# Families over the integral Bernstein Center and Tate cohomology of local
Base change lifts for ${\rm GL}_{n}(F)$

Sabyasachi Dhar and Santosh Nadimpalli

###### Abstract.

Let $p$ and $l$ be distinct odd primes, and let $F$ be a $p$-adic field. Let
$\pi$ be a generic smooth integral representation of ${\rm GL}_{n}(F)$ over an
$\overline{\mathbb{Q}}_{l}$-vector space. Let $E$ be a finite Galois extension
of $F$ with $[E:F]=l$. Let $\Pi$ be the base change lift of $\pi$ to the group
${\rm GL}_{n}(E)$. Let $\mathbb{W}^{0}(\Pi,\psi_{E})$ be the lattice of
$\overline{\mathbb{Z}}_{l}$-valued functions in the Whittaker model of $\Pi$,
with respect to a standard ${\rm Gal}(E/F)$-equivaraint additive character
$\psi_{E}:E\rightarrow\overline{\mathbb{Q}}_{l}^{\times}$. We show that the
unique generic sub-quotient of the zero-th Tate cohomology group of
$\mathbb{W}^{0}(\Pi,\psi_{E})$ is isomorphic to the Frobenius twist of the
unique generic sub-quotient of the mod-$l$ reduction of $\pi$. We first prove
a version of this result for a family of smooth generic representations of
${\rm GL}_{n}(E)$ over the integral Bernstein center of ${\rm GL}_{n}(F)$. Our
methods use the theory of Rankin-selberg convolutions and simple identities of
local $\gamma$-factors. The results of this article remove the hypothesis that
$l$ does not divide the pro-order of ${\rm GL}_{n-1}(F)$ in our previous work
[DN22].

##  1\. Introduction

Let $G$ be a reductive group defined over a number field $K$. Let $\sigma$ be
an order $l$ automorphism of $G$ defined over $K$, and let $G^{\sigma}$ be the
connected component of the fixed points of $\sigma$. In the seminal paper
[TV16], D.Treumann and A. Venkatesh established the functoriality lifting of a
mod-$l$ automorphic form on $G^{\sigma}$ to a mod-$l$ automorphic form on $G$;
here mod-$l$ automorphic form is defined as a Hecke eigenclass in the
cohomology of congruence subgroup with $k$-coefficients. At the same time,
they also made some conjectures for representation theory of $p$-adic groups,
and these conjectures predict that local (Langlands) functoriality is
compatible with Tate cohomology for the action of $\langle\sigma\rangle$ (see
[TV16, Section 6.3]). After initial evidence in the context of local cyclic
base change for depth-zero cuspidal representations due to Ronchetti
([Ron16]), Feng ([Fen24], [Fen23]) made remarkable progress towards these
conjectures–using the recent advances due to the works of V. Lafforgue and
Fargues-Scholze. In this article, using completely different line of
arguments, we approach the conjectures in [TV16, Section 6.3] in the setting
of the local Base change lifting. We use local Rankin-Selberg zeta functions
and their functional equations to study the Tate cohomology and linkage
principle. Our approach is in the spirit of local converse theorems. In our
earlier work [DN22], we assumed that the prime $l$ is large. In this article,
exploiting the theory of smooth representations in families, we remove this
hypothesis.

Let $(\Pi,W)$ be an irreducible smooth
$\overline{\mathbb{F}}_{l}$-representation of $G(F)$, where $G$ is a connected
reductive group over a $p$-adic field $F$. Say $\sigma\in{\rm Aut}(G)$ is of
order $l$ and that $\Pi^{\sigma}$ is isomorphic to $\Pi$. The Tate cohomology
group $\widehat{H}^{i}(\sigma,V)$ is a representation of $G^{\sigma}(F)$. Any
irreducible sub-quotient $\pi$ of $\widehat{H}^{i}(\sigma,W)$ is defined to be
linked with $\Pi$. In the article [TV16], the authors discovered that linkage
is compatible with local functoriality. In this paper, we prove the Linkage
conjectures in the local Base change setting of ${\rm GL}_{n}$ for all odd
$l$. The authors proved the main theorem under the restriction that $l$ does
not divide the pro-order of ${\rm GL}_{n-1}(F)$. By using the theory of Co-
Whittaker modules and integral Bernstein center, developed by Emerton and Helm
([EH14], [Hel20]) in an essential way, we remove this hypothesis on $l$.

To state the main results of this article, some notations are in order. Let
$F$ be a finite extension of $\mathbb{Q}_{p}$ and let $E/F$ be a finite Galois
extension of prime degree $l$. Let $\overline{\mathcal{K}}$ be the algebraic
closure of the fraction field of $W(\overline{\mathbb{F}}_{l})$ with ring of
integers $\mathcal{O}$. Let $(\pi,V)$ be an irreducible smooth representation
of ${\rm GL}_{n}(F)$, where $V$ is a vector space over
$\overline{\mathcal{K}}$. We assume that $(\pi,V)$ is an integral
representation, i.e., $V$ admits a ${\rm GL}_{n}(F)$-invariant
$\mathcal{O}$-lattice. Let $(\Pi,W)$ be the base change lifting of $\pi$ to
${\rm GL}_{n}(E)$. The representation $\Pi$ is isomorphic to $\Pi^{\gamma}$,
where $\gamma$ is any generator of ${\rm Gal}(E/F)$. Let us fix an isomorphism
$T:\Pi\rightarrow\Pi^{\gamma}$ such that $T^{l}=\operatorname{id}$. Note that
$\Pi$ is an integral representation of ${\rm GL}_{n}(E)$.

If $\pi$ is generic, then $\Pi$ is also a generic representation. Let us fix
an additive character $\psi_{F}$ of $F$ and let $\psi_{E}$ be the composition
$\psi_{F}\circ{\rm Tr}$, where ${\rm Tr}$ is the trace map of the extension
$E/F$. Let $\mathbb{W}(\pi,\psi_{F})$ (resp. $\mathbb{W}(\Pi,\psi_{E})$) be
the Whittaker model of $\pi$ (resp. $\Pi$). Vigneras showed that the
$\mathcal{O}$-module $\mathbb{W}^{0}(\Pi,\psi_{E})$ is a ${\rm
GL}_{n}(E)$-stable lattice and by multiplicity one results,
$\mathbb{W}^{0}(\Pi,\psi_{E})$ is stable under the Galois action of ${\rm
Gal}(E/F)$ induced by its natural action on ${\rm GL}_{n}(E)$. Note that there
exists a unique generic sub-quotient of the mod-$l$ reduction of a generic
integral representation $\pi$ of ${\rm GL}_{n}(K)$, where $K$ is a $p$-adic
field and this is denoted by $J_{l}(\pi)$. It is easy to show that the Tate
cohomology of $\widehat{H}^{0}(\mathbb{W}^{0}(\Pi,\psi_{E}))$ has a unique
generic sub-quotient as a representation of ${\rm GL}_{n}(F)$ ([DN22,
Proposition 6.3]). Let $V$ be a vector space over $\overline{\mathbb{F}}_{l}$,
and let $V^{(l)}$ be the vector space $V\otimes_{{\rm
Fr}}\overline{\mathbb{F}}_{l}$ where ${\rm Fr}$ be the Frobenius automorphism
of $\overline{\mathbb{F}}_{l}$. Ronchetti ([Ron16, Theorem 6]) proved that the
Tate cohomology group $\widehat{H}^{1}(\mathcal{L})$ is zero for any ${\rm
GL}_{n}(E)$ and ${\rm Gal}(E/F)$ stable lattice $\mathcal{L}$ in a cuspidal
representation. When $l$ does not divide the pro-order of ${\rm GL}_{n}(F)$,
the authors showed the same vanishing result for generic representations
([DN22, Corollary 8.2]). We prove the following theorem on the zeroth Tate
cohomology:

######  Theorem 1.1.

Let $l$ and $p$ be two distinct odd primes. Let $F$ be a p-adic field and let
$E/F$ be a Galois extension of degree $l$. Let $\pi$ be an integral smooth
generic representation of ${\rm GL}_{n}(F)$ over a
$\overline{\mathbb{Q}}_{l}$-vector space. Let $\Pi$ be a base change lifting
of $\pi$ to ${\rm GL}_{n}(E)$. The unique generic sub-quotient of the Tate
cohomology group $\widehat{H}^{0}(\mathbb{W}^{0}(\Pi,\psi_{E}))$ is isomorphic
to $J_{l}(\pi)^{(l)}$.

In our previous work ([DN22]), we proved the above theorem when $l$ does not
divide the pro-order of ${\rm GL}_{n-1}(F)$. We use the theory of
Rankin–Selberg zeta integrals and their functional equations. The hypothesis
on the prime $l$ in [DN22] is needed for the completeness of mod-$l$ Whittaker
models. In order to remove this hypothesis on $l$, we use the theory of smooth
representations in families due to Emerton and Helm. This approach deals with
the nilpotents which arise in the mod-$l$ Zeta integrals. So, we first prove a
version of the above theorem over families and use it to prove the above
theorem. In order to define the correct family, we needed the work of Helm on
local converse theorem, where he defines a map between the integral Bernstein
center and a certain deformation ring. This gave us the notation of Base
change lifting map between Bernstein centers.

We will begin with the description of our result on Tate cohomology of a
family of smooth representations of ${\rm GL}_{n}(E)$. Let
$\mathcal{M}_{n}(K)$ be the category of smooth
$W(\overline{\mathbb{F}}_{l})[{\rm GL}_{n}(K)]$-modules, where $K$ is a
$p$-adic field, and the center of the category $\mathcal{M}_{n}(K)$ is denoted
by $\mathcal{Z}_{n}(K)$. The primitive idempotents of $\mathcal{Z}_{n}(K)$
corresponds to an inertial class $[L,\sigma]$, where $\sigma$ is a
supercuspidal $\overline{\mathbb{F}}_{l}$ representation of the Levi-subgroup
$L$ of ${\rm GL}_{n}(K)$ ([Hel16a, Section 12]). Let $e_{\mathfrak{s}}$ be a
primitive idempotent of $\mathcal{Z}_{n}(F)$. Using mod-$l$ semisimple
Langlands correspondence of Vigneras ([Vig01, Theorem 1.6]), we get a smooth
representation

$\rho:W_{F}\rightarrow{\rm GL}_{n}(\overline{\mathbb{F}}_{l})$

associated with $(L,\sigma)$. Let $e_{\mathfrak{r}}$ be the primitive
idempotent in $\mathcal{Z}_{n}(E)$ associated with
$\operatorname{res}_{W_{E}}\rho$. Using Helm’s construction of a homomorphism
between irreducible component of Bernstein centre to a deformation ring, we
obtain a map

$z_{E/F}:e_{\mathfrak{r}}\mathcal{Z}_{n}(E)\rightarrow
e_{\mathfrak{s}}\mathcal{Z}_{n}(F)$

which interpolates the cuspidal support of Base change lifting ([AC89, Chapter
1]).

To interpolate those generic representations of ${\rm GL}_{n}(E)$ which arise
as a base change lift of representations with a given cuspidal support, we
define

$\mathcal{V}=(e_{\mathfrak{r}}\operatorname{ind}_{N_{n}(E)}^{{\rm
GL}_{n}(E)}\psi_{E}^{-1})\otimes_{z_{E/F}}e_{\mathfrak{s}}\mathcal{Z}_{n}(F).$

The module $\mathcal{V}$ is a co-Whittaker
$e_{\mathfrak{s}}\mathcal{Z}_{n}(F)[{\rm GL}_{n}(F)]$-module. The theory of
Rankin-selberg $\gamma$-factors in families show that the representation
$\mathcal{V}$ is isomorphic to $\mathcal{V}^{\gamma}$, for all $\gamma\in{\rm
Gal}(E/F)$. Thus, the space of Whittaker functions (containing functions on
${\rm GL}_{n}(E)/N_{n}(E)$ with values in $e_{\mathfrak{s}}\mathcal{Z}_{n}$)
is invariant under the action of ${\rm Gal}(E/F)$. We then prove that the Tate
cohomology group ${\widehat{H}}^{0}(\mathbb{W}(\mathcal{V},\psi_{E}))$
realises

$[e_{\mathfrak{s}}(\operatorname{ind}_{N_{n}(F)}^{{\rm
GL}_{n}(F)}\psi_{F}^{-1})\otimes_{W(\overline{\mathbb{F}}_{l})}\overline{\mathbb{F}}_{l}]^{(l)}$

as a sub-quotient. The proof of this result is similar to our previous result
in [DN22, Theorem 6.7], where we use Rankin-Selberg local zeta integrals and
their liftings from homogeneous spaces over $F$ to the corresponding spaces
over $E$.

Theorem 1.1 is proved by a specialization argument. The proofs of this theorem
and the previous version in families are inspired from the proofs of local
converse theorems, as in [JPSS79, Proposition 7.5.2] and [Hen93, Theorem 1.1].
Note that the above arguments in families and our specialization arguments
essentially use Kirillov models and completeness of Whittaker models. The
following is the essential content. Let $X_{F}$ be the space ${\rm
GL}_{n-1}(F)/N_{n-1}(F)$, and let $H\in
C_{c}(X_{F},\overline{\mathbb{F}}_{l})$. Typically, for some fixed integer $k$
and a suitably chosen Whittaker function $W$ in the Kirrilov model of $\Pi$,
the function $H$ is equal to

$[\widehat{H}^{0}(\pi_{E})(w)-\pi_{F}^{(l)}(w)]W\left(\begin{pmatrix}g&0\\\
0&1\end{pmatrix}\right)$

when ${\rm det}(g)=k$ and zero otherwise. Essentially we need to show that $H$
is zero. If not, there exists a primitive idempotent
$e_{\mathfrak{s}}\in\mathcal{Z}_{n}(F)$ and a
$e_{\mathfrak{s}}\mathcal{Z}_{n}(F)$-valued Whittaker function

$W^{\prime}\in\mathbb{W}(e_{\mathfrak{s}}\operatorname{ind}_{N_{n-1}(F)}^{{\rm
GL}_{n-1}(F)}\psi_{F}^{-1},\psi_{F}^{-1})$

such that

$\int_{X_{F}}H(g)\otimes W^{\prime}(g)\,d\mu\neq 0.$

When the ring
$e_{\mathfrak{s}}\mathcal{Z}_{n}(F)\otimes\overline{\mathbb{F}}_{l}$ is
reduced, we can simply use Rankin-Selberg theory of mod-$l$ representations to
prove the vanishing of $H$ and deduce our main theorem. This has been the main
point of our work [DN22, Theorem 6.7]. In this article, to overcome the
problems with the reducibility of the ring
$e_{\mathfrak{s}}\mathcal{Z}_{n}(F)\otimes\overline{\mathbb{F}}_{l}$, we use
our result on families.

##  2\. Integral Bernstein Center and Base change

In this section, we recall the theory of integral Bernstein centers and define
a base change map between Bernstein centers–which is compatible with the local
base change lifting of irreducible smooth representations of ${\rm GL}_{n}$.
We essentially follow the works of Emerton–Helm ([EH14]) and Helm ([Hel20]).

### 2.1.

For any finite extension $K$ of $\mathbb{Q}_{p}$, we denote by
$\mathfrak{o}_{K}$ its ring of integers and the maximal ideal of
$\mathfrak{o}_{K}$ is denoted by $\mathfrak{p}_{K}$. Let $q_{K}$ be the
cardinality of the residue field $\mathfrak{o}_{K}/\mathfrak{p}_{K}$. The Weil
group of $K$ is denoted by $W_{K}$ and its inertia subgroup is denoted by
$I_{K}$. The prime to $l$-subgroup of $I_{K}$ is denoted by $I_{K}^{(l)}$. We
denote by $G_{n}(K)$ the group ${\rm GL}_{n}(K)$ and it is equipped with the
natural topology induced from that of $K$. Let $F$ be a finite extension of
$\mathbb{Q}_{p}$ and let $E$ be a finite Galois extension of prime degree $l$,
where $p$ and $l$ are distinct odd primes. The Galois group ${\rm Gal}(E/F)$
is denoted by $\Gamma$. Let $\mathbb{F}$ be the algebraic closure of the
finite field of $l$ elements and let $W(\mathbb{F})$ be the ring of Witt
vectors; we use the notation $\Lambda$ for $W(\mathbb{F})$. Let
$\overline{\mathcal{K}}$ be the algebraic closure of the field of fraction of
$\Lambda$ with ring of integers $\mathcal{O}$. Let
$\psi_{F}:F\rightarrow\Lambda^{\times}$ be a non-trivial additive character
and let $\psi_{E}$ be the character $\psi_{F}\circ{\rm Tr}_{E/F}$, where ${\rm
Tr}_{E/F}$ denotes the trace map of the extension $E/F$.

### 2.2.

Let $R$ be a commutative ring with unity and let $G$ be the $K$-rational
points of a reductive algebraic group defined over $K$. We denote by
$\mathcal{M}_{R}(G)$ the category of smooth $R[G]$-modules (the stabilizer of
every element under the action of $G$ is open), and the Bernstein center of
$\mathcal{M}_{R}(G)$ is denoted by $\mathcal{Z}_{R}(G)$. We assume that $G$ is
the group $G_{n}(K)$. We use the notation $\mathcal{Z}_{n}^{K}$ for the ring
$\mathcal{Z}_{\Lambda}(G_{n}(K))$. Let $P$ be a parabolic subgroup of $G$. Let
$M$ be a Levi-subgroup of $P$ and let $U$ be the unipotent radical of $P$. Let
$(\sigma,W)$ be a supercuspidal representation of $M$, where $W$ is an
$\mathbb{F}$-vector space. Let $\mathfrak{s}=[M,\sigma]$ be the inertial
equivalence class containing the pair $(M,\sigma)$. Let
$\mathcal{M}_{\mathfrak{s}}(G)$ be the full subcategory of
$\mathcal{M}_{\Lambda}(G)$ consisting of all smooth $\Lambda[G]$ modules such
that any irreducible subquotient has mod-$l$ (see [Hel16a, Definition 4.12])
inertial supercuspidal support equal to $\mathfrak{s}$. Given any primitive
idempotent $e$ of $\mathcal{Z}_{n}^{K}$, there exists a unique inertial
equivalence class $\mathfrak{s}$ such that the Bernstein center of
$\mathcal{M}_{\mathfrak{s}}(G)$ is equal to $e\mathcal{Z}_{n}^{K}$. The
primitive idempotents of $\mathcal{Z}_{n}^{K}$ correspond to the inertial
classes of the form $[M,\sigma]$, where $\sigma$ is a supercuspidal
representation of $M$ over an $\mathbb{F}$-vector space.

### 2.3.

Helm in his foundational work [Hel20] described integral Bernstein center in
terms of local Galois deformation rings. He constructed a map between
connected component of the integral Bernstein center and a local Galois
deformation ring which interpolates the local Langlands correspondence. Helm’s
constructions naturally defines a map between Bernstein centers
$\mathcal{Z}_{n}^{E}$ and $\mathcal{Z}_{n}^{F}$, which interpolates the
compatibility of cuspidal support maps of base change lifting. For the cyclic
extension $E/F$, the base change lifting of smooth irreducible representations
of $G$ over complex vector spaces is characterized via certain character
identities (see [AC89, Chapter 3]). There is a relation between the local base
change lifting of irreducible smooth $\overline{\mathcal{K}}$-representations
of $G$ and the local Langlands correspondence over $\overline{\mathcal{K}}$.
Here, we use the local Langlands correspondence by fixing the normalizations
as in [EH14, Section 4.2].

#### 2.3.1.

An $l$-inertial type is a representation $\nu:I_{K}^{(l)}\rightarrow{\rm
GL}_{n}(\Lambda)$ that extends to a representation of $\mathcal{W}_{F}$. Let
$\mathfrak{s}$ be the inertial equivalence class $[M,\sigma]$. where $M$ is a
Levi subgroup of $G$ and $\sigma$ is a supercuspidal representation of $M$
over an $\mathbb{F}$-vector space. Vigneras’ construction ([Vig01, Theorem
1.6]) of mod-$l$ local Langlands correspondence attaches a unique semisimple
representation $\overline{\rho}:W_{K}\rightarrow{\rm GL}_{n}(\mathbb{F})$ with
the pair $(M,\sigma)$

$(M,\sigma)\leftrightsquigarrow\overline{\rho}\ (\text{mod}-l\ \text{LLC})$

Let $\overline{\tau}$ be an irreducible representation of $I_{K}^{(l)}$ over
an $\mathbb{F}$-vector space, and let $\tau$ be the (unique) lift of
$\overline{\tau}$ over $\Lambda$. Given any $\Lambda$-algebra $A$ and a
representation $\rho:W_{K}\rightarrow{\rm GL}_{n}(A)$, we have

$\rho_{A}\simeq\bigoplus_{[\overline{\tau}]}({\rm
Hom}_{A}(\tau,\rho_{A})\otimes\tau).$

The direct sum is over the $W_{K}$-conjugacy classes of
$I_{K}^{(l)}$-representations, denoted as $[\overline{\tau}]$. The space ${\rm
Hom}_{A}(\tau,\rho)$ is a free $A$-module and a pseudoframing of $\rho_{A}$ is
a choice of a basis for each ${\rm Hom}_{A}(\tau,\rho_{A})$. In the article
[Hel20, Section 8], the author constructed a $\Lambda$-algebra $R_{\nu}$
corresponding to an $l$-inertial type $\nu$–which is universal for the
pseudoframed deformation $\rho_{A}:W_{K}\rightarrow{\rm GL}_{n}(A)$, i.e., for
any choice of basis of ${\rm Hom}_{I_{K}^{(l)}}(\tau,\rho_{A})$ which lifts a
basis of ${\rm Hom}_{I_{K}^{(l)}}(\overline{\tau},\overline{\rho})$ for each
$W_{F}$-conjugacy classes of $I_{K}^{(l)}$-representations, denoted by
$[\overline{\tau}]$, there exists a map $R_{\nu}^{K}\rightarrow A$ such that
the pseudoframed deformation $\rho_{A}$ is obtained as a base change of the
universal pseudoframed deformation $\rho_{\nu}^{K}$. The affine scheme
corresponding to the deformation ring $R_{\nu}^{K}$ is equipped with an action
of a group $G_{\nu}^{K}$, defined in [Hel20, Section 8], which acts by change
of framing; we recall the precise definition of $G_{\nu}^{K}$ in the proof of
Lemma 2.1. This action gives a space of invariance $(R_{\nu}^{K})^{\rm inv}$,
which is infact a subalgebra of $R_{\nu}^{K}$. The space $(R_{\nu}^{K})^{\rm
inv}$ is useful to construct the base change map between the integral
Bernstein centers, as we will see in the next subsections.

#### 2.3.2.

In [HM18, Section 7], the authors constructed a map

$\mathbb{L}^{K}_{\nu}:e\mathcal{Z}_{n}^{K}\rightarrow R^{K}_{\nu}$

which is compatible with the local Langlands correspondance, i.e., for any
morphism

$x:R_{\nu}^{K}\rightarrow\overline{\mathcal{K}},$

the morphism

$x\circ\mathbb{L}_{\nu}^{K}:e\mathcal{Z}_{n}^{K}\rightarrow\overline{\mathcal{K}}$

gives the action of $e\mathcal{Z}_{n}^{K}$ on the representation
$\pi_{x}$–associated with $\rho_{x}$ via the local Langlands correspondence.
The map $\mathbb{L}_{\nu}^{K}$ identifies the algebra $e\mathcal{Z}_{n}^{K}$
with the $\mathcal{G}_{\nu}$-fixed points of $R_{\nu}^{K}$. This morphism
plays a fundamental role in proving the local Langlands in families by Emerton
and Helm ([EH14, Conjecture 1.3.1]). Let $F$ be a finite extension of
$\mathbb{Q}_{p}$ and let $E$ be a finite extension of $F$ of degree $l$ with
$l\neq p$. Let $\nu:I_{F}^{(l)}\rightarrow{\rm GL}_{n}(\Lambda)$ be an
$l$-inertial type. As the group $I_{F}^{(l)}$ is equal to $I_{E}^{(l)}$, any
pseudo-framed deformation $\rho_{A}:W_{F}\rightarrow{\rm GL}_{n}(\Lambda)$
also determines a pseudoframed deformation $\rho_{A}:W_{E}\rightarrow{\rm
GL}_{n}(\Lambda)$ by restriction and hence we get a map

$B_{E/F}:R_{\nu}^{E}\rightarrow R_{\nu}^{F}.$ (2.1)

######  Lemma 2.1.

The map $B_{E/F}$ induces a map between $(R_{\nu}^{E})^{\rm inv}$ and
$(R_{\nu}^{F})^{\rm inv}$.

###### Proof.

Let $X_{q,m}$ be the affine $\Lambda$-scheme parametrizing the pairs $(g,h)$,
where $g$, $h$ are the invertible $m$ by $m$ matrices with $ghg^{-1}=h^{q}$.
Let $R_{q,m}$ be the ring of functions of the connected compotent of $X_{q,m}$
containing the pair $({\rm Id}_{n},{\rm Id}_{n})$. Note that the
$\Lambda$-algebra $R_{\nu}^{F}$ is isomorphic to
$\bigotimes_{[\overline{\tau}]}R_{q_{\overline{\tau}},n_{\overline{\tau}}}$,
where $n_{\overline{\tau}}$ is the dimension of the space ${\rm
Hom}_{I_{F}^{(l)}}(\overline{\tau},\overline{\nu})$. The group ${\rm
GL}_{n_{\overline{\tau}}}$ acts on
$R_{q_{\overline{\tau}},n_{\overline{\tau}}}$ by change of frame. Let
$S_{E}=\\{\tau_{1},\tau_{2},\dots,\tau_{k}\\}$ (resp. $S_{F}\subset S_{E}$) be
a set of representatives for the equivalence classes of $I_{F}^{(l)}$
representations for the action of $W_{E}$ (resp. $W_{F}$). The group

$\prod_{\overline{\tau}\in S_{E}}{\rm GL}_{n_{\overline{\tau}}}$

denoted by $\mathcal{G}_{\nu}^{E}$, acts on $R^{E}_{\nu}$ via its action on
$R_{q_{\overline{\tau}},n_{\overline{\tau}}}$. Similarly the group
$\mathcal{G}_{\nu}^{F}$, isomorphic to $\prod_{\overline{\tau}\in S_{F}}{\rm
GL}_{n_{\overline{\tau}}}$ is contained in $\mathcal{G}_{\nu}^{E}$. Thus, the
image of the restriction of $B_{E/F}$ to $(R_{\nu}^{E})^{\rm inv}$ is
contained in the space of invariance $(R_{\nu}^{F})^{\rm inv}$. ∎

#### 2.3.3.

Let $e_{\mathfrak{s}}$ be a primitive idempotent of $\mathcal{Z}_{n}^{F}$
corresponding to the inertial equivalence class $\mathfrak{s}$. Fix a pair
$(L,\sigma)$ in the class $\mathfrak{s}$, where $L$ is a Levi subgroup of
$G_{n}(F)$ and $\sigma$ is an irreducible supercuspidal
$\mathbb{F}$-representation of $L$. Let $\rho_{F}$ be the $n$-dimensional
semisimple representation of $W_{F}$ associated with $\sigma$ via the mod-$l$
semisimple local Langlands correspondence ([Vig01, Theorem 1.6]). Let
$\nu:I_{F}^{(l)}\rightarrow{\rm GL}_{n}(\Lambda)$ be the $l$-inertial type
such that its mod-$l$ reduction is isomorphic to ${\rm
res}_{I_{F}^{(l)}}(\rho_{F})$. Note that the restriction ${\rm
res}_{W_{E}}(\rho_{F})$ via mod-$l$ local Langlands correspondence defines a
pair $(L^{\prime},\sigma^{\prime})$ such that its inertial equivalence class,
denoted by $\mathfrak{r}$, is independent of the choice of $(L,\sigma)$. Let
$e_{\mathfrak{r}}$ be the primitive idempotent of $\mathcal{Z}_{n}^{E}$
associated with $\mathfrak{r}$. The map $B_{E/F}$ induces a map
$z_{E/F}:e_{\mathfrak{r}}\mathcal{Z}_{n}^{E}\rightarrow
e_{\mathfrak{s}}\mathcal{Z}_{n}^{F}$ such that the following diagram commutes.

$e_{\mathfrak{r}}\mathcal{Z}_{n}^{E}$$R^{E}_{\nu}$$e_{\mathfrak{s}}\mathcal{Z}_{n}^{F}$$R^{F}_{\nu}$$\mathbb{L}_{\nu}^{E}$$z_{E/F}$$B_{E/F}$$\mathbb{L}_{\nu}^{F}$

Let $\rho_{x_{F}}:W_{F}\rightarrow{\rm GL}_{n}(\overline{\mathcal{K}})$ be the
representation corresponding to
$x_{F}:R_{\nu}^{F}\rightarrow\overline{\mathcal{K}}$ and let $\pi_{x_{F}}$ be
a smooth irreducible representation of ${\rm GL}_{n}(F)$ associated to
$\rho_{x_{F}}$ via, the local Langlands correspondence. Let $x_{E}$ be the map
$x_{F}\circ B_{E/F}:R_{\nu}^{E}\rightarrow\overline{\mathcal{K}}$, and the
representation $\pi_{x_{E}}$ is the base change lifting of $\pi_{x_{F}}$. The
action of $\mathcal{Z}_{n}^{E}$ on the representation $\pi_{x_{E}}$ factorises
through $e_{\mathfrak{r}}\mathcal{Z}_{n}^{E}$, and it is given by the
homomorphism $x_{F}\circ\mathbb{L}_{\nu}^{F}\circ z_{E/F}$.

##  3\. Rankin–Selberg convolution in families

In this section, we review the theory of Rankin-Selberg convolution over
arbitrary Noetherian $\Lambda$-algebras. We start by recalling the notion of
co-Whittaker modules. For a precise reference, see [HM18], [Mos16a].

###  3.1. Co-Whittaker modules

Fix a non-trivial additive character $\psi_{K}:K\rightarrow\Lambda^{\times}$.
For any Noetherian $\Lambda$-algebra $R$, we denote by $\psi_{K,R}$ the
composition

$K\xrightarrow{\psi_{K}}\Lambda^{\times}\rightarrow R^{\times}.$

Let $N_{n}(K)$ be the group of all unipotent upper triangular matrices in
$G_{n}(K)$. The character $\psi_{K,R}:K\rightarrow R^{\times}$ induces a
character of $N_{n}(K)$, defined as

$(x_{ij})_{i,j=1}^{n}\longmapsto\psi_{K,R}(x_{12}+x_{23}+\cdots x_{n-1,n}),$

which is denoted by $\Psi_{K,R}$. Let $(\pi,V)$ be a smooth
$R[G_{n}(K)]$-module. We denote by $V^{(n)}$ the space of
$\Psi_{K,R}$-coinvariants, which is defined as the quotient of $V$ modulo the
$R$-submodule generated by the set $\Big{\\{}\pi(x)v-(\Psi_{K,R})(x)v:x\in
N_{n}(K),v\in V\Big{\\}}$. The $R$-module $V^{(n)}$ is the $n$-th Bernstein-
Zelevinski derivative of $V$.

#### 3.1.1.

A smooth admissible $R[G_{n}(K)]$-module $(\pi,V)$ is called co-Whittaker if

  1. (1)

the $n$-th derivative $V^{(n)}$ is a free $R$-module of rank one,

  2. (2)

for any quotient $W$ of $V$ with $W^{(n)}=0$, we have $W=0$.

A co-Whittaker module $(\pi,V)$ admits a central character denoted by
$\varpi_{\pi}:F^{\times}\rightarrow R^{\times}$. By definition, the
representation $(\pi,V)$ admits an $R$-module isomorphism $V^{(n)}\simeq R$.
By Frobenius reciprocity, this isomorphism induces a $G_{n}(K)$-equivariant
homomorphism

$V\longrightarrow{\rm Ind}_{N_{n}(K)}^{G_{n}(K)}(\Psi_{K,R}).$

The image of $V$ under the above map is called the Whittaker space of $\pi$.
It is denoted by $\mathbb{W}(\pi,\psi_{K,R})$ and is independent of the choice
of isomorphism $V^{(n)}\simeq R$. The identification $V^{(n)}\simeq R$ , also
induces a $P_{n}(K)$-equivariant map

$V\longrightarrow{\rm Ind}_{N_{n}(K)}^{P_{n}(K)}(\Psi_{K,R}),$

the image of the above map is called Kirillov space of $\pi$, and it is
denoted by $\mathbb{K}(\pi,\psi_{K,R})$. The map $W\mapsto{\rm
res}_{P_{n}(K)}(W)$ gives an isomorphism
$\mathbb{W}(\pi,\psi_{K,R})\xrightarrow{\sim}\mathbb{K}(\pi,\psi_{K,R})$
([MM22, Section 4]). Moreover, the space $\mathbb{K}(\pi,\psi_{K,R})$ contains
${\rm ind}_{N_{n}(K)}^{P_{n}(K)}(\Psi_{K,R})$ as $R[P_{n}(K)]$-submodule.

#### 3.1.2.

For a co-Whittaker $R[G_{n}(K)]$-module $(\pi,V)$, the endomorphism ring ${\rm
End}_{R[G_{n}(K)]}(V)$ is equal to $R$, and the action of the integral
Bernstein center $\mathcal{Z}_{n}^{K}$ on $V$ induces a map
$f_{\pi}:\mathcal{Z}_{n}^{K}\rightarrow R$. The map $f_{\pi}$ is called the
supercuspidal support of $\pi$. The supercuspidal supports of two co-Whittaker
$R[G_{n}(K)]$-modules are the same if and only if they have the same Whittaker
spaces (see [LM20, Lemma 2.4]). Moreover, if we have two co-Whittaker
$R[G_{n}(K)]$-modules $(\pi_{1},V_{1})$ and $(\pi_{2},V_{2})$ with a
$G_{n}(K)$-equivariant surjection $V_{1}\rightarrow V_{2}$, then
$f_{\pi_{1}}=f_{\pi_{2}}$.

#### 3.1.3.

Let $W_{n}^{K}$ be the smooth $\Lambda[G_{n}(K)]$-module ${\rm
ind}_{N_{n}(K)}^{G_{n}(K)}(\Psi_{K}^{-1})$. For a primitive idempotent $e$ in
the integral Bernstein center $\mathcal{Z}_{n}^{K}$, the space $eW_{n}^{K}$ is
a co-Whittaker $e\mathcal{Z}_{n}^{K}[G_{n}(K)]$-module. The following result
is due to [Hel16b, Theorem 6.3], which specify the universal property of
$eW_{n}^{K}$.

######  Theorem 3.1.

Let $R$ be a Noetherian $\Lambda$-algebra, equipped with a
$e\mathcal{Z}_{n}^{K}$-algebra structure. Then
$eW_{n}^{K}\otimes_{e\mathcal{Z}_{n}^{K}}R$ is a co-Whittaker
$R[G_{n}(K)]$-module. Conversely, if $(\pi,V)$ is a co-Whittaker
$R[G_{n}(K)]$-module in the category $\mathcal{M}_{e}(G_{n}(K))$, then there
exists a surjection
$\beta:eW_{n}^{K}\otimes_{e\mathcal{Z}_{n}^{K}}R\rightarrow V$ such that the
induced map
$\beta^{(n)}:(eW_{n}^{K}\otimes_{e\mathcal{Z}_{n}^{K}}R)^{(n)}\rightarrow
V^{(n)}$ is an isomorphism.

We end this subsection with a version of key vanishing result, usually known
as completeness of Whittaker models, provided by G.Moss ([Mos21, Corollary
4.3]). This vanishing result will be crucially used in proving the main
results of this article.

######  Theorem 3.2.

Let $R$ be a $\Lambda$-algebra, and let $\phi$ be an element of ${\rm
ind}_{N_{n}(K)}^{G_{n}(K)}(\Psi_{K,R})$. If

$\int_{N_{n}(K)\setminus G_{n}(K)}\phi(g)\otimes W(g)\,dg=0,$

for all $W\in\mathbb{W}(eW_{n}^{K},\psi_{K}^{-1})$ and for all primitive
idempotents $e$ of $\mathcal{Z}_{n}^{K}$, then $\phi=0$.

###  3.2. Rankin-Selberg formal series

In this subsection, we recall the Rankin-Selberg gamma factors over families,
which provide the classical Rankin-Selberg gamma factors as a specialization
on $\overline{\mathcal{K}}$-points. For a reference, see [Mos16a, Section 3].

#### 3.2.1.

We now introduce some notations. Let $w_{n}$ be the matrix of $G_{n}(K)$ of
the form:

$w_{n}=\begin{pmatrix}0&&&1\\\ &&.\\\ &.\\\ 1&&&0\end{pmatrix}$

For $r\in\mathbb{Z}$, set $G_{n-1}(K)_{r}=\big{\\{}g\in G_{n-1}(K):v_{K}({\rm
det}(g))=r\big{\\}}$, where $v_{K}$ denote the normalised discrete valuation
of $K$. Let $X_{K}$ be the coset space $N_{n-1}(K)\setminus G_{n-1}(K)$. For
an integer $r$, we denote by $X_{r}^{K}$ the set of the form
$\big{\\{}N_{n-1}(K)g:g\in G_{n-1}(K)_{r}\big{\\}}$.

#### 3.2.2.

Let $A$ and $B$ be two Noetherian $\Lambda$-algebras. Let $\pi$ (resp.
$\pi^{\prime}$) be the co-Whittaker $A[G_{n}(K)]$ (resp.
$B[G_{n-1}(K)]$)-module. For any $W\in\mathbb{W}(\pi,\psi_{K,A})$ and
$W^{\prime}\in\mathbb{W}(\pi^{\prime},\psi_{K,B}^{-1})$, the integral

$c_{r}^{K}(W,W^{\prime}):=\int_{X_{r}^{K}}W\begin{pmatrix}g&0\\\
0&1\end{pmatrix}\otimes W^{\prime}(g)\,dg$

is well-defined for all integers $r$ and it is zero for $r<<0$ (see [Mos16a,
Section 3]). The formal Laurent series

$\sum_{r\in\mathbb{Z}}c_{r}^{K}(W,W^{\prime})X^{r},$

is an element of $S^{-1}((A\otimes_{\Lambda}B)[X,X^{-1}])$, where $S$ is the
multiplicative set consisting of polynomials $\sum_{r=t}^{s}a_{r}X^{r}$ with
$a_{t}$ and $a_{s}$ being units in $A\otimes_{\Lambda}B$. Now, let us consider
the functions $\widetilde{W}$ and $\widetilde{W^{\prime}}$, defined as

$\widetilde{W}(g):=W(w_{n}(g^{t})^{-1})$

and

$\widetilde{W^{\prime}}(g):=W^{\prime}(w_{n-1}(x^{t})^{-1}),$

for all $g\in G_{n}(K)$ and $x\in G_{n-1}(K)$. Making change of variables, we
have the identity:

$c_{r}^{K}(\widetilde{W},\widetilde{W^{\prime}})=c_{-r}^{K}\big{(}\pi(w_{n})W,\pi^{\prime}(w_{n-1})W^{\prime}\big{)}.$
(3.1)

####  3.2.3. Functional Equation

Given $\pi$ and $\pi^{\prime}$ as above, there is a unique element
$\gamma(X,\pi,\pi^{\prime},\psi_{K})$ in the ring of fraction
$S^{-1}((A\otimes_{\Lambda}B)[X,X^{-1}])$ such that

$\sum_{r\in\mathbb{Z}}c_{r}^{K}(\widetilde{W},\widetilde{W^{\prime}})X^{-r}=\varpi_{\pi^{\prime}}(-1)^{n-1}\gamma(X,\pi,\pi^{\prime},\psi_{K})\sum_{r\in\mathbb{Z}}c_{r}^{K}(W,W^{\prime})X^{r},$

for all $W\in\mathbb{W}(\pi,\psi_{K,A})$ and
$W^{\prime}\in\mathbb{W}(\pi^{\prime},\psi_{K,B}^{-1})$.

Let $e$ (resp. $e^{\prime}$) be the primitive idempotent in
$\mathcal{Z}_{n}^{K}$ (resp. $\mathcal{Z}_{n-1}^{K}$) such that the
supercuspidal support map $f_{\pi}$ (resp. $f_{\pi^{\prime}}$) factors through
the center $e\mathcal{Z}_{n}^{K}$ (resp. $e^{\prime}\mathcal{Z}_{n-1}^{K}$).
The gamma factor $\gamma(X,eW_{n}^{K},e^{\prime}W_{n-1}^{K},\psi_{K})$
corresponding to the pair $(eW_{n}^{K},e^{\prime}W_{n-1}^{K})$ admits the
following universal property ([Mos16b, Theorem 5.4]) in the families of co-
Whittaker modules.

######  Theorem 3.3.

Let $A$ and $B$ be two Noetherian $\Lambda$-algebras. Let $\pi$ and
$\pi^{\prime}$ be two co-Whittaker $A[G_{n}(K)]$ and $B[G_{n-1}(K)]$-modules
respectively. Let $e$ (resp. $e^{\prime}$) be the primitive idempotent of
$\mathcal{Z}_{n}^{K}$ (resp. $\mathcal{Z}_{n-1}^{K}$) such that supercuspidal
support $f_{\pi}$ (resp. $f_{\pi^{\prime}}$) factors through
$e\mathcal{Z}_{n}^{K}$ (resp. $e^{\prime}\mathcal{Z}_{n-1}^{K}$). Then

$\gamma(X,\pi,\pi^{\prime},\psi_{K})=(f_{\pi}\otimes
f_{\pi^{\prime}})\big{(}\gamma(X,eW_{n}^{K},e^{\prime}W_{n-1}^{K},\psi_{K})\big{)}.$

###  3.3. Frobenius twist

Let $R$ be a Noetherian $\mathbb{F}$-algebra. Let $(\pi,V)$ be a co-Whittaker
$R[G_{n}(K)]$-module. Let ${\rm Fr}:R\rightarrow R$ be the map $x\mapsto
x^{l}$. We denote by $V^{(l)}$ the $R$ module $V\otimes_{{\rm Fr}}R$. The
module $(\pi^{(l)},V^{(l)})$ is called the Frobenius twist of $(\pi,V)$. Let
$\mathfrak{s}$ be the inertial equivalence class such that
$\pi\in\mathcal{M}_{\mathfrak{s}}(G_{n}(K))$, and let $e$ be the primitive
idempotent of $\mathcal{Z}_{n}^{K}$, associated with $\mathfrak{s}$. The
composition

$e\mathcal{Z}_{n}^{K}\xrightarrow{f_{\pi}}R\xrightarrow{\lambda\mapsto\lambda^{l}}R,$

denoted by $f_{\pi^{(l)}}$, is the supercuspidal support of the Frobenius
twist $\pi^{(l)}$. Let $R^{\prime}$ be a Noetherian $\mathbb{F}$-algebra, and
let $(\pi^{\prime},V^{\prime})$ be the co-Whittaker
$R^{\prime}[G_{n-1}(K)]$-module. Let $\mathfrak{s}^{\prime}$ be the inertial
equivalence class such that
$\pi^{\prime}\in\mathcal{M}_{\mathfrak{s}^{\prime}}(G_{n-1}(K))$, and let
$e^{\prime}$ be the primitive idempotent in $\mathcal{Z}_{n-1}^{K}$,
associated with $\mathfrak{s^{\prime}}$. Using Theorem 3.3 and the properties
of supercuspidal supports $f_{\pi^{(l)}}$ and $f_{\pi^{\prime(l)}}$, we get
the following identity of gamma factors

$\gamma(X,\pi,\pi^{\prime},\psi_{K})^{l}=\gamma(X^{l},\pi^{(l)},\pi^{\prime(l)},\psi_{K}^{l}).$
(3.2)

In the next subsection, we set up an idenitity of the integrals on the
homeogeneous space of $F$ with those on the homogeneous space of $E$. This is
crucial for the main results.

###  3.4. Integrals on homogeneous space

As before, let $E$ be a finite Galois extension of a $p$-adic field $F$ with
degree of extension $l$, where $l$ is coprime to $p$. We denote by $\Gamma$
the Galois group ${\rm Gal}(E/F)$. Let $R$ be an $\mathbb{F}$-algebra with $p$
being invertible $R$. There is a natural action of $\Gamma$ on the coset space
$X_{E}$ and hence on the space $C_{c}^{\infty}(X_{E},R)$, consisting of smooth
and compactly supported $R$-valued functions on $X_{E}$, given by
$(\gamma.\varphi)(g):=\varphi(\gamma^{-1}g)$, for all $\gamma\in\Gamma$, $f\in
C_{c}^{\infty}(X_{E},R)$ and for all $g\in X_{E}$. We denote by
$C_{c}^{\infty}(X_{E},R)^{\Gamma}$ the space of $\Gamma$-fixed elements in
$C_{c}^{\infty}(X_{E},R)$. With these, we now deduce:

######  Proposition 3.4.

Let $d\mu_{E}$ and $d\mu_{F}$ be the Haar measures on $X_{E}$ and $X_{F}$
respectively. Then, there exists a non-zero scalar $\alpha\in\mathbb{F}$ such
that for all $\varphi\in C_{c}^{\infty}(X_{E},R)^{\Gamma}$, we have

$\int_{X_{E}}\varphi\,d\mu_{E}=\alpha\int_{X_{F}}\varphi\,d\mu_{F}.$

###### Proof.

The proof is immediate by following the arguments of [DN22, Proposition 5.2]
mutatis-mutandis. ∎

######  Remark 3.5.

The Haar measures $d\mu_{E}$ and $d\mu_{F}$ on $X_{E}$ and $X_{F}$,
respectively, are now choosen in a way that ensures $\alpha=1$. Moreover, if
$e$ is the ramification index of the extension $E$ over $F$, then for all
$k\notin\\{re:r\in\mathbb{Z}\\}$, we have

$\int_{(X_{E}^{k})^{\Gamma}}\varphi\,d\mu_{F}=0,$

and for all $k\in\\{re:r\in\mathbb{Z}\\}$, we have

$\int_{(X_{E}^{k})^{\Gamma}}\varphi\,d\mu_{F}=\int_{X_{F}^{\frac{r}{e}}}\varphi\,d\mu_{F}.$

##  4\. Tate cohomology of co-Whittaker modules

In this section, we study the compatibility of Tate cohomology with local base
change for universal co-Whittaker modules.

### 4.1.

Let $E$ be a finite Galois extension of a $p$-adic field $F$ with $[E:F]=l$,
where $l$ and $p$ are distinct primes. We fix a generator $\gamma$ of the
cyclic group ${\rm Gal}(E/F)$. For any $\Lambda$-algebra $R$ and any smooth
$R[G_{E}]$-module $V$, we denote by $V^{\gamma}$ the smooth $R[G_{E}]$-module,
where the underlying set is $V$ but the $G_{E}$-action on $V^{\gamma}$ is
twisted by $\gamma$. Note that the functor $V\mapsto V^{\gamma}$ is exact and
covariant.

###  4.2. Galois invariance

Let $e_{\mathfrak{s}}$ be a primitive idempotent of $\mathcal{Z}_{n}^{F}$, and
let $\nu$ be the $l$-inertial type corresponding to $e_{\mathfrak{s}}$. Then
there exists a primitive idempotent $e_{\mathfrak{r}}$ of
$\mathcal{Z}_{n}^{E}$, which corresponds to the same $l$-inertial type $\nu$.
We have the base change maps
$z_{E/F}:e_{\mathfrak{r}}\mathcal{Z}_{n}^{E}\rightarrow
e_{\mathfrak{s}}\mathcal{Z}_{n}^{F}$ and $B_{E/F}:R_{\nu}^{E}\rightarrow
R_{\nu}^{F}$, as defined in the subsection (2.3.3), with the following
identity

$\mathbb{L}_{\nu}^{F}\circ z_{E/F}=B_{E/F}\circ\mathbb{L}_{\nu}^{E}.$ (4.1)

We denote by $A_{E}$ and $A_{F}$ the Noetherian $\Lambda$-algebras
$e_{\mathfrak{r}}\mathcal{Z}_{n}^{E}$ and
$e_{\mathfrak{s}}\mathcal{Z}_{n}^{E}$ respectively. Note that the ring $A_{F}$
is considered as an $A_{E}$-module via the map $z_{E/F}$.

######  Lemma 4.1.

Let $\mathcal{V}$ be the co-Whittaker $A_{F}[G_{n}(E)]$-module
$e_{\mathfrak{r}}W_{n}^{E}\otimes_{A_{E}}A_{F}$. Then we have

$\mathbb{W}(\mathcal{V},\psi_{E,A_{F}})=\mathbb{W}(\mathcal{V}^{\gamma},\psi_{E,A_{F}}).$

###### Proof.

We use the local converse therorem ([LM20, Theorem 1.1]) for co-Whittaker
modules. It is enough to show that

$\gamma(X,\mathcal{V},\tau,\psi_{E})=\gamma(X,\mathcal{V}^{\gamma},\tau,\psi_{E}),$

for $\tau$ varying over all irreducible generic integral
$\overline{\mathcal{K}}$-representation of $G_{t}(E)$, for $1\leq
t\leq[\frac{n}{2}]$. For a specialization
$x:A_{F}\rightarrow\overline{\mathcal{K}}$, the $G_{n}(E)$ representation
$(e_{\mathfrak{r}}W_{n}^{E}\otimes_{A_{E}}A_{F})\otimes_{x}\overline{\mathcal{K}}$,
denoted by $\Pi_{x}$, is the base change lift of a smooth representation
$\pi_{x}$ of $G_{n}(F)$. In particular, $\Pi_{x}^{\gamma}\simeq\Pi_{x}$, for
all $\gamma\in\Gamma$. Since

$x(\gamma(X,\mathcal{V},\tau,\psi_{E}))=\gamma(X,\Pi_{x},\tau,\psi_{E}),$

and

$x(\gamma(X,\mathcal{V}^{\gamma},\tau,\psi_{E}))=\gamma(X,\Pi_{x}^{\gamma},\tau,\psi_{E}),$

we have

$x(\gamma(X,\mathcal{V},\tau,\psi_{E}))=x(\gamma(X,\mathcal{V}^{\gamma},\tau,\psi_{E})),$

for all $\overline{\mathcal{K}}$-points of $A_{F}$. The lemma now follows
since $A_{F}$ is reduced and flat over $\Lambda$. ∎

From these, we deduce:

######  Lemma 4.2.

Let $\mathcal{V}$ be the co-Whittaker module
$e_{\mathfrak{r}}W_{n}^{E}\otimes_{A_{E}}A_{F}$. Then the Whittaker space
$\mathbb{W}(\mathcal{V},\psi_{E,A_{F}})$ is invariant under the action of
${\rm Gal}(E/F)$.

###### Proof.

Since $\mathcal{V}$ is co-Whittaker, we have the $A_{F}$-module isomorphism
$\mathcal{V}^{(n)}\simeq A_{F}$. Precomposing this isomorphism with the
quotient map $\mathcal{V}\rightarrow\mathcal{V}^{(n)}$ induces the
$A_{F}$-linear map $\mathcal{W}:\mathcal{V}\rightarrow A_{F}$ with

$\mathcal{W}(n.v)=\Psi_{E,A_{F}}(n)\mathcal{W}(v),$

for all $v\in\mathcal{V}$ and $n\in N_{n}(E)$. Moreover, for any
$\gamma\in{\rm Gal}(E/F)$, we have

$\mathcal{W}(\gamma(n).v)=\Psi_{E,A_{F}}(\gamma(n))\mathcal{W}(v)=\Psi_{E,A_{F}}(n)\mathcal{W}(v).$

Therefore, the Whittaker spaces of $\mathcal{V}$ and $\mathcal{V}^{\gamma}$
with respect to the character $\psi_{E}$ are induced by the same linear map
$\mathcal{W}$. Let $W_{v}$ be an element of
$\mathbb{W}(\mathcal{V},\psi_{E,A_{F}})$. Then

$(\gamma^{-1}.W_{v})(g)=\mathcal{W}(\gamma(g).v).$

This shows that
$\gamma^{-1}.W_{v}\in\mathbb{W}(\mathcal{V}^{\gamma},\psi_{E,A_{F}})$. But
$\mathbb{W}(\mathcal{V}^{\gamma},\psi_{E,A_{F}})=\mathbb{W}(\mathcal{V},\psi_{E,A_{F}})$,
by Lemma 4.1. This completes the proof. ∎

We now prove the following theorem.

######  Theorem 4.3.

Let $\mathcal{V}$ be the co-Whittaker $A_{F}[G_{n}(E)]$-module
$e_{\mathfrak{r}}W_{n}^{E}\otimes_{A_{E}}A_{F}$, and let $\mathcal{V}_{F}$ be
the $\mathbb{F}[G_{n}(F)]$-module
$e_{\mathfrak{s}}W_{n}^{F}\otimes_{\Lambda}\mathbb{F}$. Let $R$ be the
$\mathbb{F}$-algebra $A_{F}\otimes_{\Lambda}\mathbb{F}$. Then there is a
$G_{n}(F)$-stable subspace $\mathcal{M}$ of the Tate cohomology group
$\widehat{H}^{0}(\mathbb{W}(\mathcal{V},\psi_{E,A_{F}}))$ with
$G_{n}(F)$-equivariant surjection

$\mathcal{M}\longrightarrow\mathbb{W}(\mathcal{V}_{F}^{(l)},\psi_{F,R}^{l}).$

###### Proof.

We prove the above theorem using induction on the integer $n$.

#### 4.2.1.

Let us consider the case $n=1$. Recall that $\mathcal{Z}_{1}^{K}$ is the
convolution $\Lambda$-algebra $C^{\infty}_{c}(K^{\times},\Lambda)$. Let
$P_{K}$ be the prime to $l$ part of $\mathfrak{o}_{K}^{\times}$. For any
character $\eta:P_{K}\rightarrow\Lambda^{\times}$, we get an idempotent
$e_{\eta}$ of $\mathcal{Z}_{1}^{K}$, and the $\Lambda$-algebra
$e_{\eta}\mathcal{Z}_{1}^{K}$ is equal to
$\mathcal{H}(K^{\times},P_{K},\eta)$–the endomorphism algebra of the
$\Lambda[E^{\times}]$ representation
$\operatorname{ind}_{P_{K}}^{K^{\times}}\eta$. Local class field theory gives
a character $\nu$ of $I_{K}^{(l)}$ assocaited with $\eta$. The ring
$R_{\nu}^{K}$ is isomorphic to $\Lambda[E^{\times}/P_{K}]$ and
$\mathbb{L}_{\eta}^{K}$ is the identity map. Let
$\chi:F^{\times}\rightarrow\mathbb{F}$ be a character and let $\tilde{\chi}$
be the character $\chi\circ{\rm Nr}_{E/F}$. The characters $\chi$ and
$\tilde{\chi}$ give rise to idempotents $e_{\mathfrak{s}}$ and
$e_{\mathfrak{r}}$ of $\mathcal{Z}_{1}^{F}$ and $\mathcal{Z}_{1}^{E}$
respectively. From local class field theory the map $B_{E/F}$ is induced by
the norm map ${\rm Nr}_{E/F}:E^{\times}\rightarrow F^{\times}$. The module
$e_{\mathfrak{r}}\mathcal{Z}_{1}^{E}$ also has a natural $E^{\times}$-action
and the identification

$e_{\mathfrak{r}}\mathcal{Z}_{1}^{E}\otimes_{z_{E/F}}e_{\mathfrak{s}}\mathcal{Z}_{1}^{F}\simeq
e_{\mathfrak{s}}\mathcal{Z}_{1}^{F},$

makes $e_{\mathfrak{s}}\mathcal{Z}_{1}^{F}$ an
$e_{\mathfrak{s}}\mathcal{Z}_{1}^{F}[E^{\times}]$-module. Thus, we have

$\widehat{H}^{0}(e_{\mathfrak{s}}\mathcal{Z}_{1}^{F})\xrightarrow{\sim}(e_{\mathfrak{s}}\mathcal{Z}_{1}^{F}\otimes_{\Lambda}\mathbb{F})^{(l)}.$

#### 4.2.2.

So now, we assume that the result is true for $n-1$. We denote by $\tau_{E}$
(resp. $\tau_{F}$) the action of $G_{n}(E)$ (resp. $G_{n}(F)$) on the space
$\mathcal{V}$ (resp. $\mathcal{V}_{F}$). Recall that
$\mathbb{W}(\mathcal{V},\psi_{E,A_{F}})$ is invariant under $\Gamma$-action.
Let $\Phi_{n}$ be the composite map

$\mathbb{K}(\mathcal{V},\psi_{E,A_{F}})^{\Gamma}\xrightarrow{\theta_{\ell}}{\rm
Ind}_{N_{n}(E)}^{P_{n}(E)}(\Psi_{E,R})\xrightarrow{{\rm res}_{P_{n}(F)}}{\rm
Ind}_{N_{n}(F)}^{P_{n}(F)}(\Psi_{F,R}^{l}),$

where the map $\theta_{\ell}$ is induced by the morphism $A_{F}\rightarrow R$,
sending $x$ to $x\otimes 1$. Note that the map $\Phi_{n}$ factorizes through
the Tate cohomology space
$\widehat{H}^{0}(\mathbb{K}(\mathcal{V},\psi_{E,A_{F}}))$. Let
$\mathcal{M}(\psi_{F})$ be the inverse image of the Kirillov space
$\mathbb{K}(\mathcal{V}_{F}^{(l)},\psi_{F,R}^{l})$ under $\Phi_{n}$. Note that
$\mathcal{M}(\psi_{F})$ is non-zero, and it is stable under the action of
$P_{n}(F)$ with non-zero $P_{n}(F)$-equivariant map

$\Phi_{n}:\mathcal{M}(\psi_{F})\longrightarrow\mathbb{K}(\mathcal{V}_{F}^{(l)},\psi_{F,R}^{l}).$

Let $V$ be an element of $\mathcal{M}(\psi_{F})$. We will show that

$\Phi_{n}(\overline{\tau_{E}(w_{n})}V)=\tau_{F}^{(l)}(w_{n})\Phi_{n}(V),$
(4.2)

where $\overline{\tau_{E}(w_{n})}$ is the induced action of $\tau_{E}(w_{n})$
on $\widehat{H}^{0}(\mathbb{K}(\mathcal{V},\psi_{E,A_{F}}))$.

#### 4.2.3.

Let $e_{\mathfrak{s}^{\prime}}$ be an arbitrary primitive idempotent of
$\mathcal{Z}_{n-1}^{F}$ and let $\nu^{\prime}$ be the $l$-inertial type
corresponding to $e_{\mathfrak{s}^{\prime}}$. Then there exists a primitive
idempotent $e_{\mathfrak{r}^{\prime}}$ of $\mathcal{Z}_{n-1}^{E}$, which
corresponds to the same $l$-inertial type $\nu^{\prime}$. If $A_{E}^{\prime}$
(resp. $A_{F}^{\prime}$) denotes the Noetherian $\Lambda$-algebras
$e_{\mathfrak{r}^{\prime}}\mathcal{Z}_{n-1}^{E}$ (resp.
$e_{\mathfrak{s}^{\prime}}\mathcal{Z}_{n-1}^{F}$), then we have the base
change map $z_{E/F}^{\prime}:A_{E}^{\prime}\rightarrow A_{F}^{\prime}$ with

$\mathbb{L}_{\nu^{\prime}}^{F}\circ
z_{E/F}^{\prime}=B_{E/F}^{\prime}\circ\mathbb{L}_{\nu^{\prime}}^{E}.$ (4.3)

We denote by $\mathcal{V}^{\prime}$ the co-Whittaker
$A_{F}^{\prime}[G_{n-1}(E)]$-module
$e_{\mathfrak{r}}^{\prime}W_{n-1}^{E}\otimes_{A_{E}^{\prime}}A_{F}^{\prime}$.
Let $R^{\prime}$ be the $\mathbb{F}$-algebra
$A_{F}^{\prime}\otimes_{\Lambda}\mathbb{F}$, and let
$\mathcal{V}_{F}^{\prime}$ the co-Whittaker $R^{\prime}[G_{n-1}(F)]$-module
$e_{\mathfrak{s}^{\prime}}W_{n-1}^{F}\otimes_{\Lambda}\mathbb{F}$. Using
induction hypothesis, there is a $G_{n-1}(F)$-stable subspace
$\mathcal{M}^{\prime}(\psi_{F})$ of the Tate cohomology group
$\widehat{H}^{0}(\mathbb{W}(\mathcal{V}^{\prime},\psi_{E,A_{F}^{\prime}}^{-1}))$
such that the non-zero $P_{n}(F)$-equivariant map

$\Phi_{n-1}:\mathbb{K}(\mathcal{V}^{\prime},\psi_{E,A_{F}^{\prime}}^{-1})^{\Gamma}\xrightarrow{\theta_{\ell}^{\prime}}{\rm
Ind}_{N_{n-1}(E)}^{P_{n-1}(E)}(\Psi_{E,R^{\prime}}^{-1})\xrightarrow{{\rm
res}_{P_{n-1}(F)}}{\rm
Ind}_{N_{n-1}(F)}^{P_{n-1}(F)}(\Psi_{F,R^{\prime}}^{-l}),$

gives the $G_{n}(F)$-equivariant surjection

$\mathcal{M}^{\prime}(\psi_{F})\longrightarrow\mathbb{W}(\mathcal{V}_{F}^{\prime},\psi_{F,R^{\prime}}^{-l}),$

which we also denote by $\Phi_{n-1}$. Here, the map $\theta_{\ell}^{\prime}$
is induced by the morphism $A_{F}^{\prime}\rightarrow R^{\prime}$, sending
$\alpha$ to $\alpha\otimes 1$. We denote by $\tau_{E}^{\prime}$ (resp.
$\tau_{F}^{\prime}$) the action of $G_{n-1}(E)$ (resp. $G_{n-1}(F)$) on the
space $\mathcal{V}^{\prime}$ (resp. $\mathcal{V}_{F}^{\prime}$).

#### 4.2.4.

Let $V^{\prime}$ be an element of
$\mathbb{W}(\mathcal{V}_{F}^{\prime(l)},\psi_{F,R^{\prime}}^{-l})$. Then there
exists an element $W^{\prime}$ in the Whittaker space
$\mathbb{W}(\mathcal{V}^{\prime},\psi_{E,A_{F}^{\prime}})^{\Gamma}$ such that
${\rm res}_{P_{n-1}(E)}(W^{\prime})$ is mapped to ${\rm
res}_{P_{n-1}(F)}(V^{\prime})$ under $\Phi_{n-1}$. Let $W$ be the element of
$\mathbb{W}(\mathcal{V},\psi_{E,A_{F}})^{\Gamma}$ such that ${\rm
res}_{P_{n}(E)}(W)=V$. Then, using functional equation over $E$, we get

$\sum_{k\in\mathbb{Z}}c_{k}^{E}(\widetilde{W},\widetilde{W^{\prime}})X^{-fk}=\varpi_{\tau_{E}^{\prime}}(-1)^{n-1}\gamma(X,\mathcal{V},\mathcal{V}^{\prime},\psi_{E})\sum_{k\in\mathbb{Z}}c_{k}^{E}(W,W^{\prime})X^{fk},$

where $f$ is the residue degree of the extension $E/F$. Applying the morphism
$(\theta_{\ell}\otimes\theta_{\ell}^{\prime})$, and using the identity (3.1)
and Remark 3.5, the above relation becomes

$\begin{split}\sum_{k\in\mathbb{Z}}c_{-k}^{F}\big{(}\overline{\tau_{E}(w_{n})}&\theta_{\ell}(W),\overline{\tau_{E}^{\prime}(w_{n-1})}\theta_{\ell}^{\prime}(W^{\prime})\big{)}X^{-lk}\\\
&=\varpi_{\tau_{E}^{\prime}}(-1)^{n-1}(\theta_{\ell}\otimes\theta_{\ell}^{\prime})(\gamma(X,\mathcal{V},\mathcal{V}^{\prime},\psi_{E}))\sum_{k\in\mathbb{Z}}c_{k}^{F}\big{(}\theta_{\ell}(W),V^{\prime}\big{)}X^{lk}.\end{split}$
(4.4)

By induction hypothesis, we have

$\Phi_{n-1}(\overline{\tau_{E}^{\prime}(w_{n-1})}W^{\prime})=\tau_{F}^{\prime(l)}(w_{n-1})V^{\prime}.$

Using this, it follows from (4.4) that

$\begin{split}\sum_{k\in\mathbb{Z}}c_{-k}^{F}\big{(}\overline{\tau_{E}(w_{n})}&\theta_{\ell}(W),\tau_{F}^{\prime(l)}(w_{n-1})V^{\prime}\big{)}X^{-lk}\\\
&=\varpi_{\tau_{F}^{\prime}}(-1)^{l(n-1)}(\theta_{\ell}\otimes\theta_{\ell}^{\prime})(\gamma(X,\mathcal{V},\mathcal{V}^{\prime},\psi_{E}))\sum_{k\in\mathbb{Z}}c_{k}^{F}(\theta_{\ell}(W),V^{\prime})X^{lk}.\end{split}$
(4.5)

#### 4.2.5.

Recall that $\Phi_{n}(V)$ is an element of the Kirillov space
$\mathbb{K}(\mathcal{V}_{F}^{(l)},\psi_{F,R}^{l})$. Let $U$ be the element of
$\mathbb{W}(\mathcal{V}_{F}^{(l)},\psi_{F,R}^{l})$ such that ${\rm
res}_{P_{n}(F)}(U)=\Phi_{n}(V)$. By Theorem 3.2, the assertion (4.2) is
equivalent to the following identity:

$\begin{split}\sum_{k\in\mathbb{Z}}c_{-k}^{F}\big{(}\Phi_{n}(\overline{\tau_{E}(w_{n})}\theta_{\ell}(W)),\tau_{F}^{\prime(l)}&(w_{n-1})V^{\prime}\big{)}X^{-lk}\\\
=&\sum_{k\in\mathbb{Z}}c_{-k}^{F}\big{(}\tau_{F}^{(l)}(w_{n})U,\tau_{F}^{\prime(l)}(w_{n-1})V^{\prime}\big{)}X^{-lk}\end{split}$
(4.6)

From functional equation over F, we get

$\displaystyle\sum_{k\in\mathbb{Z}}c_{k}^{F}(\widetilde{U},\widetilde{V^{\prime}})X^{-k}=\varpi_{\tau_{F}^{\prime}}(-1)^{l(n-1)}\gamma(X,\mathcal{V}_{F}^{(l)},\mathcal{V}_{F}^{\prime(l)},\psi_{F}^{l})\sum_{k\in\mathbb{Z}}c_{k}^{F}(U,V^{\prime})X^{k}.$

Using the relation (3.1) and replacing the variable $X$ by $X^{l}$ to the
above equality, we have

$\displaystyle\begin{split}\sum_{k\in\mathbb{Z}}c_{-k}^{F}\big{(}\tau_{F}^{(l)}(w_{n})U,\
&\tau_{F}^{\prime(l)}(w_{n-1})V^{\prime}\big{)}X^{-lk}\\\
=&\varpi_{\tau_{F}^{\prime}}(-1)^{l(n-1)}\gamma(X,\mathcal{V}_{F}^{(l)},\mathcal{V}_{F}^{\prime(l)},\psi_{F}^{l})\sum_{k\in\mathbb{Z}}c_{k}^{F}(U,V^{\prime})X^{lk}.\end{split}$

#### 4.2.6.

Comparing the above equation with (4.5), the equality (4.6) is now equivalent
to the following identity of gamma factors

$(\theta_{\ell}\otimes\theta_{\ell}^{\prime})(\gamma(X,\mathcal{V},\mathcal{V}^{\prime},\psi_{E}))=\gamma(X,\mathcal{V}_{F}^{(l)},\mathcal{V}_{F}^{\prime(l)},\psi_{F}^{l}).$

First, note that the base change maps $B_{E/F}$ and $B_{E/F}^{\prime}$
together with the universal property of the pairs
$(R_{\nu}^{E},\rho_{\nu}^{E})$ and
$(R_{\nu^{\prime}}^{E},\rho_{\nu^{\prime}}^{F})$ induces the isomorphisms

${\rm
res}_{\mathcal{W}_{E}}(\rho_{\nu}^{F})\simeq\rho_{\nu}^{E}\otimes_{R_{\nu}^{E}}R_{\nu}^{F}$
and ${\rm
res}_{\mathcal{W}_{E}}(\rho_{\nu^{\prime}}^{F})\simeq\rho_{\nu^{\prime}}^{E}\otimes_{R_{\nu^{\prime}}^{E}}R_{\nu^{\prime}}^{F}$.

Using these isomorphisms and the commutativity relations (4.1) and (4.3), we
have

$\displaystyle\gamma(X,\mathcal{V},\mathcal{V}^{\prime},\psi_{E})$
$\displaystyle=(z_{E/F}\circ(\mathbb{L}_{\nu}^{E})^{-1})\otimes(z_{E/F}^{\prime}\circ(\mathbb{L}_{\nu^{\prime}}^{E})^{-1})\big{(}\gamma(X,\rho_{\nu}^{E}\otimes\rho_{\nu^{\prime}}^{E},\psi_{E})\big{)}$
$\displaystyle=((\mathbb{L}_{\nu}^{F})^{-1}\circ
B_{E/F})\otimes((\mathbb{L}_{\nu^{\prime}}^{F})^{-1}\circ
B_{E/F}^{\prime})(\gamma(X,\rho_{\nu}^{E}\otimes\rho_{\nu^{\prime}}^{E},\psi_{E}))$
$\displaystyle=((\mathbb{L}_{\nu}^{F})^{-1}\otimes(\mathbb{L}_{\nu^{\prime}}^{F})^{-1})\big{(}\gamma(X,(\rho_{\nu}^{E}\otimes_{R_{\nu}^{E}}R_{\nu}^{F})\otimes(\rho_{\nu^{\prime}}^{E}\otimes_{R_{\nu^{\prime}}^{E}}R_{\nu^{\prime}}^{F}),\psi_{E})\big{)}$
$\displaystyle=((\mathbb{L}_{\nu}^{F})^{-1}\otimes(\mathbb{L}_{\nu^{\prime}}^{F})^{-1})\big{(}\gamma(X,\rho_{\nu}^{F}\otimes\rho_{\nu^{\prime}}^{F}\otimes{\rm
ind}_{\mathcal{W}_{E}}^{\mathcal{W}_{F}}(1_{E}),\psi_{F})\big{)}$
$\displaystyle=\prod_{\eta}\gamma(X,e_{\mathfrak{s}}W_{n}^{F},e_{\mathfrak{s}^{\prime}}W_{n-1}^{F}\otimes\eta,\psi_{F}),$

where $\eta$ runs over the characters of ${\rm Gal}(E/F)$. Finally, applying
the morphism $(\theta_{\ell}\otimes\theta_{\ell}^{\prime})$ and using the
identity (3.2), we get

$(\theta_{\ell}\otimes\theta_{\ell}^{\prime})(\gamma(X,\mathcal{V},\mathcal{V}^{\prime},\psi_{E}))=\gamma(X^{l},\mathcal{V}_{F}^{(l)},\mathcal{V}_{F}^{\prime(l)},\psi_{F}^{l}).$

This shows that $\Phi_{n}$ is a non-zero $G_{n}(F)$-equivariant map. Since
$\mathcal{V}_{F}^{(l)}$ is co-Whittaker $R[G_{n}(F)]$-module, the map
$\Phi_{n}$ is infact a surjection. This completes the proof. ∎

##  5\. Tate cohomology of generic representations

In this section, we prove our main result (Theorem 1.1) using Theorem 4.3. We
will continue with the notations of the preceding section.

### 5.1.

Let $\pi_{F}$ be an integral generic representation of $G_{n}(F)$ over a
$\overline{\mathcal{K}}$-vector space whose mod-$l$ inertial supercuspidal
support is $\mathfrak{s}$. Let $\pi_{E}$ be the base change lifting of
$\pi_{F}$ with mod-$l$ inertial supercuspidal support $\mathfrak{r}$. Recall
that we have the base change map

$z_{E/F}:e_{\mathfrak{r}}\mathcal{Z}_{n}^{E}\longrightarrow
e_{\mathfrak{s}}\mathcal{Z}_{n}^{F}.$

If $f_{\pi_{F}}$ (resp. $f_{\pi_{E}}$) denotes the supercuspidal support of
$\pi_{F}$ (resp. $\pi_{E}$), then we have

$f_{\pi_{E}}=f_{\pi_{F}}\circ z_{E/F}.$

Let $J_{\ell}(\pi_{F})$ be the unique generic sub-quotient of the mod-$l$
reduction $r_{\ell}(\pi_{F})$. The supercuspidal support of the
$\mathbb{F}$-representation $J_{\ell}(\pi_{F})$ is equal to $\mathfrak{s}$
([Hel16a, Proposition 4.13]). We denote by $A_{E}$ and $A_{F}$ the Noetherian
$\Lambda$-algebras $e_{\mathfrak{r}}\mathcal{Z}_{n}^{E}$ and
$e_{\mathfrak{s}}\mathcal{Z}_{n}^{F}$ respectively.

######  Theorem 5.1.

Let $F$ be a $p$-adic field and let $E$ be a finite Galois extension of $F$
with $[E:F]=l$, where $l$ and $p$ are distinct odd primes. Let $\pi_{F}$ be an
integral generic $\overline{\mathcal{K}}$-representation of $G_{n}(F)$, and
let $\pi_{E}$ be the base change lifting of $\pi_{F}$ to $G_{n}(E)$. Let
$\mathbb{W}^{0}(\pi_{E},\psi_{E,\overline{\mathcal{K}}})$ be the space of all
$\mathcal{O}$-valued functions in
$\mathbb{W}(\pi_{E},\psi_{E,\overline{\mathcal{K}}})$. Then the
$\mathbb{F}$-representation $J_{\ell}(\pi_{F})^{(l)}$ is a subquotient of the
Tate cohomology group
$\widehat{H}^{0}(\mathbb{W}^{0}(\pi_{E},\psi_{E,\overline{\mathcal{K}}}))$.

###### Proof.

The proof relies on the completeness of Whittaker models (Theorem 3.2).
Although the proof follows from the same line of arguments as in Theorem 4.3,
we provide it in detail for the purpose of the completeness. We follow the
same notations and terminologies as in Theorem 4.3.

#### 5.1.1.

Note that the lattice
$\mathbb{W}^{0}(\pi_{E},\psi_{E,\overline{\mathcal{K}}})$ is stable under the
action of $G_{n}(E)$ ([Vig04, Theorem]). Let $\Phi_{n}$ be the composite map

$\mathbb{K}^{0}(\pi_{E},\psi_{E,\overline{\mathcal{K}}})^{\Gamma}\xrightarrow{r_{\ell}}{\rm
Ind}_{N_{n}(E)}^{P_{n}(E)}(\Psi_{E,\mathbb{F}}^{l})\xrightarrow{{\rm
res}_{P_{n}(F)}}{\rm Ind}_{N_{n}(F)}^{P_{n}(F)}(\Psi_{F,\mathbb{F}}^{l}),$

where $r_{\ell}$ denotes the pointwise mod-$l$ reduction. It is clear that the
map $\Phi_{n}$ factorizes through the space
$\widehat{H}^{0}(\mathbb{K}^{0}(\pi_{E},\psi_{E,\overline{\mathcal{K}}}))$.
Let $\mathcal{N}(\psi_{F})$ be the inverse image of the Kirillov space
$\mathbb{K}(J_{\ell}(\pi_{F})^{(l)},\psi_{F,\mathbb{F}}^{l})$ under
$\Phi_{n}$. Then $\mathcal{N}(\psi_{F})$ is a non-zero $P_{n}(F)$-stable
subspace of
$\widehat{H}^{0}(\mathbb{K}^{0}(\pi_{E},\psi_{E,\overline{\mathcal{K}}}))$
with a non-zero $P_{n}(F)$-equivariant map

$\Phi_{n}:\mathcal{N}(\psi_{F})\longrightarrow\mathbb{K}(J_{\ell}(\pi_{F})^{(l)},\psi_{F,\mathbb{F}}^{l}).$

We will prove that $\mathcal{N}(\psi_{F})$ is $G_{n}(F)$-stable and the map
$\Phi_{n}$ is $G_{n}(F)$-equivariant. To be precise, let $V$ be an element of
$\mathcal{N}(\psi_{F})$. We will show that

$\Phi_{n}(\overline{\pi_{E}(w_{n})}V)=J_{\ell}(\pi_{F})^{(l)}(w_{n})\Phi_{n}(V),$

where $\overline{\pi_{E}(w_{n})}$ is the linear operator on
$\widehat{H}^{0}(\mathbb{K}(\pi_{E},\psi_{E,\overline{\mathcal{K}}}))$ induced
by $\pi_{E}(w_{n})$. Let $W$ be an element of
$\mathbb{W}^{0}(\pi_{E},\psi_{E,\overline{\mathcal{K}}})^{\Gamma}$ with ${\rm
res}_{P_{n}(E)}(W)=V$. By Theorem 3.2, the above assertion is equivalent to
the following identity:

$\sum_{k\in\mathbb{Z}}c_{k}^{F}\big{(}\Phi_{n}(\overline{\pi_{E}(w_{n})}V),V^{\prime}\big{)}X^{k}=\sum_{k\in\mathbb{Z}}c_{k}^{F}\big{(}J_{\ell}(\pi_{F})^{(l)}(w_{n})\Phi_{n}(V),V^{\prime}\big{)}X^{k},$
(5.1)

for all
$V^{\prime}\in\mathbb{W}\big{(}(e_{\mathfrak{s}^{\prime}}W_{n-1}^{F}\otimes_{\Lambda}\mathbb{F})^{(l)},\psi_{F,R^{\prime}}^{-l}\big{)}$
and for all primitive idempotents $e_{\mathfrak{s}^{\prime}}$ of
$\mathcal{Z}_{n-1}^{F}$. Here $R^{\prime}$ denotes the ring
$e_{\mathfrak{s}^{\prime}}\mathcal{Z}_{n-1}^{F}\otimes_{\Lambda}\mathbb{F}$.

#### 5.1.2.

Let $e_{\mathfrak{s}^{\prime}}$ be a primitive idempotent of
$\mathcal{Z}_{n-1}^{F}$. Then there exists a primitive idempotent
$e_{\mathfrak{r}^{\prime}}$ of $\mathcal{Z}_{n-1}^{E}$ such that we have the
base change map $z_{E/F}^{\prime}:A_{E}^{\prime}\rightarrow A_{F}^{\prime}$
with

$\mathbb{L}_{\nu^{\prime}}^{F}\circ
z_{E/F}^{\prime}=B^{\prime}_{E/F}\circ\mathbb{L}_{\nu^{\prime}}^{E},$

where $\nu^{\prime}$ is the $l$-inertial type, which corresponds to both
$e_{\mathfrak{r}^{\prime}}$ and $e_{\mathfrak{s}^{\prime}}$. As before, we
denote by $\mathcal{V}^{\prime}$ and $\mathcal{V}_{F}^{\prime}$ the co-
Whittaker modules
$e_{\mathfrak{r}^{\prime}}W_{n-1}^{E}\otimes_{A_{E}^{\prime}}A_{F}^{\prime}$
and $e_{\mathfrak{s}^{\prime}}W_{n-1}^{F}\otimes_{\Lambda}\mathbb{F}$
respectively. The action of $G_{n}(E)$ (resp. $G_{n}(F)$) on the space
$\mathcal{V}^{\prime}$ (resp. $\mathcal{V}_{F}^{\prime}$) is denoted by
$\pi_{E}^{\prime}$ (resp. $\pi_{F}^{\prime}$).

#### 5.1.3.

Let $V^{\prime}$ be an element of
$\mathbb{W}(\mathcal{V}_{F}^{\prime(l)},\psi_{F,R^{\prime}}^{-l})$. Following
the arguments of Theorem 4.3, we get an element $W^{\prime}$ in
$\mathbb{W}(\mathcal{V}^{\prime},\psi_{E,A_{F}^{\prime}}^{-1})^{\Gamma}$ such
that $W^{\prime}$ is mapped to $V^{\prime}$ under the composition

$\Phi_{n-1}:\mathbb{W}(\mathcal{V}^{\prime},\psi_{E,A_{F}^{\prime}}^{-1})^{\Gamma}\xrightarrow{\theta_{\ell}^{\prime}}{\rm
Ind}_{N_{n-1}(E)}^{P_{n-1}(E)}(\Psi_{E,R^{\prime}}^{-1})\xrightarrow{{\rm
res}_{G_{n-1}(F)}}{\rm
Ind}_{N_{n-1}(F)}^{G_{n-1}(F)}(\Psi_{F,R^{\prime}}^{-l}),$

with the following identity

$\Phi_{n-1}(\pi_{E}^{\prime}(w_{n-1})W^{\prime})=\pi_{F}^{\prime(l)}(w_{n-1})V^{\prime}.$
(5.2)

Recall that the map $\theta_{\ell}^{\prime}$ is induced by the morphism
$A_{F}^{\prime}\rightarrow R^{\prime}$, sending $x$ to $x\otimes 1$. From
functional equation over $E$, we get

$\displaystyle\sum_{k\in\mathbb{Z}}c_{-k}^{E}\big{(}\overline{\pi_{E}(w_{n})}$
$\displaystyle
r_{l}(W),\pi_{E}^{\prime}(w_{n-1})\theta_{\ell}^{\prime}(W^{\prime})\big{)}X^{-fk}$
$\displaystyle=\varpi_{\pi_{E}^{\prime}}(-1)^{n-1}(r_{\ell}\otimes\theta_{\ell}^{\prime})\big{(}\gamma(X,\pi_{E},\mathcal{V}^{\prime},\psi_{E})\big{)}\sum_{k\in\mathbb{Z}}c_{k}^{E}\big{(}r_{l}(W),V^{\prime}\big{)}X^{fk}.$

Using Remark 3.5 and the relation (5.2), it follows from the above identity
that

$\begin{split}\sum_{k\in\mathbb{Z}}c_{-k}^{F}\big{(}\overline{\pi_{E}(w_{n})}&r_{l}(W),\pi_{F}^{\prime(l)}(w_{n-1})V^{\prime}\big{)}X^{-lk}\\\
&=\varpi_{\pi_{F}^{\prime}}(-1)^{l(n-1)}(r_{\ell}\otimes\theta_{\ell}^{\prime})\big{(}\gamma(X,\pi_{E},\mathcal{V}^{\prime},\psi_{E})\big{)}\sum_{k\in\mathbb{Z}}c_{k}^{F}(r_{l}(W),V^{\prime})X^{lk}.\end{split}$
(5.3)

#### 5.1.4.

Let $U$ be an element of
$\mathbb{W}(J_{\ell}(\pi_{F})^{(l)},\psi_{F,\mathbb{F}}^{-l})$ such that ${\rm
res}_{P_{n}(F)}(U)=\Phi_{n}(V)$. The functional equation over $F$ gives

$\displaystyle\begin{split}\sum_{k\in\mathbb{Z}}c_{-k}^{F}\big{(}J_{\ell}(\pi_{F})^{(l)}&(w_{n})U,\pi_{F}^{\prime(l)}(w_{n-1})V^{\prime}\big{)}X^{-k}\\\
&=\varpi_{\pi_{F}^{\prime}}(-1)^{l(n-1)}\gamma(X,J_{\ell}(\pi_{F})^{(l)},\mathcal{V}_{F}^{\prime(l)},\psi_{F}^{l})\sum_{k\in\mathbb{Z}}c_{k}^{F}(U,V^{\prime})X^{k}\end{split}$

Replacing $X$ by $X^{l}$ to the above equation, we get

$\begin{split}\sum_{k\in\mathbb{Z}}c_{-k}^{F}\big{(}J_{\ell}(\pi_{F})^{(l)}&(w_{n})U,\pi_{F}^{\prime(l)}(w_{n-1})W^{\prime}\big{)}X^{-lk}\\\
&=\varpi_{\pi_{F}^{\prime}}(-1)^{l(n-1)}\gamma(X^{l},J_{\ell}(\pi_{F})^{(l)},\mathcal{V}_{F}^{\prime(l)},\psi_{F}^{l})\sum_{k\in\mathbb{Z}}c_{k}^{F}(U,W^{\prime})X^{lk}.\end{split}$
(5.4)

#### 5.1.5.

Comparing the relations (5.3) and (5.4), the assertion (5.1) is now equivalent
to the following identity of gamma factors

$(r_{\ell}\otimes\theta_{\ell}^{\prime})\big{(}\gamma(X,\pi_{E},\mathcal{V}^{\prime},\psi_{E})\big{)}=\gamma(X^{l},J_{\ell}(\pi_{F})^{(l)},\mathcal{V}_{F}^{\prime(l)},\psi_{F}^{l}).$

This follows from similar type of computation of gamma factors as we did in
Theorem 4.3. First, note that

$\gamma(X,\pi_{E},\mathcal{V}^{\prime},\psi_{E})=(f_{\pi_{E}}\otimes
z_{E/F}^{\prime})(\gamma(X,e_{\mathfrak{r}}W_{n}^{E},e_{\mathfrak{r}^{\prime}}W_{n-1}^{E},\psi_{E})).$

Using the relations (4.1) and (4.3) and the fact that
$f_{\pi_{E}}=f_{\pi_{F}}\circ z_{E/F}$, we get the following equality:

$\gamma(X,\pi_{E},\mathcal{V}^{\prime},\psi_{E})=\prod_{\eta}\gamma(X,\pi_{F},e_{\mathfrak{s}^{\prime}}W_{n-1}^{F}\otimes\eta,\psi_{F}),$

where $\eta$ runs over the characters of ${\rm Gal}(E/F)$. Applying the
morphism $(r_{\ell}\otimes\theta_{\ell}^{\prime})$ to the above identity, we
get

$(r_{\ell}\otimes\theta_{\ell}^{\prime})(\gamma(X,\pi_{E},\mathcal{V}^{\prime},\psi_{E}))=\gamma(X,J_{\ell}(\pi_{F}),\mathcal{V}_{F}^{\prime},\psi_{F})^{l}.$

Finally, the identity (3.3) gives

$(r_{\ell}\otimes\theta_{\ell}^{\prime})(\gamma(X,\pi_{E},\mathcal{V}^{\prime},\psi_{E}))=\gamma(X^{l},J_{\ell}(\pi_{F})^{(l)},\mathcal{V}_{F}^{\prime(l)},\psi_{F}^{l}).$

Thus, the space $\mathcal{N}(\psi_{F})$ is stable under the action of
$G_{n}(F)$ and the map $\Phi_{n}$ is surjective. Now, using [DN22, Proposition
6.3], we get that there is a unique generic subquotient of
$\widehat{H}^{0}(\mathbb{W}(\pi_{E},\psi_{E,\overline{\mathcal{K}}})$, and
this is necessarily equal to $J_{\ell}(\pi_{F})^{(l)}$. This completes the
proof. ∎

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Santosh Nadimpalli,
<EMAIL_ADDRESS><EMAIL_ADDRESS>
Sabyasachi Dhar,
<EMAIL_ADDRESS><EMAIL_ADDRESS>
Department of Mathematics and Statistics, Indian Institute of Technology
Kanpur, U.P. 208016, India.