gG_{p}^{\circ}g^{-1}\right)}\operatorname{ch}(gG_{p}^{\circ}),\ g\in G_{p}.$

  2. (2)

The lattice
$\mathcal{J}_{1}(G_{p}/G_{p}^{\circ},\mathbf{Z}[1/p])_{P_{p},\delta_{P}}$
inside $C_{c}^{\infty}(G_{p}/G_{p}^{\circ})_{P_{p},\delta_{P}}$ consists of
$\mathbf{Z}[1/p]$-linear combinations of elements of the form

$\frac{1}{\operatorname{vol}_{P_{p}}\left(P_{p}(1)\cap
gG_{p}^{\circ}g^{-1}\right)}\operatorname{ch}(gG_{p}^{\circ}),\ g\in G_{p}$

where $P_{p}(1)$ is the open subgroup of $P_{p}$ given by $\\{x\in P_{p}\ |\
\det(x)\in\mathbf{Z}_{p}^{\times},\det(x)\equiv 1\mod p\\}$.

######  Lemma 6.1.3.

The map $\Phi$ identifies the lattices

$\displaystyle C_{c}^{\infty}(P_{p}\backslash
G_{p}/G_{p}^{\circ},\mathbf{Z}[1/p])$
$\displaystyle\simeq_{\Phi}\mathcal{J}(G_{p}/G_{p}^{\circ},\mathbf{Z}[1/p])_{P_{p},\delta_{P}}$
$\displaystyle C_{c}^{\infty,1}(P_{p}\backslash
G_{p}/G_{p}^{\circ},\mathbf{Z}[1/p])$
$\displaystyle\simeq_{\Phi}\mathcal{J}_{1}(G_{p}/G_{p}^{\circ},\mathbf{Z}[1/p])_{P_{p},\delta_{P}}.$

###### Proof.

The first identification follows at once from the definitions and 6.1.1. Thus
we deal with the second one. It is straightforward to show that every double
coset $P_{p}gG_{p}^{\circ}$ is equal to
$P_{p}(z_{1}\left[\begin{smallmatrix}p^{m}&\\\
&1\end{smallmatrix}\right],z_{2})G_{p}^{\circ}$ with $m\in\mathbf{Z}$ and
$z_{i}\in Z_{p}$, or $P_{p}(z_{1}\left[\begin{smallmatrix}p^{m}&p^{-n}\\\
&1\end{smallmatrix}\right],z_{2})G_{p}^{\circ}$ with $n\in\mathbf{Z}_{\geq
1},m\in\mathbf{Z},m>-n$ and $z_{i}\in Z_{p}$. By definition of the two
lattices and the map $\Phi$, it suffices to show that

$\displaystyle\tfrac{1}{p-1}\cdot\operatorname{vol}_{P_{p}}\left(P_{p}\cap
gG_{p}^{\circ}g^{-1}\right)$
$\displaystyle=\operatorname{vol}_{P_{p}}\left(P_{p}(1)\cap
gG_{p}^{\circ}g^{-1}\right),\ g=\left(z_{1}\left[\begin{smallmatrix}p^{m}&\\\
&1\end{smallmatrix}\right],z_{2}\right)$
$\displaystyle\operatorname{vol}_{P_{p}}\left(P_{p}\cap
gG_{p}^{\circ}g^{-1}\right)$
$\displaystyle=\operatorname{vol}_{P_{p}}\left(P_{p}(1)\cap
gG_{p}^{\circ}g^{-1}\right),\
g=\left(z_{1}\left[\begin{smallmatrix}p^{m}&p^{-n}\\\
&1\end{smallmatrix}\right],z_{2}\right).$

We firstly treat the first case, in which we have $P_{p}\cap
gG_{p}^{\circ}g^{-1}=P_{p}^{\circ}\cap\left[\begin{smallmatrix}p^{m}&\\\
&1\end{smallmatrix}\right]H_{p}^{\circ}\left[\begin{smallmatrix}p^{m}&\\\
&1\end{smallmatrix}\right]^{-1}.$ A quick matrix calculation shows that this
is nothing more than $P_{p}^{\circ}$ if $m\leq 0$ and
$\left[\begin{smallmatrix}\mathbf{Z}_{p}^{\times}&p^{m}\mathbf{Z}_{p}\\\
&1\end{smallmatrix}\right]$ if $m>0$. The former has volume $1$ and the latter
has volume $\frac{1}{p^{m}}$. On the other hand, if $m\leq 0$ then
$P_{p}(1)\cap gG_{p}^{\circ}g^{-1}=P_{p}(1)\cap
P_{p}^{\circ}=\left[\begin{smallmatrix}1+p\mathbf{Z}_{p}&\mathbf{Z}_{p}\\\
&1\end{smallmatrix}\right]$ which clearly has volume $\frac{1}{p-1}$. If $m>0$
then $P_{p}(1)\cap
gG_{p}^{\circ}g^{-1}=\left[\begin{smallmatrix}1+p\mathbf{Z}_{p}&p^{m}\mathbf{Z}_{p}\\\
&1\end{smallmatrix}\right]$ which has volume $\frac{1}{(p-1)p^{m}}$ as
required. For the second case, note that $m+n$ is strictly greater than zero.
Another quick matrix computation shows that in this case the two subgroups are
identical and given by
$\left[\begin{smallmatrix}1+p^{n+m}\mathbf{Z}_{p}&\mathbf{Z}_{p}\\\
&1\end{smallmatrix}\right]$ if $m\leq 0$ and
$\left[\begin{smallmatrix}1+p^{n+m}\mathbf{Z}_{p}&p^{m}\mathbf{Z}_{p}\\\
&1\end{smallmatrix}\right]$ if $m>0$. ∎

We now let $\pi_{p,1},\pi_{p,2}$ be unramified Whittaker type
$H_{p}$-representations with $\omega_{\pi_{p,1}}\neq\omega_{\pi_{p,2}}^{-1}$.
One then has a surjective linear map by acting on the spherical vector

$C_{c}^{\infty}(G_{p}/G_{p}^{\circ})_{P_{p},\delta_{P}}[\tfrac{1}{1-\mathcal{S}_{p}}]\longrightarrow(\pi_{p,1}\otimes\pi_{p,2})_{P_{p},\delta_{P}}$

where the object on the right denotes the $\delta_{P}$-twisted
$P_{p}$-coinvariants of the representation $\pi_{p,1}\boxtimes\pi_{p,2}$
defined in a similar fashion as
$C_{c}^{\infty}(G_{p}/G_{p}^{\circ})_{P_{p},\delta_{P}}$. 6.1.1 and 4.4.1 then
imply that

$\dim\ \mathrm{Hom}_{P_{p}}(\pi_{p,1}\otimes\pi_{p,2},\delta_{P})\leq 1$

and a non-zero linear form in this space is non-vanishing on the spherical
vector $W_{\pi_{p,1}}^{\mathrm{sph}}\otimes W_{\pi_{p,2}}^{\mathrm{sph}}.$ In
our setup we can actually explicitly write down a non-zero element of this
space using zeta integrals. The construction is similar to that of Section
4.2. Indeed, the assignment

(15)
$\displaystyle\mathscr{D}_{\Pi_{p}}:\Pi_{p}:=\pi_{p,1}\otimes\pi_{p,2}\longrightarrow\mathbf{C},\
W_{1}\otimes W_{2}\mapsto\lim_{s\rightarrow
0}\frac{\int_{\mathbf{Q}_{p}^{\times}}\left(W_{1}\boxtimes
W_{2}\right)(\left[\begin{smallmatrix}a&\\\ &1\end{smallmatrix}\right])\
|a|_{p}^{s-1}\ d^{\times}a}{L(\pi_{p,1}\boxtimes\pi_{p,2},s)}$

is well-defined and gives an element of
$\operatorname{Hom}_{P_{p}}(\pi_{p,1}\otimes\pi_{p,2},\delta_{P})$, which
takes the value $L(\omega_{\pi_{p,1}\boxtimes\pi_{p,2}},0)^{-1}$ on the
normalized spherical vector. This is clearly non-zero since
$\omega_{\pi_{p,1}}\neq\omega_{\pi_{p,2}}^{-1}$.

######  Remark 6.1.4.

It is worth noting that one can now obtain equality in 3.2.4 for all
unramified Whittaker type $H_{p}$-representations,
$\pi_{p,1},\pi_{p,2},\pi_{p,3}$ with the product of their central characters
being trivial. This revolves around using Frobenius reciprocity and a property
of the mirabolic subgroup of $\operatorname{GL}_{2}$: Let
$\pi_{p,1},\pi_{p,2},\pi_{p,3}$ be unramified Whittaker type
$H_{p}$-representations with the product of their central characters being
trivial. As usual, we write $I(\chi_{p,1},\chi_{p,2})$ for the normalized
induction of $\left[\begin{smallmatrix}\chi_{p,1}&\\\
&\chi_{p,2}\end{smallmatrix}\right]$. We may assume after twisting that
$\pi_{p,3}$ has trivial central character. Hence,
$\omega_{\pi_{p,1}}\omega_{\pi_{p,2}}=1$ and $\pi_{p,3}$ is isomorphic to
$I(|\cdot|_{p}^{-a},|\cdot|_{p}^{a})$ with $a\in\mathbf{C}$ different to
$\tfrac{1}{2}$. However, using the fact that $B_{p}=Z_{p}\times P_{p}$, one
can check that

$\pi_{p,3}^{\vee}\simeq
I(|\cdot|_{p}^{a},|\cdot|_{p}^{-a})\simeq\left(\mathrm{Ind}_{P_{p}}^{H_{p}}\left(\delta_{P}^{a+1/2}\right)\right)_{1}$

where the subscript on the right denotes the eigenspace of the trivial
character with respect to the action of the center. Thus, we have a chain of
isomorphisms

$\displaystyle\mathrm{Hom}_{H_{p}}(\pi_{p,1}\otimes\pi_{p,2}\otimes\pi_{p,3},\mathbf{1})$
$\displaystyle\simeq\mathrm{Hom}_{H_{p}}\left(\pi_{p,1}\otimes\pi_{p,2},I(|\cdot|_{p}^{a},|\cdot|_{p}^{-a})\right)$
$\displaystyle\simeq\mathrm{Hom}_{P_{p}}\left(\pi_{p,1}\otimes\pi_{p,2},\delta_{P}^{a+1/2}\right)$
$\displaystyle\simeq\mathrm{Hom}_{P_{p}}\left(\pi_{p,1}\otimes\pi_{p,2}\otimes\delta_{P}^{-a+1/2},\delta_{P}\right)$

where the second isomorphism follows by Frobenius reciprocity for the non-
compact induction (e.g [31] III $2.5$) and the fact that
$\omega_{\pi_{p,1}}\omega_{\pi_{p,2}}=1$. However, as a
$P_{p}$-representation, $\pi_{p,1}\otimes\pi_{p,2}\otimes\delta_{P}^{-a+1/2}$
is isomorphic to $\pi_{p,1}\otimes\sigma_{p,2}$ where $\sigma_{p,2}$ is the
unramified Whittaker type $H_{p}$-representation given by
$\pi_{p,2}\otimes|\det|_{p}^{-a+1/2}$. Finally, as a quasi-character of
$\mathbf{Q}_{p}^{\times}$, we have
$\omega_{\pi_{p,1}}\omega_{\sigma_{p,2}}=|\cdot|_{p}^{-2a+1}$ and this non-
trivial since $a\neq\tfrac{1}{2}$. Thus we have shown that

$\mathrm{Hom}_{H_{p}}(\pi_{p,1}\otimes\pi_{p,2}\otimes\pi_{p,3},\mathbf{1})\simeq\mathrm{Hom}_{P_{p}}(\pi_{p,1}\otimes\sigma_{p,2},\delta_{P})$

where $\pi_{p,1}$ and $\sigma_{p,2}$ are unramified Whittaker type
$H_{p}$-representations with $\omega_{\pi_{p,1}}\omega_{\sigma_{p,2}}\neq 1$.
But for each such pair, as we already noted, the explicit integral
representation (15) above, gives a non-zero element of the Hom-space on the
right. Unlike [29] and [12], this approach does not distinguish between
different cases of representations within this family. Of course it does not
deal with ramified representations, but it does however give a unified proof
for the whole of family of unramified Whittaker type representations, making
full use of these two properties.

It is also worth noting that $\Lambda_{\Pi_{p}}$ and $\mathscr{D}_{\Pi_{p}}$
are related through the following commutative diagram

${{C_{c}^{\infty}(G_{p}/G_{p}^{\circ})_{P_{p},\delta_{P}}[\tfrac{1}{1-\mathcal{S}_{p}}]}}$${{(\Pi_{p})_{P_{p},\delta_{P}}}}$${{\mathbf{C}\simeq(\Pi_{p}^{\vee})^{G_{p}^{\circ}}}}$${{C_{c}^{\infty}(P_{p}\backslash
G_{p}/G_{p}^{\circ})[\tfrac{1}{1-\mathcal{S}_{p}}]}}$$\scriptstyle{\mathscr{D}_{\Pi_{p}}}$$\scriptstyle{\simeq_{\Phi}}$$\scriptstyle{\Lambda_{\Pi_{p}}}$

Passing to an integral level, we have the following result.

######  Theorem 6.1.5.

Let $\Pi_{p}:=\pi_{p,1}\otimes\pi_{p,2}$ be as above, and assume that the
spherical Hecke eigensystem $\Theta_{\Pi_{p}}$ takes values in $\mathcal{R}$
when restricted to $\mathcal{H}_{G_{p}}^{\circ}(\mathcal{R})$, for some
$\mathbf{Z}[1/p]$-algebra $\mathcal{R}$. Let $\mathcal{Z}$ be a non-zero
linear form in $\mathrm{Hom}_{P_{p}}(\Pi_{p},\delta_{P})$, normalized to take
the value $L(\omega_{\Pi_{p}},0)^{-1}$ on $W_{\pi_{p,1}}^{\mathrm{sph}}\otimes
W_{\pi_{p,2}}^{\mathrm{sph}}$. Then:

  1. (1)

The linear form $\mathcal{Z}$ is valued in $\mathcal{R}$ on the lattice

$\mathrm{span}_{\mathcal{R}}\left\\{\frac{1}{\operatorname{vol}_{P_{p}}(P_{p}\cap
gG_{p}^{\circ}g^{-1})}\ g\cdot(W_{\pi_{p,1}}^{\mathrm{sph}}\otimes
W_{\pi_{p,2}}^{\mathrm{sph}})\ |\ g\in
G_{p}\right\\}\subseteq\pi_{p,1}\otimes\pi_{p,2}.$

  2. (2)

The linear form $\mathcal{Z}$ is valued in the ideal
$\left(p-1,L(\Pi_{p},0)^{-1}\right)\subseteq\mathcal{R}$ on the lattice

$\mathrm{span}_{\mathcal{R}}\left\\{\frac{1}{\operatorname{vol}_{P_{p}}(P_{p}(1)\cap
gG_{p}^{\circ}g^{-1})}\ g\cdot(W_{\pi_{p,1}}^{\mathrm{sph}}\otimes
W_{\pi_{p,2}}^{\mathrm{sph}})\ |\ g\in
G_{p}\right\\}\subseteq\pi_{p,1}\otimes\pi_{p,2}.$

###### Proof.

By multiplicity one described above, $\mathcal{Z}$ is already completely
determined and coincides with $\mathscr{D}_{\Pi_{p}}$. The value of
$\mathscr{D}_{\Pi_{p}}$ on an element of the lattice in question can be
expressed as an $\mathcal{R}$-linear combination of elements of the form
$\mathscr{D}_{\Pi_{p}}\left(\xi\cdot(W_{\pi_{p,1}}^{\mathrm{sph}}\otimes
W_{\pi_{p,2}}^{\mathrm{sph}})\right)$ for some
$\xi\in\mathcal{J}(G_{p}/G_{p}^{\circ},\mathbf{Z}[1/p])_{P_{p},\delta_{P}}$.
Then $\xi$ corresponds to some $f$ in $C_{c}^{\infty}(P_{p}\backslash
G_{p}/G_{p}^{\circ},\mathbf{Z}[1/p])$ via $\Phi$. The first part then follows
from 4.3.8 and the commutative diagram above. For the second part, using the
same argument as for the first part, the value of $\mathscr{D}_{\Pi_{p}}$
evaluated on an element of the lattice in question is an $\mathcal{R}$-linear
combination of elements of the form $\Lambda_{\Pi_{p}}(f)$ where now $f\in
C_{c}^{\infty,1}(P_{p}\backslash G_{p}/G_{p}^{\circ},\mathbf{Z}[1/p])$. But by
4.4.1, and 4.4.5, the corresponding Hecke operator
$\mathcal{P}_{f,\mathrm{loc}}$, is given by
$\tfrac{-\mathcal{S}_{p}\mathcal{Q}_{f}^{{}^{\prime}}}{1-\mathcal{S}_{p}}$
where
$\mathcal{Q}_{f}\in(p-1,\mathcal{P}_{p}(1))\subseteq\mathcal{H}_{G_{p}}^{\circ}(\mathbf{Z}[1/p])$
. Hence by 4.2.5,
$\Lambda_{\Pi_{p}}(f)=\Lambda_{\Pi_{p}}(\mathcal{P}_{f,\mathrm{loc}}\cdot
f_{0})=\Theta_{\Pi_{p}}(\mathcal{Q}_{f})\Theta_{\Pi_{p}}(1-\mathcal{S}_{p})^{-1}\Lambda_{\Pi_{p}}(f_{0})=\Theta_{\Pi_{p}}(\mathcal{Q}_{f})$.
This gives the result. ∎

####  6.1.1. Necessity of inverting  $\displaystyle 1-\mathcal{S}_{p}$ via an
$\mathrm{Ext}$-vanishing result of Prasad

Recall 4.4.1 which states that the module $C_{c}^{\infty}(P_{p}\backslash
G_{p}/G_{p}^{\circ})[\frac{1}{1-\mathcal{S}_{p}}]$ is free of rank one over
$\mathcal{H}_{G_{p}}^{\circ}[\frac{1}{1-\mathcal{S}_{p}}]$, generated by
$\operatorname{ch}(P_{p}G_{p}^{\circ})$. A natural thing to ask is whether the
localisation at the Hecke operator  $\displaystyle 1-\mathcal{S}_{p}$ is
optimal and necessary. In other words is the module
$C_{c}^{\infty}(P_{p}\backslash G_{p}/G_{p}^{\circ})$ generated by
$\operatorname{ch}(P_{p}G_{p}^{\circ})$ over $\mathcal{H}_{G_{p}}^{\circ}$. As
the title of this section suggests, the answer is no. Indeed, if
$C_{c}^{\infty}(P_{p}\backslash G_{p}/G_{p}^{\circ})$ was cyclic as a
$\mathcal{H}_{G_{p}}^{\circ}$-module, then our earlier argument would show
that

$\dim\ \mathrm{Hom}_{P_{p}}(\pi_{p,1}\otimes\pi_{p,2},\delta_{P})\leq 1$

for every unramified Whittaker type $\pi_{p,1},\pi_{p,2}$. In particular,

$\dim\ \mathrm{Hom}_{P_{p}}(\pi_{p}\otimes\pi_{p}^{\vee},\delta_{P})\leq 1$

for every unramified irreducible principal-series $\pi_{p}$ with
$\omega_{\pi_{p}}=1$. By the same argument as the one in 6.1.4, one can deduce
that the Hom-space above is nothing more than

$\mathrm{Hom}_{H_{p}}\left(\pi_{p}\otimes\pi_{p}^{\vee}\otimes\mathrm{ind}_{B_{p}}^{H_{p}}\delta_{B}^{-1/2},\mathbf{1}\right)=\mathrm{Hom}_{\mathrm{PGL_{2}(\mathbf{Q}_{p}})}\left(\pi_{p}\otimes\pi_{p}^{\vee}\otimes\mathrm{ind}_{B_{p}}^{H_{p}}\delta_{B}^{-1/2},\mathbf{1}\right).$

It is well-known that the non-Whittaker type principal-series
$\mathrm{ind}_{B_{p}}^{H_{p}}\delta_{B}^{-1/2}$ fits into a short exact
sequence of $\mathrm{PGL_{2}(\mathbf{Q}_{p}})$-modules

$0\longrightarrow\mathbf{1}\longrightarrow\mathrm{ind}_{B_{p}}^{H_{p}}\delta_{B}^{-1/2}\longrightarrow\mathrm{St}\longrightarrow
0.$

Tensoring with $\pi_{p}$ and taking
$\mathrm{Ext}_{\mathrm{PGL}_{2}(\mathbf{Q}_{p})}^{*}\left(-,\pi_{p}\right)$,
we obtain a long exact sequence

(16) $\displaystyle
0\longrightarrow\mathrm{Hom}_{\mathrm{PGL_{2}(\mathbf{Q}_{p}})}\left(\pi_{p}\otimes\pi_{p}^{\vee}\otimes\mathrm{St},\mathbf{1}\right)$
$\displaystyle\longrightarrow\mathrm{Hom}_{\mathrm{PGL_{2}(\mathbf{Q}_{p}})}\left(\pi_{p}\otimes\pi_{p}^{\vee}\otimes\mathrm{ind}_{B_{p}}^{H_{p}}\delta_{B}^{-1/2},\mathbf{1}\right)$
$\displaystyle\longrightarrow\mathrm{Hom}_{\mathrm{PGL}_{2}(\mathbf{Q}_{p})}\left(\pi_{p}\otimes\pi_{p}^{\vee},\mathbf{1}\right)\longrightarrow\mathrm{Ext}_{\mathrm{PGL}_{2}(\mathbf{Q}_{p})}^{1}\left(\pi_{p}\otimes\mathrm{St},\pi_{p}\right).$

By ([30] Proposition $2.6$) and self-duality of $\mathrm{St}$, the
$\mathrm{Ext}$-group appearing above is naturally isomorphic to
$\mathrm{Ext}_{\mathrm{PGL}_{2}(\mathbf{Q}_{p})}^{1}\left(\pi_{p}\otimes\pi_{p}^{\vee},\mathrm{St}\right)$.
By the Schneider-Stuhler duality theorem ([30] Theorem $8.1$) and the fact
that
$\mathrm{Ext}_{\mathrm{PGL}_{2}(\mathbf{Q}_{p})}^{i}\left(\pi_{p}\otimes\pi_{p}^{\vee},\mathrm{St}\right)=0$
for $i>1$ (which is the split rank of $\mathrm{PGL_{2}})$, we have

$\mathrm{Ext}_{\mathrm{PGL}_{2}(\mathbf{Q}_{p})}^{1}\left(\pi_{p}\otimes\pi_{p}^{\vee},\mathrm{St}\right)\simeq\operatorname{Hom}_{\mathrm{PGL}_{2}(\mathbf{Q}_{p})}\left(\mathbf{1},\pi_{p}\otimes\pi_{p}^{\vee}\right).$

However, this space vanishes by ([30] Proposition $8.1$). The space
$\mathrm{Hom}_{\mathrm{PGL}_{2}(\mathbf{Q}_{p})}\left(\pi_{p}\otimes\pi_{p}^{\vee},\mathbf{1}\right)$
is at least one dimensional and
$\mathrm{Hom}_{\mathrm{PGL}_{2}(\mathbf{Q}_{p})}\left(\pi_{p}\otimes\pi_{p}^{\vee}\otimes\mathrm{St},\mathbf{1}\right)$
is precisely one dimensional, where the second fact follows from the main
result of [29]. Thus going back to (16), we finally deduce that

$\dim\
\mathrm{Hom}_{\mathrm{PGL}_{2}(\mathbf{Q}_{p})}\left(\pi_{p}\otimes\pi_{p}^{\vee}\otimes\mathrm{ind}_{B_{p}}^{H_{p}}\delta_{B}^{-1/2},\mathbf{1}\right)\geq
2$

Of course this contradicts the bound $\dim\
\mathrm{Hom}_{P_{p}}\left(\pi_{p}\otimes\pi_{p}^{\vee},\delta_{P}\right)\leq
1$. Thus the $\mathcal{H}_{G_{p}}^{\circ}$-module
$C_{c}^{\infty}(P_{p}\backslash G_{p}/G_{p}^{\circ})$ is indeed not generated
by $\operatorname{ch}(P_{p}G_{p}^{\circ})$ without localising at the Hecke
operator  $\displaystyle 1-\mathcal{S}_{p}$ .

####  6.1.2. Relation to branching laws for $\ell$-modular representations

We fix a prime $\ell\neq p$ and we identify the residue field of
$\overline{\mathbf{Z}}_{\ell}$ with $\overline{\mathbf{F}}_{\ell}$. We fix a
choice of $p^{i}$ in $\overline{\mathbf{F}}_{\ell}$ for each $i\in\mathbf{Z}$,
compatible with the projection from $\overline{\mathbf{Z}}_{\ell}$ to
$\overline{\mathbf{F}}_{\ell}$. The character $\delta_{P}$ is still given
(using different notation) by $\delta_{P}(\left[\begin{smallmatrix}a&b\\\
&1\end{smallmatrix}\right])=p^{-v_{p}(a)}$. We have algebra and module
structures on $\mathcal{H}_{G_{p}}^{\circ}(\overline{\mathbf{F}}_{\ell})$,
$C_{c}^{\infty}(P_{p}\backslash
G_{p}/G_{p}^{\circ},\overline{\mathbf{F}}_{\ell}),C_{c}^{\infty}(G_{p}/G_{p}^{\circ},\overline{\mathbf{F}}_{\ell})_{P_{p},\delta_{P}}$
by lifting to the corresponding $\overline{\mathbf{Z}}_{\ell}$-points and then
projecting back down. Clearly this is independent of the choice of lifts and
thus well-defined.

As in ([11] Section $3.3$) we say that an unramified
$\overline{\mathbf{F}}_{\ell}$-linear $H_{p}$-representation $\pi_{p}$ is non-
degenerate if $\dim\ (\pi_{p})^{H_{p}^{\circ}}=1$ and
$(\pi_{p})^{H_{p}^{\circ}}$ generates $\pi_{p}$. For such a representation we
still write $W_{\pi_{p}}^{\mathrm{sph}}$ for its spherical vector. The
irreducible $\overline{\mathbf{F}}_{\ell}$-linear unramified principal-series
are indeed non-degenerate. This is part of [28], but for
$\operatorname{GL}_{2}$, it’s been known before that as mentioned in [2]. For
the reducible cases, it is a bit more subtle compared to the zero
characteristic case (see for example [37]).

######  Theorem 6.1.6.

Suppose $\ell\nmid p(p-1)$ and let $\pi_{p,1},\pi_{p,2}$ be unramified non-
degenerate $\overline{\mathbf{F}}_{\ell}$-linear $H_{p}$-representations with
$\omega_{\pi_{p,1}}(p)\neq\omega_{\pi_{p,2}}^{-1}(p)$ in
$\overline{\mathbf{F}}_{\ell}^{\times}$. Then

$\dim_{\overline{\mathbf{F}}_{\ell}}\
\mathrm{Hom}_{P_{p}}(\pi_{p,1}\otimes\pi_{p,2},\delta_{P})\leq 1$

and any such non-zero linear form is non-vanishing on
$W_{\pi_{p,1}}^{\mathrm{sph}}\otimes W_{\pi_{p,2}}^{\mathrm{sph}}$.

###### Proof.

By fixing an isomorphism $\mathbf{C}\simeq\overline{\mathbf{Q}}_{\ell}$, it
follows from 4.4.3 that $C_{c}^{\infty}(P_{p}\backslash
G_{p}/G_{p}^{\circ},\overline{\mathbf{Z}}_{\ell})[\frac{1}{1-\mathcal{S}_{p}}]$
is cyclic as a
$\mathcal{H}_{G_{p}}^{\circ}(\overline{\mathbf{Z}}_{\ell})[\frac{1}{1-\mathcal{S}_{p}}]$-module
generated by $\operatorname{ch}(P_{p}G_{p}^{\circ})$. From this, it is clear
that $C_{c}^{\infty}(P_{p}\backslash
G_{p}/G_{p}^{\circ},\overline{\mathbf{F}}_{\ell})[\frac{1}{1-\mathcal{S}_{p}}]$
is also cyclic as a
$\mathcal{H}_{G_{p}}^{\circ}(\overline{\mathbf{F}}_{\ell})[\frac{1}{1-\mathcal{S}_{p}}]$-module
generated by $\operatorname{ch}(P_{p}G_{p}^{\circ})$. We have a commutative
square

${{C_{c}^{\infty}(P_{p}\backslash
G_{p}/G_{p}^{\circ},\overline{\mathbf{Z}}_{\ell})}}$${{C_{c}^{\infty}(G_{p}/G_{p}^{\circ},\overline{\mathbf{Z}}_{\ell})_{P_{p},\delta_{P}}}}$${{C_{c}^{\infty}(P_{p}\backslash
G_{p}/G_{p}^{\circ},\overline{\mathbf{F}}_{\ell})}}$${{C_{c}^{\infty}(G_{p}/G_{p}^{\circ},\overline{\mathbf{F}}_{\ell})_{P_{p},\delta_{P}}}}$$\scriptstyle{\nu_{\ell}}$$\scriptstyle{\tau_{\ell}}$$\scriptstyle{\Phi}$$\scriptstyle{\tilde{\Phi}}$

where $\Phi$ is the map from 6.1.1 and $\tilde{\Phi}$ is defined by first
lifting through $\nu_{\ell}$ and then applying $\Phi$ and $\tau_{\ell}$, which
is clearly well-defined. From the volume calculations in the proof of 6.1.3,
the definition of $\Phi$ and the assumption on $\ell\nmid p(p-1)$, it follows
that $\tilde{\Phi}$ is surjective. A diagram chase using injectivity of $\Phi$
also shows that $\tilde{\Phi}$ is injective, and finally
$\mathcal{H}_{G_{p}}^{\circ}(\overline{\mathbf{F}}_{\ell})$-equivariance
follows from the definitions. Thus,
$C_{c}^{\infty}(G_{p}/G_{p}^{\circ},\overline{\mathbf{F}}_{\ell})_{P_{p},\delta_{P}}[\frac{1}{1-\mathcal{S}_{p}}]$
is a cyclic
$\mathcal{H}_{G_{p}}^{\circ}(\overline{\mathbf{F}}_{\ell})[\frac{1}{1-\mathcal{S}_{p}}]$-module
generated by $\operatorname{ch}(G_{p}^{\circ})$. Of course, the linear map
$C_{c}^{\infty}(G_{p}/G_{p}^{\circ},\overline{\mathbf{F}}_{\ell})\rightarrow\Pi_{p}=\pi_{p,1}\boxtimes\pi_{p,2}$
can be defined in a completely algebraic manner via
$\operatorname{ch}(gG_{p}^{\circ})\mapsto g\cdot W_{\Pi_{p}}^{\mathrm{sph}}$.
Then the linear map

$C_{c}^{\infty}(G_{p}/G_{p}^{\circ},\overline{\mathbf{F}}_{\ell})_{P_{p},\delta_{P}}[\tfrac{1}{1-\mathcal{S}_{p}}]\longrightarrow(\Pi_{p})_{P_{p},\delta_{P}},\
\tfrac{\operatorname{ch}(gG_{p}^{\circ})}{(1-\mathcal{S}_{p})^{i}}\mapsto\tfrac{g\cdot
W_{\Pi_{p}}^{\mathrm{sph}}}{(1-\omega_{\Pi_{p}}(p))^{i}},\ g\in
G_{p},i\in\mathbf{Z}_{\geq 0}$

is surjective. By the cyclicity mentioned above and the fact that for such
modular representations the space $(\Pi_{p})^{G_{p}^{\circ}}$ is still one
dimensional, its image is generated by the spherical vector and the result
follows. ∎

######  Corollary 6.1.7.

Suppose $\ell\nmid p(p-1)$. Let $\pi_{p,1},\pi_{p,2}$ be unramified non-
degenerate $\overline{\mathbf{F}}_{\ell}$-linear representations of $H_{p}$
and let $\pi_{p,3}$ be an unramified normalized parabolic induction
$\mathrm{ind}_{B_{p}}^{H_{p}}\left[\begin{smallmatrix}\chi_{p,1}\\\
&\chi_{p,2}\end{smallmatrix}\right]$ with $(\chi_{p,1}/\chi_{p,2})(p)\neq p$
in $\overline{\mathbf{F}}_{\ell}^{\times}$. Then

$\dim_{\overline{\mathbf{F}}_{\ell}}\
\mathrm{Hom}_{H_{p}}(\pi_{p,1}\otimes\pi_{p,2}\otimes\pi_{p,3},\mathbf{1})\leq
1.$

If furthermore $\pi_{p,3}$ is also non-degenerate then any such non-zero
linear form does not vanish on the spherical vector
$W_{\pi_{p,1}}^{\mathrm{sph}}\otimes W_{\pi_{p,2}}^{\mathrm{sph}}\otimes
W_{\pi_{p,3}}^{\mathrm{sph}}$.

###### Proof.

Again by twisting we may assume that $\pi_{p,3}$ has trivial central
character, i.e. $\chi_{p,2}=\chi_{p,1}^{-1}$ and
$\omega_{\pi_{p,1}}\omega_{\pi_{p,2}}=1$. in the same manner as 6.1.4, one can
then relate the Hom-space in question to
$\mathrm{Hom}_{P_{p}}(\pi_{p,1}\otimes\sigma_{p,2},\delta_{P})$ where
$\sigma_{p,2}=\pi_{p,2}\otimes\left[(\delta_{P}^{1/2}\chi_{p,1})(\det(-))\right]$
as an $H_{p}$-representation. By assumption, we have $\chi_{p,1}(p)^{2}\neq p$
and thus the product of central characters
$(\omega_{\pi_{p,1}}\omega_{\sigma_{p,2}})(p)=\chi_{p,1}(p)^{2}\cdot
p^{-1}\neq 1$. Then the bound on the dimension follows from 6.1.6. Finally, to
show that if such a non-zero linear form exists then it doesn’t vanish on the
spherical vector (assuming $\pi_{p,3}$ is non-degenerate), one needs to trace
through the various isomorphisms of $\mathrm{Hom}$-spaces in 6.1.4 (Frobenius
reciprocity and duality) while using [37] Lemme $18$ part $(3)$ and its proof,
and the corresponding result of 6.1.6. ∎

###  6.2. Integral lattice in
$\mathcal{S}(\mathbf{Q}_{p}^{2})\otimes\pi_{p,1}\otimes\pi_{p,2}$

As usual, we let $\pi_{p,1},\pi_{p,2}$ be unramified $H_{p}$-representations
of Whittaker type. By 3.2.3 we know that a non-zero $H_{p}$-invariant period
in
$\operatorname{Hom}_{H_{p}}(\mathcal{S}(\mathbf{Q}_{p}^{2})\otimes\pi_{p,1}\otimes\pi_{p,2},\mathbf{1})$
is unique up to scalars and is non-vanishing on the unramified vector
$\operatorname{ch}(\mathbf{Z}_{p}^{2})\otimes
W_{\pi_{p,1}}^{\mathrm{sph}}\otimes W_{\pi_{p,2}}^{\mathrm{sph}}$. We can thus
rescale it so it maps $\operatorname{ch}(\mathbf{Z}_{p}^{2})\otimes
W_{\pi_{p,1}}^{\mathrm{sph}}\otimes W_{\pi_{p,2}}^{\mathrm{sph}}$ to $1$ and
we call such a period normalized. In fact we will see later, when we discuss
an application of this to global Rankin-Selberg integrals, that such a non-
zero period always exists and we will explicitly write it down. For now
though, we need only consider an abstract period of this form, without any
sort of realization of it.

######  Theorem 6.2.1.

Let $\mathcal{Z}$ be the non-zero normalized $H_{p}$-invariant period in
$\operatorname{Hom}_{H_{p}}(\mathcal{S}(\mathbf{Q}_{p}^{2})\otimes\pi_{p,1}\otimes\pi_{p,2},\mathbf{1})$.
Assume that the spherical Hecke eigensystem of $\pi_{p,1}\boxtimes\pi_{p,2}$
restricts to a morphism
$\mathcal{H}_{G_{p}}^{\circ}(\mathcal{R})\rightarrow\mathcal{R}$ for some
$\mathbf{Z}[1/p]$-algebra $\mathcal{R}\subseteq\mathbf{C}$. Then:

  1. (1)

The period $\mathcal{Z}$ is valued in $\mathcal{R}$ on the lattice

  2. (2)

The period $\mathcal{Z}$ is valued in the ideal
$\left(p-1,L(\pi_{p,1}\boxtimes\pi_{p,2},0)^{-1}\right)\subseteq\mathcal{R}$
on the lattice

  3. (3)

If $\omega_{p}:=\omega_{\pi_{p,1}}\omega_{\pi_{p,2}}\neq 1$, the period
$\mathcal{Z}$ satisfies

$L(\omega_{p},0)\mathcal{Z}(-)\in\left(p-1,L(\pi_{p,1}\boxtimes\pi_{p,2},0)^{-1}\right)\subseteq\mathcal{R}$

on the lattice

###### Proof.

The proof relies on the three parts of 5.1.1, in particular the integral
behavior of the Hecke operators $\mathcal{P}_{\delta}$ attached to integrals
elements $\delta$. To obtain the result one combines this with Proposition
4.4.1 of [26], whose proof is formal and adapts to our setup in an identical
fashion. ∎

######  Remark 6.2.2.

Notice how parts $(2)$ and $(3)$ compared to part $(1)$, of 6.2.1, have the
determinant level condition $G_{p}^{\circ}[p]$. Also, note that part $(3)$
compared to part $(2)$ allows for Schwartz functions which might not vanish at
the origin, at the expense of a potential Dirichlet $L$-factor popping up.

######  Remark 6.2.3.

The assumption on the spherical Hecke eigensystem of
$\pi_{p,1}\boxtimes\pi_{p,2}$ is a natural one to make from a global viewpoint
as we will see later when we discuss an application of this to global Rankin-
Selberg integrals for product of modular forms.

###  6.3. Integral lattice in $\pi_{p,1}\otimes\pi_{p,2}\otimes\pi_{p,3}$

For notational convenience we now let $\pi_{p,3}$ be an unramified Whittaker
type $H_{p}$-representation of the form $I(\chi_{p},|\cdot|_{p}^{-1/2})$ such
that $\omega_{\pi_{p,1}}\omega_{\pi_{p,2}}\omega_{\pi_{p,3}}=1$. From 3.2.4 we
know that a non-zero invariant period in
$\operatorname{Hom}_{H_{p}}(\pi_{p,1}\otimes\pi_{p,2}\otimes\pi_{p,3},\mathbf{1})$
is unique up to scalars and is non vanishing on the spherical vector
$W_{\pi_{p,1}}^{\mathrm{sph}}\otimes W_{\pi_{p,2}}^{\mathrm{sph}}\otimes
W_{\pi_{p,3}}^{\mathrm{sph}}$. We can thus rescale it to map this spherical
vector to $1$ and call such a period normalized. In fact by 6.1.4 (or by [12])
such a period always exists. Once again we can make use of the
$H_{p}$-equivariant map

$\mathcal{S}(\mathbf{Q}_{p}^{2})\rightarrow\pi_{p,3},\ \phi\mapsto
f_{\hat{\phi},\chi_{p}}$

constructed in [25], which we describe in this section for completeness: For a
Schwartz function $\phi\in\mathcal{S}(\mathbf{Q}_{p}^{2})$, we write
$\hat{\phi}$ for its Fourier transform

$\hat{\phi}(x,y):=\int_{\mathbf{Q}_{p}}\int_{\mathbf{Q}_{p}}e_{p}(xv-
yu)\phi(u,v)\ dudv$

where $e_{p}$ is the standard additive character of $\mathbf{Q}_{p}$ mapping
$1/p^{n}$ to $\mathrm{exp}(2\pi i/p^{n})$ (this is an explicit description of
the character $\psi$ fixed at the beginning of Section 4) and $du,dv$ denote
the usual additive Haar measure on $\mathbf{Q}_{p}$. Since $\pi_{p,3}$ is
unramified, the character $\chi_{p}$ is of the form $|\cdot|_{p}^{a}$ for some
$a\in\mathbf{C}$. Let $b=a+\tfrac{1}{2}$, then by ([25] $3.2.2$),

$f_{\phi,\chi_{p}}(h):=\lim_{s\rightarrow
0}\frac{|\det(h)|_{p}^{s+a+\tfrac{1}{2}}}{L\left(|\cdot|_{p}^{b},2s+1\right)}\int_{\mathbf{Q}_{p}^{\times}}\phi((0,x)h)|x|_{p}^{2s+b+1}\
d^{\times}x,\ h\in H_{p}$

is well-defined and gives an element of $I(\chi_{p},|\cdot|_{p}^{-1/2})$.
Furthermore, the assignment $\phi\mapsto f_{\hat{\phi},\chi_{p}}$ is
surjective, $H_{p}$-equivariant and by ([26] $3.2.5$) it maps
$\operatorname{ch}(\mathbf{Z}_{p}^{2})$ to $W_{\pi_{p,3}}^{\mathrm{sph}}$.

######  Corollary 6.3.1.

Let $\mathcal{Z}$ be the non-zero normalized invariant period in
$\operatorname{Hom}_{H_{p}}(\pi_{p,1}\otimes\pi_{p,2}\otimes\pi_{p,3},\mathbf{1})$.
Assume that the spherical Hecke eigensystem of $\pi_{p,1}\boxtimes\pi_{p,2}$
restricts to a morphism
$\mathcal{H}_{G_{p}}^{\circ}(\mathcal{R})\rightarrow\mathcal{R}$ for some
$\mathbf{Z}[1/p]$-algebra $\mathcal{R}\subseteq\mathbf{C}$. Then:

  1. (1)

The period $\mathcal{Z}$ is valued in $\mathcal{R}$ on the lattice

  2. (2)

The period $\mathcal{Z}$ is valued in the ideal
$\left(p-1,L(\pi_{p,1}\boxtimes\pi_{p,2},0)^{-1}\right)\subseteq\mathcal{R}$
on the lattice

  3. (3)

If $\omega_{p}:=\omega_{\pi_{p,1}}\omega_{\pi_{p,2}}\neq 1$, the period
$\mathcal{Z}$ satisfies

$L(\omega_{p},0)\mathcal{Z}(-)\in\left(p-1,L(\pi_{p,1}\boxtimes\pi_{p,2},0)^{-1}\right)\subseteq\mathcal{R}$

on the lattice

###### Proof.

This follows from 6.2.1 and the map induced by $f_{(\hat{-}),\chi_{p}}$ at the
level of $\mathrm{Hom}$-spaces. ∎

###  6.4. Rankin-Selberg integral for product of modular forms

####  6.4.1. The local integrals

We once again fix an additive character
$\psi:\mathbf{Q}_{p}\rightarrow\mathbf{C}^{\times}$ of conductor one. As
before, we consider the family of $G_{p}$-representations
$\pi_{p,1}\boxtimes\pi_{p,2}$ where $\pi_{p,1}$ and $\pi_{p,2}$ are unramified
$H_{p}$-representations of Whittaker type, and we identify $\pi_{p,1}$ with
$\mathcal{W}(\pi_{p,1},\psi)$ and $\pi_{p,2}$ with
$\mathcal{W}(\pi_{p,2},\overline{\psi})$.

######  Definition 6.4.1 ([17]).

For
$W_{1}\in\mathcal{W}(\pi_{p,1},\psi),W_{2}\in\mathcal{W}(\pi_{p,2},\overline{\psi})$
and $\phi\in\mathcal{S}(\mathbf{Q}_{p}^{2})$, the zeta integral
$Z(W_{1},W_{2},\phi,s)$ is given by

(17) $\displaystyle Z(\phi,W_{1},W_{2},s)=\int_{N_{p}\backslash
H_{p}}W_{1}(h)W_{2}(h)\phi((0,1)h)|\mathrm{det}(h)|_{p}^{s}\ dh$

######  Proposition 6.4.2.

  1. (1)

The zeta integral of _6.4.1_ converges for $\Re(s)$ large enough and admits
unique meromorphic continuation as a rational function of $p^{s}$.

  2. (2)

As $W_{1},W_{1}$ and $\phi$ vary, the $Z(W_{1}.W_{2},\phi,s)$ generate the
fractional ideal $L(\pi_{p,1}\boxtimes\pi_{p,2},s)\mathbf{C}[p^{s},p^{-s}]$.

###### Proof.

This is one of the main results of [17] where it is stated and proved in
greater generality. ∎

######  Proposition 6.4.3.

Let $\phi_{0}=\operatorname{ch}(\mathbf{Z}_{p}^{2})$. Then,

$Z(\phi_{0},W_{\pi_{p,1}}^{\mathrm{sph}},W_{\pi_{p,1}}^{\mathrm{sph}},s)=L(\pi_{p,2}\boxtimes\pi_{p,2},s).$

###### Proof.

This is well know and the strategy is that of [10] $2.2.2$, using [36] to
evaluate the spherical Whittaker functions in terms of Satake parameters. ∎

######  Definition 6.4.4.

Let $\pi_{p,1},\pi_{p,2}$ be as above. The linear form
$\mathcal{Z}:\mathcal{S}(\mathbf{Q}_{p}^{2})\otimes\pi_{p,1}\otimes\pi_{p,2}\rightarrow\mathbf{C}$
is given by

$\mathcal{Z}(\phi\otimes W_{1}\otimes W_{1})=\lim_{s\rightarrow
0}\frac{Z(\phi,W_{1},W_{2},s)}{L(\pi_{p,1}\boxtimes\pi_{p,2},s)}.$

By 6.4.2, the linear form $\mathcal{Z}$ is well defined. Additionally, 6.4.3
shows that $\mathcal{Z}$ takes the value $1$ on $\phi_{0}\otimes
W_{\pi_{p,1}}^{\mathrm{sph}}\otimes W_{\pi_{p,2}}^{\mathrm{sph}}$. A standard
change of variables also shows that $\mathcal{Z}$ is in fact
$H_{p}$-invariant. Putting everything together, the linear form $\mathcal{Z}$
gives a non-zero invariant period in the space
$\operatorname{Hom}_{H_{p}}(\mathcal{S}(\mathbf{Q}_{p}^{2})\otimes\pi_{p,1}\otimes\pi_{p,2},\mathbf{1})$,
which by 3.2.3, is in fact a basis.

####  6.4.2. The global integral

We fix a prime $\ell$ and an isomorphism
$\iota:\mathbf{C}\simeq\overline{\mathbf{Q}}_{\ell}$. We let $f_{1}$ and
$f_{2}$ be two normalised cuspidal Hecke eigenforms in $S_{k}(\Gamma_{1}(N))$
with $k\geq 2$ and we write $\epsilon_{i}$ for the Nebentype of $f_{i}$. To
these we can associate two irreducible admissible automorphic representations
of $H(\mathbf{A})$ ([8]), which we call
$\pi_{1}=\pi_{\iota,f_{1}}=\otimes_{p}^{{}^{\prime}}\pi_{p,1}$ and
$\pi_{2}=\pi_{\iota,f_{2}}=\otimes_{p}^{{}^{\prime}}\pi_{p,2}$, with
normalisation as in [27]. It is well-known that each local component is
generic, and it is unramified if and only if $p$ doesn’t divide $\infty$ or
$N$. We write $\omega_{1}$ and $\omega_{2}$ for their respective central
characters. We also write $L=L_{f_{1},f_{2}}$ for the number field associated
to $f_{1}$ and $f_{2}$, given by adjoining the Hecke eigenvalues of $f_{1}$
and $f_{2}$ to $\mathbf{Q}$. We write
$\mathbf{L}=\mathbf{L}_{\iota,f_{1},f_{2}}$ for a choice of $\ell$-adic number
field containing the image of $L$ under $\iota$.

To a Schwartz function $\Phi\in\mathcal{S}(\mathbf{A}^{2})$, one can associate
an Eisenstein series using Godement-Jacquet theory ([16]), which is given by

$\displaystyle E(g,s;\Phi)=\sum_{\gamma\in B(\mathbf{Q})\backslash
H(\mathbf{Q})}|\det(\gamma
g)|_{p}^{s}\int_{\mathbf{A}^{\times}}\Phi((0,x)\gamma
g)\omega_{1}(x)\omega_{2}(x)|x|_{p}^{2s}\ d^{\times}x.$

For cusp forms $\varphi_{1}\in\pi_{1}$ and $\varphi_{2}\in\pi_{2}$, one has
the global Rankin-Selbger integral as in [18], which is given by

$\displaystyle
I(s,\Phi,\varphi_{1},\varphi_{2})=\int_{Z(\mathbf{A})H(\mathbf{Q})\backslash
H(\mathbf{A})}\varphi_{1}(g)\varphi_{2}(g)E(g,s;\Phi)\ dg.$

By [18], this is a meromorphic function of $s\in\mathbf{C}$ and satisfies a
functional equation. We let $S$ denote the finite set of places
$S=\\{\infty,p|\ell N\\}$, and write $L^{S}(\pi_{1}\boxtimes\pi_{2},s)$ for
the partial $L$-function

$\displaystyle L^{S}(\pi_{1}\boxtimes\pi_{2},s)=\prod_{p\not\in
S}L(\pi_{p,1}\boxtimes\pi_{p,2},s)$

where the local $L$-factors are as in 6.4.2. Finally, we note that the local
zeta integral

$Z(\Phi_{p},W_{p,1},W_{p,2},s)$

of 6.4.1 still exists for the infinite place and satisfies similar properties
([15]).

######  Theorem 6.4.5.

Let $\varphi_{i}\simeq\otimes_{p}W_{p,i}$ be decomposable cusp forms, with
$W_{p,i}=W_{\pi_{p,i}}^{\mathrm{sph}}$ for $p\not\in S$. Let
$g=(g_{1},g_{2})\in G(\mathbf{A}^{\infty})$, and suppose that
$\Phi=\otimes_{p}\Phi_{p}$ is also decomposable. Write

$\varepsilon_{\ell,f_{1},f_{2}}\left(\Phi,g_{1},g_{2}\right):=\lim_{s\rightarrow
0}\frac{I(s,\Phi,g_{1}\varphi_{1},g_{2}\varphi_{2})}{L^{S}(\pi_{1}\boxtimes\pi_{2},s)\cdot\prod_{p\in
S}Z(\Phi_{p},g_{1,p}W_{p,1},g_{2,p}W_{p,2},s)}\in\overline{\mathbf{Q}}_{\ell}.$

The following are true:

  1. (1)

If $\Phi_{p}$ is valued in
$\operatorname{vol}_{H_{p}}(\mathrm{Stab}_{H_{p}}(\Phi_{p})\cap
g_{p}G_{p}^{\circ}g_{p}^{-1})^{-1}\mathcal{O}_{\mathbf{L}}$ for $p\not\in S$,
then

$\varepsilon_{\ell,f_{1},f_{2}}(\Phi,g_{1},g_{2})\in\mathcal{O}_{\mathbf{L}}.$

  2. (2)

Suppose that the assumption of part $(1)$ holds and let $S_{0}$ denote a
finite set of places away from $S$ for which: $p\equiv
1\mod\ell,(\epsilon_{1}\epsilon_{2})(p)\neq 1$ and $\Phi_{p}$ is valued in
$\operatorname{vol}_{H_{p}}(\mathrm{Stab}_{H_{p}}(\Phi_{p})\cap
g_{p}G_{p}^{\circ}[p]g_{p}^{-1})^{-1}\mathcal{O}_{\mathbf{L}}$ (note the
addition of the determinant level $G_{p}^{\circ}[p]$ instead of
$G_{p}^{\circ}$). Then,

$\left(\prod_{\begin{subarray}{c}p\in S_{0}\ \mathrm{s.t}:\\\
\Phi_{p}\notin\mathcal{S}_{0}(\mathbf{Q}_{p}^{2})\end{subarray}}L(\epsilon_{1}\epsilon_{2},0)^{-1}\right)\cdot\varepsilon_{\ell,f_{1},f_{2}}(\Phi,g_{1},g_{2})\in\prod_{\begin{subarray}{c}p\in
S_{0}\end{subarray}}\left(p-1,L(\pi_{p,1}\boxtimes\pi_{p,2},0)^{-1}\right)\subseteq\mathcal{O}_{\mathbf{L}}.$

In particular, if the level of $f_{1},f_{2}$ satisfies
$(\\#(\mathbf{Z}/N\mathbf{Z})^{\times},\ell)=1$, then this holds without the
bracketed product on the left.

###### Proof.

Firstly note that by the works of [18], [17] and [15], the function inside the
limit admits unique analytic continuation as an entire function of $s$, and
thus the limit makes sense. We write $S^{{}^{\prime}}$ for the finite set of
places outside $S$, consisting of $p$ such that $g_{p}\not\in G_{p}^{\circ}$,
or $\Phi_{p}\neq\operatorname{ch}(\mathbf{Z}_{p}^{2})$. By Section $4$ of
[18], the limit in question coincides with

$\prod_{p\in S^{{}^{\prime}}}\left(\lim_{s\rightarrow
0}\frac{Z(\Phi_{p},g_{1,p}W_{p,1},g_{2,p}W_{p,2},s)}{L(\pi_{p,1}\boxtimes\pi_{p,2},s)}\right)=\prod_{p\in
S^{{}^{\prime}}}\mathcal{Z}(\Phi_{p}\otimes g_{1,p}W_{p,1}\otimes
g_{2,p}W_{p,2})$

where $\mathcal{Z}$ is as in 6.4.4. We are now in a position to apply 6.2.1 at
each prime $p\in S^{{}^{\prime}}$, with the $\mathbf{Z}[1/p]$-algebra
$\mathcal{R}$ taken to be $\mathcal{O}_{\mathbf{L}}$. Thus, it suffices to
show that the local condition on the spherical Hecke eigensystem of each local
component $\pi_{p,1}\boxtimes\pi_{p,2}$ is satisfied. Indeed, this follows
similarly to Theorem $8.2.1$ in [11] using as mentioned before, the
normalizations of [27]. If the last assumption on the level of $f_{1},f_{2}$
holds then the bracketed integral on the left is a unit in
$\mathcal{O}_{\mathbf{L}}$ which concludes the proof. ∎

##  Appendix A Structure of $C_{c}^{\infty}(P_{p}\backslash
G_{p}/G_{p}^{\circ})[\tfrac{1}{1-\mathcal{S}_{p}}]$

This appendix is dedicated to the proof of 4.4.1. As such we refer the reader
back to Section 4 for notation and definitions.

######  Theorem A.0.1.

The module $C_{c}^{\infty}(P_{p}\backslash
G_{p}/G_{p}^{\circ})[\tfrac{1}{1-\mathcal{S}_{p}}]$ is free of rank one over
$\mathcal{H}_{G_{p}}^{\circ}[\tfrac{1}{1-\mathcal{S}_{p}}]$, generated by the
characteristic function $f_{0}:=\operatorname{ch}(P_{p}G_{p}^{\circ})$. In
particular, for $f\in C_{c}^{\infty}(P_{p}\backslash G_{p}/G_{p}^{\circ})$, we
have
$f=\tfrac{-\mathcal{S}_{p}\mathcal{Q}_{f}^{{}^{\prime}}}{1-\mathcal{S}_{p}}\cdot
f_{0}$, where $\mathcal{Q}_{f}$ is the Hecke operator described in _Section
4.4_.

###### Proof.

We firstly show that the module in question is cyclic with $f_{0}$ as a
generator. By (11), it suffices to show that the intersection of the kernels
of the linear forms $\Lambda_{\Pi_{p}}$ as we vary the representation
$\Pi_{p}$, is trivial. For an element in this intersection, we have
$\Lambda_{\Pi_{p}}(f)=\Theta_{\Pi_{p}}(\mathcal{Q}_{f})=0$ for every
$\Pi_{p}$. By the usual density argument, this implies that $\mathcal{Q}_{f}$
is identically zero in $\mathcal{H}_{G_{p}}^{\circ}$. We now claim that
$\mathcal{Q}_{f}=0$ implies that $f=0$. Firstly, for $m,n\in\mathbf{Z}_{\geq
0}$ and $r,t\in\mathbf{Z}$, we put
$f_{m,n}^{r,t}:=\operatorname{ch}(P_{p}p^{r}(p^{t}\left[\begin{smallmatrix}p^{m}&p^{-n}\\\
&1\end{smallmatrix}\right],1)G_{p}^{\circ})$ in
$C_{c}^{\infty}(P_{p}\backslash G_{p}/G_{p}^{\circ})$. We can always write $f$
as

$f=\left(\sum_{i=1}^{N}a_{i}f_{m_{i},0}^{r_{i},t_{i}}\right)+\left(\sum_{j=1}^{M}b_{j}f_{\mu_{j},\nu_{j}}^{\rho_{j},\tau_{j}}\right)$

where:

  * •

$\nu_{j}>0$ for all $j$

  * •

$|m_{0}|\leq|m_{1}|\leq\dots\leq|m_{N}|$; if $|m_{i}|=|m_{i+1}|$ then
$r_{i}\leq r_{i+1}$; if $|m_{i}|=|m_{i+1}|$ and $r_{i}=r_{i+1}$ then
$t_{i}\leq t_{i+1}$.

  * •

$|\mu_{0}|+\nu_{0}\leq|\mu_{1}|+\nu_{1}\leq\dots\leq|\mu_{M}|+\nu_{M}$; if
$|\mu_{j}|+\nu_{j}=|\mu_{j+1}|+\nu_{j+1}$ then $|\mu_{j}|\leq|\mu_{j+1}|$; if
$|\mu_{j}|+\nu_{j}=|\mu_{j+1}|+\nu_{j+1}$ and $|\mu_{j}|=|\mu_{j+1}|$ then
$\rho_{j}\leq\rho_{j+1}$; if
$|\mu_{j}|+\nu_{j}=|\mu_{j+1}|+\nu_{j+1},|\mu_{j}|=|\mu_{j+1}|$ and
$\rho_{j}=\rho_{j+1}$ then $\tau_{j}\leq\tau_{j+1}$.

By 4.3.6 we have

(18)
$\displaystyle\mathcal{Q}_{f}=\left(\sum_{i=1}^{N}a_{i}\mathscr{V}_{r_{i},t_{i}}\mathscr{P}_{m_{i}}\right)+\left(\sum_{j=1}^{M}b_{j}\mathscr{V}_{\rho_{j},\tau_{j}}\left(\mathscr{P}_{\mu_{j}}+\mathscr{Q}_{\mu_{j},\nu_{j}}\mathcal{P}_{p}(1)\right)\right)=0$

in $\mathcal{H}_{G_{p}}^{\circ}=\mathbf{C}[\mathcal{S}_{p,1}^{\pm
1},\mathcal{T}_{p,1},\mathcal{S}_{p,2}^{\pm 1},\mathcal{T}_{p,2}]$. We write
(18) as

(19)
$\displaystyle\left(\sum_{i=1}^{N}a_{i}\mathscr{V}_{r_{i},t_{i}}\mathscr{P}_{m_{i}}\right)+\left(\sum_{j=1}^{M}b_{j}\mathcal{V}_{\rho_{j},\tau_{j}}\mathscr{P}_{\mu_{j}}\right)+\left(\sum_{j=1}^{M}b_{j}\mathscr{V}_{\rho_{j},\tau_{j}}\mathscr{Q}_{\mu_{j},\nu_{j}}\mathcal{P}_{p}(1)\right)=0$

If the rightmost sum in (19) is non-zero, then $\mathcal{P}_{p}(1)$ divides
$\left(\sum_{i=1}^{N}a_{i}\mathscr{V}_{r_{i},t_{i}}\mathscr{P}_{m_{i}}\right)+\left(\sum_{j=1}^{M}b_{j}\mathscr{V}_{\rho_{j},\tau_{j}}\mathscr{P}_{\mu_{j}}\right)$.
We now claim that this cannot happen. In particular, we show that
$\mathcal{P}_{p}(1)$ never divides a non-zero finite sum of the form $\sum c\
(\mathcal{S}_{p,1}\mathcal{S}_{p,2})^{s}\mathcal{S}_{p,1}^{k}\mathscr{P}_{\lambda}$.
Consider an arbitrary sum of this form, which we write as

(20)
$\displaystyle\Delta:=\left(p^{-n/2}\mathcal{P}_{p,1}^{(n)}(1)\sum_{i,j}c_{i,j}(\mathcal{S}_{p,1}\mathcal{S}_{p,2})^{s_{i}}\mathcal{S}_{p,1}^{k_{j}}\right)$
$\displaystyle+\left(p^{-n/2}\mathcal{P}_{p,2}^{(n)}(1)\sum_{i,j}d_{i,j}(\mathcal{S}_{p,1}\mathcal{S}_{p,2})^{\tilde{s}_{i}}\mathcal{S}_{p,1}^{\tilde{k}_{j}}\right)$
(21) $\displaystyle+\text{terms of this form involving
}\scalebox{0.9}{\mbox{$\displaystyle\mathcal{P}_{p,1}^{(e_{1})}(1),\mathcal{P}_{p,2}^{(e_{2})}(1)$}}\text{
with }e_{1},e_{2}<n.$

It is straightforward to compute, using the explicit description of the
spherical Hecke eigensystem of the representations $\Pi_{p}$, that

  * •

$\mathcal{P}_{p}(1)=1-\frac{1}{p}\mathcal{T}_{p,1}\mathcal{T}_{p,2}-\frac{1}{p}\mathcal{S}_{p,1}\mathcal{S}_{p,2}\mathcal{T}_{p,1}\mathcal{T}_{p,2}+\frac{1}{p}\mathcal{S}_{p,1}\mathcal{T}_{p,2}^{2}+\frac{1}{p}\mathcal{S}_{p,2}\mathcal{T}_{p,1}^{2}-2\mathcal{S}_{p,1}\mathcal{S}_{p,2}+\mathcal{S}_{p,1}^{2}\mathcal{S}_{p,2}^{2}$

  * •

$\frac{1}{p^{n/2}}\mathcal{P}_{p,1}^{(n)}(1)=\frac{1}{p^{\lfloor
n/2\rfloor}}\mathcal{T}_{p,1}^{n-1}-\frac{1}{p^{\lfloor
n/2\rfloor}}\mathcal{S}_{p,1}\mathcal{T}_{p,2}\mathcal{T}_{p,1}^{n-2}+O(\mathcal{T}_{p,1}^{n-3})$

  * •

$\frac{1}{p^{n/2}}\mathcal{P}_{p,2}^{(n)}(1)=\frac{1}{p^{\lfloor
n/2\rfloor}}\mathcal{T}_{p,2}^{n-1}-\frac{1}{p^{\lfloor
n/2\rfloor}}\mathcal{S}_{p,2}\mathcal{T}_{p,1}\mathcal{T}_{p,2}^{n-2}+O(\mathcal{T}_{p,2}^{n-3})$

where $O(\mathcal{T}_{p,i}^{e})$ denotes a linear combination of terms of the
form $\mathcal{S}_{p,1}^{a}\mathcal{S}_{p,2}^{b}\mathcal{T}_{p,i}^{c}$ with
$c\leq e$. From this, by looking at the degrees of the $\mathcal{T}_{p,i}$
terms, we can assume that that $n\geq 3$ in (20), since otherwise the non-
divisibility claim holds trivially. It is also important to note that $\Delta$
has no monomials containing $\mathcal{T}_{p,1}^{a}\mathcal{T}_{p,2}^{b}$ with
both $a,b\geq 2$. If $\mathcal{P}_{p}(1)$ divides $\Delta$ in
$\mathcal{H}_{G_{p}}^{\circ}$, then it is not hard to see that we can write

$\Delta=\left(A_{1}\mathcal{T}_{p,1}^{n-3}+A_{2}\mathcal{T}_{p,1}^{n-4}\mathcal{T}_{p,2}+\dots+A_{n-3}\mathcal{T}_{p,1}\mathcal{T}_{p,2}^{n-4}+A_{n-2}\mathcal{T}_{p,2}^{n-3}+C\right)\mathcal{P}_{p}(1)$

where $A_{i}\in\mathbf{C}[\mathcal{S}_{p,1}^{\pm 1},\mathcal{S}_{p,2}^{\pm
1}]$ and $C$ is a $\mathbf{C}[\mathcal{S}_{p,1}^{\pm 1},\mathcal{S}_{p,2}^{\pm
1}]$-linear combination of monomials containing only
$\mathcal{T}_{p,1}^{a}\mathcal{T}_{p,2}^{b}$ with $a+b<n-3$.
Comparing coefficients of $\mathcal{T}_{p,1}^{n-1}$ and
$\mathcal{T}_{p,2}^{n-1}$ in $\Delta$, we get the relations

(22) $\displaystyle A_{1}=\frac{1}{p^{\lfloor
n/2\rfloor-1}}\sum_{i,j}c_{i,j}\mathcal{S}_{p,1}^{s_{i}+k_{j}}\mathcal{S}_{p,2}^{s_{i}-1}\
\text{ and }\ A_{n-2}=\frac{1}{p^{\lfloor
n/2\rfloor-1}}\sum_{i,j}d_{i,j}\mathcal{S}_{p,1}^{\tilde{s}_{i}+\tilde{k}_{j}-1}\mathcal{S}_{p,2}^{\tilde{s}_{i}}.$

The monomials that contribute to $\frac{1}{p^{\lfloor
n/2\rfloor}}\mathcal{S}_{p,1}\mathcal{T}_{p,2}\mathcal{T}_{p,1}^{n-2}$ in
$\Delta$ are $A_{1}\mathcal{T}_{p,1}^{n-3}$ and
$A_{2}\mathcal{T}_{p,1}^{n-4}\mathcal{T}_{p,2}$. Similarly, the monomials
contributing to $\frac{1}{p^{\lfloor
n/2\rfloor}}\mathcal{S}_{p,2}\mathcal{T}_{p,1}\mathcal{T}_{p,2}^{n-2}$ in
$\Delta$ are $A_{n-2}\mathcal{T}_{p,2}^{n-3}$ and
$A_{n-3}\mathcal{T}_{p,1}\mathcal{T}_{p,2}^{n-4}$. Comparing coefficients of
$\mathcal{T}_{p,2}\mathcal{T}_{p,1}^{n-2}$ and
$\mathcal{T}_{p,1}\mathcal{T}_{p,2}^{n-2}$, we get the relations

(23)
$\displaystyle-\frac{1}{p}(1+\mathcal{S}_{p,1}\mathcal{S}_{p,2})A_{1}+\frac{1}{p}\mathcal{S}_{p,2}A_{2}$
$\displaystyle=\frac{-1}{p^{\lfloor
n/2\rfloor}}\sum_{i,j}c_{i,j}\mathcal{S}_{p,1}^{s_{i}+k_{j}+1}\mathcal{S}_{p,2}^{s_{i}}$
$\displaystyle\frac{1}{p}(1+\mathcal{S}_{p,1}\mathcal{S}_{p,2})A_{n-2}+\frac{1}{p}\mathcal{S}_{p,1}A_{n-3}$
$\displaystyle=\frac{-1}{p^{\lfloor
n/2\rfloor}}\sum_{i,j}d_{i,j}\mathcal{S}_{p,1}^{\tilde{s}_{i}+\tilde{k}_{j}}\mathcal{S}_{p,2}^{\tilde{s}_{i}+1}.$

Combining (22) and (23) and simplifying, we obtain the relations

(24) $\displaystyle A_{1}=\mathcal{S}_{p,2}A_{2}\ \text{ and }\
A_{n-2}=\mathcal{S}_{p,2}A_{n-3}.$

We observe that for $n=3$ or $4$, (24) already implies our claim that
$\mathcal{P}_{p}(1)|\Delta$ if and only if $\Delta=0$. Thus we may assume that
$n\geq 5$. Comparing coefficients of
$\mathcal{T}_{p,1}^{a}\mathcal{T}_{p,2}^{b}$ with $a+b=n-1$ and $a,b\geq 2$
(which as we remarked earlier, they vanish), we obtain the following system of
equations:

(25)
$\displaystyle\mathcal{S}_{p,1}A_{1}-(1+\mathcal{S}_{p,1}\mathcal{S}_{p,2})A_{2}+\mathcal{S}_{p,2}A_{3}$
$\displaystyle=0$
$\displaystyle\mathcal{S}_{p,1}A_{2}-(1+\mathcal{S}_{p,1}\mathcal{S}_{p,2})A_{3}+\mathcal{S}_{p,2}A_{4}$
$\displaystyle=0$ $\displaystyle\vdots$
$\displaystyle\mathcal{S}_{p,1}A_{n-4}-(1+\mathcal{S}_{p,1}\mathcal{S}_{p,2})A_{n-3}+\mathcal{S}_{p,2}A_{n-2}$
$\displaystyle=0.$

A lengthy but elementary substitution exercise, together with (24), yields
that

$A_{2}=\mathcal{S}_{p,1}\mathcal{S}_{p,2}A_{2}\ \text{for}\ n=5$
$A_{2}=\mathcal{S}_{p,2}^{n-5}A_{n-3}\ \text{and}\
A_{n-3}=\mathcal{S}_{p,1}^{n-5}A_{2}\ \text{for}\ n\geq 6.$

Combining this with (24) and (25), we deduce that $A_{1}=\dots=A_{n-2}=0$,
which finishes the proof of our first claim.
Thus, (19) now implies that
$S:=\sum_{j=1}^{M}b_{j}\mathscr{V}_{\rho_{j},\tau_{j}}\mathscr{Q}_{\mu_{j},\nu_{j}}$
is identically zero. We claim that $b_{M}=0$, and thus inductively, all the
$b_{i}=0$. To show this, we use the explicit expression of $\mathscr{V}$ and
$\mathscr{Q}$ found in 4.3.6. The leading monomial of
$\mathscr{V}_{\rho_{M},\tau_{M}}\mathscr{Q}_{\mu_{M},\nu_{M}}$ is given up to
a scalar multiple, by

$\begin{dcases}(\mathcal{S}_{p,1}\mathcal{S}_{p,2})^{\rho_{M}}\mathcal{S}_{p,1}^{\tau_{M}}\mathcal{T}_{p,1}^{\mu_{M}+\nu_{M}-1}\mathcal{T}_{p,2}^{\nu_{M}-1}&,\
\text{if}\ \mu_{M}\geq 0\\\
(\mathcal{S}_{p,1}\mathcal{S}_{p,2})^{\rho_{M}}\mathcal{S}_{p,1}^{\tau_{M}}\mathcal{T}_{p,1}^{\nu_{M}-1}\mathcal{T}_{p,2}^{|\mu_{M}|+\nu_{M}-1}&,\
\text{if}\ \mu_{M}<0.\end{dcases}$

Firstly, we assume that $\mu_{M}=0$. If
$|\mu_{M-1}|+\nu_{M-1}<|\mu_{M}|+\nu_{M}=\nu_{M}$, then the result follows at
once. by considering powers of $\mathcal{T}_{p,1}$. If
$|\mu_{M-1}|+\nu_{M-1}=|\mu_{M}|+\nu_{M}=\nu_{M}$, then we must have
$|\mu_{M-1}|\leq|\mu_{M}|=0$ and hence $\mu_{M-1}=0$ which then implies that
$\nu_{M-1}=\nu_{M}$. The result then follows by considering
$\rho_{M},\tau_{M}$ and $\rho_{M-1},\tau_{M-1}$ since in this case, the
equality
$(|\mu_{M}|+\nu_{M},|\mu_{M}|,\rho_{M},\tau_{M})=(|\mu_{M-1}|+\nu_{M-1},|\mu_{M-1}|,\rho_{M-1},\tau_{M-1})$
implies that
$f_{\mu_{M},\nu_{M}}^{\rho_{M},\tau_{M}}=f_{\mu_{M-1},\nu_{M-1}}^{\rho_{M-1},\tau_{M-1}}$.
We now assume that $|\mu_{M}|>0$. If
$|\mu_{M-1}|+\nu_{M-1}<|\mu_{M}|+\nu_{M-1}$ then the result follows again by
considering the exponents of $\mathcal{T}_{p,1}$ or $\mathcal{T}_{p,2}$
depending on the parity of $\mu_{M}$. Assume instead that
$|\mu_{M-1}|+\nu_{M-1}=|\mu_{M}|+\nu_{M-1}$, in which case we have
$|\mu_{M-1}|\leq|\mu_{M}|.$ Then

$\nu_{M}=\nu_{M-1}+|\mu_{M-1}|-|\mu_{M}|\leq\nu_{M-1}.$

Let $K\in\\{1,\dots,M-1\\}$ be the smallest integer for which
$|\mu_{K}|+\nu_{K}=|\mu_{K+1}|+\nu_{K}=\dots=|\mu_{M-1}|+\nu_{M-1}=|\mu_{M}|+\nu_{M}.$
Then

$|\mu_{K}|\leq\dots\leq|\mu_{M-1}|\leq|\mu_{M}|\text{ and
}\nu_{M}\leq\nu_{M-1}\leq\dots\leq\nu_{K}.$

Without loss of generality, assume that $\mu_{M}>0$. We now show that we may
also assume
$b_{M}\mathscr{V}_{\rho_{M},\tau_{M}}\mathscr{Q}_{\mu_{M},\nu_{M}}$ is the
only term of $S$ with a monomial which is a scalar multiple of

$(\mathcal{S}_{p,1}\mathcal{S}_{p,2})^{\rho_{M}}\mathcal{S}_{p,1}^{\tau_{M}}\mathcal{T}_{p,1}^{\mu_{M}+\nu_{M}-1}\mathcal{T}_{p,1}^{\nu_{M}-1}$

and hence $b_{M}=0$: Any other term containing such monomial, has to come from
some $\mathscr{Q}_{\mu_{L},\nu_{L}}$, with $L\in\\{K,\dots,M-1\\}$ and
$\mu_{L}\geq 0$. Let $L_{1}>L_{2}>\dots>L_{d}$ be all such values of $L$. The
leading monomial of
$b_{L_{d}}\mathscr{V}_{\rho_{L_{d}},\tau_{L_{d}}}\mathscr{Q}_{\mu_{L_{d}},\nu_{L_{d}}}$
is a scalar multiple of

$(\mathcal{S}_{p,1}\mathcal{S}_{p,2})^{\rho_{L_{d}}}\mathcal{S}_{p,1}^{\tau_{L_{d}}}\mathcal{T}_{p,1}^{\mu_{L_{d}}+\nu_{L_{d}}-1}\mathcal{T}_{p,2}^{\nu_{L_{d}}-1}.$

Since in this case $f_{\mu_{L_{i}},\nu_{L_{i}}}^{\rho_{L_{i}},\tau_{L_{i}}}$
is completely determined by
$(|\mu_{L_{i}}|+\nu_{L_{i}},|\mu_{L_{i}}|,\rho_{L_{i}},\tau_{L_{i}})$, or
equivalently
$(|\mu_{L_{i}}|+\nu_{L_{i}},\nu_{L_{i}},\rho_{L_{i}},\tau_{L_{i}})$, it
follows that
$b_{L_{d}}\mathscr{V}_{\rho_{L_{d}},\tau_{L_{d}}}\mathscr{Q}_{\mu_{L_{d}},\nu_{L_{d}}}$
is the only term in $S$ containing a monomial which is a scalar multiple of
$(\mathcal{S}_{p,1}\mathcal{S}_{p,2})^{\rho_{L_{d}}}\mathcal{S}_{p,1}^{\tau_{L_{d}}}\mathcal{T}_{p,1}^{\mu_{L_{d}}+\nu_{L_{d}}-1}\mathcal{T}_{p,2}^{\nu_{L_{d}}-1}$.
Hence $b_{L_{d}}=0$ and continuing like this we conclude that
$b_{L_{1}}=\dots=b_{L_{d}}=0$. Thus our assumption is justified. But then
$b_{M}=0$ which is what we initially wanted to prove. This proves our second
claim, i.e $b_{j}=0$ for all $j=1,\dots,M$. Thus (19) now implies that

$T:=\sum_{i=1}^{N}a_{i}\mathscr{V}_{r_{i},t_{i}}\mathscr{P}_{m_{i}}=0.$

Our last claim is that the coefficients $a_{i}$ are also zero: We do this by
showing that $a_{N}=0$. Recall that

$\mathscr{P}_{m_{i}}=\frac{1}{p^{m_{i}}}\begin{cases}\mathcal{P}_{p,1}^{(m_{i}+1)}(1),\
&\mathrm{if}\ m_{i}\geq 0\\\ \mathcal{P}_{p,2}^{(|m_{i}|+1)}(1),\
&\mathrm{if}\ m_{i}<0\end{cases}$

Suppose first that $|m_{N}|\geq 2$. Looking at the explicit expressions for
$\mathcal{P}_{p,1}^{(n)}(1)$ and $\mathcal{P}_{p,2}^{(n)}(1)$ it follows that
$a_{N}\mathscr{V}_{r_{N},t_{N}}\mathscr{P}_{m_{N}}$ is the only term of $T$
that contains a monomial which is a scalar multiple of

$(\mathcal{S}_{p,1}\mathcal{S}_{p,2})^{r_{N}}\mathcal{S}_{p,1}^{t_{N}}\mathcal{T}_{p,1}^{m_{N}}$

and hence $a_{N}=0$. Now suppose that $|m_{N}|=1$. If $0=|m_{N-1}|<1$ we are
clearly done by considering once again exponents of $\mathcal{T}_{p,1}$ or
$\mathcal{T}_{p,2}$. If $|m_{N-1}|=1$, we let $K\in\\{1,\dots,N-1\\}$ be the
smallest integer for which

$|m_{K}|=|m_{K+1}|=\dots=|m_{N-1}|=|m_{N}|=1.$

Then, we have

$\displaystyle T$
$\displaystyle=\left(\sum_{i=1}^{K-1}a_{i}(\mathcal{S}_{p,1}\mathcal{S}_{p,2})^{r_{i}}\mathcal{S}_{p,1}^{t_{i}}\right)\mathscr{P}_{0}+\left(\sum_{i=K}^{N}a_{i}(\mathcal{S}_{p,1}\mathcal{S}_{p,2})^{r_{i}}\mathcal{S}_{p,1}^{t_{i}}\mathscr{P}_{m_{i}}\right),\
\text{where}\ m_{i}=\pm 1$
$\displaystyle=\left(\sum_{i=1}^{K-1}a_{i}(\mathcal{S}_{p,1}\mathcal{S}_{p,2})^{r_{i}}\mathcal{S}_{p,1}^{t_{i}}\right)\mathscr{P}_{0}+\left(\sum_{i\in
I_{1}}a_{i}(\mathcal{S}_{p,1}\mathcal{S}_{p,2})^{r_{i}}\mathcal{S}_{p,1}^{t_{i}}\right)\mathscr{P}_{1}+\left(\sum_{i\in
I_{-1}}a_{i}(\mathcal{S}_{p,1}\mathcal{S}_{p,2})^{r_{i}}\mathcal{S}_{p,1}^{t_{i}}\right)\mathscr{P}_{-1}.$

where $I_{1}\cup I_{-1}=\\{K,\dots,N\\}$, $i\in I_{1}$ if and only if
$m_{1}=1$ and $i\in I_{-1}$ if and only if $m_{i}=-1$. We write

$C_{1}:=\sum_{i\in
I_{1}}a_{i}(\mathcal{S}_{p,1}\mathcal{S}_{p,2})^{r_{i}}\mathcal{S}_{p,1}^{t_{i}}\
\text{ and }\ C_{-1}:=\sum_{i\in
I_{-1}}a_{i}(\mathcal{S}_{p,1}\mathcal{S}_{p,2})^{r_{i}}\mathcal{S}_{p,1}^{t_{i}}.$

Recall that

$\displaystyle\mathscr{P}_{0}=1-\mathcal{S}_{p,1}\mathcal{S}_{p,2},\
\mathscr{P}_{1}=-\frac{1}{p}\mathcal{S}_{p,1}\mathcal{T}_{p,2}+\frac{1}{p}\mathcal{T}_{p,1},\
\mathscr{P}_{-1}=\frac{1}{p}\mathcal{S}_{p,2}\mathcal{T}_{p,1}+\frac{1}{p}\mathcal{T}_{p,2}.$

Since $T=0$, we must have

$-\frac{C_{1}}{p}\mathcal{S}_{p,1}\mathcal{T}_{p,2}+\frac{C_{1}}{p}\mathcal{T}_{p,1}-\frac{C_{-1}}{p}\mathcal{S}_{p,2}\mathcal{T}_{p,1}+\frac{C_{-1}}{p}\mathcal{T}_{p,2}=0.$

From this, we deduce that

$C_{1}=C_{-1}\mathcal{S}_{p,2}\ \text{ and }\ C_{-1}=C_{1}\mathcal{S}_{p,1}.$

It follows at once that $C_{1}=C_{-1}=0.$ Since for $i\in I_{1}$,
$f_{m_{i},0}^{r_{i},t_{i}}=f_{1,0}^{r_{i},t_{i}}$ is completely determined by
$(r_{i},t_{i})$ we conclude that $a_{i}=0$ for every $i\in I_{1}$ and in an
identical fashion $a_{i}=0$ for every $i\in I_{-1}$. This deals with the case
where $|m_{N}|=1$. Finally, if $m_{N}=0$, then

$T=\left(\sum_{i=1}^{N}a_{i}(\mathcal{S}_{p,1}\mathcal{S}_{p,2})^{r_{i}}\mathcal{S}_{p,1}^{t_{i}}\right)\mathscr{P}_{0}=0$

and $f_{m_{i},0}^{r_{i},t_{i}}=f_{0,0}^{r_{i},t_{i}}$ is once again determined
by $(r_{i},t_{i})$. Thus the claim follows.
Putting everything together, we have shown that $\mathscr{Q}_{f}=0$ implies
that $f=0$ and as a result, we deduce that

$\bigcap_{\Pi_{p}}\mathrm{ker}(\Lambda_{\Pi_{p}})=0.$

This is exactly what we wanted to prove in order to establish the cyclicity
statement. For freeness, note that
$\Lambda_{\Pi_{p}}:C_{c}^{\infty}(P_{p}\backslash
G_{p}/G_{p}^{\circ})\rightarrow(\Pi_{p}^{\vee})^{G_{p}^{\circ}}$, for
$\omega_{\pi_{p,1}}\neq\omega_{\pi_{p,2}}^{-1}$, can be extended naturally to
a Hecke equivariant map

$C_{c}^{\infty}(P_{p}\backslash
G_{p}/G_{p}^{\circ})[\tfrac{1}{1-\mathcal{S}_{p}}]\longrightarrow(\Pi_{p}^{\vee})^{G_{p}^{\circ}},\
\tfrac{f}{(1-\mathcal{S}_{p})^{r}}\mapsto\tfrac{\Lambda_{\Pi_{p}}(f)}{\Theta_{\Pi_{p}}(1-\mathcal{S}_{p}^{-1})^{r}}.$

We just proved that every such element $\tfrac{f}{(1-\mathcal{S}_{p})^{r}}$ is
given by
$\tfrac{-\mathcal{S}_{p}\mathcal{Q}_{f}^{{}^{\prime}}}{(1-\mathcal{S}_{p})^{r+1}}\cdot
f_{0}.$ Thus freeness follows from Hecke equivariance and the fact that this
slightly shrinked family of representations $\Pi_{p}$ is still dense in
$\mathrm{Spec}(\mathcal{H}_{G_{p}}^{\circ})$. ∎

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Mathematics Institute, Zeeman Building, University of Warwick, Coventry CV4
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