2. l 0 : GL r (F SL r+ (F, g. g Our metaplectic n-fold cover GL r (F with c = 0 is the preimage of l 0 (GL r (F via the canonical projection SL r+ (F SL r+ (F. Then GL r (F is defined by a 2-cocycle σ r : GL r (F GL r (F µ n. For arbitrary parameter c {0,..., n }, we define the twisted cocycle σ (c r σ r (c (g, g = σ r (g, g (det(g, det(g c F for g, g GL r (F, where recall from the notation section that (, F is the n th order Hilbert symbol for F. The metaplectic cover with a parameter c is defined by this cocycle. In [KP], the metaplectic (c cover with parameter c is denoted by GL r (F but we avoid this notation. This is because later we will introduce the notation GL r (F, which has a completely different meaning. Also we suppress the superscript (c from the notation of the cocycle and always agree that the parameter c is fixed throughout the paper. By carefully studying Matsumoto s construction, Banks, Levy, and Sepanski ([BLS] gave an explicit description of the 2-cocycle σ r and shows that their 2-cocycle is block-compatible in the following sense: For the standard (r,..., r k -parabolic of GL r, so that its Levi M is of the form GL r GL rk which is embedded diagonally into GL r, we have (2.2 = σ r ( g k σ ri (g i, g i i= g k, i<j k g g k by (det(g i, det(g j F (det(g i, det(g j c F, for all g i, g i GL r i (F. (See [BLS, Theorem, 3]. Strictly speaking in [BLS] only the case c = 0 is considered but one can derive the above formula using the bilinearity of the Hilbert symbol. This 2-cocycle generalizes the well-known cocycle given by Kubota [Kub] for the case r = 2. Also we should note that if r =, this cocycle is trivial. Note that GL r (F is not the F -rational points of an algebraic group, but this notation seems to be standard. Let us list some other important properties of the cocycle σ r, which we will use in this paper. Proposition 2.3. Let B = T N B be the Borel subgroup of GL r where T is the maximal torus and N B the unipotent radical. The cocycle σ r satisfies the following properties: i j

7 METAPLECTIC TENSOR PRODUCTS 7 (0 σ r (g, g σ r (gg, g = σ r (g, g g σ r (g, g for g, g, g GL r. ( σ r (ng, g n = σ r (g, g for g, g GL r and n, n N B, and so in particular σ r (ng, n = σ r (n, g n =. (2 σ r (gn, g = σ r (g, ng for g, g GL r and n N B. (3 σ r (η, t = ( t j, t i for η M and t = diag(t,..., t r T, where Φ + is the set of α=(i,j Φ + ηα<0 positive roots and each root α Φ + is identified with a pair of integers (i, j with i < j r as usual. (4 σ r (t, t = (t i, t j (det(t, det(t c for t = diag(t,..., t r, t = diag(t,..., t r T. i<j (5 σ r (t, η = for t T and η M. Proof. The first one is simply the definition of 2-cocycle and all the others are some of the properties of σ r listed in [BLS, Theorem 7, p.53]. We need to recall how this cocycle is constructed. As mentioned earlier, Matsumoto constructed SL r+ (F. It is shown in [BLS] that SL r+ (F is defined by a cocycle σ SLr+ which satisfies the block-compatibility in a much stronger sense as in [BLS, Theorem 7, 2, p45]. (Note that our SL r+ corresponds to G of [BLS]. Then the cocycle σ r is defined by σ r (g, g = σ SLr+ (l(g, l(g (det(g, det(g F (det(g, det(g c F, where l is the embedding defined by ( g (2.4 l : GL r (F SL r+ (F, g det(g. See [BLS, p.46]. (Note the difference between this embedding and the one in (2.. This is the reason we have the extra Hilbert symbol in the definition of σ r. Since we would like to emphasize the cocycle being used, we denote GL r (F by σ GLr (F when the cocycle σ is used. Namely σ GLr (F is the group whose underlying set is and the group law is defined by σ GLr (F = GL r (F µ n = {(g, ξ : g GL r (F, ξ µ n }, (g, ξ (g, ξ = (gg, σ r (g, g ξξ. To use the block-compatible 2-cocycle of [BLS] has obvious advantages. In particular, it has been explicitly computed and, of course, it is block-compatible. Indeed, when we consider purely local problems, we always assume that the cocycle σ r is used. However it does not allow us to construct the global metaplectic cover GL r (A. Namely one cannot define the adelic block-combatible 2-cocycle simply by taking the product of the local block-combatible 2-cocycles over all the places. Namely for g, g GL r (A, the product σ r,v (g v, g v v is not necessarily finite. This can be already observed for the case r = 2. (See [F, p.25]. For this reason, we will use a different 2-cocycle τ r which works nicely with the global metaplectic cover GL r (A. To construct such τ r, first assume F is non-archimedean. It is known that an open compact subgroup K splits in GL r (F, and moreover if n F =, we have K = GL r (O F. (See [KP,

8 8 SHUICHIRO TAKEDA Proposition 0..2]. Also for k, k K, a property of the Hilbert symbol gives (det(k, det(k F =. Hence one has a continuous map s r : GL r (F µ n such that σ r (k, k s r (ks r (k = s r (kk for all k, k K. Then define our 2-cocycle τ r by (2.5 τ r (g, g := σ r (g, g sr(gs r (g s r (gg for g, g GL r (F. If F is archimedean, we set τ r = σ r. The choice of s r and hence τ r is not unique. However when n F =, there is a canonical choice with respect to the splitting of K in the following sense: Assume that F is such that n F =. Then the Hilbert symbol (, F is trivial on O F O F, and hence, when restricted to GL r(o F GL r (O F, the cocycle σ r is the restriction of σ SLr+ to the image of the embedding l. Now it is known that the compact group SL r+ (O F also splits in SL r+ (F, and hence there is a map s r : SL r+ (F µ n such that the section SL r+ (F SL r+ (F given by (g, s r (g is a homomorphism on SL r+ (O F. (Here we are assuming SL r+ (F is realized as SL r+ (F µ n as a set and the group structure is defined by the cocycle σ SLr+. Moreover s r SLr+(O F is determined up to twists by the elements in H (SL r+ (O F, µ n = Hom(SL r+ (O F, µ n. But Hom(SL r+ (O F, µ n = because SL r+ (O F is a perfect group and µ n is commutative. Hence s r SLr+(O F is unique. (See also [KP, p. 43] for this matter. We choose s r so that (2.6 s r GLr(O F = s r l(glr(o F. With this choice, we have the commutative diagram (2.7 σ GLr (O F SLr+ (O F k (k, s r(k K SL r+ (O F, k (k, s r(k where the top arrow is (g, ξ (l(g, ξ, the bottom arrow is l, and all the arrows can be seen to be homomorphisms. This choice of s r will be crucial for constructing the metaplectic tensor product of automorphic representations. Also note that the left vertical arrow in the above diagram is what is called the canonical lift in [KP] and denoted by κ there. (Although we do not need this fact in this paper, if r = 2 one can show that τ r can be chosen to be block compatible, which is the cocycle used in [F]. Using τ r, we realize GL r (F as as a set and the group law is given by Note that we have the exact sequence GL r (F = GL r (F µ n, (g, ξ (g, ξ = (gg, τ r (g, g ξξ. 0 µ n GLr (F p GL r (F 0 given by the obvious maps, where we call p the canonical projection. We define a set theoretic section κ : GL r (F GL r (F, g (g,. Note that κ is not a homomorphism. But by our construction of the cocycle τ r, κ K is a homomorphism if F is non-archimedean and K is a sufficiently small open compact subgroup. Moreover if n F =, one has K = GL r (O F.

9 METAPLECTIC TENSOR PRODUCTS 9 Also we define another set theoretic section s r : GL r (F GL r (F, g (g, s r (g where s r (g is as above, and then we have the isomorphism GL r (F σ GLr (F, which gives rise to the commutative diagram (g, ξ (g, s r (gξ, GL r (F σ GLr (F s r GL r (F g (g, of set theoretic maps. Also note that the elements in the image s r (GL r (F multiply via σ r in the sense that for g, g GL r (F, we have (2.8 (g, s r (g (g, s r (g = (gg, σ r (g, g s r (gg. Let us mention Lemma 2.9. Assume F is non-archimedean with n F =. We have (2.0 κ T K = s r T K, κ W = s r W, κ NB K = s r NB K, where W is the Weyl group and K = GL r (O F. In particular, this implies s r T K = s r W = s r NB K =. Proof. See [KP, Proposition 0.I.3]. Remark 2.. Though we do not need this fact in this paper, it should be noted that s r splits the Weyl group W if and only if (, F =. So in particular it splits W if n F =. See [BLS, 5]. If P is a parabolic subgroup of GL r whose Levi is M = GL r GL rk, we often write for the metaplectic preimage of M(F. Next let and M(F = GL r (F GL rk (F GL r (F = {g GL r (F : det g F n }, GL r (F its metaplectic preimage. Also we define and often denote its preimage by M (F = {(g,..., g k M(F : det g i F n } M (F = GL r (F GL r k (F. The group M (F is a normal subgroup of finite index. Indeed, we have the exact sequence (2.2 M (F M(F F n \F F n \F, }{{} k times where the third map is given by (diag(g,..., g k, ξ (det(g,..., det(g k. We should mention the explicit isomorphism F n \F F n \F M (F \ M(F defined as follows: First for each i {,..., k}, define a map ι i : F GL ri by ( a (2.3 ι i (a =. I ri

10 0 SHUICHIRO TAKEDA Then the map given by ι (a (a,..., a k (, ιk(ak is a homomorphism. Clearly the map is well-defined and -. Moreover this is surjective because each element g i GL ri is written as and g i ι i (det(g i n GL r i. The following should be mentioned. g i = g i ι i (det(g i n ι i (det(g i n Lemma 2.4. The groups F n, M (F and M (F are closed subgroups of F, M(F and M(F, respectively. Proof. It is well-known that F n is closed and of finite index in F. Hence the group F n \F F n \F is discrete, in particular Hausdorff. But both M (F \ M(F and M (F \M(F are, as topological groups, isomorphic to this Hausdorff space. This completes the proof. Remark 2.5. If F = C, clearly M (F = M(F. If F = R, then necessarily n = 2 and GL (2 r (R consists of the elements of positive determinants, which is usually denoted by GL + r (R. Accordingly one may denote GL r (R and M (R by GL + r (R and M + (R respectively. Both GL + r (R and GL r (R share the identity component, and hence they have the same Lie algebra. The same applies to M + (R and M(R. Let us mention the following important fact. Let Z GLr (F GL r (F be the center of GL r (F. Then its metaplectic preimage Z GLr (F is not the center of GL r (F in general. (It might not be even commutative for n > 2. The center, which we denote by Z GLr (F, is (2.6 Z GLr (F = {(ai r, ξ : a r +2rc F n, ξ µ n } = {(ai r, ξ : a F n d, ξ µn }, where d = gcd(r + 2c, n. (The second equality is proven in [GO, Lemma ]. Note that Z GLr (F is a closed subgroup. Let π be an admissible representation of a subgroup H GL r (F, where H is the metaplectic preimage of a subgroup H GL r (F. We say π is genuine if each element (, ξ H acts as multiplication by ξ, where we view ξ as an element of C in the natural way The global metaplectic cover GL r (A. In this subsection we consider the global metaplectic group. So we let F be a number field which contains all the n th roots of unity and A the ring of adeles. Note that if n > 2, then F must be totally complex. We shall define the n-fold metaplectic cover GL r (A of GL r (A. (Just like the local case, we write GL r (A even though it is not the adelic points of an algebraic group. The construction of GL r (A has been done in various places such as [KP, FK]. First define the adelic 2-cocycle τ r by τ r (g, g := v τ r,v (g v, g v,

11 METAPLECTIC TENSOR PRODUCTS for g, g GL r (A, where τ r,v is the local cocycle defined in the previous subsection. By definition of τ r,v, we have τ r,v (g v, g v = for almost all v, and hence the product is well-defined. We define GL r (A to be the group whose underlying set is GL r (A µ n and the group structure is defined via τ r as in the local case, i.e. (g, ξ (g, ξ = (gg, τ r (g, g ξξ, for g, g GL r (A, and ξ, ξ µ n. Just as the local case, we have 0 µ n GLr (A p GL r (A 0, where we call p the canonical projection. Define a set theoretic section κ : GL r (A GL r (A by g (g,. It is well-known that GL r (F splits in GL r (A. However the splitting is not via κ. In what follows, we will see that the splitting is via the product of all the local s r. Let us start with the following product formula of σ r. Proposition 2.7. For g, g GL r (F, we have σ r,v (g, g = for almost all v, and further σ r,v (g, g =. v Proof. From the explicit description of the cocycle σ r,v (g, g given at the end of 4 of [BLS], one can see that σ r,v (g, g is written as a product of Hilbert symbols of the form (t, t Fv for t, t F. This proves the first part of the proposition. The second part follows from the product formula for the Hilbert symbol. Proposition 2.8. If g GL r (F, then we have s r,v (g = for almost all v, where s r,v is the map s r,v : GL(F v µ n defining the local section s r : GL(F v GL r (F v. Proof. By the Bruhat decomposition we have g = bwb for some b, b B(F and w W. Then for each place v s r,v (g = s r,v (bwb = σ r,v (b, wb s r,v (bs r,v (wb /τ r,v (b, wb by (2.5 = σ r,v (b, wb s r,v (bσ r,v (w, b s r,v (ws r,v (b /τ r,v (w, b τ r,v (b, wb again by (2.5. By the previous proposition, σ r,v (b, wb = σ r,v (w, b = for almost all v. By (2.0 we know s r,v (b = s r,v (w = s r,v (b = for almost all v. Finally by definition of τ r,v, τ r,v (w, b = τ r,v (b, wb = for almost all v. This proposition implies that the expression s r (g := v s r,v (g makes sense for all g GL r (F, and one can define the map s r : GL r (F GL r (A, g (g, s r (g. Moreover, this is a homomorphism because of Proposition 2.7 and (2.8. Unfortunately, however, the expression v s r,v(g v does not make sense for every g GL r (A because one does not know whether s r,v (g v = for almsot all v. Yet, we have

12 2 SHUICHIRO TAKEDA Proposition 2.9. The expression s r (g = v s r,v(g v makes sense when g is in GL r (F or N B (A, so s r is defined on GL r (F and N B (A. Moreover, s r is indeed a homomorphism on GL r (F and N B (A. Also if g GL r (F and n N B (A, both s r (gn and s r (ng make sense and further we have s r (gn = s r (gs r and s r (ng = s r s r (g. Proof. We already know s r (g is defined and s r is a homomorphism on GL r (F. Also s r is defined thanks to (2.0 and s r is a homomorphism on N B (A thanks to Proposition 2.3 (. Moreover for all places v, we have σ r,v (g v, n v = again by Proposition 2.3 (. Hence for all v, s r,v (gn v = s r,v (gs r,v (n v /τ r,v (g, n v. For almost all v, the right hand side is. Hence the global s r (gn is defined. Also this equality shows that s r (gn = s r (gs r. The same argument works for ng. If H GL r (A is a subgroup on which s r is not only defined but also a group homomorphism, we write H := s r (H. In particular we have (2.20 GL r (F := s r (GL r (F and N B (A := s r (N B (A. We define the groups like Let us mention GL r (A, M(A, M (A, etc completely analogously to the local case. Lemma 2.2. The groups A n, M (A and M (A are closed subgroups of A, M(A and M(A, respectively. Proof. That A n and M (A are closed follows from the following lemma together with Lemma 2.4. Once one knows M (A is closed, one will know M (A is closed because it is the preimage of the closed M (A under the canonical projection, which is continuous. Lemma Let G be an algebraic group over F and G(A its adelic points. Let H G(A be a subgroup such that H is written as H = v H v (algebraically where for each place v, H v := H G(F v is a closed subgroup of G(F v. Then H is closed. Proof. Let (x i i I be a net in H that converges in G(A, where I is some index set. Let g = lim i I x i. Assume g / H. Then there exists a place w such that g w / H w. Since H w is closed, the set U w := G(F w \H w is open. Then there exists an open neighborhood U of g of the form U = v U v, where U v is some open neighborhood of g v and at v = w, U v = U w. But for any i I, x i / U because x i,w / U w, which contradicts the assumption that g = lim i I x i. Hence g H, which shows H is closed. Just like the local case, the preimage Z GLr (A of the center Z GLr (A of GL r (A is in general not the center of GL r (A but the center, which we denote by Z GLr (A, is Z GLr (A = {(ai r, ξ : a r +2rc A n, ξ µ n } = {(ai r, ξ : a A n d, ξ µn }, where d = gcd(r + 2c, n. The center is a closed subgroup of GL r (A. 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 κ(k v = κ(gl r (O Fv for all v with v n and v. If 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 = for almost all v, we have the surjection (2.23 ρ : v GLr (F v GL r (A, Π v(g v, ξ v (Π vg v, Π v ξ v,

13 METAPLECTIC TENSOR PRODUCTS 3 where the product Π v ξ v is literary the product inside µ n. This is a group homomorphism because τ r = v τ r,v and the groups GL r (A and GL r (F v are defined, respectively, by τ r and τ r,v. We have GLr (F v / ker ρ = GL r (A, v where ker ρ consists of the elements of the form (, ξ with ξ v µ n and Π v ξ v =. Let π be a representation of H GLr (A where H is the metaplectic preimage of a subgroup H GL r (A. Just like the local case, we call π genuine if (, ξ H(A acts as multiplication by ξ for all ξ µ n. Also we have the notion of automorphic representation as well as automorphic form on GL r (A or M(A. In 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.7]. Hence if π is an automorphic representation of GL r (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, for all (g, ξ GL r (A (or M(A. In particular every automorphic form in the space of a genuine automorphic representation is genuine. Suppose we are given a collection of irreducible admissible representations π v of GL r (F v such that π v is κ(k v -spherical for almost all v. Then we can form an irreducible admissible representation of v GL 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 admissible representation of GL r (A, it is written as vπ v where π v is an irreducible admissible representation of GL r (F v, and for almost all v, π v is κ(k v -spherical. (To see it, view π as a representation of the restricted product v GL r (F v by pulling it back by ρ as in (2.23 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. Finally in this section, let us mention that we define (2.24 GL r (F := GL r (F GL r (A, namely GL r (F = {g GL r (F : det g A n }. But since F contains µ n, one can easily show that GL r (F = {g GL r (F : det g F n }. (See, for example, [AT, Chap. 9, Theorem ]. Also for n = 2, this is a consequence of the Hasse- Minkowski theorem. Similarly we define M (F = M(F M (A. 3. The metaplectic cover M of the Levi M Both locally and globally, one cannot show the cocycle τ r has the block-compatibility as in (2.2 (except when r = 2. Yet, in order to define the metaplectic tensor product, it seems to be necessary to have the block-compatibility of the cocycle. To get round it, we will introduce another cocycle τ M, but this time it is a cocycle only on the Levi M, and will show that τ M is cohomologous to the restriction τ r M M of τ r to M M both for the local and global cases.

14 4 SHUICHIRO TAKEDA 3.. The cocycle τ M. In this subsection, we assume that all the groups are over F if F is local and over A if F is global, and suppress it from our notation. We define the cocycle τ M : M M µ n, by τ M ( g g k, g = g k k τ ri (g i, g i i, det(g j i<j k(det(g (det(g i, det(g j c, i j i= where (, is the local or global Hilbert symbol. Note that the definition makes sense both locally and globally. Moreover the global τ M is the product of the local ones. We define the group c M to be c M = M µn as a set and the group structure is given by τ M. The superscript c is for compatible. One advantage to work with c M is that each GLri embeds into c M via the natural map (g i, ξ ( I r + +r i g i, ξ. I ri++ +rk Indeed, the cocycle τ M is so chosen that we have this embedding. Also recall our notation M = GL r GL r k, and M = GL r GL r k. We define c M analogously to c M, namely the group structure of c M is defined via the cocycle τ M. Of course, c M is a subgroup of c M. Note that each GL r i naturally embeds into c M as above. Lemma 3.. The subgroups GL r i and GL r j in c M commute pointwise for i j. Proof. Locally or globally, it suffices to show τ M (g i, g j = τ M (g j, g i for g i GL r i and g j GL r j. But the block-compatibility of the 2-cocycle τ M, we have τ M (g i, g j = τ ri (g i, I rj τ rj (I rj, g j =, and similarly have τ M (g j, g i =. Lemma 3.2. There is a surjection given by the map whose kernel is so that c M = GL r GL r GL r k ((g, ξ,..., (g k, ξ k ( g c M g k, ξ ξ k, K P := {((, ξ,..., (, ξ k : ξ ξ k = }, GL r k /K P. Proof. The block-compatibility of τ M guarantees that the map is indeed a group homomorphism. The description of the kernel is immediate.

15 METAPLECTIC TENSOR PRODUCTS The relation between τ M and τ r. Note that for the group M (instead of c M, the group structure is defined by the restriction of τ r to M M, and hence each GL ri might not embed into GL r in the natural way because of the possible failure of the block-compatibility of τ r unless r = 2. To make explicit the relation between c M and M, the discrepancy between τm and τ r M M (which we denote simply by τ r has to be clarified. Local case: Assume F is local. Then we have τ M ( g, g g =σ r ( g g k g k, g k g k k i= s ri (g i s ri (g i s ri (g i g i, so τ M and σ r M M are cohomologous via the function k i= s r i. Here recall from Section 2.2 that the map s ri : GL ri µ n relates τ ri with σ ri by σ ri (g i, g i = τ ri (g i, g i s ri (g i, g i s ri (g i s ri (g i, for g i, g i GL r i. Moreover if n F =, s ri is chosen to be canonical in the sense that (2.6 is satisfied. The block-compatibility of σ r implies for m = g τ r (m, m g k s r (mm s r (ms r (m = σ r(m, m = τ M (m, m and m = (3.3 ŝ M (m = we have g k i= s ri (g i g i s ri (g i s ri (g i,. Hence if we define ŝ M : M µ n by g k k i= s r i (g i, s r (m (3.4 τ M (m, m = τ r (m, m ŝm (mŝ M (m ŝ M (mm, namely τ r and τ M are cohomologous via ŝ M. Therefore we have the isomorphism α M : c M M, (m, ξ (m, ŝm (mξ. The following lemma will be crucial later for showing that the global τ M is also cohomologous to τ r M(A M(A. Lemma 3.5. Assume F is such that n F =. Then for all k M(O F, we have ŝ M (k =. Proof. First note that if k, k M(O F, then τ r (k, k = τ M (k, k = and so by (3.4 we have ŝ M (kk = ŝ M (kŝ M (k,

16 6 SHUICHIRO TAKEDA i.e. ŝ M is a homomorphism on M M (O F. Hence it suffices to prove the lemma only for the elements k M(O F of the form k = I r + +r i k i I ri++ +rk where k i GL ri is in the i th place on the diagonal. Namely we need to prove s ri (k i s r (k =. In what follows, we will show that this follows from the canonicality of s r and s ri, and the fact that the cocycle for SL r+ is block-compatible in a very strong sense as in [BLS, Lemma 5, Theorem 7 2, p.45]. Recall from (2.6 that s r has been chosen to satisfy s r GLr(O F = s r l(glr(o F, where s r is the map on SL r+ (F that makes the diagram (2.7 commute. Similarly for s ri with r replaced by r i. Let us write ( gi l i : GL ri (F SL ri+(f, g i det(g i for the embedding that is used to define the cocycle σ ri. Define the embedding ( I r+ +r i A b F : SL ri+(f SL r+ (F, A b c d I ri++ +r k, c d where A is a r i r i -block and accordingly b is r i, c is r i and d is. Note that this embedding is chosen so that we have (3.6 F (l i (k i = l(k. By the block compatibility of σ SLr+ we have σ SLr+ F (SLri + F (SL ri + = σ SLri +. This is nothing but [BLS, Lemma 5, 2]. (The reader has to be careful in that the image F (SL ri+ is not a standard subgroup in the sense defined in [BLS, p.43] if one chooses the set of simple roots of SL r+ in the usual way. One can, however, choose differently so that F (SL ri+ is indeed a standard subgroup. And all the results of [BLS, 2] are totally independent of the choice of. This implies the map (g i, ξ (F (g i, ξ for (g i, ξ SL ri+ is a homomorphism. Hence the canonical section SL r+ (O F SL r+ (F, which is given by g (g, s r (g, restricts to the canonical section SL ri+(o F SL ri+(f, which is given by g i (g i, s ri (g i. Namely we have the commutative diagram g i (g i, s ri (g i SL ri+(o F SL ri+(o F (g, ξ (F (g, ξ F SLr+ (O F g (g, s r(g SL r+ (O F, where all the maps are homomorphisms. In particular, we have (3.7 s r (F (g i = s ri (g i,

17 METAPLECTIC TENSOR PRODUCTS 7 for all g i SL ri+(o F. Thus s r (k = s r (l(k by (2.6 = s r (F (l i (k i by (3.6 = s ri (l i (k i by (3.7 = s ri (k i by (2.6 with r replaced by r i. The lemma has been proven. Global case: Assume F is a number field. We define ŝ M : M(A µ n by ŝ M ( v m v := v ŝ Mv (m v for v m v M(A. The product is finite thanks to Lemma 3.5. Since both of the cocycles τ r and τ M are the products of the corresponding local ones, one can see that the relation (3.4 holds globally as well. Thus analogously to the local case, we have the isomorphism α M : c M(A M(A, (m, ξ (m, ŝm (mξ. Lemma 3.8. The splitting of M(F into c M(A is given by s M : M(F c M(A, g g k ( g g k, k s i (g i. i= Proof. For each i the splitting s ri : GL ri (F GL ri (A is given by g i (g i, s ri (g i, where GL ri (A is defined via the cocycle τ ri. The lemma follows by the block-compatibility of τ M and the product formula for the Hilbert symbol. Just like the case of GL r (A, the section s M as in this lemma cannot be defined on all of M(A even set theoretically because the expression i s r i (g i does not make sense to all diag(g,..., g k M(A. So we only have a partial set theoretic section But analogously to Proposition 2.9, we have s M : M(A c M(A. Proposition 3.9. The partial section s M is defined on both M(F and N M (A, where N M (A is the unipotent radical of the Borel subgroup of M, and moreover it gives rise to a group homomorphism on each of these subgroups. Also for m M(F and n N M (A, both s M (mn and s M (nm are defined and further s M (mn = s M (ms M and s M (nm = s M s M (m. Proof. This follows from Proposition 2.9 applied to each GL ri (A together with the block-compatibility of the cocycle τ M. (Note that one also needs to use the fact that for all g, g in the subgroup generated by M(F and N M (A, we have (det(g, det(g A =. This splitting is related to the splitting s r : GL r (F GL r (A by

18 8 SHUICHIRO TAKEDA Proposition 3.0. We have the following commutative diagram: Proof. For m = g g k c M(A α M s M M(F M(F, we have GLr (A s r GL r (F. k k α M (s M (m = α M (m, s ri (g i = (m, ŝ M (m s ri (g i = (m, s r (m = s r (m, i= where for the elements in M(F, all of s ri and s r are defined globally, and the second equality follows from the definition of ŝ M as in (3.3. This proposition implies Corollary 3.. Assume π is an automorphic subrepresentation of c M(A. The representation of M(A defined by π α M is also automorphic. Proof. If π is realized in a space V of automorphic forms on c M(A, then π α M is realized in the space of functions of the form f α M for f V. The automorphy follows from the commutativity of the diagram in the above lemma. i= The following remark should be kept in mind for the rest of the paper. Remark 3.2. The results of this subsection essentially show that we may identify c M (locally or globally with M. We may even pretend that the cocycle τ r has the block-compatibility property. We need to make the distinction between c M and M only when we would like to view the group M as a subgroup of GLr. For most part of this paper, however, we will not have to view M as a subgroup of GL r. Hence we suppress the superscript c from the notation and always denote c M simply by M, when there is no danger of confusion. Accordingly, we denote the partial section s M simply by s The center Z M of M. In this subsection F is either local or global, and accordingly we let R = F or A as in the notation section. And all the groups are over R. For any group H (metaplectic or not, we denote its center by Z H. In particular for each group H GL r, we let Z H = center of H. For the Levi part M = GL r GL r2 GL r, we of course have I r Z M = { a a k I rk : a i R }. But for the center Z M of M, we have Z M Z M,

19 METAPLECTIC TENSOR PRODUCTS 9 in general, and indeed Z M might not be even commutative. In what follows, we will describe Z M in detail. For this purpose, let us start with Lemma 3.3. Assume F is local. Then for each g GL r (F and a F, we have Proof. First let us note that if we write σ r = σ (c r σ r (g, ai r σ r (ai r, g = (det(g, a r +2cr. to emphasize the parameter c, then σ r (c (g, ai r σ r (c (ai r, g = σ r (0 (g, ai r σ r (0 (ai r, g (det(g, a r 2c because (a r, det(g = (det(g, a r. Hence it suffices to show the lemma for the case c = 0. But this can be done by using the recipe provided by [BLS]. Namely let g = ntηn for n, n N B, t T and η M. Then σ r (g, ai r = σ r (ntηn, ai r = σ r (tη, n ai r by Proposition 2.3 ( and (2 = σ r (tη, ai r by n ai r = ai r n and Proposition 2.3 ( = σ r (t, ηai r σ r (η, ai r σ r (t, η by Proposition 2.3 (0 = σ r (t, ai r ησ r (η, ai r by Proposition 2.3 (5 = σ r (tai r, ησ r (t, ai r σ r (ai r, η σ r (η, ai r by Proposition 2.3 (0 = σ r (t, ai r σ r (η, ai r by Proposition 2.3 (5. Now by Proposition 2.3 (3, σ(η, ai r is a product of ( a, a s, which is. Hence by using Proposition 2.3 (4, we have r σ r (g, ai r = σ r (t, ai r = (t i, a r i. By an analogous computation, one can see Using (a, t i = (t i, a, one can see σ r (ai r, g = σ r (ai r, t = σ r (g, ai r σ r (ai r, g = i= r (a, t i i. i= r (t i, a r. But this is equal to (det(g, a r because det(g = r i= t i. Note that this lemma immediately implies that the center Z GLr of GLr is indeed as in (2.6, though a different proof is provided in [KP]. Also with this lemma, we can prove Proposition 3.4. Both locally and globally, the center is described as Z M I r = { Z M a a k I rk i= : a r +2cr i R n and a a r mod R n }.

20 20 SHUICHIRO TAKEDA Proof. First assume F is local. Let m = diag(g,..., g k M and a = diag(a I r,..., a k I rk. It suffices to show σ r (m, aσ r (a, m = if and only if all a i are as in the proposition. But σ r (m, aσ r (a, m r = σ ri (g i, a i I ri σ ri (a i I ri, g i = i= r i<j r i<j r (det(g i, a rj j (det(g i, a rj i j (a ri (det(g i, a rj σ ri (g i, a i I ri σ ri (a i I ri, g i i= i j r = (det(g i, a ri +2cri i (det(g i, a rj+2crj j i= i j r r = (det(g i, a i a rj+2crj j, i= j= i, det(g j i j j +2c (a ri j c i, det(g j c where for the third equality we used the above lemma with r replaced by r i. Now assume a is such that (a,. Then the above product must be for any m. In particular, Z M choose m so that g j = for all i j. Then we must have (det(g i, a r i j= arj+2crj j = for all g i GL ri. This implies i= i j= a i r j= a rj+2crj j F n for all i. Since this holds for all i, one can see a i a j F n for all i j, which implies a a r mod F n. But if a a r mod F n, then r r r r (det(g i, a a rj+2crj j = (det(g i, a i a rj+2crj i = i= j= r (det(g i, a r +2cr i. This must be equal to for any choice of g i, which gives a r +2cr i F n. Conversely if a is of the form as in the proposition, one can see that σ r (m, aσ r (a, m = r i= (det(g i, a r i j= arj+2crj j = for any m. The global case follows from the local one because locally by using (2.5 and am = ma, one can see σ r (m, aσ r (a, m = if and only if τ r (m, aτ r (a, m =, and the global τ r is the product of local ones. Lemma 3.3 also implies Lemma 3.5. Both locally and globally, Z GLr i= commutes with GL r pointwise. Proof. The local case is an immediate corollary of Lemma 3.3 because if g GL r the lemma implies σ r (g, ai r = σ r (ai r, g. Hence by (2.5, locally τ r (g, ai r = τ r (ai r, g for all g GL r and a F. Since the global τ r is the product of the local ones, the global case also follows.

21 METAPLECTIC TENSOR PRODUCTS 2 Let us mention that in particular, if n = 2 and r = even, then Z GLr center of GL r. This fact is used crucially in [T]. GL r and Z GLr is the It should be mentioned that this description of the center Z M easily implies (3.6 Z GLr M = Z M M. Also we have Proposition 3.7. Both locally and globally, the groups Z M and M commute pointwise, which gives (3.8 Z M = Z M M, and hence (3.9 Z GLr Z M = Z GLr ( Z M M = Z M (Z GLr M. Proof. By the block compatibility of the cocycle τ M, one can see that an element of the form I r ( a, ξ commutes with all the elements in M if and only if each (a i I ri, ξ commutes with all the elements in GL r i a k I rk. But this is always the case by the above lemme (with r replaced by r i. This proves the proposition. If F is global, we define Z M (F = Z M (A s(m(f, where recall that s : M(F M(A is the section that splits M(F. Similarly we define groups like Z GLr (F, M (F, etc. Namely in general for any subgroup H M(A, we define the F -rational points H(F of H by (3.20 H(F := H s(m(f The abelian subgroup A M. Again in this subsection, F is local or global, and R = F or A. As we mentioned above, the preimage Z M of the center Z M of the Levi M might not be even commutative. For later purposes, we let A M be a closed abelian subgroup of Z M containing the center Z GLr. Namely A M is a closed abelian subgroup such that Z GLr A M Z M. We let A M :=, p(a M where p is the canonical projection. If F is global, we always assume (A is chosen compatibly A M with the local (F A M v in the sense that we have A M (A = v AM (F v. Note that if A M (F v (hence A M (F v is closed, then A M (A (hence A M (A is closed by Lemma Of course there are many different choices for A M. But we would like to choose A M so that the following hypothesis is satisfied:

22 22 SHUICHIRO TAKEDA Hypothesis (. Assume F is global. The image of M(F in the quotient A M (AM (A\M(A is discrete in the quotient topology. The author does not know if one can always find such A M for general n. But at least we have Proposition 3.2. If n = 2, the above hypothesis is satisfied for a suitable choice of A M. For n > 2, if d = gcd(n, r + 2cr is such that n divides nr i /d for all i =,..., k, (which is the case, for example, if d =, then the above hypothesis is satisfied with A M = Z M. Proof. This is proven in Appendix A. We believe that for any reasonable choice of A M the above hypothesis is always satisfied, but the author does not know how to prove it at this moment. This is a bit unfortunate in that this subtle technical issue makes the main theorem of the paper conditional when n > 2. However if n = 2, our main results are complete, and this is the only case we need for our applications to symmetric square L-functions in [T, T2], which is the main motivation for the present work. Let us mention that the group A M (AM (A (for any choice of A M is a normal subgroup of M(A, and hence the quotient A M (AM (A\M(A is a group. Accordingly, if the hypothesis is satisfied, the image of M(F in the quotient is a discrete subgroup and hence closed. Also we have following the convention as in (3.20, and we set A M (F = A M (A s(m(f. A M (F = p(a M (F. 4. On the local metaplectic tensor product In this section we first review the local metaplectic tensor product of Mezo [Me] and then extend his theory further, first by proving that the metaplectic tensor product behaves in the expected way under the Weyl group action, and second by establishing the compatibility of the metaplectic tensor product with parabolic inductions. Hence in this section, all the groups are over a local (not necessarily non-archimedean field F unless otherwise stated. Accordingly, we assume that our metaplectic group is defined by the block-compatible cocycle σ r of [BLS], and hence by GL r we actually mean σ GLr. 4.. Mezo s metaplectic tensor product. Let π,, π k be irreducible genuine representations of GL r,..., GL rk, respectively. The construction of the metaplectic tensor product takes several steps. First of all, for each i, fix an irreducible constituent π i Then we have π i GL r i = g m i g (π i, of the restriction π i GL r i of π i to GL r i. where g runs through a finite subset of GL ri, m i is a positive multiplicity and g (π i is the representation twisted by g. Then we construct the tenor product representation of the group GL r π π k GL r k. Note that this group is merely the direct product of the groups GL r i. implies that this tensor product representation The genuineness of the representations π,..., π k