Local Rankin--Selberg integrals for Speh representations

We construct analogues of Rankin--Selberg integrals for Speh representations of the general linear group over a $p$-adic field. The integrals are in terms of the Shalika model and are expected to be the local counterparts of (suitably regularized) global integrals involving square-integrable automorphic forms and Eisenstein series on the general linear group over a global field. We relate the local integrals to the classical ones studied by Jacquet--Piatetski-Shapiro--Shalika. We also introduce a unitary structure for Speh representation on the Shalika model, as well as various other models including Zelevinsky's degenerate Whittaker model.


Introduction
The theory of Rankin-Selberg integrals for GL n × GL n , studied by Jacquet, Piatetski-Shapiro and Shalika in a series of papers starting in the late 1970s (notably [JPSS83]), is a basic tool in the theory of automorphic forms with an abundance of applications. The theory is based on global zeta integrals (which involve Eisenstein series in the case n = n) that unfold to adelic integrals of Whittaker-Fourier coefficients of cuspidal representations. By local multiplicity one, these integrals factorize into a product of local zeta integrals pertaining to generic representations and their Whittaker models.
The purpose of this paper is to study a modification of the local Rankin-Selberg integrals in the equal-rank case for a class of representations Sp(π, m) where π is an irreducible generic E. M. Lapid and Z. Mao the standard module whose image is the Speh representation induces its unitary structure -a fact that is true in general for any unitarizable representation on a reductive group (cf. [KZ77,§ 4]). (In the case at hand we will explicitly relate this unitary structure to that on the Shalika model.) However, the semidefiniteness of the intertwining operator is far from obvious -in fact it is equivalent to unitarizability, which is known to be a difficult problem in general, as is evident from the work of Vogan and many others. Another realization of the inner product is obtained by using global theory to embed Speh representations as local constituents of automorphic forms in the discrete spectrum of GL mn over the adeles [Spe83]. Finally, in the m = 2 case one can also realize a Speh representation in the discrete spectrum of L 2 (H\GL 2n ) where H is the symplectic group of rank n [Smi18, LO19]. However, there is no such analogue for m > 2.
In principle, the new local integrals are the local counterpart of certain global integrals, just as in the classical case. However, in addition to Eisenstein series, these global integrals involve automorphic forms in the discrete spectrum, rather than cusp forms, and they unfortunately do not converge (for any value of s). It should be possible (for instance by using the recent work of Zydor [Zyd19]) to carry out a regularization procedure to make sense of these integrals and to justify the unfolding procedure. However, we will not discuss this aspect in the paper. Nor we will discuss the archimedean case, for which we expect many of our results to hold without change.
The main new results of this paper are in § § 4 and 5. The unitary structure for Speh representations (and more generally, Sp(π, m) for unitarizable generic π), on their various models, is given in Theorem 4.3. The new zeta integrals are defined in § 5. The convergence, unramified computation and local functional equations are stated in Theorem 5.1.
We now give some more details about the contents of the paper. In § 2 we first introduce some notation and recall Zelevinsky's classification of irreducible representations of the general linear group over a local non-archimedean field F . We then introduce the class of mhomogeneous representations, which includes the usual Speh representations and which is the main focus of the paper. In terms of Zelevinsky's classification, they simply correspond to multisegments consisting of segments of length m, where m 1 is a fixed integer parameter. The case m = 1 exactly corresponds to generic representations (i.e. the classical theory). In § 3 we introduce the models pertaining to m-homogeneous representations, following Moeglin-Waldspurger. (In order to use their results, we assume from § 3 onward that F is of characteristic 0. As was pointed out to us by Dmitry Gourevitch, this assumption can be lifted. Details will appear elsewhere.) We also introduce the transition maps between the models. They are given by integrals which entail no convergence issues. Finally, we introduce the Kirillov-Shalika model which is the analogue of the classical Kirillov model for generic representations. In § 4 we introduce a family of bilinear forms on a pair of models of m-homogeneous representations.
In the case where the two representations are in duality, these bilinear forms specialize to an invariant pairing, at least under some restrictions. In the unitarizable case this gives rise to a manifestly positive invariant unitary structure. The invariance is proved by induction on m using Bernstein's theorem on invariant distributions with respect to the mirabolic subgroup.
In § 5 we define the local Rankin-Selberg integrals for m-homogeneous representations using their Shalika models. Applying the transition maps, we can express these integrals in terms of the Zelevinsky model. Hence, we get their rationality in q s , the unramified computation and functional equations. In § 6 we obtain more information about the poles of the zeta functions and relate them to the above-mentioned bilinear forms and, in particular, to the invariant pairing. In § 7 we go back to the Kirillov-Shalika model and analyze in detail the case of Speh representations of GL 4 pertaining to supercuspidal representations of GL 2 . We study the asymptotic behavior 910 Local Rankin-Selberg integrals for Speh representations of a function in the Kirillov-Shalika model. At this stage, it is hard to tell whether the result is representative of the general case or merely a low-rank fluke. In § 8 we write an informal global expression, modeled after the classical Rankin-Selberg integrals, whose regularization is expected to unfold to the local integrals studied in the paper. The regularization is necessary as the integral does not converge. (It would also eliminate extraneous terms in the unfolding procedure.) However, we do not discuss the regularization procedure and only give a purely heuristic argument. Finally, in Appendix A we relate the pairing of § 4 to that induced by the intertwining operator on the standard module.

Notation
Throughout the paper, fix a non-archimedean local field F with ring of integers O and absolute value |·|. In principle it should be possible to deal with the archimedean case as well with proper adjustments, but we do not consider this case here. From § 3 onward, F is assumed to be of characteristic 0.
If H is an algebraic group over F , we often also use H to denote H(F ). We will consider complex, smooth representations of finite length of the groups GL n (F ), n 0. We denote the set of irreducible representations of GL n (F ) (up to equivalence) by Irr GL n and set Irr = n 0 Irr GL n . We write Irr GL 0 = {1}. (In contrast, the one-dimensional trivial character of GL 1 will be denoted by 1 F * .) The subset of supercuspidal (respectively, squareintegrable, essentially square-integrable, tempered, generic) representations will be denoted by Irr cusp (respectively, Irr sqr , Irr esqr , Irr tmp , Irr gen ). Thus, Irr cusp ⊂ Irr sqr ⊂ Irr esqr and Irr esqr , Irr tmp ⊂ Irr gen .
By convention 1 ∈ Irr tmp but 1 / ∈ Irr esqr . Let π be a representation of GL n (F ). We denote by π ∨ the contragredient of π and by soc(π) the socle of π (the maximal semisimple subrepresentation of π). If π is non-zero, then we write deg π = n, the degree of π. For any character ω of F * (i.e. ω ∈ Irr GL 1 ) we denote by πω the representation obtained from π by twisting by the character ω • det. For instance, π|·| is the twist of π by |det|. We also write J P (π) for the (normalized) Jacquet module of π with respect to a parabolic subgroup P of GL n , defined over F . If τ ∈ Irr GL n , then we write τ π if τ occurs as a subquotient of π, that is, if τ occurs in the Jordan-Hölder sequence of π. If τ occurs with multiplicity one in the Jordan-Hölder sequence of π, then we write τ unq π.
If π 1 , . . . , π k are representations of GL n 1 (F ), . . . , GL n k (F ) respectively, then we denote the representation parabolically induced from π 1 ⊗ · · · ⊗ π k (normalized induction), with respect to the standard parabolic subgroup of block upper triangular matrices, by π 1 × · · · × π k and refer to it as the product representation. We also use the notation Ind G H and ind G H to denote induction and induction with compact support (both normalized) from a subgroup H of G.
For any set A we denote by M(A) the free commutative monoid generated by A, considered as an ordered monoid. Thus, an element of M(A) (a multiset of A) is a finite (possibly empty) formal sum of element of A.

Zelevinsky classification
We recall the well-known results and terminology of [Zel80].
If every segment that occurs in m 1 is unlinked with every segment that occurs in m 2 , By identifying an irreducible supercuspidal representation with a singleton segment we view M(Irr cusp ) as a submonoid of M(SEG). The map Z restricts to a bijection M(Irr cusp ) → Irr gen . (2) An element of M(Irr cusp ) is called a cuspidal datum. We write c(Z(m)) = c(m). The resulting map c : Irr → M(Irr cusp ) 912 Local Rankin-Selberg integrals for Speh representations is the supercuspidal support (which of course can be defined without reference to the Zelevinsky classification). The restriction of c to Irr gen is the inverse of (2). For any segment ∆ = ∆ (l) ρ|·| −1/2 denote either the segment obtained by removing the endpoint e(∆) of ∆ if l > 1 or the empty set otherwise.

Ladder representations
A multisegment m is called a (strict) ladder if it can be written as m = ∆ 1 + · · · + ∆ k where ∆ i+1 ≺ ∆ i for all i = 1, . . . , k − 1. The corresponding irreducible representation Z(m) is called a ladder representation.
The Jacquet module of ladder representations was described in [KL12]. The following lemma is an immediate consequence.
Lemma 2.2 [KL12]. Let π = Z(m) be a ladder representation and P a maximal parabolic subgroup. The following statements hold.
Strictly speaking, the results of [KL12] are stated in terms of the Langlands classification. However, they are also valid in the form above (for the Zelevinsky classification) by either repeating the arguments, or using the Zelevinsky involution.

m-homogeneous representations 2
From now on let m, n 1 be integers and G = GL mn . We say that σ ∈ Irr G is m-homogeneous if σ = Z(∆ 1 + · · · + ∆ k ) where each ∆ i is of length m. (If m = 1 this simply means that σ is generic.) We denote by Irr m -hmgns G the set of irreducible m-homogeneous representations of G. For any π = Z({ρ 1 } + · · · + {ρ k }) ∈ Irr gen , define The following result is clear.
Lemma 2.3. The map π → Sp(π, m) defines a bijection between Irr gen GL n and Irr m -hmgns G.
Remark 2.4. The notion of m-homogeneous representations is very close to the concept of 'representations of type (n, m)' introduced in [CFGK19] and studied further in [CFK18]. The difference is that we only consider irreducible representations and emphasize the roles of the Moeglin-Waldspurger models.
(i) σ i := Sp(π i , m) is a ladder representation for all i.

Local Rankin-Selberg integrals for Speh representations
Moreover, let Q be the maximal standard parabolic subgroup of type ((m − 1)n, n) and denote by J Q (σ) ;d the direct summand of J Q (σ) pertaining to the supercuspidal data d ∈ M(Irr cusp ) in the second (GL n ) factor. Then Remark 2.8. For m = 1, π i is essentially a discrete series. This is not the case for m > 1 in general.
Proof. Write π = Z( i∈I {ρ i }) with ρ i ∈ Irr cusp and let l 1. We say that a subset J of I is an l-chain if it can be written, necessarily uniquely, as J = {i 1 , . . . , i r } where for all j = 1, . . . , r − 1 we have ρ i j = ρ i j+1 |·| α j with α j ∈ {1, . . . , l}. (For example, for a 1-chain, ρ ir , . . . , ρ i 1 is a segment.) Clearly, J is an l-chain if and only if Z( j∈J ∆ (l) ρ j ) is a ladder representation. We say that two partitions of I are equivalent if one can be obtained from the other by applying a permutation τ of I such that ρ τ (i) = ρ i for all i. It is easy to see that for any l 1 there exists a partition P (l) (I) of I consisting of l-chains, such that for any J, J ∈ P (l) (I) at least one of the following conditions holds: (iii) for every j ∈ J and j ∈ J the segments ∆ Moreover, P (l) (I) is unique up to equivalence. Indeed, P (l) (I) can be defined inductively by taking a maximal l-chain J of I (with respect to inclusion) together with the partition P (l) (I \J). It follows from this description that if l m, then up to equivalence, P (l) (J) = {J ∈ P (l) (I) : J ⊂ J} for any J ∈ P (m) (I) and, in particular, P (l) (I) is a refinement of P (m) (I).
This concludes the proof of the proposition. 2 Remark 2.9. It can be shown that up to permutation, σ 1 , . . . , σ t are the unique ladder representations such that σ = σ 1 × · · · × σ t . We will not need to use this fact.
By Frobenius reciprocity and [LM16, Corollary 4.10], we have the following corollary.
Corollary 2.10. For any π ∈ Irr gen GL n , By induction on m, we get the following result.

The models
3.1 Definition of models Throughout this section, fix π ∈ Irr gen GL n , let σ = Sp(π, m) ∈ Irr G and let P = P σ = P m,n = M U be the standard parabolic subgroup of G of type m (n, . . . , n). LetŪ = t U be the opposite of U . Fix a non-trivial character ψ of F . Let Ψ be the function on G given by

Local Rankin-Selberg integrals for Speh representations
We denote the restriction of Ψ to a subset A of G by Ψ A . Let N = N mn (respectively,N = t N ) be the group of upper (respectively, lower) unitriangular matrices in G. Then Ψ N is a (degenerate) character on N that is trivial on U and non-degenerate on N M := N ∩ M . Recall that by (3), is, the Zelevinsky model of σ. By Corollaries 2.10 and 2.11, for any W Ze ∈ W Ψ N (σ), we have where π ⊗m = m π ⊗ · · · ⊗ π, δ P is the modulus character of P and δ = δ is a particular case of more general models considered in [MW87] (for any reductive group). Let us recall the setup. Let g = Mat nm,nm be the Lie algebra of G over F . For any cocharacter ϕ of the diagonal torus T , let g = j∈Z g ϕ j be the corresponding grading Consider the nilpotent nm × nm matrix J m,n consisting of m lower triangular Jordan blocks of size n × n each. We say that ϕ is of type (m, n) if Ad(ϕ(s))J m,n = s −1 J m,n , or equivalently, if λ ϕ i − λ ϕ i+1 = 1 for all i not dividing n. If ϕ is of type (m, n), then Ψ Uϕ is a character of U ϕ . By [MW87] (in particular, § II.2) we obtain the following theorem. 3 Theorem 3.1 [MW87]. Suppose that ϕ is of type (m, n). Then the space In the setting of [MW87] the data pertaining to Theorem 3.1 is the pair (ϕ 2 , J m,n ). (The more general context of [MW87] applies to cocharacters which are not necessarily even. However, we will not discuss them here.)

E. M. Lapid and Z. Mao
We denote by W Ψ Uϕ (σ) the image of σ in Ind G Uϕ Ψ Uϕ . It consists of functions that are left equivariant (with respect to some character) under the centralizer of Ψ Uϕ in P ϕ .
Clearly, any ϕ of type (m, n) is determined by the m-tuple (λ ϕ n , λ ϕ 2n , . . . , λ ϕ mn ). We consider (m − 1)n + 1 cocharacters ϕ 0 , . . . , ϕ (m−1)n of T of type (m, n) such that λ ϕ i nk − λ ϕ i n(k+1) = max(0, nk − i), k = 1, . . . , m − 1. (Up to a cocharacter of the center of G, the cocharacter ϕ 2 (m−1)n corresponds to the SL 2 -triple pertaining to J m,n .) For simplicity we write n and 0 r < n, then U i consists of the matrices whose n × n blocks A j,k satisfy: (There is no constraint on A j,k if j < k and d + 2 < k.) In particular, U 0 = N while U (m−1)n consists of the matrices whose difference from the identity matrix is strictly upper triangular in each n × n block. Also, In analogy with the case m = 2 we will refer to W Ψ U (σ) as the Shalika model of σ. We caution, however, that in the literature, this terminology usually refers to the image of τ ∈ Irr GL 2n (possibly generic) in Ind GL 2n S ψ S under a non-trivial intertwining operator, if it exists (in which case it is unique up to a scalar [JR96]), where S is the Shalika group S = ( g g ) In X In : g ∈ GL n , X ∈ Mat n,n and ψ S is the character on S given by ψ(tr X). In the case at hand, any W Sh ∈ W Ψ U (σ) automatically satisfies an equivariance property under the centralizer of Ψ U in P (which is conjugate to S in the case m = 2), which justifies our terminology. In general, even for m = 2, Letting G act on right on the vector space F mn of row vectors with standard basis e 1 , . . . , e mn , P is the stabilizer of the flag (span{e nj−k : j = 1, . . . , m, k = 0, . . . , i − 1}) i=0,...,n .
We denote by κ : n GL m × · · · × GL m → M the isomorphism such that the ith copy of GL m acts on span{e nj+i : If X is a matrix over F , then we write X for the maximum of the absolute value of its entries.
Lemma 3.2. Suppose that W Sh ∈ W Ψ U (σ). Then there exists C > 0 with the following property.
Proof. It is enough to consider the case g = κ(g 1 , . . . , g n ) ∈ M . Assume that W Sh (g) = 0. Fix 1 i < n. For any X ∈ Mat m,m (F ), let Y ∈ U be the matrix such that Y nj+i,nk+i+1 = X j+1,k+1 for all 0 j, k < m and all other non-diagonal entries of Y are zero. Then W Sh (gY ) = 918 Local Rankin-Selberg integrals for Speh representations It consists of the matrices in G whose n × n blocks are all scalar matrices. Let ι : GL m → M Ψ be the resulting identification.
In general, write i = nd + r, 0 r < n. Then the reductive part of The unipotent radical of M Ψ i consists of the matrices whose n×n blocks A j,k satisfy the following requirements.
This group is trivial if i is divisible by n, and is of dimension d + 1 otherwise.
is a subgroup of G which contains U i and U i+1 as normal subgroups and the quotients (ii) We have a short exact sequence where c i denotes the map u → Ψ([·, u]) and P D denotes the Pontryagin dual. Dually, where c i is defined by the same formula as c i .

E. M. Lapid and Z. Mao
Proof. For any j, k, we have Therefore, U i ⊂ P i+1 and U i+1 ⊂ P i . Hence, U i and U i+1 normalize each other, so that U i · U i+1 is a subgroup of G that contains U i and U i+1 as normal subgroups. The equalities (6) are now consists of the upper unitriangular matrices whose n × n blocks A j,k satisfy the following requirements.
This group is of dimension d + 1. (It coincides with the unipotent radical of M Ψ i+1 unless i + 1 is divisible by n.) In the rest of the section we endow various unipotent subgroups of G with Haar measures. Thanks to the choice of basis e 1 , . . . , e mn , the Lie algebra of any of these unipotent groups has a natural basis as a vector space over F . Our convention will be to take the measure corresponding to the product measure where the Haar measure on F is the one which is self-dual with respect to ψ.
The following is a special case of [GGS16] (see also [GGS17]). For future reference and in order to be self-contained we provide the (elementary) proof. We refer the reader to [GGS16,GGS17] for a more thorough discussion about interplay between models.
Proposition 3.5. For any i = 0, . . . , (m − 1)n − 1, the map . Its inverse is given by In both cases the integrands are compactly supported.
. It follows from Lemma 3.3 and the smoothness of W i that W i | U i+1 is compactly supported modulo U i ∩ U i+1 and that, for any u ∈ U i , . By similar reasoning as before, W i+1 | U i is compactly supported modulo U i ∩ P Ψ i+1 . By Lemma 3.3 and Fourier inversion, the map (9) defines a G-equivariant left inverse to T i . Since the spaces are irreducible, it is also a right inverse. 2 920 Local Rankin-Selberg integrals for Speh representations Remark 3.6. Suppose that σ is unramified, ψ has conductor O and W i ∈ W Ψ U i (σ) is the unramified vector such that W i (e) = 1. Then T i W i (e) = 1. This follows immediately from the proof of Proposition 3.5.

Model transition part II
We now introduce a subgroup of G that will play an important role in what follows. Let D = D m,n be the joint stabilizer of the vectors e jn , j = 1, . . . , m, in G and let The following lemma is straightforward.
Hence, we can rewrite (9) as where the integrand is compactly supported.
Proof. Let g = ank ∈ G with a = diag(a 1 , . . . , a nm ), n ∈ N and k ∈ G(O). It is well known and easy to prove that if g ∈N , then g max i=1,...,mn | nm j=i a j |. On the other hand, it is also easy to see that if g ∈ D, then |a jn | 1 for j = 1, . . . , m. Thus, if g ∈ D and W Ze (g) = 0, then by the support condition for Whittaker functions we get |a i | C 1 for all i where C 1 depends only on W Ze . By the above, if moreover g ∈N , then g is bounded in terms of W Ze as required.

E. M. Lapid and Z. Mao
Proof. It is enough to evaluate χ i on an element ι(t) where t = diag(t 1 , . . . , t m ) is in the diagonal torus of GL m . Note that ι(t) lies in the center Z M of M . Writing W i = N ∩U i W Ze (u·) du with W Ze ∈ W Ψ N (σ) (the integrand is compactly supported by Lemma 3.8), the required relation follows from the equality ∈ SL m (alternating signs on the non-principal diagonal) andw m,n = ι(w m ). By Lemma 3.9, we have W Sh (w m,n g) = W Sh (g) for any W Sh ∈ W Ψ U (σ).
Lemma 3.11. The inverse of T is given by where the integrand is compactly supported.
Proof. From Proposition 3.5 we only need to check that the integrand is compactly supported. Assume that W Sh = T W Ze . By Remark 3.10, the integral equals By Lemma 3.8 the integrand is compactly supported in v,ū. Thus, the integrand on the righthand side of (11) is compactly supported.
Lemma 3.14. For any i = 0, . . . , (m − 1)n − 1, the map is an isomorphism between Ind D D∩U i Ψ U i and Ind D D∩U i+1 Ψ U i+1 , whose inverse is given by As in the proof of Proposition 3.5, by Lemma 3.7 the function is the Fourier transform of the function W i Ψ −1 U i+1 | U i ∩U \U i+1 ∩U at c i (u). The first claim follows by Fourier inversion. Suppose where Ω is a compact subset of D. Fix g ∈ Ω. It follows from the above that The last part now follows from the fact that ind D U Ψ U is irreducible. 2 From Lemma 3.14, Proposition 3.5 and Corollary 3.13 we obtain the following result.
We will prove a special case in Corollary 4.4 below. We do not know whether, in general, the restriction of σ to Q is of finite length. (See Proposition 7.1 for a very special case.) Recall that in the case m = 1 this is known (for any π ∈ Irr, not necessarily generic) using the theory of derivatives of Bernstein and Zelevinsky [BZ76,BZ77]. It would be very interesting to have an analogous theory for m > 1.

Unitary structure
We take the unnormalized Tamagawa measure on GL r with respect to ψ, that is, the Haar measure associated to the standard gauge form ( i,j=1,...,r dg i,j )/(det g r ) on GL r and the self-dual Haar measure on F with respect to ψ. Following our convention on Haar measures for unipotent groups (see § 3.2), we obtain a (right) Haar measure on the F -points of any algebraic group whose reductive part is a product of general linear groups. This will cover all algebraic groups considered here.
Throughout this section let π, π ∈ Irr gen GL n and let σ = Sp(π, m) and σ = Sp(π , m). We will work with the models considered in the previous section.
For any 0 i (m − 1)n and s ∈ C, we define a bilinear form on and for W Sh ∈ W Ψ U (σ), W Sh ∈ W Ψ −1 U (σ ), It follows from Lemma 3.2 that |det| is bounded above on the support of W Sh | D . Hence, if B Sh (W Sh , W Sh , s) converges absolutely at s 0 ∈ R, then it converges absolutely for any s with Re s s 0 .
A similar statement holds for any B i , although we will not use it. We also write B i (W i , W i ) = B(W i , W i , 0) assuming the latter is well defined (either as a convergent integral, or by analytic continuation), in which case it is D-invariant.
In general, we do not know whether B i (·, ·) is always defined. (See § 6 and in particular Example 6.5 for further discussion.) Proposition 4.1. The integral defining B i (W i , W i , s) converges for Re s + e(π) + e(π ) + 1 > 0. Moreover, for all 0 i < (m − 1)n, Finally, there exist W i ∈ W Ψ U i (σ) and W i ∈ W Local Rankin-Selberg integrals for Speh representations Remark 4.2. In Proposition 6.2 below we prove that B i (W i , W i , s) admits meromorphic continuation in s to a rational function in q s .
Proof. First note that the last statement follows from Corollary 3.15. Next, we show the convergence of the integral defining B 0 . Upon twisting π and π by |·| (s+e(π )−e(π))/2 and |·| (s+e(π)−e(π ))/2 respectively and using the inequality |xy| (|x| 2 + |y 2 |)/2, we may assume without loss of generality that π = π, W Ze = W Ze and s = 0. Thus, we need to show the convergence of provided that e(π) > − 1 2 . In fact, we show a slightly stronger assertion, namely the convergence of for any 0 Φ ∈ S(Mat m,nm (F )) where η ∈ Mat m,nm (F ) is the matrix whose ith row is e ni , i = 1, . . . , m. Note that the stabilizer of η under the right G-action on Mat m,nm (F ) is D. Since the modulus character of D is |det| m , (17) is formally well defined and can be rewritten as We may identify the vector space Mat m,nm (F ) with Mat m,m (F n ). Observe that, for any l = diag(g 1 , . . . , g m ) ∈ M , g ∈ G, we have |det l| (m−1)/2 U D \U Φ(ηulg) du =Φ g (e n g 1 , . . . , e n g m )δ (l) whereΦ g ∈ S((F n ) m ) is the functioñ where the integral is taken over the n m 2 -dimensional affine space of upper triangular F n -valued m × m-matrices whose diagonal entries are v 1 , . . . , v m . Thus, (18) is equal to g (e n g 1 , . . . , e n g m ) m i=1 |det g i | i dg 1 · · · dg m |det g| m dg where l = diag(g 1 , . . . , g m ) ∈ M . Thus, by (5b) the inner integral is a finite linear combination of products of Rankin-Selberg integrals for π ×π at i, i = 1, . . . , m. The assumption that e(π) > − 1 2 guarantees that these Rankin-Selberg integrals converge. Since the outer integral is a finite sum, we obtain the convergence of (17).

E. M. Lapid and Z. Mao
Moreover, by the unitarity of the Fourier transform and the argument of Proposition 3.5 (cf. (10)) we have where the integrals are absolutely convergent. (We can also write the integrals as It follows that if at least one of integrals converges, then so does the other and We can now conclude the convergence for all i and the identity (16) since they clearly reduce to the case i = 0. 2 Theorem 4.3. Suppose that π = π ∨ (or equivalently, σ = σ ∨ ) and π is (AT) (see § 2.1). Then In particular, if π ∈ Irr gen GL n is unitarizable, then B i gives a unitary structure on W Ψ U i (σ).
Proof. By Proposition 4.1 B i (·, ·) is well defined and not identically zero. To show invariance it suffices to consider i = 0. We use induction on m. The case m = 1 (in which D is the standard mirabolic subgroup) is well known and follows from Bernstein's theorem [Ber84]. For the induction step, let m > 1 and let Q be the subgroup of the standard maximal parabolic subgroup of G of type ((m − 1)n, n) consisting of the matrices whose lower right n × n corner is upper unitriangular. Write Here we consider GL (m−1)n (and hence, D m−1,n ) as a subgroup of G. (Note that δ D = |det| m while δ D∩Q = |det| n+m−1 .) By (5a) and the induction hypothesis, the inner integral is left-(Q , |det| n−1 )-equivariant in g. Hence, we can replace the domain of outer integration by Q \D 1,mn where D 1,mn is the standard mirabolic subgroup of G (the stabilizer of e mn ). (Note that δ D 1,mn = |det| and δ Q = |det| n .) It follows that B 0 (·, ·) is D 1,mn -invariant. By Bernstein's theorem, it is G-invariant as required. 2 We immediately deduce a special case of Conjecture 3.16.

926
Local Rankin-Selberg integrals for Speh representations Corollary 4.4. Conjecture 3.16 holds for any π ∈ Irr (AT ) GL n . In particular, it holds for any unitarizable π ∈ Irr gen GL n .
In view of Theorem 4.3 and Bernstein's theorem, it is natural to make following related conjecture.
Perhaps even more is true.
Conjecture 4.7. For any m-homogeneous σ, σ ∈ Irr G, there is a unique up to scalar Dinvariant bilinear form on σ × σ .
(We do not know whether this is known even in the case m = 1.)

Statement of the result
We write an analogue of the Rankin-Selberg integral for σ × σ on the Shalika model as follows. Recall that η ∈ Mat m,nm (F ) is the matrix whose ith row is e ni , i = 1, . . . , m, so that D is the stabilizer of η in G. For any W Sh ∈ W Ψ U (σ), W Sh ∈ W Ψ −1 U (σ ), Φ ∈ S(Mat m,nm (F )), consider This expression was already considered in some form in the proof of Proposition 4.1. Note that in the case n = 1 (where U = 1) Z reduces to the generalized Tate integral for (a character of) GL m considered by Godement and Jacquet [GJ72].
For any k, let Theorem 5.1. The integral Z(W Sh , W Sh , Φ, s) has the following properties.
where c is a measure-theoretic constant (depending only on F , m and n). (iv) We have a local functional equation We will prove the theorem below by relating Z(W Sh , W Sh , Φ, s) to the usual Rankin-Selberg integrals.

A result of Jacquet, Piatetski-Shapiro and Shalika
Recall the GL n × GL n local Rankin-Selberg integrals studied by Jacquet, Piatetski-Shapiro and Shalika [JPSS83]. They are given by Nn (π ), Φ ∈ S(F n ) and s ∈ C. The integral converges for Re s + e(π) + e(π ) > 0 and admits a meromorphic continuation in s to a rational function in q s . The quotient is a Laurent polynomial in q s which can be made non-zero at any given s ∈ C by an appropriate choice of W , W , Φ. Moreover, we have a functional equation andΦ is the Fourier transform of Φ given bŷ Φ(y) =  (g 1 , . . . , g m ) ∈ M . This is a linear combination of products Nn (π ) and Φ i ∈ S(F n ). Thus, the integral defining Z M (W, W ,Φ, (s 1 , . . . , s m )) is absolutely convergent provided that Re s i + e(π) + e(π ) > 0 for all i.
Moreover, we have a functional equation

Proof of the theorem
The fulcrum for Theorem 5.1 is the following proposition.
By Proposition 4.1 we get

E. M. Lapid and Z. Mao
We write this as The required identity now follows from (19). For convergence, as in the proof of Proposition 4.1, we may assume that Φ 0, s ∈ R, π =π and W 2 = W 1 , so that all the integrands considered above are non-negative. Therefore, the manipulations are justified for s + 2e(π) + 1 > m by (23). 2 Proposition 5.2 immediately implies the first part of Theorem 5.1 (absolute convergence). In view of Remark 3.6, Proposition 5.2 also reduces the second and third parts of Theorem 5.1 (analyticity and unramified computation) to the analogous statements for the usual Rankin-Selberg integrals.
Remark 5.3. If σ and σ are unramified, then However, in general for π, π ∈ Irr gen GL n , the equality does not always hold.
Finally, we prove the functional equation (last part of Theorem 5.1). For any W Ze ∈ W Ψ N (σ), define W Ze ∈ W Ψ −1 N (σ ∨ ) by W Ze (g) = W Ze (w nm t g −1 ). Then T W Ze = T ( W Ze ). Note that w mn = diag( m w n , . . . , w n )w m,n where w m,n = ι(w m ); write g = w m,n t g −1 , g ∈ G. Then, for any g ∈ G, we have and by Fourier inversion The last part of Theorem 5.1 therefore follows from Proposition 5.2 and the functional equation (24) using the change of variable g → g in the integral on the right-hand side of (25).
This finishes the proof of Theorem 5.1.

More analytic results
In this section we prove some more analytic properties of the zeta integrals defined in the previous section, as well as the bilinear forms of § 4. Some of these properties are well known in the case m = 1. However, there are also some new phenomena.

Relation between zeta integrals and B Sh
Recall that Q = D M Ψ , δ Q | D = δ D = |det| m and δ Q | M Ψ = 1. Hence, we can write Z(W Sh , W Sh , Φ, s) as |det l| s dl |det p| s−m dp |det g| s dg.
Using Lemma 3.9 and the identification ι : GL m → M Ψ , we get where, for any character ω of F * , f Φ,ω,s (g) = GLm Φ g (l)ω(det l)|det l| ns dl |det g| s and Φ g ∈ S(Mat m,m (F )) is given by Φ g (X) = Φ(µ(X)g) where µ(X) ∈ Mat m×nm is the matrix whose ith row is m j=1 X i,j e nj . Note that Φ → f Φ,ω,s is an intertwining map from S(Mat m,nm (F )) ⊗ |det| s to Ind G Q ν s where ν s is the character on Q such that ν s | D = |det| s−m/2 and ν s • ι = ω −1 • det.
Proof. This follows from Corollary 3.13 and (26) by taking W Sh such that W Sh | D is supported in U Ω for a small neighborhood Ω of e and Φ supported in a small neighborhood of η. Similarly, let ord Z (s) = ord Z;σ,σ (s) 0 be the maximal order of pole of Z(W Sh , W Sh , Φ, ·) at s as we vary W ∈ W Ψ U (σ), W ∈ W Ψ −1 U (σ ) and Φ ∈ S(Mat m,nm (F )). By Lemma 6.1, we have ord Z (s) 0 for all s. We can sharpen this as follows.
Proof. It is enough to prove the meromorphic continuation for i = (m − 1)n, that is, for B Sh . This case follows from equality (26). Indeed, taking ω = ω π ω π and Φ to be the characteristic function of a small neighborhood of η, f Φ,ω,s is supported in QΩ for a small neighborhood Ω of e and hence Z(W Sh , W Sh , Φ, s) is a non-zero constant multiple of B Sh (W Sh , W Sh , s − m). We also get that ord B (s − m) ord Z (s) for all s.
On the other hand, f Φ,ω,s (g) is a generalized Tate integral with respect to GL m , and hence Remark 6.3. Note that if π and π are tempered, then it follows from Theorem 5.1 part (ii) that ord Z (s) = 0 unless Re s ∈ 1 2 Z and Re s m. Thus, in general, many poles of f Φ,ωπω π ,s do not contribute a pole for Z(·, ·, ·, s).
Remark 6.4. In general, we do not know what precisely is the fractional ideal of Z[q ±s ] generated by If both π and π are unitarizable, then we expect that this ideal is generated by m−1 i=0 L(s − i, π × π ), that is, part (ii) of Theorem 5.1 is tight in this case.
6.2 More results in the (AT) case Proposition 6.6. Suppose that π is (AT) and let π = π ∨ . Then, for any W Sh ∈ W Ψ U (σ), where both sides are well defined.
Proof. By the first part of Theorem 5.1, the integral defining Z(W Sh , W Sh , Φ, m) is absolutely convergent. Moreover, since the modulus function of D is |det| m , we can write For π = π ∨ , by Theorem 4.3 we get From the functional equations (22) we deduce the following corollary.
Corollary 6.7. Suppose that π is (AT) and let π = π ∨ . Then ord Z (0) is equal to the order of the zero of the product of γ-factors on the right-hand side of (22) at s = 0.

Local Rankin-Selberg integrals for Speh representations
Under mild assumptions, we can give a lower bound for the real part of the first location of a pole.
Proof. Indeed, taking W Sh = W Sh and Φ 0, the right-hand side of (25) is a power series in q −s with non-negative coefficients a k which vanish for k 0. Assume to the contrary that Z(W Sh , W Sh , Φ, s) is holomorphic throughout Re s m−1. Then the power series would converge at s = m − 1. However, the integral on the right-hand side of (25) diverges at s = m − 1 since it contains F * Φg (λe n , e n , . . . , e n )|λ| s−m+1 dλ as an inner integral. We obtain a contradiction. 2 Corollary 6.10. Suppose n, m > 1, π = π and ω π is unitary. Then any B i admits a pole in the right half plane Re s −1. Hence, there exists W Sh ∈ W Ψ U (π) such that the integral defining B Sh (W Sh , W Sh , s) diverges for all s −1. In particular, U M Ψ \G |W Sh (g)| 2 dg diverges.
Proof. Indeed, by Lemma 6.9 we have ord Z (s) > 0 for some s with Re s m − 1. Hence, by Proposition 6.2, ord B (s − m) > 0 for that s (since n, m > 1). Therefore, the integral defining B Sh (W Sh , W Sh , s) diverges for all s −1 (cf. (15)). In particular, Our final result in this section is the following lemma.
Lemma 6.11. Suppose that π is (AT) and let π = π ∨ . Then ord Z (0) = ord B (−m) + 1. In particular, if π is supercuspidal, then B is holomorphic at s = −m (cf. Example 6.8). Moreover, let Then there exists a constant c (depending only on F , m and n) such that and Φ ∈ S(Mat m,nm (F )).
(Recall that the latter diverges for W Sh = W Sh if m > 1.) 7. The case n = m = 2 Given σ = Sp(π, m), it is natural to ask what is the asymptotic behavior of a function in W Ψ U (σ) or (what is essentially the same thing) in K ψ (σ). In the case n = 2 or if n = 3 and m = 2, P Ψ is a spherical subgroup of G and the problem can in principle be analyzed by the methods of [SV17]. We will only treat here the case where m = n = 2 and π is supercuspidal, in a self-contained way, without appealing to the general results of [SV17]. For n > 2 and m > 1 (excluding the case n = 3 and m = 2), P Ψ is no longer a spherical subgroup and the problem seems to be more difficult than the analogous problem for W Ψ N (σ). We have little to say about it. We note that in the case where n = 2 and σ is unramified, an explicit formula for the unramified W Sh was given by Sato [Sat05]. This is a special case of a formula of Sakellaridis [Sak06]. In general, it would be an interesting problem to obtain such an explicit formula in the unramified case for any m, n. Once again, this goes beyond the scope of [Sak13].

934
Local Rankin-Selberg integrals for Speh representations Recall that in the case at hand, Q = P = P w where w = , and that ω Ψ π is the character of P Ψ whose restriction to U is Ψ U and whose composition with ι is ω π • det. Also, a b c d = max(|a|, |b|, |c|, |d|).
Proof. First note that property (29) determines ϕ uniquely (if it exists). It then also follows that if (29) is satisfied, then A necessarily intertwines the Q-action. Moreover, if (30) is satisfied, then ϕ = 0 if and only if L is compactly supported modulo P Ψ . Also note that in relation (29) it is enough to consider g 2 = 1 since both sides are (GL diag 2 , ω π • det)-equivariant. (For simplicity write g = g 1 .) Recall that, by Lemma 3.2, there exists a constant C 1 > 1 such that L(κ(g, 1)) = 0 unless g C 1 . Suppose that L = T (W Ze )| Q . Write g = u (y) diag(t 1 , t 2 )k where u (y) = 1 y 1 with k ∈ GL 2 (O). We claim that there exists C 3 such that for all x ∈ F such that |x| > C 3 |t 2 |. Indeed, writē where the superscript denotes conjugation. Our claim follows since wκ(g, 1)w −1 = diag(g, I 2 ). Next, we show that there exists a compact set C of GL 2 (F ) such that if g C 1 and g / ∈ C, then both sides of (31) vanish if |x| C 3 |t 2 |.
First, note that the condition g C 1 means that |t 1 |, |t 2 |, |t 2 y| C 1 . Now Therefore, if the right-hand side of (31) is non-zero, then by the supercuspidality of π, x and t 1 t −1 2 are confined to a compact subset of F * . Since |x| C 3 |t 2 |, we infer that t 2 belongs to a compact set of F * , and hence also t 1 . Finally, |y| is bounded since |t 2 y| C 1 . Hence, g belongs to a compact set. 935

E. M. Lapid and Z. Mao
On the other hand,ū (x)κ(g, 1) ∈ N diag(t 1 , 1, t 2 , 1)ū(t −1 2 x)K and since |t −1 2 x| C 3 we infer from the supercuspidality of π that if the left-hand side of (31) is non-zero, then t 1 , t 2 belong to a compact subset of F * . As before, g belongs to a compact set. Our claim follows.
In conclusion, (31) holds for all x ∈ F provided that g C 1 and g / ∈ C. Integrating (31) over x ∈ F , we conclude that if g C 1 and g / ∈ C, then By (27), this is equal to MC ϕ (g, 1) where ϕ ∈ W Ψ N M (π 1/2 ⊗ π −1/2 ) is the restriction of W Ze (·w) to M . Thus, (29) and (30) hold. In view of Corollary 3.13, this proves the proposition. (Note that W Ze → ϕ is Q-equivariant since w conjugates P to Q.) 2 Remark 7.2. It follows from (the proof of) Proposition 7.1 that there exists a non-zero W Ze ∈ W Ψ N (σ) that vanishes on M (in which case T W Ze (·w)| Q ∈ K ψ (σ) is compactly supported modulo P Ψ ). This can be also shown directly by realizing σ as the image of the intertwining operator π 1/2 × π −1/2 → π −1/2 × π 1/2 and taking the image of a suitable vector in π 1/2 × π −1/2 that is supported in the big cell.
Proof. We may assume without loss of generality that π is unitary. Then B(W Sh , W Sh , s − 1) = GL 2 |det g| s−1 |W Sh (κ(g, 1))| 2 dg. By Proposition 7.1, the analytic properties are governed by those of GL 2 : g 1 |det g| s−1 |MC ϕ (g, 1)| 2 dg, which can be written as Thus, the poles are simple and are confined to q 2s = 1. If q s = 1, then the residue is clearly non-zero. If q s = −1, then the residue is a constant multiple of where ω is the non-trivial quadratic unramified character of F * . Thus, by (28)

Global heuristics
Let F be a number field with ring of adeles A. We consider G = GL nm as a group over F and write G(A) 1 = {g ∈ G(A) : |det g| = 1}. As before, let Q be the stabilizer of span{e ni : i = 1, . . . , m} in G -a maximal non-standard parabolic subgroup of G of type ((n − 1)m, m). For any Φ ∈ S(Mat m,nm (A)) and a Hecke character ω of F * \A * , consider the degenerate normalized Eisenstein series that is given by for Re s 0 (more precisely, Re s > m if ω is unitary) where, as in § 6.1, Φ g (l)ω(det l)|det l| ns dl |det g| s and Φ g ∈ S(Mat m,m (A)) is given by Φ g (X) = Φ(µ(X)g), where µ(X) ∈ Mat m,nm (A) is the matrix whose ith row is m j=1 X i,j e nj . By the method of Tate's thesis (which goes back to Riemann), E Φ,ω admits a meromorphic continuation with finitely many (simple) poles and a functional equation As before, let P = M U be the standard maximal parabolic subgroup of G of type ( We extend |·| M to a left-U (A)-and right-K-invariant function |·| P on G(A) where K is the standard maximal compact subgroup of G(A). For any x = (x 1 , . . . , x m ) ∈ R m >0 and λ = (λ 1 , . . . , λ m ) ∈ C m , we write x λ = i x λ i i . Let π = ⊗π v be an irreducible cuspidal representation of GL n (A). Let φ : G(A) → C be a smooth function such that, for all g ∈ G(A), the function l ∈ M (A) → δ P (l) −1/2 φ(lg) belongs to the space of m π ⊗ · · · ⊗ π. The Eisenstein series exists and is a square-integrable automorphic form on G(F )\G(A) 1 which is non-zero for a suitable φ. As we vary φ, we obtain an irreducible automorphic representation of G(A) whose local components are Sp(π v , m). (It is well known that as we vary over π and m 1, these representations furnish the entire automorphic discrete spectrum of the general linear group 937 E. M. Lapid and Z. Mao [MW89].) Similarly, let π be another irreducible cuspidal representation of GL n (A) and let φ and ϕ be analogous functions with respect to π .
Formally, we would have liked to consider the integral where ω = ω π ω π . For m = 1, this is of course the classical Rankin-Selberg integral. Unfortunately, for m > 1 this integral does not converge as none of the functions that appear in the integrand is rapidly decreasing. A suitable regularization (in the spirit of [Zag81] or later accounts) is therefore needed in order to make sense of (33). We will not pursue this matter here. Instead, we will be content with a purely heuristic argument, anticipating what a possible regularization of (33) would yield.
As in the case m = 1, we unfold (formally) expression (33). For any i = 1, . . . , m, let Q i = L i V i be the stabilizer of the flag (span{e nj−k : j = 1, . . . , m, k = 0, . . . , r − 1}) r=1,...,i in G. Thus, Q 1 = Q ⊃ Q 2 ⊃ · · · ⊃ Q n−1 = Q n = P and L i GL m(n−i) × L i with L i i GL m × · · · × GL m . Let p i : Q i → L i be the resulting projection and let Q i be the inverse image of GL m diagonally embedded in L i . In particular, Q 1 = Q 1 = Q and Q n = M Ψ U . Note that, for all i = 1, . . . , n − 1, Q i+1 is the stabilizer in Q i of the character Ψ V i and V i /V i−1 is abelian (and can be identified with Mat m,(n−i)m ), where for consistency we let V 0 = 0.
In the first step we unfold (33) to write it as The Pontryagin dual of the compact abelian group V 1 (F )\V 1 (A) is isomorphic to Mat m,(n−1)m (F ). We consider only the contribution from the non-degenerate χ, that is, those corresponding to matrices of rank m (anticipating that the degenerate ones will not contribute, either by the cuspidality of π or by the regularization procedure itself). The non-degenerate characters form a single orbit under Q = Q 1 , namely the orbit of Ψ V 1 , and the stabilizer of Ψ V 1 is Q 2 . We thus get which we write as Local Rankin-Selberg integrals for Speh representations Once again, we expand the inner integral according to characters of the compact abelian group V 2 (A)/V 1 (A)V 2 (F ) and consider only the non-degenerate characters. Continuing in this way, we get, for k = 1, . . . , n, For k = n, we obtain , ω π • det)-equivariant (taking into account the identification ι : GL m → M Ψ ). Therefore, up to a volume factor, we get This integral (which actually converges for Re s > m if ω is unitary) is Eulerian. Let S be a finite set of places of F containing all the archimedean ones such that, for all v / ∈ S, ϕ and ϕ are G(O v )-invariant (and, in particular, π v and π v are unramified), ψ v has conductor O v , Φ is invariant under translation by Mat m,mn (O v ) and Φ(X) = 0 unless X v ∈ Mat m,mn (O v ). Using (26) and Theorem 5.1 part (iii), up to a measure-theoretic constant, the integral (34) is equal to where L S (s, π×π ) is the partial Rankin-Selberg L-function and, for any W Sh ∈ W Ψ U (Sp(π S , m)) and W Sh ∈ W Ψ −1 U (Sp(π S , m)), which is essentially the product over v ∈ S of the integrals considered in § 5. (We tacitly assume that much of the analysis of § § 3-5 carries over to the archimedean case.) be the standard module which admits σ = Sp(π, m) as the Langlands quotient. We realize Π in the subspace W Ψ N (Π) of Ind G N Ψ N consisting of functions W such that l ∈ M → δ −1/2 P (l)δ −1 (l)W (lg) ∈ W Ψ N M (π ⊗m ) for all g ∈ G. Define an intertwining operator on W Ψ N (Π) by wherew m,n is as in Remark 3.10. The integral defining M W is absolutely convergent and its image is Then the bilinear form converges absolutely and defines a G-invariant pairing on In the rest of the appendix we prove the following identity.
The identity will follow from a series of identities proved below. For i = 1, . . . , m − 1, let U i be the unipotent radical of the standard parabolic subgroup P i of G of type (in, n, n, . . . , n). LetŪ i = t U i be its opposite.
Lemma A.2. Let τ ∈ Irr gen GL n . Then, for any W Ze ∈ W Ψ N (Sp(τ, m)), we have Proof. Let W i Sh = T (i−1)n W Ze ∈ W Ψ U (i−1)n (Sp(τ, m)). Recall that U (i−1)n is the subgroup of P i consisting of matrices whose n × n blocks A j,k satisfy: -A j,j is upper unitriangular for all j = 1, . . . , m; -A j,k is strictly upper triangular if j = k and j, k i; -A j,k = 0 if j > k and j > i.
(There are no conditions on A j,k if k > j and k > i.) The inverse transform in Proposition 3.5 gives We may replace the domain of integration by (N ∩ U (i−1)n ∩ GL in )\(N D ∩ GL in ) where GL in is embedded in G by h → h I (m−i)n . Let U i = U (i−1)n ∩ GL in = U ∩ GL in and D i = D ∩ GL in . Thus, the above integral can be taken over N ∩ U i \N ∩ D i , and by Lemma 3.11 the integrand is compactly supported.
The expression on the left-hand side of (A.3) is The same argument as in Lemma 3.8 shows that the function W i Sh (uū) is compactly supported in u uniformly in u. Thus, the above double integral is absolutely convergent. Changing the order of integration and making a change of variable inū, we get Notice that the partial integration over U ∩Ū i ⊂ D ∩Ū i is the composition of the transforms T j defined in Proposition 3.5 for j = (i − 1)n, . . . , (m − 1)n − 1. Thus, the above expression is where W Sh = T W Ze . By Lemma 3.9, W Sh (ι(ŵ i )g) = W Sh (g). The above expression becomes Now Lemma 3.11 gives (A.3). For the second part, we only need to note that, for allv ∈Ū i , we have W Ze (ι(ŵ i )v) = W Ze (ι(ŵ i )). 2 WriteŪ as a (semidirect) product of abelian groupsŪ 2Ū3 · · ·Ū m , whereŪ i consists of the elementsū inŪ such thatū j,k = δ j,k if j n(i − 1) or j > ni. For brevity, for any i = 1, . . . , m, we denote the iterated integral Ū 2 ∩D Ū 3 ∩D

· · ·
Ū i ∩D f (ū i · · ·ū 3ū2 ) dū i · · · dū 3 dū 2 (assuming convergence) by where p i : GL in Ū i → GL in is the projection. Now writev =v 1v2 wherev 1 ∈Ū i ∩ D and v 2 ∈Ū ∩ D i−1 and note that p i (ū) ∈ GL in ∩Ū normalizesŪ i ∩ D. Therefore, (A.4) is equal to Here we can interchange the order of integration as l is integrated over a fixed compact set, by the choice of W . The claim now follows from Lemma A.3 and (A.1). 2 Let W Ψ −1 N (σ ∨ ) be the subspace of W Ψ −1 N (σ ∨ ) consisting of the functions W ∨ Ze such that W ∨ Ze | D is contained in ind D N D Ψ N and W ∨ Ze | D is supported in PŪ ∩ D.
Lemma A.5. For any W Ze ∈ W Ψ N (σ) and W ∨ Ze ∈ W Ψ −1 N (σ ∨ ) , we have Proof. Note that PŪ ∩ D = U D D MŪD . Thus, the left-hand side is where we made a change of variablev → l −1ū−1v l. By the condition on W ∨ Ze | D and Lemma 3.8, the integrand is compactly supported, which justifies the previous steps. Applying Lemma A.2 for both integrals overŪ D , we get the required statement. 2 Since (A.2) holds up to a scalar, in order to conclude Proposition A.1, it suffices, in view of Lemmas A.4 and A.5, to show the existence of W ∈ W Ψ N (Π) and W ∨ Ze ∈ W Ψ −1 N (σ ∨ ) such that the right-hand side of (A.5) is non-zero. By Corollary 3.15, given φ ∈ S(Ū D ) and W ∈ ind This finishes the proof of Proposition A.1.
Remark A.6. Let us return to the setup of § 8. It is well known that the Petersson inner product of cusp forms in π factorizes as the product over v of the Bernstein inner product on the Whittaker model of π v . Now let ϕ be as in (32). The Ψ N th Fourier coefficient of ϕ is the Ψ N M th Whittaker coefficient of the constant term of ϕ, which is given by the iterated residue M −1 of the global intertwining operator. Proposition A.1 (assumed to work in the archimedean case as well) gives a factorization of the square of the Petersson norm of ϕ in terms of the local inner product (14) on the Zelevinsky model of Sp(π v , m). Indeed, by the Maass-Selberg relations, the Petersson inner product is given by M −1 and Proposition A.1 will reduce the statement to the classical case.