# Shalika models on ${{\rm GU}}_{2,2}$ and the degree 5 $L$-function of ${{\rm
GSp}}_{4}$

Antonio Cauchi and Armando Gutierrez Terradillos

###### Abstract.

We consider a Rankin–Selberg integral which unfolds to a Shalika period for
the unitary similitude group ${{\rm GU}}_{2,2}$ attached to a quadratic
extension $E/{\bf Q}$. We show that it represents the degree 5 $L$-function of
${{\rm GSp}}_{4}$ twisted by the quadratic character of $E/{\bf Q}$ of
cuspidal automorphic representations which contribute to the theta
correspondence for the pair $(\mathrm{P}{{\rm GSp}}_{4},\mathrm{P}{{\rm
GU}}_{2,2})$. Along the way, we obtain a local criterion to determine the
multiplicity of local Shalika models for unramified representations of ${{\rm
GU}}_{2,2}$ at unramified inert primes and prove a Casselman-Shalika formula
for them.

A.C. was financially supported by the NSERC grant RGPIN-2018-04392 and
Concordia Horizon postdoc fellowship n.8009. Both authors were also supported
by the European Research Council (ERC) under the European Union’s Horizon 2020
research and innovation programme (grant agreement No. 682152).

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
  3. 3 Uniqueness of local Shalika functionals at inert primes
  4. 4 A Casselman-Shalika formula for the Shalika model of ${\rm GU}_{2,2}$
  5. 5 Proof of Theorem 4.5
  6. 6 The zeta integral

##  1\. Introduction

Automorphic $L$-functions are one of the cornerstones of the Langlands
program. They provide a bridge between number theory and representation
theory, dispensing a profound link between the arithmetic of algebraic objects
such as elliptic curves and harmonic analysis on reductive groups. Automorphic
$L$-functions are defined by the data of a cuspidal automorphic representation
$\pi$ on a reductive group $G$ and of a finite dimensional representation $r$
of the Langlands group ${}^{L}G$. Their zeros or poles are often linked to the
non-vanishing of certain periods of $\pi$ and to the participation of $\pi$ as
a Langlands functorial transfer or theta lift of an automorphic representation
of another reductive group. There are mainly two approaches to study the
analytic properties of automorphic $L$-functions: via Rankin–Selberg integrals
or by the Langlands–Shahidi method. In this article, we present an integral
representation of a twist of the degree $5$ $L$-function of ${{\rm GSp}}_{4}$,
which gives a new instance of the curious phenomenon where a Rankin–Selberg
integral of a cusp form on one group is used to represent an $L$-function of a
different cusp form on another group.

Precisely, we consider the integral

$I(\varphi,s):=\int_{{\rm GSp}_{4}({\bf Q})Z_{{\rm GSp}_{4}}({\bf
A})\backslash{\rm GSp}_{4}({\bf A})}E^{*}(s,g)\varphi(g)dg,$

where $E^{*}(s,g)$ is the normalised degenerate Siegel Eisenstein series for
${\rm GSp}_{4}$ and $\varphi$ is a cusp form in the space of a cuspidal
representation $\Pi$ of ${\rm GU}_{2,2}({\bf A})$ with trivial central
character. This integral is a “quasi-split analogous” to the integral for the
group ${\rm SO}_{3,3}$ studied in [4] and can be regarded as the degenerate
companion of the integral for ${\rm SO}_{4,2}$ studied in [35] and [30]. The
integral $I(\varphi,s)$ unfolds to the unitary Shalika period of $\varphi$;
this fact led us to perform a detailed analysis of local Shalika models for
unramified representations of ${{\rm GU}}_{2,2}$.

Shalika periods first appeared in the study of the analytic properties of the
partial exterior square $L$-function on ${\rm GL}_{2n}$. More precisely, a
celebrated theorem of Jacquet and Shalika (cf. [24]) asserts that the partial
exterior square $L$-function of a cuspidal automorphic representation $\pi$ of
${\rm GL}_{2n}$ has a simple pole at $s=1$ if and only if the Shalika period
does not vanish on the space of $\pi$. A similar statement for the unitary
group ${{\rm GU}}_{2,2}$, which is closely connected to ${\rm GL}_{4}$, was
studied by Morimoto and Furusawa in [13]. In the subsequent work [29], using
the exceptional isomorphism $\mathrm{PGU}_{2,2}\simeq\mathrm{PGSO}_{4,2}$,
Morimoto characterized the existence of unitary Shalika periods in terms of
the theta correspondence for the pair $({{\rm GSp}}_{4},\mathrm{GSO}_{4,2})$.
The results of [13] and [29] can be summarized as follows. Let $\Pi$ be a
cuspidal automorphic representation of ${\rm GU}_{2,2}({\bf A})$ with trivial
central character, which, using the aforementioned exceptional isomorphism,
can be considered as an automorphic representation of ${\rm PGSO}_{4,2}$. The
following are equivalent:

  1. (1)

$\Pi$ is globally generic and the partial twisted exterior square $L$-function
$L^{\Sigma}(s,\Pi,\wedge_{t}^{2})$ has a simple pole at $s=1$.

  2. (2)

A certain unitary Shalika period does not vanish on $\Pi$.

  3. (3)

$\Pi$ comes from a globally generic cuspidal automorphic representation
$\sigma$ of ${\rm GSp}_{4}$ with trivial central character, by means of a
theta correspondence for $({\rm GSp}_{4},{\rm GSO}_{4,2})$.

If any of these holds, one has that

$L^{\Sigma}(s,\Pi,\wedge^{2}_{t})=L^{\Sigma}(s,\sigma,{\rm
std}\otimes\chi_{E/{\bf Q}})\zeta^{\Sigma}(s),$

where $L^{\Sigma}(s,\sigma,{\rm std}\otimes\chi_{E/{\bf Q}})$ denotes the
partial degree 5 $L$-function for ${\rm GSp}_{4}$ twisted by the quadratic
Hecke character associated to the imaginary quadratic extension $E/{\bf Q}$
which defines ${\rm GU}_{2,2}$. By the above characterization, the integral
$I(\varphi,s)$ is identically zero if $\Pi$ does not come from ${\rm
GSp}_{4}$.

The purpose of this article is two-fold. On the one hand, we obtain a
criterion to determine the multiplicity of local Shalika models for
irreducible generic unramified representations of ${\rm GU}_{2,2}({\bf
Q}_{p})$ in the image of the local theta correspondence. When these models are
unique, we prove an explicit Casselman-Shalika formula for them. On the other
hand, we use this formula to relate $I(\varphi,s)$ to the partial $L$-function
$L^{\Sigma}(s,\sigma,{\rm std}\otimes\chi_{E/{\bf Q}})$.

We start by describing our local contributions, which could be of independent
interest. Let $p$ be inert in $E/{\bf Q}$ and let $\sigma_{p}$ be an
irreducible generic unramified representation of ${\rm GSp}_{4}({\bf Q}_{p})$
with trivial central character. Note that $\sigma_{p}$ is is necessarily a
principal series ${\rm Ind}_{B_{{\rm GSp}_{4}}({\bf Q}_{p})}^{{\rm
GSp}_{4}({\bf Q}_{p})}\chi$. We denote by $g_{\chi}\in{\rm Sp}_{4}({\bf C})$
its Frobenius conjugacy class. We let $\Pi_{p}$ be the small theta lift of the
generic irreducible factor of ${\sigma_{p}}_{|{{\rm GSp}_{4}^{+}({\bf
Q}_{p})}}$ by means of the theta correspondence for the dual pair $({\rm
GSp}_{4}^{+}({\bf Q}_{p}),{\rm GSO}_{4,2}({\bf Q}_{p}))$ studied by [29].
Using the exceptional isomorphism, $\Pi_{p}$ can be seen as an irreducible
unramified representation of ${\rm GU}_{2,2}({\bf Q}_{p})$ with trivial
central character. If we let $S$ be the Shalika subgroup defined by (4) and
$\chi_{\eta,p}$ be the $p$-component of the character $\chi_{\eta}$ on $S$
defined in §2.4, a unitary Shalika functional for $\Pi_{p}$ is a linear
functional $\Lambda_{p,\mathcal{S}}:\Pi_{p}\to{\bf C}$, such that for $s\in
S({\bf Q}_{p})$ and $v$ in the space of $\Pi_{p}$ we have

$\Lambda_{p,\mathcal{S}}(\Pi_{p}(s)v)=\chi_{\eta,p}(s)\Lambda_{p,\mathcal{S}}(v).$

We fix a spherical vector $\phi_{0}$ of $\Pi_{p}$, which we normalise as in
§4.2. Then, in view of Theorem 1.2, we are interested in a formula for the
values $\mathcal{S}_{p}(g):=\Lambda_{p,\mathcal{S}}(\Pi_{p}(g)\phi_{0})$ as
$g$ varies. By using properties of the local theta correspondence and adapting
the methods of [33], we prove the following.

######  Theorem 1.1 (Theorems 3.8 & 4.5).

Let $\sigma_{p}$ and $\Pi_{p}$ be as above. Suppose that $\Pi_{p}$ is generic.

  1. (1)

There exists a unique unitary Shalika functional for $\Pi_{p}$ up to constant;

  2. (2)

For all $n\geq 0$, a suitably normalised spherical Shalika functional
satisfies

$\mathcal{S}_{p}\left(\left(\begin{smallmatrix}p^{n}I_{2}&\\\
&I_{2}\end{smallmatrix}\right)\right)=\frac{p^{-2n}}{1+p^{-1}}\mathcal{A}\left((-1)^{n}e^{\check{\rho}+n(\check{\alpha}_{1}+\check{\alpha}_{2})}\prod_{\alpha\in\Phi^{+,s}_{{\rm
Sp}_{4}}}(1+p^{-1}e^{-\check{\alpha}})\right)(g_{\chi})(\mathcal{A}(e^{\check{\rho}})(g_{\chi}))^{-1}.$

We refer to §4 for further explanation on the notation used in the formula.
When $p$ is split in $E$, the analogous result follows from the work of [33]
and the fact that $\mathrm{PGU}_{2,2}({\bf Q}_{p})\simeq\mathrm{PGL}_{4}({\bf
Q}_{p})$. We would like to remark that, despite the fact that the theta
correspondence between $({\rm GSp}_{4}^{+}({\bf Q}_{p}),{\rm GSO}_{4,2}({\bf
Q}_{p}))$ is not functorial at inert primes, our formula still expresses the
Shalika functional of the small theta lift of $\sigma_{p}$ in terms of its
Frobenius conjugacy class. This can serve as a _local motivation_ for Theorem
1.2 below.

Let us now consider the Rankin-Selberg integral $I(\varphi,s)$, with $\varphi$
a cusp form in a globally generic representation $\Pi$ which is in the image
of the global theta correspondence studied by [29]. Thanks to the unfolding
and the uniqueness of the Shalika model at the unramified places of $\Pi$, we
reduce the calculation of $I(\varphi,s)$ to the ones of the corresponding
local zeta integrals. At unramified places these are calculated by using the
Casselman-Shalika formulas of [33] and of Theorem 1.1.

######  Theorem 1.2 (Theorem 6.3).

Let $\Pi$ be an irreducible cuspidal automorphic representation of ${\rm
GU}_{2,2}({\bf A})$ with trivial central character. Suppose that $\Pi$ appears
in the theta lift of $\sigma$ in the sense of [29, Theorem B], where $\sigma$
is a globally generic irreducible cuspidal automorphic representation of ${\rm
GSp}_{4}({\bf A})$ with trivial central character. We have

$I(\varphi,s)=I_{\Sigma}(s)L^{\Sigma}(s,\sigma,{\rm std}\otimes\chi_{E/{\bf
Q}}),$

where $\Sigma$ is a finite set of primes containing the ramified primes for
$\Pi$, $\sigma$, and $E/{\bf Q}$, and $I_{\Sigma}(s)$ is the integral over the
places in $\Sigma$ and infinity. Furthermore, the partial $L$-function
$L^{\Sigma}(s,\sigma,{\rm std}\otimes\chi_{E/{\bf Q}})$ extends to a
holomorphic function for all $s$.

A few remarks on Theorem 1.2 are in order. One novelty of our integral
representation of the degree 5 $L$-function of $\mathrm{GSp}_{4}$ is that it
might be used to address motivic questions. Indeed, since $\mathrm{GSp}_{4}$
and $\mathrm{GU}_{2,2}$ have Shimura varieties, our integral seems to have
applications to the study of the conjectural “standard” degree 5 motive
associated to cohomological cuspidal automorphic representations of
$\mathrm{GSp}_{4}$, such as the construction of an Euler system for the
corresponding Galois representation, similar in spirit to the one given in
work in progress by Skinner and Sangiovanni Vincentelli for the symmetric
square of a modular form. Interestingly, our result adapts to other settings,
such as for the standard $L$-function of cusp forms of the split exceptional
group $G_{2}$, which will be explored in forthcoming work.

Since the theta lift of $\sigma$ to ${\rm GSO}_{4,2}$ is non-zero and cuspidal
by hypothesis, Rallis tower property implies that the theta lift of $\sigma$
to ${\rm GSO}_{3,1}$ and ${\rm GSO}_{2,0}$ is zero. Thus, by using [26,
Theorem 7.2.5], one could deduce that the partial $L$-function
$L^{\Sigma}(s,\sigma,{\rm std}\otimes\chi_{E/{\bf Q}})$ is entire. By
analysing $I(\varphi,s)$ at the points where $E^{*}(s,g)$ might obtain a pole,
we deduce that $I(\varphi,s)$ is indeed holomorphic at any $s$. Here we use
the fact that unramified generic representations on ${\rm GL}_{4}({\bf
Q}_{p})$ are not ${\rm Sp}_{4}({\bf Q}_{p})$-distinguished and the vanishing
of an integral which involves the degenerate Klingen Eisenstein series on
${\rm GSp}_{4}$. The latter integral is intimately related to the construction
of [9].

Acknowledgements. The project started when the two authors were both working
at the Universitat Politecnica de Catalunya; we thus kindly thank Victor
Rotger for his support. We are indebted to Wee Teck Gan, who suggested to
prove uniqueness of Shalika models by means of the local theta correspondence.
The first named author would also like to thank Patrick Allen, Mathilde
Gerbelli-Gauthier, Aaron Pollack, Martí Roset Julià, and Giovanni Rosso for
discussions related to this project.

##  2\. Preliminaries

###  2.1. Groups

####  2.1.1. Definitions

Denote by $J_{2}$ the $2\times 2$ anti-diagonal matrix with entries 1 in the
anti-diagonal and let $J=\left(\begin{smallmatrix}&J_{2}\\\
-J_{2}&\end{smallmatrix}\right)$. Fix an imaginary quadratic field $E={\bf
Q}(\delta)$, with $\delta=\sqrt{d}$ and ring of integers ${\mathcal{O}}_{E}$.
Define $\mathbf{G}$ to be the group scheme over ${\bf Q}$ given by

$\mathbf{G}(R):={{\rm GU}}_{2,2}(R)=\\{(g,m_{g})\in{\rm GL}_{4}(R\otimes_{{\bf
Q}}E)\times\mathbb{G}_{\rm m}(R):\bar{g}^{t}Jg=m_{g}J\\},$

where $\bar{\bullet}$ denotes the non-trivial automorphism of order 2 of
$E/{\bf Q}$ and $R$ denotes any ${\bf Q}$-algebra. Denote by
$\nu:\mathbf{G}\to\mathbb{G}_{\rm m},\;g\mapsto m_{g}$ the similitude
character. Its kernel is the group ${\rm U}_{2,2}$. Define the subgroup
$\mathbf{G}^{\star}$ of $\mathbf{G}$ given by

$\mathbf{G}^{\star}(R)=\\{(g,m_{g})\in{{{\rm GU}}}_{2,2}\;:\;{\rm
det}(g)=m_{g}^{2}\\}.$

Note that $\mathbf{G}^{\star}$ is a form of ${{\rm GSpin}}_{6}$. Inside
$\mathbf{G}^{\star}$, we have the subgroup

$\mathbf{H}(R):={{\rm GSp}}_{4}(R)=\\{(g,m_{g})\in{\rm
GL}_{4}(R)\times\mathbb{G}_{\rm m}(R):g^{t}Jg=m_{g}J\\}.$

Denote by $\iota$ the natural embedding

$\iota:\mathbf{H}\hookrightarrow\mathbf{G},\;(g,m_{g})\mapsto(g,m_{g}),$

induced by the inclusion ${\rm GL}_{4}\hookrightarrow{\rm Res}_{E/{\bf Q}}{\rm
GL}_{4}$.

Finally, define the similitude orthogonal groups ${{\rm GO}}_{2+i,i}$ for
$i=0,1,2$ as the group scheme over ${\bf Q}$ whose points are

${{\rm GO}}_{2+i,i}(R)=\\{(g,\nu_{g})\in{\rm
GL}_{2+2i}(R)\times\mathbb{G}_{\rm m}(R):g^{t}S_{i}g=\nu_{g}S_{i}\\},$

where

$S_{0}=\left(\begin{smallmatrix}2&\\\
&-2d\end{smallmatrix}\right),\;S_{1}=\left(\begin{smallmatrix}&&1\\\
&S_{0}&\\\ 1&&\end{smallmatrix}\right),\;S_{2}=\left(\begin{smallmatrix}&&1\\\
&S_{1}&\\\ 1&&\end{smallmatrix}\right).$

Moreover let ${{\rm GSO}}_{2+i,i}$ be the subgroup

${{\rm GSO}}_{2+i,i}(R)=\\{(g,\nu_{g})\in{{\rm GO}}_{2+i,i}(R):{\rm
det}(g)=\nu_{g}^{1+i}\\}.$

Notice that ${{\rm GSO}}_{2,0}({\bf Q})\simeq E^{\times}$; we identify these
groups in a way so that the character $(g,\nu_{g})\mapsto\nu_{g}$ is given by
the norm map $N_{E/{\bf Q}}$. Throughout the paper, we denote
$\mathbf{G}^{\prime}:={{\rm GSO}}_{4,2}$. Finally, for any ${\bf Q}$-algebra
$R$, define

$\mathbf{H}^{+}(R):=\\{h\in\mathbf{H}(R)\,:\,m_{h}=\nu_{g}\text{ for some
}g\in\mathbf{G}^{\prime}(R)\\}.$

Notice that $\mathbf{H}^{+}({\bf Q})$ is normal in $\mathbf{H}({\bf Q})$ with
quotient isomorphic to ${\bf Q}^{\times}/N_{E/{\bf Q}}(E^{\times})$.

####  2.1.2. Parabolic subgroups

Let $B_{\mathbf{G}}=T_{\mathbf{G}}U_{\mathbf{G}}$ denote the upper-triangular
Borel subgroup of $\mathbf{G}$, with $T_{\mathbf{G}}$ the diagonal torus

$\left\\{\left(\begin{smallmatrix}a&&&\\\ &b&&\\\ &&\nu\bar{b}^{-1}&\\\
&&&\nu\bar{a}^{-1}\end{smallmatrix}\right)\;:a,b\in{\rm Res}_{E/{\bf
Q}}\mathbb{G}_{\rm m},\;\nu\in\mathbb{G}_{\rm m}\right\\}.$

The modulus character $\delta_{B_{\mathbf{G}}}:B_{\mathbf{G}}({\bf
Q}_{p})\to{\bf C}^{\times}$ is given by

$\left(\begin{smallmatrix}a&&&\\\ &b&&\\\ &&\nu\bar{b}^{-1}&\\\
&&&\nu\bar{a}^{-1}\end{smallmatrix}\right)\mapsto\tfrac{|a\bar{a}|^{3}|b\bar{b}|}{|\nu|^{4}}.$

We let
$B_{\mathbf{H}}=T_{\mathbf{H}}U_{\mathbf{H}}=B_{\mathbf{G}}\cap\mathbf{H}$ be
the upper triangular Borel subgroup of $\mathbf{H}$ and let $P_{G}=M_{G}N_{G}$
be the standard Siegel parabolic of $G\in\\{\mathbf{H},\mathbf{G}\\}$. For
instance,

$M_{\mathbf{H}}=\\{m=\left(\begin{smallmatrix}g&\\\ &\mu
J_{2}{}^{t}g^{-1}J_{2}\end{smallmatrix}\right),\;g\in{\rm
GL}_{2},\mu\in\mathbb{G}_{\rm m}\\},$
$N_{\mathbf{H}}=\left\\{\left(\begin{smallmatrix}1&&a&x\\\ &1&y&a\\\ &&1&\\\
&&&1\end{smallmatrix}\right),\;\text{ with }x,y,a\in\mathbb{G}_{\rm
a}\right\\}.$

Denote by $\delta_{P_{\mathbf{H}}}:P_{\mathbf{H}}({\bf Q}_{p})\to{\bf
C}^{\times}$ the modulus character of $P_{\mathbf{H}}$ given by

$\displaystyle\delta_{P_{\mathbf{H}}}$
$\displaystyle:\left(\begin{smallmatrix}g&\star\\\ &\mu
I_{2}^{\prime}{}^{t}g^{-1}I_{2}^{\prime}\end{smallmatrix}\right)\mapsto\left|\tfrac{{\rm
det}(g)}{\mu}\right|^{3}.$

Finally, we let $B_{\mathbf{G}^{\prime}}$ be the upper-triangular Borel of
$\mathbf{G}^{\prime}$ with Levi the maximal diagonal torus

$T_{\mathbf{G}^{\prime}}=\left\\{\left(\begin{smallmatrix}a&&&&\\\ &b&&&\\\
&&x&&\\\ &&&\nu b^{-1}&\\\ &&&&\nu
a^{-1}\end{smallmatrix}\right)\;:\,a,b\in\mathbb{G}_{\rm m},\;x\in{\rm
Res}_{E/{\bf Q}}\mathbb{G}_{\rm m},\;\nu=N_{E/{\bf Q}}(x)\right\\}.$

####  2.1.3. An exceptional isomorphism

Recall that we have an exceptional isomorphism
$j:\mathbf{P}\mathbf{G}\simeq\mathbf{P}\mathbf{G}^{\prime}$, where the letter
$\mathbf{P}$ stands for taking the quotient by the center. In what follows, we
sketch the construction of this isomorphism as done in [29, pp. 33–34] and
refer to _loc.cit._ for further details. Define the six dimensional ${\bf
Q}$-vector space111This differs from the one defined in [29] because of the
different choices of Hermitian forms defining ${\rm GU}_{2,2}$. Conjugating by
${{\big{(}\begin{smallmatrix}J_{2}&\\\ &I\end{smallmatrix}\big{)}}}$, one gets
the vector space introduced in _loc.cit._.

$\mathcal{V}:=\left\\{v(x_{1},x_{2},x_{3},x_{4},x_{5},x_{6}):=\begin{pmatrix}0&-x_{1}&x_{2}&-x_{3}+\delta
x_{4}\\\ x_{1}&0&x_{3}+\delta x_{4}&x_{5}\\\ -x_{2}&-x_{3}-\delta
x_{4}&0&x_{6}\\\ x_{3}-\delta
x_{4}&-x_{5}&-x_{6}&0\end{pmatrix}\;|\;x_{i}\in{\bf Q}\right\\},$

endowed with the bilinear form $\Psi:\mathcal{V}\times\mathcal{V}\to{\bf Q}$
defined by

$\Psi(v_{1},v_{2})={\rm
Tr}(v_{1}J^{\prime}\overline{v_{2}}^{t}J^{\prime}),\;\text{with
}J^{\prime}={{\big{(}\begin{smallmatrix}&J_{2}\\\
J_{2}&\end{smallmatrix}\big{)}}}.$

Choose the basis of $\mathcal{V}$ given by $\\{v_{i}\\}_{i}$, where
$v_{i}=v(e_{i})$ for $i=1,\cdots,6$, with $\\{e_{i}\\}_{i}$ denoting the
standard basis of ${\bf Q}^{6}$. This induces an isomorphism ${\rm
GO}(\mathcal{V},\Psi)\simeq{\rm GO}_{4,2}$. The homomorphism

$\displaystyle\phi:\;\mathbf{G}^{\star}$ $\displaystyle\to{\rm
GO}(\mathcal{V},\Psi),$ (1) $\displaystyle g$ $\displaystyle\mapsto(v\mapsto
gv^{t}g),$

is well defined and satisfies the relation

$\nu_{\phi(g)}=m_{g}^{2}.$

Hence, $\phi$ induces a homomorphism
$\Phi:\;\mathbf{G}^{\star}\to\mathbf{G}^{\prime}$. Given $\alpha\in
E^{\times}$, define $r_{\alpha}=\left(\begin{smallmatrix}\alpha&&&\\\ &1&&\\\
&&1&\\\ &&&\overline{\alpha}^{-1}\end{smallmatrix}\right)$. There is an action
of $E^{\times}$ on $\mathcal{V}$ via the rule

$\alpha\cdot v=\overline{\alpha}r_{\alpha}vr_{\alpha},\,\text{for }\alpha\in
E^{\times},v\in\mathcal{V}.$ (2)

On the other hand, $E^{\times}$ acts on $g\in\mathbf{G}^{\star}$ by
$\alpha\cdot g=r_{\alpha}gr_{\alpha}^{-1}$. This action allows us to consider
the semidirect product $\mathbf{G}^{\star}\rtimes E^{\times}$. Then, using the
action (2), $\Phi$ extends to a surjective homomorphism
$\Gamma:\;\mathbf{G}^{\star}\rtimes E^{\times}\to\mathbf{PG}^{\prime}$.
Finally, the surjective homomorphism

$\displaystyle\Xi:\;\mathbf{G}^{\star}\rtimes E^{\times}$
$\displaystyle\to\mathbf{P}\mathbf{G}$ $\displaystyle(g,\alpha)$
$\displaystyle\mapsto gr_{\alpha}$

satisfies the property that $\mathrm{ker}(\Gamma)=\mathrm{ker}(\Xi)$ and hence
we can define
$j:=\Gamma^{*}\circ(\Xi^{*})^{-1}:\;\mathbf{P}\mathbf{G}\simeq\mathbf{P}\mathbf{G}^{\prime}$.
Finally, by lifting any diagonal element of $\mathbf{P}\mathbf{G}$ to
$\mathbf{G}^{\star}\rtimes E^{\times}$ and calculating the action (via (1) and
(2)) of its lift on the basis $\\{v_{i}\\}_{i}$ of $\mathcal{V}$, we obtain
the following.

######  Lemma 2.1.

The isomorphism $j:\mathbf{P}\mathbf{G}\simeq\mathbf{P}\mathbf{G}^{\prime}$
sends

$j:\left(\begin{smallmatrix}a&&&\\\ &b&&\\\ &&\nu\bar{b}^{-1}&\\\
&&&\nu\bar{a}^{-1}\end{smallmatrix}\right)\mapsto\left(\begin{smallmatrix}N_{E/{\bf
Q}}(ab)&&&&\\\ &\nu N_{E/{\bf Q}}(a)&&&\\\ &&\nu\bar{a}b&&\\\ &&&\nu N_{E/{\bf
Q}}(b)&\\\ &&&&\nu^{2}\end{smallmatrix}\right),$

with $a,b\in{\rm Res}_{E/{\bf Q}}\mathbb{G}_{\rm m}$ and
$\nu\in\mathbb{G}_{\rm m}$.

###  2.2. Root system and Weyl groups

####  2.2.1. Root system

Write any element of $T_{\mathbf{H}}$ as
$t(a,b,\nu)=\left(\begin{smallmatrix}a&&&\\\ &b&&\\\ &&\nu b^{-1}&\\\ &&&\nu
a^{-1}\end{smallmatrix}\right)$ and let $\alpha_{1},\alpha_{2},\alpha_{0}\in
X^{*}(T_{\mathbf{H}})$ be the characters

$\alpha_{1}(t(a,b,\nu))=a,\,\alpha_{2}(t(a,b,\nu))=b,\,\alpha_{0}(t(a,b,\nu))=\nu.$

Furthermore, let $\check{\alpha}_{1},\check{\alpha}_{2},\check{\alpha}_{0}\in
X_{*}(T_{\mathbf{H}})$ be the cocharacters

$\check{\alpha}_{1}(x)=t(x,1,1),\,\check{\alpha}_{2}(x)=t(1,x,1),\,\check{\alpha}_{0}(x)=t(1,1,x).$

We denote by $\langle\,,\,\rangle:X^{*}(T_{\mathbf{H}})\times
X_{*}(T_{\mathbf{H}})\to{\bf Z}$ the canonical pairing given by composition.
Our choice of Borel subgroup $B_{\mathbf{H}}$ fixes the set of positive roots
for $\mathbf{P}\mathbf{H}$ (and $\mathbf{H}$); explicitly,

$\Phi_{\mathbf{P}\mathbf{H}}^{+}=\\{\alpha_{1}-\alpha_{2},\alpha_{1}+\alpha_{2}-\alpha_{0},2\alpha_{1}-\alpha_{0},2\alpha_{2}-\alpha_{0}\\}.$

The set of simple positive roots consists of the short root
$\alpha_{1}-\alpha_{2}$ and the long one $2\alpha_{2}-\alpha_{0}$. We also
denote by $\Phi_{\mathbf{P}\mathbf{H}}^{+,s}$, resp.
$\Phi_{\mathbf{P}\mathbf{H}}^{+,l}$, the set of positive short, resp. long,
roots. We have
$\Phi_{\mathbf{P}\mathbf{H}}^{+,l}=\\{2\alpha_{1}-\alpha_{0},2\alpha_{2}-\alpha_{0}\\}$.

To each root $\alpha$, we denote by $\check{\alpha}$ the corresponding coroot.
There is a homomorphism $x_{\alpha}:{\rm SL}_{2}\to\mathbf{P}\mathbf{H}$ such
that $x_{\alpha}\left(\left(\begin{smallmatrix}1&c\\\
&1\end{smallmatrix}\right)\right)\subseteq N^{\alpha}$, the latter being the
one parameter unipotent subgroup of $\alpha$. Then,
$\check{\alpha}=x_{\alpha}\circ\iota$, with $\iota:{\rm
GL}_{1}\to\mathrm{SL}_{2}$ the embedding of the diagonal torus, is such that
$\langle\alpha,\check{\alpha}\rangle=2$. Explicitly,

$\Phi_{\mathbf{P}\mathbf{H}}^{\vee,+}=\\{\check{\alpha}_{1}-\check{\alpha}_{2},\check{\alpha}_{1}+\check{\alpha}_{2},\check{\alpha}_{1},\check{\alpha}_{2}\\}$

is the set of positive coroots.

Recall that the dual group of $\mathbf{H}$ is ${}^{L}\mathbf{H}={\rm
GSpin}_{5}({\bf C})$, thus via an exceptional isomorphism we have

${}^{L}\mathbf{P}\mathbf{H}={\rm Spin}_{5}({\bf C})\simeq{\rm Sp}_{4}({\bf
C}).$

We thus can identify the set of positive roots of $\mathbf{P}\mathbf{H}$ with
the set of positive coroots of ${\rm Sp}_{4}$ as (see the proof of [32, Lemma
2.3.1])

$\\{\alpha_{1}-\alpha_{2},\alpha_{1}+\alpha_{2}-\alpha_{0},2\alpha_{1}-\alpha_{0},2\alpha_{2}-\alpha_{0}\\}\to\\{\check{\alpha}_{2},\check{\alpha}_{1},\check{\alpha}_{1}+\check{\alpha}_{2},\check{\alpha}_{1}-\check{\alpha}_{2}\\}.$

Note that the set $\Phi_{\mathbf{P}\mathbf{H}}^{+,l}$ is sent to the set
$\Phi^{\vee,+,s}_{{\rm
Sp}_{4}}=\\{\check{\alpha}_{1}+\check{\alpha}_{2},\check{\alpha}_{1}-\check{\alpha}_{2}\\}$
of coroots corresponding to the short roots for ${\rm Sp}_{4}$.

####  2.2.2. Weyl groups

Let $p$ be a prime and let $G_{/{\bf Q}_{p}}$ be an algebraic group. Given a
maximal split subtorus $A\subset G({\bf Q}_{p})$, we let the Weyl group be
$W_{G}=N_{G}(A)/C_{G}(A)$, where $N_{G}(A)$, resp. $C_{G}(A)$, denotes the
normaliser, resp. centralizer, of $A$ in $G({\bf Q}_{p})$. When $G$ is either
$\mathbf{H}_{/{\bf Q}_{p}}$ or $\mathbf{G}_{/{\bf Q}_{p}}$ with $p$ inert in
$E$, $A$ can be taken to be $T_{\mathbf{H}}({\bf Q}_{p})$. We then have that
$W_{\mathbf{H}}\simeq W_{\mathbf{G}}$. The Weyl group $W_{\mathbf{H}}$ is
generated by the reflections $s_{1},s_{2}$ associated to the positive roots
$\alpha_{1}-\alpha_{2}$, $2\alpha_{2}-\alpha_{0}$ respectively. We choose
coset representatives

$s_{1}=\left(\begin{smallmatrix}&1&&\\\ 1&&&\\\ &&&1\\\
&&1&\end{smallmatrix}\right),\;\;s_{2}=\left(\begin{smallmatrix}1&&&\\\
&&1&\\\ &-1&&\\\ &&&1\end{smallmatrix}\right).$

Then $W_{\mathbf{H}}$ is of order eight and its elements are
$\\{1,s_{1},s_{2},s_{1}s_{2},s_{2}s_{1},s_{1}s_{2}s_{1},s_{2}s_{1}s_{2},s_{1}s_{2}s_{1}s_{2}\\}$.
We will denote the longest Weyl element by $\omega_{0}$.

Given a character $\chi$ of $T_{G}({\bf Q}_{p})$, we let
${}^{\omega}\chi(t):=\chi(\omega^{-1}t\omega)$, for any $\omega\in W_{G}$.

###  2.3. Dual groups and $L$-functions

Here and throughout the article, if $G_{/{\bf Q}_{p}}$ is an algebraic group
with a Borel subgroup $B_{G}$, we will adopt the notation $I_{G}(\chi)$ for
the normalised induction ${\rm Ind}_{B_{G}({\bf Q}_{p})}^{G({\bf
Q}_{p})}(\chi)$.

####  2.3.1. The twisted exterior square representation

Recall that the dual group of $\mathbf{G}$ is

${}^{L}\mathbf{G}=({\rm GL}_{4}({\bf C})\times{\rm GL}_{1}({\bf
C}))\rtimes{\rm Gal}(E/{\bf Q}),$

with the action of the non-trivial element $\tau\in{\rm Gal}(E/{\bf Q})$ given
by (cf. [2, §1.8(c)])

$(g,\lambda)\mapsto(\Phi_{4}{}^{t}g^{-1}\Phi_{4},\lambda{\rm det}(g)),\text{
where }\Phi_{4}=\left(\begin{smallmatrix}&&&1\\\ &&-1&\\\ &1&&\\\
-1&&&\end{smallmatrix}\right).$

The standard representation of ${\rm GL}_{4}({\bf C})$ on ${\bf C}^{4}$, with
basis $\\{e_{1},e_{2},e_{3},e_{4}\\}$, induces the representation $\wedge^{2}$
on $\wedge^{2}{\bf C}^{4}\simeq{\bf C}^{6}$ by the rule
$(\wedge^{2}g)\cdot(e_{i}\wedge e_{j})=g\cdot e_{i}\wedge g\cdot e_{j}$.
Consider the six dimensional representation

${\rm GL}_{4}({\bf C})\times{\rm GL}_{1}({\bf C})\to{\rm GL}_{6}({\bf C})$

given by $(g,\lambda)\mapsto(\wedge^{2}g)\lambda$. Notice that both
representations $\wedge^{2}$ and $\wedge^{2}\circ\tau$ have same highest
weight $(1,1,0,0,0)$. Hence they are isomorphic, i.e. there exists $A\in{\rm
GL}_{6}({\bf C})$ such that
$(\wedge^{2}\circ\tau)(g,\lambda)=A^{-1}(\wedge^{2}(g,\lambda))A$. Choose $A$
with ${\rm Tr}(A)>0$ and extend $\wedge^{2}$ to ${}^{L}\mathbf{G}$ by sending

$(g,\lambda,1)\mapsto(\wedge^{2}g)\lambda,\,(1,1,\tau)\mapsto A.$

The resulting representation, denoted by
$\wedge_{t}^{2}:{}^{L}\mathbf{G}\to{\rm GL}_{6}({\bf C})$, is called the
twisted exterior square representation.

Finally, recall that the dual group of $\mathbf{P}\mathbf{G}$ is the derived
subgroup of ${}^{L}\mathbf{G}$, namely

${}^{L}\mathbf{P}\mathbf{G}={\rm SL}_{4}({\bf C})\rtimes{\rm Gal}(E/{\bf Q}).$

We still denote by $\wedge_{t}^{2}:{}^{L}\mathbf{P}\mathbf{G}\to{\rm
GL}_{6}({\bf C})$ the corresponding twisted exterior square representation.

####  2.3.2. The twisted exterior square $L$-function

We now define the partial twisted exterior square $L$-function of a cuspidal
automorphic representation $\Pi$ of $\mathbf{P}\mathbf{G}({\bf A})$. Note that
an automorphic representation of $\mathbf{P}\mathbf{G}({\bf A})$ is the same
as one of $\mathbf{G}({\bf A})$ with trivial central character, and we will
use the two notions interchangeably. Suppose that $p$ is a prime for which
$\Pi_{p}$ is an unramified representation of $\mathbf{G}({\bf Q}_{p})$.

  * •

Suppose that $p$ is inert and let $E_{p}$ be the completion at
$p\mathcal{O}_{E}$ of $E$. Any unramified character of $T_{\mathbf{G}}({\bf
Q}_{p})$ is of the form $\xi=\xi_{1}\otimes\xi_{2}\otimes\xi_{0}$, where
$\xi_{1},\xi_{2}$ are unramified characters of $E^{\times}_{p}$ and $\xi_{0}$
is an unramified character of ${\bf Q}^{\times}_{p}$, given by

$\xi:\left(\begin{smallmatrix}a&&&\\\ &b&&\\\ &&\nu\bar{b}^{-1}&\\\
&&&\nu\bar{a}^{-1}\end{smallmatrix}\right)\mapsto\xi_{1}(a)\xi_{2}(b)\xi_{0}(\nu).$

If $\Pi_{p}$ is an unramified irreducible subquotient of the (normalised)
induction $I_{\mathbf{G}}(\xi)$, the Frobenius conjugacy class $g_{\xi}$ of
$\Pi_{p}$ is represented by the element

$g_{\xi}=({\rm
diag}(\xi_{1}(p),\xi_{2}(p),1,1),\xi_{0}(p),\tau)\in{}^{L}\mathbf{G}.$

Note that, since $\Pi_{p}$ has trivial central character, one can choose a
representative of the conjugacy class in ${}^{L}\mathbf{P}\mathbf{G}$.

  * •

Suppose that $p$ splits as $\mathfrak{p}\bar{\mathfrak{p}}$ in $E$. Then the
isomorphism $E\otimes_{\bf Q}{\bf Q}_{p}\simeq E_{\mathfrak{p}}\times
E_{\bar{\mathfrak{p}}}$ induces an isomorphism ${\rm GL}_{4}(E\otimes_{\bf
Q}{\bf Q}_{p})\simeq{\rm GL}_{4}(E_{\mathfrak{p}})\times{\rm
GL}_{4}(E_{\bar{\mathfrak{p}}})$. If $g$ maps to $(g_{1},g_{2})$, then
$\bar{g}$ is sent to $(g_{2},g_{1})$. Hence, we have

$\displaystyle\mathbf{G}({\bf Q}_{p})\simeq\\{(g_{1},g_{2},m)\in{\rm
GL}_{4}(E_{\mathfrak{p}})\times{\rm GL}_{4}(E_{\bar{\mathfrak{p}}})\times{\rm
GL}_{1}({\bf Q}_{p})\,:\,g_{2}=mJ{}^{t}g_{1}^{-1}J^{-1}\\}$

After fixing an isomorphism $\iota_{\mathfrak{p}}:E_{\mathfrak{p}}\simeq{\bf
Q}_{p}$, we thus have

$\displaystyle\mathbf{G}({\bf Q}_{p})\simeq{\rm GL}_{4}({\bf Q}_{p})\times{\rm
GL}_{1}({\bf Q}_{p}),\,(g_{1},g_{2},m)\mapsto(g_{1},m).$ (3)

This let us identify $\Pi_{p}$ with an unramified representation $\pi_{p}$ of
${\rm GL}_{4}({\bf Q}_{p})$ with trivial central character. Its Frobenius
conjugacy class is then represented by

$g_{\xi}=({\rm
diag}(\xi_{1}(p),\xi_{2}(p),\xi_{3}(p),\xi_{4}(p)),1,1)\in{}^{L}\mathbf{P}\mathbf{G},$

where $\xi=\otimes_{i}\xi_{i}$ is an unramified character of the diagonal
torus of ${\rm GL}_{4}({\bf Q}_{p})$, with $\prod\xi_{i}=1$, such that
$\pi_{p}$ appears as the unique unramified subquotient of $I_{{\rm
GL}_{4}}(\xi)$.

Define the local Euler factor of the twisted exterior square $L$-function for
$\Pi_{p}$ by

$L(s,\Pi_{p},\wedge^{2}_{t}):=\frac{1}{{\rm
det}(1-p^{-s}\wedge^{2}_{t}(g_{\xi}))}.$

######  Remark 2.2.

At a split prime $p$, the Euler factor $L(s,\Pi_{p},\wedge^{2}_{t})$ coincides
with the exterior square Euler factor $L(s,\pi_{p},\wedge^{2})$.

Let $S$ be a finite set of primes containing all the ramified primes of $\Pi$
and of $E/{\bf Q}$, then we define

$L^{S}(s,\Pi,\wedge^{2}_{t}):=\prod_{p\not\in S}L(s,\Pi_{p},\wedge^{2}_{t}).$

####  2.3.3. The degree 5 $L$-function on ${\rm PGSp}_{4}$

Recall that ${}^{L}\mathbf{P}\mathbf{H}={\rm Spin}_{5}({\bf C})\simeq{\rm
Sp}_{4}({\bf C})$. Under this exceptional isomorphism, the standard
representation $V_{4}$ of ${\rm Sp}_{4}({\bf C})$ coincides with the spin
representation of ${\rm Spin}_{5}({\bf C})$. The exterior square
$\wedge^{2}V_{4}={\rm std}\oplus\mathbf{1}$, where ${\rm std}:{\rm
Sp}_{4}({\bf C})\to{\rm GL}_{5}({\bf C})$ is the irreducible representation of
highest weight $(1,1)$ and gives the composition of the projection ${\rm
Spin}_{5}({\bf C})\to{\rm SO}_{5}({\bf C})$ with the standard representation
of ${\rm SO}_{5}({\bf C})$. To it we can associate the degree 5 $L$-function
on ${\rm PGSp}_{4}$ (which is often called the standard $L$-function on ${\rm
PGSp}_{4}$) and its twist by the Hecke character $\chi_{E/{\bf Q}}:{\bf
A}^{\times}/{\bf Q}^{\times}\to\\{\pm 1\\}$ associated to the imaginary
quadratic extension $E/{\bf Q}$. Recall that the $p$-component of
$\chi_{E/{\bf Q}}$ is

$\chi_{E_{p}/{\bf Q}_{p}}(x)=\begin{cases}-1&\text{if }E_{p}\text{ is a field
and }x\not\in N_{E_{p}/{\bf Q}_{p}}(E_{p}^{\times})\\\
1&\text{otherwise,}\end{cases}$

where we have denoted $E_{p}:={\bf Q}_{p}\otimes_{\bf Q}E$.

Let $\sigma$ be a cuspidal automorphic representation of $\mathbf{H}({\bf A})$
with trivial central character. Suppose that $p$ is a prime for which
$\sigma_{p}$ is unramified, then $\sigma_{p}$ has Frobenius conjugacy class in
${\rm Sp}_{4}({\bf C})$.

######  Remark 2.3.

If the smooth representation $\sigma_{p}$ appears as the unramified
constituent of the (normalised) induction
$I_{\mathbf{H}}(\chi)=\mathrm{Ind}_{B_{\mathbf{H}}({\bf
Q}_{p})}^{\mathbf{H}({\bf Q}_{p})}\chi$, with
$\chi=\chi_{1}\otimes\chi_{2}\otimes\chi_{0}$, where
$\chi_{0},\chi_{1},\chi_{2}$ are unramified characters of ${\bf
Q}^{\times}_{p}$, given by

$\chi:\left(\begin{smallmatrix}a&&&\\\ &b&&\\\ &&\lambda b^{-1}&\\\ &&&\lambda
a^{-1}\end{smallmatrix}\right)\mapsto\chi_{1}(a)\chi_{2}(b)\chi_{0}(\lambda),$

then its Frobenius conjugacy class is represented by the element

$g_{\chi}={\rm
diag}((\chi_{1}\chi_{2}\chi_{0})(p),(\chi_{1}\chi_{0})(p),(\chi_{2}\chi_{0})(p),\chi_{0}(p))\in{}^{L}\mathbf{P}\mathbf{H}.$

Define the local Euler factor of the degree 5 $L$-function twisted by
$\chi_{E_{p}/{\bf Q}_{p}}$ for $\sigma_{p}$ as follows:

$L(s,\sigma_{p},{\rm std}\otimes\chi_{E_{p}/{\bf Q}_{p}}):=\frac{1}{{\rm
det}(1-\chi_{E_{p}/{\bf Q}_{p}}(p)p^{-s}{\rm std}(g_{\chi}))}.$

######  Remark 2.4.

A simple calculation shows that

${\rm det}(1-\chi_{E_{p}/{\bf Q}_{p}}(p)p^{-s}{\rm
std}(g_{\chi}))=(1-\chi_{E_{p}/{\bf
Q}_{p}}(p)p^{-s})\prod_{i=1}^{2}(1-(\chi_{E_{p}/{\bf
Q}_{p}}\chi_{i})(p)p^{-s})(1-(\chi_{E_{p}/{\bf
Q}_{p}}\chi_{i}^{-1})(p)p^{-s}).$

Let $S$ be a finite set of primes containing all the ramified primes of
$\sigma$ and of $E/{\bf Q}$, then we define

$L^{S}(s,\sigma,{\rm std}\otimes\chi_{E/{\bf Q}}):=\prod_{p\not\in
S}L(s,\sigma_{p},{\rm std}\otimes\chi_{E_{p}/{\bf Q}_{p}}).$

###  2.4. Shalika models and a theta correspondence

Throughout this paper we will consider the Shalika subgroup of $\mathbf{G}$,
which is defined as follows. For any ${\bf Q}$-algebra $R$, $S$ is given by

$S(R)=\left\\{\left(\begin{smallmatrix}a&b&&\\\ c&d&&\\\ &&a&-b\\\
&&-c&d\end{smallmatrix}\right)\left(\begin{smallmatrix}1&&\alpha&x\\\
&1&y&\overline{\alpha}\\\ &&1&\\\
&&&1\end{smallmatrix}\right)\;:\;\left(\begin{smallmatrix}a&b\\\
c&d\end{smallmatrix}\right)\in{\rm
GL}_{2}(R),x,y\in\mathbb{G}_{a}(R),\;\alpha\in{\rm Res}_{E/{\bf
Q}}\mathbb{G}_{a}(R)\right\\}.$ (4)

Note that the Shalika group $S$ is isomorphic to ${\rm GL}_{2}N_{\mathbf{G}}$,
where $N_{\mathbf{G}}$ is the unipotent radical of the Siegel of $\mathbf{G}$,
while ${\rm GL}_{2}$ embeds into the Levi of the Siegel parabolic
$M_{\mathbf{H}}$ of $\mathbf{H}$ via the embedding $h\mapsto
h^{*}:=\left(\begin{smallmatrix}h&\\\ &{\rm
det}(h)J_{2}{}^{t}h^{-1}J_{2}\end{smallmatrix}\right)$. Notice that the
modulus character $\delta_{S}$ of $S$ is trivial.

We now fix a non-degenerate character $\psi:{\bf Q}\backslash{\bf A}\to{\bf
C}^{\times}$. Given $\eta\in E^{\times}$ such that $\bar{\eta}=-\eta$, we
define

$\chi_{\eta}:N_{\mathbf{G}}({\bf Q})\backslash N_{\mathbf{G}}({\bf
A})\longrightarrow{\bf C}^{\times},\,\left(\begin{smallmatrix}1&&\alpha&x\\\
&1&y&\overline{\alpha}\\\ &&1&\\\
&&&1\end{smallmatrix}\right)\mapsto\psi(\eta\alpha-\eta\overline{\alpha}).$

By a slight abuse of notation, denote also by $\chi_{\eta}$ the character on
$S({\bf Q})\backslash S({\bf A})$ given by $h^{*}n\mapsto\chi_{\eta}(n)$.

Let $\Pi$ be a cuspidal automorphic representation of $\mathbf{G}({\bf A})$
with trivial central character.

######  Definition 2.5.

For a cusp form $\varphi$ in the space of $\Pi$, the _unitary_ Shalika period
is defined by

$\mathcal{S}_{\varphi}^{\eta}(g):=\int_{{\bf A}^{\times}{\rm GL}_{2}({\bf
Q})\backslash{\rm GL}_{2}({\bf A})}\int_{N_{\mathbf{G}}({\bf Q})\backslash
N_{\mathbf{G}}({\bf A})}\varphi\left(nh^{*}g\right)\chi_{\eta}^{-1}(n)dndg.$

The representation $\Pi$ is said to have a Shalika model if there exist
$\varphi$ in $\Pi$ and $g\in\mathbf{G}({\bf A})$ such that
$\mathcal{S}_{\varphi}^{\eta}(g)\neq 0$.

In [13], Furusawa and Morimoto characterized the representations $\Pi$ with
unitary Shalika model in terms of the existence of a pole at $s=1$ of their
partial twisted exterior square $L$-function. Namely, they showed the
following.

######  Theorem 2.6 ([13, Theorem 4.1]).

Let $\Pi$ be an irreducible cuspidal automorphic representation of
$\mathbf{G}({\bf A})$ with trivial central character. Then the following two
conditions are equivalent:

  1. (1)

The Shalika period $\mathcal{S}_{\varphi}^{\eta}(1)$ does not vanish on the
space of $\Pi$.

  2. (2)

$\Pi$ is globally generic and the partial twisted exterior square $L$-function
$L^{S}(s,\Pi,\wedge^{2}_{t})$ has a simple pole at $s=1$.

Motivated by this, in [29], Morimoto gave a characterization of cuspidal
automorphic representations of $\mathbf{G}({\bf A})$ with Shalika models by
means of the theta correspondence $\Theta$ of the pair $(\mathbf{H}^{+}({\bf
A}),\mathbf{G}^{\prime}({\bf A}))$. Crucially, Morimoto employed the
exceptional isomorphism
$j:\mathbf{P}\mathbf{G}\simeq\mathbf{P}\mathbf{G}^{\prime}$ to obtain the
following.

######  Theorem 2.7 ([29, Theorem 3.6]).

Let $\Pi$ be an irreducible cuspidal automorphic representation of
$\mathbf{G}({\bf A})$ with trivial central character. Then $\Pi$ has a Shalika
model if and only if $\Pi$ appears as a subquotient of $\Theta(\sigma^{+})$,
for an irreducible cuspidal automorphic representation $\sigma^{+}$ of
$\mathbf{H}^{+}({\bf A})$ which appears as a subrepresentation of a globally
generic cuspidal automorphic representation $\sigma$ of $\mathbf{H}({\bf A})$
with trivial central character.

Thanks to a detailed study of the local theta correspondence, Morimoto
obtained the following relation.

######  Corollary 2.8.

Suppose that $\Pi$ appears in $\Theta(\sigma^{+})$, where $\sigma^{+}$ is an
irreducible $\mathbf{H}^{+}({\bf A})$-component of a globally generic
irreducible cuspidal automorphic representation $\sigma$ of $\mathbf{H}({\bf
A})$ with trivial central character. Then, for big enough ${\rm Re}(s)$,

$L^{S}(s,\Pi,\wedge^{2}_{t})=L^{S}(s,\sigma,{\rm std}\otimes\chi_{E/{\bf
Q}})\zeta^{S}(s),$

where $S$ is a finite set of places which contains all the ramified primes of
$\Pi$, $\sigma$, and $E/{\bf Q}$.

###### Proof.

This is [29, (3.6)] and it follows from comparing the Euler factors at
$p\not\in S$. For the convenience of the reader, we sketch a proof of it.
Suppose that $p\not\in S$ is inert in $E$ and that $\Pi_{p}$ is an unramified
irreducible subquotient of $I_{\mathbf{G}}(\xi)$. If we write
$\Pi_{p}=\theta(\sigma_{p}^{+})$, for $\sigma_{p}$ an unramified generic
principal series $I_{\mathbf{H}}(\chi)$, then by [29, Theorem 6.21(3)] we have
that (see the proof of Proposition 3.4)

$\xi_{1}=\chi_{0}\circ N_{E_{p}/{\bf Q}_{p}},\,\xi_{2}=(\chi_{2}\chi_{0})\circ
N_{E_{p}/{\bf Q}_{p}},\,\xi_{0}=\chi_{1}\chi_{E_{p}/{\bf Q}_{p}}.$

Therefore, the Frobenius conjugacy class $g_{\xi}=({\rm
diag}(\chi_{1}^{-1}\chi_{2}^{-1}(p),\chi_{1}^{-1}\chi_{2}(p),1,1),\chi_{1}(p)\chi_{E_{p}/{\bf
Q}_{p}}(p),\tau)$. As in [25, p. 12], we thus have that
$\wedge^{2}_{t}(g_{\xi})=\chi_{E_{p}/{\bf Q}_{p}}(p)\cdot{\rm
diag}(\chi_{1}(p),\chi_{2}^{-1}(p),\chi_{2}(p),\chi_{1}^{-1}(p),-1,1)$, hence

$\displaystyle L(s,\Pi_{p},\wedge^{2}_{t})$
$\displaystyle=[(1+\chi_{E_{p}/{\bf Q}_{p}}(p)p^{-s})(1-\chi_{E_{p}/{\bf
Q}_{p}}(p)p^{-s})\prod_{i=1}^{2}(1-\chi_{E_{p}/{\bf
Q}_{p}}\chi_{i}(p)p^{-s})(1-\chi_{E_{p}/{\bf
Q}_{p}}\chi_{i}^{-1}(p)p^{-s})]^{-1}$
$\displaystyle=(1-p^{-s})^{-1}L(s,\sigma_{p},{\rm std}\otimes\chi_{E_{p}/{\bf
Q}_{p}}),$

as $\chi_{E_{p}/{\bf Q}_{p}}(p)=-1$. The proof in the split case is similar. ∎

######  Remark 2.9.

Let $\sigma$ and $\sigma^{\prime}$ be two irreducible automorphic
representations of $\mathbf{H}({\bf A})$ which contain $\sigma^{+}$ as
$\mathbf{H}^{+}({\bf A})$ subrepresentation. Then, at each prime $p$ that
splits in $E$ we have that $\sigma_{p}^{\prime}\simeq\sigma_{p}$, while at
each inert prime $p$ for $E$ one has that $\sigma_{p}^{\prime}$ is isomorphic
either to $\sigma_{p}$ or to $\sigma_{p}\otimes\chi_{E_{p}/{\bf Q}_{p}}$. This
implies that $\sigma$ and $\sigma^{\prime}$ are isomorphic as
$\mathbf{H}^{+}({\bf A})$-representations. In particular, the assertions of
the two results above do not depend on a choice of $\sigma$.

When $\Pi$ has a Shalika model, $\mathcal{S}_{\varphi}^{\eta}$ defines an
intertwining map

$\Pi\hookrightarrow\mathrm{Ind}_{S({\bf A})}^{\mathbf{G}({\bf
A})}\chi_{\eta}.$

Thus, at each finite place $p$ the local component $\Pi_{p}$ has a local
Shalika model in the sense that we have a $\mathbf{G}({\bf
Q}_{p})$-equivariant injection

$\Pi_{p}\hookrightarrow\mathrm{Ind}_{S({\bf Q}_{p})}^{\mathbf{G}({\bf
Q}_{p})}\chi_{\eta,p}.$

Notice that this is equivalent to the existence of a _Shalika_ functional
$\Lambda_{p,\mathcal{S}}:\Pi_{p}\to{\bf C}$ such that for $s\in S({\bf
Q}_{p})$ and $v$ in the space of $\Pi_{p}$, we have

$\Lambda_{p,\mathcal{S}}(\Pi_{p}(s)v)=\chi_{\eta,p}(s)\Lambda_{p,\mathcal{S}}(v).$

When $p$ splits in $E$, the isomorphism (3) shows that
$\mathbf{P}\mathbf{G}({\bf Q}_{p})\simeq\mathrm{PGL}_{4}({\bf Q}_{p})$. Hence
we identify the the space of Shalika functionals on $\Pi_{p}$ with the space
Shalika functionals on an admissible representation of $\mathrm{PGL}_{4}({\bf
Q}_{p})$ (cf Lemma 4.1 below). Then, by a result of Jacquet–Rallis, we have
the following.

######  Proposition 2.10 ([23, Proposition 6.1]).

If $p$ is split in $E$, then

${\rm dim}_{\bf C}\,{\rm Hom}_{S({\bf Q}_{p})}(\Pi_{p},\chi_{\delta,p})\leq
1.$

By the results of [33], if $\Pi_{p}$ is unramified, the space of Shalika
functionals is one-dimensional if and only if $\Pi_{p}$ is generic and its
Frobenius conjugacy class lies in ${\rm Sp}_{4}({\bf C})$ (cf. [33, Corollary
5.4] and the discussion at the end of §8 of _loc.cit._). This is motivated by
the relation between Shalika functionals for $\mathrm{PGL}_{4}$ and the image
of the functorial lifts from $\mathrm{PGSp}_{4}$ to $\mathrm{PGL}_{4}$. In the
following section, using the calculations of [29] on the local theta
correspondence, we prove an analogous result in the case when $p$ is inert in
$E$.

##  3\. Uniqueness of local Shalika functionals at inert primes

Throughout this section we let $p$ be a prime which is inert in $E$. In what
follows, we discuss the uniqueness of Shalika models for unramified generic
representations of $\mathbf{P}\mathbf{G}({\bf Q}_{p})$ which lie in the image
of the local theta correspondence studied by [29]. We crucially use the
explicit computations of [29] of the local theta lifts from
$\mathbf{P}\mathbf{H}^{+}({\bf Q}_{p})$ to $\mathbf{P}\mathbf{G}({\bf
Q}_{p})$.

###  3.1. Generic unramified representations of $\mathbf{P}\mathbf{H}^{+}({\bf
Q}_{p})$

An irreducible generic unramified representation $\sigma$ of
$\mathbf{P}\mathbf{H}({\bf Q}_{p})$ is necessarily a principal series
$I_{\mathbf{H}}(\chi)$, for a unique $W_{\mathbf{H}}$-orbit of characters
$\chi=\chi_{1}\otimes\chi_{2}\otimes\chi_{0}$, where
$\chi_{1},\chi_{2},\chi_{0}$ are unramified smooth characters such that
$\chi_{1}\chi_{2}\chi_{0}^{2}=1$ and $|\cdot|^{\pm
1}\not\in\\{\chi_{1},\chi_{2},\chi_{1}\chi_{2},\chi_{1}\chi_{2}^{-1}\\}$ (cf.
[32, Tables A.1 & A.13]).

######  Lemma 3.1.

The restriction $I_{\mathbf{H}^{+}}(\chi)$ of $I_{\mathbf{H}}(\chi)$ to
$\mathbf{H}^{+}({\bf Q}_{p})$ is irreducible if and only if

$\displaystyle\mathbf{(ND)}$ $\displaystyle
I_{\mathbf{H}}(\chi)\otimes\chi_{E_{p}/{\bf Q}_{p}}\not\simeq
I_{\mathbf{H}}(\chi),$

where $\chi_{E_{p}/{\bf Q}_{p}}$ is regarded as a representation of
$\mathbf{H}({\bf Q}_{p})$ factoring through the symplectic multiplier
$\nu:\mathbf{H}({\bf Q}_{p})\to{\bf Q}_{p}^{\times}$.

###### Proof.

By [18, Lemma 2.1], the restriction $I_{\mathbf{H}^{+}}(\chi)$ of
$I_{\mathbf{H}}(\chi)$ is a finite direct sum of irreducible representations

$I_{\mathbf{H}^{+}}(\chi)=\oplus_{i=1}^{M}\pi_{i}^{m},$

with each factor $\pi_{i}$ irreducible and inequivalent, of same multiplicity
$m$ such that $m^{2}M$ equals the number of one-dimensional characters of
$\mathbf{H}({\bf Q}_{p})$ in the collection

$X_{\mathbf{H}^{+}}(\chi):=\\{\text{ one-dimensional character }\psi\text{ of
}\mathbf{H}({\bf Q}_{p})\;:\;\psi_{|\mathbf{H}^{+}({\bf Q}_{p})}=1\text{ and
}I_{\mathbf{H}}(\chi)\otimes\psi\simeq I_{\mathbf{H}}(\chi)\\}.$

By [1, Theorem 1.4], $m=1$ and thus, [18, Corollary 2.2] implies that
$\mathbf{H}({\bf Q}_{p})/\mathbf{H}^{+}({\bf Q}_{p})\simeq{\bf Z}/2{\bf Z}$
acts transitively on the set $\\{\pi_{1},\pi_{M}\\}$. Thus $M\leq 2$, with
$M=2$ if $\chi_{E_{p}/{\bf Q}_{p}}\in X_{\mathbf{H}^{+}}(\chi)$. ∎

Here, $\mathbf{(ND)}$ stands for non-dihedral. By [16, (H3) & Proposition
2.2], $\mathbf{(ND)}$ can be checked at the level of Satake parameters.
Precisely, we have the following.

######  Lemma 3.2.

We have that $I_{\mathbf{H}}(\chi)\otimes\chi_{E_{p}/{\bf Q}_{p}}\simeq
I_{\mathbf{H}}(\chi)$ if and only if $\chi_{E_{p}/{\bf
Q}_{p}}\in\\{\chi_{1},\chi_{2}\\}$.

###### Proof.

It follows from explicitly comparing the Weyl orbits of the two Satake
parameters. ∎

Now suppose that $I_{\mathbf{H}^{+}}(\chi)$ is reducible. By the proof of
Lemma 3.1, it decomposes as the direct sum of two irreducible representations,
which can be characterized as follows. Contrary to what happens for
$\mathbf{H}({\bf Q}_{p})$, there are two orbits of the maximal torus of
$\mathbf{H}^{+}({\bf Q}_{p})$ on the generic characters of
$U_{\mathbf{H}}({\bf Q}_{p})$. Let $\psi$, $\psi_{-}$ be representatives of
these two orbits, with $\psi$ equal to the character $\psi_{N}$ of [29, §6.2].
Then, $I_{\mathbf{H}^{+}}(\chi)$ decomposes as a direct sum

$I_{\mathbf{H}^{+}}(\chi)=\sigma_{+}\oplus\sigma_{-},$

with $\sigma_{+}$, resp. $\sigma_{-}$, generic with respect to $\psi$, resp.
$\psi_{-}$ (cf. [29, Lemma 4.2(3)]). In loc.cit. these components are
conveniently described as inductions of the Klingen parabolic of
$\mathbf{H}^{+}({\bf Q}_{p})$ as follows. If we let $Q_{\mathbf{H}}$ denote
standard Klingen parabolic, with Levi ${\rm GL}_{2}\times{\rm GL}_{1}$, then
its restriction $Q_{\mathbf{H}^{+}}$ to $\mathbf{H}^{+}({\bf Q}_{p})$ has Levi
${\rm GL}_{2}^{+}({\bf Q}_{p})\times{\bf Q}_{p}^{\times}$, where

${\rm GL}_{2}^{+}({\bf Q}_{p}):=\\{g\in{\rm GL}_{2}({\bf Q}_{p})\,:\,{\rm
det}(g)\in{\rm N}_{E_{p}/{\bf Q}_{p}}(E_{p}^{\times})\\}.$

Let $B_{2}$ denote the upper-triangular Borel of ${\rm GL}_{2}^{+}({\bf
Q}_{p})$ and denote by $\psi_{2}$, resp. $\psi_{2,-}$, the restriction of
$\psi$, resp. $\psi_{-}$, to the unipotent radical of $B_{2}$. Following [29,
§4.2] (or adapting Lemmas 3.1 and 3.2 to ${\rm GL}_{2}$), any irreducible
principal series of ${\rm GL}_{2}({\bf Q}_{p})$ whose restriction to ${\rm
GL}_{2}^{+}({\bf Q}_{p})$ is reducible is of the form $\varphi\cdot{\rm
Ind}_{B_{{\rm GL}_{2}}({\bf Q}_{p})}^{{\rm GL}_{2}({\bf
Q}_{p})}(\chi_{E_{p}/{\bf Q}_{p}},1)$, with $\varphi$ a character of ${\bf
Q}_{p}^{\times}$. Let ${\rm Ind}_{B_{{\rm GL}_{2}}({\bf Q}_{p})}^{{\rm
GL}_{2}({\bf Q}_{p})}(\chi_{E_{p}/{\bf Q}_{p}},1)^{+}$, resp. ${\rm
Ind}_{B_{{\rm GL}_{2}}({\bf Q}_{p})}^{{\rm GL}_{2}({\bf
Q}_{p})}(\chi_{E_{p}/{\bf Q}_{p}},1)^{-}$, be the irreducible ${\rm
GL}_{2}^{+}({\bf Q}_{p})$-constituent of ${\rm Ind}_{B_{{\rm GL}_{2}}({\bf
Q}_{p})}^{{\rm GL}_{2}({\bf Q}_{p})}(\chi_{E_{p}/{\bf Q}_{p}},1)$, which is
generic with respect to $\psi_{2}$, resp. $\psi_{2,-}$. Then, by induction in
stages we have that

$I_{\mathbf{H}}(\chi)={\rm Ind}_{Q_{\mathbf{H}}({\bf Q}_{p})}^{\mathbf{H}({\bf
Q}_{p})}(\chi_{1},\chi_{0}\cdot{\rm Ind}_{B_{{\rm GL}_{2}}({\bf Q}_{p})}^{{\rm
GL}_{2}({\bf Q}_{p})}(\chi_{2},1)),$

and, if $\chi_{2}=\chi_{E_{p}/{\bf Q}_{p}}$,
$I_{\mathbf{H}^{+}}(\chi)=\sigma_{+}\oplus\sigma_{-}$, with

$\displaystyle\sigma_{+}$ $\displaystyle={\rm
Ind}_{Q_{\mathbf{H}^{+}}}^{\mathbf{H}^{+}({\bf
Q}_{p})}(\chi_{1},\chi_{0}\cdot{\rm Ind}_{B_{{\rm GL}_{2}}({\bf Q}_{p})}^{{\rm
GL}_{2}({\bf Q}_{p})}(\chi_{E_{p}/{\bf Q}_{p}},1)^{+}),$
$\displaystyle\sigma_{-}$ $\displaystyle={\rm
Ind}_{Q_{\mathbf{H}^{+}}}^{\mathbf{H}^{+}({\bf
Q}_{p})}(\chi_{1},\chi_{0}\cdot{\rm Ind}_{B_{{\rm GL}_{2}}({\bf Q}_{p})}^{{\rm
GL}_{2}({\bf Q}_{p})}(\chi_{E_{p}/{\bf Q}_{p}},1)^{-}).$

###  3.2. Local theta correspondence for $(\mathbf{P}\mathbf{H}^{+}({\bf
Q}_{p}),$$\mathbf{P}\mathbf{G}({\bf Q}_{p}))$

By the exceptional isomorphism of Lemma 2.1, we consider the theta
correspondence studied in [29] as a correspondence for
$(\mathbf{P}\mathbf{H}^{+}({\bf Q}_{p}),\mathbf{P}\mathbf{G}({\bf Q}_{p}))$.
Let $\Omega$ be the induced Weil representation for this pair of groups (cf.
[29, p. 68]). For any irreducible admissible representation $\sigma$ of
$\mathbf{P}\mathbf{H}^{+}({\bf Q}_{p})$ (resp. of $\mathbf{P}\mathbf{G}({\bf
Q}_{p})$), the maximal $\sigma$-isotypic quotient of $\Omega$ has the form
$\sigma\boxtimes\Theta(\sigma)$, with $\Theta(\sigma)$ a smooth representation
of $\mathbf{P}\mathbf{G}({\bf Q}_{p})$ (resp. of
$\mathbf{P}\mathbf{H}^{+}({\bf Q}_{p})$). $\Theta(\sigma)$ is called the big
theta lift of $\sigma$. By [29, Theorem 6.21], the representation
$\Theta(\sigma)$ has a unique irreducible quotient $\theta(\sigma)$, which is
referred to as the small theta lift of $\sigma$, if it is non-zero. In ibid.
the small theta lift of principal series is calculated explicitly. Before
stating this result, we need the following lemma.

######  Lemma 3.3.

Let
$\xi^{\prime}=\xi^{\prime}_{1}\otimes\xi^{\prime}_{2}\otimes\xi^{\prime}_{0}$
be a character of $T_{\mathbf{P}\mathbf{G}^{\prime}}({\bf Q}_{p})$ defined by

$\xi^{\prime}:\left(\begin{smallmatrix}a&&&&\\\ &b&&&\\\ &&x&&\\\ &&&\mu
b^{-1}&\\\ &&&&\mu
a^{-1}\end{smallmatrix}\right)\mapsto\xi^{\prime}_{1}(a)\xi^{\prime}_{2}(b)\xi^{\prime}_{0}(x),\;\forall
a,b\in{\bf Q}_{p}^{\times},x\in E_{p}^{\times}\,(\text{and }\mu=N_{E_{p}/{\bf
Q}_{p}}(x)).$

The character $\xi:=\xi^{\prime}\circ j$ of $T_{\mathbf{P}\mathbf{G}}({\bf
Q}_{p})$ is given by $\left(\begin{smallmatrix}a&&&\\\ &b&&\\\
&&\nu\bar{b}^{-1}&\\\
&&&\nu\bar{a}^{-1}\end{smallmatrix}\right)\mapsto\xi_{1}(a)\xi_{2}(b)\xi_{0}(\nu)$,
where

$\displaystyle\xi_{1}(a)$
$\displaystyle=(\xi^{\prime}_{1}\xi^{\prime}_{2}\circ N_{E_{p}/{\bf
Q}_{p}})(a)\xi^{\prime}_{0}(\bar{a}),$ $\displaystyle\xi_{2}(b)$
$\displaystyle=\xi^{\prime}_{1}\circ N_{E_{p}/{\bf
Q}_{p}}(b)\xi^{\prime}_{0}(b),$ $\displaystyle\xi_{0}(\nu)$
$\displaystyle=\xi^{\prime}_{2}\xi^{\prime}_{0}(\nu).$

###### Proof.

Recall that, by Lemma 2.1, we have that

$j:\left(\begin{smallmatrix}a&&&\\\ &b&&\\\ &&\nu\bar{b}^{-1}&\\\
&&&\nu\bar{a}^{-1}\end{smallmatrix}\right)\mapsto\left(\begin{smallmatrix}N_{E_{p}/{\bf
Q}_{p}}(ab)&&&&\\\ &\nu N_{E_{p}/{\bf Q}_{p}}(a)&&&\\\ &&\nu\bar{a}b&&\\\
&&&\nu N_{E_{p}/{\bf Q}_{p}}(b)&\\\ &&&&\nu^{2}\end{smallmatrix}\right),$

with $a,b\in E_{p}^{\times}$ and $\nu\in{\bf Q}_{p}^{\times}$. Thus,
$\xi:=\xi^{\prime}\circ j$ is explicitly given by

$\xi:\left(\begin{smallmatrix}a&&&\\\ &b&&\\\ &&\nu\bar{b}^{-1}&\\\
&&&\nu\bar{a}^{-1}\end{smallmatrix}\right)\mapsto\xi^{\prime}_{1}(N_{E_{p}/{\bf
Q}_{p}}(ab))\xi^{\prime}_{2}(\nu N_{E_{p}/{\bf
Q}_{p}}(a))\xi^{\prime}_{0}(\nu\bar{a}b).$

Regrouping the terms of the latter as

$\xi^{\prime}_{1}(N_{E_{p}/{\bf Q}_{p}}(a))\xi^{\prime}_{2}(N_{E_{p}/{\bf
Q}_{p}}(a))\xi^{\prime}_{0}(\bar{a})\xi^{\prime}_{1}(N_{E_{p}/{\bf
Q}_{p}}(b))\xi^{\prime}_{0}(b)\xi^{\prime}_{2}(\nu)\xi^{\prime}_{0}(\nu),$

we get the result. ∎

######  Proposition 3.4.

Let $\sigma$ be an irreducible $\mathbf{H}^{+}({\bf Q}_{p})$-factor of an
irreducible unramified principal series $I_{\mathbf{H}}(\chi)$ of
$\mathbf{P}\mathbf{H}({\bf Q}_{p})$. Then, we have the following:

  1. (1)

If $I_{\mathbf{H}^{+}}(\chi)$ is irreducible, then
$\Theta(\sigma)=I_{\mathbf{G}}(\xi)$, with

$\xi_{1}=\chi_{0}\circ N_{E_{p}/{\bf Q}_{p}},\,\xi_{2}=(\chi_{2}\chi_{0})\circ
N_{E_{p}/{\bf Q}_{p}},\,\xi_{0}=\chi_{1}\chi_{E_{p}/{\bf Q}_{p}}.$

Moreover, $\theta(\sigma)$ is the unique irreducible unramified quotient of
$I_{\mathbf{G}}(\xi)$ and it equals to $\Theta(\sigma)$ if it is generic.

  2. (2)

If $I_{\mathbf{H}^{+}}(\chi)$ is reducible then, up to translating by the Weyl
action, we fall in one of the following two cases:

    1. (a)

suppose that $\chi_{2}=\chi_{E_{p}/{\bf Q}_{p}}$:

      1. (i)

If $\sigma$ is generic with respect to $\psi$, then
$\Theta(\sigma)=I_{\mathbf{G}}(\xi)$, with

$\xi_{1}=\xi_{2}=\chi_{0}\circ N_{E_{p}/{\bf
Q}_{p}},\,\xi_{0}=\chi_{1}\chi_{E_{p}/{\bf Q}_{p}}.$

Moreover, $\theta(\sigma)$ is the unique irreducible unramified quotient of
$I_{\mathbf{G}}(\xi)$ and it equals to $\Theta(\sigma)$ if it is generic.

      2. (ii)

If $\sigma$ is generic w.r.t. $\psi_{-}$ (but not w.r.t. $\psi$), then
$\theta(\sigma)=0$.

    2. (b)

suppose that $\chi_{1}=\chi_{E_{p}/{\bf Q}_{p}}$ and $\chi_{2}=\mathbf{1}$:
then

$\Theta(\sigma)=\theta(\sigma)=\begin{cases}I_{\mathbf{G}}(\xi)_{\rm
gen}&\text{ if }\sigma\text{ is generic w.r.t. }\psi,\\\
I_{\mathbf{G}}(\xi)_{\rm ng}&\text{ if }\sigma\text{ is generic w.r.t.
}\psi_{-},\end{cases}$

where $\xi_{1}=\xi_{2}=\chi_{0}\circ N_{E_{p}/{\bf
Q}_{p}},\,\xi_{0}=\mathbf{1},$ and $I_{\mathbf{G}}(\xi)_{\rm gen}$ (resp.
$I_{\mathbf{G}}(\xi)_{\rm ng}$) is the generic (resp. non-generic) constituent
of $I_{\mathbf{G}}(\xi)$. In particular, $\theta(\sigma)$ is not unramified
when $\sigma$ is generic w.r.t. $\psi$.

###### Proof.

The theta lift of $\sigma$ is calculated explicitly in [29, Theorem 6.21(3)].
According to the classification of the non-discrete series of
$\mathbf{H}^{+}({\bf Q}_{p})$ given in [29, Lemma 4.2], $\sigma$ falls in one
of the cases (3a), (3b), or (3d) (the remaining case (3c) is excluded as we
are supposing $I_{\mathbf{H}}(\chi)$ to be irreducible). Let’s treat the cases
separately.

  * (3d)

Let $\sigma=I_{\mathbf{H}^{+}}(\chi)$, i.e. $I_{\mathbf{H}^{+}}(\chi)$ is
irreducible. It is generic with respect to both $\psi$ and $\psi_{-}$. By the
last case of [29, Theorem 6.21(3)], $\theta(\sigma)$ is the unique irreducible
unramified quotient of

$\mathrm{Ind}^{\mathbf{G}^{\prime}({\bf Q}_{p})}_{B_{\mathbf{G}^{\prime}}({\bf
Q}_{p})}\left(\chi_{1}^{-1}\chi_{E_{p}/{\bf
Q}_{p}},\chi_{2}^{-1}\chi_{E_{p}/{\bf Q}_{p}},(\chi_{1}\chi_{2}\chi_{0})\circ
N_{E_{p}/{\bf Q}_{p}}\right).$

Using the exceptional isomorphism $j:\mathbf{P}\mathbf{G}({\bf
Q}_{p})\simeq\mathbf{P}\mathbf{G}^{\prime}({\bf Q}_{p})$ and Lemma 3.3, this
principal series can be written as $I_{\mathbf{G}}(\xi)$, with

$\displaystyle\xi_{1}$ $\displaystyle=\chi_{0}\circ N_{E_{p}/{\bf Q}_{p}},$
$\displaystyle\xi_{2}$ $\displaystyle=(\chi_{2}\chi_{0})\circ N_{E_{p}/{\bf
Q}_{p}},$ $\displaystyle\xi_{0}$ $\displaystyle=\chi_{1}\chi_{E_{p}/{\bf
Q}_{p}}.$

Moreover, if we assume that $\theta(\sigma)$ is generic, by [27, Theorem 2.7],
$\Theta(\sigma)=I_{\mathbf{G}}(\xi)$ is irreducible (and hence equal to
$\theta(\sigma)$).

  * (3a)

Suppose that $I_{\mathbf{H}^{+}}(\chi)=\sigma_{+}\oplus\sigma_{-}$, with
$\chi_{2}=\chi_{E_{p}/{\bf Q}_{p}}$. By the case of [29, Lemma 4.2 3(a)] in
[29, Theorem 6.21(3)], $\theta(\sigma_{-})=0$, while $\theta(\sigma_{+})$ is
the unique irreducible unramified quotient of $I_{\mathbf{G}}(\xi)$, with
$\xi=(\chi_{0}\circ N_{E_{p}/{\bf Q}_{p}},\chi_{0}\circ N_{E_{p}/{\bf
Q}_{p}},\chi_{1}\chi_{E_{p}/{\bf Q}_{p}})$. As in the previous case, if
$\theta(\sigma_{+})$ is generic then [27, Theorem 2.7] implies that
$\Theta(\sigma_{+})=I_{\mathbf{G}}(\xi)$ is irreducible and so equal to
$\theta(\sigma_{+})$.

  * (3b)

Suppose that $I_{\mathbf{H}^{+}}(\chi)=\sigma_{+}\oplus\sigma_{-}$, with
$\chi_{1}=\chi_{E_{p}/{\bf Q}_{p}}$ and $\chi_{2}=\mathbf{1}$. By the case of
[29, Lemma 4.2 3(b)] in [29, Theorem 6.21(3)], we have that, if
$\xi_{1}=\xi_{2}=\chi_{0}\circ N_{E_{p}/{\bf Q}_{p}},\,\xi_{0}=\mathbf{1},$

$I_{\mathbf{G}}(\xi)=I_{\mathbf{G}}(\xi)_{\rm gen}\oplus
I_{\mathbf{G}}(\xi)_{\rm ng}$ (5)

with $I_{\mathbf{G}}(\xi)_{\rm gen}$ (resp. $I_{\mathbf{G}}(\xi)_{\rm ng}$)
denoting the generic (resp. non-generic) constituent of $I_{\mathbf{G}}(\xi)$,
and

$\displaystyle\Theta(\sigma_{+})$
$\displaystyle=\theta(\sigma_{+})=I_{\mathbf{G}}(\xi)_{\rm gen},$
$\displaystyle\Theta(\sigma_{-})$
$\displaystyle=\theta(\sigma_{-})=I_{\mathbf{G}}(\xi)_{\rm ng}.$

Notice that $I_{\mathbf{G}}(\xi)_{\rm gen}$ is not unramified as, by [27,
Theorem 2.7], it would contradict (5) otherwise.

∎

In the next section, we relate the uniqueness of the Shalika functional for
$\theta(\sigma)$ to the irreducibility of the the big theta lift
$\Theta(\theta(\sigma))$, which is calculated explicitly in the proof of [29,
Theorem 6.21]. In the following Lemma, we summarize the result for the cases
relevant to us.

######  Lemma 3.5.

Let $\sigma$ be the irreducible $\psi$-generic factor of an irreducible
unramified principal series $I_{\mathbf{H}}(\chi)$ of $\mathbf{H}({\bf
Q}_{p})$, with trivial central character. Suppose that $\theta(\sigma)$ is
generic and unramified. Then $\Theta(\theta(\sigma))=\sigma$.

###### Proof.

According to the classification of the irreducible non-discrete series
representations given in [29, Lemma 4.2], Proposition 3.4 described the local
theta correspondence from $\mathbf{P}\mathbf{H}({\bf Q}_{p})$ to
$\mathbf{P}\mathbf{G}^{\prime}({\bf Q}_{p})$ for representations falling in
cases $3(a)$, $3(b)$ and $3(d)$ of loc.cit, with case $3(c)$ excluded because
of the irreducibility of $I_{\mathbf{H}}(\chi)$. By the hypothesis we impose
here on $\theta(\sigma)$, the Proposition above rules out the case $3(b)$ and
shows that $\theta(\sigma)$ is an irreducible principal series in the
remaining two cases. Before we go on, notice the following.

  1. (1)

Via the exceptional isomorphism

${{\rm GSO}}_{3,1}({\bf Q}_{p})\simeq\left({\rm GL}_{2}(E_{p})\times{\bf
Q}_{p}^{\times}\right)/\\{(a^{-1},N_{E_{p}/{\bf Q}_{p}}(a))\,:\,a\in
E_{p}^{\times}\\},$

an irreducible representation of ${{\rm GSO}}_{3,1}({\bf Q}_{p})$ can be
denoted by $\pi(\rho,\varphi)$ for an irreducible representation $\rho$ of
${\rm GL}_{2}(E_{p})$ and a character $\chi$ of ${\bf Q}_{p}^{\times}$ such
that the central character of $\rho$ equals to $\varphi\circ N_{E_{p}/{\bf
Q}_{p}}$. Then, if $\tau$ and $\varphi$ are characters of $E_{p}^{\times}$ and
${\bf Q}_{p}^{\times}$ respectively, (cf. [29, p. 51]) shows the equality

$\pi\left({\rm Ind}_{B_{{\rm GL}_{2}}(E_{p})}^{{\rm
GL}_{2}(E_{p})}\left(\tau,\tau^{\sigma}\cdot(\varphi\circ N_{E_{p}/{\bf
Q}_{p}})\right),\varphi\cdot\tau_{|_{{\bf Q}_{p}^{\times}}}\right)={\rm
Ind}_{B_{{{\rm GSO}}_{3,1}}({\bf Q}_{p})}^{{{\rm GSO}}_{3,1}({\bf
Q}_{p})}(\varphi,\tau),$

with $\tau^{\sigma}(a)=\tau(\bar{a})$.

  2. (2)

If we let $P_{\mathbf{G}^{\prime}}$ be the standard parabolic of
$\mathbf{G}^{\prime}$ with Levi ${{\rm GSO}}_{3,1}\times{\rm GL}_{1}$,
combining (1) with [29, Theorem 6.21], we have

$\displaystyle\theta(\sigma)=\mathrm{Ind}_{P_{\mathbf{G}^{\prime}}({\bf
Q}_{p})}^{{\mathbf{G}^{\prime}}({\bf
Q}_{p})}\left(\chi_{1}^{-1}\chi_{E_{p}/{\bf
Q}_{p}},\pi\left(((\chi_{1}\chi_{2}\chi_{0})\circ N_{E_{p}/{\bf
Q}_{p}})\cdot\mathrm{Ind}_{B_{{\rm GL}_{2}}(E_{p})}^{{\rm
GL}_{2}(E_{p})}(1,\chi_{2}^{-1}\circ N_{E_{p}/{\bf
Q}_{p}}),\chi_{1}\chi_{E_{p}/{\bf Q}_{p}}\right)\right).$ (6)

Indeed, this follows by doing induction in stages and by the fact that
$\theta(\sigma)$ is equal to

$\mathrm{Ind}^{\mathbf{G}^{\prime}({\bf Q}_{p})}_{B_{\mathbf{G}^{\prime}}({\bf
Q}_{p})}\left(\chi_{1}^{-1}\chi_{E_{p}/{\bf
Q}_{p}},\chi_{2}^{-1}\chi_{E_{p}/{\bf Q}_{p}},(\chi_{1}\chi_{2}\chi_{0})\circ
N_{E_{p}/{\bf Q}_{p}}\right).$

  3. (3)

The representation $\pi\left(((\chi_{1}\chi_{2}\chi_{0})\circ N_{E_{p}/{\bf
Q}_{p}})\cdot\mathrm{Ind}_{B_{{\rm GL}_{2}}(E_{p})}^{{\rm
GL}_{2}(E_{p})}(1,\chi_{2}^{-1}\circ N_{E_{p}/{\bf
Q}_{p}}),\chi_{1}\chi_{E_{p}/{\bf Q}_{p}}\right)$, appearing in (6), is in the
image of the theta lift from ${\rm GL}_{2}^{+}({\bf Q}_{p})$. Indeed, by [29,
Corollary 6.11 & Lemma 6.12], it is equal to $\theta(\sigma_{0}^{+})$ with
$\sigma_{0}^{+}$ the $\psi_{2}$-generic irreducible component of

$\sigma_{0}={\rm Ind}_{B_{{\rm GL}_{2}}({\bf Q}_{p})}^{{\rm GL}_{2}({\bf
Q}_{p})}(\chi_{1}\chi_{2}\chi_{0},\chi_{1}\chi_{0}).$

We can now calculate the big theta lift of $\theta(\sigma)$. Using [29, p. 75
(1)], we get

$\Theta(\theta(\sigma))^{*}=\mathrm{Hom}_{\mathbf{G}^{\prime}({\bf
Q}_{p})}\left(\Omega,\theta(\sigma)\right),$

and the latter space has been studied in [29, Propositions 6.19 & 6.20] and
calculated explicitly in the proof of Proposition 6.21 of loc.cit. for the
cases of interest here, i.e. 3$(a)$ and 3$(d)$ (see p. 80 of [29]). For the
convenience of the reader, we sketch the calculation. Suppose that both
$\chi_{1}=\chi_{2}=1$. Then, because of the trivial central character
condition, $\chi_{0}$ is quadratic. As, by hypothesis, it is also unramified
$\chi_{0}$ is forcely $\chi_{E_{p}/{\bf Q}_{p}}$. In this case, we now use
[29, Proposition 6.20] to deduce the result. Firstly, if
$Q_{\mathbf{G}^{\prime}}$ is the standard parabolic of $\mathbf{G}^{\prime}$
with Levi ${\rm GL}_{2}\times{\rm Res}_{E/{\bf Q}}({\rm GL}_{1})$, we can
write $\theta(\sigma)$ as

$\displaystyle\mathrm{Ind}_{Q_{\mathbf{G}^{\prime}}({\bf
Q}_{p})}^{{\mathbf{G}^{\prime}}({\bf Q}_{p})}\left(\chi_{E_{p}/{\bf
Q}_{p}}\cdot\mathrm{Ind}_{B_{{\rm GL}_{2}}({\bf Q}_{p})}^{{\rm GL}_{2}({\bf
Q}_{p})}(1,1),1\right).$ (7)

Using (7) and [29, Proposition 6.20], we get an injection
$\Theta(\theta(\sigma))^{*}\hookrightarrow\sigma^{*}$, which induces an
isomorphism on the contragradients
$\Theta(\theta(\sigma))^{\vee}\simeq\sigma^{\vee}$ as $\sigma$ is irreducible.
Suppose that either $\chi_{1}$ or $\chi_{2}$ is not trivial. Without loss of
generality, up to using the Weyl action, we assume $\chi_{1}\neq 1$. By [29,
Proposition 6.19(1)] and point (3) above, we have that

$\Theta(\theta(\sigma))={\rm Ind}_{Q_{\mathbf{H}^{+}}}^{\mathbf{H}^{+}({\bf
Q}_{p})}(\chi_{1},\chi_{0}\cdot{\rm Ind}_{B_{{\rm GL}_{2}}({\bf Q}_{p})}^{{\rm
GL}_{2}({\bf Q}_{p})}(\chi_{2},1)^{+}),$

which is equal to $\sigma$ as explained in §3.1. ∎

######  Remark 3.6.

Note that if we relax the hypotheses of Lemma 3.5 by allowing either $\sigma$
or $\theta(\sigma)$ to be non-generic, then $\Theta(\theta(\sigma))$ is not
necessarily $\sigma$.

###  3.3. Shalika functionals

We now study the local multiplicity of Shalika models for generic unramified
representations in the image of the theta correspondence. We are thankful to
Wee Teck Gan for explaining to us how to relate the uniqueness of Shalika
functionals to the one of Whittaker models by means of the local theta
correspondence.

######  Proposition 3.7.

Let $\sigma$ and $\Pi$ be irreducible admissible representations of
$\mathbf{P}\mathbf{H}^{+}({\bf Q}_{p})$ and $\mathbf{P}\mathbf{G}({\bf
Q}_{p})$, for which

${\rm Hom}_{\mathbf{P}\mathbf{H}^{+}({\bf
Q}_{p})\times\mathbf{P}\mathbf{G}({\bf Q}_{p})}(\Omega,\sigma\boxtimes\Pi)\neq
0.$

Then we have the following inclusion

${\rm Hom}_{U_{\mathbf{H}}({\bf Q}_{p})}(\sigma,\psi)\subseteq{\rm
Hom}_{U_{\mathbf{H}}({\bf Q}_{p})}(\Theta(\Pi),\psi)\simeq{\rm Hom}_{S({\bf
Q}_{p})}(\Pi,\chi_{\delta,p}).$

In particular, if $\Theta(\Pi)$ is irreducible

${\rm dim}\,{\rm Hom}_{S({\bf Q}_{p})}(\Pi,\chi_{\delta,p})\leq 1.$

###### Proof.

As $\Theta(\Pi)\twoheadrightarrow\sigma$, we have the first inclusion.
Moreover,

${\rm Hom}_{U_{\mathbf{H}}({\bf Q}_{p})}(\Theta(\Pi),\psi)\simeq{\rm
Hom}_{U_{\mathbf{H}}({\bf Q}_{p})\times\mathbf{P}\mathbf{G}({\bf
Q}_{p})}(\Omega,\psi\boxtimes\Pi)\simeq{\rm Hom}_{\mathbf{P}\mathbf{G}({\bf
Q}_{p})}(\Omega_{\psi},\Pi),$

where $\Omega_{\psi}$ denotes the twisted Jacquet module of $\Omega$ with
respect to the generic datum $(U_{\mathbf{H}}({\bf Q}_{p}),\psi)$ of
$\mathbf{P}\mathbf{H}^{+}({\bf Q}_{p})$. By [29, Proposition 6.6(2)],
$\Omega_{\psi}$ is isomorphic to ${\rm ind}_{S({\bf Q}_{p})}^{\mathbf{G}({\bf
Q}_{p})}(\chi_{\delta,p})$ as a $\mathbf{G}({\bf Q}_{p})$-representation222By
the exceptional isomorphism introduced above, the Shalika subgroup of
$\mathbf{P}\mathbf{G}^{\prime}({\bf Q}_{p})$ and its character $\psi^{\prime}$
of Definition [29, Definition 6.5] agree with $S({\bf Q}_{p})$ and
$\chi_{\delta,p}$ (cf. [29, (3.5)])., hence

${\rm Hom}_{\mathbf{P}\mathbf{G}({\bf Q}_{p})}(\Omega_{\psi},\Pi)\simeq{\rm
Hom}_{\mathbf{P}\mathbf{G}({\bf Q}_{p})}({\rm ind}_{S({\bf
Q}_{p})}^{\mathbf{G}({\bf Q}_{p})}(\chi_{\delta,p}),\Pi)\simeq{\rm
Hom}_{S({\bf Q}_{p})}(\chi_{\delta,p},\Pi)\simeq{\rm Hom}_{S({\bf
Q}_{p})}(\Pi^{\vee},\chi_{\delta,p}^{-1}).$

It remains to show that

${\rm Hom}_{S({\bf Q}_{p})}(\Pi,\chi_{\delta,p})\simeq{\rm Hom}_{S({\bf
Q}_{p})}(\Pi^{\vee},\chi_{\delta,p}^{-1}).$

This fact, which is already implicit in [29, Theorem 6.9], can be proved as
follows. Moeglin, Vigneras, and Waldspurger in [28, Ch. 4] show that, if $V$
is the $E_{p}$-module with Hermitian form $\langle\,,\,\rangle_{V}$ that
defines $\mathbf{P}\mathbf{G}({\bf Q}_{p})$, there exists an element $h\in{\rm
GL}_{{\bf Q}_{p}}(V)$ such that $\langle hx,hy\rangle_{V}=\langle
y,x\rangle_{V}$ and such that $\overline{g}=hgh^{-1}$. Let $\Pi^{\dagger}$ be
the representation of $\mathbf{G}({\bf Q}_{p})$ given by the same space as
$\Pi$ and with action $\Pi^{\dagger}(x)=\Pi(hxh^{-1})$. Then a linear form
$\Lambda:\Pi\to{\bf C}$ such that for $s\in S({\bf Q}_{p})$ and $v$ in the
space of $\Pi$

$\Lambda(\Pi(s)v)=\chi_{\delta,p}(s)\Lambda(v),$

induces a linear form on $\Pi^{\dagger}$ such that

$\Lambda(\Pi^{\dagger}(s)v)=\chi_{\delta,p}(hsh^{-1})\Lambda(v)=\chi_{\delta,p}^{-1}(s)\Lambda(v).$

This identifies

${\rm Hom}_{S({\bf Q}_{p})}(\Pi,\chi_{\delta,p})\simeq{\rm Hom}_{S({\bf
Q}_{p})}(\Pi^{\dagger},\chi_{\delta,p}^{-1}).$

The desired isomorphism follows from the fact that
$\Pi^{\vee}\simeq\Pi^{\dagger}$ (cf. [28, Ch. 4, §II.1]). Finally, when
$\Theta(\Pi)$ is irreducible, $\Theta(\Pi)=\sigma$ and the upper bound on the
dimension of the space of Shalika functionals follows from the uniqueness of
Whittaker models. ∎

######  Theorem 3.8.

Let $\sigma$ and $\Pi$ be irreducible generic unramified representations of
$\mathbf{P}\mathbf{H}^{+}({\bf Q}_{p})$ and $\mathbf{P}\mathbf{G}({\bf
Q}_{p})$, for which $\Pi=\theta(\sigma)$. Then

${\rm Hom}_{U_{\mathbf{H}}({\bf Q}_{p})}(\sigma,\psi)\simeq{\rm Hom}_{S({\bf
Q}_{p})}(\Pi,\chi_{\delta,p}).$

In particular, $\Pi$ admits a unique non-trivial Shalika functional up to
constant.

###### Proof.

It follows from Proposition 3.7 and Lemma 3.5. ∎

##  4\. A Casselman-Shalika formula for the Shalika model of ${\rm GU}_{2,2}$

In what follows, we describe the main technical ingredients used in the
calculation of the unramified local zeta integrals. We first recall the
Casselman-Shalika formula given in [33] for the Shalika model of unramified
representations of ${\rm GL}_{4}({\bf Q}_{p})$. Then we present a similar
formula for unramified representations on ${\rm GU}_{2,2}({\bf Q}_{p})$, with
$p$ inert in $E$, which lie in the image of the local theta correspondence.
The proof of the latter, which is the first main result of the article, is
based on the methods of [33] and will be described in §5.

###  4.1. The split case

Let $\Pi$ be an irreducible unramified smooth representation of
$\mathbf{G}({\bf Q}_{p})$ with trivial central character. Recall that, when
$p$ is split in $E$, $\mathbf{G}({\bf Q}_{p})$ is isomorphic to
$\mathbf{GL}_{4}({\bf Q}_{p})\times\mathbf{GL}_{1}({\bf Q}_{p})$ via (3) and
we can identify $\Pi$ with an unramified representation $\pi$ of ${\rm
GL}_{4}({\bf Q}_{p})$ with trivial central character (cf. §2.3.2). We
conveniently compose the isomorphism (3) with conjugation by the diagonal
matrix $w=({\rm diag}(1,1,-1,1),-1)\in{\rm GL}_{4}({\bf Q}_{p})\times{\rm
GL}_{1}({\bf Q}_{p})$. This has the effect of identifying the subgroups

${\rm GL}_{2}({\bf Q}_{p})=\left\\{\left(\begin{smallmatrix}g&\\\ &{\rm
det}(g)I_{2}^{\prime}{}^{t}g^{-1}I_{2}^{\prime}\end{smallmatrix}\right)\right\\}\simeq\left\\{\left(\left(\begin{smallmatrix}g&\\\
&g\end{smallmatrix}\right),{\rm det}(g)\right)\right\\},$ $N_{\mathbf{G}}({\bf
Q}_{p})\simeq\left\\{u(X)=\left(\begin{smallmatrix}I&X\\\
&I\end{smallmatrix}\right),X\in M_{2\times 2}({\bf Q}_{p})\right\\},$

such that, if $\eta$ is mapped to $-1$ via
$\iota_{\mathfrak{p}}:E_{\mathfrak{p}}\simeq{\bf Q}_{p}$, the character
$\chi_{\eta,p}:N_{\mathbf{G}}({\bf Q}_{p})\to{\bf C}^{\times}$ corresponds to
the character

$u(X)\mapsto\psi_{p}({\rm Tr}(X)).$

######  Lemma 4.1.

Having a Shalika functional $\Lambda_{p,\mathcal{S}}:\Pi\to{\bf C}$ is
equivalent to having a functional $\Lambda_{p,\mathcal{S}}^{\rm
split}:\pi\to{\bf C}$, such that, for all $v\in\pi$, we have

$\Lambda_{p,\mathcal{S}}^{\rm split}(\left(\begin{smallmatrix}g&\\\
&g\end{smallmatrix}\right)u(X)\cdot v)=\psi_{p}({\rm
Tr}(X))\Lambda_{p,\mathcal{S}}^{\rm split}(v),\,\forall g\in{\rm GL}_{2}({\bf
Q}_{p}),\forall X\in M_{2\times 2}({\bf Q}_{p}).$

###### Proof.

This follows immediately from the identifications above. ∎

The functional of Lemma 4.1 defines a _classical_ Shalika functional for
$\pi$. Suppose that the representation $\pi$ appears as the unique unramified
irreducible factor of a principal series $I_{{\rm GL}_{4}}(\chi)$, where
$\chi=(\chi_{1},\chi_{2},\chi_{3},\chi_{4})$ is an unramified character of the
diagonal torus of ${\rm GL}_{4}({\bf Q}_{p})$ such that
$\prod_{i}\chi_{i}=\mathbf{1}$.

By Proposition 2.10, there exists at most one non-trivial Shalika functional
$\Lambda_{p,\mathcal{S}}^{\rm split}$ on $\pi$, up to constant. By [33], there
exists a non-trivial Shalika functional if and only if $\pi=I_{{\rm
GL}_{4}}(\chi)$ and $\chi$ is of the form (or a Weyl translate of)
$(\chi_{1},\chi_{2},\chi_{2}^{-1},\chi_{1}^{-1})$ (cf. [33, Corollary 5.4] and
the discussion at the end of §8 of _loc.cit._). Therefore, we can assume that
the Satake parameters $g_{\chi}\in{\rm Sp}_{4}({\bf C})$ (cf. [33, §5]). Under
this assumption, the Shalika functional $\Lambda_{p,\mathcal{S}}^{\rm split}$
is uniquely defined up to a scaling factor. We therefore normalise it
according to [33, Theorem 2.1]. Before explaining this, we introduce some
extra notation.

We let $\Gamma$, resp. $W$, denote the Weyl groups $W_{{\rm Sp}_{4}}$,
$W_{{\rm GL}_{4}}$ of ${\rm Sp}_{4}({\bf Q}_{p})$, ${\rm GL}_{4}({\bf Q}_{p})$
respectively; we also let $\Phi_{{\rm GL}_{4}}$, $\Phi_{{\rm Sp}_{4}}$ denote
the root systems for ${\rm GL}_{4}$ and ${\rm Sp}_{4}$. The embedding ${\rm
Sp}_{4}\hookrightarrow{\rm GL}_{4}$ identifies $\Gamma$ as a subgroup of the
Weyl group of ${\rm GL}_{4}$ and induces a surjection

$\Phi_{{\rm GL}_{4}}\twoheadrightarrow\Phi_{{\rm Sp}_{4}},$

which is one-to-one onto the set of long roots $\Phi_{{\rm Sp}_{4}}^{l}$ and
two-to-one on the set of short roots $\Phi_{{\rm Sp}_{4}}^{s}$. For each
$\alpha\in\Phi_{{\rm Sp}_{4}}$, we denote by $\alpha$, $\tilde{\alpha}$ the
(possibly equal) elements in the pre-image of it. Finally, a superscript $+$
over $\Phi$ will denote positive roots.

For each root $\alpha$, $\check{\alpha}$ denotes the corresponding coroot.
Then, the notation $e^{\check{\alpha}}(g_{\chi})$ stands for
$\chi(g_{\alpha})$, where, if $\alpha:{\rm
diag}(t_{1},t_{2},t_{3},t_{4})\mapsto t_{i}t_{j}^{-1}$, $g_{\alpha}$ is equal
to the diagonal matrix with $p$ on the $i$-th line, $p^{-1}$ on the $j$-th
line, and $1$’s otherwise. We can finally give the following.

######  Definition 4.2.

We define

$\mathcal{S}_{p}^{\rm split}(g):=\Lambda_{p,\mathcal{S}}^{\rm
split}(g\cdot\phi_{0}),$

where $\phi_{0}$ is the spherical vector of $\pi$ such that $\phi_{0}(1)=1$
(with $\phi_{0}$ seen as an element of $I_{{\rm GL}_{4}}(\chi)$), and
$\Lambda_{p,\mathcal{S}}^{\rm split}(\cdot)$ is the Shalika functional such
that

$\Lambda_{p,\mathcal{S}}^{\rm
split}(\phi_{0})=\frac{\mathcal{Q}}{1+p^{-1}}\cdot\frac{e^{-\check{\rho}}\prod_{\beta\in\Phi_{{\rm
GL}_{4}}^{+}}(1-e^{-\check{\beta}})}{\prod_{\alpha\in\Phi_{{\rm
Sp}_{4}}^{+}}(1-p^{-1}e^{-\check{\alpha}})\mathcal{A}\left(e^{\check{\rho}}\right)}(g_{\chi}),$

where $\mathcal{Q}=\sum_{\omega\in W}({\rm Iw}_{{\rm GL}_{4}}\omega{\rm
Iw}_{{\rm GL}_{4}}:{\rm Iw}_{{\rm GL}_{4}})^{-1}$,
$\check{\rho}=\tfrac{1}{2}\sum_{\alpha\in\Phi_{{\rm
Sp}_{4}}^{+}}\check{\alpha}$, and $\mathcal{A}$ denotes the alternator

$\mathcal{A}(\cdot)=\sum_{\omega\in\Gamma}(-1)^{\ell(\omega)}\omega(\cdot).$

######  Remark 4.3.

As explained in the proof of [33, Theorem 2.1], the factor
${\prod_{\alpha\in\Phi_{{\rm
Sp}_{4}}^{+}}(1-p^{-1}e^{-\check{\alpha}})}(g_{\chi})$ appearing in the
denominator of the formula above would be zero if $I_{{\rm GL}_{4}}(\chi)$ was
reducible.

The following is a special case of [33, Theorem 2.1] \- the statement in
_loc.cit._ contains a small typo, however we invite the reader to consult [33,
(78)] where the correct formula is given.

######  Theorem 4.4.

For all $n\geq 0$, we have

$\mathcal{S}_{p}^{\rm split}\left(\left(\begin{smallmatrix}p^{n}I_{2}&\\\
&I_{2}\end{smallmatrix}\right)\right)=\frac{p^{-2n}}{1+p^{-1}}\cdot\mathcal{A}\Big{(}e^{\check{\rho}+n(\check{\alpha}_{1}+\check{\alpha}_{2})}\cdot\prod_{\alpha\in\Phi_{{\rm
Sp}_{4}}^{+,s}}(1-p^{-1}e^{-\check{\alpha}})\Big{)}(g_{\chi})({\mathcal{A}\left(e^{\check{\rho}}\right)}(g_{\chi}))^{-1}.$

We’d like to note that a similar formula was already proved in [4], as, via
the isomorphism between $\mathrm{PGL}_{4}$ and $\mathrm{PGSO}_{3,3}$, Shalika
models coincide with the “Whittaker-orthogonal” models considered by [4].

###  4.2. The inert case

Let $p$ be a prime inert over $E$ and let $\Pi=I_{\mathbf{G}}(\xi)$ be an
irreducible generic unramified representation of $\mathbf{G}({\bf Q}_{p})$
with trivial central character. We suppose that $\Pi$ is the local theta lift
from $\mathbf{H}^{+}({\bf Q}_{p})$ of the generic component of
$I_{\mathbf{H}^{+}}(\chi)$, such that $\chi_{1}\chi_{2}\chi_{0}^{2}=1$ and

$\displaystyle|\cdot|^{\pm
1}\not\in\\{\chi_{1},\chi_{2},\chi_{1}\chi_{2},\chi_{1}\chi_{2}^{-1}\\}.$ (8)

By Proposition 3.4, we have

$\displaystyle\xi=(\chi_{0}\circ N_{E_{p}/{\bf Q}_{p}},\chi_{2}\chi_{0}\circ
N_{E_{p}/{\bf Q}_{p}},\chi_{1}\chi_{E_{p}/{\bf Q}_{p}}).$ (9)

Theorem 3.8 asserts that there exists a unique Shalika functional
$\Lambda_{p,\mathcal{S}}$ for $\Pi$ up to constant. Let $\phi_{0}$ be the
spherical vector of $\Pi$ such that $\phi_{0}(1)=1$. The goal of this
subsection is to give an explicit expression for
$\mathcal{S}_{p}\left(\left(\begin{smallmatrix}p^{n}I_{2}&\\\
&I_{2}\end{smallmatrix}\right)\right)$, where we denote

$\mathcal{S}_{p}(g):=\Lambda_{p,\mathcal{S}}(g\cdot\phi_{0}).$

Recall that the relative root system of $\mathbf{P}\mathbf{G}_{/{\bf Q}_{p}}$
is equal to the root system of $\mathbf{P}\mathbf{H}_{/{\bf Q}_{p}}$, allowing
us to express the resulting formula in terms of the root system of ${\rm
Sp}_{4}$. This, together with the equality (9) given by the theta
correspondence, let us relate spherical Shalika functionals on
$\mathbf{P}\mathbf{G}$ to $L$-factors for $\mathbf{P}\mathbf{H}$. If we let
$i:X^{*}(T_{{\rm Sp}_{4}}({\bf C}))\simeq X_{*}(T_{\mathbf{P}\mathbf{H}}({\bf
C}))$ be the isomorphism induced by the duality between
$\mathbf{P}\mathbf{H}({\bf C})$ and ${\rm Sp}_{4}({\bf C})$, then for every
root $\alpha$ for ${\rm Sp}_{4}({\bf C})$, with $\check{\alpha}$ the
corresponding coroot, we can write
$e^{\check{\alpha}}(g_{\chi})=\chi(i(\alpha)(p))$, where $g_{\chi}$ is the
Frobenius conjugacy class of $I_{\mathbf{H}}(\chi)$.

######  Theorem 4.5.

For all $n\geq 0$, a suitably normalised spherical Shalika functional
satisfies

$\mathcal{S}_{p}\left(\left(\begin{smallmatrix}p^{n}I_{2}&\\\
&I_{2}\end{smallmatrix}\right)\right)=\frac{p^{-2n}}{1+p^{-1}}\mathcal{A}\left((-1)^{n}e^{\check{\rho}+n(\check{\alpha}_{1}+\check{\alpha}_{2})}\prod_{\alpha\in\Phi^{+,s}_{{\rm
Sp}_{4}}}(1+p^{-1}e^{-\check{\alpha}})\right)(g_{\chi})(\mathcal{A}(e^{\check{\rho}})(g_{\chi}))^{-1}.$

Here, we have again denoted
$\mathcal{A}(\cdot)=\sum_{\omega}(-1)^{\ell(\omega)}\omega(\cdot)$. The proof
of Theorem 4.5, which occupies all the next section, follows the strategy of
the proof of [33, Theorem 2.1]. We would like to point out that most of the
steps follow verbatim the ones of _loc.cit._ , with main differences the proof
of Proposition 5.11 and the calculations around Proposition 5.18 (e.g. Lemma
5.19).

##  5\. Proof of Theorem 4.5

###  5.1. An auxiliary computation

The first goal of this section is to compute the representatives of the double
quotient $S({\bf Q}_{p})\setminus\mathbf{G}({\bf Q}_{p})/B_{\mathbf{G}}({\bf
Q}_{p})$ and, in particular, to show that the group $S({\bf Q}_{p})$ acts on
the flag $\mathbf{G}({\bf Q}_{p})/B_{\mathbf{G}}({\bf Q}_{p})$ with an open
orbit. We denote by $E_{p}={\bf Q}_{p}(\delta)$ the unique unramified
quadratic extension of ${\bf Q}_{p}$. Let $P_{\mathbf{G}}\simeq
M_{\mathbf{G}}N_{\mathbf{G}}$ denote the Siegel parabolic of $\mathbf{G}$ and
recall that $S({\bf Q}_{p})\hookrightarrow P_{\mathbf{G}}({\bf Q}_{p})$
induced by the inclusion ${\rm GL}_{2}({\bf Q}_{p})\hookrightarrow{\rm
GL}_{2}(E_{p})\times{\rm GL}_{1}({\bf Q}_{p}),\,g\mapsto(g,{\rm det}(g))$.

######  Lemma 5.1.

We have the following decomposition:

$\mathbf{G}({\bf
Q}_{p})=\bigsqcup_{w\in{W_{\delta}}}\bigsqcup_{\tilde{w}\in{}^{M}W_{\mathbf{H}}}S({\bf
Q}_{p})w\tilde{w}B_{\mathbf{G}}({\bf Q}_{p}),$

where $W_{\delta}=\left\\{id,w_{\delta}=\left(\begin{smallmatrix}1&&&\\\
\delta&1&&\\\ &&1&\\\ &&-\overline{\delta}&1\end{smallmatrix}\right)\right\\}$
and ${}^{M}W_{\mathbf{H}}:=\\{id,s_{2},s_{2}s_{1},s_{2}s_{1}s_{2}\\}$, with
$s_{1}=(1\,2),\,s_{2}=(2\,3)$ introduced in §2.2.2, is the set of minimal
length representatives (Kostant representatives) of
$W_{M_{\mathbf{H}}}\backslash W_{\mathbf{H}}$.

###### Proof.

Using the Bruhat decomposition, we have that

$\mathbf{G}({\bf Q}_{p})=\bigsqcup_{\tilde{w}\in\tilde{W}}P_{\mathbf{G}}({\bf
Q}_{p})\tilde{w}B_{\mathbf{G}}({\bf Q}_{p}),$

where $\tilde{W}$ is the set of Konstant representatives of
$W_{M_{\mathbf{G}}}\backslash W_{\mathbf{G}}$, with $W_{M_{\mathbf{G}}}$ the
Weyl group associated to the Levi of the Siegel subgroup $P_{\mathbf{G}}$.
Since the root datum of $\mathbf{H}_{/{\bf Q}_{p}}$ equals to the relative
root datum of $\mathbf{G}_{/{\bf Q}_{p}}$, we have
$W_{M_{\mathbf{G}}}\backslash W_{\mathbf{G}}=W_{M_{\mathbf{H}}}\backslash
W_{\mathbf{H}}$ and thus the disjoint union above is indexed by the elements
in the set of Kostant representatives for the quotient
$W_{M_{\mathbf{H}}}\backslash W_{\mathbf{H}}$. Thus any $g\in\mathbf{G}({\bf
Q}_{p})$ can be expressed as $m\cdot u\cdot\tilde{w}\cdot b$ for a unique
$\tilde{w}\in{}^{M}W_{\mathbf{H}}$ and some $b\in B_{\mathbf{G}}({\bf
Q}_{p})$, $m\cdot u=\left(\begin{smallmatrix}h&\\\ &\nu
J{}^{t}\bar{h}^{-1}J\end{smallmatrix}\right)\left(\begin{smallmatrix}I&X\\\
&I\end{smallmatrix}\right)\in P_{\mathbf{G}}({\bf Q}_{p})$.

Write $m$ as the product of
$m^{\prime}\lambda_{\nu}=\left(\begin{smallmatrix}h&\\\
&J{}^{t}\bar{h}^{-1}J\end{smallmatrix}\right)\left(\begin{smallmatrix}I&\\\
&\nu I\end{smallmatrix}\right)$, then we have that

$\displaystyle g$
$\displaystyle=m^{\prime}\lambda_{\nu}u\tilde{w}b=m^{\prime}\lambda_{\nu}u\lambda_{\nu}^{-1}\lambda_{\nu}\tilde{w}b=m^{\prime}u^{\prime}\lambda_{\nu}\tilde{w}b=m^{\prime}u^{\prime}\tilde{w}\lambda_{\nu}^{\prime}b\in{\rm
GL}_{2}(E_{p})N_{\mathbf{G}}({\bf Q}_{p})\tilde{w}B_{\mathbf{G}}({\bf
Q}_{p}),$

where we have denoted $u^{\prime}=\lambda_{\nu}u\lambda_{\nu}^{-1}\in
N_{\mathbf{G}}({\bf Q}_{p})$ and
$\lambda_{\nu}^{\prime}=\tilde{w}\lambda_{\nu}\tilde{w}^{-1}\in
T_{\mathbf{G}}({\bf Q}_{p})$. This implies that

$\displaystyle S({\bf Q}_{p})\backslash\mathbf{G}({\bf
Q}_{p})/B_{\mathbf{G}}({\bf Q}_{p})$
$\displaystyle=\bigsqcup_{\tilde{w}\in{}^{M}W_{\mathbf{H}}}S({\bf
Q}_{p})\backslash P_{\mathbf{G}}({\bf Q}_{p})\tilde{w}B_{\mathbf{G}}({\bf
Q}_{p})/B_{\mathbf{G}}({\bf Q}_{p})$
$\displaystyle=\bigsqcup_{\tilde{w}\in{}^{M}W_{\mathbf{H}}}{\rm GL}_{2}({\bf
Q}_{p})\backslash{\rm GL}_{2}(E_{p})N_{\mathbf{G}}({\bf
Q}_{p})\tilde{w}B_{\mathbf{G}}({\bf Q}_{p})/B_{\mathbf{G}}({\bf Q}_{p}).$

Finally, recall that ${\rm GL}_{2}({\bf Q}_{p})$ acts on
$\mathbb{P}^{1}_{E_{p}}={\rm GL}_{2}(E_{p})/B_{{\rm GL}_{2}}(E_{p})$ with two
orbits, one closed and one open with generators given by the identity and
$\left(\begin{smallmatrix}1&\\\ \delta&1\end{smallmatrix}\right)$
respectively. Since any $\tilde{w}$ is defined by the condition
$\tilde{w}^{-1}\Phi_{M_{\mathbf{G}}}^{+}\subset\Phi^{+}$, i.e.
$\tilde{w}^{-1}B_{{\rm GL}_{2}}(E_{p})\tilde{w}\subset B_{\mathbf{G}}({\bf
Q}_{p})$, we use this fact to deduce that

$\mathbf{G}({\bf Q}_{p})=\bigsqcup_{w\in
W_{\delta}}\bigsqcup_{\tilde{w}\in{}^{M}W_{\mathbf{H}}}S({\bf
Q}_{p})w\tilde{w}B_{\mathbf{G}}({\bf Q}_{p}),$

which completes the proof. ∎

Following Lemma 5.1, we now calculate the stabiliser of each orbit for the
action of $S({\bf Q}_{p})$ on the flag $\mathbf{G}({\bf
Q}_{p})/B_{\mathbf{G}}({\bf Q}_{p})$. Before doing so, we denote by
$T_{\delta}$ the maximal non-split torus of ${\rm GL}_{2}({\bf Q}_{p})$ given
by

$T_{\delta}:=\\{\left(\begin{smallmatrix}a&b\\\
b\delta^{2}&a\end{smallmatrix}\right)\,:\,(a,b)\in{\bf
Q}_{p}^{2}\smallsetminus\\{(0,0)\\}\\},$

which we see inside $S({\bf Q}_{p})$ via the embedding of ${\rm GL}_{2}({\bf
Q}_{p})\hookrightarrow S({\bf Q}_{p})$.

######  Proposition 5.2.

The group $S({\bf Q}_{p})$ acts on the flag $\mathbf{G}({\bf
Q}_{p})/B_{\mathbf{G}}({\bf Q}_{p})$, with an open orbit
$\mathcal{O}_{w_{\delta}s_{2}s_{1}s_{2}}$ and seven closed ones, where

  * •

$\mathcal{O}_{\rm id}={\rm Stab}_{id}\backslash S({\bf Q}_{p})$, with ${\rm
Stab}_{\rm id}=B_{{\rm GL}_{2}}({\bf Q}_{p})N_{\mathbf{G}}({\bf Q}_{p})$.

  * •

$\mathcal{O}_{s_{2}}={\rm Stab}_{s_{2}}\backslash S({\bf Q}_{p})$, with

${\rm Stab}_{s_{2}}=B_{{\rm GL}_{2}}({\bf
Q}_{p})\left\\{\left(\begin{smallmatrix}1&&\alpha&x\\\ &1&&\bar{\alpha}\\\
&&1&\\\ &&&1\end{smallmatrix}\right)\,:\,x\in{\bf Q}_{p},\alpha\in
E_{p}\right\\}.$

  * •

$\mathcal{O}_{s_{2}s_{1}}={\rm Stab}_{s_{2}s_{1}}\backslash S({\bf Q}_{p})$,
with

${\rm Stab}_{s_{2}s_{1}}=B_{{\rm GL}_{2}}({\bf
Q}_{p})\left\\{\left(\begin{smallmatrix}1&&&x\\\ &1&&\\\ &&1&\\\
&&&1\end{smallmatrix}\right)\,:\,x\in{\bf Q}_{p}\right\\}.$

  * •

$\mathcal{O}_{s_{2}s_{1}s_{2}}={\rm Stab}_{s_{2}s_{1}s_{2}}\backslash S({\bf
Q}_{p})$, with

${\rm Stab}_{s_{2}s_{1}s_{2}}=B_{{\rm GL}_{2}}({\bf Q}_{p}).$

  * •

$\mathcal{O}_{w_{\delta}}={\rm Stab}_{w_{\delta}}\backslash S({\bf Q}_{p})$,
with

${\rm Stab}_{w_{\delta}}=T_{\delta}N_{\mathbf{G}}({\bf Q}_{p}).$

  * •

$\mathcal{O}_{w_{\delta}s_{2}}={\rm Stab}_{w_{\delta}s_{2}}\backslash S({\bf
Q}_{p})$, with

${\rm
Stab}_{w_{\delta}s_{2}}=T_{\delta}\left\\{\left(\begin{smallmatrix}1&&\alpha&x\\\
&1&\delta\alpha+\bar{\delta}\bar{\alpha}-\delta\bar{\delta}x&\bar{\alpha}\\\
&&1&\\\ &&&1\end{smallmatrix}\right)\,:\,x\in{\bf Q}_{p},\alpha\in
E_{p}\right\\}.$

  * •

$\mathcal{O}_{w_{\delta}s_{2}s_{1}}={\rm
Stab}_{w_{\delta}s_{2}s_{1}}\backslash S({\bf Q}_{p})$, with

${\rm
Stab}_{w_{\delta}s_{2}s_{1}}=T_{\delta}\left\\{\left(\begin{smallmatrix}1&&\bar{\delta}x&x\\\
&1&\delta\bar{\delta}x&\delta x\\\ &&1&\\\
&&&1\end{smallmatrix}\right)\,:\,x\in{\bf Q}_{p}\right\\}.$

  * •

$\mathcal{O}_{w_{\delta}s_{2}s_{1}s_{2}}={\rm
Stab}_{w_{\delta}s_{2}s_{1}s_{2}}\backslash S({\bf Q}_{p})$, with

${\rm Stab}_{w_{\delta}s_{2}s_{1}s_{2}}=T_{\delta}.$

###### Proof.

By Lemma 5.1, a set of representatives for the $S({\bf Q}_{p})$-orbits of
$\mathbf{G}({\bf Q}_{p})/B_{\mathbf{G}}({\bf Q}_{p})$ is given by
$\\{w\tilde{w}\\}$ with $w$ and $\tilde{w}$ varying in $W_{\delta}$ and
${}^{M}W_{\mathbf{H}}$ respectively. The stabilizer of each element $v$ of
this set is given by $\mathrm{Stab}_{v}:=S({\bf Q}_{p})\cap
vB_{\mathbf{G}}({\bf Q}_{p})v^{-1}$. We start by calculating the stabiliser of
each element $\tilde{w}\in{}^{M}W_{\mathbf{H}}$. Since
$\Phi_{M_{\mathbf{G}}}^{+}\subset\tilde{w}\Phi^{+}$, we have
$\mathrm{Stab}_{\tilde{w}}=B_{{\rm GL}_{2}}({\bf
Q}_{p})(N_{\mathbf{G}}\cap\tilde{w}U_{\mathbf{G}}\tilde{w}^{-1})({\bf
Q}_{p})$, where recall that $U_{\mathbf{G}}$ denotes the unipotent radical of
the upper triangular Borel parabolic of $\mathbf{G}$ and $B_{{\rm GL}_{2}}$
embeds into $S$ via the map $b\mapsto\left(\begin{smallmatrix}b&\\\ &{\rm
det}(b)J{}^{t}b^{-1}J\end{smallmatrix}\right)$. Therefore $\mathrm{Stab}_{\rm
id}=B_{{\rm GL}_{2}}({\bf Q}_{p})N_{\mathbf{G}}({\bf Q}_{p})$. Since $s_{2}$
sends the $\mathrm{Sp}_{4}$-positive roots
$\Phi^{+}_{\mathrm{Sp}_{4}}=\\{\alpha_{1}-\alpha_{2},\alpha_{1}+\alpha_{2},2\alpha_{1},2\alpha_{2}\\}$
to
$\\{\alpha_{1}+\alpha_{2},\alpha_{1}-\alpha_{2},2\alpha_{1},-2\alpha_{2}\\}$,
$\mathrm{Stab}_{s_{2}}=B_{{\rm GL}_{2}}({\bf
Q}_{p})\left\\{\left(\begin{smallmatrix}1&&\alpha&x\\\ &1&&\bar{\alpha}\\\
&&1&\\\ &&&1\end{smallmatrix}\right)\,:\,x\in{\bf Q}_{p},\alpha\in
E_{p}\right\\}$. Similarly, $s_{2}s_{1}$, resp. $s_{2}s_{1}s_{2}$, sends
$\Phi^{+}_{\mathrm{Sp}_{4}}$ to
$\\{-\alpha_{1}-\alpha_{2},\alpha_{1}-\alpha_{2},-2\alpha_{2},2\alpha_{1}\\}$,
resp. to
$\\{\alpha_{1}-\alpha_{2},-\alpha_{1}-\alpha_{2},-2\alpha_{2},-2\alpha_{1}\\}$,
hence $\mathrm{Stab}_{s_{2}s_{1}}=B_{{\rm GL}_{2}}({\bf
Q}_{p})\left\\{\left(\begin{smallmatrix}1&&&x\\\ &1&&\\\ &&1&\\\
&&&1\end{smallmatrix}\right)\,:\,x\in{\bf Q}_{p}\right\\},$ while
$\mathrm{Stab}_{s_{2}s_{1}s_{2}}=B_{{\rm GL}_{2}}({\bf Q}_{p})$. We are now
left with calculating the stabiliser of the orbits associated to the elements
$w_{\delta}\tilde{w}$. Conjugating by $w_{\delta}^{-1}$ gives that
$w_{\delta}^{-1}\mathrm{Stab}_{w_{\delta}\tilde{w}}w_{\delta}=w_{\delta}^{-1}S({\bf
Q}_{p})w_{\delta}\cap\tilde{w}B_{\mathbf{G}}({\bf Q}_{p})\tilde{w}^{-1}$.
Given $g=mu=\left(\begin{smallmatrix}a&b&&\\\ c&d&&\\\ &&a&-b\\\
&&-c&d\end{smallmatrix}\right)\left(\begin{smallmatrix}1&&\alpha&x\\\
&1&y&\overline{\alpha}\\\ &&1&\\\ &&&1\end{smallmatrix}\right)$,

$w_{\delta}^{-1}gw_{\delta}=w_{\delta}^{-1}mw_{\delta}w_{\delta}^{-1}uw_{\delta}=\left(\begin{smallmatrix}a+b\delta&b&&\\\
c^{\prime}&d-b\delta&&\\\ &&a+b\bar{\delta}&-b\\\
&&-\bar{c}^{\prime}&d-\bar{\delta}b\end{smallmatrix}\right)\left(\begin{smallmatrix}1&&\alpha-\bar{\delta}x&x\\\
&1&y^{\prime}&\overline{\alpha}-\delta x\\\ &&1&\\\
&&&1\end{smallmatrix}\right),$

with $c^{\prime}=c-b\delta^{2}+\delta(d-a)$ and
$y^{\prime}=y+\delta\bar{\delta}x-\delta\alpha-\bar{\delta}\bar{\alpha}$.
Using again that $\Phi_{M_{\mathbf{G}}}^{+}\subset\tilde{w}\Phi^{+}$,
$w_{\delta}^{-1}mw_{\delta}\in\tilde{w}B_{\mathbf{G}}({\bf
Q}_{p})\tilde{w}^{-1}$ if $c^{\prime}=0$, i.e.

$c=b\delta^{2}+\delta(a-d).$

Since $c\in{\bf Q}_{p}$, this further implies that $a-d=0$, hence $m$ equals
to $\left(\begin{smallmatrix}a&b&&\\\ b\delta^{2}&a&&\\\ &&a&-b\\\
&&-b\delta^{2}&a\end{smallmatrix}\right)$. When $\tilde{w}=id$,
$w_{\delta}^{-1}uw_{\delta}\in B_{\mathbf{G}}({\bf Q}_{p})$, which implies
that

${\rm Stab}_{w_{\delta}}=\left\\{\left(\begin{smallmatrix}a&b&&\\\
b\delta^{2}&a&&\\\ &&a&-b\\\
&&-b\delta^{2}&a\end{smallmatrix}\right)\,:\,\left(\begin{smallmatrix}a&b\\\
b\delta^{2}&a\end{smallmatrix}\right)\in{\rm GL}_{2}({\bf
Q}_{p})\right\\}N_{\mathbf{G}}({\bf Q}_{p})=T_{\delta}N_{\mathbf{G}}({\bf
Q}_{p}).$

When $\tilde{w}=s_{2}$,
$w_{\delta}^{-1}uw_{\delta}\in\tilde{w}U_{\mathbf{G}}\tilde{w}^{-1}$ if
$y^{\prime}=0$, because
$N_{\mathbf{G}}\cap\tilde{w}U_{\mathbf{G}}\tilde{w}^{-1}=\left(\begin{smallmatrix}1&&\star&\star\\\
&1&&\star\\\ &&1&\\\ &&&1\end{smallmatrix}\right)$. Thus

${\rm
Stab}_{w_{\delta}s_{2}}=T_{\delta}\left\\{\left(\begin{smallmatrix}1&&\alpha&x\\\
&1&\delta\alpha+\bar{\delta}\bar{\alpha}-\delta\bar{\delta}x&\bar{\alpha}\\\
&&1&\\\ &&&1\end{smallmatrix}\right)\right\\}.$

Finally, when $\tilde{w}=s_{2}s_{1}$, resp. $\tilde{w}=s_{2}s_{1}s_{2}$,
$N_{\mathbf{G}}\cap\tilde{w}U_{\mathbf{G}}\tilde{w}^{-1}$ equals to
$\left(\begin{smallmatrix}1&&&\star\\\ &1&&\\\ &&1&\\\
&&&1\end{smallmatrix}\right)$, resp. $\\{I_{4}\\}$. From this, one deduces the
last two cases. ∎

The calculations presented so far in this section can be employed, using
Mackey theory, to calculate the multiplicity of Shalika functionals for
principal series of $\mathbf{PG}({\bf Q}_{p})$, which are not necessarily in
the image of the theta correspondence:

######  Proposition 5.3.

Let $\Pi=I_{\mathbf{G}}(\xi)$ be an irreducible generic unramified
representation of $\mathbf{G}({\bf Q}_{p})$ with trivial central character.
Then $\Pi$ has a non-trivial Shalika functional if any of the following holds:

  1. (1)

$\xi_{1}(\alpha)\xi_{2}(\bar{\alpha})\xi_{0}(\alpha\bar{\alpha})=1,$
$\forall\alpha\in E_{p}^{\times}$;

  2. (2)

$\xi_{1}(a)\xi_{0}(ad)=\xi_{2}^{-1}(a)$, $\forall a,d\in{\bf Q}_{p}^{\times}$;

  3. (3)

$\xi_{1}(a)\xi_{0}(ad)=\xi_{2}^{-1}(d)$, $\forall a,d\in{\bf Q}_{p}^{\times}$.

Moreover, if only one of the conditions above holds, $\Pi$ admits a unique
non-trivial Shalika functional up to constant.

######  Remark 5.4.

Note that, if $\Pi$ is in the image of the local theta correspondence,
condition (1) is automatically satisfied.

The proof of this Proposition, which uses rather standard arguments (cf.
[31]), is elementary but very lengthy. As it is not used in the article, we
omit it here and refer to the second author’s PhD thesis for full details.

###  5.2. Shalika functionals and distributions

From now on, we assume that the representation $I_{\mathbf{G}}(\xi)$ is
irreducible and it is in the image of the local theta correspondence from
$\mathbf{H}^{+}$ (see equality (9)). This, together with [27, Theorem 2.7],
shows that the Shalika functional exists and it is unique (see Theorem 3.8).
Recall that $w_{\rm op}$ denotes the representative of the open orbit of
$S({\bf Q}_{p})\setminus\mathbf{G}({\bf Q}_{p})/B_{\mathbf{G}}({\bf Q}_{p})$.
It was described explicitly in Proposition 5.2. Consider the group
$\tilde{S}({\bf Q}_{p}):=w_{\rm op}^{-1}S({\bf Q}_{p})w_{\rm op}$. Then
$B_{\mathbf{G}}({\bf Q}_{p})\cdot\tilde{S}({\bf Q}_{p})$ is Zariski open in
$G({\bf Q}_{p})$. Set the character

$\displaystyle\chi_{\tilde{S},p}:\;\tilde{S}({\bf Q}_{p})\to{\bf
C}^{\times},\;s\mapsto\chi_{\delta,p}(w_{\rm op}sw_{\rm op}^{-1}).$

The map $\mathrm{Ind}_{S({\bf Q}_{p})}^{\mathbf{G}({\bf
Q}_{p})}\chi_{\delta,p}\to\mathrm{Ind}_{\tilde{S}({\bf
Q}_{p})}^{\mathbf{G}({\bf Q}_{p})}\chi_{\tilde{S},p}$, given by sending
$f\mapsto f_{\tilde{S}}(g):=f(w_{\rm op}g)$, is an isomorphism of
$\mathbf{G}({\bf Q}_{p})$-representations. The composition of the above map
with the map $I_{\mathbf{G}}(\xi)\to\mathrm{Ind}_{S({\bf
Q}_{p})}^{\mathbf{G}({\bf Q}_{p})}\chi_{\delta,p}$ associated to the Shalika
functional $\Lambda_{p,\mathcal{S}}$ results in a $\mathbf{G}({\bf
Q}_{p})$-intertwining map of the form

$I_{\mathbf{G}}(\xi)\to\mathrm{Ind}_{\tilde{S}({\bf Q}_{p})}^{\mathbf{G}({\bf
Q}_{p})}\chi_{\tilde{S},p}.$

As in [33], working with this realisation of the Shalika model will clarify
some of the computations. The associated Shalika functional is given by

$\Lambda_{p,\tilde{S}}(g\cdot v):=\Lambda_{p,\mathcal{S}}((w_{\rm op}g)\cdot
v).$

The spherical vector in this Shalika model associated to $\tilde{S}({\bf
Q}_{p})$ is then

$\tilde{\mathcal{S}}_{p}(g):=\Lambda_{p,\tilde{S}}(g\cdot\phi_{0})=\Lambda_{p,\mathcal{S}}((w_{\rm
op}gw_{\rm op}^{-1})\cdot\phi_{0}),$ (10)

where the last equality follows since $w_{\rm op}\in\mathbf{G}({\bf Z}_{p})$.

Consider the canonical surjective map
$P_{\xi}:\;\mathcal{C}^{\infty}_{c}(\mathbf{G}({\bf Q}_{p}))\to
I_{\mathbf{G}}(\xi)$ defined by

$\displaystyle\phi$ $\displaystyle\mapsto
P_{\xi}(\phi)(x):=\int_{B_{\mathbf{G}}({\bf
Q}_{p})}\xi^{-1}(b)\delta_{B_{\mathbf{G}}}^{1/2}(b)\phi(bx)db,$ (11)

where $db$ is the Haar measure of $B_{\mathbf{G}}({\bf Q}_{p})$ normalised
according to the Iwasawa decomposition, i.e. $dg=dbdk$, with $dg$ the Haar
measure of $\mathbf{G}({\bf Q}_{p})$ satisfying $\mathrm{vol}(\mathbf{G}({\bf
Z}_{p}))=1$. This induces the injective map

$P_{\xi}^{*}:\;I_{\mathbf{G}}(\xi)^{*}\to\mathcal{D}(\mathbf{G}({\bf
Q}_{p})),$

where $I_{\mathbf{G}}(\xi)^{*}$ denotes the dual of $I_{\mathbf{G}}(\xi)$ and
$\mathcal{D}(\mathbf{G}({\bf Q}_{p}))$ are the distributions on
$\mathbf{G}({\bf Q}_{p})$. Using $P_{\xi}^{*}$, we can then associate to the
Shalika functional $\Lambda_{p,\tilde{S}}$ on $I_{\mathbf{G}}(\xi)$ a
distribution $\Delta_{\tilde{S},\xi}$, with the following property

$\Delta_{\tilde{S},\xi}(bg\tilde{s})=\xi^{-1}(b)\delta_{B_{\mathbf{G}}}^{1/2}(b)\chi_{\tilde{S},p}^{-1}(\tilde{s})\Delta_{\tilde{S},\xi}(g),$

for all $b\in B_{\mathbf{G}}({\bf Q}_{p})$, $g\in\mathbf{G}({\bf Q}_{p})$, and
$\tilde{s}\in\tilde{S}({\bf Q}_{p})$.

If we denote $g_{n}:=\left(\begin{smallmatrix}p^{n}I_{2}&\\\
&I_{2}\end{smallmatrix}\right)$, where $n\geq 0$, notice that
$\mathcal{S}_{p}(g_{n})=\tilde{\mathcal{S}}_{p}(g_{-n})$. To prove Theorem
4.5, we calculate $\tilde{\mathcal{S}}_{p}(g_{-n})$ explicitly. As in [33,
§4], using the work of Casselman (cf. [6]) and Hironaka (cf. [21]), one can
express (cf. [33, (38)])

$\tilde{\mathcal{S}}_{p}(g_{-n})=Q^{-1}\sum_{\omega\in
W_{\mathbf{H}}}\prod_{\begin{subarray}{c}\alpha\in\Phi_{\mathbf{P}\mathbf{H}}^{+}\\\
\omega\alpha>0\end{subarray}}c_{\alpha}(\xi)T^{*}_{\omega^{-1}}\Delta_{\tilde{S},^{\omega}{\xi}}(R_{g_{-n}}\mathbf{1}_{\rm
Iw}),$ (12)

where $\mathcal{Q}=\sum_{\omega\in W_{\mathbf{H}}}({\rm Iw}\omega{\rm Iw}:{\rm
Iw})^{-1}$. Here we have used the following notation:

  1. (1)

For each $\omega\in W_{\mathbf{G}}$, we have the intertwining map
$T_{\omega}:I_{\mathbf{G}}(\xi)\to I_{\mathbf{G}}({}^{\omega}\xi)$ (cf. [5,
§3.7]) and its pullback $T_{\omega}^{*}:\mathcal{D}(\mathbf{G}({\bf
Q}_{p}))_{{}^{\omega}\xi^{-1}}\to\mathcal{D}(\mathbf{G}({\bf
Q}_{p}))_{\xi^{-1}}$ (cf. [21, §1.2]),

  2. (2)

${\rm Iw}$ is the Iwahori subgroup of $\mathbf{G}({\bf Q}_{p})$ and
$\mathbf{1}_{{\rm Iw}}$ denotes its characteristic function,

  3. (3)

$R_{g_{-n}}$ is the right translate action of $g_{-n}\in\mathbf{G}({\bf
Q}_{p})$ on $\mathcal{C}^{\infty}_{c}(\mathbf{G}({\bf Q}_{p}))$,

  4. (4)

the constants $c_{\alpha}(\xi)$ are defined as in [21, (1.12)] (cf. §5.2.1
below).

Theorem 4.5 will follow from the uniqueness of the Shalika functional and from
a direct calculation of each term of the right hand side of (12) explicitly.

####  5.2.1. Definition of $c_{\alpha}(\xi)$

In what follows, we define the constants $c^{G}_{\alpha}(\xi)$, where
$G_{/{\bf Q}_{p}}$ is either $\mathbf{H}$ or $\mathbf{G},$ $\xi$ is a
character of the maximal torus of $G({\bf Q}_{p})$, and where $\alpha$ runs
through the positive roots $\Phi^{+}_{G}$ induced by our choice of the upper-
triangular Borel subgroup of $G$. Here, we closely follow [8, §3].

For the following discussion, we let $M_{\alpha}$ be the ${\bf Q}_{p}$-points
of the Levi subgroup associated to a positive root $\alpha\in\Phi^{+}_{G}$ and
denote by $M^{der}_{\alpha}$ the simply connected cover of its derived
subgroup. In our case, as $G$ varies in $\\{\mathbf{H},\mathbf{G}\\}$,
$M^{der}_{\alpha}$ is either ${\rm SL}_{2}({\bf Q}_{p})$ or ${\rm
SL}_{2}(E_{p})$.

######  Definition 5.5.

Let $F$ be either ${\bf Q}_{p}$ or $E_{p}$, with ring of integers
$\mathcal{O}_{F}$ and $\mathfrak{p}$ its maximal ideal. Consider the group
${\rm SL}_{2}$ defined over $F$ with maximal diagonal torus $T_{{\rm SL}_{2}}$
and define the following coset of $T_{{\rm SL}_{2}}(F)/T_{{\rm
SL}_{2}}(\mathcal{O}_{F})$

$a=\left(\begin{smallmatrix}\mathfrak{p}-\mathfrak{p}^{2}&\\\
&(\mathfrak{p}-\mathfrak{p}^{2})^{-1}\end{smallmatrix}\right).$

Given $\alpha\in\Phi^{+}_{G}$, $a_{\alpha}$ denotes the image of a
representative of $a$ via the map $x_{\alpha}:M_{\alpha}^{der}\to G({\bf
Q}_{p})$.

If $\alpha$ is such that $M_{\alpha}^{der}={\rm SL}_{2}(F)$, we denote
$q_{\alpha}=|\mathcal{O}_{F}/\mathfrak{p}\mathcal{O}_{F}|$.

######  Definition 5.6.

Given $\alpha\in\Phi^{+}_{G}$ and $\omega\in W_{G}$, we define

$\displaystyle
c_{\alpha}^{G}(\xi)=\frac{1-q_{\alpha}^{-1}\xi(a_{\alpha})}{1-\xi(a_{\alpha})},\;\;c^{G}_{\omega}(\xi)=\prod_{\begin{subarray}{c}\alpha\in\Phi^{+}_{G}\\\
\omega\alpha<0\end{subarray}}c_{\alpha}^{G}(\xi).$

If $G=\mathbf{G}$, we simply denote them by $c_{\alpha}(\xi)$ and
$c_{\omega}(\xi)$ respectively.

######  Example 5.7.

  * •

If $G=\mathbf{G}$, we have that
$M_{2\alpha_{2}-\alpha_{0}}^{der}\\!\\!=\\!{\rm SL}_{2}({\bf Q}_{p})$ and
$M_{\alpha_{1}-\alpha_{2}}^{der}\\!\\!=\\!{{\rm SL}}_{2}(E_{p})$. Then
$a_{\alpha_{1}-\alpha_{2}}=\left(\begin{smallmatrix}p&&&\\\ &p^{-1}&&\\\
&&p&\\\ &&&p^{-1}\end{smallmatrix}\right)$ and
$a_{2\alpha_{2}-\alpha_{0}}=\left(\begin{smallmatrix}1&&&\\\ &p&&\\\
&&p^{-1}&\\\ &&&1\end{smallmatrix}\right)$. Furthermore, by [8, p. 219], we
obtain $q_{2\alpha_{2}-\alpha_{0}}=p$ and $q_{\alpha_{1}-\alpha_{2}}=p^{2}$,
implying that $[{\rm Iw}s_{1}{\rm Iw}:{\rm Iw}]=p^{2}$ and $[{\rm Iw}s_{2}{\rm
Iw}:{\rm Iw}]=p$, where $s_{1},s_{2}$ are the Weyl elements corresponding to
$\alpha_{1}-\alpha_{2}$ and $2\alpha_{2}-\alpha_{0}$, respectively.

  * •

When $G=\mathbf{H}$, which is split, for any root $\alpha\in\Phi_{G}^{+}$ we
have $q_{\alpha}=p$ and $a_{\alpha}$ is the image of $p$ under the coroot
$\check{\alpha}$ (cf. [6, Remarks 1.1 and 3.3]).

###  5.3. First reduction: removing the dependence on $n$

This subsection, which closely follows [33, §5 & §6], relies on the uniqueness
of the Shalika functional given by Theorem 3.8.

######  Lemma 5.8.

For any $\omega\in W_{\mathbf{H}}$ and $\phi\in
I_{\mathbf{G}}({}^{\omega}\xi)$ with $\mathrm{supp}(\phi)\subset
B_{\mathbf{G}}({\bf Q}_{p})\tilde{S}({\bf Q}_{p})$ we have

$\displaystyle\Lambda_{p,\tilde{S}}(\phi)=\int_{B_{\mathbf{G}}({\bf
Q}_{p})\cap\tilde{S}({\bf Q}_{p})\setminus\tilde{S}({\bf
Q}_{p})}\phi(h)\chi_{\tilde{S}}^{-1}(h)dh.$ (13)

###### Proof.

The proof is a verbatim translation of the one of [33, Corollary 5.5]. ∎

In the integral above, $dh$ denotes the Haar measure on $\mathbf{G}({\bf
Q}_{p})$ so that the Iwahori subgroup ${\rm Iw}$ has volume equal to $1$. This
corresponds to choosing the normalisation on the Shalika functional by
imposing

$\displaystyle\Lambda_{p,\tilde{S}}(P_{\xi}(\mathbf{1}_{\rm Iw}))=1.$ (14)

Notice that, if $f\in\mathcal{C}^{\infty}_{c}(\mathbf{G}({\bf Q}_{p}))$ is
chosen so that $P_{\xi}(f)\in I_{\mathbf{G}}(\xi)$ has its support contained
in $B_{\mathbf{G}}({\bf Q}_{p})\cdot\tilde{S}({\bf Q}_{p})$, Lemma 5.8 can be
rephrased as

$\Delta_{\tilde{S},\xi}(f)=\int_{B_{\mathbf{G}}({\bf Q}_{p})\tilde{S}({\bf
Q}_{p})}\xi^{-1}\delta_{B_{\mathbf{G}}}^{1/2}(b_{h})f(h)\chi_{\tilde{S}}^{-1}(\tilde{s}_{h})dh,$
(15)

where we have written $h=b_{h}\cdot\tilde{s}_{h}$, with $b_{h}\in
B_{\mathbf{G}}({\bf Q}_{p})$ and $\tilde{s}_{h}\in\tilde{S}({\bf Q}_{p})$. Let
us point out the fact that here we are using that $B_{\mathbf{G}}({\bf
Q}_{p})\tilde{S}({\bf Q}_{p})$ is dense in $\mathbf{G}({\bf Q}_{p})$. From now
on, we identify the Shalika distribution $\Delta_{\tilde{S},\xi}$ as the one
defined by (15). Using this expression, we obtain the following.

######  Proposition 5.9.

For every $n\geq 0$, we have

$\tilde{\mathcal{S}}_{p}(g_{-n})=Q^{-1}\sum_{\omega\in
W_{\mathbf{H}}}\prod_{\begin{subarray}{c}\alpha\in\Phi_{\mathbf{P}\mathbf{H}}^{+}\\\
\omega\alpha>0\end{subarray}}c_{\alpha}(\xi)({}^{\omega}\xi^{-1}\delta^{1/2}_{B_{\mathbf{G}}})(g_{n})T^{*}_{\omega^{-1}}\Delta_{\tilde{S},^{\omega}{\xi}}(\mathbf{1}_{\rm
Iw}).$ (16)

###### Proof.

The proof of the statement follows closely the one of [33, (48)]; therefore we
sketch it and invite the curious reader to consult either [33, §6] or the
second author’s PhD thesis for further details.

First, notice that $R_{g_{-n}}\mathbf{1}_{\mathrm{Iw}}$ has support inside
$B_{\mathbf{G}}({\bf Q}_{p})\cdot\tilde{S}({\bf Q}_{p})$ (as for [33, Lemma
5.1]). Hence, we can apply the formula (15) to obtain

$\Delta_{\tilde{S},\xi}(R_{g_{-n}}\mathbf{1}_{\mathrm{Iw}})=\xi^{-1}\delta^{1/2}_{B_{\mathbf{G}}}(g_{n}),$

which is proved as in [33, Corollary 6.1]. By uniqueness of the Shalika
functional (see Theorem 3.8), we now have that
$T^{*}_{\omega^{-1}}\Delta_{\tilde{S},{}^{\omega}\xi}$ is a constant multiple
of $\Delta_{\tilde{S},\xi}$, which implies, as in [33, Corollary 6.2], the
equality

$T^{*}_{\omega^{-1}}\Delta_{\tilde{S},{}^{\omega}\xi}(R_{g_{-n}}\mathbf{1}_{\mathrm{Iw}})=({}^{\omega}\xi^{-1}\delta^{1/2}_{B_{\mathbf{G}}})(g_{n})T^{*}_{\omega^{-1}}\Delta_{\tilde{S},{}^{\omega}\xi}(\mathbf{1}_{\mathrm{Iw}}).$

Plugging this into (12), we get the desired formula. ∎

###  5.4. Second reduction: Convergence of the Shalika functional on all of
its domain

Thanks to Proposition 5.9, we have reduced the proof of Theorem 4.5 to an
explicit evaluation of
$T^{*}_{\omega^{-1}}\Delta_{\tilde{S},^{\omega}{\xi}}(\mathbf{1}_{\rm Iw})$
for each $\omega\in W_{\mathbf{H}}$. Since, by uniqueness of the Shalika
functional, $T^{*}_{\omega^{-1}}\Delta_{\tilde{S},{}^{\omega}\xi}$ is a
constant multiple of $\Delta_{\tilde{S},\xi}$, it is tempting to use (15) to
evaluate $T^{*}_{\omega^{-1}}\Delta_{\tilde{S},^{\omega}{\xi}}(\mathbf{1}_{\rm
Iw})$. To do so we need to first show that the period integral of Lemma 5.8
converges absolutely for all $\phi\in I_{\mathbf{G}}(\xi)$, so that we can
evaluate the formula (15) at the characteristic function $\mathbf{1}_{{\rm
Iw},\omega}$ of ${\rm Iw}\,\omega\,{\rm Iw}$.

We first consider the following auxiliary calculation. Recall that we have
denoted by $T_{\delta}$ the maximal non-split torus of ${\rm GL}_{2}({\bf
Q}_{p})$ which is equal to the stabilizer of the open orbit $w_{\rm op}$ (cf.
Proposition 5.2)

######  Lemma 5.10.

We have an isomorphism

$T_{\delta}\setminus{\rm GL}_{2}({\bf Q}_{p})\simeq T_{1}({\bf
Q}_{p})\overline{N}_{{\rm GL}_{2}}({\bf Q}_{p}),$

where $T_{1}({\bf Q}_{p})\overline{N}_{{\rm GL}_{2}}({\bf
Q}_{p})=\left\\{t(y)\bar{n}(x)=\left(\begin{smallmatrix}y&\\\
&1\end{smallmatrix}\right)\left(\begin{smallmatrix}1&\\\
x&1\end{smallmatrix}\right),\;y\in{\bf Q}_{p}^{\times},\;x\in{\bf
Q}_{p}\right\\}$.

###### Proof.

Recall that the group ${\rm GL}_{2}({\bf Q}_{p})$ acts on
$\mathbb{P}^{1}_{E_{p}}\simeq{\rm GL}_{2}(E_{p})/B_{{\rm GL}_{2}}(E_{p})$ with
two orbits, generated by the elements $(1:0)$ and $(1:\delta)$. Every
$g=\left(\begin{smallmatrix}a&b\\\ c&d\end{smallmatrix}\right)\in
T_{\delta}\setminus{\rm GL}_{2}({\bf Q}_{p})$ acts on $(1:\delta)$ by

$\left(\begin{smallmatrix}a&b\\\
c&d\end{smallmatrix}\right)\cdot(1:\delta)=(a+b\delta:c+d\delta).$

Denote by $O_{\delta}$ the corresponding orbit. It consists of elements
$(m:n)$ such that $mn\neq 0$ and $mn^{-1}\not\in{\bf Q}_{p}$. Since
$T_{\delta}\setminus{\rm GL}_{2}({\bf Q}_{p})$ acts transitively on
$O_{\delta}$, we have $T_{\delta}\setminus{\rm GL}_{2}({\bf Q}_{p})\simeq
O_{\delta}$ as topological spaces. On the other hand, the subgroup $T_{1}({\bf
Q}_{p})\overline{N}_{{\rm GL}_{2}}({\bf
Q}_{p})=\left\\{\left(\begin{smallmatrix}y&\\\
&1\end{smallmatrix}\right)\left(\begin{smallmatrix}1&\\\
x&1\end{smallmatrix}\right)\right\\}\subset{\rm GL}_{2}({\bf Q}_{p})$ acts on
$O_{\delta}$ with one orbit, with representative $(1:\delta)$. Concretely, if
$\left(\begin{smallmatrix}y&\\\ x&1\end{smallmatrix}\right)\in T_{1}({\bf
Q}_{p})\overline{N}_{{\rm GL}_{2}}({\bf Q}_{p})T_{1}({\bf Q}_{p})$, we have

$\left(\begin{smallmatrix}y&\\\
x&1\end{smallmatrix}\right)(1:\delta)=(y:x+\delta),$

which implies that the stabilizer of $(1\\!:\\!\delta)$ is just the identity
matrix; hence $T_{1}({\bf Q}_{p})\overline{N}_{{\rm GL}_{2}}({\bf
Q}_{p})\simeq O_{\delta}$. ∎

We now can study the convergence of the period integral of Lemma 5.8 for all
$\phi\in I_{\mathbf{G}}(\xi)$. Since $\xi$ is an unramified character, we can
write $\xi_{1}=|\cdot|^{z_{1}}\circ{\rm N}_{E_{p}/{\bf Q}_{p}}$,
$\xi_{2}=|\cdot|^{z_{2}}\circ{\rm N}_{E_{p}/{\bf Q}_{p}}$, and
$\xi_{0}=|\cdot|^{z_{0}}$, where $z_{1},z_{2},z_{0}$ are complex numbers such
that $|\cdot|^{\sum 2z_{i}}=1$. In particular, ${\rm Re}(z_{0})=-{\rm
Re}(z_{1})-{\rm Re}(z_{2})$.

######  Proposition 5.11.

Suppose that ${\rm Re}(z_{1})>{\rm Re}(z_{2})+1>1$. The period integral of
Lemma 5.8 converges absolutely for every $\phi\in I_{\mathbf{G}}(\xi)$ and
thus represents a Shalika functional.

###### Proof.

As $I_{\mathbf{G}}(\xi)$ is unramified, it suffices to prove the proposition
when $\phi$ is equal to the spherical vector $\phi_{0}$, which we therefore
assume from now on. In order to simplify the calculation, after conjugating by
$w_{\rm op}$ (using (10)), the convergence of the integral for $\tilde{S}$
follows from the one over the Shalika group $S$ given by

$\displaystyle\Lambda_{p,\mathcal{S}}(\phi_{0})$
$\displaystyle:=\int_{T_{\delta}\setminus S({\bf Q}_{p})}\\!\\!\phi_{0}(w_{\rm
op}^{-1}s)\chi_{\delta,p}^{-1}(s)ds$ $\displaystyle=\int_{{\bf
Q}_{p}^{\times}}\int_{{\bf Q}_{p}}\int_{N_{\mathbf{G}}({\bf
Q}_{p})}\\!\\!\\!\\!\\!\\!\phi_{0}(\omega^{-1}w_{\delta}^{-1}t(y)\bar{n}(x)n)\chi_{\delta,p}^{-1}(n)|y|^{-1}dndxd^{\times}y,$

where, for the second equality, we have used the isomorphism of Lemma 5.10 and
denoted $\omega:=s_{2}s_{1}s_{2}$; we have also used the fact that the right
Haar measure $ds$ corresponds to the right Haar measure on $T_{1}({\bf
Q}_{p})\overline{N}_{{\rm GL}_{2}}({\bf Q}_{p})$, which is
$|y|^{-1}dxd^{\times}y$ (up to a volume factor coming from the normalisation
of $ds$ chosen in §5.3).

Notice that $w_{\delta}^{-1}t(y)\bar{n}(x)=\iota\left(\begin{smallmatrix}y&\\\
x-y\delta&1\end{smallmatrix}\right)=t(y)\iota\left(\begin{smallmatrix}1&\\\
x-y\delta&1\end{smallmatrix}\right)$ where $\iota:{\rm
GL}_{2}(E_{p})\hookrightarrow P_{\mathbf{G}}({\bf Q}_{p})$, so that the
integral becomes equal to

$\displaystyle\Lambda_{p,\mathcal{S}}(\phi_{0})$ $\displaystyle=\int_{{\bf
Q}_{p}^{\times}}\int_{{\bf Q}_{p}}\int_{N_{\mathbf{G}}({\bf
Q}_{p})}\\!\\!\\!\\!\\!\\!\phi_{0}(\omega^{-1}t(y)\iota\left(\begin{smallmatrix}1&\\\
x-y\delta&1\end{smallmatrix}\right)n)\chi_{\delta,p}^{-1}(n)|y|^{-1}dndxd^{\times}y$
$\displaystyle=\int_{{\bf Q}_{p}^{\times}}\int_{{\bf
Q}_{p}}\int_{N_{\mathbf{G}}({\bf
Q}_{p})}\\!\\!\\!\\!\\!\\!\phi_{0}(t(y)\omega^{-1}\iota\left(\begin{smallmatrix}1&\\\
x-y\delta&1\end{smallmatrix}\right)n)\chi_{\delta,p}^{-1}(n)|y|^{-1}dndxd^{\times}y$
$\displaystyle=\int_{{\bf Q}_{p}^{\times}}\int_{{\bf
Q}_{p}}\int_{N_{\mathbf{G}}({\bf
Q}_{p})}\\!\\!\\!\\!\\!\\!\phi_{0}(\omega^{-1}\iota\left(\begin{smallmatrix}1&\\\
x-y\delta&1\end{smallmatrix}\right)n)\chi_{\delta,p}^{-1}(n)(\xi_{1}\xi_{0})(y)dndxd^{\times}y,$

where in the last equality we have used that
$\phi_{0}(t(y)g)=\xi\delta_{B_{\mathbf{G}}}^{1/2}(t(y))\phi_{0}(g)$. As
$\phi_{0}$ is fixed by the right translation action of $\mathbf{G}({\bf
Z}_{p})$ and since conjugation by the Weyl element $s_{1}$ yields
$s_{1}\iota\left(\begin{smallmatrix}1&\\\
x-y\delta&1\end{smallmatrix}\right)s_{1}=\iota\left(\begin{smallmatrix}1&x-y\delta\\\
&1\end{smallmatrix}\right)$, the integral simplifies further as

$\int_{{\bf Q}_{p}^{\times}}\int_{{\bf Q}_{p}}\int_{N_{\mathbf{G}}({\bf
Q}_{p})}\\!\\!\\!\\!\\!\\!\phi_{0}(\omega_{0}^{-1}\iota\left(\begin{smallmatrix}1&x-y\delta\\\
&1\end{smallmatrix}\right)n)\chi_{\delta,p}^{-1}(n)(\xi_{1}\xi_{0})(y)dndxd^{\times}y,$

where we have denoted $\omega_{0}:=s_{2}s_{1}s_{2}s_{1}$, which is a
representative of the longest Weyl element of $W_{\mathbf{G}}$. Since
$\xi_{1}=|\cdot|^{z_{1}}\circ{\rm N}_{E_{p}/{\bf Q}_{p}}$,
$\xi_{2}=|\cdot|^{z_{2}}\circ{\rm N}_{E_{p}/{\bf Q}_{p}}$, and
$\xi_{0}=|\cdot|^{z_{0}}$, we have that
$(\xi_{1}\xi_{0})(y)=|y|^{2z_{1}+z_{0}}$ and the integral above is dominated
absolutely by

$\int_{{\bf Q}_{p}^{\times}}\int_{{\bf Q}_{p}}\int_{N_{\mathbf{G}}({\bf
Q}_{p})}\\!\left|\phi_{0}(\omega_{0}^{-1}\iota\left(\begin{smallmatrix}1&x-y\delta\\\
&1\end{smallmatrix}\right)n)\right||y|^{{\rm Re}(z_{1})-{\rm
Re}(z_{2})}dndxd^{\times}y,$

where we have used that $|\chi_{\delta,p}^{-1}(n)|=1$. Furthermore, since
$\\{0\\}\in{\bf Q}_{p}$ is a measure $0$ set of ${\bf Q}_{p}$ and
$\phi_{0}\left(\omega_{0}\iota\left(\begin{smallmatrix}1&x+y\delta\\\
&1\end{smallmatrix}\right)n\right)$ is a continuous function in all $y\in{\bf
Q}_{p}$, if ${\rm Re}(z_{1})>{\rm Re}(z_{2})+1$ the latter extends to

$\int_{{\bf Q}_{p}}\int_{{\bf Q}_{p}}\int_{N_{\mathbf{G}}({\bf
Q}_{p})}\\!\left|\phi_{0}(\omega_{0}^{-1}\iota\left(\begin{smallmatrix}1&x-y\delta\\\
&1\end{smallmatrix}\right)n)\right||y|^{{\rm Re}(z_{1})-{\rm
Re}(z_{2})-1}dndxdy.$

We now show that this integral converges. We write it as the sum of two
integrals $I_{1}+I_{2}$ by splitting the domain of $x,y$ into
$\\{x,y\,:\,x-\delta y\in\mathcal{O}_{p}\\}\cup\\{x,y\,:\,x-\delta
y\not\in\mathcal{O}_{p}\\}$ and analyze each separately. The first integral
equals to

$\displaystyle I_{1}:=\iint_{\mathcal{O}_{p}}\\!\\!\\!\cdots dxdy$
$\displaystyle=\sum_{n\geq 0}p^{-n({\rm Re}(z_{1})-{\rm
Re}(z_{2})-1)}\int_{\Omega_{n}}\int_{{\bf Z}_{p}}\int_{N_{\mathbf{G}}({\bf
Q}_{p})}\\!\left|\phi_{0}(\omega_{0}^{-1}\iota\left(\begin{smallmatrix}1&x-y\delta\\\
&1\end{smallmatrix}\right)n)\right|dndxdy$ $\displaystyle\leq\sum_{n\geq
0}p^{-n({\rm Re}(z_{1})-{\rm Re}(z_{2})-1)}\cdot\int_{U_{\mathbf{G}}({\bf
Q}_{p})}\\!\left|\phi_{0}(\omega_{0}^{-1}u)\right|du,$

where we denoted $\Omega_{n}=\\{y\,:\,|y|=p^{-n}\\}$. The latter converges
because ${\rm Re}(z_{1})>{\rm Re}(z_{2})+1$ and because the intertwining
operator

$T_{\omega_{0}}(\phi_{0})(1)=\int_{U_{\mathbf{G}}({\bf
Q}_{p})}\\!\phi_{0}(\omega_{0}^{-1}u)du$

converges absolutely as long as ${\rm Re}(z_{1})>{\rm Re}(z_{2})>0$ (cf. [34,
(2.3)]). The second integral

$\displaystyle I_{2}$ $\displaystyle=\iint_{\\{x-\delta
y\not\in\mathcal{O}_{p}\\}}\int_{N_{\mathbf{G}}({\bf
Q}_{p})}\\!\left|\phi_{0}(\omega_{0}^{-1}\iota\left(\begin{smallmatrix}1&x-y\delta\\\
&1\end{smallmatrix}\right)n)\right||y|^{{\rm Re}(z_{1})-{\rm
Re}(z_{2})-1}dndxdy$ $\displaystyle=\iint_{\\{x-\delta
y\not\in\mathcal{O}_{p}\\}}\int_{N_{\mathbf{G}}({\bf
Q}_{p})}\\!\left|\phi_{0}(\omega_{0}^{-1}\iota\left(\begin{smallmatrix}x-y\delta&\\\
1&(x-y\delta)^{-1}\end{smallmatrix}\right)\iota\left(\begin{smallmatrix}(x-y\delta)^{-1}&1\\\
-1&\end{smallmatrix}\right)n)\right||y|^{{\rm Re}(z_{1})-{\rm
Re}(z_{2})-1}dndxdy$ $\displaystyle=\iint_{\\{x-\delta
y\not\in\mathcal{O}_{p}\\}}\int_{N_{\mathbf{G}}({\bf
Q}_{p})}\\!\left|\phi_{0}(\iota\left(\begin{smallmatrix}(x-y\delta)^{-1}&-1\\\
&x-y\delta\end{smallmatrix}\right)\omega_{0}^{-1}n)\right||y|^{{\rm
Re}(z_{1})-{\rm Re}(z_{2})-1}dndxdy$ $\displaystyle=\iint_{\\{x-\delta
y\not\in\mathcal{O}_{p}\\}}\int_{N_{\mathbf{G}}({\bf
Q}_{p})}\\!\left|\phi_{0}(\omega_{0}^{-1}n)\right||y^{-1}x^{2}-y\delta^{2}|^{1-{\rm
Re}(z_{1})+{\rm Re}(z_{2})}dndxdy,$ (17)

where for the third equality we have made the change of variable
$n\mapsto\iota\left(\begin{smallmatrix}(x-y\delta)^{-1}&1\\\
-1&\end{smallmatrix}\right)n\iota\left(\begin{smallmatrix}(x-y\delta)^{-1}&1\\\
-1&\end{smallmatrix}\right)^{-1}$ and we have used that
$\left(\begin{smallmatrix}(x-y\delta)^{-1}&1\\\
-1&\end{smallmatrix}\right)\in{\rm GL}_{2}(\mathcal{O}_{p})$; for the fourth,
we used that

$\phi_{0}(\iota\left(\begin{smallmatrix}(x-y\delta)^{-1}&-1\\\
&x-y\delta\end{smallmatrix}\right)g)=\xi_{1}^{-1}\xi_{2}(x-y\delta)|x^{2}-\delta^{2}y^{2}|\phi_{0}(g).$

Notice that

$|y^{-1}x^{2}-y\delta^{2}|=\begin{cases}|y|&\text{ if }v_{p}(y)\leq
v_{p}(x)\\\ |x^{2}y^{-1}|&\text{ if }v_{p}(y)>v_{p}(x).\end{cases}$

In particular, $|y^{-1}x^{2}-y\delta^{2}|<1$ because $x-\delta
y\not\in\mathcal{O}_{p}$. Therefore

$\eqref{badintegralfinal}\leq\sum_{n\geq 1}p^{-n({\rm Re}(z_{1})-{\rm
Re}(z_{2})-1)}\cdot\int_{N_{\mathbf{G}}({\bf
Q}_{p})}\\!\left|\phi_{0}(\omega^{-1}n)\right|dn,$

and the latter converges as above because of our hypotheses. ∎

######  Remark 5.12.

Recall that in the proof above we have used the explicit representatives
$s_{1}$, $s_{2}$ fixed in §2.1.1 of the two Weyl elements in $W_{\mathbf{G}}$
associated to the roots $\alpha_{1}-\alpha_{2}$ and $2\alpha_{2}-\alpha_{0}$,
respectively.

We now proceed as in [33, Proposition 7.2] to show the following.

######  Corollary 5.13.

Let $D$ be the set of unramified characters of $\mathbf{P}\mathbf{G}({\bf
Q}_{p})$ of the form (9), for which (8) holds. There exists a unique non-zero
Shalika functional on $I_{\mathbf{G}}(\xi)$, which satisfies (14), for almost
all $\xi\in D$. Moreover, if $\\{\phi_{\xi}\\}_{\xi\in D}$ is a rational
family of functions $\phi_{\xi}\in I_{\mathbf{G}}(\xi)$, then the period
integral $\Lambda_{p,\tilde{S}}(\phi_{\xi})$ of (13) is a rational function on
$\xi\in D$.

###### Proof.

We apply Bernstein’s Theorem (cf. [17, (12.2)]) to the system of equations
given by

  * •

$\Lambda_{p,\tilde{S}}^{\xi}$ is a Shalika functional on
$I_{\mathbf{G}}(\xi)$, for all $\xi\in D$,

  * •

$\Lambda_{p,\tilde{S}}^{\xi}$ satisfies (14).

Indeed, by Proposition 5.11, there exists a unique normalised Shalika
functional $\Lambda_{p,\tilde{S}}^{\xi}$, given by the period integral of
Lemma 5.8, for $\xi\in D$, which satisfies that ${\rm Re}(z_{1})>{\rm
Re}(z_{2})+1$ and ${\rm Re}(z_{2})>0$. The latter conditions define an open
subset of $D$ \- here we are identifying $D$ with the variety ${\bf C}^{2}$ by
sending $\xi\mapsto(z_{1},z_{2})$. We can thus apply Bernstein’s theorem to
deduce that there exists a unique non-zero Shalika functional
$\Lambda_{p,\tilde{S}}^{\xi}$ which satisfies (14) for almost all $\xi\in D$.
This has the further property that, if $\\{\phi_{\xi}\\}_{\xi\in D}$ is a
rational family of functions $\phi_{\xi}\in I_{\mathbf{G}}(\xi)$, then
$\Lambda_{p,\tilde{S}}^{\xi}(\phi_{\xi})$ is a rational function of $\xi\in
D$. This concludes the proof. ∎

######  Remark 5.14.

  1. (1)

Let ${\bf C}[D]$ denote the algebra of regular functions on $D$ and let ${\bf
C}(D)$ be its quotient field. Then, in the statement above, by a rational
function $f$ on $\xi\in D$ we mean an element of $X\otimes_{\bf C}{\bf C}(D)$,
where $X$ is the space of locally constant functions on $\mathbf{G}({\bf
Z}_{p})$ that are left-invariant under $B_{\mathbf{G}}({\bf Z}_{p})$.

  2. (2)

Notice that the exact same argument as in Corollary 5.13 shows the well-known
statement that, if $\\{\phi_{\xi}\\}_{\xi\in D}$ is a rational family of
functions $\phi_{\xi}\in I_{\mathbf{G}}(\xi)$, then the period integral
$T_{\omega}(\phi_{\xi})$ is a rational function on $\xi\in D$, for $\omega\in
W_{\mathbf{G}}$.

###  5.5. Third reduction: the right hand side of (16)

As discussed above, $T^{*}_{\omega^{-1}}\Delta_{\tilde{S},{}^{\omega}\xi}$ is
a constant multiple of $\Delta_{\tilde{S},\xi}$; we will now use the integral
expression for the Shalika model in Proposition 5.11 to calculate the constant
explicitly for $\omega$ ranging through a set of generators of
$W_{\mathbf{H}}$. Then Corollary 5.13 will be employed to extend the result
outside the region of convergence.

Before we go on with the calculations, we need to introduce some notation. For
$\alpha\in\Phi_{\mathbf{G}}^{+}$, denote by $N^{\alpha}\subset U_{\mathbf{G}}$
the one parameter unipotent subgroup associated to it. Explicitly, if $\alpha$
equals $2\alpha_{2}-\alpha_{0}$, resp. $\alpha_{1}-\alpha_{2}$, we let

$x_{2\alpha_{2}-\alpha_{0}}:\mathbf{G}_{\rm a}\to
N^{2\alpha_{2}-\alpha_{0}},\,y\mapsto\left(\begin{smallmatrix}1&&&\\\ &1&y&\\\
&&1&\\\ &&&1\end{smallmatrix}\right),$ $x_{\alpha_{1}-\alpha_{2}}:{\rm
Res}_{\mathcal{O}_{p}/{\bf Z}_{p}}\mathbf{G}_{\rm a}\to
N^{\alpha_{1}-\alpha_{2}},\,y\mapsto\left(\begin{smallmatrix}1&y&&\\\ &1&&\\\
&&1&-\bar{y}\\\ &&&1\end{smallmatrix}\right).$

We have the following decomposition.

######  Lemma 5.15.

  1. (1)

The element $s_{2}x_{2\alpha_{2}-\alpha_{0}}(y)\in
s_{2}N^{2\alpha_{2}-\alpha_{0}}({\bf Z}_{p})$ factors as

$\left(\begin{smallmatrix}1&&&\\\ &&1&\\\ &-1&-y&\\\
&&&1\end{smallmatrix}\right)=\left(\begin{smallmatrix}1&&&\\\ &-y^{-1}&1&\\\
&&-y&\\\ &&&1\end{smallmatrix}\right)\left(\begin{smallmatrix}1&&&\\\ &1&&\\\
&y^{-1}&1&\\\ &&&1\end{smallmatrix}\right)\in B_{\mathbf{G}}({\bf
Q}_{p})\tilde{S}({\bf Q}_{p}),$

if $y\neq 0$.

  2. (2)

The element $s_{1}x_{\alpha_{1}-\alpha_{2}}(y)\in
s_{1}N^{\alpha_{1}-\alpha_{2}}({\bf Z}_{p})$ factors as

$\displaystyle\left(\begin{smallmatrix}&1&&\\\ 1&y&&\\\ &&&1\\\
&&1&-\overline{y}\end{smallmatrix}\right)=\left(\begin{smallmatrix}-(y\overline{y}+b_{y})^{-1}&\overline{y}(y\overline{y}+b_{y})^{-1}&&\\\
&1&&\\\ &&(y\overline{y}+b_{y})^{-1}&y(y\overline{y}+b_{y})^{-1}\\\
&&&-1\end{smallmatrix}\right)\left(\begin{smallmatrix}\overline{y}&-b_{y}&&\\\
1&y&&\\\ &&y&b_{y}\\\ &&-1&\overline{y}\end{smallmatrix}\right)\in
B_{\mathbf{G}}({\bf Q}_{p})\tilde{S}({\bf Q}_{p}),$

where $y=a_{y}+\delta b_{y}$, if $y\overline{y}+b_{y}\neq 0$.

###### Proof.

It follows from a direct matrix calculation. ∎

To compute the ratio of $T^{*}_{\omega^{-1}}\Delta_{\tilde{S},{}^{\omega}\xi}$
by $\Delta_{\tilde{S},\xi}$, it is enough to evaluate it at the element
$\mathbf{1}_{\rm Iw}$. Recall that, thanks to our normalisation (14),
$\Delta_{\tilde{S},\xi}(\mathbf{1}_{\rm Iw})=1$.

Finally, recall that the character $\xi$ is of the form (9):

$\xi=(\chi_{0}\circ N_{E_{p}/{\bf Q}_{p}},\chi_{2}\chi_{0}\circ N_{E_{p}/{\bf
Q}_{p}},\chi_{1}\chi_{E_{p}/{\bf Q}_{p}}),$

for some regular unramified character $\chi$ of $T_{\mathbf{H}}({\bf Q}_{p})$.

######  Proposition 5.16.

We obtain the following equality:

$\displaystyle
T^{*}_{s_{2}^{-1}}\Delta_{\tilde{S},{}^{s_{2}}\xi}(\mathbf{1}_{{\rm
Iw}})=-\chi^{-1}(a_{\alpha_{1}-\alpha_{2}})c_{2\alpha_{2}-\alpha_{0}}(\xi),$

where we recall that $c_{\alpha_{1}-\alpha_{2}}(\xi)$ is the constant defined
in Definition 5.6.

###### Proof.

By [6, Theorem 3.4], we have

$\displaystyle T_{s_{2}^{-1}}(P_{{}^{s_{2}}\xi}(\mathbf{1}_{\rm
Iw}))=(c_{2\alpha_{2}-\alpha_{0}}({}^{s_{2}}\xi)-1)P_{{}^{s_{2}}\xi}(\mathbf{1}_{{\rm
Iw}})+p^{-1}P_{{}^{s_{2}}\xi}(\mathbf{1}_{{\rm Iw}s_{2}{\rm Iw}}).$ (18)

Applying $\Delta_{\tilde{S},\xi}$ to (18) and recalling that
$\Delta_{\tilde{S},\xi}(\mathbf{1}_{\rm Iw})=1$, we get

$\displaystyle
T^{*}_{s_{2}^{-1}}\Delta_{\tilde{S},{}^{s_{2}}\xi}(\mathbf{1}_{{\rm
Iw}})=c_{2\alpha_{2}-\alpha_{0}}({}^{s_{2}}\xi)-1+p^{-1}\Delta_{\tilde{S},\xi}(\mathbf{1}_{{\rm
Iw}s_{2}{\rm Iw}}).$ (19)

We are thus reduced to compute $\Delta_{\tilde{S},\xi}(\mathbf{1}_{{\rm
Iw}s_{2}{\rm Iw}})$. To do so, we first express every element of ${\rm
Iw}s_{2}{\rm Iw}$ as a product of the form $B_{\mathbf{G}}({\bf
Q}_{p})\tilde{S}({\bf Q}_{p})$. We proceed as in [33, Proposition 8.1]. Using
the Iwahori decomposition of ${\rm Iw}$ we find that ${\rm Iw}s_{2}{\rm
Iw}=B_{\mathbf{G}}({\bf Z}_{p})\overline{U}_{\mathbf{G}}(p{\bf
Z}_{p})s_{2}B_{\mathbf{G}}({\bf Z}_{p})\overline{U}_{\mathbf{G}}(p{\bf
Z}_{p})$, with $\overline{U}_{\mathbf{G}}$ the unipotent radical of lower-
triangular matrices. Isolating the one parameter subgroup
$\overline{N}^{\alpha}$ of $\overline{U}_{\mathbf{G}}$ associated to the root
$-\alpha$, with $\alpha=2\alpha_{2}-\alpha_{0}$, we can write it as

$\displaystyle B_{\mathbf{G}}({\bf Z}_{p})\overline{N}^{\alpha}(p{\bf
Z}_{p})\overline{U}_{\mathbf{G}}^{\widehat{\alpha}}(p{\bf
Z}_{p})s_{2}B_{\mathbf{G}}({\bf Z}_{p})\overline{N}^{\alpha}(p{\bf
Z}_{p})\overline{U}_{\mathbf{G}}^{\widehat{\alpha}}(p{\bf Z}_{p}),$

where $\overline{U}_{\mathbf{G}}^{\widehat{\alpha}}$ is the subgroup of
$\overline{U}_{\mathbf{G}}$ generated by the one parameter subgroups of all
the negative roots apart from $-\alpha$. Notice that
$s_{2}^{-1}\overline{U}_{\mathbf{G}}^{\widehat{\alpha}}(p{\bf
Z}_{p})s_{2}=\overline{U}_{\mathbf{G}}^{\widehat{\alpha}}(p{\bf Z}_{p})$,
while $s_{2}^{-1}\overline{N}^{\alpha}(p{\bf Z}_{p})s_{2}=N^{\alpha}(p{\bf
Z}_{p})$, hence

$\displaystyle{\rm Iw}s_{2}{\rm Iw}$ $\displaystyle=B_{\mathbf{G}}({\bf
Z}_{p})s_{2}N^{\alpha}(p{\bf
Z}_{p})\overline{U}_{\mathbf{G}}^{\widehat{\alpha}}(p{\bf
Z}_{p})B_{\mathbf{G}}({\bf Z}_{p})\overline{N}^{\alpha}(p{\bf
Z}_{p})\overline{U}_{\mathbf{G}}^{\widehat{\alpha}}(p{\bf Z}_{p})$
$\displaystyle=B_{\mathbf{G}}({\bf Z}_{p})s_{2}N^{\alpha}(p{\bf
Z}_{p})B_{\mathbf{G}}^{\widehat{\alpha}}({\bf Z}_{p})N^{\alpha}({\bf
Z}_{p})\overline{U}_{\mathbf{G}}^{\widehat{\alpha}}(p{\bf Z}_{p})$
$\displaystyle=B_{\mathbf{G}}({\bf Z}_{p})s_{2}N^{\alpha}({\bf
Z}_{p})\overline{U}_{\mathbf{G}}^{\widehat{\alpha}}(p{\bf Z}_{p}),$

where we wrote $B_{\mathbf{G}}^{\widehat{\alpha}}({\bf
Z}_{p})=T_{\mathbf{G}}({\bf Z}_{p})U_{\mathbf{G}}^{\widehat{\alpha}}({\bf
Z}_{p})$, and used that $\overline{U}_{\mathbf{G}}^{\widehat{\alpha}}(p{\bf
Z}_{p})B_{\mathbf{G}}({\bf Z}_{p})\overline{N}^{\alpha}(p{\bf
Z}_{p})=B_{\mathbf{G}}({\bf
Z}_{p})\overline{U}_{\mathbf{G}}^{\widehat{\alpha}}(p{\bf Z}_{p})$. The above
equality respects the Haar measures on both sides up to the volume $[{\rm
Iw}s_{2}{\rm Iw}:{\rm Iw}]$, which is equal to $p$ (see Example 5.7).

Notice that $\overline{U}_{\mathbf{G}}^{\widehat{\alpha}}(p{\bf Z}_{p})\subset
B_{\mathbf{G}}^{\widehat{\alpha}}({\bf Z}_{p})\tilde{S}({\bf Z}_{p})$. This
fact follows from an explicit computation and the fact that $\tilde{S}({\bf
Z}_{p})$ contains $\overline{N}_{\mathbf{G}}({\bf Z}_{p})$, where
$\overline{N}_{\mathbf{G}}$ is the unipotent radical opposite to
$N_{\mathbf{G}}$. Using this inclusion, we further get

$\displaystyle{\rm Iw}s_{2}{\rm Iw}$ $\displaystyle=B_{\mathbf{G}}({\bf
Z}_{p})s_{2}N^{\alpha}({\bf
Z}_{p})\overline{U}_{\mathbf{G}}^{\widehat{\alpha}}(p{\bf Z}_{p})$
$\displaystyle\subset B_{\mathbf{G}}({\bf Z}_{p})s_{2}N^{\alpha}({\bf
Z}_{p})B_{\mathbf{G}}^{\widehat{\alpha}}({\bf Z}_{p})\tilde{S}({\bf Z}_{p})$
$\displaystyle=B_{\mathbf{G}}({\bf Z}_{p})s_{2}N^{\alpha}({\bf
Z}_{p})\tilde{S}({\bf Z}_{p}).$

Now, assume for a moment that $\xi$ satisfies the hypothesis of Proposition
5.11. Then, the Shalika distribution is given by the formula (15). The
inclusion above, together with the left $B_{\mathbf{G}}({\bf
Z}_{p})$-invariance and right $\tilde{S}({\bf Z}_{p})$-invariance of
$\xi^{-1}\delta_{B_{\mathbf{G}}}^{1/2}(b(x))\chi_{\tilde{S}}^{-1}(\tilde{s}(x))$,
gives

$\Delta_{\tilde{S},\xi}(\mathbf{1}_{{\rm Iw}s_{2}{\rm
Iw}})=p\cdot\int_{s_{2}{N}^{2\alpha_{2}-\alpha_{0}}({\bf
Z}_{p})}\xi^{-1}\delta_{B_{\mathbf{G}}}^{1/2}(b(x))\chi_{\tilde{S}}^{-1}(\tilde{s}(x))dx,$
(20)

where, because of (14), the Haar measure $dx$ is normalised such that
$\mathrm{vol}(B_{\mathbf{G}}({\bf Z}_{p}))=\mathrm{vol}(\tilde{S}({\bf
Z}_{p}))=1$, while $p$ comes from the volume factor $[{\rm Iw}s_{2}{\rm
Iw}:{\rm Iw}]$. Lemma 5.17 below calculates (20); in particular, we have

$\Delta_{\tilde{S},\xi}(\mathbf{1}_{{\rm Iw}s_{2}{\rm
Iw}})=p-1-\xi(a_{2\alpha_{2}-\alpha_{0}}).$

Plugging this into (19), we get

$T^{*}_{s_{2}^{-1}}\Delta_{\tilde{S},{}^{s_{2}}\xi}(\mathbf{1}_{{\rm
Iw}})=c_{2\alpha_{2}-\alpha_{0}}({}^{s_{2}}\xi)-p^{-1}(1+\xi(a_{2\alpha_{2}-\alpha_{0}})).$

Since

$c_{2\alpha_{2}-\alpha_{0}}({}^{s_{2}}\xi)=\frac{1-p^{-1}\xi^{-1}(a_{2\alpha_{2}-\alpha_{0}})}{1-\xi^{-1}(a_{2\alpha_{2}-\alpha_{0}})},$

we can rearrange the expression as

$T^{*}_{s_{2}^{-1}}\Delta_{\tilde{S},{}^{s_{2}}\xi}(\mathbf{1}_{{\rm
Iw}})=-\xi(a_{2\alpha_{2}-\alpha_{0}})c_{2\alpha_{2}-\alpha_{0}}(\xi)=-\chi^{-1}(a_{\alpha_{1}-\alpha_{2}})c_{2\alpha_{2}-\alpha_{0}}(\xi),$

where we have used (9) to identify
$\xi(a_{2\alpha_{2}-\alpha_{0}})=\chi^{-1}(a_{\alpha_{1}-\alpha_{2}})$. This
proves the result for the set of characters which satisfy the hypothesis of
Proposition 5.11. Since $T_{s_{2}^{-1}}(P_{{}^{s_{2}}\xi}(\mathbf{1}_{\rm
Iw}))$ is a rational function on $\xi$ (cf. Remark 5.14), we can apply
Corollary 5.13 to deduce that
$\Lambda_{p,\tilde{S}}\left(T_{s_{2}^{-1}}(P_{{}^{s_{2}}\xi}(\mathbf{1}_{\rm
Iw}))\right)$ is a rational function on $\xi$ and we can so extend the
expression to every $\xi$.

∎

######  Lemma 5.17.

Suppose that $\xi$ satifies the hypothesis of Proposition 5.11; then

$\int_{s_{2}N^{2\alpha_{2}-\alpha_{0}}({\bf
Z}_{p})}(\xi^{-1}\delta^{1/2}_{B_{\mathbf{G}}})(b(x))\chi_{\tilde{S}}^{-1}(\tilde{s}(x))dx=1-p^{-1}-p^{-1}\xi(a_{2\alpha_{2}-\alpha_{0}}).$

###### Proof.

Given $x=s_{2}x_{2\alpha_{2}-\alpha_{0}}(y)\in
s_{2}{N}^{2\alpha_{2}-\alpha_{0}}({\bf Z}_{p})$, Lemma 5.15(1) shows that, if
$y\neq 0$,

$(\xi^{-1}\delta^{1/2}_{B_{\mathbf{G}}})(b(x))\chi_{\tilde{S}}^{-1}(\tilde{s}(x))=|y|^{-1}\xi_{2}(y)\chi_{\delta,p}^{-1}(w_{\rm
op}\tilde{s}(x)w_{\rm
op}^{-1})=|y|^{2z_{2}-1}\psi^{-1}_{p}(2\delta^{2}y^{-1}).$

Since $N^{2\alpha_{2}-\alpha_{0}}({\bf Z}_{p})\simeq{\bf Z}_{p}$, the
integral, away from $\\{0\\}$, reduces to

$\int_{s_{2}N^{2\alpha_{2}-\alpha_{0}}({\bf
Z}_{p})}(\xi^{-1}\delta^{1/2}_{B_{\mathbf{G}}})(b(x))\chi_{\tilde{S}}^{-1}(\tilde{s}(x))dx=\int_{{\bf
Z}_{p}}|y|^{2z_{2}-1}\psi^{-1}_{p}(2\delta^{2}y^{-1})dy,$

where we recall that $dy$ is the additive Haar measure of ${\bf Z}_{p}$. The
previous integral is equal to

$\sum_{j=0}^{\infty}p^{-j(2z_{2}-1)}\int_{(p)^{j}-(p)^{j+1}}\psi_{p}^{-1}(-2\delta^{2}y^{-1})dy.$
(21)

Since the conductor of the character
$y\mapsto\psi_{p}^{-1}(-2\delta^{2}y^{-1})$ is ${\bf Z}_{p}$, the only two
integrals of (21) that contribute to the sum are the ones with $j=0$ and
$j=1$. They are equal to $1-p^{-1}$ and $-p^{-2}$ respectively. Thus the
integral is equal to
$1-p^{-1}-p^{-(2z_{2}+1)}=1-p^{-1}-p^{-1}\xi(a_{2\alpha_{2}-\alpha_{0}})$,
where for the latter equality we have conveniently used the matrix
$a_{2\alpha_{2}-\alpha_{0}}$ introduced in Example 5.7. ∎

For the following discussion, we fix an isomorphism $\mathcal{O}_{p}\simeq{\bf
Z}_{p}^{2}$ of ${\bf Z}_{p}$-modules, for which the topology on
$\mathcal{O}_{p}$ corresponds to the product topology on ${\bf Z}_{p}^{2}$, in
a way that the additive Haar measure $dy$ of $E_{p}$ gets mapped to $dadb$,
where $da$ and $db$ are the additive Haar measures of ${\bf Q}_{p}$ giving
${\bf Z}_{p}$ measure $1$.

######  Proposition 5.18.

We obtain the following equality:

$\displaystyle
T^{*}_{s_{1}^{-1}}\Delta_{\tilde{S},{}^{s_{1}}\xi}(\mathbf{1}_{{\rm
Iw}})=-\chi^{-1}(a_{2\alpha_{2}-\alpha_{0}})c_{\alpha_{1}-\alpha_{2}}(\xi)\frac{1+p^{-1}\chi(a_{2\alpha_{2}-\alpha_{0}})}{1+p^{-1}\chi^{-1}(a_{2\alpha_{2}-\alpha_{0}})}.$

###### Proof.

We prove the formula following the same strategy as in Proposition 5.16. In
particular, by [6, Theorem 3.4], we have

$\displaystyle T_{s_{1}^{-1}}(P_{{}^{s_{1}}\xi}(\mathbf{1}_{\rm
Iw}))=(c_{\alpha_{1}-\alpha_{2}}({}^{s_{1}}\xi)-1)P_{{}^{s_{1}}\xi}(\mathbf{1}_{{\rm
Iw}})+p^{-2}P_{{}^{s_{1}}\xi}(\mathbf{1}_{{\rm Iw}s_{1}{\rm Iw}}).$ (22)

Applying $\Delta_{\tilde{S},\xi}$ to (22), we get

$\displaystyle
T^{*}_{s_{1}^{-1}}\Delta_{\tilde{S},{}^{s_{1}}\xi}(\mathbf{1}_{{\rm
Iw}})=c_{\alpha_{1}-\alpha_{2}}({}^{s_{1}}\xi)-1+p^{-2}\Delta_{\tilde{S},\xi}(\mathbf{1}_{{\rm
Iw}s_{1}{\rm Iw}}).$ (23)

To calculate $\Delta_{\tilde{S},\xi}(\mathbf{1}_{{\rm Iw}s_{1}{\rm Iw}})$
explicitly we use two facts. Firstly, we express every element of ${\rm
Iw}s_{1}{\rm Iw}$ as a product of the form $B_{\mathbf{G}}({\bf
Q}_{p})\tilde{S}({\bf Q}_{p})$. Secondly, we initially assume that $\xi$
satisfies the hypothesis of Proposition 5.11 to use formula (15) for the
Shalika distribution and then extend the result for all $\xi$ by using
Corollary 5.13.

In the exact same way as in Proposition 5.16, we get that ${\rm Iw}s_{1}{\rm
Iw}\subset B_{\mathbf{G}}({\bf Z}_{p})s_{1}N^{\alpha}({\bf
Z}_{p})\tilde{S}({\bf Z}_{p})$. Thus, if $\xi$ satisfies the hypothesis of
Proposition 5.11, we get

$\Delta_{\tilde{S},\xi}(\mathbf{1}_{{\rm Iw}s_{1}{\rm
Iw}})=p^{2}\cdot\int_{s_{1}{N}^{\alpha_{1}-\alpha_{2}}({\bf
Z}_{p})}\xi^{-1}\delta_{B_{\mathbf{G}}}^{1/2}(b(x))\chi_{\tilde{S}}^{-1}(\tilde{s}(x))dx,$
(24)

where the Haar measure $dx$ is normalised such that
$\mathrm{vol}(B_{\mathbf{G}}({\bf Z}_{p}))=\mathrm{vol}(\tilde{S}({\bf
Z}_{p}))=1$, and $p^{2}$ comes from the volume factor $[{\rm Iw}s_{1}{\rm
Iw}:{\rm Iw}]$. By Lemma 5.19 below, we get

$\Delta_{\tilde{S},\xi}(\mathbf{1}_{{\rm Iw}s_{1}{\rm
Iw}})=-1+(1-p)\frac{1-p\chi(a_{2\alpha_{2}-\alpha_{0}})}{1+\chi(a_{2\alpha_{2}-\alpha_{0}})}=-1+p(p-1)\frac{1-p^{-1}\chi^{-1}(a_{2\alpha_{2}-\alpha_{0}})}{1+\chi^{-1}(a_{2\alpha_{2}-\alpha_{0}})}.$

Hence, (23) becomes