Test vectors for Rankin-Selberg $L$-functions

TL;DR

This paper constructs explicit test vectors for local Rankin-Selberg $L$-functions associated with pairs of generic representations of $GL_n imes GL_m$ over $p$-adic fields, facilitating the computation of local zeta integrals.

Contribution

It introduces a unipotent averaging method to produce Whittaker functions that serve as test vectors for local $L$-functions, expressed explicitly in terms of Langlands parameters.

Findings

Constructed non-zero local zeta integrals using unipotent averaging.

Expressed zeta integrals explicitly via Langlands parameters.

Provided conditions under which Whittaker functions serve as test vectors.

Abstract

We study the local zeta integrals attached to a pair of generic representations $(\pi,\tau)$ of $GL_n\times GL_m$, $n>m$, over a $p$-adic field. Through a process of unipotent averaging we produce a pair of corresponding Whittaker functions whose zeta integral is non-zero, and we express this integral in terms of the Langlands parameters of $\pi$ and $\tau$. In many cases, these Whittaker functions also serve as a test vector for the associated Rankin-Selberg (local) $L$-function.

Equations

$$\Psi (s;W,W^{\prime })=\underset{U_m(F)\backslash {\mathrm{GL}}_m(F)}{\int }W\left(\begin{array}{cc}h& \\ & I_{n-m}\end{array}\right)W^{\prime }(h)\Vert \det h{\Vert }^{s-\frac{n-m}{2}}dh,$$

$$\mathcal{W}(\pi ,\psi )\otimes \mathcal{W}(\tau ,{\psi }^{-1})⟶\mathbb{C}(q^{-s})$$

$$W\otimes W^{\prime }\mapsto \Psi (s;W,W^{\prime }),$$

$$L(s,\pi\times\tau)=\prod_{i=1}^{n}\prod_{j=1}^{m}\bigl{(}1-\alpha_{i}\gamma_{j}q^{-s}\bigr{)}^{-1}.$$

$$\Psi (s;W,W^{\prime })=L(s,\pi \times \tau ).$$

$$L(s,\pi ⊠\tau )=\underset{U_m(F)\backslash {\mathrm{GL}}_m(F)}{\int }{W}_{{\xi }^0}\left(\begin{array}{cc}h& \\ & I_{n-m}\end{array}\right)W_{{\phi }^0}(h)\Vert \det h{\Vert }^{s-\frac{n-m}{2}}dh$$

$$N_n=\left\{\left(\begin{array}{cc}{I}_{n-1}& \ast \\ & 1\end{array}\right)\right\}.$$

$$P_n(R)=\left\{\left(\begin{array}{cc}h& y\\ & 1\end{array}\right):h\in {\mathrm{GL}}_{n-1}(R),y\in R^{n-1}\right\}\cong {\mathrm{GL}}_{n-1}(R)⋉N_n(R).$$

$$u\mapsto \psi \left(\sum _{i=1}^{n-1}{u}_{i,i+1}\right)\text{for }u\in U_n(F)$$

$$\operatorname{Hom}_{\operatorname{GL}_{n}(F)}\bigl{(}V_{\pi},\operatorname{Ind}_{\operatorname{U}_{n}(F)}^{\operatorname{GL}_{n}(F)}\psi\bigr{)}\neq 0.$$

$$W_v(ug)$$

$$W_v(g)$$

$$K_1({\mathfrak{p}}^f)$$

$$K_0({\mathfrak{p}}^f)$$

$$t(a_1,\dots ,a_{n-1})=\left(\begin{array}{ccccc}{a}_1{a}_2\dots a_{n-1}& & & & \\ & a_1{a}_2\dots a_{n-2}& & & \\ & & \ddots & & \\ & & & a_1& \\ & & & & 1\end{array}\right).$$

$$\underset{U_n(F)\backslash {\mathrm{GL}}_n(F)}{\int }f(g)dg=\underset{Z_n(F)\times A_n(F)\times {\mathrm{GL}}_n(\mathfrak{o})}{\int }f(zak){\delta }_{B_n}(a{)}^{-1}dzdadk,$$

$${\delta }_{B_n}(a)=\prod _{i=1}^{n-1}\Vert a_i{\Vert }^{i(n-i)}.$$

$$W_{\xi^{0}}\bigl{(}t(a_{1},\ldots,a_{n-1})\bigr{)}\neq 0\implies a_{1},\ldots,a_{n-1}\in\mathfrak{o}.$$

$$ϵ(s,\pi ,\psi )=ϵ(\pi ,\psi )q^{m(\pi )(\frac{1}{2}-s)}$$

$${\chi }_{\pi }(g)=\{\begin{array}{ll}1& \text{if }m(\pi )=0,\\ {\omega }_{\pi }(g_{n,n})& \text{if }m(\pi )>0.\end{array}$$

$$\pi(g)\xi^{0}=\chi_{\pi}(g)\xi^{0}\quad\text{for all }g\in K_{0}\bigl{(}\mathfrak{p}^{m(\pi)}\bigr{)}.$$

$$L(s,\pi )=\prod _{i=1}^n(1-{\alpha }_i{q}^{-s}{)}^{-1}.$$

$$L(s,\pi \times \tau )=\prod _{i=1}^n\prod _{j=1}^m(1-{\alpha }_i{\gamma }_j{q}^{-s}{)}^{-1}.$$

$${\mathrm{Ind}}_Q^{{\mathrm{GL}}_n(F)}(\eta ,\eta \Vert \cdot \Vert ,\dots ,\eta \Vert \cdot {\Vert }^{b-1}).$$

$$\xi ={\mathrm{Ind}}_P^{{\mathrm{GL}}_n(F)}({\tau }_1\otimes \cdots \otimes {\tau }_r)$$

$$L(s,\pi \times (\tau ⊞{\tau }^{\prime }))$$

$$L(s,(\pi ⊞{\pi }^{\prime })\times \tau )$$

$$L(s,\pi \times \tau )=P(q^{-s})L(s,\pi ⊠\tau )$$

$$L(s,\pi \otimes \chi )=L(s,\pi \times \chi ),$$

$$L(s,\pi \times \tau )$$

$$=L(s+m-1+b-1,\eta \otimes \chi ).$$

$$L(s,\pi ⊠\tau )=\{\begin{array}{ll}{\prod }_{j=0}^{m-1}L(s+j+b-1,\eta \otimes \chi )& \text{if }m\le n,\\ {\prod }_{i=0}^{b-1}L(s+m-1+i,\eta \otimes \chi )& \text{if }m>n.\end{array}$$

Full text

A. R. B. was partially supported by EPSRC Grant EP/K034383/1. M. L. was supported by a Royal Society University Research Fellowship. No data were created in the course of this study.

1. introduction

Let $F$ be a non-archimedean local field with ring of integers $\mathfrak{o}$ and residue field of cardinality $q$. For $m<n$, let $\pi$ and $\tau$ be irreducible admissible representations of $\operatorname{GL}_{n}(F)$ and $\operatorname{GL}_{m}(F)$, respectively. We fix an additive character $\psi$ of $F$ with conductor $\mathfrak{o}$ and assume that $\pi$ and $\tau$ are generic relative to $\psi$.

Recall that the local zeta integral $\Psi(s;W,W^{\prime})$ is defined by

[TABLE]

where $W\in\mathcal{W}(\pi,\psi)$ and $W^{\prime}\in\mathcal{W}(\tau,\psi^{-1})$ are Whittaker functions in the corresponding Whittaker spaces, and $\operatorname{U}_{m}$ is the group of unipotent matrices. It converges for $\Re(s)\gg 1$, and the collection of such zeta integrals spans a fractional ideal $\mathbb{C}[q^{s},q^{-s}]L(s,\pi\boxtimes\tau)$ of the ring $\mathbb{C}[q^{s},q^{-s}]$. We may choose the generator to satisfy $1/L(s,\pi\boxtimes\tau)\in\mathbb{C}[q^{-s}]$ and $\lim_{s\to\infty}L(s,\pi\boxtimes\tau)=1$, and this gives the local Rankin–Selberg factor attached to the pair $(\pi,\tau)$ in [5, §2.7].

In particular, if we define a map

[TABLE]

via

[TABLE]

then there is an element in $\mathcal{W}(\pi,\psi)\otimes\mathcal{W}(\tau,\psi^{-1})$ that maps to $L(s,\pi\boxtimes\tau)$. However, a priori this element need not be a pure tensor. In this paper, we produce a pure tensor $W\otimes W^{\prime}$ for which the associated zeta integral is explicitly computable and non-zero. The precise result that we prove is the following.

Theorem 1.1**.**

Let $\{\alpha_{i}\}_{i=1}^{n}$ and $\{\gamma_{j}\}_{j=1}^{m}$ denote the Langlands parameters of $\pi$ and $\tau$, respectively, and let $L(s,\pi\times\tau)$ be the naive Rankin–Selberg $L$-factor defined by

[TABLE]

Then there is a pair $(W,W^{\prime})\in\mathcal{W}(\pi,\psi)\times\mathcal{W}(\tau,\psi^{-1})$, described explicitly in §3, such that

[TABLE]

When $\Psi(s;W,W^{\prime})=L(s,\pi\boxtimes\tau)$, the pair $(W,W^{\prime})$ is called a test vector for $(\pi,\tau)$. Hence the theorem produces a test vector whenever $L(s,\pi\times\tau)=L(s,\pi\boxtimes\tau)$—for instance, if either $\pi$ or $\tau$ is unramified or if $L(s,\pi\boxtimes\tau)=1$. In general, one has $L(s,\pi\times\tau)=P(q^{-s})L(s,\pi\boxtimes\tau)$ for a non-zero polynomial $P\in\mathbb{C}[X]$ (see Lemma 2.1).

The overview of our method is as follows. Let $\xi^{0}$ (resp. $\varphi^{0}$) denote the “essential vector” in the space of $\pi$ (resp. $\tau$), and let $W_{\xi^{0}}\in\mathcal{W}(\pi,\psi)$ (resp. $W_{\varphi^{0}}\in\mathcal{W}(\tau,\psi^{-1})$) be the associated essential Whittaker functions, as described in detail in §2. When $\tau$ is unramified, it follows from [10, Corollary 3.3] that

[TABLE]

for a suitable normalization of the measure on $\operatorname{U}_{m}(F)\backslash\operatorname{GL}_{m}(F)$. When $m=n-1$, the above equality is part of the characterization of the essential vector in [4, 6]; the fact that it holds for any $m<n$ is the result of a concrete realization of essential functions in [10]. On the other hand, if $\tau$ is ramified then the local integral in (1.2) vanishes. Through a process of unipotent averaging (see (3.1) below), we modify $W_{\xi^{0}}$ to obtain a Whittaker function $W\in\mathcal{W}(\pi,\psi)$ such that the resulting zeta integral $\Psi(s;W,W_{\varphi^{0}})$ equals $cL(s,\pi\times\tau)$ for a non-zero number $c\in\mathbb{C}$, depending on the conductor of $\tau$, its central character $\omega_{\tau}$, and $\psi$. Setting $W^{\prime}=c^{-1}W_{\varphi^{0}}$, we obtain the required pair $(W,W^{\prime})$.

We mention some related results in the literature. First, if $\pi$ and $\tau$ are discrete series representations, then the existence of a test vector $(W,W^{\prime})$ was shown in [9], but the Whittaker function $W^{\prime}$ there is taken to be in a larger space, namely the Whittaker space associated to the standard module of $\tau$. Second, the so-called local Birch lemma, arising in the context of $\mathfrak{p}$-adic interpolation of special values of twisted Rankin–Selberg (global) $L$-functions, is also related. It concerns evaluation of a local integral in the special case that $\pi$ is unramified and $\tau$ is the twist of an unramified representation by a character with non-trivial conductor; see [8, Proposition 3.1] and [7, Theorem 2.1]. The approach in [8] is similar to ours in that it also uses a process of unipotent averaging in order to modify the Whittaker function on the larger general linear group. We can of course apply Theorem 1.1 to their setup: Since $L(s,\pi\times\tau)=1$ in this case, the pair $(W,W_{\varphi^{0}})$ described above has the property that $\Psi(s;W,W_{\varphi^{0}})$ is an explicit constant (independent of $s$).

Finally, note that one can obtain a global version of Theorem 1.1 by combining the test vectors at all (finite) places. In work in progress, we study the analogous question over an archimedean local field.

Acknowledgements

The second author (M. K.) would like to thank A. Raghuram for pointing him to [8].

2. preliminaries

Let $\mathfrak{p}$ be the unique maximal ideal in $\mathfrak{o}$. We fix a generator $\varpi$ of $\mathfrak{p}$ with absolute value $\|\varpi\|=q^{-1}$. Let $\operatorname{v}:F^{\times}\rightarrow\mathbb{Z}$ denote the valuation map, and extend it to fractional ideals in the usual way. For any $n>1$, let $\operatorname{B}_{n}=\operatorname{T}_{n}\operatorname{U}_{n}$ be the Borel subgroup of $\operatorname{GL}_{n}$ consisting of upper triangular matrices; let $\operatorname{P}_{n}^{\prime}\supset\operatorname{B}_{n}$ be the standard parabolic subgroup of type $(n-1,1)$ with Levi decomposition $\operatorname{P}_{n}^{\prime}=\operatorname{M}_{n}\operatorname{N}_{n}$. Then $\operatorname{M}_{n}\cong\operatorname{GL}_{n-1}\times\operatorname{GL}_{1}$ and

[TABLE]

Also, we write $\operatorname{Z}_{n}$ to denote the center consisting of scalar matrices and $\operatorname{A}_{n}\subset\operatorname{T}_{n}$ to denote the subtorus consisting of diagonal matrices with lower-right entry $1$.

If $R$ is any $F$-algebra and $H$ is any algebraic $F$-group, we write $H(R)$ to denote the corresponding group of $R$-points. Let $\operatorname{P}_{n}(R)\subset\operatorname{P}_{n}^{\prime}(R)$ denote the mirabolic subgroup consisting of matrices whose last row is of the form $(0,\ldots,0,1)$, i.e.,

[TABLE]

The character

[TABLE]

defines a generic character of $\operatorname{U}_{n}(F)$, and by abuse of notation we continue to denote this character by $\psi$. Further, for any algebraic subgroup $V\subseteq\operatorname{U}_{n}$, $\psi$ defines a character of $V(F)$ via restriction. In particular, we may consider the character $\psi|_{\operatorname{N}_{n}(F)}$; its stabilizer in $\operatorname{M}_{n}(F)$ is then $\operatorname{P}_{n-1}(F)$, where we regard $\operatorname{P}_{n-1}$ as a subgroup of $\operatorname{M}_{n}$ via $h\mapsto\begin{pmatrix}h&\\ &1\end{pmatrix}$.

An irreducible representation $(\pi,V_{\pi})$ of $\operatorname{GL}_{n}(F)$ is said to be generic if

[TABLE]

By Frobenius reciprocity, this means that there is a non-zero linear form $\lambda:V_{\pi}\to\mathbb{C}$ satisfying $\lambda(\pi(u)v)=\psi(u)\lambda(v)$ for $v\in V_{\pi}$, $u\in\operatorname{U}_{n}(F)$. It is known (see [3]) that for a generic $\pi$ the space of such linear functionals, or equivalently the space $\operatorname{Hom}_{\operatorname{GL}_{n}(F)}(V_{\pi},\operatorname{Ind}_{\operatorname{U}_{n}(F)}^{\operatorname{GL}_{n}(F)}\psi)$, is of dimension $1$. Let $\mathcal{W}(\pi,\psi)$ denote the Whittaker model of $\pi$, viz. the space of functions $W_{v}$ on $\operatorname{GL}_{n}(F)$ defined by $W_{v}(g)=\lambda(\pi(g)v)$ for $v\in V_{\pi}$. Then $\mathcal{W}(\pi,\psi)$ is independent of the choice of $\lambda$, and for $u\in\operatorname{U}_{n}(F)$, $g\in\operatorname{GL}_{n}(F)$,

[TABLE]

We will consider certain compact open subgroups of $\operatorname{GL}_{n}(F)$; namely, for any integer $f\geq 0$, set

[TABLE]

so that $K_{1}(\mathfrak{p}^{f})$ is a normal subgroup of $K_{0}(\mathfrak{p}^{f})$, with quotient $K_{0}(\mathfrak{p}^{f})/K_{1}(\mathfrak{p}^{f})\cong(\mathfrak{o}/\mathfrak{p}^{f})^{\times}$.

Next we introduce our choice of measures. For $n\geq 1$ we normalize the Haar measure on $\operatorname{GL}_{n}(F)$ and $\operatorname{GL}_{n}(\mathfrak{o})$ so that $\operatorname{vol}(\operatorname{GL}_{n}(\mathfrak{o}))=1$, and we fix the Haar measure on $\operatorname{U}_{n}(F)$ for which $\operatorname{vol}(\operatorname{U}_{n}(F)\cap\operatorname{GL}_{n}(\mathfrak{o}))=1$. From these, we obtain a right-invariant measure on $\operatorname{U}_{n}(F)\backslash\operatorname{GL}_{n}(F)$. We may make this explicit using the Iwasawa decomposition. For instance, let $dx$ be the Haar measure on $F$ such that $\mathfrak{o}$ has unit volume, and let $d^{\times}x$ be the multiplicative measure on $F^{\times}$ such that $\operatorname{vol}(\mathfrak{o}^{\times})=1$, i.e., $d^{\times}x=\frac{q}{q-1}\frac{dx}{\|x\|}$. Let $dz$ and $da$ be the corresponding measures on the center $\operatorname{Z}_{n}(F)\cong F^{\times}$ and the subtorus $\operatorname{A}_{n}(F)\cong(F^{\times})^{n-1}$, respectively. We fix the isomorphism $(F^{\times})^{n-1}\cong\operatorname{A}_{n}(F)$ via $(a_{1},\ldots,a_{n-1})\mapsto a=t(a_{1},\ldots,a_{n-1})$, where

[TABLE]

Then $da=d^{\times}a_{1}\,d^{\times}a_{2}\cdots d^{\times}a_{n-1}$. If $f\in C_{c}^{\infty}(\operatorname{GL}_{n}(F))$ is $\operatorname{U}_{n}(F)$-invariant on the left, we then have the integration formula

[TABLE]

where $\delta_{\operatorname{B}_{n}}$ is the modulus character, defined so that

[TABLE]

Next we review the notion of conductor and the theory of the essential vector associated to an irreducible, admissible, generic representation $\pi$. According to [4] (see also [6]), there is a unique positive integer $m(\pi)$ such that the space of K_{1}\bigl{(}\mathfrak{p}^{m(\pi)}\bigr{)}-fixed vectors is $1$-dimensional. Further, as alluded to in the introduction, by loc. cit. there is a unique vector $\xi^{0}$ in this space, called the essential vector, with the associated essential function $W_{\xi^{0}}\in\mathcal{W}(\pi,\psi)$ satisfying the condition $W_{\xi^{0}}\begin{pmatrix}gh&\\ &1\end{pmatrix}=W_{\xi^{0}}\begin{pmatrix}g&\\ &1\end{pmatrix}$ for all $h\in\operatorname{GL}_{n-1}(\mathfrak{o})$ and $g\in\operatorname{GL}_{n-1}(F)$. Since $\operatorname{U}_{n}$ acts via $\psi$ on the left, it follows that

[TABLE]

If $\pi$ is unramified, let $W_{\pi}^{0,\psi}\in\mathcal{W}(\pi,\psi)$ denote the normalized spherical function [6, p. 2]. If $m(\pi)=0$ then by uniqueness of essential functions, one has the equality $W_{\xi^{0}}=W_{\pi}^{0,\psi}$. The integral ideal $\mathfrak{p}^{m(\pi)}$ is called the conductor of $\pi$. In passing, we mention that the integer $m(\pi)$ can also be characterized as the degree of the monomial in the local $\epsilon$-factor $\epsilon(s,\pi,\psi)$ [4], i.e. so that

[TABLE]

for some $\epsilon(\pi,\psi)\in\mathbb{C}^{\times}$.

A crucial property of the conductor is that $K_{0}(\mathfrak{p}^{m}(\pi))$ acts on the space of $K_{1}(\mathfrak{p}^{m(\pi)})$-fixed vectors via the central character $\omega_{\pi}$ (cf. [2, Section 8]). Precisely, for g=(g_{i,j})\in K_{0}\bigl{(}\mathfrak{p}^{m(\pi)}\bigr{)}, define

[TABLE]

It is shown in loc. cit. that $\chi_{\pi}$ is a character of $K_{0}(\mathfrak{p}^{m(\pi)})$ trivial on $K_{1}(\mathfrak{p}^{m(\pi)})$, and

[TABLE]

We end this section by recalling the definition of conductor of a multiplicative character $\chi$ of $F^{\times}$. If $\chi$ is trivial on $\mathfrak{o}^{\times}$ then the conductor of $\chi$ is $\mathfrak{o}$; otherwise, the conductor is $\mathfrak{p}^{n}$, where $n\geq 1$ is the least integer such that $\chi$ is trivial on $1+\mathfrak{p}^{n}$.

2.1. Rankin–Selberg $L$-functions

In this subsection alone we drop the assumption that $m<n$ and allow $(m,n)$ to be an arbitrary pair of positive integers. For $\pi$ and $\tau$ irreducible, admissible, generic representations of $\operatorname{GL}_{n}(F)$ and $\operatorname{GL}_{m}(F)$, respectively, let $L(s,\pi\boxtimes\tau)$ be as defined in [5]. When $m<n$, $L(s,\pi\boxtimes\tau)$ is defined as in the introduction. For $m>n$, one defines $L(s,\pi\boxtimes\tau)=L(s,\tau\boxtimes\pi)$. For $m=n$, the defining local integrals are different and involve a Schwartz function on $F^{n}$; see loc. cit.

Next we elaborate on the definition of the naive Rankin–Selberg $L$-factor, $L(s,\pi\times\tau)$, introduced in Theorem 1.1. By definition, the $L$-function $L(s,\pi)$ is of the form $P_{\pi}(q^{-s})^{-1}$, where $P_{\pi}\in\mathbb{C}[X]$ has degree at most $n$ and satisfies $P_{\pi}(0)=1$. We may then find $n$ complex numbers $\{\alpha_{i}\}_{i=1}^{n}$ (allowing some of them to be zero) satisfying

[TABLE]

We call the set $\{\alpha_{i}\}$ the Langlands parameters of $\pi$; if $\pi$ is spherical, they agree with the usual Satake parameters. Let $\{\gamma_{j}\}_{j=1}^{m}$ be the Langlands parameters of $\tau$, and set

[TABLE]

Of course, $L(s,\pi\times\tau)=L(s,\pi\boxtimes\tau)$ if both $\pi$ and $\tau$ are spherical.

In the following lemma we describe the connection between $L(s,\pi\times\tau)$ and $L(s,\pi\boxtimes\tau)$. To that end, we first recall the classification of irreducible admissible representations of $\operatorname{GL}_{n}(F)$. Let $\mathcal{A}_{n}$ denote the set of equivalence classes of such representations, and put $\mathcal{A}=\bigcup\mathcal{A}_{n}$. The essentially square-integrable representations of $\operatorname{GL}_{n}(F)$ have been classified by Bernstein and Zelevinsky, and they are as follows. If $\sigma$ is an essentially square-integrable representation of $\operatorname{GL}_{n}(F)$, then there is a divisor $a\mid n$ and a supercuspidal representation $\eta$ of $\operatorname{GL}_{a}(F)$ such that if $b=\frac{n}{a}$ and $\operatorname{Q}$ is the standard (upper) parabolic subgroup of $\operatorname{GL}_{n}(F)$ of type $(a,\ldots,a)$, then $\sigma$ can be realized as the unique quotient of the (normalized) induced representation

[TABLE]

The integer $a$ and the class of $\eta$ are uniquely determined by $\sigma$. In short, $\sigma$ is parametrized by $b$ and $\eta$, and we denote this by $\sigma=\sigma_{b}(\eta)$; further, $\sigma$ is square-integrable (also called “discrete series”) if and only if the representation $\eta\|\cdot\|^{\frac{b-1}{2}}$ of $\operatorname{GL}_{a}(F)$ is unitary.

Now, let $\operatorname{P}$ be an upper parabolic subgroup of $\operatorname{GL}_{n}(F)$ of type $(n_{1},\ldots,n_{r})$. For each $i=1,\ldots,r$, let $\tau_{i}^{0}$ be a discrete series representation of $\operatorname{GL}_{n_{i}}(F)$. Let $(s_{1},\ldots,s_{r})$ be a sequence of real numbers satisfying $s_{1}\geq\cdots\geq s_{r}$, and put $\tau_{i}=\tau_{i}^{0}\otimes\|\cdot\|^{s_{i}}$ (an essentially square-integrable representation). Then the induced representation

[TABLE]

is said to be a representation of $\operatorname{GL}_{n}(F)$ of Langlands type. If $\tau\in\mathcal{A}_{n}$, then it is well known that it is uniquely representable as the quotient of an induced representation of Langlands type. We write $\tau=\tau_{1}\boxplus\cdots\boxplus\tau_{r}$ to denote this realization of $\tau$. Thus one obtains a sum operation on the set $\mathcal{A}$ [5, §9.5]. It follows easily from the definition that $L(s,\pi\times\tau)$ is bi-additive, i.e.

[TABLE]

for all $\pi,\pi^{\prime},\tau,\tau^{\prime}\in\mathcal{A}$. The local factor $L(s,\pi\boxtimes\tau)$ is also bi-additive in the above sense, by [5, §9.5, Theorem].

Lemma 2.1**.**

Let $m$ and $n$ be positive integers, and consider $\pi\in\mathcal{A}_{n}$, $\tau\in\mathcal{A}_{m}$. Then

[TABLE]

for a polynomial $P\in\mathbb{C}[X]$ (depending on $\pi$ and $\tau$) satisfying $P(0)=1$.

Proof.

Since $\pi$ and $\tau$ are sums of essentially square-integrable representations and $L(s,\pi\times\tau)$ and $L(s,\pi\boxtimes\tau)$ are both additive with respect to $\boxplus$, it suffices to prove the lemma for a pair $(\pi,\tau)$ of essentially square-integrable representations. In particular, assume $\pi=\sigma_{b}(\eta)$ as above.

We proceed by induction on $m$. If $m=1$ then $\tau=\chi$ is a quasi-character of $F^{\times}$ and $L(s,\pi\boxtimes\tau)=L(s,\pi\otimes\chi)$, where $\pi\otimes\chi$ is the representation of $\operatorname{GL}_{n}(F)$ defined by $g\mapsto\pi(g)\chi(\det g)$. If $\chi$ is unramified, then

[TABLE]

and consequently $P=1$. On the other hand, if $\chi$ is ramified then $L(s,\pi\times\chi)=1$, and the assertion follows since $L(s,\pi\otimes\chi)^{-1}$ is a polynomial in $q^{-s}$.

We now assume $m>1$ and $\tau$ is an essentially square-integrable representation of $\operatorname{GL}_{m}(F)$, say $\tau=\sigma_{b^{\prime}}(\eta^{\prime})$, where $\eta^{\prime}\in\mathcal{A}_{a^{\prime}}$ is supercuspidal and $a^{\prime}b^{\prime}=m$. Then the standard $L$-factor $L(s,\tau)$ is given by $L(s,\tau)=L(s+b^{\prime}-1,\eta^{\prime})$ [5]. Therefore, $L(s,\tau)=1$ unless $a^{\prime}=1$ and $\eta^{\prime}=\chi$ is an unramified quasi-character of $F^{\times}$. On the other hand, if $L(s,\tau)=1$, then $L(s,\pi\times\tau)=1$ and the assertion of the lemma follows. Hence we may assume $\tau=\sigma_{m}(\chi)$ for an unramified quasi-character $\chi$ of $F^{\times}$, in which case

[TABLE]

On the other hand, it follows from [5, §8.2, Theorem] that

[TABLE]

From (2.6) and (2.7), one sees that the ratio $\frac{L(s,\pi\times\tau)}{L(s,\pi\boxtimes\tau)}$ is a polynomial in $q^{-s}$, thus proving the lemma. ∎

Corollary 2.2**.**

If $L(s,\pi\boxtimes\tau)=1$ then either $L(s,\pi)=1$ or $L(s,\tau)=1$.

Proof.

If $L(s,\pi\boxtimes\tau)=1$ then Lemma 2.1 implies that $L(s,\pi\times\tau)$ is a polynomial in $q^{-s}$, and hence must be $1$. This in turn implies the conclusion. ∎

3. the main calculation

Recall that $\xi^{0}$ and $\varphi^{0}$ are the essential vectors of $\pi$ and $\tau$, respectively. Here we construct a pair $(W,W^{\prime})\in\mathcal{W}(\pi,\psi)\times\mathcal{W}(\tau,\psi^{-1})$ as in Theorem 1.1. Let ${\mathfrak{n}}$, $\mathfrak{q}$ and $\mathfrak{c}$ denote the conductors of $\pi$, $\tau$ and $\omega_{\tau}$, respectively. If $\tau$ is an unramified representation of $\operatorname{GL}_{m}(F)$, then by (1.2) we have

[TABLE]

Thus, in this case we can take $(W,W^{\prime})=(W_{\xi^{0}},W_{\varphi^{0}})$.

Let us assume from now on that $\tau$ is ramified, meaning $\operatorname{v}(\mathfrak{q})>0$. Since $\mathfrak{c}\supseteq\mathfrak{q}$, we have $\operatorname{v}(\mathfrak{c})\leq\operatorname{v}(\mathfrak{q})$. Consider $\beta=(\beta_{1},\ldots,\beta_{m})\in F^{m}$, with $\beta_{i}\in\mathfrak{q}^{-1}$ for $i=1,\ldots,m$, and let $u(\beta)$ denote the $n\times n$ matrix with $1$s on the diagonal and $\beta^{t}$ embedded above the diagonal in the $(m+1)^{\text{st}}$ column. Let $\xi^{0}_{\beta}$ denote the vector $\xi_{\beta}^{0}=\pi(u(\beta))\xi^{0}$, and define

[TABLE]

(When $m=1$ we understand there to be one summand, so that $\overline{\xi}=\xi^{0}_{(\varpi^{-\operatorname{v}(\mathfrak{c})})}$.) We will now calculate $\Psi(s;W_{\overline{\xi}},W_{\varphi^{0}})$, which by linearity equals

[TABLE]

Put $K=\operatorname{GL}_{m}(\mathfrak{o})$. By (2.2), for fixed $\beta=(\beta_{1},\ldots,\beta_{m})$, we have

[TABLE]

where $(k_{1},\ldots,k_{m})$ is the bottom row of the matrix $k$. Here we have used the fact that the function $h\mapsto W_{\xi^{0}}\begin{pmatrix}h&\\ &I_{n-m}\end{pmatrix}$, $h\in\operatorname{GL}_{m}(F)$, is right $K$-invariant. Now, performing the average over $\beta_{j}\in\mathfrak{q}^{-1}/\mathfrak{o}$ for each $j<m$, we see that $\Psi(s;W_{\overline{\xi}},W_{\varphi^{0}})$ equals

[TABLE]

By (2.4) we have $W_{\xi^{0}}(t(a_{1},\ldots,a_{n-1}))=0$ unless $a_{1},\ldots,a_{n-1}\in\mathfrak{o}$. In view of (2.1), it follows that the integrand vanishes unless $z$ is integral.

Note that

[TABLE]

For $r\in\mathbb{Z}_{\geq 0}$, put

[TABLE]

where $\mathcal{G}(\omega_{\tau},\psi,y)$ denotes the Gauss sum

[TABLE]

Then we have

[TABLE]

Suppose that $\omega_{\tau}$ is ramified, so that $\operatorname{v}(\mathfrak{c})>0$. Then $\mathcal{G}(\omega_{\tau},\psi,\varpi^{r-\operatorname{v}(\mathfrak{c})}k_{m})$ vanishes unless $\operatorname{v}(\varpi^{r-\operatorname{v}(\mathfrak{c})}k_{m})=-\operatorname{v}(\mathfrak{c})$, which implies $\operatorname{v}(k_{m})=-r$. Since $k_{m}$ is integral, it follows that $r=0$ is the only contributing term to $\Psi(s;W_{\overline{\xi}},W_{\varphi^{0}})$, so that

[TABLE]

Moreover, since $k_{m}\in\mathfrak{o}^{\times}$, we have $\mathcal{G}(\omega_{\tau},\psi,\varpi^{-\operatorname{v}(\mathfrak{c})}k_{m})=\omega_{\tau}(k_{m})^{-1}\mathcal{G}(\omega_{\tau},\psi,\varpi^{-\operatorname{v}(\mathfrak{c})})$, and thus

[TABLE]

where

[TABLE]

Suppose now that $\omega_{\tau}$ is unramified, so that $\mathfrak{c}=\mathfrak{o}$ and $m>1$. Then $\mathcal{G}(\omega_{\tau},\psi,\varpi^{r-\operatorname{v}(\mathfrak{c})}k_{m})=1$ for all $r$. If $r>0$ then $K_{0}(\mathfrak{q}\mathfrak{p}^{-r}\cap\mathfrak{o})\supsetneq K_{0}(\mathfrak{q})$; since the conductor of $\tau$ is $\mathfrak{q}$, it follows from [4, Theorem 5.1] that

[TABLE]

which in turn implies that $\Psi_{r}=0$. Hence only the $r=0$ term contributes, and again we arrive at (3.2).

It remains only to identify the integral over $\operatorname{A}_{m}(F)$.

Lemma 3.1**.**

When $\tau$ is ramified, we have

[TABLE]

Proof.

Let $\alpha=(\alpha_{1},\ldots,\alpha_{n})$ and $(\gamma_{1},\ldots,\gamma_{m})$ denote the Langlands parameters of $\pi$ and $\tau$, respectively. Since $\tau$ is ramified, we may take $\gamma_{m}=0$. We set $\gamma_{m+1}=\ldots=\gamma_{n}=0$ and write $\gamma=(\gamma_{1},\ldots,\gamma_{n})$.

Writing $a=t(a_{1},\ldots,a_{m-1})$ as in (2.1), by (2.4) we see that $W_{\varphi^{0}}(a)$ vanishes unless each $a_{i}$ is integral. Setting

[TABLE]

the integral in question may be written as

[TABLE]

where

[TABLE]

On the other hand, by [11, Theorem 4.1], we have

[TABLE]

where $s_{\lambda}$ denotes the Schur polynomial

[TABLE]

(see [1, Theorem 38.1]). Thus the integral becomes

[TABLE]

where the last line follows from (2.3) and the fact that $\gamma_{m}=\ldots=\gamma_{n}=0$.

By the Cauchy identity [1, Theorem 43.3], for sufficiently small $x\in\mathbb{C}$, we have

[TABLE]

Therefore,

[TABLE]

∎

Finally, we take $W=W_{\overline{\xi}}$ and $W^{\prime}=c^{-1}W_{\varphi^{0}}$ to conclude the proof of Theorem 1.1.