# Ordinary parts and local-global compatibility at $\ell=p$

Bence Hevesi<EMAIL_ADDRESS>Department of Mathematics
King’s College London
Strand, London WC2R 2LS, United Kingdom

###### Abstract.

We prove local-global compatibility results at $\ell=p$ for the torsion
automorphic Galois representations constructed by Scholze, generalising the
work of Caraiani–Newton. In particular, we verify, up to a nilpotent ideal,
the local-global compatibility conjecture at $\ell=p$ of Gee–Newton in the
case of imaginary CM fields under some technical assumptions.

The key new ingredient is a local-global compatibility result for $Q$-ordinary
self-dual automorphic representations for arbitrary parabolic subgroups.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
  3. 3 $Q$-ordinary Hida theory and $Q$-ordinary parts of Bruhat strata
  4. 4 $Q$-ordinary parts in characteristic $0$
  5. 5 A torsion local-global compatibility conjecture
  6. 6 Proof of Theorem 5.7
  7. A Reminder on Bernstein–Zelevinsky and Langlands classifications

##  1\. Introduction

In this article, we are concerned with the question of local-global
compatibility of the global Langlands correspondence at $\ell=p$ for
$\textnormal{GL}_{n}$ over imaginary CM number fields. Verifying local-global
compatibility at $\ell=p$, among other things, proved to be essential in
establishing modularity lifting results. In the case of imaginary CM fields,
this problem has seen a breakthrough when the authors of [All+23] laid out and
executed a strategy to attack it. Another breakthrough came very recently with
the novel work of Caraiani–Newton. In [CN23], they introduced many new ideas
to the method, yielding a robust argument, and significantly stronger results.
In this article, we follow and seemingly push the ideas of [CN23] close to
their limit. As a result, in characteristic $0$ in the residually irreducible
case, under some genericity assumption, we prove that the automorphic Galois
representations constructed in [Har+16], [Sch15] satisfy the expected local-
global compatibility at $\ell=p$, up to only bounding the nilpotence of the
monodromy (see Theorem 1.1 below). Moreover, our results go beyond
characteristic $0$ automorphic Galois representations and yield local-global
compatibility for the torsion automorphic Galois representations constructed
in [Sch15] as well, vastly generalising the results of [All+23], and [CN23]
(see Theorem 1.4).

We now describe our results in more detail. Let $F$ be a number field and fix
a prime number $p$. Consider a regular algebraic cuspidal automorphic
representation (RACAR) $\pi$ of $\textnormal{GL}_{n}(\mathbf{A}_{F})$ and an
isomorphism $t:\overline{\mathbf{Q}}_{p}\cong\mathbf{C}$. Langlands
reciprocity predicts the existence of a continuous semisimple Galois
representation

$r_{t}(\pi):\textnormal{Gal}(\overline{F}/F)\to\textnormal{GL}_{n}(\overline{\mathbf{Q}}_{p})$

determined by the property that, for each finite place $v\notin
S:=\\{v\mid\pi_{v}\textnormal{ is ramified}\\}\cup\\{v|p\\}$ in $F$,
$r_{t}(\pi)|_{\textnormal{Gal}(\overline{F}_{v}/F_{v})}$ is unramified,
admitting an isomorphism

$\textnormal{WD}(r_{t}(\pi)|_{\textnormal{Gal}(\overline{F}_{v}/F_{v})})^{F-ss}\cong
t^{-1}\textnormal{rec}(\pi_{v}\otimes|\det|_{v}^{\frac{1-n}{2}}).$

Moreover, $r_{t}(\pi)$ is expected to satisfy local-global compatibility also
at places in $S$ governed by the shape of $\pi$ at $S$ and at $\infty$. In
particular, for a place $v$ of $F$ above $p$, it is predicted that
$r_{t}(\pi)$ is de Rham at $v$ with labelled Hodge–Tate weights that can be
read off from the infinitesimal character of $\pi_{\infty}$, and it further
admits an isomorphism

$\textnormal{WD}(r_{t}(\pi)|_{\textnormal{Gal}(\overline{F}_{v}/F_{v})})^{F-ss}\cong
t^{-1}\textnormal{rec}(\pi_{v}\otimes|\det|_{v}^{\frac{1-n}{2}}).$

Assuming that $F$ is an imaginary CM field, the representations $r_{t}(\pi)$
have been constructed in [Har+16], and [Sch15]. We prove the following, making
progress on verifying local-global compatibility at $p$ for the automorphic
Galois representations of [Har+16].

######  Theorem 1.1.

Let $F$ be an imaginary CM field, $t:\overline{\mathbf{Q}}_{p}\cong\mathbf{C}$
be any isomorphism, and $\pi$ be a regular algebraic cuspidal automorphic
representation of $\textnormal{GL}_{n}(\mathbf{A}_{F})$ of weight
$\lambda\in(\mathbf{Z}_{+}^{n})^{{\rm{Hom}}(F,\mathbf{C})}$. Assume that

  * •

$\overline{r_{t}(\pi)}$ is irreducible and decomposed generic in the sense of
[CS19] (see Definition 2.24).

Then, for any $p$-adic place $v|p$ of $F$, and
$\iota:F_{v}\hookrightarrow\overline{\mathbf{Q}}_{p}$, $r_{t}(\pi)$ is de Rham
at $v$, with labelled $\iota$-Hodge–Tate weights
$\lambda_{t\circ\iota,n}<...<\lambda_{t\circ\iota,1}+n-1$ and we have

(1.1)
$\textnormal{WD}(r_{t}(\pi)|_{\textnormal{Gal}(\overline{F}_{v}/F_{v})})^{F-ss}\preceq
t^{-1}\textnormal{rec}(\pi_{v}\otimes|\det|_{v}^{\frac{1-n}{2}}).$

######  Remark 1.2.

For the notion of "$\preceq$" in 1.1, see §2.6. It in particular means that we
have an isomorphism

$\textnormal{WD}(r_{t}(\pi)|_{\textnormal{Gal}(\overline{F}_{v}/F_{v})})^{ss}\cong(t^{-1}\textnormal{rec}(\pi_{v}\otimes|\det|_{v}^{\frac{1-n}{2}}))^{ss}$

and additionally says that, in some precise sense, the monodromy of the LHS is
at least as nilpotent as the monodromy of the RHS. As a consequence, assuming
that monodromy of $\textnormal{rec}(\pi_{v}\otimes|\det|_{v}^{\frac{1-n}{2}})$
is in fact $0$, we obtain an isomorphism

$\textnormal{WD}(r_{t}(\pi)|_{\textnormal{Gal}(\overline{F}_{v}/F_{v})})^{F-ss}\cong
t^{-1}\textnormal{rec}(\pi_{v}\otimes|\det|_{v}^{\frac{1-n}{2}}).$

######  Remark 1.3.

With the additional assumption that $\pi$ is also conjugate self-dual, local-
global compatibility of $r_{t}(\pi)$ is known in full generality at each place
by work of many people (see [Shi20] for a survey on this).

Beyond the self-dual case, analogous local-global compatibility at places
prime-to-$p$ was proved in [Var14]. However, having a better understanding of
the monodromy is proved to be a rather subtle task. Nevertheless, there is
work of Allen–Newton [AN21] that deals with the case of weight $0$ automorphic
representations of $\textnormal{GL}_{2}(\mathbf{A}_{F})$.

At $\ell=p$, such local-global compatibility results were first established in
[All+23] where they treated the Fontaine–Laffaille and ordinary cases. Their
so-called "degree shifting argument" is essentially the only known way to
attack local-global compatibility questions without self-duality conditions
and relies heavily on the deep vanishing result for unitary Shimura varieties
of [CS19].111This is where the decomposed generic assumption of our results
originates. As we will discuss, it is an important feature of their work that
they also prove local-global compatibility for Galois representations attached
to genuine torsion Hecke eigenclasses. Presence of such torsion classes is one
of the main subtleties of our setup.

Very recently, there is also work of A’Campo, [ACa23], where he proves that
$r_{t}(\pi)$ is de Rham and specifies the Hodge–Tate weights. Although his
argument is a generalisation of the (Fontaine–Laffaille) degree shifting
argument, it only works in characteristic $0$.

Finally, Caraiani–Newton [CN23] go further and prove that automorphic Galois
representations coming from Hecke eigenclasses with hyperspecial level at $p$
must be crystalline with the correct Hodge–Tate weights (cf. loc. cit. Theorem
4.3.1). The novelty of their work lies in a clever combination of a
generalisation of the Fontaine–Laffaille and ordinary degree shifting
arguments. This adds up to a very robust method that keeps track of torsion as
well. Our work generalises theirs to deal with arbitrary level and to also
keep track of local Hecke operators at $p$.

We deduce Theorem 1.1 from a more general result that also deals with the
torsion automorphic Galois representations constructed in [Sch15]. In this
generality, formulating a precise conjecture already requires some work that
we summarise now assuming that $F$ is an imaginary CM field and for details we
point the reader to §5.

Let $K\subset\textnormal{GL}_{n}(\mathbf{A}_{F}^{\infty})$ be a good compact
open subgroup (in the sense of [All+23], §2.1.1) with
$K_{p}=\prod_{v}\textnormal{GL}_{n}(\mathcal{O}_{F_{v}})$ and let $X_{K}$ be
the corresponding locally symmetric space for $\textnormal{GL}_{n_{/F}}$. Let
$E\subset\overline{\mathbf{Q}}_{p}$ be a large enough finite field extension
so that the images of all embeddings
$F\hookrightarrow\overline{\mathbf{Q}}_{p}$ lie in $E$, and set $\mathcal{O}$
to be its ring of integers, and $\varpi\in\mathcal{O}$ be a choice of
uniformiser. Given a highest weight vector $\lambda=(\lambda_{v})_{v|p}$ for
$(\textnormal{Res}_{F/\mathbf{Q}}\textnormal{GL}_{n})_{E}$, and a so-called
Weil–Deligne inertial type222This is simply, for each $p$-adic place $v$ of
$F$, a choice of an isomorphism class of a pair of an inertial type and a
monodromy operator. $\tau=(\tau_{v})_{v|p}$ at $p$ (see §2.6 for a
definition), we obtain an $\mathcal{O}$-local system
$\mathcal{V}_{(\lambda,\tau)}$ on $X_{K}$. The corresponding Betti cohomology
groups $H^{\ast}(X_{K},\mathcal{V}_{(\lambda,\tau)})$ admit an action of an
abstract Hecke algebra
$\mathbf{T}^{\lambda,\tau}=\mathbf{T}\otimes_{\mathcal{O}}\mathfrak{z}_{\lambda,\tau}^{\circ}$.
Here $\mathbf{T}$ is the usual abstract spherical Hecke algebra (over
$\mathcal{O}$) acting at unramified places and
$\mathfrak{z}_{\lambda,\tau}^{\circ}=\otimes_{v|p}\mathfrak{z}_{\lambda_{v},\tau_{v}}^{\circ}$
is an $\mathcal{O}$-flat algebra consisting of Hecke operators at $p$
admitting a natural identification
$\mathfrak{z}_{\lambda_{v},\tau_{v}}^{\circ}[1/p]\cong\mathfrak{z}_{\tau_{v}}$
where the latter denotes the Bernstein centre corresponding to $\tau_{v}$.
Denote by $\mathbf{T}^{\lambda,\tau}(K)$ the quotient of
$\mathbf{T}^{\lambda,\tau}$ acting faithfully on
$H^{\ast}(X_{K},\mathcal{V}_{(\lambda,\tau)})$. Given a maximal ideal
$\mathfrak{m}\subset\mathbf{T}^{\lambda,\tau}(K)$, Scholze constructed a
continuous Galois representation

$\overline{\rho}_{\mathfrak{m}}:\textnormal{Gal}(\overline{F}/F)\to\textnormal{GL}_{n}(\mathbf{T}^{\lambda,\tau}(K)/\mathfrak{m})$

and, assuming that $\overline{\rho}_{\mathfrak{m}}$ is absolutely irreducible
(in other words, $\mathfrak{m}$ is non-Eisenstein), a lift to a continuous
representation

$\rho_{\mathfrak{m}}:\textnormal{Gal}(\overline{F}/F)\to\textnormal{GL}_{n}(\mathbf{T}^{\lambda,\tau}(K)_{\mathfrak{m}}/I),$

where $I\subset\mathbf{T}^{\lambda,\tau}(K)_{\mathfrak{m}}$ is a nilpotent
ideal with $I^{4}=0$ (for the bound on the nilpotence degree, see [NT16]).
Both of these representations are determined by insisting that they satisfy
local-global compatibility at prime-to-$p$ unramified places.

As explained by the work of Calegari–Geraghty ([CG18]), to prove automorphy
lifting theorems in the positive defect333For the definition of the defect of
a group, we point the reader for instance to [GN20], §2.1. case (such as the
case of $\textnormal{GL}_{n}$ over an imaginary CM field), it is essential to
verify local-global compatibility of $\rho_{\mathfrak{m}}$. The reason why
this does not follow from characteristic $0$ local-global compatibility is
that, as we move away from the defect $0$ case (such as the case of
$\textnormal{GL}_{2}$ over a totally real field or, more generally, the case
of Shimura varieties), the cohomology
$H^{\ast}(X_{K},\mathcal{V}_{(\lambda,\tau)})_{\mathfrak{m}}$ typically
contains genuine torsion.444To support this sentence mathematically, we invite
the reader to look at for instance [BV13] where, in the defect $1$ case (such
as the case of $\textnormal{GL}_{2}$ over an imaginary quadratic field), they
prove exponential growth of torsion in the cohomology of locally symmetric
spaces as we deepen the level while the space of characteristic $0$
automorphic forms contributing to the cohomology stays small. In particular,
one seeks an integral version of local-global compatibility. Such a conjecture
was formulated in [CG18] in the Fontaine–Laffaille case (cf. loc. cit.
Conjecture B) and indeed, under some assumptions, this was proved in [All+23].
The special feature of the Fontaine–Laffaille case is that there the integral
$p$-adic Hodge theory is as well-behaved as it could be. As a consequence, one
can make a conjecture that is reminiscent of the characteristic $0$ one.
Similarly, the ordinary case of [All+23] is again special enough to make a
more naive yet valid prediction. However, as one moves away from these two
cases, it is less clear what one should ask for. The first signs (known to the
author) of a general conjecture (in the potentially crystalline case) appeared
in [Car+16] and a precise formulation (in the crystalline case) can be found
in [GN20], Conjecture 5.1.12. Caraiani and Newton recently proved a large part
of the conjecture of Gee–Newton (cf. [CN23], Theorem 1.3).

The two key ideas in [GN20], Conjecture 5.1.12 are to use Kisin’s potentially
semistable ($p$-torsion free) local deformation rings and that semisimple
local Langlands can be interpolated over the generic fibre of these
deformation rings. In particular, we obtain a workable notion of being
"torsion potentially semistable of a given weight and inertial type".
Moreover, by $p$-adically rescaling, we can test compatibility of the action
of Hecke operators at $p$ using the interpolation map.

Namely, for $v|p$ setting
$\rho_{v}:=\rho_{\mathfrak{m}}|_{\textnormal{Gal}(\overline{F}_{v}/F_{v})}$,
an $\mathcal{O}$-flat quotient
$R^{\lambda_{v},\preceq\tau_{v}}_{\overline{\rho}_{v}}$ of the unrestricted
local framed deformation ring $R^{\Box}_{\overline{\rho}_{v}}$ is constructed
in [Kis07]. It is characterised by the property that its $\overline{E}$-points
$\rho:\textnormal{Gal}(\overline{F}_{v}/F_{v})\to\textnormal{GL}_{n}(\overline{E})$
are exactly those lifts of $\overline{\rho}_{v}$ that are potentially
semistable with Hodge–Tate weights determined by $\lambda_{v}$ (by the usual
$\rho$-shift) and such that the Weil–Deligne inertial type of
$\textnormal{WD}(\rho)$ is bounded by $\tau_{v}$. Moreover, on the generic
fiber one can construct a map

$\eta:\mathfrak{z}_{\tau_{v}}\to
R^{\lambda_{v},\preceq\tau_{v}}_{\overline{\rho}_{v}}[1/p]$

interpolating the (Tate-normalised) semisimple local Langlands correspondence.
Although $\eta$ does not necessarily send
$\mathfrak{z}_{\lambda_{v},\tau_{v}}^{\circ}$ into
$R^{\lambda_{v},\preceq\tau_{v}}_{\overline{\rho}_{v}}$, we can introduce the
subring

$\mathfrak{z}_{\lambda_{v},\tau_{v}}^{\circ,\textnormal{int}}:=\eta^{-1}(R_{\overline{\rho}_{v}}^{\lambda_{v},\preceq\tau_{v}})\cap\mathfrak{z}_{\lambda_{v},\tau_{v}}^{\circ}\subset\mathfrak{z}_{\lambda_{v},\tau_{v}}^{\circ}.$

This ring still has the property
$\mathfrak{z}_{\lambda_{v},\tau_{v}}^{\circ,\textnormal{int}}[1/p]=\mathfrak{z}_{\tau_{v}}$.
Local-global compatibility can then be phrased by asking that there exists a
(necessarily unique) dotted arrow making the diagram

(1.2)
${R^{\Box}_{\overline{\rho}_{v}}}$${\mathbf{T}^{\lambda,\tau}(K)_{\mathfrak{m}}/I}$${R^{\lambda_{v},\preceq\tau_{v}}_{\overline{\rho}_{v}}}$${\mathfrak{z}_{\lambda_{v},\tau_{v}}^{\circ,\textnormal{int}}}$${R^{\lambda_{v},\preceq\tau_{v}}_{\overline{\rho}_{v}}[1/p]}$${\mathfrak{z}_{\tau_{v}}}$$\scriptstyle{\rho_{v}}$nat$\scriptstyle{\eta\mid_{\mathfrak{z}_{\lambda_{v},\tau_{v}}^{\circ,\textnormal{int}}}}$$\scriptstyle{\eta}$

commutative. Here the map nat denotes the natural map towards the faithful
Hecke algebra (along the inclusion
$\mathfrak{z}_{\lambda_{v},\tau_{v}}^{\circ,\textnormal{int}}\subset\mathfrak{z}_{\lambda_{v},\tau_{v}}^{\circ}$)
and the inclusions in the bottom are the ones induced by inverting $p$. Our
main result is then (roughly) as follows (cf. Theorem 5.7).

######  Theorem 1.4.

Let $F$ be an imaginary CM field that contains an imaginary quadratic field
$F_{0}$ in which $p$ splits with totally real subfield $F^{+}$. Let $\bar{v}$
be a $p$-adic place in $F^{+}$. Assume the following:

  1. (i)

There is a $p$-adic place $\bar{v}^{\prime}$ of $F^{+}$ such that
$\bar{v}\neq\bar{v}^{\prime}$ and

$\sum_{\bar{v}^{\prime\prime}\neq\bar{v},\bar{v}^{\prime}}[F^{+}_{\bar{v}^{\prime\prime}}:\mathbf{Q}_{p}]\geq\frac{1}{2}[F^{+}:\mathbf{Q}]$

where the sum runs over $p$-adic places of $F^{+}$;

  2. (ii)

$\overline{\rho}_{\mathfrak{m}}$ is decomposed generic.

Then, up to possibly enlarging $I$, there is a (necessarily unique) dotted
arrow making (1.2) commutative.

######  Remark 1.5.

The actual statement puts further mild constrains on $F$. This is so that we
can appeal to the unconditional base change results of [Shi14] and these
constraints are already present in [Sch15].

Assumption i) is essential to our methods and already appears in [All+23], and
[CN23]. In particular, this rules out the case of $F$ being an imaginary
quadratic field. Using Theorem 1.1, it seems plausible to weaken this
assumption, for instance allowing one to prove the theorem for $n=2$, and
$[F:\mathbf{Q}]=4$.

Finally, assumption ii) on $\overline{\rho}_{\mathfrak{m}}$ is crucial to be
able to use the vanishing results of [CS19].

######  Remark 1.6.

We point out that Theorem 1.4 yields a new result even in the crystalline case
since it also matches the crystalline Frobenius with the Frobenius at $v$
under local Langlands. Namely, when $\tau_{v}$ is assumed to be trivial (i.e.
when the inertial type is the trivial representation of the inertia subgroup
and the monodromy is the $0$ matrix), [CN23] proves the existence of a dotted
arrow making the top triangle of (1.2) commute. Note that in the crystalline
case $\mathfrak{z}_{\tau_{v}}\cong E[T_{1},...,T_{n}^{\pm 1}]$ is the usual
spherical Hecke algebra, and
$\mathfrak{z}_{\lambda_{v},\tau_{v}}^{\circ}\subset\mathfrak{z}_{\tau_{v}}$ is
some suitable subalgebra acting on
$H^{\ast}(X_{K},\mathcal{V}_{(\lambda,\tau)})$. Therefore, the commutativity
of the lower triangle of (1.2) roughly says that, beyond
$\rho_{\mathfrak{m}}|_{\textnormal{Gal}(\overline{F}_{v}/F_{v})}$ being
crystalline with the right Hodge–Tate weights, its crystalline Frobenius too
is compatible with the local Langlands correspondence (i.e. with the map
$\eta$). In particular, in characteristic $0$, for a given automorphic
representation $\pi$, we prove that the crystalline Frobenius on the filtered
$\varphi$-module associated with
$r_{t}(\pi)|_{\textnormal{Gal}(\overline{F}_{v}/F_{v})}$ coincides with the
Frobenius on
$t^{-1}\textnormal{rec}(\pi_{v}\otimes|\det|^{\frac{1-n}{2}}_{v})$.

Although in general for proving small $R=\mathbf{T}$ results it often suffices
to prove factorisation through the right potentially semistable deformation
rings at $p$, we remark that understanding the crystalline Frobenius is for
instance essential to make the big $R=\mathbf{T}$ results of [GN20], §5
unconditional.

In the rest of the introduction, we briefly explain our approach to Theorem
1.4 focusing on what it adds to the proof of [CN23]. The first observation in
the proof of loc. cit. is that to ensure the existence of a dotted arrow
making the top triangle of 1.2 commute, it suffices to construct a
characteristic $0$ lift of $\rho_{v}$ mod $\varpi^{m}$ with the right $p$-adic
Hodge theoretic properties for every $m\in\mathbf{Z}_{\geq 1}$. More
precisely, assume for a second that $\tau$ is trivial and fix integers $m\geq
1$, $q\geq 0$. Setting
$A:=\mathbf{T}(H^{q}(X_{K},\mathcal{V}_{\lambda}/\varpi^{m})_{\mathfrak{m}})$,
Caraiani and Newton construct a finite flat $\mathcal{O}$-algebra
$\widetilde{A}$,555$\widetilde{A}$ is a faithful Hecke algebra for some
$P$-ordinary middle degree (Betti) cohomology of some $U(n,n)$ Shimura variety
where $P$ is the Siegel parabolic subgroup in $U(n,n)$. a so-called degree
shifting map

$\overline{\mathcal{S}}:\widetilde{A}\to A,$

and a lift
$\widetilde{\rho}_{v}:\textnormal{Gal}(\overline{F}_{v}/F_{v})\to\textnormal{GL}_{n}(\widetilde{A})$
of
$\rho_{v}:\textnormal{Gal}(\overline{F}_{v}/F_{v})\to\textnormal{GL}_{n}(A)$
under $\overline{\mathcal{S}}$. Moreover, after inverting $p$, the lift
$\widetilde{\rho}_{v}$ becomes crystalline with the expected Hodge–Tate
weights. Then a simple diagram chase gives the factorisation of the top
triangle in 1.2 for
$\rho_{v}:\textnormal{Gal}(\overline{F}_{v}/F_{v})\to\textnormal{GL}_{n}(A)$.
Taking limit over $m$ then finishes the proof. In particular, their proof
consists of two main steps. The first of which is constructing the transfer
map $\overline{\mathcal{S}}$ (that purely takes place on the automorphic side)
descending the Satake transfer for the parabolic subgroup $P$ in $U(n,n)$. The
other step for them is proving a $P$-ordinary local-global compatibility
result in characteristic $0$ for conjugate self-dual automorphic Galois
representations (cf. [CN23], Theorem 3.1.2).

For us to conclude, we need an enrichment of their output. Namely, now set
$\tau$ to be arbitrary, and consider
$A^{\lambda,\tau}:=\mathbf{T}^{\lambda,\tau}(H^{q}(X_{K},\mathcal{V}_{(\lambda,\tau)})_{\mathfrak{m}})$
that is now naturally a $\mathfrak{z}_{\lambda_{v},\tau_{v}}^{\circ}$-algebra.
We too construct a finite flat $\mathcal{O}$-algebra
$\widetilde{A}^{\lambda_{v},\tau_{v}}$ by similarly taking the faithful Hecke
algebra of some $P$-ordinary middle degree cohomology of a $U(n,n)$-Shimura
variety. However, in our case we prove that it is equipped with the structure
of a $\mathfrak{z}_{\lambda_{v},\tau_{v}}^{\circ}$-algebra. We then perform
the degree shifting but we further carefully make sure that the obtained map
$\overline{\mathcal{S}}^{v}$ is a map of
$\mathfrak{z}_{\lambda_{v},\tau_{v}}^{\circ}$-algebras. This requires some
extension of Hida theory for torsion Betti cohomology, and a study of how
Hecke operators at $p$ behave under Poincaré duality. The other ingredient in
the proof of Theorem 1.4 is a strong enough $P$-ordinary local-global
compatibility result in characteristic $0$ for conjugate self-dual automorphic
representations. Here our main result is as follows (cf. Theorem 4.27 and
Theorem 4.30).

######  Theorem 1.7.

Let $\pi$ be a regular algebraic conjugate self-dual cuspidal automorphic
representation of $\textnormal{GL}_{n}(\mathbf{A}_{F})$ of weight
$\lambda\in(\mathbf{Z}^{n}_{+})^{{\rm{Hom}}(F,\mathbf{C})}$,
$t:\overline{\mathbf{Q}}_{p}\cong\mathbf{C}$ be a fixed identification and
$v|p$ be a $p$-adic place of $F$. Let $Q\subset\textnormal{GL}_{n}$ be the
standard parabolic subgroup corresponding to a partition $n=n_{1}+...+n_{t}$,
and denote by $M$ its Levi quotient. Assume that $t^{-1}\pi_{v}$ is
$Q$-ordinary of weight $\lambda_{v}$.

Then the $Q$-ordinary part $(t^{-1}\pi_{v})^{Q\textnormal{-ord}}$ is an
irreducible smooth $\overline{\mathbf{Q}}_{p}$-representation of
$M(F_{v})=\textnormal{GL}_{n_{1}}(F_{v})\times...\times\textnormal{GL}_{n_{t}}(F_{v})$.
Write $(t^{-1}\pi_{v})^{Q\textnormal{-ord}}=\pi_{1}\otimes...\otimes\pi_{t}$.
Then there is moreover an isomorphism

$r_{t}(\pi)|_{\textnormal{Gal}(\overline{F}_{v}/F_{v})}\sim\begin{pmatrix}\rho_{1}&\ast&...&\ast\\\
0&\rho_{2}&...&\ast\\\ .&.&.&.\\\ .&.&.&.\\\ 0&...&0&\rho_{t}\end{pmatrix}$

where, for $1\leq j\leq t$,

$\rho_{j}:\textnormal{Gal}(\overline{F}_{v}/F_{v})\to\textnormal{GL}_{n_{j}}(\overline{\mathbf{Q}}_{p})$

is potentially semistable such that, for every embedding
$\iota:F_{v}\hookrightarrow\overline{\mathbf{Q}}_{p}$, the labelled
$\iota$-Hodge–Tate weights of $\rho_{j}$ are given by

$\lambda_{t\circ\iota,n+1-(n_{1}+...+n_{j})}+n_{1}+...+n_{j}-1>...>\lambda_{t\circ\iota,n+1-(n_{1}+...+n_{j-1}+1)}+n_{1}+...+n_{j-1}$

and we have an isomorphism

$\textnormal{WD}(\rho_{j})^{F-ss}\cong\textnormal{rec}(\pi_{j}\otimes|\det|^{n_{1}+...+n_{j-1}}_{v}\otimes|\det|_{v}^{\frac{1-n_{j}}{2}}).$

With Theorem 1.7, and the degree shifting map $\overline{\mathcal{S}}^{v}$ in
hand, the arguments of [CN23] (most notably the determinant argument of loc.
cit. §3.2) produce a lift
$\widetilde{\rho}_{v}:\textnormal{Gal}(\overline{F}_{v}/F_{v})\to\textnormal{GL}_{n}(\widetilde{A}^{\lambda_{v},\tau_{v}})$
of $\rho_{v}$ under $\overline{\mathcal{S}}^{v}$. Thanks to Theorem 1.7, there
is a dotted arrow making a diagram like 1.2 commute for
$\widetilde{\rho}_{v}$. Then an easy diagram chase using that
$\overline{\mathcal{S}}^{v}$ is a
$\mathfrak{z}_{\lambda_{v},\tau_{v}}^{\circ}$-algebra map proves the same for
$\rho_{v}$ valued in $A^{\lambda,\tau}$. Passing to the limit over $m$ then
allows one to conclude.

######  Remark 1.8.

Theorem 1.7 generalises the ordinary local-global compatibility result
[Ger18], Corollary 2.7.8 (see also [Tho15], Theorem 2.4) to the case of a
general standard parabolic subgroup. The main extra difficulty is having a
generalisation of [Ger18], Lemma 5.4 2) for $Q$-ordinary parts for a general
standard parabolic subgroup. This is the first part of Theorem 1.7 and is a
question purely in the realm of smooth representations of $p$-adic reductive
groups. This part of the theorem we prove in a rather general setup in §4.2
that might be of independent interest.

######  Remark 1.9.

As it does not require any significant extra work, we in fact also produce the
lifts $\widetilde{\rho}_{v}$ in a greater generality where the $A$ now plays
the role of a faithful Hecke algebra for $Q$-ordinary Betti cohomology for
$\textnormal{GL}_{n/F}$. Although we do not investigate it further, coupled
with a suitable $Q$-ordinary local-global compatibility conjecture using some
potentially semistable $Q$-ordinary deformation rings, it seems to have the
potential to prove integral $Q$-ordinary local-global compatibility.

The organisation of the paper is as follows. In Section 2, we collect the
preliminaries on the cohomology of locally symmetric spaces and introduce the
necessary local systems and the corresponding Hecke actions for the group
$\textnormal{GL}_{n/F}$ and the quasi-split unitary group $\widetilde{G}$. In
Section 3, we recollect and generalise Hida theory in the Betti setting for
general standard parabolic subgroups. In Section 4, we revisit and further
develop a theory of ordinary parts for locally algebraic representations.
Moreover, we finish the section with proving Theorem 1.7. In Section 5, we
formulate a torsion local-global compatibility conjecture following [GN20]. In
Section 6, we execute the strategy outlined in the introduction to prove
Theorem 1.4 and conclude the section with deducing Theorem 1.1. In the
Appendix, we make a brief recollection on the smooth representation theory of
$\textnormal{GL}_{n}$ over a $p$-adic field and prove a couple of relevant
technical lemmas that are only used at the end of §4.2.

### Acknowledgements

First and foremost, I would like to express my gratitude to my advisor Ana
Caraiani for suggesting this project, generously sharing her knowledge and
ideas on the subject and for spending much of her time listening to my ideas.
This work would not have been completed without her support. I am also
indebted to James Newton for the many illuminating conversations and for
constantly showing interest in my work. I am grateful to my other advisor Fred
Diamond for providing plenty of feedback on this work during our meetings and
asking even more great questions. I thank Lambert A’Campo, Ana Caraiani, Fred
Diamond, James Newton and Matteo Tamiozzo for comments on an earlier draft of
this paper.

This work was supported by the Engineering and Physical Sciences Research
Council [EP/S021590/1]. The EPSRC Centre for Doctoral Training in Geometry and
Number Theory (The London School of Geometry and Number Theory), University
College London, King’s College London and Imperial College London.

## Notation and Conventions

Given a number field $F$, we will denote by $S(F)$ its set of finite places
and by $S_{p}(F)$ its set of $p$-adic places. We set $G_{F}$ to be the
absolute Galois group $\textnormal{Gal}(\overline{F}/F)$ and for a finite set
$S\subset S(F)$ we denote by $G_{F,S}$ the quotient of $G_{F}$ corresponding
to the maximal Galois extension of $F$, unramified outside $S$. We denote by
$\mathbf{A}_{F}$ the ring of adeles of $F$ and for $S$ a finite set of places
of $F$ we denote by $\mathbf{A}_{F}^{S}$ its prime-to-$S$ part and by
$\mathbf{A}_{F,S}$ its $S$-part. For $v\in S(F)$, set $F_{v}$ to be the
$v$-adic completion of $F$, fix a choice of uniformiser $\varpi_{v}$ and set
$k_{v}:=\mathcal{O}_{F_{v}}/\varpi_{v}$ to be its residue field. Set
$G_{F_{v}}:=\textnormal{Gal}(\overline{F}_{v}/F_{v})$. Moreover,
$I_{F_{v}}\subset G_{F_{v}}$ will denote its inertia subgroup and set
$\textnormal{Frob}_{v}\in G_{F_{v}}/I_{F_{v}}$ to be the geometric Frobenius.

For $G$ a reductive group over a number field $F$ and a finite set $S\subset
S(F)$, we will denote by $G^{S}:=G(\mathbf{A}_{F}^{S\cup\infty})$ and by
$G_{S}:=G(\mathbf{A}_{F,S})$. Moreover, if $S=S_{p}(F)$, we only write
$G^{p}$, respectively $G_{p}$.

Let $G$ be a linear algebraic group over $\mathcal{O}_{L}$ where
$\mathcal{O}_{L}$ is the ring of integers of a finite extension
$L/\mathbf{Q}_{p}$. Let $\varpi_{L}\in\mathcal{O}_{L}$ be a choice of
uniformiser. For $n\in\mathbf{Z}_{\geq 0}$, denote $\ker(G(\mathcal{O}_{L})\to
G(\mathcal{O}_{L}/\varpi_{L}^{n}))$ by $G^{n}$.

For $G$ a reductive group over a finite extension $L/\mathbf{Q}_{p}$ with
parabolic subgroup $Q=M\ltimes N\subset G$, we denote by
$\delta_{Q}:M(L)\to\mathbf{Q}^{\times}$ the corresponding modulus character
$x\mapsto|\det(\textnormal{ad}(x)|\textnormal{Lie}N)|_{L}$. For a smooth
representation $\sigma$ of $M(L)$, we denote by
$\textnormal{n-Ind}_{Q(L)}^{G(L)}\sigma$ the normalised parabolic induction
$\textnormal{Ind}_{Q(L)}^{G(L)}\delta_{Q}^{1/2}\sigma$. For a smooth
$\overline{\mathbf{Q}}_{p}$-representation $\pi$ of $G(L)$, we will denote by
$J_{Q}(\pi)$ its unnormalised Jacquet module associated with $Q$. In general
we denote by c-Ind compact induction for smooth representations.

Set $W_{L}$ to be the Weil group of $L$ and write
$\textnormal{Art}_{L}:L^{\times}\xrightarrow{\sim}W_{L}^{\textnormal{ab}}$ for
the Artin map of local class field theory normalised by sending uniformisers
to lifts of the geometric Frobenius.

We denote by $\textnormal{rec}_{L}$ the local Langlands correspondence for
$L$. If it’s clear from the context, we will just write rec instead. Moreover,
for $\pi$ an irreducible admissible $\textnormal{GL}_{n}(L)$-representation,
we set
$\textnormal{rec}^{T}(\pi)=\textnormal{rec}(\pi\otimes|\cdot|_{L}^{\frac{1-n}{2}})$.
Then $\textnormal{rec}^{T}$ commutes with $\textnormal{Aut}(\mathbf{C})$ and
therefore $\textnormal{rec}^{T}$ makes sense over $\overline{\mathbf{Q}}_{p}$
by choosing an abstract isomorphism
$t:\overline{\mathbf{Q}}_{p}\cong\mathbf{C}$. In the literature, this is often
called the Tate normalisation of the local Langlands correspondence.

We fix an algebraic closure $\overline{\mathbf{Q}}_{p}/\mathbf{Q}_{p}$ and
denote by $\textnormal{val}_{p}$ the $p$-adic valuation normalised by setting
$\textnormal{val}_{p}(p)=1$. By convention, the $\iota$-Hodge–Tate weights of
the $p$-adic cyclotomic character $\epsilon:G_{L}\to\mathbf{Z}_{p}^{\times}$
are $-1$.

For an $\ell$-adic Galois representation
$\rho:G_{L}\to\textnormal{GL}_{n}(\overline{\mathbf{Q}}_{\ell})$ with
$\ell\neq p$, we denote by $\textnormal{WD}(\rho)$ the associated Weil–Deligne
representation. For a de Rham $p$-adic Galois representation
$\rho:G_{L}\to\textnormal{GL}_{n}(\overline{\mathbf{Q}}_{p})$, we denote by
$\textnormal{WD}(\rho)$ the associated Weil–Deligne representation provided by
the recipe of Fontaine.

For a Weil–Deligne representation $(r,N)$, we denote by $(r,N)^{F-ss}$ its
Frobenius semisimplification and by $(r,N)^{ss}$ its semisimplification.

For a ring $R$, we denote by $D(R)$ the derived category of left $R$-modules
and by $D^{+}(R)$ resp. $D^{b}(R)$ the bounded below resp. bounded derived
category. Given moreover a locally profinite group $G$, we will denote by
$\textnormal{Mod}_{\textnormal{sm}}(R[G])$ the category of smooth
$R[G]$-modules and denote by $D_{\textnormal{sm}}(R[G])$ its derived category
and by $D^{+}_{\textnormal{sm}}(R[G])$ resp. $D^{b}_{\textnormal{sm}}(R[G])$
its bounded below resp. bounded derived category.

For $G/L$ a split reductive group with a choice of a Borel subgroup $B$ and a
maximal torus $T$, denote by $w_{0}^{G}$ the longest element in the Weyl group
$W_{G}:=W(G,T)$. For a standard parabolic subgroup $Q\subset G$ with Levi
decomposition $M\ltimes N$, set $W^{Q}$ to be the set of minimal length
representatives of $W_{G}/W_{Q}$. We denote by $w_{0}^{Q}$ the longest element
in $W^{Q}$ that is, in fact, given by $w_{0}^{G}w_{0}^{M}$. Similar notations
apply to ${}^{Q}W$. Moreover, for another standard parabolic subgroup
$Q^{\prime}\subset G$ with Levi decomposition $M^{\prime}\ltimes N^{\prime}$,
denote by ${}^{Q^{\prime}}W^{Q}$ the set of minimal length representatives of
$W_{Q^{\prime}}\backslash W_{G}/W_{Q}$.

##  2\. Preliminaries

We collect the necessary preliminaries regarding locally symmetric spaces and
their cohomology following [CN23], §2. The main differences compared to the
setup of [All+23], §2 are the use of different infinite level locally
symmetric spaces, which take into account the profinite topology on the adelic
part.

###  2.1. Locally symmetric spaces

Consider a number field $F$ and a connected linear algebraic group $G$ over
$F$, with a model over $\mathcal{O}_{F}$, still denoted by $G$. One can then
associate with $\textnormal{Res}_{F/\mathbf{Q}}G$ its symmetric space $X^{G}$,
a homogeneous $G(F\otimes_{\mathbf{Q}}\mathbf{R})$-space, as is defined in
[BS73], §2. By [BS73], Lemma 2.1, $X^{G}$ is unique up to isomorphism of
homogeneous $G(F\otimes_{\mathbf{Q}}\mathbf{R})$-spaces.

Given a good compact open subgroup $K_{G}\subset G(\mathbf{A}^{\infty}_{F})$
in the sense of [All+23], §2.1.1, we can form the corresponding locally
symmetric space

$X_{K_{G}}:=G(F)\backslash(X^{G}\times G(\mathbf{A}_{F}^{\infty})/K_{G}),$

a smooth orientable Riemannian manifold. Borel–Serre then construct a partial
compactification $\overline{X}^{G}$ of $X^{G}$ (cf. [BS73], §7.1) and form the
compactified locally symmetric space

$\overline{X}_{K_{G}}:=G(F)\backslash(\overline{X}^{G}\times
G(\mathbf{A}_{F}^{\infty})/K_{G}),$

an orientable compact smooth manifold with corners and with interior
$X_{K_{G}}$. We will denote the corresponding boundary by $\partial
X^{G}:=\overline{X}^{G}\setminus X^{G}$ resp. $\partial
X_{K_{G}}:=\overline{X}_{K_{G}}\setminus X_{K_{G}}$.

Following [CN23], we define the sets

$\mathfrak{X}_{G}:=\varprojlim_{K_{G}}X_{K_{G}},\textnormal{
}\overline{\mathfrak{X}}_{G}:=\varprojlim_{K_{G}}\overline{X}_{K_{G}},\textnormal{
}\partial\mathfrak{X}_{G}=\varprojlim_{K_{G}}\partial X_{K_{G}}$

where the limits run over good subgroups of $G(\mathbf{A}_{F}^{\infty})$, and
make them into topological spaces by endowing them with the projective limit
topology. The latter two then become compact Hausdorff spaces, being
projective limits of such. They all are equipped with the natural continuous
right action of $G(\mathbf{A}_{F}^{\infty})$. Moreover, the induced action of
any good subgroup $K_{G}\subset G(\mathbf{A}_{F}^{\infty})$ on
$\overline{\mathfrak{X}}_{G}$, and $\partial\mathfrak{X}_{G}$ is free in the
sense of [NT16], Definition 2.23. As explained in [CN23], §2.1.1, we can also
introduce these spaces as the quotient spaces

$\mathfrak{X}_{G}=G(F)\setminus X^{G}\times
G(\mathbf{A}_{F}^{\infty}),\textnormal{
}\overline{\mathfrak{X}}_{G}=G(F)\setminus\overline{X}^{G}\times
G(\mathbf{A}_{F}^{\infty}),\textnormal{ and }$
$\partial\mathfrak{X}_{G}=G(F)\setminus\partial X^{G}\times
G(\mathbf{A}_{F}^{\infty})$

where the group $G(\mathbf{A}_{F}^{\infty})$ is equipped with its locally
profinite topology. We remind the reader that, before [CN23], the more common
choice was to use the discrete topology of $G(\mathbf{A}_{F}^{\infty})$
instead (see [NT16]). They yield different spaces of course, but the produced
finite level cohomologies (with the induced Hecke actions) compare well (cf.
[CN23], Lemma 2.1.5). We denote by
$j:\mathfrak{X}_{G}\hookrightarrow\overline{\mathfrak{X}}_{G}$ the natural
open immersion.

For later use we introduce some further notation. Set $S$ to be a finite set
of finite places of $F$ and $K^{S}_{G}\subset
G(\mathbf{A}^{S\cup\\{\infty\\}}_{F})$ be a compact open subgroup that extends
to some good subgroup of $G(\mathbf{A}_{F}^{\infty})$. We then set

$\overline{X}_{K_{G}^{S}}:=\overline{\mathfrak{X}}_{G}/K_{G}^{S}=\varprojlim_{K_{G,S}}\overline{X}_{K_{G}^{S}K_{G,S}},\textnormal{
and }\partial
X_{K_{G}^{S}}:=\partial\mathfrak{X}_{G}/K_{G}^{S}=\varprojlim_{K_{G,S}}\partial
X_{K_{G}^{S}K_{G,S}}$

where the limits run over compact open subgroups $K_{G,S}\subset
G(\mathbf{A}_{F}^{\infty})$ such that $K_{G}^{S}K_{G,S}$ is good.

###  2.2. Coefficient systems at $S$ and Hecke operators away from $S$

For $S$ a finite set of finite places of $F$, we set
$G^{S}:=G(\mathbf{A}_{F}^{S\cup\\{\infty\\}})$ and
$G_{S}:=G(\mathbf{A}_{F,S})$, and similarly, for a good subgroup $K_{G}\subset
G(\mathbf{A}^{\infty}_{F})$, we set $K^{S}_{G}:=\prod_{v\notin S}K_{G,v}$ and
$K_{G,S}:=\prod_{v\in S}K_{G,v}$.

Let $R$ be a commutative ring. Given a smooth $R[K_{G,S}]$-module
$\mathcal{V}$, the formalism of [All+23] attaches to it a $G^{S}\times
K_{G,S}$-equivariant sheaf $\mathcal{V}\in\textnormal{Sh}_{G^{S}\times
K_{G,S}}(\mathfrak{X}_{G},R)$ resp.
$\mathcal{V}\in\textnormal{Sh}_{G^{S}\times
K_{G,S}}(\overline{\mathfrak{X}}_{G},R)$ in the sense of [Sch98] and [NT16],
§2.4. Namely, one inflates $\mathcal{V}$ to get an element of
$\textnormal{Mod}_{\textnormal{sm}}(R[G^{S}\times K_{G,S}])$ which, by [NT16],
Lemma 2.25, is equivalent to the category $\textnormal{Sh}_{G^{S}\times
K_{G,S}}(\ast,R)$ and we then pull it back along $f:\mathfrak{X}_{G}\to\ast$
resp. $\overline{f}:\overline{\mathfrak{X}}_{G}\to\ast$. As explained in
[Sch98], the categories $\textnormal{Sh}_{G^{S}\times
K_{G,S}}(\mathfrak{X}_{G})$ and $\textnormal{Sh}_{G^{S}\times
K_{G,S}}(\overline{\mathfrak{X}}_{G})$ have enough injectives and both $f_{!}$
and $\overline{f}_{!}=\overline{f}_{\ast}$ land in
$\textnormal{Sh}_{G^{S}\times
K_{G,S}}(\ast,R)\cong\textnormal{Mod}_{\textnormal{sm}}(R[G^{S}\times
K_{G,S}])$. Therefore, we see that both

$R\Gamma(\overline{\mathfrak{X}}_{G},j_{!}\mathcal{V})=R(\overline{f}_{\ast}\circ
j_{!})\mathcal{V}=Rf_{!}\mathcal{V}$

and

$R\Gamma(\overline{\mathfrak{X}}_{G},\mathcal{V})=R\overline{f}_{\ast}\mathcal{V}=R\overline{f}_{!}\mathcal{V}$

lie in $D^{+}_{\textnormal{sm}}(R[G^{S}\times K_{G,S}])$. By taking derived
invariants, we end up with

$R\Gamma(K_{G},R\Gamma(\overline{\mathfrak{X}}_{G},\mathcal{V}))\textnormal{
and }R\Gamma(K_{G},R\Gamma(\overline{\mathfrak{X}}_{G},j_{!}\mathcal{V}))$

in $D^{+}(\mathcal{H}(G^{S},K^{S}_{G})\otimes_{\mathbf{Z}}R)$ where
$\mathcal{H}(G^{S},K^{S}_{G}):=\mathbf{Z}[K_{G}^{S}\backslash
G^{S}/K_{G}^{S}]$ is the prime-to-$S$ Hecke algebra i.e., the additive group
of integer valued $K_{G}^{S}$-biinvariant functions on $G^{S}$ equipped with
the convolution product.

On the other hand, by descent (cf. [NT16], Lemma 2.24), $\mathcal{V}$ and
$j_{!}\mathcal{V}$ give rise to sheaves on $\overline{X}_{K_{G}}$, which, by
abuse of notation, we denote by the same letter. Then, by combining the fact
that $j$ is a homotopy equivalence and [CN23], Proposition 2.1.3, we obtain
natural isomorphisms

$R\Gamma(X_{K_{G}},\mathcal{V})\cong
R\Gamma(\overline{X}_{K_{G}},\mathcal{V})\cong
R\Gamma(K_{G},R\Gamma(\overline{\mathfrak{X}}_{G},\mathcal{V}))^{\sim}$

and

$R\Gamma_{c}(X_{K_{G}},\mathcal{V}):=R\Gamma(\overline{X}_{K_{G}},j_{!}\mathcal{V})\cong
R\Gamma(K_{G},R\Gamma(\overline{\mathfrak{X}}_{G},j_{!}\mathcal{V}))^{\sim}$

in $D^{+}(R)$, where $(-)^{\sim}$ denotes the forgetful functor. In
particular, we have a morphism

$\mathcal{H}(G^{S},K_{G}^{S})\otimes_{\mathbf{Z}}R\to\textnormal{End}_{D^{+}(R)}(R\Gamma_{(c)}(X_{K_{G}},\mathcal{V})).$

The same observations also apply to $R\Gamma(\partial X_{K_{G}},\mathcal{V})$.

Finally, assuming that $R$ is Noetherian and $K_{G}^{\prime}\subset K_{G}$ is
a good normal subgroup, one sees, by writing down the complex explicitly in
terms of a well-chosen simplicial complex, that
$R\Gamma_{(c)}(X_{K^{\prime}_{G}},\mathcal{V})$ is in fact a perfect object in
$D^{+}(R[K_{G}/K^{\prime}_{G}])$ (cf. [CN23], Lemma 2.1.6).

We now turn our attention to completed cohomology following the viewpoint of
[CN23]. For this we fix $m\in\mathbf{Z}_{\geq 1}$ and set
$R=\mathcal{O}/\varpi^{m}$ where $\mathcal{O}$ is the ring of integers of a
finite field extension $E/\mathbf{Q}_{p}$. Moreover, let $S\subset
S_{p}:=S_{p}(F)$ be a subset of the set of $p$-adic places of $F$.

######  Definition 2.1.

Given
$\mathcal{V}\in\textnormal{Mod}_{\textnormal{sm}}(\mathcal{O}/\varpi^{m}[K_{G,S_{p}}])$,
we define its completed cohomology (with compact support) at $S$ of level
$K^{S}_{G}$ to be

$\pi(K^{S}_{G},\mathcal{V}):=R\Gamma(K^{S}_{G},R\Gamma(\overline{\mathfrak{X}}_{G},\mathcal{V}))^{\sim}\cong
R\Gamma(\overline{X}_{K^{S}_{G}},\mathcal{V})\in
D^{+}_{\textnormal{sm}}(\mathcal{O}/\varpi^{m}[K_{G,S}])$

resp.

$\pi_{c}(K^{S}_{G},\mathcal{V}):=R\Gamma(K^{S}_{G},R\Gamma(\overline{\mathfrak{X}}_{G},j_{!}\mathcal{V}))^{\sim}\cong
R\Gamma(\overline{X}_{K^{S}_{G}},j_{!}\mathcal{V})\in
D^{+}_{\textnormal{sm}}(\mathcal{O}/\varpi^{m}[K_{G,S}]).$

Note that if $\mathcal{V}$ is inflated from an element
$\mathcal{V}^{S}\in\textnormal{Mod}_{\textnormal{sm}}(\mathcal{O}/\varpi^{m}[K_{G,S_{p}\backslash
S}])$, then the $S$-completed cohomology complexes in fact lie in
$D^{+}_{\textnormal{sm}}(\mathcal{O}/\varpi^{m}[G_{S}])$.

For any set of finite places $S_{p}\subset T$,
$\mathcal{H}(G^{T},K_{G}^{T})\otimes_{\mathbf{Z}}\mathcal{O}/\varpi^{m}$ acts
on completed cohomology (with compact support) as it does so on
$R\Gamma(K^{S}_{G},R\Gamma(\overline{\mathfrak{X}}_{G},(j_{!})\mathcal{V}))$.
Moreover, one similarly defines boundary completed cohomology

$\pi_{\partial}(K^{S}_{G},\mathcal{V}):=R\Gamma(\partial
X_{K^{S}_{G}},\mathcal{V})\in
D^{+}_{\textnormal{sm}}(\mathcal{O}/\varpi^{m}[K_{G,S}]).$

Note that [CN23], Lemma 2.1.7 justifies the use of the term completed
cohomology i.e., it shows that, after taking cohomology groups, we get back
Emerton’s completed cohomology as defined in [Eme06a].

Finally, we have the usual phenomenon of completed cohomology at $S$ being
independent of the weight at $S$ (cf. [CN23], Lemma 2.1.8). To state this more
precisely, let
$\mathcal{V}\in\textnormal{Mod}_{\textnormal{sm}}(\mathcal{O}/\varpi^{m}[K_{G,S_{p}\setminus
S}\times\Delta_{S}])$ where $\Delta_{S}\subset G_{S}$ is a submonoid
containing a compact open subgroup $U_{S}\subset G_{S}$ and assume that
$\mathcal{V}$ is $\mathcal{O}/\varpi^{m}$-flat. We can then view $\mathcal{V}$
as a $G^{T}\times K_{G,T\setminus S}\times U_{S}$-equivariant sheaf on
$\overline{\mathfrak{X}}_{G}$.

######  Lemma 2.2.

We have canonical isomorphisms

$R\Gamma(\overline{\mathfrak{X}}_{G},\mathcal{V})\cong
R\Gamma(\overline{\mathfrak{X}}_{G},\mathcal{O}/\varpi^{m})\otimes_{\mathcal{O}/\varpi^{m}}\mathcal{V},$

and

$R\Gamma(\partial\mathfrak{X}_{G},\mathcal{V})\cong
R\Gamma(\partial\mathfrak{X}_{G},\mathcal{O}/\varpi^{m})\otimes_{\mathcal{O}/\varpi^{m}}\mathcal{V}$

in $D^{+}_{\textnormal{sm}}(\mathcal{O}/\varpi^{m}[G^{T}\times K_{G,T\setminus
S}\times U_{S}])$.

###### Proof.

This is [CN23], Lemma 2.1.8. ∎

Following [CN23], we define the object

$R\Gamma(\overline{\mathfrak{X}}_{G},\mathcal{V}):=R\Gamma(\overline{\mathfrak{X}}_{G},\mathcal{O}/\varpi^{m})\otimes_{\mathcal{O}/\varpi^{m}}\mathcal{V}\in
D^{+}_{\textnormal{sm}}(\mathcal{O}/\varpi^{m}[G^{T}\times K_{G,T\setminus
S}\times\Delta_{S}]).$

In particular, we can endow $R\Gamma(X_{K_{G}},\mathcal{V})$ with a natural
$\mathcal{H}(G^{T},K^{T}_{G})\otimes\mathcal{H}(\Delta_{S},K_{G,S})$-action.

We also have a version of Lemma 2.2 with $\mathcal{V}$ being a complex of
sheaves with bounded cohomology. Namely, consider $\mathcal{V}\in
D^{b}_{\textnormal{sm}}(\mathcal{O}/\varpi^{m}[K_{G,S_{p}\setminus S}])$ and
denote also by $\mathcal{V}$ the associated object in
$D^{b}(\textnormal{Sh}_{G^{S_{p}\setminus S}\times K_{G,S_{p}\setminus
S}}(\overline{\mathfrak{X}}_{G},\mathcal{O}/\varpi^{m}))$. One can then makes
sense of the derived tensor product

$R\Gamma(\overline{\mathfrak{X}}_{G},\mathcal{O}/\varpi^{m})\otimes_{\mathcal{O}/\varpi^{m}}^{\mathbf{L}}-:D^{b}_{\textnormal{sm}}(\mathcal{O}/\varpi^{m}[K_{G,S_{p}\setminus
S}])\to D^{b}_{\textnormal{sm}}(\mathcal{O}/\varpi^{m}[G^{S_{p}\setminus
S}\times K_{G,S_{p}\setminus S}])$

as explained on page 13 of [CN23]. Then loc. cit. Lemma 2.1.9 shows that there
is a canonical isomorphism

$R\Gamma(\overline{\mathfrak{X}}_{G},\mathcal{O}/\varpi^{m})\otimes_{\mathcal{O}/\varpi^{m}}^{\mathbf{L}}\mathcal{V}\xrightarrow{\sim}R\Gamma(\overline{\mathfrak{X}}_{G},\mathcal{V})$

in $D^{b}_{\textnormal{sm}}(G^{S_{p}\setminus S}\times K_{G,S_{p}\setminus
S})$.

We finally define cohomology complexes with $\mathcal{O}$-coefficients by
taking homotopy limit. For this we start with an $\mathcal{O}[K_{G,S}]$-module
$\mathcal{V}$, finite free as an $\mathcal{O}$-module. We set

$R\Gamma_{(c)}(X_{K_{G}},\mathcal{V}):=\varprojlim_{m}R\Gamma_{(c)}(X_{K_{G}},\mathcal{V}/\varpi^{m})\in
D^{+}(\mathcal{O}),$

where the projective limit is understood as a homotopy limit. One can endow
these complexes with an action of the Hecke algebra
$\mathcal{H}(G^{S},K_{G}^{S})\otimes_{\mathbf{Z}}\mathcal{O}$ (see the
footnote on Page 14 of [CN23] for a discussion on how it relates to the Hecke
action defined in [NT16]).

The next lemma explains that after taking cohomology we get back classical
Betti cohomology with $\mathcal{O}$-coefficients. Such an argument will be
used at several places whenever the Mittag–Leffler condition holds.

######  Lemma 2.3.

Let $K_{G}\subset G(\mathbf{A}^{\infty}_{F})$ be a good subgroup, $S$ be a
finite set of finite places of $F$ and $\mathcal{V}$ an
$\mathcal{O}[K_{G,S}]$-module, finite free as an $\mathcal{O}$-module. Then,
for every $j\in\mathbf{Z}_{\geq 0}$, we have a natural identification

$H^{j}(R\Gamma_{(c)}(X_{K_{G}},\mathcal{V}))\cong
H^{j}_{(c)}(X_{K_{G}},\mathcal{V}).$

###### Proof.

By [Aut13], 0CQE, it suffices to prove that the higher inverse limit

$R^{1}\varprojlim_{m}H^{j}_{(c)}(X_{K_{G}},\mathcal{V}/\varpi^{m})$

vanishes. This vanishing is ensured once we prove that the inverse system

$\\{H^{j}_{(c)}(X_{K_{G}},\mathcal{V}/\varpi^{m})\\}_{m\geq 0}$

satisfies Mittag-Leffler. To see this, we note that, by [CN23], Lemma 2.1.6,
all the appearing cohomology groups are finite. Therefore, the images of all
the transition maps in the inverse system must stabilise after finitely many
steps. ∎

We now recall the definition of the unnormalised Satake transform. Assume that
$G$ is reductive and $P=M\rtimes N\subset G$ is a parabolic subgroup with its
Levi decomposition. Given a good subgroup $K_{G}\subset
G(\mathbf{A}_{F}^{\infty})$, set $K_{P}=K_{G}\cap P(\mathbf{A}^{\infty}_{F})$,
$K_{N}=K_{G}\cap N(\mathbf{A}^{\infty}_{F})$ and
$K_{M}=\textnormal{im}(K_{P}\to M(\mathbf{A}_{F}^{\infty}))$. We call $K_{G}$
decomposed with respect to $P=M\rtimes N$, if $K_{P}=K_{M}\rtimes K_{N}$;
equivalently, if $K_{M}=K_{G}\cap M(\mathbf{A}^{\infty}_{F})$.

Assume $K_{G}$ is decomposed with respect to $P=M\rtimes N$ and let $S$ be a
finite set of finite places such that, for $v\notin S$, $K_{G,v}$ is a
hyperspecial maximal compact in $G(F_{v})$. Then we have the usual maps on
Hecke algebras

$r_{P}:\mathcal{H}(G^{S},K_{G}^{S})\to\mathcal{H}(P^{S},K_{P}^{S})\textnormal{
and }r_{M}:\mathcal{H}(P^{S},K_{P}^{S})\to\mathcal{H}(M^{S},K_{M}^{S})$

given by "restriction to $P$" and "integration along $N$", respectively (cf.
[NT16], 2.2.3, 2.2.4). We then can define $\mathcal{S}:=r_{M}\circ r_{P}$, the
unnormalised Satake transform.

###  2.3. Hecke algebras of types

We will make use of the Hecke algebra and Bernstein centre action at $p$ with
respect to locally algebraic types. For this we briefly recall the content of
Appendix A.1 and A.4 of [Vig04]. First we consider the following general
setup. Let $R$ be a commutative ring, $G$ be a locally profinite group,
$K\subset G$ an open subgroup and $\sigma$ be a smooth $R[K]$-module, finitely
generated over $R$.

Given $\sigma$, we can define the corresponding Hecke algebra

$\mathcal{H}(\sigma):=\textnormal{End}_{R[G]}(\textnormal{c-Ind}_{K}^{G}\sigma).$

For $\pi\in\textnormal{Mod}_{\textnormal{sm}}(R[G])$, $\mathcal{H}(\sigma)$
acts on the space of invariants

${\rm{Hom}}_{R[G]}(\textnormal{c-Ind}_{K}^{G}\sigma,\pi)={\rm{Hom}}_{R[K]}(\sigma,\pi)=\sigma^{\vee}\otimes_{R[K]}\pi$

on the right. We also note that $\mathcal{H}(\sigma)$ can be identified with
the convolution algebra of compactly supported functions
$f:G\to\textnormal{End}_{R}(\sigma)$ satisfying
$f(k_{1}gk_{2})=\sigma(k_{1})f(g)\sigma(k_{2})$ for every $k_{1},k_{2}\in K$
and $g\in G$. The isomorphism is realised by acting with the convolution
algebra on $\textnormal{c-Ind}_{K}^{G}\sigma$ via convolution. From this
description it is clear that $\mathcal{H}(\sigma)$ is spanned over $R$ by
elements represented by pairs $[g,\psi]$ where $g\in G$ and
$\psi\in\textnormal{End}_{R}(\sigma)$ such that

(2.1) $\sigma(k)\circ\psi=\psi\circ\sigma(g^{-1}kg)$

for every $k\in K\cap gKg^{-1}$. More precisely, such a pair gives rise to the
function $G\to\textnormal{End}_{R}(\sigma)$ supported on $KgK$ that sends $g$
to $\psi$. Moreover, under the mentioned identification, $[g,\psi]$ acts on
$\phi\in{\rm{Hom}}_{R[K]}(\sigma,\pi)$ by the formula

$\phi\cdot[g,\psi]:v\mapsto\sum_{i}\pi(g_{i})^{-1}\phi([g,\psi](g_{i})v)$

for $v\in\sigma$ and $KgK=\coprod_{i}Kg_{i}$.

Note that we have an anti-involution

$\mathcal{H}(\sigma)\xrightarrow{\sim}\mathcal{H}(\sigma^{\vee}),$
$[g,\psi]\mapsto[g^{-1},\psi^{t}].$

Moreover, $[h,\chi]=[g^{-1},\psi^{t}]\in\mathcal{H}(\sigma^{\vee})$ acts on
$\sum_{j}f_{j}\otimes
p_{j}\in\sigma^{\vee}\otimes_{R[K]}\pi=(\sigma^{\vee}\otimes_{R}\pi)^{K}$ via
the formula

$[h,\chi]\cdot(\sum_{j}f_{j}\otimes
p_{j})=\sum_{i}\left(\sum_{j}[h,\chi](h_{i})f_{j}\otimes\pi(h_{i})(p_{j})\right)=$
$=\sum_{i}\left(\sum_{j}[g^{-1},\psi^{t}](g_{i}^{-1})f_{j}\otimes\pi(g_{i}^{-1})(p_{j})\right)$

for
$\coprod_{i}h_{i}K=KhK=Kg^{-1}K=\coprod_{i}g_{i}^{-1}K=(\coprod_{i}Kg_{i})^{-1}$.
This gives rise to a left action of $\mathcal{H}(\sigma^{\vee})$ on
$\sigma^{\vee}\otimes_{R[K]}\pi$. Moreover, as the computation shows, the
anti-isomorphism
$\mathcal{H}(\sigma)\xrightarrow{\sim}\mathcal{H}(\sigma^{\vee})$ intertwines
the two actions under the identification
${\rm{Hom}}_{R[K]}(\sigma,\pi)\cong\sigma^{\vee}\otimes_{R[K]}\pi$.

We are now ready to equip the cohomology of locally symmetric spaces with a
Hecke action at $p$. For this, we revisit the setup of §2.1, and $G$ will
again denote a connected linear algebraic group $G$ over $F$ admitting a model
over $\mathcal{O}_{F}$.

######  Lemma 2.4.

Let $S\subset S(F)$ be a finite set of finite places, $K_{G}\subset
G(\mathbf{A}^{\infty}_{F})$ a good subgroup, and
$\sigma\in\textnormal{Mod}_{\textnormal{sm}}(\mathcal{O}/\varpi^{m}[K_{G,S}])$,
finite free as an $\mathcal{O}/\varpi^{m}$-module. Then the diagram of derived
functors

${{D^{+}\textnormal{Sh}_{G(\mathbf{A}_{F}^{\infty})}(\overline{\mathfrak{X}}_{G},\mathcal{O}/\varpi^{m})}}$${{D^{+}_{\textnormal{sm}}(\mathcal{O}/\varpi^{m}[G(\mathbf{A}_{F}^{\infty})])}}$${{D^{+}(\mathcal{H}(G^{S},K_{G}^{S})\times\mathcal{H}(\sigma^{\vee}))}}$${{D^{+}\textnormal{Sh}_{K_{G}}(\overline{\mathfrak{X}}_{G},\mathcal{O}/\varpi^{m})}}$${{D^{+}\textnormal{Sh}(\overline{X}_{K_{G}},\mathcal{O}/\varpi^{m})}}$${{D^{+}(\mathcal{O}/\varpi^{m})}}$$\scriptstyle{R\Gamma(\overline{\mathfrak{X}}_{G},-)}$forgetdescent$\scriptstyle{R\textnormal{Hom}_{\mathcal{O}/\varpi^{m}[K_{G}]}(\sigma,-)}$$\scriptstyle{R\Gamma(\overline{X}_{K_{G}},-\otimes_{\mathcal{O}/\varpi^{m}}\sigma^{\vee})}$forget

is commutative.

###### Proof.

Throughout the proof, we repeatedly use [Wei94], Corollary 10.8.3 without
further mention. According to [NT16], Lemma 2.28,
$\Gamma(\overline{\mathfrak{X}}_{G},-)$ preserves injectives. Moreover, the
forgetful functor

$(-)^{\sim}:D^{+}\textnormal{Sh}_{G(\mathbf{A}_{F}^{\infty})}(\overline{\mathfrak{X}}_{G},\mathcal{O}/\varpi^{m})\to
D^{+}\textnormal{Sh}_{K_{G}}(\overline{\mathfrak{X}}_{G},\mathcal{O}/\varpi^{m})$

is exact and preserves injectives by [Sch98], §3, Corollary 3.888Note that a
running assumption in [Sch98] is that the ring of coefficients is
$\mathbf{C}$. However, one sees that the proof of loc. cit. goes through
without a change also with $\mathcal{O}/\varpi^{m}$-coefficients. In
particular, we get that the composition of the functors on the top of the
square is naturally isomorphic to the functor

$R\Gamma(K_{G},R\Gamma(\overline{\mathfrak{X}}_{G},(-)^{\sim})\otimes_{\mathcal{O}/\varpi^{m}}\sigma^{\vee}):D^{+}\textnormal{Sh}_{G(\mathbf{A}_{F}^{\infty})}(\overline{\mathfrak{X}}_{G},\mathcal{O}/\varpi^{m})\to
D^{+}(\mathcal{O}/\varpi^{m}).$

On the other hand, the argument of Lemma 2.2 shows that we have a natural
isomorphism

$R\Gamma(\overline{\mathfrak{X}}_{G},-)\otimes_{\mathcal{O}/\varpi^{m}}\sigma^{\vee}\cong
R\Gamma(\overline{\mathfrak{X}}_{G},-\otimes_{\mathcal{O}/\varpi^{m}}\sigma^{\vee})$

of derived functors
$D^{+}\textnormal{Sh}_{K_{G}}(\overline{\mathfrak{X}}_{G},\mathcal{O}/\varpi^{m})\to
D^{+}_{\textnormal{sm}}(\mathcal{O}/\varpi^{m}[K_{G}])$.

Finally, we note that the descent functor respects tensor products since the
pullback functor (its inverse) does. We then obtain a natural isomorphism of
derived functors

$R\Gamma(K_{G},R\Gamma(\overline{\mathfrak{X}}_{G},-\otimes_{\mathcal{O}/\varpi^{m}}\sigma^{\vee}))\cong
R\Gamma(\overline{X}_{K_{G}},-\otimes_{\mathcal{O}/\varpi^{m}}\sigma^{\vee}).$

By putting these observations together, we can conclude. ∎

We spell out a generalisation of [All+23], Proposition 2.2.22 that will be
used to make twisting arguments in the proof of local-global compatibility
analogous to the ones in loc. cit., Corollary 4.4.8. Let
$G=\textnormal{GL}_{n,F}$, and
$K\subset\textnormal{GL}_{n}(\mathbf{A}_{F}^{\infty})$ be a good subgroup, and
$\chi:G_{F}\to\mathcal{O}^{\times}$ be a continuous character such that
$\chi\circ\textnormal{Art}_{F_{v}}$ is trivial on $\det(K_{v})$ for each
finite place $v\notin T$ of $F$. Let
$\sigma\in\textnormal{Mod}_{\textnormal{sm}}(\mathcal{O}/\varpi^{m}[K_{p}])$,
finite free as an $\mathcal{O}/\varpi^{m}$-module. Set
$\sigma_{\chi}:K_{p}\to\mathcal{O}^{\times}$ to be the continuous character
defined by

$(k_{v})_{v\in S_{p}}\mapsto\prod_{v\in
S_{p}}\chi(\textnormal{Art}_{F_{v}}(\det(k_{v}))).$

Define the isomorphism of $\mathcal{O}$-algebras

$f_{\chi}:\mathcal{H}(G^{T},K^{T})\otimes_{\mathbf{Z}}\mathcal{H}(\sigma^{\vee})\to\mathcal{H}(G^{T},K^{T})\otimes_{\mathbf{Z}}\mathcal{H}(\sigma^{\vee}\otimes\sigma_{\chi^{-1}})$

sending a function
$f:\textnormal{GL}_{n}(\mathbf{A}_{F})\to\textnormal{End}(\sigma^{\vee})$
lying in the source of $f_{\chi}$ to the function
$f_{\chi}(f):g\mapsto\chi(\textnormal{Art}_{F}(\textnormal{det}(g)))^{-1}f(g)$.

######  Lemma 2.5.

Let $K\subset\textnormal{GL}_{n}(\mathbf{A}_{F}^{\infty})$ be a good subgroup,
and $\chi:G_{F}\to\mathcal{O}^{\times}$ be a continuous character such that
$\chi\circ\textnormal{Art}_{F_{v}}$ is trivial on $\det(K_{v})$ for each
finite place $v\notin T$ of $F$. Let
$\sigma\in\textnormal{Mod}_{\textnormal{sm}}(\mathcal{O}/\varpi^{m}[K_{p}])$,
finite free as an $\mathcal{O}/\varpi^{m}$-module. Then there is an
isomorphism

$R\Gamma(X_{K},\sigma^{\vee})\cong
R\Gamma(X_{K},\sigma^{\vee}\otimes\sigma_{\chi^{-1}})$

in $D^{+}(\mathcal{O}/\varpi^{m})$ that is
$\mathcal{H}(G^{T},K^{T})\otimes_{\mathbf{Z}}\mathcal{H}(\sigma^{\vee})$-equivariant
when we consider its usual action on the left and the one induced by pre-
composition with $f_{\chi}$ on the right.

###### Proof.

To see this, we introduce some constructions. For any
$\textnormal{GL}_{n}(\mathbf{A}_{F}^{\infty})$-equivariant sheaf
$\mathcal{F}\in\textnormal{Sh}_{\textnormal{GL}_{n}(\mathbf{A}_{F}^{\infty})}(\overline{\mathfrak{X}}_{\textnormal{GL}_{n}},\mathcal{O}/\varpi^{m})$,
consider the map

(2.2)
$\Gamma(\overline{\mathfrak{X}}_{\textnormal{GL}_{n}},\mathcal{F})\to\Gamma(\overline{\mathfrak{X}}_{\textnormal{GL}_{n}},\mathcal{F}),$
$s\mapsto s^{\chi}$

defined by the formula
$s^{\chi}((x,g)):=\chi(\textnormal{Art}_{F}(\det(g)))s((x,g)).$ An easy
computation shows that 2.2 becomes
$\textnormal{GL}_{n}(\mathbf{A}_{F}^{\infty})$-equivariant when we twist the
target by the character $g\mapsto\chi(\textnormal{Art}_{F}(\det(g)))^{-1}$. In
particular, 2.2 descends to a map

(2.3)
${\rm{Hom}}_{\mathcal{O}/\varpi^{m}[K]}(\sigma,\Gamma(\overline{\mathfrak{X}}_{\textnormal{GL}_{n}},\mathcal{F}))\to{\rm{Hom}}_{\mathcal{O}/\varpi^{m}[K]}(\sigma\otimes\sigma_{\chi},\Gamma(\overline{\mathfrak{X}}_{\textnormal{GL}_{n}},\mathcal{F})).$

After unravelling the definition of the Hecke action, one sees that this
amounts to saying that the induced map 2.3 satisfies the desired Hecke-
equivariance of the lemma.

We can conclude by choosing an injective resolution
$\mathcal{O}/\varpi^{m}\to\mathcal{I}^{\bullet}$ in the category
$\textnormal{Sh}_{\textnormal{GL}_{n}(\mathbf{A}_{F}^{\infty})}(\overline{\mathfrak{X}}_{\textnormal{GL}_{n}},\mathcal{O}/\varpi^{m})$
and applying the previous observation with the choice of
$\mathcal{F}=\mathcal{I}^{i}$ for every $i\in\mathbf{Z}_{\geq 0}$. Indeed,
this is because, by Lemma 2.4, applying the forgetful functor to

${\rm{Hom}}_{\mathcal{O}/\varpi^{m}[K]}(\sigma,\Gamma(\overline{\mathfrak{X}}_{\textnormal{GL}_{n}},\mathcal{I}^{\bullet}))\textnormal{,
and
}{\rm{Hom}}_{\mathcal{O}/\varpi^{m}[K]}(\sigma\otimes\sigma_{\chi},\Gamma(\overline{\mathfrak{X}}_{\textnormal{GL}_{n}},\mathcal{I}^{\bullet}))$

computes $R\Gamma(X_{K},\sigma^{\vee})$, and
$R\Gamma(X_{K},\sigma^{\vee}\otimes\sigma_{\chi^{-1}})$, respectively.
Moreover, the Hecke actions come from these identifications. ∎

###  2.4. Hecke equivariance of Poincaré duality

In the proof of our local-global compatibility results, we appeal to the
Poincaré duality isomorphism for the cohomology of the locally symmetric
spaces attached to $\textnormal{GL}_{n/F}$. To keep track of the Hecke algebra
actions during this process, it is crucial to verify that Poincaré duality is
equivariant with respect to suitable Hecke actions on the two sides. This is
already checked for instance in [NT16], Proposition 3.7 for unramified prime-
to-$p$ places. We will need a version of this Hecke equivariancy for the Hecke
algebra actions at $p$. This will need slightly more care and will be handled
in this subsection.

We return to our setup in Section 2.2. We let $S\subset S_{p}(F)$ to be a set
of $p$-adic places, $K\subset G(\mathbf{A}_{F}^{\infty})$ a good subgroup. Let
$\sigma\in\textnormal{Mod}_{\textnormal{sm}}(\mathcal{O}/\varpi^{m}[K_{S}])$,
which we also assume to be finite free over $\mathcal{O}/\varpi^{m}$. Then, by
the previous subsection, for any $G_{S}$-equivariant sheaf $\mathcal{G}$ on
$\overline{X}_{K^{S}}$ in the sense of [NT16], Section 2.4, the complex

$R\textnormal{Hom}_{\mathcal{O}/\varpi^{m}[K_{S}]}(\sigma,R\Gamma(\overline{X}_{K^{S}},\mathcal{G}))$

lies in
$D^{+}(\mathcal{H}(\sigma)^{\textnormal{op}})=D^{+}(\mathcal{H}(\sigma^{\vee}))$.
Then, if we let $\pi:\overline{X}_{K^{S}}\to\overline{X}_{K}$ to be the
natural projection and $\overline{f}:\overline{X}_{K^{S}}\to\ast$, Lemma 2.4
implies that there is an induced morphism of algebras

$\mathcal{H}(\sigma^{\vee})\to\textnormal{End}_{D^{+}(\mathcal{O}/\varpi^{m})}(R\Gamma(\overline{X}_{K},\pi_{\ast}^{K_{S}}(\mathcal{G}\otimes_{\mathcal{O}/\varpi^{m}}\overline{f}^{\ast}\sigma^{\vee})).$

To prove Hecke equivariancy of Poincaré duality for
$R\Gamma(X_{K},\sigma^{\vee})$, we give a different description of this action
as in [NT16], Lemma 2.19. As we are working with places above $p$, which are
not assumed to be unramified, we need to refine the constructions of [NT16].

We pick an element $[g,\psi]\in\mathcal{H}(\sigma)^{\textnormal{op}}$ as in
the previous subsection. We define an action of this element on cohomology.
Let $K^{\prime}:=K^{S}K^{\prime}_{S}$ where $K^{\prime}_{S}=K_{S}\cap
g^{-1}K_{S}g$. By abuse of notation, whenever $K^{\prime\prime}_{S}\subset
K_{S}$ is a compact open subgroup, we denote by $\pi$ the projection
$\overline{X}_{K^{S}}\to\overline{X}_{K^{S}K^{\prime\prime}_{S}}$. We consider
the correspondence

${\overline{X}_{K^{\prime}}}$${\overline{X}_{K}}$${\overline{X}_{K}}$$\scriptstyle{p_{1}}$$\scriptstyle{p_{2}}$

where $p_{1}$ is the natural projection and $p_{2}$ is given by the map
$\overline{X}_{K^{\prime}}\xrightarrow[]{\cdot g^{-1}}\overline{X}_{K^{S}\cdot
gK^{\prime}_{S}g^{-1}}$ followed by the natural projection. Then an easy
computation shows that, by the intertwining property 2.1 of $\psi$, it
descends to a map

$p_{1}^{\ast}\pi_{\ast}^{K_{S}}\overline{f}^{\ast}\sigma=\pi_{\ast}^{K^{\prime}_{S}}\overline{f}^{\ast}\sigma\xrightarrow[]{\psi}(g^{-1})^{\ast}\pi_{\ast}^{gK^{\prime}_{S}g^{-1}}\overline{f}^{\ast}\sigma=p_{2}^{\ast}\pi_{\ast}^{K_{S}}\overline{f}^{\ast}\sigma,$

giving a cohomological correspondence. For any $G_{S}$-equivariant sheaf of
$\mathcal{O}/\varpi^{m}$-modules $\mathcal{G}$ on $\overline{X}_{K^{S}}$, we
get an induced map

$\Psi:R\Gamma(\overline{X}_{K^{\prime}},p_{2}^{\ast}\pi_{\ast}^{K_{S}}(\mathcal{G}\otimes_{\mathcal{O}/\varpi^{m}}\overline{f}^{\ast}\sigma^{\vee}))\cong
R{\rm{Hom}}_{\textnormal{Sh}(\overline{X}_{K^{\prime}},\mathcal{O}/\varpi^{m})}(p_{2}^{\ast}\pi_{\ast}^{K_{S}}\overline{f}^{\ast}\sigma,p_{2}^{\ast}\pi_{\ast}^{K_{S}}\mathcal{G})\to$
$R{\rm{Hom}}_{\textnormal{Sh}(\overline{X}_{K^{\prime}},\mathcal{O}/\varpi^{m})}(p_{1}^{\ast}\pi_{\ast}^{K_{S}}\overline{f}^{\ast}\sigma,p_{1}^{\ast}\pi_{\ast}^{K_{S}}\mathcal{G})\cong
R\Gamma(\overline{X}_{K^{\prime}},p_{1}^{\ast}\pi_{\ast}^{K_{S}}(\mathcal{G}\otimes_{\mathcal{O}/\varpi^{m}}\overline{f}^{\ast}\sigma^{\vee})),$

where the map in the middle is induced by the map $\psi$ in the first
component and by the multiplication
$g^{-1}:p_{2}^{\ast}\pi_{\ast}^{K_{S}}\mathcal{G}\cong
p_{1}^{\ast}\pi_{\ast}^{K_{S}}\mathcal{G}$ in the second component.

We then obtain an endomorphism

$\theta([g,\psi])\in\textnormal{End}_{D^{+}(\mathcal{O}/\varpi^{m})}(R\Gamma(\overline{X}_{K},\pi_{\ast}^{K_{S}}(\mathcal{G}\otimes_{\mathcal{O}/\varpi^{m}}\overline{f}^{\ast}\sigma^{\vee})))$

defined by

$R\Gamma(\overline{X}_{K},\pi_{\ast}^{K_{S}}(\mathcal{G}\otimes_{\mathcal{O}/\varpi^{m}}\overline{f}^{\ast}\sigma^{\vee}))\xrightarrow{p_{2}^{\ast}}R\Gamma(\overline{X}_{K^{\prime}},p_{2}^{\ast}\pi_{\ast}^{K_{S}}(\mathcal{G}\otimes_{\mathcal{O}/\varpi^{m}}\overline{f}^{\ast}\sigma^{\vee}))\xrightarrow{\Psi}$
$\to
R\Gamma(\overline{X}_{K^{\prime}},p_{1}^{\ast}\pi_{\ast}^{K_{S}}(\mathcal{G}\otimes_{\mathcal{O}/\varpi^{m}}\overline{f}^{\ast}\sigma^{\vee}))=R\Gamma(\overline{X}_{K},p_{1,{\ast}}p_{1}^{\ast}\pi_{\ast}^{K_{S}}(\mathcal{G}\otimes_{\mathcal{O}/\varpi^{m}}\overline{f}^{\ast}\sigma^{\vee}))\xrightarrow{\textnormal{tr}}$
$\to
R\Gamma(\overline{X}_{K},\pi_{\ast}^{K_{S}}(\mathcal{G}\otimes_{\mathcal{O}/\varpi^{m}}\overline{f}^{\ast}\sigma^{\vee}))$

where the last map is the trace map coming from the canonical map induced by
the adjoint pair $(p_{1,\ast}=p_{1,!},p_{1}^{\ast}=p_{1}^{!})$.

######  Lemma 2.6.

The endomorphism $\theta([g,\psi])$ of
$R\Gamma(\overline{X}_{K},\pi_{\ast}^{K_{S}}(\mathcal{G}\otimes_{\mathcal{O}/\varpi^{m}}\overline{f}^{\ast}\sigma^{\vee}))$
coincides with the one induced by $[g,\psi]$ via the recipe of Lemma 2.4.

###### Proof.

Note that we have a natural isomorphism

$R\Gamma(\overline{X}_{K},\pi_{\ast}^{K_{S}}(\mathcal{G}\otimes_{\mathcal{O}/\varpi^{m}}\overline{f}^{\ast}\sigma^{\vee}))\cong
R{\rm{Hom}}_{\textnormal{Sh}(\overline{X}_{K},\mathcal{O}/\varpi^{m})}(\pi_{\ast}^{K_{S}}\overline{f}^{\ast}\sigma,\pi_{\ast}^{K_{S}}\mathcal{G})$

in $D^{+}(\mathcal{O}/\varpi^{m})$. Indeed, this follows from the fact that
the functor

$\pi_{\ast}^{K_{S}}:\textnormal{Sh}_{K_{S}}(\overline{X}_{K^{S}},\mathcal{O}/\varpi^{m})\xrightarrow{}\textnormal{Sh}(\overline{X}_{K},\mathcal{O}/\varpi^{m})$

is an equivalence of categories with inverse $\pi^{\ast}$ and pullbacks
commute with tensor products. We obtain analogous descriptions of the other
two complexes appearing in the definition of $\theta([g,\psi])$.

We now pick an injective resolution $\mathcal{G}\to\mathcal{I}^{\bullet}$ in
$D^{+}(\textnormal{Sh}_{G_{S}}(\overline{X}_{K^{S}},\mathcal{O}/\varpi^{m}))$.
Notice that, by [Sch98], §3, Corollary 3, it gives rise to an injective
resolution in
$D^{+}(\textnormal{Sh}_{K_{S}}(\overline{X}_{K^{S}},\mathcal{O}/\varpi^{m}))$,
and
$D^{+}(\textnormal{Sh}_{K_{S}^{\prime}}(\overline{X}_{K^{S}},\mathcal{O}/\varpi^{m}))$.
Moreover, since the functors $\pi_{\ast}^{K_{S}}$,
$p_{1}^{\ast}\pi_{\ast}^{K_{S}}$, and $p_{2}^{\ast}\pi_{\ast}^{K_{S}}$ are all
equivalences, $\pi_{\ast}^{K_{S}}\mathcal{I}^{\bullet}$,
$p_{1}^{\ast}\pi_{\ast}^{K_{S}}\mathcal{I}^{\bullet}$, and
$p_{2}^{\ast}\pi_{\ast}^{K_{S}}\mathcal{I}^{\bullet}$ are injective
resolutions. If we combine this with the discussion of the previous paragraph,
we see that

$\Gamma(\overline{X}_{K},\pi_{\ast}^{K_{S}}(\mathcal{I}^{\bullet}\otimes_{\mathcal{O}/\varpi^{m}}\overline{f}^{\ast}\sigma^{\vee}))={\rm{Hom}}_{K_{S}}(\sigma,\mathcal{I}^{\bullet}(\overline{X}_{K^{S}}))$

computes
$R\Gamma(\overline{X}_{K},\pi_{\ast}^{K_{S}}(\mathcal{G}\otimes_{\mathcal{O}/\varpi^{m}}\overline{f}^{\ast}\sigma^{\vee}))$.
Analogous observations apply to the other two complexes appearing in the
definition of $\theta([g,\psi])$. We reduced the lemma to comparing the two
actions in

$\textnormal{End}_{\mathcal{O}/\varpi^{m}}({\rm{Hom}}_{K_{S}}(\sigma,\mathcal{I}^{i}(\overline{X}_{K^{S}})))$

for every $i\in\mathbf{Z}$. The lemma then follows from the concrete
description of the effect of the trace map on global sections and an easy
unravelling of the definitions. ∎

######  Remark 2.7.

Note that our discussion applies also to the case of
$R\Gamma_{c}(X_{K},\sigma^{\vee})$. To see this, denote by
$j:\mathfrak{X}_{G}\hookrightarrow\overline{\mathfrak{X}}_{G}$ the natural
open immersion and by $\overline{f}:\overline{\mathfrak{X}}_{G}\to\ast$ the
projection to the point. Then the claim follows from the fact that we have an
isomorphism

$j_{!}j^{\ast}\overline{f}^{\ast}\sigma^{\vee}\cong
j_{!}\mathcal{O}/\varpi^{m}\otimes_{\mathcal{O}/\varpi^{m}}\overline{f}^{\ast}\sigma^{\vee}$

of $G(\mathbf{A}_{F}^{\infty})$-equivariant sheaves. This identification
follows from an equivariant version of [KS94], Proposition 2.5.13.

######  Corollary 2.8.

Given
$\sigma\in\textnormal{Mod}_{\textnormal{sm}}(\mathcal{O}/\varpi^{m}[K_{S}])$,
finite free as an $\mathcal{O}/\varpi^{m}$-module. The Verdier duality
isomorphism

$R{\rm{Hom}}_{\mathcal{O}/\varpi^{m}}(R\Gamma_{c}(X_{K},\sigma^{\vee}),\mathcal{O}/\varpi^{m})\cong
R\Gamma(X_{K},\sigma)[\dim_{\mathbf{R}}X_{K}]$

is equivariant with respect to the natural left action of
$\mathcal{H}(\sigma)$ on the right and the one induced by the anti-isomorphism

$\mathcal{H}(\sigma)\xrightarrow{\sim}\mathcal{H}(\sigma^{\vee}),$
$[g,\psi]\mapsto[g^{-1},\psi^{t}]$

on the left.

###### Proof.

Just as in the proof of [NT16], Proposition 3.7, this follows from the
functoriality of Verdier duality and Lemma 2.6 taking into account that
passing to duals interchanges pullbacks with traces in the definition of
$\theta([g,\psi])$. ∎

###  2.5. The quasi-split unitary group

From now on we specialise to our setup of interest. The two group we will be
interested in are the quasi-split unitary group $U(n,n)$ and the general
linear group appearing as its Levi subgroup. In particular, we fix an integer
$n\geq 2$ and an imaginary CM field $F$ with maximal totally real subfield
$F^{+}\subset F$. Denote by $c\in\textnormal{Gal}(F/F^{+})$ its complex
conjugation and set $\overline{S}_{p}:=S_{p}(F^{+})$ resp. $S_{p}:=S_{p}(F)$.
Consider the $2n\times 2n$ matrix

$J_{n}:=\begin{pmatrix}0&\Psi_{n}\\\ -\Psi_{n}&0\end{pmatrix}$

where $\Psi_{n}$ denotes the $n\times n$ matrix with $1$’s on the anti-
diagonal and $0$’s elsewhere. We then set $\widetilde{G}/\mathcal{O}_{F^{+}}$
to be the group scheme that, for an $\mathcal{O}_{F^{+}}$-algebra $R$, has
$R$-points given by

$\widetilde{G}(R)=\\{g\in\textnormal{GL}_{2n}(R\otimes_{\mathcal{O}_{F^{+}}}\mathcal{O}_{F})\mid{}^{t}gJ_{n}g^{c}=J_{n}\\}$

where ${}^{t}(-)$ denotes the transpose matrix. This is an integral model of
the quasi-split unitary group $U(n,n)/F^{+}$, a form of
$\textnormal{GL}_{2n}$, splitting after base change to $F$. In particular, it
becomes reductive after base change to $\mathcal{O}_{F^{+}_{\bar{v}}}$ for
$\bar{v}$ a finite place of $F^{+}$ which is unramified in $F$.

We let $P\subset\widetilde{G}$ to be the Siegel parabolic consisting of block
upper triangular matrices with blocks of size $n\times n$. Let $P=G\ltimes U$
be a Levi decomposition such that $G$ is given by the closed subgroup of block
diagonal matrices. Then $G$ can be identified with
$\textnormal{Res}_{\mathcal{O}_{F}/\mathcal{O}_{F^{+}}}\textnormal{GL}_{n}$ as
in [NT16], Lemma 5.1.999We use their choice of identification. We will write
$\widetilde{X}$ for the symmetric space $X^{\widetilde{G}}$ and
$\widetilde{X}_{\widetilde{K}}$ for the associated locally symmetric space for
a good subgroup
$\widetilde{K}\subset\widetilde{G}(\mathbf{A}_{F^{+}}^{\infty})$. Similarly,
we denote by $X$ the symmetric space $X^{G}$ and write $X_{K}$ for the
associated locally symmetric space for a good subgroup $K\subset
G(\mathbf{A}^{\infty}_{F^{+}})=\textnormal{GL}_{n}(\mathbf{A}_{F}^{\infty})$.

Write $T\subset\widetilde{B}\subset\widetilde{G}$ for the subgroup consisting,
respectively, of the diagonal and upper triangular matrices of
$\widetilde{G}$. These form a maximal torus and a Borel subgroup of
$\widetilde{G}$. Moreover, $B=\widetilde{B}\cap G\subset G$ is the Borel
subgroup of upper triangular matrices.

Recall that, for a place $\bar{v}$ of $F^{+}$ splitting in $F$, a choice of
place $v\mid\bar{v}$ in $F$ gives a canonical isomorphism
$\iota_{v}:\widetilde{G}(F^{+}_{\bar{v}})\cong\textnormal{GL}_{2n}(F_{v})$.
Indeed, there is an isomorphism $F^{+}_{\bar{v}}\otimes_{F^{+}}F\cong
F_{v}\times F_{v^{c}}$ and $\iota_{v}$ is the projection to the first factor
of the natural inclusion
$\widetilde{G}(F^{+}_{\bar{v}})\subset\textnormal{GL}_{2n}(F_{v})\times\textnormal{GL}_{2n}(F_{v^{c}})$.
Under $\iota_{v}$, $P(F^{+}_{\bar{v}})$ is identified with the standard
parabolic subgroup $P_{(n,n)}(F_{v})\subset\textnormal{GL}_{2n}(F_{v})$ of
block upper triangular matrices of type $(n,n)$ and $G(F^{+}_{\bar{v}})$ with
its standard Levi subgroup of block diagonal matrices. Similarly,
$\widetilde{B}(F^{+}_{\bar{v}})$ is identified with the subgroup of upper
triangular matrices and $T(F^{+}_{\bar{v}})$ with the diagonal matrices.
Moreover, for any parabolic subgroup
$\widetilde{B}_{F^{+}_{\bar{v}}}\subset\widetilde{Q}\subset
P_{F^{+}_{\bar{v}}}$, $\widetilde{Q}(F^{+}_{\bar{v}})$ is identified with a
standard parabolic subgroup
$P_{(n_{1},...,n_{t})}(F_{v})\subset\textnormal{GL}_{2n}(F_{v})$ where
$(n_{1},...,n_{t})$ refines $(n,n)$. Let
$\widetilde{Q}=\widetilde{M}\ltimes\widetilde{N}$ its Levi decomposition and
set $M=G\cap\widetilde{M}$.

Note that, since the inclusion

$G(F^{+}_{\bar{v}})=\textnormal{GL}_{n}(F_{v})\times\textnormal{GL}_{n}(F_{v^{c}})\hookrightarrow\textnormal{GL}_{2n}(F_{v})$

under $\iota_{v}$ is given by

(2.4) $(A,B)\mapsto\begin{pmatrix}(\Psi_{n}{}^{t}B^{-1}\Psi_{n})^{c}&0\\\
0&A\end{pmatrix},$

we have $M(F^{+}_{\bar{v}})=M_{(n_{k+1},...,n_{t})}(F_{v})\times
M_{(n_{k},...,n_{1})}(F_{v^{c}})\hookrightarrow G(F^{+}_{\bar{v}})$ where
$1\leq k\leq t$ is so that $n_{1}+...+n_{k}=n$. We set
$\theta_{n}:\textnormal{GL}_{n}(F_{v^{c}})\cong\textnormal{GL}_{n}(F_{v})$ to
be the map $B\mapsto(\Psi_{n}{}^{t}B^{-1}\Psi_{n})^{c}$ above.

###  2.6. Inertial local Langlands for $\textnormal{GL}_{n}$

Set $L=F_{v}$ for some $v\in S_{p}(F)$. According to [BD84], the category
$\textnormal{Mod}_{\textnormal{sm}}(\overline{E}[G(L)])$ admits a direct sum
decomposition

$\oplus_{\Omega}\textnormal{Mod}_{\textnormal{sm}}(\overline{E}[G(L)])[\Omega]$

into so-called Bernstein blocks. In terms of the local Langlands
correspondence, two irreducible representation
$\pi_{1},\pi_{2}\in\textnormal{Mod}_{\textnormal{sm}}(\overline{E}[G(L)])$
correspond to the same Bernstein block if and only if

$\textnormal{rec}(\pi_{1})^{ss}|_{I_{L}}\cong\textnormal{rec}(\pi_{2})^{ss}|_{I_{L}},$

and a general representation
$\pi\in\textnormal{Mod}_{\textnormal{sm}}(\overline{E}[G(L)])$ lies in
$\Omega$ if each of its Jordan–Hölder constituents does. Moreover, the centre
$\mathfrak{z}_{\Omega}$ of the category
$\textnormal{Mod}_{\textnormal{sm}}(\overline{E}[G(L)])[\Omega]$ is called the
Bernstein centre possessing the following property. Being the centre of the
Bernstein block, it acts on each object lying in $\Omega$. In particular, for
any irreducible $\pi$ lying in $\Omega$, the natural action induces a
character $\chi_{\pi}:\mathfrak{z}_{\Omega}\to\overline{E}$. Then, given a
pair of irreducible objects
$\pi_{1},\pi_{2}\in\textnormal{Mod}_{\textnormal{sm}}(\overline{E}[G(L)])[\Omega]$,
$\chi_{\pi_{1}}=\chi_{\pi_{2}}$ if and only if $\pi_{1}$ and $\pi_{2}$ have
the same supercuspidal support (cf. [BD84]).101010For a brief overview of
these results stated with more care, see [Hel16], §3.

Work of Bushnell–Kutzko shows that, given any Bernstein block $\Omega$, there
is always a pair $(J,\sigma)$ of a compact open $J\subset G(L)$ and an
irreducible $\overline{E}$-representation of $J$ such that
$\pi\in\textnormal{Mod}_{\textnormal{sm}}(\overline{E}[G(L)])$ lies in
$\Omega$ if and only if it is generated by its $\sigma$-isotypic vectors. Such
a pair is then called a semisimple Bushnell–Kutzko type for the block
$\Omega$. Given a semisimple Bushnell–Kutzko type $(J,\sigma)$, [BK98],
Theorem 4.3 shows that taking $\sigma$-invariants sets up an equivalence of
categories

$\textnormal{Mod}_{\textnormal{sm}}(\overline{E}[G(L)])[\Omega]\xrightarrow{\sim}\textnormal{Mod}(\mathcal{H}(\sigma)),$
$\pi\mapsto{\rm{Hom}}_{J}(\sigma,\pi)\cong{\rm{Hom}}_{G(L)}(\textnormal{c-Ind}_{J}^{G(L)}\sigma,\pi)$

where
$\mathcal{H}(\sigma):=\textnormal{End}_{G(L)}(\textnormal{c-Ind}_{J}^{G(L)}\sigma)$.
In particular, we see that the action of the Bernstein centre on
$\textnormal{c-Ind}_{J}^{G(L)}\sigma$ identifies $\mathfrak{z}_{\Omega}$ with
the centre $Z(\mathcal{H}(\sigma))$ of $\mathcal{H}(\sigma)$. Finally, given a
Bernstein block $\Omega$, after possibly enlarging $E$, it admits a canonical
model over $E$ (i.e. a direct summand of
$\textnormal{Mod}_{\textnormal{sm}}(E[G(L)])$) with centre
$\mathfrak{z}_{\Omega,E}$, a model of $\mathfrak{z}_{\Omega}$ and, for any
type $(J,\sigma)$ with a model $\sigma_{E}$ over $E$, we still have
$\mathfrak{z}_{\Omega,E}\cong Z(\mathcal{H}(\sigma_{E}))$ (cf. [Car+16],
§3.13). In particular, from now on for a given Bernstein block $\Omega$, we
always assume that $E$ is sufficiently large so the block is already defined
over $E$.

In [SZ99], the authors refine the Bernstein decomposition of
$\textnormal{Mod}_{\textnormal{sm}}(\overline{E}[G(L)])$ to a "stratification"
of the category and construct a type theory with respect to this
stratification as we will discuss now. We state their results in terms of
local Langlands, following [BC09], §6.5.

######  Definition 2.9.

We define a Weil–Deligne inertial type (for $L$) to be an isomorphism class of
pairs $\tau=(\rho_{\tau},N_{\tau})$ such that
$\rho_{\tau}:I_{L}\to\textnormal{GL}_{n}(E)$ is a representation of the
inertia subgroup $I_{L}\subset W_{L}$ with open kernel, $N_{\tau}\in M_{n}(E)$
is a nilpotent matrix such that there exists a Weil–Deligne representation
$(r,N)$ of $L$ and an isomorphism $(r,N)|_{I_{L}}\cong(\rho_{\tau},N_{\tau})$.

Given a nilpotent matrix $N\in M_{n}(E)$, its Jordan normal form gives rise to
a partition $P_{N}$ of $n$. A partition $P$ can be viewed uniquely as a
decreasing function $P:\mathbf{Z}_{>0}\to\mathbf{Z}_{\geq 0}$ with finite
support (where $P$ is a partition of $\sum_{i\in\mathbf{Z}_{>0}}P(i)$). Given
two nilpotent matrices $N_{1},N_{2}\in M_{n}(E)$, we write $N_{1}\preceq
N_{2}$ if and only $\sum_{1\leq i\leq k}P_{N_{1}}(i)\leq\sum_{1\leq i\leq
k}P_{N_{2}}(i)$ for every $k\in\mathbf{Z}_{\geq 1}$. We record the following
observation (cf. [BC09], Proposition 7.8.1).

######  Proposition 2.10.

Let $N_{1},N_{2}\in M_{n}(E)$ be two nilpotent matrices. Then $N_{1}\preceq
N_{2}$ if and only if, for all $i\in\mathbf{Z}_{\leq 1}$, we have

$\textnormal{rank}(N^{i}_{1})\leq\textnormal{rank}(N^{i}_{2}).$

######  Definition 2.11.

Given two Weil–Deligne inertial types $\tau_{1},\tau_{2}$ for $L$, we write
$\tau_{1}\preceq\tau_{2}$ if $\rho_{\tau_{1}}\cong\rho_{\tau_{2}}$ and
$N_{\tau_{1}}\preceq N_{\tau_{2}}$. Moreover, given two Weil–Deligne
representations $r_{1},r_{2}$ for $L$, we write $r_{1}\preceq r_{2}$ if
$r_{1}|_{I_{L}}\preceq r_{2}|_{I_{L}}$.

######  Theorem 2.12 ([SZ99]).

Let $\tau$ be a Weil–Deligne inertial type. Then there is a smooth irreducible
$E$-representation $\sigma(\tau)$ of $G(\mathcal{O}_{L})$ such that, for $\pi$
any irreducible smooth representation of $G(L)$, the following hold.

  1. (i)

If $\pi|_{G(\mathcal{O}_{L})}$ contains $\sigma(\tau)$, then
$\textnormal{rec}(\pi)|_{I_{L}}\preceq\tau$;

  2. (ii)

if $\textnormal{rec}(\pi)|_{I_{L}}\cong\tau$, then $\pi|_{G(\mathcal{O}_{L})}$
contains $\sigma(\tau)$ with multiplicity one;

  3. (iii)

if $\textnormal{rec}(\pi)|_{I_{L}}\preceq\tau$ and $\pi$ is generic, then
$\pi|_{G(\mathcal{O}_{L})}$ contains $\sigma(\tau)$.

###### Proof.

See [Hun+18], Theorem 2.5.4 and the references therein. ∎

We point out that the Theorem makes no mention about the uniqueness of
$\sigma(\tau)$. In particular, whenever we consider an inertial Weil–Deligne
type $\tau$, we fix a choice of such $\sigma(\tau)$.

######  Remark 2.13.

Notice that when $\tau=(\rho_{\tau},N_{\tau})$ is so that $N_{\tau}=0$, we
obtain the potentially crystalline inertial local Langlands of [Car+16],
Theorem 3.7.

Note that each Weil–Deligne inertial type $\tau$ gives rise to an inertial
type $\rho_{\tau}$ in the classical sense and henceforth to a Bernstein block
$\Omega$. Moreover, if $(J,\sigma)$ is a type for $\Omega$, then, by
construction, $\sigma(\tau)$ is a direct summand of
$\textnormal{c-Ind}_{J}^{G(\mathcal{O}_{L})}\sigma$. In particular,
$\mathfrak{z}_{\Omega}$ acts on
$\textnormal{c-Ind}_{G(\mathcal{O}_{L})}^{G(L)}\sigma(\tau)$. One sees that
this, in fact, is a faithful action,111111In fact, [Pyv20], Theorem 7.1 shows
that $\mathcal{H}(\sigma(\tau))$ is a free module over
$\mathfrak{z}_{\Omega}$. yielding an injection
$\mathfrak{z}_{\Omega}\hookrightarrow\mathfrak{z}_{\tau}:=Z(\mathcal{H}(\sigma(\tau)))$.

###  2.7. Local systems on $\widetilde{X}_{\widetilde{K}}$ and $X_{K}$

We now introduce the local systems we will be working with. These will be
constructed from locally algebraic representations
$\sigma_{\textnormal{alg}}\otimes\sigma_{\textnormal{sm}}$ where the algebraic
part $\sigma_{\textnormal{alg}}$ will encode the weight of the automorphic
representations considered (i.e. their shape at $\infty$) and the smooth part
$\sigma_{\textnormal{sm}}$ will pin down their inertial type at $p$.

Our integral coefficient systems will depend on the choice of a prime $p$. We
further set $E/\mathbf{Q}_{p}$ to be our coefficient field, a sufficiently
large subfield of $\overline{\mathbf{Q}}_{p}$ finite over $\mathbf{Q}_{p}$
such that
$\textnormal{Hom}(F,E)=\textnormal{Hom}(F,\overline{\mathbf{Q}}_{p})$, and
denote by $\mathcal{O}$ its ring of integers. Fix a choice of uniformiser
$\varpi\in\mathcal{O}$ as well.

We start by introducing our locally algebraic representations for $G$. For
each $p$-adic place $v$ of $F$, we consider a standard (possibly non-proper)
parabolic subgroup $Q_{v}\subset\textnormal{GL}_{n}$ with standard Levi
decomposition $Q_{v}=M_{v}\ltimes N_{v}$ and consider the corresponding
parahoric subgroup scheme $\prescript{v}{}{\mathcal{Q}}$ of
$\textnormal{GL}_{n}$. For $S\subset S_{p}$, we set $Q_{S}:=\prod_{v\in
S}Q_{v}$ and $Q_{p}:=Q_{S_{p}}$. Set
$\mathcal{Q}_{v}:=\prescript{v}{}{\mathcal{Q}}(\mathcal{O}_{F_{v}})\subset\textnormal{GL}_{n}(\mathcal{O}_{F_{v}})$
to be the corresponding parahoric subgroup. For a finite set of $p$-adic
places $S\subset S_{p}$, we set $\mathcal{Q}_{S}:=\prod_{v\in
S}\mathcal{Q}_{v}\subset\textnormal{GL}_{n}(\widehat{\mathcal{O}}_{F,S})$.
Moreover, for any $v\in S_{p}$, and integers $c\geq b\geq 0$ with $c\geq 1$,
we denote by $\mathcal{Q}_{v}(b,c)\subset\mathcal{Q}_{v}$ the subgroup of
matrices which are block upper triangular modulo $\varpi_{v}^{c}$ and block
unipotent modulo $\varpi_{v}^{b}$, where $\varpi_{v}\in\mathcal{O}_{F_{v}}$ is
some choice of uniformiser. Extend the definition in the obvious way to define
$\mathcal{Q}_{S}(b,c)$. In particular, we have
$\mathcal{Q}_{v}(0,1)=\mathcal{Q}_{v}$. Note that $\mathcal{Q}_{v}(b,c)$
admits an Iwahori decomposition
$\overline{N}_{v}^{c}M_{v}^{b}N_{v}^{0}$121212Recall from Notation and
Conventions that for $H$ being any of the groups $\overline{N}_{v},M_{v},$ or
$N_{v}$, and $n\in\mathbf{Z}_{\leq 0}$, $H^{n}$ denotes the subgroup of
matrices of $H(\mathcal{O}_{F_{v}})$ that reduce to the identity modulo
$\varpi_{v}^{n}$. and therefore the formalism of [All+23], 2.1.9 applies.

Such parahoric subgroups will be our level subgroups for which we introduce
locally algebraic representations. These representations will then yield local
systems convenient for the development of $Q_{p}$-ordinary Hida theory. After
taking $Q_{p}$-ordinary parts, the cohomology of these local systems will
encode integral $Q_{p}$-ordinary automorphic representations with prescribed
weights at $\infty$ and inertial types at $p$. When $Q_{p}$ is taken to be
$\prod_{v}\textnormal{GL}_{n}$, this will simply mean pinning down the weight
of the automorphic and the inertial type at $p$ of the whole automorphic
representation.

We first take care of the algebraic part. As usual, the character group of
$(\textnormal{Res}_{F^{+}/\mathbf{Q}}T)_{E}=(\textnormal{Res}_{F/\mathbf{Q}}T_{n})_{E}$,
for $T_{n}\subset\textnormal{GL}_{n}$ the subgroup of diagonal matrices, can
be identified with $(\mathbf{Z}^{n})^{{\rm{Hom}}(F,E)}$. A character
$\lambda=(\lambda_{\iota,i})\in(\mathbf{Z}^{n})^{\textnormal{Hom}(F,E)}$ will
then be
$(\textnormal{Res}_{F^{+}/\mathbf{Q}}B)_{E}=(\textnormal{Res}_{F/\mathbf{Q}}B_{n})_{E}$-dominant
if and only if, for every $\iota\in{\rm{Hom}}(F,E)$, we have

$\lambda_{\iota,1}\geq...\geq\lambda_{\iota,n}.$

Then, given $\lambda\in(\mathbf{Z}^{n}_{+})^{{\rm{Hom}}(F,E)}$, highest weight
theory provides an integral representation of $\prod_{\iota:F\hookrightarrow
E}\textnormal{GL}_{n}(\mathcal{O})$. This will simply be the representation
constructed for instance in [Ger18], §2.2. More precisely, if
$B_{n}\subset\textnormal{GL}_{n}$ is the standard Borel of upper triangular
matrices, $w_{0,n}$ denotes the longest element of the Weyl group of
$\textnormal{GL}_{n}$ and $\iota\in{\rm{Hom}}(F,E)$, we consider the algebraic
induction

$\xi_{\lambda_{\iota}}:=(\textnormal{Ind}_{B_{n}}^{\textnormal{GL}_{n}}w_{0,n}\lambda_{\iota})_{/\mathcal{O}}:=\\{f\in\mathcal{O}[\textnormal{GL}_{n}]\mid
f(bg)=(w_{0,n}\lambda_{\iota})(b)f(g)$ $\textnormal{ for every }\mathcal{O}\to
R,b\in B_{n}(R),g\in\textnormal{GL}_{n}(R)\\},$

and set $\mathcal{V}_{\lambda_{\iota}}:=\xi_{\lambda_{\iota}}(\mathcal{O})$,
$V_{\lambda_{\iota}}:=\mathcal{V}_{\lambda_{\iota}}\otimes_{\mathcal{O}}E$. We
then set
$\mathcal{V}_{\lambda}:=\otimes_{\iota,\mathcal{O}}\mathcal{V}_{\lambda_{\iota}}$
and $V_{\lambda}:=\mathcal{V}_{\lambda}\otimes_{\mathcal{O}}E$. Note that
$V_{\lambda}$ is the highest weight representation of
$(\textnormal{Res}_{F^{+}/\mathbf{Q}}G)_{E}=(\textnormal{Res}_{F/\mathbf{Q}}\textnormal{GL}_{n})_{E}\cong\prod_{\iota:F\hookrightarrow
E}\textnormal{GL}_{n,E}$ of highest weight $\lambda$, a finite dimensional
$E$-representation. Moreover, $\mathcal{V}_{\lambda}\subset V_{\lambda}$ is a
$G(\mathcal{O})$-stable $\mathcal{O}$-lattice. In particular, for every
$m\in\mathbf{Z}_{\geq 1}$, $\mathcal{V}_{\lambda}/\varpi^{m}$ is a smooth
$\mathcal{O}/\varpi^{m}[\prod_{v\in
S_{p}}\textnormal{GL}_{n}(\mathcal{O}_{F_{v}})]$-module131313In particular, it
becomes a smooth $\mathcal{O}/\varpi^{m}[\mathcal{Q}_{S_{p}}]$-module. under
the product of diagonal embeddings
$\textnormal{GL}_{n}(\mathcal{O}_{F_{v}})\hookrightarrow\prod_{\iota:F_{v}\hookrightarrow
E}\textnormal{GL}_{n}(\mathcal{O})$, finite free over
$\mathcal{O}/\varpi^{m}$, and the formalism of §2.2 applies.

For a dominant weight
$\lambda_{v}\in(\mathbf{Z}^{n}_{+})^{{\rm{Hom}}(F_{v},E)}$, set
$\mathcal{V}_{w_{0}^{Q_{v}}\lambda_{v}}=\otimes_{\iota:F_{v}\hookrightarrow
E}\mathcal{V}_{w_{0}^{Q_{v}}\lambda_{\iota}}$ to be the representation of
$\prod_{\iota:F_{v}\hookrightarrow E}M_{v}(\mathcal{O})$ associated with
$w_{0}^{Q_{v}}\lambda_{v}$ by the previous procedure where
$w_{0}^{Q_{v}}=w_{0}^{M_{v}}w_{0}^{\textnormal{GL}_{n}}$ denotes the product
of the longest Weyl group element of $M_{v}$, and of $\textnormal{GL}_{n}$.
Concretely, if we assume that $Q_{v}=P_{(n_{1},..,n_{k})}$, then
$w_{0}^{Q_{v}}\lambda_{v}=(\lambda_{v}^{n_{1}},...,\lambda_{v}^{n_{k}})$,
where

$\lambda_{v}^{n_{i}}=(\lambda_{\iota}^{n_{i}})_{\iota:F_{v}\hookrightarrow
E}=(\lambda_{\iota,n+1-(n_{1}+...+n_{i})},...,\lambda_{\iota,n+1-(n_{1}+...+n_{i-1}+1)})_{\iota}\in(\mathbf{Z}^{n_{i}}_{+})^{\textnormal{Hom}(F_{v},E)},$

and

$\mathcal{V}_{w_{0}^{Q_{v}}\lambda_{v}}=\mathcal{V}_{\lambda^{n_{1}}_{v}}\otimes...\otimes\mathcal{V}_{\lambda^{n_{k}}_{v}}.$

We then have the analogue of [CN23], Lemma 2.1.12 with identical proof.

######  Lemma 2.14.

The natural $\prod_{\iota:F_{v}\hookrightarrow
E}Q_{v}(\mathcal{O})$-equivariant map

$\mathcal{V}_{\lambda_{v}}\to\mathcal{V}_{w_{0}^{Q_{v}}\lambda_{v}}$

given by evaluation of functions at the identity is a surjection.

We now turn to the smooth part which will be given by inflating to parahoric
level the types of Schneider–Zink. Therefore, the smooth part will in fact
depend on the choice of $Q_{v}$. For $v\in S_{p}$ consider a standard
parabolic $Q_{v}$ and assume that it corresponds to a partition
$(n_{1},...,n_{k})$ of $n$.

######  Definition 2.15.

We call a tuple $\underline{\tau_{v}}=(\tau_{v,i})_{i=1,...,k}$ consisting of
($E$-valued) $n_{i}$-dimensional Weil–Deligne inertial types $\tau_{v,i}$ for
$F_{v}$ to be an inertial type of type $Q_{v}$.

Given an inertial type $\underline{\tau_{v}}$ of type $Q_{v}$, Schneider–Zink
provides a smooth irreducible $E$-representation $\sigma(\tau_{v,i})$ of
$\textnormal{GL}_{n_{i}}(\mathcal{O}_{F_{v}})$ for every $i=1,...,k$ (cf.
Theorem 2.12). In particular, we obtain an irreducible smooth
$E$-representation

$\sigma(\underline{\tau_{v}}):=\otimes_{i=1,...,k}\sigma(\tau_{v,i})$

of $M_{v}^{0}=M_{v}(\mathcal{O}_{F_{v}})$. We fix a choice of
$M_{v}^{0}$-stable $\mathcal{O}$-lattice
$\sigma(\underline{\tau_{v}})^{\circ}\subset\sigma(\underline{\tau_{v}})$. We
set $c_{v}\geq 0$ to be the smallest integer such that
$M_{v}^{c_{v}}=\textnormal{ker}(M_{v}^{0}\to\prod_{v\in
S_{p}}M_{v}(\mathcal{O}_{F_{v}}/\varpi^{c_{v}}))$ acts trivially on
$\sigma(\underline{\tau_{v}})^{\circ}$. Then $\mathcal{Q}_{v}(0,c_{v})$ acts
on $\sigma(\underline{\tau_{v}})^{\circ}$ by sending

$\begin{pmatrix}A_{1}&.&.&.&\ast\\\ .&A_{2}&&&.\\\ .&&.&&.\\\ .&&&.&.\\\
\ast&.&.&.&A_{k}\end{pmatrix}\in\mathcal{Q}_{v}(0,c_{v})$

to $\sigma(\underline{\tau_{v}})^{\circ}(A_{1},...,A_{k})$. Since the
corresponding map

$\mathcal{Q}_{v}(0,c_{v})\to
M_{v}^{0}/M_{v}^{c_{v}}=M_{v}(\mathcal{O}_{F_{v}}/\varpi_{v}^{c_{v}})$

is easily checked to be a group homomorphism, this indeed defines a group
action. We then denote by $\widetilde{\sigma(\underline{\tau_{v}})}^{\circ}$
the representation $\sigma(\underline{\tau_{v}})^{\circ}$ viewed as an
$\mathcal{O}[\mathcal{Q}_{v}(0,c_{v})]$-module.

We then set our locally algebraic representation associated with the data
$(Q_{S_{p}},\lambda,\underline{\tau}):=(Q_{v},\lambda_{v},\underline{\tau_{v}})_{v\in
S_{p}}$ to be

$\mathcal{V}_{(\lambda,\underline{\tau})}^{Q_{p}}:=\mathcal{V}_{\lambda}\otimes_{\mathcal{O}}(\bigotimes_{v\in
S_{p},\mathcal{O}}\widetilde{\sigma(\underline{\tau_{v}})}^{\circ})^{\vee}.$

This gives rise to a $\mathcal{O}/\varpi^{m}[\mathcal{Q}_{p}(0,c_{p})]$-module
for $c_{p}:=\max\\{c_{v}\\}_{v\in S_{p}}$. By abuse of notation, we will use
the same notation for the induced local systems on $X_{K},\overline{X}_{K}$
and $\partial X_{K}$, respectively, for $K\subset
G(\mathbf{A}_{F^{+}}^{\infty})$ any good subgroup with
$K_{p}\subset\mathcal{Q}_{p}(0,c_{p})$.

Note that when, at each place $v\in S_{p}$, $Q_{v}=\textnormal{GL}_{n}$, these
local systems simplify. In this case we will abbreviate the notation to
$\mathcal{V}_{(\lambda,\tau)}=\mathcal{V}_{\lambda}\otimes_{\mathcal{O}}\sigma(\tau)^{\circ,\vee}$.
We denote the corresponding locally algebraic type by
$\sigma(\lambda,\tau)^{\circ}:=(\mathcal{V}_{(\lambda,\tau)})^{\vee}$. By
Lemma 2.4, the corresponding mod $\varpi^{m}$ cohomology groups are obtained
by taking invariants of completed cohomology

$R\Gamma_{(c)}(X_{K},\mathcal{V}_{(\lambda,\tau)}/\varpi^{m})\cong
R{\rm{Hom}}_{\mathcal{O}/\varpi^{m}[K_{p}]}(\sigma(\lambda,\tau)^{\circ}/\varpi^{m},\pi_{(c)}(K^{p},\mathcal{O}/\varpi^{m}))\cong$
$R{\rm{Hom}}_{\mathcal{O}/\varpi^{m}[G_{p}]}(\textnormal{c-Ind}_{K_{p}}^{G_{p}}\sigma(\lambda,\tau)^{\circ}/\varpi^{m},\pi_{(c)}(K^{p},\mathcal{O}/\varpi^{m})).$

As a consequence, we get a natural right action of the Hecke algebra
$\mathcal{H}(\sigma(\lambda,\tau)^{\circ})=\textnormal{End}_{G_{p}}(\textnormal{c-Ind}_{K_{p}}^{G_{p}}\sigma(\lambda,\tau)^{\circ})$
on $R\Gamma_{(c)}(X_{K},\mathcal{V}_{(\lambda,\tau)}/\varpi^{m})$. In
particular, as we will explain in 5.3, we obtain a natural action of an
"integral Bernstein centre"
$\mathfrak{z}_{\lambda,\tau}^{\circ}:=\mathcal{H}(\sigma(\lambda,\tau)^{\circ})\cap\mathfrak{z}_{\Omega}$.
In fact, we will see that we have a similar description for general $Q$ after
taking $Q$-ordinary parts due to independence of level and weight, yielding
natural Hecke actions at $p$.

We now turn to treating the case of $\widetilde{G}$. Here we will only take
ordinary parts at a certain subset $\overline{S}\subset\overline{S}_{p}$ where
we wish to prove local-global compatibility using the degree shifting argument
and will vary the level at the rest of the places. Therefore, we fix such a
set $\overline{S}\subset\overline{S}_{p}$ of $p$-adic places. From now on, we
assume that every $\overline{v}\in\overline{S}_{p}$ splits in $F$ and fix a
choice of place $v\mid\overline{v}$ in $F$. This fixes a lift
$\iota:F\hookrightarrow E$ for every embedding
$\overline{\iota}:F^{+}\hookrightarrow E$. Therefore, it induces an
identification

$(\textnormal{Res}_{F^{+}/\mathbf{Q}}\widetilde{G})_{E}=\prod_{\textnormal{Hom}(F^{+},E)}\textnormal{GL}_{2n,E}$

and an identification of the character group of
$(\textnormal{Res}_{F^{+}/\mathbf{Q}}T)_{E}$ with
$(\mathbf{Z}^{2n})^{{\rm{Hom}}(F^{+},E)}$. This identifies a weight
$\lambda=(\lambda_{\iota,i})\in(\mathbf{Z}^{n})^{{\rm{Hom}}(F,E)}$ with
$\tilde{\lambda}=(\tilde{\lambda}_{\overline{\iota},i})$ where

$\tilde{\lambda}_{\overline{\iota}}=(-w_{0}^{\textnormal{GL}_{n}}\lambda_{\iota
c},\lambda_{\iota})=(-\lambda_{\iota c,n},...,-\lambda_{\iota
c,1},\lambda_{\iota,1},...,\lambda_{\iota,n}).$

Note that the
$(\textnormal{Res}_{F^{+}/\mathbf{Q}}\widetilde{B})_{E}$-dominant weights are
precisely given by $(\mathbf{Z}^{2n}_{+})^{{\rm{Hom}}(F^{+},E)}$. For such
weights we can therefore define $\mathcal{V}_{\tilde{\lambda}}\subset
V_{\tilde{\lambda}}$. For every $m\in\mathbf{Z}_{\geq 1}$, we then obtain a
smooth
$\mathcal{O}/\varpi^{m}[\prod_{\bar{v}\in\overline{S}_{p}}\widetilde{G}(\mathcal{O}_{F^{+}_{\bar{v}}})]$-module
$\mathcal{V}_{\tilde{\lambda}}/\varpi^{m}$, finite free as an
$\mathcal{O}/\varpi^{m}$-module. These cover the algebraic parts of our
locally algebraic representations. In general, if for $\bar{v}\in\overline{S}$
given a tuple
$(Q_{v^{\prime}},\lambda_{v^{\prime}},\underline{\tau_{v^{\prime}}})_{v^{\prime}\mid\bar{v}}$
as before such that the weight $\tilde{\lambda}_{\bar{v}}$ associated with
$(\lambda_{v},\lambda_{v^{c}})$ is dominant, we introduce a tuple
$(\widetilde{Q}_{\bar{v}},\tilde{\lambda}_{\bar{v}},\underline{\tau_{\bar{v}}}=(\underline{\tau_{v}},\underline{\tau_{v^{c}}}))$
where
$\widetilde{Q}_{\bar{v}}=\widetilde{M}_{\bar{v}}\ltimes\widetilde{N}_{\bar{v}}\subset\widetilde{G}_{F^{+}_{\bar{v}}}$
is the standard parabolic subgroup with $\widetilde{Q}_{\bar{v}}\cap
G_{F^{+}_{\bar{v}}}=Q_{v}\times
Q_{v^{c}}\subset\textnormal{GL}_{n,F_{v}}\times\textnormal{GL}_{n,F_{v^{c}}}$.
As before, we denote by $\prescript{\bar{v}}{}{\widetilde{\mathcal{Q}}}$ the
corresponding parahoric subgroup scheme of
$\widetilde{G}_{F^{+}_{\bar{v}}}\cong^{\iota_{v}}\textnormal{GL}_{2n,F_{v}}$.
We set
$\widetilde{\mathcal{Q}}_{\bar{v}}=\prescript{\bar{v}}{}{\widetilde{\mathcal{Q}}}(\mathcal{O}_{F^{+}_{\bar{v}}})$
with Iwahori decomposition
$\overline{\widetilde{N}}_{\bar{v}}^{1}\widetilde{M}_{\bar{v}}^{0}\widetilde{N}_{\bar{v}}^{0}$.
We define the identification

$\iota_{v}^{w_{0}}:\textnormal{GL}_{n}(F_{v})\times\textnormal{GL}_{n}(F_{v^{c}})\xrightarrow{\sim}\textnormal{GL}_{n}(F_{v})\times\textnormal{GL}_{n}(F_{v}),$
$(A,B)\mapsto(A,\theta_{n}B).$

We then set
$\widetilde{Q}_{\bar{v}}^{w_{0}}=\widetilde{M}_{\bar{v}}^{w_{0}}\ltimes\widetilde{N}_{\bar{v}}^{w_{0}}\subset\widetilde{G}_{F^{+}_{\bar{v}}}$
to be the standard parabolic subgroup associated with the Levi
$\iota_{v}^{w_{0}}\widetilde{M}_{\bar{v}}\subset\textnormal{GL}_{2n,F_{v}}$
under the identification $\iota_{v}$. Set

$\tilde{\sigma}(\underline{\tau_{\bar{v}}})^{\circ}:=\sigma(\underline{\tau_{v}})^{\circ}\otimes_{\mathcal{O}}(\theta_{n}^{-1})^{\ast}\sigma(\underline{\tau_{v^{c}}})^{\circ}\in\textnormal{Mod}_{\textnormal{sm}}(\mathcal{O}[\widetilde{M}_{\bar{v}}^{w_{0},0}]).$

Note that here
$\theta_{n}:\textnormal{GL}_{n}(F_{v^{c}})\cong\textnormal{GL}_{n}(F_{v})$, as
previously defined, is given by the map
$B\mapsto(\Psi_{n}{}^{t}B^{-1}\Psi_{n})^{c}$ and we are applying the pullback
along $\theta^{-1}_{n}$. For $\bar{v}\in\overline{S}$, define

$\mathcal{V}^{\widetilde{Q}_{\bar{v}}^{w_{0}}}_{(\tilde{\lambda}_{\bar{v}},\underline{\tau_{\bar{v}}})}:=\mathcal{V}_{\tilde{\lambda}_{\bar{v}}}\otimes_{\mathcal{O}}\tilde{\sigma}\widetilde{(\underline{\tau_{\overline{v}}})^{\circ}}^{\vee}$

where $\tilde{\lambda}_{\bar{v}}$ have the obvious meaning of considering the
weights corresponding to all the embeddings inducing the given place.

Finally, given a tuple
$(\widetilde{Q}_{\overline{S}},\tilde{\lambda}_{\overline{S}},\underline{\tau_{\overline{S}}})=(\widetilde{Q}_{\bar{v}},\tilde{\lambda}_{\bar{v}},\underline{\tau}_{\bar{v}})_{\bar{v}\in\overline{S}}$
coming from a tuple $(Q_{p},\lambda,\underline{\tau})$ as above, and a
dominant weight
$\tilde{\lambda}\in(\mathbf{Z}^{2n}_{+})^{{\rm{Hom}}(F^{+},E)}$ for
$\widetilde{G}$ extending $\tilde{\lambda}_{\overline{S}}$, we set

$\mathcal{V}_{(\tilde{\lambda},\underline{\tau})}^{\widetilde{Q}_{\overline{S}}^{w_{0}}}:=(\bigotimes_{\bar{v}\notin\overline{S},\mathcal{O}}\mathcal{V}_{\tilde{\lambda}_{\bar{v}}})\otimes_{\mathcal{O}}(\bigotimes_{\bar{v}\in\overline{S},\mathcal{O}}\mathcal{V}^{\widetilde{Q}_{\bar{v}}^{w_{0}}}_{(\tilde{\lambda}_{\bar{v}},\underline{\tau_{\bar{v}}})}),$

a locally algebraic $\mathcal{O}$-representation of
$\left(\prod_{\bar{v}\in\overline{S}_{p}\setminus\overline{S}}\widetilde{G}(\mathcal{O}_{F^{+}_{\bar{v}}})\right)\times\widetilde{\mathcal{Q}}^{w_{0}}_{\overline{S}}(0,c_{p})$
for an appropriate integer $c_{p}\geq 1$ as before. In particular,
$\mathcal{V}_{(\tilde{\lambda},\underline{\tau})}^{\widetilde{Q}_{\overline{S}}^{w_{0}}}/\varpi^{m}$
becomes a smooth
$\left(\prod_{\bar{v}\in\overline{S}_{p}\setminus\overline{S}}\widetilde{G}(\mathcal{O}_{F^{+}_{\bar{v}}})\right)\times\widetilde{\mathcal{Q}}^{w_{0}}_{\overline{S}}(0,c_{p})$-module.
When, for every $\bar{v}\in\overline{S}$, the parabolic at $\bar{v}$ is the
Siegel one, we will abbreviate the notation to
$\mathcal{V}^{\overline{S}}_{(\tilde{\lambda},\tau)}$. Again, by abuse of
notation we will denote identically the local systems they induce on locally
symmetric spaces.

We also introduce a local system on $\widetilde{X}_{\widetilde{K}}$ suitable
for a dual version of the degree shifting argument in §6.1. For this we assume
now that, for each $\bar{v}\in\overline{S}$,
$\tilde{\lambda}_{\bar{v}}:=(\lambda_{v},-w_{0}^{\textnormal{GL}_{n}}\lambda_{v^{c}})$
is dominant for $\widetilde{G}$.161616Note that this identification
$(\mathbf{Z}^{n})^{{\rm{Hom}}(F,E)}\cong(\mathbf{Z}^{2n})^{{\rm{Hom}}(F^{+},E)}$
is not the one used in the previous paragraph. Then, for
$\bar{v}\in\overline{S}$, we set

$\tilde{\sigma}(\underline{\tau_{\bar{v}}})^{\circ,w_{0}}:=(\theta_{n}^{-1})^{\ast}\sigma(\underline{\tau_{v^{c}}})^{\circ}\otimes_{\mathcal{O}}\sigma(\underline{\tau_{v}})^{\circ}\in\textnormal{Mod}_{\textnormal{sm}}(\mathcal{O}[\widetilde{M}_{\bar{v}}^{0}]),$

and

$\mathcal{V}_{(\tilde{\lambda}_{\bar{v}},\underline{\tau_{\bar{v}}})}^{\widetilde{Q}_{\bar{v}},w_{0}^{P}}:=\mathcal{V}_{w_{0}^{P}\tilde{\lambda}_{\bar{v}}}\otimes_{\mathcal{O}}(\tilde{\sigma}(\underline{\tau_{\bar{v}}})^{\circ,w_{0}})^{\sim,\vee}\in\textnormal{Mod}(\mathcal{O}[\widetilde{\mathcal{Q}}(0,c_{p})]).$

For any dominant weight
$\widetilde{\lambda}\in(\mathbf{Z}^{2n}_{+})^{{\rm{Hom}}(F^{+},E)}$ extending
$\widetilde{\lambda}_{\overline{S}}$, we can then set

$\mathcal{V}_{(\tilde{\lambda},\underline{\tau})}^{\widetilde{Q}_{\overline{S}},w_{0}^{P}}:=(\bigotimes_{\bar{v}\notin\overline{S},\mathcal{O}}\mathcal{V}_{\tilde{\lambda}_{\bar{v}}})\otimes_{\mathcal{O}}(\bigotimes_{\bar{v}\in\overline{S},\mathcal{O}}\mathcal{V}^{\widetilde{Q}_{\bar{v}},w_{0}^{P}}_{(\tilde{\lambda}_{\bar{v}},\underline{\tau_{\bar{v}},})})$

an $\mathcal{O}$-representation of
$\left(\prod_{\bar{v}\in\overline{S}_{p}\setminus\overline{S}}\widetilde{G}(\mathcal{O}_{F^{+}_{\bar{v}}})\right)\times\widetilde{\mathcal{Q}}_{\overline{S}}(0,c_{p})$.
We denote identically the corresponding local systems induced on locally
symmetric spaces for $\widetilde{G}$.

###  2.8. Explicit Hecke operators

Next we spell out the explicit formula for the usual unramified Hecke
operators and $U_{p}$-operators as for instance in [NT16], [All+23] and
[CN23]. Fix for any $v$ finite place of $F$ a uniformiser
$\varpi_{v}\in\mathcal{O}_{F_{v}}$. We start by introducing the usual explicit
Hecke operators at unramified places. Let $v$ be a finite place of $F$, and
$1\leq i\leq n$ an integer. Write
$T_{v,i}\in\mathcal{H}(\textnormal{GL}_{n}(F_{v}),\textnormal{GL}_{n}(\mathcal{O}_{F_{v}}))$
for the double coset operator

$T_{v,i}=[\textnormal{GL}_{n}(\mathcal{O}_{F_{v}})\textnormal{diag}(\varpi_{v},...,\varpi_{v},1,...,1)\textnormal{GL}_{n}(\mathcal{O}_{F_{v}})]$

where $\varpi_{v}$ appears exactly $i$ times in the diagonal. We define the
polynomial

$P_{v}(X)=X^{n}-T_{v,1}X^{n-1}+...+(-1)^{i}q_{v}^{i(i-1)/2}T_{v,i}X^{n-i}+...$
$+q_{v}^{n(n-1)/2}T_{v,n}\in\mathcal{H}(\textnormal{GL}_{n}(F_{v}),\textnormal{GL}_{n}(\mathcal{O}_{F_{v}}))[X]$

where recall that $q_{v}=|\mathcal{O}_{F_{v}}/\varpi_{v}|$. Note that
$P_{v}(X)$ corresponds to the characteristic polynomial of the Frobenius
element acting on $\textnormal{rec}^{T}_{F_{v}}(\pi_{v})$ for $\pi_{v}$ any
unramified representation of $\textnormal{GL}_{n}(F_{v})$.

If $\bar{v}$ is a finite place of $F^{+}$, unramified in $F$, $v$ is a choice
of place of $F$ above it, and $1\leq j\leq 2n$ is an integer, then we denote
by
$\widetilde{T}_{v,j}\in\mathcal{H}(\widetilde{G}(F^{+}_{\bar{v}}),\widetilde{G}(\mathcal{O}_{F^{+}_{\bar{v}}}))\otimes_{\mathbf{Z}}\mathbf{Z}[q_{\bar{v}}^{-1}]$
the operator denoted by $T_{G,v,j}$ in [NT16], Proposition-Definition 5.2. In
particular, $q_{v}^{-j(2n-j)/2}\widetilde{T}_{v,j}$ is the operator
corresponding to the $i$th symmetric polynomial in $2n$ variables under the
dual map on Hecke algebras corresponding to the unramified endoscopic transfer
from $\widetilde{G}(F^{+}_{\bar{v}})$ to $\textnormal{GL}_{2n}(F_{v})$ (where
we also apply the (normalised) Satake isomorphism). We then define the
polynomial

$\widetilde{P}_{v}(X)=X^{2n}-\widetilde{T}_{v,1}X^{2n-1}+...+(-1)^{j}q_{v}^{j(j-1)/2}X^{2n-j}+...$
$+q_{v}^{2n(2n-1)/2}\widetilde{T}_{v,2n}\in\mathcal{H}(\widetilde{G}(F^{+}_{\bar{v}}),\widetilde{G}(\mathcal{O}_{F^{+}_{\bar{v}}}))\otimes_{\mathbf{Z}}\mathbf{Z}[q_{\bar{v}}^{-1}][X].$

This then corresponds to the characteristic polynomial of the Frobenius
element acting on $\textnormal{rec}^{T}_{F_{v}}(\pi_{v})$, where $\pi_{v}$ is
the base change with respect to $F_{v}/F^{+}_{\bar{v}}$ of any unramified
representation $\sigma_{\bar{v}}$ of $\widetilde{G}(F^{+}_{\bar{v}})$.

We finally describe the effect of the unnormalised Satake transform
$\mathcal{S}=r_{G}\circ r_{P}$ (for the notation, see the end of section 2.2)
at unramified places. We use the following notation: for $f(X)$ a polynomial
of degree $d$, with constant term a unit $a_{0}$, set
$f^{\vee}(X):=a_{0}^{-1}X^{d}f(X^{-1})$. Therefore, $f^{\vee}(X)$ is the monic
polynomial with zeroes given by the inverse of the zeroes of $f(X)$. We then
have the following.

######  Proposition 2.16.

Let $v$ be a finite place of $F$, unramified above the place $\bar{v}$ of
$F^{+}$. Then the unnormalised Satake transform

$\mathcal{S}:\mathcal{H}(\widetilde{G}(F^{+}_{\bar{v}}),\widetilde{G}(\mathcal{O}_{F^{+}_{\bar{v}}}))\to\mathcal{H}(G(F^{+}_{\bar{v}}),G(\mathcal{O}_{F^{+}_{\bar{v}}}))$

sends $\widetilde{P}_{v}(X)$ to
$P_{v}(X)q_{v}^{n(2n-1)}P_{v^{c}}^{\vee}(q_{v}^{1-2n}X)$.

###### Proof.

This follows from the explicit formula for $\mathcal{S}$ given in [NT16],
Proposition-Definition 5.3. ∎

We now turn to discussing the $U_{p}$-operators we will consider. As before,
we assume that every $p$-adic place $\bar{v}\in\overline{S}_{p}$ splits in $F$
and we fix a choice of $v|\bar{v}$ in $S_{p}$. We start with the case of $G$.
Consider a tuple
$(Q_{S_{p}},\lambda,\underline{\tau})=(Q_{v},\lambda_{v},\underline{\tau_{v}})_{v\in
S_{p}}$ with Levi decomposition $Q_{v}=M_{v}\ltimes N_{v}$. Set $X_{Q_{v}}$ to
be the set of $B_{n}$-dominant cocharacters in $X_{\ast}(Z(M_{v}))$.
Concretely, if $Q_{v}=P_{(n_{1},...,n_{k})}\subset\textnormal{GL}_{n}$, then
it identifies in $X_{\ast}(T_{n})^{+}=\mathbf{Z}^{n}_{+}$ with the elements
with jumps only at the indices $n_{1}+...+n_{j}$ for $1\leq j\leq k$. For
$c\geq 1$, we then define the subset
$\Delta_{\mathcal{Q}_{v}}(c)\subset\textnormal{GL}_{n}(F_{v})$ given by

$\Delta_{\mathcal{Q}_{v}}(c):=\coprod_{\nu\in
X_{Q_{v}}}\mathcal{Q}_{v}(0,c)\nu(\varpi_{v})\mathcal{Q}_{v}(0,c).$

By [CN23], Lemma 2.1.15, the elements $\nu(\varpi_{v})$ are
$\mathcal{Q}_{v}(0,c)$-positive in the sense of loc. cit.. In particular,
$\Delta_{\mathcal{Q}_{v}}(c)$ forms a monoid and
$\Delta_{M_{v}}^{+}:=\Delta_{\mathcal{Q}_{v}}(c)\cap M_{v}(F_{v})\subset
M_{v}(F_{v})^{+}=\\{m\in M_{v}(F_{v})\mid mN^{0}_{v}m^{-1}\subset
N^{0}_{v}\textnormal{ and
}m^{-1}\overline{N}^{1}_{v}m\subset\overline{N}^{1}\\}$ so the formalism of
[All+23], 2.1.9 applies. Set $\Delta_{M_{v}}$ to be the group generated by
$\Delta_{M_{v}}^{+}$. One proves that the map

$\mathcal{H}(\Delta^{+}_{M_{v}},M^{0}_{v})\to\mathcal{H}(\Delta_{\mathcal{Q}_{v}}(c),\mathcal{Q}_{v}(0,c)),$
$[M^{0}_{v}\nu(\varpi_{v})M^{0}_{v}]\mapsto[\mathcal{Q}_{v}\nu(\varpi_{v})\mathcal{Q}_{v}]$

is an isomorphism of Hecke algebras (cf. [BK98], Corollary 6.12). In
particular, the latter is commutative.

We introduce our distinguished element, the "$U_{p}$-operator at $v$" in our
Hecke algebras. Namely, if $Q_{v}=P_{(n_{1},...,n_{k})}$, set

$u_{v}^{Q_{v}}:=\textnormal{diag}(\varpi^{k-1},...,\varpi^{k-1},\varpi^{k-2},...,\varpi,1,...,1)\in\Delta_{\mathcal{Q}_{v}}$

where, for $1\leq j\leq k$, $\varpi^{j-1}$ appears exactly $n_{j}$ times.
Then, for $c\geq 1$, we denote by
$U_{v}^{Q_{v}}\in\mathcal{H}(\Delta_{\mathcal{Q}_{v}}(c),\mathcal{Q}_{v}(0,c))$
the double coset operator
$[\mathcal{Q}_{v}(0,c)u_{v}^{Q}\mathcal{Q}_{v}(0,c)]$. Note that
$\Delta_{M_{v}}=\Delta_{M_{v}}^{+}[u_{v}^{Q_{v},\pm 1}]$.

We introduce the usual $\lambda$-twisted action of
$\Delta_{\mathcal{Q}_{v}}(c_{p})$171717For the integer $c_{p}\geq 1$
introduced in the previous section. on
$\mathcal{V}_{(\lambda_{v},\underline{\tau_{v}})}^{Q_{v}}$ which will then
yield our action of $U_{p}$-operators on the corresponding cohomology groups
by the formalism of [All+23], 2.1.9. Define the character
$\alpha_{\lambda}^{Q_{v}}:\Delta_{\mathcal{Q}_{v}}\to E^{\times}$ by setting
it to be trivial on $\mathcal{Q}_{v}$ and, for $\nu\in X_{Q_{v}}$, sending
$\nu(\varpi_{v})$ to

$\prod_{\iota:F_{v}\hookrightarrow
E}\iota(\varpi_{v})^{\langle\nu,w_{0}^{G}\lambda_{\iota}\rangle}.$

We then view $\mathcal{V}^{Q_{v}}_{(\lambda_{v},\underline{\tau_{v}})}$ as an
$\mathcal{O}[\Delta_{\mathcal{Q}_{v}}(c_{p})]$-module by inflating the
$\mathcal{Q}_{v}(0,c_{p})$-action on the factor
$(\widetilde{\sigma(\underline{\tau_{v}})}^{\circ})^{\vee}$ and acting on the
factor $\mathcal{V}_{\lambda_{v}}$ by the recipe

$g\cdot^{\lambda,Q_{v}}x:=\alpha_{\lambda}^{Q_{v}}(g)^{-1}g\cdot x$

where $-\cdot-$ is the usual action of $g\in\Delta_{\mathcal{Q}_{v}}(c_{p})$
on $x\in\mathcal{V}_{\lambda_{v}}$.

######  Lemma 2.17.

The $\mathcal{O}[\Delta_{\mathcal{Q}_{v}}(c_{p})]$-module structure on
$\mathcal{V}^{Q_{v}}_{(\lambda_{v},\underline{\tau_{v}})}$ makes sense i.e.,
the twisted action of $\Delta_{\mathcal{Q}_{v}}(c_{p})$ on $V_{\lambda_{v}}$
preserves the lattice $\mathcal{V}_{\lambda_{v}}$.

###### Proof.

This follows from the fact that $V_{\lambda_{v}}$ has lowest weight
$w_{0}^{G}\lambda_{v}$. More precisely, one uses [Ger18], Lemma 2.2 to
conclude. ∎

As mentioned, the formalism of [All+23], 2.1.9 applies here. In particular,
for $S_{p}\subset T$ any finite set of finite places, and a choice of good
subgroup $K\subset\textnormal{GL}_{n}(\mathbf{A}^{\infty}_{F})$ with
$K_{v}=\mathcal{Q}_{v}(b,c)$ for some $0\leq b\leq c$ with $c_{p}\leq c$, for
every $v\in S_{p}$, we have a canonical homomorphism

$\mathcal{H}(G^{T},K^{T})\otimes_{\mathbf{Z}}\mathcal{H}(\Delta_{\mathcal{Q}_{p}(c_{p})},K_{p})\to\textnormal{End}_{D^{+}(\mathcal{O})}(R\Gamma(X_{K},\mathcal{V}^{Q_{p}}_{(\lambda,\underline{\tau})})).$

In particular, the above constructed $U_{p}$-operators act on the cohomology
complex $R\Gamma(X_{K},\mathcal{V}_{(\lambda,\underline{\tau})}^{Q_{p}})$.

For later use, set $\mathbf{T}^{T}(K,\lambda,\underline{\tau})$ to be the
image of $\mathcal{H}(G^{T},K^{T})\otimes_{\mathbf{Z}}\mathcal{O}$ inside the
ring
$\textnormal{End}_{D^{+}(\mathcal{O})}(R\Gamma(X_{K},\mathcal{V}^{Q_{p}}_{(\lambda,\underline{\tau})}))$,
a finite $\mathcal{O}$-algebra.

Next we consider the case of $\widetilde{G}$. Fix a subset of $p$-adic places
$\overline{S}\subset\overline{S}_{p}$ and, for each
$\bar{v}\in\overline{S}_{p}$, fix a choice $v\mid\bar{v}$ in $S_{p}$. Consider
a tuple
$(\widetilde{Q}_{\overline{S}},\tilde{\lambda}_{\overline{S}},\underline{\tau_{\overline{S}}}):=(\widetilde{Q}_{\bar{v}},\tilde{\lambda}_{\bar{v}},\underline{\tau_{\bar{v}}})_{\bar{v}\in\overline{S}}$
as before. We can then similarly define the set of $\widetilde{B}$-dominant
cocharacters $X_{\widetilde{Q}_{\bar{v}}^{w_{0}}}\subset
X_{\ast}(Z(\widetilde{M}_{\bar{v}}^{w_{0}}))$ and the corresponding open
submonoids
$\widetilde{\Delta}_{\widetilde{\mathcal{Q}}^{w_{0}}_{\bar{v}}}(c)\subset\widetilde{G}(F^{+}_{\bar{v}})$.
The rest of the conclusions of [CN23], Lemma 2.1.15 will again apply. We
further set
$\widetilde{\Delta}_{\widetilde{\mathcal{Q}}^{w_{0}}_{\overline{S}}}(c_{p}):=\prod_{\bar{v}\in\overline{S}}\widetilde{\Delta}_{\widetilde{\mathcal{Q}}^{w_{0}}_{\bar{v}}}(c_{p})$.
Again, we introduce a notation for our $U_{p}$-operator in
$\mathcal{H}(\widetilde{\Delta}_{\widetilde{\mathcal{Q}}^{w_{0}}_{\bar{v}}}(c_{p}),\widetilde{\mathcal{Q}}^{w_{0}}_{\bar{v}}(0,c_{p}))$.
For $v|\bar{v}$, we set
$u_{\bar{v}}^{\widetilde{Q}_{v}^{w_{0}}}:=\iota_{v}^{-1}u_{v}^{\iota_{v}\widetilde{Q}^{w_{0}}_{\bar{v}}}$
where
$\iota_{\bar{v}}\widetilde{Q}^{w_{0}}_{v}\subset\textnormal{GL}_{2n}/F_{v}$ is
the standard parabolic subgroup corresponding to
$\widetilde{Q}^{w_{0}}_{\bar{v}}$ under $\iota_{v}$. Then set
$U_{\bar{v}}^{\widetilde{Q}_{\bar{v}}^{w_{0}}}\in\mathcal{H}(\widetilde{\Delta}_{\widetilde{\mathcal{Q}}^{w_{0}}_{\bar{v}}}(c_{p}),\widetilde{\mathcal{Q}}^{w_{0}}_{\bar{v}}(0,c_{p}))$
to be the double coset operator
$[\widetilde{\mathcal{Q}}^{w_{0}}_{\bar{v}}(0,c)u_{v}^{\widetilde{Q}^{w_{0}}_{\bar{v}}}\widetilde{\mathcal{Q}}^{w_{0}}_{\bar{v}}(0,c_{p})]$.
We note that as the notation suggests,
$U_{\bar{v}}^{\widetilde{Q}^{w_{0}}_{\bar{v}}}$ is independent of the choice
of $v|\bar{v}$.

Moreover, by the exact same recipe as before, we equip
$\mathcal{V}_{(\widetilde{\lambda},\underline{\tau})}^{\widetilde{Q}_{\overline{S}}}$
with an
$\widetilde{\Delta}_{\widetilde{\mathcal{Q}}^{w_{0}}_{\overline{S}}}(c_{p})$-module
structure extending the natural action of
$\widetilde{\mathcal{Q}}^{w_{0}}_{\overline{S}}(0,c_{p})$. This yields a
canonical homomorphism of algebras

$\mathcal{H}(\widetilde{G}^{T},\widetilde{K}^{T})\otimes_{\mathbf{Z}}\mathcal{H}(\widetilde{\Delta}_{\widetilde{\mathcal{Q}}^{w_{0}}_{\overline{S}}}(c_{p}),\widetilde{K}_{\overline{S}})\to\textnormal{End}_{D^{+}(\mathcal{O})}(R\Gamma(\widetilde{X}_{\widetilde{K}},\mathcal{V}_{(\tilde{\lambda},\underline{\tau})}^{\widetilde{Q}^{w_{0}}_{\overline{S}}}))$

for any finite set of finite places $S_{p}\subset T$ with $T=T^{c}$ and good
subgroup $\widetilde{K}\subset\widetilde{G}(\mathbf{A}_{F^{+}}^{\infty})$ with
$\widetilde{K}_{\bar{v}}=\widetilde{\mathcal{Q}}_{\bar{v}}^{w_{0}}(b,c^{\prime})$
for some $0\leq b\leq c^{\prime}$ with $c_{p}\leq c^{\prime}$ for every
$\bar{v}\in\overline{S}$. Therefore, the constructed $U_{p}$-operators act on
the complex
$R\Gamma(\widetilde{X}_{\widetilde{K}},\mathcal{V}_{(\tilde{\lambda},\underline{\tau})}^{\widetilde{Q}^{w_{0}}_{\overline{S}}})$.

We finally set
$\mathbf{T}^{T}(\widetilde{K},\tilde{\lambda},\underline{\tau})$ to be the
corresponding faithful quotient of
$\mathcal{H}(\widetilde{G}^{T},\widetilde{K}^{T})\otimes_{\mathbf{Z}}\mathcal{O}$.

###  2.9. Automorphic Galois representations and the middle degree cohomology

In this subsection, we recall the well-known results which we will need later
about Galois representations attached to (mod $p$) automorphic forms. We also
state the consequence of the vanishing results of Caraiani–Scholze that serves
as the key ingredient for proving local-global compatibility in the style of
[All+23].

Recall that $\pi$ is called a regular algebraic conjugate self-dual cuspidal
automorphic representation (RACSDCAR) of $\textnormal{GL}_{n}(\mathbf{A}_{F})$
if it is a regular algebraic cuspidal automorphic representation (RACAR) of
$\textnormal{GL}_{n}(\mathbf{A}_{F})$ satisfying $\pi^{c}\cong\pi^{\vee}$
where $(.)^{\vee}$ denotes the contragradient of the representation.

By work of many people, such automorphic representations admit associated
Galois representations satisfying local-global compatibility at every place.

######  Theorem 2.18.

[HT01, TY07, Shi11, CH13, Clo13, Bar+12, Bar+11, Car12, Car14] Let $\pi$ be a
RACSDCAR of $\textnormal{GL}_{n}(\mathbf{A}_{F})$ of weight
$\lambda\in(\mathbf{Z}_{+}^{n})^{{\rm{Hom}}(F,\mathbf{C})}$. Then for any
isomorphism $t:\overline{\mathbf{Q}}_{p}\xrightarrow{\sim}\mathbf{C}$ there is
a continuous semisimple Galois representation

$r_{t}(\pi):G_{F}\to\textnormal{GL}_{n}(\overline{\mathbf{Q}}_{p})$

satisfying the following conditions:

  1. (i)

We have an isomorphism $r_{t}(\pi)^{c}\cong r_{t}(\pi)^{\vee}(1-n)$.

  2. (ii)

For each $p$-adic place $v$ of $F$, $r_{t}(\pi)|_{G_{F_{v}}}$ is potentially
semistable and for each embedding
$\iota:F_{v}\hookrightarrow\overline{\mathbf{Q}}_{p}$ we have

$\textnormal{HT}_{\iota}(r_{t}(\pi)|_{G_{F_{v}}})=\\{\lambda_{t\circ\iota,n},\lambda_{t\circ\iota,n-1}+1,...,\lambda_{t\circ\iota,1}+n-1\\}.$

  3. (iii)

For each finite place $v$ of $F$, we have

$\textnormal{WD}(r_{t}(\pi)|_{G_{F_{v}}})^{F-ss}\cong\textnormal{rec}^{T}(t^{-1}\pi_{v}).$

Once combined with Shin’s base change result [Shi14] for automorphic
representations of $\widetilde{G}$, we obtain the following.

######  Theorem 2.19.

Assume that $F$ contains an imaginary quadratic field and that $\tilde{\pi}$
is a cuspidal automorphic representation of
$\widetilde{G}(\mathbf{A}_{F^{+}})$ that is $\xi$-cohomological for some
irreducible algebraic representation $\xi$ of
$\widetilde{G}_{\mathbf{C}}\cong\prod_{{\rm{Hom}}(F^{+},\mathbf{C})}\textnormal{GL}_{2n,\mathbf{C}}$.
For any isomorphism $t:\overline{\mathbf{Q}}_{p}\xrightarrow{\sim}\mathbf{C}$,
there is a continuous, semisimple Galois representation

$r_{t}(\tilde{\pi}):G_{F}\to\textnormal{GL}_{2n}(\overline{\mathbf{Q}}_{p})$

satisfying the following conditions:

  1. (i)

For each place $\ell\neq p$ which is unramified in $F$ and above which
$\tilde{\pi}$ is unramified, and for each place $v\mid\ell$ of $F$,
$r_{t}(\tilde{\pi})|_{G_{F_{v}}}$ is unramified and the characteristic
polynomial of $r_{t}(\textnormal{Frob}_{v})$ is equal to the image of
$\widetilde{P}_{v}(X)$ in $\overline{\mathbf{Q}}_{p}[X]$ corresponding to the
base change of $t^{-1}(\tilde{\pi}_{\bar{v}})$.

  2. (ii)

For each place $v|p$ of $F$, $r_{t}(\tilde{\pi})$ is potentially semistable,
and for each $\iota:F\hookrightarrow\overline{\mathbf{Q}}_{p}$, we have

$\textnormal{HT}_{\iota}(r_{t}(\tilde{\pi}))=\\{\tilde{\lambda}_{\iota,1}+2n-1,\tilde{\lambda}_{\iota,2}+2n-2,...,\tilde{\lambda}_{\iota,2n}\\},$

where
$\tilde{\lambda}\in(\mathbf{Z}^{2n}_{+})^{{\rm{Hom}}(F,\overline{\mathbf{Q}}_{p})}$
is the highest weight of the representation $t^{-1}(\xi\otimes\xi)^{\vee}$ of
$\textnormal{GL}_{2n}$ over $\overline{\mathbf{Q}}_{p}$.

  3. (iii)

If $F_{0}\subset F$ is an imaginary quadratic field and $\ell$ is a prime
(possibly $\ell=p)$ that splits in $F_{0}$, then for each place $v|\ell$ of
$F$ lying above a place $\bar{v}$ of $F^{+}$, there is an isomorphism

$\textnormal{WD}(r_{t}(\tilde{\pi})|_{G_{F_{v}}})^{F-ss}\cong\textnormal{rec}^{T}(\tilde{\pi}_{\bar{v}}\circ\iota_{v}).$

###### Proof.

For the proof see [All+23], Theorem 2.3.3 and the references therein. ∎

######  Theorem 2.20.

Let $\mathfrak{m}\subset\mathbf{T}^{T}(K,\lambda,\underline{\tau})$ be a
maximal ideal. Suppose that the finite set of places $T$ of $F$ satisfies the
following condition.

  * •

Let $v\notin T$ be a finite place of $F$, and let $\ell$ be its residue
characteristic. Then either $T$ contains no $\ell$-adic places of $F$ and
$\ell$ is unramified in $F$, or there is an imaginary quadratic field
$F_{0}\subset F$ in which $\ell$ splits.

Then there exists a continuous semisimple Galois representation

$\bar{\rho}_{\mathfrak{m}}:G_{F,T}\to\textnormal{GL}_{n}(\mathbf{T}^{T}(K,\lambda,\underline{\tau})/\mathfrak{m})$

such that for each finite place $v\notin T$ of $F$, the characteristic
polynomial of $\bar{\rho}_{\mathfrak{m}}(\textnormal{Frob}_{v})$ is equal to
the image of $P_{v}(X)$ in
$(\mathbf{T}^{T}(K,\lambda,\underline{\tau})/\mathfrak{m})[X]$.

###### Proof.

This is proved the same way as [All+23], Theorem 2.3.5. ∎

######  Definition 2.21.

We say that a maximal ideal
$\mathfrak{m}\subset\mathbf{T}^{T}(K,\lambda,\underline{\tau})$ is non-
Eisenstein if $\bar{\rho}_{\mathfrak{m}}$ is absolutely irreducible.

######  Theorem 2.22.

Let $\mathfrak{m}\subset\mathbf{T}^{T}(K,\lambda,\underline{\tau})$ be a non-
Eisenstein maximal ideal.181818We are implicitly assuming that $T$ satisfies
the condition as in Theorem 2.20. There exists an integer $N\geq 1$, depending
only on $n$ and $[F:\mathbf{Q}]$, an ideal
$I\subset\mathbf{T}^{T}(K,\lambda,\underline{\tau})$ satisfying $I^{N}=0$, and
a continuous homomorphism

$\rho_{\mathfrak{m}}:G_{F,T}\to\textnormal{GL}_{n}(\mathbf{T}^{T}(K,\lambda,\underline{\tau})/I)$

such that, for each finite place $v\notin T$ of $F$, the characteristic
polynomial of $\rho_{\mathfrak{m}}(\textnormal{Frob}_{v})$ is equal to the
image of $P_{v}(X)$ in $(\mathbf{T}^{T}(K,\lambda,\underline{\tau})/I)[X]$.

###### Proof.

This is proved the same way as [Sch15], Corollary 5.4.4. ∎

######  Theorem 2.23.

Let
$\widetilde{\mathfrak{m}}\subset\widetilde{\mathbf{T}}^{T}(\widetilde{K},\tilde{\lambda},\underline{\tau})$
be a maximal ideal. Then there is a continuous, semisimple Galois
representation

$\bar{\rho}_{\widetilde{\mathfrak{m}}}:G_{F,T}\to\textnormal{GL}_{2n}(\widetilde{\mathbf{T}}^{T}(\widetilde{K},\tilde{\lambda},\underline{\tau})/\widetilde{\mathfrak{m}})$

such that for each finite place $v\notin T$ of $F$, the characteristic
polynomial of $\bar{\rho}_{\widetilde{\mathfrak{m}}}(\textnormal{Frob}_{v})$
is given by the image of $\widetilde{P}_{v}(X)$ in
$(\widetilde{\mathbf{T}}^{T}(\widetilde{K},\tilde{\lambda},\underline{\tau})/\widetilde{\mathfrak{m}})[X]$.

###### Proof.

The same proof applies as in [CN23], Theorem 2.1.26 noting that the first step
is to pass to deep enough level where
$\mathcal{V}_{(\tilde{\lambda},\underline{\tau})}^{\widetilde{Q}^{w_{0}}_{\overline{S}}}/\varpi$
is trivialised. ∎

Finally, we discuss the key technical condition we need to have access to the
vanishing result of Caraiani–Scholze.

######  Definition 2.24.

A continuous representation
$\bar{\rho}:G_{F}\to\textnormal{GL}_{m}(\mathcal{O}/\varpi)$ is called
decomposed generic if there exists a prime $\ell\neq p$ such that the
following are satisfied:

  1. (i)

$\ell$ splits completely in $F$;

  2. (ii)

for every place $v|\ell$ of $F$, the representation $\bar{\rho}|_{G_{F_{v}}}$
is unramified and the eigenvalues $\alpha_{1},...,\alpha_{m}$ of
$\bar{\rho}(\textnormal{Frob}_{v})$ satisfy $\alpha_{i}/\alpha_{j}\neq\ell$
for $i\neq j$.

######  Remark 2.25.

As explained in [All+23], Lemma 4.3.2, once we know that $\bar{\rho}$ is
decomposed generic, an argument using Chebotarev’s density theorem shows that
there are infinitely many choices of $\ell$ as in Definition 2.24.

######  Theorem 2.26.

[CS19, Kos21] Let
$\widetilde{\mathfrak{m}}\subset\widetilde{\mathbf{T}}^{T}(\widetilde{K},\tilde{\lambda},\underline{\tau})$
be a maximal ideal such that the associated Galois representation
$\bar{\rho}_{\widetilde{\mathfrak{m}}}$ is decomposed generic. If we set
$d=\textnormal{dim}_{\mathbf{C}}\widetilde{X}_{\widetilde{K}}$, we have a
$\widetilde{\mathbf{T}}^{T}$-equivariant diagram

$H^{d}(\widetilde{X}_{\widetilde{K}},\mathcal{V}_{(\tilde{\lambda},\underline{\tau})}^{\widetilde{Q}^{w_{0}}_{\overline{S}}}[1/p])_{\widetilde{\mathfrak{m}}}\hookleftarrow
H^{d}(\widetilde{X}_{\widetilde{K}},\mathcal{V}_{(\tilde{\lambda},\underline{\tau})}^{\widetilde{Q}^{w_{0}}_{\overline{S}}})_{\widetilde{\mathfrak{m}}}\twoheadrightarrow
H^{d}(\partial\widetilde{X}_{\widetilde{K}},\mathcal{V}_{(\tilde{\lambda},\underline{\tau})}^{\widetilde{Q}_{\overline{S}}^{w_{0}}})_{\widetilde{\mathfrak{m}}}.$

###### Proof.

This follows from the main result of [CS19] as explained in [All+23], Theorem
4.3.3 except the extra conditions appearing in [CS19] that
$[F^{+}:\mathbf{Q}]\geq 2$ and that the length of
$\bar{\rho}_{\widetilde{\mathfrak{m}}}$ is at most $2$. These conditions were
removed in [Kos21]. ∎

##  3\. $Q$-ordinary Hida theory and $Q$-ordinary parts of Bruhat strata

In the first part of the section, based on [All+23], §5.2 and [CN23], §2.2, we
spell out $Q$-ordinary Hida theory for the Betti cohomology of the locally
symmetric space $X_{\widetilde{K}}$ and $X_{K}$, where $Q$ will be an
arbitrary standard parabolic subgroup. Finally, we close the section with a
computation of $Q$-ordinary parts of certain Bruhat strata of parabolic
induction. This is completely analogous to [All+23], §5.3 and [CN23], §2.3.
More precisely, the contents of §3.1, §3.2, and §3.4 are carried out for the
Borel subgroup in [All+23], §5.2, and for the Siegel parabolic subgroup in
[CN23], §2.2. Moreover, the content of §3.5 has been worked out for the Borel
subgroup in [All+23], §5.3, and for the Siegel parabolic subgroup in [CN23],
§2.3. Moreover, Hida theory with dual coefficients for the opposite parabolic
is considered in [CN23] that we generalise in §3.3. A seemingly new result in
§3.3 is a $Q$-ordinary and Hecke equivariant Verdier duality with
$\mathcal{O}/\varpi^{m}$-coefficients (see Proposition 3.21).

###  3.1. Ordinary parts of smooth representations

In this text we use several incarnations of ordinary parts. To track the
representation theory throughout our arguments, it is often crucial to take
the point of view of Emerton [Eme10], [Eme10a] on taking ordinary parts. We
recollect here his approach. In fact, following [All+23], we consider a
modified version that is more convenient to do homological algebra with and
coincides with the original definition on admissible representations which
exhaust all the objects which we will consider here.

The setup for the subsection is as follows. We let $L/\mathbf{Q}_{p}$ be a
finite field extension, $G/L$ a connected reductive group. Set $Q\subset G$ to
be a parabolic subgroup with a Levi decomposition $Q=M\ltimes N$ and denote by
$\overline{Q}=M\ltimes\overline{N}$ the opposite parabolic subgroup. Denote by
$Z_{M}\subset M$ the centre of the Levi factor. Assume that
$\mathcal{Q}\subset G(L)$ is a compact open subgroup which admits an Iwahori
decomposition

$\overline{N}^{1}\times M^{0}\times
N^{0}\xrightarrow[]{\sim}\mathcal{Q}\xleftarrow[]{\sim}N^{0}\times
M^{0}\times\overline{N}^{1}$

with respect to $Q$ in the sense of [All+23], §2.1.9. Let

$...\subset\mathcal{Q}(b,b)\subset...\subset\mathcal{Q}(1,1)\subset\mathcal{Q}(0,1)=\mathcal{Q}\subset
G(L)$

be a cofinal family of compact open subgroups of $G(L)$ such that each
$\mathcal{Q}(b,b)$ is normal in $\mathcal{Q}$ and $\mathcal{Q}(b,b)$ admits an
Iwahori decomposition

$\overline{N}^{b}M^{b}N^{0}\xrightarrow[]{\sim}\mathcal{Q}(b,b)$

with respect to $Q$. Then, by setting
$\mathcal{Q}(b,c)=\overline{N}^{c}M^{b}N^{0}$ for $1\leq b\leq c$, one checks
that we get a compact open subgroup of $\mathcal{Q}$ (admitting an Iwahori
decomposition).

######  Example 3.1.

For $G=\textnormal{GL}_{n,L}$, and $Q=M\ltimes N\subset G$ a parabolic
subgroup standard with respect to the Borel subgroup of upper triangular
matrices, we can consider the corresponding parahoric group scheme
$\mathcal{Q}^{\textnormal{sch}}\subset\textnormal{GL}_{n}$. Then
$\mathcal{Q}:=\mathcal{Q}^{\textnormal{sch}}(\mathcal{O}_{L})$ admits an
Iwahori decomposition $\overline{N}^{1}M^{0}N^{0}$, where
$M^{0}=M(\mathcal{O}_{L})$, $N^{0}=N(\mathcal{O}_{L})$, and $\overline{N}^{1}$
is
$\textnormal{ker}(\overline{N}(\mathcal{O}_{L})\to\overline{N}(\mathcal{O}_{L}/\varpi_{L})$.
If $Q$ is the parabolic subgroup corresponding to the partition
$(n_{1},...,n_{t})$ of $n$, $\mathcal{Q}$ will be the subgroup of matrices in
$G(\mathcal{O}_{L})$ that are upper block triangular of type
$(n_{1},...,n_{t})$ modulo $\varpi_{L}$. For integers $0\leq b\leq c$ with
$c\geq 1$, we can then set $\mathcal{Q}(b,c)$ to be the subgroup of matrices
in $G(\mathcal{O}_{L})$ that are block upper triangular of type
$(n_{1},...,n_{t})$ modulo $\varpi_{L}^{c}$, and block unipotent of type
$(n_{1},...,n_{t})$ modulo $\varpi_{L}^{b}$.

However, we consider this more general setup since it allows us to for
instance set $M^{1}$ to be $\textnormal{ker}(M(\mathcal{O}_{L})\to
M(\mathcal{O}_{L}/\varpi_{L}^{d})$ for some arbitrary integer $d\geq 0$.
Moreover, we have the freedom to choose $N^{0}$ to be other compact open
subgroups in $N(\mathcal{O}_{L})$ that are preserved under conjugation by
$M^{0}$. All of this allows the formalism to be more flexible and saves some
space in introducing notations and applying it in later sections.

######  Remark 3.2.

Note that an easy argument shows that $M^{0}$ normalises $N^{0}$ and each
$\overline{N}^{c}$ for $c\geq 1$.

We set

$M^{+}:=\\{m\in M(L)\mid mN^{0}m^{-1}\subset N^{0}\textnormal{ and
}m^{-1}\overline{N}^{1}m\subset\overline{N}^{1}\\}$

and define $Z^{+}_{M}:=Z_{M}(L)\cap M^{+}$. By Remark 3.2, we see that in fact
both $M^{+}\subset M(L)$ and $Z_{M}^{+}\subset Z_{M}(L)$ are open submonoids.
We make the following assumption that will be satisfied in our context.

######  Hypothesis 3.3.

For any $m\in M^{+}$, and $c\geq 1$, we have
$m^{-1}\overline{N}^{c}m\subset\overline{N}^{c}$.

Denote by $M^{+}\times_{Z_{M}^{+}}Z_{M}(L)$ the quotient of the monoid
$M^{+}\times Z_{M}(L)$ by the equivalence relation generated by
$(mz^{+},z)\sim(m,z^{+}z)$ for $m\in M^{+}$, $z^{+}\in Z_{M}^{+}$ and $z\in
Z_{M}(L)$.

######  Lemma 3.4.

The morphism of monoids $M^{+}\times Z_{M}(L)\to M(L)$ given by multiplication
factors through $M^{+}\times_{Z_{M}^{+}}Z_{M}(L)$ and descends to an
isomorphism.

###### Proof.

This is essentially Proposition 3.3.6 of [Eme06]. Since our definition of
$M^{+}$ is slightly different, we provide a sketch of proof here.

We start by picking $z_{p}\in Z_{M}^{+}$ such that
$\\{z_{p}^{k}N^{0}z_{p}^{-k}\\}_{k\geq 0}$ forms a basis of neighborhoods of
$0$. Such a $z_{p}\in Z_{M}^{+}$ exists by [Eme10a], Lemma 3.1.3, (2),(3) and
Lemma 3.1.4, (3). Note that $z_{p}^{-1}$ and $Z_{M}^{+}$ generate $Z_{M}(L)$
as a monoid. Indeed, given $z\in Z_{M}(L)$, then for $k\geq 0$ large enough,
$z_{p}^{k}z$ will satisfy the assumption of loc. cit. Lemma 3.1.3, (3) and
Lemma 3.1.4, (3), and, in particular, will lie in $Z_{M}^{+}$.

We further see that $M^{+}$ and $z_{p}^{-1}$ generate $M(L)$ as a monoid. To
see this, pick $m\in M(L)$ and assume that we have

$mN^{0}m^{-1}\subset z_{p}^{-k}N^{0}z_{p}^{k}\textnormal{ and
}m^{-1}\overline{N}^{1}m\subset z_{p}^{k}\overline{N}^{1}z_{p}^{-k}$

for a large enough integer $k\geq 0$. Such $k$ always exists by loc. cit.
Lemma 3.1.3, (2) and Lemma 3.1.4, (2). Then $z_{p}^{k}m=mz_{p}^{k}$ clearly
lies in $M^{+}$.

One can now conclude just as in the proof of [Eme06], Proposition 3.3.6. ∎

From now on, we fix a $z_{p}\in Z_{M}^{+}$ as in the proof of Lemma 3.4. We
set $\Delta_{M}^{+}\subset M^{+}$ to be any open submonoid containing $M^{0}$
and $z_{p}$. Moreover, for $c\geq 1$, we set
$\mathcal{Q}^{+}(0,c)=\overline{N}^{c}M^{+}N^{0}$ and
$\Delta_{\mathcal{Q}}(c)=\overline{N}^{c}\Delta_{M}^{+}N^{0}$. Then [All+23],
Lemma 2.1.10 says that $\mathcal{Q}^{+}(0,c)$ is also a monoid, open in
$G(L)$. Moreover,
$\mathcal{Q}\mathcal{Q}^{+}(0,c)\mathcal{Q}=\mathcal{Q}^{+}(0,c)$, and
$\mathcal{Q}^{+}\cap M(L)=M^{+}$. The same conclusion holds for
$\Delta_{\mathcal{Q}}(c)$ too, since the assumption
$M^{0}\subset\Delta_{M}^{+}$ ensures that the proof of loc. cit. Lemma 2.1.10
applies. Moreover, set $Q^{+}=\mathcal{Q}^{+}(0,c)\cap Q(L)=M^{+}\ltimes
N^{0}$ and $\Delta_{Q}=\Delta_{\mathcal{Q}}(c)\cap Q(L)=\Delta_{M}^{+}\ltimes