Rankin-Selberg local factors modulo $\ell$

After extending the theory of Rankin-Selberg local factors to pairs of $\ell$-modular representations of Whittaker type, of general linear groups over a non-archimedean local field, we study the reduction modulo $\ell$ of $\ell$-adic local factors and their relation to these $\ell$-modular local factors. While the $\ell$-modular local $\gamma$-factor we associate to such a pair turns out to always coincide with the reduction modulo $\ell$ of the $\ell$-adic $\gamma$-factor of any Whittaker lifts of this pair, the local $L$-factor exhibits a more interesting behaviour; always dividing the reduction modulo-$\ell$ of the $\ell$-adic $L$-factor of any Whittaker lifts, but with the possibility of a strict division occurring. In our main results, we completely describe $\ell$-modular $L$-factors in the generic case. We obtain two simple to state nice formulae: Let $\pi,\pi'$ be generic $\ell$-modular representations; then, writing $\pi_b,\pi'_b$ for their banal parts, we have \[L(X,\pi,\pi')=L(X,\pi_b,\pi_b').\] Using this formula, we obtain the inductivity relations for local factors of generic representations. Secondly, we show that \[L(X,\pi,\pi')=\mathbf{GCD}(r_{\ell}(L(X,\tau,\tau'))),\] where the divisor is over all integral generic $\ell$-adic representations $\tau$ and $\tau'$ which contain $\pi$ and $\pi'$, respectively, as subquotients after reduction modulo $\ell$.


Introduction
Let F be a locally compact non-archimedean local field of residual characteristic p and residual cardinality q, and let R be an algebraically closed field of characteristic ℓ prime to p. In this article, following Jacquet-Piatetskii-Shapiro-Shalika in [7] for complex representations, we associate local Rankin-Selberg integrals to pairs of R-representations of Whittaker type ρ and ρ ′ of GL n (F ) and GL m (F ), and show that they define L-factors L(ρ, ρ ′ , X) and satisfy a functional equation defining local γ-factors.
In particular, we define local factors for ℓ-modular representations. The theory of ℓ-modular representations of GL n (F ) was developed by Vignéras in [15], culminating in her ℓ-modular local Langlands correspondence for GL n (F ), c.f. [18], which is characterised initially on supercuspidal ℓ-modular representations by compatibility with the ℓ-adic local Langlands correspondence. The possibility of characterising such a correspondence with ℓ-modular invariants forms part of the motivation for this work. Indeed, already for GL 2 (F ) this is an interesting question, answered in this special case by Vignéras in [17].
We show that an L-factor attached to ℓ-adic representations of Whittaker type is equal to the inverse of a polynomial with coefficients in Z ℓ , allowing us to define a natural reduction modulo-ℓ map on the set of ℓ-adic L-factors. Furthermore, for ℓ-modular representations π and π ′ of Whittaker type of GL n (F ) and GL m (F ), there exist ℓ-adic representations τ and τ ′ of Whittaker type of GL n (F ) and GL m (F ) which stabilise natural Z ℓ -lattices Λ and Λ ′ in their respective spaces such that the ℓ-modular representations induced by the actions of τ and τ ′ on Λ ⊗ Z ℓ F ℓ and Λ ′ ⊗ Z ℓ F ℓ are isomorphic to π and π ′ . Our first main result is a comparison between the L-factors and local γ-factors defined by these two reduction modulo-ℓ maps. Theorem.
2. Let θ be an ℓ-adic character of F . The local γ-factor associated to L(π, π ′ , X) and to the reduction modulo-ℓ of θ is equal to the reduction modulo-ℓ of the local γ-factor associated to L(τ, τ ′ , X) and θ.
A particularly interesting case is the L-factor associated to a pair of irreducible cuspidal representations of GL n (F ). For R-representations ρ and ρ ′ of GL n (F ), we write n(ρ, ρ ′ ) for the number of unramified characters χ of GL n (F ) such that ρ ≃ χ ⊗ (ρ ′ ) ∨ . Let τ and τ ′ be integral cuspidal ℓ-adic representations of GL n (F ) and let π and π ′ denote their reductions modulo-ℓ. In our second main result we examine the L-factor L(π, π ′ , X). Theorem.
This work further develops the theory of ℓ-modular local L-factors of Mínguez in [10]. In particular, we use his results on Tate L-factors modulo-ℓ. Recently, Moss in [13] has studied Lfactors attached to representations of GL n (F )× GL 1 (F ) in a more general setting, and has given partial results concerning the GL n (F )×GL n−1 (F ) convolution in [14]. In a further investigation, we intend to study the local factors associated to generic segments, in terms of the local factors associated to cuspidal representations, as well as the inductivity relation satisfied by the local factors.

Preliminaries
Before embarking on the study of local L-factors in positive characteristic, we introduce the basic theory and background on representations of the general linear group. In particular, starting with results given in the standard reference [15], we show how integration behaves with respect to group decompositions. Indeed, this deserves checking as not all formulae follow from mimicking the proofs in the characteristic zero setting, due to the presence of compact open subgroups of measure zero. Additionally, we review the theory of ℓ-adic and ℓ-modular representations of Whittaker type and reduction modulo-ℓ, drawing on results originally in [16], but our exposition will be influenced by the recent generalisation to inner forms of general linear groups in [11].

Notations
Let F be a locally compact non-archimedean local field of residual characteristic p with absolute value | |. Let o denote the ring of integers in F , p = ̟o the unique maximal ideal of o, and q the cardinality of k = o/p.
Let R be a commutative ring with identity of characteristic ℓ not equal to p. If R contains a square root of q, we fix such a choice q 1/2 .
Let M n,m = Mat(n, m, F ), M n = Mat(n, n, F ), η be the row vector (0, . . . , 0, 1) ∈ M 1,n , G n = GL(n, F ). We write ν for the character | | • det. Let G k n = {g ∈ G n , |g| = q −k } (and more generally X k = X ∩ G k n for X ⊂ G n ), B n the Borel subgroup of upper triangular matrices, A n the diagonal torus, N n the unipotent radical of B n .
We fix a character θ from (F, +) to R × and, by abuse of notation, we will denote by θ the character If λ is a partition of n, P λ is the standard parabolic subgroup of G n attached to it, M λ the standard Levi factor of P λ , and N λ its unipotent radical. If t + r = n, we let and H t,r = G t U t,r . By restriction, θ defines a character of U t,r . We let P n = H n−1,1 denote the mirabolic subgroup of G n .
We denote by w n the antidiagonal matrix of G n with ones on the second diagonal, and if n = r+t, we denote by w t,r the matrix diag(I t , w r ). Notice that our notations are different from those of [8] for U r,t , H t,r , and w t,r .
If φ ∈ C ∞ c (F n ), we denote by φ its Fourier transform with respect to the θ-self-dual R-Haar measure dx on F n satisfying dx(o) = q −s/2 , where s satisfies that θ | p s is trivial, but θ | p s−1 is not.
For G a locally profinite group, we let R R (G) denote the abelian category of smooth Rrepresentations of G. All R-representations henceforth considered are assumed to be smooth. For π an R-representation with central character, for example an R-representation of G n parabolically induced from an irreducible R-representation, we denote its central character by c π .
Let Q ℓ be an algebraic closure of the ℓ-adic numbers, Z ℓ its ring of integers, and F ℓ its residue field which is an algebraic closure of the finite field of ℓ elements. By an ℓ-adic representation of G we mean a representation of G on a Q ℓ -vector space, and by an ℓ-modular representation of G we mean a representation of G on a F ℓ -vector space. For H a closed subgroup of G, we write Ind G H for the functor of normalised smooth induction from R R (H) to R R (G), and write ind G H for the functor of normalised smooth induction with compact support.
We assume that our choice of square roots of q in F ℓ and Q ℓ are compatible; in the sense that the former is the reduction modulo-ℓ of the latter, which is chosen in Z ℓ .

R-Haar Measures
Let R be a commutative ring with identity of characteristic ℓ and let G be a locally profinite group which admits a compact open subgroup of pro-order invertible in R. We let f is locally constant and compactly supported}, (we sometimes write this, more simply, as C ∞ c (G) according to the context). A left (resp. right ) R-Haar measure on G is a non-zero linear form on C ∞ c (G, R) which is invariant under left (resp. right) translation by G. If µ is a left (or right) R-Haar measure on G and f ∈ C ∞ c (G, R), we write By [15, I 2.4], for each compact open subgroup K of G of pro-order invertible in R there exists a unique left R-Haar measure µ such that µ(K) = 1. The volume µ(K ′ ) = µ(1 K ′ ) of a compact open subgroup K ′ of G is equal to zero if and only if the pro-order of K ′ is equal to zero in R.
In the present work, the modulus character of G is the unique character δ G : G → R × such that, if µ is a left R-Haar measure on G, δ G µ is a right R-Haar measure on G. More generally, if H is a closed subgroup of G, we let δ = δ −1 G | H δ H , and be the space of functions from G to R, fixed on the right by a compact open subgroup of G, compactly supported modulo H, and which transform by δ under H on the left (we sometimes write this as C ∞ c (H\G, δ)).
for dh a right R-Haar measure on H. It is proved in [15, I 2.8] that the map f → f H is surjective, and that there is a unique, up to an invertible scalar, non-zero linear form d H\G g on C ∞ c (H\G, δ, R), which is right invariant under G. We call such a non-zero linear form on C ∞ c (H\G, δ, R) a δ-quasi-invariant quotient measure on H\G and, for f ∈ C ∞ c (H\G, δ, R), we write For the remainder of this section, let G denote a unimodular locally profinite group. Suppose that B is a closed subgroup of G, K is a compact open subgroup of G such that G = BK, and K 1 is a normal compact open subgroup of K with pro-order prime to ℓ.
Lemma 2.2. Let dg be an R-Haar measure on G. There exist a right R-Haar measure db on B and a right K-invariant measure dk on K ∩ B\K such that, for all f ∈ C ∞ c (G, R), we have Proof. We observe first that the map φ → φ | K is a vector space isomorphism between C ∞ c (B\G, δ B , R) and C ∞ c ((K ∩ B)\K, R). It is injective because G = BK. To show surjectivity, we recall that the characteristic functions 1 (K∩B)kU , with U a compact subgroup of K of pro-order invertible in R and k ∈ K, span C ∞ c ((K ∩ B)\K, R). Moreover, 1 B kU belongs to C ∞ c (B\G, δ B , R), and a computation shows Remark 2.3. Let K n = GL n (o F ). By the Iwasawa decomposition, we have G n = B n K n . Let µ Gn be an R-Haar measure on G n . If ℓ = 0, or more generally ℓ ∤ q − 1, then, for all f ∈ C ∞ c (G n , R), we have for good choices of a left R-Haar measure db on B n and an R-Haar measure dk on K n . As noticed by Mínguez in [10], this is no longer true in general. More precisely, it is not true when ℓ | q − 1 as the restriction of an R-Haar measure on K to C ∞ c ((K n ∩ B n )\K n ) is zero. That is why we use a right invariant measure on K ∩ B\K in Lemma 2.2.
Let K be a compact group, K 1 an open subgroup of K of pro-order invertible in R and P be a closed subgroup of K. Proof. To prove this, we introduce the R-Haar measure λ : f → µ(f P ) on K. By computation, as in the proof of Lemma 2.2, we have 1 P sK 1 = dp(P ∩ sK 1 s −1 )1 P sK 1 = dp(P ∩ K 1 )1 P sK 1 , for s in K. In particular, with t = 1/dp(P ∩ K 1 ), we have This implies where t ′ = tλ(K 1 ) ∈ R * , and the result follows.
Remark 2.5. An example of this we need is when K = K n , K 1 = K n,1 the pro-p unipotent radical of K n , P = P n ∩ K n , and R is of positive characteristic ℓ not equal to p. In this case, Let Q, L and N be closed subgroups of G such that Q = LN and L normalises N . Suppose that there exists a compact open subgroup K 1 of G of pro-order invertible in R such that Let dl be an R-Haar measure on L and dn be an R-Haar measure on N .
Lemma 2.6. Let f ∈ C ∞ c (Q, R). There exists a unique right R-Haar measure dq on Q such that Proof. As L normalises N , we see that L N f (nl)dldn is a right Q-invariant linear form on C ∞ c (Q, R). But as Q ∩ K 1 = (N ∩ K 1 )(M ∩ K 1 ), it is easy to see that L N 1 Q∩K 1 (nl)dldn is non-zero.
Remark 2.7. A typical instance is when G = G n , Q = LN is a standard parabolic subgroup of G n , and K 1 = K n,1 .
We have the following corollary to Lemmata 2.2 and 2.6.
Corollary 2.8. Let dg be an R-Haar measure on G. There exist an R-Haar measure da on A n , and a right K n -invariant measure dk on (K n ∩ B n )\K n such that, for all f ∈ C ∞ c (N n \G n , R), we have From Iwasawa decomposition, we also have G n = P n Z n K n . We use the following integration formula, which is proved in a similar fashion.
Corollary 2.9. Let dg be an R-Haar measure on G. There exist an R-Haar measure dz on Z n , a δ-quasi-invariant quotient measure dp on N n \P n , and a right K n -invariant measure dk on Henceforth, equalities involving integrals will be true only up to the correct normalisation of measures.

Derivatives
Henceforth, we suppose that R is an algebraically closed field. Following [2], we define the following exact functors: , extension by the trivial representation twisted by ν 2. The identity functor 1 : We recall the classification of irreducible R-representations of P n .
Let π be an R-representation of G n . The zeroth derivative π (0) of π is π. Let τ = π | Pn and set Lemma 2.12 ([1]). Let π be an R-representation of finite length. Then the dimension of π (n) is finite and equal to the dimension of Hom Nn (π, θ).
The derivatives of a product are given by the Leibniz rule. ). Suppose π is an R-representation of G n and ρ is an R-representation of G m , then (π × ρ) (k) has a filtration with successive quotients π (i) × ρ (k−i) , for 0 i k.
Finally, we will use several times the following proposition.
Proposition 2.14 ([2, Proposition 3.7]). Let ρ and ρ ′ be R-representations of G n and τ and τ ′ be R-representations of P m , we have

Parabolic induction, integral structures and reduction modulo-ℓ
Let Q be a parabolic subgroup of G n with Levi factor L. We write i Gn Q for the functor of normalised parabolic induction from R R (L) to R R (G n ). If τ = π 1 ⊗ π 2 ⊗ · · · ⊗ π r is a smooth Rrepresentation of L (m 1 ,...,mr) with i m i = n, we will use the product notation π 1 × π 2 × · · · × π r for the induced representation i Gn P (m 1 ,...,mr ) (τ ). An R-representation of G n is called cuspidal if it is irreducible and it does not appear as a subrepresentation of any parabolically induced representation.
An ℓ-adic representation (π, V ) of G is called integral if it has finite length, and if V contains a G-stable Z ℓ -lattice Λ. Such a lattice is called an integral structure in π. A character is integral if and only if it takes values in Z ℓ . By [15,II 4.12], a cuspidal representation is integral if and only if its central character is integral.
If π is an integral ℓ-adic representation with integral structure Λ, then π defines an ℓ-modular representation on the space Λ ⊗ Z ℓ F ℓ . By the Brauer-Nesbitt principle [19,Theorem 1], the semisimplification, in the Grothendieck group of finite length ℓ-modular representations, of (π, Λ ⊗ Z ℓ F ℓ ) is independent of the choice of integral structure in π and we call this semisimple representation r ℓ (π) the reduction modulo-ℓ of π. We say that an ℓ-modular representation π lifts to an integral ℓ-adic representation τ if r ℓ (τ ) ≃ π, we will only really use this notion of lift when π is irreducible.
Let H be a closed subgroup of G, σ be an integral ℓ-adic representation of H, and Λ be an integral structure in σ. By [15, I 9

Representations of Whittaker type
Before defining representations of Whittaker type we recall that the irreducible R-representations of G n satisfying Hom Nn (π, θ) = 0 are called generic.
ρ are said to be equivalent if they have the same length, and ν a ρ ≃ ν a ′ ρ ′ . Hence, as noticed in [11, 7.2], the segment [a, b] ρ identifies with the cuspidal pair and the equivalence relation on segments is the restriction of the classical isomorphism equivalence relation on cuspidal pairs. To such a segment ∆, in [11, Definition 7.5] the authors associate a certain quotient L(∆) of ν a ρ × ν a+1 ρ × · · · × ν b ρ. The representation L(∆) in fact determines ∆, as its normalised Jacquet module with respect to the opposite of N (m,...,m) is equal to r ∆ = (M (m,...,m) , ν a ρ ⊗ ν a+1 ρ ⊗ · · · ⊗ ν b ρ) according to [11,Lemma 7.14]. The conclusion of this is that the objects ∆, L(∆), and r ∆ determine one another, and hence we call L(∆) a segment and, by abuse of notation, we write ∆ for L(∆), in order to lighten notations. We say that ∆ precedes ∆ ′ if we can extract from the sequence (ν a ρ, . . . , a subsequence which is a segment of length strictly larger than both the length of ∆ and the length of ∆ ′ . We say ∆ and ∆ ′ are linked if ∆ precedes ∆ ′ or ∆ ′ precedes ∆. Let ρ be a cuspidal R-representation of G m . In [11], a positive integer e(ρ) is attached to ρ, For k ∈ N, we denote by St(k, ρ) the normalised generalised Steinberg representation associated with ρ, i.e. the unique generic subquotient of ν (1−k)/2 ρ×· · ·×ν (k−1)/2 ρ. By [11,Remarque 8.14], the representation St(k, ρ) is equal to the segment ∆ associated to the sequence [(1 − k)/2, (k − 1)/2] ρ if and only if k < e(ρ). In this case, we say that ∆ = St(k, ρ) is a generic segment. Note that, our notation St(k, ρ) differs from that used in [11] and [16] (more precisely, our St(k, ρ) corresponds to St(k, ν (1−k)/2 ρ) in those references).
We will study L-factors of representations of Whittaker type.
Let π be a representation of Whittaker type. By Lemmata 2.12 and 2.13, the space Hom Nn (π, θ) is of dimension 1, and we denote by W (π, θ) the Whittaker model of π, i.e. W (π, θ) denotes the image of π in Ind Gn Nn (θ). Note that, a representation of Whittaker type may not be irreducible, however, it is of finite length. In fact, thanks to the results of Zelevinsky (c.f. [20]) in the ℓ-adic setting, and by [16,Theorem 5.7] (a more detailed proof of which can be found in [11, Theorem 9.10 and Corollary 9.12]) in the ℓ-modular setting, the irreducible representations of Whittaker type of G n are exactly the generic representations.
Remark 2.16. According to [16,Theorem V.7], if π = ∆ 1 × · · · × ∆ t is a representation of Whittaker type of G n , then π is irreducible if and only if the segments ∆ i and ∆ j are unlinked, for all i, j ∈ {1, . . . t} with i = j.
If π is a smooth representation of G n , we denote by π the representation g → π( t g −1 ) of G n . Let τ be an ℓ-adic irreducible representation of G n , then τ ≃ τ ∨ , by [6]. Hence when ∆ is an ℓ-modular generic segment of G n , it lifts to an ℓ-adic segment D according to the discussion before Definition 2.15, and as D ≃ D ∨ , we deduce by reduction modulo-ℓ, c.f. [15, I 9.7], that ∆ ≃ ∆ ∨ . If π = ∆ 1 × · · · × ∆ t is a representation of G n of Whittaker type, we have π = ∆ t × · · · × ∆ 1 = ∆ ∨ t × · · · × ∆ ∨ 1 , and we deduce that π is also of Whittaker type. In order to state the functional equation for L-factors of representations of Whittaker type, we will need the following lemma: Lemma 2.17. Let π be a representation of Whittaker type of G n , then π is of Whittaker type and the map W → W , where W (g) = W (w n t g −1 ), is an R-vector space isomorphism between W (π, θ) and W ( π, θ −1 ).

Rankin-Selberg local factors for representations of Whittaker type
The theory of derivatives ([1] and [2]) being valid in positive characteristic (see Subsection 2.3) and equipped with the theory of R-Haar measures (see Subsection 2.2), means we can now safely follow [7] to define L-factors and ε-factors. However, as we do not have a Langlands' quotient theorem at our disposal, which would allow us to associate to an irreducible representation of G n , a unique representation with an injective Whittaker model lying above it, we restrict our attention to representations of Whittaker type (see Subsection 2.5).

Definition of the L-factors
We first recall the asymptotics of Whittaker functions obtained in [8, Proposition 2.2]. We write Z i for subgroup {diag(tI i , I n−i ), t ∈ F × } of G n . The diagonal torus A n of G n is the direct product Z 1 × · · · × Z n .
Lemma 3.1. Let π be a representation of Whittaker type of G n . For each i between 1 and n − 1, there is a finite family X i (π) of characters of Z i , such that if W is a Whittaker function in W (π, θ), then its restriction W (z 1 , . . . , z n−1 ) to A n−1 = Z 1 × · · · × Z n−1 is a sum of functions of the form The proof of Jacquet-Piatetskii-Shapiro-Shalika in [op. cit.] applies mutatis mutandis for ℓmodular representations.
Remark 3.2. For 1 ≤ i ≤ n − 1, we can take X i (π) to be the family of central characters (restricted to Z i ) of the irreducible components of the (non-normalised) Jacquet module π N i,n−i . We denote by X n (π) the singleton {ω π }. We denote by E i (π), the family of central characters (restricted to Z i ) of the irreducible components of the normalised Jacquet module π N i,n−i , for 1 ≤ i ≤ n−1, and let E n (π) = X n (π). The family E i (π) is obtained from X i (π) by multiplication by an unramified character of Z i , in particular, if R = Q ℓ , the characters in E i (π) are integral if and only if those in X i (π) are integral.
Proposition 3.3. Let π be a representation of Whittaker type of G n , and π ′ a representation of Whittaker type of G m , for m ≤ n.
Then for every k ∈ Z, the coefficient is well-defined, and vanishes for k << 0. When m = n − 1, we will simply write c k (W, W ′ ) for c k (W, W ′ ; 0). • The case n = m. Under the same notation as Proposition 3.3, we define the following formal Laurent series • The case m ≤ n − 1. Under the same notation as Proposition 3.3, we define the following formal Laurent series When m = n − 1, we will simply write I(W, W ′ , X) for I(W, W ′ , X; 0).
The L-factors we study are defined by the following theorem.
Theorem 3.5. Let π be a representation of Whittaker type of G n , and π ′ a representation of Whittaker type of G m , for 1 ≤ m ≤ n.
• If n = m, the R-submodule spanned by the Laurent series I(W, W ′ , φ, X) as W varies in W (π, θ), W ′ varies in W (π ′ , θ −1 ), and φ varies in C ∞ c (F n ), is a fractional ideal of R[X ±1 ], and it has a unique generator which is an Euler factor L(π, π ′ , X).
• If 1 ≤ m ≤ n−1, fix j between 0 and n−m−1. The R-submodule spanned by the Laurent series I(W, W ′ , X; j) as W varies in W (π, θ), W ′ varies in W (π ′ , θ −1 ), is a fractional ideal of R[X ±1 ], is independent of j, and it has a unique generator which is an Euler factor L(π, π ′ , X).
Proof. We treat the case m ≤ n − 2, the case m ≥ n − 1 is totally similar. First we want to prove that our formal series in fact belong to R(X). In this case, the coefficient c k (W, W ′ ; j) is equal to which, by smoothness of W and W ′ , we can write as a finite sum it is thus enough to check that belongs to R(X). Following the proof of [7] we see that, by Lemma 3.1, they belong to 1 P (X) R[X ±1 ], where P (X) is a suitable power of the product over the unramified characters χ i 's in E i (π) for 1 ≤ i ≤ n and the unramified characters µ j 's in E j (π ′ ) for 1 ≤ j ≤ m of the Tate L factors L(χ i µ j , X). By [10], this factor is equal to 1 if R = F ℓ , and q ≡ 1[ℓ], and is equal to 1/(1 − χ i µ j (̟)X) otherwise. The other properties follow immediately from [7].
The proof of Theorem 3.5 implies the following corollary.
Proof. By our assertion at the end of the proof of Theorem 3.5, the polynomial Q = 1/L(π, π ′ , X) divides (in R[X ±1 ], hence in R[X]) a power of the product P of the polynomials 1/L(χ i µ j , X) over the set of unramified characters χ i in E i (π) for 1 ≤ i ≤ n and unramified characters µ j 's in E j (π ′ ) for 1 ≤ j ≤ m. We already noticed that P must be 1 if R = F ℓ and q ≡ 1[ℓ], which proves our assertion in this case. In general, P belongs to Z ℓ [X], with constant term 1, as so do the polynomials 1/L(χ i µ j , X). Let

The functional equation
We have defined Rankin-Selberg L-factors of pairs of representations of Whittaker type, we now need to show that these satisfy a local functional equation. By identifying F n with M 1,n , the space C ∞ c (F n ) provides a smooth representation ρ of G n , with G n acting by right translation. We also denote by ρ the action by right translation of G n on any space of functions. For a ∈ R[X ±1 ], we denote by χ a the character in Hom(G n , R[X ±1 ] × ) defined as: g → a v(det(g)) , in particular ν = χ q 1/2 is the absolute value of the determinant.
Let π be a representation of Whittaker type of G n , and π ′ be a representation of Whittaker type of G m . If m = n, we write for the space of R-linear maps, L : for the space of R-linear maps, L : π × π → R[X ±1 ], satisfying for all W ∈ W (π, θ), W ′ ∈ W (π ′ , θ −1 ), h ∈ G k m , and u ∈ U m+1,n−m−1 . We denote by C ∞ c,0 (F n ) the subspace of C ∞ c (F n ) which is the kernel of the evaluation map Ev 0 : φ → φ(0).
The proof in the complex case of Jacquet-Piatetskii-Shapiro-Shalika in [7] is long. Some results obtained in the complex case [op. cit.] using invariant distributions can be obtained quicker using derivatives which is how we proceed.
We have an exact sequence of representations of G n We tensor this sequence by π ⊗ π ′ and, as π ⊗ π ′ is flat as an R-vector space, we obtain By considering central characters, it is clear that the space Hom Gn (π ⊗ π ′ , χ X ) = 0. Applying Hom Gn ( , χ X ) which is left exact, we obtain that Hom is of rank at most 1. Now, we have an isomorphism between C ∞ c,0 (F n ) with G n acting via right translation and ind Gn Pn (δ

1/2
Pn ), hence we have by Frobenius reciprocity. Now, by the theory of derivatives (see Subsection 2.3), π and π ′ , as P n -modules, are of finite length, with irreducible subquotients of the form (Φ + ) k Ψ + (ρ), for ρ an irreducible representation of G n−k−1 and k between 0 and n − 1. Moreover, (Φ + ) n−1 Ψ + (1) appears with multiplicity 1, as a submodule. By Proposition 2.14, the space is zero, except when j = k, in which case it is isomorphic to Hom G k (ρ ⊗ ρ ′ , χ X ν −1 )). If ρ and ρ ′ are irreducible and k ≥ 1, by considering central characters, the space ]. This ends the proof in the case n = m, as R[X ±1 ] is principal.
We now consider the case m ≤ n − 1. Again, the space D(π, π ′ ) is nonzero as it contains the map (W, W ′ ) → I(W, W ′ , X)/L(π, π ′ , X), we will show that it injects into R[X ±1 ], which will prove the statement. Let L be in D(π, π ′ ), by definition, the map L factors through τ × π ′ , where τ is the quotient of π by its subspace spanned by π(u)W − θ(u)W for u ∈ U m+1,n−m−1 and W ∈ W (π, θ). Hence τ is nothing other than the space of (Φ + ) n−m−1 (π). Taking into account the normalisation in the definition of the derivatives, we obtain the following injection: We next prove the following lemma.
is a P m+1 -module of finite length, and as π is of Whittaker type, it contains (Φ + ) m−1 Ψ + (1) as a submodule, the latter's multiplicity being 1 as a composition factor. By the theory of derivatives, all of the other irreducible subquotients are either of the form Ψ + (σ), with σ an irreducible representation of G m , or of the form Φ + (σ), with σ an irreducible representation of P m of the form Proof of the Lemma. As π ′ is of Whittaker type, its restriction to P m is of finite length, with irreducible subquotients of the form (Φ + ) m−k−1 Ψ + (µ), for µ an irreducible representation of G k . Moreover, the representation (Φ + ) m−k−1 Ψ + (1) occurs with multiplicity 1, and is a submodule. If σ is an irreducible representation of P m of the form (Φ + ) m−j−1 Ψ + (σ ′ ), with σ ′ a representation of G j , for some j ≥ 1, then is zero by Proposition 2.14 if j = k (in particular if k = 1). Moreover, if k = j, by the same Proposition, we have which is zero, by considering central characters. Hence we have proved the first part of the lemma. If σ = (Φ + ) m−1 Ψ + (1), reasoning as above, we see at once that , the latter space being isomorphic to R[X ±1 ], and this completes the proof of the lemma.
All in all, we deduce that , and this ends the proof of the proposition.
Remark 3.10. Notice that all the injections defined in the proof of Proposition 3.7 are in fact isomorphisms. This could be viewed directly, or we can simply see that after composing all of them we obtain an isomorphism.
We are now in a position to state the local functional equation and define the Rankin-Selberg ε-factor of a pair of representations of Whittaker type. We recall that an invertible element of R[X ±1 ] is an element of the form cX k , for c in R × , and k in Z.
Corollary 3.11. Let π be a representation of Whittaker type of G n , and π ′ be a representation of Whittaker type of G m .
Proof. It is a consequence of Proposition 3.7 if n = m, and if m ≤ n − 1 with j = 0, as the functionals on both sides of the equality belong, respectively, to D(π, π ′ , C ∞ c (F n )) and D(π, π ′ ). For j = 0, it follows from the case j = 0 as in the complex setting, c.f. [8].

Compatibility with reduction modulo-ℓ
Let π = ∆ 1 × · · · × ∆ t be an ℓ-modular representation of Whittaker type of G n . As in Section 2.5, for 1 i t, we can choose integral ℓ-adic segments D i and integral structures Λ i in D i such that Λ i ⊗ Z ℓ F ℓ ≃ ∆ i . Moreover, the ℓ-adic representation τ = D 1 × · · · × D t is an integral representation of Whittaker type, and Λ = Λ 1 × · · · × Λ t is an integral structure in τ satisfying We denote by W e (τ, θ) the functions in W (τ, θ) with integral values. We will need the following result concerning integral structures in ℓ-adic representations of Whittaker type.
Proof. If τ is generic, it is shown in [19,Theorem 2] that W e (τ, θ) is an integral structure in W (τ, θ), such that W (π, r ℓ (θ)) = W e (τ, θ)⊗ Z ℓ F ℓ . We use this result together with the properties of parabolic induction with respect to lattices, and a result from [4] about the explicit description of Whittaker functionals on induced representations.
A function f in τ = W (D 1 , θ) × · · · × W (D t , θ), by defintion of parabolic induction, is a map from G n to W (D, θ), i.e. for g ∈ G n , f (g) ∈ W (D, θ) identifies with a map from M to Q ℓ , so we can view f as a map of two variables from G n × M to Q ℓ , and similarly, we can view the elements in π = W (∆ 1 , θ) × · · · × W (∆ t , θ) as maps from G n × M to F ℓ . In [4,Corollary 2.3], it is shown (for minimal parabolics, but their method works for general parabolics), that there is a Weyl element w in G n , such that if one takes f ∈ τ , then there is a compact open subgroup U f of U which satisfies that for any compact subgroup U ′ of U containing U f , the integral This assertion is also true for π with the same proof, for the same choice of w, we write µ(h) = U h(wu, 1 M )r ℓ (θ) −1 (u)du for h ∈ π. Both λ and µ are nonzero Whittaker functionals on τ and π, respectively, and λ sends L to Z ℓ for a correct normalisation of du. We can moreover suppose, for correct normalisations of the ℓ-adic and the ℓ-modular Haar measures du, that µ = r ℓ (λ). The surjective map w : τ → W (τ, θ) which takes f to W f , defined by W f (g) = λ(τ (g)f ), sends L to W e (τ, θ). Similarly, for h ∈ π, if we set W h (g) = µ(π(g)h), then the map π → W (π, r ℓ (θ)), taking h to W h , is surjective, and we have r ℓ (W f ) = W r ℓ (f ) . From this, we obtain that W = w(L) is a sublattice of W e (τ, θ) (see [15], I 9.3), and r ℓ (W) = W (π, θ).
Let π and π ′ are integral ℓ-adic representations of Whittaker type of G n and G m . By Corollary 3.6, we already know that L(π, π ′ , X) is the inverse of a polynomial with integral coefficients, even without the integrality assumption. With the integrality assumption, we now consider the associated ε-factor.
Lemma 3.13. The factor ε(π, π ′ , θ, X) is of the form cX k , for c a unit in Z ℓ .
If P is an element of Z ℓ [X] with nonzero reduction modulo-ℓ, we write r ℓ (P −1 ) for (r ℓ (P )) −1 . We now prove our first main result.
Let π and π ′ be ℓ-modular representations of Whittaker type of G n and G m . Let τ and τ ′ be ℓ-adic representations of Whittaker type τ and τ ′ of G n and G m with integral structures W e (τ, θ) and W e (τ ′ , θ), as in Lemma 3.12.
Proof. We give the proof for m ≤ n − 1, and j = 0, the other cases being similar. By definition, one can write L(π, π ′ , X) as a finite sum i I(W i , W ′ i , X), for W i ∈ W (π, θ) and W ′ i ∈ W (π ′ , θ −1 ). By Theorem 3.12, there are Whittaker functions W i,e ∈ W e (τ, µ) and . By Remark 2.1, we have L(π, π ′ , X) = r ℓ ( i I(W i,e , W ′ i,e , X)). As i I(W i,e , W ′ i,e , X) belongs to we obtain that L(π, π ′ , X) belongs to r ℓ (L(τ, τ ′ , X))F ℓ [X ±1 ]. This proves the first assertion. The equality for γ factors follows the functional equation, and Remark 2.1.

L-factors of pairs of cuspidal representations.
We introduce the terminology of [5] on exceptional poles of Rankin-Selberg L-functions. We will not, however, make full use of this machinery in the following, as we will specialise to L-factors of pairs of cuspidal representations. In this case, following [7] or [5] is completely equivalent. However, for a further inquiry of L-factors of generic segments, we will use the full theory. As we intend to study in more detail the L-factors of representations of Whittaker type in the near future, we already introduce the terminology in this section.

Exceptional poles
We now recall the notion of exceptional pole, due to Cogdell and Piatetski-Shapiro ( [5]). Let R be an algebraically closed field. Let π and π ′ be a pair of R-representations of Whittaker type of G n , W ∈ W (π, θ) and W ′ ∈ W (π ′ , θ −1 ), and φ ∈ C ∞ c (F n ). Suppose that x as is a pole of order d of L(π, π ′ , X). As R[X] is a principal ideal domain, we can write the partial fraction expansion of I(W, W ′ , φ, X) at x as The map T x is a non-zero trilinear from W (π, θ) × W (π ′ , θ −1 ) × C ∞ c (F n ) to R.
Definition 4.1. Let π and π ′ be R-representations of Whittaker type of G n and G m , and let W and W ′ belong to W (π, θ) and W (π ′ , θ −1 ), respectively.
• If n = m, we say that a pole x in R of the rational map L(π, π ′ , X) is an exceptional pole if and only if the trilinear form T x , vanishes on W (π, θ) × W (π ′ , θ −1 ) × C ∞ c,0 (F n ), i.e. admits a factorisation of the form T x (W, W ′ , φ) = B x (W, W ′ )φ(0).
• If m < n, we say the local factor L(π, π ′ , X) has no exceptional poles.
Thanks to a change of variable in I(W, W ′ , φ, X), one sees that if x is an exceptional pole of L(π, π ′ , X), then B x satisfies for any g in G n . We deduce the following property.
Proposition 4.2. If π and π ′ are irreducible R-representations of Whittaker type of G n and G m , i.e. generic R-representations, and x is an exceptional pole of L(π, π ′ , X), then π ′ ∨ ≃ χ −1 x π.
Remark 4.3. When π and π ′ are complex or ℓ-adic representations, the converse of Proposition 4.2 is true, and is proved in [9,Proposition 4.6]. This is no longer the case for ℓ-modular representations, in cases when q n ≡ 1[ℓ], as we shall see later. In fact, we can already see this when q ≡ 1[ℓ], as L(π, π ′ , X) is always equal to 1 in this case. This makes the case q n ≡ 1[ℓ] pathological for the computation of L factors of pairs.
We now introduce an auxiliary Euler factor.
As before, we have the following result.
Remark 4.6. When π is cuspidal, as W | Pn has compact support modulo N n (hence W | G n−1 has compact support mod N n−1 ), the Laurent series I (0) (W, W ′ , X) only has finitely many nonzero terms, hence is a Laurent polynomial. In particular, the factor L (0) (W, W ′ , X) is equal to 1.
The following property follows from Corollary 2.9.
Remark 4.8. When π and π ′ are complex or ℓ-adic representations, the factor L (0) (π, π ′ , X) has simple poles, as L(c π c π ′ , X n ) does. This latter assertion is not true for ℓ-modular representations, when n is not prime to ℓ. The first assertion is not true either, as we shall see for example when q n ≡ 1[ℓ]. In any case, we always have L(π, π ′ , X) = L (0) (π, π ′ , X)L (0) (π, π ′ , X), and L (0) (π, π ′ , X) can be expressed in terms of the L-factors of the derivatives of π and π ′ (see [5], at least for generic segments, this fact remains true modulo-ℓ). So in the characteristic zero case, the factor L (0) (π, π ′ , X) is well understood, as it has simple poles, and those are the exceptional poles, which can be determined thanks to Proposition 4.2 and Remark 4.3. In the ℓ-modular setting, we already said that in Remark 4.3 that the converse of Proposition 4.2 is no longer true in general. Moreover, L (0) (π, π ′ , X) might have poles which are not simple anymore. These are the two sources of complications modulo-ℓ, the first being the most problematic.

L-factors of pairs for cuspidal representations
We will study in more detail the L-factors of pairs of cuspidal representations. We will express such factors in terms of the Tate L-factors of the unramified characters fixing these cuspidal representations. Before we can do this, we need to recall the following result of Bernstein in [3] for ℓ-adic representations. Theorem 4.9. Let π and π ′ be irreducible ℓ-adic representations of G n , if π ′ ≃ π ∨ (i.e. Hom Gn (π ⊗ π ′ , Q ℓ ) = {0}), Then we have Hom Pn (π ⊗ π ′ , Q ℓ ) = Hom Gn (π ⊗ π ′ , Q ℓ ), i.e. any P n -invariant bilinear pairing between π and π ′ is in fact G n -invariant.
Let ρ and ρ ′ be cuspidal representations of G n and G m , with m ≤ n. First, we observe that if m < n, as the restriction to P n of any W in W (ρ, θ) has compact support modulo N n (c.f. [17]), the integrals of the form I(W, W ′ , X) for W ′ ∈ W (ρ ′ , θ) are in fact in R[X ±1 ]. In particular, if m < n, then L(ρ, ρ ′ , X) is trivial. Proposition 4.10. If m < n, then L(ρ, ρ ′ , X) is equal to 1.
Hence the interesting case is when n = m. We will use the following ℓ-modular version of Bernstein's result.
Proof. Let τ be a cuspidal ℓ-adic representation of G n with reduction modulo-ℓ equal to π. Any W and W ′ lift to V and V ′ in W e (τ, θ) and W e (τ ∨ , θ −1 ). As C : (V, V ′ ) → Nn\Pn V (p)V ′ (p)dp is P n -invariant, it is G n -invariant by theorem 4.9. But C takes integral values on W e (τ, θ) × W e (τ ∨ , θ −1 ), and r ℓ (C) = B, hence B is G n -invariant.
Let R be Q ℓ or F ℓ , and let R u (G 1 ) denote the set of unramified R-characters of G 1 . Let τ be a integral cuspidal ℓ-adic representation of G n , and π be the reduction modulo-ℓ of τ . Denote by R(τ ) and R(π) the respective cyclic subgroups of R u (G 1 ) fixing τ and π by twisting, and denote by n(τ ) and n(π) their respective orders. We recall, that by looking at the central characters of τ and π, the integers n(τ ) and n(π) both divide n. It follows, from the Bushnell-Kutzko construction of all irreducible cuspidal representations via types given in [15,III 5], that the map r ℓ from R(τ ) to R(π) is surjective, with kernel the ℓ-part R ℓ (τ ) of R(τ ). Hence we can write ℓ dπ = |R(τ )|/|R(π)|, with d π the multiplicity of ℓ as a factor of |R(τ )|, which is independant from τ (c.f. [12,Remark 3.21] for more details about these assertions).
By Proposition 4.2, we know that if x is a pole of L(π, π ′ , X), then it will be of the form x = χ(̟) for some χ ∈ R(π, π ′ ). In this case, we want to compute the order of x as a pole of L(π, π ′ , X). We can suppose that π ′ ≃ π ∨ , and look at the pole at x = 1.