# Inequalities governing the peripheral spike activity

Willy Wong<EMAIL_ADDRESS>Edward S. Rogers Department of Electrical
and Computer Engineering and Institute of Biomedical Engineering, University
of Toronto, Toronto, ON M5S3G4, Canada

###### Abstract

A fundamental inequality governing the spike activity of peripheral neurons is
derived and tested against auditory data. This inequality states that the
steady-state firing rate below saturation lies between the arithmetic and
geometric means of the spontaneous and peak activities during adaptation.
Implications towards the development of computational auditory models are
explored.

##  I  Introduction

Rate coding of auditory neurons is the first step in the processing of sound
by the nervous system [kiang1965discharge, heil2015basic]. The mechanism of
transduction from sound to neural response is an important but unsolved
problem, e.g. meddis2010computational, rudnicki2015modeling.

A recent paper showed that intensity coding in many modalities and animal
species follows a simple equation [wong2021consilience]. This equation governs
the three fixed points in a typical adaptation curve: The steady-state
activity after long term stimulation (SS) equals the geometric mean of the
spontaneous activity (SR) and the peak activity (PR) just after stimulus
onset. That is,

$\text{SS}=\sqrt{\text{PR}\times\text{SR}}$ (1)

This relationship was tested against 250 measurements of adaptation across
many sensory modalities (hearing, vision, taste, smell, touch, etc.), animals
of different phyla, and in data spanning nearly 100 years since the original
recordings of Edgar Adrian [adrian1926impulsesa, adrian1926impulsesb]. This
suggests that (1) is a fundamental equation of neurophysiology.

While (1) was found to work in different modalities, hearing is where the
testing of adaptation can be carried out most rigorously. There are a number
of reasons for this. Adaptation involves repeated measurements from stimuli
held constant. The signal can be controlled to a high degree of accuracy
unlike in other modalities like temperature, taste and smell, e.g.
baylin1979temporal. Secondly, measurements of visual ganglion cells often
involve lateral contributions from other units making it difficult to probe
single unit responses to light [enroth1973adaptation]. Thus, in many respects,
the auditory system is ideal for systematic studies in adaptation.

There are a number of auditory studies that have explored how adaptation
changes as a function of stimulus parameters. When (1) is obeyed across
different stimulus intensities, the predicted relationship between SS and PR
is a straight-line on a double-log plot with slope of 1/2. Several studies
were found to obey this relationship, e.g. fay1978coding, westerman1984rapid,
yates1985very, sumner2012auditory. See wong2021consilience for more details.

##  II Violations to the geometric law

Consilience can be used to prove the generality of (1)
[wilson2000consilience]. When scores of unrelated measurements obey the same
law under vastly different conditions, this constitutes strong evidence that
the law is obeyed widely. However, consilience is less useful in the
determination of precise mathematical relationships.

One example would be the adaptation measurements reported in smith1975short.
Virtually all of the data show a strong linear relationship between peak
versus steady-state activity (see panels a-g in Fig. 1). This indicates a
violation of (1). The paper goes one step further to propose a remarkably
simple equation governing the peak and steady-states:
$\text{SS}=\text{PR}/2.5$. While this equation appears to fit the data well,
it cannot be the correct functional relationship between the steady-state and
peak activity for the following reason: A unit that undergoes adaptation to a
stimulus level of zero intensity must satisfy the condition that both PR and
SS equal the spontaneous activity or SR. A more sensible expression relating
SS to PR that can account for the straight line dependence will be derived
later.

The geometric law was first predicted theoretically in wong2020rate. We will
quickly review this theory before deriving a new inequality which will
encompass both the geometric law as well violations to this law. Implications
of the inequality to the development of computational auditory models are also
discussed.

##  III A theoretical basis of the peripheral activity

An approach to understanding the rate coding of sensory peripheral neurons in
terms of measurement uncertainty or entropy was most recently expounded in
wong2020rate following developments in norwich1977information,
norwich1993information, norwich1995universal, wong1997physics,
wong2013perceptual. The idea behind the theory is straightforward: The sensory
receptor samples the stimulus repeatedly and averages these samples. By the
central limit theorem, the uncertainty in the sample mean is normally
distributed when sample size is large. Calculating the Shannon entropy under
finite resolution yields an equation relating entropy to the variance of the
measurements and the sample size. Three final steps are required to obtain an
equation governing the rate response. First, the association of the variance
with the mean of the stimulus (i.e. the strength of stimulation) via a power
function; second, the sample size approaches its optimal value by a first-
order relaxation process; third, the association of firing rate with the
measurement uncertainty or entropy. This last equation in particular seeks to
change the very nature of how we understand the sensory coding of intensity
[norwich1993information].

Written out in full, the equations are

$\displaystyle F=kH$ (2) $\displaystyle
H=\dfrac{1}{2}\log\left(1+\dfrac{\beta\left(I+\delta I\right)^{p}}{m}\right)$
(3) $\displaystyle\dfrac{dm}{dt}=-a(m-m_{eq})$ (4) $\displaystyle
m_{eq}=(I+\delta I)^{p/2}$ (5)

where $F$ is the firing rate of the peripheral neuron, $H$ the entropy, $I$
the stimulus magnitude and $t$ the duration of the stimulation. $m$ is the
sample size and $m_{eq}$ its optimal or equilibrium value. The other terms are
parameters which are assumed to be positive and constant for a given unit.
These equations were developed without consideration of underlying mechanism,
derived in full in wong2020rate and tested successfully against a number of
experimental paradigms including experiments conducted on auditory units. The
evaluation of (2-5) requires knowledge of the five parameters
$k,\beta,p,\delta I,a$ which are obtained by fitting the equations to data.
But their precise determination can be difficult as the equations are prone to
overfitting.

Adaptation is defined as the condition where a unit is initially in silence
and in steady-state; a stimulus is turned on and held constant for $t\geq 0$
[benda2014spike]. We can solve for the adaptation response as follows. The
process is first described conceptually. A change in the stimulus level brings
about measurement uncertainty and the sample size is increased as a
consequence. The optimal sample size is calculated from (5) which is dependent
on intensity, and solving the differential equation in (4) gives the change in
sample size as a function of time. Since the numerator of the argument inside
the logarithm in (3) is constant, increasing the value of $m$ results in a
monotonic fall in uncertainty $H$ (and therefore $F$) giving rise to the decay
in spike activity observed in adaptation. Next we proceed to a mathematical
description.

A constant level of stimulus implies that $m_{eq}$ in (5) is constant, i.e.
$m_{eq}=(I+\delta I)^{p/2}$ where $I$ is the level of stimulation. From this,
we solve for the sample size $m$ as a function of time from (4) to obtain

$m(t)=m\left(0\right)e^{-at}+m_{eq}\left(1-e^{-at}\right)$ (6)

Due to the continuity condition of (4), $m(0)$ equals the value of $m$ just
prior to the stimulus being turned on. For $t\leq 0$, we have $I=0$ and the
unit is in steady-state. Therefore, $m(0)=\delta I^{p/2}$. Substituting $m$
into (3) and (2) is all that is needed to solve the neuronal response to a
constant stimulus.

The solutions of these equations to a step function as well as to many other
input types (square wave, sinusoidal, ramp, etc) compare well to experimental
measurements and are detailed in wong2020rate. The response can also be
carried out numerically using just a few simple lines of computer code (see
Appendix A). There are, however, several drawbacks to this approach. The
predicted response falls short in two key aspects: (a) There is no saturation
of response at high intensities; (b) The predictions are purely deterministic
and do not account for the stochastic nature of the response.

Next we turn to the derivation of several inequalities governing the predicted
response of adaptation activity.

##  IV Inequalities concerning peripheral activity

We begin by defining $x=\beta\delta I^{p/2}$ and $y=\beta\left(I+\delta
I\right)^{p}/\delta I^{p/2}$. Note that $0\leq x\leq y$ due to the fact that
all of the parameters are positive and $I\geq 0$. We also set $k/2=1$ as it
has no bearing on the results.

From this, the spike activity $F$ at arbitrary time $t\geq 0$ is obtained by
substituting (6) into (3) and (2) to obtain

$\displaystyle F$
$\displaystyle=\frac{1}{2}k\log\left(1+\dfrac{\beta\left(I+\delta
I\right)^{p}}{\delta I^{p/2}e^{-at}+\left(I+\delta
I\right)^{p/2}\left(1-e^{-at}\right)}\right)$ (7)
$\displaystyle=\log\left(1+\dfrac{xy}{\alpha
x+\left(1-\alpha\right)\sqrt{xy}}\right)$ (8)

where $\alpha=e^{-at}$. Since $t\geq 0$, both $\alpha$ and $1-\alpha$ are non-
negative numbers. Observe that

$F\geq\log\left(1+\dfrac{xy}{\alpha\sqrt{xy}+\left(1-\alpha\right)\sqrt{xy}}\right)=\log\left(1+\sqrt{xy}\right)$
(9)

and

$F\leq\log\left(1+\dfrac{xy}{\alpha
x+\left(1-\alpha\right)x}\right)=\log\left(1+y\right)$ (10)

where $x\leq\sqrt{xy}\leq y$ was used. We recognize the lower bound in (9) to
be the activity as $t\rightarrow\infty$ in (8) and the upper bound in (10) to
be the activity as $t\rightarrow 0$. Moreover, since $\log\left(1+y\right)$ is
the largest attainable value of (8), we formally define this as the peak
activity $\text{PR}=\log\left(1+y\right)$ and the steady-state activity
$\text{SS}=\log\left(1+\sqrt{xy}\right)$. Thus, the equations show that $F$ is
bounded by both the peak and steady-state activities during adaptation.

Next we show that the steady-state activity obeys the inequality

$\sqrt{\text{PR}\times\text{SR}}\leq\text{SS}\leq\frac{\text{PR}+\text{SR}}{2}$
(11)

where $\text{SR}=\log\left(1+x\right)$ is the spontaneous activity obtained by
substituting $\lim\limits_{I\rightarrow 0}y=x$ into the definition of SS. That
is, the steady-state activity is bounded by the geometric and arithmetic mean
of the spontaneous and peak activities. This remarkable result is independent
of intensity or the choice of any of the parameters in the equation
$k,\beta,p,\delta I,a$.

First, we establish the upper bound for SS:


# An application of a theorem of Emerton to mod $p$ representations of
$\mathrm{GL}_{2}$

Yongquan HU

###### Abstract.

Let $p$ be a prime and $L$ be a finite extension of $\mathbb{Q}_{p}$. We study
the ordinary parts of $\mathrm{GL}_{2}(L)$-representations arised in the mod
$p$ cohomology of Shimura curves attached to indefinite division algebras
which splits at a finite place above $p$. The main tool of the proof is a
theorem of Emerton [10].

###### Contents

  1. 1 Introduction
  2. 2 Representation theoretic preparation
    1. 2.1 Principal series
    2. 2.2 Computation of $H^{1}(K_{1}/Z_{1},\pi)$
    3. 2.3 Ordinary part
    4. 2.4 A definition
  3. 3 Local results
    1. 3.1 Construction of Breuil-Paškūnas
    2. 3.2 The case $f=2$
  4. 4 Global results
    1. 4.1 A theorem of Emerton
    2. 4.2 Application
  5. 5 Appendix

##  1\. Introduction

Let $p$ be a prime number and $L$ a finite extension of $\mathbb{Q}_{p}$. Let
$G=\mathrm{GL}_{2}(L)$. If $L=\mathbb{Q}_{p}$, the work of Barthel-Livné [1]
and of Breuil [3] gave a complete classification of irreducible smooth
representations of $G$ over $\overline{\mathbb{F}}_{p}$ with a central
character (this last restriction is now removed by Berger [2]). However, when
$F\neq\mathbb{Q}_{p}$, the situation is much more complicated and a large part
of the theory is still mysterious. The main difficulty lies in the study of
supersingular representations, which are irreducible smooth representations of
$G$ which do not arise as subquotients of parabolic inductions. For example,
when $[L:\mathbb{Q}_{p}]=2$, Schraen has shown that supersingular
representations are not of finite presentation [20] (similar result was first
proved in [15] if $L$ is a finite extension of $\mathbb{F}_{p}[[t]]$).

Though the general theory of smooth representations of $G$ could be very
weird, when $L$ is unramified over $\mathbb{Q}_{p}$, Breuil and Paškūnas were
able to construct some ‘nicer’ ones in [5]. Precisely, they constructed
families of admissible smooth representations (by local methods) which are
related to two dimensional continuous
$\overline{\mathbb{F}}_{p}$-representations of
$\mathrm{Gal}(\overline{\mathbb{Q}}_{p}/L)$ via the Buzzard-Diamond-Jarvis
conjecture [7]. Recent work of Emerton-Gee-Savitt [11] shows that this
construction is indeed very important and tightly related to the mod $p$ local
Lancorollaryglands program.

To state our main result we need some notations. Let $F$ be a totally real
field and $D$ a quaternion algebra with center $F$ which splits at exactly one
infinite place. One can associate to $D$ a system of Shimura curves
$(X_{U})_{U}$ indexed by open compact subgroups of
$(D\otimes_{\mathbb{Q}}\mathbb{A}_{f})^{\times}$ (where $\mathbb{A}_{f}$
denotes the ring of finite adèles of $\mathbb{Q}$), which are projective and
smooth over $F$. Put

$S^{D}(\overline{\mathbb{F}}_{p}):=\varinjlim
H^{1}_{\textrm{\'{e}t}}(X_{U,\overline{\mathbb{Q}}},\overline{\mathbb{F}}_{p})$

where the inductive limit is taken over all open compact subgroups of
$(D\otimes_{\mathbb{Q}}\mathbb{A}_{f})^{\times}$. Let
$\overline{\rho}:\mathrm{Gal}(\overline{\mathbb{Q}}/F)\rightarrow\mathrm{GL}_{2}(\overline{\mathbb{F}}_{p})$
be an irreducible, continuous, totally odd representation. Assume moreover
$\overline{\rho}$ is modular, which means that

$\pi^{D}(\overline{\rho}):=\mathrm{Hom}_{\mathrm{Gal}(\overline{\mathbb{Q}}/F)}(\overline{\rho},S^{D}(\overline{\mathbb{F}}_{p}))$

is non-zero. A conjecture of Buzzard, Diamond and Jarvis [7] says that the
space $\pi^{D}(\overline{\rho})$ decomposes as a restricted tensor product

$\pi^{D}(\overline{\rho})\cong{\bigotimes}_{w}^{\prime}\pi_{w}$

where each factor $\pi_{w}$ is an admissible smooth representation of
$(D\otimes_{F}F_{w})^{\times}$ and depends only on the restriction of
$\overline{\rho}$ at $w$. Note that, when $w|p$, the local factor $\pi_{w}$ is
supposed to be the right representation in the mod $p$ local Langlands (or
Jacquet-Langlands) program, and many important properties about it have been
proved, see e.g. [14], [4], [11].

In this article we prove some extra property about $\pi_{w}$, $w|p$, when the
restriction of $\overline{\rho}$ at $w$ is reducible indecomposable and
generic (see §3), and when $F$ is unramified at $w$. In this case, it is hoped
that $\pi_{w}$ has a filtration of length $f:=[F_{w}:\mathbb{Q}_{p}]$ of the
form (where $(\pi_{i})_{i}$ denote the graded pieces of the filtration)

(1.1) $\pi_{0}\ \textbf{---}\ \pi_{1}\ \textbf{---}\ \cdots\ \textbf{---}\
\pi_{f}$

such that $\pi_{i}$ is a principal series if $i\in\\{0,f\\}$ and supersingular
otherwise. Our main theorem is as follows.

######  Theorem 1.1.

Assume the decomposition
$\pi^{D}(\overline{\rho})\cong\otimes^{\prime}_{w}\pi_{w}$ holds. Let $w$ be a
place above $p$. Assume that $D$ splits at $w$, and $\overline{\rho}_{w}$ is
reducible indecomposable and generic. Assume that $F$ is unramified at $w$ and
$[F_{w}:\mathbb{Q}_{p}]\geq 2$. Then $\pi_{w}$ contains a unique sub-
representation $\pi$ which is of length 2 and which fits into an exact
sequence

$0\rightarrow\pi_{0}\rightarrow\pi\rightarrow\pi_{1}\rightarrow 0$

such that $\pi_{0}$ is a principal series and $\pi_{1}$ is a supersingular
representation.

In other words, we prove that $\pi_{w}$ contains the first two graded pieces
in (1.1). The main observation in the proof of Theorem 1.1 is a theorem of
Emerton [10], which allows us to control the ordinary part of $\pi_{w}$.

Back to the local situation, we get the following consequence.

######  Corollary 1.2.

Let $L$ be a finite unramified extension of $\mathbb{Q}_{p}$ of degree $f\geq
2$. There exist a principal series $\pi_{0}$ and a supersingular
representation $\pi_{1}$ such that

$\mathrm{Ext}^{1}_{G}(\pi_{1},\pi_{0})\neq 0.$

If moreover $[L:\mathbb{Q}_{p}]=2$, and if $\pi_{2}$ denotes the $f$-th smooth
dual of $\pi_{0}$ (see [17, Prop. 5.4]), then

$\mathrm{Ext}^{1}_{G}(\pi_{2},\pi_{1})\neq 0.$

The existence of extensions in Corollary 1.2 is not known before. Remark that
such extensions do not exist when $L=\mathbb{Q}_{p}$ ([19]). We also prove, in
§5, that when $L$ is a local field of characteristic $p$ there is no non-
trivial extension of a principal series by a supersingular representation.

The paper is organized as follows. In §2 we prove some necessary results about
smooth $\overline{\mathbb{F}}_{p}$-representations of $G$, especially the
structure of (irreducible) principal series. In §3, we recall the construction
of [5] and show, under certain assumption, that the representations they
constructed behave well. In §4, we show that this assumption is satisfied in
certain global situation. In the appendix (§5), we show the counterpart of
Theorem 1.1 is false if $L$ is of characteristic $p$.

##  2\. Representation theoretic preparation

In this section, $L$ denotes a finite extension of $\mathbb{Q}_{p}$, with ring
of integers $\mathcal{O}_{L}$, maximal ideal $\mathfrak{p}_{L}$, and residue
field (identified with) $\mathbb{F}_{q}=\mathbb{F}_{p^{f}}$. Fix a uniformiser
$\varpi$ of $L$; we take $\varpi=p$ when $L$ is unramified over
$\mathbb{Q}_{p}$. For $\lambda\in\mathbb{F}_{q}$,
$[\lambda]\in\mathcal{O}_{L}$ denotes its Teichmüller lifting.

Let $G=\mathrm{GL}_{2}(L)$, and define the following subgroups of $G$:

$P=\begin{pmatrix}{*}&{*}\\\ {0}&{*}\end{pmatrix},\ \
\overline{P}=\begin{pmatrix}{*}&{0}\\\ {*}&{*}\end{pmatrix},\ \
N=\begin{pmatrix}{1}&{*}\\\ {0}&{1}\end{pmatrix},\ \
T=\begin{pmatrix}{*}&{0}\\\ {0}&{*}\end{pmatrix},$

and let $Z$ be the center of $G$.

Let $K=\mathrm{GL}_{2}(\mathcal{O}_{L})$ and $K_{1}$ be the kernel of the
reduction morphism $K\twoheadrightarrow\mathrm{GL}_{2}(\mathbb{F}_{q})$. Let
$I\subset K$ denote the (upper) Iwahori subgroup and $I_{1}\subset I$ be the
pro-$p$-Iwahori subgroup. Let $N_{0}=N\cap K$, $T_{0}=T\cap K$ and
$T_{1}=T\cap K_{1}$.

We call a weight an irreducible representation of $K$ over
$\overline{\mathbb{F}}_{p}$, which is always an inflation of an irreducible
representation of $\mathrm{GL}_{2}(\mathbb{F}_{q})$. A weight is isomorphic to
([1, Prop. 1])

$\mathrm{Sym}^{r_{0}}\overline{\mathbb{F}}_{p}^{2}\otimes_{\overline{\mathbb{F}}_{p}}(\mathrm{Sym}^{r_{1}}\overline{\mathbb{F}}_{p}^{2})^{\mathrm{Fr}}\otimes\cdots\otimes_{\overline{\mathbb{F}}_{p}}(\mathrm{Sym}^{r_{f-1}}\overline{\mathbb{F}}_{p}^{2})^{\mathrm{Fr}^{f-1}}\otimes_{\overline{\mathbb{F}}_{p}}{\det}^{a}$

where $0\leq r_{i}\leq q-1$, $0\leq a\leq q-2$ and
$\mathrm{Fr}:\bigl{(}\begin{smallmatrix}{a}&{b}\\\
{c}&{d}\end{smallmatrix}\bigl{)}\mapsto\bigl{(}\begin{smallmatrix}{a^{p}}&{b^{p}}\\\
{c^{p}}&{d^{p}}\end{smallmatrix}\bigl{)}$ is the Frobenius on
$\mathrm{GL}_{2}(\mathbb{F}_{q})$. We denote this representation by
$(r_{0},...,r_{f-1})\otimes{\det}^{a}$.

###  2.1. Principal series

By the work of Barthel-Livné [1], smooth irreducible
$\overline{\mathbb{F}}_{p}$-representations111Although we choose to work with
$\overline{\mathbb{F}}_{p}$-representations in this section, all the results
hold true if we replace $\overline{\mathbb{F}}_{p}$ by a sufficiently large
finite extension of $\mathbb{F}_{p}$. of $G$ with a central character fall
into four classes:

  1. (i)

one-dimensional representations, i.e. characters.

  2. (ii)

principal series $\mathrm{Ind}_{P}^{G}\chi$, with $\chi\neq\chi^{s}$ where
$\chi$ is a smooth character of $T$ inflated to $P$ and $\chi^{s}$ is the
character of $T$ defined as $\chi^{s}(\bigl{(}\begin{smallmatrix}{a}&{0}\\\
{0}&{d}\end{smallmatrix}\bigl{)})=\chi(\bigl{(}\begin{smallmatrix}{d}&{0}\\\
{0}&{a}\end{smallmatrix}\bigl{)})$.

  3. (iii)

special series, i.e. twists of the Steinberg representation $\mathrm{Sp}$.

  4. (iv)

supersingular representations.

In this paper, only the second and fourth classes will be involved. When we
talk about a principal series $\mathrm{Ind}_{P}^{G}\chi$, we implicitly mean
$\chi\neq\chi^{s}$. Recall first the following result from [5, §8].

######  Proposition 2.1.

Let $\pi$ be a principal series of $G$.

(i) We have an isomorphism of $\overline{\mathbb{F}}_{p}$-vector spaces
$\mathrm{Hom}(L^{\times},\overline{\mathbb{F}}_{p})\cong\mathrm{Ext}^{1}_{G,\zeta}(\pi,\pi)$.222
$\mathrm{Ext}^{1}_{G,\zeta}$ means that we consider extensions with a central
character and $\zeta$ is the central character of $\pi$.

(ii) Assume $L\neq\mathbb{Q}_{p}$. Let $\pi^{\prime}$ be a smooth irreducible
non-supersingular representation of $G$, then
$\mathrm{Ext}^{1}_{G}(\pi^{\prime},\pi)=0$ except when $\pi^{\prime}\cong\pi$.

###### Proof.

(i) follows from [5, Cor. 8.2(ii)], noting that $\pi$ is irreducible by our
convention, hence the two cases excluded in _loc. cit._ can not happen. (ii)
follows from [5, Thm.8.1] using [5, Thm. 7.16(i)]. ∎

We recall the construction of the isomorphism in Proposition 2.1(i). Let
$\delta\in\mathrm{Hom}(L^{\times},\overline{\mathbb{F}}_{p})$. We lift
$\delta$ to a homomorphism of $P$ to $\overline{\mathbb{F}}_{p}$ via
$P\twoheadrightarrow L^{\times}$ given by
$\bigl{(}\begin{smallmatrix}{a}&{b}\\\ {0}&{d}\end{smallmatrix}\bigl{)}\mapsto
ad^{-1}$. Write $\pi=\mathrm{Ind}_{P}^{G}\chi$ and let $\epsilon_{\delta}$ be
the extension

(2.1) $0\rightarrow\chi\rightarrow\epsilon_{\delta}\rightarrow\chi\rightarrow
0$

corresponding to $\delta$. Explicitly, $\epsilon_{\delta}$ has a basis
$\\{v,w\\}$ with the action of $P$ represented by
$\bigl{(}\begin{smallmatrix}{\chi}&{\chi\delta}\\\
{0}&{\chi}\end{smallmatrix}\bigl{)}$: $b\cdot v=\chi(b)v$ and $b\cdot
w=\chi(b)w+\chi(b)\delta(b)v$ for $b\in P$. Then, inducing to $G$ we obtain an
exact sequence

(2.2)
$0\rightarrow\mathrm{Ind}_{P}^{G}\chi\rightarrow\mathrm{Ind}_{P}^{G}\epsilon_{\delta}\rightarrow\mathrm{Ind}_{P}^{G}\chi\rightarrow
0$

which is the element in $\mathrm{Ext}^{1}_{G}(\pi,\pi)$ corresponding to
$\delta$. Since by definition $\delta$ is trivial on $Z$, the representation
$\mathrm{Ind}_{P}^{G}\epsilon_{\delta}$ has a central character.

Taking $K_{1}$-invariants, (2.2) induces a long exact sequence

$0\rightarrow(\mathrm{Ind}_{P}^{G}\chi)^{K_{1}}\rightarrow(\mathrm{Ind}_{P}^{G}\epsilon_{\delta})^{K_{1}}\rightarrow(\mathrm{Ind}_{P}^{G}\chi)^{K_{1}}\overset{\partial_{\delta}}{\rightarrow}H^{1}(K_{1}/Z_{1},\mathrm{Ind}_{P}^{G}\chi).$

######  Proposition 2.2.

Notations are as above. The map $\partial_{\delta}$ is zero if and only if
$\delta$ is an unramified homomorphism, i.e. $\delta$ is trivial on
$\mathcal{O}_{L}^{\times}$. Moreover, if $\partial_{\delta}$ is nonzero, then
it is an injection.

###### Proof.

First, if $\delta$ is unramified, then it splits when restricted to $T\cap K$,
hence the sequence (2.2) also splits when restricted to $K$. This shows that
$\partial_{\delta}=0$ in this case.

Now assume $\delta$ is ramified. It suffices to show that the inclusion
$(\mathrm{Ind}_{P}^{G}\chi)^{K_{1}}\hookrightarrow(\mathrm{Ind}_{P}^{G}\epsilon_{\delta})^{K_{1}}$
is an equality. By definition, $\mathrm{Ind}_{P}^{G}\epsilon_{\delta}$
identifies with the vector space of smooth functions
$f:G\rightarrow\overline{\mathbb{F}}_{p}v\oplus\overline{\mathbb{F}}_{p}w$
such that $f(pg)=p\cdot f(g)$, where $p\in P$, $g\in G$, and $\\{v,w\\}$ is
the chosen basis of $\epsilon_{\delta}$ as above; the action of $G$ on
$\mathrm{Ind}_{P}^{G}\epsilon_{\delta}$ is given by $(g^{\prime}\cdot
f)(g):=f(gg^{\prime})$. Let $f$ be such a function which is fixed by $K_{1}$.
We need to show that $f(g)\in\overline{\mathbb{F}}_{p}v$ for any $g\in G$.
Using the decomposition

$G=PK=P\begin{pmatrix}{0}&{1}\\\
{1}&{0}\end{pmatrix}K_{1}\coprod\Bigl{(}\coprod_{\lambda\in\mathbb{F}_{q}}P\begin{pmatrix}{1}&{0}\\\
{[\lambda]}&{1}\end{pmatrix}K_{1}\Bigr{)},$

it suffices to check this for $g=\bigl{(}\begin{smallmatrix}{0}&{1}\\\
{1}&{0}\end{smallmatrix}\bigl{)}$ and $g=\bigl{(}\begin{smallmatrix}{1}&{0}\\\
{[\lambda]}&{1}\end{smallmatrix}\bigl{)}$, $\lambda\in\mathbb{F}_{q}$. If
$h=\bigl{(}\begin{smallmatrix}{1+\varpi a}&{0}\\\
{0}&{1}\end{smallmatrix}\bigl{)}\in T_{1}$ and
$g=\bigl{(}\begin{smallmatrix}{1}&{0}\\\
{[\lambda]}&{1}\end{smallmatrix}\bigl{)}$, we have (as $h\cdot f=f$):

$f(g)=(h\cdot f)(g)=f\Bigl{(}\begin{pmatrix}{1+\varpi a}&{0}\\\
{[\lambda](1+\varpi
a)}&{1}\end{pmatrix}\Bigr{)}=h\cdot\Bigl{[}f\Bigl{(}\begin{pmatrix}{1}&{0}\\\
{[\lambda](1+\varpi a)}&{1}\end{pmatrix}\Bigr{)}\Bigr{]}=h\cdot[f(g)]$

where the last equality holds because $f$ is fixed by $K_{1}$. This shows that
the vector $f(g)$ is fixed by $T_{1}$, hence lies in
$\overline{\mathbb{F}}_{p}v$ because $\delta$ is non-trivial on $T_{1}$.
Similar (and simpler) argument works for
$g=\bigl{(}\begin{smallmatrix}{0}&{1}\\\ {1}&{0}\end{smallmatrix}\bigl{)}$.
This finishes the proof. ∎

Since $K_{1}$ is normal in $K$, the space $H^{1}(K_{1}/Z_{1},\pi)$ can be
viewed naturally as a $K$-representation.

######  Corollary 2.3.

Let
$d:=\dim_{\overline{\mathbb{F}}_{p}}\mathrm{Hom}(1+\mathfrak{p}_{L},\overline{\mathbb{F}}_{p})$.
There is a $K$-equivariant embedding $(\pi^{K_{1}})^{\oplus d}\hookrightarrow
H^{1}(K_{1}/Z_{1},\pi)$.

###### Proof.

The association $\delta\mapsto\partial_{\delta}$ defines an
$\overline{\mathbb{F}}_{p}$-linear map

$\mathrm{Hom}(F^{\times},\overline{\mathbb{F}}_{p})\rightarrow\mathrm{Hom}_{K}(\pi^{K_{1}},H^{1}(K_{1}/Z_{1},\pi)).$

By Proposition 2.2, its kernel is the subspace of unramified homomorphisms, so
we obtain an exact sequence

$0\rightarrow\mathrm{Hom}^{\rm
un}(L^{\times},\overline{\mathbb{F}}_{p})\rightarrow\mathrm{Hom}(L^{\times},\overline{\mathbb{F}}_{p})\rightarrow\mathrm{Hom}_{K}(\pi^{K_{1}},H^{1}(K_{1}/Z_{1},\pi))$

(where $\mathrm{Hom}^{\rm un}$ means unramified homomorphisms). The assertion
follows from Proposition 2.2 and that
$\mathrm{Hom}(\mathcal{O}_{L}^{\times},\overline{\mathbb{F}}_{p})\cong\mathrm{Hom}(1+\mathfrak{p}_{L},\overline{\mathbb{F}}_{p})$.
∎

Let $\pi$ be a principal series. For later use, we recall some basic results
about its $I_{1}$-invariants. By [1, Thm. 34], $\mathrm{soc}_{K}\pi$ is not
necessarily irreducible in general, but it contains a _unique_ irreducible
sub-representation which is of dimension $\geq 2$, which we denote by
$\sigma$. The decomposition $G=PI_{1}\coprod P\Pi I_{1}$ (where
$\Pi:=\bigl{(}\begin{smallmatrix}{0}&{1}\\\
{\varpi}&{0}\end{smallmatrix}\bigl{)}$) implies that $\pi^{I_{1}}$ is always
2-dimensional, spanned over $\overline{\mathbb{F}}_{p}$ by $f_{1},f_{2}$
characterized as follows:

(2.3) $f_{1}(\mathrm{Id})=1,\ f_{1}(\Pi)=0;\ \ \ f_{2}(\mathrm{Id})=0,\
f_{2}(\Pi)=1.$

Here $\mathrm{Id}$ denotes the identity matrix of $G$. It is easy to see that
$\langle K.f_{2}\rangle=\sigma$ and
$\sum_{\lambda\in\mathbb{F}_{q}}\bigl{(}\begin{smallmatrix}{\varpi}&{[\lambda]}\\\
{0}&{1}\end{smallmatrix}\bigl{)}f_{2}=\chi(\bigl{(}\begin{smallmatrix}{1}&{0}\\\
{0}&{\varpi}\end{smallmatrix}\bigl{)})f_{2}$. Extending $\sigma$ to be a
$KZ$-representation by letting $\bigl{(}\begin{smallmatrix}{\varpi}&{0}\\\
{0}&{\varpi}\end{smallmatrix}\bigl{)}$ act trivially and setting
$\lambda:=\chi(\bigl{(}\begin{smallmatrix}{1}&{0}\\\
{0}&{\varpi}\end{smallmatrix}\bigl{)})$, we get
$\pi\cong\mathrm{c}\textrm{-}\mathrm{Ind}_{KZ}^{G}\sigma/(T-\lambda)\otimes\chi^{\prime}\circ\det$,
for some (uniquely determined) character
$\chi^{\prime}:L^{\times}\rightarrow\overline{\mathbb{F}}_{p}^{\times}$. Here
we have used the formula (5.1) (see §5 below) for the action of $T$.

###  2.2. Computation of $H^{1}(K_{1}/Z_{1},\pi)$

Assume in this subsection that $L$ is _unramified_ over $\mathbb{Q}_{p}$ of
degree $f$. Although it is not always needed, we assume for convenience
_$f\geq 2$_.

Define the following elements in the completed Iwasawa algebra
$\overline{\mathbb{F}}_{p}[[N_{0}]]$:

$X_{i}:=\sum_{\mu\in\mathbb{F}_{q}}\mu^{-p^{i}}\begin{pmatrix}{1}&{[\mu]}\\\
{0}&{1}\end{pmatrix}\in\overline{\mathbb{F}}_{p}[[N_{0}]].$

A similar proof as that of [20, Prop. 2.13] shows that
$\overline{\mathbb{F}}_{p}[[N_{0}]]\cong\overline{\mathbb{F}}_{p}[[X_{0},...,X_{f-1}]]$.
Let $\tau=(s_{0},...,s_{f-1})\otimes{\det}^{b}$ be a weight. We can view
$\tau$ as an $\overline{\mathbb{F}}_{p}[[N_{0}]]$-module. A direct
generalization of [20, Prop. 2.14], using [1, Lem. 2], shows that $\tau$ is
isomorphic to
$\overline{\mathbb{F}}_{p}[[X_{0},...,X_{f-1}]]/(X_{0}^{s_{0}+1},...,X_{f-1}^{s_{f-1}+1})$
as a module over
$\overline{\mathbb{F}}_{p}[[N_{0}]]\cong\overline{\mathbb{F}}_{p}[[X_{0},...,X_{f-1}]]$.
Precisely, if $w\in\bigl{(}\begin{smallmatrix}{0}&{1}\\\
{1}&{0}\end{smallmatrix}\bigl{)}\cdot\tau^{N_{0}}$ is non-zero (such a vector
is unique up to a scalar), then $\tau$ is generated by $w$ as an
$\overline{\mathbb{F}}_{p}[[N_{0}]]$-module.

######  Lemma 2.4.

The $\overline{\mathbb{F}}_{p}$-vector space
$\mathrm{Ext}^{1}_{N_{0}}(\tau,\overline{\mathbb{F}}_{p})$ is of dimension
$f$.

###### Proof.

We have the identification

$\mathrm{Ext}^{1}_{N_{0}}(\tau,\overline{\mathbb{F}}_{p})\cong\mathrm{Ext}^{1}_{\overline{\mathbb{F}}_{p}[[N_{0}]]}(M,\overline{\mathbb{F}}_{p}),$

where $M$ denotes
$\overline{\mathbb{F}}_{p}[[X_{0},...,X_{f-1}]]/(X_{0}^{s_{0}+1},...,X_{f-1}^{s_{f-1}+1})$.
The assertion then follows from the theory of Koszul complex ([6, §1.6]).
Explicitly, for each $0\leq j\leq f-1$, the extension

$0\rightarrow\overline{\mathbb{F}}_{p}e_{j}\rightarrow\overline{\mathbb{F}}_{p}[[X_{0},...,X_{f-1}]]/(X_{0}^{r_{0}+1},...,X_{j}^{r_{j}+2},...,X_{f-1}^{r_{f-1}+1})\rightarrow
M\rightarrow 0,$

is non split (where $e_{j}$ is sent to $X_{j}^{r_{j}+1}$), and they form a
basis of
$\mathrm{Ext}^{1}_{\overline{\mathbb{F}}_{p}[[N_{0}]]}(M,\overline{\mathbb{F}}_{p})$.
∎

We need take into account of the action of $\mathcal{H}$, where

$\mathcal{H}:=\bigl{\\{}\bigl{(}\begin{smallmatrix}{[\lambda]}&{0}\\\
{0}&{[\mu]}\end{smallmatrix}\bigl{)}:\lambda,\mu\in\mathbb{F}_{q}^{\times}\bigr{\\}}\subset
K.$

Note that the order of $\mathcal{H}$ is prime to $p$ and $\mathcal{H}$
normalizes $N_{0}$.

######  Proposition 2.5.

Let $\tau=(s_{0},...,s_{f-1})\otimes{\det}^{b}$ be as above. Let $\psi$ be a
character of $\mathcal{H}$ such that
$\mathrm{Ext}^{1}_{\mathcal{H}N_{0}}(\tau,\psi)\neq 0$. Then there exists
$0\leq j\leq f-1$ such that $\psi=\psi_{\tau}^{s}\alpha^{(s_{j}+1)p^{j}}$,
where $\psi_{\tau}$ is the character corresponding to the action of
$\mathcal{H}$ on $\tau^{N_{0}}$. Moreover,
$\mathrm{Ext}^{1}_{\mathcal{H}N_{0}}(\tau,\psi)$ is of dimension 1.

###### Proof.

It is easy to see that if $V$ is an
$\overline{\mathbb{F}}_{p}[[\mathcal{H}N_{0}]]$-module, and $w\in V$ is an
$\mathcal{H}$-eigenvector of character $\chi$, then $X_{j}w$ is also an
$\mathcal{H}$-eigenvector, but of character $\chi\alpha^{p^{j}}$. So by the
proof of Lemma 2.4, the characters $\psi$ of $\mathcal{H}$ such that
$\mathrm{Ext}^{1}_{\mathcal{H}N_{0}}(\tau,\psi)\neq 0$ are the characters
$\\{\psi_{\tau}^{s}\alpha^{(s_{j}+1)p^{j}},0\leq j\leq f-1\\}$. ∎

From now on, we assume $\pi=\mathrm{Ind}_{P}^{G}\chi$ is a principal series
satisfying:

(H) the $K$-socle of $\pi$ is irreducible, and if
$(r_{0},...,r_{f-1})\otimes{\det}^{a}$ is the socle, then $0\leq r_{i}\leq
p-2$.

######  Remark 1.

The first condition of (H) amounts to demand that if we write
$\chi=\eta_{1}\otimes\eta_{2}$ for
$\eta_{i}:L^{\times}\rightarrow\overline{\mathbb{F}}_{p}^{\times}$, then
$\eta_{1}\eta_{2}^{-1}$ is a ramified character.

Define a set of weights, depending on (the $K$-socle of) $\pi$, as follows:
for $0\leq j\leq f-1$, let
$\sigma_{j}(\pi):=(s_{0},...,s_{f-1})\otimes{\det}^{b}$ where $s_{i}=r_{i}$
for $i\notin\\{j-1,j\\}$ and $s_{j-1}=p-2-r_{j-1}$, $s_{j}=r_{j}+1$, $b\equiv
a+p^{j-1}(r_{j-1}+1)-p^{j}\ (\mathrm{mod}\ q-1)$. They are well defined under
the condition (H) and the assumption $f\geq 2$.

######  Proposition 2.6.

Let $\pi=\mathrm{Ind}_{P}^{G}\chi$ be a principal series satisfying (H) and
denote $\sigma=\mathrm{soc}_{K}\pi=(r_{0},...,r_{f-1})\otimes{\det}^{a}$. Let
$\tau$ be a weight. Then $\mathrm{Ext}^{1}_{K/Z_{1}}(\tau,\pi|_{K})\neq 0$ if
and only if one of the following holds

  1. (i)

$\tau\cong\sigma_{j}(\pi)$ for some $0\leq j\leq f-1$; in this case
$\dim\mathrm{Ext}^{1}_{K}(\sigma_{j}(\pi),\pi|_{K})=1$.

  2. (ii)

$\tau\cong\sigma$; in this case
$\dim\mathrm{Ext}^{1}_{K/Z_{1}}(\sigma,\pi|_{K})=f$.

###### Proof.

Iwasawa decomposition implies that
$(\mathrm{Ind}_{P}^{G}\chi)|_{K}\cong\mathrm{Ind}_{P\cap K}^{K}(\chi|_{T\cap
K})$. To simplify the notation, we write $\psi=\chi|_{P\cap K}$ for its
restriction to $P\cap K$. By Shapiro’s lemma, we have an isomorphism
$\mathrm{Ext}^{1}_{K}(\tau,\pi|_{K})\cong\mathrm{Ext}^{1}_{P\cap
K}(\tau,\psi)$. Consider a non-split extension of ${P\cap K}$-representations

(2.4) $0\rightarrow\psi\rightarrow V\rightarrow\tau\rightarrow 0.$

First assume $V$ remains non-split when restricted to $\mathcal{H}N_{0}$, so
that $\mathrm{Ext}^{1}_{\mathcal{H}N_{0}}(\tau,\psi)\neq 0$. If we write
$\tau=(s_{0},...,s_{f-1})\otimes{\det}^{b}$, then Proposition 2.5 implies that
$\psi=\psi_{\tau}^{s}\alpha^{(s_{j}+1)p^{j}}$ for some $0\leq j\leq f-1$.
Using the relation $\psi^{s}=(r_{0},...,r_{f-1})\otimes{\det}^{a}$, a simple
calculation shows that $\psi_{\tau}=\psi_{\sigma_{j+1}(\pi)}$, hence
$\tau\cong\sigma_{j+1}(\pi)$. That is we are in case (i) of the theorem. Again
by Proposition 2.5, $\mathrm{Ext}^{1}_{P\cap K}(\tau,\psi)$ has dimension
$\leq 1$, so the same holds for $\mathrm{Ext}^{1}_{K}(\tau,\pi|_{K})$. To
conclude in this case, it suffices to construct a non-zero element in
$\mathrm{Ext}^{1}_{K}(\sigma_{j+1}(\pi),\pi|_{K})$. In fact, [5, Cor.5.6] says
that $\mathrm{Ext}^{1}_{K}(\sigma_{j+1}(\pi),\sigma)$ is non-zero and has
dimension 1. In view of the exact sequence

$\mathrm{Hom}_{K}(\sigma_{j+1}(\pi),\pi/\sigma)\rightarrow\mathrm{Ext}^{1}_{K}(\sigma_{j+1}(\pi),\sigma)\rightarrow\mathrm{Ext}^{1}_{K}(\sigma_{j+1}(\pi),\pi|_{K})$

we are reduced to show $\mathrm{Hom}_{K}(\sigma_{j+1}(\pi),\pi/\sigma)=0$. If
the later space were non-zero, we would get an inclusion
$\Sigma\hookrightarrow\pi$, where $\Sigma$ denotes the unique non-split
extension of $\sigma_{j+1}(\pi)$ by $\sigma$. But $K_{1}$ acts trivially on
$\Sigma$ (see [5, Cor. 5.6]), we would get an inclusion
$\Sigma\hookrightarrow\pi^{K_{1}}\cong\mathrm{Ind}_{P(\mathbb{F}_{q})}^{\mathrm{GL}_{2}(\mathbb{F}_{q})}\psi$,
which contradicts [5, Thm. 2.4].

Now assume $V$ is split when restricted to $\mathcal{H}N_{0}$ and choose a
$\mathcal{H}N_{0}$-splitting $s:\tau\hookrightarrow V$. This implies that $V$
is fixed by $\bigl{(}\begin{smallmatrix}{1}&{\mathfrak{p}_{L}}\\\
{0}&{1}\end{smallmatrix}\bigl{)}$ since both $\psi$ and $\tau$ are. If $n\in
N_{0}$ and $h\in T_{1}$, a simple calculation shows that $hn=n^{\prime}nh$,
for some $n^{\prime}\in\bigl{(}\begin{smallmatrix}{1}&{\mathfrak{p}_{L}}\\\
{0}&{1}\end{smallmatrix}\bigl{)}$, therefore the actions on $V$ of $T_{1}$ and
$N_{0}$ commute.

Claim: if $x\in\tau$ lies in the radical of $\tau$ (as
$N_{0}$-representation), then $s(x)\in V$ is fixed by $T_{1}$.

Proof of Claim. Let $w\in\bigl{(}\begin{smallmatrix}{0}&{1}\\\
{1}&{0}\end{smallmatrix}\bigl{)}\tau^{N_{0}}$ be a non-zero vector. We have
seen that $w$ generates $\tau$ as an $N_{0}$-representation. The condition
that $x$ lies in the radical of $\tau$ is equivalent to that there exists a
finite set of elements $n_{i}\in N_{0}$ such that $x=\sum_{i}(n_{i}-1)w$. Let
$x$ be such an element and assume there exists $h\in T_{1}$ with
$(h-1)s(x)\neq 0$. The remark above implies that

(2.5) $(h-1)s(x)=\sum_{i}(h-1)(n_{i}-1)s(w)=\sum_{i}(n_{i}-1)(h-1)s(w).$

In particular, $(h-1)s(w)$ is non-zero. But, this vector lies in the
underlying space of $\psi$ on which $N_{0}$ acts trivially, so the equality
(2.5) forces that $(h-1)s(x)=0$ as $n_{i}\in N_{0}$, contradiction. The claim
follows.

By the claim, the extension (2.4) is the pullback of a (non-split) exact
sequence

$0\rightarrow\psi\rightarrow W\rightarrow\tau/\mathrm{rad}(\tau)\rightarrow 0$

on which $N_{0}$ acts trivially but $T_{1}$ acts non trivially. This forces
that $\psi=\psi_{\tau}^{s}$, so that $\tau\cong\sigma$ (since
$\psi^{s}=\chi_{\sigma}$) and we are in case (ii) of the theorem. Moreover,
because $T_{1}/Z_{1}\cong
1+\mathfrak{p}_{L}\cong\mathcal{O}_{L}\cong\mathbb{Z}_{p}^{f}$ (since $L$ is
unramified) , the space $\mathrm{Ext}^{1}_{T_{1}/Z_{1}}(\psi,\psi)$ has
dimension $f$.

To conclude in this case, we need show
$\dim\mathrm{Ext}^{1}_{K/Z_{1}}(\sigma,\pi|_{K})$ has dimension $\geq f$, but
it follows from Corollary 2.3. ∎

######  Remark 2.

When $L=\mathbb{Q}_{p}$, the dimension part of Proposition 2.6(ii) is not
always true, cf. [5, Thm. 7.16(iii)].

The above proof has the following consequence.

######  Corollary 2.7.

Let $\pi$ be as in Proposition 2.6 and $\tau=\sigma_{j}(\pi)$ for some $0\leq
j\leq f-1$. Let $0\rightarrow\pi\rightarrow E\rightarrow\tau\rightarrow 0$ be
the unique non-split $K$-extension. Then the induced sequence
$0\rightarrow\pi^{I_{1}}\rightarrow
E^{I_{1}}\rightarrow\tau^{I_{1}}\rightarrow 0$ is exact.

###### Proof.

With notations in the proof of Proposition 2.6, the extension $E$ comes from
the extension $\Sigma$ of $\tau$ by $\sigma$. The result follows from the
corresponding statement for $\Sigma$, see [5, Prop. 4.13]. ∎

The next lemma will be used in the proof of Proposition 3.2.

######  Lemma 2.8.

Let $\pi$ be a principal series satisfying (H) and $\sigma$ be its $K$-socle.
Assume that $V$ is a smooth representation of $G$ such that
$\pi\hookrightarrow V$. Assume that $\mathrm{Hom}_{K}(\sigma,V|_{K})$ is
1-dimensional and $\mathrm{Hom}_{K}(\sigma,V/\pi)\neq 0$. Then $V$ contains a
sub-$G$-representation $V^{\prime}$ which is a non-split extension of $\pi$ by
$\pi$.

###### Proof.

The assumption $\dim\mathrm{Hom}_{K}(\sigma,V)=1$ means that $\sigma$ appears
in $\mathrm{soc}_{K}V$ with multiplicity one and is contained in $\pi$.
Therefore $V$ contains a sub-$K$-representation $E$ which fits into a non-
split extension

$0\rightarrow\pi|_{K}\rightarrow E\rightarrow\sigma\rightarrow 0.$

We first describe the extension $E$ more explicitly. Proposition 2.2 implies a
surjective morphism
$\mathrm{Ext}^{1}_{G,\zeta}(\pi,\pi)\twoheadrightarrow\mathrm{Ext}^{1}_{K,\zeta}(\sigma,\pi|_{K})$,
given by pullback via $\sigma\hookrightarrow\pi|_{K}$. Choose an extension

$0\rightarrow\pi\rightarrow\mathrm{Ind}_{P}^{G}\epsilon_{\delta}\rightarrow\pi\rightarrow
0$

which lifts $E$. Choose a basis $\\{v,w\\}$ of $\epsilon_{\delta}$ as in §2.1
and define two elements of $\mathrm{Ind}_{P}^{G}\epsilon_{\delta}$ as follows:
let $f_{v}\in\pi\subset\mathrm{Ind}_{P}^{G}\epsilon_{\delta}$ be the vector
$f_{2}$ defined in (2.3), and $f_{w}$ be the element characterized by (write
$\Pi=\bigl{(}\begin{smallmatrix}{0}&{1}\\\ {p}&{0}\end{smallmatrix}\bigl{)}$):

$\mathrm{Supp}(f_{w})=P\Pi I_{1}=P\Pi N_{0},\ \ f_{w}(b\Pi n)=b\cdot w,\
\forall b\in P,\forall n\in N_{0}.$

Here $\cdot$ means the action of $P$ on $\epsilon_{\delta}$. It is easy to see
that $f_{w}$ is well defined and fixed by $N_{0}$. If $h\in T_{1}$, then

$hf_{w}=f_{w}+\delta(\Pi h\Pi^{-1})f_{v}.$

Moreover, the image of $f_{w}$ in (the quotient) $\pi$ lies in
$\sigma=\mathrm{soc}_{K}\pi$, see §2.1; that is, $f_{w}$ itself lies in $E$.

Now view $f_{w}$ as a vector in $V$ via the inclusion $E\subset V$. Consider
the operator
$S:=\sum_{\lambda\in\mathbb{F}_{q}}\bigl{(}\begin{smallmatrix}{p}&{[\lambda]}\\\
{0}&{1}\end{smallmatrix}\bigl{)}\in\overline{\mathbb{F}}_{p}[G]$. Because
$f_{w}$ is fixed by $N_{0}$, we get by Lemma 5.1(iii),

$h(S(f_{w}))=S(h(f_{w}))=Sf_{w}+\delta(\Pi h\Pi^{-1})Sf_{v}=Sf_{w}+\delta(\Pi
h\Pi^{-1})\lambda f_{v}.$

Hence $f_{w}^{\prime}:=S(f_{w})-\lambda f_{w}$ is fixed by $T_{1}$. By Lemma
5.1(iv), for $n$ large enough, $S^{n}f_{w}^{\prime}$ is fixed by $I_{1}$.
Moreover, by the proof of [18, Lem. 4.1], $S^{n+1}f_{w}^{\prime}$ generates an
irreducible $K$-representation which is isomorphic to $\sigma$. Since $\sigma$
appears with multiplicity one in the $K$-socle of $V$ by assumption, we deduce
that $S^{n+1}f_{w}^{\prime}\in\overline{\mathbb{F}}_{p}f_{v}$. In particular,
$S^{n+2}f_{w}=\lambda S^{n+1}f_{w}$ in $V/\pi$, showing that $S^{n+1}f_{w}$
(which is non-zero) generates a principal series in $V/\pi$, which must be
isomorphic to $\pi$. ∎

###  2.3. Ordinary part

In this subsection $L$ is a finite extension of $\mathbb{Q}_{p}$ of degree
$n$.

Recall that Emerton has defined a functor, called ordinary parts and denoted
by $\mathrm{Ord}_{P}$, from the category of admissible smooth
$\overline{\mathbb{F}}_{p}$-representations of $G$ to the category of
admissible smooth $\overline{\mathbb{F}}_{p}$-representations of $T$. Let
$\mathbb{R}^{i}\mathrm{Ord}_{P}$ be its right derived functors for $i\geq 1$.
It follows from [9, Prop. 3.6.1] and [12] that
$\mathbb{R}^{i}\mathrm{Ord}_{P}$ vanishes for $i\geq n+1$.

Write $G_{L}=\mathrm{Gal}(\overline{\mathbb{Q}}_{p}/L)$. Let
$\epsilon:G_{L}\rightarrow\mathbb{Z}_{p}^{\times}$ be $p$-adic the cyclotomic
character ad $\omega$ be its reduction modulo $p$. View them as characters of
$L^{\times}$ via the local Artin map normalized in such a way that
uniformizers of $L$ are sent to geometric Frobenii. Denote by $\alpha$ the
character $\omega\otimes\omega^{-1}:T\rightarrow\mathbb{F}_{p}^{\times}$.

######  Proposition 2.9.

(i) If $U$ is an admissible smooth representation of $G$ and $V$ is a smooth
representation of $G$, then

$\mathrm{Hom}_{G}(\mathrm{Ind}_{\overline{P}}^{G}U,V)\cong\mathrm{Hom}_{T}(U,\mathrm{Ord}_{P}(V)).$

(ii) There is a canonical isomorphism $\mathbb{R}^{n}\mathrm{Ord}_{P}(V)\cong
V_{N}\otimes\alpha^{-1}$, where $\pi_{N}$ is the space of coinvariants (i.e.
the usual Jacquet module of $V$ with respect to $P$).

(iii) We have $\mathrm{Ord}_{P}(\mathrm{Ind}_{{P}}^{G}U)\cong U^{s}$ and
$\mathbb{R}^{n}\mathrm{Ord}_{P}(\mathrm{Ind}_{{P}}^{G}U)\cong
U\otimes\alpha^{-1}$.

(iv) If $\pi$ is an absolutely irreducible supersingular representation of $G$
over $\overline{\mathbb{F}}_{p}$, then
$\mathrm{Ord}_{P}(\pi)=\mathbb{R}^{n}\mathrm{Ord}_{P}(\pi)=0$.

###### Proof.

(i) is [8, Theorem 4.4.6]. (ii) is [9, Prop.3.6.2]. (iii) follows from [8,
Cor. 4.3.5] and [9, Prop. 3.6.2], using the natural isomorphism
$\mathrm{Ind}_{P}^{G}U\cong\mathrm{Ind}_{\overline{P}}^{G}U^{s}$. For (iv),
the first assertion follows from (i); the second follows from (ii). To see
this, let $\pi(N)\subset\pi$ be the subspace spanned by vectors of the form
$(n-1)v$, for all $n\in N$ and $v\in\pi$, so that $\pi_{N}=\pi/\pi(N)$. It is
easily checked that $\pi(N)$ is stable under the group $P$ and non-zero. The
result follows from the main result of [18], which says that $\pi|_{P}$ is
irreducible. ∎

###  2.4. A definition

We recall a definition due to Emerton [10, §3.6], which plays a crucial role
below.

We normalize the local Artin map $L^{\times}\hookrightarrow G_{L}^{\rm ab}$ in
such a way that uniformizers of $L$ are sent to geometric Frobenii.

Denote by $S$ the following subtorus of $T$

$S:=\Bigl{\\{}\begin{pmatrix}{a}&{0}\\\ {0}&{1}\end{pmatrix}|a\in
L^{\times}\Bigr{\\}}\subset T$

so that $S\cong L^{\times}$. The composite of this isomorphism with the local
Artin map defines an injection $\iota:S\hookrightarrow G_{L}^{\rm ab}$, and
hence an anti-diagonal embedding

(2.6) $S\hookrightarrow G_{L}^{\rm ab}\times S,\ \ s\mapsto(\iota(s),s^{-1}).$

######  Definition 1.

Let $V$ be a representation of $G_{L}\times S$.

(i) Let $V^{\rm ab}$ be the maximal sub-object of $V$ on which $G_{L}$ acts
through its maximal abelian quotient $G_{L}^{\rm ab}$. This is a $G_{L}\times
S$-sub-representation of $V$.

(ii) Let $V^{\mathrm{ab},S}$ be the subspace of $V^{\mathrm{ab}}$ consisting
of $S$-fixed vectors, where $S$ acts through the anti-diagonal embedding (2.6)
and the action of $G_{L}^{\rm ab}\times S$ on $V^{\rm ab}$.

The space $V^{\mathrm{ab},S}$ is stable under the action of $G_{L}$ and, of
course, this action factors through $G_{L}^{\rm ab}$.

######  Lemma 2.10.

Let $V$ be a representation of $G_{L}\times S$. Assume that the action of
$G_{L}$ on $V/V^{\mathrm{ab},S}$ factors through $G_{L}^{\rm ab}$. Then, for
any subquotient $W$ of $V$ (as $G_{L}\times S$-representations), the action of
$G_{L}$ on $W/W^{\mathrm{ab},S}$ also factors through $G_{L}^{\mathrm{ab}}$.

###### Proof.

By definition, if $H_{L}$ denotes the kernel of the quotient map
$G_{L}\twoheadrightarrow G_{L}^{\rm ab}$, then $V^{\rm ab}=V^{H_{L}}$ and
$V^{\mathrm{ab},S}=(V^{H_{L}})^{S}$. If $W$ is a sub-$G_{L}\times
S$-representation of $V$, we deduce that $V^{\mathrm{ab},S}\cap
W=W^{\mathrm{ab},S}$, hence a $G_{L}$-equivariant inclusion
$W/W^{\mathrm{ab},S}\hookrightarrow V/V^{\mathrm{ab},S}$ and the result holds
in this case. If $W$ is a quotient of $V$, the result is obvious. The general
case follows from this. ∎

##  3\. Local results

We keep notations of Section 2. Assume $L$ is _unramified_ over
$\mathbb{Q}_{p}$ of degree $f\geq 2$.

###  3.1. Construction of Breuil-Paškūnas

Let
$\overline{\rho}:\mathrm{Gal}(\overline{\mathbb{Q}}_{p}/L)\rightarrow\mathrm{GL}_{2}(\overline{\mathbb{F}}_{p})$
be a continuous representation. Assume $\overline{\rho}$ is reducible and
_generic_ in the sense of [5, §11], i.e. $\overline{\rho}$ is of the form

$\overline{\rho}\cong\begin{pmatrix}{\mathrm{nr}(\mu)\omega_{f}^{(r_{0}+1)+\cdots+p^{f-1}(r_{f-1}-1)}}&{*}\\\
{0}&{\mathrm{nr}(\mu^{-1})}\end{pmatrix}\otimes\eta$

where $\mu\in\overline{\mathbb{F}}_{p}^{\times}$, $r_{i}\in\\{0,...,p-3\\}$
with $(r_{0},...,r_{f-1})\neq(0,...,0),(p-3,...,p-3)$, $\omega_{f}$ is Serre’s
fundamental character of level $f$, and
$\eta:\mathrm{Gal}(\overline{\mathbb{Q}}_{p}/L)\rightarrow\overline{\mathbb{F}}_{p}^{\times}$
is a continuous character.

To $\overline{\rho}$ is associated a set of weights, called Serre weights and
denoted by $\mathscr{D}(\overline{\rho})$, as follows (see [5, §11] or [7]).
First, the genericity condition on $\overline{\rho}$ implies that
$\overline{\rho}$ is in the category of Fontaine-Lafaille [13]. Writing down
the associated Fontaine-Lafaille module, we define a subset
$J_{\overline{\rho}}$ of $\mathcal{S}:=\\{0,...,f-1\\}$ which we refer to [16,
§2] for its precise definition. We recall that $J_{\overline{\rho}}$ measures
how far $\overline{\rho}$ is from splitting, in the sense that
$J_{\overline{\rho}}=\mathcal{S}$ if and only if $\overline{\rho}$ is split.
Second, we define $\mathscr{D}(x_{0},...,x_{f-1})$ to be the set of $f$-tuples
$\tau=(\tau_{0}(x_{0}),...,\tau_{f-1}(x_{f-1}))$ satisfying the following
conditions:

  1. (i)

$\tau_{i}(x_{i})\in\\{x_{i},x_{i}+1,p-2-x_{i},p-3-x_{i}\\}$

  2. (ii)

if $\tau_{i}(x_{i})\in\\{x_{i},x_{i}+1\\}$, then
$\tau_{i+1}(x_{i+1})\in\\{x_{i+1},p-2-x_{i+1}\\}$

  3. (iii)

if $\tau_{i}(x_{i})\in\\{p-2-x_{i},p-3-x_{i}\\}$, then
$\tau_{i+1}(x_{i+1})\in\\{p-3-x_{i+1},x_{i+1}+1\\}$

  4. (iv)

if $\tau_{i}(x_{i})\in\\{p-3-x_{i},x_{i}+1\\}$, then $i\in
J_{\overline{\rho}}$

with the conventions $x_{f}:=x_{0}$ and $\tau_{f}(x_{f}):=\tau_{0}(x_{0})$.
Then $\mathscr{D}(\overline{\rho})$ can be explicitly described as

$\mathscr{D}(\overline{\rho})=\bigl{\\{}(\tau_{0}(r_{0}),...,\tau_{f-1}(r_{f-1}))\otimes{\det}^{e(\tau)(r_{0},...,r_{f-1})},\
\tau\in\mathscr{D}(x_{0},...,x_{f-1})\bigr{\\}},$

where $e(\tau)(x_{0},...,x_{f-1})$ is defined as in [5, §4]. Remark that there
are $2^{|J_{\overline{\rho}}|}$ elements in $\mathscr{D}(\overline{\rho})$,
and it always contains the weight
$\sigma_{0}:=(r_{0},...,r_{f-1})\otimes\eta\circ\det$. For
$\sigma\in\mathscr{D}(\overline{\rho})$ which corresponds to
$\tau\in\mathcal{D}(x_{0},...,x_{f-1})$, we set

$\ell(\sigma):=\mathrm{Card}\\{i\in\mathcal{S};\tau_{i}(x_{i})\in\\{p-2-x_{i},p-3-x_{i}\\}\\},$

and call it the _length_ of $\sigma$.

Let $D_{0}(\overline{\rho})$ be the maximal representation of
$\mathrm{GL}_{2}(\mathbb{F}_{q})$ such that

  1. (i)

the $\mathrm{GL}_{2}(\mathbb{F}_{q})$-socle of $D(\overline{\rho})$ is
$\oplus_{\tau\in\mathscr{D}(\overline{\rho})}\tau$

  2. (ii)

each Serre weight $\tau\in\mathscr{D}(\overline{\rho})$ occurs exactly once in
$D(\overline{\rho})$.

Let $D_{1}(\overline{\rho})=D_{0}(\overline{\rho})^{I_{1}}$ with the induced
action of $I$ and we choose an action of
$\Pi=\bigl{(}\begin{smallmatrix}{0}&{1}\\\ {p}&{0}\end{smallmatrix}\bigl{)}$
such that $\Pi^{2}$ is the identity. The amalgam structure of $G$ (more
precisely, of $\mathrm{SL}_{2}(L)$) then allows Breuil and Paškūnas to
construct a family of smooth admissible representations of $G$ over
$\overline{\mathbb{F}}_{p}$, with $K$-socle being
$\oplus_{\sigma\in\mathscr{D}(\overline{\rho})}\sigma$. The construction is as
follows (see [5]). We first embed $K$-equivariantly $D_{0}(\overline{\rho})$
inside an injective envelope
$\Omega:=\mathrm{Inj}_{K}(\oplus_{\sigma\in\mathscr{D}(\overline{\rho})}\sigma)$.
Then using the decomposition of $\Omega|_{I}$ we can give an action of
$\bigl{(}\begin{smallmatrix}{0}&{1}\\\ {p}&{0}\end{smallmatrix}\bigl{)}$ on
$\Omega$ which is compatible with the one on $D_{1}(\overline{\rho})$ via the
embedding we have chosen. In such a way, a theorem of Ihara allows us to get a
smooth action of $G$ on $\Omega$ and we let $V$ be the sub-representation
generated by $D_{0}(\overline{\rho})$. In particular, $V$ depends on the
choice of the action of $\bigl{(}\begin{smallmatrix}{0}&{1}\\\
{p}&{0}\end{smallmatrix}\bigl{)}$ on $\Omega$, and actually there are quite a
lot of such choices. Finally, we twist $V$ by $\eta\circ\det$ to recover the
determinant of $\overline{\rho}$. Denote by $\mathcal{V}(\overline{\rho})$ the
family of representations $V$ obtained in this way.

It is expected that such a $V\in\mathcal{V}(\overline{\rho})$ has length $f$,
or, at least, there exists one $V\in\mathcal{V}(\overline{\rho})$ which has
length $f$. Precisely, we hope that $V$ has a filtration of length $f$ of the
form (where $(\pi_{i})_{i}$ denote the graded pieces of the filtration)

$\pi_{0}\ \textbf{---}\ \pi_{1}\ \textbf{---}\ \cdots\ \textbf{---}\ \pi_{f}$

such that $\pi_{i}$ is a principal series if $i\in\\{0,f\\}$ and supersingular
otherwise. This is the case when $\overline{\rho}$ is (reducible) split; in
this case, $V$ is a direct sum of $\pi_{i}$’s, $0\leq i\leq f$ (see [5, §19]).
However, it is not even known whether such a $V$ has finite length if
$\overline{\rho}$ is non-split, except for the case $L=\mathbb{Q}_{p}$ (cf.
[8, Conj. 2.3.7]).

In the following, we assume $\overline{\rho}$ is _non-split_.

######  Lemma 3.1.

For any $V\in\mathcal{V}(\overline{\rho})$, the $G$-socle of $V$ is an
irreducible principal series and isomorphic to
$\mathrm{c}\textrm{-}\mathrm{Ind}_{KZ}^{G}\sigma_{0}/(T-\lambda)$ for some
$\lambda\in\overline{\mathbb{F}}_{p}^{\times}$.

###### Proof.

Let $\pi$ be an irreducible sub-representation of $V$. Let
$\sigma\in\mathscr{D}(\overline{\rho})$ be a Serre weight which is contained
in $\mathrm{soc}_{K}\pi$. The proof of [5, Thm. 15.4(ii)] shows that we can go
from $D_{0,\sigma}(\overline{\rho})^{I_{1}}$ to
$D_{0,\sigma_{0}}(\overline{\rho})^{I_{1}}$ using $\chi\mapsto\chi^{s}$ (with
notations there). In particular, we see that $\sigma_{0}$ is also contained in
$\pi$, hence $\langle G.\sigma_{0}\rangle\subset\pi$. But, by construction,
$\langle G\cdot\sigma_{0}\rangle$ is an (irreducible) principal series, so
that $\pi=\langle G\cdot\sigma_{0}\rangle$. The last assertion follows from
[1, Thm. 30]. ∎

######  Remark 3.

In Lemma 3.1, any $\lambda\in\overline{\mathbb{F}}_{p}^{\times}$ could happen.
In fact, the construction in [5] does not take into account of the whole
information of $\overline{\rho}$. For the representations arising from the
cohomology of Shimura curves, $\lambda$ is uniquely determined by
$\overline{\rho}$ (see [4] or §4).

######  Proposition 3.2.

Let $V\in\mathcal{V}(\overline{\rho})$. Assume that $\mathrm{Ord}_{P}(V)$ is
one-dimensional. Then $V$ contains a sub-representation $\pi$ which is of
length 2 and fits into an exact sequence

$0\rightarrow\pi_{0}\rightarrow\pi\rightarrow\pi_{1}\rightarrow 0$

such that $\pi_{0}$ is a principal series and $\pi_{1}$ is a supersingular
representation. Moreover, $\pi_{1}$ is uniquely determined (by $V$) in the
following two cases:

(i) either $\mathscr{D}(\overline{\rho})=\\{\sigma_{0}\\}$, in which case
$\mathrm{soc}_{K}(\pi_{1})=\oplus_{\sigma\in\mathscr{D}(\overline{\rho}^{\rm
ss}),\ell(\sigma)=1}\sigma$;

(ii) or $V^{I_{1}}=D_{1}(\overline{\rho})$, in which case we only have an
inclusion

(3.1) $\bigoplus_{\sigma\in\mathscr{D}(\overline{\rho}^{\rm
ss}),\ell(\sigma)=1}\sigma\subseteq\mathrm{soc}_{K}(\pi_{1}),$

which is an equality when $f=2$.

######  Remark 4.

It is easy to check that the set $\\{\sigma\in\mathcal{\overline{\rho}^{\rm
ss}}:\ell(\sigma)=1\\}$ is exactly the set $\\{\sigma_{j}(\pi_{0}):0\leq j\leq
f-1\\}$ (well defined thanks to the genericity condition on
$\overline{\rho}$).

###### Proof.

Lemma 3.1 implies that the $G$-socle of $V$ is a principal series; we denote
it by $\pi_{0}$. By assumption, $\mathrm{Ord}_{P}(V)$ is one dimensional. We
claim that $V/\pi_{0}$ does not admit principal series as a sub-
representation. In fact, if $\pi^{\prime}\hookrightarrow V/\pi_{0}$ is a
principal series, then $\mathrm{Ext}^{1}_{G}(\pi^{\prime},\pi_{0})\neq 0$
since $\pi_{0}$ is the $G$-socle of $V$. But Proposition 2.1 implies that
$\pi^{\prime}\cong\pi_{0}$, and the corresponding extension is of the form
(2.2) for certain
$\delta\in\mathrm{Hom}(L^{\times},\overline{\mathbb{F}}_{p})$. The claim
follows from the assumption on $\mathrm{Ord}_{P}(V)$ using Proposition
2.9(iii). By [22, Thm. 2]$,V/\pi_{0}$ is still a smooth admissible
representation, hence it contains at least one irreducible sub-representation;
by the claim it must be supersingular. This shows the first assertion of the
proposition.

We define $\pi_{1}$ and determine its $K$-socle under assumptions (i) or (ii).
First assume $\mathscr{D}(\overline{\rho})=\\{\sigma_{0}\\}$. By construction
recalled above, $V$ sits inside a certain $\Omega$ such that $\Omega|_{K}$ is
an injective envelope of $\sigma_{0}$. Proposition 2.6 then implies that

$\mathrm{soc}_{K}(\Omega/\pi_{0})=\bigl{(}\bigoplus_{j=0}^{f-1}\sigma_{0}\bigr{)}\oplus\bigl{(}\bigoplus_{j=0}^{f-1}\sigma_{j}(\pi_{0})\bigr{)}.$

By Lemma 2.8, we deduce that $\mathrm{soc}_{K}(V/\pi_{0})$ does not contain
$\sigma_{0}$ so that

$\mathrm{soc}_{K}(V/\pi_{0})\subseteq\bigl{(}\bigoplus_{j=0}^{f-1}\sigma_{j}(\pi_{0})\bigr{)}.$

We claim that $V/\pi_{0}$ contains a unique irreducible sub-representation. In
fact, let $\pi_{1}\hookrightarrow V/\pi_{0}$ be an irreducible sub-
representation and let $\sigma$ be a weight contained in the $K$-socle of
$\pi_{1}$. Then $\sigma$ is of the form $\sigma_{j}(\pi_{0})$ for some $0\leq
j\leq f-1$. We check that all other $\sigma_{j}(\pi_{0})$’s are also contained
in $\pi_{1}$ under the process $\chi\mapsto\chi^{s}$ (argument as in [5, Thm.
15.4]). To do this, we may assume $j=1$ so that
$\sigma=(p-2-r_{0},r_{1}+1,r_{2},...,r_{f-1})$ (up to twist). If we let
$I(\sigma_{0},\sigma^{[s]})$ be the unique sub-representation of
$D_{0}(\overline{\rho})$ with cosocle $\sigma^{[s]}$ (see [5, Cor. 3.12]),
where $\sigma^{[s]}:=(r_{0}+1,p-2-r_{1},p-1-r_{2},...,p-1-r_{f-1})$ (up to a
twist uniquely determined by $\sigma$) is as in [5, page 9]. Using [5, Cor.
4.11], $\sigma_{0}(\pi_{0})$ occurs in $I(\sigma_{0},\sigma^{[s]})$ as an
irreducible constituent. This shows that $\sigma_{0}(\pi)$ also occurs in the
$K$-socle of $\pi$. Repeating this argument gives the result and shows that
$\mathrm{soc}_{K}\pi_{1}=\oplus_{j=0}^{f-1}\sigma_{j}(\pi_{0})=\oplus_{\sigma\in\mathscr{D}(\overline{\rho}),\ell(\sigma)=1}\sigma$
by Remark 4.

Now assume (ii) $V^{I_{1}}=D_{1}(\overline{\rho})$. By the construction, $V$
sits inside certain $\Omega\in\mathrm{Mod}_{G}^{\rm sm}$ such that
$\Omega|_{K}$ is an injective envelope of
$\oplus_{\sigma\in\mathscr{D}(\overline{\rho})}\sigma$. Since
$\mathrm{soc}_{K}(\pi_{0})=\sigma_{0}$, we can $K$-equivariantly decompose
$\Omega=\oplus_{\sigma\in\mathscr{D}(\overline{\rho})}\Omega_{\sigma}$ so that
$\mathrm{soc}_{K}(\Omega_{\sigma})=\sigma$ for each $\sigma$ and that
$\pi_{0}$ is contained in $\Omega_{\sigma_{0}}$. Therefore
$\Omega/\pi_{0}=(\Omega_{\sigma_{0}}/\pi_{0})\oplus(\oplus_{\sigma\neq\sigma_{0}}\Omega_{\sigma})$
and Proposition 2.6 then implies that

$\mathrm{soc}_{K}(V/\pi_{0})\subseteq\mathrm{soc}_{K}(\Omega/\pi_{0})=\bigl{(}\bigoplus_{j=0}^{f-1}\sigma_{0}\bigr{)}\oplus\bigl{(}\bigoplus_{j=0}^{f-1}\sigma_{j}(\pi_{0})\bigr{)}\oplus\bigl{(}\bigoplus_{\sigma\in\mathscr{D}(\overline{\rho}),\sigma\neq\sigma_{0}}\sigma\bigr{)}.$

Here, although $\sigma_{j}(\pi_{0})$ is possibly isomorphic to some
$\sigma\in\mathscr{D}(\overline{\rho})$ in view of Remark 4 (automatically of
length 1), we use $\sigma_{j}(\pi)$ to emphasize that it is a sub-
representation of $\Omega_{\sigma_{0}}/\pi_{0}$ and use
$\sigma\in\mathscr{D}(\overline{\rho})$ to emphasize that it is contained in
$\Omega_{\sigma}$. Lemma 2.8 implies that $V/\pi_{0}$ does not admit
$\sigma_{0}$ as a sub-$K$-representation, hence

$\bigoplus_{\sigma\in\mathscr{D}(\overline{\rho}),\sigma\neq\sigma_{0}}\sigma\subseteq\mathrm{soc}_{K}(V/\pi_{0})\subseteq\bigl{(}\bigoplus_{j=0}^{f-1}\sigma_{j}(\pi_{0})\bigr{)}\oplus\bigl{(}\bigoplus_{\sigma\in\mathscr{D}(\overline{\rho}),\sigma\neq\sigma_{0}}\sigma\bigr{)}.$

Let $\sigma\in\mathscr{D}(\overline{\rho})$ such that $\ell(\sigma)=1$. We
know $\sigma=\sigma_{j}(\pi_{0})$ for some $0\leq j\leq f-1$ by Remark 4. Let
$\Sigma$ be the unique non-split extension of $\sigma_{j}(\pi_{0})$ by
$\sigma_{0}$; then
$\Sigma^{I_{1}}=\sigma_{0}^{I_{1}}\oplus\sigma_{j}(\pi_{0})^{I_{1}}$ is
$2$-dimensional by [5, Prop. 4.13]. We have $\mathrm{Hom}_{K}(\Sigma,V)=0$; if
not, we would have $\Sigma\hookrightarrow V$ and therefore
$\Sigma\oplus\sigma\hookrightarrow V$ which would contradict the assumption
$V^{I_{1}}=D_{1}(\overline{\rho})$ which is multiplicity free ([5, Cor.
13.5]). In all we get

$\mathrm{soc}_{K}(V/\pi_{0})\subseteq\bigl{(}\bigoplus_{\sigma\in\mathscr{D}(\overline{\rho}^{\rm
ss}),\ell(\sigma)=1}\sigma\bigr{)}\oplus\bigl{(}\bigoplus_{\sigma\in\mathscr{D}(\overline{\rho}),\ell(\sigma)\geq
2}\sigma\bigr{)}.$

As in case (i), we show that $V/\pi_{0}$ admits a unique irreducible
sub-$G$-representation. For this, we show that any sub-representation
$\pi_{1}$ of $V/\pi_{0}$ contains
$\oplus_{\sigma\in\mathscr{D}(\overline{\rho}^{\rm ss}),\ell(\sigma)=1}\sigma$
in its $K$-socle. Indeed, if $\pi_{1}$ contains some
$\sigma\in\mathscr{D}(\overline{\rho})$ with $\ell(\sigma)\geq 2$, then the
same argument as in the proof of [5, Thm. 15.4], using the process
$\chi\mapsto\chi^{s}$, shows that $\pi_{1}$ also contains another
$\sigma^{\prime}\in\mathscr{D}(\overline{\rho})$ with
$\ell(\sigma^{\prime})<\ell(\sigma)$. Remark that, although it is not
necessary for us, we can guarantee that $\sigma^{\prime}$ belongs to
$\mathscr{D}(\overline{\rho})$, not just to $\mathscr{D}(\overline{\rho}^{\rm
ss})$. So we may assume $\ell(\sigma)=1$ at the beginning. Still using the
process $\chi\mapsto\chi^{s}$, we claim that $\pi_{1}$ contains all other
$\sigma^{\prime}\in\mathscr{D}(\overline{\rho}^{\rm ss})$ with
$\ell(\sigma^{\prime})=1$. The argument is similar to that of case (i) above
but slightly different, as follows.

Let $D_{0,0}(\overline{\rho}):=D_{0}(\overline{\rho})\cap\pi_{0}$ and let
$D_{0,1}(\overline{\rho})$ be the maximal sub-$K$-representation of
$D_{0}(\overline{\rho})/D_{0,0}(\overline{\rho})$ which does not contain any
element of $\mathscr{D}(\overline{\rho}^{\rm ss})$ of length $\geq 2$. By [16,
Lem. 5.1], the $K$-socle of $D_{0,1}(\overline{\rho})$ is exactly
$\oplus_{\sigma\in\mathscr{D}(\overline{\rho}^{\rm ss}),\ell(\sigma)=1}$. By
[5, Thm. 13.1], $D_{0,1}(\overline{\rho})$ is a sub-$K$-representation of
$D_{0,1}(\overline{\rho}^{\rm ss})$ (see [5, Thm. 15.4(ii)] for this
notation), since it does not contain any element of
$\mathscr{D}(\overline{\rho}^{\rm ss})$ of length $0$ or $\geq 2$. Moreover,
$D_{0,1}(\overline{\rho})^{I_{1}}$ is stable under the action of
$\bigl{(}\begin{smallmatrix}{0}&{1}\\\ {p}&{0}\end{smallmatrix}\bigl{)}$ and
compatible with that of $D_{0,1}(\overline{\rho})^{I_{1}}$ because they are
both multiplicity free. In all, we are reduced to check the claim in the case
$\overline{\rho}=\overline{\rho}^{\rm ss}$, which is done in [5, Thm.
1.54(ii)].

To conclude, we just let $\pi_{1}$ be the sub-representation of $V/\pi_{0}$
generated by $\oplus_{\sigma\in\mathscr{D}(\overline{\rho}^{\rm
ss}),\ell(\sigma)=1}$. The inclusion (3.1) follows from the claim and the
equality when $f=2$ is obvious. ∎

######  Remark 5.

In [11], it is shown that for some $V\in\mathcal{V}(\overline{\rho})$ coming
from cohomology of Shimura curves, the condition (ii) of Proposition 3.2 is
verified. Moreover, for such a $V$, it is hoped that (3.1) is always an
equality.

###  3.2. The case $f=2$

We have seen that non-split $G$-extensions of a supersingular representation
by a principal series exist (under the conditions which will be checked in
§4). In this section, we deduce from this the existence of non-split
extensions of the converse type in the case $f=2$, namely extensions of the
form

$0\rightarrow\pi_{1}\rightarrow*\rightarrow\pi_{2}\rightarrow 0$

with $\pi_{1}$ supersingular and $\pi_{2}$ a principal series.

We keep notations in the previous subsection.

######  Proposition 3.3.

Assume $f=2$. Let $\pi_{0}=\mathrm{Ind}_{{P}}^{G}\chi$ and $\pi_{1}$ be a
supersingular representation. If $\mathrm{Ext}^{1}_{G}(\pi_{1},\pi_{0})\neq
0$, then
$\mathrm{Ext}^{1}_{G}(\mathrm{Ind}_{{P}}^{G}\chi^{s}\alpha,\pi_{1})\neq 0$.

###### Proof.

Let

(3.2) $0\rightarrow\pi_{0}\rightarrow V\rightarrow\pi_{1}\rightarrow 0$

be a non-split extension. Apply the functor $\mathrm{Ord}_{P}$ and using Lemma
3.4 below, we get a surjection

$\mathbb{R}^{1}\mathrm{Ord}_{P}\pi_{1}\twoheadrightarrow\mathbb{R}^{2}\mathrm{Ord}_{P}\pi_{0}.$

By Proposition 2.9(ii) (as $f=2$), we know
$\mathbb{R}^{2}\mathrm{Ord}_{P}\pi_{0}\cong\chi\alpha^{-1}$, hence
$\mathbb{R}^{1}\mathrm{Ord}_{P}\pi_{1}$ is non-zero and admits a quotient
isomorphic to $\chi\alpha^{-1}$. Because there is no non-trivial extension
between two non-isomorphic $T$-characters, we deduce that
$\mathbb{R}^{1}\mathrm{Ord}_{P}\pi_{1}$ also contains a sub-character
isomorphic to $\chi\alpha^{-1}$. The assertion follows from the long exact
sequence [9, (3.7.5)] which implies that
$\mathrm{Ext}^{1}_{G}(\mathrm{Ind}_{\overline{P}}^{G}\chi\alpha^{-1},\pi_{1})\cong\mathrm{Hom}_{T}(\chi\alpha^{-1},\mathbb{R}^{1}\mathrm{Ord}_{P}(\pi_{1}))\neq
0$. ∎

######  Lemma 3.4.

We have $\mathbb{R}^{2}\mathrm{Ord}_{P}V=0$.

###### Proof.

By assumption $f=2$, Proposition 2.9(iii) implies that
$\mathbb{R}^{2}\mathrm{Ord}_{P}W=W_{N}\otimes\alpha^{-1}$ for any
$G$-representation $W$. The sequence (3.2) induces an exact sequence
$(\pi_{0})_{N}\rightarrow V_{N}\rightarrow(\pi_{1})_{N}\rightarrow 0$. Since
$(\pi_{1})_{N}=0$ by Proposition 2.9(iv), it suffices to prove that the map
$(\pi_{0})_{N}\rightarrow V_{N}$ is zero. If not, then
$V_{N}\cong(\pi_{0})_{N}$ as $(\pi_{0})_{N}$ is one dimensional, and the
adjunction formula (see [8, §3.6]) implies that

$\mathrm{Hom}_{G}(V,\mathrm{Ind}_{P}^{G}(\pi_{0})_{N})=\mathrm{Hom}_{G}(V,\pi_{0})\neq
0,$

As a consequence, the extension (3.2) splits, giving the desired
contradiction. ∎

We have remarked that when $\overline{\rho}$ is reducible, we hope that any
$V\in\mathcal{V}(\overline{\rho})$ is a successive extension of the
irreducible representations $(\pi_{i})_{0\leq i\leq 2}$, with
$\pi_{0},\pi_{2}$ being principal series. It is easily checked that if we
write $\pi_{0}=\mathrm{Ind}_{P}^{G}\chi$ for suitable $\chi$, then
$\pi_{2}=\mathrm{Ind}_{P}^{G}\chi^{s}\alpha$, which is compatible with
Proposition 3.3.

######  Remark 6.

Note that $\pi_{2}$ is exactly the $f$-th smooth dual of $\pi_{0}$, in the
sense of [17, Def. 3.12 & Prop. 5.4].

##  4\. Global results

We prove the main result of this article in this section.

Let $F$ be a totally real number field. For each finite place $v$ of $F$,
denote by $F_{v}$ the completion of $F$ at $v$. Write
$G_{F}=\mathrm{Gal}(\overline{F}/F)$ and
$G_{F_{v}}=\mathrm{Gal}(\overline{F_{v}}/F_{v})$, and we identify $G_{F_{v}}$
with a subgroup of $G_{F}$ by fixing an embedding
$\overline{F}\hookrightarrow\overline{F_{v}}$. We fix a finite extension $E$
of $\mathbb{Q}_{p}$, which serves as the coefficient field and is allowed to
be enlarged. Write $\mathcal{O}_{E}$ for the ring of integers of $E$, $k_{E}$
its residue field, and $\varpi_{E}$ a fixed uniformizer.

###  4.1. A theorem of Emerton

Let $D$ be a quaternion algebra over $F$ which splits at exactly one infinite
place denoted by $\tau$. Fix an isomorphism
$D_{\tau}\overset{\mathrm{def}}{=}D\otimes_{F,\tau}\mathbb{R}\cong\mathrm{M}_{2}(\mathbb{R})$.
Let $D_{f}^{\times}=(D\otimes_{\mathbb{Q}}\mathbb{A}_{f})^{\times}$.

For any open compact subgroup $U\subset D_{f}^{\times}$, let $X_{U}$ be the
projective smooth algebraic curve over $F$ associated to $U$ and consider the
étale cohomology with coefficients in $A$

$H^{1}_{\textrm{\'{e}t}}(X_{U,\overline{\mathbb{Q}}},A),$

where $A$ denotes one of $E$, $\mathcal{O}_{E}$, or
$\mathcal{O}_{E}/\varpi_{E}^{s}$ for some $s>0$. For two open compact
subgroups $V\subseteq U$ of $D_{f}^{\times}$, we have natural morphisms of
algebraic curves $X_{V}\rightarrow X_{U}$ defined over $F$, which induces a
$\mathrm{Gal}(\overline{F}/F)$-equivariant map

$H^{1}_{\textrm{\'{e}t}}(X_{U,\bar{\mathbb{Q}}},A)\rightarrow
H^{1}_{\textrm{\'{e}t}}(X_{V,\bar{\mathbb{Q}}},A).$

Define

$S^{D}(A):=\varinjlim H^{1}_{\textrm{\'{e}t}}(X_{U,\bar{\mathbb{Q}}},A)$

where the limit is taken over all the open compact subgroups $U\subset
D_{f}^{\times}$. It carries a continuous action of
$\mathrm{Gal}(\overline{F}/F)$ and a smooth admissible action of
$D_{f}^{\times}$ commuting with each other.

Let $\overline{\rho}:G_{F}\rightarrow\mathrm{GL}_{2}(k_{E})$ be an
irreducible, continuous, totally odd representation. Assume that
$\overline{\rho}$ is modular, in the sense that
$\mathrm{Hom}_{G_{F}}(\overline{\rho},S^{D}(k_{E}))\neq 0$. Let $\Sigma$ be a
finite set of finite places of $F$ which contains all those places dividing
$p$, or at which $U_{w}$ is not maximal, or $D$ or $\overline{\rho}$ is
ramified. Define as usual the Hecke operators

$T_{v}=[\mathrm{GL}_{2}(\mathcal{O}_{F_{v}})\begin{pmatrix}{\varpi_{v}}&{0}\\\
{0}&{1}\end{pmatrix}\mathrm{GL}_{2}(\mathcal{O}_{F_{v}})],\ \
S_{v}=[\mathrm{GL}_{2}(\mathcal{O}_{F_{v}})\begin{pmatrix}{\varpi_{v}}&{0}\\\
{0}&{\varpi_{v}}\end{pmatrix}\mathrm{GL}_{2}(\mathcal{O}_{F_{v}})]$

and let $\mathbb{T}^{\Sigma}(U)$ denote the commutative $A$-algebra generated
by $T_{v}$ and $S_{v}$ for $v\notin\Sigma$. We let
$\mathfrak{m}_{\overline{\rho}}^{\Sigma}=\mathfrak{m}_{\overline{\rho}}^{\Sigma}(U)$
denote the maximal ideal of $\mathbb{T}^{\Sigma}$ corresponding to
$\overline{\rho}$, i.e. satisfying

$T_{v}\ \textrm{mod}\
\mathfrak{m}_{\overline{\rho}}^{\Sigma}\equiv\textrm{trace}(\overline{\rho}(\mathrm{Frob}_{v}))\
\ \mathrm{and}\ \ \textbf{N}(v)S_{v}\ \textrm{mod}\
\mathfrak{m}_{\overline{\rho}}^{\Sigma}\equiv\det(\overline{\rho}(\mathrm{Frob}_{v}))$

for all $v\notin\Sigma$. Let

$S^{D}(U,k_{E})[\mathfrak{m}_{\overline{\rho}}^{\Sigma}]=\\{f\in
S^{D}(U,k_{E})|Tf=0\ \mathrm{for\ all\
}T\in\mathfrak{m}_{\overline{\rho}}^{\Sigma}\\}.$

By [7, Lemma 4.6], it is independent of $\Sigma$, so denote it by
$S^{D}(U,k_{E})[\mathfrak{m}_{\overline{\rho}}]$. We can consider the direct
limit over $U$ of the spaces $S^{U}(U,k_{E})[\mathfrak{m}_{\overline{\rho}}]$
which yields $S^{D}(k_{E})[\mathfrak{m}_{\overline{\rho}}]$. Recall the
following conjecture due to Buzzard, Diamond and Jarvis [7, Conj. 4.9].

######  Conjecture 1.

The representation $S^{D}(k_{E})[\mathfrak{m}_{\overline{\rho}}]$ of
$G_{F}\times D_{f}^{\times}$ is isomorphic to a restricted tensor product

(4.1)
$S^{D}(k_{E})[\mathfrak{m}_{\overline{\rho}}]\cong\overline{\rho}\otimes(\otimes_{w}^{\prime}\pi_{w}),$

where $\pi_{w}$ is a smooth admissible representation of $D_{w}^{\times}$ such
that

  * $\bullet$

if $w$ does not divide $p$, then $\pi_{w}$ is the representation attached to
$\overline{\rho}_{w}:=\overline{\rho}|_{G_{F_{w}}}$ by the modulo $\ell$ local
Langlands and Jacquet-Langlands correspondence, see [7, §4];

  * $\bullet$

if $w|p$, then $\pi_{w}\neq 0$; moreover if $F$ and $D$ are unramified at $v$,
and $\sigma$ is an irreducible $k_{E}$-representation of
$\mathrm{GL}_{2}(\mathcal{O}_{F_{w}})$, then

$\mathrm{Hom}_{\mathrm{GL}_{2}(\mathcal{O}_{F_{w}})}(\sigma,\pi_{w})\neq 0\
\Longleftrightarrow\sigma\in W(\overline{\rho}_{w}).$

Here $W(\overline{\rho}_{w})$ is a certain set of Serre weights associated to
$\overline{\rho}_{w}$ which, when $\overline{\rho}_{w}$ is generic, coincides
with $\mathscr{D}(\overline{\rho}_{w})$ (in §3) up to normalisation.

From now on, assume that $D$ splits at some finite place $v$ lying above $p$.
For $U^{v}$ an open subgroup of $\prod_{w\neq v}\mathcal{O}_{D_{w}}^{\times}$,
we write

$S^{D}(U^{v},A):=\varinjlim S^{D}(U^{v}U_{v},A)$

where the inductive limit is taken over all compact open subgroups $U_{v}$ of
$D_{v}^{\times}\cong\mathrm{GL}_{2}(F_{v})$. The following result is due to
Emerton (see Definition 1 for the notation).

######  Theorem 4.1.

For any $n\geq 0$, the action of $\mathrm{Gal}(\overline{\mathbb{Q}}_{p}/F)$
on the cokernel of the embedding

$\mathrm{Ord}_{P}\bigl{(}S^{D}(U^{v},k_{E})_{\mathfrak{m}_{\overline{\rho}}}\bigr{)}^{\mathrm{ab},S_{v}}\hookrightarrow\mathrm{Ord}_{P}\bigl{(}S^{D}(U^{v},k_{E})_{\mathfrak{m}_{\overline{\rho}}}\bigr{)}$

factors through $G_{F_{v}}^{\rm ab}$.

###### Proof.

This is Theorem 5.6.11 of [10] in our setting. The proof of Emerton works
equally in this case. In fact, Lemmas 5.6.7 and 5.6.8 _loc. cit._ hold true,
with the only change being to replace the absolute value $|\ |_{p}$ by $|\
|_{v}$, the absolute value on $F_{v}$ normalised as
$|\varpi_{v}|_{v}:=q_{v}^{-1}$. We also need

$S^{D}(U^{v},\mathcal{O}_{E})_{\mathfrak{m}_{\overline{\rho}}}^{N_{0,v}}/\varpi_{L}S^{D}(U^{v},\mathcal{O}_{E})_{\mathfrak{m}_{\overline{\rho}}}^{N_{0,v}}\rightarrow
S^{D}(U^{v},k_{E})_{\mathfrak{m}_{\overline{\rho}}}^{N_{0,v}}$

to be surjective to apply Lemma 5.6.3 _loc.cit._. This is a consequence, by
taking inductive limit over $r$, of the isomorphisms (provided $r$ is large
enough so that $U^{v}I_{r,v}$ is neat)

$S^{D}(U^{v},\mathcal{O}_{E})_{\mathfrak{m}_{\overline{\rho}}}^{I_{r,v}}/\varpi_{L}S^{D}(U^{v},\mathcal{O}_{E})_{\mathfrak{m}_{\overline{\rho}}}^{I_{r,v}}\rightarrow
S^{D}(U^{v},k_{E})_{\mathfrak{m}_{\overline{\rho}}}^{I_{r,v}}.$

Here, we denote by $I_{r,v}$ the open subgroup of
$\mathrm{GL}_{2}(\mathcal{O}_{F_{v}})$ defined as
$\\{g\equiv\bigl{(}\begin{smallmatrix}{1}&{*}\\\
{0}&{1}\end{smallmatrix}\bigl{)}\mod\varpi_{v}^{r}\\}$. ∎

###  4.2. Application

Assuming the existence of the decomposition as in Conjecture 1, we get the
following information about the local factor $\pi_{v}$ (for $v|p$) when
$\overline{\rho}|_{G_{F_{v}}}$ is reducible with scalar endomorphisms.

######  Theorem 4.2.

Assume the decomposition (4.1) holds. If for some $v|p$, $\overline{\rho}_{v}$
is reducible indecomposable and isomorphic to
$\bigl{(}\begin{smallmatrix}{\psi_{1}}&{*}\\\
{0}&{\psi_{2}}\end{smallmatrix}\bigl{)}$, with $\psi_{1}\neq\psi_{2}$, then
$\mathrm{Ord}_{P}(\pi_{D,v}(\overline{\rho}))$ is an admissible semi-simple
$T_{v}$-representation.

###### Proof.

The natural inclusion
$S^{D}(U^{v},k_{E})[\mathfrak{m}_{\overline{\rho}}]\hookrightarrow
S^{D}(U^{v},k_{E})_{\mathfrak{m}_{\overline{\rho}}}$ induces an inclusion
$\mathrm{Ord}_{P}(S^{D}(U^{v},k_{E})[\mathfrak{m}_{\overline{\rho}}])\hookrightarrow\mathrm{Ord}_{P}(S^{D}(U^{v},k_{E})_{\mathfrak{m}_{\overline{\rho}}})$,
hence Lemma 2.10 and Theorem 4.1 imply that the action of $G_{F_{v}}$ on the
cokernel of

$\bigl{(}\mathrm{Ord}_{P}(S^{D}(U^{v},k_{E})[\mathfrak{m}_{\overline{\rho}}])\bigr{)}^{\mathrm{ab},S_{v}}\hookrightarrow\mathrm{Ord}_{P}(S^{D}(U^{v},k_{E})[\mathfrak{m}_{\overline{\rho}}])$

factors through $G_{F_{v}}^{\rm ab}$.

Assume that the decomposition (4.1) holds. Then, as a representation of
$G_{F_{v}}\times\mathrm{GL}_{2}(F_{v})$,
$S^{D}(U^{v},k_{E})[\mathfrak{m}_{\overline{\rho}}]$ is isomorphic to
$(\overline{\rho}_{v}\otimes\pi_{v})^{\oplus r}$ for some integer $r\geq 1$.
Apply Lemma 2.10 again, we deduce that the action of $G_{F_{v}}$ on the
cokernel of

$\mathrm{Ord}_{P}(\overline{\rho}_{v}\otimes\pi_{v})^{\mathrm{ab},S_{v}}\hookrightarrow\mathrm{Ord}_{P}(\overline{\rho}_{v}\otimes\pi_{v})$

factors through $G_{F_{v}}^{\rm ab}$.

Now assume that $\mathrm{Ord}_{P}(\pi_{v})$ is not semi-simple. Because there
is no non-trivial extension between two non-isomorphic $k_{E}$-characters of
$T_{v}$, there must exist some character $\chi:T_{v}\rightarrow
k_{E}^{\times}$ and some non-trivial extension as (2.1)

$0\rightarrow\chi\rightarrow\epsilon_{\delta}\rightarrow\chi\rightarrow 0$

such that $\epsilon_{\delta}$ appears as a _subquotient_ of
$\mathrm{Ord}_{P}(\pi_{v})$. This implies that
$\mathrm{Ord}_{P}(\overline{\rho}_{v}\otimes\pi_{v})$, which is equal to
$\overline{\rho}_{v}\otimes\mathrm{Ord}_{P}(\pi_{v})$, has a $G_{F_{v}}\times
T_{v}$-equivariant, hence $G_{F_{v}}\times S_{v}$-equivariant, subquotient of
the form $\overline{\rho}_{v}\otimes\epsilon_{\delta}$. Applying Lemma 2.10 to
it shows that the action of $G_{F_{v}}$ on the cokernel of

(4.2)
$(\overline{\rho}_{v}\otimes\epsilon_{\delta})^{\mathrm{ab},S_{v}}\hookrightarrow\overline{\rho}_{v}\otimes\epsilon_{\delta}$

factors through $G_{F_{v}}^{\rm ab}$. We claim that
$(\overline{\rho}_{v}\otimes\epsilon_{\delta})^{\mathrm{ab},S_{v}}$ is at most
$1$-dimensional over $k_{E}$, while
$\overline{\rho}_{v}\otimes\epsilon_{\delta}$ is $4$-dimensional, and the
cokernel of (4.2) admits a quotient isomorphic to
$\overline{\rho}_{v}\otimes\chi$. Because $\overline{\rho}_{v}$ is reducible
and indecomposable by assumption, the action of $G_{F_{v}}$ on
$\overline{\rho}_{v}\otimes\chi$ does not factor through $G_{F_{v}}^{\rm ab}$,
hence a contradiction which shows that $\mathrm{Ord}_{P}(\pi_{v})$ is semi-
simple.

To verify the claim, we choose a basis $\\{v_{1},v_{2}\\}$ of
$\overline{\rho}_{v}$ over $k_{E}$ such that $g\cdot v_{1}=\psi_{1}(g)v_{1}$
for $g\in G_{F_{v}}$; also choose a basis $\\{w_{1},w_{2}\\}$ of
$\epsilon_{\delta}$ such that $s\cdot w_{1}=\chi(s)w_{1}$ and $s\cdot
w_{2}=\chi(s)(w_{2}+\delta(s)w_{1})$ for $s\in T_{v}$. It is clear that
$V^{\rm ab}\subset\psi_{1}\otimes\epsilon_{\delta}$, as $\overline{\rho}_{v}$
is indecomposable and $\psi_{1}\neq\psi_{2}$. Since the action of $S_{v}$ on
$\overline{\rho}_{v}\otimes\epsilon_{\delta}$ is via the anti-diagonal
embedding $S_{v}\hookrightarrow G_{F_{v}}^{\rm ab}\times S_{v}$,
$s\mapsto(\iota(s),s^{-1})$, we get

$s\cdot(v_{1}\otimes w_{1})=\psi_{1}(\iota(s))v_{1}\otimes\chi(s^{-1})w_{1}$
$s\cdot(v_{1}\otimes
w_{2})=\psi_{1}(\iota(s))v_{1}\otimes[\chi(s^{-1})(w_{2}+\delta(s^{-1})w_{1})].$

Because $\delta:F_{v}^{\times}\cong S_{v}\rightarrow k_{E}$ is non-trivial and
the extension $\epsilon_{\delta}$ admits a central character, there exists
$s\in S_{v}$ such that $\delta(s^{-1})=-\delta(s)\neq 0$, hence

$(\overline{\rho}_{v}\otimes\epsilon_{\delta})^{\mathrm{ab},S_{v}}\subseteq
k_{E}(v_{1}\otimes w_{1}).$

The claim follows easily from this. ∎

Combined with results proved in §3, we get the following corollary.

######  Corollary 4.3.

Keep assumptions in Theorem 4.2. Assume moreover that $\overline{\rho}_{v}$ is
generic and $F_{v}$ is unramified over $\mathbb{Q}_{p}$ of degree $\geq 2$.
Then the $\mathrm{GL}_{2}(F_{v})$-representation $\pi_{v}$ contains a sub-
representation $\pi$ which is of length $2$ and fits into a non-split exact
sequence

$0\rightarrow\pi_{0}\rightarrow\pi\rightarrow\pi_{1}\rightarrow 0$

with $\pi_{0}$ a principal series and $\pi_{1}$ supersingular. If moreover
$[F_{v}:\mathbb{Q}_{p}]=2$, and if $\pi_{2}$ denotes the $f_{v}$-th smooth
dual of $\pi_{0}$, then
$\mathrm{Ext}^{1}_{\mathrm{GL}_{2}(F_{v})}(\pi_{2},\pi_{1})\neq 0$.

###### Proof.

Under our assumptions together with the genericity condition of
$\overline{\rho}_{v}$, we can apply [4, Thm. 3.7.1] to get that
$\mathrm{soc}_{\mathrm{GL}_{2}(\mathcal{O}_{F_{v}})}(\pi_{v})$ is of
multiplicity 1. Then Theorem 4.2 implies that $\mathrm{Ord}_{P_{v}}(\pi_{v})$
is one-dimensional, so that the result follows from Propositions 3.2 and 3.3
and Remark 6. ∎

We can get rid of the assumption (4.1) in Corollary 4.3 as follows (note
however that then $\pi_{v}$ does not make sense).

###### Proof of Corollary 1.2.

In [4], Breuil and Diamond explicitly constructed a certain
$k_{E}$-representation $\pi_{v}^{\prime}$ of $\mathrm{GL}_{2}({F_{v}})$, and
showed that $\pi_{v}^{\prime}$ must be the local factor $\pi_{v}$ if (4.1)
holds (see [4, Cor. 3.7.4]). We have a
$G_{F_{v}}\times\mathrm{GL}_{2}(F_{v})$-equivariant embedding
$\overline{\rho}_{v}\otimes\pi_{v}^{\prime}\hookrightarrow
S^{D}(U^{v},k_{E})[\mathfrak{m}_{\overline{\rho}}]$, so that
$\mathrm{Ord}_{P}(\pi_{v}^{\prime})$ is semi-simple by the same proof of
Theorem 4.2. Moreover, under genericity assumption on $\overline{\rho}_{v}$,
[4, Thm. 3.7.1] says that the $K_{v}$-socle of $\pi_{v}^{\prime}$ is of
multiplicity one, hence $\mathrm{Ord}_{P}(\pi_{v}^{\prime})$ is of dimension
1. We deduce that $\pi_{v}^{\prime}$ contains a non-split extension of
$\pi_{1}$ by $\pi_{0}$. The last assertion of Corollary 1.2 is deduced as in
Corollary 4.3. ∎

##  5\. Appendix

In this section, we prove, contrast to Proposition 3.3, that there is no non-
trivial extension of principal series by supersingular representations, when
$L$ is a local field of characteristic $p$. Notations are the same as in
Section 2.

We fix a uniformizer $\varpi$ of $L$. For an irreducible smooth representation
$\sigma$ of $K$, we view it as a representation of $KZ$ by letting
$\bigl{(}\begin{smallmatrix}{\varpi}&{0}\\\
{0}&{\varpi}\end{smallmatrix}\bigl{)}$ act trivially. Consider the compact
induction $\mathrm{c}\textrm{-}\mathrm{Ind}_{KZ}^{G}\sigma$ and recall that
the $\overline{\mathbb{F}}_{p}$-algebra
$\mathrm{End}_{G}(\mathrm{c}\textrm{-}\mathrm{Ind}_{KZ}^{G}\sigma)$ is
isomorphic to $\overline{\mathbb{F}}_{p}[T]$ where $T$ is a certain Hecke
operator (normalized as in [1]). To describe the action of $T$, for $g\in G$
and $v\in\sigma$, denote by
$[g,v]\in\mathrm{c}\textrm{-}\mathrm{Ind}_{KZ}^{G}\sigma$ the function
supported on $KZg^{-1}$ and such that $[g,v](g^{-1})=v$. Let
$v_{0}\in\sigma^{I_{1}}$ be a non-zero vector and recall that $\mathrm{Id}$
denotes the identity matrix of $\mathrm{GL}_{2}(L)$, then the action of $T$ on
$\mathrm{c}\textrm{-}\mathrm{Ind}_{KZ}^{G}\sigma$ is characterized by the
formula (and by the $G$-equivariance of $T$):

(5.1)
$T([\mathrm{Id},v_{0}])=\sum_{\lambda\in\mathbb{F}_{q}}\Bigl{[}\begin{pmatrix}{\varpi}&{[\lambda]}\\\
{0}&{1}\end{pmatrix},v_{0}\Bigr{]}+\epsilon(\sigma)[\Pi,v_{0}]$

where $\epsilon(\sigma)=1$ if $\sigma$ is of dimension $1$ and
$\epsilon(\sigma)=0$ otherwise.

In [15, Def. 2.7] is defined an operator by the formula:

$S=\displaystyle\sum\limits_{\lambda\in\mathbb{F}_{q}}\begin{pmatrix}{\varpi}&{[\lambda]}\\\
{0}&{1}\end{pmatrix}\in\overline{\mathbb{F}}_{p}[G]$

which acts on any $G$-representation. By (5.1) we get immediately

(5.2) $T([\mathrm{Id},v_{0}])=S([\mathrm{Id},v_{0}])$

when $\sigma$ is of dimension $\geq 2$. Set $S^{1}=S$ and by induction
$S^{n}=S\circ S^{n-1}$ for $n\geq 1$. We then have $S^{n+m}=S^{n}\circ
S^{m}=S^{m}\circ S^{n}$. Recall some useful properties of $S$.

######  Lemma 5.1.

Let $\pi$ be a smooth representation of $G$ and let $v\in\pi$.

(i) If $v$ is an eigenvector of $\mathcal{H}$, then so is $Sv$ for the same
eigencharacter.

(ii) If $v$ is fixed by $N_{0}$, then so is $Sv$.

(iii) If $v$ is fixed by $N_{0}$, then so is $h\cdot v$ for $h\in T_{1}$, and
we have the formula $h\cdot Sv=S(h\cdot v)$ for all $h\in T_{1}$.

(iv) If $v$ is fixed by
$\bigl{(}\begin{smallmatrix}{1+\mathfrak{p}_{L}}&{\mathcal{O}_{L}}\\\
{\mathfrak{p}_{L}^{n+1}}&{1+\mathfrak{p}_{L}}\end{smallmatrix}\bigl{)}$ with
$n\geq 1$, then $Sv$ is fixed by
$\bigl{(}\begin{smallmatrix}{1+\mathfrak{p}_{L}}&{\mathcal{O}_{L}}\\\
{\mathfrak{p}_{L}^{n}}&{1+\mathfrak{p}_{L}}\end{smallmatrix}\bigl{)}$. In
particular, if $v$ is fixed by $I_{1}$, then so is $Sv$.

(v) Assume that $L$ is of characteristic $p$ and $\pi$ is a supersingular
representation of $G$. Then there exists an integer $n\gg 0$ such that
$S^{n}v=0$.

###### Proof.

(i), (ii) and (iii) follow from an easy calculation, see [15, Lem. 2.8] for
the details. (iv) is proved in [15, §4.2, (4.4)]. (v) is just [15, Thm. 5.1],
which requires $L$ to be of characteristic $p$. ∎

######  Theorem 5.2.

Assume that $L$ is of characteristic $p$. Let $\pi_{1}$ be a supersingular
representation and $\pi_{2}$ be a principal series or a special series of $G$.
Then $\mathrm{Ext}^{1}_{G}(\pi_{2},\pi_{1})=0$.

###### Proof.

Consider an extension of $G$-representations

$0\rightarrow\pi_{1}\rightarrow V\rightarrow\pi_{2}\rightarrow 0.$

Because $\pi_{2}$ is a principal series or a special series, it always
contains a sub-$K$-representation of dimension $\geq 2$, say $\sigma$, and
$\pi_{2}$ is a quotient of
$\mathrm{c}\textrm{-}\mathrm{Ind}_{KZ}^{G}\sigma/(T-\lambda)\otimes\chi\circ\det$
for some $\lambda\in\overline{\mathbb{F}}_{p}^{\times}$ and
$\chi:L^{\times}\rightarrow\overline{\mathbb{F}}_{p}^{\times}$, see [1, Thm.
33]. Let $\bar{w}\in\sigma^{I_{1}}\subset\pi_{2}$ be a non-zero vector, which
is unique up to a scalar and is automatically an eigenvector of $\mathcal{H}$.
Choose a lifting $w\in V$ of $\bar{w}$ arbitrarily. Since the order of
$\mathcal{H}$ is prime to $p$, we may choose $w$ so that it is an eigenvector
of $\mathcal{H}$. Denote by $M=\langle N_{0}.w\rangle\subset V$ the
sub-$N_{0}$-representation generated by $w$ and choose vectors
$v_{i}\in\pi_{1}$, where $i$ runs over a finite set, such that $\\{w,v_{i}\\}$
forms a basis of $M$. Then by [15, Lem. 5.2], for any $n>0$, $\langle
N_{0}.S^{n}w\rangle$ is spanned by the vectors $\\{S^{n}w,S^{n}v_{i}\\}$. But,
since $M$ is finite dimensional and $\pi_{1}$ is supersingular, Lemma 5.1(v)
implies that $S^{n}v_{i}=0$ for $n\gg 0$, hence $S^{n}w$ is fixed by $N_{0}$
for such $n$. Since
$S\bar{w}\overset{(\ref{equation-S=T})}{=}T\bar{w}=\lambda\bar{w}$ in
$\pi_{2}$ and $\lambda\neq 0$, $\frac{1}{\lambda^{n}}S^{n}w$ is still a
lifting of $\bar{w}$, so we may assume the chosen lifting $w$ is fixed by
$N_{0}$. Since $\bar{w}$ is fixed by $I_{1}$, we have (in particular)
$(h-1)\bar{w}=0$ for any $h\in T_{1}$, hence $(h-1)w\in\pi_{1}$. By Lemma
5.1(v), there exists $n_{h}\gg 0$ such that $S^{n_{h}}(h-1)w=0$. The
representation $V$ being smooth, $\langle T_{1}.w\rangle$ is finite
dimensional, so we may find $n$ large enough so that $S^{n}(h-1)w=0$ for all
$h\in T_{1}$. But, Lemma 5.1(iii) implies

$0=S^{n}(h-1)w=(h-1)S^{n}w,$

so that $S^{n}w$ is fixed by $I_{1}\cap P$. Again, by Lemma 5.1(iv), up to
enlarge $n$, $S^{n}w$ is fixed by $I_{1}$. Replacing $w$ by
$\frac{1}{\lambda^{n}}S^{n}w$, we obtain a lifting $w$ of $\bar{w}$ which is
fixed by $I_{1}$.

Next, consider the vector $(S-\lambda)w$ which belongs to $\pi_{1}$ as its
image in $\pi_{2}$ is zero. By Lemma 5.1(v) again, there exists $n\gg 0$ such
that $S^{n}(S-\lambda)w=0$. Replacing $w$ by $\frac{1}{\lambda^{n}}S^{n}w$, we
get a lifting $w$ of $\bar{w}$ satisfying $Sw=\lambda w$.

Summarizing, we obtain a lifting $w\in V^{I_{1}}$ of $\bar{w}$ which is an
eigenvector of $\mathcal{H}$ and satisfies $Sw=\lambda w$. By the proof of
[18, Lem. 4.1]

$\tau:=\langle K\cdot w\rangle\subset V$

is irreducible and isomorphic to $\sigma$. Moreover, the fact $Sw=\lambda w$
implies that the $G$-morphism
$\mathrm{c}\textrm{-}\mathrm{Ind}_{KZ}^{G}\tau\otimes\chi\circ\det\rightarrow
V$ (here $\chi$ is the character appeared at the beginning of the proof)
induced by Frobenius reciprocity must factor through (note that $Sw=Tw$ by
(5.2))

$\phi:\mathrm{c}\textrm{-}\mathrm{Ind}_{KZ}^{G}\tau/(T-\lambda)\otimes\chi\circ\det\rightarrow
V.$

This shows that $V$ contains $\pi_{1}$ as a sub-representation and hence
splits. ∎

Acknowledgements The author would like to thank V. Paškūnas for some useful
discussion and C. Breuil for several comments on the first version of the
paper.

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