# Finiteness properties of the category of mod $p$ representations of
$\operatorname{GL}_{2}(\mathbb{Q}_{p})$

Vytautas Paškūnas  and  Shen-Ning Tung

(Date: .)

###### Abstract.

We establish Bernstein-centre type of results for the category of mod $p$
representations of $\operatorname{GL}_{2}(\mathbb{Q}_{p})$. We treat all the
remaining open cases, which occur when $p$ is $2$ or $3$. Our arguments carry
over for all primes $p$. This allows us to remove the restrictions on the
residual representation at $p$ in Lue Pan’s recent proof of the Fontaine–Mazur
conjecture for Hodge–Tate representations of
$\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ with equal Hodge–Tate
weights.

###### Contents

  1. 1 Introduction
    1. 1.1 A sketch of the proof
    2. 1.2 What is left to do?
    3. 1.3 Acknowledgements.
  2. 2 Endomorphism rings
  3. 3 Pseudorepresentations
  4. 4 Representations of $\operatorname{GL}_{2}(\mathbb{Q}_{p})$
    1. 4.1 Blocks
    2. 4.2 Quotient category
    3. 4.3 The centre
    4. 4.4 Colmez’s Montreal functor
    5. 4.5 Banach space representations
  5. 5 Density
    1. 5.1 Capture
    2. 5.2 Locally algebraic vectors in $\Pi(P)$
  6. 6 Main results
    1. 6.1 Finiteness
    2. 6.2 Banach space representations
    3. 6.3 The centre
    4. 6.4 Complements
  7. 7 Application to Hecke eigenspaces
  8. A Normality of $R^{\mathrm{ps}}[1/p]$

##  1\. Introduction

Recently, Lue Pan has given a new proof of the Fontaine–Mazur conjecture for
Hodge–Tate representations of
$\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ with equal Hodge–Tate
weights, [38]. One of the ingredients in his proof are the Bernstein-centre
type of results for the category of mod $p$ representations of
$\operatorname{GL}_{2}(\mathbb{Q}_{p})$ proved in [39], [41]. This caused him
to impose some restrictions on the Galois representations at $p$, when $p=2$
and $p=3$. In this paper we remove these restrictions by proving the required
finiteness results in these remaining cases, see Theorem 7.3.

Let $L$ be a finite extension of $\mathbb{Q}_{p}$ with ring of integers
$\mathcal{O}$ and residue field $k$. Let
$G=\operatorname{GL}_{2}(\mathbb{Q}_{p})$ and let
$\operatorname{Mod}^{\mathrm{sm}}_{G}(\mathcal{O})$ be the category of smooth
$G$-representations on $\mathcal{O}$-torsion modules. Let
$\operatorname{Mod}^{\mathrm{l.fin}}_{G}(\mathcal{O})$ be the full subcategory
of $\operatorname{Mod}^{\mathrm{sm}}_{G}(\mathcal{O})$ consisting of
representations, which are equal to the union of their subrepresentations of
finite length. This is equivalent to the requirement that every finitely
generated subrepresentation is of finite length and we call such
representations locally finite. We fix a character
$\zeta:Z\rightarrow\mathcal{O}^{\times}$ of the centre of $G$, and let
$\operatorname{Mod}^{\mathrm{l.fin}}_{G,\zeta}(\mathcal{O})$ be the full
subcategory of $\operatorname{Mod}^{\mathrm{l.fin}}_{G}(\mathcal{O})$
consisting of representations with central character $\zeta$. The category
$\operatorname{Mod}^{\mathrm{l.fin}}_{G,\zeta}(\mathcal{O})$ is locally finite
and by general results of Gabriel [31] decomposes as a product

$\operatorname{Mod}^{\mathrm{l.fin}}_{G,\zeta}(\mathcal{O})\cong\prod_{\mathfrak{B}}\operatorname{Mod}^{\mathrm{l.fin}}_{G,\zeta}(\mathcal{O})_{\mathfrak{B}}$

of indecomposable subcategories, called blocks. Moreover, each block is anti-
equivalent to the category of pseudo-compact modules over a pseudo-compact
ring $E_{\mathfrak{B}}$. The centre of the ring $E_{\mathfrak{B}}$, which we
denote by $Z_{\mathfrak{B}}$, is naturally isomorphic to the centre of the
category
$\operatorname{Mod}^{\mathrm{l.fin}}_{G,\zeta}(\mathcal{O})_{\mathfrak{B}}$,
which by definition is the ring of natural transformations of the identity
functor. This means that $Z_{\mathfrak{B}}$ acts functorially on every object
in $\operatorname{Mod}^{\mathrm{l.fin}}_{G,\zeta}(\mathcal{O})$. The
finiteness result in the title is an analogue of a result of Bernstein in the
theory of smooth representations of $p$-adic groups on $\mathbb{C}$-vector
spaces, [5, Proposition 3.3]:

######  Theorem 1.1.

The ring $Z_{\mathfrak{B}}$ is noetherian and $E_{\mathfrak{B}}$ is a finitely
generated $Z_{\mathfrak{B}}$-module.

To prove the theorem we use in an essential way the direct connection between
the $\operatorname{GL}_{2}(\mathbb{Q}_{p})$ representations and the
representations of the absolute Galois group of $\mathbb{Q}_{p}$, which we
denote by $\mathcal{G}_{\mathbb{Q}_{p}}$, discovered by Colmez in [19], via
his celebrated Montreal functor $\check{\mathbf{V}}$, which we review in
section 4.4.

For each fixed block
$\operatorname{Mod}^{\mathrm{l.fin}}_{G,\zeta}(\mathcal{O})_{\mathfrak{B}}$
there is a finite extension $L^{\prime}$ of $L$ with ring of integers
$\mathcal{O}^{\prime}$, such that
$\operatorname{Mod}^{\mathrm{l.fin}}_{G,\zeta}(\mathcal{O})_{\mathfrak{B}}\otimes_{\mathcal{O}}\mathcal{O}^{\prime}$
decomposes into a finite product of indecomposable subcategories, each of
which remain indecomposable after a further extension of scalars. Such
absolutely indecomposable blocks have been classified in [40] and they
correspond to semi-simple representations
$\bar{\rho}:\mathcal{G}_{\mathbb{Q}_{p}}\rightarrow\operatorname{GL}_{2}(k)$,
which are either absolutely irreducible or a direct sum of two characters.
This bijection realizes the semi-simple mod $p$ local Langlands correspondence
established in a visionary paper of Breuil, [9].

If $\bar{\rho}$ is absolutely irreducible then
$\operatorname{Mod}^{\mathrm{l.fin}}_{G,\zeta}(\mathcal{O})_{\mathfrak{B}_{\bar{\rho}}}$
contains only one irreducible object $\pi$, satisfying
$\check{\mathbf{V}}(\pi^{\vee})\cong\bar{\rho}$, where $\vee$ denotes
Pontryagin dual. Moreover, $\pi$ is not a subquotient of any parabolically
induced representation; such representations are called supersingular.

If $\bar{\rho}=\chi_{1}\oplus\chi_{2}$, where
$\chi_{1},\chi_{2}:\mathcal{G}_{\mathbb{Q}_{p}}\rightarrow k^{\times}$ are
characters, then the irreducible objects in
$\operatorname{Mod}^{\mathrm{l.fin}}_{G,\zeta}(\mathcal{O})_{\mathfrak{B}_{\bar{\rho}}}$
are the irreducible subquotients of the representation

$(\operatorname{Ind}^{G}_{B}\chi_{1}\otimes\chi_{2}\omega^{-1})_{\mathrm{sm}}\oplus(\operatorname{Ind}^{G}_{B}\chi_{2}\otimes\chi_{1}\omega^{-1})_{\mathrm{sm}},$

where we consider $\chi_{1}$, $\chi_{2}$ as characters of
$\mathbb{Q}_{p}^{\times}$ via the Artin map
$\operatorname{Art}_{\mathbb{Q}_{p}}:\mathbb{Q}_{p}^{\times}\rightarrow\mathcal{G}_{\mathbb{Q}_{p}}^{\mathrm{ab}}$
and $\omega(x)=x|x|\pmod{p}$ for all $x\in\mathbb{Q}_{p}^{\times}$ corresponds
to the cyclotomic character modulo $p$, see section 4.1 for an explicit list.

All the blocks, except when $p$ is either $2$ or $3$ and
$\bar{\rho}=\chi\oplus\chi\omega$, have been well understood in [39], [41].
These exceptional blocks are the main focus of this paper, however our
arguments work for all $p$ and all blocks.

The action of $Z_{\mathfrak{B}_{\bar{\rho}}}$ induces a functorial ring
homomorphism

$c_{\tau}:Z_{\mathfrak{B}_{\bar{\rho}}}\rightarrow\operatorname{End}_{G}(\tau),$

for every object $\tau$ in
$\operatorname{Mod}^{\mathrm{l.fin}}_{G,\zeta}(\mathcal{O})_{\mathfrak{B}_{\bar{\rho}}}$.
Since $\check{\mathbf{V}}$ is a functor, it induces a ring homomorphism

$\operatorname{End}_{G}(\tau)\rightarrow\operatorname{End}^{\mathrm{cont}}_{\mathcal{G}_{\mathbb{Q}_{p}}}(\check{\mathbf{V}}(\tau^{\vee}))^{\mathrm{op}},\quad\varphi\mapsto\check{\mathbf{V}}(\varphi).$

We denote the action of $\mathcal{G}_{\mathbb{Q}_{p}}$ on
$\check{\mathbf{V}}(\tau^{\vee})$ by $\rho_{\check{\mathbf{V}}(\tau^{\vee})}$.

Let $R^{\mathrm{ps},\zeta\varepsilon}_{\operatorname{tr}\bar{\rho}}$ be the
universal deformation ring parameterising pseudorepresentations lifting
$\operatorname{tr}\bar{\rho}$ with determinant $\zeta\varepsilon$, where
$\varepsilon$ is the $p$-adic cyclotomic character. We may evaluate the
universal pseudorepresentation $T:\mathcal{G}_{\mathbb{Q}_{p}}\rightarrow
R^{\mathrm{ps},\zeta\varepsilon}_{\operatorname{tr}\bar{\rho}}$ at
$g\in\mathcal{G}_{\mathbb{Q}_{p}}$ to obtain an element $T(g)\in
R^{\mathrm{ps},\zeta\varepsilon}_{\operatorname{tr}\bar{\rho}}$.

######  Theorem 1.2.

There is an $\mathcal{O}$-algebra homomorphism

(1)
$\upsilon:R^{\mathrm{ps},\zeta\varepsilon}_{\operatorname{tr}\bar{\rho}}\rightarrow
Z_{\mathfrak{B}_{\bar{\rho}}},$

satisfying the following compatibility property with the Colmez’s functor. For
all
$\tau\in\operatorname{Mod}^{\mathrm{l.fin}}_{G,\zeta}(\mathcal{O})_{\mathfrak{B}_{\bar{\rho}}}$
and all $g\in\mathcal{G}_{\mathbb{Q}_{p}}$ we have

$\check{\mathbf{V}}(c_{\tau}(\upsilon(T(g))))=\rho_{\check{\mathbf{V}}(\tau^{\vee})}(g)+\rho_{\check{\mathbf{V}}(\tau^{\vee})}(g^{-1})\zeta\varepsilon(g),$

in
$\operatorname{End}_{\mathcal{G}_{\mathbb{Q}_{p}}}^{\mathrm{cont}}(\check{\mathbf{V}}(\tau^{\vee}))$.

The construction of the map (1) is the main point of the paper. Outside the
exceptional cases (1) has been established in [39], [41] using a different
argument to the present paper. Our main result is

######  Theorem 1.3.

The map (1) makes $Z_{\mathfrak{B}_{\bar{\rho}}}$ and
$E_{\mathfrak{B}_{\bar{\rho}}}$ into finitely generated
$R^{\mathrm{ps},\zeta\varepsilon}_{\operatorname{tr}\bar{\rho}}$-modules.

Since $R^{\mathrm{ps},\zeta\varepsilon}_{\operatorname{tr}\bar{\rho}}$ is
known to be noetherian by the work of Chenevier [18], Theorem 1.3 implies
Theorem 1.1. Moreover, Theorems 1.3, 1.2 are sufficient to remove the
restrictions in Lue Pan’s paper. Further, we can reprove most of the results
concerning Banach space representations in [39], see section 6.2 and Corollary
6.16.

To give a flavour of the results on Banach space representations, we will
explain the following special case. Let
$\operatorname{Ban}^{\mathrm{adm}}_{G,\zeta}(L)$ be the category of admissible
unitary $L$-Banach space representations of $G$ with central character
$\zeta$. This category is abelian, [44]. By [21], Colmez’s functor induces a
bijection between the equivalence classes of absolutely irreducible non-
ordinary $\Pi\in\operatorname{Ban}^{\mathrm{adm}}_{G,\zeta}(L)$ and absolutely
irreducible Galois representations
$\rho:\mathcal{G}_{\mathbb{Q}_{p}}\rightarrow\operatorname{GL}_{2}(L)$. We
show that there are no extensions between $\Pi$ and other irreducible
representations in $\operatorname{Ban}^{\mathrm{adm}}_{G,\zeta}(L)$. Hence
$\operatorname{Ban}^{\mathrm{adm}}_{G,\zeta}(L)^{\mathrm{fl}}_{\Pi}$ is a
direct summand of
$\operatorname{Ban}^{\mathrm{adm}}_{G,\zeta}(L)^{\mathrm{fl}}$, where the
superscript $\mathrm{fl}$ indicates finite length and subscript $\Pi$
indicates that all the irreducible subquotients are isomorphic to $\Pi$. We
show in Corollary 6.16 that $\check{\mathbf{V}}$ induces an anti-equivalence
of categories between
$\operatorname{Ban}^{\mathrm{adm}}_{G,\zeta}(L)^{\mathrm{fl}}_{\Pi}$ and the
category of modules of finite length over $R^{\zeta\varepsilon}_{\rho}$, the
universal deformation ring of $\rho$ parameterizing deformations of $\rho$
with determinant equal to $\zeta\varepsilon$ to local artinian $L$-algebras.
Such results were known before for $p\geq 5$, [39] and under assumptions on
the reduction modulo $p$ of a $G$-invariant lattice in $\Pi$ if $p=2$ or
$p=3$, [41].

Let
$R^{\mathrm{ps},\zeta\varepsilon}_{\operatorname{tr}\bar{\rho}}\llbracket\mathcal{G}_{\mathbb{Q}_{p}}\rrbracket/J$
be the largest quotient of
$R^{\mathrm{ps},\zeta\varepsilon}_{\operatorname{tr}\bar{\rho}}\llbracket\mathcal{G}_{\mathbb{Q}_{p}}\rrbracket$
such that the Cayley–Hamilton theorem holds for the universal
pseudorepresentation with determinant $\zeta\varepsilon$ lifting
$\operatorname{tr}\bar{\rho}$. Such algebras have been studied by
Bellaïche–Chenevier [3], Chenevier [18]; they play a key role in this paper.
The subscript $\mathrm{tf}$ will indicate the maximal $\mathcal{O}$-torsion
free quotient. Our second main result asserts that

######  Theorem 1.4.

The essential image of
$\operatorname{Mod}^{\mathrm{l.fin}}_{G,\zeta}(\mathcal{O})_{\mathfrak{B}_{\bar{\rho}}}$
under $\check{\mathbf{V}}$ is anti-equivalent to the category of pseudo-
compact
$(R^{\mathrm{ps},\zeta\varepsilon}_{\operatorname{tr}\bar{\rho}}\llbracket\mathcal{G}_{\mathbb{Q}_{p}}\rrbracket/J)_{\mathrm{tf}}$-modules.
The map (1) induces an isomorphism

(2)
$R^{\mathrm{ps},\zeta\varepsilon}_{\operatorname{tr}\bar{\rho}}[1/p]\overset{\cong}{\longrightarrow}Z_{\mathfrak{B}_{\bar{\rho}}}[1/p].$

Moreover, if $p\neq 2$ then
$Z_{\mathfrak{B}_{\bar{\rho}}}=(R^{\mathrm{ps},\zeta\varepsilon}_{\operatorname{tr}\bar{\rho}_{\mathfrak{B}}})_{\mathrm{tf}}$
and if $p=2$ then the cokernel of (1) is killed by $2$.

######  Corollary 1.5.

$Z_{\mathfrak{B}_{\bar{\rho}}}$ is a complete local noetherian
$\mathcal{O}$-algebra with residue field $k$. It is $\mathcal{O}$-torsion free
and $Z_{\mathfrak{B}_{\bar{\rho}}}[1/p]$ is normal.

If we are not in the exceptional cases then Theorem 1.3 is proved in [39],
[41] essentially by first computing $E_{\mathfrak{B}}$ and then its centre
$Z_{\mathfrak{B}}$. Moreover, it is proved there that (1) is an isomorphism
and
$R^{\mathrm{ps},\zeta\varepsilon}_{\operatorname{tr}\bar{\rho}}\llbracket\mathcal{G}_{\mathbb{Q}_{p}}\rrbracket/J$
is $\mathcal{O}$-torsion free. The argument in this paper, sketched in section
1.1 below, is different: we do not compute $E_{\mathfrak{B}}$.

Our original strategy for proving (2) in this paper was to use an argument of
Gabber in Appendix to [34]. We showed that
$R^{\mathrm{ps},\zeta\varepsilon}_{\operatorname{tr}\bar{\rho}}[1/p]$ is
normal, $Z_{\mathfrak{B}}[1/p]$ is reduced, and (1) induces a bijection on
maximal spectra
$\operatorname{m-Spec}Z_{\mathfrak{B}}[1/p]\rightarrow\operatorname{m-Spec}R^{\mathrm{ps},\zeta\varepsilon}_{\operatorname{tr}\bar{\rho}}[1/p]$
and isomorphism of the residue fields. However, this is replaced by a
different argument in the final version, which also proves the first part of
Theorem 1.4. One important ingredient in the proof are results of Colmez,
Dospinescu and VP [21], [22], which imply that the universal framed
deformation of $\bar{\rho}$ with determinant $\zeta\varepsilon$ lies in the
image of $\check{\mathbf{V}}$. We show in the appendix A that
$(R^{\mathrm{ps},\zeta\varepsilon}_{\operatorname{tr}\bar{\rho}}\llbracket\mathcal{G}_{\mathbb{Q}_{p}}\rrbracket/J)_{\mathrm{tf}}$
acts faithfully on this representation using the theory of Cayley–Hamilton
algebras, [43].

The normality of the ring
$R^{\mathrm{ps},\zeta\varepsilon}_{\operatorname{tr}\bar{\rho}}[1/p]$ is
proved in the appendix, where we show if the generic fibre of the framed
deformation ring $R^{\square}_{\bar{\rho}}[1/p]$ is normal then
$R^{\mathrm{ps},\zeta\varepsilon}_{\operatorname{tr}\bar{\rho}}[1/p]$ and the
corresponding rigid analytic space
$(\operatorname{Spf}R^{\mathrm{ps},\zeta\varepsilon}_{\operatorname{tr}\bar{\rho}})^{\mathrm{rig}}$
are normal. The same applies to the deformation rings without the fixed
determinant condition. In fact, we prove this statement not only for
$\mathcal{G}_{\mathbb{Q}_{p}}$, but for any pro-finite group satisfying
Mazur’s finiteness condition at $p$. We then show that
$R^{\square}_{\bar{\rho}}[1/p]$ is normal for all $2$-dimensional semi-simple
representations of $\mathcal{G}_{\mathbb{Q}_{p}}$; the hard cases are
precisely those corresponding to the exceptional blocks, if $p=2$ then the
assertion has been shown in [22], if $p=3$ then we give a proof in the
appendix based on Böckle’s explicit description of the framed deformation ring
in [7]. We note that the argument of [22] has been generalized by Iyengar [33]
showing that when $\bar{\rho}$ is the trivial $d$-dimensional representation
of the absolute Galois group of a $p$-adic field, containing $4$-th root of
unity if $p=2$, then $R^{\square}_{\bar{\rho}}[1/p]$ is normal, so our result
also apply in this setting. We expect111This has now been proved in [8,
Corollary 4.22]. the rings $R^{\square}_{\bar{\rho}}[1/p]$ to be normal for
any $d$-dimensional representation $\bar{\rho}$ of $\mathcal{G}_{F}$, where
$F$ is a finite extension of $\mathbb{Q}_{p}$.

###  1.1. A sketch of the proof

We will now explain the construction of the map (1). To fix ideas we will
discuss a special case $p=2$, $\bar{\rho}=\mathbf{1}\oplus\omega$ and
$\zeta=\mathbf{1}$. Since the cyclotomic character is trivial modulo $2$,
$\bar{\rho}=\mathbf{1}\oplus\mathbf{1}$. The corresponding block has $2$
irreducible representations: the trivial $\mathbf{1}$ and the smooth Steinberg
representation $\operatorname{Sp}$. Instead of working with representations on
$\mathcal{O}$-torsion modules it is more convenient to use Pontryagin duality
and work with representations of $G$ on compact $\mathcal{O}$-modules. We
denote by $\operatorname{\mathfrak{C}}(\mathcal{O})_{\mathfrak{B}}$ the
category anti-equivalent to
$\operatorname{Mod}^{\mathrm{l.fin}}_{G,\zeta}(\mathcal{O})_{\mathfrak{B}}$
under the Pontryagin duality.

Let $P_{\mathbf{1}^{\vee}}$ and $P_{\operatorname{Sp}^{\vee}}$ be projective
envelopes of $\mathbf{1}^{\vee}$ and $\operatorname{Sp}^{\vee}$ in
$\operatorname{\mathfrak{C}}(\mathcal{O})_{\mathfrak{B}}$, respectively. Then
$P_{\mathfrak{B}}:=P_{\mathbf{1}^{\vee}}\oplus P_{\operatorname{Sp}^{\vee}}$
is a projective generator of
$\operatorname{\mathfrak{C}}(\mathcal{O})_{\mathfrak{B}}$ and by results of
Gabriel [31] the category
$\operatorname{\mathfrak{C}}(\mathcal{O})_{\mathfrak{B}}$ is equivalent to the
category of pseudo-compact modules of
$E_{\mathfrak{B}}:=\operatorname{End}_{\operatorname{\mathfrak{C}}(\mathcal{O})}(P_{\mathfrak{B}})$.
The equivalence is induced by the functors

$N\mapsto\operatorname{Hom}_{\operatorname{\mathfrak{C}}(\mathcal{O})}(P_{\mathfrak{B}},N),\quad\mathrm{m}\mapsto\mathrm{m}\operatorname{\widehat{\otimes}}_{E_{\mathfrak{B}}}P_{\mathfrak{B}}.$

The centre $Z_{\mathfrak{B}}$ of the category
$\operatorname{\mathfrak{C}}(\mathcal{O})_{\mathfrak{B}}$ is naturally
isomorphic to the centre of the ring $E_{\mathfrak{B}}$.

Colmez’s functor kills all objects on which
$\operatorname{SL}_{2}(\mathbb{Q}_{p})$ acts trivially, and these form a thick
subcategory. Thus $\check{\mathbf{V}}$ factors through the quotient category,
which we denote by $\mathfrak{Q}(\mathcal{O})_{\mathfrak{B}}$ and let
$\mathcal{T}:\operatorname{\mathfrak{C}}(\mathcal{O})_{\mathfrak{B}}\rightarrow\mathfrak{Q}(\mathcal{O})_{\mathfrak{B}}$
be the quotient functor. Moreover, $\check{\mathbf{V}}$ induces an equivalence
of categories between $\mathfrak{Q}(\mathcal{O})_{\mathfrak{B}}$ and its
essential image under $\check{\mathbf{V}}$. To prove this one needs to show
that $\check{\mathbf{V}}$ sends non-split extensions to non-split extensions,
such arguments are originally due to Colmez, in the case $p=2$ this has been
shown by the second author in his thesis, [47].

Since $\mathcal{T}(\mathbf{1}^{\vee})=0$,
$\mathcal{T}(\operatorname{Sp}^{\vee})$ is the only irreducible object in
$\mathfrak{Q}(\mathcal{O})_{\mathfrak{B}}$ up to isomorphism. Moreover, it is
shown in [39] that $\mathcal{T}(P_{\operatorname{Sp}^{\vee}})$ is a projective
envelope of $\mathcal{T}(\operatorname{Sp}^{\vee})$, and $\mathcal{T}$ induces
an isomorphism

$E_{\mathfrak{B}}^{\prime}:=\operatorname{End}_{\operatorname{\mathfrak{C}}(\mathcal{O})}(P_{\operatorname{Sp}^{\vee}})\cong\operatorname{End}_{\mathfrak{Q}(\mathcal{O})}(\mathcal{T}P_{\operatorname{Sp}^{\vee}}).$

Since $\mathcal{T}(\operatorname{Sp}^{\vee})$ is the only irreducible in
$\mathfrak{Q}(\mathcal{O})_{\mathfrak{B}}$,
$\mathcal{T}(P_{\operatorname{Sp}^{\vee}})$ is a projective generator of
$\mathfrak{Q}(\mathcal{O})_{\mathfrak{B}}$, and thus
$\mathfrak{Q}(\mathcal{O})_{\mathfrak{B}}$ is equivalent to the category of
pseudo-compact $E_{\mathfrak{B}}^{\prime}$-modules. This implies that
$\mathcal{T}(P_{\operatorname{Sp}^{\vee}})$ and hence, by equivalence of
categories, $\check{\mathbf{V}}(P_{\operatorname{Sp}^{\vee}})$ are flat over
$E_{\mathfrak{B}}^{\prime}$. Since
$\check{\mathbf{V}}(\operatorname{Sp}^{\vee})$ is a one dimensional
representation of $\mathcal{G}_{\mathbb{Q}_{p}}$, in fact the trivial
representation with our normalizations, an application of Nakayama’s lemma
shows that $\check{\mathbf{V}}(P_{\operatorname{Sp}^{\vee}})$ is a free
$E_{\mathfrak{B}}^{\prime}$-module of rank $1$. The action of
$\mathcal{G}_{\mathbb{Q}_{p}}$ on
$\check{\mathbf{V}}(P_{\operatorname{Sp}^{\vee}})$ commutes with the action of
$E_{\mathfrak{B}}^{\prime}$ and so induces a homomorphism

$\alpha:\mathcal{O}\llbracket\mathcal{G}_{\mathbb{Q}_{p}}\rrbracket\rightarrow\operatorname{End}_{E_{\mathfrak{B}}^{\prime}}(\check{\mathbf{V}}(P_{\operatorname{Sp}^{\vee}}))\cong(E_{\mathfrak{B}}^{\prime})^{\mathrm{op}}.$

We show that this map is surjective. In general the argument is carried out in
section 2 in an abstract setting and then in Proposition 4.18 we verify that
the conditions of the abstract setting are satisfied. However, in the special
case under the consideration the argument is easier: since
$\check{\mathbf{V}}$ induces an equivalence of categories between
$\mathfrak{Q}(\mathcal{O})_{\mathfrak{B}}$ and its essential image, the
$E_{\mathfrak{B}}^{\prime}$-cosocle and
$\mathcal{O}\llbracket\mathcal{G}_{\mathbb{Q}_{p}}\rrbracket$-cosocle of
$\check{\mathbf{V}}(P_{\operatorname{Sp}^{\vee}})$ coincide. This implies that
there is $v\in\check{\mathbf{V}}(P_{\operatorname{Sp}^{\vee}})$, which is a
generator of $\check{\mathbf{V}}(P_{\operatorname{Sp}^{\vee}})$ both as
$E_{\mathfrak{B}}^{\prime}$\- and as
$\mathcal{O}\llbracket\mathcal{G}_{\mathbb{Q}_{p}}\rrbracket$-module. This
implies that $\alpha$ is surjective.

We also show in section 3 that the natural map

$\beta:\mathcal{O}\llbracket\mathcal{G}_{\mathbb{Q}_{p}}\rrbracket\rightarrow
R^{\mathrm{ps},\zeta\varepsilon}_{\operatorname{tr}\bar{\rho}}\llbracket\mathcal{G}_{\mathbb{Q}_{p}}\rrbracket/J$

is surjective, where
$R^{\mathrm{ps},\zeta\varepsilon}_{\operatorname{tr}\bar{\rho}}\llbracket\mathcal{G}_{\mathbb{Q}_{p}}\rrbracket/J$
is the largest quotient of
$R^{\mathrm{ps},\zeta\varepsilon}_{\operatorname{tr}\bar{\rho}}\llbracket\mathcal{G}_{\mathbb{Q}_{p}}\rrbracket$
such that the Cayley–Hamilton theorem holds for the universal
pseudorepresentation with determinant $\zeta\varepsilon$ lifting
$\operatorname{tr}\bar{\rho}$.

The idea is to show that $\operatorname{Ker}\alpha$ contains
$\operatorname{Ker}\beta$, since this implies that the action of
$\mathcal{G}_{\mathbb{Q}_{p}}$ on
$\check{\mathbf{V}}(P_{\operatorname{Sp}^{\vee}})$ induces surjections:

(3)
$\mathcal{O}\llbracket\mathcal{G}_{\mathbb{Q}_{p}}\rrbracket\twoheadrightarrow
R^{\mathrm{ps},\zeta\varepsilon}_{\operatorname{tr}\bar{\rho}}\llbracket\mathcal{G}_{\mathbb{Q}_{p}}\rrbracket/J\twoheadrightarrow
E_{\mathfrak{B}}^{\prime}.$

This is proved using the results of Berger–Breuil [4] on universal unitary
completions of locally algebraic principal series and density arguments
already used in [21] and also in [46]. Morally, the argument should be that
$\check{\mathbf{V}}(P_{\operatorname{Sp}^{\vee}})$ injects into the product of
all $2$-dimensional crystabelline representations of
$\mathcal{G}_{\mathbb{Q}_{p}}$ with mod $p$ reduction isomorphic to
$\bar{\rho}$ and determinant $\zeta\varepsilon$, then an element in
$\operatorname{Ker}\beta$ would kill this product and hence
$\check{\mathbf{V}}(P_{\operatorname{Sp}^{\vee}})$. In reality the argument is
technically a bit more complicated: we also have to consider deformations of
such representations to local artinian $L$-algebras, see section 5.

Wang-Erickson has proved in [48] that
$R^{\mathrm{ps},\zeta\varepsilon}_{\operatorname{tr}\bar{\rho}}\llbracket\mathcal{G}_{\mathbb{Q}_{p}}\rrbracket/J$
is a finitely generated
$R^{\mathrm{ps},\zeta\varepsilon}_{\operatorname{tr}\bar{\rho}}$-module. The
surjection (3) implies that the image of
$R^{\mathrm{ps},\zeta\varepsilon}_{\operatorname{tr}\bar{\rho}}$ is contained
in the centre of $E_{\mathfrak{B}}^{\prime}$, which we denote by
$Z_{\mathfrak{B}}^{\prime}$. Moreover, both $E_{\mathfrak{B}}^{\prime}$ and
$Z_{\mathfrak{B}}^{\prime}$ are finite
$R^{\mathrm{ps},\zeta\varepsilon}_{\operatorname{tr}\bar{\rho}}$-modules. We
show that (3) induces an isomorphism

$(R^{\mathrm{ps},\zeta\varepsilon}_{\operatorname{tr}\bar{\rho}}\llbracket\mathcal{G}_{\mathbb{Q}_{p}}\rrbracket/J)_{\mathrm{tf}}\overset{\cong}{\longrightarrow}E_{\mathfrak{B}}^{\prime}$

by showing that the universal framed deformation of $\bar{\rho}$ with
determinant $\zeta\varepsilon$ lies in the image of $\check{\mathbf{V}}$ using
[21], [22] and
$(R^{\mathrm{ps},\zeta\varepsilon}_{\operatorname{tr}\bar{\rho}}\llbracket\mathcal{G}_{\mathbb{Q}_{p}}\rrbracket/J)_{\mathrm{tf}}$
acts faithfully on it, see Proposition A.11. The assertions about the centre
in Theorem 1.4 with $Z_{\mathfrak{B}}^{\prime}$ instead of $Z_{\mathfrak{B}}$
are proved by studying the centre of
$(R^{\mathrm{ps},\zeta\varepsilon}_{\operatorname{tr}\bar{\rho}}\llbracket\mathcal{G}_{\mathbb{Q}_{p}}\rrbracket/J)_{\mathrm{tf}}$.
This argument is carried out in the appendix for $d$-dimensional
representations of any profinite group, satisfying Mazur’s finiteness
condition at $p$.

We then transfer this result from $E_{\mathfrak{B}}^{\prime}$ and
$Z_{\mathfrak{B}}^{\prime}$ to $E_{\mathfrak{B}}$ and $Z_{\mathfrak{B}}$. Let
$M_{\mathfrak{B}}$ be the kernel of
$P_{\mathfrak{B}}\rightarrow(P_{\mathfrak{B}})_{\operatorname{SL}_{2}(\mathbb{Q}_{p})}$.
We show that

$\operatorname{End}_{\operatorname{\mathfrak{C}}(\mathcal{O})}(M_{\mathfrak{B}})\cong\operatorname{End}_{\operatorname{\mathfrak{C}}(\mathcal{O})}(P_{\mathfrak{B}})=E_{\mathfrak{B}}$

by examining various exact sequences and showing that certain
$\operatorname{Ext}$-groups vanish. We also show that the cosocle of
$M_{\mathfrak{B}}$ is a direct sum of finitely many copies of
$\operatorname{Sp}^{\vee}$. Thus $M_{\mathfrak{B}}$ is a quotient of
$(P_{\mathfrak{B}}^{\prime})^{\oplus n}$ for some $n\geq 1$. This allows to
conclude that
$\operatorname{End}_{\operatorname{\mathfrak{C}}(\mathcal{O})}(M_{\mathfrak{B}})$
and its centre are finitely generated $Z_{\mathfrak{B}}^{\prime}$-modules,
which finishes the proof of Theorem 1.3. The arguments showing finiteness of
$E_{\mathfrak{B}}$ and $Z_{\mathfrak{B}}$ over $Z_{\mathfrak{B}}^{\prime}$ are
carried out in section 4.3. Then with some more effort we are able to show
that $Z_{\mathfrak{B}}^{\prime}=Z_{\mathfrak{B}}$, see Corollary 6.15.

###  1.2. What is left to do?

Although we believe that our results will suffice for most number theoretic
applications, for example [38], to complete the programme started in [39] one
would have to compute the ring $E_{\mathfrak{B}}$ in the exceptional cases.
This will be harder than [39, Section 10.5], which is already quite involved.
We expect that the map in (3) induces isomorphisms

$R^{\mathrm{ps},\zeta\varepsilon}_{\operatorname{tr}\bar{\rho}}\llbracket\mathcal{G}_{\mathbb{Q}_{p}}\rrbracket/J\overset{\cong}{\longrightarrow}E_{\mathfrak{B}}^{\prime},\quad
R^{\mathrm{ps},\zeta\varepsilon}_{\operatorname{tr}\bar{\rho}}\overset{\cong}{\longrightarrow}Z_{\mathfrak{B}}$

This is known to hold for all blocks except for the exceptional ones. Theorem
1.4 implies that to prove this result it would be enough to show that
$R^{\mathrm{ps},\zeta\varepsilon}_{\operatorname{tr}\bar{\rho}}\llbracket\mathcal{G}_{\mathbb{Q}_{p}}\rrbracket/J$
is $\mathcal{O}$-torsion free and for the second isomorphism in the case $p=3$
it would be enough to show222This follows from [8, Corollary 5.11]. that
$R^{\mathrm{ps},\zeta\varepsilon}_{\operatorname{tr}\bar{\rho}}$ is
$\mathcal{O}$-torsion free.

It seems likely that using the results of this paper one can remove the
restriction on the prime $p$ in Lue Pan’s work [37] on the Fontaine–Mazur
conjecture in the residually reducible case, which generalizes the work of
Skinner–Wiles. We hope to return to these questions in future work.

###  1.3. Acknowledgements.

VP would like to thank Toby Gee for organizing a seminar on Lue Pan’s paper
[38] and for the invitation to participate. Lue Pan’s paper was a major source
of motivation to deal with the exceptional cases. VP would like to thank
Patrick Allen and Gebhard Böckle for stimulating correspondence. Parts of the
paper were written, when SNT was visiting Academia Sinica. SNT would like to
thank Carl Wang-Erickson for stimulating correspondence. The authors would
like to thank Gabriel Dospinescu and Toby Gee for their detailed comments on
an earlier draft.

##  2\. Endomorphism rings

Let $E$ be a pseudo-compact $\mathcal{O}$-algebra and let $\mathrm{PC}(E)$ be
the category of left pseudo-compact $E$-modules, see [12], [31, §IV.3]. Let
$\operatorname{Irr}(E)$ be the set of equivalence classes of irreducible
objects in $\mathrm{PC}(E)$.

Let $M$ be in $\mathrm{PC}(E)$. We assume that we are given a continuous
$E$-linear action of a profinite group $\mathcal{G}$ on $M$, which makes $M$
into a pseudo-compact module over the completed group algebra
$\mathcal{O}\llbracket\mathcal{G}\rrbracket$. The action induces a
homomorphism of $\mathcal{O}$-algebras
$\mathcal{O}\llbracket\mathcal{G}\rrbracket\rightarrow\operatorname{End}_{E}^{\mathrm{cont}}(M)$.
In this section we will study, when this map is surjective and its kernel.

If $N$ is a pseudo-compact $E$-module, which is finitely generated as an
$E$-module, then we may present $N$ as

$\prod_{i\in I}E\rightarrow E^{\oplus n}\rightarrow N\rightarrow 0.$

By applying $\operatorname{Hom}^{\mathrm{cont}}_{E}(\ast,M)$ we obtain an
exact sequence

$0\rightarrow\operatorname{Hom}^{\mathrm{cont}}_{E}(N,M)\rightarrow M^{\oplus
n}\rightarrow\oplus_{i\in I}M.$

We thus may identify $\operatorname{Hom}^{\mathrm{cont}}_{E}(N,M)$ with a
closed submodule of $M^{\oplus n}$, which makes
$\operatorname{Hom}^{\mathrm{cont}}_{E}(N,M)$ into a pseudo-compact left
$\mathcal{O}\llbracket\mathcal{G}\rrbracket$-module.

######  Lemma 2.1.

Let $N$ be a finitely generated projective $E$-module and let $\mathrm{m}$ be
a right pseudo-compact $\mathcal{O}\llbracket\mathcal{G}\rrbracket$-module.
Then the natural map

(4)
$\mathrm{m}\operatorname{\widehat{\otimes}}_{\mathcal{O}\llbracket\mathcal{G}\rrbracket}\operatorname{Hom}^{\mathrm{cont}}_{E}(N,M)\rightarrow\operatorname{Hom}^{\mathrm{cont}}_{E}(N,\mathrm{m}\operatorname{\widehat{\otimes}}_{\mathcal{O}\llbracket\mathcal{G}\rrbracket}M)$

is an isomorphism.

###### Proof.

Since $N$ is finitely generated and projective, we may present it as

$F\overset{e}{\longrightarrow}F\rightarrow N\rightarrow 0,$

where $F\cong E^{\oplus n}$ and $e$ is an idempotent. In particular, $N$ is a
pseudo-compact $E$-module. The map in (4) is induced by a continuous bilinear
map

$(v,\phi)\mapsto[w\mapsto v\operatorname{\widehat{\otimes}}\phi(w)].$

It is an isomorphism if $N=F$. The general case follows by applying the
idempotent $e$ to the isomorphism obtained for $N=F$. ∎

######  Lemma 2.2.

Let $\\{\rho_{i}\\}_{i\in I}$ be a family of pairwise distinct absolutely
irreducible right pseudo-compact
$\mathcal{O}\llbracket\mathcal{G}\rrbracket$-modules. Then the map

(5) $\mathcal{O}\llbracket\mathcal{G}\rrbracket\rightarrow\prod_{i\in
I}\operatorname{End}_{k}(\rho_{i})^{\mathrm{op}}$

is surjective.

###### Proof.

Since $\mathcal{G}$ is profinite each $\rho_{i}$ is a finite dimensional
$k$-vector space. Thus
$\varphi_{i}:\mathcal{O}\llbracket\mathcal{G}\rrbracket\rightarrow\operatorname{End}_{k}(\rho_{i})^{\mathrm{op}}$,
given by the action, is continuous for the discrete topology on the target.
Since $\rho_{i}$ is absolutely irreducible $\varphi_{i}$ is surjective.
Moreover, $\operatorname{Ker}\varphi_{i}$ is an open maximal two-sided ideal
of $\mathcal{O}\llbracket\mathcal{G}\rrbracket$. If $i\neq j$ then
$\rho_{i}\not\cong\rho_{j}$ and thus
$\operatorname{Ker}\varphi_{i}+\operatorname{Ker}\varphi_{j}=\mathcal{O}\llbracket\mathcal{G}\rrbracket$.
This implies that for every finite subset $F$ of $I$ the map
$\mathcal{O}\llbracket\mathcal{G}\rrbracket\rightarrow\prod_{i\in
F}\operatorname{End}_{k}(\rho_{i})^{\mathrm{op}}$ is surjective. Thus the
image of (5) is dense for the product topology on the target. On the other
hand (5) is a continuous map between pseudo-compact $\mathcal{O}$-modules and
thus its image is closed, which implies surjectivity. ∎

If $M$ is in $\mathrm{PC}(E)$ then we let $\mathfrak{r}(M)$ be the
intersection of open maximal submodules of $M$. Then $\mathfrak{r}(E)$ is a
closed two-sided ideal of $E$ and $\mathfrak{r}(M)$ is the closure of
$\mathfrak{r}(E)M$ inside $M$.

######  Proposition 2.3.

Let us assume that the following hold

  1. (1)

$M$ is a finitely generated projective $E$-module;

  2. (2)

$M/\mathfrak{r}(M)=\operatorname{cosoc}_{\mathcal{G}}M$;

  3. (3)

for all $S\in\operatorname{Irr}(E)$, such that

$\rho_{S}:=\operatorname{Hom}_{E}^{\mathrm{cont}}(M,S)\neq 0,$

$\dim_{k}\rho_{S}$ is finite and $\rho_{S}$ is an absolutely irreducible
representation of $\mathcal{G}$;

  4. (4)

if $S,S^{\prime}\in\operatorname{Irr}(E)$ and $S\not\cong S^{\prime}$ then
$\operatorname{Hom}_{\mathcal{G}}(\rho_{S},\rho_{S^{\prime}})=0$.

Then the map
$\mathcal{O}\llbracket\mathcal{G}\rrbracket\rightarrow\operatorname{End}_{E}^{\mathrm{cont}}(M)$
is surjective.

###### Proof.

For $S\in\operatorname{Irr}(E)$ we let $\overline{M}_{S}$ be the smallest
quotient of $M$ such that

$\rho_{S}=\operatorname{Hom}_{E}^{\mathrm{cont}}(M,S)=\operatorname{Hom}_{E}^{\mathrm{cont}}(\overline{M}_{S},S).$

It follows from (3) that the above subspaces are finite dimensional. Thus
$\overline{M}_{S}\cong S^{\oplus d}$ such that
$d\cdot\dim_{k}\operatorname{End}_{E}^{\mathrm{cont}}(S)=\dim_{k}\rho_{S}$.
Since $S$ is irreducible $\operatorname{End}_{E}^{\mathrm{cont}}(S)$ is a skew
field. It acts on $\rho_{S}$ and this action commutes with the action of
$\mathcal{G}$. Since $\rho_{S}$ is absolutely irreducible we conclude that
$\operatorname{End}_{E}^{\mathrm{cont}}(S)=k$ and $\dim_{k}\rho_{S}=d$. Thus
$\operatorname{End}^{\mathrm{cont}}_{E}(\overline{M}_{S})\cong M_{d}(k)$ and
thus does not have non-trivial two-sided ideals. Hence, the natural right
action of $\operatorname{End}_{E}(\overline{M}_{S})$ on $\rho_{S}$ induces an
injective ring homomorphism

(6)
$\operatorname{End}_{E}^{\mathrm{cont}}(\overline{M}_{S})\rightarrow\operatorname{End}_{k}(\rho_{S})^{\mathrm{op}},$

which has to be surjective as both $k$-vector spaces have dimension equal to
$d^{2}$.

The isomorphism
$M/\mathfrak{r}(M)\cong\prod_{S\in\operatorname{Irr}(E)}\overline{M}_{S}$
induces an isomorphism

$\operatorname{End}_{E}^{\mathrm{cont}}(M/\mathfrak{r}(M))\cong\prod_{S\in\operatorname{Irr}(E)}\operatorname{End}_{E}^{\mathrm{cont}}(\overline{M}_{S}).$

Since $\rho_{S}\not\cong\rho_{S^{\prime}}$ if $S\not\cong S^{\prime}$, it
follows from Lemma 2.2 together with the isomorphism (6) that the action of
$\mathcal{O}\llbracket\mathcal{G}\rrbracket$ on $M/\mathfrak{r}(M)$ induces a
surjection

(7)
$\mathcal{O}\llbracket\mathcal{G}\rrbracket\twoheadrightarrow\operatorname{End}_{E}^{\mathrm{cont}}(M/\mathfrak{r}(M)).$

Since $M$ is projective and $M/\mathfrak{r}(M)$ is pro-semisimple we have

(8)
$\operatorname{End}^{\mathrm{cont}}_{E}(M)\twoheadrightarrow\operatorname{Hom}^{\mathrm{cont}}_{E}(M,M/\mathfrak{r}(M))\cong\operatorname{End}_{E}^{\mathrm{cont}}(M/\mathfrak{r}(M)).$

If $\mathrm{m}$ is an irreducible right pseudo-compact
$\mathcal{O}\llbracket\mathcal{G}\rrbracket$-module and $\mathfrak{a}$ is its
annihilator then $\mathcal{O}\llbracket\mathcal{G}\rrbracket/\mathfrak{a}$ is
a finite dimensional simple $k$-algebra, and thus
$\mathcal{O}\llbracket\mathcal{G}\rrbracket/\mathfrak{a}$ is semi-simple as a
left $\mathcal{O}\llbracket\mathcal{G}\rrbracket$-module, and thus
$(\mathcal{O}\llbracket\mathcal{G}\rrbracket/\mathfrak{a})\operatorname{\widehat{\otimes}}_{\mathcal{O}\llbracket\mathcal{G}\rrbracket}M$
is semi-simple as a left $\mathcal{O}\llbracket\mathcal{G}\rrbracket$-module.
Hence, the surjection
$M\twoheadrightarrow(\mathcal{O}\llbracket\mathcal{G}\rrbracket/\mathfrak{a})\operatorname{\widehat{\otimes}}_{\mathcal{O}\llbracket\mathcal{G}\rrbracket}M$
factors through as

$M\twoheadrightarrow\operatorname{cosoc}_{\mathcal{G}}M\twoheadrightarrow(\mathcal{O}\llbracket\mathcal{G}\rrbracket/\mathfrak{a})\operatorname{\widehat{\otimes}}_{\mathcal{O}\llbracket\mathcal{G}\rrbracket}M.$

Moreover, the maps become isomorphisms after applying
$\mathrm{m}\operatorname{\widehat{\otimes}}_{\mathcal{O}\llbracket\mathcal{G}\rrbracket}$.
Thus

(9)
$\mathrm{m}\operatorname{\widehat{\otimes}}_{\mathcal{O}\llbracket\mathcal{G}\rrbracket}M\cong\mathrm{m}\operatorname{\widehat{\otimes}}_{\mathcal{O}\llbracket\mathcal{G}\rrbracket}\operatorname{cosoc}_{\mathcal{G}}M\cong\mathrm{m}\operatorname{\widehat{\otimes}}_{\mathcal{O}\llbracket\mathcal{G}\rrbracket}M/\mathfrak{r}(M),$

as $M/\mathfrak{r}(M)=\operatorname{cosoc}_{\mathcal{G}}M$ by assumption.
Lemma 2.1 together with (9) imply that by applying
$\mathrm{m}\operatorname{\widehat{\otimes}}_{\mathcal{O}\llbracket\mathcal{G}\rrbracket}$
to (8) we obtain isomorphisms

(10)
$\begin{split}\mathrm{m}\operatorname{\widehat{\otimes}}_{\mathcal{O}\llbracket\mathcal{G}\rrbracket}\operatorname{End}^{\mathrm{cont}}_{E}(M)&\overset{\cong}{\longrightarrow}\operatorname{Hom}^{\mathrm{cont}}_{E}(M,\mathrm{m}\operatorname{\widehat{\otimes}}_{\mathcal{O}\llbracket\mathcal{G}\rrbracket}M)\\\
&\overset{\cong}{\longrightarrow}\operatorname{Hom}^{\mathrm{cont}}_{E}(M,\mathrm{m}\operatorname{\widehat{\otimes}}_{\mathcal{O}\llbracket\mathcal{G}\rrbracket}M/\mathfrak{r}(M))\\\
&\overset{\cong}{\longrightarrow}\mathrm{m}\operatorname{\widehat{\otimes}}_{\mathcal{O}\llbracket\mathcal{G}\rrbracket}\operatorname{Hom}^{\mathrm{cont}}_{E}(M,M/\mathfrak{r}(M))\\\
&\overset{\cong}{\longrightarrow}\mathrm{m}\operatorname{\widehat{\otimes}}_{\mathcal{O}\llbracket\mathcal{G}\rrbracket}\operatorname{End}^{\mathrm{cont}}_{E}(M/\mathfrak{r}(M)),\end{split}$

where the $\mathcal{G}$-action on $\operatorname{End}^{\mathrm{cont}}_{E}(M)$
and $\operatorname{End}^{\mathrm{cont}}_{E}(M/\mathfrak{r}(M))$ is given by
$(g.\varphi)(v):=g(\varphi(v))$.

If the map
$\mathcal{O}\llbracket\mathcal{G}\rrbracket\rightarrow\operatorname{End}^{\mathrm{cont}}_{E}(M)$
is not surjective then its cokernel is a non-zero left pseudo-compact
$\mathcal{O}\llbracket\mathcal{G}\rrbracket$-module and thus will have an
irreducible quotient $\mathrm{m}^{\prime}$. If we let
$\mathrm{m}=\operatorname{Hom}_{k}(\mathrm{m}^{\prime},k)$ with the right
$\mathcal{O}\llbracket\mathcal{G}\rrbracket$-action then
$\mathrm{m}\operatorname{\widehat{\otimes}}_{\mathcal{O}\llbracket\mathcal{G}\rrbracket}\mathrm{m}^{\prime}$
is non-zero, as the evaluation map
$\mathrm{m}\operatorname{\widehat{\otimes}}_{\mathcal{O}\llbracket\mathcal{G}\rrbracket}\mathrm{m}^{\prime}\twoheadrightarrow
k$ is non-zero. By construction the composition

$\mathrm{m}\operatorname{\widehat{\otimes}}_{\mathcal{O}\llbracket\mathcal{G}\rrbracket}\mathcal{O}\llbracket\mathcal{G}\rrbracket\rightarrow\mathrm{m}\operatorname{\widehat{\otimes}}_{\mathcal{O}\llbracket\mathcal{G}\rrbracket}\operatorname{End}^{\mathrm{cont}}_{E}(M)\twoheadrightarrow\mathrm{m}\operatorname{\widehat{\otimes}}_{\mathcal{O}\llbracket\mathcal{G}\rrbracket}\mathrm{m}^{\prime}$

is the zero map. Thus
$\mathrm{m}\operatorname{\widehat{\otimes}}_{\mathcal{O}\llbracket\mathcal{G}\rrbracket}\mathcal{O}\llbracket\mathcal{G}\rrbracket\rightarrow\mathrm{m}\operatorname{\widehat{\otimes}}_{\mathcal{O}\llbracket\mathcal{G}\rrbracket}\operatorname{End}^{\mathrm{cont}}_{E}(M)$
cannot be surjective. However, the commutative diagram

${\mathrm{m}\operatorname{\widehat{\otimes}}_{\mathcal{O}\llbracket\mathcal{G}\rrbracket}\mathcal{O}\llbracket\mathcal{G}\rrbracket}$${\mathrm{m}\operatorname{\widehat{\otimes}}_{\mathcal{O}\llbracket\mathcal{G}\rrbracket}\operatorname{End}^{\mathrm{cont}}_{E}(M)}$${\mathrm{m}\operatorname{\widehat{\otimes}}_{\mathcal{O}\llbracket\mathcal{G}\rrbracket}\mathcal{O}\llbracket\mathcal{G}\rrbracket}$${\mathrm{m}\operatorname{\widehat{\otimes}}_{\mathcal{O}\llbracket\mathcal{G}\rrbracket}\operatorname{End}^{\mathrm{cont}}_{E}(M/\mathfrak{r}(M))}$$\scriptstyle{=}$$\scriptstyle{\cong}$$\scriptstyle{\eqref{surj3}}$$\scriptstyle{\eqref{surj1}}$

implies that the top horizontal arrow is surjective, yielding a contradiction.
∎

We remind the reader that as a consequence of the topological Nakayama’s
lemma, see Lemma 0.3.3 in Exposé $VII_{B}$ in [32], the following holds:

######  Lemma 2.4.

Let $N$ be a pseudo-compact left $E$-module. Then $N$ is projective in
$\mathrm{PC}(E)$ if and only if the functor
$\mathrm{m}\mapsto\mathrm{m}\operatorname{\widehat{\otimes}}_{E}N$ from the
category of right pseudo-compact $E$-modules to the category of abelian groups
is exact. In this case, $N\twoheadrightarrow N/\mathfrak{r}(N)$ is a
projective envelope of $N/\mathfrak{r}(N)$.

######  Corollary 2.5.

If in addition to the assumptions of Proposition 2.3 we assume that
$M/\mathfrak{r}(M)\cong E/\mathfrak{r}(E)$ as $E$-modules then $M$ is a free
$E$-module of rank $1$ and the action of
$\mathcal{O}\llbracket\mathcal{G}\rrbracket$ on $M$ induces a surjection

$\mathcal{O}\llbracket\mathcal{G}\rrbracket\twoheadrightarrow
E^{\mathrm{op}},$

which is uniquely determined up to a conjugation by $E^{\times}$.

We will now give a characterisation of the kernel of
$\mathcal{O}\llbracket\mathcal{G}\rrbracket\rightarrow\operatorname{End}_{E}^{\mathrm{cont}}(M)$
in favourable settings.

Let $\mathrm{m}$ be a finite dimensional $L$-vector space with continuous
$\mathcal{O}$-linear action of $E$ on the right. The image of $E$ in
$\operatorname{End}_{L}(\mathrm{m})$ is a compact $\mathcal{O}$-module and
thus $E$ stabilises an $\mathcal{O}$-lattice $\mathrm{m}^{0}$ in $\mathrm{m}$.
The action of $\mathcal{O}\llbracket\mathcal{G}\rrbracket$ on $M$ induces a
continuous left action of $\mathcal{O}\llbracket\mathcal{G}\rrbracket$ on
$\mathrm{m}^{0}\operatorname{\widehat{\otimes}}_{E}M$ and hence on

$\mathrm{m}\otimes_{E}M=(\mathrm{m}^{0}\otimes_{E}M)[1/p]=(\mathrm{m}^{0}\operatorname{\widehat{\otimes}}_{E}M)[1/p].$

######  Lemma 2.6.

Let $\\{\mathrm{m}_{i}\\}_{i\in I}$ be a family of finite dimensional
$L$-vector spaces with continuous right $\mathcal{O}$-linear action of $E$.
For each $i\in I$, let $\mathfrak{a}_{i}$ be the $E$-annihilator of
$\mathrm{m}_{i}$ and let $\mathfrak{b}_{i}$ be the
$\mathcal{O}\llbracket\mathcal{G}\rrbracket$-annihilator of
$\mathrm{m}_{i}\otimes_{E}M$. If $M$ is a free $E$-module of finite rank and
$\bigcap_{i\in I}\mathfrak{a}_{i}=0$ then

$\operatorname{Ker}(\mathcal{O}\llbracket\mathcal{G}\rrbracket\rightarrow\operatorname{End}_{E}^{\mathrm{cont}}(M))=\bigcap_{i\in
I}\mathfrak{b}_{i}.$

###### Proof.

For each $i\in I$, let $\mathfrak{c}_{i}$ be the
$\operatorname{End}_{E}^{\mathrm{cont}}(M)$-annihilator of
$\mathrm{m}_{i}\otimes_{E}M$. Since $\mathfrak{b}_{i}$ is the preimage of
$\mathfrak{c}_{i}$ in $\mathcal{O}\llbracket\mathcal{G}\rrbracket$, it is
enough to show that $\bigcap_{i\in I}\mathfrak{c}_{i}=0$.

Let $w_{1},\ldots,w_{n}$ be an $E$-basis of $M$. Then we may identify
$\operatorname{End}_{E}^{\mathrm{cont}}(M)$ with $M_{n}(E)$ by mapping
$\varphi$ to the matrix $(a_{kj})$, given by

$\varphi(w_{k})=\sum_{j=1}^{n}a_{kj}w_{j}$

for all $1\leq k\leq n$. If $v\in\mathrm{m}_{i}$ then

$v\operatorname{\widehat{\otimes}}\varphi(w_{k})=\sum_{j=1}^{n}(va_{kj})\operatorname{\widehat{\otimes}}w_{j}.$

Thus $\varphi$ annihilates $\mathrm{m}_{i}\otimes_{E}M$ if and only if
$va_{kj}=0$ for all $v\in\mathrm{m}_{i}$ and all $1\leq k,j\leq n$, which is
equivalent to $\varphi\in M_{n}(\mathfrak{a}_{i})$. Since $\bigcap_{i\in
I}\mathfrak{a}_{i}=0$ we have $\bigcap_{i\in I}M_{n}(\mathfrak{a}_{i})=0$ and
thus $\bigcap_{i\in I}\mathfrak{c}_{i}=0$. ∎

##  3\. Pseudorepresentations

Let $\mathcal{G}$ be a profinite group and let $\bar{\rho}$ be a continuous
semi-simple representation of $\mathcal{G}$ on a $2$-dimensional $k$-vector
space. We fix a continuous group homomorphism
$\psi:\mathcal{G}\rightarrow\mathcal{O}^{\times}$ lifting $\det\bar{\rho}$.
Let $D^{\mathrm{ps},\psi}$ be the functor from the category of augmented
artinian $\mathcal{O}$-algebras with residue field $k$ to the category of
sets, which maps $(A,\mathfrak{m}_{A})$ to the set of continuous functions
$t:\mathcal{G}\rightarrow A$ satisfying the following conditions:

  * •

$t(1)=2$;

  * •

$t(g)\equiv\operatorname{tr}\bar{\rho}(g)\pmod{\mathfrak{m}_{A}},\quad\forall
g\in\mathcal{G}$;

  * •

$t(gh)=t(hg),\quad\forall g,h\in\mathcal{G}$;

  * •

$\psi(g)t(g^{-1}h)-t(g)t(h)+t(gh)=0,\quad\forall g,h\in\mathcal{G}$.

The data $(t,\psi)$ determines a continuous polynomial law of homogeneous
degree $2$ on $\mathcal{G}$, see [18, Lemma 1.9]. This deformation problem is
pro-representable by a local $\mathcal{O}$-algebra $R^{\mathrm{ps},\psi}$ with
residue field $k$, complete with respect to profinite topology. We denote by
$T:\mathcal{G}\rightarrow R^{\mathrm{ps},\psi}$ the universal deformation. We
extend it $R^{\mathrm{ps},\psi}$-linearly to a continuous function
$T:R^{\mathrm{ps},\psi}\llbracket\mathcal{G}\rrbracket\rightarrow
R^{\mathrm{ps},\psi}$. The homomorphism
$\psi:\mathcal{G}\rightarrow\mathcal{O}^{\times}$ induces a continuous
$\mathcal{O}$-algebra homomorphism
$\psi:\mathcal{O}\llbracket\mathcal{G}\rrbracket\rightarrow\mathcal{O}$, which
we extend $R^{\mathrm{ps},\psi}$-linearly to a continuous
$R^{\mathrm{ps},\psi}$-algebra homomorphism
$\psi:R^{\mathrm{ps},\psi}\llbracket\mathcal{G}\rrbracket\rightarrow
R^{\mathrm{ps},\psi}$. Let $J$ be the closed two-sided ideal of
$R^{\mathrm{ps},\psi}\llbracket\mathcal{G}\rrbracket$ generated by
$a^{2}-T(a)a+\psi(a)$ for all $a\in
R^{\mathrm{ps},\psi}\llbracket\mathcal{G}\rrbracket$.

######  Proposition 3.1.

The ring homomorphism $\mathcal{O}\llbracket\mathcal{G}\rrbracket\rightarrow
R^{\mathrm{ps},\psi}\llbracket\mathcal{G}\rrbracket/J$ is surjective.

###### Proof.

Let $\overline{R}$ and $C$ be the images of $R^{\mathrm{ps},\psi}$ and
$\mathcal{O}\llbracket\mathcal{G}\rrbracket$ in
$R^{\mathrm{ps},\psi}\llbracket\mathcal{G}\rrbracket/J$, respectively. Since
$R^{\mathrm{ps},\psi}$, $\mathcal{O}\llbracket\mathcal{G}\rrbracket$ and
$R^{\mathrm{ps},\psi}\llbracket\mathcal{G}\rrbracket/J$ are pseudo-compact
$\mathcal{O}$-modules $\overline{R}$ and $C$ are closed subrings of
$R^{\mathrm{ps},\psi}\llbracket\mathcal{G}\rrbracket/J$. It is enough to show
that $C$ contains $\overline{R}$, since in this case we deduce that $C$
contains the image of $R^{\mathrm{ps},\psi}\llbracket\mathcal{G}\rrbracket$,
which is equal to $R^{\mathrm{ps},\psi}\llbracket\mathcal{G}\rrbracket/J$.

Let $B=\overline{R}\cap C$ and let $\mathfrak{m}_{B}$ be the intersection of
$B$ with the maximal ideal of $\overline{R}$. Then $B$ is a closed subring of
$\overline{R}$. This implies that $B$ is complete for the profinite topology.
If $x\in\mathfrak{m}_{B}$ then $1+x$ has an inverse in $B$ given by the
geometric series. Since $B$ is an $\mathcal{O}$-algebra and the residue field
of $\overline{R}$ is $k$, we conclude that $(B,\mathfrak{m}_{B})$ is a local
ring with residue field $k$.

Let $\overline{T}$ be the specialisation of $T$ along
$R^{\mathrm{ps},\psi}\twoheadrightarrow\overline{R}$. The relation
$g^{2}-\overline{T}(g)g+\psi(g)=0$ in
$R^{\mathrm{ps},\psi}\llbracket\mathcal{G}\rrbracket/J$ implies that
$\overline{T}(g)=g+g^{-1}\psi(g)$. Thus $\overline{T}$ takes values in $B$.
The universal property of $R^{\mathrm{ps},\psi}$ implies that there is a
continuous homomorphism of $\mathcal{O}$-algebras
$\varphi:R^{\mathrm{ps},\psi}\rightarrow B$, such that
$\varphi(T(g))=\overline{T}(g)$ for all $g\in\mathcal{G}$. Using the universal
property of $R^{\mathrm{ps},\psi}$ again, we conclude that if we compose
$\varphi$ with the inclusion $B\subset\overline{R}$ we get back the surjection
$R^{\mathrm{ps},\psi}\twoheadrightarrow\overline{R}$ that we have started
with. Thus $B=\overline{R}$. ∎

######  Remark 3.2.

The proposition does not hold if we do not fix the determinant or consider
representations $\bar{\rho}$ of dimension bigger than $2$. Counterexamples may
be obtained with $\mathcal{G}=\mathbb{Z}_{p}$ and $\bar{\rho}$ trivial
representation of $\mathcal{G}$ on an $n$-dimensional $k$-vector space, using
[18, Ex. 1.7(i), 1.11(i)].

##  4\. Representations of $\operatorname{GL}_{2}(\mathbb{Q}_{p})$

Let $G$ be a $p$-adic analytic group and let $Z$ be its center. We let
$\operatorname{Mod}^{\mathrm{sm}}_{G}(\mathcal{O})$ be the category of smooth
representations of $G$ on $\mathcal{O}$-torsion modules. Pontryagin duality,
$\pi\mapsto\pi^{\vee}:=\operatorname{Hom}_{\mathcal{O}}(\pi,L/\mathcal{O})$
equipped with the compact open topology, induces an antiequivalence of
categories between $\operatorname{Mod}^{\mathrm{sm}}_{G}(\mathcal{O})$ and the
category $\operatorname{Mod}^{\mathrm{pro}}_{G}(\mathcal{O})$ of linearly
compact $\mathcal{O}$-modules with a continuous $G$-action [28, Lemma 2.2.7].
The inverse is given by $M\mapsto
M^{\vee}:=\operatorname{Hom}^{\mathrm{cont}}_{\mathcal{O}}(M,L/\mathcal{O})$.
In particular, if $G$ is compact then
$\operatorname{Mod}^{\mathrm{pro}}_{G}(\mathcal{O})$ is the category of
linearly compact $\mathcal{O}\llbracket G\rrbracket$-modules, where
$\mathcal{O}\llbracket G\rrbracket$ is the completed group algebra. We define
$\operatorname{Mod}^{\mathrm{sm}}_{G}(k)$ and
$\operatorname{Mod}^{\mathrm{pro}}_{G}(k)$ the same way with $\mathcal{O}$
replaced by $k$. Moreover for a continuous character
$\zeta:Z\rightarrow\mathcal{O}^{\times}$, adding the subscript $\zeta$ in any
of the above categories indicates the corresponding full subcategory of
$G$-representations on which $Z$ acts by $\zeta$. Denote
$\operatorname{Mod}^{\mathrm{l.fin}}_{G,\zeta}(\mathcal{O})$ the full
subcategory of $\operatorname{Mod}^{\mathrm{sm}}_{G,\zeta}(\mathcal{O})$
consisting of representations in
$\operatorname{Mod}^{\mathrm{sm}}_{G,\zeta}(\mathcal{O})$ which are equal to
the union of their subrepresentations of finite length. We let
$\mathfrak{C}(\mathcal{O})$ be the full subcategory of
$\operatorname{Mod}^{\mathrm{pro}}_{G}(\mathcal{O})$ antiequivalent to
$\operatorname{Mod}^{\mathrm{l.fin}}_{G,\zeta}(\mathcal{O})$ under the
Pontryagin duality.

###  4.1. Blocks

From now on, we will assume $G=\operatorname{GL}_{2}(\mathbb{Q}_{p})$. Every
irreducible object $\pi$ of
$\operatorname{Mod}^{\mathrm{sm}}_{G}(\mathcal{O})$ is killed by $\varpi$ and
hence is an object of $\operatorname{Mod}^{\mathrm{sm}}_{G}(k)$.

Let $\operatorname{Irr}_{G,\zeta}$ be the set of irreducible representations
in $\operatorname{Mod}^{\mathrm{sm}}_{G,\zeta}(k)$. We write
$\pi\leftrightarrow\pi^{\prime}$ if $\pi\cong\pi^{\prime}$ or
$\operatorname{Ext}^{1}_{G,\zeta}(\pi,\pi^{\prime})\neq 0$ or
$\operatorname{Ext}^{1}_{G,\zeta}(\pi^{\prime},\pi)\neq 0$, where
$\operatorname{Ext}^{1}_{G,\zeta}(\pi,\pi^{\prime})$ is the Yoneda extension
group of $\pi^{\prime}$ by $\pi$ in
$\operatorname{Mod}^{\mathrm{sm}}_{G,\zeta}(k)$. We write
$\pi\sim\pi^{\prime}$ if there exists
$\pi_{1},\cdots,\pi_{n}\in\operatorname{Irr}_{G,\zeta}$ such that
$\pi\cong\pi_{1}$, $\pi^{\prime}\cong\pi_{n}$ and
$\pi_{i}\leftrightarrow\pi_{i+1}$ for $1\leq i\leq n-1$. The relation $\sim$
is an equivalence relation on $\operatorname{Irr}_{G,\zeta}$. A block is an
equivalence class of $\sim$.

Barthel-Livné [2] and Breuil [9] have classified the absolutely irreducible
smooth representations $\pi$ admitting a central character. The blocks
containing an absolutely irreducible representation have been determined in
[40, Corollary 6.2]. There are the following cases:

  1. (i)

$\mathfrak{B}=\\{\pi\\}$ with $\pi$ supersingular;

  2. (ii)

$\mathfrak{B}=\\{(\operatorname{Ind}_{B}^{G}\chi_{1}\otimes\chi_{2}\omega^{-1})_{\mathrm{sm}},(\operatorname{Ind}_{B}^{G}\chi_{2}\otimes\chi_{1}\omega^{-1})_{\mathrm{sm}}\\}$
with $\chi_{2}\chi_{1}^{-1}\neq\mathbf{1},\omega^{\pm 1}$;

  3. (iii)

$p>2$ and
$\mathfrak{B}=\\{(\operatorname{Ind}_{B}^{G}\chi\otimes\chi\omega^{-1})_{\mathrm{sm}}\\}$;

  4. (iv)

$p\geq 5$ and
$\mathfrak{B}=\\{\mathbf{1},\operatorname{Sp},(\operatorname{Ind}^{G}_{B}\omega\otimes\omega^{-1})_{\mathrm{sm}}\\}\otimes\chi\circ\det$;

  5. (v)

$p=3$ and
$\mathfrak{B}=\\{\mathbf{1},\operatorname{Sp},\omega\circ\det,\operatorname{Sp}\otimes\omega\circ\det\\}\otimes\chi\circ\det$;

  6. (vi)

$p=2$ and
$\mathfrak{B}=\\{\mathbf{1},\operatorname{Sp}\\}\otimes\chi\circ\det$;

where $\chi$, $\chi_{1}$, $\chi_{2}:\mathbb{Q}_{p}^{\times}\rightarrow
k^{\times}$ are smooth characters and
$\omega:\mathbb{Q}_{p}^{\times}\rightarrow k^{\times}$ is the character
$\omega(x)=x|x|\pmod{\varpi}$, and $\operatorname{Sp}$ is the Steinberg
representation defined by the exact sequence

$0\rightarrow\mathbf{1}\rightarrow(\operatorname{Ind}_{B}^{G}\mathbf{1})_{\mathrm{sm}}\rightarrow\operatorname{Sp}\rightarrow
0.$

If $\pi\in\operatorname{Irr}_{G,\zeta}$ is not absolutely irreducible then
there is a finite extension $k^{\prime}$ of $k$, such that
$\pi\otimes_{k}k^{\prime}$ is a finite direct sum of absolutely irreducible
representations, see [39, Proposition 5.11], so no information is lost by
working with absolutely irreducible representations.

Given a block $\mathfrak{B}$, we denote by $\pi_{\mathfrak{B}}$ the direct sum
of all representations in $\mathfrak{B}$ and let $P_{\mathfrak{B}}$ be a
projective envelope of $\pi_{\mathfrak{B}}^{\vee}$ in
$\mathfrak{C}(\mathcal{O})$. Then
$E_{\mathfrak{B}}=\operatorname{End}_{\mathfrak{C}(\mathcal{O})}(P_{\mathfrak{B}})$
is a pseudo-compact $\mathcal{O}$-algebra. We denote the centre of
$E_{\mathfrak{B}}$ by $Z_{\mathfrak{B}}$.

By [39, Corollary 5.35], the category $\mathfrak{C}(\mathcal{O})$ decomposes
into a direct product of subcategories

(11)
$\mathfrak{C}(\mathcal{O})\cong\prod_{\mathfrak{B}\in\operatorname{Irr}_{G,\zeta}/\sim}\mathfrak{C}(\mathcal{O})_{\mathfrak{B}},$

where the objects of $\mathfrak{C}(\mathcal{O})_{\mathfrak{B}}$ are those $M$
in $\mathfrak{C}(\mathcal{O})$ such that for every irreducible subquotient $S$
of $M$, $S^{\vee}$ lies in $\mathfrak{B}$. Moreover, the category
$\mathfrak{C}(\mathcal{O})_{\mathfrak{B}}$ is equivalent to the category of
compact right $E_{\mathfrak{B}}$-modules and the centre of
$\mathfrak{C}(\mathcal{O})_{\mathfrak{B}}$ is isomorphic to $Z_{\mathfrak{B}}$
[39, Proposition 5.45].

######  Lemma 4.1.

If $\mathfrak{B}$ contains an absolutely irreducible representation then
$Z_{\mathfrak{B}}$ is a local pseudo-compact $\mathcal{O}$-algebra with
residue field $k$.

###### Proof.

If $\pi$ is absolutely irreducible then $\operatorname{End}_{G}(\pi)=k$ and
thus the action of $Z_{\mathfrak{B}}$ on $\pi$ defines a homomorphism of
$\mathcal{O}$-algebras $c_{\pi}:Z_{\mathfrak{B}}\rightarrow k$. If
$\pi,\pi^{\prime}\in\mathfrak{B}$ are distinct and there is a non-split
extension $0\rightarrow\pi\rightarrow\tau\rightarrow\pi^{\prime}\rightarrow 0$
then $\operatorname{End}_{G}(\tau)=k$ and we conclude that
$c_{\pi}=c_{\tau}=c_{\pi^{\prime}}$. Using the transitivity property of the
relation $\sim$ on $\operatorname{Irr}_{G,\zeta}$, we conclude that
$c_{\pi}=c_{\pi^{\prime}}$ for all $\pi,\pi^{\prime}\in\mathfrak{B}$. It
follows from the proof of [31, Proposition IV.4.12] that the Jacobson radical
of $Z_{\mathfrak{B}}$ consists of elements that kill all the irreducible
representations. Thus $\operatorname{Ker}c_{\pi}$ is the maximal ideal of
$Z_{\mathfrak{B}}$ with residue field $k$. The last assertion follows from the
fact that $Z_{\mathfrak{B}}$ is closed in $E_{\mathfrak{B}}$ and [31,
Proposition IV.4.13]. ∎

######  Lemma 4.2.

Let $P$ be projective in $\mathfrak{C}(\mathcal{O})$. Then
$P^{\operatorname{SL}_{2}(\mathbb{Q}_{p})}=0$.

###### Proof.

Let $J$ be the Pontryagin dual of $P$. Then $J$ is injective in
$\operatorname{Mod}^{\mathrm{l.fin}}_{G,\zeta}(\mathcal{O})$ and the assertion
is equivalent to $J_{\operatorname{SL}_{2}(\mathbb{Q}_{p})}=0$. Let $N$ be the
unipotent subgroup $\left(\begin{smallmatrix}1&\mathbb{Q}_{p}\\\
0&1\end{smallmatrix}\right)$. Then by [29, Proposition 3.6.2] and [30,
Corollary 3.12], we have $J_{N}=0$. Thus
$J_{\operatorname{SL}_{2}(\mathbb{Q}_{p})}=0$ and the lemma follows.
Alternatively, one can deduce the statement from Proposition 4.3 below. ∎

######  Proposition 4.3.

Let $P$ and $M$ be in $\operatorname{\mathfrak{C}}(\mathcal{O})$, such that
$P$ is projective and $M/\varpi M$ is of finite length. Then
$\operatorname{Hom}_{\operatorname{\mathfrak{C}}(\mathcal{O})}(M,P)=0$.

###### Proof.

Let $K^{\prime}$ be a compact open pro-$p$ subgroup of
$\operatorname{SL}_{2}(\mathbb{Q}_{p})$ such that $K^{\prime}\cap Z=\\{1\\}$.
Then $P$ is projective in
$\operatorname{Mod}^{\mathrm{pro}}_{K^{\prime}}(\mathcal{O})$ by [30,
Corollary 3.10] and thus $P\cong\prod_{i\in I}\mathcal{O}\llbracket
K^{\prime}\rrbracket$ for some index set $I$. Thus it is enough to show that
$\operatorname{Hom}_{\mathcal{O}\llbracket
K^{\prime}\rrbracket}^{\mathrm{cont}}(M,\mathcal{O}\llbracket
K^{\prime}\rrbracket)=0$. Topological Nakayama’s lemma for compact
$\mathcal{O}$-modules implies that it is enough to show that
$\operatorname{Hom}_{k\llbracket
K^{\prime}\rrbracket}^{\mathrm{cont}}(M/\varpi M,k\llbracket
K^{\prime}\rrbracket)=0$. Since $M/\varpi M$ is of finite length, it is enough
to show that $\operatorname{Hom}_{k\llbracket
K^{\prime}\rrbracket}(\pi^{\vee},k\llbracket K^{\prime}\rrbracket)=0$ for
every $\pi\in\mathfrak{B}$. This follows from [42, Lemma 5.16]. Note that if
$K_{n}=1+M_{2}(2p^{n}\mathbb{Z}_{p})$ and
$K_{n}^{\prime}=K_{n}\cap\operatorname{SL}_{2}(\mathbb{Q}_{p})$ then
$K_{n}=K_{n}^{\prime}(Z\cap K_{n})$ and so $\pi^{K_{n}}=\pi^{K^{\prime}_{n}}$
as $Z\cap K_{n}$ acts trivially on $\pi$, and so the argument in [42, Lemma
5.16] carries over to the restriction of $\pi$ to
$\operatorname{SL}_{2}(\mathbb{Q}_{p})$. ∎

We will denote by $\operatorname{Ord}_{B}$ Emerton’s functor of ordinary
parts, [28].

######  Lemma 4.4.

Let $\pi\hookrightarrow J$ be an injective envelope of $\pi\in\mathfrak{B}$ in
$\operatorname{Mod}^{\mathrm{l.fin}}_{G,\zeta}(\mathcal{O})$. If $\pi$ is
supersingular then $\operatorname{Ord}_{B}J=0$. If $\pi$ is an irreducible
subquotient of $(\operatorname{Ind}_{\overline{B}}^{G}\chi)_{\mathrm{sm}}$ for
some character $\chi:T\rightarrow k^{\times}$, where $\overline{B}$ is the
subgroup of lower triangular matrices, then $\operatorname{Ord}_{B}J$ is
isomorphic to an injective envelope of $\chi$ in
$\operatorname{Mod}^{\mathrm{l.fin}}_{T,\zeta}(\mathcal{O})$, and also in
$\operatorname{Mod}^{\mathrm{sm}}_{T,\zeta}(\mathcal{O})$.

###### Proof.

Since $J$ is injective and $\operatorname{Ord}_{B}$ is adjoint to parabolic
induction, which is an exact functor, $\operatorname{Ord}_{B}J$ is injective
in $\operatorname{Mod}^{\mathrm{l.fin}}_{T,\zeta}(\mathcal{O})$. Thus it is a
direct sum of injective envelopes of characters of $T$. Moreover,

(12)
$\operatorname{Hom}_{T}(\chi,\operatorname{Ord}_{B}J)\cong\operatorname{Hom}_{G}((\operatorname{Ind}_{\overline{B}}^{G}\chi)_{\mathrm{sm}},J).$

Since $J$ is an injective envelope of $\pi$, this group is non-zero if and
only if $\pi$ is a subquotient of
$(\operatorname{Ind}_{\overline{B}}^{G}\chi)_{\mathrm{sm}}$, in which case the
dimension of the spaces in (12) is equal to the multiplicity with which $\pi$
occurs as a subquotient of
$(\operatorname{Ind}_{\overline{B}}^{G}\chi)_{\mathrm{sm}}$. If $\pi$ is
supersingular then it does not occur as a subquotient of principal series and
thus $\operatorname{Ord}_{B}J=0$. Otherwise, it follows from [2] that there is
a unique character $\chi$, such that the multiplicity is non-zero, in which
case it is equal to $1$. This proves the first assertion. The same (albeit
easier) proof as in [39, Proposition 5.16] implies that every injective object
in $\operatorname{Mod}^{\mathrm{l.fin}}_{T,\zeta}(\mathcal{O})$ is also
injective in $\operatorname{Mod}^{\mathrm{sm}}_{T,\zeta}(\mathcal{O})$. ∎

######  Lemma 4.5.

Let $\pi\hookrightarrow J$ be an injective envelope of $\pi\in\mathfrak{B}$ in
$\operatorname{Mod}^{\mathrm{l.fin}}_{G,\zeta}(\mathcal{O})$. If $\pi$ is not
a character then $J^{\operatorname{SL}_{2}(\mathbb{Q}_{p})}=0$, otherwise
$(J^{\operatorname{SL}_{2}(\mathbb{Q}_{p})})^{\vee}$ is nonzero and finitely
generated over $\mathcal{O}$.

###### Proof.

If $J^{\operatorname{SL}_{2}(\mathbb{Q}_{p})}$ is non-zero then $\pi\cap
J^{\operatorname{SL}_{2}(\mathbb{Q}_{p})}$ is non-zero as $\pi\hookrightarrow
J$ is essential. Since $\pi$ is absolutely irreducible, we deduce that $\pi$
is a character and $\pi$ is the $G$-socle of
$J^{\operatorname{SL}_{2}(\mathbb{Q}_{p})}$. Note that the action of $G$ on
$J^{\operatorname{SL}_{2}(\mathbb{Q}_{p})}$ factors through

$G/Z\operatorname{SL}_{2}(\mathbb{Q}_{p})\cong\mathbb{Q}_{p}^{\times}/(\mathbb{Q}_{p}^{\times})^{2}\cong\begin{cases}\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/2\mathbb{Z}&\text{if
}p>2\\\
\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/2\mathbb{Z}&\text{if
}p=2.\end{cases}$

It follows that $J^{\operatorname{SL}_{2}(\mathbb{Q}_{p})}[\varpi]$ is a
finite dimensional $k$-vector space. Dually this implies that
$(J^{\operatorname{SL}_{2}(\mathbb{Q}_{p})})^{\vee}/\varpi(J^{\operatorname{SL}_{2}(\mathbb{Q}_{p})})^{\vee}$
is finite-dimensional over $k$ and the lemma follows from Nakayama’s lemma. ∎

######  Lemma 4.6.

Let $J$ be injective in
$\operatorname{Mod}^{\mathrm{l.fin}}_{G,\zeta}(\mathcal{O})$ and
$\kappa\in\operatorname{Mod}^{\mathrm{l.fin}}_{T,\zeta}(\mathcal{O})$ be such
that $\kappa^{\vee}$ is a finitely generated $\mathcal{O}$-module. Then
$\operatorname{Hom}_{T}(\operatorname{Ord}_{B}J,\kappa)=0$.

###### Proof.

The following is an analogue of Proposition 4.3 in an easier setting. Let
$T_{0}$ to be an open torsion-free pro-$p$ subgroup of
$T\cap\operatorname{SL}_{2}(\mathbb{Q}_{p})$. Since $\operatorname{Ord}_{B}J$
is injective in $\operatorname{Mod}^{\mathrm{sm}}_{T,\zeta}(\mathcal{O})$ by
Lemma 4.4 it is injective in
$\operatorname{Mod}^{\mathrm{sm}}_{T_{0}}(\mathcal{O})$, as restriction to
$T_{0}$ is adjoint to $\operatorname{c-Ind}_{ZT_{0}}^{T}$, which is exact.
Thus the Pontryagin dual of $\operatorname{Ord}_{B}J$ is isomorphic to
$\prod_{i\in I}\mathcal{O}\llbracket T_{0}\rrbracket$ for some index set $I$.
Hence, it suffices to show that

$\operatorname{Hom}^{\mathrm{cont}}_{\mathcal{O}\llbracket
T_{0}\rrbracket}(\kappa^{\vee},\mathcal{O}\llbracket T_{0}\rrbracket)=0.$

This is clear as $\mathcal{O}\llbracket T_{0}\rrbracket$ is isomorphic to the
ring of formal power series in one variable and $\kappa^{\vee}$ is a finitely
generated $\mathcal{O}$-module. ∎

######  Lemma 4.7.

Let $\tau$ be a smooth representation of $G$ whose irreducible subquotients
consists of characters in $\mathfrak{B}$. Then
$\operatorname{SL}_{2}(\mathbb{Q}_{p})$ acts trivially on $\tau$ and there is
an exact sequence

(13)
$0\rightarrow\tau\rightarrow(\operatorname{Ind}^{G}_{B}\tau)_{\mathrm{sm}}\rightarrow
Q\rightarrow 0.$

Moreover, the irreducible subquotients of $Q$ are twists of
$\operatorname{Sp}$ by a character. In particular,
$\operatorname{Hom}_{G}(\tau,Q)=0$.

###### Proof.

Since $\operatorname{SL}_{2}(\mathbb{Q}_{p})$ acts trivially on $\tau$ by [47,
Lemma 1.2.1], the unipotent radical of $B$ acts trivially on $\tau$. Thus
$(\operatorname{Ind}^{G}_{B}\tau|_{B})_{\mathrm{sm}}$ coincides with the
parabolic induction. Since the map
$\tau\rightarrow(\operatorname{Ind}^{G}_{B}\tau)_{\mathrm{sm}}$ defined by
$v\mapsto(g\mapsto gv)$ is $G$-equivariant and injective, we obtain the first
assertion.

To show the second assertion, we choose an increasing and exhaustive
filtration $\\{R^{j}\\}_{j\geq 0}$ of $\tau$ such that $R^{j}/R^{j-1}$ is a
character for each $j$. Then we have
$R^{j}/R^{j-1}\hookrightarrow(\operatorname{Ind}^{G}_{B}R^{j}/R^{j-1})_{\mathrm{sm}}$
with quotient isomorphic to a twist of $\operatorname{Sp}$ by a character.
Thus the second assertion follows from the exactness of
$(\operatorname{Ind}^{G}_{B}-)_{\mathrm{sm}}$. ∎

######  Lemma 4.8.

Let $J$ be injective in
$\operatorname{Mod}^{\mathrm{l.fin}}_{G,\zeta}(\mathcal{O})$ and let $\tau$
and $Q$ be as in Lemma 4.7. If $\tau^{\vee}$ is finitely generated over
$\mathcal{O}$ then

$\operatorname{Hom}_{G}(J,\tau)=\operatorname{Hom}_{G}(J,(\operatorname{Ind}^{G}_{B}\tau)_{\mathrm{sm}})=\operatorname{Hom}_{G}(J,Q)=0.$

###### Proof.

Since $\tau^{\vee}$ is finitely generated over $\mathcal{O}$, $\tau[\varpi]$
is a finite $k$-vector space, thus

$(\operatorname{Ind}^{G}_{B}\tau)_{\mathrm{sm}}^{\vee}/\varpi\cong(\operatorname{Ind}^{G}_{B}\tau[\varpi])_{\mathrm{sm}}^{\vee}$

is of finite length in $\operatorname{\mathfrak{C}}(\mathcal{O})$ and the
assertion follows from Proposition 4.3. ∎

######  Proposition 4.9.

Let $J$ and $\tau$ be as in Lemma 4.8. Then
$\operatorname{Ext}^{1}_{G,\zeta}(J,\tau)=0$.

###### Proof.

Consider an exact sequence $0\rightarrow\tau\rightarrow I\rightarrow
J\rightarrow 0$. Applying $\operatorname{Ord}_{B}$ to it, we get an exact
sequence

$\displaystyle 0$
$\displaystyle\rightarrow\operatorname{Ord}_{B}\tau\rightarrow\operatorname{Ord}_{B}I\rightarrow\operatorname{Ord}_{B}J\rightarrow\mathbb{R}^{1}\operatorname{Ord}_{B}\tau\rightarrow\mathbb{R}^{1}\operatorname{Ord}_{B}I\rightarrow\mathbb{R}^{1}\operatorname{Ord}_{B}J$

of smooth $T$-representations. It is proved in [30] that the functors
$H^{i}\operatorname{Ord}_{B}$ in [29] coincide with the derived functors
$\mathbb{R}^{i}\operatorname{Ord}_{B}$. Moreover,
$H^{1}\operatorname{Ord}_{B}$ coincides with the $N$-coinvariants twisted by
the character $\alpha^{-1}$, where $\alpha(\bigl{(}\begin{smallmatrix}a&0\\\
0&d\end{smallmatrix}\bigr{)})=\omega(ad^{-1})$, by [29, Proposition 3.6.2].

Since $\operatorname{SL}_{2}(\mathbb{Q}_{p})$ acts trivially on $\tau$, we
have $\operatorname{Ord}_{B}\tau=0$ and
$\mathbb{R}^{1}\operatorname{Ord}_{B}\tau\cong\tau\otimes\alpha^{-1}$. Thus
the above exact sequence reduces to

(14)
$0\rightarrow\operatorname{Ord}_{B}I\rightarrow\operatorname{Ord}_{B}J\rightarrow\tau\otimes\alpha^{-1}\rightarrow
I_{N}\otimes\alpha^{-1}\rightarrow 0.$

Since the middle map is zero by Lemma 4.6, the map $\tau\hookrightarrow I$
induces an isomorphism $\tau\cong I_{N}$ of $T$-representations and hence

(15)
$\operatorname{Hom}_{G}(I,(\operatorname{Ind}^{G}_{B}\tau)_{\mathrm{sm}})\cong\operatorname{Hom}_{G}(\tau,(\operatorname{Ind}^{G}_{B}\tau)_{\mathrm{sm}}).$

by adjunction formula.

It follows from Lemma 4.8 that the first $3$ terms in the long exact sequence
obtained by applying $\operatorname{Hom}_{G}(J,-)$ to (13) are zero. Thus by
applying
$\operatorname{Hom}_{G}(I,-)\rightarrow\operatorname{Hom}_{G}(\tau,-)$ to (13)
we obtain the following commutative diagram:

${\operatorname{Hom}_{G}(I,\tau)}$${\operatorname{Hom}_{G}(\tau,\tau)}$${\operatorname{Hom}_{G}(I,(\operatorname{Ind}^{G}_{B}\tau)_{\mathrm{sm}})}$${\operatorname{Hom}_{G}(\tau,(\operatorname{Ind}^{G}_{B}\tau)_{\mathrm{sm}})}$${\operatorname{Hom}_{G}(I,Q)}$${\operatorname{Hom}_{G}(\tau,Q)}$$\scriptstyle{\sim}$$\scriptstyle{\eqref{equa:Frob}}$

with exact columns. Last part of Lemma 4.7 says that
$\operatorname{Hom}_{G}(\tau,Q)=0$ and hence $\operatorname{Hom}_{G}(I,Q)=0$.
Thus all the maps in the top square of the diagram are isomorphisms. The
preimage of $\operatorname{id}_{\tau}\in\operatorname{Hom}_{G}(\tau,\tau)$ in
$\operatorname{Hom}_{G}(I,\tau)$ splits the exact sequence
$0\rightarrow\tau\rightarrow I\rightarrow J\rightarrow 0$. This proves the
proposition. ∎

###  4.2. Quotient category

Let $\mathfrak{T}(\mathcal{O})$ be the full subcategory of
$\mathfrak{C}(\mathcal{O})$ whose objects have trivial
$\operatorname{SL}_{2}(\mathbb{Q}_{p})$-action. By [39, Lemma 10.25] (for
$p>2$) and [47, Lemma 1.2.1] (for $p=2$), $\mathfrak{T}(\mathcal{O})$ is a
thick subcategory of $\mathfrak{C}(\mathcal{O})$ and hence we may consider the
quotient category
$\mathfrak{Q}(\mathcal{O}):=\mathfrak{C}(\mathcal{O})/\mathfrak{T}(\mathcal{O})$.
Let
$\mathcal{T}:\mathfrak{C}(\mathcal{O})\rightarrow\mathfrak{Q}(\mathcal{O})$ be
the quotient functor; we note that $\mathcal{T}$ is the identity on objects.
It is shown in [39, §10.3] that $\mathfrak{Q}(\mathcal{O})$ is an abelian
category with enough projectives and $\mathcal{T}$ is an exact functor.

For a block $\mathfrak{B}$, we denote

$P_{\mathfrak{B}}^{\prime}=\bigoplus_{\begin{subarray}{c}\pi\in\mathfrak{B}\\\
\pi^{\operatorname{SL}_{2}(\mathbb{Q}_{p})}=0\end{subarray}}P_{\pi^{\vee}},\quad
E_{\mathfrak{B}}^{\prime}=\operatorname{End}_{\mathfrak{C}(\mathcal{O})}(P_{\mathfrak{B}}^{\prime}),\quad
Z_{\mathfrak{B}}^{\prime}=Z(E_{\mathfrak{B}}^{\prime}),$

where $P_{\pi^{\vee}}$ is a projective envelope of $\pi^{\vee}$ in
$\mathfrak{C}(\mathcal{O})$.

######  Proposition 4.10.

Let $\mathfrak{B}$ be a block.

  1. (1)

$\mathcal{T}P_{\mathfrak{B}}^{\prime}$ is a projective object of
$\mathfrak{Q}(\mathcal{O})$ and
$E_{\mathfrak{B}}^{\prime}\cong\operatorname{End}_{\mathfrak{Q}(\mathcal{O})}(\mathcal{T}P_{\mathfrak{B}}^{\prime})$.

  2. (2)

The functor
$M\mapsto\operatorname{Hom}_{\mathfrak{Q}(\mathcal{O})}(\mathcal{T}P_{\mathfrak{B}}^{\prime},M)$
defines an equivalence of categories between
$\mathfrak{Q}(\mathcal{O})_{\mathfrak{B}}$ and the category of pseudo-compact
right $E_{\mathfrak{B}}^{\prime}$-modules, with the inverse given by
$\mathrm{m}\mapsto\mathrm{m}\operatorname{\widehat{\otimes}}_{E_{\mathfrak{B}}^{\prime}}\mathcal{T}P_{\mathfrak{B}}^{\prime}$.

###### Proof.

See [39, Lemma 10.27] for the first assertion and [31, §IV.4 Theorem 4,
Corollary 1, Corollary 5] for the second assertion. ∎

###  4.3. The centre

In this section we will prove key results towards showing that
$E_{\mathfrak{B}}$ and $Z_{\mathfrak{B}}$ are finite over
$Z_{\mathfrak{B}}^{\prime}$. If $\mathfrak{B}$ is of type (i), (ii) or (iii)
then we have $P_{\mathfrak{B}}=P_{\mathfrak{B}}^{\prime}$ and thus
$Z_{\mathfrak{B}}=Z_{\mathfrak{B}}^{\prime}$.

######  Lemma 4.11.

Let $E$ be a ring with centre $Z$ and let $\mathrm{m}$ be a finitely generated
(right) $E$-module. Assume that $Z$ is noetherian and $E$ is (module) finite
over $Z$. Then $\operatorname{End}_{E}(\mathrm{m})$ and its centre
$Z(\operatorname{End}_{E}(\mathrm{m}))$ are noetherian and finite over $Z$.

###### Proof.

Since $\mathrm{m}$ is finitely generated over $E$, there is a surjection
$E^{\oplus n}\twoheadrightarrow\mathrm{m}$ for some $n$. This induces an
injection
$\operatorname{End}_{E}(\mathrm{m})\hookrightarrow\operatorname{Hom}_{E}(E^{\oplus
n},\mathrm{m})$ and a surjection
$M_{n}(E^{\mathrm{op}})\cong\operatorname{End}_{E}(E^{\oplus
n})\twoheadrightarrow\operatorname{Hom}_{E}(E^{\oplus n},\mathrm{m})$ of
$Z$-modules. Since $M_{n}(E^{\mathrm{op}})$ is finite over $E^{\mathrm{op}}$,
it is finite and noetherian over $Z$. It follows that
$\operatorname{Hom}_{E}(E^{\oplus n},\mathrm{m})$ and thus
$\operatorname{End}_{E}(\mathrm{m})$, which can be identified with a
$Z$-submodule of $\operatorname{Hom}_{E}(E^{\oplus n},\mathrm{m})$, are finite
and noetherian over $Z$. This proves the lemma. ∎

######  Lemma 4.12.

Let
$M_{\mathfrak{B}}=\ker(P_{\mathfrak{B}}\rightarrow(P_{\mathfrak{B}})_{\operatorname{SL}_{2}(\mathbb{Q}_{p})})$.
Then the following hold:

  1. (1)

$(M_{\mathfrak{B}})_{\operatorname{SL}_{2}(\mathbb{Q}_{p})}=0$.

  2. (2)

For all $\pi\in\mathfrak{B}$,
$\operatorname{Hom}_{\mathfrak{C}(\mathcal{O})}(M_{\mathfrak{B}},\pi^{\vee})$
is finite dimensional over $k$.

###### Proof.

It suffices to consider the block $\mathfrak{B}$ of type $(iv)$, $(v)$ and
$(vi)$, otherwise $M_{\mathfrak{B}}=P_{\mathfrak{B}}$ and the assertion is
trivial. Let $J_{\mathfrak{B}}$ be the Pontryagin dual of $P_{\mathfrak{B}}$.
Then we have an exact sequence $0\rightarrow
J_{\mathfrak{B}}^{\operatorname{SL}_{2}(\mathbb{Q}_{p})}\rightarrow
J_{\mathfrak{B}}\rightarrow M_{\mathfrak{B}}^{\vee}\rightarrow 0$. By applying
$\operatorname{Hom}_{\operatorname{GL}_{2}(\mathbb{Q}_{p})}(\pi,-)$, we get a
long exact sequence

$\displaystyle 0$
$\displaystyle\rightarrow\operatorname{Hom}_{G}(\pi,J_{\mathfrak{B}}^{\operatorname{SL}_{2}(\mathbb{Q}_{p})})\rightarrow\operatorname{Hom}_{G}(\pi,J_{\mathfrak{B}})\rightarrow\operatorname{Hom}_{G}(\pi,M_{\mathfrak{B}}^{\vee})$
$\displaystyle\rightarrow\operatorname{Ext}^{1}_{G,\zeta}(\pi,J_{\mathfrak{B}}^{\operatorname{SL}_{2}(\mathbb{Q}_{p})})\rightarrow\operatorname{Ext}^{1}_{G,\zeta}(\pi,J_{\mathfrak{B}})=0,$

where $\operatorname{Ext}^{1}_{G,\zeta}$ is the extension group in
$\operatorname{Mod}^{\mathrm{l.fin}}_{G,\zeta}(\mathcal{O})$.

If $\pi\in\mathfrak{B}$ is a character, then both
$\operatorname{Hom}_{G}(\pi,J_{\mathfrak{B}}^{\operatorname{SL}_{2}(\mathbb{Q}_{p})})$
and $\operatorname{Hom}_{G}(\pi,J_{\mathfrak{B}})$ are $1$-dimensional over
$k$, and
$\operatorname{Ext}^{1}_{G,\zeta}(\pi,J_{\mathfrak{B}}^{\operatorname{SL}_{2}(\mathbb{Q}_{p})})=0$
by [47, Lemma 1.2.1]. Thus
$\operatorname{Hom}_{G}(\pi,M_{\mathfrak{B}}^{\vee})=0$ and the first
assertion follows (see [20, Lemma III.40] for another proof).

If $\pi\in\mathfrak{B}$ is not a character, then
$\operatorname{Hom}_{G}(\pi,J_{\mathfrak{B}}^{\operatorname{SL}_{2}(\mathbb{Q}_{p})})=0$
and $\operatorname{Hom}_{G}(\pi,J_{\mathfrak{B}})$ is $1$-dimensional over
$k$. Since $\pi$ is killed by $\varpi$, for the second assertion it is enough
to show that
$\operatorname{Ext}^{1}_{G,\zeta}(\pi,(J_{\mathfrak{B}}[\varpi])^{\operatorname{SL}_{2}(\mathbb{Q}_{p})})$
is finite dimensional over $k$. This holds, since
$(J_{\mathfrak{B}}[\varpi])^{\operatorname{SL}_{2}(\mathbb{Q}_{p})}$ is of
finite length by Lemma 4.5 and if $\chi$ is a character then
$\operatorname{Ext}^{1}_{G,\zeta}(\pi,\chi\circ\det)$ is finite dimensional. ∎

######  Corollary 4.13.

There is a surjection $(P_{\mathfrak{B}}^{\prime})^{\oplus
n}\twoheadrightarrow M_{\mathfrak{B}}$ for some $n\in\mathbb{N}$.

###### Proof.

This follows from Lemma 4.12, which shows that the cosocle of
$M_{\mathfrak{B}}$ contains no characters and each irreducible representation
in $\mathfrak{B}$ appears with finite multiplicity. ∎

######  Lemma 4.14.

Both
$\operatorname{Hom}_{\mathfrak{C}(\mathcal{O})}(P_{\mathfrak{B}}/M_{\mathfrak{B}},P_{\mathfrak{B}})$
and
$\operatorname{Ext}^{1}_{\mathfrak{C}(\mathcal{O})}(P_{\mathfrak{B}}/M_{\mathfrak{B}},P_{\mathfrak{B}})$
are equal to zero.

###### Proof.

The first assertion follows since $\operatorname{SL}_{2}(\mathbb{Q}_{p})$ acts
trivially on $P_{\mathfrak{B}}/M_{\mathfrak{B}}$ and
$P_{\mathfrak{B}}^{\operatorname{SL}_{2}(\mathbb{Q}_{p})}=0$ (see Lemma 4.2),
and the second assertion follows from Proposition 4.9 applied to
$J=P_{\mathfrak{B}}^{\vee}$ and
$\tau=J_{\mathfrak{B}}^{\operatorname{SL}_{2}(\mathbb{Q}_{p})}$. It follows
from Lemma 4.5 that $\tau^{\vee}$ is a finitely generated
$\mathcal{O}$-module. ∎

######  Proposition 4.15.

There is a natural isomorphism
$E_{\mathfrak{B}}\cong\operatorname{End}_{\mathfrak{C}(\mathcal{O})}(M_{\mathfrak{B}})$.
In particular, $Z_{\mathfrak{B}}\cong
Z(\operatorname{End}_{\mathfrak{C}(\mathcal{O})}(M_{\mathfrak{B}}))$.

###### Proof.

Let $\phi\in
E_{\mathfrak{B}}=\operatorname{End}_{\mathfrak{C}(\mathcal{O})}(P_{\mathfrak{B}})$.
Then the composition
$M_{\mathfrak{B}}\xrightarrow{\phi}P_{\mathfrak{B}}\twoheadrightarrow
P_{\mathfrak{B}}/M_{\mathfrak{B}}$ is the zero map since
$\operatorname{SL}_{2}(\mathbb{Q}_{p})$ acts trivially on
$P_{\mathfrak{B}}/M_{\mathfrak{B}}\cong(P_{\mathfrak{B}})_{\operatorname{SL}_{2}(\mathbb{Q}_{p})}$
and $(M_{\mathfrak{B}})_{\operatorname{SL}(\mathbb{Q}_{p})}=0$ by Lemma 4.12.
Thus $\phi$ maps $M_{\mathfrak{B}}$ to $M_{\mathfrak{B}}$ and restriction to
$M_{\mathfrak{B}}$ induces a ring homomorphism
$E_{\mathfrak{B}}\rightarrow\operatorname{End}_{\mathfrak{C}(\mathcal{O})}(M_{\mathfrak{B}})$.

Applying the functor
$\operatorname{Hom}_{\mathfrak{C}(\mathcal{O})}(M_{\mathfrak{B}},-)$ to the
exact sequence $0\rightarrow M_{\mathfrak{B}}\rightarrow
P_{\mathfrak{B}}\rightarrow P_{\mathfrak{B}}/M_{\mathfrak{B}}\rightarrow 0$,
we get the exact sequence

$0\rightarrow\operatorname{End}_{\mathfrak{C}(\mathcal{O})}(M_{\mathfrak{B}})\rightarrow\operatorname{Hom}_{\mathfrak{C}(\mathcal{O})}(M_{\mathfrak{B}},P_{\mathfrak{B}})\rightarrow\operatorname{Hom}_{\mathfrak{C}(\mathcal{O})}(M_{\mathfrak{B}},P_{\mathfrak{B}}/M_{\mathfrak{B}}).$

Since the last term is equal to zero by Lemma 4.12, we obtain
$\operatorname{End}_{\mathfrak{C}(\mathcal{O})}(M_{\mathfrak{B}})\cong\operatorname{Hom}_{\mathfrak{C}(\mathcal{O})}(M_{\mathfrak{B}},P_{\mathfrak{B}})$.
On the other hand, by applying the functor
$\operatorname{Hom}_{\mathfrak{C}(\mathcal{O})}(-,P_{\mathfrak{B}})$ to the
same short exact sequence, we get the exact sequence

$\displaystyle 0$
$\displaystyle\rightarrow\operatorname{Hom}_{\mathfrak{C}(\mathcal{O})}(P_{\mathfrak{B}}/M_{\mathfrak{B}},P_{\mathfrak{B}})\rightarrow\operatorname{End}_{\mathfrak{C}(\mathcal{O})}(P_{\mathfrak{B}})\rightarrow\operatorname{Hom}_{\mathfrak{C}(\mathcal{O})}(M_{\mathfrak{B}},P_{\mathfrak{B}})$
$\displaystyle\rightarrow\operatorname{Ext}^{1}_{\mathfrak{C}(\mathcal{O})}(P_{\mathfrak{B}}/M_{\mathfrak{B}},P_{\mathfrak{B}}).$

Since
$\operatorname{Hom}_{\mathfrak{C}(\mathcal{O})}(P_{\mathfrak{B}}/M_{\mathfrak{B}},P_{\mathfrak{B}})=0$
and
$\operatorname{Ext}^{1}_{\mathfrak{C}(\mathcal{O})}(P_{\mathfrak{B}}/M_{\mathfrak{B}},P_{\mathfrak{B}})=0$
by Lemma 4.14, we deduce
$\operatorname{End}_{\mathfrak{C}(\mathcal{O})}(P_{\mathfrak{B}})\cong\operatorname{Hom}_{\mathfrak{C}(\mathcal{O})}(M_{\mathfrak{B}},P_{\mathfrak{B}})$
and the proposition follows. ∎

######  Corollary 4.16.

There is a natural surjective homomorphism

$Z_{\mathfrak{B}}^{\prime}\rightarrow
Z(\operatorname{End}_{\mathfrak{C}(\mathcal{O})}(M_{\mathfrak{B}}))\cong
Z_{\mathfrak{B}}.$

###### Proof.

Note that we have
$(M_{\mathfrak{B}})^{\operatorname{SL}_{2}(\mathbb{Q}_{p})}=0$ by Lemma 4.2
and $(M_{\mathfrak{B}})_{\operatorname{SL}_{2}(\mathbb{Q}_{p})}=0$ by Lemma
4.12. It follows from [39, Lemma 10.26] that the functor $\mathcal{T}$ induces
an isomorphism

$\displaystyle\operatorname{End}_{\mathfrak{C}(\mathcal{O})}(M_{\mathfrak{B}})$
$\displaystyle\cong\operatorname{End}_{\mathfrak{Q}(\mathcal{O})}(\mathcal{T}M_{\mathfrak{B}}),\quad\varphi\mapsto\mathcal{T}\varphi.$

Since $Z_{\mathfrak{B}}^{\prime}$ is the centre of
$\mathfrak{Q}(\mathcal{O})_{\mathfrak{B}}$ it acts on
$\mathcal{T}M_{\mathfrak{B}}$ and this action induces a homomorphism
$Z_{\mathfrak{B}}^{\prime}\rightarrow
Z(\operatorname{End}_{\mathfrak{Q}(\mathcal{O})}(\mathcal{T}M_{\mathfrak{B}}))$,
which we may compose with an isomorphism above to obtain a homomorphism
$Z_{\mathfrak{B}}^{\prime}\rightarrow
Z(\operatorname{End}_{\mathfrak{C}(\mathcal{O})}(M_{\mathfrak{B}}))$. Since
$Z_{\mathfrak{B}}$ is the centre of
$\operatorname{\mathfrak{C}}(\mathcal{O})_{\mathfrak{B}}$ a similar argument
shows that $Z_{\mathfrak{B}}^{\prime}$ is a $Z_{\mathfrak{B}}$-algebra and the
surjection $(P_{\mathfrak{B}}^{\prime})^{\oplus n}\twoheadrightarrow
M_{\mathfrak{B}}$ in Corollary 4.13 is $Z_{\mathfrak{B}}$-equivariant. It
induces a $Z_{\mathfrak{B}}$-equivariant surjection
$(\mathcal{T}P_{\mathfrak{B}}^{\prime})^{\oplus
n}\twoheadrightarrow\mathcal{T}M_{\mathfrak{B}}$. We deduce that the map

$Z_{\mathfrak{B}}^{\prime}\rightarrow
Z(\operatorname{End}_{\mathfrak{C}(\mathcal{O})}(M_{\mathfrak{B}}))$

is a homomorphism of $Z_{\mathfrak{B}}$-algebras. Proposition 4.15 implies
that the composition

$Z_{\mathfrak{B}}\rightarrow Z_{\mathfrak{B}}^{\prime}\rightarrow
Z(\operatorname{End}_{\mathfrak{C}(\mathcal{O})}(M_{\mathfrak{B}}))\cong
Z_{\mathfrak{B}}$

is the identity map, which implies that the homomorphism is surjective. ∎

###  4.4. Colmez’s Montreal functor

Let $\mathcal{G}_{\mathbb{Q}_{p}}$ be the absolute Galois group of
$\mathbb{Q}_{p}$. We will consider $\zeta$ as a character of
$\mathcal{G}_{\mathbb{Q}_{p}}$ via local class field theory, normalized so
that the uniformizers correspond to geometric Frobenius. Let
$\varepsilon:\mathcal{G}_{\mathbb{Q}_{p}}\rightarrow\mathbb{Z}_{p}^{\times}$
be the $p$-adic cyclotomic character.

Colmez [19] has defined an exact and covariant functor $\mathbf{V}$ from the
category of smooth, finite length representations of $G$ on
$\mathcal{O}$-torsion modules with a central character to the category of
continuous finite-length representations of $\mathcal{G}_{\mathbb{Q}_{p}}$ on
$\mathcal{O}$-torsion modules. This functor is modified in [39, §5.7] to an
exact covariant functor

$\check{\mathbf{V}}:\mathfrak{C}(\mathcal{O})\rightarrow\operatorname{Mod}^{\mathrm{pro}}_{\mathcal{G}_{\mathbb{Q}_{p}}}(\mathcal{O})$

as follows. Let $M$ be in $\mathfrak{C}(\mathcal{O})$, if it is of finite
length, we define
$\check{\mathbf{V}}(M):=\mathbf{V}(M^{\vee})^{\vee}(\zeta\varepsilon)$ where
$\vee$ denotes the Pontryagin dual. For general
$M\in\mathfrak{C}(\mathcal{O})$, we may write $M\cong\varprojlim M_{i}$, with
$M_{i}$ of finite length in $\mathfrak{C}(\mathcal{O})$ and define
$\check{\mathbf{V}}(M):=\varprojlim\check{\mathbf{V}}(M_{i})$. With this
normalization, we have

  * •

$\check{\mathbf{V}}(\pi^{\vee})=0$ if $\pi\cong\chi\circ\det$;

  * •

$\check{\mathbf{V}}(\pi^{\vee})=\chi_{1}$ if
$\pi\cong(\operatorname{Ind}^{G}_{B}\chi_{1}\otimes\chi_{2})_{\mathrm{sm}}$;

  * •

$\check{\mathbf{V}}(\pi^{\vee})=\chi$ if
$\pi\cong\operatorname{Sp}\otimes\chi\circ\det$;

  * •

$\check{\mathbf{V}}(\pi^{\vee})=\mathbf{V}(\pi)$ is a $2$-dimensional
absolutely irreducible Galois representation if $\pi$ is supersingular.

The functor $\check{\mathbf{V}}$ induces a bijection
$\mathfrak{B}\mapsto\bar{\rho}_{\mathfrak{B}}$ between blocks containing an
absolutely irreducible representation and equivalence classes of semi-simple
representations
$\bar{\rho}:\mathcal{G}_{\mathbb{Q}_{p}}\rightarrow\operatorname{GL}_{2}(k)$,
such that all irreducible summands of $\bar{\rho}$ are absolutely irreducible.
The representation $\bar{\rho}_{\mathfrak{B}}$ can be described explicitly
according to the classification of blocks given in section 4.1: in case (i)
$\bar{\rho}_{\mathfrak{B}}=\check{\mathbf{V}}(\pi^{\vee})$ is absolute
irreducible, in case (ii) $\bar{\rho}_{\mathfrak{B}}=\chi_{1}\oplus\chi_{2}$,
in cases (iii) and (vi) $\bar{\rho}_{\mathfrak{B}}=\chi\oplus\chi$, in cases
(iv) and (v) $\bar{\rho}_{\mathfrak{B}}=\chi\oplus\chi\omega$.

Since the functor
$\check{\mathbf{V}}:\mathfrak{C}(\mathcal{O})\rightarrow\operatorname{Mod}^{\mathrm{pro}}_{\mathcal{G}_{\mathbb{Q}_{p}}}(\mathcal{O})$
kills characters and hence every object in $\mathfrak{T}(\mathcal{O})$, it
factors through
$\mathcal{T}:\mathfrak{C}(\mathcal{O})\rightarrow\mathfrak{Q}(\mathcal{O})$.
We denote
$\check{\mathbf{V}}:\mathfrak{Q}(\mathcal{O})\rightarrow\operatorname{Mod}^{\mathrm{pro}}_{\mathcal{G}_{\mathbb{Q}_{p}}}(\mathcal{O})$
by the same letter.

######  Proposition 4.17.

For each block $\mathfrak{B}$, the functor $\check{\mathbf{V}}$ induces an
equivalence of categories between $\mathfrak{Q}(\mathcal{O})_{\mathfrak{B}}$
and its essential image in
$\operatorname{Mod}^{\mathrm{pro}}_{\mathcal{G}_{\mathbb{Q}_{p}}}(\mathcal{O})$.

###### Proof.

This is due to [39] for cases (i)-(iv), [46, Proposition 2.8] for case (v) and
[47, Proposition 1.3.2] for case (iv). ∎

######  Proposition 4.18.

The map
$\mathcal{O}\llbracket\mathcal{G}_{\mathbb{Q}_{p}}\rrbracket\rightarrow\operatorname{End}_{E_{\mathfrak{B}}^{\prime}}^{\mathrm{cont}}(\check{\mathbf{V}}(P_{\mathfrak{B}}^{\prime}))$
is surjective. Moreover, if $\mathfrak{B}$ is supersingular then
$\operatorname{End}_{E_{\mathfrak{B}}^{\prime}}^{\mathrm{cont}}(\check{\mathbf{V}}(P_{\mathfrak{B}}^{\prime}))\cong
M_{2}((E_{\mathfrak{B}}^{\prime})^{\mathrm{op}})$, and
$\operatorname{End}_{E_{\mathfrak{B}}^{\prime}}^{\mathrm{cont}}(\check{\mathbf{V}}(P_{\mathfrak{B}}^{\prime}))\cong(E_{\mathfrak{B}}^{\prime})^{\mathrm{op}}$,
otherwise.

###### Proof.

It suffices to show that $M=\check{\mathbf{V}}(P_{\mathfrak{B}}^{\prime})$
satisfies all four conditions in Proposition 2.3. We note that the functor
$\mathrm{m}\mapsto\mathrm{m}\operatorname{\widehat{\otimes}}_{E_{\mathfrak{B}}^{\prime}}\check{\mathbf{V}}(P_{\mathfrak{B}}^{\prime})$
is exact by the equivalence of categories in Proposition 4.10 (2), the
exactness of $\check{\mathbf{V}}$ and the isomorphism

$\check{\mathbf{V}}(\mathrm{m}\operatorname{\widehat{\otimes}}_{E_{\mathfrak{B}}^{\prime}}P_{\mathfrak{B}}^{\prime})\cong\mathrm{m}\operatorname{\widehat{\otimes}}_{E_{\mathfrak{B}}^{\prime}}\check{\mathbf{V}}(P_{\mathfrak{B}}^{\prime})$

see the proof of [39, Lemma 5.53]. Thus by Lemma 2.4,
$\check{\mathbf{V}}(P_{\mathfrak{B}}^{\prime})$ is a projective
$E_{\mathfrak{B}}^{\prime}$-module. Let $\mathfrak{r}$ be the Jacobson radical
of $E_{\mathfrak{B}}^{\prime}$. Since

$(E_{\mathfrak{B}}^{\prime}/\mathfrak{r})\operatorname{\widehat{\otimes}}_{E_{\mathfrak{B}}^{\prime}}\check{\mathbf{V}}(P_{\mathfrak{B}}^{\prime})\cong\check{\mathbf{V}}((E_{\mathfrak{B}}^{\prime}/\mathfrak{r})\operatorname{\widehat{\otimes}}_{E_{\mathfrak{B}}^{\prime}}P_{\mathfrak{B}}^{\prime})$

is either $1$\- or $2$-dimensional $k$-vector space, topological Nakayama’s
lemma implies that $\check{\mathbf{V}}(P_{\mathfrak{B}}^{\prime})$ is a
finitely generated $E_{\mathfrak{B}}^{\prime}$-module, and thus part (1) of
Proposition 2.3 holds. Proposition 4.17 implies that part (2) of Proposition
2.3 holds.

If $\mathrm{m}$ is a right pseudo-compact $E_{\mathfrak{B}}^{\prime}$-module
then [12, Lemma 2.4] implies that

(16)
$\operatorname{Hom}_{\mathcal{O}}^{\mathrm{cont}}(\mathrm{m}\operatorname{\widehat{\otimes}}_{E_{\mathfrak{B}}^{\prime}}\check{\mathbf{V}}(P_{\mathfrak{B}}^{\prime}),k)\cong\operatorname{Hom}_{E_{\mathfrak{B}}^{\prime}}^{\mathrm{cont}}(\check{\mathbf{V}}(P_{\mathfrak{B}}^{\prime}),\operatorname{Hom}_{\mathcal{O}}^{\mathrm{cont}}(\mathrm{m},k)).$

Since in our situation $E_{\mathfrak{B}}^{\prime}$ is a compact
$\mathcal{O}$-algebra, the irreducible (left or right)
$E_{\mathfrak{B}}^{\prime}$-modules are finite dimensional vector spaces with
discrete topology, thus the map
$\mathrm{m}\mapsto\mathrm{m}^{*}:=\operatorname{Hom}_{k}(\mathrm{m},k)$
induces a bijection between irreducible left and irreducible right
$E_{\mathfrak{B}}^{\prime}$-modules. Moreover, if $\mathrm{m}$ is an
irreducible right $E_{\mathfrak{B}}^{\prime}$-module then it follows from (16)
that
$\rho_{\mathrm{m}^{*}}\cong(\mathrm{m}\operatorname{\widehat{\otimes}}_{E_{\mathfrak{B}}^{\prime}}\check{\mathbf{V}}(P_{\mathfrak{B}}^{\prime}))^{*}$,
thus parts (3) and (4) of Proposition 2.3 follow from Proposition 4.17.

If $\mathfrak{B}$ is supersingular then $\mathfrak{B}$ contains only one
irreducible $\pi$, which is not a character. Thus
$P_{\mathfrak{B}}=P_{\mathfrak{B}}^{\prime}$,
$E_{\mathfrak{B}}=E_{\mathfrak{B}}^{\prime}$ is a local ring and
$k\operatorname{\widehat{\otimes}}_{E_{\mathfrak{B}}}P_{\mathfrak{B}}=\pi^{\vee}$.
Thus
$k\operatorname{\widehat{\otimes}}_{E_{\mathfrak{B}}}\check{\mathbf{V}}(P_{\mathfrak{B}})\cong\check{\mathbf{V}}(\pi^{\vee})\cong\bar{\rho}_{\mathfrak{B}}$
is an absolutely irreducible $2$-dimensional representation. Since
$E_{\mathfrak{B}}$ is a local ring, we deduce that
$\check{\mathbf{V}}(P_{\mathfrak{B}})$ is a free $E_{\mathfrak{B}}$-module of
rank $2$. Thus
$\operatorname{End}^{\mathrm{cont}}_{E_{\mathfrak{B}}}(\check{\mathbf{V}}(P_{\mathfrak{B}}))\cong
M_{2}((E_{\mathfrak{B}}^{\prime})^{\mathrm{op}})$.

If $\mathfrak{B}$ is of type (iii) of (vi) then the block in the quotient
category contains only one irreducible object and Colmez’s functor maps it to
a $1$-dimensional $\mathcal{G}_{\mathbb{Q}_{p}}$-representation. The same
argument as in the supersingular case shows that $E_{\mathfrak{B}}^{\prime}$
is a local ring and $\check{\mathbf{V}}(P_{\mathfrak{B}}^{\prime})$ is a free
$E_{\mathfrak{B}}^{\prime}$-module of rank $1$, and thus
$\operatorname{End}_{E_{\mathfrak{B}}^{\prime}}^{\mathrm{cont}}(\check{\mathbf{V}}(P_{\mathfrak{B}}^{\prime}))=(E_{\mathfrak{B}}^{\prime})^{\mathrm{op}}$.

If $\mathfrak{B}$ is of type (ii), (iv) or (v) then
$\mathfrak{Q}(\mathcal{O})_{\mathfrak{B}}$ contains exactly two irreducible
objects and Colmez’s functor sends them to distinct $1$-dimensional
$\mathcal{G}_{\mathbb{Q}_{p}}$-representations $\chi_{1}$, $\chi_{2}$. It
follows from Corollary 2.5 that
$\check{\mathbf{V}}(P_{\mathfrak{B}}^{\prime})$ is a free
$E_{\mathfrak{B}}^{\prime}$-module of rank $1$, and thus
$\operatorname{End}_{E_{\mathfrak{B}}^{\prime}}^{\mathrm{cont}}(\check{\mathbf{V}}(P_{\mathfrak{B}}^{\prime}))=(E_{\mathfrak{B}}^{\prime})^{\mathrm{op}}$.
∎

###  4.5. Banach space representations

Let $\operatorname{Ban}^{\mathrm{adm}}_{G,\zeta}(L)$ be the category of
admissible unitary $L$-Banach space representations [44, §3] on which $Z$ acts
by the character $\zeta$. We note that
$\operatorname{Ban}^{\mathrm{adm}}_{G,\zeta}(L)$ is an abelian category [44,
Theorem 3.5]. Any $\Pi\in\operatorname{Ban}^{\mathrm{adm}}_{G,\zeta}(L)$ has
an open, bounded and $G$-invariant lattice $\Theta$ and
$\Theta\otimes_{\mathcal{O}}k$ is an admissible smooth $k$-representation of
$G$. Let $\Theta^{d}=\operatorname{Hom}_{\mathcal{O}}(\Theta,\mathcal{O})$ be
the Schikhof dual of $\Theta$ endowed with the topology of pointwise
convergence. Then $\Theta^{d}$ is an object of
$\operatorname{Mod}^{\mathrm{pro}}_{G}(\mathcal{O})$, [39, Lemma 4.4]. If
$\Theta^{d}$ is in $\mathfrak{C}(\mathcal{O})$ then $\Xi^{d}$ is in
$\mathfrak{C}(\mathcal{O})$ for every open bounded $G$-invariant lattice $\Xi$
in $\Pi$, since $\Theta$ and $\Xi$ are commensurable and
$\mathfrak{C}(\mathcal{O})$ is closed under subquotients [39, Lemma 4.6].

If $\Pi\in\operatorname{Ban}^{\mathrm{adm}}_{G,\zeta}(L)$, then we let

$\check{\mathbf{V}}(\Pi)=\check{\mathbf{V}}(\Theta^{d})\otimes_{\mathcal{O}}L,$

where $\Theta$ is any open bounded $G$-invariant lattice in $\Pi$. Then
$\check{\mathbf{V}}$ is exact and contravariant on
$\operatorname{Ban}^{\mathrm{adm}}_{G,\zeta}(L)$.

##  5\. Density

###  5.1. Capture

Let $G=\operatorname{GL}_{2}(\mathbb{Q}_{p})$ and
$K=\operatorname{GL}_{2}(\mathbb{Z}_{p})$. Write $Z$ for the center of $G$ and
$Z(K)$ for the center of $K$. Let $\psi:Z(K)\rightarrow\mathcal{O}^{\times}$
be a continuous character. We identify $Z$ with $\mathbb{Q}_{p}^{\times}$ and
$Z(K)$ with $\mathbb{Z}_{p}^{\times}$ via the map
$\left(\begin{smallmatrix}x&0\\\ 0&x\end{smallmatrix}\right)\mapsto x$.

######  Lemma 5.1.

Let $\\{V_{i}\\}_{i\in I}$ be a family of continuous representations of $K$ on
finite dimensional $L$-vector spaces with central character $\psi$, and let
$M\in\operatorname{Mod}^{\mathrm{pro}}_{K,\psi}(\mathcal{O})$ be
$\mathcal{O}$-torsion free. The following conditions are equivalent.

  1. (1)

The smallest quotient $M\twoheadrightarrow Q$ such that
$\operatorname{Hom}^{\mathrm{cont}}_{\mathcal{O}\llbracket
K\rrbracket}(Q,V_{i}^{*})\cong\operatorname{Hom}^{\mathrm{cont}}_{\mathcal{O}\llbracket
K\rrbracket}(M,V_{i}^{*})$ for all $i\in I$ is equal to $M$.

  2. (2)

The intersection of the kernels of all
$\phi\in\operatorname{Hom}^{\mathrm{cont}}_{\mathcal{O}\llbracket
K\rrbracket}(M,V^{*}_{i})$ for each $i\in I$ is equal to zero.

  3. (3)

The image of the evaluation map

$\bigoplus_{i\in
I}\operatorname{Hom}_{K}(V_{i},\Pi(M))\otimes_{L}V_{i}\rightarrow\Pi(M)$

is a dense subspace, where
$\Pi(M):=\operatorname{Hom}^{\mathrm{cont}}_{\mathcal{O}}(M,L)$ is an L-Banach
space equipped with the supremum norm.

###### Proof.

See [21, Lemma 2.7, Lemma 2.10]. ∎

######  Definition 5.2.

We say that $\\{V_{i}\\}_{i\in I}$ captures $M$ if it satisfies one of the
equivalent conditions above.

Since $1+p\mathbb{Z}_{p}$ (resp. $1+4\mathbb{Z}_{2}$) is a free pro-$p$ group
of rank 1 if $p>2$ (resp. $p=2$), there is a smooth non-trivial character
$\chi:\mathbb{Z}_{p}^{\times}\rightarrow L^{\times}$ and a continuous
character $\eta_{0}:\mathbb{Z}_{p}^{\times}\rightarrow L^{\times}$ such that
$\psi=\chi\eta_{0}^{2}$. Let $e$ be the smallest integer such that $\chi$ is
trivial on $1+p^{e}\mathbb{Z}_{p}$. Let

$J=\left(\begin{smallmatrix}\mathbb{Z}_{p}^{\times}&\mathbb{Z}_{p}\\\
p^{e}\mathbb{Z}_{p}&\mathbb{Z}_{p}^{\times}\end{smallmatrix}\right),$

and let $\chi\otimes\mathbf{1}:J\rightarrow L^{\times}$ be the character which
sends $\left(\begin{smallmatrix}a&b\\\ c&d\end{smallmatrix}\right)$ to
$\chi(a)$. Then the representation
$\tau=\operatorname{Ind}^{K}_{J}(\chi\otimes\mathbf{1})$ is a principal series
type. That is, for an irreducible smooth $\overline{L}$-representation $\pi$
of $G$, we have $\operatorname{Hom}_{L[K]}(\tau,\pi)\neq 0$ if and only if
$\pi\cong(\operatorname{Ind}^{G}_{B}\psi_{1}\otimes\psi_{2})_{\mathrm{sm}}$,
where $B$ is a Borel subgroup and $\psi_{1}|_{\mathbb{Z}_{p}^{\times}}=\chi$
and $\psi_{2}|_{\mathbb{Z}_{p}^{\times}}=\mathbf{1}$ [11, §A 2.2].

######  Proposition 5.3.

The family

$\\{\operatorname{Ind}^{K}_{J}(\chi\otimes\mathbf{1})\otimes\operatorname{Sym}^{2a}L^{2}\otimes(\det)^{-a}\otimes\eta\eta_{0}\circ\det\\}_{a\in\mathbb{N},\eta},$

where $\eta$ runs over all the characters with $\eta^{2}=\mathbf{1}$, captures
every projective object in
$\operatorname{Mod}^{\mathrm{pro}}_{K,\psi^{-1}}(\mathcal{O})$.

###### Proof.

See [41, Proposition 2.7]. ∎

We will denote the family of representations in the above proposition by
$\\{V_{i}\\}_{i\in I}$. We note that each $V_{i}$ is a twist of a locally
algebraic representation by a unitary character $\eta_{0}$, which might not be
locally algebraic. However, twisting by its inverse will get us to a locally
algebraic situation, which is sufficient for all arguments that follow.

###  5.2. Locally algebraic vectors in $\Pi(P)$

Let $\zeta:Z\rightarrow\mathcal{O}^{\times}$ be a continuous character and
$\psi=\zeta|_{K}$. Let $P$ be a projective object in
$\mathfrak{C}(\mathcal{O})$ and
$E=\operatorname{End}_{\mathfrak{C}(\mathcal{O})}(P)$. In particular, $P$ is a
torsionfree compact linear-topological $\mathcal{O}$-module. Let

$\Pi(P):=\operatorname{Hom}^{\mathrm{cont}}_{\mathcal{O}}(P,L)$

with the topology induced by the supremum norm. Then we have
$E[1/p]\cong\operatorname{End}^{\mathrm{cont}}_{G}(\Pi(P))$.

If $V$ is a continuous representation of $K$ on a finite dimensional
$L$-vector space, then

(17)
$\operatorname{Hom}_{K}(V,\Pi(P))\cong\operatorname{Hom}^{\mathrm{cont}}_{\mathcal{O}\llbracket
K\rrbracket}(P,V^{*}).$

Since $P$ is projective in
$\operatorname{Mod}^{\mathrm{pro}}_{K,\psi^{-1}}(\mathcal{O})$ by [30,
Corollary 3.10], the family of finite dimensional $K$-representations
associated to $\psi$ in Proposition 5.3, which we denote by $\\{V_{i}\\}_{i\in
I}$, captures $P$. We view $V_{i}$ as a representation of $KZ$ by letting
$\left(\begin{smallmatrix}\varpi&0\\\ 0&\varpi\end{smallmatrix}\right)$ act by
$\zeta(\varpi)$.

######  Proposition 5.4.

For each $i\in I$, we let
$A_{i}:=\operatorname{End}_{G}(\operatorname{c-Ind}^{G}_{KZ}V_{i})$. Then

  1. (1)

$A_{i}$ is isomorphic to $L[T]$;

  2. (2)

$\operatorname{c-Ind}^{G}_{KZ}V_{i}$ is flat over $A_{i}$.

###### Proof.

We may write
$V_{i}=\operatorname{Ind}^{K}_{J}(\chi\otimes\mathbf{1})\otimes_{L}W_{i}$,
where the action of $KZ$ on $W_{i}$ extends to an action of $G$, see
Proposition 5.3. Then

$\operatorname{c-Ind}^{G}_{KZ}V_{i}\cong\operatorname{c-Ind}^{G}_{KZ}(\operatorname{Ind}^{K}_{J}(\chi\otimes\mathbf{1}))\otimes_{L}W_{i}.$

Here we view $\operatorname{Ind}^{K}_{J}(\chi\otimes\mathbf{1})$ as a
representation of $KZ$ by letting $\left(\begin{smallmatrix}\varpi&0\\\
0&\varpi\end{smallmatrix}\right)$ act on
$\operatorname{Ind}^{K}_{J}(\chi\otimes\mathbf{1})$ by
$\zeta(\varpi)\zeta^{-1}_{W_{i}}(\varpi)$, where $\zeta_{W_{i}}$ is the
central character of $W_{i}$. Since the restriction of $W_{i}$ to any compact
open subgroup of $G$ remains absolutely irreducible, the above isomorphism
induces an isomorphism of $L$-algebras

$\operatorname{End}_{G}(\operatorname{c-Ind}^{G}_{KZ}V_{i})\cong\operatorname{End}_{G}(\operatorname{c-Ind}^{G}_{KZ}(\operatorname{Ind}^{K}_{J}(\chi\otimes\mathbf{1}))).$

Thus we may assume that $W_{i}$ is the trivial representation.

By [14], the $K$-type $\operatorname{Ind}^{K}_{J}(\chi\otimes\mathbf{1})$ is a
$G$-cover of the $K_{M}$-type $\chi\otimes\mathbf{1}$, where
$M=\mathbb{Q}_{p}^{\times}\times\mathbb{Q}_{p}^{\times}$ and
$K_{M}=\mathbb{Z}_{p}^{\times}\times\mathbb{Z}_{p}^{\times}$. Thus there is an
algebra isomorphism

$j_{M}:\operatorname{End}_{M}(\operatorname{c-Ind}^{M}_{K_{M}}(\chi\otimes\mathbf{1}))\xrightarrow{\sim}\operatorname{End}_{G}(\operatorname{c-Ind}^{G}_{J}(\chi\otimes\mathbf{1}))$

such that for each
$f\in\operatorname{End}_{M}(\operatorname{c-Ind}^{M}_{K_{M}}\chi\otimes\mathbf{1})$,
we have $\operatorname{Supp}(j_{M}f)=J\operatorname{Supp}(f)J$. It follows
that

$L[T]\xrightarrow{\sim}\operatorname{End}_{M}(\operatorname{c-Ind}^{M}_{K_{M}Z}(\chi\otimes\mathbf{1}))\xrightarrow[j_{M}]{\sim}A_{i},$

where $T$ maps to an element in $A_{i}$ supported at
$JZ\left(\begin{smallmatrix}p&0\\\ 0&1\end{smallmatrix}\right)JZ$. Here we
view $\chi\otimes\mathbf{1}$ as a representation of $K_{M}Z$ by letting
$\left(\begin{smallmatrix}\varpi&0\\\ 0&\varpi\end{smallmatrix}\right)$ act by
$\zeta(\varpi)$. This shows the first assertion.

To prove the second assertion, it suffices to show that
$\operatorname{c-Ind}^{G}_{KZ}V_{i}$ is torsion free since $A_{i}$ is a PID.
After tensoring with $\overline{L}$, this is equivalent to
$\operatorname{c-Ind}^{G}_{KZ}V_{i}$ has no $T-\lambda$ torsion, which is
easily seen using the fact that the functions in
$\operatorname{c-Ind}^{G}_{KZ}V_{i}$ are compactly supported. ∎

In particular, Frobenius reciprocity gives

(18)
$\operatorname{Hom}_{K}(V_{i},\Pi(P))\cong\operatorname{Hom}_{G}(\operatorname{c-Ind}^{G}_{KZ}V_{i},\Pi(P)).$

Hence, $\operatorname{Hom}_{K}(V_{i},\Pi(P))$ is naturally an $A_{i}$-module
and we may transport the action of $A_{i}$ onto
$\operatorname{Hom}^{\mathrm{cont}}_{\mathcal{O}\llbracket
K\rrbracket}(P,V_{i}^{*})$ via (17).

If $V$ is a continuous representation of $K$ on a finite dimensional
$L$-vector space and if $\Theta$ is an open, bounded $K$-invariant lattice in
$V$, let $|\cdot|$ be the norm on $V^{*}$ given by
$|\ell|:=\sup_{v\in\Theta}|\ell(v)|$, so that
$\Theta^{d}=\operatorname{Hom}_{\mathcal{O}}(\Theta,\mathcal{O})$ is the unit
ball in $V^{*}$ with respect to $|\cdot|$. The topology on
$\operatorname{Hom}^{\mathrm{cont}}_{\mathcal{O}\llbracket
K\rrbracket}(P,V^{*})$ is given by the norm $\lVert\phi\rVert:=\sup_{v\in
P}|\phi(v)|$, and $\operatorname{Hom}^{\mathrm{cont}}_{\mathcal{O}\llbracket
K\rrbracket}(P,\Theta^{d})$ is the unit ball in this Banach space.

######  Proposition 5.5.

For all $i\in I$ the submodule

$\operatorname{Hom}^{\mathrm{cont}}_{\mathcal{O}\llbracket
K\rrbracket}(P,V_{i}^{*})_{\mathrm{l.fin}}:=\\{\phi\in\operatorname{Hom}^{\mathrm{cont}}_{\mathcal{O}\llbracket
K\rrbracket}(P,V_{i}^{*}):\ell_{A_{i}}(A_{i}\phi)<\infty\\}$

is dense in $\operatorname{Hom}^{\mathrm{cont}}_{\mathcal{O}\llbracket
K\rrbracket}(P,V_{i}^{*})$, where $\ell_{A_{i}}(A_{i}\phi)$ is the length of
$A_{i}\phi$ as an $A_{i}$-module.

###### Proof.

See [21, Proposition 2.19]. ∎

######  Proposition 5.6.

Let $\mathfrak{m}$ be a maximal ideal of $A_{i}$ and let $\Pi$ be a completion
of $A_{i}/\mathfrak{m}^{n}\otimes_{A_{i}}\operatorname{c-Ind}^{G}_{KZ}V_{i}$
with respect to a $G$-invariant norm. Then $\Pi$ is the universal unitary
completion of
$A_{i}/\mathfrak{m}^{n}\otimes_{A_{i}}\operatorname{c-Ind}^{G}_{KZ}V_{i}$.
Moreover, the action of $A_{i}$ on
$A_{i}/\mathfrak{m}^{n}\otimes_{A_{i}}\operatorname{c-Ind}^{G}_{KZ}V_{i}$
extends to a continuous action of $A_{i}$ on $\Pi$.

###### Proof.

Let $\Pi^{\circ}$ be a $G$-invariant $\mathcal{O}$-lattice of $\Pi$. Then

$\Theta:=\Pi^{\circ}\cap(A_{i}/\mathfrak{m}^{n}\otimes_{A_{i}}\operatorname{c-Ind}^{G}_{KZ}V_{i})$

is a $G$-invariant $\mathcal{O}$-lattice of
$A_{i}/\mathfrak{m}^{n}\otimes_{A_{i}}\operatorname{c-Ind}^{G}_{KZ}V_{i}$. By
[27, Proposition 1.17], it suffices to show that $\Theta$ is of finite type
over $\mathcal{O}[G]$.

By Proposition 5.4 (i), we have $\kappa(\mathfrak{m}):=A_{i}/\mathfrak{m}\cong
L[T]/f(T)$, where $f(T)\in L[T]$ is an irreducible polynomial, is a finite
extension of $L$. Define a finite, increasing, exhaustive filtration
$\\{R^{j}\\}_{n\geq j\geq 0}$ of
$A_{i}/\mathfrak{m}^{n}\otimes_{A_{i}}\operatorname{c-Ind}^{G}_{KZ}V_{i}$ by
$G$-invariant $A_{i}$-submodules
$R^{j}=\mathfrak{m}^{n-j}/\mathfrak{m}^{n}\otimes_{A_{i}}\operatorname{c-Ind}^{G}_{KZ}V_{i}$.
Then we have

$R^{j}/R^{j-1}\cong\kappa(\mathfrak{m})\otimes_{A_{i}}\operatorname{c-Ind}^{G}_{KZ}V_{i}$

for each $j$ and $\\{\Theta^{j}:=\Theta\cap R^{j}\\}_{n\geq j\geq 0}$ is a
finite, increasing, exhaustive filtration of $\Theta$ such that $\Theta^{j}$
is a $G$-invariant $\mathcal{O}$-lattice of $R^{j}$ for each $j$. Moreover,
$\Theta^{j}/\Theta^{j-1}$ gives rise to a $G$-invariant $\mathcal{O}$-lattice
of $R^{j}/R^{j-1}\cong
A_{i}/\mathfrak{m}\otimes_{A_{i}}\operatorname{c-Ind}^{G}_{KZ}V_{i}$, and thus
is finitely generated over $\mathcal{O}[G]$ by the proof of [4, Theorem
4.3.1]. This implies that $\Theta$ is finitely generated over $\mathcal{O}[G]$
and the first assertion follows.

If $\phi\in A_{i}$ then $\phi(\Theta)\subset\varpi^{n}\Theta$ for some
$n\in\mathbb{Z}$, as $\Theta$ is finitely generated over $\mathcal{O}[G]$.
This implies the second assertion. ∎

We denote the universal unitary completion of
$A_{i}/\mathfrak{m}^{n}\otimes_{A_{i}}\operatorname{c-Ind}^{G}_{KZ}V_{i}$ by
$\Pi_{i,\mathfrak{m},n}$. If $n=1$ then $\Pi_{i,\mathfrak{m},1}$ is the
universal unitary completion of
$\kappa(\mathfrak{m})\otimes_{A_{i}}\operatorname{c-Ind}^{G}_{KZ}V_{i}$
studied in [4, Theorem 4.3.1] and [10, Proposition 2.2.1].

######  Corollary 5.7.

If $\phi\in\operatorname{Hom}^{\mathrm{cont}}_{\mathcal{O}\llbracket
K\rrbracket}(P,V_{i}^{*})_{\mathrm{l.fin}}$ is such that $A_{i}\phi\cong
A_{i}/\mathfrak{m}^{n}$ for a maximal ideal $\mathfrak{m}$ of $A_{i}$, then
$\phi$ induces an injection $\Pi_{i,\mathfrak{m},n}\hookrightarrow\Pi(P)$.
Moreover, $\Pi_{i,\mathfrak{m},n}$ admits a filtration of length $n$ such that
each graded piece is isomorphic to the universal unitary completion
$\Pi_{i,\mathfrak{m},1}$ of
$\kappa(\mathfrak{m})\otimes_{A_{i}}\operatorname{c-Ind}^{G}_{KZ}V_{i}$.

###### Proof.

The assumption $A_{i}\phi\cong A_{i}/\mathfrak{m}^{n}$ implies that
$A_{i}/\mathfrak{m}^{n}\otimes_{A_{i}}\operatorname{c-Ind}^{G}_{KZ}V_{i}$
injects into $\Pi(P)$. The first assertion follows immediately from
Proposition 5.6. To show the second assertion, we let $\\{R^{j}\\}_{n\geq
j\geq 0}$ be the filtration of
$A_{i}/\mathfrak{m}^{n}\otimes_{A_{i}}\operatorname{c-Ind}^{G}_{KZ}V_{i}$
defined in Proposition 5.6. Let $\Pi^{j}$ be the closure of $R^{j}$ in
$\Pi_{i,\mathfrak{m},n}$; then $\Pi^{n}=\Pi_{i,\mathfrak{m},n}$. Since
$\mathfrak{m}R^{j}=R^{j-1}$ we have $\mathfrak{m}\Pi^{j}=\Pi^{j-1}$ and hence
$\Pi^{j}=\mathfrak{m}^{n-j}\Pi_{i,\mathfrak{m},n}$. If $\Pi^{j}=\Pi^{j-1}$
then $\Pi^{j}=\mathfrak{m}\Pi^{j}$ and hence
$\Pi^{j}=\mathfrak{m}^{j}\Pi^{j}=\mathfrak{m}^{n}\Pi_{i,\mathfrak{m},n}=0$.
Since $\Pi^{j}\neq 0$ for $1\leq j\leq n$ we conclude that
$\Pi^{j}\neq\Pi^{j-1}$ for $1\leq j\leq n$. Moreover, $\Pi^{j}/\Pi^{j-1}$
contains $R^{j}/R^{j-1}\cong
A_{i}/\mathfrak{m}\otimes_{A_{i}}\operatorname{c-Ind}^{G}_{KZ}V_{i}$ as a
dense subspace, and thus is isomorphic to $\Pi_{i,\mathfrak{m},1}$. This
proves the corollary. ∎

Note that the image of any
$\phi\in\operatorname{Hom}_{G}(A_{i}/\mathfrak{m}^{n}\otimes_{A_{i}}\operatorname{c-Ind}^{G}_{KZ}V_{i},\Pi(P))$
is isomorphic to
$A_{i}/\mathfrak{m}^{k}\otimes_{A_{i}}\operatorname{c-Ind}^{G}_{KZ}V_{i}$ for
some $0\leq k\leq n$. Hence it induces an injection
$\Pi_{i,\mathfrak{m},k}\hookrightarrow\Pi(P)$ by Corollary 5.7.

######  Proposition 5.8.

Let $P$ be a projective object in $\mathfrak{C}(\mathcal{O})$. Then the image
of the evaluation map

$\bigoplus_{i\in
I}\bigoplus_{\mathfrak{m},n}\operatorname{Hom}^{\mathrm{cont}}_{G}(\Pi_{i,\mathfrak{m},n},\Pi(P))\otimes_{L}\Pi_{i,\mathfrak{m},n}\rightarrow\Pi(P),$

where $\mathfrak{m}$ runs through maximal ideals of $A_{i}$ and
$n\in\mathbb{N}$, is a dense subspace.

###### Proof.

Let $\Pi$ be the closure of the image of the evaluation map and $M$ the image
of $P$ under
$\operatorname{Hom}^{\mathrm{cont}}_{L}(\Pi(P),L)\twoheadrightarrow\operatorname{Hom}^{\mathrm{cont}}_{L}(\Pi,L)$.
Then we have

$\displaystyle\operatorname{Hom}^{\mathrm{cont}}_{\mathcal{O}\llbracket
K\rrbracket}(P,V_{i}^{*})_{\mathrm{l.fin}}$
$\displaystyle\cong\oplus_{\mathfrak{m}}\operatorname{Hom}^{\mathrm{cont}}_{\mathcal{O}\llbracket
K\rrbracket}(P,V_{i}^{*})[\mathfrak{m}^{\infty}]$
$\displaystyle\cong\oplus_{\mathfrak{m}}\operatorname{Hom}_{K}(V_{i},\Pi(P))[\mathfrak{m}^{\infty}]$
$\displaystyle\cong\oplus_{\mathfrak{m}}\operatorname{Hom}_{G}(\operatorname{c-Ind}^{G}_{KZ}V_{i},\Pi(P))[\mathfrak{m}^{\infty}]$
$\displaystyle\cong\oplus_{\mathfrak{m}}\varinjlim_{n}\operatorname{Hom}_{G}(\operatorname{c-Ind}^{G}_{KZ}V_{i},\Pi(P))[\mathfrak{m}^{n}]$
$\displaystyle\cong\oplus_{\mathfrak{m}}\varinjlim_{n}\operatorname{Hom}_{G}(A_{i}/\mathfrak{m}^{n}\otimes_{A_{i}}\operatorname{c-Ind}^{G}_{KZ}V_{i},\Pi(P))$
$\displaystyle\cong\oplus_{\mathfrak{m}}\varinjlim_{n}\operatorname{Hom}^{\mathrm{cont}}_{G}(\Pi_{i,\mathfrak{m},n},\Pi(P)).$

Here the first isomorphism is due to the fact that any module $M$ over a
commutative ring $A$, such that every finitely generated submodule is of
finite length, admits a decomposition
$M\cong\oplus_{\mathfrak{m}}M[\mathfrak{m}^{\infty}]$ with $\mathfrak{m}$
running through maximal ideals of $A$, the second isomorphism is given by
(17), and the last isomorphism is due to Corollary 5.7. Similarly, we have

$\displaystyle\operatorname{Hom}^{\mathrm{cont}}_{\mathcal{O}\llbracket
K\rrbracket}(M,V_{i}^{*})_{\mathrm{l.fin}}$
$\displaystyle\cong\oplus_{\mathfrak{m}}\operatorname{Hom}^{\mathrm{cont}}_{\mathcal{O}\llbracket
K\rrbracket}(M,V_{i}^{*})[\mathfrak{m}^{\infty}]$
$\displaystyle\cong\oplus_{\mathfrak{m}}\operatorname{Hom}_{K}(V_{i},\Pi)[\mathfrak{m}^{\infty}]$
$\displaystyle\cong\oplus_{\mathfrak{m}}\operatorname{Hom}_{G}(\operatorname{c-Ind}^{G}_{KZ}V_{i},\Pi)[\mathfrak{m}^{\infty}]$
$\displaystyle\cong\oplus_{\mathfrak{m}}\varinjlim_{n}\operatorname{Hom}_{G}(A_{i}/\mathfrak{m}^{n}\otimes_{A_{i}}\operatorname{c-Ind}^{G}_{KZ}V_{i},\Pi)$
$\displaystyle\cong\oplus_{\mathfrak{m}}\varinjlim_{n}\operatorname{Hom}^{\mathrm{cont}}_{G}(\Pi_{i,\mathfrak{m},n},\Pi)$
$\displaystyle=\oplus_{\mathfrak{m}}\varinjlim_{n}\operatorname{Hom}^{\mathrm{cont}}_{G}(\Pi_{i,\mathfrak{m},n},\Pi(P)).$

Thus by Proposition 5.5, we have
$\operatorname{Hom}^{\mathrm{cont}}_{\mathcal{O}\llbracket
K\rrbracket}(P,V_{i}^{*})\cong\operatorname{Hom}^{\mathrm{cont}}_{\mathcal{O}\llbracket
K\rrbracket}(M,V_{i}^{*})$ for each $i\in I$. Combining this with Lemma 5.1,
we deduce the proposition. ∎

######  Corollary 5.9.

Let
$\mathrm{m}_{i,\mathfrak{m},n}:=\operatorname{Hom}_{\mathfrak{C}(\mathcal{O})}(P,\Theta^{d})\otimes_{\mathcal{O}}L$,
where $\Theta$ is an open bounded $G$-invariant lattice in
$\Pi_{i,\mathfrak{m},n}$. Then

$\bigcap_{i\in I}\bigcap_{\mathfrak{m},n}\mathfrak{a}_{i,\mathfrak{m},n}=0,$

where
$\mathfrak{a}_{i,\mathfrak{m},n}:=\operatorname{ann}_{E}\bigl{(}\mathrm{m}_{i,\mathfrak{m},n}\bigr{)}$.

###### Proof.

For $\Pi\in\operatorname{Ban}^{\mathrm{adm}}_{G,\zeta}(L)$ with a
$G$-invariant $\mathcal{O}$-lattice $\Theta$, we have

$\operatorname{Hom}_{\mathfrak{C}(\mathcal{O})}(P,\Theta^{d})\otimes_{\mathcal{O}}L\cong\operatorname{Hom}^{\mathrm{cont}}_{G}(\Pi,\Pi(P)).$

Thus the evaluation map in Proposition 5.8 induces an $E[1/p]$-homomorphism

$\bigoplus_{i\in
I}\bigoplus_{\mathfrak{m},n}\mathrm{m}_{i,\mathfrak{m},n}\otimes_{L}\Pi_{i,\mathfrak{m},n}\rightarrow\Pi(P)$

with a dense image. Since $E[1/p]$ acts faithfully on the RHS of the map, it
acts faithfully on the LHS as well. This proves the corollary since the
$E$-action on LHS factors through the quotient $E/\bigcap_{i\in
I}\bigcap_{\mathfrak{m},n}\mathfrak{a}_{i,\mathfrak{m},n}$. ∎

##  6\. Main results

Given a block $\mathfrak{B}$, we have defined $\pi_{\mathfrak{B}}$,
$P_{\mathfrak{B}}$, $E_{\mathfrak{B}}$ in §4.1 and
$P_{\mathfrak{B}}^{\prime}$, $E_{\mathfrak{B}}^{\prime}$ in §4.2. We assume
that all irreducibles in $\mathfrak{B}$ are absolutely irreducible. This can
be achieved by replacing $k$ with a finite extension. Let
$\bar{\rho}_{\mathfrak{B}}$ be the 2-dimensional semi-simple Galois
representation of $\mathcal{G}_{\mathbb{Q}_{p}}$ over $k$ defined by
$\check{\mathbf{V}}(\pi_{\mathfrak{B}}^{\vee})$ in cases (i), (ii), (iv), (v)
and by a direct sum of two copies of
$\check{\mathbf{V}}(\pi_{\mathfrak{B}}^{\vee})$ in cases (iii), (vi), see
section 4.4 for an explicit description. We write
$R^{\mathrm{ps},\zeta\varepsilon}_{\operatorname{tr}\bar{\rho}_{\mathfrak{B}}}$
for the universal pseudodeformation ring of
$\operatorname{tr}\bar{\rho}_{\mathfrak{B}}$ with a fixed determinant
$\zeta\varepsilon$. This ring is noetherian by [18, Proposition F]. We let
$T:\mathcal{G}_{\mathbb{Q}_{p}}\rightarrow
R^{\mathrm{ps},\zeta\varepsilon}_{\operatorname{tr}\bar{\rho}_{\mathfrak{B}}}$
be the universal object, see section 3.

###  6.1. Finiteness

Let $\\{V_{i}\\}_{i\in I}$ be a family of $K$-representations defined in
Proposition 5.3 and let
$A_{i}=\operatorname{End}_{G}(\operatorname{c-Ind}^{G}_{KZ}V_{i})$. For each
$i\in I$, a maximal ideal $\mathfrak{m}$ of $A_{i}$ and $n\in\mathbb{N}$, we
write $\Pi_{i,\mathfrak{m},n}$ for the universal unitary completion of
$A_{i}/\mathfrak{m}^{n}\otimes_{A_{i}}\operatorname{c-Ind}^{G}_{KZ}V_{i}$.

######  Lemma 6.1.

Assume $\Pi_{i,\mathfrak{m},n}$ is a subrepresentation of
$\Pi(P_{\mathfrak{B}}^{\prime})$. Then
$\check{\mathbf{V}}(\Pi_{i,\mathfrak{m},n})$ is a finite free
$A_{i}/\mathfrak{m}^{n}$-module of rank equal to
$\dim_{A_{i}/\mathfrak{m}}(\check{\mathbf{V}}(\Pi_{i,\mathfrak{m},1}))\leq 2$.
Moreover,

  1. (i)

if
$\mathop{\mathrm{rank}}\nolimits_{A_{i}/\mathfrak{m}^{n}}\check{\mathbf{V}}(\Pi_{i,\mathfrak{m},n})=2$
then $\check{\mathbf{V}}(\Pi_{i,\mathfrak{m},n})$ is a deformation to
$A_{i}/\mathfrak{m}^{n}$ of the absolutely irreducible $2$-dimensional
$L$-representation $\check{\mathbf{V}}(\Pi_{i,\mathfrak{m},1})$ of
$\mathcal{G}_{\mathbb{Q}_{p}}$;

  2. (ii)

if
$\mathop{\mathrm{rank}}\nolimits_{A_{i}/\mathfrak{m}^{n}}\check{\mathbf{V}}(\Pi_{i,\mathfrak{m},n})=1$
then the action of $\mathcal{G}_{\mathbb{Q}_{p}}$ on
$\check{\mathbf{V}}(\Pi_{i,\mathfrak{m},n})$ is given by a
$(A_{i}/\mathfrak{m}^{n})^{\times}$-valued character lifting
$\check{\mathbf{V}}(\Pi_{i,\mathfrak{m},1})$.

###### Proof.

Since $V_{i}$ is a principal series type, the
$\mathcal{G}_{\mathbb{Q}_{p}}$-module

$\check{\mathbf{V}}(\Pi_{i,\mathfrak{m},n})/\mathfrak{m}\check{\mathbf{V}}(\Pi_{i,\mathfrak{m},n})\xrightarrow{\sim}\check{\mathbf{V}}(\Pi_{i,\mathfrak{m},1})$

has dimension $r\leq 2$ over $\kappa(\mathfrak{m}):=A_{i}/\mathfrak{m}$ by [4,
Theorem 4.3.1] and [10, Proposition 2.2.1]. Nakayama’s lemma implies that we
have a surjection

$(A_{i}/\mathfrak{m}^{n})^{\oplus
r}\twoheadrightarrow\check{\mathbf{V}}(\Pi_{i,\mathfrak{m},n}).$

Proposition 5.7 and exactness of $\check{\mathbf{V}}$ imply that
$\check{\mathbf{V}}(\Pi_{i,\mathfrak{m},n})$ has length $nr$ as
$A_{i}/\mathfrak{m}^{n}$-module. Hence the surjection is an isomorphism and
the lemma follows. ∎

######  Lemma 6.2.

Under the same assumptions as in Lemma 6.1, there is a natural map
$\theta_{i,\mathfrak{m},n}:R^{\mathrm{ps},\zeta\varepsilon}_{\operatorname{tr}\bar{\rho}_{\mathfrak{B}}}\rightarrow
A_{i}/\mathfrak{m}^{n}$, which induces a map

$R^{\mathrm{ps},\zeta\varepsilon}_{\operatorname{tr}\bar{\rho}_{\mathfrak{B}}}\llbracket\mathcal{G}_{\mathbb{Q}_{p}}\rrbracket/J\rightarrow\operatorname{End}_{A_{i}/\mathfrak{m}^{n}}(\check{\mathbf{V}}(\Pi_{i,\mathfrak{m},n})).$

###### Proof.

If $\check{\mathbf{V}}(\Pi_{i,\mathfrak{m},n})$ is of rank $2$ over
$A_{i}/\mathfrak{m}^{n}$, it follows from Lemma 6.1 (i) that
$\check{\mathbf{V}}(\Pi_{i,\mathfrak{m},n})$ is a deformation of the
2-dimensional Galois representation
$\check{\mathbf{V}}(\Pi_{i,\mathfrak{m},1})$ to $A_{i}/{\mathfrak{m}}^{n}$
with determinant $\zeta\varepsilon$. It follows from [18, §4.1, Theorem 3.17]
that there is an $\mathcal{O}$-algebra map
$\theta_{i,\mathfrak{m},n}:R^{\mathrm{ps},\zeta\varepsilon}_{\operatorname{tr}\bar{\rho}_{\mathfrak{B}}}\rightarrow
A_{i}/\mathfrak{m}^{n}$ such that the specialization of $T$ along
$\theta_{i,\mathfrak{m},n}$ is equal to
$\operatorname{tr}_{A_{i}/{\mathfrak{m}}^{n}}\check{\mathbf{V}}(\Pi_{i,\mathfrak{m},n})$.
This map induces a homomorphism of $\mathcal{O}$-algebras

$R^{\mathrm{ps},\zeta\varepsilon}_{\operatorname{tr}\bar{\rho}_{\mathfrak{B}}}\llbracket\mathcal{G}_{\mathbb{Q}_{p}}\rrbracket\rightarrow\operatorname{End}_{A_{i}/\mathfrak{m}^{n}}(\check{\mathbf{V}}(\Pi_{i,\mathfrak{m},n})),$

and Cayley–Hamilton for $M_{2}(A_{i}/\mathfrak{m}^{n})$ implies that $J$ lies
in the kernel of this map.

If $\check{\mathbf{V}}(\Pi_{i,\mathfrak{m},n})$ is given by a character
$\chi_{i,\mathfrak{m},n}:\mathcal{G}_{\mathbb{Q}_{p}}\rightarrow(A_{i}/\mathfrak{m}^{n})^{\times}$,
then the same argument applies to the representation
$\chi_{i,\mathfrak{m},n}\oplus\chi_{i,\mathfrak{m},n}^{-1}\zeta\varepsilon$,
so that we get a map

$R^{\mathrm{ps},\zeta\varepsilon}_{\operatorname{tr}\bar{\rho}_{\mathfrak{B}}}\llbracket\mathcal{G}_{\mathbb{Q}_{p}}\rrbracket/J\rightarrow\operatorname{End}_{A_{i}/\mathfrak{m}^{n}}(\chi_{i,\mathfrak{m},n}\oplus\chi_{i,\mathfrak{m},n}^{-1}\zeta\varepsilon).$

Its image commutes with the idempotent which projects onto the direct summand
$\chi_{i,\mathfrak{m},n}$. Hence, the image is contained in
$\operatorname{End}_{A_{i}/\mathfrak{m}^{n}}(\chi_{i,\mathfrak{m},n})\times\operatorname{End}_{A_{i}/\mathfrak{m}^{n}}(\chi_{i,\mathfrak{m},n}^{-1}\zeta\varepsilon)$
and we may project to
$\operatorname{End}_{A_{i}/\mathfrak{m}^{n}}(\chi_{i,\mathfrak{m},n})$ to
obtain the required homomorphism. ∎

In Propositions 4.18 and 3.1 we have established surjections

$\alpha:\mathcal{O}\llbracket\mathcal{G}_{\mathbb{Q}_{p}}\rrbracket\twoheadrightarrow\operatorname{End}_{E_{\mathfrak{B}}^{\prime}}^{\mathrm{cont}}(\check{\mathbf{V}}(P_{\mathfrak{B}}^{\prime})),\quad\beta:\mathcal{O}\llbracket\mathcal{G}_{\mathbb{Q}_{p}}\rrbracket\twoheadrightarrow
R^{\mathrm{ps},\zeta\varepsilon}_{\operatorname{tr}\bar{\rho}_{\mathfrak{B}}}\llbracket\mathcal{G}_{\mathbb{Q}_{p}}\rrbracket/J.$

######  Theorem 6.3.

The above maps induce a surjection

$R^{\mathrm{ps},\zeta\varepsilon}_{\operatorname{tr}\bar{\rho}_{\mathfrak{B}}}\llbracket\mathcal{G}_{\mathbb{Q}_{p}}\rrbracket/J\twoheadrightarrow\operatorname{End}_{E_{\mathfrak{B}}^{\prime}}^{\mathrm{cont}}(\check{\mathbf{V}}(P_{\mathfrak{B}}^{\prime})).$

In particular, $E_{\mathfrak{B}}^{\prime}$ and $Z_{\mathfrak{B}}^{\prime}$ are
finite over
$R^{\mathrm{ps},\zeta\varepsilon}_{\operatorname{tr}\bar{\rho}_{\mathfrak{B}}}$
and hence noetherian.

###### Proof.

For the first part we have to show that
$\operatorname{Ker}\beta\subset\operatorname{Ker}\alpha$. Let
$M=\check{\mathbf{V}}(P_{\mathfrak{B}}^{\prime})$ and
$\mathrm{m}_{i,\mathfrak{m},n}:=\mathrm{m}(\Pi_{i,\mathfrak{m},n})$ with
$i,\mathfrak{m},n$ as in Corollary 5.9. Then the assumptions in Lemma 2.6 are
satisfied by Corollary 5.9. It follows that the kernel of $\alpha$ is given by
$\cap_{i\in I}\cap_{\mathfrak{m},n}\mathfrak{b}_{i,\mathfrak{m},n}$, where
$\mathfrak{b}_{i,\mathfrak{m},n}$ is the
$\mathcal{O}\llbracket\mathcal{G}_{\mathbb{Q}_{p}}\rrbracket$-annihilator of

$\mathrm{m}_{i,\mathfrak{m},n}\otimes_{E_{\mathfrak{B}}^{\prime}}\check{\mathbf{V}}(P_{\mathfrak{B}}^{\prime})\cong\check{\mathbf{V}}(\mathrm{m}_{i,\mathfrak{m},n}\otimes_{E_{\mathfrak{B}}^{\prime}}P_{\mathfrak{B}}^{\prime})\cong\check{\mathbf{V}}(\Pi_{i,\mathfrak{m},n}).$

Since the action of
$\mathcal{O}\llbracket\mathcal{G}_{\mathbb{Q}_{p}}\rrbracket$ on
$\check{\mathbf{V}}(\Pi_{i,\mathfrak{m},n})$ factors through
$R^{\mathrm{ps},\zeta\varepsilon}_{\operatorname{tr}\bar{\rho}_{\mathfrak{B}}}\llbracket\mathcal{G}_{\mathbb{Q}_{p}}\rrbracket/J$
by Lemma 6.2, $\mathfrak{b}_{i,\mathfrak{m},n}$ contains the kernel of $\beta$
and hence $\operatorname{Ker}\beta\subset\operatorname{Ker}\alpha$.

The second assertion is a consequence of the first assertion and the
finiteness of
$R^{\mathrm{ps},\zeta\varepsilon}_{\operatorname{tr}\bar{\rho}_{\mathfrak{B}}}\llbracket\mathcal{G}_{\mathbb{Q}_{p}}\rrbracket/J$
over
$R^{\mathrm{ps},\zeta\varepsilon}_{\operatorname{tr}\bar{\rho}_{\mathfrak{B}}}$
[48, Proposition 3.6]. ∎

######  Corollary 6.4.

$E_{\mathfrak{B}}$ and $Z_{\mathfrak{B}}$ are finite over
$R^{\mathrm{ps},\zeta\varepsilon}_{\operatorname{tr}\bar{\rho}_{\mathfrak{B}}}$
and hence noetherian.

###### Proof.

By [39, Lemma 10.26], we have

$\displaystyle\operatorname{End}_{\mathfrak{C}(\mathcal{O})}(M_{\mathfrak{B}})$
$\displaystyle\cong\operatorname{End}_{\mathfrak{Q}(\mathcal{O})}(\mathcal{T}M_{\mathfrak{B}}),$
$\displaystyle\operatorname{Hom}_{\mathfrak{C}(\mathcal{O})}(P_{\mathfrak{B}}^{\prime},M_{\mathfrak{B}})$
$\displaystyle\cong\operatorname{Hom}_{\mathfrak{Q}(\mathcal{O})}(\mathcal{T}P_{\mathfrak{B}}^{\prime},\mathcal{T}M_{\mathfrak{B}}),$

since
$M_{\mathfrak{B}}^{\operatorname{SL}_{2}(\mathbb{Q}_{p})}=(M_{\mathfrak{B}})_{\operatorname{SL}_{2}(\mathbb{Q}_{p})}=(P_{\mathfrak{B}}^{\prime})_{\operatorname{SL}_{2}(\mathbb{Q}_{p})}=0$.
Let
$\mathrm{m}:=\operatorname{Hom}_{\mathfrak{C}(\mathcal{O})}(P_{\mathfrak{B}}^{\prime},M_{\mathfrak{B}})$,
which is a finitely generated right $E_{\mathfrak{B}}^{\prime}$-module by
Corollary 4.13. Moreover, we have

$\operatorname{End}_{E_{\mathfrak{B}}^{\prime}}(\mathrm{m})\cong\operatorname{End}_{\mathfrak{Q}(\mathcal{O})}(\mathcal{T}M_{\mathfrak{B}})\cong\operatorname{End}_{\mathfrak{C}(\mathcal{O})}(M_{\mathfrak{B}})$

by Proposition 4.10 (2) and the isomorphism above. Theorem 6.3 implies that
the conditions of Lemma 4.11 are satisfied with $E=E_{\mathfrak{B}}^{\prime}$
and $Z=Z_{\mathfrak{B}}^{\prime}$. Thus
$\operatorname{End}_{\mathfrak{C}(\mathcal{O})}(M_{\mathfrak{B}})$ and its
centre are finite over $Z_{\mathfrak{B}}^{\prime}$, and thus finite over
$R^{\mathrm{ps},\zeta\varepsilon}_{\operatorname{tr}\bar{\rho}_{\mathfrak{B}}}$
by Theorem 6.3, and hence noetherian. This implies the corollary since
$\operatorname{End}_{\mathfrak{C}(\mathcal{O})}(M_{\mathfrak{B}})\cong
E_{\mathfrak{B}}$ and
$Z(\operatorname{End}_{\mathfrak{C}(\mathcal{O})}(M_{\mathfrak{B}}))\cong
Z_{\mathfrak{B}}$ by Proposition 4.15. ∎

######  Remark 6.5.

Since $Z_{\mathfrak{B}}$ is noetherian the $\mathfrak{m}$-adic topology
coincides with the linearly compact topology in Lemma 4.1.

Let us spell out the properties of the map
$R^{\mathrm{ps},\zeta\varepsilon}_{\operatorname{tr}\bar{\rho}_{\mathfrak{B}}}\rightarrow
Z_{\mathfrak{B}}$ constructed in Corollary 6.4. Since $Z_{\mathfrak{B}}$ acts
functorially on every object in
$\operatorname{\mathfrak{C}}(\mathcal{O})_{\mathfrak{B}}$, the homomorphism
$R^{\mathrm{ps},\zeta\varepsilon}_{\operatorname{tr}\bar{\rho}_{\mathfrak{B}}}\rightarrow
Z_{\mathfrak{B}}$ induces a functorial ring homomorphism

$c_{M}:R^{\mathrm{ps},\zeta\varepsilon}_{\operatorname{tr}\bar{\rho}_{\mathfrak{B}}}\rightarrow\operatorname{End}_{\operatorname{\mathfrak{C}}(\mathcal{O})}(M),$

for every object $M$ in $\operatorname{\mathfrak{C}}(\mathcal{O})$. Since
$\check{\mathbf{V}}$ is a functor, it induces a ring homomorphism

$\operatorname{End}_{\operatorname{\mathfrak{C}}(\mathcal{O})}(M)\rightarrow\operatorname{End}^{\mathrm{cont}}_{\mathcal{G}_{\mathbb{Q}_{p}}}(\check{\mathbf{V}}(M)),\quad\varphi\mapsto\check{\mathbf{V}}(\varphi).$

We denote the action of $\mathcal{G}_{\mathbb{Q}_{p}}$ on
$\check{\mathbf{V}}(M)$ by $\rho_{\check{\mathbf{V}}(M)}$. Finally, for all
$g\in\mathcal{G}_{\mathbb{Q}_{p}}$ we may evaluate the universal
pseudorepresentation $T:\mathcal{G}_{\mathbb{Q}_{p}}\rightarrow
R^{\mathrm{ps},\zeta\varepsilon}_{\operatorname{tr}\bar{\rho}_{\mathfrak{B}}}$
at $g\in\mathcal{G}_{\mathbb{Q}_{p}}$ to obtain an element $T(g)\in
R^{\mathrm{ps},\zeta\varepsilon}_{\operatorname{tr}\bar{\rho}_{\mathfrak{B}}}$.

######  Proposition 6.6.

For each $M\in\operatorname{\mathfrak{C}}(\mathcal{O})_{\mathfrak{B}}$ and
each $g\in\mathcal{G}_{\mathbb{Q}_{p}}$

$\check{\mathbf{V}}(c_{M}(T(g)))=\rho_{\check{\mathbf{V}}(M)}(g)+\rho_{\check{\mathbf{V}}(M)}(g^{-1})\zeta\varepsilon(g),$

in
$\operatorname{End}_{\mathcal{G}_{\mathbb{Q}_{p}}}^{\mathrm{cont}}(\check{\mathbf{V}}(M))$.

###### Proof.

Since $g^{2}-T(g)g+\zeta\varepsilon(g)=0$ in
$R^{\mathrm{ps},\zeta\varepsilon}_{\operatorname{tr}\bar{\rho}_{\mathfrak{B}}}\llbracket\mathcal{G}_{\mathbb{Q}_{p}}\rrbracket/J$,
the equality $T(g)\operatorname{id}=g+\zeta\varepsilon(g)g^{-1}$ holds in that
ring. The rest is just unravelling the definitions. ∎

Since $P_{\mathfrak{B}}$ is a projective generator for
$\operatorname{\mathfrak{C}}(\mathcal{O})_{\mathfrak{B}}$, the functor

$N\mapsto\mathrm{m}(N):=\operatorname{Hom}_{\operatorname{\mathfrak{C}}(\mathcal{O})}(P_{\mathfrak{B}},N)$

induces an equivalence of categories between
$\operatorname{\mathfrak{C}}(\mathcal{O})_{\mathfrak{B}}$ and the category of
right pseudo-compact $E_{\mathfrak{B}}$-modules. The inverse functor is given
by
$\mathrm{m}\mapsto\mathrm{m}\operatorname{\widehat{\otimes}}_{E_{\mathfrak{B}}}P_{\mathfrak{B}}$.

######  Corollary 6.7.

For $N$ in $\operatorname{\mathfrak{C}}(\mathcal{O})_{\mathfrak{B}}$ the
following assertions are equivalent:

  1. (1)

there is a surjection $P_{\mathfrak{B}}^{\oplus n}\twoheadrightarrow N$ for
some $n\geq 1$;

  2. (2)

$\mathrm{m}(N)$ is a finitely generated $E_{\mathfrak{B}}$-module;

  3. (3)

$\mathrm{m}(N)$ is a finitely generated
$R^{\mathrm{ps},\zeta\varepsilon}_{\operatorname{tr}\bar{\rho}_{\mathfrak{B}}}$-module;

  4. (4)

$k\operatorname{\widehat{\otimes}}_{R^{\mathrm{ps},\zeta\varepsilon}_{\operatorname{tr}\bar{\rho}_{\mathfrak{B}}}}N$
is of finite length in $\operatorname{\mathfrak{C}}(\mathcal{O})$;

  5. (5)

the cosocle of $N$ in $\operatorname{\mathfrak{C}}(\mathcal{O})$ is of finite
length.

The equivalent conditions hold if $N$ is finitely generated over
$\mathcal{O}\llbracket H\rrbracket$ for a compact open subgroup $H$ of $G$.

###### Proof.

(1) implies (2) since $\mathrm{m}$ is exact. (2) implies (3) by Corollary 6.4.
Since

$k\operatorname{\widehat{\otimes}}_{R^{\mathrm{ps},\zeta\varepsilon}_{\operatorname{tr}\bar{\rho}_{\mathfrak{B}}}}\mathrm{m}(N)\cong\mathrm{m}(k\operatorname{\widehat{\otimes}}_{R^{\mathrm{ps},\zeta\varepsilon}_{\operatorname{tr}\bar{\rho}_{\mathfrak{B}}}}N),$

and the functor $\mathrm{m}$ is an anti-equivalence (3) implies (4). Let
$N\twoheadrightarrow\operatorname{cosoc}(N)$ be the cosocle of $N$. Since the
maximal ideal of
$R^{\mathrm{ps},\zeta\varepsilon}_{\operatorname{tr}\bar{\rho}_{\mathfrak{B}}}$
acts trivially on every semi-simple object, the surjection factors through

$k\operatorname{\widehat{\otimes}}_{R^{\mathrm{ps},\zeta\varepsilon}_{\operatorname{tr}\bar{\rho}_{\mathfrak{B}}}}N\twoheadrightarrow\operatorname{cosoc}(N),$

and so (4) implies (5). If $\operatorname{cosoc}(N)$ is of finite length then
there is a surjection $\pi_{\mathfrak{B}}^{\oplus
n}\twoheadrightarrow\operatorname{cosoc}(N)$ for some $n\geq 1$. Since
$P_{\mathfrak{B}}$ is projective there is a map
$\varphi:P_{\mathfrak{B}}^{\oplus n}\rightarrow N$ lifting
$(P_{\mathfrak{B}})^{\oplus n}\twoheadrightarrow\pi_{\mathfrak{B}}^{\oplus
n}\twoheadrightarrow\operatorname{cosoc}(N)$. The cokernel of $\varphi$ will
have zero cosocle and hence $\varphi$ is surjective, so that (5) implies (1).

If $N$ is finitely generated over $\mathcal{O}\llbracket H\rrbracket$, which
we may assume to be pro-$p$, then $((N/\varpi N)^{\vee})^{H}$ is a finite
dimensional $k$-vector space, and hence the $G$-socle of $N^{\vee}$ is of
finite length, which dually implies that (5) holds. ∎

Since every irreducible in $\mathfrak{B}$ is admissible, its Pontryagin dual
is finitely generated over $\mathcal{O}\llbracket H\rrbracket$ for any compact
open subgroup $H$ of $G$. It follows from Corollary 6.7 (4) that
$k\operatorname{\widehat{\otimes}}_{R^{\mathrm{ps},\zeta\varepsilon}_{\operatorname{tr}\bar{\rho}_{\mathfrak{B}}}}P_{\mathfrak{B}}$
is also a finitely generated $\mathcal{O}\llbracket H\rrbracket$-module. This
implies that the assumptions made in section 4 of [39] are satisfied with the
category $\operatorname{\mathfrak{C}}(\mathcal{O})$ in [39, §4] equal to
$\operatorname{\mathfrak{C}}(\mathcal{O})_{\mathfrak{B}}$, and we will record
some consequences below.

###  6.2. Banach space representations

The category $\operatorname{Ban}^{\mathrm{adm}}_{G,\zeta}(L)$ decomposes into
a direct sum of categories [39, Proposition 5.36]:

(19)
$\operatorname{Ban}^{\mathrm{adm}}_{G,\zeta}(L)\cong\bigoplus_{\mathfrak{B}\in\operatorname{Irr}_{G,\zeta}/\sim}\operatorname{Ban}^{\mathrm{adm}}_{G,\zeta}(L)_{\mathfrak{B}}$

where the objects of
$\operatorname{Ban}^{\mathrm{adm}}_{G,\zeta}(L)_{\mathfrak{B}}$ are those
$\Pi$ in $\operatorname{Ban}^{\mathrm{adm}}_{G,\zeta}(L)$ such that for every
open bounded $G$-invariant lattice $\Theta$ in $\Pi$ the irreducible
subquotients of $\Theta\otimes_{\mathcal{O}}k$ lie in $\mathfrak{B}$. This
condition is equivalent to require $\Theta^{d}$ to be an object of
$\operatorname{\mathfrak{C}}(\mathcal{O})_{\mathfrak{B}}$.

As in the previous subsection we fix a block $\mathfrak{B}$ consisting of
absolutely irreducible representations. Let
$\operatorname{Mod}^{\mathrm{fg}}_{E_{\mathfrak{B}}[1/p]}$ be the category of
finitely generated right $E_{\mathfrak{B}}[1/p]$-modules. The functor

$\mathrm{m}:\operatorname{Ban}^{\mathrm{adm}}_{G,\zeta}(L)_{\mathfrak{B}}\rightarrow\operatorname{Mod}^{\mathrm{fg}}_{E_{\mathfrak{B}}[1/p]},\quad\Pi\mapsto\mathrm{m}(\Pi):=\operatorname{Hom}_{\operatorname{\mathfrak{C}}(\mathcal{O})}(P_{\mathfrak{B}},\Theta^{d})\otimes_{\mathcal{O}}L,$

where $\Theta$ is any open bounded $G$-invariant lattice in $\Pi$, is exact,
contravariant and fully faithful by [39, Lemma 4.45]. Moreover, it induces an
anti-equivalence of categories

(20)
$\mathrm{m}:\operatorname{Ban}^{\mathrm{adm}}_{G,\zeta}(L)_{\mathfrak{B}}^{\mathrm{fl}}\overset{\cong}{\longrightarrow}\operatorname{Mod}^{\mathrm{fl}}_{E_{\mathfrak{B}}[1/p]},$

where the superscript $\mathrm{fl}$ indicates the subcategories of objects of
finite length in the respective categories, see [39, Theorem 4.34]. We write
anti-equivalence instead of equivalence to indicate that $\mathrm{m}$ is
contravariant.

If $\mathfrak{m}$ is a maximal ideal of
$R^{\mathrm{ps},\zeta\varepsilon}_{\operatorname{tr}\bar{\rho}_{\mathfrak{B}}}[1/p]$
then we let
$\operatorname{Ban}^{\mathrm{adm}}_{G,\zeta}(L)_{\mathfrak{B},\mathfrak{m}}^{\mathrm{fl}}$
be the full subcategory of $\operatorname{Ban}^{\mathrm{adm}}_{G,\zeta}(L)$
consisting of finite length Banach space representations, which are killed by
some power of $\mathfrak{m}$. The functor $\mathrm{m}$ induces an anti-
equivalence between this category and the category of
$E_{\mathfrak{B}}[1/p]$-modules of finite length, which are killed by a power