$r=\lvert\\{\tau\in\Sigma_{F}\mid\omega_{x,\mathcal{R},\tau}=1\\}\rvert$. Then

$\dim S^{\dagger}(K^{p})[\chi^{S}\otimes\delta]=2^{r}\dim S^{\rm
cl}(K^{p})[\chi^{S}\otimes\delta]\neq 0.$

###### Proof.

The assumptions imply that the character $\delta$ is locally algebraic and
that it factors as $\delta=\delta_{\lambda}\delta_{\mathcal{R}}$ for some
$\lambda\in X^{*}(\underline{T})^{+}$ and some unramified character
$\delta_{\mathcal{R}}$. By Breuil’s adjunction formula [Bre15, Théorème 4.3]
(see also [BHS19, eq. (5.5)]) and [BHS19, Lemma 5.2.3] we have

$\displaystyle S^{\dagger}(K^{p})[\chi^{S}\otimes\delta]$
$\displaystyle=\operatorname{\mathrm{Hom}}_{U(\mathfrak{g})}(M(\lambda),\Pi^{\mathrm{la}}[\chi^{S}])^{N_{0}}[\mathfrak{m}_{\delta_{\mathcal{R}}}],$
$\displaystyle S^{\rm cl}(K^{p})[\chi^{S}\otimes\delta]$
$\displaystyle=\operatorname{\mathrm{Hom}}_{U(\mathfrak{g})}(L(\lambda),\Pi^{\mathrm{la}}[\chi^{S}])^{N_{0}}[\mathfrak{m}_{\delta_{\mathcal{R}}}].$

In particular, by Lemma 7.24, these spaces are indentified with the dual
vector spaces of the fiber of $\mathcal{M}_{\infty,x,\mathcal{R}}(M(\lambda))$
resp. of $\mathcal{M}_{\infty,x,\mathcal{R}}(L(\lambda))$ at $k(x)$. Thus, as
$m_{x}=\dim\mathcal{M}_{\infty,x,\mathcal{R}}(L(\lambda))\otimes k(x)$, the
result is a direct corollary of Theorem 7.19 (and Proposition 7.6). ∎

We can also deduce the following corollary on the structure of the completed
cohomology $\Pi$ (see Definition 5.2), which is a representation of
$G:=U(\mathbb{Q}_{p})$. Let $\mathfrak{gl}_{3}$ be the Lie algebra (over $L$)
of the group $\operatorname{\mathrm{GL}}_{3}$ and for a dominant $\lambda$ we
consider the extension

$ N(\lambda)=\left[L(s_{1}\cdot\lambda)\oplus L(s_{2}\cdot\lambda)-
L(\lambda)\right]\in\operatorname{\mathrm{Ext}}_{\mathcal{O}}^{1}(L(\lambda),L(s_{1}\cdot\lambda)\oplus
L(s_{2}\cdot\lambda)),$

which is non trivial when mapped in each of
$\operatorname{\mathrm{Ext}}_{\mathcal{O}}^{1}(L(\lambda),L(s_{i}\cdot\lambda))$,
for $i=1,2$. This extension is the quotient of the Verma module $M(\lambda)$
by $M(s_{1}s_{2}\cdot\lambda)+M(s_{2}s_{1}\cdot\lambda)$.

As before we consider the Lie algebra

$\mathfrak{g}={\rm
Lie}\big{(}\underline{G}_{L}\simeq(\operatorname{\mathrm{Res}}_{F\otimes_{\mathbb{Q}}\mathbb{Q}_{p}/\mathbb{Q}_{p}}\operatorname{\mathrm{GL}}_{3})\times_{\mathbb{Q}_{p}}
L\big{)}\simeq \prod_{\tau\in\Sigma_{F}}\mathfrak{gl}_{3}$

with Borel $\mathfrak{b}\simeq\prod_{\tau}\mathfrak{b}_{\tau}$. Associated to
a dominant weight $\lambda=(\lambda_{\tau})_{\tau}\in
X^{*}(\underline{T})^{+}$ and
$w_{\mathcal{R}}=(w_{\mathcal{R},\tau})_{\tau\in\Sigma_{F}}\in W$ we define
the object

$N(\lambda,w_{\mathcal{R}})=\left(\operatorname*{\mathchoice{\raisebox{-10.00012pt}{\leavevmode\resizebox{14.81496pt}{10.00012pt}{\hbox{\raisebox{0.0pt}{$\displaystyle\boxtimes$}}}}}{\raisebox{-10.00012pt}{\leavevmode\resizebox{14.81496pt}{10.00012pt}{\hbox{\raisebox{0.0pt}{$\textstyle\boxtimes$}}}}}{\raisebox{-7.00009pt}{\leavevmode\resizebox{11.35814pt}{7.00009pt}{\hbox{\raisebox{0.0pt}{$\scriptstyle\boxtimes$}}}}}{\raisebox{-5.00006pt}{\leavevmode\resizebox{9.05354pt}{5.00006pt}{\hbox{\raisebox{0.0pt}{$\scriptscriptstyle\boxtimes$}}}}}}_{\tau:w_{\mathcal{R},\tau}\neq
1}L(\lambda_{\tau})\boxtimes\operatorname*{\mathchoice{\raisebox{-10.00012pt}{\leavevmode\resizebox{14.81496pt}{10.00012pt}{\hbox{\raisebox{0.0pt}{$\displaystyle\boxtimes$}}}}}{\raisebox{-10.00012pt}{\leavevmode\resizebox{14.81496pt}{10.00012pt}{\hbox{\raisebox{0.0pt}{$\textstyle\boxtimes$}}}}}{\raisebox{-7.00009pt}{\leavevmode\resizebox{11.35814pt}{7.00009pt}{\hbox{\raisebox{0.0pt}{$\scriptstyle\boxtimes$}}}}}{\raisebox{-5.00006pt}{\leavevmode\resizebox{9.05354pt}{5.00006pt}{\hbox{\raisebox{0.0pt}{$\scriptscriptstyle\boxtimes$}}}}}}_{\tau:w_{\mathcal{R},\tau}=1}N(\lambda_{\tau})\right)$

of the category
$\mathcal{O}_{\chi_{\lambda}}=\operatorname*{\mathchoice{\raisebox{-10.00012pt}{\leavevmode\resizebox{14.81496pt}{10.00012pt}{\hbox{\raisebox{0.0pt}{$\displaystyle\boxtimes$}}}}}{\raisebox{-10.00012pt}{\leavevmode\resizebox{14.81496pt}{10.00012pt}{\hbox{\raisebox{0.0pt}{$\textstyle\boxtimes$}}}}}{\raisebox{-7.00009pt}{\leavevmode\resizebox{11.35814pt}{7.00009pt}{\hbox{\raisebox{0.0pt}{$\scriptstyle\boxtimes$}}}}}{\raisebox{-5.00006pt}{\leavevmode\resizebox{9.05354pt}{5.00006pt}{\hbox{\raisebox{0.0pt}{$\scriptscriptstyle\boxtimes$}}}}}}_{\tau}\mathcal{O}_{\chi_{\lambda_{\tau}}}^{\mathfrak{g}l_{3},\mathfrak{b}_{\tau}}$.
We also define

$
S(\lambda,w_{\mathcal{R}})=\operatorname*{\mathchoice{\raisebox{-10.00012pt}{\leavevmode\resizebox{14.81496pt}{10.00012pt}{\hbox{\raisebox{0.0pt}{$\displaystyle\boxtimes$}}}}}{\raisebox{-10.00012pt}{\leavevmode\resizebox{14.81496pt}{10.00012pt}{\hbox{\raisebox{0.0pt}{$\textstyle\boxtimes$}}}}}{\raisebox{-7.00009pt}{\leavevmode\resizebox{11.35814pt}{7.00009pt}{\hbox{\raisebox{0.0pt}{$\scriptstyle\boxtimes$}}}}}{\raisebox{-5.00006pt}{\leavevmode\resizebox{9.05354pt}{5.00006pt}{\hbox{\raisebox{0.0pt}{$\scriptscriptstyle\boxtimes$}}}}}}_{\tau}S(\lambda_{\tau},w_{\mathcal{R},\tau})\in\mathcal{O}_{\chi_{\lambda}},$

where

$S(\lambda_{\tau},w_{\mathcal{R},\tau})=\left\\{\begin{array}[]{ccc}\bigoplus_{w\leqslant
w_{\mathcal{R},\tau}w_{0}}L(w\cdot\lambda_{\tau})&\text{if
}w_{\mathcal{R},\tau}\neq 1\\\ \bigoplus_{\ell(w)\neq
1}L(w\cdot\lambda_{\tau})\bigoplus N(\lambda_{\tau})&\text{if
}w_{\mathcal{R},\tau}=1\end{array}\right.,$

so that $S(\lambda,w_{\mathcal{R}})=\bigoplus_{w\leqslant
w_{\mathcal{R}}w_{0}}L(w\cdot\lambda)$ if $w_{\mathcal{R},\tau}\neq 1$ for all
$\tau$, and

$ N(\lambda,w_{\mathcal{R}})\subset S(\lambda,w_{\mathcal{R}}),$

otherwise.

If $M$ is a $U(\mathfrak{g})$-module, we denote
$\operatorname{\mathrm{Hom}}_{E}(M,E)$ the $U(\mathfrak{g})$-module with
underlying vector space $\operatorname{\mathrm{Hom}}_{E}(M,E)$ and action of
$\mathfrak{r}\in U(\mathfrak{g})$ given by

$
(\mathfrak{r}\cdot\phi)(m):=\phi(\dot{\mathfrak{r}}m),\quad\phi\in\operatorname{\mathrm{Hom}}_{E}(M,E),m\in
M,$

where $\mathfrak{r}\mapsto\dot{\mathfrak{r}}$ is the anti-involution of
$U(\mathfrak{g})$ extending $-1$ on $\mathfrak{g}$. We denote
$\overline{\underline{B}}$ the Borel opposite to $\underline{B}$, whose Lie
algebra is $\overline{\mathfrak{b}}$ with $\overline{\mathfrak{n}}$ its
nilpotent radical. We then denote
$B=\underline{B}(\mathbb{Q}_{p}),\overline{B}=\overline{\underline{B}}(\mathbb{Q}_{p})$
and $\delta_{B}$ the modulus character of $B$. We then denote
$M^{\prime}:=\operatorname{\mathrm{Hom}}_{E}(M,E)^{\overline{\mathfrak{n}}^{\infty}}$
the vectors which are killed by a finite power of $\overline{\mathfrak{n}}$.
If $M=\bigoplus_{\lambda\in X^{*}(\underline{T})_{L}}M_{\lambda}
\in\mathcal{O}^{\mathfrak{g},\mathfrak{b}}$, then
$M^{\prime}\in\mathcal{O}^{\mathfrak{g},\overline{\mathfrak{b}}}$. Finally
recall that if $M\in\mathcal{O}^{\mathfrak{g},\overline{\mathfrak{b}}}$ and
$\delta$ is a smooth character of $\underline{T}(\mathbb{Q}_{p})$, then Orlik-
Strauch constructed (see [OS10] or also [Bre16])

$\mathcal{F}_{\overline{B}}^{G}(M,\delta),$

which is a locally analytic representation of $G$. In particular, locally
analytic principal series are of this form : if
$M=M(\lambda)^{\vee}\in\mathcal{O}^{\mathfrak{g},\mathfrak{b}}$, then

$
\mathcal{F}_{\overline{B}}^{G}((M(\lambda)^{\vee})^{\prime},\delta)=\operatorname{ind}_{\overline{B}}^{G}(\delta_{\lambda}\delta)^{\mathrm{la}}.$
(16)

Let
$\rho:\operatorname{\mathrm{Gal}}_{E}\longrightarrow\operatorname{\mathrm{GL}}_{n}(L)$
be a crystalline, Hodge-Tate regular and $\varphi$-generic autodual
representation satisfying Hypothesis 5.9 such that
$\Pi[\mathfrak{m}_{\rho}]\neq 0$ where $\mathfrak{m}_{\rho}$ is the ideal of
$\mathbb{T}^{S}\otimes L$ associated to $\rho$. Let $\mathcal{R}$ a choice of
refinement and $\delta_{\mathcal{R}}$ the associated unramified character.
Denote
$\lambda=(\lambda_{\tau})_{\tau}\coloneqq\operatorname{HT}(\rho)-\delta_{G}\in
X^{*}(T)^{+}$ the (dominant) algebraic character associated to $\rho$ as
before, where
$\operatorname{HT}(\rho)=(h_{1,\tau}>\cdots>h_{n,\tau})_{\tau\in\Sigma_{F}}\in
X^{*}(\underline{T})$ gives the Hodge-Tate weights of $\rho$. As
$\Pi[m_{\rho}]\neq 0$ and $\rho$ satisfies Hypothesis 5.9, it corresponds to a
point $x\in\mathcal{X}_{\infty}(L)$. Denote
$w_{\rho,\mathcal{R}}=(w_{\rho,\mathcal{R},\tau})_{\tau\in\Sigma_{F}}$ and
$m_{\rho}:=m_{x}\geqslant 1$ as in Section 7.1.

######  Corollary 7.26.

For $\rho,\lambda,\mathcal{R}$ as above and all $w\leqslant
w_{\mathcal{R}}w_{0}$, we have

$\dim\operatorname{\mathrm{Hom}}_{G}(\operatorname{ind}_{\overline{B}}^{G}(\delta_{w\cdot\lambda}\delta_{\mathcal{R}}\delta_{B}^{-1})^{\rm
la},\Pi^{\mathrm{la}}[\mathfrak{m}_{\rho}])=m_{\rho}.$

###### Proof.

By [Bre15, Proposition 4.2] and [BHS19, Lemma 5.2.3], we have, for all
$M\in\mathcal{O}$

$\displaystyle
\operatorname{\mathrm{Hom}}_{U(\mathfrak{g})}(M,\Pi^{\mathrm{la}}[\mathfrak{m}_{\rho}])^{N^{0}}[\mathfrak{m}_{\delta_{\mathcal{R}}}]\simeq\operatorname{\mathrm{Hom}}_{G(\mathbb{Q}_{p})}(\mathcal{F}_{\overline{B}}^{G}(M^{\prime},\delta_{\mathcal{R}}\delta_{B}^{-1}),\Pi^{\mathrm{la}}[\mathfrak{m}_{\rho}])$
$\displaystyle\simeq\operatorname{\mathrm{Hom}}_{(\mathfrak{g},\overline{B}_{p})}(M\otimes_{L}C^{\infty}_{c}(N_{B}(L),\delta_{\mathcal{R}}),\Pi[\mathfrak{m}_{\rho}]).$

Thus, using equation (16) and Lemma 7.24 we deduce that the statement is
equivalent to

$
\dim\operatorname{\mathrm{Hom}}_{U(\mathfrak{g})}(M(w\cdot\lambda)^{\vee},\Pi^{\mathrm{la}}[\mathfrak{m}_{\rho})[\mathfrak{m}_{\delta}]
=\dim\mathcal{M}_{\infty,x,\mathcal{R}}(M(w\cdot\lambda)^{\vee})\otimes
k(x)=m_{x},$

which is Theorem 7.13. ∎

######  Corollary 7.27.

For $\rho,\lambda,\mathcal{R}$ as before, we have an injection of
$(\mathfrak{g},B(L))$-modules

$
\left(S(\lambda,w_{\rho,\mathcal{R}})\otimes_{L}C^{\infty}_{c}(N_{B}(L),\delta_{\mathcal{R}})\right)^{\oplus
m_{\rho}}\hookrightarrow\Pi^{\mathrm{la}}[\mathfrak{m}_{\rho}],$

or, equivalently, an injection of $G$-representations

$\mathcal{F}_{\overline{B}}^{G}(S(\lambda,w_{\rho,\mathcal{R}})^{\prime},\delta_{\mathcal{R}}\delta_{B}^{-1})^{\oplus
m_{\rho}}\subset\Pi[\mathfrak{m}_{\rho}].$

Moreover each map from
$\mathcal{F}_{\overline{B}}^{G}(M(w\cdot\lambda)^{\prime},\delta_{\mathcal{R}}\delta_{B}^{-1})$
to $\Pi[\mathfrak{m}_{\rho}]$ factors through the previous representation
$\mathcal{F}_{\overline{B}}^{G}(S(\lambda,w_{\rho,\mathcal{R}})^{\prime},\delta_{\mathcal{R}}\delta_{B}^{-1})$.

###### Proof.

The two statements about injections are equivalent and each of the $m_{\rho}$
asserted maps comes from a section of

$
\operatorname{\mathrm{Hom}}_{U(\mathfrak{g})}(S(\lambda,w_{\rho,\mathcal{R}}),\Pi^{\mathrm{la}}[\mathfrak{m}_{\rho}])^{N^{0}}[\mathfrak{m}_{\delta_{\mathcal{R}}}],$

by the adjunction recalled in the proof of the previous corollary.

We already know, by [BHS19], that for all $w\leqslant
w_{\rho,\mathcal{R}}w_{0}$ we have, in previously used notations

$
\dim\operatorname{\mathrm{Hom}}_{U(\mathfrak{g})}(L(w\cdot\lambda),\Pi^{\mathrm{la}}[\mathfrak{m}_{\rho}])^{N^{0}}[\mathfrak{m}_{\delta_{\mathcal{R}}}]=m_{x}=m_{\rho}.$

Moreover, for each $w_{\rho,\mathcal{R}}w_{0}\geqslant w$ with
$w_{\rho,\mathcal{R},\tau}\neq 1$ if $w_{\tau}=1$, we have

$\displaystyle m_{\rho}$
$\displaystyle=\dim\operatorname{\mathrm{Hom}}_{U(\mathfrak{g})}(M(w\cdot\lambda),\Pi^{\mathrm{la}}[\mathfrak{m}_{\rho}])^{N^{0}}[\mathfrak{m}_{\delta_{\mathcal{R}}}]$
$\displaystyle=\dim\operatorname{\mathrm{Hom}}_{U(\mathfrak{g})}(L(w\cdot\lambda),\Pi^{\mathrm{la}}[\mathfrak{m}_{\rho}])^{N^{0}}[\mathfrak{m}_{\delta_{\mathcal{R}}}],$

by Corollary 7.8. Thus, for those $w$, all maps from $M(w\cdot\lambda)$
factors through $L(w\cdot\lambda)$.

So we really need to take care of the direct factors of
$S(\lambda,w_{\rho,\mathcal{R}})$ where a factor $N(\lambda_{\tau})$ appears.
Such a factors is of the form

$\operatorname*{\mathchoice{\raisebox{-10.00012pt}{\leavevmode\resizebox{14.81496pt}{10.00012pt}{\hbox{\raisebox{0.0pt}{$\displaystyle\boxtimes$}}}}}{\raisebox{-10.00012pt}{\leavevmode\resizebox{14.81496pt}{10.00012pt}{\hbox{\raisebox{0.0pt}{$\textstyle\boxtimes$}}}}}{\raisebox{-7.00009pt}{\leavevmode\resizebox{11.35814pt}{7.00009pt}{\hbox{\raisebox{0.0pt}{$\scriptstyle\boxtimes$}}}}}{\raisebox{-5.00006pt}{\leavevmode\resizebox{9.05354pt}{5.00006pt}{\hbox{\raisebox{0.0pt}{$\scriptscriptstyle\boxtimes$}}}}}}_{\tau\in
I_{1}}L(w_{\tau}\cdot\lambda_{\tau})\boxtimes\operatorname*{\mathchoice{\raisebox{-10.00012pt}{\leavevmode\resizebox{14.81496pt}{10.00012pt}{\hbox{\raisebox{0.0pt}{$\displaystyle\boxtimes$}}}}}{\raisebox{-10.00012pt}{\leavevmode\resizebox{14.81496pt}{10.00012pt}{\hbox{\raisebox{0.0pt}{$\textstyle\boxtimes$}}}}}{\raisebox{-7.00009pt}{\leavevmode\resizebox{11.35814pt}{7.00009pt}{\hbox{\raisebox{0.0pt}{$\scriptstyle\boxtimes$}}}}}{\raisebox{-5.00006pt}{\leavevmode\resizebox{9.05354pt}{5.00006pt}{\hbox{\raisebox{0.0pt}{$\scriptscriptstyle\boxtimes$}}}}}}_{\tau\in
I_{2}}N(\lambda_{\tau}),\quad\Sigma=I_{1}\sqcup I_{2},$

and is a quotient of $M(w\cdot\lambda)$ where $w=(w_{\tau})$ with $w_{\tau}=1$
if $\tau\in I_{2}$, and even of $M_{I}(w\cdot\lambda)$ where $I=\\{
s_{1,\tau}|\tau\in I_{1}\text{ such that }
w_{\tau}=1=w_{\rho,\mathcal{R},\tau}\\}$.

We first prove that any map from $M_{I}(w\cdot\lambda)$ has to factor through
$S(\lambda,w_{\rho,\mathcal{R}})$ and more precisely through the previous
factor.

Choose $\tau_{0}\in I_{2}$ so that
$w_{\rho,\mathcal{R},\tau_{0}}=w_{\tau_{0}}=1$ and for $i=1,2$ let
$s_{i}^{\tau_{0}}\in W$ with $(s_{i}^{\tau_{0}})_{\tau}=w_{\tau}=$ if
$\tau\neq\tau_{0}$, and $w_{\tau_{0}}=s_{i}$. Then

$ M_{I}(s_{i}^{\tau_{0}}\cdot\lambda)\subset M_{I}(w\cdot\lambda).$

Moreover, $M_{I}(s_{i}^{\tau_{0}}\cdot\lambda)$ has a quotient

$Q_{i}^{\tau_{0}}:=\operatorname*{\mathchoice{\raisebox{-10.00012pt}{\leavevmode\resizebox{14.81496pt}{10.00012pt}{\hbox{\raisebox{0.0pt}{$\displaystyle\boxtimes$}}}}}{\raisebox{-10.00012pt}{\leavevmode\resizebox{14.81496pt}{10.00012pt}{\hbox{\raisebox{0.0pt}{$\textstyle\boxtimes$}}}}}{\raisebox{-7.00009pt}{\leavevmode\resizebox{11.35814pt}{7.00009pt}{\hbox{\raisebox{0.0pt}{$\scriptstyle\boxtimes$}}}}}{\raisebox{-5.00006pt}{\leavevmode\resizebox{9.05354pt}{5.00006pt}{\hbox{\raisebox{0.0pt}{$\scriptscriptstyle\boxtimes$}}}}}}_{\tau\in
I_{1}}L(\lambda_{\tau})\boxtimes
L(s_{i}\cdot\lambda_{\tau_{0}})\boxtimes\operatorname*{\mathchoice{\raisebox{-10.00012pt}{\leavevmode\resizebox{14.81496pt}{10.00012pt}{\hbox{\raisebox{0.0pt}{$\displaystyle\boxtimes$}}}}}{\raisebox{-10.00012pt}{\leavevmode\resizebox{14.81496pt}{10.00012pt}{\hbox{\raisebox{0.0pt}{$\textstyle\boxtimes$}}}}}{\raisebox{-7.00009pt}{\leavevmode\resizebox{11.35814pt}{7.00009pt}{\hbox{\raisebox{0.0pt}{$\scriptstyle\boxtimes$}}}}}{\raisebox{-5.00006pt}{\leavevmode\resizebox{9.05354pt}{5.00006pt}{\hbox{\raisebox{0.0pt}{$\scriptscriptstyle\boxtimes$}}}}}}_{\tau\in
I_{2}:\tau\neq\tau_{0}}M(w_{\tau}\cdot\lambda_{\tau}),$

and we first prove that maps from $M_{I}(s_{i}^{\tau_{0}}\cdot\lambda)$ into
$\Pi^{\mathrm{la}}[\mathfrak{m}_{\rho}]$ factors through $Q_{i}^{\tau_{0}}$.
This is equivalent to proving that
$\mathcal{M}_{\infty,x,\mathcal{R}}(M(s_{i}^{\tau_{0}}))\otimes
k(x)\longrightarrow\mathcal{M}_{\infty,x,\mathcal{R}}(M(\lambda))\otimes k(x)$
factors through $\mathcal{M}_{\infty,x,\mathcal{R}}(Q_{i}^{\tau_{0}})\otimes
k(x)$. By Corollary 7.23, this is equivalent to the same question for
$\mathcal{B}_{x}$.

######  Claim 7.28.

If $G=G_{1}\times G_{2}$, $\lambda=(\lambda_{1},\lambda_{2})$ is an algebraic
weight and
$\mathcal{O}_{\chi_{\lambda}}=\mathcal{O}_{\chi_{\lambda_{1}}}\boxtimes\mathcal{O}_{\chi_{\lambda_{2}}}$,
then

$\mathcal{B}_{G}(M_{1}\boxtimes
M_{2})=\mathcal{B}_{G_{1}}(M_{1})\boxtimes\mathcal{B}_{G_{2}}(M_{2}),$

where $\mathcal{B}_{H}$ is Bezrukavnikov’s functor of Theorem 7.4 for the
group $H$, under the obvious isomorphism of Steinberg varieties

$ X_{G}=X_{G_{1}} \times X_{G_{2}}.$

###### Proof.

This follows from the very construction of Bezrukvanikov’s functor. The
functors $\mathcal{B}_{G}$ is even defined on the larger category
$D^{b}(\operatorname{Perv}_{\underline{N}}(\underline{G}/\underline{N}))$ and
compatible with its monoidal structure. Then, by Theorem 7.4, we know that
dual Vermas are sent to the structure sheaves of the respective components by
$\mathcal{B}_{G}$, and similarly for $\mathcal{B}_{G_{i}}$. In particular we
have an isomorphism

$
\mathcal{B}_{G}(M((w_{1},w_{2})\cdot(\lambda_{1},\lambda_{2}))^{\vee})\simeq\mathcal{B}_{G_{1}}(M(w_{1}\cdot\lambda_{1})^{\vee})\boxtimes\mathcal{B}_{G_{2}}(M(w_{2}\cdot\lambda_{2})^{\vee}).$

As the functors $\mathcal{B}_{G}$ and
$\mathcal{B}_{G_{1}}\boxtimes\mathcal{B}_{G_{2}}$ are both monoïdal and
triangulated, using translation functors we deduce that $\mathcal{B}_{G}$ and
$\mathcal{B}_{G_{1}}\boxtimes\mathcal{B}_{G_{2}}$ are isomorphic on projective
objects. Thus by the same proof of Lemma 7.21 and Lemma 7.23, we deduce the
isomorphism of functors on $\mathcal{O}_{\chi_{\lambda}}$. ∎

So we want to prove that
$\mathcal{B}_{x}(M_{I}(s_{i}^{\tau_{0}}\cdot\lambda))\otimes
k(x)\longrightarrow\mathcal{B}_{x}(M_{I}(w\cdot\lambda))\otimes k(x)$ factors
through $Q_{i}^{\tau_{0}}$. By the previous claim, it suffices to show one
$\tau$ at a time using
$M(s_{i}^{\tau_{0}}\cdot\lambda)=M(s_{i}\cdot\lambda_{\tau_{0}})\boxtimes
M_{I}(w^{\tau_{0}}\cdot\lambda^{\tau_{0}})$. Thus when $\tau\neq\tau_{0}$ this
is obvious for $\tau\in I_{2}$ and reduces to freeness of $X_{3,I,w_{\tau}}$
when $\tau\in I_{1}$ as before. For $\tau=\tau_{0}$ this amount to show that,
for $k\neq\ell\in\\{1,2\\}$ the map

$ \mathcal{B}_{x}(M(s_{k}s_{\ell}\cdot\lambda_{\tau_{0}}))\otimes
k(x)\longrightarrow \mathcal{B}_{x}(M(s_{i}\cdot\lambda_{\tau_{0}}))\otimes
k(x),$

vanishes. But as $\mathcal{B}_{x}(M(s_{i}\cdot\lambda_{\tau_{0}}))$ is free of
rank 1, this is obvious. Thus, we have a factorisation through
$Q_{i}^{\tau_{0}}$ for all $\tau_{0},i=1,2$, thus

$\mathcal{M}_{\infty,x,\mathcal{R}}(M_{I}(w\cdot\lambda))\otimes
k(x)=\mathcal{M}_{\infty,x,\mathcal{R}}(M)\otimes k(x),$

where

$M=\operatorname*{\mathchoice{\raisebox{-10.00012pt}{\leavevmode\resizebox{14.81496pt}{10.00012pt}{\hbox{\raisebox{0.0pt}{$\displaystyle\boxtimes$}}}}}{\raisebox{-10.00012pt}{\leavevmode\resizebox{14.81496pt}{10.00012pt}{\hbox{\raisebox{0.0pt}{$\textstyle\boxtimes$}}}}}{\raisebox{-7.00009pt}{\leavevmode\resizebox{11.35814pt}{7.00009pt}{\hbox{\raisebox{0.0pt}{$\scriptstyle\boxtimes$}}}}}{\raisebox{-5.00006pt}{\leavevmode\resizebox{9.05354pt}{5.00006pt}{\hbox{\raisebox{0.0pt}{$\scriptscriptstyle\boxtimes$}}}}}}_{\tau:w_{\rho,\mathcal{R},\tau}\neq
1}L(\lambda_{\tau})\boxtimes\operatorname*{\mathchoice{\raisebox{-10.00012pt}{\leavevmode\resizebox{14.81496pt}{10.00012pt}{\hbox{\raisebox{0.0pt}{$\displaystyle\boxtimes$}}}}}{\raisebox{-10.00012pt}{\leavevmode\resizebox{14.81496pt}{10.00012pt}{\hbox{\raisebox{0.0pt}{$\textstyle\boxtimes$}}}}}{\raisebox{-7.00009pt}{\leavevmode\resizebox{11.35814pt}{7.00009pt}{\hbox{\raisebox{0.0pt}{$\scriptstyle\boxtimes$}}}}}{\raisebox{-5.00006pt}{\leavevmode\resizebox{9.05354pt}{5.00006pt}{\hbox{\raisebox{0.0pt}{$\scriptscriptstyle\boxtimes$}}}}}}_{\tau:w_{\rho,\mathcal{R},\tau}=1}\underbrace{M(\lambda_{\tau})/(M(s_{1}s_{2}\cdot\lambda_{\tau})+M(s_{2}s_{1}\cdot\lambda_{\tau}))}_{N(\lambda_{\tau})}.$

Now we prove the last part of the statement, i.e. any map from (the Orlik-
Strauch induction of) a Verma $M(w\cdot\lambda)$ to
$\Pi^{\mathrm{la}}[\mathfrak{m}_{\rho}]$ will factor through
$S(\lambda,w_{\mathcal{R}})$. Assume given a map in
$\operatorname{\mathrm{Hom}}_{U(\mathfrak{g})}(M(w\cdot\lambda),\Pi^{\mathrm{la}}[\mathfrak{m}_{\rho}])^{N^{0}}[\mathfrak{m}_{\delta_{\mathcal{R}}}]$,
and let $I_{1}=\\{\tau\in\Sigma|w_{\rho,\mathcal{R},\tau}=w_{\tau}=1\\}$ and
$I_{2}$ its complement, then by the previous argument the map factors through

$\operatorname*{\mathchoice{\raisebox{-10.00012pt}{\leavevmode\resizebox{14.81496pt}{10.00012pt}{\hbox{\raisebox{0.0pt}{$\displaystyle\boxtimes$}}}}}{\raisebox{-10.00012pt}{\leavevmode\resizebox{14.81496pt}{10.00012pt}{\hbox{\raisebox{0.0pt}{$\textstyle\boxtimes$}}}}}{\raisebox{-7.00009pt}{\leavevmode\resizebox{11.35814pt}{7.00009pt}{\hbox{\raisebox{0.0pt}{$\scriptstyle\boxtimes$}}}}}{\raisebox{-5.00006pt}{\leavevmode\resizebox{9.05354pt}{5.00006pt}{\hbox{\raisebox{0.0pt}{$\scriptscriptstyle\boxtimes$}}}}}}_{\tau\in
I_{1}}L(w_{\tau}\cdot\lambda_{\tau})\boxtimes\operatorname*{\mathchoice{\raisebox{-10.00012pt}{\leavevmode\resizebox{14.81496pt}{10.00012pt}{\hbox{\raisebox{0.0pt}{$\displaystyle\boxtimes$}}}}}{\raisebox{-10.00012pt}{\leavevmode\resizebox{14.81496pt}{10.00012pt}{\hbox{\raisebox{0.0pt}{$\textstyle\boxtimes$}}}}}{\raisebox{-7.00009pt}{\leavevmode\resizebox{11.35814pt}{7.00009pt}{\hbox{\raisebox{0.0pt}{$\scriptstyle\boxtimes$}}}}}{\raisebox{-5.00006pt}{\leavevmode\resizebox{9.05354pt}{5.00006pt}{\hbox{\raisebox{0.0pt}{$\scriptscriptstyle\boxtimes$}}}}}}_{\tau\in
I_{2}}N(\lambda_{\tau}).$

As any quotient of this module is a sub-representation of
$S(\lambda,w_{\rho,\mathcal{R}})$, we have that any map

$\mathcal{F}_{\overline{B}}^{G}(M(w\cdot\lambda)^{*},\delta_{\mathcal{R}}\delta_{B}^{-1})\rightarrow\Pi[\mathfrak{m}_{\rho}]$

factors through
$\mathcal{F}_{\overline{B}}^{G}(S(\lambda,w_{\rho,\mathcal{R}})^{*},\delta_{\mathcal{R}}\delta_{B}^{-1})$.

We now prove the injective part. As $S(\lambda,w_{\rho,\mathcal{R}})$ is the
direct sum of terms of the form, for $w\in W$,

$M_{w,I_{1},I_{2}}=\operatorname*{\mathchoice{\raisebox{-10.00012pt}{\leavevmode\resizebox{14.81496pt}{10.00012pt}{\hbox{\raisebox{0.0pt}{$\displaystyle\boxtimes$}}}}}{\raisebox{-10.00012pt}{\leavevmode\resizebox{14.81496pt}{10.00012pt}{\hbox{\raisebox{0.0pt}{$\textstyle\boxtimes$}}}}}{\raisebox{-7.00009pt}{\leavevmode\resizebox{11.35814pt}{7.00009pt}{\hbox{\raisebox{0.0pt}{$\scriptstyle\boxtimes$}}}}}{\raisebox{-5.00006pt}{\leavevmode\resizebox{9.05354pt}{5.00006pt}{\hbox{\raisebox{0.0pt}{$\scriptscriptstyle\boxtimes$}}}}}}_{\tau\in
I_{1}}L(w_{\tau}\cdot\lambda_{\tau})\boxtimes\operatorname*{\mathchoice{\raisebox{-10.00012pt}{\leavevmode\resizebox{14.81496pt}{10.00012pt}{\hbox{\raisebox{0.0pt}{$\displaystyle\boxtimes$}}}}}{\raisebox{-10.00012pt}{\leavevmode\resizebox{14.81496pt}{10.00012pt}{\hbox{\raisebox{0.0pt}{$\textstyle\boxtimes$}}}}}{\raisebox{-7.00009pt}{\leavevmode\resizebox{11.35814pt}{7.00009pt}{\hbox{\raisebox{0.0pt}{$\scriptstyle\boxtimes$}}}}}{\raisebox{-5.00006pt}{\leavevmode\resizebox{9.05354pt}{5.00006pt}{\hbox{\raisebox{0.0pt}{$\scriptscriptstyle\boxtimes$}}}}}}_{\tau\in
I_{2}}N(\lambda_{\tau}),\quad\Sigma=I_{1}\sqcup I_{2},$

where $I_{2}\subset\\{ \tau\in\Sigma|w_{\rho,\mathcal{R},\tau}
=w_{\tau}=1\\}$, we first prove that the direct sum of $m_{\rho}$ copies of
(the Orlik-Strauch induction of) each term $M_{w,I_{1},I_{2}}$ injects in
$\Pi^{\mathrm{la}}[m_{\rho}]$. First remark

$
\dim\operatorname{\mathrm{Hom}}_{U(\mathfrak{g})}(M_{w,I_{1},I_{2}},\Pi^{\mathrm{la}}[\mathfrak{m}_{\rho}])[\mathfrak{m}_{\delta_{\mathcal{R}}}]=2^{|I_{2}|}m_{\rho},$

by the previous factorisation of $M_{I}(w\cdot\rho)$ for some $I\subset I_{1}$
and the computation using $\mathcal{B}_{x}$ (and using Proposition 7.6). Now
each quotient of $M_{w,I_{1},I_{2}}$ is of the form

$ M_{w,I_{1}\cup J,I_{2} \backslash
J}=\operatorname*{\mathchoice{\raisebox{-10.00012pt}{\leavevmode\resizebox{14.81496pt}{10.00012pt}{\hbox{\raisebox{0.0pt}{$\displaystyle\boxtimes$}}}}}{\raisebox{-10.00012pt}{\leavevmode\resizebox{14.81496pt}{10.00012pt}{\hbox{\raisebox{0.0pt}{$\textstyle\boxtimes$}}}}}{\raisebox{-7.00009pt}{\leavevmode\resizebox{11.35814pt}{7.00009pt}{\hbox{\raisebox{0.0pt}{$\scriptstyle\boxtimes$}}}}}{\raisebox{-5.00006pt}{\leavevmode\resizebox{9.05354pt}{5.00006pt}{\hbox{\raisebox{0.0pt}{$\scriptscriptstyle\boxtimes$}}}}}}_{\tau\in
I_{1}\cup
J}L(w_{\tau}\cdot\lambda_{\tau})\boxtimes\operatorname*{\mathchoice{\raisebox{-10.00012pt}{\leavevmode\resizebox{14.81496pt}{10.00012pt}{\hbox{\raisebox{0.0pt}{$\displaystyle\boxtimes$}}}}}{\raisebox{-10.00012pt}{\leavevmode\resizebox{14.81496pt}{10.00012pt}{\hbox{\raisebox{0.0pt}{$\textstyle\boxtimes$}}}}}{\raisebox{-7.00009pt}{\leavevmode\resizebox{11.35814pt}{7.00009pt}{\hbox{\raisebox{0.0pt}{$\scriptstyle\boxtimes$}}}}}{\raisebox{-5.00006pt}{\leavevmode\resizebox{9.05354pt}{5.00006pt}{\hbox{\raisebox{0.0pt}{$\scriptscriptstyle\boxtimes$}}}}}}_{\tau\in
I_{2}\backslash J}N(\lambda_{\tau}),$

for some $J\subset I_{2}$ (remark that if $\tau\in I_{2},w_{\tau}=1$), and
thus

$ \dim\operatorname{\mathrm{Hom}}_{U(\mathfrak{g})}(M_{w,I_{1}\cup
J,I_{2}\backslash
J},\Pi^{\mathrm{la}}[\mathfrak{m}_{\rho}])[\mathfrak{m}_{\delta_{\mathcal{R}}}]=2^{|I_{2}|-|J|}m_{\rho}.$

We deduce that the dimension of homomorphisms modulo those which factors
through a strict quotient is

$ 2^{|I_{2}|}m_{\rho}-\sum_{\emptyset\neq J\subset
I_{2}}(-1)^{|J|+1}2^{|I_{2}|-|J|}m_{\rho}=m_{\rho}.$

In particular there are $m_{\rho}$ independent injective maps from
$\mathcal{F}_{\overline{B}}^{G}(M_{w,I_{1},I_{2}}^{\prime},\delta_{\mathcal{R}})$
to $\Pi^{\mathrm{la}}$. Now when $w$ and $I_{1},I_{2}$ varies, these objects
have distincts irreducible in their socle. Thus the direct sum of all those
maps

$
\bigoplus_{w,I_{1},I_{2}}\mathcal{F}_{\overline{B}}^{G}(M_{w,I_{1},I_{2}}^{\prime},\delta_{\mathcal{R}})^{\oplus
m_{\rho}}=\mathcal{F}_{\overline{B}}^{G}(S(\lambda,w_{\rho,\mathcal{R}})^{\prime},\delta_{\mathcal{R}}\delta_{B}^{-1})^{\oplus
m_{\rho}},$

injects into $\Pi[\mathfrak{m}_{\rho}]$.∎

######  Remark 7.29.

In particular, for each $\tau$ such that $w_{\mathcal{R},\tau}=1$ we deduce
the injection of the locally analytic representation

$
\mathcal{F}_{\overline{B}_{\tau}}^{G_{\tau}}(N(\lambda_{\tau})^{\prime},\delta_{\mathcal{R},\tau}\delta_{B_{\tau}}^{-1})=[{\rm
LA}_{s_{1}}\oplus{\rm LA}_{s_{2}}-{\rm LALG}],$

as representation of $\operatorname{\mathrm{GL}}_{3}(F_{\tau})$ (acting
through $\tau$), where

$ {\rm
LALG}:=L(\lambda_{\tau})\otimes_{L}\operatorname{ind}_{\overline{B}_{\tau}}^{G_{\tau}}(\delta_{\mathcal{R},\tau}\delta_{B_{\tau}}^{-1}),$

is an irreducible locally algebraic representation which appears in cosocle,
where

$ {\rm
LA}_{s}=\mathcal{F}_{\overline{B}_{\tau}}^{G_{\tau}}(L(s\cdot\lambda_{\tau})^{\prime},\delta_{\mathcal{R},\tau}\delta_{B_{\tau}}^{-1})$

is the irreducible, non-locally algebraic, socle of the locally analytic
principal series

$ {\rm
LA}_{s}\subset\operatorname{ind}_{\overline{B}_{\tau}}^{G_{\tau}}(\delta_{s\cdot\lambda_{\tau}}\delta_{\mathcal{R},\tau}\delta_{B_{\tau}}^{-1})^{\mathrm{la}}.$

In this case, the locally algebraic representation ${\rm LALG}$ appears with
multiplicity $m_{\rho}$ in the socle by the main result of [BHS19], but also
with multiplicity $m_{\rho}$ as an higher order Jordan-Hölder factor, namely,
in the cosocle of the previous
$\mathcal{F}_{\overline{B}_{\tau}}^{G_{\tau}}(N(\lambda_{\tau})^{\prime},\delta_{\mathcal{R},\tau}\delta_{B_{\tau}}^{-1})$.

##  8 Existence of very critical classical modular forms

In this section we show the existence of a classical form $f$ satisfying the
hypothesis of Theorem 1.2. The main difficulty is to find a form satisfying
the Taylor-Wiles hypothesis, which is moreover completely critical at $p$
(i.e. $w_{\rho_{f},\mathcal{R}}=1$).

For a finite extension $F$ of $\mathbb{Q}_{p}$, we denote by
$\operatorname{\mathrm{rec}}_{F}:F^{\times}\rightarrow{\rm
Gal}_{F}^{\mathrm{ab}}$ the local reciprocity map sending a uniformizer of $F$
on a geometric Frobenius. If $K$ is a number field we denote by
$\operatorname{\mathrm{Art}}_{K}$ the Artin reciprocity map
$\mathbb{A}_{K}^{\times}/K^{\times}\rightarrow{\rm Gal}_{K}^{\mathrm{ab}}$
such that, for any finite place $v$ of $K$ the precomposition of
$\operatorname{\mathrm{Art}}_{K}$ with the inclusion
$K_{v}^{\times}\hookrightarrow\mathbb{A}_{K}^{\times}$ is
$\operatorname{\mathrm{rec}}_{K_{v}}$. If $\Psi$ is a character of
$\mathbb{A}_{K}^{\times}/K^{\times}$ and $v$ is a finite place of $K$ such
that $\Psi_{v}$ is unramified, we write $\Psi(v)$ for the evaluation of
$\Psi_{v}$ at an uniformizer of $F_{v}^{\times}$. First, we remark the
following,

######  Lemma 8.1.

Let $K/\mathbb{Q}_{p}$ be a finite extension and let $\rho_{p}:{\rm
Gal}_{K}\longrightarrow\operatorname{\mathrm{GL}}_{n}(\overline{\mathbb{Q}}_{p})$
be a crystalline representation with regular Hodge–Tate weights such that
there exists a refinement $F_{\bullet}\subset D_{\rm cris}(\rho_{p})$ which
contains the Hodge filtration. We moreover assume that the eigenvalues of the
linearization of the crystalline Frobenius on $D_{\rm cris}(\rho_{p})$ are
pairwise distinct. Then $\rho_{p}$ is a split sum of characters.

###### Proof.

This is a simple application of weak admissibility. Up to extending scalars,
we can assume that $D=D_{\rm cris}(\rho_{p})=\bigoplus_{\tau}D_{\tau}$ is
split, and is a filtered $\varphi$-module. We consider the linearization
$\varphi^{f}_{\tau}$ of the Frobenius on $D_{\tau}$, where
$f=[K_{0}:\mathbb{Q}_{p}]$. We write ${\rm Fil}^{\bullet}D_{\tau}$ for the
filtration on $D_{\tau}$ induced by the Hodge-filtration on $D$. The
assumption is that the Hodge filtration on $D$ is $\varphi$-stable i.e. there
is a full flag of $K\otimes\overline{\mathbb{Q}}_{p}$-modules $F_{\bullet}$,
stable under $\varphi$, such that, for all $\tau$, if
$k_{1}^{\tau}\leqslant\dots\leqslant k^{\tau}_{n}$ are the (opposite)
$\tau$-Hodge-Tate weights (with multiplicities) then $F_{i,\tau}\subset{\rm
Fil}^{k_{n-i+1}}D_{\tau}$. Denote the eigenvalues of $\varphi^{f}$ on
$F_{i,\tau}$ by $(\varphi_{1},\dots,\varphi_{i})$. Thus by weak admissibility,

$\frac{1}{f}(v(\varphi_{1})+\dots+v(\varphi_{i}))\geqslant\sum_{\tau}\sum_{k=1}^{i}k^{\tau}_{n+1-k}.$

Now, if $G_{i}$ is a complementary $\varphi$-stable subspace of $F_{i}$ in $D$
(which exists due to the assumptions on the eigenvalues of $\varphi^{f}$),
then we see directly that the $\tau$-Hodge-Tate weights of $G_{i}$ are
$k^{\tau}_{1},\dots,k^{\tau}_{n-i}$. Thus by weak admissibility again,

$\frac{1}{f}(v(\varphi_{i+1})+\dots+v(\varphi_{n}))\geqslant\sum_{\tau}\sum_{k=1}^{n-i}k^{\tau}_{k}.$

But by weak admissibility of $D$, the endpoints of both polygons gives

$\frac{1}{f}(v(\varphi_{1})+\dots+v(\varphi_{n}))=\sum_{\tau}\sum_{i}k^{\tau}_{i}.$

Thus both $G_{i}$ and $F_{i}$ are weakly admissible, thus admissible, thus
$\rho_{p}$ splits accordingly. As this is true for all $i$, we get the Lemma.
∎

It follows that, when $n=3$, an eigenform $f$ as in Theorem 1.2 has a split
representation at $p$. In the case of modular forms, it was asked by Greenberg
(see the work of Ghate and Vatsal [Gha04], [GV04]) if a cuspform whose
representation is split at $p$ is necessarily a CM form. The natural
generalization of this question to $\operatorname{\mathrm{GL}}_{3}$ would
suggest that we cannot find a form $f$ to apply Theorem 1.2 with very large
image. Fortunately, we can construct an analog of a CM form for ${\rm GL}_{3}$
(more precisely for $U(3)$) which still has adequate image modulo $p$.

###  8.1 Choosing a Hecke character

Let $E$ be a CM field with totally real subfield $E^{+}=F$ and let
$F^{\prime}$ be a totally real field disjoint from $E$, such that
$F^{\prime}/\mathbb{Q}$ is Galois and such that $[F^{\prime}:\mathbb{Q}]=3$.
Set $K=EF^{\prime}$. This is a CM field. We moreover assume that all the
ramified primes of $K/E$ lie above split primes in $E/E^{+}$. Choose two
distinct primes $p$ and $\ell$ such that $\ell$ is totally split in
$K=EF^{\prime}$ and primes above $p$ in $E^{+}=F$ are totally split in $K$.
Moreover assume $p>8(=2(n+1)$ when $n=3$) and $\zeta_{p}\notin E$.

######  Example 8.2.

  1. 1.

The easiest choice is $F^{\prime}=\mathbb{Q}(\zeta_{7})^{+}$ and
$E=\mathbb{Q}(i\sqrt{3})$ so that $7$ is split in $E$. For this $F^{\prime}$,
we can also choose $E=\mathbb{Q}(i,\sqrt{3})$, with maximal totally real
subfield $E^{+}=\mathbb{Q}(\sqrt{3})$ so that $E/E^{+}$ is unramified
everywhere.

  2. 2.

The second easiest choice for $F^{\prime}$ is
$F^{\prime}=\mathbb{Q}(\zeta_{9})^{+}$. In this case we can choose
$E=\mathbb{Q}(i\sqrt{5})$.

  3. 3.

If $E=\mathbb{Q}(i)$, we can choose $F^{\prime}=\mathbb{Q}(\alpha)$ with
$\alpha$ a root of $X^{3}-X^{2}-4X-1$, which has discriminant $13^{2}$.

  4. 4.

If $F^{\prime}=\mathbb{Q}(\alpha)$ and $E=\mathbb{Q}(i)$, we can choose any
prime $p>8,\ell$ congruent to $1,5,21,25\pmod{52}$, like $5,53,73,...$. In
particular in that case we better should exclude $p=13$ as in the early
version [Bel10] (who knows?).

  5. 5.

If $F^{\prime}=\mathbb{Q}(\zeta_{7})^{+}$ and $E=\mathbb{Q}(i\sqrt{3})$, we
can choose any prime congruent to $1,13\pmod{21}$ like $13,43,97...$.

  6. 6.

If $F^{\prime}=\mathbb{Q}(\zeta_{7})^{+}$ and $E=\mathbb{Q}(i,\sqrt{3})$, we
can take any prime $\ell\equiv 1,13\pmod{84}$ like $13,97,169...$ and $p\equiv
1,13\pmod{21}$ like $13,43,97...$.

  7. 7.

If we really want to use $p=13$ and that $p=13$ is inert in $F=E^{+}$, and if
we want moreover $E/E^{+}$ to be unramified everywhere, we can choose
$E=\mathbb{Q}(i,\sqrt{7})$ with
$F^{\prime}=\mathbb{Q}(\beta)\subset\mathbb{Q}(\zeta_{43})$ as $43$ is split
in $\mathbb{Q}(i,\sqrt{7})/\mathbb{Q}(\sqrt{7})$, with $\beta$ a root of
$X^{3}-X^{2}-14X-8$.

In the following we say that a weight
$\underline{k}\in\mathbb{Z}^{\operatorname{\mathrm{Hom}}(K,\mathbb{C})}$ is
_very regular_ if, for $\tau_{1}\neq\tau_{2}$ in
$\operatorname{\mathrm{Hom}}(K,\mathbb{C})$, we have $\lvert
k_{\tau_{1}}-k_{\tau_{2}}\rvert\geqslant 2$.

Let $\Psi$ be an algebraic Hecke character of $\mathbb{A}_{K}^{\times}$ with
algebraic very regular weight $\underline{k}=(k_{v})_{v|\infty}$, such that
$\Psi^{c}=\Psi^{\vee}$ and such that $\Psi$ is unramified both at $p$ and
$\ell$. Choose an isomorphism
$\iota:\mathbb{C}\simeq\overline{\mathbb{Q}_{p}}$. We moreover assume that

$(\Psi,p)$

if $\mathfrak{p}|p$ in $E$, we have
$\Psi(v)\Psi(v^{\prime})^{-1}\notin\\{1,p\\}$ for $v\neq v^{\prime}$ places of
$K$ dividing $\mathfrak{p}$.

$(\Psi,\ell)$

There exists $\lambda|\ell$ in $E$, and $\lambda^{\prime}|\lambda$ in
$E(\zeta_{p})$, such that for all $v_{1}\neq v_{2}$ places of $K$ dividing
$\lambda$, if $v_{1}^{\prime},v_{2}^{\prime}$ are the corresponding places
above $\lambda^{\prime}$ in $K(\zeta_{p})$,
$\iota(\Psi(v_{1}^{\prime})\pmod{\mathfrak{m}_{\overline{\mathbb{Q}}_{p}}}\neq\iota(\Psi(v_{2}^{\prime}))\pmod{\mathfrak{m}_{\overline{\mathbb{Q}}_{p}}}$.

Consider moreover the following hypothesis on $\Psi$ :

$(\Psi,Ram)$

If $v$ is a place of $K$ such that $\Psi$ is ramified at $v$, then $v$ divides
a prime which is totally split in $K/\mathbb{Q}$.

Let
$\Psi_{p}:\mathbb{A}_{K}^{\times}\longrightarrow\overline{\mathbb{Q}}_{p}^{\times}$
be the $p$-adic realization of $\Psi$ and $\iota$, and $\psi_{p}:{\rm
Gal}_{K}\rightarrow\overline{\mathbb{Q}}_{p}^{\times}$ such that
$\psi_{p}=\Psi_{p}\circ\operatorname{\mathrm{Art}}_{K}$. It is a Galois
representation satisfying $\psi_{p}^{\vee}=\psi_{p}^{c}$.

###  8.2 Galois induction

######  Definition 8.3.

We denote by $\rho$ the induced Galois representation

$\rho=\operatorname{ind}_{{\rm Gal}_{K}}^{{\rm Gal}_{E}}\psi_{p}=\\{f:{\rm
Gal}_{E}\longrightarrow\overline{\mathbb{Z}_{p}}^{\times}|
f(gk)=\psi_{p}^{-1}(k)f(g)\forall g\in{\rm Gal}_{E},k\in{\rm Gal}_{K}\\},$

where the action of $g\in{\rm Gal}_{E}$ is given by $(g\cdot
f)(x)=f(g^{-1}x)$.

Then $\rho$ is a three dimensional Galois representation since $[K:E]$ is
Galois of degree 3. We claim the following

######  Lemma 8.4.

  1. 1.

The representation $\overline{\rho}:=\rho\otimes\overline{\mathbb{F}_{p}}$ is
absolutely irreducible, in particular $\rho$ is absolutely irreducible.

  2. 2.

The representation
$\overline{\rho}(\operatorname{\mathrm{Gal}}_{E(\zeta_{p})})$ is adequate.

  3. 3.

The representation $\rho$ is polarized, i.e. $\rho^{c}\simeq\rho^{\vee}$.

  4. 4.

The representation $\rho_{\operatorname{\mathrm{Gal}}_{E_{v}}}$ is split,
$\varphi$-generic, Hodge–Tate regular for any $v|p$ in $E$,

  5. 5.

If $v$ is a place of $E$ such that $\rho$ is ramified at $v$, then
$\operatorname{\mathrm{Hom}}_{\operatorname{\mathrm{Gal}}_{E_{v}}}(\rho_{v},\rho_{v}(1))=0$.

###### Proof.

We will actually prove that $\overline{\rho}({\rm Gal}_{E(\zeta_{p})})$ acts
absolutely irreducibly, which will imply point 1 and point 2 will follow by
[Tho12] Lemma 2.4. To prove point 1, remark that if we denote by
$\sigma\in{\rm Gal}_{E}$ a lift of a generator of the Galois group
$\operatorname{\mathrm{Gal}}(K/E)=<\overline{\sigma}>=\mathbb{Z}/3\mathbb{Z}$,
then $\rho$ has a basis given by $f,\sigma\cdot f,\sigma^{2}\cdot f$, where
$f$ is the function

$ f:{\rm Gal}_{E}={\rm Gal}_{K}\coprod\sigma{\rm Gal}_{K}\coprod\sigma^{2}{\rm
Gal}_{K}\longrightarrow\overline{\mathbb{Z}}_{p}^{\times},k\in{\rm
Gal}_{K}\mapsto\psi_{p}^{-1}(k),\sigma k,\sigma^{2}k\mapsto 0.$

Then $\sigma^{3}\cdot f=\psi_{p}(\sigma^{3})f.$ Thus, after restricting to
${\rm Gal}_{K}$, there is an isomorphism $\rho_{|{\rm
Gal}_{K}}\simeq\psi_{p}\oplus\psi_{p}^{\sigma}\oplus\psi_{p}^{\sigma^{2}}$,
where $\psi_{p}^{\sigma}=\psi_{p}(\sigma^{-1}\cdot\sigma)$. We reduce mod $p$,
where we have a similar reduction after restricting to ${\rm Gal}_{K}$.
Because of the hypothesis $(\Psi,\ell)$ away from $p$, we have that
$\overline{\rho}_{{\rm Gal}_{E(\zeta_{p})_{\lambda^{\prime}}}}$, for
$\lambda^{\prime}|\ell$, is the sum of three distinct characters. Moreover the
group ${\rm Gal}_{E}$ acts transitively on these three eigenspaces. Therefore
this representation is absolutely irreducible. To prove point 3, we compute
$\rho^{\vee}$. By [CR81, Prop. 10.28], we have an isomorphism

$\rho^{\vee}\simeq\operatorname{\mathrm{Ind}}_{{\rm Gal}_{K}}^{{\rm
Gal}_{E}}\psi_{p}^{-1}=\operatorname{\mathrm{Ind}}_{{\rm Gal}_{K}}^{{\rm
Gal}_{E}}\psi_{p}^{c}\simeq\rho^{c}.$

Let us prove 4. As $p$ is totally split in $K/F$, we have for $v|p$ in $E$,
${\rm Gal}_{E_{v}}\subset{\rm Gal}_{K}$ so that $\rho_{|{\rm
Gal}_{E_{v}}}\simeq\psi_{p,v}\oplus\psi_{p,v}^{\sigma}\oplus\psi_{p,v}^{\sigma^{2}}$.
As the group ${\rm Gal}_{E}$ acts transitively on the three places of $K$ over
$v$, we have $\rho_{|{\rm
Gal}_{E_{v}}}\simeq\bigoplus_{v^{\prime}|v}\psi_{p,v^{\prime}}$. Therefore
$\rho_{|{\rm Gal}_{E_{v}}}$ is crystalline and the eigenvalues of the
Frobenius endomorphism of $D_{\mathrm{cris}}(\rho_{|{\rm Gal}_{E_{v}}})$ are
the $\Psi(v^{\prime})$ for $v^{\prime}|v$ in $K$. It follows from hypothesis
$(\Psi,p)$ that $\rho_{|{\rm Gal}_{E_{v}}}$ is $\varphi$-generic. Moreover the
Hodge–Tate weights of $\rho_{|{\rm Gal}_{E_{v}}}$ corresponds to the algebraic
(infinitesimal) weight of $\Psi$, which was assumed regular so that
$\rho_{{\rm Gal}_{E_{v}}}$ is Hodge–Tate regular.

Finally we prove 5. Let $v$ be a place of $E$ such that $\rho_{v}$ is
ramified. Then either $v$ is ramified in $K/E$ or $\Psi_{v}$ is ramified.
Assume in a first time that $\Psi_{v}$ is ramified. Then $(\Psi,Ram)$ implies
that $v$ divides a prime of $\mathbb{Q}$ which is totally split in $K$. In
particular, $v$ is split in $K/E$. As above, we have
$\rho_{v}\simeq\bigoplus_{v^{\prime}|v}\psi_{p,v^{\prime}}$ with
$\psi_{p,v^{\prime}}=\Psi_{v^{\prime}}\circ\operatorname{\mathrm{rec}}_{K_{v^{\prime}}}^{-1}$
as $v^{\prime}\nmid p$. Therefore it follows from Lemma 8.5 below that
$\operatorname{\mathrm{Hom}}_{{\rm Gal}_{E_{v}}}(\rho_{v},\rho_{v}(1))=0$.

Now assume that $v$ is non split in $K$. As $K/E$ is Galois there is a unique
place $w$ of $K$ over $v$ and $\rho_{v}\simeq\operatorname{\mathrm{Ind}}_{{\rm
Gal}_{K_{w}}}^{{\rm Gal}_{E_{v}}}\psi_{p,w}$. By Frobenius reciprocity, we
have

$\operatorname{\mathrm{Hom}}_{{\rm
Gal}_{E_{v}}}(\rho_{v},\rho_{v}(1))\simeq\operatorname{\mathrm{Hom}}_{{\rm
Gal}_{K_{w}}}(\psi_{p,w}\oplus\psi_{p,w}^{\sigma}\oplus\psi_{p,w}^{\sigma^{2}},\psi_{p,w}\chi_{\mathrm{cyc}|K_{w}}).$

Assume that $\psi_{p,w}=\psi_{p,w}^{\sigma}\chi_{\mathrm{cyc}|K_{w}}$. As
$\chi_{\mathrm{cyc}|K_{w}}=\chi_{\mathrm{cyc}|K_{w}}^{\sigma}$, we deduce
$\psi_{p,w}^{\sigma}=\psi_{p,w}^{\sigma^{2}}\chi_{\mathrm{cyc}|K_{w}}$ and
$\psi_{p,w}^{\sigma^{2}}=\psi_{p,w}^{\sigma^{3}}\chi_{\mathrm{cyc}|K_{w}}=\psi_{p,w}\chi_{\mathrm{cyc}|K_{w}}$
so that $\psi_{p,w}=\psi_{p,w}\chi_{\mathrm{cyc}|K_{w}}^{3}$ which is false.
We prove similarly than
$\psi_{p,w}\neq\psi_{p,w}^{\sigma^{2}}\chi_{\mathrm{cyc}|K_{w}}$ and deduce
$\operatorname{\mathrm{Hom}}_{{\rm Gal}_{E_{v}}}(\rho_{v},\rho_{v}(1))=0$. If
$\Psi_{w}\neq\Psi_{w}\circ\sigma$, then the characters
$\Psi_{w},\Psi_{w}\circ\sigma,\Psi\circ\sigma^{2}$ are pairwise distinct and
$\rho_{v}$ is irreducible so that $\operatorname{\mathrm{Hom}}_{{\rm
Gal}_{E_{v}}}(\rho_{v},\rho_{v}(1))=0$. If $\Psi_{w}=\Psi_{w}\circ\sigma$,
then $\rho_{v}$ is not irreducible, but clearly
$\operatorname{\mathrm{Hom}}_{{\rm Gal}_{E_{v}}}(\rho_{v},\rho_{v}(1))=0$ (as
$\psi_{p|{\rm Gal}_{E_{v}}}\neq\psi_{p|{\rm Gal}_{E_{v}}}(1)$). ∎

######  Lemma 8.5.

Let $\Psi:\mathbb{A}_{K}^{\times}/K^{\times}$ be an algebraic Hecke character
of very regular weight $\underline{k}$. Then, if $\ell$ is a prime number
which is totally split in $K$, then
$\Psi_{v}\neq\Psi_{w}\lvert\cdot\rvert_{w}$ for all places $v,w$ of $K$
dividing $\ell$.

###### Proof.

Let $\Psi$ and $\ell$ be as in the statement. Fix
$\iota:\mathbb{C}\simeq\overline{\mathbb{Q}}_{\ell}$ and let
$\lvert\cdot\rvert_{\ell}$ be the unique absolute value on
$\overline{\mathbb{Q}}_{\ell}$ extending the one on $\mathbb{Q}_{\ell}$. Let
$\Psi_{\iota}$ be the continuous character
$\mathbb{A}_{K}^{\times}/K^{\times}K_{\infty}^{\times}\rightarrow\overline{\mathbb{Q}}_{\ell}^{\times}$
defined by

$\Psi_{\iota,w}(x_{w})=\begin{cases}\Psi_{w}(x_{w})&\text{if}\
w\not|\ell,w\not|\infty\\\ 1&\text{if}\ w|\infty\\\
\iota(\Psi_{w}(w_{w}))\prod_{{\tau\in\operatorname{\mathrm{Hom}}(K_{w},\overline{\mathbb{Q}}_{\ell}),\tau|w}}\tau(x_{w})^{k_{\iota^{-1}\tau}}&\text{if}\
w|\ell,\end{cases}$

where $\tau|w$ means that $|.|_{\ell}\circ\tau$ extends the absolute value
given by $w$ on $K$, and
$(k_{\sigma})_{\sigma\in\operatorname{\mathrm{Hom}}(K,\mathbb{C})}$ is the
weight of $\Psi$. As the group
$\mathbb{A}_{K}^{\times}/K^{\times}K_{\infty}^{\times}$ is compact, we have
$\operatorname{\mathrm{Im}}(\Psi_{\iota})\subset\overline{\mathbb{Z}}_{\ell}^{\times}$.
As $\ell$ is totally split $\iota$ induces a bijection between $\\{v|\ell\\}$
and $\operatorname{\mathrm{Hom}}(K,\mathbb{C})$. Let $v$ be a place of $K$
dividing $\ell$ corresponding to $\tau$ (i.e. $|.|_{\ell}\circ\iota^{-1}\tau$
extends $|.|_{v}$) and denote $k_{v}\coloneqq k_{\iota^{-1}\tau}$. We have

$\lvert\iota{\Psi_{v}(\ell)}\tau(\ell)^{k_{v}}\rvert_{\ell}=1$

so that $\lvert\iota(\Psi_{v}(\ell))\rvert=l^{k_{v}}$. As $\ell$ is a
uniformizer of $K_{v}$, for any $v|\ell$, the result follows. ∎

###  8.3 Construction of an explicit set of Hecke characters

In this subsection we explain one way to find a $\Psi$ as before, satisfying
hypothesis $(\Psi,p),(\Psi,\ell),(\Psi,Ram)$. Fix $E$ a CM extension, with
$E^{+}=F$ its maximal totally real subfield, so that $[E:E^{+}]=2$. Fix also
$F^{\prime}$ disjoint from $E$, a totally real degree 3 Galois extension of
$\mathbb{Q}$. Choose $p,\ell$ two primes with are totally split in
$K:=EF^{\prime}$ such that $p>8$. The following Lemma is a more precise
version of [CHT08, Lem. 4.1.1].

######  Lemma 8.6.

Let $F$ be a number field. Let $S$ be a finite set of places of $F$. Let
$\chi_{S}$ be an unramified continuous character
$F_{S}^{\times}\coloneqq\prod_{v\in
S}F_{v}^{\times}\rightarrow\mathbb{C}^{\times}$ of finite order. Let $T$ be a
set of finite places of $F$, disjoint from $S$ and of Dirichlet density $1$.
Then there exists a continuous character
$\chi:\mathbb{A}_{F}^{\times}/F^{\times}\rightarrow\mathbb{C}^{\times}$ of
finite order such that $\chi_{|F_{S}^{\times}}=\chi_{S}$ and the ramification
places of $\chi$ are in $T$.

###### Proof.

Let $U^{S}$ be the product of the $\mathcal{O}_{F_{v}}^{\times}$ for $v\notin
S$. Then $F^{\times}\cap U^{S}$ is a finitely generated subgroup of
$F^{\times}$. Let us write $m$ for the order of the finite cyclic group
$\chi_{S}(F^{\times}\cap U^{S})$. It follows from the proof of Theorem 1 in
[Che51] that we can find finitely many places $w_{1},\dots,w_{r}$ in $T$ such
that the subgroup of $F^{\times}\cap U^{S}$ congruent to $1$ modulo
$\mathfrak{p}_{w_{1}},\dots,\mathfrak{p}_{w_{r}}$ is contained in
$(F^{\times}\cap U^{S})^{m}$. We conclude as in the proof of [CHT08, Lem.
4.1.1] choosing for $U$ the product of the $U_{v}$ for $v$ not in $S$ nor
$\\{w_{1},\dots,w_{r}\\}$ and a small enough subgroup at $w_{1},\dots,w_{r}$.
∎

######  Lemma 8.7.

Let $K$ be an (imaginary) CM field with totally real subfield $K^{+}$ and
complex conjugacy $c$. Denote
$\psi:\mathbb{A}_{K}^{\times}/K^{\times}\rightarrow\mathbb{C}^{\times}$ be a
continuous character. Assume that there exists a finite set $S$ of places of
$K$ which are split in $K/K^{+}$ and such that $\psi_{v}^{-1}=\psi_{cv}$ for
$v\in S$. Moreover, assume that $S$ contains the Archimedean places. Let $T$
be a finite set of places of $K$ that contains $S$ and is stable under $c$,
such that $\psi$ is unramified outside of $T$. Then there exists a Hecke
character
$\widetilde{\psi}:\mathbb{A}_{K}^{\times}/K^{\times}\rightarrow\mathbb{C}^{\times}$
such that $\widetilde{\psi}^{-1}=\widetilde{\psi}^{c}$ and
$\widetilde{\psi}_{v}=\psi_{v}$ for $v\in S$ and such that
$\widetilde{\psi}_{v}$ is unramified outside of $T$.

###### Proof.

Let $\theta=\psi\circ N_{K/K^{+}}$. As $S$ contains the Archimedean places,
the character $\theta$ is trivial at Archimedean places and is therefore a
character of finite order. Let $U_{T}\subset\prod_{v\in T\smallsetminus
S}K_{v}^{\times}$ be a compact open subgroup such that $\theta_{|U_{T}}$ is
trivial and such that $c(U_{T})=U_{T}$. Let

$U=\big{(}\prod_{v\notin T}\mathcal{O}_{K_{v}}^{\times}\big{)}\cdot
U_{T}\cdot\big{(}\prod_{v\in S}K_{v}^{\times}\big{)}.$

We have an injection of compact groups

$N_{K/K^{+}}(\mathbb{A}_{K}^{\times})/(N_{K/K^{+}}(\mathbb{A}_{K}^{\times})\cap
K^{\times}U)\hookrightarrow\mathbb{A}_{K}^{\times}/K^{\times}U.$

Under our hypothesis, the character
$\psi_{|N_{K/K^{+}}(\mathbb{A}_{K}^{\times})}$ is trivial on
$(N_{K/K^{+}}(\mathbb{A}_{K}^{\times})\cap K^{\times}U)$. Therefore it extends
to a character $\alpha$ of finite order of $\mathbb{A}_{K}^{\times}$ trivial
on $K^{\times}U$. We thus have $\psi\circ N_{K/K^{+}}=\alpha\circ
N_{K/K^{+}}$. It is easy to check that the character
$\widetilde{\psi}=\psi\alpha^{-1}$ satisfies our requirements. ∎

######  Proposition 8.8.

For each choice of fields $E$ and $F^{\prime}$ and places $p$ and $\ell$ and
very regular weight $\underline{k}$ as above there exists a Hecke character
$\Psi:\mathbb{A}_{K}^{\times}/K^{\times}\rightarrow\mathbb{C}^{\times}$
satisfying $(\Psi,p)$, $(\Psi,\ell)$ and $(\Psi,Ram)$ and such that
$\Psi^{-1}=\Psi^{c}$.

###### Proof.

Let $\underline{k}$ be a very regular weight. It follows from [Sch88], Section
0.3, that there exists a Hecke character $\Psi_{0}$ of
$\mathbb{A}_{K}^{\times}/K^{\times}$ with weight $\underline{k}$. Using Lemma
8.6, we can construct a Hecke character $\theta$ of finite order such that,
setting $\Psi_{1}=\Psi_{0}\theta$, we have

  * •

the character $\Psi_{1}$ satisfies $(\Psi_{1},p)$ and $(\Psi_{1},\ell)$ ;

  * •

there exists finitely many primes $\ell_{1},\dots,\ell_{r}$, different from
$p$ and $\ell$, which are totally split in $K$ and such that $\Psi_{1}$ is
only ramified at places dividing $\ell_{1},\dots,\ell_{r}$ ;

  * •

we have $\Psi_{1,w}^{-1}=\Psi_{1,cw}$ for any place $w$ of $K$ dividing $\ell$
or $p$.

Now it follows from Lemma 8.7 that there exists a Hecke character $\Psi$ of
$\mathbb{A}_{K}^{\times}/K^{\times}$ such that

  * •

$\Psi^{-1}=\Psi^{c}$ ;

  * •

$\Psi_{v}=\Psi_{1,v}$ if $v$ is a place of $K$ dividing $p$ or $\ell$ ;

  * •

$\Psi$ is ramified only at places dividing $\ell_{1},\dots,\ell_{r}$.∎

###  8.4 Automorphic Induction and base change

Let $\Psi$ and $\rho$ as in subsection 8.1 and let $U$ denote the unitary
group in three variables for $E/E^{+}$ that is compact at infinity and quasi-
split at all finite places. We need to find an automorphic form for $U$ whose
associated Galois representation is induced representation $\rho$ from 8.3.

######  Proposition 8.9.

There exists an automorphic representation $\Pi$ of
$\operatorname{\mathrm{GL}}_{3,E}$, cuspidal, cohomological at infinity,
unramified at $\ell$ and $p$, polarized, whose associated Galois
representation is given by $\rho$.

###### Proof.

This is the content of [Hen12] Théorème 3 (as $K/E$ cyclic of degree 3) for
the existence of the automorphic representation, Théorème 5 for the
compatibility with the local correspondence at $\ell$ and $p$ and at infinity
(cf. the following remark of [Hen12]). Polarization can be checked after base
change of the automorphic induction to $K$, where it follows as
$\Psi^{c}=\Psi^{\vee}$, and as $\Psi\neq\Psi^{\sigma}$ for
$\sigma\in\operatorname{\mathrm{Gal}}(K/E)$ such that $\sigma\neq 1$.
Moreover, the automorphic induction is also cuspidal (Theorem 2 of [Hen12]). ∎

######  Conjecture 8.10.

There exists a cohomological, cuspidal, automorphic representation $\pi$ of
$U$ whose base change to $\operatorname{\mathrm{GL}}_{3,E}$ is $\Pi$.

######  Proposition 8.11.

If $E/E^{+}$ is everywhere unramified (e.g. for $E=\mathbb{Q}(i,\sqrt{3})$ or
$\mathbb{Q}(i,\sqrt{7})$), then the previous conjecture is true.

###### Proof.

This is [Lab11] Theorem 5.4. ∎

######  Proposition 8.12.

If $E$ is quadratic imaginary, then the previous conjecture is true.

###### Proof.

By [Mor10] Corollary 8.5.3 (ii), there exists $\pi^{\prime}$ an automorphic
representation for $GU(3)$ associated to $\Pi\times 1$, which is automorphic
for $\operatorname{\mathrm{GL}}_{3}\times\operatorname{\mathrm{GL}}_{1}$. By
[HS22] Lemma A.7 (based on [HT01]), there exists $\pi$, an automorphic
representation of $U(3)$ associated to $\pi^{\prime}$. ∎

######  Corollary 8.13.

If $E$ is quadratic imaginary or if $E/E^{+}$ is everywhere unramified, then
there exists a classical form on $U(3)$ satisfying the hypothesis of Theorem
1.2.

###### Proof.

Let $\pi$ be the automorphic representation of $U$ considered above, and let
$f\in\pi$ be an eigenform for the Hecke operators away from a set $S$ of bad
places of $\pi$. Then $\rho_{f}=\rho_{\pi}=\rho$ is crystalline at $p$ and
$\varphi$-generic. In particular it has $3!=6$ refinements which are
automorphic and split at $p$. Hence there exists an automorphic refinement
$\mathcal{R}$ of $f$ with relative position $w_{\mathcal{R}}=1$ with respect
to the Hodge filtration. In particular, for this choice of a refinement, there
exists a refined classical modular form $f^{\prime}$ satisfying all hypothesis
of Theorem 1.2. But, by Lemma 8.4(5) we know that $f$ gives, for all $v\in
S\backslash S_{p}$, a point of $\mathcal{X}_{\overline{\rho_{v}}}^{\square}$
which satisfies $\operatorname{\mathrm{Hom}}_{{\rm
Gal}_{E_{v}}}(\rho_{v},\rho_{v}(1))=0$. When $v$ splits in $E/E^{+}$, such a
$v$ is a smooth point by [All16] Prop 1.2.2. ∎

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## Index