# On a refined local converse theorem for ${\mathrm{SO}}(4)$

Pan Yan  Department of Mathematics, University of Arizona, Tucson, AZ 85721
USA<EMAIL_ADDRESS>and  Qing Zhang  School of Mathematics and
Statistics, Huazhong University of Science and Technology, Wuhan, 430074,
China<EMAIL_ADDRESS>
###### Abstract.

Recently, Hazeltine-Liu, and independently Haan-Kim-Kwon, proved a local
converse theorem for ${\mathrm{SO}}_{2n}(F)$, which says that, up to an outer
automorphism of ${\mathrm{SO}}_{2n}(F)$, an irreducible generic representation
of ${\mathrm{SO}}_{2n}(F)$ is uniquely determined by its twisted gamma factors
by generic representations of ${\mathrm{GL}}_{k}(F)$ for $k=1,\dots,n$. In
this paper, we prove a “refined” version of a local converse theorem for the
group ${\mathrm{SO}}_{4}$ over a $p$-adic field $F$, namely, we show that a
generic supercuspidal representation $\pi$ of ${\mathrm{SO}}_{4}(F)$ is
uniquely determined by its ${\mathrm{GL}}_{1}$, ${\mathrm{GL}}_{2}$ twisted
local gamma factors and a twisted exterior square local gamma factor of $\pi$.

###### Key words and phrases:

gamma factors, Howe vectors, refined local converse theorem

###### 2010 Mathematics Subject Classification:

11F70, 22E50

##  1\. Introduction

Let $F$ be a local field. Let $G_{n}$ denote the split classical groups
${\mathrm{SO}}_{2n+1},{\mathrm{Sp}}_{2n}$ or quasi-split classical groups
${\mathrm{U}}_{2n},{\mathrm{U}}_{2n+1}$ over $F$. Here in the quasi-split
case, the unitary groups are defined with respect to a fixed quadratic
extension $E/F$. Given an irreducible generic representation $\pi$ of
$G_{n}(F)$ and an irreducible generic representation $\tau$ of
${\mathrm{GL}}_{n}(F)$ (in the split case) or ${\mathrm{GL}}_{n}(E)$ (in the
unitary group case), one can associate a local gamma factor
$\gamma(s,\pi\times\tau,\psi)$ using either Langlands-Shahidi method, or
appropriate Rankin-Selberg integrals; see [Kaplan] for a nice survey on the
definition of these gamma factors using Rankin-Selberg method. Here $\psi$ is
a fixed additive character of $F$. The local converse theorems for these
groups, see [JS, Chai, JL, Morimoto, Sp(2r), U(2r+1), Jo] for various cases,
asserts that the representation $\pi$ is uniquely determined by the family
$\gamma(s,\pi\times\tau,\psi)$ as $\tau$ runs over all irreducible generic
representations of ${\mathrm{GL}}_{m}(F)$ (or ${\mathrm{GL}}_{m}(E)$) for all
$m$ with $1\leq m\leq n$. See [LZ] for more references. However, a similar
converse theorem for ${\mathrm{SO}}_{2n}$ is false in general because of the
existence of an outer automorphism, which is defined by conjugation of an
element $c\in{\mathrm{O}}_{2n}(F)-{\mathrm{SO}}_{2n}(F)$ (for an explicit
matrix form of $c$, see Section 3.4). In fact, the twisted gamma factors of
${\mathrm{SO}}_{2n}(F)\times{\mathrm{GL}}_{k}(F)$ cannot distinguish $\pi$ and
$c\cdot\pi$ [HL:p-adic, HL:finite, HKK]. Here $c\cdot\pi$ is the
representation of ${\mathrm{SO}}_{2n}(F)$ defined by
$(c\cdot\pi)(g)=\pi(cgc^{-1})$. Thus if one only uses the twisted gamma
factors of ${\mathrm{SO}}_{2n}(F)\times{\mathrm{GL}}_{k}(F)$, the best
possible result one could expect is that these gamma factors can determine
$\pi$ up to the conjugation by $c$. This is actually the main result of
[HL:p-adic] and [HKK]. A similar phenomenon is also known over finite fields,
see [HL:finite]. This phenomenon is consistent with the work of Arthur on the
local Langlands correspondence [Arthur].

Let $\pi$ be an irreducible generic representation of ${\mathrm{SO}}_{2n}(F)$.
Since $\pi$ and $c\cdot\pi$ are generic with respect to the same Whittaker
datum, they cannot be in the same $L$-packet unless $\pi\cong c\cdot\pi$ by
the uniqueness of generic element (with respect to a fixed Whittaker datum) in
an $L$-packet as proved in [Atobe]. It is thus desirable to find additional
invariants to distinguish $\pi$ and $c\cdot\pi$ if $\pi\ncong c\cdot\pi.$

In this paper, for the small group ${\mathrm{SO}}_{4}(F)$, we construct an
additional twisted local gamma factor
$\gamma(s,\pi,\wedge^{2}_{+}\times\eta,\psi)$, where $\wedge^{2}_{+}$ is a
constituent of $\wedge^{2}$ of the dual group
${\mathrm{SO}}_{4}({\mathbb{C}})$ (see explanation below), and $\eta$ is a
character of $F^{\times}$. We show that, the twisted gamma factors of
${\mathrm{SO}}_{4}\times{\mathrm{GL}}_{k}$ for $k=1,2$, plus the gamma factor
$\gamma(s,\pi,\wedge^{2}_{+}\times\eta,\psi)$ can determine the representation
$\pi$ uniquely. Namely, we show the following

######  Theorem 1.1.

Let $F$ be a $p$-adic field. Let $\pi,\pi^{\prime}$ be two irreducible
$\psi$-generic supercuspidal representations of ${\mathrm{SO}}_{4}(F)$ with
the same central character. If
$\gamma(s,\pi\times\tau,\psi)=\gamma(s,\pi^{\prime}\times\tau,\psi)$ and
$\gamma(s,\pi,\wedge^{2}_{+}\times\eta,\psi)=\gamma(s,\pi^{\prime},\wedge^{2}_{+}\times\eta,\psi)$
for any irreducible generic representation $\tau$ of ${\mathrm{GL}}_{k}(F)$
with $k=1,2$ and any quasi-character $\eta$ of $F^{\times}$, then
$\pi\cong\pi^{\prime}.$

Let us now explain the representation $\wedge^{2}_{+}$ of
${\mathrm{SO}}_{4}({\mathbb{C}})$. In fact, such a representation can be
defined for ${\mathrm{SO}}_{2n}({\mathbb{C}})$ where $n$ is any positive
integer, as explained below. Let $W={\mathbb{C}}^{2n}$ and let $\\{e_{i},1\leq
i\leq 2n\\}$ be the standard basis of $W$. Consider the bilinear form $Q$ on
$W$ defined by

$Q(e_{i},e_{j})=\begin{cases}1,&i+j=2n+1,\\\ 0,&i+j\neq 2n+1.\end{cases}$

In other words, the bilinear form $Q$ is given by
$Q(v_{1},v_{2})=v_{1}J_{2n}v_{2}^{t}$, where $v_{i}\in W$ is viewed as a row
vector, and $J_{2n}=\begin{pmatrix}&&1\\\ &\iddots&\\\
1&&\end{pmatrix}\in{\mathrm{GL}}_{2n}$. Using the bilinear form $Q$, we can
realize the groups ${\mathrm{O}}_{2n}({\mathbb{C}})$ and
${\mathrm{SO}}_{2n}({\mathbb{C}})$ as

${\mathrm{O}}_{2n}({\mathbb{C}})=\left\\{{g\in{\mathrm{GL}}_{2n}({\mathbb{C}}):Q(gw_{1},gw_{2})=Q(w_{1},w_{2})\text{
for all }w_{1},w_{2}\in W}\right\\},$

and

${\mathrm{SO}}_{2n}({\mathbb{C}})=\left\\{{g\in{\mathrm{O}}_{2n}({\mathbb{C}}):\det(g)=1}\right\\}.$

We fix an isomorphism $\wedge^{2n}(W)\to{\mathbb{C}}$ given by

$e_{1}\wedge e_{2}\wedge\dots\wedge e_{2n}\mapsto 1.$

Then the bilinear form $Q$ defines an isomorphism

$\varphi_{2}:\wedge^{n}(W^{*})\to\wedge^{n}(W)$

and the wedge product
$\wedge^{n}(W)\times\wedge^{n}(W)\to\wedge^{2n}(W)\to\wedge^{2n}(W)\cong{\mathbb{C}}$
determines an isomorphism

$\varphi_{1}:\wedge^{n}(W)\to\wedge^{n}(W^{*}).$

Denote $\rho=\varphi_{1}\circ\varphi_{2}$. Then we can check that
$\rho^{2}={\mathrm{id}}$. Let $W_{j}\subset\wedge^{n}(W)$ be the
$j$-eigenspace of $\rho$ for $j\in\left\\{{\pm 1}\right\\}$. In more explicit
words, given

${\mathcal{E}}=\sum e_{i_{1}}\wedge e_{i_{2}}\wedge\dots\wedge
e_{i_{n}}\in\wedge^{n}(W),$

we have

$\varphi_{2}({\mathcal{E}})=\frac{1}{m}e_{2n+1-i_{1}}\wedge
e_{2n+1-i_{2}}\dots\wedge
e_{2n+1-i_{n}}\in\wedge^{n}(W)\cong\wedge^{n}(W^{*}),$

where $m$ is the number of summands of pure wedges in ${\mathcal{E}}$. Then
${\mathcal{E}}\in W_{j}$ if and only if

${\mathcal{E}}\wedge\varphi_{2}({\mathcal{E}})=je_{1}\wedge
e_{2}\wedge\dots\wedge e_{2n},j\in\left\\{{\pm 1}\right\\}.$

Since for any $g\in{\mathrm{GL}}_{2n}({\mathbb{C}})$, we have
$(ge_{i_{1}})\wedge(ge_{i_{2}})\wedge\dots\wedge(ge_{i_{2n}})=\det(g)e_{i_{1}}\wedge\dots\wedge
e_{i_{2n}}$, so each $W_{j}$ is invariant under the action of
${\mathrm{SO}}_{2n}({\mathbb{C}})$. Denote the corresponding action of
${\mathrm{SO}}_{2n}({\mathbb{C}})$ on $W_{j}$ by $\wedge^{n}_{+}$ (resp.
$\wedge^{n}_{-}$) if $j=1$ (reps. $j=-1$). Then we have

$\wedge^{n}=\wedge^{n}_{+}\oplus\wedge^{n}_{-}.$

In fact, by [FH]*Theorem 19.2, both $\wedge^{n}_{+}$ and $\wedge^{n}_{-}$ are
irreducible. Actually, one can also check that
$\wedge^{n}_{-}=c\cdot\wedge^{n}_{-}$, where $c$ denotes the outer
automorphism of ${\mathrm{SO}}_{2n}({\mathbb{C}})$. Note that, the definition
of $\Lambda^{n}_{+}$ depends on the choice of the isomorphism
$\wedge^{2n}(W)\to{\mathbb{C}}$. Thus there is no canonical choice of
$\Lambda_{+}^{n}.$

We assume that $F$ is a $p$-adic field and $W_{F}^{\prime}$ is the Weil-
Deligne group of $F$. Let $\pi$ be an irreducible representation
${\mathrm{SO}}_{2n}(F)$ and let $\sigma$ be an irreducible representation of
${\mathrm{GL}}_{k}(F)$ for some positive integer $k$, then we write
$\gamma(s,\pi,\wedge^{n}_{\pm}\times\sigma,\psi):=\gamma(s,(\wedge^{n}_{\pm}\circ\phi_{\pi})\otimes\phi_{\sigma},\psi)$
by abuse of notation, where $\psi$ is a fixed nontrivial additive character of
$F$, and $\phi_{\pi}:W_{F}^{\prime}\to{\mathrm{SO}}_{2n}({\mathbb{C}})$ (resp.
$\phi_{\sigma}:W_{F}^{\prime}\to{\mathrm{GL}}_{k}({\mathbb{C}})$) is the local
Langlands parameter of $\pi$ (resp. $\sigma$) as in [Arthur]. Note that there
is also no canonical choice of $\phi_{\pi}$ because of the existence of the
outer automorphism $c$. Thus it is desirable to give pure representation
theoretic definition of these local gamma factors
$\gamma(s,\pi,\wedge^{n}_{+}\times\sigma,\psi)$.

In this article, for the small group ${\mathrm{SO}}_{4}$ case, given an
automorphic cuspidal generic representation $\pi$ of
${\mathrm{SO}}_{4}({\mathbb{A}})$ where ${\mathbb{A}}$ is the ring of adeles
of a global field $F$, and a character $\eta$ of
$F^{\times}\backslash{\mathbb{A}}^{\times}$, we construct a global integral
which is Eulerian and at an unramified place, it represents the local
$L$-function $L(s,\pi_{v},\wedge^{2}_{+}\times\eta_{v})$. The local gamma
factors $\gamma(s,\pi_{v},\wedge^{2}_{+}\times\eta_{v},\psi_{v})$ are then
constructed using the local functional equations of those local integrals. It
turns out that the new local gamma factor can determine the values of certain
Whittaker functions of $\pi$ on the Bruhat cells, at which only the sum of
Whittaker functions of $\pi$ and $c\cdot\pi$ can be determined using the
${\mathrm{GL}}_{k}$-twisted gamma factors, as shown in [HL:p-adic, HKK].
Theorem 1.1 follows from this consideration and the results of [HL:p-adic,
HKK] easily.

Although we can only handle the small rank case, we expect that our result can
shed some light on the general problem: find enough gamma factors for generic
representations of ${\mathrm{SO}}_{2n}(F)$ so that they can distinguish a
representation $\pi$ of ${\mathrm{SO}}_{2n}(F)$ and its $c$-conjugate
$c\cdot\pi$ if $\pi\ncong c\cdot\pi$. Indeed, we form the following refined
local converse conjecture for ${\mathrm{SO}}_{2n}(F)$.

######  Conjecture 1.2.

Let $F$ be a $p$-adic field. Let $\pi,\pi^{\prime}$ be two irreducible
$\psi$-generic representations of ${\mathrm{SO}}_{2n}(F)$ with the same
central character. Assume that

$\gamma(s,\pi\times\tau,\psi)=\gamma(s,\pi^{\prime}\times\tau,\psi),\gamma(s,\pi,\wedge^{n}_{+}\times\sigma,\psi)=\gamma(s,\pi^{\prime},\wedge^{n}_{+}\times\sigma,\psi),$

for all irreducible generic representations $\tau,\sigma$ of
${\mathrm{GL}}_{k}(F)$ with $1\leq k\leq n$, then we have
$\pi\cong\pi^{\prime}$.

In fact, in the ${\mathrm{SO}}_{4}$ case, we only used the gamma factors of
$\wedge^{2}_{+}$ twisted by ${\mathrm{GL}}_{1}$. Thus it is possible that many
twisted gamma factors in Conjecture 1.2 are redundant in the general case. The
requirement on twisting by ${\mathrm{GL}}_{k}$ with $1\leq k\leq n$ in the
above conjecture is just an analogue of the Jacquet’s local converse
conjecture as proved in [Chai, JL].

Moreover, in the above Conjecture 1.2, the gamma factors
$\gamma(s,\pi,\wedge^{n}_{+}\times\sigma,\psi)$ could be replaced by
$\gamma(s,\pi,\wedge^{n}_{-}\times\sigma,\psi)$. In fact, we expect that
$\gamma(s,\pi,\wedge^{n}_{-}\times\sigma,\psi)=\gamma(s,c\cdot\pi,\wedge^{n}_{+}\times\sigma,\psi),$
since $\wedge^{n}_{-}\cong c\cdot\wedge^{n}_{+}$. Thus the corresponding gamma
factors for the representation $\wedge^{n}_{-}$ won’t give more information if
one understands $\gamma(s,\pi,\wedge^{n}_{+}\otimes\sigma,\psi)$ for every
$\pi$ and $\sigma$.

If $\pi$ is an unramified representation of ${\mathrm{SO}}_{2n}(F)$ in general
position (which means its Satake parameters are in general position), it is
easy to see that the gamma factors (in this case, they are just $L$-factors)
$\gamma(s,\pi,\wedge^{n}_{+},\psi)$ can distinguish $\pi$ and $c\cdot\pi$.
Thus if one of $\pi,\pi^{\prime}$ is unramified in general position, the above
conjecture follows from the results of [HL:p-adic, HKK]. But for general
representations $\pi$ of ${\mathrm{SO}}_{2n}(F)$ with $n\geq 3$, we don’t know
how to construct the gamma factors
$\gamma(s,\pi,\wedge^{n}_{+}\times\sigma,\psi)$ from the representation
theoretic side at this moment.

The paper is organized as follows. In Section 2, we introduce the local zeta
integral and compute it at unramified places, and use it to define the gamma
factor $\gamma(s,\pi,\wedge^{2}_{+}\times\eta,\psi)$. We also construct a
global integral and show that it is Eulerian and its local piece is the local
zeta integral we study in this paper. In Section 3 we review the theory of
partial Bessel functions as well as a local converse theorem of Hazeltine and
Liu. Finally, in Section 4, we prove Theorem 1.1.

## Acknowledgement

The authors would like to thank Alex Hazeltine and Baiying Liu for sending us
their manuscripts [HL:finite, HL:p-adic] and for helpful communications. The
idea of this paper was initiated when the second named author was a graduate
student at Ohio State University. We thank our advisor Jim Cogdell for his
constant encouragement and support.

## Notations

Let $F$ be a $p$-adic field. For an algebraic group $\mathbf{G}$ we denote its
group of $F$-points by $G(F)$ or simply by $G$. For a positive integer $r,$
let $J_{r}=\begin{pmatrix}&&1\\\ &\iddots&\\\
1&&\end{pmatrix}\in{\mathrm{GL}}_{r}(F)$. We realize the special orthogonal
group ${\mathrm{SO}}_{r}$ as the group of all $r\times r$ matrices which
satisfy ${}^{t}\\!gJ_{r}g=J_{r}$. In this paper we will focus on the case when
$r=4$.

Let $B$ denote the upper triangular Borel subgroup of ${\mathrm{SO}}_{4}$.
Then we have the Levi decomposition $B=TU$ where $T$ is the torus and $U$ is
the unipotent subgroup. More specifically, we have

$T=\left\\{{t(a_{1},a_{2}):=\begin{pmatrix}a_{1}&&&\\\ &a_{2}&&\\\
&&a_{2}^{-1}&\\\ &&&a_{1}^{-1}\end{pmatrix},a_{1},a_{2}\in
F^{\times}}\right\\},$

and

$U=\left\\{{u(x,y):={\bf{x}}_{\alpha}(x){\bf{x}}_{\beta}(y)=\begin{pmatrix}1&x&&\\\
&1&&\\\ &&1&-x\\\ &&&1\end{pmatrix}\begin{pmatrix}1&&y&\\\ &1&&-y\\\ &&1&\\\
&&&1\end{pmatrix},x,y\in F}\right\\}.$

Let $\bar{U}$ be the opposite of $U$, i.e.,

$\overline{U}=\left\\{{\bar{u}(x,y):={\bf{x}}_{-\alpha}(x){\bf{x}}_{-\beta}(y)=\begin{pmatrix}1&&&\\\
x&1&&\\\ &&1&\\\ &&-x&1\end{pmatrix}\begin{pmatrix}1&&&\\\ &1&&\\\ y&&1&\\\
&-y&&1\end{pmatrix},x,y\in F}\right\\}.$

Let $P=MN$ be the Siegel parabolic subgroup of ${\mathrm{SO}}_{4}$, with the
Levi subgroup $M\cong{\mathrm{GL}}_{2}$ and unipotent radical
$N=\left\\{{{\bf{x}}_{\beta}(y),y\in F}\right\\}$. For
$h\in{\mathrm{GL}}_{2}$, we let $h^{*}=J_{2}{}^{t}\\!h^{-1}J_{2}$ and denote

${\bf{m}}(h):=\begin{pmatrix}h&\\\ &h^{*}\end{pmatrix}\in M.$

Consider the roots $\alpha,\beta$ of ${\mathrm{SO}}_{4}$ defined by

$\alpha(t(a_{1},a_{2}))=a_{1}/a_{2},\beta(t(a_{1},a_{2}))=a_{1}a_{2}.$

Then the set $\Delta=\left\\{{\alpha,\beta}\right\\}$ is the set of simple
roots of ${\mathrm{SO}}_{4}$. Let
${\mathbf{W}}=\left\\{{1,s_{\alpha},s_{\beta},s_{\alpha}s_{\beta}}\right\\}$
be the Weyl group of ${\mathrm{SO}}_{4}$. In matrix form, we can take

$s_{\alpha}=\begin{pmatrix}&1&&\\\ -1&&&\\\ &&&-1\\\ &&1&\end{pmatrix}\textrm{
and }s_{\beta}=\begin{pmatrix}&&1&\\\ &&&-1\\\ -1&&&\\\ &1&&\end{pmatrix}.$

Let $\left\\{{e_{1},e_{2},e_{3},e_{4}}\right\\}$ be the standard basis of the
dimension 4 quadratic space $(W={\mathbb{C}}^{4},Q)$ which defines
${\mathrm{SO}}_{4}({\mathbb{C}})$ with matrix
$\left\\{{Q(e_{i},e_{j})}\right\\}=J_{4}$. The subspace
$W_{1}\subset\wedge^{2}(W)$ as in the introduction is spanned by
${\mathcal{B}}=\left\\{{e_{1}\wedge e_{3},e_{1}\wedge e_{4}+e_{3}\wedge
e_{2},e_{2}\wedge e_{4}}\right\\}$. Let $\wedge^{2}_{+}$ be the action of
${\mathrm{SO}}_{4}({\mathbb{C}})$ on $W_{1}$. For a torus element
${\mathrm{diag}}(a,b,b^{-1},a^{-1})\in{\mathrm{SO}}_{4}({\mathbb{C}})$, we
have

$\wedge^{2}_{+}({\mathrm{diag}}(a,b,b^{-1},a^{-1}))={\mathrm{diag}}(ab^{-1},1,a^{-1}b),$

with respect to the ordered basis ${\mathcal{B}}$ of $W_{1}$.

Let $\pi$ be an irreducible representation of ${\mathrm{SO}}_{4}(F)$ over $F$.
Let $\varpi$ be a uniformizer of $F$ and let $q$ be the cardinality of the
residue field of $F$. We can consider the $L$-function
$L(s,\pi,\wedge^{2}_{+})$. In particular, if $\pi$ is unramified with Satake
parameter
${\mathrm{diag}}(a,b,b^{-1},a^{-1})\in{\mathrm{SO}}_{4}(\mathbb{C})$, we have

$L(s,\pi,\wedge^{2}_{+})=\frac{1}{(1-q^{-s})(1-ab^{-1}q^{-s})(1-a^{-1}bq^{-s})}.$

If in addition $\eta$ is a character of $F^{\times}$, we then have

$L(s,\pi,\wedge^{2}_{+}\times\eta)=\frac{1}{(1-\eta(\varpi)q^{-s})(1-ab^{-1}\eta(\varpi)q^{-s})(1-a^{-1}b\eta(\varpi)q^{-s})}.$

Consider

$c=\begin{pmatrix}1&&&\\\ &&1&\\\ &1&&\\\ &&&1\end{pmatrix},$

which is an element in ${\mathrm{O}}_{4}-{\mathrm{SO}}_{4}$ and it is used to
define the outer automorphism on ${\mathrm{SO}}_{4}$.

##  2\. The local zeta integral and gamma factor for $\wedge^{2}_{+}$

Let $F$ be a non-archimedean local field and $\psi$ be a fixed additive
character of $F$. Let ${\mathfrak{o}}_{F}$ be the ring of integers of $F$,
${\mathfrak{p}}$ the maximal ideal of ${\mathfrak{o}}_{F}$, $\varpi$ be a
fixed uniformizer of $F$ and $q$ the cardinality of
${\mathfrak{o}}_{F}/{\mathfrak{p}}$. Let $(\cdot,\cdot)_{F}$ be the local
Hilbert symbol. In this section, given a generic irreducible unramified
representation $\pi$ of ${\mathrm{SO}}_{4}(F)$ and a character $\eta$ of
$F^{\times}$, we construct a local zeta integral which represents the
$L$-function $L(s,\pi,\wedge^{2}_{+}\times\eta)$. The integral is pretty much
like the integral of ${\mathrm{SL}}_{2}\times{\mathrm{GL}}_{1}$ constructed in
[GPSR], which is not surprising considering that
${\mathrm{SO}}_{4}({\mathbb{C}})$ is just
${\mathrm{SL}}_{2}({\mathbb{C}})\times{\mathrm{SL}}_{2}({\mathbb{C}})/\left\\{{\pm
1}\right\\}.$

###  2.1. Weil representations of $\widetilde{\mathrm{SL}}_{2}$

We recall the theory of the Weil representation of
$\widetilde{\mathrm{SL}}_{2}(F)$, the metaplectic double cover of
${\mathrm{SL}}_{2}(F)$. Recall that the product on
$\widetilde{\mathrm{SL}}_{2}(F)$ is given by

$(g_{1},\zeta_{1})(g_{2},\zeta_{2})=(g_{1}g_{2},\zeta_{1}\zeta_{2}c(g_{1},g_{2})),$

where $c:{\mathrm{SL}}_{2}(F)\times{\mathrm{SL}}_{2}(F)\rightarrow\left\\{{\pm
1}\right\\}$ is defined by

$c(g_{1},g_{2})=({\bf{x}}(g_{1}),{\bf{x}}(g_{2}))_{F}(-{\bf{x}}(g_{1}){\bf{x}}(g_{2}),{\bf{x}}(g_{1}g_{2}))_{F},$

where

${\bf{x}}\begin{pmatrix}a&b\\\
c&d\end{pmatrix}=\left\\{\begin{array}[]{lll}c,&c\neq 0,\\\
d,&c=0.\end{array}\right.$

For a subgroup $A$ of ${\mathrm{SL}}_{2}(F)$, we denote by $\widetilde{A}$ the
preimage of $A$ in $\widetilde{\mathrm{SL}}_{2}(F)$, which is a subgroup of
$\widetilde{\mathrm{SL}}_{2}(F)$. For an element $g\in{\mathrm{SL}}_{2}(F)$,
we sometimes also let $g$ denote the element $(g,1)$ inside
$\widetilde{\mathrm{SL}}_{2}(F)$.

A representation $\pi$ of $\widetilde{\mathrm{SL}}_{2}(F)$ is called genuine
if $\pi(\zeta g)=\zeta\pi(g)$ for all $g\in\widetilde{\mathrm{SL}}_{2}(F)$ and
$\zeta\in\mu_{2}$. Let $\omega_{\psi}$ be the Weil representation of
$\widetilde{\mathrm{SL}}_{2}(F)$ realized on ${\mathcal{S}}(F)$, the space of
Bruhat-Schwartz functions on $F$. For any $f\in{\mathcal{S}}(F)$, the action
is given by the following formulas:

$\displaystyle\left(\omega_{\psi}\begin{pmatrix}&1\\\
-1&\end{pmatrix}\right)f(x)$ $\displaystyle=\gamma(\psi)\hat{f}(x),$
$\displaystyle\left(\omega_{\psi}\begin{pmatrix}1&b\\\
&1\end{pmatrix}\right)f(x)$ $\displaystyle=\psi(bx^{2})f(x),b\in F,$
$\displaystyle\left(\omega_{\psi}\begin{pmatrix}a&\\\
&a^{-1}\end{pmatrix}\right)f(x)$
$\displaystyle=|a|^{1/2}\frac{\gamma(\psi)}{\gamma(\psi_{a})}f(ax),a\in
F^{\times},$

and

$\omega_{\psi}(\zeta)f(x)=\zeta f(x),\zeta\in\mu_{2}.$

Several notations need to be explained. Here
$\hat{f}(x)=\int_{F}f(y)\psi(2xy)dy$ is the Fourier transform of $f(x)$ and
the Haar measure $dy$ is normalized so that $(\hat{f})^{\hat{~}{}}(x)=f(-x)$.
The constant $\gamma(\psi)$ is the Weil index associated to the character
$\psi$. For $a\in F^{\times}$, the character $\psi_{a}$ is defined by
$\psi_{a}(x)=\psi(ax)$.

Let $T_{2}$ be the torus of ${\mathrm{SL}}_{2}(F)$, which consists of elements
of the form $t_{{\mathrm{SL}}_{2}}(a)={\mathrm{diag}}(a,a^{-1})$ where $a\in
F^{\times}$. The product in $\widetilde{T}_{2}$ is given by the Hilbert
symbol, i.e.,

${\mathrm{diag}}(a,a^{-1}){\mathrm{diag}}(b,b^{-1})={\mathrm{diag}}(ab,(ab)^{-1})(a,b)_{F}.$

The function

$\mu_{\psi}(a)=\frac{\gamma(\psi)}{\gamma(\psi_{a})}$

satisfies

$\mu_{\psi}(a)\mu_{\psi}(b)=\mu_{\psi}(ab)(a,b)_{F},$

and thus defines a genuine character of $\widetilde{T}_{2}$. All of the above
facts in this subsection can be found in [GPS80, Section 1].

###  2.2. The local zeta integral

Denote $p:\widetilde{\mathrm{SL}}_{2}(F)\rightarrow{\mathrm{SL}}_{2}(F)$ the
projection. For a character $\eta$ of $F^{\times}$ and $s\in{\mathbb{C}}$,
consider the character $\eta_{s}$ of $\widetilde{T}_{2}$ defined by
$\eta_{s}((t_{{\mathrm{SL}}_{2}}(a),\zeta))=\eta_{s}(a)=\eta(a)|a|^{s}$. Then
$\eta_{s}$ factor through ${\mathrm{SL}}_{2}(F)$ and hence is not genuine. The
product $\mu_{\psi}\eta_{s}$ is a genuine character of $\widetilde{T}_{2}.$
Let $P_{2}=T_{2}N_{2}$ be the upper triangular parabolic subgroup of
${\mathrm{SL}}_{2}$ with unipotent $N_{2}$. Then
$\widetilde{P}_{2}=\widetilde{T}_{2}\ltimes N_{2}$. Extend
$\mu_{\psi}\eta_{s}$ to a character of $\widetilde{P}_{2}$ by letting the
action of $N_{2}$ be trivial. We consider the induced representation

$I(s,\eta,\psi)={\mathrm{Ind}}_{\widetilde{P}_{2}}^{\widetilde{\mathrm{SL}}_{2}}(\mu_{\psi}\eta_{s-1/2}).$

A section $f_{s}\in I(s,\eta,\psi)$ satisfies

$f_{s}((t_{{\mathrm{SL}}_{2}}(a),\zeta)\tilde{h})=\zeta\mu_{\psi}(a)\eta(a)|a|^{s+1/2}f_{s}(\tilde{h}).$

Note that for any $\phi\in{\mathcal{S}}(F)$ and $f_{s}\in I(s,\eta,\psi)$, the
function

$\omega_{\psi^{-1}}(\tilde{h})\phi(1)f_{s}(\tilde{h})$

on $\widetilde{\mathrm{SL}}_{2}(F)$ is trivial on $\mu_{2}$, and thus defines
a function on ${\mathrm{SL}}_{2}$.

Let $\psi_{U}$ be the generic character of $U$ defined by

$\psi_{U}(u(x,y))=\psi\left(x-2y\right).$

The choice of $\psi_{U}$ defined above is to make the integrals for
${\mathrm{SO}}_{4}\times{\mathrm{GL}}_{1}$ and
${\mathrm{SO}}_{4}\times{\mathrm{GL}}_{2}$ easier; see [Kaplan, HL:finite,
HL:p-adic]. In fact, if we fix a nontrivial additive character $\psi_{0}$ of
$F$ in priori and take $\psi(x)=\psi_{0}(\frac{1}{4}x)$, the character
$\psi_{U}$ is the same as those defined in [Kaplan, HL:finite, HL:p-adic]. Let
$(\pi,V_{\pi})$ be a $\psi_{U}$-generic representation of
${\mathrm{SO}}_{4}(F)$. For $W\in{\mathcal{W}}(\pi,\psi_{U})$,
$\phi\in{\mathcal{S}}(F)$ and $f_{s}\in I(s,\eta,\psi^{-1})$, we consider the
local zeta integral

$\Psi(W,\phi,f_{s})=\int_{N_{2}\backslash{\mathrm{SL}}_{2}(F)}W\left({\bf{m}}(h)\right)(\omega_{\psi^{-1}}(h)\phi)(1)f_{s}(h)dh.$

Formally this is well-defined because for any $n\in N_{2}$, we have
$W({\bf{m}}(nh))=\psi(n)W({\bf{m}}(h))$ and
$\omega_{\psi^{-1}}(nh)\phi(1)=\psi^{-1}(n)\omega_{\psi^{-1}}(h)\phi(1)$.

######  Lemma 2.1.

The local zeta integral $\Psi(W,\phi,f_{s})$ is absolutely convergent for
${\mathrm{Re}}(s)\gg 0$ and defines a meromorphic function of $q^{-s}$.
Moreover, we can choose $W,\phi,f_{s}$ such that $\Psi(W,\phi,f_{s})$ is a
nonzero constant.

###### Proof.

The first assertion follows from the asymptotic behavior of $W$ and we omit
the details. See [GPSR] for a proof of similar results. The second assertion
will be proved by (4.1) (in the proof of Proposition 4.3). ∎

###  2.3. The local functional equation

Recall that $N=\left\\{{{\bf{x}}_{\beta}(y):y\in F}\right\\}$. Denote
$\psi_{N}=\psi_{U}|_{N}$. For $A\in{\mathrm{GL}}_{2}(F)$, we have

${\bf{m}}(A){\bf{x}}_{\beta}(y){\bf{m}}(A)^{-1}={\bf{x}}_{\beta}(\det(A)y).$

Thus the stabilizer of $\psi_{N}$ in $M\cong{\mathrm{GL}}_{2}(F)$ is
${\mathrm{SL}}_{2}(F)$. From this relation, it is easy to check the following

######  Lemma 2.2.

For any $n\in N$ and $h\in\widetilde{\mathrm{SL}}_{2}(F)$, we have

$W({\bf{m}}(p(h))n)=\psi_{N}(n)W({\bf{m}}(p(h))).$

We now prove the following results.

######  Proposition 2.3.

Except for a finite number of $q^{-s}$, there is at most one trilinear form
$T$ on ${\mathcal{W}}(\pi,\psi_{U})\times\omega_{\psi^{-1}}\times
I(s,\eta,\psi^{-1})$ such that

$T(\pi(n)W,\phi,f_{s})=\psi_{N}(n)T(W,\phi,f_{s}),\forall n\in N$

and

$T(\pi({\bf{m}}(p(h)))W,\omega_{\psi^{-1}}(h)\phi,r(h)f_{s})=T(W,\phi,f_{s}),\forall
h\in\widetilde{\mathrm{SL}}_{2}(F).$

Here $r(h)$ denotes the right translation by $h$.

###### Proof.

A trilinear form $T$ as in the proposition defines an element in

$\displaystyle{\mathrm{Hom}}_{\widetilde{\mathrm{SL}}_{2}}(\pi_{N,\psi_{N}}\otimes\omega_{\psi^{-1}}\otimes
I(s,\eta,\psi^{-1}),{\mathbb{C}})$ $\displaystyle=$
$\displaystyle{\mathrm{Hom}}_{\widetilde{P}_{2}}(\pi_{N,\psi_{N}}\otimes\omega_{\psi^{-1}},\widetilde{\eta}_{s}^{-1}),$

where $\pi_{N,\psi_{N}}$ is the Jacquet module of $\pi$ with respect to
$(N,\psi_{N})$ and it is viewed as a representation of
$\widetilde{\mathrm{SL}}_{2}$ by the projection $p$, and
$\widetilde{\eta}_{s}=\mu_{\psi^{-1}}\eta_{s-1/2}$.

We consider the representation $\pi_{N,\psi_{N}}$ of ${\mathrm{SL}}_{2}$. We
have the exact sequence

$0\rightarrow\oplus_{\alpha\in
F^{\times}/F^{\times,2}}{\mathrm{ind}}_{N_{2}}^{P_{2}}((\pi_{N,\psi_{N}})_{N_{2},\psi_{\alpha}})\rightarrow\pi_{N,\psi_{N}}\rightarrow(\pi_{N,\psi_{N}})_{N_{2}}\rightarrow
0.$

Recall that $\psi_{\alpha}$ denotes the character of $F$ defined by
$\psi_{\alpha}(x)=\psi(\alpha x)$. This exact sequence follows from a simple
application of the general theory of [BZ]; see [U11]*(1.1) for a proof of a
similar situation. Since the Jacquet functor preserve admissibility and sends
a finitely generated representation to a finitely generated representation,
$\pi_{N,\psi_{N}}$ has finite length as a representation of
${\mathrm{SL}}_{2}(F)$ and $(\pi_{N,\psi_{N}})_{N_{2}}$ is finite dimensional.
By the uniqueness of Whittaker model, we have
$\dim(\pi_{N,\psi_{N}})_{N_{2},\psi_{\alpha}}\leq 1$ and by our assumption, we
have

$\dim(\pi_{N,\psi_{N}})_{N_{2},\psi}=\dim\pi_{U,\psi_{U}}=1.$

Thus after excluding a finite number of $q^{-s}$, we have

$\displaystyle{\mathrm{Hom}}_{\widetilde{P}_{2}}(\pi_{N,\psi_{N}}\otimes\omega_{\psi^{-1}},\tilde{\eta}_{s})$
$\displaystyle=\oplus_{\alpha\in
F^{\times}/F^{\times,2}}{\mathrm{Hom}}_{\widetilde{P}_{2}}({\mathrm{ind}}_{N_{2}}^{P_{2}}(\psi_{\alpha})\otimes\omega_{\psi^{-1}},\widetilde{\eta}_{s})$
$\displaystyle=\oplus_{\alpha\in
F^{\times}/F^{\times,2}}{\mathrm{Hom}}_{P_{2}}({\mathrm{ind}}_{N_{2}}^{P_{2}}(\psi_{\alpha}),\omega_{\psi}\otimes\widetilde{\eta}_{s}^{-1})$
$\displaystyle=\oplus_{\alpha\in
F^{\times}/F^{\times,2}}{\mathrm{Hom}}_{N_{2}}(\psi_{\alpha},\omega_{\psi}\otimes\widetilde{\eta}_{s}^{-1}).$

The result follows from the fact that
$(\omega_{\psi}\otimes\widetilde{\eta}_{s}^{-1})_{N_{2},\psi_{\alpha}}=0$ if
$\alpha\neq 1$ and
$\dim(\omega_{\psi}\otimes\widetilde{\eta}_{s}^{-1})_{N_{2},\psi_{\alpha}}=1$,
which could be checked easily from the Weil representation formulas. ∎

Let $M_{s}:I(s,\eta,\psi^{-1})\to I(1-s,\eta^{-1},\psi^{-1})$ be the standard
intertwining operator defined by

$M_{s}(f_{s})(g)=\int_{N_{2}}f_{s}(w_{2}ng)dn,$

where $w_{2}=\begin{pmatrix}0&1\\\ -1&0\end{pmatrix}\in{\mathrm{SL}}_{2}(F)$.
Consider the character $\chi$ of $F^{\times}$ defined by $\chi(a)=(a,-1)_{F}$.

######  Corollary 2.4.

There is a meromorphic function
$\gamma(s,\pi,\wedge^{2}_{+}\times\chi\eta,\psi)$ such that

$\Psi(W,\phi,M_{s}(f_{s}))=\gamma(s,\pi,\wedge^{2}_{+}\times\chi\eta)\Psi(W,\phi,f_{s}),$

for all $W\in{\mathcal{W}}(\pi,\psi),\phi\in{\mathcal{S}}(F)$ and $f_{s}\in
I(s,\eta,\psi)$.

###### Proof.

Using Lemma 2.2, we can check that both trilinear forms
$(W,\phi,f_{s})\mapsto\Psi(W,\phi,f_{s})$ and
$(W,\phi,f_{s})\mapsto\Psi(W,\phi,M_{s}(f_{s}))$ satisfy the conditions in
Proposition 2.3. Thus by the uniqueness of such trilinear forms, these two
trilinear forms are proportional. Denote this proportion by
$\gamma(s,\pi,\wedge^{2}_{+}\times\chi\eta,\psi)$ temporarily. By Lemma 2.1,
$\gamma(s,\pi,\wedge^{2}_{+}\times\chi\eta,\psi)$ is a meromorphic function of
$s$. ∎

The notation $\gamma(s,\pi,\wedge^{2}_{+}\times\chi\eta,\psi)$ will be
justified by the unramified calculation in the next subsection.

###  2.4. Unramified calculation

Let $\pi$ be an unramified representation of ${\mathrm{SO}}_{4}(F)$ with
Satake parameter
${\mathrm{diag}}(a,b,b^{-1},a^{-1})\in{\mathrm{SO}}_{4}({\mathbb{C}})$. Let
$W^{0}$ be the unramified Whittaker function for $\pi$ normalized so that
$W^{0}(I_{4})=1$. Let $a_{k}=t(\varpi^{k},\varpi^{-k})$. By the Casselman-
Shalika formula [CS]*Theorem 5.4, we have

$W^{0}(a_{k})=\frac{q^{-k}}{(a-b)(ab-1)}(a^{k+2}b^{-k+1}-a^{-k+1}b^{k+2}-a^{k+1}b^{-k}+a^{-k}b^{k+1})$

if $k\geq 0$, and $W^{0}(a_{k})=0$ if $k<0$. For
$h\in{\mathrm{SL}}_{2}(F)=N_{2}T_{2}K_{2}$, with
$K_{2}={\mathrm{SL}}_{2}({\mathfrak{o}}_{F})$, we can write
$h=n{\mathrm{diag}}(a,a^{-1})k$. The Haar measure on the quotient
$N_{2}\backslash{\mathrm{SL}}_{2}(F)$ is $dh=|a|^{-2}dkda.$ We also assume
that $\psi$ is unramified. Thus we have $\mu_{\psi^{-1}}(u)=1$ for
$u\in{\mathfrak{o}}_{F}^{\times}$. Let $\phi^{0}\in{\mathcal{S}}(F)$ be the
characteristic function of ${\mathfrak{o}}_{F}$ and let $f_{s}^{0}\in
I(s,\eta,\psi^{-1})$ be the function such that $f(k)=1$ for $k\in K_{2}$.

We have

$f_{s}^{0}({\mathrm{diag}}(\varpi^{k},\varpi^{-k}))=\mu_{\psi^{-1}}(\varpi^{k})\eta(\varpi)^{k}|\varpi^{k}|^{s+1/2},$

and

$(\omega_{\psi^{-1}}({\mathrm{diag}}(\varpi^{k},\varpi^{-k}))\phi^{0})(1)=|\varpi^{k}|^{1/2}\mu_{\psi^{-1}}(\varpi^{k}).$

Notice that

$\mu_{\psi^{-1}}(\varpi^{k})\mu_{\psi^{-1}}(\varpi^{k})=\mu_{\psi^{-1}}(\varpi^{2k})(\varpi^{k},\varpi^{k})=(\varpi^{k},\varpi^{k})=(\varpi,-1)^{k}=\chi(\varpi)^{k}.$

Thus, we obtain

$\displaystyle\Psi(W^{0},\phi^{0},f_{s}^{0})$
$\displaystyle=\sum_{k=0}^{\infty}W^{0}(a_{k})\eta(\varpi)^{k}q^{-ks}\chi(\varpi)^{k}$
$\displaystyle=\frac{1}{(a-b)(ab-1)}\sum_{k}(a^{k+2}b^{-k+1}-a^{-k+1}b^{k+2}-a^{k+1}b^{-k}+a^{-k}b^{k+1})t^{k}$
$\displaystyle=\frac{1+t}{(1-ab^{-1}t)(1-a^{-1}bt)}$
$\displaystyle=\frac{1-t^{2}}{(1-t)(1-ab^{-1}t)(1-a^{-1}bt)},$

where $t=\chi(\varpi)\eta(\varpi)q^{-s}.$ Hence we get

$\Psi(W^{0},\phi^{0},f_{s}^{0})=\frac{L(s,\pi,\wedge^{2}_{+}\times\chi\eta)}{L(2s,\eta^{2})}.$

We summarize the above calculation in the following

######  Proposition 2.5.

Assume that $\pi$ is an unramified representation of ${\mathrm{SO}}_{4}(F)$
with Satake parameter ${\mathrm{diag}}(a,b,b^{-1},a^{-1})$,
$W^{0}\in{\mathcal{W}}(\pi,\psi)$ is the unramified Whittaker function such
that $W^{0}(I_{4})=1$, $\phi^{0}\in{\mathcal{S}}(F)$ is the characteristic
function of ${\mathfrak{o}}_{F}$ and $f_{s}^{0}\in I(s,\eta,\psi^{-1})$ is the
function such that $f_{s}^{0}(k)=1$ for
$k\in{\mathrm{SL}}_{2}({\mathfrak{o}}_{F})$. Then we have

$\Psi(W^{0},\phi^{0},f_{s}^{0})=\frac{L(s,\pi,\wedge^{2}_{+}\times\chi\eta)}{L(2s,\eta^{2})}.$

###  2.5. A global integral

In this subsection, we present a global integral so that it is Eulerian and
its local piece at a local place is the the local zeta integral defined in
$\S$2.2. In this subsection, $F$ is a global field such that its
characteristic is not $2$. Let ${\mathbb{A}}$ be the ring of adeles of $F$.
Let $\psi$ be a fixed additive character of $F\backslash{\mathbb{A}}$.

Let $\eta$ be a quasi-character of
$F^{\times}\backslash{\mathbb{A}}^{\times}$, and $s\in{\mathbb{C}}$. We
consider the global induced representation

$I(s,\eta,\psi^{-1})={\mathrm{Ind}}_{\widetilde{P}_{2}({\mathbb{A}})}^{\widetilde{\mathrm{SL}}_{2}({\mathbb{A}})}(\mu_{\psi^{-1}}\eta_{s-1/2}).$

Here $\mu_{\psi^{-1}}$ is a global version of the local $\mu_{\psi^{-1}}$
defined in $\S$2.1. For a standard section $f_{s}\in I(s,\eta,\psi^{-1})$, we
consider the Eisenstein series

$E(s,f_{s})(h)=\sum_{\gamma\in
B_{2}(F)\backslash{\mathrm{SL}}_{2}(F)}f_{s}(\gamma h).$

It is standard that $E(s,f_{s})$ converges for ${\mathrm{Re}}(s)\gg 0$ and has
a meromorphic extension to ${\mathbb{C}}$. There is also a global Weil
representation $\omega_{\psi}$ of $\widetilde{\mathrm{SL}}_{2}({\mathbb{A}})$
on ${\mathcal{S}}({\mathbb{A}})$, the space of Bruhat-Schwatz functions on
${\mathbb{A}}$. For $\phi\in{\mathcal{S}}({\mathbb{A}})$, we consider the
theta series on $\widetilde{\mathrm{SL}}_{2}({\mathbb{A}})$:

$\theta(\phi)(g)=\sum_{x\in F}(\omega_{\psi}(g)\phi)(x).$

We have

$\theta(\phi)(g)=\theta(\phi)_{0}(g)+\sum_{a\in
F^{\times}}|a|^{-1/2}\mu_{\psi}(a)^{-1}(\omega_{\psi}(t_{{\mathrm{SL}}_{2}}(a)g)\phi)(1).$

Let $\varphi$ be a cusp form on
${\mathrm{SO}}_{4}(F)\backslash{\mathrm{SO}}_{4}({\mathbb{A}})$, $f_{s}$ be a
standard section in $I(s,\eta,\psi^{-1})$, $\theta$ be a theta series on
$\widetilde{\mathrm{SL}}_{2}({\mathbb{A}})$. We consider the integral

$Z(s,\varphi,\theta,f_{s})=\int_{{\mathrm{SL}}_{2}(F)\backslash{\mathrm{SL}}_{2}({\mathbb{A}})}\int_{N(F)\backslash
N({\mathbb{A}})}\varphi(n{\bf{m}}(g))\psi^{\prime}(n)dn\theta(g)E(s,g,f_{s})dg.$

By the definition of $E$, we get

$\displaystyle Z(s,\varphi,\theta,f_{s})$
$\displaystyle=\int_{B_{2}(F)\backslash{\mathrm{SL}}_{2}({\mathbb{A}})}\int_{N(F)\backslash
N({\mathbb{A}})}\varphi(n{\bf{m}}(g))\psi^{\prime}(n)dn\theta(g)f_{s}(g)dg.$

Now plugging the definition of $\theta$, and note that the term $\theta_{0}$
contributes zero because $\phi$ is a cusp form. Thus

$\displaystyle Z(s,\varphi,\theta,f_{s})$
$\displaystyle=\int_{N_{2}(F)\backslash{\mathrm{SL}}_{2}({\mathbb{A}})}\int_{N(F)\backslash
N({\mathbb{A}})}\varphi(n{\bf{m}}(g))\psi^{\prime}(n)dn(\omega_{\psi}(g)\phi)(1)f_{s}(g)dg$
$\displaystyle=\int_{N_{2}({\mathbb{A}})\backslash{\mathrm{SL}}_{2}({\mathbb{A}})}\int_{N_{2}(F)\backslash
N_{2}({\mathbb{A}})}\int_{N(F)\backslash
N({\mathbb{A}})}\varphi(n{\bf{m}}(u){\bf{m}}(g))\psi^{\prime}(n)dn(\omega_{\psi}(ug)\phi)(1)f_{s}(ug)dudg$
$\displaystyle=\int_{N_{2}({\mathbb{A}})\backslash{\mathrm{SL}}_{2}({\mathbb{A}})}\int_{N_{2}(F)\backslash
N_{2}({\mathbb{A}})}\int_{N(F)\backslash
N({\mathbb{A}})}\varphi(n{\bf{m}}(u){\bf{m}}(g))\psi^{\prime}(n)dn\psi(u)du(\omega_{\psi}(g)\phi)(1)f_{s}(g)dg$
$\displaystyle=\int_{N_{2}({\mathbb{A}})\backslash{\mathrm{SL}}_{2}({\mathbb{A}})}\int_{U(F)\backslash
U({\mathbb{A}})}\varphi(u{\bf{m}}(g))\psi_{U}(u)du(\omega_{\psi}(g)\phi)(1)f_{s}(g)dg$
$\displaystyle=\int_{N_{2}({\mathbb{A}})\backslash{\mathrm{SL}}_{2}({\mathbb{A}})}W_{\varphi}^{\psi_{U}^{-1}}({\bf{m}}(g))(\omega_{\psi}(g)\phi)(1)f_{s}(g)dg.$

Thus the global integral is Eulerian and its local piece is exactly the local
integral defined in §2.2.

##  3\. Partial Bessel functions

Write $G={\mathrm{SO}}_{4}(F)$ and $Z=\left\\{{\pm I_{4}}\right\\}$ the center
of $G$. In this section, we assume that $\psi$ is an unramified character of
$F$. For a character $\omega$ of $G$, let $C_{c}^{\infty}(G,\omega)$ be the
space of compactly supported smooth functions $f$ on $G$ such that
$f(zg)=\omega(z)f(g)$ for all $z\in Z,g\in G$. Moreover, denote
$C^{\infty}(G,\psi_{U},\omega)$ the space of functions $W$ on $G$ such that
$W(zug)=\omega(z)\psi_{U}(u)W(g)$ for all $z\in Z,u\in U,g\in G$. Here the
smoothness of $W$ means that for each $W$, there exists an open compact
subgroup $K$ such that $W(gk)=W(g)$ for all $g\in G,k\in K$.

###  3.1. Howe vectors

We recall the theory of Howe vectors as developed in [Ba95]. For a positive
integer $m$, let $K_{m}=(1+{\mathrm{Mat}}_{4\times
4}({\mathfrak{p}}^{m}))\cap{\mathrm{SO}}_{4}(F)$. Define a character
$\tau_{m}$ of $K_{m}$ by

$\tau_{m}(k)=\psi(\varpi^{-2m}(k_{12}-2k_{13})).$

One can check that $\tau_{m}$ is indeed a character of $K_{m}$.

Let $d_{m}=t(\varpi^{-2m},1).$ Consider the subgroup
$H_{m}=d_{m}K_{m}d_{m}^{-1}$. Define $\psi_{m}(h)=\tau_{m}(d_{m}^{-1}hd_{m})$
for $h\in H_{m}$. Let $U_{m}=U\cap H_{m}$. We then have

$U_{m}=\left\\{{u(x,y),x,y\in{\mathfrak{p}}^{-m}}\right\\}$

and $\psi_{m}|_{U_{m}}=\psi_{U}|_{U_{m}}$.

For a positive integer $m$ and an element $W\in C^{\infty}(G,\psi_{U},\omega)$
with $W(1)=1$, following [Ba95], we consider

(3.1)
$W_{m}(g)=\frac{1}{{\mathrm{vol}}(U_{m})}\int_{U_{m}}W(gu)\psi_{m}^{-1}(u)du.$

For a fixed $W$, let $C$ be a positive integer such that $W$ is invariant
under the right translation by $K_{C}$, then a function $W_{m}$ with $m\geq C$
is called a Howe vector. It is known that $W_{m}(1)=1$. In particular,
$W_{m}\neq 0$. Moreover, for $m\geq C$, we have

(3.2) $W_{m}(ugh)=\psi_{U}(u)\psi_{m}(h)W_{m}(g),\forall u\in U,h\in
H_{m},g\in G.$

Because of (3.2), the functions $W_{m}$, $m\geq C$, are called partial Bessel
functions. For a proof of the statements in this subsection, see [Ba95].

###  3.2. Partial Bessel function and Bruhat order

Let $\pi$ be an irreducible generic supercuspidal representation of $G$ with
central character $\omega$. Let ${\mathcal{M}}(\pi)$ be the space of matrix
coefficients of $\pi$. Then we have ${\mathcal{M}}(\pi)\subset
C_{c}^{\infty}(G)$. For $f\in{\mathcal{M}}(\pi)$, we consider

$W^{f}(g)=\int_{U}f(ug)\psi^{-1}_{U}(u)du.$

Note that the above integral makes sense because $Ug$ is closed in $G$ and $f$
has compact support in $G$. We have $W^{f}\in C^{\infty}(G,\psi_{U},\omega)$.
Moreover, since $\pi$ is generic, there exists an $f\in{\mathcal{M}}(\pi)$
such that $W^{f}(1)=1$. For a positive integer $m$, we can consider the
function

${\mathcal{B}}_{m}(g,f):=(W^{f})_{m}(g),g\in G.$

For a Weyl element $w\in{\mathbf{W}}$ of $G$, we denote $C(w)=BwB$. Recall the
Bruhat order on ${\mathbf{W}}$ is defined as $w_{1}\geq w_{2}$ if and only if
$C(w)\subset\overline{C(w_{1})}$. For $w\in{\mathbf{W}}$, we consider the open
set

$\Omega_{w}=\cup_{w^{\prime}\geq w}C(w)$

of $G$. For $w\in{\mathbf{W}}$, we consider $A_{w}=\left\\{{t\in
T|\gamma(t)=1,\forall\gamma\in\Delta\textrm{ with }w\gamma>0}\right\\}.$ Note
that the Bruhat order is particularly simple for ${\mathrm{SO}}_{4}$. We
recall the following result of [CST], specializing to our case when
$G={\mathrm{SO}}_{4}$.

######  Lemma 3.1 ([CST]*Lemma 5.13).

Let $w\in{\mathbf{W}}$, $m>0$ and $f\in C_{c}^{\infty}(\Omega_{w},\omega)$.
Suppose that ${\mathcal{B}}_{m}(wa,f)=0$ for all $a\in A_{w}$. Then there
exists a function $f_{0}\in C_{c}^{\infty}(\Omega_{w}-C(w),\omega)$ such that
for sufficiently large $m$ depending only on $f$, we have
${\mathcal{B}}_{m}(g,f)={\mathcal{B}}_{m}(g,f_{0})$ for all $g\in G$.

###  3.3. Several preparation results

In this subsection, we collect several preparation results which will be used
in the proof of our local converse theorem.

Recall that $N_{2}=\\{{\bf{n}}(x)=\left(\begin{smallmatrix}1&x\\\
&1\end{smallmatrix}\right),x\in F\\}$ is the upper triangular unipotent
subgroup of ${\mathrm{SL}}_{2}(F)$. Let
$\overline{N}_{2}=\\{\bar{\bf{n}}(x)=\left(\begin{smallmatrix}1&\\\
x&1\end{smallmatrix}\right),x\in F\\}$ be the lower triangular unipotent
subgroup of ${\mathrm{SL}}_{2}(F)$. Let
$N_{2,m}=\begin{pmatrix}1&{\mathfrak{p}}^{-m}\\\ &1\end{pmatrix}$ and
$\overline{N}_{2,m}=\begin{pmatrix}1&\\\ {\mathfrak{p}}^{3m}&1\end{pmatrix}$.
Note that $\overline{N}_{2}$ and $N_{2}$ splits in
$\widetilde{\mathrm{SL}}_{2}(F)$. Moreover, for any $g_{1}\in N_{2}$ and
$g_{2}\in\overline{N}_{2}$, we have $c(g_{1},g_{2})=1$. In fact, if
$g_{1}={\bf{n}}(y)$ and $g_{2}=\bar{\bf{n}}(x)$ with $x\neq 0$, we have
${\bf{x}}(g_{1})=1$ and ${\bf{x}}(g_{2})=x$, and thus

$c(g_{1},g_{2})=(1,x)_{F}(-x,x)_{F}=1.$

For an integer $i$, we consider the following section $f_{s}^{i}\in
I(s,\eta,\psi^{-1})$ defined by

$f_{s}^{i}((g,\epsilon))=\left\\{\begin{array}[]{lll}\epsilon\gamma_{\psi^{-1}}(a)\eta_{s+1/2}(a),&\textrm{
if }g=\begin{pmatrix}a&b\\\ &a^{-1}\end{pmatrix}\bar{n}(x),\textrm{ with }a\in
F^{\times},b\in F,\epsilon\in\left\\{{\pm
1}\right\\},x\in{\mathfrak{p}}^{3i},\\\ 0,&\textrm{
otherwise.}\end{array}\right.$

Note that the support of $f_{s}^{i}$ is $\widetilde{B}_{2}\overline{N}_{2,i}$,
which is open in $\widetilde{\mathrm{SL}}_{2}(F)$. Thus $f_{s}^{i}$ is well-
defined.

######  Lemma 3.2 ([CZ]*Lemma 3.8).

  1. (1)

There exists an integer $i_{2}$ such that for all $i\geq i_{2}$, $f_{s}^{i}$
defines a section in $I(s,\eta,\psi^{-1})$.

  2. (2)

Let $X$ be an open compact subset of $N_{2}$, then there exists an integer
$I(X,\eta)$ such that for all $i\geq I(X,\eta)$, we have

$\tilde{f}_{s}^{i}(w_{2}x)={\mathrm{vol}}(\overline{N}_{2,i})=q^{-3i}$

for all $x\in X$, where $\tilde{f}_{s}^{i}=M_{s}(f_{s}^{i})$ and
$w_{2}=\begin{pmatrix}&1\\\ -1&\end{pmatrix}$.

Let $\phi^{m}$ be the characteristic function of $1+{\mathfrak{p}}^{m}$. We
have the following

######  Lemma 3.3 ([CZ]*Lemma 3.9).

We have

  1. (1)

$\omega_{\psi^{-1}}(n)\phi^{m}=\psi^{-1}(n)\phi^{m}$ for all $n\in N_{2,m}$.

  2. (2)

$\omega_{\psi^{-1}}(\bar{n})\phi^{m}=\phi^{m}$ for all
$\bar{n}\in\overline{N}_{2,m}.$

  3. (3)

$\omega_{\psi^{-1}}(w_{2})\phi^{m}(a)=\gamma(\psi^{-1})\psi^{-1}(2a)q^{-m}$.

###  3.4. A local converse theorem of Hazeltine-Liu and Haan-Kim-Kwon

In this subsection, we recall the local converse theorem of Hazeltine and Liu
for the split group ${\mathrm{SO}}_{2n}(F)$, which is realized by $J_{2n}$,
namely,
${\mathrm{SO}}_{2n}(F)=\left\\{{g\in{\mathrm{SL}}_{2n}(F):g^{t}J_{2n}g=J_{2n}}\right\\}$.
The outer automorphism $c$ of split ${\mathrm{SO}}_{2n}$ can be realized by
the matrix

$c=\begin{pmatrix}I_{n-1}&&\\\ &J_{2}&\\\ &&I_{n-1}\end{pmatrix}.$

Given an irreducible generic representation $\pi$ of ${\mathrm{SO}}_{2n}(F)$
and an irreducible generic representation $\tau$ of ${\mathrm{GL}}_{k}(F)$,
one can associate a local gamma factor $\gamma(s,\pi\times\tau,\psi)$; see
[Kaplan]. In [HL:p-adic] and [HKK], Hazeltine-Liu and Haan-Kim-Kwon proved the
following local converse theorem for ${\mathrm{SO}}_{2n}(F)$ independently.

######  Theorem 3.4 ([HL:p-adic, HKK]).

Let $\pi,\pi^{\prime}$ be two irreducible $\psi$-generic representations of
${\mathrm{SO}}_{2n}(F)$ with the same central character. If
$\gamma(s,\pi\times\tau,\psi)=\gamma(s,\pi^{\prime}\times\tau,\psi)$ for any
irreducible generic representation $\tau$ of ${\mathrm{GL}}_{k}(F)$ with
$1\leq k\leq n$, then either $\pi\cong\pi^{\prime}$ or $\pi\cong
c\cdot\pi^{\prime}$.

##  4\. On a refined local converse theorem for ${\mathrm{SO}}_{4}$

In this section, we prove Theorem 1.1.

###  4.1. A preparation step

Let $\pi$ and $\pi^{\prime}$ be two irreducible $\psi$-generic supercuspidal
representations of ${\mathrm{SO}}_{4}(F)$ with the same central character
$\omega$. Take $f\in{\mathcal{M}}(\pi)$ (resp.
$f^{\prime}\in{\mathcal{M}}(\pi^{\prime})$) such that
$W^{f}(1)=W^{f^{\prime}}(1)=1$. Note that there are many such choices of $f$
and $f^{\prime}$. We then can consider the Howe vectors $W^{f}_{m}$ (resp.
$W^{f^{\prime}}_{m}$) and partial Bessel functions ${\mathcal{B}}_{m}(g,f)$
(resp. ${\mathcal{B}}_{m}(g,f)$). We will show that
${\mathcal{B}}_{m}(g,f)={\mathcal{B}}_{m}(g,f^{\prime})$ under the assumption
of Theorem 1.1.

We start from the following lemma, which is a direct application of Lemma 3.1.

######  Lemma 4.1.

There exist functions $f_{\alpha}\in
C_{c}^{\infty}(\Omega_{s_{\alpha}},\omega)$ and $f_{\beta}\in
C_{c}^{\infty}(\Omega_{s_{\beta}},\omega)$ such that

${\mathcal{B}}_{m}(g,f)-{\mathcal{B}}_{m}(g,f^{\prime})={\mathcal{B}}_{m}(g,f_{\alpha})+{\mathcal{B}}_{m}(g,f_{\beta})$

for $m$ large enough, which only depend on $f,f^{\prime}$.

###### Proof.

Recall that for $w\in{\mathbf{W}}$, we have defined $A_{w}$ in Section 3.2.
Write the trivial element in ${\mathbf{W}}$ by 1. Then $A_{1}$ is the center
$Z$ of ${\mathrm{SO}}_{4}(F)$. By the assumption on the central character, we
have ${\mathcal{B}}_{m}(z,f-f^{\prime})=0$ for all $z\in Z$ and $m$ large.
Thus by Lemma 3.1, there is a function $f_{0}\in
C_{c}^{\infty}(\Omega_{1}-C(1),\omega)$ such that

${\mathcal{B}}_{m}(g,f-f^{\prime})={\mathcal{B}}_{m}(g,f_{0}).$

Note that $\Omega_{1}-C(1)=\Omega_{s_{\alpha}}\cup\Omega_{s_{\beta}}$. Thus a
partition of unity argument will give functions $f_{\alpha}\in
C_{c}^{\infty}(\Omega_{s_{\alpha}},\omega)$ and $f_{\beta}\in
C_{c}^{\infty}(\Omega_{s_{\beta}},\omega)$ such that

${\mathcal{B}}_{m}(g,f-f^{\prime})={\mathcal{B}}_{m}(g,f_{0})={\mathcal{B}}_{m}(g,f_{\alpha})+{\mathcal{B}}_{m}(g,f_{\beta}),$

for $m$ large. ∎

######  Lemma 4.2.

  1. (1)

For $m$ large and for any $t\in T$ and $r\in F$, we have

${\mathcal{B}}_{m}(ts_{\alpha}{\bf{x}}_{\alpha}(r),f_{\beta})=0.$

  2. (2)

For $m$ large and for any $t\in T$ and $r\in F$ with
${\bf{x}}_{\alpha}(r)\notin U_{m}$, we have

${\mathcal{B}}_{m}(ts_{\alpha}{\bf{x}}_{\alpha}(r),f_{\alpha})=0.$

###### Proof.

(1) Note that $ts_{\alpha}{\bf{x}}_{\alpha}(r)\in Bs_{\alpha}B$ while
$\Omega_{s_{\beta}}=Bs_{\beta}B\cup Bs_{\alpha}s_{\beta}B$. Thus
$ts_{\alpha}{\bf{x}}_{\alpha}(r)\notin\Omega_{s_{\beta}}$. The result follows.

(2) Note that $Bs_{\alpha}B$ is closed in
$\Omega_{s_{\alpha}}=Bs_{\alpha}B\cup Bs_{\alpha}s_{\beta}B$. Thus
$\mathrm{supp}(f_{\alpha})\cap Bs_{\alpha}B$ is compact. Note that the map

$B\times F\to Bs_{\alpha}B$ $(b,r)\to bs_{\alpha}{\bf{x}}_{\alpha}(r)$

is an isomorphism. Thus there exists an open compact subset $B_{c}\subset B$
and $F_{c}\subset F$ such that if
$f_{\alpha}(bs_{\alpha}{\bf{x}}_{\alpha}(r))\neq 0$, then $b\in B_{c}$ and
$r\in F_{c}$. We take $m$ large enough so that
${\bf{x}}_{\alpha}(F_{c})\subset U_{m}$. We remind the reader that $U_{m}$ was
defined in Section 3.1. Note that this choice of $m$ only depends on
$f_{\alpha}$. We then have

$\displaystyle{\mathcal{B}}_{m}(ts_{\alpha}{\bf{x}}_{\alpha}(r),f_{\alpha})$
$\displaystyle=\frac{1}{{\mathrm{vol}}(U_{m})}\int_{U\times
U_{m}}\psi_{U}^{-1}(uu^{\prime})f_{\alpha}(uts_{\alpha}{\bf{x}}_{\alpha}(r)u^{\prime})dudu^{\prime}.$

For $u^{\prime}\in U_{m}$, we can write
$u^{\prime}={\bf{x}}_{\alpha}(s_{1}){\bf{x}}_{\beta}(s_{2})$. Note that
$s_{\alpha}{\bf{x}}_{\beta}(s_{2})=u^{\prime\prime}s_{\alpha}$ for some
$u^{\prime\prime}\in U$. Thus

$f_{\alpha}(uts_{\alpha}{\bf{x}}_{\alpha}(r)u^{\prime})=f_{\alpha}(utu^{\prime\prime}s_{\alpha}{\bf{x}}_{\alpha}(r+s_{1})).$

Note that if ${\bf{x}}_{\alpha}(r)\notin U_{m}$, then
${\bf{x}}_{\alpha}(r+s_{1})\notin U_{m}$ for ${\bf{x}}_{\alpha}(s_{1})\in
U_{m}$. Thus $f_{\alpha}(uts_{\alpha}{\bf{x}}_{\alpha}(r)u^{\prime})=0$ for
any $u^{\prime}\in U_{m}$ under the assumption ${\bf{x}}_{\alpha}(r)\notin
U_{m}$. Thus
${\mathcal{B}}_{m}(ts_{\alpha}{\bf{x}}_{\alpha}(r),f_{\alpha})=0.$ ∎

######  Proposition 4.3.

Assume that
$\gamma(s,\pi,\wedge^{2}_{+}\times\eta,\psi)=\gamma(s,\pi^{\prime},\wedge^{2}_{+}\times\eta,\psi)$
for every quasi-character $\eta$ of $F^{\times}$, then we have

${\mathcal{B}}_{m}(g,f)={\mathcal{B}}_{m}(g,f^{\prime}),\forall g\in
Bs_{\alpha}B,$

for $m$ large depending on $f$ and $f^{\prime}$.

The proof given below is indeed similar to the proof of [CZ]*Theorem 3.10. For
completeness, we still give the full details.

###### Proof.

Write $W_{m}$ for the Whittaker function
$(W^{f})_{m}={\mathcal{B}}_{m}(\cdot,f)$ or
$(W^{f^{\prime}})_{m}={\mathcal{B}}_{m}(\cdot,f^{\prime})$. For a quasi-
character $\eta$ of $F^{\times}$, we consider $f_{s}^{i}\in
I(s,\eta,\psi^{-1})$ defined in Lemma 3.2 and $\phi^{m}\in{\mathcal{S}}(F)$,
the characteristic function of $1+{\mathfrak{p}}^{m}$ as in Lemma 3.3. Then we
consider the integral $\Psi(W_{m},\phi^{m},f_{s}^{i})$. This integral could be
computed on the dense subset $N_{2}\backslash
N_{2}T_{2}\overline{N}_{2}\subset N_{2}\backslash{\mathrm{SL}}_{2}$. Notice
that ${\bf{m}}(\bar{\bf{n}}_{2}(x))={\bf{x}}_{-\alpha}(x)$ and
${\bf{m}}(t_{{\mathrm{SL}}_{2}}(a))=t(a,a^{-1})$ with
$t_{{\mathrm{SL}}_{2}}(a)={\mathrm{diag}}(a,a^{-1})\in{\mathrm{SL}}_{2}(F)$.
We have

$\displaystyle\Psi(W_{m},\phi^{m},f_{s}^{i})$
$\displaystyle=\int_{F^{\times}\times
F}W_{m}(t(a,a^{-1}){\bf{x}}_{-\alpha}(x))(\omega_{\psi^{-1}}(t_{{\mathrm{SL}}_{2}}(a){\bf{n}}_{2}(x)))\phi^{m}(1)f_{s}^{i}(t_{{\mathrm{SL}}_{2}}(a)\bar{\bf{n}}_{2}(x))|a|^{-2}dxda$
$\displaystyle=\int_{F^{\times}{\mathfrak{p}}^{3i}}W_{m}(t(a,a^{-1}){\bf{x}}_{-\alpha}(x))(\omega_{\psi^{-1}}(t_{{\mathrm{SL}}_{2}}(a)\bar{\bf{n}}_{2}(x)))\phi^{m}(1)\gamma_{\psi^{-1}}(a)\eta_{s+1/2}(a)|a|^{-2}dxda.$

For $i\geq m$, we have ${\bf{x}}_{-\alpha}(x)\in H_{m}$. Thus by (3.2) and
Lemma 3.3, we have

$W_{m}(t_{{\mathrm{SL}}_{2}}(a)\bar{\bf{n}}_{2}(x))=W_{m}(t_{{\mathrm{SL}}_{2}}(a)),(\omega_{\psi^{-1}}(t_{{\mathrm{SL}}_{2}}(a)\bar{\bf{n}}_{2}(x)))\phi^{m}(1)=(\omega_{\psi^{-1}}(t_{{\mathrm{SL}}_{2}}(a)))\phi^{m}(1).$

Since
$(\omega_{\psi^{-1}}(t_{{\mathrm{SL}}_{2}}(a)))\phi^{m}(1)=\gamma_{\psi^{-1}}(a)|a|^{1/2}\phi^{m}(a)$
and $\gamma_{\psi^{-1}}(a)\gamma_{\psi^{-1}}(a)=(a,a)=(a,-1)=\chi(a)$, we get

(4.1) $\displaystyle\Psi(W_{m},\phi^{m},f_{s}^{i})$
$\displaystyle=q^{-3i}\int_{F^{\times}}W_{m}(t(a,a^{-1}))(\omega_{\psi^{-1}}(t_{{\mathrm{SL}}_{2}}(a)))\phi^{m}(1)\gamma_{\psi^{-1}}(a)\eta_{s+1/2}(a)|a|^{-2}da$
$\displaystyle=q^{-3i}\int_{1+{\mathfrak{p}}^{m}}W_{m}(t(a,a^{-1}))\chi(a)\eta_{s-1}(a)da$
$\displaystyle=q^{-3i-m},$

if $m\geq\mathrm{cond}(\chi\eta)$. In particular, we have

(4.2)
$\Psi((W^{f})_{m},\phi^{m},f_{s}^{i})=\Psi((W^{f^{\prime}})_{m},\phi^{m},f_{s}^{i}).$

By the assumption on the local gamma factors and the local functional
equation, (4.2) implies that

(4.3)
$\Psi((W^{f})_{m},\phi_{m},\widetilde{f}_{s}^{i})=\Psi((W^{f^{\prime}})_{m},\phi_{m},\widetilde{f}_{s}^{i}).$

The integral $\Psi(W_{m},\phi^{m},\widetilde{f}_{s}^{i})$ can be taken on the
open dense subset $N_{2}\backslash N_{2}T_{2}w_{2}N_{2}$ of
$N_{2}\backslash{\mathrm{SL}}_{2}$. Notice that ${\bf{m}}(w_{2})=s_{\alpha}$
and ${\bf{m}}({\bf{n}}(x))={\bf{x}}_{\alpha}(x)$. We have

$\displaystyle\Psi(W_{m},\phi^{m},\tilde{f}_{s}^{i})$
$\displaystyle=\int_{F^{\times}\times
F}W_{m}(t(a,a^{-1})s_{\alpha}{\bf{x}}_{\alpha}(x))(\omega(t_{{\mathrm{SL}}_{2}}(a)w_{2}{\bf{n}}_{2}(x)))\phi^{m}(1)\tilde{f}_{s}^{i}(t_{{\mathrm{SL}}_{2}}(a)w_{2}{\bf{n}}_{2}(x))|a|^{-2}dxda$
$\displaystyle=\int_{F^{\times}\times{\mathfrak{p}}^{-m}}W_{m}(t(a,a^{-1})w_{2}{\bf{x}}_{\alpha}(x))(\omega_{\psi^{-1}}(t_{{\mathrm{SL}}_{2}}(a)w_{2}{\bf{n}}_{2}(x)))\phi^{m}(1)\tilde{f}_{s}^{i}(t_{{\mathrm{SL}}_{2}}(a)w_{2}{\bf{n}}_{2}(x))|a|^{-2}dxda$
$\displaystyle+\int_{F^{\times}(F-{\mathfrak{p}}^{-m})}W_{m}(t(a,a^{-1})s_{\alpha}{\bf{x}}_{\alpha}(x))(\omega_{\psi^{-1}}(t_{{\mathrm{SL}}_{2}}(a)w_{2}{\bf{n}}_{2}(x)))\phi^{m}(1)\tilde{f}_{s}^{i}(t_{{\mathrm{SL}}_{2}}(a)w_{2}{\bf{n}}_{2}(x))|a|^{-2}dxda.$

By Lemma 4.1 and Lemma 4.2, for $m$ large, we have

$(W^{f})_{m}(t(a,a^{-1})s_{\alpha}{\bf{x}}_{\alpha}(x))=(W^{f^{\prime}})_{m}(t(a,a^{-1})s_{\alpha}{\bf{x}}_{\alpha}(x)),\forall
x\in F-{\mathfrak{p}}^{m}.$

Thus for $i$ large, by Lemma 3.2 and Lemma 3.3, we get

$\displaystyle\Psi((W^{f})_{m},\phi^{m},f_{s}^{i})-\Psi((W^{f^{\prime}})_{m},\phi^{m},f_{s}^{i})$
$\displaystyle=$
$\displaystyle\int_{F^{\times}\times{\mathfrak{p}}^{-m}}{\mathcal{B}}_{m}(t(a,a^{-1})s_{\alpha}{\bf{n}}_{2}(x),f-f^{\prime})(\omega_{\psi^{-1}}(t_{{\mathrm{SL}}_{2}}(a)w_{2}{\bf{n}}_{2}(x)))\phi^{m}(1)\tilde{f}_{s}^{i}(t_{{\mathrm{SL}}_{2}}(a)w_{2}{\bf{n}}_{2}(x))dxda$
$\displaystyle=$ $\displaystyle
q^{-3i+m}\int_{F^{\times}}{\mathcal{B}}_{m}(t(a,a^{-1})s_{\alpha}),f-f^{\prime})\omega_{\psi^{-1}}(w_{2})\phi^{m}(a)\chi(a)\eta_{-s-1}(a)da$
$\displaystyle=$ $\displaystyle
q^{-3i}\gamma(\psi^{-1})\int_{F^{\times}}{\mathcal{B}}_{m}(t(a,a^{-1})s_{\alpha}),f-f^{\prime})\psi^{-1}(2a)\chi(a)\eta_{-s-1}(a)da.$

Thus (4.3) implies that

$\int_{F^{\times}}{\mathcal{B}}_{m}(t(a,a^{-1})s_{\alpha}),f-f^{\prime})\psi^{-1}(2a)\chi(a)\eta_{-s-1}(a)da=0.$

Note that this equation is true for any quasi-character $\eta$ of $F^{\times}$
by assumption. Thus by the inverse Mellin transform, we get that

${\mathcal{B}}_{m}(t(a,a^{-1})s_{\alpha}),f-f^{\prime})\psi^{-1}(2a)=0,\forall
a\in F^{\times}.$

Since $\psi^{-1}(2a)\neq 0$, we then get

(4.4) ${\mathcal{B}}_{m}(t(a,a^{-1})s_{\alpha}),f-f^{\prime})=0,\forall a\in
F^{\times}.$

Note that $A_{s_{\alpha}}=\left\\{{t(a,a^{-1}),a\in F^{\times}}\right\\}$.
Lemma 3.1 implies that
${\mathcal{B}}_{m}(ts_{\alpha},f)={\mathcal{B}}_{m}(ts_{\alpha},f^{\prime})$
for all $t\in T$ and for possibly larger $m$. Finally, notice that any element
in $Bs_{\alpha}B$ can be written as the form $ts_{\alpha}{\bf{x}}_{\alpha}(r)$
for some $r\in F$. This finishes the proof. ∎

###  4.2. The group ${\mathrm{GSO}}_{4}$

First, we introduce a closely related group, the orthogonal similitude group
$\mathrm{GO}_{4}(F)$, which is realized as
$\mathrm{GO}_{4}(F)=\\{g\in{\mathrm{GL}}_{4}(F):g^{t}J_{4}g=\lambda(g)J_{4},\,\lambda(g)\in
F^{\times}\\}$. Since $\det(g)^{2}=\lambda(g)^{4}$, it has two connected
components depending on whether $\det(g)/\lambda(g)^{2}$ is $1$ or $-1$. We
denote the identity component by ${\mathrm{GSO}}_{4}(F)$, i.e.,
${\mathrm{GSO}}_{4}(F)=\\{g\in{\mathrm{GL}}_{4}(F):g^{t}J_{4}g=\lambda(g)J_{4},\,\lambda(g)\in
F^{\times},\det(g)=\lambda(g)^{2}\\}$. Note that
${\mathrm{SO}}_{4}(F)=\\{g\in{\mathrm{GSO}}_{4}(F):\lambda(g)=1\\}$. Note that
$c\in{\mathrm{G}}{\mathrm{O}}_{4}(F)-{\mathrm{GSO}}_{4}(F)$ and thus the
conjugation by $c$ is an outer automorphism on ${\mathrm{GSO}}_{4}(F)$.

The two simple roots also define embeddings
$\iota_{\alpha},\iota_{\beta}:{\mathrm{GL}}_{2}(F)\to{\mathrm{GSO}}_{4}(F)$
which can be described explicitly as

$\iota_{\alpha}\left(\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\right)=\begin{pmatrix}a&b&0&0\\\ c&d&0&0\\\ 0&0&a&-b\\\
0&0&-c&d\end{pmatrix},\,\,\iota_{\beta}\left(\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\right)=\begin{pmatrix}a&0&b&0\\\ 0&a&0&-b\\\ c&0&d&0\\\
0&-c&0&d\end{pmatrix}.$

Notice that $\lambda(\iota_{\alpha}(g))=\det(g)$ and
$\lambda(\iota_{\beta}(g))=\det(g)$ for $g\in{\mathrm{GL}}_{2}(F)$. The
embeddings $\iota_{\alpha}$ and $\iota_{\beta}$ define a homomorphism

${\mathrm{GL}}_{2}(F)\times{\mathrm{GL}}_{2}(F)\to{\mathrm{GSO}}_{4}(F)$
$(g,h)\mapsto\iota_{\alpha}(g)\iota_{\beta}(h),$

which gives an isomorphism

${\mathrm{GSO}}_{4}(F)\cong({\mathrm{GL}}_{2}(F)\times{\mathrm{GL}}_{2}(F))/\Delta(F^{\times}),$

where $\Delta(F^{\times})=\\{(aI_{2},a^{-1}I_{2}):a\in
F^{\times}\\}\subset{\mathrm{GL}}_{2}(F)\times{\mathrm{GL}}_{2}(F)$. Thus an
irreducible representation of ${\mathrm{GSO}}_{4}(F)$ is of the form
$\pi_{1}\boxtimes\pi_{2}$ with $\omega_{\pi_{1}}=\omega_{\pi_{2}}$, where
$\pi_{i}$ is an irreducible representation of ${\mathrm{GL}}_{2}(F)$ and
$\omega_{\pi_{i}}$ is the central character of $\pi_{i}$.

Note that the outer automorphism $c$ on ${\mathrm{GSO}}_{4}(F)$ satisfies

$c\iota_{\alpha}(g)c=\iota_{\beta}(g),\forall g\in{\mathrm{GL}}_{2}(F).$

Thus for an irreducible representation $\Pi=\pi_{1}\boxtimes\pi_{2}$ of
${\mathrm{GSO}}_{4}(F)$ with irreducible representations $\pi_{1},\pi_{2}$ of
${\mathrm{GL}}_{2}(F)$ with $\omega_{\pi_{1}}=\omega_{\pi_{2}}$, we have

$c\cdot\Pi=\pi_{2}\boxtimes\pi_{1}.$

###  4.3. Proof of Theorem 1.1

###### Proof of Theorem 1.1.

By assumption and Theorem 3.4, either $\pi\cong\pi^{\prime}$ or $\pi\cong
c\cdot\pi^{\prime}$. If $\pi\cong\pi^{\prime}$, we are done. From now on, we
assume that $\pi\cong c\cdot\pi^{\prime}$. From this condition, we get that
$c\cdot\pi\cong\pi^{\prime}$. We will show that $\pi\cong
c\cdot\pi\cong\pi^{\prime}$ under the assumption of Theorem 1.1.

We fix the notations as in $\S$4.1. By the assumption
$\gamma(s,\pi,\wedge^{2}_{+}\times\eta)=\gamma(s,\pi^{\prime},\wedge^{2}_{+}\times\eta)$
for all quasi-character $\eta$ of $F^{\times}$ and Proposition 4.3, we have

(4.5) $(W^{f})_{m}(ts_{\alpha})=(W^{f^{\prime}})_{m}(ts_{\alpha}),\forall t\in
T.$

By [GK]*§2, there exists an irreducible representation $\Pi$ of
${\mathrm{GSO}}_{4}(F)$ such that $\Pi|_{{\mathrm{SO}}_{4}(F)}$ contains $\pi$
as a direct summand. By the uniqueness of Whittaker functional of $\Pi$, we
know that $\pi$ is the unique $\psi_{U}$-generic irreducible direct summand of
$\Pi|_{{\mathrm{SO}}_{4}(F)}$.

We can write $\Pi=\pi_{1}\boxtimes\pi_{2}$ for irreducible representations
$\pi_{1},\pi_{2}$ of ${\mathrm{GL}}_{2}(F)$ with
$\omega_{\pi_{1}}=\omega_{\pi_{2}}$. Note that ${\mathrm{GSO}}_{4}(F)$ and
${\mathrm{SO}}_{4}(F)$ share the same maximal unipotent subgroup. Assume that
$\Lambda\in{\mathrm{Hom}}_{U}(\pi,\psi_{U})$ is the unique $\psi_{U}$
Whittaker functional (up to scaler) of $\pi$. Then the map
$\Pi\to\pi\to\psi_{U}$ gives a nonzero Whittaker functional of $\Pi$. By abuse
of notations, we also denote this Whittaker functional by $\Lambda$. For
$v\in\pi\subset\Pi$, we can consider $\widetilde{W}_{v}(g)=\Lambda(\Pi(g)v)$.
Note that $\widetilde{W}_{v}|_{\pi}$ gives a Whittaker function of $\pi$ and
any Whittaker function of $\pi$ is of this form. We write $\widetilde{W}^{f}$
the corresponding Whittaker function on ${\mathrm{GSO}}_{4}(F)$ such that
$\widetilde{W}^{f}|_{{\mathrm{SO}}_{4}(F)}=W^{f}$. For a positive integer $m$,
if we define

(4.6)
$\widetilde{W}^{f}_{m}(g)=\frac{1}{{\mathrm{vol}}(U_{m})}\int_{U_{m}}\widetilde{W}^{f}(gu)\psi_{U}^{-1}(u)du,g\in{\mathrm{GSO}}_{4}(F),$

then $\widetilde{W}^{f}_{m}|_{{\mathrm{SO}}_{4}(F)}=W^{f}_{m}.$

On the other hand, by the decomposition $\Pi=\pi_{1}\boxtimes\pi_{2}$ and
uniqueness of Whittaker functionals, the Whittaker functional $\Lambda$ on
$\Pi$ has the form $\Lambda=\lambda_{1}\boxtimes\lambda_{2}$ for
$\lambda_{i}\in{\mathrm{Hom}}_{U_{i}}(\pi_{i},\psi_{U}|_{U_{i}})$, where
$U_{1}=\left\\{{{\bf{x}}_{\alpha}(r):r\in F}\right\\}$ and
$U_{2}=\left\\{{{\bf{x}}_{\beta}(r):r\in F}\right\\}$ are the corresponding
maximal unipotent subgroups in $\iota_{\alpha}({\mathrm{GL}}_{2}(F))$ and
$\iota_{\beta}({\mathrm{GL}}_{2}(F))$. Then ${\mathcal{W}}(\Pi,\psi_{U})$ is
spanned by pure tensors of the form $W^{1}\boxtimes W^{2}$ with
$W^{i}\in{\mathcal{W}}(\pi_{i},\psi_{U}|_{U_{i}}),$ where $(W^{1}\boxtimes
W^{2})(\iota_{\alpha}(g)\iota_{\beta}(h))=W^{1}(g)W^{2}(h)$.

We now assume that $f\in{\mathcal{M}}(\pi)$ is chosen to be a pure tensor
$f=f_{1}\boxtimes f_{2}$ for some $f_{i}\in{\mathcal{M}}(\pi_{i})$, namely,

$f(\iota_{\alpha}(g)\iota_{\beta}(h))=f_{1}(g)f_{2}(h).$

Of course, we also assume that $W^{f}(1)=1$. Then we have
$\widetilde{W}^{f}=W^{1}\boxtimes W^{2}$ for
$W^{i}(g)=\int_{U_{i}}f_{i}(ug)\psi_{U}^{-1}(u)du$, which is a Whittaker
function of $\pi_{i}$. Using (4.6), we see that
$\widetilde{W}^{f}_{m}=W^{1}_{m}\boxtimes W^{2}_{m}$, where
$W^{i}_{m}(g):=\frac{1}{{\mathrm{vol}}(U_{i}\cap H_{m})}\int_{U_{i}\cap
H_{m}}W^{i}_{m}(gu)\psi_{U}^{-1}(u)du$, is a Howe vector for $\pi_{i}$.

By assumption, we have $\pi^{\prime}\cong c\cdot\pi$ and thus
$\Pi^{\prime}:=c\cdot\Pi=\pi_{2}\boxtimes\pi_{1}$ is an irreducible
representation of ${\mathrm{GSO}}_{4}(F)$ such that
$\Pi^{\prime}_{{\mathrm{SO}}_{4}(F)}$ contains $\pi^{\prime}$ as the unique
$\psi_{U}$-generic direct summand. Note that $f^{\prime}:=f_{2}\boxtimes
f_{1}\in{\mathcal{M}}(\Pi^{\prime})$ satisfies $W^{f^{\prime}}(1)=1$.
Moreover, by (3.1) and (4.6), we have
$\widetilde{W}^{f^{\prime}}_{m}|_{{\mathrm{SO}}_{4}(F)}=(W^{f^{\prime}})_{m}$
and $\widetilde{W}^{f^{\prime}}_{m}=W^{2}_{m}\boxtimes W^{1}_{m}$.

By (4.5), we have
$\widetilde{W}^{f}_{m}(g)=\widetilde{W}^{f^{\prime}}_{m}(g)$, for $g\in
Bs_{\alpha}B$ and $m$ large enough. Notice that
$Bs_{\alpha}B\subset\iota_{\alpha}({\mathrm{GL}}_{2})$. The above discussion
shows that

(4.7) $W_{m}^{1}(g)=W_{m}^{2}(g),\forall g\in
B_{{\mathrm{GL}}_{2}}s_{\alpha}B_{{\mathrm{GL}}_{2}},$

where $B_{{\mathrm{GL}}_{2}}$ is the upper triangular subgroup of
${\mathrm{GL}}_{2}(F)$. Since $\omega_{\pi_{1}}=\omega_{\pi_{2}}$, we have
$W_{m}^{1}(z)=W_{m}^{2}(z)$ for all $z\in Z_{{\mathrm{GL}}_{2}}$. A simple
application of an analogue of Lemma 3.1 in the ${\mathrm{GL}}_{2}$-case (see
[CST]*Lemma 5.13) shows that $W_{m}^{1}(g)=W_{m}^{2}(g)$ for all $g\in
B_{{\mathrm{GL}}_{2}}$. By Bruhat decomposition and (4.7), we get

$W_{m}^{1}(g)=W_{m}^{2}(g),\forall g\in{\mathrm{GL}}_{2}(F),$

when $m$ is large enough. By uniqueness of Whittaker functional, we get that
$\pi_{1}\cong\pi_{2}$. This implies that $\Pi^{\prime}\cong
c\cdot\Pi\cong\Pi$. Since $\pi$ (resp. $\pi^{\prime}$) is the unique
$\psi_{U}$-generic direct summand of $\Pi$ (resp. $\Pi^{\prime}$), we get that
$\pi\cong\pi^{\prime}$. This concludes the proof. ∎

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