N^{0}$. We finally set $\Delta_{M}$ to be the monoid generated by
$\Delta_{M}^{+}$ and $z_{p}^{-1}$.

Given
$\pi\in\textnormal{Mod}_{\textnormal{sm}}(\mathcal{O}/\varpi^{m}[\Delta_{Q}])$,
we can consider the Hecke action of $\Delta_{M}^{+}$ on the space of
$N^{0}$-invariants $\Gamma(N^{0},\pi)$. Namely, for $m\in\Delta_{M}^{+}$ and
$v\in\Gamma(N^{0},\pi)$, the action is given by

$m\cdot v:=\sum_{n\in N^{0}/mN^{0}m^{-1}}nmv.$

We then obtain a left exact functor

$\Gamma(N^{0},-):\textnormal{Mod}_{\textnormal{sm}}(\mathcal{O}/\varpi^{m}[\Delta_{Q}])\to\textnormal{Mod}_{\textnormal{sm}}(\mathcal{O}/\varpi^{m}[\Delta_{M}^{+}])$

and, in particular, a derived functor202020Here we use [All+23], Lemma 5.2.4
to see that both categories considered are abelian with enough injectives.

$R\Gamma(N^{0},-):D^{+}_{\textnormal{sm}}(\mathcal{O}/\varpi^{m}[\Delta_{Q}])\to
D^{+}_{\textnormal{sm}}(\mathcal{O}/\varpi^{m}[\Delta_{M}^{+}]).$

We also have the exact localisation functor

$(-)^{Q\textnormal{-ord}}:\textnormal{Mod}_{\textnormal{sm}}(\mathcal{O}/\varpi^{m}[\Delta_{M}^{+}])\to\textnormal{Mod}_{\textnormal{sm}}(\mathcal{O}/\varpi^{m}[\Delta_{M}])$

induced by the inclusion $\Delta_{M}^{+}\subset\Delta_{M}$. We then obtain the
functor of "taking $Q$-ordinary parts at infinite level"

$D^{+}_{\textnormal{sm}}(\mathcal{O}/\varpi^{m}[\Delta_{Q}])\to
D^{+}_{\textnormal{sm}}(\mathcal{O}/\varpi^{m}[\Delta_{M}]),$ $\pi\mapsto
R\Gamma(N^{0},\pi)^{Q\textnormal{-ord}}.$

We now consider several versions of taking $Q$-ordinary parts at finite level
and will compare them. To work in the generality we need, we will take
invariants with respect to general "types" and not only the trivial
representation. In particular, set $\sigma$ to be a smooth
$\mathcal{O}/\varpi^{m}[M^{0}]$-modules, finite free over
$\mathcal{O}/\varpi^{m}$. We will often abuse the notation and confuse
$\sigma$ with $\textnormal{Inf}_{M^{0}}^{M^{0}\ltimes N^{0}}\sigma$. We define
the Hecke algebras
$\mathcal{H}(\sigma)^{\Delta_{M}^{+}}\subset\mathcal{H}(\sigma)^{+}\subset\mathcal{H}(\sigma)$
as the subalgebra generated by functions supported on $\Delta_{M}^{+}$,
respectively on $M^{+}$. Finally, note that $[z_{p},\textnormal{id}]$ lies
$\mathcal{H}(\sigma)^{\Delta_{M}^{+}}$ and is a central element. Combined with
Lemma 3.4, it easily implies that
$\mathcal{H}(\sigma)^{+}[[z_{p},\textnormal{id}]^{-1}]=\mathcal{H}(\sigma)$.
We set
$\mathcal{H}(\sigma)^{\Delta_{M}}=\mathcal{H}(\sigma)^{\Delta_{M}^{+}}[[z_{p},\textnormal{id}]^{-1}]$.

The first candidate for ordinary parts at level $\sigma$ is as follows. For
$\pi\in D^{+}_{\textnormal{sm}}(\mathcal{O}/\varpi^{m}[\Delta_{Q}])$ we apply
the functor

$R{\rm{Hom}}_{\mathcal{O}/\varpi^{m}[M^{0}]}(\sigma^{\vee},-):D^{+}_{\textnormal{sm}}(\mathcal{O}/\varpi^{m}[\Delta_{M}])\to
D^{+}_{\textnormal{sm}}(\mathcal{H}(\sigma)^{\Delta_{M}})$

to $R\Gamma(N^{0},\pi)^{Q\textnormal{-ord}}$. Here the functor
$R{\rm{Hom}}_{\mathcal{O}/\varpi^{m}[M^{0}]}(\sigma^{\vee},-)$ is constructed
by taking the usual left Hecke action of $\mathcal{H}(\sigma)^{\Delta_{M}}$ on
the space of $\sigma^{\vee}$-invariants.

In order to define the other candidate, we introduce the functor

${\rm{Hom}}_{\mathcal{O}/\varpi^{m}[M^{0}\ltimes
N^{0}]}(\sigma^{\vee},-):\textnormal{Mod}_{\textnormal{sm}}(\mathcal{O}/\varpi^{m}[\Delta_{Q}])\to\textnormal{Mod}(\mathcal{H}(\sigma)^{\Delta_{M}^{+}})$

where the target is the category of left
$\mathcal{H}(\sigma)^{\Delta_{M}^{+}}$-modules. We spell out the definition of
${\rm{Hom}}_{\mathcal{O}/\varpi^{m}[M^{0}\ltimes N^{0}]}(\sigma^{\vee},-)$.
Let $\pi\in\textnormal{Mod}_{\textnormal{sm}}(\Delta_{Q})$, pick
$[m,\psi]\in\mathcal{H}(\sigma)^{\Delta_{M}^{+}}$ and
$\phi\in{\rm{Hom}}_{\mathcal{O}/\varpi^{m}[M^{0}\ltimes
N^{0}]}(\sigma^{\vee},\pi)$. The action is defined by setting

$[m,\psi]\cdot\phi:v\mapsto\sum_{n\tilde{m}\in N^{0}\rtimes
M^{0}/m(N^{0}\rtimes M^{0})m^{-1}\cap N^{0}\rtimes
M^{0}}\pi(n\tilde{m}m)\phi(\psi^{t}\circ\sigma^{\vee}((n\tilde{m})^{-1})v).$

Note that we have identifications of sets

(3.1) $N^{0}\rtimes M^{0}/\left(m(N^{0}\rtimes M^{0})m^{-1}\cap N^{0}\rtimes
M^{0}\right)\cong$ $\left\\{(n,\tilde{m})\mid n\in
N^{0}/\tilde{m}mN^{0}(\tilde{m}m)^{-1},\tilde{m}\in M^{0}/mM^{0}m^{-1}\cap
M^{0}\right\\}\cong$ $(M^{0}/mM^{0}m^{-1}\cap M^{0})\times
N^{0}/mN^{0}m^{-1}.$

In particular, we obtain the following lemma.

######  Lemma 3.5.

We have a natural equivalence of functors

${\rm{Hom}}_{\mathcal{O}/\varpi^{m}[M^{0}]}(\sigma^{\vee},-)\circ\Gamma(N^{0},-)\cong{\rm{Hom}}_{\mathcal{O}/\varpi^{m}[M^{0}\ltimes
N^{0}]}(\sigma^{\vee},-):\textnormal{Mod}_{\textnormal{sm}}(\mathcal{O}/\varpi^{m}[\Delta_{Q}])\to$
$\to\textnormal{Mod}(\mathcal{H}(\sigma)^{\Delta_{M}^{+}}).$

In particular, for $\pi\in
D^{+}_{\textnormal{sm}}(\mathcal{O}/\varpi^{m}[\Delta_{Q}])$, we have a
natural isomorphism

$R{\rm{Hom}}_{\mathcal{O}/\varpi^{m}[M^{0}]}(\sigma^{\vee},R\Gamma(N^{0},\pi))\cong
R{\rm{Hom}}_{\mathcal{O}/\varpi^{m}[M^{0}\ltimes N^{0}]}(\sigma^{\vee},\pi)$

in $D^{+}(\mathcal{H}(\sigma)^{\Delta_{M}^{+}})$.

###### Proof.

As the underlying $\mathcal{O}/\varpi^{m}$-modules of the two functors
evaluated on some
$\pi\in\textnormal{Mod}_{\textnormal{sm}}(\mathcal{O}/\varpi^{m}[\Delta_{Q}])$
clearly coincide, to see the first part, we only have to check that the Hecke
actions match up. This follows from the definitions and the identifications
3.1.

The second part follows from [Wei94], Corollary 10.8.3 as soon as we verify
the fact that $\Gamma(N^{0},-)$ carries injectives to
${\rm{Hom}}_{\mathcal{O}/\varpi^{m}[M^{0}]}(\sigma^{\vee},-)$-acyclics. To see
this we argue just as in the proof of [All+23], Lemma 5.2.7, (1). Namely, we
have the exact forgetful functors

$\textnormal{Mod}_{\textnormal{sm}}(\mathcal{O}/\varpi^{m}[\Delta_{Q}])\xrightarrow[]{\alpha}\textnormal{Mod}_{\textnormal{sm}}(\mathcal{O}/\varpi^{m}[M^{0}\ltimes
N^{0}]),$
$\textnormal{Mod}_{\textnormal{sm}}(\mathcal{O}/\varpi^{m}[\Delta_{M}^{+}])\xrightarrow[]{\beta}\textnormal{Mod}_{\textnormal{sm}}(\mathcal{O}/\varpi^{m}[M^{0}])\textnormal{
and}$
$\textnormal{Mod}(\mathcal{H}(\sigma)^{\Delta_{M}^{+}})\xrightarrow[]{\gamma}\textnormal{Mod}(\mathcal{O}/\varpi^{m}).$

Moreover, [All+23], Lemma 5.2.4 says that $\alpha$ and $\beta$ preserve
injectives. Now pick an injective
$\mathcal{I}\in\textnormal{Mod}_{\textnormal{sm}}(\mathcal{O}/\varpi^{m}[\Delta_{Q}])$.
We would like to show that, for $i\geq 1$,

$R^{i}{\rm{Hom}}_{\mathcal{O}/\varpi^{m}[M^{0}]}(\sigma^{\vee},\Gamma(N^{0},\mathcal{I}))=0.$

As the previous discussion shows, we can equivalently verify that

$\gamma
R^{i}{\rm{Hom}}_{\mathcal{O}/\varpi^{m}[M^{0}]}(\sigma^{\vee},\Gamma(N^{0},\mathcal{I}))=R^{i}{\rm{Hom}}_{\mathcal{O}/\varpi^{m}[M^{0}]}(\sigma^{\vee},\Gamma(N^{0},\alpha\mathcal{I})).$

However, $\Gamma(N^{0},-)$, as a functor
$\textnormal{Mod}_{\textnormal{sm}}(\mathcal{O}/\varpi^{m}[M^{0}\ltimes
N^{0}])\to\textnormal{Mod}_{\textnormal{sm}}(\mathcal{O}/\varpi^{m}[M^{0}])$,
preserves injectives as it possesses an exact left adjoint given by inflation.
Therefore, $\alpha\mathcal{I}$ being injective,
$R^{i}{\rm{Hom}}_{\mathcal{O}/\varpi^{m}[M^{0}]}(\sigma^{\vee},\Gamma(N^{0},\alpha\mathcal{I}))$
vanishes. ∎

We further consider the functor

$(-)^{Q\textnormal{-ord}}:\textnormal{Mod}(\mathcal{H}(\sigma)^{\Delta_{M}^{+}})\to\textnormal{Mod}(\mathcal{H}(\sigma)^{\Delta_{M}})$

defined by localising along
$\mathcal{H}(\sigma)^{\Delta_{M}^{+}}\subset\mathcal{H}(\sigma)^{\Delta_{M}}$.

######  Lemma 3.6.

The functor

$(-)^{Q\textnormal{-ord}}:\textnormal{Mod}_{\textnormal{sm}}(\mathcal{O}/\varpi^{m}[\Delta_{M}^{+}])\to\textnormal{Mod}_{\textnormal{sm}}(\mathcal{O}/\varpi^{m}[\Delta_{M}])$

sends injectives to
${\rm{Hom}}_{\mathcal{O}/\varpi^{m}[M^{0}]}(\sigma^{\vee},-)$-acyclics.

###### Proof.

Set $\Delta_{M}^{\prime+}$ to be the monoid generated by $M^{0}$ and $z_{p}$.
Then
$\mathcal{H}(\Delta_{M}^{\prime+},M^{b})=\mathcal{O}/\varpi^{m}[\Delta_{M}^{\prime+}/M^{b}]$
is Noetherian for every integer $b\geq 0$ and the proof of [All+23], Lemma
5.2.7, (2) applies and we get that $(-)^{Q\textnormal{-ord}}$, as a functor

$\textnormal{Mod}_{\textnormal{sm}}(\mathcal{O}/\varpi^{m}[\Delta_{M}^{\prime+}])\to\textnormal{Mod}_{\textnormal{sm}}(\mathcal{O}/\varpi^{m}[\Delta_{M}^{\prime}]),$

even preserves injectives.

In particular, proving that the forgetful functor

$\textnormal{Mod}_{\textnormal{sm}}(\mathcal{O}/\varpi^{m}[\Delta_{M}^{+}])\to\textnormal{Mod}_{\textnormal{sm}}(\mathcal{O}/\varpi^{m}[\Delta_{M}^{\prime+}])$

preserves injectives would allow us to conclude by running an argument similar
to that of the second part of the proof of Lemma 3.5. To see that the
forgetful functor preserves injectives, we note that it admits
$\mathcal{O}/\varpi^{m}[\Delta_{M}^{+}]\otimes_{\mathcal{O}/\varpi^{m}[\Delta_{M}^{\prime+}]}-$
as a left adjoint. Therefore, it’s enough to see that the latter functor is
exact. This follows from the fact that
$\mathcal{O}/\varpi^{m}[\Delta_{M}^{+}]$ is free as a right
$\mathcal{O}/\varpi^{m}[\Delta_{M}^{\prime+}]$-module with set of generators
given by a well-chosen set of representatives of
$\Delta_{M}/\Delta_{M}^{\prime}$. ∎

Then our other candidate for ordinary parts of some $\pi\in
D^{+}_{\textnormal{sm}}(\mathcal{O}/\varpi^{m}[\Delta_{Q}])$ is the
composition $R{\rm{Hom}}_{\mathcal{O}/\varpi^{m}[M^{0}\ltimes
N^{0}]}(\sigma^{\vee},\pi)^{Q\textnormal{-ord}}$. We see that the two
candidates in fact coincide.

######  Corollary 3.7.

Given $\pi\in D^{+}_{\textnormal{sm}}(\mathcal{O}/\varpi^{m}[\Delta_{Q}])$, we
have a natural isomorphism

$R{\rm{Hom}}_{\mathcal{O}/\varpi^{m}[M^{0}]}(\sigma^{\vee},R\Gamma(N^{0},\pi)^{Q\textnormal{-ord}})\cong
R{\rm{Hom}}_{\mathcal{O}/\varpi^{m}[M^{0}\ltimes
N^{0}]}(\sigma^{\vee},\pi)^{Q\textnormal{-ord}}$

in $D^{+}(\mathcal{H}(\sigma)^{\Delta_{M}})$.

###### Proof.

Exactness of $(-)^{Q\textnormal{-ord}}$, Lemma 3.6, and an argument just as in
the proof of [All+23], Lemma 5.2.6 implies that we have

$R{\rm{Hom}}_{\mathcal{O}/\varpi^{m}[M^{0}]}(\sigma^{\vee},-)\circ(-)^{Q\textnormal{-ord}}\cong$
$\cong
R({\rm{Hom}}_{\mathcal{O}/\varpi^{m}[M^{0}]}(\sigma^{\vee},-)\circ(-)^{Q\textnormal{-ord}})$
$\cong
R((-)^{Q\textnormal{-ord}}\circ{\rm{Hom}}_{\mathcal{O}/\varpi^{m}[M^{0}]}(\sigma^{\vee},-))$
$\cong(-)^{Q\textnormal{-ord}}\circ
R{\rm{Hom}}_{\mathcal{O}/\varpi^{m}[M^{0}]}(\sigma^{\vee},-).$

In particular, for $\pi\in
D^{+}_{\textnormal{sm}}(\mathcal{O}/\varpi^{m}[\Delta_{Q}])$, we get a natural
isomorphism

$R{\rm{Hom}}_{\mathcal{O}/\varpi^{m}[M^{0}]}(\sigma^{\vee},R\Gamma(N^{0},\pi)^{Q\textnormal{-ord}})\cong
R{\rm{Hom}}_{\mathcal{O}/\varpi^{m}[M^{0}]}(\sigma^{\vee},R\Gamma(N^{0},\pi))^{Q\textnormal{-ord}}.$

As a consequence, Lemma 3.5 implies that there is a natural isomorphism

$R{\rm{Hom}}_{\mathcal{O}/\varpi^{m}[M^{0}]}(\sigma^{\vee},R\Gamma(N^{0},\pi)^{Q\textnormal{-ord}})\cong
R{\rm{Hom}}_{\mathcal{O}/\varpi^{m}[M^{0}\ltimes
N^{0}]}(\sigma^{\vee},\pi)^{Q\textnormal{-ord}}.$

∎

We now further assume that the action of $M^{1}$ on $\sigma$ is trivial. Note
that by the second paragraph of Example 3.1 it is not a serious restriction as
we are free to change $M^{1}$ to be a smaller compact open subgroup, and
$\sigma$ is assumed to be finite and free over $\mathcal{O}/\varpi^{m}$. For
$c\geq 1$, set $\widetilde{\sigma}$ to be the smooth
$\mathcal{O}/\varpi^{m}[\mathcal{Q}(0,c)]$-module defined by the map
$\mathcal{Q}(0,c)\to\mathcal{Q}(0,c)/\mathcal{Q}(c,c)\cong M^{0}/M^{c}$ i.e.,
for $\bar{n}mn\in\mathcal{Q}(0,c)$, we set
$\widetilde{\sigma}(\bar{n}mn)=\sigma(m)$. Consider the subalgebras
$\mathcal{H}(\widetilde{\sigma})^{\Delta_{\mathcal{Q}}(c)}\subset\mathcal{H}(\widetilde{\sigma})^{+}\subset\mathcal{H}(\widetilde{\sigma})$
generated by functions supported on $\Delta_{\mathcal{Q}}(c)$ and
$\mathcal{Q}^{+}(0,c)$, respectively. We then have the following observation.

######  Lemma 3.8.

For any $c\geq 1$, there is an isomorphism of algebras

$t_{c}:\mathcal{H}(\sigma)^{\Delta_{M}^{+}}\to\mathcal{H}(\widetilde{\sigma})^{\Delta_{\mathcal{Q}}(c)}$

such that, for any
$\pi\in\textnormal{Mod}_{\textnormal{sm}}(\mathcal{O}/\varpi^{m}[\Delta_{\mathcal{Q}}(c)])$,
the inclusion

${\rm{Hom}}_{\mathcal{O}/\varpi^{m}[\mathcal{Q}(0,c)]}(\widetilde{\sigma}^{\vee},\pi)\hookrightarrow{\rm{Hom}}_{\mathcal{O}/\varpi^{m}[M^{0}]}(\sigma^{\vee},\Gamma(N^{0},\pi))$

intertwines the action of $\mathcal{H}(\sigma)^{\Delta_{M}^{+}}$ on the source
via $t_{c}$ with the action of $\mathcal{H}(\sigma)^{\Delta_{M}^{+}}$ on the
target.

###### Proof.

Recall that $\mathcal{H}(\sigma)^{\Delta_{M}^{+}}$ is generated as an
$\mathcal{O}/\varpi^{m}$-module by elements of the form $[m,\psi]$ with
$m\in\Delta_{M}^{+}$. For such an element, we set
$t_{c}([m,\psi]):\Delta_{\mathcal{Q}}\to\textnormal{End}_{\mathcal{O}/\varpi^{m}}(\widetilde{\sigma})$
to be the function that is supported on $\mathcal{Q}(0,c)m\mathcal{Q}(0,c)$
and sends $m$ to $\psi$. Then one verifies using the existence of an Iwahori
decomposition that $t_{c}([m,\psi])$ indeed lies in
$\mathcal{H}(\widetilde{\sigma})^{\Delta_{G}(c)}$ i.e., we have
$\widetilde{\sigma}(q)\circ\psi=\psi\circ\widetilde{\sigma}(m^{-1}qm)$ for
$q\in\mathcal{Q}(0,c)\cap m\mathcal{Q}(0,c)m^{-1}$. This visibly yields an
isomorphism
$t_{c}:\mathcal{H}(\sigma)^{\Delta_{M}^{+}}\xrightarrow{\sim}\mathcal{H}(\widetilde{\sigma})^{\Delta_{\mathcal{Q}}(c)}$
of $\mathcal{O}/\varpi^{m}$-modules.

Before proving that $t_{c}$ also respects the algebra structure, we check the
last claim. To do this, we pick
$\phi\in{\rm{Hom}}_{\mathcal{O}/\varpi^{m}[\mathcal{Q}(0,c)]}(\widetilde{\sigma}^{\vee},\pi)\subset{\rm{Hom}}_{\mathcal{O}/\varpi^{m}[M^{0}]}(\sigma^{\vee},\Gamma(N^{0},\pi))$,
and compute, first the action of $[m,\psi]$,

$[m,\psi]\cdot\phi:v\mapsto\sum_{\tilde{m}\in M^{0}/mM^{0}m^{-1}\cap
M^{0}}\pi^{N^{0}}(\tilde{m}m)\phi(\psi^{t}\circ\sigma^{\vee}(\tilde{m}^{-1})v)=$
$\sum_{\tilde{m}\in M^{0}/mM^{0}m^{-1}\cap M^{0}}\pi(\tilde{m})\sum_{n\in
N^{0}/mN^{0}m^{-1}}\pi(nm)\phi(\psi^{t}\circ\widetilde{\sigma}^{\vee}(n^{-1}\tilde{m}^{-1})v)=$
$\sum_{(\tilde{m},n)\in(M^{0}/mM^{0}m^{-1}\cap
M^{0})\times(N^{0}/mN^{0}m^{-1})}\pi(\tilde{m}nm)\phi(\psi^{t}\circ\widetilde{\sigma}^{\vee}(n^{-1}\tilde{m}^{-1})v).$

On the other hand, we have

$t_{c}([m,\psi])\cdot\phi:v\mapsto\sum_{q\in\mathcal{Q}(0,c)/\mathcal{Q}(0,c)\cap
m\mathcal{Q}(0,c)m^{-1}}\pi(qm)\phi(\psi^{t}\circ\widetilde{\sigma}^{\vee}(q^{-1})v).$

Therefore, it suffices to prove that the inclusion $M^{0}\ltimes
N^{0}\hookrightarrow\mathcal{Q}(0,c)$ descends to a bijection

$(M^{0}/mM^{0}m^{-1}\cap
M^{0})\times(N^{0}/mN^{0}m^{-1})\cong\mathcal{Q}(0,c)/\mathcal{Q}(0,c)\cap
m\mathcal{Q}(0,c)m^{-1}.$

This follows from the Iwahori decomposition, the fact that
$m\in\Delta_{M}^{+}\subset M^{+}$ and 3.1.

We now easily deduce that $t_{c}$ respects the algebra structure. Indeed, note
that if we set
$\pi:=\textnormal{c-Ind}_{\mathcal{Q}(0,c)}^{G(L)}\widetilde{\sigma}^{\vee}$,
$\mathcal{H}(\widetilde{\sigma})^{\Delta_{\mathcal{Q}(0,c)}}$ acts faithfully
on
${\rm{Hom}}_{\mathcal{O}/\varpi^{m}[\mathcal{Q}(0,c)]}(\widetilde{\sigma}^{\vee},\pi)$.
In particular, $t_{c}$ must be an algebra homomorphism as, by the previous
paragraph, $t_{c}$ yields an algebra action of
$\mathcal{H}(\sigma)^{\Delta_{M}^{+}}$ on
${\rm{Hom}}_{\mathcal{O}/\varpi^{m}[\mathcal{Q}(0,c)]}(\widetilde{\sigma}^{\vee},\pi)$,
that factors through the faithful algebra action of the target of $t_{c}$. ∎

As a consequence of Lemma 3.8, for every $c\geq 1$, we get a left exact
functor

${\rm{Hom}}_{\mathcal{O}/\varpi^{m}[\mathcal{Q}(0,c)]}(\widetilde{\sigma}^{\vee},-):\textnormal{Mod}_{\textnormal{sm}}(\mathcal{O}/\varpi^{m}[\Delta_{\mathcal{Q}}])\to\textnormal{Mod}(\mathcal{H}(\sigma)^{\Delta_{M}^{+}}).$

######  Lemma 3.9.

For any $c\geq 1$, there is a natural isomorphism

${\rm{Hom}}_{\mathcal{O}/\varpi^{m}[\mathcal{Q}(0,c)]}(\widetilde{\sigma}^{\vee},-)^{Q\textnormal{-ord}}\cong{\rm{Hom}}_{\mathcal{O}/\varpi^{m}[M^{0}\ltimes
N^{0}]}(\sigma^{\vee},-)^{Q\textnormal{-ord}}$

of functors

$\textnormal{Mod}_{\textnormal{sm}}(\mathcal{O}/\varpi^{m}[\Delta_{\mathcal{Q}}])\to\textnormal{Mod}(\mathcal{H}(\sigma)^{\Delta_{M}}).$

###### Proof.

By the exactness of $(-)^{Q\textnormal{-ord}}$, for
$\pi\in\textnormal{Mod}_{\textnormal{sm}}(\mathcal{O}/\varpi^{m}[\Delta_{\mathcal{Q}}])$,
we have an inclusion

(3.2)
${\rm{Hom}}_{\mathcal{O}/\varpi^{m}[\mathcal{Q}(0,c)]}(\widetilde{\sigma}^{\vee},\pi)^{Q\textnormal{-ord}}\hookrightarrow{\rm{Hom}}_{\mathcal{O}/\varpi^{m}[M^{0}\ltimes
N^{0}]}(\sigma^{\vee},\pi)^{Q\textnormal{-ord}}$

and, by Lemma 3.8, it is $\mathcal{H}(\sigma)^{\Delta_{M}}$-equivariant. In
particular, we are left to check that 3.2 is a bijection. Given
$\phi\in{\rm{Hom}}_{\mathcal{O}/\varpi^{m}[M^{0}\ltimes
N^{0}]}(\sigma^{\vee},\pi)$, by smoothness of $\pi$ and the fact that
$\sigma^{\vee}$ is finite free as an $\mathcal{O}/\varpi^{m}$-module, $\phi$
lies in
${\rm{Hom}}_{\mathcal{O}/\varpi^{m}[\mathcal{Q}(0,c^{\prime})]}(\widetilde{\sigma}^{\vee},\pi)$
for some $c^{\prime}>c$. By induction, it suffices to prove that, for some
integer $k\geq 1$,

$[z_{p},\textnormal{id}]^{k}\cdot\phi\in{\rm{Hom}}_{\mathcal{O}/\varpi^{m}[\mathcal{Q}(0,c^{\prime}-1)]}(\widetilde{\sigma}^{\vee},\pi).$

This, for instance, is proved in [Eme10], Lemma 3.3.1. ∎

The following Corollary then summarises the observations of the subsection.

######  Corollary 3.10.

Given $\pi\in
D^{+}_{\textnormal{sm}}(\mathcal{O}/\varpi^{m}[\Delta_{\mathcal{Q}}])$, we
have natural isomorphisms

$R{\rm{Hom}}_{\mathcal{O}/\varpi^{m}[M^{0}]}(\sigma^{\vee},R\Gamma(N^{0},\pi)^{Q\textnormal{-ord}})\cong
R{\rm{Hom}}_{\mathcal{O}/\varpi^{m}[M^{0}\ltimes
N^{0}]}(\sigma^{\vee},\pi)^{Q\textnormal{-ord}}\cong$ $\cong
R{\rm{Hom}}_{\mathcal{O}/\varpi^{m}[\mathcal{Q}(0,c)]}(\widetilde{\sigma}^{\vee},\pi)^{Q\textnormal{-ord}}$

in $D^{+}(\mathcal{H}(\sigma)^{\Delta_{M}})$.

###  3.2. $Q$-ordinary Hida theory for $G$

We revisit the setup from §2.7. In particular, $F$ will be a CM field and
$G=\textnormal{Res}_{\mathcal{O}_{F}/\mathcal{O}_{F^{+}}}\textnormal{GL}_{n}$.
Moreover, we fix a tuple
$(Q_{p},\lambda,\underline{\tau})=(Q_{v},\lambda_{v},\underline{\tau_{v}})_{v\in
S_{p}}$ as in §2.7. We will work with good subgroups $K\subset
G(\mathbf{A}_{F^{+}}^{\infty})$ with $K^{p}$ being fixed and $K_{p}$ of the
form

$\mathcal{Q}_{p}(b,c)=\prod_{v\in S_{p}}\mathcal{Q}_{v}(b,c)\subset\prod_{v\in
S_{p}}\textnormal{GL}_{n}(\mathcal{O}_{F_{v}})$

for $c\geq b\geq 0$ with $c\geq c_{p}$ where $\mathcal{Q}_{v}(b,c)$ is the
parahoric level subgroup corresponding to $Q_{v}$, just as before. We denote
such a good subgroup by $K(b,c)\subset G(\mathbf{A}_{F^{+}}^{\infty})$. Recall
that, given a local system $\mathcal{V}_{(\lambda,\underline{\tau})}^{Q_{p}}$
as in §2.7, $\mathcal{H}(\Delta_{\mathcal{Q}_{p}}(c),\mathcal{Q}_{p}(b,c))$
acts on $R\Gamma(X_{K(b,c)},\mathcal{V}_{(\lambda,\underline{\tau})}^{Q_{p}})$
via endomorphisms in $D^{+}(\mathcal{O})$ (cf. §2.8).212121We emphasise that
here $\Delta_{\mathcal{Q}_{p}}(c)$ denotes the monoid introduced in §2.8. In
particular, for $v\in S_{p}$, the corresponding $U_{p}$-operator
$U_{v}^{Q_{v}}$ acts on the mentioned complex. Following [KT17], §2.4, we set

$R\Gamma(X_{K(b,c)},\mathcal{V}_{(\lambda,\underline{\tau})}^{Q_{p}})^{Q_{p}\textnormal{-ord}}$

to be the maximal direct summand of
$R\Gamma(X_{K(b,c)},\mathcal{V}_{(\lambda,\underline{\tau})}^{Q_{p}})$ on
which $U_{v}^{Q_{v}}$ acts invertibly for each $v\in S_{p}$. This will then be
an object of $D^{+}(\mathcal{O}[\mathcal{Q}_{p}(0,c)/\mathcal{Q}_{p}(b,c)])$
with an action of the spherical Hecke algebra $\mathbf{T}^{T}$.

On the other hand, we can apply the formalism of §3.1 with the parabolic
subgroup $Q_{p}\subset G_{p}$, compact opens given by $\mathcal{Q}_{p}(b,c)$,
open submonoid of $G_{p}$ given by $\Delta_{M_{p}}^{+}$ (from §2.8) and
$\sigma$ being the trivial $\Delta_{M_{p}}^{+}$-module. Note that in this case
$\mathcal{H}(\sigma)^{\Delta_{M_{p}}^{+}}=\mathcal{H}(\Delta_{M_{p}^{+}},M_{p}^{0})\otimes_{\mathbf{Z}}\mathcal{O}/\varpi^{m}=\mathcal{O}/\varpi^{m}[\Delta_{M_{p}}^{+}/M_{p}^{0}]$,
$\mathcal{H}(\sigma)^{\Delta_{M_{p}}}=\mathcal{O}/\varpi^{m}[\Delta_{M_{p}}/M_{p}^{0}]$
and
$\mathcal{H}(\widetilde{\sigma})^{\Delta_{\mathcal{Q}(c)}}\cong\mathcal{H}(\Delta_{\mathcal{Q}_{p}},\mathcal{Q}_{p}(0,c))\otimes_{\mathbf{Z}}\mathcal{O}/\varpi^{m}$.
One notes that, since the cohomology groups of
$R\Gamma(X_{K(b,c)},\mathcal{V}_{(\lambda,\underline{\tau})}^{Q_{p}}/\varpi^{m})$
are finite $\mathcal{O}/\varpi^{m}$-modules, the notions of taking
$Q_{p}$-ordinary parts in the sense of [KT17] and in the sense of [Eme10]
coincide. For an argument see the proof of [All+23], Proposition 5.2.15. In
other words, we have the following.

######  Lemma 3.11.

For any $m\geq 1$, $c\geq b\geq 0$ with $c\geq c_{p}$, there is a natural
$\mathbf{T}^{T}$-equivariant isomorphism

$R\Gamma(X_{K(b,c)},\mathcal{V}_{(\lambda,\underline{\tau})}^{Q_{p}}/\varpi^{m})^{Q_{p}\textnormal{-ord}}\xrightarrow{\sim}R\Gamma(\mathcal{Q}_{p}(b,c),\pi(K^{p},\mathcal{V}_{(\lambda,\underline{\tau})}^{Q_{p}}/\varpi^{m}))^{Q_{p}\textnormal{-ord}}$

in $D^{+}(\mathcal{O}/\varpi^{m}[M_{p}^{0}/M_{p}^{b}])$, induced by the
natural isomorphism

$R\Gamma(X_{K(b,c)},\mathcal{V}_{(\lambda,\underline{\tau})}^{Q_{p}}/\varpi^{m})\cong
R\Gamma(\mathcal{Q}_{p}(b,c),\pi(K^{p},\mathcal{V}_{(\lambda,\underline{\tau})}^{Q_{p}}/\varpi^{m})).$

We now highlight the two important features of Hida theory for Betti
cohomology. The first is usually referred to as the property of independence
of level. Consider completed cohomology

$\pi(K^{p},\mathcal{V}_{(\lambda,\underline{\tau})}^{Q_{p}}/\varpi^{m})\in
D^{+}_{\textnormal{sm}}(\mathcal{O}/\varpi^{m}[\Delta_{\mathcal{Q}_{p}}(c_{p})]).$

Recall that it is equipped with an action of $\mathbf{T}^{T}$.

######  Definition 3.12.

We set $Q_{p}$-ordinary completed cohomology to be

$\pi^{Q_{p}\textnormal{-ord}}(K^{p},\mathcal{V}_{(\lambda,\underline{\tau})}^{Q_{p}}/\varpi^{m}):=R\Gamma(N^{0}_{p},\pi(K^{p},\mathcal{V}_{(\lambda,\underline{\tau,N})}^{Q_{p}}/\varpi^{m}))^{Q_{p}\textnormal{-ord}}\in
D^{+}_{\textnormal{sm}}(\mathcal{O}/\varpi^{m}[\Delta_{M_{p}}]).$

By Corollary 3.10 and Lemma 3.11, for integers $m\geq 1$, $c\geq b\geq 0$ with
$c\geq c_{p}$, we have

$R\Gamma(M_{p}^{b},\pi^{Q_{p}\textnormal{-ord}}(K^{p},\mathcal{V}_{(\lambda,\underline{\tau})}^{Q_{p}}/\varpi^{m}))\cong
R\Gamma(X_{K(b,c)},\mathcal{V}_{(\lambda,\underline{\tau})}^{Q_{p}}/\varpi^{m})^{Q_{p}\textnormal{-ord}}$

in $D^{+}(\mathcal{O}/\varpi^{m}[M^{0}_{p}/M_{p}^{b}])$. As an immediate
consequence, we can deduce the independence of level property.

######  Corollary 3.13.

(Independence of level) For integers $m\geq 1$, and $c\geq b\geq 0$ with
$c\geq c_{p}$, the natural $\mathbf{T}^{T}$-equivariant morphism

$R\Gamma(X_{K(b,\max\\{c_{p},b\\})},\mathcal{V}_{(\lambda,\underline{\tau})}^{Q_{p}}/\varpi^{m})^{Q_{p}\textnormal{-ord}}\to$
$R\Gamma(X_{K(b,c)},\mathcal{V}_{(\lambda,\underline{\tau})}^{Q_{p}}/\varpi^{m})^{Q_{p}\textnormal{-ord}}$

is an isomorphism in $D^{+}(\mathcal{O}/\varpi^{m}[M^{0}_{p}/M_{p}^{b}])$.

Note that the analogous statement also holds for compactly supported and
boundary cohomology.

As a consequence of independence of level (or rather the statements behind its
proof), we can deduce that the $Q_{p}$-ordinary part of the cohomology
complexes
$R\Gamma(X_{K(0,c)},\mathcal{V}_{(\lambda,\underline{\tau})}^{Q_{p}})$ is
given by taking invariants of $Q_{p}$-ordinary completed cohomology with
respect to the corresponding smooth types of the Levi subgroup.

######  Corollary 3.14.

For every integer $c\geq c_{p}$, we have a natural
$\mathbf{T}^{T}$-equivariant isomorphism

$R\Gamma(X_{K(0,c)},\mathcal{V}_{(\lambda,\underline{\tau})}^{Q_{p}}/\varpi^{m})^{Q_{p}\textnormal{-ord}}\cong$
$\cong
R{\rm{Hom}}_{\mathcal{O}/\varpi^{m}[M^{0}_{p}]}(\sigma(\underline{\tau})^{\circ}/\varpi^{m},\pi^{Q_{p}\textnormal{-ord}}(K^{p},\mathcal{V}_{\lambda}/\varpi^{m}))$

in $D^{+}(\mathcal{O}/\varpi^{m})$ induced by Corollary 3.10, Lemma 3.11 and
Lemma 2.2.

###### Proof.

By Corollary 3.10 and Lemma 3.11, we have a natural isomorphism

$R\Gamma(X_{K(0,c)},\mathcal{V}_{(\lambda,\underline{\tau,N})}^{Q_{p}}/\varpi^{m})^{Q_{p}\textnormal{-ord}}\cong
R\Gamma(M^{0}_{p},\pi^{Q_{p}\textnormal{-ord}}(K^{p},\mathcal{V}_{(\lambda,\underline{\tau})}/\varpi^{m}))$

in $D^{+}(\mathcal{O}/\varpi^{m})$. Recall that by definition we have

(3.3)
$\pi^{Q_{p}\textnormal{-ord}}(K^{p},\mathcal{V}_{(\lambda,\underline{\tau})}^{Q_{p}}/\varpi^{m})=R\Gamma(N^{0}_{p},\pi(K^{p},\mathcal{V}_{\lambda}/\varpi^{m})\otimes_{\mathcal{O}/\varpi^{m}}\widetilde{\sigma(\underline{\tau})}^{\circ,\vee}/\varpi^{m})^{Q_{p}\textnormal{-ord}}.$

Since
$-\otimes_{\mathcal{O}/\varpi^{m}}\widetilde{\sigma(\underline{\tau})}^{\circ,\vee}/\varpi^{m}$
is exact and has
$-\otimes_{\mathcal{O}/\varpi^{m}}\widetilde{\sigma(\underline{\tau})}^{\circ}/\varpi^{m}$
as an exact left adjoint, [Wei94], Corollary 10.8.3 applies and 3.3 is
naturally isomorphic to

$\pi^{Q_{p}\textnormal{-ord}}(K^{p},\mathcal{V}_{\lambda}/\varpi^{m})\otimes_{\mathcal{O}/\varpi^{m}}\sigma(\underline{\tau})^{\circ,\vee}/\varpi^{m}=$
$={\rm{Hom}}_{\mathcal{O}/\varpi^{m}}(\sigma(\underline{\tau})^{\circ}/\varpi^{m},\pi^{Q_{p}\textnormal{-ord}}(K^{p},\mathcal{V}_{(\lambda,\underline{\tau})}^{Q_{p}}/\varpi^{m}))$

in $D^{+}_{\textnormal{sm}}(\mathcal{O}/\varpi^{m}[\Delta_{M_{p}}])$ when
$\sigma(\underline{\tau})^{\circ}$ is viewed as an
$\mathcal{O}/\varpi^{m}[\Delta_{M_{p}}]$-module via inflation from
$M^{0}_{p}$. Another application of [Wei94], Corollary 10.8.3 to

${\rm{Hom}}_{\mathcal{O}/\varpi^{m}[M_{p}^{0}]}(\sigma(\underline{\tau})^{\circ}/\varpi^{m},-)=\Gamma(M^{0}_{p},-)\circ{\rm{Hom}}_{\mathcal{O}/\varpi^{m}}(\sigma(\underline{\tau})^{\circ}/\varpi^{m},-)$

finishes the proof. ∎

We now turn to discussing the second feature of Hida theory called
independence of weight. Recall the representation

$\mathcal{V}_{w_{0}^{Q_{p}}\lambda}=\otimes_{v\in
S_{p}}\mathcal{V}_{w_{0}^{Q_{v}}\lambda_{v}}\in\textnormal{Mod}(\mathcal{O}[\prod_{v\in
S_{p}}M_{v}^{0}])$

from Lemma 2.14. View it as a $\Delta_{M_{p}}$-module via inflation.

######  Proposition 3.15.

(Independence of weight) For any integers $m\geq 1$, $c\geq c_{p}$, subset
$\overline{S}\subset\overline{S}_{p}$, and complex $\pi\in
D^{+}_{\textnormal{sm}}(\mathcal{O}/\varpi^{m}[\Delta_{\mathcal{Q}_{\overline{S}}}(c)])$,
the map introduced in Lemma 2.14 induces an isomorphism

$R\Gamma(N^{0}_{\overline{S}},\pi\otimes_{\mathcal{O}/\varpi^{m}}\mathcal{V}_{\lambda_{\overline{S}}}/\varpi^{m})^{Q_{\overline{S}}\textnormal{-ord}}\xrightarrow{\sim}R\Gamma(N^{0}_{\overline{S}},\pi)^{Q_{\overline{S}}\textnormal{-ord}}\otimes_{\mathcal{O}/\varpi^{m}}\mathcal{V}_{w_{0}^{Q_{\overline{S}}}\lambda_{\overline{S}}}/\varpi^{m}$

in
$D^{+}_{\textnormal{sm}}(\mathcal{O}/\varpi^{m}[\Delta_{M_{\overline{S}}}])$.
In particular, we have a natural isomorphism

$\pi^{Q_{p}\textnormal{-ord}}(K^{p},\mathcal{V}_{\lambda}/\varpi^{m})\xrightarrow{\sim}\pi^{Q_{p}\textnormal{-ord}}(K^{p},\mathcal{O}/\varpi^{m})\otimes_{\mathcal{O}/\varpi^{m}}\mathcal{V}_{w_{0}^{Q_{p}}\lambda}/\varpi^{m}$

in $D^{+}_{\textnormal{sm}}(\mathcal{O}/\varpi^{m}[\Delta_{M_{p}}])$.

###### Proof.

The same argument as in the proof of [CN23], Proposition 2.2.15 applies here.
∎

We now can deduce that
$R\Gamma(X_{K(0,c_{p})},\mathcal{V}_{(\lambda,\underline{\tau})}^{Q_{p}}/\varpi^{m})^{Q_{p}\textnormal{-ord}}$
admits a natural Hecke action at $p$ corresponding to the data
$(\lambda,\underline{\tau})$. Namely, we set

$\sigma(\lambda,\underline{\tau})^{\circ}:=\mathcal{V}_{w_{0}^{Q_{p}}\lambda}^{\vee}\otimes_{\mathcal{O}}\sigma(\underline{\tau})^{\circ},$

a locally algebraic $\mathcal{O}$-representation of $M^{0}_{p}$.

######  Corollary 3.16.

We have a natural isomorphism

$R\Gamma(X_{K(0,c_{p})},\mathcal{V}_{(\lambda,\underline{\tau})}^{Q_{p}}/\varpi^{m})^{Q_{p}\textnormal{-ord}}\cong$
$\cong
R{\rm{Hom}}_{\mathcal{O}/\varpi^{m}[M^{0}_{p}]}(\sigma(\lambda,\underline{\tau})^{\circ}/\varpi^{m},\pi^{Q_{p}\textnormal{-ord}}(K^{p},\mathcal{O}/\varpi^{m}))$

in $D^{+}(\mathcal{O}/\varpi^{m})$ induced by Corollary 3.14, and Proposition
3.15. In particular, we have an induced algebra homomorphism

$\mathcal{H}(\sigma(\lambda,\underline{\tau})^{\circ,\vee})\otimes_{\mathcal{O}}\mathcal{O}/\varpi^{m}\to\textnormal{End}_{D^{+}(\mathcal{O}/\varpi^{m})}(R\Gamma(X_{K(0,c_{p})},\mathcal{V}_{(\lambda,\underline{\tau})}^{Q_{p}}/\varpi^{m})^{Q_{p}\textnormal{-ord}}).$

###### Proof.

The first part of the statement follows immediately from Corollary 3.14 and
Proposition 3.15.

For the second part, note that the formalism of §3.1 with the choice
$M^{+}_{p}$ for the role of $\Delta_{M_{p}}^{+}$ implies that
$\pi^{Q_{p}\textnormal{-ord}}(K^{p},\mathcal{O}/\varpi^{m})$ can be viewed as
an object in $D^{+}_{\textnormal{sm}}(\mathcal{O}/\varpi^{m}[M_{p}])$. We then
have

$R{\rm{Hom}}_{\mathcal{O}/\varpi^{m}[M^{0}_{p}]}(\sigma(\lambda,\underline{\tau})^{\circ}/\varpi^{m},\pi^{Q_{p}\textnormal{-ord}}(K^{p},\mathcal{O}/\varpi^{m}))\in
D^{+}(\mathcal{H}(\sigma(\lambda,\underline{\tau})^{\circ,\vee}/\varpi^{m})).$

Moreover, the forgetful functor

$\textnormal{Mod}_{\textnormal{sm}}(\mathcal{O}/\varpi^{m}[G_{p}])\to\textnormal{Mod}_{\textnormal{sm}}(\mathcal{O}/\varpi^{m}[\Delta_{M_{p}}^{+}\ltimes
N^{0}_{p}])$

sends injectives to $\Gamma(N^{0}_{p},-)$-acyclics by [Eme10a], Proposition
2.1.11. Consequently, an application of [Wei94], Corollary 10.8.3 shows that
$\pi^{Q_{p}\textnormal{-ord}}(K^{p},\mathcal{O}/\varpi)\in
D^{+}(\textnormal{Mod}_{\textnormal{sm}}[\Delta_{M_{p}}])$ can be computed by
applying $R\Gamma(N^{0}_{p},-)^{Q_{p}\textnormal{-ord}}$ to the object
$\pi(K^{p},\mathcal{O}/\varpi^{m})\in
D^{+}_{\textnormal{sm}}(\mathcal{O}/\varpi^{m}[G_{p}])$ followed by an
application of the forgetful functor
$\textnormal{Mod}_{\textnormal{sm}}(\mathcal{O}/\varpi^{m}[M_{p}])\to\textnormal{Mod}_{\textnormal{sm}}(\mathcal{O}/\varpi^{m}[\Delta_{M_{p}}])$.
On the other hand, [Eme10a], Proposition 2.1.2 shows that the forgetful
functor

$\textnormal{Mod}_{\textnormal{sm}}(\mathcal{O}/\varpi^{m}[M_{p}])\to\textnormal{Mod}_{\textnormal{sm}}(\mathcal{O}/\varpi^{m}[M^{0}_{p}])$

preserves injectives. In particular, another application of [Wei94], Corollary
10.8.3 yields an algebra homomorphism

$\mathcal{H}(\sigma(\lambda,\underline{\tau})^{\circ,\vee}/\varpi^{m})\to\textnormal{End}_{D^{+}(\mathcal{O}/\varpi^{m})}(R\Gamma(X_{K(0,c_{p})},\mathcal{V}_{(\lambda,\underline{\tau})}^{Q_{p}}/\varpi^{m})^{Q_{p}\textnormal{-ord}}).$

Moreover, we have a natural morphism

$\mathcal{H}(\sigma(\lambda,\underline{\tau})^{\circ,\vee})\to\mathcal{H}(\sigma(\lambda,\underline{\tau})^{\circ,\vee}/\varpi^{m})$

induced by the short exact sequence

$0\to\textnormal{c-Ind}_{M^{0}_{p}}^{M_{p}}\sigma(\lambda,\underline{\tau})^{\circ,\vee}\xrightarrow{\varpi^{m}\cdot}\textnormal{c-Ind}_{M^{0}_{p}}^{M_{p}}\sigma(\lambda,\underline{\tau})^{\circ,\vee}\to\textnormal{c-Ind}_{M^{0}_{p}}^{M_{p}}(\sigma(\lambda,\underline{\tau})^{\circ,\vee}/\varpi^{m})\to
0$

and the proof is finished. ∎

We note that the constructed action of
$\mathcal{H}(\sigma,\underline{\tau})^{\circ,\vee}$ on
$R\Gamma(X_{K(0,c_{p})},\mathcal{V}_{(\lambda,\underline{\tau})}^{Q_{p}}/\varpi^{m})^{Q_{p}\textnormal{-ord}}$
admits another description. Namely, by independence of level, we can pass to
level $K(0,m)$ and then, by independence of weight, we get an identification

$R\Gamma(X_{K(0,c_{p})},\mathcal{V}_{(\lambda,\underline{\tau})}^{Q_{p}}/\varpi^{m})^{Q_{p}\textnormal{-ord}}\cong
R\Gamma(X_{K(0,m)},\widetilde{\sigma(\lambda,\underline{\tau})^{\circ,\vee}/\varpi^{m}})^{Q_{p}\textnormal{-ord}}$

in $D^{+}(\mathcal{O}/\varpi^{m})$. On the other hand, we have

$R\Gamma(X_{K(0,m)},\widetilde{\sigma(\lambda,\underline{\tau})^{\circ,\vee}/\varpi^{m}})\cong
R{\rm{Hom}}_{Q_{p}(0,m)}(\widetilde{\sigma(\lambda,\underline{\tau})^{\circ}/\varpi^{m}},\pi(K^{p},\mathcal{O}/\varpi^{m}))$

in $D^{+}(\mathcal{O}/\varpi^{m})$ and the latter naturally lives in
$D^{+}(\mathcal{H}(\sigma(\lambda,\underline{\tau})^{\circ,\vee}/\varpi^{m})^{+})$
according to §3.1. Therefore, we get a natural action of
$\mathcal{H}(\sigma(\lambda,\underline{\tau})^{\circ,\vee}/\varpi^{m})^{+}$ on
the complex
$R\Gamma(X_{K(0,m)},\widetilde{\sigma(\lambda,\underline{\tau})^{\circ,\vee}/\varpi^{m}})$
and, using Corollary 3.10, one sees that the induced action of
$\mathcal{H}(\sigma(\lambda,\underline{\tau})^{\circ,\vee}/\varpi^{m})$ on the
complex
$R\Gamma(X_{K(0,c_{p})},\mathcal{V}_{(\lambda,\underline{\tau})}^{Q_{p}}/\varpi^{m})^{Q_{p}\textnormal{-ord}}$
is the one constructed in Corollary 3.16. An upshot of this observation is
that this way the Hecke action can be described using the formalism of §2.4.

###  3.3. Hida theory with dual coefficients

Given a tuple $(Q_{p},\lambda,\underline{\tau})$ as in §3.2, and an integer
$m\geq 1$, we also discuss $\overline{Q}_{p}$-ordinary Hida theory for
$\mathcal{V}_{\lambda,\underline{\tau}}^{Q_{p},\vee}/\varpi^{m}$ on $X_{K}$
where $K\subset G(\mathbf{A}_{F^{+}}^{\infty})$ is a good subgroup such that
$K_{p}=\mathcal{Q}_{p}(0,\widetilde{c})$ with $\widetilde{c}\geq c_{p}$. This
is developed by applying the formalism of §3.1 with parabolic subgroup
$\overline{Q}_{p}=M_{p}\ltimes\overline{N}_{p}\subset G_{p}$,
$N^{0}:=\overline{N}_{p}^{\widetilde{c}}$, $M^{b}:=M^{b}_{p}$,
$\overline{N}^{c}:=N_{p}^{c}$, $\Delta_{M}^{+}:=(\Delta_{M_{p}}^{+})^{-1}$,
$z_{p}:=u_{p}^{Q_{p},-1}$, and
$\sigma:=\sigma(\lambda,\underline{\tau})^{\circ,\vee}$. We can then introduce
$\overline{Q}_{p}$-ordinary parts of the level $K(0,\widetilde{c})$ cohomology
of $\mathcal{V}_{(\lambda,\underline{\tau})}^{Q_{p},\vee}/\varpi^{m}$ by
inverting the Hecke operator attached to $u_{p}^{Q_{p},-1}$. We can also
introduce $\overline{Q}_{p}$-ordinary completed cohomology

$\pi^{\overline{Q}_{p}\textnormal{-ord}}(K^{p},\mathcal{V}_{(\lambda,\tau)}^{Q_{p},\vee}/\varpi^{m}):=R\Gamma(\overline{N}^{\widetilde{c}}_{p},\pi(K^{p},\mathcal{V}_{(\lambda,\tau)}^{Q_{p},\vee}/\varpi^{m}))^{\overline{Q}_{p}\textnormal{-ord}}\in
D^{+}_{\textnormal{sm}}(\mathcal{O}/\varpi^{m}[\Delta_{M_{p}}]).$

Given this setup, the formalism of §3.1 combined with the short exact sequence

$0\to\mathcal{V}_{w_{0}^{Q_{p}}\lambda}^{\vee}\to\mathcal{V}_{\lambda}^{\vee}\to\mathcal{K}_{\lambda}^{\vee}\to
0$

induced by taking duals of the surjection of Lemma 2.14 yields the
independence of weight property for dual coefficients.

######  Proposition 3.17.

(Independence of weight) For any integer $m\geq 1$, there is a natural
isomorphism

$\pi^{\overline{Q}_{p}\textnormal{-ord}}(K^{p},\mathcal{V}_{\lambda}^{\vee}/\varpi^{m})\xrightarrow{\sim}\pi^{\overline{Q}_{p}\textnormal{-ord}}(K^{p},\mathcal{O}/\varpi^{m})\otimes_{\mathcal{O}/\varpi^{m}}\mathcal{V}_{w_{0}^{Q_{p}}\lambda}^{\vee}/\varpi^{m}$

in $D^{+}_{\textnormal{sm}}(\mathcal{O}/\varpi^{m}[\Delta_{M_{p}}])$.

######  Remark 3.18.

Note that even though the definition of $\overline{Q}_{p}$-ordinary completed
cohomology seems to depend on the choice of $\tilde{c}\geq c_{p}$, it in fact
is independent of this choice up to natural isomorphism. To see this, consider
two integers $\tilde{c}_{1},\tilde{c}_{2}\geq c_{p}$. Then, by independence of
weight for $i=1,2$, we have natural isomorphisms

$R\Gamma(\overline{N}_{p}^{\tilde{c}_{i}},\pi(K^{p},\mathcal{V}_{(\lambda,\tau)}^{Q_{p},\vee}/\varpi^{m}))^{\overline{Q}_{p}\textnormal{-ord}}\cong
R\Gamma(\overline{N}_{p}^{\tilde{c}_{i}},\pi(K^{p},\mathcal{O}/\varpi^{m}))^{\overline{Q}_{p}\textnormal{-ord}}\otimes_{\mathcal{O}/\varpi^{m}}\sigma(\lambda,\underline{\tau})^{\circ}/\varpi^{m}$

in $D^{+}_{\textnormal{sm}}(\mathcal{O}/\varpi^{m}[\Delta_{M_{p}}])$. In
particular, we can reduce the question to one with trivial coefficients. In
that case, $\pi(K^{p},\mathcal{O}/\varpi^{m})$ in fact lies in
$D^{+}_{\textnormal{sm}}(\mathcal{O}/\varpi^{m}[\prod_{v\in
S_{p}}\overline{Q}_{v}(F_{v})])$ and the proof of [Eme10], Proposition 3.1.12
shows that there is a natural isomorphism

(3.4)
$R\Gamma(\overline{N}_{p}^{\tilde{c}_{1}},\pi(K^{p},\mathcal{O}/\varpi^{m}))^{\overline{Q}_{p}\textnormal{-ord}}\cong
R\Gamma(\overline{N}_{p}^{\tilde{c}_{2}},\pi(K^{p},\mathcal{O}/\varpi^{m}))^{\overline{Q}_{p}\textnormal{-ord}}$

in $D^{+}_{\textnormal{sm}}(\mathcal{O}/\varpi^{m}[M_{p}])$, showing the
claim. We note that the same argument shows that instead of
$\overline{N}^{\tilde{c}}_{p}$, we could have taken any compact open
$\overline{N}_{p}^{\circ}\subset\overline{N}_{p}^{c_{p}}$ that is preserved
under conjugation by $M^{0}_{p}$.

As a consequence of Remark 3.18, we obtain an independence of level property
for dual coefficients.

######  Proposition 3.19 (Independence of level).

For integers $m\geq 1$, and $\tilde{c}\geq c_{p}$, the natural
$\mathbf{T}^{T}$-equivariant morphism

$R\Gamma(X_{K(0,c_{p})},\mathcal{V}_{(\lambda,\underline{\tau})}^{Q_{p},\vee}/\varpi^{m})^{\overline{Q}_{p}\textnormal{-ord}}\to
R\Gamma(X_{K(0,\tilde{c})},\mathcal{V}_{(\lambda,\underline{\tau})}^{Q_{p},\vee}/\varpi^{m})^{\overline{Q}_{p}\textnormal{-ord}}$

is an isomorphism in $D^{+}(\mathcal{O}/\varpi^{m})$.

Due to the independence of weight property, and Corollary 3.10, we can
introduce the relevant Hecke action at $p$.

######  Corollary 3.20.

We have a natural isomorphism

$R\Gamma(X_{K(0,\widetilde{c})},\mathcal{V}_{(\lambda,\underline{\tau})}^{Q_{p},\vee}/\varpi^{m})^{\overline{Q}_{p}\textnormal{-ord}}\cong$
$\cong
R{\rm{Hom}}_{\mathcal{O}/\varpi^{m}[M^{0}_{p}]}(\sigma(\lambda,\underline{\tau})^{\circ,\vee}/\varpi^{m},\pi^{\overline{Q}_{p}\textnormal{-ord}}(K^{p},\mathcal{O}/\varpi^{m}))$

in $D^{+}(\mathcal{O}/\varpi^{m})$. In particular, we have a natural algebra
homomorphism

$\mathcal{H}(\sigma(\lambda,\underline{\tau})^{\circ})\otimes_{\mathcal{O}}\mathcal{O}/\varpi^{m}\to\textnormal{End}_{D^{+}(\mathcal{O}/\varpi^{m})}(R\Gamma(X_{K(0,\widetilde{c})},\mathcal{V}_{(\lambda,\underline{\tau})}^{Q_{p},\vee}/\varpi^{m})^{\overline{Q}_{p}\textnormal{-ord}}).$

###### Proof.

The proof is identical to that of Corollary 3.16. Namely, by applying
Corollary 3.10, and the obvious analogue of Lemma 3.11, we get an isomorphism

$R\Gamma(X_{K(0,\widetilde{c})},\mathcal{V}_{(\lambda,\underline{\tau})}^{Q_{p},\vee}/\varpi^{m})^{\overline{Q}_{p}\textnormal{-ord}}\cong
R\Gamma(M^{0}_{p},\pi^{\overline{Q}_{p}\textnormal{-ord}}(K^{p},\mathcal{V}_{(\lambda,\underline{\tau})}^{Q_{p},\vee}/\varpi^{m})).$

Then the isomorphism we seek is induced by the dual of the surjection

(3.5)
$\mathcal{V}_{(\lambda,\underline{\tau})}^{Q_{p}}/\varpi^{m}\to\widetilde{\sigma(\lambda,\underline{\tau})^{\circ,\vee}/\varpi^{m}}.$

∎

We mention that the introduced Hecke action once again has a slightly
different description just as explained at the end of §3.2.

Finally, we deduce Hecke-equivariance of Poincaré duality for $Q_{p}$-ordinary
cohomology with
$\mathcal{V}_{(\lambda,\underline{\tau})}^{Q_{p}}/\varpi^{m}$-coefficients.

######  Proposition 3.21.

The Verdier duality isomorphism

$R{\rm{Hom}}_{\mathcal{O}/\varpi^{m}}(R\Gamma_{c}(X_{K(0,\widetilde{c})},\mathcal{V}_{(\lambda,\underline{\tau})}^{Q_{p},\vee}/\varpi^{m}),\mathcal{O}/\varpi^{m})\cong
R\Gamma(X_{K(0,\widetilde{c})},\mathcal{V}_{(\lambda,\underline{\tau})}^{Q_{p}}/\varpi^{m})[\dim_{\mathbf{R}}X_{K}]$

induces an isomorphism

$R{\rm{Hom}}_{\mathcal{O}/\varpi^{m}}(R\Gamma_{c}(X_{K(0,\widetilde{c})},\mathcal{V}_{(\lambda,\underline{\tau})}^{Q_{p},\vee}/\varpi^{m})^{\overline{Q}_{p}\textnormal{-ord}},\mathcal{O}/\varpi^{m})\cong$
$\cong
R\Gamma(X_{K(0,\widetilde{c})},\mathcal{V}_{(\lambda,\underline{\tau})}^{Q_{p}}/\varpi^{m})^{Q_{p}\textnormal{-ord}}[\dim_{\mathbf{R}}X_{K}]$

in $D^{+}(\mathcal{O}/\varpi^{m})$. Moreover, the latter isomorphism is
equivariant with respect to the natural left action of
$\mathcal{H}(\sigma(\lambda,\underline{\tau})^{\circ,\vee})\otimes_{\mathcal{O}}\mathcal{O}/\varpi^{m}$
on the RHS and the one induced by the anti-isomorphism

$\mathcal{H}(\sigma(\lambda,\underline{\tau})^{\circ,\vee}/\varpi^{m})\xrightarrow{\sim}\mathcal{H}(\sigma(\lambda,\underline{\tau})^{\circ}/\varpi^{m}),$
$[g,\psi]\mapsto[g^{-1},\psi^{t}]$

on the LHS.

###### Proof.

The first part follows from applying Corollary 2.8 with
$\sigma=\mathcal{V}_{(\lambda,\underline{\tau})}^{Q_{p}}/\varpi^{m}$ and
noting that
$U_{p}^{Q_{p}}=[u_{p}^{Q_{p}},u_{p}^{Q_{p}}\cdot(-)]\in\mathcal{H}(\mathcal{V}_{(\lambda,\underline{\tau})}^{Q_{p}}/\varpi^{m})$.

To see the second part, we reduce the question to the case when $\tilde{c}\geq
m$ using independence of level (cf. Corollary 3.13, Proposition 3.19). In
particular,
$\widetilde{\sigma(\lambda,\underline{\tau})^{\circ,\vee}/\varpi^{m}}$ makes
sense as a representation of $\mathcal{Q}(0,\widetilde{c})$! Then, by
independence of weight and naturality of Verdier duality applied to the
$\mathcal{Q}(0,\widetilde{c})$-equivariant surjection

(3.6)
$\mathcal{V}_{(\lambda,\underline{\tau})}^{Q_{p}}/\varpi^{m}\to\widetilde{\sigma(\lambda,\underline{\tau})^{\circ,\vee}/\varpi^{m}},$

the mentioned Verdier duality isomorphism for ordinary parts is also induced
by the Verdier duality isomorphism

(3.7)
$R{\rm{Hom}}_{\mathcal{O}/\varpi^{m}}(R\Gamma_{c}(X_{K(0,\widetilde{c})},\widetilde{\sigma(\lambda,\underline{\tau})^{\circ}/\varpi^{m}}),\mathcal{O}/\varpi^{m})\cong
R\Gamma(X_{K(0,\widetilde{c})},\widetilde{\sigma(\lambda,\underline{\tau})^{\circ,\vee}/\varpi^{m}}).$

However, the Verdier duality isomorphism 3.7 satisfies the desired Hecke-
equivariance by Corollary 2.8 and Corollary 3.10.222222Note that at this step
we used that our Hecke action at $p$ matches up with the Hecke action
constructed at the end of §3.2. ∎

###  3.4. $\widetilde{Q}$-ordinary Hida theory for $\widetilde{G}$

As everything mentioned in the previous two subsections applies verbatim for
the group $\widetilde{G}$ at split $p$-adic places of $F^{+}$, we will only
set up the notations and explain the relevant results. We revisit the setup of
the corresponding part of §2.7. Fix a subset of $p$-adic places
$\overline{S}\subset\overline{S}_{p}$ and a lift $v\mid\bar{v}$ for each
$\bar{v}\in\overline{S}_{p}$ in $S_{p}$. Consider a tuple
$(Q_{p},\lambda,\underline{\tau})$ as previously and consider a corresponding
tuple
$(\widetilde{Q}_{\overline{S}},\tilde{\lambda}_{\overline{S}},\underline{\tau}):=(\widetilde{Q}_{\bar{v}},\tilde{\lambda}_{\bar{v}},\underline{\tau_{\bar{v}}})_{\bar{v}\in\overline{S}}$
as in §2.7. Further set $\tilde{\lambda}$ to be some extension of
$\tilde{\lambda}$ to a
$(\textnormal{Res}_{F^{+}/\mathbf{Q}}\widetilde{B})_{E}$-dominant weight of
$(\textnormal{Res}_{F^{+}/\mathbf{Q}}\widetilde{G})_{E}$. We then similarly
form the parahoric level subgroups

$\widetilde{\mathcal{Q}}_{\overline{S}}(b,c)=\prod_{\bar{v}\in\overline{S}}\widetilde{\mathcal{Q}}_{\bar{v}}(b,c)\subset\prod_{\bar{v}\in\overline{S}}\widetilde{G}(\mathcal{O}_{F^{+}_{\bar{v}}})$

for integers $0\leq b\leq c$ with $c\geq c_{p}$. For the rest of the
subsection, we fix a prime-to-$\overline{S}$ good level subgroup
$\widetilde{K}^{\overline{S}}\subset\widetilde{G}(\mathbf{A}_{F^{+}}^{\overline{S}\cup\\{\infty\\}})$
and set $\widetilde{K}(b,c)$ to be the good subgroup
$\widetilde{K}^{\overline{S}}\widetilde{\mathcal{Q}}_{\overline{S}}(b,c)\subset\widetilde{G}(\mathbf{A}_{F^{+}}^{\infty})$.
We then freely borrow the notation of §2.8. In particular, we have an open
submonoid
$\widetilde{\Delta}_{\widetilde{\mathcal{Q}}^{w_{0}}_{\overline{S}}}(c_{p})\subset\widetilde{G}_{\overline{S}}$,
and can set
$\widetilde{\Delta}_{\widetilde{M}_{\overline{S}}^{w_{0}}}^{+}:=\widetilde{\Delta}_{\widetilde{\mathcal{Q}}^{w_{0}}_{\overline{S}}}(c_{p})\cap\widetilde{M}^{w_{0}}_{\overline{S}}\subset\widetilde{M}_{\overline{S}}^{w_{0},+},$
and
$\widetilde{\Delta}_{\widetilde{M}_{\overline{S}}^{w_{0}}}:=\widetilde{\Delta}_{\widetilde{M}_{\overline{S}}^{w_{0}}}^{+}[u_{\bar{v}}^{\widetilde{Q}^{w_{0}}_{v},\pm
1}\mid\bar{v}\in\overline{S}]$.

For $c\geq b\geq 0$ with $c\geq c_{p}$, the Hecke algebra
$\mathcal{H}(\widetilde{\Delta}_{\widetilde{\mathcal{Q}}_{\overline{S}}^{w_{0}}}(c),\widetilde{\mathcal{Q}}_{\overline{S}^{w_{0}}}(b,c))$
acts on
$R\Gamma(\widetilde{X}_{\widetilde{K}(b,c)},\mathcal{V}_{(\tilde{\lambda},\underline{\tau})}^{\widetilde{Q}_{\overline{S}}^{w_{0}}})$
via endomorphisms in $D^{+}(\mathcal{O})$. In particular, we introduce the
$\widetilde{Q}^{w_{0}}_{\overline{S}}$-ordinary parts of the complex
$R\Gamma(\widetilde{X}_{\widetilde{K}(b,c)},\mathcal{V}_{(\tilde{\lambda},\underline{\tau})}^{\widetilde{Q}_{\overline{S}}^{w_{0}}})^{\widetilde{Q}_{\overline{S}}^{w_{0}}\textnormal{-ord}}$
as the maximal direct summand on which each
$U_{\bar{v}}^{\widetilde{Q}_{\bar{v}}^{w_{0}}}$ acts invertibly.

On the other hand, the formalism of §3.1 can be applied with the choices
$Q=\widetilde{Q}_{\overline{S}}^{w_{0}}$,
$N^{0}=\widetilde{N}_{\overline{S}}^{w_{0},0}$,
$M^{b}=\widetilde{M}^{w_{0},b}_{\overline{S}}$,
$\overline{N}^{c}=\overline{\widetilde{N}}_{\overline{S}}^{w_{0},c}$, and
$\sigma$ being the trivial
$\widetilde{\Delta}_{\widetilde{M}_{\overline{S}}^{w_{0}}}^{+}$-module. One
can then compare the two constructions and see that the analogue of Lemma 3.11
holds.

We can further introduce $\widetilde{Q}_{\overline{S}}^{w_{0}}$-ordinary
completed cohomology

$\pi^{\widetilde{Q}_{\overline{S}}^{w_{0}}\textnormal{-ord}}(\widetilde{K}^{\overline{S}},\mathcal{V}_{(\tilde{\lambda},\underline{\tau})}^{\widetilde{Q}_{\overline{S}}^{w_{0}}}/\varpi^{m}):=$
$:=R\Gamma(\widetilde{N}^{w_{0},0}_{\overline{S}},\pi(\widetilde{K}^{\overline{S}},\mathcal{V}_{(\tilde{\lambda},\underline{\tau,N})}^{\widetilde{Q}_{\overline{S}}^{w_{0}}}/\varpi^{m}))^{\widetilde{Q}_{\overline{S}}^{w_{0}}\textnormal{-ord}}\in
D^{+}_{\textnormal{sm}}(\mathcal{O}/\varpi^{m}[\widetilde{\Delta}_{\widetilde{M}_{\overline{S}}^{w_{0}}}]).$

The analogues of the independence of level and weight property hold once
again. To emphasise the normalisations and introduce the necessary notations
for later, we spell out the statement of the latter.

######  Proposition 3.22 (Independence of weight).

For any integer $m\geq 1$, there is a natural isomorphism

$\pi^{\widetilde{Q}_{\overline{S}}^{w_{0}}\textnormal{-ord}}(\widetilde{K}^{\overline{S}},\mathcal{V}_{\tilde{\lambda}}/\varpi^{m})\xrightarrow{\sim}\pi^{\widetilde{Q}_{\overline{S}}^{w_{0}}\textnormal{-ord}}(\widetilde{K}^{\overline{S}},\mathcal{V}_{\tilde{\lambda}^{\overline{S}}})\otimes_{\mathcal{O}/\varpi^{m}}\mathcal{V}_{w_{0}^{\widetilde{Q}_{\overline{S}}^{w_{0}}}\widetilde{\lambda}_{\overline{S}}}/\varpi^{m}$

in
$D^{+}_{\textnormal{sm}}(\mathcal{O}/\varpi^{m}[\widetilde{\Delta}_{\widetilde{M}_{\overline{S}}^{w_{0}}}])$.

Here we denoted by $\widetilde{\lambda}^{\overline{S}}$ the dominant weight
for $\widetilde{G}$ that is trivial at $\overline{S}$ and coincides with
$\widetilde{\lambda}$ outside $\overline{S}$. Similarly,
$\widetilde{\lambda}_{\overline{S}}$ is the dominant weight for
$\widetilde{G}$ that is trivial outside $\overline{S}$ and coincides with
$\widetilde{\lambda}$ at $\overline{S}$. Also, analogously to the previous
section, we view
$\mathcal{V}_{w_{0}^{\widetilde{Q}_{\overline{S}}^{w_{0}}}\widetilde{\lambda}_{\overline{S}}}/\varpi^{m}$
as an
$\mathcal{O}/\varpi^{m}[\widetilde{\Delta}_{\widetilde{M}_{\overline{S}}^{w_{0}}}]$-module
via inflation from $\widetilde{M}^{w_{0},0}_{\overline{S}}$.

We can then also deduce the analogues of Corollary 3.14 and Corollary 3.16. In
particular,
$R\Gamma(\widetilde{X}_{\widetilde{K}(0,c)},\mathcal{V}_{(\tilde{\lambda},\underline{\tau})}^{\widetilde{Q}_{\overline{S}}^{w_{0}}}/\varpi^{m})^{\widetilde{Q}_{\overline{S}}^{w_{0}}\textnormal{-ord}}$
admits a natural Hecke action at $p$. Namely, set

$\tilde{\sigma}(\tilde{\lambda}_{\overline{S}},\underline{\tau}_{\overline{S}})^{\circ}:=\mathcal{V}_{w_{0}^{\widetilde{Q}_{\overline{S}}^{w_{0}}}\widetilde{\lambda}_{\overline{S}}}^{\vee}\otimes_{\mathcal{O}}\tilde{\sigma}(\underline{\tau}_{\overline{S}})^{\circ}\in\textnormal{Mod}(\mathcal{O}[\widetilde{M}^{w_{0},0}_{\overline{S}}]).$

Then there is a natural algebra homomorphism

$\mathcal{H}(\tilde{\sigma}(\tilde{\lambda}_{\overline{S}},\underline{\tau}_{\overline{S}})^{\circ,\vee})\otimes_{\mathcal{O}}\mathcal{O}/\varpi^{m}\to\textnormal{End}_{D^{+}(\mathcal{O}/\varpi^{m})}(R\Gamma(\widetilde{X}_{\widetilde{K}(0,c)},\mathcal{V}_{(\tilde{\lambda},\underline{\tau})}^{\widetilde{Q}_{\overline{S}}^{w_{0}}}/\varpi^{m})^{\widetilde{Q}_{\overline{S}}^{w_{0}}\textnormal{-ord}}).$

######  Remark 3.23.

We note that if we set $\widetilde{S}:=\\{v\mid\bar{v}\in\overline{S}\\}$, we
have
$\mathcal{V}_{w_{0}^{\widetilde{Q}^{w_{0}}_{\overline{S}}}\tilde{\lambda}_{\overline{S}}}=\mathcal{V}_{w_{0}^{Q_{\widetilde{S}}}\lambda_{\widetilde{S}}}\otimes_{\mathcal{O}}\mathcal{V}_{-w_{0}^{Q_{\widetilde{S}^{c}}}\lambda_{\widetilde{S}^{c}}},$
and, by the proof of [CN23], Lemma 2.2.14, we get

$\tilde{\sigma}(\tilde{\lambda}_{\overline{S}},\underline{\tau}_{\overline{S}})^{\circ}=\sigma(\lambda_{\widetilde{S}},\underline{\tau}_{\widetilde{S}})^{\circ}\otimes_{\mathcal{O}}(\theta_{n}^{-1})^{\ast}\sigma(\lambda_{\widetilde{S}^{c}},\underline{\tau}_{\widetilde{S}^{c}})^{\circ}\in$
$\in\textnormal{Mod}(\mathcal{O}[\left(\prod_{v\in\widetilde{S}}M_{v}(\mathcal{O}_{F_{v}})\right)\times\left(\prod_{v^{c}\in\widetilde{S}^{c}}(\theta_{n}M_{v^{c}})(\mathcal{O}_{F_{v}})\right)].$

Finally, one can develop $\overline{\widetilde{Q}}_{\overline{S}}$-ordinary
Hida theory for
$\mathcal{V}_{(\tilde{\lambda},\underline{\tau})}^{\widetilde{Q}_{\overline{S}},w_{0}^{P},\vee}/\varpi^{m}$
(see the end of §2.7 for the notation) just as in §3.3. In particular, one
arrives to Hecke-equivariance of Poincaré duality at $p$ for ordinary
cohomology of $\widetilde{X}_{\widetilde{K}}$.

###  3.5. Ordinary parts of the Bruhat stratification

Here we again closely follow [CN23] on computing the ordinary parts of certain
Bruhat strata of parabolic inductions in the derived category. As we often
would not need serious changes in the proofs, we sometimes only state the
results we need and indicate how to deduce them from the arguments of [All+23]
and [CN23].

As in [CN23], 2.3.1, we restrict ourselves to a completely local setup. Let
$L/\mathbf{Q}_{p}$ be a finite field extension with ring of integers
$\mathcal{O}_{L}$ and a choice of uniformiser $\varpi_{L}$. Let $G/L$ be a
split connected reductive group with a split maximal torus $T\subset G$ and
Weyl group $W=W(G,T)$. Fix a Borel subgroup $T\subset B$ and two standard
parabolic subgroups $B\subset Q_{1},Q_{2}\subset G$ with Levi decomposition
$Q_{1}=M_{1}\ltimes N_{1}$ and $Q_{2}=M_{2}\ltimes N_{2}$, respectively. We
denote by $W_{Q_{i}}$ the Weyl group of $M_{i}$ and by
${}^{Q_{1}}W^{Q_{2}}\subset W$ the set of minimal length representatives of
$W_{Q_{1}}\backslash W/W_{Q_{2}}$. For $w\in W$ we denote its length by
$\ell(w)\in\mathbf{Z}_{\geq 0}$. Recall that $G(L)$ admits a stratification
(with respect to its $p$-adic topology) called the Bruhat
stratification232323For a reference in this generality, see [Hau18], Lemma
2.1.2. However, note that loc. cit. considers the opposite parabolic
$\overline{Q}_{1}$ on the left instead, so the closure relations are reversed
there.

$G(L)=\coprod_{w\in\prescript{Q_{1}}{}{W}^{Q_{2}}}Q_{1}(L)wQ_{2}(L)=\coprod_{w\in\prescript{Q_{1}}{}{W}^{Q_{2}}}S_{w}$

with closure relations given by the Bruhat order

$\overline{S_{w}}=\coprod_{w\geq
w^{\prime}\in\prescript{Q_{1}}{}{W}^{Q_{2}}}S_{w^{\prime}}.$

In particular, for $i\in\mathbf{Z}_{\geq 0}$, the subset

$G_{\geq i}:=\coprod_{\ell(w)\geq i}S_{w}\subset G(L)$

is open.

We first recall how the Bruhat stratification provides a "stratification" of
the exact functor

$\textnormal{Ind}_{Q_{1}(L)}^{G(L)}:D^{+}_{\textnormal{sm}}(\mathcal{O}/\varpi^{m}[Q_{1}(L)])\to
D^{+}_{\textnormal{sm}}(\mathcal{O}/\varpi^{m}[G(L)])$

in a suitable sense. We define the functor

$I_{\geq
i}:\textnormal{Mod}_{\textnormal{sm}}(\mathcal{O}/\varpi^{m}[Q_{1}(L)]))\to\textnormal{Mod}_{\textnormal{sm}}(\mathcal{O}/\varpi^{m}[Q_{2}(L)])$

which sends $\pi$ to the $Q_{2}(L)$-stable subspace of functions in
$\textnormal{Ind}_{Q_{1}(L)}^{G(L)}\pi$ which are supported at $G_{\geq i}$.
For $w\in\prescript{Q_{1}}{}{W}^{Q_{2}}$, we can further define the functor

$I_{w}:\textnormal{Mod}_{\textnormal{sm}}(\mathcal{O}/\varpi^{m}[Q_{1}(L)]))\to\textnormal{Mod}_{\textnormal{sm}}(\mathcal{O}/\varpi^{m}[Q_{2}(L)]))$

which sends $\pi$ to the set of compactly supported functions $f:S_{w}\to\pi$
which are left invariant with respect to elements of $Q_{1}(L)$ equipped with
the left $Q_{2}(L)$-action given by right multiplication on the source.
Finally, for $w\in\prescript{Q_{1}}{}{W}^{Q_{2}}$, consider also the open
subspace $S_{w}^{\circ}:=Q_{1}(L)wM_{2}(L)N_{2}(\mathcal{O}_{L})\subset
S_{w}$. We then define the functor
$I_{w}^{\circ}:\textnormal{Mod}_{\textnormal{sm}}(\mathcal{O}/\varpi^{m}[Q_{1}(L)])\to\textnormal{Mod}_{\textnormal{sm}}(\mathcal{O}/\varpi^{m}[M_{2}(L)^{+}\ltimes
N_{2}(\mathcal{O}_{L})])$ by setting $I_{w}^{\circ}(\pi)\subset I_{w}(\pi)$ to
be the subset of functions supported on $S^{\circ}_{w}$. Each of these
functors are exact as the argument of [All+23], Proposition 5.3.1 shows.

For every $i\in\mathbf{Z}_{\geq 0}$ and $\pi\in
D^{+}_{\textnormal{sm}}(\mathcal{O}/\varpi^{m}[Q_{1}(L)])$, the natural
inclusion of functors $I_{\geq i+1}\subset I_{\geq i}$ induces a distinguished
triangle

(3.8) $I_{\geq i+1}(\pi)\to I_{\geq i}(\pi)\to\oplus_{\ell(w)=i}I_{w}(\pi)\to
I_{\geq i+1}(\pi)[1]$

in $D^{+}_{\textnormal{sm}}(\mathcal{O}/\varpi^{m}[Q_{2}(L)])$. For the proof
of this, see [Hau18], Lemma 2.2.1.

Given a
$\sigma\in\textnormal{Mod}_{\textnormal{sm}}(\mathcal{O}/\varpi^{m}[M_{2}(\mathcal{O}_{L}])$,
finite free over $\mathcal{O}/\varpi^{m}$, we consider the functor

$R{\rm{Hom}}_{\mathcal{O}/\varpi^{m}[M_{2}(\mathcal{O}_{L})]}(\sigma,R\Gamma(N_{2}(\mathcal{O}_{L}),-)^{Q_{2}\textnormal{-ord}}):D^{+}_{\textnormal{sm}}(\mathcal{O}/\varpi^{m}[Q_{2}(L)])\to
D^{+}(\mathcal{H}(\sigma^{\vee})).$

We can apply this functor to 3.8 to get a distinguished triangle

$R{\rm{Hom}}_{\mathcal{O}/\varpi^{m}[M_{2}(\mathcal{O}_{L})]}(\sigma,R\Gamma(N_{2}(\mathcal{O}_{L}),I_{\geq
i+1}(\pi))^{Q_{2}\textnormal{-ord}})\to$ $\to
R{\rm{Hom}}_{\mathcal{O}/\varpi^{m}[M_{2}(\mathcal{O}_{L})]}(\sigma,R\Gamma(N_{2}(\mathcal{O}_{L}),I_{\geq
i}(\pi))^{Q_{2}\textnormal{-ord}})\to$
$\to\oplus_{\ell(w)=i}R{\rm{Hom}}_{\mathcal{O}/\varpi^{m}[M_{2}(\mathcal{O}_{L})]}(\sigma,R\Gamma(N_{2}(\mathcal{O}_{L}),I_{w}(\pi))^{Q_{2}\textnormal{-ord}})\to$
$\to
R{\rm{Hom}}_{\mathcal{O}/\varpi^{m}[M_{2}(\mathcal{O}_{L})]}(\sigma,R\Gamma(N_{2}(\mathcal{O}_{L}),I_{\geq
i+1}(\pi))^{Q_{2}\textnormal{-ord}})[1].$

The argument of [CN23], Proposition 2.3.4 shows that taking long exact
sequence of cohomology of the distinguished triangle above gives a
$\mathcal{H}(\sigma^{\vee})$-equivariant short exact sequence

$0\to
R^{j}{\rm{Hom}}_{\mathcal{O}/\varpi^{m}[M_{2}(\mathcal{O}_{L})]}(\sigma,R\Gamma(N_{2}(\mathcal{O}_{L}),I_{\geq
i+1}(\pi))^{Q_{2}\textnormal{-ord}})\to$ $\to
R^{j}{\rm{Hom}}_{\mathcal{O}/\varpi^{m}[M_{2}(\mathcal{O}_{L})]}(\sigma,R\Gamma(N_{2}(\mathcal{O}_{L}),I_{\geq
i}(\pi))^{Q_{2}\textnormal{-ord}})\to$
$\to\oplus_{\ell(w)=i}R^{j}{\rm{Hom}}_{\mathcal{O}/\varpi^{m}[M_{2}(\mathcal{O}_{L})]}(\sigma,R\Gamma(N_{2}(\mathcal{O}_{L}),I_{w}(\pi))^{Q_{2}\textnormal{-ord}})\to
0$

for every $j\in\mathbf{Z}_{\geq 0}$. To be more precise, the proof of
Proposition 2.3.4 in loc. cit. relies on their Lemma 2.3.5. We state the
obvious generalisation we need. However, we omit the proof as it can be proved
just as the version in loc. cit.

######  Lemma 3.24.

For any $i\in\mathbf{Z}_{\geq 0}$, there are decompositions

$G_{\geq i}=U_{1}^{m}\coprod U_{2}^{m}$

into open and closed subsets, indexed by $m\in\mathbf{Z}_{\geq 1}$, that are
$Q_{1}(L)$-invariant on the left and $Q_{2}(\mathcal{O}_{L})$-invariant on the
right such that

$G_{\geq i+1}=\bigcup_{m\geq 1}U_{1}^{m}.$

To apply this line of argument, we compute

$R\Gamma(N_{2}(\mathcal{O}_{L}),I_{w}(\pi))^{Q_{2}\textnormal{-ord}}$

for a class of $w\in{}^{Q_{1}}W^{Q_{2}}$. First we note the following lemma
that is essentially [All+23], Lemma 5.3.4 (see also [CN23], Lemma 2.3.6).

######  Lemma 3.25.

Let $w\in{}^{Q_{1}}W^{Q_{2}}$ such that $wM_{2}(L)w^{-1}\subset M_{1}(L)$.
Then:

  1. (i)

$I_{w}^{\circ}$ takes injectives to
$\Gamma(N_{2}(\mathcal{O}_{L}),-)$-acyclics.

  2. (ii)

Let $\pi\in D^{+}_{\textnormal{sm}}(\mathcal{O}/\varpi^{m})[Q_{1}(L)])$. Then
there is a natural isomorphism

$R\Gamma(N_{2}(\mathcal{O}_{L}),I_{w}^{\circ}(\pi))^{Q_{2}\textnormal{-ord}}\xrightarrow[]{\sim}R\Gamma(N_{2}(\mathcal{O}_{L}),I_{w}(\pi))^{Q_{2}\textnormal{-ord}}.$

###### Proof.

Note that the assumption on $w$ implies that

$S_{w}^{\circ}=Q_{1}(L)wN_{2}(\mathcal{O}_{L}).$

Knowing this, the proof of [All+23], Lemma 5.3.4 applies verbatim. ∎

If $w\in{}^{Q_{1}}W^{Q_{2}}$ such that $wM_{2}(L)w^{-1}\subset M_{1}(L)$, we
define

$N_{2,w}^{\circ}:=Q_{1}(L)\cap wN_{2}(\mathcal{O}_{L})w^{-1},$

a compact subgroup of $Q_{1}(L)$. We define the functor

$\Gamma(N_{2,w}^{\circ},-):\textnormal{Mod}_{\textnormal{sm}}(\mathcal{O}/\varpi^{m}[Q_{1}(L)])\to\textnormal{Mod}_{\textnormal{sm}}(\mathcal{O}/\varpi^{m}[M_{2}(L)^{+}])$

where an element $m\in M_{2}(L)^{+}$ acts on a $v\in\pi^{N_{2,w}^{\circ}}$ by
the formula

$m\cdot v:=\sum_{n\in
N_{2,w}^{\circ}/wmw^{-1}N_{2,w}^{\circ}wm^{-1}w^{-1}}nwmw^{-1}\cdot v$

where on the right we consider the natural action of $wmw^{-1}\in Q_{1}(L)$.
One checks easily that our assumption on $w$ ensures that this formula makes
sense meaning that we have $wmw^{-1}N_{2,w}^{\circ}wm^{-1}w^{-1}\subset
N_{2,w}^{\circ}$.

######  Lemma 3.26.

Let $w\in{}^{Q_{1}}W^{Q_{2}}$ such that $wM_{2}(L)w^{-1}\subset M_{1}(L)$ and
consider $\pi\in D^{+}_{\textnormal{sm}}(\mathcal{O}/\varpi^{m}[Q_{1}(L)])$.
Then we have a natural isomorphism

$R\Gamma(N_{2}(\mathcal{O}_{L}),I_{w}^{\circ}(\pi))\cong
R\Gamma(N_{2,w}^{\circ},\pi)$

in $D^{+}_{\textnormal{sm}}(\mathcal{O}/\varpi^{m}[M_{2}(L)^{+}])$.

###### Proof.

Again, it can be proved by running the proof of the analogous statement
[All+23], Lemma 5.3.5, keeping in mind our assumption on $w$. For the reader’s
convenience, we recall the argument here.

By the first part of Lemma 3.25, it suffices to give a natural isomorphism of
underived functors

$\Gamma(N_{2}(\mathcal{O}_{L}),I_{w}^{\circ}(-))\cong\Gamma(N_{2,w}^{\circ},-).$

For $\pi\in\textnormal{Mod}_{\textnormal{sm}}(Q_{1}(L))$, the map will send an
$N_{2}(\mathcal{O}_{L})$-invariant function

$f:Q_{1}(L)wN_{2}(\mathcal{O}_{L})\to\pi$

to $f(w)\in\pi^{N_{2,w}^{\circ}}$. This visibly gives an isomorphism of
underlying $\mathcal{O}/\varpi^{m}$-modules with inverse sending
$v\in\pi^{N_{2,w}^{\circ}}$ to $f_{v}:pwn\mapsto p\cdot v$.

We are left with checking that this defines an $M_{2}(L)^{+}$-equivariant map.
In other words, for $f\in\Gamma(N_{2}(\mathcal{O}_{L}),I_{w}^{\circ}(\pi))$
and $m\in M_{2}(L)^{+}$, we need to see that

(3.9) $\sum_{n\in
N_{2}(\mathcal{O}_{L})/mN_{2}(\mathcal{O}_{L})m^{-1}}f(wnm)=\sum_{\tilde{n}\in
N_{2,w}^{\circ}/wmw^{-1}N_{2,w}^{\circ}(wmw^{-1})^{-1}}\tilde{n}wmw^{-1}f(w).$

Note that the association

(3.10) $N_{2,w}^{\circ}/wmw^{-1}N_{2,w}^{\circ}(wmw^{-1})^{-1}\to
N_{2}(\mathcal{O}_{L})/mN_{2}(\mathcal{O}_{L})m^{-1},$ $\tilde{n}\mapsto
w^{-1}\tilde{n}w$

is injective. Moreover, for $n=w^{-1}\tilde{n}w$ lying in the image, we have
$f(wnm)=\tilde{n}wmw^{-1}f(w)$. In particular, to see that 3.9 holds, it
suffices to prove that $f(wnm)\neq 0$ only if $n$ lies in the image of 3.10.
So assume that $wnm\in S^{\circ}_{w}=Q_{1}(L)wN_{2}(\mathcal{O}_{L})$.
Accordingly, we write it in the form $wnm=qwn^{\prime}$ with $q\in Q_{1}(L)$
and $n^{\prime}\in N_{2}(\mathcal{O}_{L})$. On the other hand,

$n=w^{-1}qwn^{\prime}m^{-1}=(w^{-1}qwm^{-1})(mn^{\prime}m^{-1})$

and our assumptions imply that $mn^{\prime}m^{-1}\in N_{2}(\mathcal{O}_{L})$.
Therefore, we get

$w^{-1}qwm^{-1}=w^{-1}(qwm^{-1}w^{-1})w\in
w^{-1}N_{2,w}^{\circ}w=w^{-1}Q_{1}(L)w\cap N_{2}(\mathcal{O}_{L}).$

∎

######  Corollary 3.27.

Let $w\in{}^{Q_{1}}W^{Q_{2}}$ such that $wM_{2}(L)w^{-1}\subset M_{1}(L)$ and
$\pi\in D^{+}_{\textnormal{sm}}(\mathcal{O}/\varpi^{m}[Q_{1}(L)])$. There is a
natural isomorphism

$R\Gamma(N_{2}(\mathcal{O}_{L}),I_{w}(\pi))^{Q_{2}\textnormal{-ord}}\cong
R\Gamma(N_{2,w}^{\circ},\pi)^{Q_{2}\textnormal{-ord}}$

in $D^{+}_{\textnormal{sm}}(\mathcal{O}/\varpi^{m}[M_{2}(L)])$.

Finally, we compute $R\Gamma(N_{2,w}^{\circ},\pi)^{Q_{2}\textnormal{-ord}}$ as
in [All+23], Lemma 5.3.7. For the rest of the subsection, we assume that
$G=\textnormal{GL}_{n}$ with $T=T_{n}$, the torus consisting of diagonal
matrices and $B=B_{n}$ the Borel consisting of upper triangular matrices. We
also introduce some further notation. Given $w\in{}^{Q_{1}}W^{Q_{2}}$ such
that $wM_{2}(L)w^{-1}\subset M_{1}(L)$, set $Q_{w}:=wQ_{2}w^{-1}\cap
M_{1}\subset M_{1}$ with Levi quotient $M_{w}=wM_{2}w^{-1}$ and unipotent
radical $N_{w}=wN_{2}w^{-1}\cap M_{1}$. Consider the character

$\chi_{w}:M_{2}(L)\to\mathcal{O}^{\times},$
$m\mapsto\frac{\textnormal{Norm}_{L/\mathbf{Q}_{p}}\det_{L}(\textnormal{Ad}(m^{w})\mid_{\textnormal{Lie}N_{w}(L)})^{-1}}{|\textnormal{Norm}_{L/\mathbf{Q}_{p}}\det_{L}(\textnormal{Ad}(m^{w})\mid_{\textnormal{Lie}N_{w}(L)})|_{p}}.$

Introduce the equivalence of categories

$\tau_{w}:\textnormal{Mod}_{\textnormal{sm}}(\mathcal{O}/\varpi^{m}[M_{2}(L)])\to\textnormal{Mod}_{\textnormal{sm}}(\mathcal{O}/\varpi^{m}[wM_{2}(L)w^{-1}])$

sending $\pi$ to $\tau_{w}(\pi)$ with underlying
$\mathcal{O}/\varpi^{m}$-module $\pi$ but with the twisted action
$\tau_{w}(\pi)(m)=\pi(w^{-1}mw)$. Finally, set
$N_{w}^{\circ}=N_{w}(\mathcal{O}_{L})$ and
$N_{2,w,N_{1}}^{\circ}:=N_{2,w}^{\circ}\cap N_{1}(L)$.

######  Lemma 3.28.

Assume that $G=\textnormal{GL}_{n}$ with $T=T_{n}$, $B=B_{n}$ and let
$w\in{}^{Q_{1}}W^{Q_{2}}$ such that $wM_{2}(L)w^{-1}\subset M_{1}(L)$. Then,
for any $\pi\in D^{+}_{\textnormal{sm}}(\mathcal{O}/\varpi^{m}[M_{1}(L)])$,
there is a natural isomorphism between

$R\Gamma(N_{2,w}^{\circ},\textnormal{Inf}_{M_{1}(L)}^{Q_{1}(L)}\pi)^{Q_{2}\textnormal{-ord}}$

and

$\mathcal{O}/\varpi^{m}(\chi_{w})\otimes_{\mathcal{O}/\varpi^{m}}\tau_{w}^{-1}R\Gamma(N_{w}^{\circ},\pi)^{Q_{w}\textnormal{-ord}}[-\textnormal{rk}_{\mathbf{Z}_{p}}N_{2,w,N_{1}}^{\circ}]$

in $D^{+}_{\textnormal{sm}}(\mathcal{O}/\varpi^{m}[M_{2}(L)])$.

###### Proof.

Before starting the proof, we note that the argument is just an obvious
generalisation of the proof of [All+23], Lemma 5.3.7 and so we kept the
structure of their argument and sometimes will refer to it as loc. cit.

We set the monoid $M_{2}(L)^{+}\ltimes_{w}N_{2,w}^{\circ}$ to be
$M_{2}(L)^{+}\times N_{2,w}^{\circ}$ with the action
$(m,1)(1,n)=(1,m^{w}n(m^{w})^{-1})(m,1)$. Consider the short exact sequence

$0\to N_{2,w,N_{1}}^{\circ}\to N_{2,w}^{\circ}\to N_{w}^{\circ}\to 0$

and note that it is obviously equivariant for the $M_{2}(L)^{+}$-action on
each groups via $m\mapsto wmw^{-1}$. Then just as in loc. cit., we basically
write
$R\Gamma(N_{2,w}^{\circ},\textnormal{Inf}_{M_{1}(L)}^{Q_{1}(L)}-)^{Q_{2}\textnormal{-ord}}$
as $"R\Gamma(N_{w}^{\circ},-)^{Q_{w}\textnormal{-ord}}\circ
R\Gamma(N_{2,w,N_{1}}^{\circ},\textnormal{Inf}_{M_{1}(L)}^{Q_{1}(L)}-)^{Q_{1}\textnormal{-ord}}"$.
To make this precise, we need to introduce some functors. Denote by

$\textnormal{Res}^{w}:\textnormal{Mod}_{\textnormal{sm}}(\mathcal{O}/\varpi^{m}[M_{1}(L)])\to\textnormal{Mod}_{\textnormal{sm}}(\mathcal{O}/\varpi^{m}[M_{2}(L)^{+}\ltimes_{w}N_{2,w}^{\circ}])$

the composite of $\textnormal{Inf}_{M_{1}(L)}^{Q_{1}(L)}$ with the functor
that sends $\pi\in\textnormal{Mod}_{\textnormal{sm}}(Q_{1}(L))$ to itself as
an $\mathcal{O}/\varpi^{m}$-module with the action
$\textnormal{Res}^{w}(\pi)(mn)=\pi(m^{w}n)$. Further set

$\alpha:\textnormal{Mod}_{\textnormal{sm}}(\mathcal{O}/\varpi^{m}[M_{2}(L)^{+}\ltimes_{w}N_{w}^{\circ}])\to\textnormal{Mod}_{\textnormal{sm}}(\mathcal{O}/\varpi^{m}[M_{w}(L)^{+}\ltimes
N_{w}^{\circ}])$

to be the functor defined by composing the equivalence

$\textnormal{Mod}_{\textnormal{sm}}(\mathcal{O}/\varpi^{m}[M_{2}(L)^{+}\ltimes_{w}N_{w}^{\circ}])\xrightarrow{\sim}\textnormal{Mod}_{\textnormal{sm}}[wM_{2}(L)^{+}w^{-1}\ltimes
N_{w}^{\circ}])$

with the localisation

$\textnormal{Mod}_{\textnormal{sm}}(\mathcal{O}/\varpi^{m}[wM_{2}(L)^{+}w^{-1}\ltimes
N_{w}^{\circ}])\to\textnormal{Mod}_{\textnormal{sm}}(\mathcal{O}/\varpi^{m}[M_{w}(L)^{+}\ltimes
N_{w}^{\circ}])$

induced by the inclusion $wM_{2}(L)^{+}w^{-1}\subset M_{w}(L)^{+}$. Analogous
construction defines a functor $\beta$ such that they fit into a commutative
diagram (up to natural equivalence)

${\textnormal{Mod}_{\textnormal{sm}}(\mathcal{O}/\varpi^{m}[M_{2}(L)^{+}\ltimes_{w}N_{w}^{\circ}])}$${\textnormal{Mod}_{\textnormal{sm}}(\mathcal{O}/\varpi^{m}[M_{w}(L)^{+}\ltimes
N_{w}^{\circ}])}$${\textnormal{Mod}_{\textnormal{sm}}(\mathcal{O}/\varpi^{m}[M_{2}(L)^{+}])}$${\textnormal{Mod}_{\textnormal{sm}}(\mathcal{O}/\varpi^{m}[M_{w}(L)^{+}])}$$\scriptstyle{\alpha}$$\scriptstyle{\Gamma(N_{w}^{\circ},-)}$$\scriptstyle{\Gamma(N_{w}^{\circ},-)}$$\scriptstyle{\beta}$

where the vertical arrows are defined the usual way by considering the
corresponding Hecke actions. A reasoning analogous to the proof of Lemma 3.6
shows that $\alpha$ takes injectives to $\Gamma(N_{w}^{\circ},-)$-acyclics.
Therefore, by checking things on underived functors, we have

$R\Gamma(N_{2,w}^{\circ},\textnormal{Inf}_{M_{1}(L)}^{Q_{1}(L)}\pi)^{Q_{2}\textnormal{-ord}}\cong$
$\tau_{w}^{-1}(\beta
R\Gamma(N_{w}^{\circ},R\Gamma(N_{2,w,N_{1}}^{\circ},\textnormal{Res}^{w}(\pi)))^{Q_{w}\textnormal{-ord}}\cong$
$\tau_{w}^{-1}(R\Gamma(N_{w}^{\circ},\alpha
R\Gamma(N_{2,w,N_{1}}^{\circ},\textnormal{Res}^{w}\pi)))^{Q_{w}\textnormal{-ord}}.$

Since $N_{2,w,N_{1}}^{\circ}$ acts trivially on $\textnormal{Res}^{w}\pi$, we
get

$R\Gamma(N_{2,w,N_{1}}^{\circ},\textnormal{Res}^{w}\pi)\cong\textnormal{Res}^{w}\pi\otimes_{\mathcal{O}/\varpi^{m}}R\Gamma(N_{2,w,N_{1}}^{\circ},\mathcal{O}/\varpi^{m})$

in
$D^{+}_{\textnormal{sm}}(\mathcal{O}/\varpi^{m}[M_{2}(L)^{+}\ltimes_{w}N_{w}^{\circ}])$.
In particular, to compute $\alpha
R\Gamma(N_{2,w,N_{1}}^{\circ},\textnormal{Res}^{w}\pi)$, it suffices to
compute $\alpha R\Gamma(N_{2,w,N_{1}}^{\circ},\mathcal{O}/\varpi^{m})$.

Claim: We have natural isomorphisms

$\alpha
R\Gamma(N_{2,w,N_{1}}^{\circ},\mathcal{O}/\varpi^{m})\cong\tau_{w}\mathcal{O}/\varpi^{m}(\chi_{w})[-\textnormal{rk}_{\mathbf{Z}_{p}}N_{2,w,N_{1}}^{\circ}]\cong$
$\alpha\mathcal{O}/\varpi^{m}(\chi_{w})[-\textnormal{rk}_{\mathbf{Z}_{p}}N_{2,w,N_{1}}^{\circ}]$

in $D^{+}_{\textnormal{sm}}(\mathcal{O}/\varpi^{m}[M_{w}(L)^{+}\ltimes
N_{w}^{\circ}])$ where, by abuse of notation we consider $\chi_{w}$ as an
$M_{2}(L)^{+}\ltimes_{w}N_{w}^{\circ}$-module by inflation.

###### Proof of claim.

Assume that $Q_{1}=P_{(n_{1},...,n_{t})}$. Then we can write

$N_{1}(\mathcal{O}_{L})\cong\prod_{1\leq i<j\leq t}N_{i,j}^{\circ}$

where $N_{i,j}^{\circ}\subset N_{1}(\mathcal{O}_{L})$ corresponds to the
entries lying in $[n_{i-1}+1,n_{i}]\times[n_{j-1}+1,n_{j}]$ with the
convention that $n_{0}=0$. This induces an isomorphism

$N_{2,w,N_{1}}^{\circ}\cong\prod_{1\leq i<j\leq t}N_{2,w,ij}^{\circ}.$

Set $r_{i,j}:=\textnormal{rk}_{\mathbf{Z}_{p}}N_{2,w,ij}^{\circ}$ and, for
$1\leq k\leq t$,

$z_{p,k}:=\textnormal{diag}(p,...,p,1,...,1)$

where the first $n_{1}+...+n_{k}$ entries in the diagonal are given by $p$ and
the rest by $1$. Note that $z_{p,k}$ is in the centre of $M_{1}(L)$, so it
lies in $M_{w}(L)^{+}$ and it is invertible there, in particular, its Hecke
action on $\alpha R\Gamma(N_{2,w,N_{1}}^{\circ},\mathcal{O}/\varpi^{m})$ is
invertible.

Moreover, Künneth formula gives

$R\Gamma(N_{2,w,N_{1}}^{\circ},\mathcal{O}/\varpi^{m})\cong\bigotimes_{1\leq
i<j\leq t}R\Gamma(N_{2,w,ij}^{\circ},\mathcal{O}/\varpi^{m})$

and the Hecke action of $z_{p,k}$ is given by multiplying by the scalar

$[N_{2,w,N_{1}}^{\circ}:z_{p,k}N_{2,w}^{\circ}z_{p,k}^{-1}]=p^{\sum_{i<k<j}r_{i,j}}$

the tensor product of maps

$m_{i,j}^{k}:R\Gamma(N_{2,w,ij}^{\circ},\mathcal{O}/\varpi^{m})\to
R\Gamma(N_{2,w,ij}^{\circ},\mathcal{O}/\varpi^{m})$

induced by multiplication by $p$ on $N_{2,w,ij}^{\circ}$ if $i<k<j$ and by
$\textnormal{id}:N_{2,w,ij}^{\circ}\to N_{2,w,ij}^{\circ}$ otherwise. In the
first case this means that, for $0\leq d_{i,j}\leq r_{i,j}$,
$H^{d_{i,j}}(m_{i,j}^{k})$ is multiplication by $p^{-d_{i,j}}$ and it is
multiplication by $1$ otherwise. In particular,
$\alpha(\bigotimes_{i<j}H^{d_{i,j}}(N_{2,w,ij}^{\circ},\mathcal{O}/\varpi^{m}))\neq
0$ only when $d_{i,j}=r_{i,j}$ for each $1\leq i<j\leq t$. Note that
$\sum_{i<j}r_{i,j}=\textnormal{rk}_{\mathbf{Z}_{p}}N_{2,w,N_{1}}^{\circ}$.
Therefore, we have

$\alpha R\Gamma(N_{2,w,N_{1}}^{\circ},\mathcal{O}/\varpi^{m})\cong\alpha
H^{\textnormal{rk}_{\mathbf{Z}_{p}}N_{2,w,N_{1}}^{\circ}}(N_{2,w,N_{1}}^{\circ},\mathcal{O}/\varpi^{m})[-\textnormal{rk}_{\mathbf{Z}_{p}}N_{2,w,N_{1}}^{\circ}].$

Moreover, just as in the proof of [Hau16], Proposition 3.1.8, using [Eme10a],
Proposition 3.5.6 and the description of the corestriction map on top degree
cohomology (cf. [Eme10a], Lemma 3.5.10), we see that the latter is, as an
$\mathcal{O}/\varpi^{m}[M_{w}(L)^{+}]$-module, given by
$\alpha\mathcal{O}/\varpi^{m}(\chi_{w})[-\textnormal{rk}_{\mathbf{Z}_{p}}N_{2,w,N_{1}}^{\circ}]$.
∎

Putting everything together, we get

$R\Gamma(N_{2,w}^{\circ},\textnormal{Inf}_{M_{1}(L)}^{Q_{1}(L)}\pi)^{Q_{2}\textnormal{-ord}}\cong$
$\tau_{w}^{-1}R\Gamma(N_{w}^{\circ},\alpha(\mathcal{O}/\varpi^{m}(\chi_{w})\otimes\textnormal{Res}^{w}\pi))^{Q_{w}\textnormal{-ord}}[-\textnormal{rk}_{\mathbf{Z}_{p}}N_{2,w,N_{1}}^{\circ}]\cong$
$\cong\tau_{w}^{-1}(\tau_{w}(\mathcal{O}/\varpi^{m}(\chi_{w})))\otimes_{\mathcal{O}/\varpi^{m}}\tau_{w}^{-1}R\Gamma(N_{w}^{\circ},\pi)^{Q_{w}\textnormal{-ord}}[-\textnormal{rk}_{\mathbf{Z}_{p}}N_{2,w,N_{1}}^{\circ}]\cong$
$\mathcal{O}/\varpi^{m}(\chi_{w})\otimes_{\mathcal{O}/\varpi^{m}}\tau_{w}^{-1}R\Gamma(N_{w}^{\circ},\pi)^{Q_{w}\textnormal{-ord}}[-\textnormal{rk}_{\mathbf{Z}_{p}}N_{2,w,N_{1}}^{\circ}].$

∎

Combining the results of the subsection, we obtain the following.

######  Corollary 3.29.

Assume that $G=\textnormal{GL}_{n}$, $T=T_{n}$ and $B=B_{n}$. Let
$w\in{}^{Q_{1}}W^{Q_{2}}$ such that $wM_{2}(L)w^{-1}\subset M_{1}(L)$. Then,
for every $\pi\in D^{+}_{\textnormal{sm}}(\mathcal{O}/\varpi^{m}[M_{1}(L)])$,
$\sigma\in\textnormal{Mod}_{\textnormal{sm}}(\mathcal{O}/\varpi^{m}[M_{2}(\mathcal{O}_{L})])$
finite free as an $\mathcal{O}/\varpi^{m}$-module, and $j\in\mathbf{Z}_{\geq
0}$, the group

$R^{j}{\rm{Hom}}_{\mathcal{O}/\varpi^{m}[M_{2}(\mathcal{O}_{L})]}(\sigma,R\Gamma(N_{2}(\mathcal{O}_{L}),\textnormal{Ind}_{Q_{1}(L)}^{G(L)}\pi)^{Q_{2}\textnormal{-ord}})$

admits

$R^{j-\textnormal{rk}_{\mathbf{Z}_{p}}N_{2,w,N_{1}}^{\circ}}{\rm{Hom}}_{\mathcal{O}/\varpi^{m}[M_{2}(\mathcal{O}_{L})]}(\sigma,\mathcal{O}/\varpi^{m}(\chi_{w})\otimes_{\mathcal{O}/\varpi^{m}}\tau_{w}^{-1}R\Gamma(N_{w}^{\circ},\pi)^{Q_{w}\textnormal{-ord}})$

as a $\mathcal{H}(\sigma)$-equivariant subquotient.

For the reader’s convenience, we spell out the case of interest for our
application in proving local-global compatibility. For this we introduce some
notation that hopefully makes it easier to motivate how Corollary 3.29 will be
applied. In particular, consider $\widetilde{G}=\textnormal{GL}_{2n}$, and set
$P=P_{(n,n)}\subset\widetilde{G}$ to be the Siegel parabolic with Levi
decomposition $P=G\ltimes U$. Moreover, set $\widetilde{Q}\subset P$ to be any
standard242424Standard with respect to the Borel of upper triangular matrices.
parabolic subgroup with Levi decomposition
$\widetilde{M}\ltimes\widetilde{N}$. Set $\widetilde{Q}\cap G=Q_{c}\times
Q\subset G=\textnormal{GL}_{n}\times\textnormal{GL}_{n}$ and denote their Levi
decompositions by $M_{c}\ltimes N_{c}$, and $M\ltimes N$, respectively. Note
that $\widetilde{M}=M_{c}\times M$. Denote by $\widetilde{Q}^{w_{0}}\subset
P\subset\widetilde{G}$ the standard parabolic subgroup with Levi decomposition
$\widetilde{M}^{w_{0}}\ltimes\widetilde{N}^{w_{0}}$ where
$\widetilde{M}^{w_{0}}=M\times
M_{c}\subset\textnormal{GL}_{n}\times\textnormal{GL}_{n}$. Pick
$\widetilde{\sigma}=\sigma\otimes\sigma_{c}\in\textnormal{Mod}_{\textnormal{sm}}(\mathcal{O}/\varpi^{m}[\widetilde{M}^{w_{0}}(\mathcal{O}_{L}])=\textnormal{Mod}_{\textnormal{sm}}(\mathcal{O}/\varpi^{m}[M(\mathcal{O}_{L})\times
M_{c}(\mathcal{O}_{L})])$, finite free as an $\mathcal{O}/\varpi^{m}$-module.
Finally, note that $w_{0}^{P}\in{}^{P}W^{\widetilde{Q}^{w_{0}}}$. Then a
direct application of Corollary 3.29 with $Q_{1}=P$,
$Q_{2}=\widetilde{Q}^{w_{0}}$, and $w=w_{0}^{P}$ gives.

######  Corollary 3.30.

For every $\pi\in D^{+}_{\textnormal{sm}}(\mathcal{O}/\varpi^{m}[G(L)])$, and
$j\in\mathbf{Z}_{\geq 0}$, the group

$R^{j}{\rm{Hom}}_{\mathcal{O}/\varpi^{m}[\widetilde{M}^{w_{0}}(\mathcal{O}_{L})]}(\widetilde{\sigma},R\Gamma(\widetilde{N}^{w_{0}}(\mathcal{O}_{L}),\textnormal{Ind}_{P(L)}^{\widetilde{G}(L)}\pi)^{\widetilde{Q}^{w_{0}}\textnormal{-ord}})$

admits

$R^{j}{\rm{Hom}}_{\mathcal{O}/\varpi^{m}[M(\mathcal{O}_{L})\times
M_{c}(\mathcal{O}_{L})]}(\sigma\otimes\sigma_{c},R\Gamma(N(\mathcal{O}_{L})\times
N_{c}(\mathcal{O}_{L}),\tau_{w_{0}^{P}}^{-1}\pi)^{Q\times
Q_{c}\textnormal{-ord}})$

as a $\mathcal{H}(\sigma)\times\mathcal{H}(\sigma_{c})$-equivariant
subquotient.

We also deduce a dual statement computing $\overline{\widetilde{Q}}$-ordinary
parts of a Bruhat stratum of $\textnormal{Ind}_{P(L)}^{\widetilde{G}(L)}\pi$.

######  Corollary 3.31.

For every $\pi\in D^{+}_{\textnormal{sm}}(\mathcal{O}/\varpi^{m}[G(L)])$, and
integer $j\in\mathbf{Z}_{\geq 0}$, the group

$R^{j}{\rm{Hom}}_{\mathcal{O}/\varpi^{m}[\widetilde{M}(\mathcal{O}_{L})]}(\tau_{w_{0}^{P}}\widetilde{\sigma},R\Gamma(\overline{\widetilde{N}}^{1},\textnormal{Ind}_{P(L)}^{\widetilde{G}(L)}\pi)^{\overline{\widetilde{Q}}\textnormal{-ord}})$

admits

$R^{j}{\rm{Hom}}_{\mathcal{O}/\varpi^{m}[M_{c}(\mathcal{O}_{L})\times
M(\mathcal{O}_{L})]}(\sigma_{c}\otimes\sigma,R\Gamma(\overline{N}^{1}_{c}\times\overline{N}^{1},\pi)^{\overline{Q_{c}}\times\overline{Q}\textnormal{-ord}})$

as a $\mathcal{H}(\sigma_{c})\times\mathcal{H}(\sigma)$-equivariant
subquotient.

###### Proof.

We reduce it to Corollary 3.29. Set $\tilde{z}_{p}:=u_{p}^{\widetilde{Q}}\in
Z_{\widetilde{M}}^{+}$ to be the usual element defining the $U_{p}$-operator
with respect to $\widetilde{Q}$ and
$\widetilde{w}_{0}:=w_{0}^{\widetilde{Q}}$. Set
$\widetilde{Q}^{\tilde{w}_{0}}=\widetilde{M}^{\tilde{w}_{0}}\ltimes\widetilde{N}^{\tilde{w}_{0}}$
to be the standard parabolic subgroup with Levi subgroup
$\tilde{w}_{0}^{-1}\widetilde{M}\tilde{w}_{0}$. Note that
$\overline{N}^{\circ}:=\tilde{z}_{p}\widetilde{w}_{0}\widetilde{N}^{\tilde{w}_{0}}(\mathcal{O}_{L})(\tilde{z}_{p}\widetilde{w}_{0})^{-1}\subset\overline{\widetilde{N}}^{1}$
and, by Remark 3.18, we have a natural
($\mathcal{H}(\tau_{w_{0}^{P}}\widetilde{\sigma})$-equivariant) isomorphism

$R^{j}{\rm{Hom}}_{\mathcal{O}/\varpi^{m}[\widetilde{M}(\mathcal{O}_{L})]}\left(\tau_{w_{0}^{P}}\widetilde{\sigma},R\Gamma(\overline{\widetilde{N}}^{1},\textnormal{Ind}_{P(L)}^{\widetilde{G}(L)}\pi)^{\overline{\widetilde{Q}}\textnormal{-ord}}\right)\cong$

$\cong
R^{j}{\rm{Hom}}_{\mathcal{O}/\varpi^{m}[\widetilde{M}(\mathcal{O}_{L})]}\left(\tau_{w_{0}^{P}}\widetilde{\sigma},R\Gamma(\overline{N}^{\circ},\textnormal{Ind}_{P(L)}^{\widetilde{G}(L)}\pi)^{\overline{\widetilde{Q}}\textnormal{-ord}}\right).$

Moreover, multiplication by $\tilde{z}_{p}\widetilde{w}_{0}$ sets up a
$\mathcal{H}(\tau_{w_{0}}^{P}\widetilde{\sigma})$-equivariant isomorphism
between the latter and

(3.11)
$R^{j}{\rm{Hom}}_{\mathcal{O}/\varpi^{m}[\widetilde{M}^{\tilde{w}_{0}}(\mathcal{O}_{L})]}\left(\tau_{w_{0}^{Q\times
Q_{c}}}\widetilde{\sigma},R\Gamma(\widetilde{N}^{\tilde{w}_{0}}(\mathcal{O}_{L}),\textnormal{Ind}_{P(L)}^{\widetilde{G}(L)}\pi)^{\widetilde{Q}^{\tilde{w}_{0}}\textnormal{-ord}}\right)$

where the Hecke action on 3.11 is through the isomorphism
$\mathcal{H}(\tau_{w_{0}^{P}}\widetilde{\sigma})\cong\mathcal{H}(\tau_{w_{0}^{Q\times
Q_{c}}}\widetilde{\sigma})$ induced by $\tilde{w}_{0}$. Set
$N^{\tilde{w}_{0}}\times
N^{\tilde{w}_{0}}_{c}:=\widetilde{N}^{\tilde{w}_{0}}\cap G$ and note that
$N^{\tilde{w}_{0}}$, respectively $N^{\tilde{w}_{0}}_{c}$ is the unipotent
radical corresponding to $M^{\tilde{w}_{0}}:=w_{0}^{Q}Mw_{0}^{Q,-1}$,
respectively $M^{\tilde{w}_{0}}_{c}:=w_{0}^{Q_{c}}M_{c}w_{0}^{Q_{c},-1}$. Then
Corollary 3.29 shows that 3.11 admits

(3.12) $R^{j}{\rm{Hom}}_{\mathcal{O}/\varpi^{m}[M^{\tilde{w}_{0},0}_{c}\times
M^{\tilde{w}_{0},0}]}\left(\tau_{w_{0}^{Q_{c}}}\sigma_{c}\times\tau_{w_{0}^{Q}}\sigma,R\Gamma(N^{\tilde{w}_{0}}_{c}(\mathcal{O}_{L})\times
N^{\tilde{w}_{0}}(\mathcal{O}_{L}),\pi)^{Q^{\tilde{w}_{0}}_{c}\times
Q^{\tilde{w}_{0}}\textnormal{-ord}}\right)$

as a $\mathcal{H}(\tau_{w_{0}^{Q\times Q_{c}}}\widetilde{\sigma})$-equivariant
subquotient. Note that
$\widetilde{w}_{0}=(w_{0}^{Q_{c}},w_{0}^{Q})w_{0}^{P}=w_{0}^{P}(w_{0}^{Q},w_{0}^{Q_{c}})$.
Therefore, multiplication by $(w_{0}^{Q_{c}}u_{p}^{Q_{c}},w_{0}^{Q}u_{p}^{Q})$
and another application of Remark 3.18 sets up a
$\mathcal{H}(\widetilde{\sigma})\cong\mathcal{H}(\tau_{\tilde{w}_{0}}\widetilde{\sigma})$-equivariant
isomorphism between 3.12 and

$R^{j}{\rm{Hom}}_{\mathcal{O}/\varpi^{m}[M_{c}(\mathcal{O}_{L})\times
M(\mathcal{O}_{L})]}\left(\tau_{w_{0}^{P}}\widetilde{\sigma},R\Gamma(\overline{N}^{1}_{c}\times\overline{N}^{1},\pi)^{\overline{Q}_{c}\times\overline{Q}\textnormal{-ord}}\right)$
$\cong R^{j}{\rm{Hom}}_{\mathcal{O}/\varpi^{m}[M_{c}(\mathcal{O}_{L})\times
M(\mathcal{O}_{L})]}(\sigma_{c}\otimes\sigma,R\Gamma(\overline{N}^{1}_{c}\times\overline{N}^{1},\pi)^{\overline{Q_{c}}\times\overline{Q}\textnormal{-ord}}).$

∎

##  4\. $Q$-ordinary parts in characteristic $0$

In this section, we discuss the notion of taking ordinary parts of smooth
admissible representations of a $p$-adic reductive group that arise as local
components of cohomological automorphic representations (see [Ger18], §5.1 for
instance). In fact, we take a slightly more involved approach and define
ordinary parts of locally algebraic representations in terms of Emerton’s
slope $0$ part. It will allow us to easily compare it to the corresponding
Jacquet module and to our previous notion of ordinary parts. The latter will
be useful in the endgame of proving our local-global compatibility results. We
will then prove our main characteristic $0$ result regarding $Q$-ordinary
parts of $Q$-ordinary locally algebraic representations of
$\textnormal{GL}_{n}$. Finally, we close the section with deducing
$Q$-ordinary local-global compatibility for regular algebraic cuspidal
automorphic representations of $\textnormal{GL}_{n}$ in the conjugate self-
dual case.

###  4.1. Ordinary parts of locally algebraic representations

We briefly summarise the notion of ordinary parts for locally algebraic
representations as introduced in [BD20], §4.3 (see also [Eme11], §5.6) and how
it compares to Emerton’s ordinary part functor from [Eme10] when the
representation admits an invariant lattice. We revisit the setup of §3.1 and
without further notice will use the introduced notation. However, we further
assume that $G$ is split over $L$ and fix a choice of maximal torus $T$ and a
Borel subgroup $B$ containing it. Finally, $Q$ is now assumed to be standard
with respect to $B$. Recall that our coefficient field is $E/\mathbf{Q}_{p}$,
a finite extension, large enough, so that
$[L:\mathbf{Q}_{p}]={\rm{Hom}}(L,E)$. Then, given a
$(\textnormal{Res}_{L/\mathbf{Q}_{p}}B)_{E}$-dominant weight
$\lambda\in{\rm{Hom}}((\textnormal{Res}_{L/\mathbf{Q}_{p}}T)_{E},\mathbf{G}_{m,E})\cong\oplus_{\iota:L\hookrightarrow
E}{\rm{Hom}}(T_{E},\mathbf{G}_{m,E})$, set
$V_{\lambda}=\otimes_{\iota}V_{\lambda_{\iota}}$ to be the corresponding
absolutely irreducible algebraic $E$-representation of $G(L)$. Note that the
dual $V_{\lambda}^{\vee}$ is isomorphic to $V_{\lambda^{\vee}}$ where
$\lambda^{\vee}:=-w_{0}^{G}\lambda$.

######  Definition 4.1.

An $E$-representation $\Pi$ of $G(L)$ is called locally algebraic of weight
$\lambda$ if it is locally $V_{\lambda^{\vee}}$-algebraic in the sense of
[Eme17], Definition 4.2.1 such that the smooth vectors
${\rm{Hom}}(V_{\lambda^{\vee}},\Pi)_{\textnormal{sm}}=\varinjlim_{K\to
1}{\rm{Hom}}_{K}(V_{\lambda^{\vee}},\Pi)$ form an admissible smooth
representation of $G(L)$.

We note that our definition admits a more intrinsic formulation (cf. [Eme17]
Definition 6.3.9, Proposition 6.3.10).

######  Remark 4.2.

Note that any locally algebraic $E$-representation $\Pi$ of weight $\lambda$
is of the form $\pi\otimes V_{\lambda^{\vee}}$ with $\pi$ a smooth admissible
$E$-representation (cf. [Eme17], Proposition 4.2.4). Even better, the functor
${\rm{Hom}}(V_{\lambda^{\vee}},-)_{\textnormal{sm}}$ sets up a natural
equivalence between the category of locally algebraic $E$-representations of
$G(L)$ of weight $\lambda$ and the category of smooth admissible
$E$-representations of $G(L)$. However, the reason one might want to appeal to
the notion of locally algebraic representations is that it yields a different
notion of $p$-adic integrality. Indeed, this is our motivation to work with
locally algebraic representations.

######  Remark 4.3.

Note that, $V_{\lambda}$ being an algebraic $E$-representation of
$\textnormal{Res}_{L/\mathbf{Q}_{p}}G$,

$V_{\lambda}^{N^{0}}\cong V_{\lambda}^{N(L)}$

as subspaces of $V_{\lambda}$. Moreover, the latter is the absolutely
irreducible representation of $M(L)$ associated with $\lambda$ viewed as a
$(\textnormal{Res}_{L/\mathbf{Q}_{p}}B\cap M)_{E}$-dominant weight for
$(\textnormal{Res}_{L/\mathbf{Q}_{p}}M)_{E}$ (see [Cab84]). In particular, the
former is naturally an $M(L)$-representation. As a corollary, one sees that
for a locally algebraic $E$-representation $\Pi=\pi\otimes V_{\lambda^{\vee}}$
of weight $\lambda$, we have an induced identification
$\Pi^{N^{0}}\cong\pi^{N^{0}}\otimes V_{\lambda^{\vee}}^{N(L)}$. Moreover,
under this isomorphism, the Hecke action of $M^{+}$ on $\Pi^{N^{0}}$ coincides
with the $M^{+}$-action on $\pi^{N^{0}}\otimes V_{\lambda^{\vee}}^{N(L)}$
given by the usual Hecke action on the first factor and the natural action on
the algebraic part.

We now introduce the notion of finite slope and slope $0$ parts of
$\Pi^{N^{0}}$. For $b\geq 1$, consider the finite dimensional $E$-vector space

$\Pi_{b}:=\pi^{\mathcal{Q}(b,b)}\otimes
V_{\lambda^{\vee}}^{N(L)}\subset\Pi^{N^{0}}.$

By Hypothesis 3.3, it is a $Z_{M}^{+}$-invariant subspace. Denote by $B_{b}$
the $E$-subalgebra of $\textnormal{End}_{E}(\Pi_{b})$ generated by
$Z_{M}^{+}$. This is an Artinian $E$-algebra and as such, it decomposes into a
product of local Artinian $E$-algebras indexed by its maximal ideals. We then
say that a maximal ideal $\mathfrak{m}\subset B_{b}$ is of finite slope if the
image of $Z_{M}^{+}$ in $B_{b}$ is disjoint from $\mathfrak{m}$. For such an
$\mathfrak{m}$, the composition

$Z_{M}^{+}\to B_{b}\twoheadrightarrow
B_{b}/\mathfrak{m}\hookrightarrow\overline{\mathbf{Q}}_{p}$

lands in $\overline{\mathbf{Q}}_{p}^{\times}$. We further say that a finite
slope maximal ideal $\mathfrak{m}\subset B_{b}$ is of slope zero if the
composition lands in $\overline{\mathbf{Z}}_{p}^{\times}$. Considering the
corresponding factors of $B_{b}$ induces $Z_{M}^{+}$-equivariant
decompositions

(4.1)
$\Pi_{b}\cong(\Pi_{b})_{\textnormal{fs}}\oplus(\Pi_{b})_{\textnormal{null}}\cong(\Pi_{b})_{0}\oplus(\Pi_{b})_{>0}$

into finite slope and slope $0$ parts and their complements. Note that
$(\Pi_{b})_{0}\subset(\Pi_{b})_{\textnormal{fs}}$ and that the
$Z_{M}^{+}$-action uniquely extends to a $Z_{M}(L)$-action on both by
definition. The decompositions in 4.1 are easily checked to be compatible when
we vary $b$. Therefore, by passing to the colimit over $b\geq 1$, we get
$Z_{M}^{+}$-equivariant decompositions

$\Pi^{N^{0}}\cong(\Pi^{N^{0}})_{\textnormal{fs}}\oplus(\Pi^{N^{0}})_{\textnormal{null}}\cong(\Pi^{N^{0}})_{0}\oplus(\Pi^{N^{0}})_{>0}.$

Moreover, in the colimit, the decompositions are preserved by the
$M^{+}$-action. In particular, Lemma 3.4 and Remark 4.3 shows that
$(\Pi^{N^{0}})_{\textnormal{fs}}$ and $(\Pi^{N^{0}})_{0}$ become locally
algebraic $E$-representations of $M(L)$ of weight
$w_{0}^{M}w_{0}^{G}\lambda=w_{0}^{Q}\lambda$.

First, we compare the finite slope part with the classical (unnormalised)
Jacquet functor. This is essentially the theory of canonical liftings that
goes back to Casselman (cf. [Cas95]).

######  Proposition 4.4.

We have a natural $M(L)$-equivariant isomorphism

$(\Pi^{N^{0}})_{\textnormal{fs}}\otimes\delta_{Q}\cong J_{Q}(\pi)\otimes
V_{\lambda^{\vee}}^{N(L)}$

induced by the natural $M^{+}$-equivariant (surjective) map

$\Pi^{N^{0}}\otimes\delta_{Q}\to J_{Q}(\pi)\otimes V_{\lambda^{\vee}}^{N(L)}.$

###### Proof.

This can be deduced from [Eme06], §4.3. We sketch an argument here. It is an
easy consequence of the definitions that
$(\Pi^{N^{0}})_{\textnormal{fs}}=(\pi^{N^{0}})_{\textnormal{fs}}\otimes
V_{\lambda^{\vee}}^{N(L)}$ and
$(\Pi^{N^{0}})_{\textnormal{null}}=(\pi^{N^{0}})_{\textnormal{null}}\otimes
V_{\lambda^{\vee}}^{N(L)}$. Moreover, one sees that
$(\pi^{\mathcal{Q}(b,b)})_{\textnormal{null}}\subset\pi^{\mathcal{Q}(b,b)}$ is
the subspace of vectors on which, for some choice of $z_{p}\in
Z_{M}^{+}$252525Hence for any choice of such $z_{p}$. as in Lemma 3.4, the
action of $z_{p}$ is nilpotent. Using this description of
$(\pi^{N^{0}})_{\textnormal{null}}$, one sees that
$(\pi^{N^{0}})_{\textnormal{null}}$ is exactly the kernel of the natural
$\pi^{N^{0}}\to J_{Q}(\pi)$. Moreover, [Eme06], Proposition 4.3.4 i) shows
that $\pi^{N^{0}}\otimes\delta_{Q}\to J_{Q}(\pi)$ is an $M^{+}$-equivariant
surjection (see also [Cas95], Theorem 3.3.3 and Lemma 4.1.1). Note that in
[Eme06] we don’t see the appearance of $\delta_{Q}$. This is due to the fact
that they work with a different normalisation of the Hecke action on
$\pi^{N^{0}}$ (see loc. cit. Definition 3.4.1). Combining these observations,
we get the proposition. ∎

######  Remark 4.5.

To avoid confusion, we point out that what is denoted by $J_{Q}$ in [Eme06] is
Emerton’s locally analytic Jacquet functor. It can be applied to a certain
class of locally analytic representations $\Pi$ of $G(L)$ and defined by the
analogous formula $(\Pi^{N^{0}})_{\textnormal{fs}}$ for a suitable notion of
finite slope parts in this setup. Without elaborating on it any further, we
just note that it is easily extracted from [Eme06], §4.3 that the
constructions of [BD20], §4.3 are all compatible with the ones of Emerton. In
other words, for $\Pi$ a locally algebraic $E$-representation of $G(L)$ we
have $J_{Q}(\Pi)\cong(\Pi^{N^{0}})_{\textnormal{fs}}\otimes\delta_{Q}$ where
the former is in the sense of Emerton and the latter is in the sense of
Breuil–Ding. Again, the character $\delta_{Q}$ appears because of the
different normalisations of the Hecke action. From now on, we freely use the
notation $J_{Q}(\Pi)$ for $\Pi$ a locally algebraic representation to denote
its Jacquet module in the sense of Emerton. In light of Proposition 4.4, it
recovers the classical Jacquet functor.

We now turn to discussing $(\Pi^{N^{0}})_{0}$. We would like to compare it
with Emerton’s ordinary parts for smooth admissible
$\mathcal{O}/\varpi^{m}[G(L)]$-modules (cf. [Eme10]). In order to have a
chance to compare the two notions, we assume that our $\Pi$ is also a unitary
representation of $G(L)$. By this we mean that there is an
$\mathcal{O}$-lattice $\Pi^{\circ}\subset\Pi$262626In other words, an
$\mathcal{O}$-submodule that spans $\Pi$ over $E$ and contains no $E$-line.
that is invariant under the action of $G(L)$. The given lattice induces
$\mathcal{O}$-lattices $\Pi_{b}^{\circ}:=\Pi^{\circ}\cap\Pi_{b}\subset\Pi_{b}$
which are necessarily finite and free as $\mathcal{O}$-modules. Then, for an
integer $b\geq 1$, set $A_{b}$ to be the $\mathcal{O}$-subalgebra of
$\textnormal{End}_{\mathcal{O}}(\Pi_{b}^{\circ})$ generated by $Z_{M}^{+}$.
This is a finite $\mathcal{O}$-algebra and, in particular, we have
$A_{b}\cong\prod_{\mathfrak{n}}(A_{b})_{\mathfrak{n}}$, where the propduct
runs over maximal ideals in $A_{b}$. Call a maximal ideal $\mathfrak{n}\subset
A_{b}$ ordinary if the image of $Z_{M}^{+}$ in $A_{b}$ is disjoint from
$\mathfrak{n}$. This gives a $Z_{M}^{+}$-equivariant decomposition

$\Pi_{b}^{\circ}\cong(\Pi_{b}^{\circ})_{\textnormal{ord}}\oplus(\Pi_{b}^{\circ})_{\textnormal{nonord}}$

and, just as before, the $Z_{M}^{+}$-action on
$(\Pi_{b}^{\circ})_{\textnormal{ord}}$ extends uniquely to a
$Z_{M}(L)$-action. By passing to the colimit over $b\geq 1$, we obtain an
$M^{+}$-equivariant decomposition

$\Pi^{\circ,N^{0}}\cong\textnormal{Ord}_{Q}^{\textnormal{lalg}}(\Pi^{\circ})\oplus\textnormal{NOrd}_{Q}^{\textnormal{lalg}}(\Pi^{\circ}).$

Therefore, $\textnormal{Ord}_{Q}^{\textnormal{lalg}}(\Pi^{\circ})$ is
naturally a representation of $M(L)$. Then [BD20], Lemma 4.10 shows that we
have a natural isomorphism

$\textnormal{Ord}_{Q}^{\textnormal{lalg}}(\Pi^{\circ})\otimes_{\mathcal{O}}E\cong(\Pi^{N^{0}})_{0}$

of locally algebraic $E$-representations of $M(L)$. In particular, the former
is independent of the choice lattice $\Pi^{\circ}\subset\Pi$ and $N^{0}$ and
the latter is unitary. Motivated by this, introduce the following notation.

######  Definition 4.6.

Let $\Pi$ be a locally algebraic $E$-representation of $G(L)$ of weight
$\lambda$. We then define its $Q$-ordinary part

$\textnormal{Ord}_{Q}^{\textnormal{lalg}}(\Pi):=(\Pi^{N^{0}})_{0},$

a locally algebraic $E$ representation of $M(L)$ of weight $w_{0}^{Q}\lambda$.

In the rest of the section, we will justify that when $\Pi$ is assumed to be
unitary our locally algebraic $Q$-ordinary parts does behave like an ordinary
part. In the next section we will see that in fact under some a priori milder
assumption it still behaves like an ordinary part functor.

From now on, for the rest of the section, we assume that $\Pi$ is unitary and
choose an $\mathcal{O}$-lattice $\Pi^{\circ}$. For an integer $m\geq 1$, we
can also introduce analogous constructions for $\Pi^{\circ}_{b}/\varpi^{m}$ to
get a $Z_{M}^{+}$-equivariant decomposition
$\Pi_{b}^{\circ}/\varpi^{m}\cong(\Pi_{b}^{\circ}/\varpi^{m})_{\textnormal{ord}}\oplus(\Pi_{b}^{\circ}/\varpi^{m})_{\textnormal{nonord}}$.
The natural map then induces an isomorphism (cf. [BD20], 4.16)

(4.2)
$(\Pi_{b}^{\circ})_{\textnormal{ord}}/\varpi^{m}\xrightarrow{\sim}(\Pi_{b}^{\circ}/\varpi^{m})_{\textnormal{ord}}.$

On the other hand, $\Pi^{\circ}/\varpi^{m}$ is a smooth admissible
$\mathcal{O}/\varpi^{m}[G(L)]$-module so one can take its $Q$-ordinary part in
the sense of Emerton. Then [BD20], Lemma 4.14 shows that we have a natural
injection

(4.3) $\varinjlim_{b\geq
1}(\Pi_{b}^{\circ}/\varpi^{m})_{\textnormal{ord}}\hookrightarrow\textnormal{Ord}_{Q}(\Pi^{\circ}/\varpi^{m})$

where the latter is Emerton’s $Q$-ordinary part (cf. [Eme10], Definition
3.1.7).272727We note here that since we are in an admissible situation,
Emerton’s ordinary part simplifies to localising
$(\Pi^{\circ}/\varpi^{m})^{N^{0}}$ along
$\mathcal{O}[z_{p}]\hookrightarrow\mathcal{O}[z_{p}^{\pm 1}]$ for any $z_{p}$
as in Lemma 3.4 (see [Eme10a], Lemma 3.2.1). However, due to the presence of
group cohomology, 4.3 is not necessarily surjective. Combining 4.2 and 4.3,
one gets (cf. [BD20], Lemma 4.16) a natural $M(L)$-equivariant injection

(4.4)
$\textnormal{Ord}_{Q}^{\textnormal{lalg}}(\Pi^{\circ})\hookrightarrow\textnormal{Ord}_{Q}(\Pi^{\circ}):=\varprojlim_{m}\textnormal{Ord}_{Q}(\Pi^{\circ}/\varpi^{m}).$

######  Remark 4.7.

The map 4.4 is clearly not surjective in general: The source is a locally
algebraic representation, the target is a lattice in an $E$-Banach space
representation. A more sensible question to ask is whether it has dense image
or not. One notes that it does have dense image as long as 4.3 is surjective,
see [BD20], Remark 4.15 and Remark 4.17.

Given a unitary admissible $E$-Banach space representation $\Pi$ of $G(L)$,
with a unit ball $\Pi^{\circ}$, then
$\varprojlim_{m}\textnormal{Ord}_{Q}(\Pi^{\circ}/\varpi^{m})$ is what Emerton
calls $\textnormal{Ord}_{Q}(\Pi^{\circ})$ in [Eme10]. Moreover, one can then
set
$\textnormal{Ord}_{Q}(\Pi):=\textnormal{Ord}_{Q}(\Pi^{\circ})\otimes_{\mathcal{O}}E$.282828One
checks easily that it is independent of the choice of unit ball. We borrow his
notation. We conclude the subsection with discussing a corollary of the
adjunction property of [BD20], §4.4. Before stating the corollary, we
introduce the following notation. Let $\Pi$ be any $E$-representation of
$G(L)$ and $W$ be an algebraic $E$-representation of $G(L)$. We then denote by
$\Pi^{W\textnormal{-lalg}}\subset\Pi$ the $G(L)$-invariant subspace of locally
$W$-algebraic vectors in $\Pi$ in the sense of [Eme17], Proposition-Definition
4.2.6.

######  Proposition 4.8.

Let $\Pi$ be a unitary admissible $E$-Banach space representation of $G(L)$.
Then, for any choice of $(\textnormal{Res}_{L/\mathbf{Q}_{p}}B)_{E}$-dominant
weight $\lambda$, we have natural $M(L)$-equivariant identification

$\textnormal{Ord}_{Q}(\Pi)^{V_{\lambda^{\vee}}^{N(L)}\textnormal{-lalg}}\cong\textnormal{Ord}_{Q}^{\textnormal{lalg}}(\Pi^{V_{\lambda^{\vee}}\textnormal{-lalg}}).$

###### Proof.

Before proving the proposition, we first remark that the statement indeed
makes sense. In other words, we need to check that
$\Pi^{V_{\lambda^{\vee}}\textnormal{-lalg}}$ is indeed a locally algebraic
representation (of weight $\lambda$) in the sense of this text. Note that the
functor of taking locally $V_{\lambda^{\vee}}$-algebraic vectors factors
through the functor $\Pi\mapsto\Pi^{\textnormal{la}}$ of taking locally
analytic vectors. By [Eme17], Proposition 6.2.2 and Proposition 6.2.4, we see
that $\Pi^{\textnormal{la}}$ is an admissible locally analytic representation
of $G(L)$. By loc. cit. Proposition 6.3.6, we see that
$\Pi^{\textnormal{la},V_{\lambda^{\vee}}\textnormal{-lalg}}\cong\Pi^{V_{\lambda^{\vee}}\textnormal{-lalg}}$
is again an admissible locally analytic representation. Finally, the claim
follows from loc. cit. Proposition 6.3.10 by noting that
$\textnormal{End}_{E[G(L)]}(\textnormal{V}_{\lambda^{\vee}})=E$.

Now the proposition easily follows from [BD20] §4 as we explain now. We first
apply [BD20], Lemma 4.18 with the choice
$V=\Pi^{V_{\lambda^{\vee}}\textnormal{-lalg}}\subset W=\Pi$,
$W^{\circ}=\Pi^{\circ}$ and $V^{\circ}=V\cap W^{\circ}$. It gives an injection

(4.5)
$\textnormal{Ord}_{Q}^{\textnormal{lalg}}(\Pi^{V_{\lambda^{\vee}}\textnormal{-lalg}})\hookrightarrow\textnormal{Ord}_{Q}(\Pi).$

As
$\textnormal{Ord}_{Q}^{\textnormal{lalg}}(\Pi^{V_{\lambda^{\vee}}\textnormal{-lalg}})$
is a locally algebraic representation of $M(L)$ of weight $w_{0}^{Q}\lambda$,
it necessarily lands in
$\textnormal{Ord}_{Q}(\Pi)^{V_{\lambda^{\vee}}^{N(L)}\textnormal{-lalg}}$.

For the other inclusion, note that $\textnormal{Ord}_{Q}(\Pi)$ is again a
unitary admissible $E$-Banach space representation by [Eme10], Theorem 3.4.8.
Therefore, we see that
$\textnormal{Ord}_{Q}(\Pi)^{V_{\lambda^{\vee}}^{N(L)}\textnormal{-lalg}}\cong\pi\otimes
V_{\lambda^{\vee}}^{N(L)}$ is a locally algebraic representation of weight
$w_{0}^{Q}\lambda$. Consider the natural inclusion

$\iota_{v}:\pi\otimes
V_{\lambda^{\vee}}^{N(L)}\cong\textnormal{Ord}_{Q}(\Pi)^{V_{\lambda^{\vee}}^{N(L)}\textnormal{-lalg}}\hookrightarrow\textnormal{Ord}_{Q}(\Pi).$

Then [BD20], Proposition 4.21 applied to $f:=\iota_{v}$ yields a map

$\textnormal{Ind}_{\overline{Q}(L)}^{G(L)}\pi\otimes V_{\lambda^{\vee}}\to\Pi$

and $\iota_{v}$ can be reconstructed by precomposing the map

(4.6)
$\textnormal{Ord}_{Q}^{\textnormal{lalg}}(\textnormal{Ind}_{\overline{Q}(L)}^{G(L)}\pi\otimes
V_{\lambda^{\vee}})\to\textnormal{Ord}_{Q}(\Pi)$

induced by loc. cit. Lemma 4.18 with the inclusion

$\pi\otimes
V_{\lambda^{\vee}}^{N(L)}\hookrightarrow\textnormal{Ord}_{Q}^{\textnormal{lalg}}(\textnormal{Ind}_{\overline{Q}(L)}^{G(L)}\pi\otimes
V_{\lambda^{\vee}})$

of loc. cit. Lemma 4.19. But 4.6 factors through 4.5 and the proof is
finished. ∎

###  4.2. Ordinary parts of weakly admissible representations

We introduce the notion of weakly admissible locally algebraic
representations, an a priori weaker integrality notion than unitariness. We
then prove that the notion of being $Q$-ordinary behaves well for such
representations. For the rest of the section, we assume that
$G=\textnormal{GL}_{n}$, $T$ is the torus of diagonal matrices and $B$ is the
Borel subgroup of upper-triangular matrices. In particular,
$Q=P_{(n_{1},...,n_{t})}$ for a partition $n_{1}+...+n_{t}=n$. Fix a choice of
uniformiser $\varpi_{L}\in\mathcal{O}_{L}$ and recall that $e$ denotes the
ramification index of $L$. Moreover, for integers $0\leq b\leq c$ with $c\geq
1$, let $\mathcal{Q}(b,c)\subset G(\mathcal{O}_{L})$ be the corresponding
parahoric subgroup associated with $Q$. Then, by the Iwahori decomposition,
$\mathcal{Q}(b,c)=\overline{N}^{c}M^{b}N^{0}$. For $1\leq i\leq n$, set

$u_{p}^{(i)}:=\textnormal{diag}(\varpi_{L},...,\varpi_{L},1,...,1)$

where the first $i$ elements are given by $\varpi_{L}$ and the rest by $1$.
Then $u_{p}^{(n_{1})}\cdot....\cdot u_{p}^{(n_{1}+...+n_{t-1})}\in Z_{M}^{+}$
is our choice for the role of $z_{p}$ from the proof of Lemma 3.4. For $x\in
L^{\times}$, set $\langle x\rangle:=\textnormal{diag}(x,...,x)$.

######  Definition 4.9.

Let $\Pi$ be a locally algebraic $E$-representation of $G(L)$ of weight
$\lambda$. We say that $\Pi$ is weakly admissible (of weight $\lambda$) if,
for any standard parabolic subgroup $Q^{\prime}=M^{\prime}\ltimes
N^{\prime}\subset G$ and locally algebraic character
$\chi:Z_{M^{\prime}}(L)\to\overline{\mathbf{Q}}_{p}^{\times}$, such that
${\rm{Hom}}_{Z_{M^{\prime}}(L)}(\chi,J_{Q^{\prime}}(\Pi))\neq 0$, we have

$\mid\chi(z)\delta_{Q^{\prime}}^{-1}(z)\mid_{p}\leq 1,$

for every $z\in Z_{M^{\prime}}^{+}$. We denote the category of such by
$\textnormal{Mod}_{E}^{\textnormal{wa},\lambda}(G(L))$.

######  Remark 4.10.

This is what Hu refers to as Emerton’s condition in [Hu09], where he proves
the "easy" direction of the Breuil-Schneider conjecture.292929In fact, he asks
for this condition to be satisfied for any parabolic subgroup
$Q^{\prime}\subset G$, but as it is explained in loc. cit. Remarque 2.11, it
suffices to check it for standard parabolic subgroups. The main result
(Théorème 1.2) of [Hu09] shows that, at least when $\Pi$ is of the form
$\pi\otimes V_{\lambda^{\vee}}$ with $\pi$ irreducible and generic, $\Pi$ is
weakly admissible if and only if $\textnormal{rec}^{T}(\pi)$ comes from a de
Rham Galois representation with Hodge–Tate weights given by the usual
$\rho$-shift of $\lambda$. The latter means that the
$(\varphi,N,G_{L})$-module associated with $\textnormal{rec}^{T}(\pi)$ admits
a filtration with jumps given by the $\rho$-shift of $\lambda$ that makes it a
weakly admissible filtered $(\varphi,N,G_{L})$-module.303030To avoid
confusion, note that the original formulation of the Breuil-Schneider
conjecture uses different normalisations. Namely, one instead asks whether the
representation $\pi(n-1)\otimes V_{\lambda^{\vee}+n-1}$ admits a unitary
completion. As the character $(|\det|\det)^{n-1}$ is unitary, regardless of
the mismatch, the two different normalisations yield equivalent conjectures.
The way Hu applies it to the Breuil-Schneider conjecture is that if $\Pi$ is
unitary then one deduces easily that it must be weakly admissible (see
[Eme06], Lemma 4.4.2).

Using Emerton’s condition to define weak admissibility has the advantage of
being easy to keep track of while proving results regarding weakly admissible
representations. However, we record here an equivalent condition that has the
advantage that in the case of our examples appearing in this text it will
clearly be satisfied.

######  Lemma 4.11.

Let $\Pi=\pi\otimes V_{\lambda^{\vee}}$ be a locally algebraic
$E$-representation of G(L) of weight $\lambda$ and
$P_{(n_{1}^{\prime},...,n_{h}^{\prime})}=Q^{\prime}=M^{\prime}\ltimes
N^{\prime}\subset G$ a standard parabolic subgroup. The following two
conditions are equivalent:

  1. (i)

For each $1\leq i\leq h$, the generalised eigenvalues of the double coset
operator $[N^{\prime 0}u_{p}^{(n_{1}+...+n_{i})}N^{\prime 0}]$ acting on
$\Pi^{N^{\prime 0}}$ lie in $\overline{\mathbf{Z}}_{p}$.

  2. (ii)

For any locally algebraic character
$\chi:Z_{M}(L)\to\overline{\mathbf{Q}}_{p}^{\times}$ with

${\rm{Hom}}_{Z_{M}(L)}(\chi,J_{Q^{\prime}}(\Pi))\neq 0,$

we have

$|\chi(z)\delta_{Q^{\prime}}^{-1}(z)|_{p}\leq 1$

for every $z\in Z_{M}^{+}$.

In particular, $\Pi$ is weakly admissible if and only if it satisfies $i)$ for
every choice of standard parabolic subgroup $Q^{\prime}\subset G$.

###### Proof.

We first make a few observations about condition $ii)$. Note that $\chi$ is of
the form $\chi_{\textnormal{sm}}\otimes\chi_{\textnormal{alg}}$. Moreover,
$J_{Q^{\prime}}(\Pi)=J_{Q^{\prime}}(\pi)\otimes V_{(w_{0}^{Q}\lambda)^{\vee}}$
and $V_{(w_{0}^{Q}\lambda)^{\vee}}$ has constant central character sending
$u_{p}^{(n^{\prime}_{1}+...+n^{\prime}_{i})}$ to

$-w_{0}^{G}\lambda(u_{p}^{(n^{\prime}_{1}+...+n^{\prime}_{i})})=\prod_{\iota:L\hookrightarrow
E}\iota(\varpi_{L})^{-(\lambda_{\iota,n}+...+\lambda_{\iota,n+1-(n_{1}+...+n_{i})})}.$

Therefore, $\chi$ is forced to be a smooth twist of $-w_{0}^{G}\lambda$.

Moreover, after extending $\pi$ to a
$\overline{\mathbf{Q}}_{p}$-representation, we can apply [Cas95], Proposition
2.1.9 to get a direct sum decomposition

(4.7)
$J_{Q^{\prime}}(\pi)=\bigoplus_{\chi_{\textnormal{sm}}:Z_{M^{\prime}}(L)\to\overline{\mathbf{Q}}_{p}^{\times}}J_{Q^{\prime}}(\pi)_{\chi_{\textnormal{sm}}}$

of the $M(L)$-representation into generalised eigenspaces with respect to all
the possible central characters. Here we used that both base change to
$\overline{\mathbf{Q}}_{p}$ and $J_{Q^{\prime}}$ preserves admissibility. In
particular, we see that $J_{Q^{\prime}}(\pi)_{\chi_{\textnormal{sm}}}\neq 0$
if and only if
${\rm{Hom}}_{Z_{M^{\prime}}(L)}(\chi_{\textnormal{sm}}(-w_{0}^{G}\lambda),J_{Q^{\prime}}(\Pi))\neq
0$.

We also note that the decomposition 4.7 is a refinement of the decomposition
of $J_{Q^{\prime}}(\pi)$ into generalised eigenspaces with respect to the
actions of the $u_{p}^{(n^{\prime}_{1}+...+n^{\prime}_{i})}$’s. On the other
hand, it is easy to check that $\chi|_{Z_{M^{\prime}}^{0}}$ and
$\delta_{Q^{\prime}}|_{Z_{M^{\prime}}^{0}}$ necessarily land in
$\overline{\mathbf{Z}}_{p}^{\times}$. So the inequality
$|\chi(z)\delta_{Q^{\prime}}^{-1}(z)|_{p}\leq 1$ is satisfied for all $z\in
Z_{M^{\prime}}^{+}$ if and only if it is satisfied for
$z=u_{p}^{(n^{\prime}_{1}+...+n^{\prime}_{i})}$ for all $1\leq i\leq h$.

Finally, we finish the proof by recalling that the natural map $\Pi^{N^{\prime
0}}\otimes\delta_{Q^{\prime}}\to J_{Q^{\prime}}(\Pi)$ is
$Z_{M^{\prime}}^{+}$-equivariant, and its kernel is the subspace on which the
$U_{p}$-operators act nilpotently. ∎

######  Remark 4.12.

As was promised, using Lemma 4.11, we can find a large source of examples of
weakly admissible representations. Namely, given a number field $F$, we can
consider any regular algebraic cuspidal automorphic representation $\pi$ of
$\textnormal{GL}_{n}(\mathbf{A}_{F})$ of weight $\lambda$. Given a $p$-adic
place $v\in S_{p}(F)$, set $L=F_{v}$, and look at the component $\pi_{v}$, a
smooth admissible $\mathbf{C}$-representation of $G(L)$. After fixing an
identification $t:\overline{\mathbf{Q}}_{p}\cong\mathbf{C}$, we can find a
model of $t^{-1}\pi_{v}$ over some large enough field extension
$E/\mathbf{Q}_{p}$. Since $t^{-1}\pi_{v}$ can then be found as a
$\textnormal{GL}_{n}(F_{v})$-equivariant direct summand in
$(\varinjlim_{K_{v}\subset\textnormal{GL}_{n}(\mathcal{O}_{F_{v}})}H^{\ast}(X_{K^{v}K_{v}},\mathcal{V}_{\lambda}))\otimes_{\mathcal{O}}E$
and the normalised $U_{p}$-operators already act integrally on the latter by
[CN23], Lemma 2.1.17, we see that Lemma 4.11, i) is satisfied for any standard
parabolic subgroup, making $t^{-1}\pi_{v}\otimes_{E}V_{\lambda^{\vee}}$ into a
weakly admissible $E$-representation of $G(L)$.

######  Definition 4.13.

Let $\Pi$ be a weakly admissible $E$-representation of $G(L)$ of weight
$\lambda$. We say that $\Pi$ is $Q$-ordinary if
$\textnormal{Ord}_{Q}^{\textnormal{lalg}}(\Pi)\neq 0$. We denote the
subcategory of such by
$\textnormal{Mod}^{Q\textnormal{-ord},\lambda}_{E}(G(L))\subset\textnormal{Mod}_{E}^{\textnormal{wa},\lambda}(G(L))$.

######  Remark 4.14.

We also note that both the notion of weakly admissible and the notion of
ordinary generalises to any split reductive group over $L$ and we will be
speaking of weakly admissible and ordinary representations of Levi subgroups
$M(L)$ of $G(L)$ without further explanation.

We would also like to talk about parabolic induction for locally algebraic
representations.

######  Definition 4.15.

Let $\Pi_{M}=\pi_{M}\otimes V_{(w_{0}^{G}\lambda)^{\vee}}$ be a weakly
admissible $E$-representation of $M(L)$ of weight $w_{0}^{Q}\lambda$. We then
set its parabolic induction to be

$\textnormal{Ind}_{Q}^{G}(\Pi_{M}):=(\textnormal{Ind}_{Q(L)}^{G(L)}\pi)\otimes
V_{\lambda^{\vee}},$

a locally algebraic $E$-representation of $G(L)$ of weight $\lambda$.

Note that $\textnormal{Ind}_{Q}^{G}$ can only be applied to locally algebraic
representations with weight of the form $w_{0}^{Q}\lambda$.

The key result of the section is the following theorem.

######  Theorem 4.16.

Let $\Pi_{M}$ be a weakly admissible $E$-representation of $M(L)$ of weight
$w_{0}^{Q}\lambda$. Then $\textnormal{Ind}_{Q}^{G}(\delta_{Q}\otimes\Pi_{M})$
is a weakly admissible $E$-representation of $G(L)$ of weight $\lambda$ and we
have a natural isomorphism

(4.8)
$\textnormal{Ord}_{Q}^{\textnormal{lalg}}(\textnormal{Ind}_{Q}^{G}(\delta_{Q}\otimes\Pi_{M}))\cong\textnormal{Ord}_{M}^{\textnormal{lalg}}(\Pi_{M}).$

In particular, if $\Pi_{M}$ is $M$-ordinary, then
$\textnormal{Ind}_{Q}^{G}(\delta_{Q}\otimes\Pi_{M})$ is $Q$-ordinary and if
$\Pi_{M}$ is also absolutely irreducible, we have
$\textnormal{Ord}_{Q}^{\textnormal{lalg}}(\textnormal{Ind}_{Q}^{G}(\delta_{Q}\otimes\Pi_{M}))\cong\Pi_{M}.$

Before starting the proof, we prove some preliminary technical lemmas. The
proof of these lemmas and the proof of Theorem 4.16 was intentionally written
in a way so that it clearly generalises to split reductive groups other than
$\textnormal{GL}_{n}$. However, we kept the assumption $G=\textnormal{GL}_{n}$
for the whole section as later we will appeal to the specific features of the
representation theory of $\textnormal{GL}_{n}(L)$, such as the
Bernstein–Zelevinsky classification.

Introduce the following notation. Denote by $\Sigma_{G}$ the set of roots of
$(G,T)$ and by $\Sigma_{G}^{+}\subset\Sigma_{G}$ the subset of $B$-dominant
roots. Given a standard parabolic subgroup $M^{\prime}\ltimes
N^{\prime}=Q^{\prime}\subset G$, analogous notations apply to $M^{\prime}$.
Further denote by $\Sigma_{N^{\prime}}\subset\Sigma_{G}^{+}$ the roots
corresponding to $N^{\prime}$. Note that
$\Sigma_{G}^{+}=\Sigma_{M^{\prime}}^{+}\coprod\Sigma_{N^{\prime}}$ and
$\Sigma_{G}=-\Sigma_{N}\coprod\Sigma_{M^{\prime}}\coprod\Sigma_{N^{\prime}}$.

######  Lemma 4.17.

Let $M\ltimes N=Q,M^{\prime}\ltimes N^{\prime}=Q^{\prime}\subset G$ be two
standard parabolic subgroups, $\nu\in X_{\ast}(Z_{M^{\prime}})$ be a
cocharacter that is moreover $B$-dominant. Consider an element
$w\in{}^{Q^{\prime}}W^{Q}$ and define the standard parabolic subgroups
$M_{w^{-1}}\ltimes N_{w^{-1}}=Q_{w^{-1}}^{M}:=M\cap w^{-1}Q^{\prime}w\subset
M$ and $M_{w}^{\prime}\ltimes
N_{w}^{\prime}=Q_{w}^{M^{\prime}}:=M^{\prime}\cap wQw^{-1}\subset M^{\prime}$.
Then $w^{-1}\nu$ is in $X_{\ast}(Z_{M_{w^{-1}}})$ and is a $B\cap M$-dominant
cocharacter. In particular, the isomorphism
$M_{w}^{\prime}(L)\xrightarrow{\sim}M_{w^{-1}}(L)$, $m\mapsto w^{-1}mw$
restricts to a map $Z_{M^{\prime}}^{+}\to Z_{M_{w^{-1}}}^{+}$.313131Note that
$M_{w^{-1}}$ is considered as a Levi subgroup in $M$ and $Z_{M_{w^{-1}}}^{+}$
has its meaning accordingly.

###### Proof.

To see that $w^{-1}\nu$ lies in $X_{\ast}(Z_{M_{w^{-1}}})$, note that, by
[Hau18], Lemma 2.1.1, (ii), we have $M_{w^{-1}}=M\cap w^{-1}M^{\prime}w$ and
$M_{w}^{\prime}=M^{\prime}\cap wMw^{-1}$. In particular, conjugation by
$w^{-1}$ restricts to an isomorphism
$M_{w}^{\prime}\xrightarrow{\sim}M_{w^{-1}}$ from which the claim follows.

To verify that $w^{-1}\nu$ is $B\cap M$-dominant, we have to check that for
any $\alpha\in\Sigma_{M}^{+}$, we have $\langle w^{-1}\nu,\alpha\rangle\geq
0$. Equivalently, we prove that $\langle\nu,w\alpha\rangle\geq 0$. Since
$w\in{}^{Q^{\prime}}W^{Q}$, we know that $w\alpha\in\Sigma_{G}^{+}$ so the
inequality follows from the fact that $\nu$ is $B$-dominant.

For the last claim of the lemma, note that any $z\in Z_{M^{\prime}}^{+}$ must
be of the form $\nu(\varpi_{L})\cdot z^{\prime}$ for some $B$-dominant
cocharacter $\nu\in X_{\ast}(Z_{M^{\prime}})$ and $z^{\prime}\in
Z_{M^{\prime}}^{0}$. ∎

######  Lemma 4.18.

Let $M\ltimes N=Q$, $M^{\prime}\ltimes N^{\prime}=Q^{\prime}$,
$w\in{}^{Q^{\prime}}W^{Q}$, $Q_{w^{-1}}^{M}=M_{w^{-1}}\ltimes N_{w^{-1}}$ and
$Q_{w}^{M^{\prime}}=M^{\prime}_{w}\ltimes N^{\prime}_{w}$ as in Lemma 4.17.
Further consider that standard parabolic subgroups
$Q_{w^{-1}}=M_{w^{-1}}\ltimes(N_{w^{-1}}N)$,
$Q^{\prime}_{w}=M^{\prime}_{w}\ltimes(N^{\prime}_{w}N^{\prime})\subset G$.
Then, for any $z\in Z_{M^{\prime}}^{+}$, we have

$\delta_{Q^{\prime}_{w}}(z)\leq\delta_{Q_{w^{-1}}}(w^{-1}zw).$

Moreover, if $w\neq 1$, we can find $z\in Z_{M^{\prime}}^{+}$ such that
$\delta_{Q^{\prime}_{w}}(z)<\delta_{Q_{w^{-1}}}(w^{-1}zw)$.

###### Proof.

Recall that we have

$\delta_{Q^{\prime}_{w}}(z)=\left|\prod_{\alpha\in\Sigma_{N^{\prime}_{w}}\coprod\Sigma_{N^{\prime}}}\alpha(z)\right|_{L}$

and

$\delta_{Q_{w^{-1}}}(w^{-1}zw)=\left|\prod_{\alpha\in\Sigma_{N_{w^{-1}}}\coprod\Sigma_{N}}\alpha(w^{-1}zw)\right|_{L}=\left|\prod_{\alpha\in\Sigma_{N_{w^{-1}}}\coprod\Sigma_{N}}w\alpha(z)\right|_{L}.$

We also note that $w\in{}^{Q^{\prime}}W^{Q}$ means that we have

  1. (i)

$w^{-1}(-\Sigma_{G}^{+})\cap\Sigma_{G}^{+}\subset\Sigma_{N}$, and

  2. (ii)

$(-\Sigma_{G}^{+})\cap w(\Sigma_{G}^{+})\subset-\Sigma_{N^{\prime}}$.

Moreover, by [Hau18], Lemma 2.1.1, (ii), $w$ sets up a bijection between
$\Sigma_{M_{w^{-1}}}$ and $\Sigma_{M_{w}^{\prime}}$. In particular, we have

$w(\Sigma_{N_{w^{-1}}}\coprod\Sigma_{N})\subset\Sigma_{N^{\prime}_{w}}\coprod\Sigma_{N^{\prime}}\coprod-\Sigma_{N^{\prime}}.$

From this the first part of the lemma follows.

For the last part, we assume that $1\neq w$ and would like to find $z\in
Z_{M^{\prime}}^{+}$ such that
$\delta_{Q^{\prime}_{w}}(z)<\delta_{Q_{w^{-1}}}(w^{-1}zw)$. Note that if we
found an $\alpha\in\Sigma_{N}$ such that $w\alpha\in-\Sigma_{N^{\prime}}$,
then for any choice of a $B$-dominant cocharacter $\nu\in
X_{\ast}(Z_{M^{\prime}})$ with $\langle\nu,w\alpha\rangle<0$, $\nu(\varpi)$
would be a suitable choice for $z$. The existence of such an $\alpha$ is the
content of [AHV19], Lemma 5.13. ∎

###### Proof of Theorem 4.16.

We first prove that $\textnormal{Ind}_{Q}^{G}(\delta_{Q}\otimes\Pi_{M})$ is
weakly admissible. To do so, we fix a standard parabolic subgroup
$Q^{\prime}=M^{\prime}\ltimes N^{\prime}\subset G$ and write
$\Pi_{M}=\pi_{M}\otimes V_{-w_{0}^{G}\lambda}$. We would like to understand
the $Z_{M^{\prime}}^{+}$-action on
$J_{Q^{\prime}}(\textnormal{Ind}_{Q}^{G}(\delta_{Q}\otimes\Pi_{M}))$. By
applying the geometric lemma (cf. [BZ77], Lemma 2.12) to
$J_{Q^{\prime}}(\textnormal{Ind}_{Q(L)}^{G(L)}(\delta_{Q}\otimes\pi_{M}))$, we
obtain an $M^{\prime}(L)$-equivariant filtration of
$J_{Q^{\prime}}(\textnormal{Ind}_{Q}^{G}(\delta_{Q}\otimes\Pi_{M}))$ with
subquotients given by

$I_{w}^{Q^{\prime}}:=\biggl{(}\delta_{Q^{\prime}}^{1/2}\textnormal{Ind}_{Q_{w}^{M^{\prime}}}^{M^{\prime}}\Bigl{(}\delta_{Q_{w}^{M^{\prime}}}^{1/2}w^{\ast}\bigl{(}\delta_{Q_{w^{-1}}^{M}}^{-1/2}J_{Q_{w^{-1}}^{M}}(\delta_{Q}^{1/2}\pi_{M})\bigr{)}\Bigr{)}\biggr{)}\otimes
V_{(w_{0}^{Q}\lambda)^{\vee}}$

for $w\in{}^{Q^{\prime}}W^{Q}$ where $M^{\prime}_{w}\ltimes
N_{w}^{\prime}=Q_{w}^{M^{\prime}}:=M^{\prime}\cap wQw^{-1}\subset M^{\prime}$,
$M_{w^{-1}}\ltimes N_{w^{-1}}=Q_{w^{-1}}^{M}:=M\cap w^{-1}Q^{\prime}w$, and
$w^{\ast}(-)$ denotes the pullback along the isomorphism
$M_{w^{-1}}(L)\xrightarrow{\sim}M_{w}^{\prime}(L)$, $m\mapsto
wmw^{-1}$.333333For the fact that it is indeed an isomorphism, see [Hau18],
Lemma 2.1.1, (ii). To check weak admissibility, it suffices to see that for
any $w\in{}^{Q^{\prime}}W^{Q}$, any locally algebraic character
$\chi=\chi_{\textnormal{sm}}(-w_{0}^{G}\lambda):Z_{M^{\prime}}(L)\to\overline{\mathbf{Q}}_{p}^{\times}$
with a $Z_{M^{\prime}}(L)$-equivariant embedding $\chi\hookrightarrow
I_{w}^{Q^{\prime}}$ and any $z\in Z_{M^{\prime}}^{+}$, we have

(4.9)
$\textnormal{val}_{p}(\chi_{\textnormal{sm}}(z))\geq\textnormal{val}_{p}(w_{0}^{G}\lambda(z))+\textnormal{val}_{p}(\delta_{Q^{\prime}}(z)).$

Fix such a Weyl group element $w$, character $\chi$, and embedding
$\chi\hookrightarrow I_{w}^{Q^{\prime}}$. Set
$I_{w}^{Q^{\prime},\textnormal{sm}}$ to be the smooth part of
$I_{w}^{Q^{\prime}}$. Fix a $z\in Z_{M^{\prime}}^{+}$ and compute

$I_{w}^{Q^{\prime},\textnormal{sm}}(z)=\delta_{Q^{\prime}}^{1/2}(z)\delta_{Q_{w}^{M^{\prime}}}^{1/2}(z)\delta_{Q_{w^{-1}}^{M}}^{-1/2}(w^{-1}zw)J_{Q_{w^{-1}}^{M}}(\delta_{Q}^{1/2}\pi_{M})(w^{-1}zw)=$
$=\delta_{Q^{\prime}}^{1/2}(z)\delta_{Q_{w^{-1}}^{M}}^{-1/2}(w^{-1}zw)\delta_{Q}^{1/2}(w^{-1}zw)J_{Q_{w^{-1}}^{M}}(\pi_{M})(w^{-1}zw).$

Here we used that $z$ is a central element in $M^{\prime}(L)$. Since
$w\in{}^{Q^{\prime}}W^{Q}$, Lemma 4.17 shows that $w^{-1}zw\in
Z_{M_{w^{-1}}}^{+}$. In particular, by weak admissibility of $\Pi_{M}$, we see
that for any $Z_{M_{w^{-1}}}(L)$-equivariant embedding
$\widetilde{\chi}=\widetilde{\chi}_{\textnormal{sm}}(-w_{0}^{G}\lambda)\hookrightarrow
J_{Q_{w^{-1}}^{M}}(\Pi_{M})=J_{Q_{w^{-1}}^{M}}(\pi_{M})\otimes
V_{-w_{0}^{G}\lambda}$, we have

$\textnormal{val}_{p}(\widetilde{\chi}_{\textnormal{sm}}(w^{-1}zw))\geq\textnormal{val}_{p}(\delta_{Q_{w^{-1}}^{M}}(w^{-1}zw))+\textnormal{val}_{p}(w_{0}^{G}\lambda(w^{-1}zw)).$

Our chosen embedding $\chi\hookrightarrow I_{w}^{Q^{\prime}}$ then induces a
$w^{-1}Z_{M^{\prime}}(L)w$-equivariant embedding
$\widetilde{\chi}^{\prime}_{\textnormal{sm}}:=(w^{-1})^{\ast}(\chi_{\textnormal{sm}}\delta_{Q^{\prime}}^{-1/2})\delta_{Q_{w^{-1}}^{M}}^{1/2}\delta_{Q}^{-1/2}\hookrightarrow
J_{Q_{w^{-1}}^{M}}(\pi_{M})$. As a consequence, there exists a character
$\widetilde{\chi}:=\widetilde{\chi}_{\textnormal{sm}}(-w_{0}^{G}\lambda):Z_{M_{w^{-1}}}(L)\to\overline{\mathbf{Q}}_{p}^{\times}$
with an embedding $\widetilde{\chi}\hookrightarrow
J_{Q_{w^{-1}}^{M}}(\Pi_{M})$ such that
$\widetilde{\chi}_{\textnormal{sm}}|_{w^{-1}Z_{M^{\prime}}(L)w}=\widetilde{\chi}_{\textnormal{sm}}^{\prime}$.
Pick any such $\widetilde{\chi}$ and compute

(4.10)
$\textnormal{val}_{p}\bigl{(}\chi_{\textnormal{sm}}(z)\delta_{Q^{\prime}}^{-1/2}(z)\delta_{Q_{w^{-1}}^{M}}^{1/2}(w^{-1}zw)\delta_{Q}^{-1/2}(w^{-1}zw)\bigr{)}=\textnormal{val}_{p}(\widetilde{\chi}_{\textnormal{sm}}(w^{-1}zw))\geq$
$\geq\textnormal{val}_{p}(\delta_{Q_{w^{-1}}^{M}}(w^{-1}zw))+\textnormal{val}_{p}(w_{0}^{G}\lambda(w^{-1}zw))\geq\textnormal{val}_{p}(\delta_{Q_{w^{-1}}^{M}}(wzw^{-1}))+\textnormal{val}_{p}(w_{0}^{G}\lambda(z)).$

Here the last inequality follows from the fact that $\lambda$ is
$(\textnormal{Res}_{L/\mathbf{Q}_{p}}B)_{E}$-dominant. After rearranging 4.10,
we get

$\textnormal{val}_{p}(\chi_{\textnormal{sm}}(z))\geq\frac{1}{2}\Bigl{(}\textnormal{val}_{p}(\delta_{Q_{w^{-1}}^{M}}(w^{-1}zw))+\textnormal{val}_{p}(\delta_{Q}(w^{-1}zw))-\textnormal{val}_{p}(\delta_{Q^{\prime}}(z))\Bigr{)}+$
$+\textnormal{val}_{p}(\delta_{Q^{\prime}}(z))+\textnormal{val}_{p}(w_{0}^{G}\lambda(z)).$

In particular, it suffices to prove that

$\textnormal{val}_{p}\Bigl{(}\delta_{Q_{w^{-1}}^{M}}(w^{-1}zw)\delta_{Q}(w^{-1}zw)\delta_{Q^{\prime}}^{-1}(z)\Bigr{)}\geq
0.$

To see this, denote by $Q_{w^{-1}}\subset G$ the standard parabolic subgroup
$M_{w^{-1}}\ltimes(N_{w^{-1}}N)$ and by $Q^{\prime}_{w}\subset G$ the standard
parabolic subgroup $M_{w}^{\prime}\ltimes(N^{\prime}_{w}N^{\prime})$. Then one
has $\delta_{Q^{\prime}}(z)=\delta_{Q^{\prime}_{w}}(z)$ and
$\delta_{Q_{w^{-1}}^{M}}(w^{-1}zw)\delta_{Q}(w^{-1}zw)=\delta_{Q_{w^{-1}}}(w^{-1}zw)$.
In particular, we only need to see that
$\delta_{Q^{\prime}_{w}}(z)\leq\delta_{Q_{w^{-1}}}(w^{-1}zw)$. This is the
content of the first part of Lemma 4.18. We conclude that
$\textnormal{Ind}_{Q}^{G}(\delta_{Q}\otimes\Pi_{M})$ is indeed weakly
admissible.

We note that we essentially already proved the rest of the theorem, too.
Namely, recall that we have an $M(L)$-equivariant split injection

(4.11)
$\textnormal{Ord}_{Q}^{\textnormal{lalg}}(\textnormal{Ind}_{Q}^{G}(\delta_{Q}\otimes\Pi_{M}))\xhookrightarrow{\oplus}J_{Q}(\textnormal{Ind}_{Q}^{G}(\delta_{Q}\otimes\Pi_{M}))\otimes\delta_{Q}^{-1}$

with image given by the slope $0$ part (cf. §4.1). By the geometric lemma
applied to the RHS of 4.11, we get a surjection

(4.12)
$J_{Q}(\textnormal{Ind}_{Q}^{G}(\delta_{Q}\otimes\Pi_{M}))\otimes\delta_{Q}^{-1}\twoheadrightarrow
I_{1}^{Q}\otimes\delta_{Q}^{-1}=\Pi_{M}$

with its kernel $K$ admitting a filtration with subquotients given by
$I_{w}^{Q}\otimes\delta_{Q}^{-1}$ for $1\neq w\in{}^{Q}W^{Q}$. In particular,
if we prove that, for any $1\neq w\in{}^{Q}W^{Q}$, the slope zero part
$(\delta_{Q}^{-1}\otimes I_{w}^{Q})_{0}$ is trivial, then we immediately
obtain that 4.8 holds. Indeed, we then can conclude by applying slope $0$ part
to the short exact sequence

$0\to K\to
J_{Q}(\textnormal{Ind}_{Q}^{G}(\delta_{Q}\otimes\Pi_{M}))\otimes\delta_{Q}^{-1}\to\Pi_{M}\to
0.$

By the proof of Lemma 4.11, to prove the claimed description of $K$, it is
enough to check that for any $1\neq w\in{}^{Q}W^{Q}$ and any locally algebraic
character
$\chi=\chi_{\textnormal{sm}}(-w_{0}^{G}\lambda):Z_{M}(L)\to\overline{\mathbf{Q}}_{p}^{\times}$
with an embedding $\chi\hookrightarrow I_{w}\otimes\delta_{Q}^{-1}$, there is
a $z\in Z_{M}^{+}$ such that

$\textnormal{val}_{p}(\chi_{\textnormal{sm}}(z))>\textnormal{val}_{p}(w_{0}^{G}\lambda(z)).$

Based on the previous paragraph of the proof, to see this, we only need to
prove that we can find $z\in Z_{M}^{+}$ such that
$\delta_{Q_{w}}(z)<\delta_{Q_{w^{-1}}}(w^{-1}zw)$. This is the content of the
second part of Lemma 4.18. ∎

Theorem 4.16 shows that we obtain a functor

$\textnormal{Ind}_{Q}^{G}(-\otimes\delta_{Q}):\textnormal{Mod}_{E}^{\textnormal{wa},w_{0}^{Q}\lambda}(M(L))\to\textnormal{Mod}_{E}^{\textnormal{wa},\lambda}(G(L))$

that restricts to a functor

$\textnormal{Mod}_{E}^{M\textnormal{-ord},w_{0}^{Q}\lambda}(M(L))\to\textnormal{Mod}_{E}^{Q\textnormal{-ord},\lambda}(G(L)).$

Furthermore, we already constructed a functor

$\textnormal{Ord}_{Q}^{\textnormal{lalg}}(-):\textnormal{Mod}_{E}^{\textnormal{wa},\lambda}(G(L))\to\textnormal{Mod}_{E}^{\textnormal{wa},w_{0}^{Q}\lambda}(M(L)).$

We introduce the notion of $Z$-integral weakly admissible representations
which, by Theorem 4.16, form the essential image of
$\textnormal{Ord}_{Q}^{\textnormal{lalg}}$.

######  Definition 4.19.

Call a weakly admissible $E$-representation $\Pi$ of $G(L)$ (of weight
$\lambda$) $Z$-integral if
$\textnormal{Ord}_{G}^{\textnormal{lalg}}(\Pi)=\Pi$. We denote the
corresponding subcategory by
$\textnormal{Mod}_{E}^{Z\textnormal{-int},\lambda}(G(L))\subset\textnormal{Mod}_{E}^{\textnormal{wa},\lambda}(G(L))$.

######  Lemma 4.20.

The pair of functors
$(\textnormal{Ord}_{Q}^{\textnormal{lalg}},\textnormal{Ind}_{Q}^{G}(-\otimes\delta_{Q}))$,
between the categories $\textnormal{Mod}_{E}^{\textnormal{wa},\lambda}(G(L))$
and $\textnormal{Mod}_{E}^{Z\textnormal{-int},w_{0}^{Q}\lambda}(M(L))$, forms
an adjoint pair. The association in one direction sends, for
$\Pi\in\textnormal{Mod}_{E}^{\textnormal{wa},\lambda}(G(L))$,
$\Pi_{M}\in\textnormal{Mod}_{E}^{\textnormal{wa},w_{0}^{Q}\lambda}(M(L))$, a
map $f:\Pi\to\textnormal{Ind}_{Q}^{G}(\Pi_{M}\otimes\delta_{Q})$ to the
composition
$\textnormal{Ord}_{Q}^{\textnormal{lalg}}(\Pi)\xrightarrow{g_{1}}\textnormal{Ord}_{Q}^{\textnormal{lalg}}(\textnormal{Ind}_{Q}^{G}(\Pi_{M}\otimes\delta_{Q}))\xrightarrow{g_{2}}\Pi_{M}$
where $g_{1}$ is induced by applying
$\textnormal{Ord}_{Q}^{\textnormal{lalg}}$ to $f$ and $g_{2}$ is the
isomorphism of Theorem 4.16.

###### Proof.

By Frobenius reciprocity, maps of the form
$f:\Pi\to\textnormal{Ind}_{Q}^{G}(\Pi_{M}\otimes\delta_{Q})$ are in natural
bijection with maps of the form $g:J_{Q}(\Pi)\to\Pi_{M}\otimes\delta_{Q}$.
Since $\Pi_{M}$ is $Z$-integral, any such $g$ must factor through the split
surjection

(4.13)
$J_{Q}(\Pi)\twoheadrightarrow\textnormal{Ord}_{Q}^{\textnormal{lalg}}(\Pi)\otimes\delta_{Q}.$

Moreover, from the obtained map
$\textnormal{Ord}_{Q}^{\textnormal{lalg}}(\Pi)\otimes\delta_{Q}\to\Pi_{M}\otimes\delta_{Q}$,
$g$ can be recovered by precomposing it with 4.13. Finally, twisting by
$\delta_{Q}^{-1}$ certainly sets up a bijection of Hom sets.

That this association coincides with the one in the statement follows from an
easy diagram chase, the definition of Frobenius reciprocity and the definition
of the isomorphism 4.8. ∎

We record an immediate corollary that was the main motivation to prove Theorem
4.16.

######  Corollary 4.21.

Let $\Pi$ be an absolutely irreducible $Q$-ordinary $E$-representation of
$G(L)$ of weight $\lambda$. Then
$\textnormal{Ord}_{Q}^{\textnormal{lalg}}(\Pi)$ is an absolutely irreducible
locally algebraic $E$-representation of $M(L)$ of weight $w_{0}^{Q}\lambda$.

###### Proof.

Since $J_{Q}(\Pi)$ is of finite finite length, at the very least, so is
$\textnormal{Ord}_{Q}^{\textnormal{lalg}}(\Pi)$. After possibly enlarging $E$,
we can pick an absolutely irreducible quotient

(4.14)
$\textnormal{Ord}_{Q}^{\textnormal{lalg}}(\Pi)\twoheadrightarrow\Pi_{M}.$

In particular, $\Pi_{M}$ is $Z$-integral. Adjunction then gives a non-trivial
map $g:\Pi\to\textnormal{Ind}_{Q}^{G}(\Pi_{M}\otimes\delta_{Q})$ that must
then be an embedding. By Lemma 4.20, we recover 4.14 by applying
$\textnormal{Ord}_{Q}^{\textnormal{lalg}}(-)$ to $g$ and postcomposing it with
4.8. In particular, due to left exactness of
$\textnormal{Ord}_{Q}^{\textnormal{lalg}}$, 4.14 must also be an injection. ∎

As a consequence, for any absolutely irreducible object
$\Pi\in\textnormal{Mod}_{E}^{Q\textnormal{-ord},\lambda}(G(L))$, there is a
unique absolutely irreducible
$\Pi_{M}\in\textnormal{Mod}_{E}^{\textnormal{wa},w_{0}^{Q}\lambda}(M(L))$ such
that $\Pi$ embeds into $\textnormal{Ind}_{Q}^{G}(\Pi_{M}\otimes\delta_{Q})$.
We can then make the following definition.

######  Definition 4.22.

Given an absolutely irreducible
$\Pi\in\textnormal{Mod}_{E}^{Q\textnormal{-ord},\lambda}(G(L))$, we say that
$\Pi_{M}\in\textnormal{Mod}_{E}^{\textnormal{wa},w_{0}^{Q}\lambda}(M(L))$ is
its $Q$-ordinary support if it is the (necessarily unique) absolutely
irreducible representation in
$\textnormal{Mod}_{E}^{\textnormal{wa},w_{0}^{Q}\lambda}(M(L))$ with an
embedding
$\Pi\hookrightarrow\textnormal{Ind}_{Q}^{G}(\Pi_{M}\otimes\delta_{Q})$.

A very useful feature of the notion of the $Q$-ordinary support is that it can
be read off from $\Pi$ by applying
$\textnormal{Ord}_{Q}^{\textnormal{lalg}}(-)$.

For the rest of the subsection, we will be occupied with understanding the
relation between $\Pi$ and its $Q$-ordinary support in terms of the
Bernstein–Zelevinsky and Langlands classifications. In particular, it is this
point from which we make use of the assumption that $G=\textnormal{GL}_{n}$.
For a quick review on the Bernstein–Zelevinsky classification and the relevant
notations used in the rest of the section, we refer to the appendix.

######  Lemma 4.23.

Let $\Pi=\pi\otimes V_{\lambda^{\vee}}$ be an absolutely irreducible weakly
admissible $E$-representation of $G(L)$ of weight $\lambda$ that is
$Z$-integral. Then, for any
$\pi_{\textnormal{sc}}\in\textnormal{SC}(\pi)$343434Recall that we denote by
$\textnormal{SC}(\pi)$ the supercuspidal support of $\pi$., we have

(4.15) $\frac{1}{e}\sum_{\iota:L\hookrightarrow
E}(\lambda_{\iota,n}+\frac{1-n}{2})\leq\frac{\textnormal{val}_{p}(\pi_{\textnormal{sc}}(\langle\varpi\rangle))}{\textnormal{deg}(\pi_{\textnormal{sc}})}\leq\frac{1}{e}\sum_{\iota:L\hookrightarrow
E}(\lambda_{\iota,1}+\frac{n-1}{2}).$

###### Proof.

Assume that $\pi=Z(\Delta_{1},...\Delta_{k})$ for an ordered multiset of
segments $\underline{\Delta}:=(\Delta_{1},...,\Delta_{k})$ with
$\Delta_{i}:=\Delta(\pi_{i},r_{i})$ for $i=1,...,k$. Denote by
$Q_{\underline{\Delta}}\subset G$ the corresponding standard parabolic
subgroup. For $\Delta=\Delta(\sigma,r)$, set

$v_{\Delta}:=\frac{\sum_{i=0}^{r-1}\textnormal{val}_{p}(\sigma(i)(\langle\varpi\rangle))}{r\deg(\sigma)}=\frac{\textnormal{val}_{p}(\sigma(\langle\varpi\rangle))}{\deg(\sigma)}+\frac{[L:\mathbf{Q}_{p}]}{e}\frac{1-r}{2},$

the arithmetic mean value of the numbers
$\frac{\textnormal{val}_{p}(\pi_{\textnormal{sc}}(\langle\varpi\rangle))}{\deg(\pi_{\textnormal{sc}})}$
for $\pi_{\textnormal{sc}}\in\textnormal{SC}(Z(\Delta))$.

Note that we can assume that $\underline{\Delta}$ is ordered so that

(4.16) $(v_{\Delta_{1}},-r_{1})\leq...\leq(v_{\Delta_{k}},-r_{k})$

with respect to the lexicographic ordering. Similarly, we can assume that
$\underline{\Delta}$ is ordered so that

(4.17) $(v_{\Delta_{1}},r_{1})\leq...\leq(v_{\Delta_{k}},r_{k}).$

Indeed, if $\Delta_{i}$ and $\Delta_{j}$ are linked for some $1\leq i\neq
j\leq k$, say $\Delta_{i}$ precedes $\Delta_{j}$, then
$v_{\Delta_{j}}<v_{\Delta_{i}}$ and, by well-orderedness of the multiset of
segments in the sense of Bernstein–Zelevinsky, we must also have $j<i$.
Otherwise, if $\Delta_{i}$ and $\Delta_{j}$ are not linked, for instance if
$v_{\Delta_{i}}=v_{\Delta_{j}}$, we have the freedom of choosing the order.

If we choose $\underline{\Delta}$ so that 4.16 is satisfied, we see that

$v_{\Delta_{1}}+\frac{[L:\mathbf{Q}_{p}]}{e}\frac{1-r_{1}}{2}=\frac{\textnormal{val}_{p}(\pi_{1}(r_{1}-1)(\langle\varpi\rangle))}{\deg(\pi_{1})}\leq\frac{\textnormal{val}_{p}(\pi_{\textnormal{sc}}(\langle\varpi\rangle))}{\deg(\pi)}$

for any $\pi_{\textnormal{sc}}\in\textnormal{SC}(\pi)$. Similarly, if we
choose $\underline{\Delta}$ so that 4.17 is satisfied, we see that

$\frac{\textnormal{val}_{p}(\pi_{\textnormal{sc}}(\langle\varpi\rangle))}{\deg(\pi_{\textnormal{sc}})}\leq
v_{\Delta_{k}}+\frac{r_{k}-1}{2}=\frac{\textnormal{val}_{p}(\pi_{k})(\langle\varpi\rangle))}{\deg(\pi_{k})}$

for any $\pi_{\textnormal{sc}}\in\textnormal{SC}(\pi)$. In particular, it
suffices to prove that, for any choice of ordering $\underline{\Delta}$, we
have

(4.18)
$\frac{\textnormal{val}_{p}((\delta_{Q_{\underline{\Delta}}}^{1/2}w_{0}^{G}\lambda)(u_{p}^{(\deg(\Delta_{1}))}))}{\deg(\Delta_{1})}\leq
v_{\Delta_{1}}+\frac{[L:\mathbf{Q}_{p}]}{e}\frac{1-r_{1}}{2}$

and

(4.19)