# The mod-$p$ representation theory of the metaplectic cover of
$\operatorname{GL}_{2}(\mathbf{Q}_{p})$

Robin Witthaus<EMAIL_ADDRESS>
###### Abstract.

Half-integral weight modular forms are naturally viewed as automorphic forms
on the so-called metaplectic covering of
$\operatorname{GL}_{2}(\mathbf{A}_{\mathbf{Q}})$ – a central extension by the
roots of unity $\mu_{2}$ in $\mathbf{Q}$. For an odd prime number $p$, we give
a complete classification of the smooth irreducible genuine mod-$p$
representations of the corresponding covering of
$\operatorname{GL}_{2}(\mathbf{Q}_{p})$ by showing that the functor of taking
pro-$p$-Iwahori-invariants and its left adjoint define a bijection onto the
set of simple right modules of the pro-$p$ Iwahori Hecke algebra. As an
application of our investigation of the irreducible subquotients of the
universal module over the spherical Hecke algebra depending on some weight, we
prove that being of finite length is equivalent to being finitely generated
and admissible. Finally, we explain a relation to locally algebraic
irreducible unramified genuine principal series representations.

###### Contents

  1. 1 Introduction
    1. 1.1 Results
    2. 1.2 Conventions and notations
  2. 2 The metaplectic covering group
    1. 2.1 Hilbert symbol
    2. 2.2 Definition and first properties
    3. 2.3 Genuine representations and further notations
  3. 3 The pro-$p$ Iwahori Hecke algebra
    1. 3.1 Generators and relations
    2. 3.2 Decomposing the pro-$p$ Iwahori Hecke algebra
    3. 3.3 The Hecke algebra of a regular orbit
      1. 3.3.1 Simple modules
    4. 3.4 The Hecke algebra of a non-regular orbit
      1. 3.4.1 Simple modules
    5. 3.5 Extensions of simple modules
      1. 3.5.1 Extensions of supersingular modules
      2. 3.5.2 Extensions of principal modules
  4. 4 Genuine representation theory
    1. 4.1 Preliminaries
    2. 4.2 Weights
    3. 4.3 Principal series representations
      1. 4.3.1 Smooth irreducible genuine $\widetilde{T}$-representations
      2. 4.3.2 Irreducibility of principal series representations
      3. 4.3.3 From principal Hecke modules to principal series representations
      4. 4.3.4 Extensions of principal series representations
    4. 4.4 Supersingular representations
      1. 4.4.1 Basic lemma
      2. 4.4.2 From supersingular Hecke modules to supersingular representations
      3. 4.4.3 …and back
  5. 5 Spherical Hecke algebras
    1. 5.1 Computational preparations
    2. 5.2 Jordan-Hölder sequence
    3. 5.3 Howe’s theorem
  6. 6 Relation to locally algebraic representations
    1. 6.1 Savin’s local Shimura correspondence
    2. 6.2 Genuine principal series in characteristic $0$
    3. 6.3 Mod-$p$ reduction

##  1\. Introduction

The Shimura correspondence [28] is a map $\operatorname{Sh}$ from the space of
half-integral weight cusp forms to the space of even-integral weight modular
forms, satisfying $\operatorname{Sh}\circ
T_{p^{2}}=T_{p}\circ\operatorname{Sh}$ for each prime $p$, where $T_{p^{2}}$
is the half-integral weight Hecke operator defined by Shimura and $T_{p}$ is
the usual integral weight Hecke operator, see also [25]. This correspondence
has been reformulated in terms of automorphic representations by Gelbart and
Piatetski-Shapiro [12] as well as, independently and more generally, by
Flicker [11]: letting $\mathbf{A}_{\mathbf{Q}}$ denote the ring of rational
adèles, they consider the metaplectic cover

$1\to\mu_{2}\to\widetilde{\operatorname{GL}}_{2}(\mathbf{A}_{\mathbf{Q}})\to\operatorname{GL}_{2}(\mathbf{A}_{\mathbf{Q}})\to
1,$

which is a central topological extension by the roots of unity $\mu_{2}=\\{\pm
1\\}$ in $\mathbf{Q}$. It is constructed from corresponding local metaplectic
covers

$1\to\mu_{2}\to\widetilde{\operatorname{GL}}_{2}(\mathbf{Q}_{p})\to\operatorname{GL}_{2}(\mathbf{Q}_{p})\to
1$

as $p$ ranges over all places, see [13]. These local extensions are defined
using the degree-two Hilbert symbol, and they split over the maximal compact
open subgroup if $2<p<\infty$. The quadratic reciprocity law implies that the
global covering splits over $\operatorname{GL}_{2}(\mathbf{Q})$, and Shimura’s
correspondence is then incarnated as a map

$\left\\{\begin{array}[]{c}\text{Genuine complex}\\\ \text{automorphic
representations}\\\ \text{of
$\widetilde{\operatorname{GL}}_{2}(\mathbf{A}_{\mathbf{Q}})$}\end{array}\right\\}\to\left\\{\begin{array}[]{c}\text{Complex}\\\
\text{automorphic representations}\\\ \text{of
$\operatorname{GL}_{2}(\mathbf{A}_{\mathbf{Q}})$}\end{array}\right\\},$

where “genuine” means that the subgroup $\mu_{2}$ acts non-trivially. This map
is defined by means of local Shimura correspondences relating those smooth
irreducible admissible complex representations of
$\widetilde{\operatorname{GL}}_{2}(\mathbf{Q}_{p})$ which are genuine to those
of $\operatorname{GL}_{2}(\mathbf{Q}_{p})$ (here we restrict to finite places
$p$).

Given a prime $\ell$, it is natural to ask whether there exists an $\ell$-adic
analogue of the local Shimura correspondence for the group
$\widetilde{\operatorname{GL}}_{2}(\mathbf{Q}_{p})$, attaching an irreducible
admissible unitary $\ell$-adic Banach space representation of
$\operatorname{GL}_{2}(\mathbf{Q}_{p})$ to a genuine one of the covering
group. If $\ell\neq p$, then passing to smooth vectors still remembers a large
part of the representation (the smooth vectors will be dense). In fact,
Vignéras [29] proved that taking smooth vectors and remembering the
commensurability class of the smooth part of an invariant unit ball sets up an
equivalence onto a suitable category. Since smooth representations do not see
the topology of the coefficient field, one may then try to use the existing
complex local correspondence to define an $\ell$-adic one. The case $\ell=p$
is more subtle (and therefore more exciting to study) as there are usually no
smooth vectors at all and it is not clear how to possibly formulate a $p$-adic
local Shimura correspondence. By passing to the mod-$p$ reduction of an
invariant unit ball, any such correspondence should give rise to a semi-simple
mod-$p$ local Shimura correspondence (for $p>2$), which seems easier to
attack. Hence, instead of directly looking for a $p$-adic correspondence, it
is reasonable to first study the mod-$p$ situation. In order to do this, it
will be necessary—or at least very useful—to have a good understanding of the
smooth irreducible admissible mod-$p$ representations of
$\widetilde{\operatorname{GL}}_{2}(\mathbf{Q}_{p})$. On such an object, the
central subgroup $\mu_{2}$ either acts trivially, i.e. the representation
descends to $\operatorname{GL}_{2}(\mathbf{Q}_{p})$, or it acts via the non-
trivial character in which case the representation is called genuine.

The smooth irreducible $\overline{\mathbf{F}}_{p}$-linear representations of
$\operatorname{GL}_{2}(\mathbf{Q}_{p})$ (admitting a central character) are
well-understood by the work of Barthel–Livné [1] and Breuil [4]: Letting
$Z\cong\mathbf{Q}_{p}^{\times}$ denote the center of
$\operatorname{GL}_{2}(\mathbf{Q}_{p})$, their approach is based on the
spherical Hecke algebras
$\operatorname{End}_{\operatorname{GL}_{2}(\mathbf{Q}_{p})}(\operatorname{c-Ind}^{\operatorname{GL}_{2}(\mathbf{Q}_{p})}_{\operatorname{GL}_{2}(\mathbf{Z}_{p})Z}(\sigma))$,
where $\sigma$ is an irreducible representation of
$\operatorname{GL}_{2}(\mathbf{F}_{p})$ viewed as a representation of
$\operatorname{GL}_{2}(\mathbf{Z}_{p})Z$ via inflation and letting the central
element $p$ act via a fixed non-zero scalar. Namely, Barthel and Livné show
that the Hecke algebra is a polynomial ring $\overline{\mathbf{F}}_{p}[T]$ in
a single Hecke operator $T$, corresponding to the double coset
$\operatorname{GL}_{2}(\mathbf{Z}_{p})Z\left(\begin{smallmatrix}1&0\\\
0&p^{-1}\end{smallmatrix}\right)\operatorname{GL}_{2}(\mathbf{Z}_{p})Z$ in a
suitable sense, and they observe that any of the irreducible objects admitting
a central character is a quotient of the cokernel of a linear polynomial in
$T$, for some $\sigma$. For $\lambda\in\overline{\mathbf{F}}_{p}^{\times}$,
they compute the cokernel $\operatorname{coker}(T-\lambda)$ to be, up to semi-
simplification, equal to a principal series, which is generically irreducible
and otherwise of length two; they called the remaining irreducible
representations, i.e. those arising as a quotient of $\operatorname{coker}(T)$
for some $\sigma$, supersingular. It was Breuil who proved the cokernel of $T$
to be in fact irreducible by computing the subspace of vectors fixed by the
standard pro-$p$ Iwahori subgroup $I_{1}$, implying that any smooth
irreducible $\overline{\mathbf{F}}_{p}$-representation of
$\operatorname{GL}_{2}(\mathbf{Q}_{p})$ admitting a central character is
admissible – Berger [2] later proved the existence of central characters for
any smooth irreducible $\overline{\mathbf{F}}_{p}$-representation of
$\operatorname{GL}_{2}(\mathbf{Q}_{p})$.

The consideration of $I_{1}$-invariants was studied more generally by Vignéras
[30] who proved that passing to $I_{1}$-invariants defines a bijection between
the set of isomorphism classes of smooth irreducible (admissible)
$\overline{\mathbf{F}}_{p}$-representations of
$\operatorname{GL}_{2}(\mathbf{Q}_{p})$ and the set of isomorphism classes of
(finite dimensional) simple right modules of the pro-$p$ Iwahori Hecke algebra
$\mathcal{H}_{\operatorname{GL}_{2}(\mathbf{Q}_{p})}=\operatorname{End}_{\operatorname{GL}_{2}(\mathbf{Q}_{p})}(\operatorname{c-Ind}^{\operatorname{GL}_{2}(\mathbf{Q}_{p})}_{I_{1}}(\mathbf{1}))$,
where $\mathbf{1}$ denotes the trivial
$\overline{\mathbf{F}}_{p}^{\times}$-valued character of $I_{1}$. We emphasize
that her result relies on Breuil’s previously mentioned computation. Ollivier
[20] took this further by establishing that, at least after fixing a central
character, the adjoint pair
$(-\otimes_{\mathcal{H}_{\operatorname{GL}_{2}(\mathbf{Q}_{p})}}\operatorname{c-Ind}^{\operatorname{GL}_{2}(\mathbf{Q}_{p})}_{I_{1}}(\mathbf{1}),(-)^{I_{1}})$
defines an equivalence between the category of right
$\mathcal{H}_{\operatorname{GL}_{2}(\mathbf{Q}_{p})}$-modules and the category
of smooth $\overline{\mathbf{F}}_{p}$-representations of
$\operatorname{GL}_{2}(\mathbf{Q}_{p})$ generated by their $I_{1}$-invariants.

On the other hand, to the best of the author’s knowledge, the smooth genuine
mod-$p$ representation theory of
$\widetilde{\operatorname{GL}}_{2}(\mathbf{Q}_{p})$ has not yet been
investigated. The aim of this work is to fill this gap in the literature and
prove a complete classification result for the irreducible objects. It is the
point of view of Vignéras and Ollivier that we will use to tackle this task.
We also obtain a clear picture in terms of the approach of Barthel-Livné in
the genuine setting. Despite not finding any existing results for the covering
of the general linear group, we point out that Peskin [24] studied the smooth
mod-$p$ representation theory of the metaplectic covering of
$\operatorname{SL}_{2}(\mathbf{Q}_{p})$ by considering spherical Hecke
algebras; however, it only allowed her to obtain a classification up the
supersingular representations. We are optimistic that, by studying the
restriction of the irreducible
$\widetilde{\operatorname{GL}}_{2}(\mathbf{Q}_{p})$-representations to the
covering of $\operatorname{SL}_{2}(\mathbf{Q}_{p})$, one should be able to
recover and complete her classification.

Finally, we note that Breuil [5] computed, depending on small weights, the
mod-$p$ reduction of locally algebraic unramified principal series, the
unitary completions of which were later [3] shown to be irreducible admissible
$p$-adic Banach space representations of
$\operatorname{GL}_{2}(\mathbf{Q}_{p})$.

The metaplectic counterpart of this task is interesting for two reasons: 1) it
might shed some light on how to “lift” a mod-$p$ Shimura correspondence to a
$p$-adic one and 2) his computations led Breuil to a formulation of a semi-
simple mod-$p$ local Langlands correspondence (LLC) for
$\operatorname{GL}_{2}(\mathbf{Q}_{p})$, which ultimately led to a $p$-adic
LLC [7], and so a possible semi-simple mod-$p$ metaplectic LLC for
$\operatorname{GL}_{2}(\mathbf{Q}_{p})$ might give some valuable information
about a $p$-adic version.

We check that Breuil’s computations also go through in our setting if the
weight is at most $p-1$. While we do not attempt to formulate a mod-$p$
Shimura correspondence, we do establish a relation between the smooth genuine
mod-$p$ representation theory of the metaplectic covering group of
$\operatorname{GL}_{2}(\mathbf{Q}_{p})$ and Galois representations in [31].

###  1.1. Results

Fix an odd prime number $p$ and let $\overline{\mathbf{F}}_{p}$ be our field
of coefficients.111In the main text we work over an arbitrary algebraically
closed field of characteristic $p$. Put
$G=\operatorname{GL}_{2}(\mathbf{Q}_{p})$ so that the metaplectic covering
takes the form

$1\to\mu_{2}\to\widetilde{G}\to G\to 1.$

It splits over the maximal compact open subgroup
$K=\operatorname{GL}_{2}(\mathbf{Z}_{p})$ and thus uniquely over the pro-$p$
group $I_{1}$, so that we can unambiguously identify
$\widetilde{I}_{1}=I_{1}\times\mu_{2}$ as groups. Let
$\iota\colon\mu_{2}\hookrightarrow\mathbf{F}_{p}^{\times}\subset\overline{\mathbf{F}}_{p}^{\times}$
denote the non-trivial character, and let
$\operatorname{Mod}^{\operatorname{sm}}_{\widetilde{G},\iota}(\overline{\mathbf{F}}_{p})$
denote the category of smooth $\overline{\mathbf{F}}_{p}$-representations of
$\widetilde{G}$, which are genuine, i.e. on which the central subgroup
$\mu_{2}$ acts via $\iota$. Let
$\mathcal{H}=\operatorname{End}_{\widetilde{G}}(\operatorname{c-Ind}^{\widetilde{G}}_{\widetilde{I}_{1}}(\iota))$
be the genuine version of the pro-$p$ Iwahori Hecke algebra. Denoting by
$\operatorname{Mod}_{\mathcal{H}}$ the category of right
$\mathcal{H}$-modules, we have the adjoint pair $(\mathcal{T},\mathcal{I})$,

$\mathcal{T}(-)=(-)\otimes_{\mathcal{H}}\operatorname{c-Ind}^{\widetilde{G}}_{\widetilde{I}_{1}}(\iota)\colon\operatorname{Mod}_{\mathcal{H}}\rightleftarrows\operatorname{Mod}^{\operatorname{sm}}_{\widetilde{G},\iota}(\overline{\mathbf{F}}_{p})\nobreak\mspace{6.0mu}{:}\nonscript\mkern-3.0mu\mathpunct{}\mspace{2.0mu}(-)^{I_{1}}=\mathcal{I}(-).$

######  Theorem 1.1.

The adjoint pair $(\mathcal{T},\mathcal{I})$ defines a bijection

$\left\\{\begin{array}[]{c}\text{Simple right}\\\
\text{$\mathcal{H}$-modules}\end{array}\right\\}\cong\left\\{\begin{array}[]{c}\text{Smooth
irreducible genuine}\\\ \text{$\overline{\mathbf{F}}_{p}$-representations of
$\widetilde{G}$}\end{array}\right\\},$

where both sides are considered up to isomorphism.

This result is a consequence of theorems 4.31, 4.54 and 5.17, the last of
which says that being of finite length as a smooth
$\widetilde{G}$-representation is equivalent to being finitely generated and
admissible, as well as a careful analysis of the objects appearing on the
left-hand side. In order to describe the latter, we follow the approach of
Vignéras in [30] and decompose the Hecke algebra into a direct sum of smaller
algebras whose centers we compute to be either isomorphic to
$\overline{\mathbf{F}}_{p}[X,Y,Z^{\pm 1}]/(XY)$ (regular case) or to
$\overline{\mathbf{F}}_{p}[\mathfrak{Z},Z^{\pm 1}]$ (non-regular case). We
show that a simple $\mathcal{H}$-module is finite dimensional and thus admits
a central character. It will be called supersingular if it is killed by $X$
and $Y$ (resp. by $\mathfrak{Z}$), otherwise we call the module principal.
This leads to the following result, which is a summary of propositions 3.27,
3.28 and 3.36.

######  Theorem 1.2.

The simple right $\mathcal{H}$-modules fall into two subclasses:

  1. (a)

Principal modules

  2. (b)

Supersingular modules.

Each of these subclasses is in itself described explicitly.

As a consequences of the two theorems, we obtain an explicit description of
the smooth irreducible genuine $\overline{\mathbf{F}}_{p}$-representations of
$\widetilde{G}$. In particular, there are only two kinds of such objects:
principal series and supersingular representations (i.e. those not occuring as
a subquotient of a principal series). From the point of view of modules over
the pro-$p$ Iwahori Hecke algebra, the supersingular objects naturally come in
pairs. Moreover, it turns out that a genuine principal series is always
irreducible; they are in bijective correspondence with pairs of smooth
characters on the subgroup $S$ of squares in $\mathbf{Q}_{p}^{\times}$: We
have an equality of groups $\widetilde{B}_{2}=B_{2}\times\mu_{2}$, where
$B_{2}$ is the subgroup of $G$ consisting of those upper-triangular matrices
having elements of even valuation on the diagonal, and the correspondence is
given by
$(\chi_{1},\chi_{2})\mapsto\operatorname{Ind}^{\widetilde{G}}_{\widetilde{B}_{2}}(\chi_{1}^{\prime}\otimes\chi_{2}^{\prime}\boxtimes\iota)$
for any extension $\chi_{i}^{\prime}$ of $\chi_{i}$ to the subgroup
$p^{2\mathbf{Z}}\mathbf{Z}_{p}^{\times}$.

As for the spherical Hecke algebras, having fixed an explicit splitting
$\widetilde{K}\cong K\times\mu_{2}$, we may form the smooth irreducible
genuine $\overline{\mathbf{F}}_{p}$-representation
$\operatorname{Sym}^{r}(\overline{\mathbf{F}}_{p}^{2})\boxtimes\iota$ of
$\widetilde{K}$ for $0\leq r\leq p-1$. The center $Z(\widetilde{G})$ being
equal to the product $Z^{\operatorname{sq}}\times\mu_{2}$, we may extend it to
a smooth genuine $\widetilde{K}Z(\widetilde{G})$-representation by letting the
central element $p^{2}$ act trivially. Inside the spherical Hecke algebra
$\operatorname{End}_{\widetilde{G}}(\operatorname{c-Ind}^{\widetilde{G}}_{\widetilde{K}Z(\widetilde{G})}(\operatorname{Sym}^{r}(\overline{\mathbf{F}}_{p}^{2})\boxtimes\iota))$
one then has the Hecke operator $\widetilde{T}$ corresponding to the double
coset represented by a lift of $\left(\begin{smallmatrix}1&0\\\
0&p^{-2}\end{smallmatrix}\right)$, which also exists in characteristic $0$,
i.e. if $\overline{\mathbf{F}}_{p}$ is replaced by $\overline{\mathbf{Q}}_{p}$
or $\overline{\mathbf{Z}}_{p}$. We summarize propositions 5.2, 6.7 and Theorem
5.13 as follows.

######  Theorem 1.3.

Assume that $0\leq r\leq p-1$ and let $a_{p}\in\overline{\mathbf{Z}}_{p}$.

  1. (i)

The spherical Hecke algebra
$\operatorname{End}_{\widetilde{G}}(\operatorname{c-Ind}^{\widetilde{G}}_{\widetilde{K}Z(\widetilde{G})}(\operatorname{Sym}^{r}(\overline{\mathbf{F}}_{p}^{2})\boxtimes\iota))=\overline{\mathbf{F}}_{p}[\widetilde{T}]$
is a polynomial ring in the Hecke operator $\widetilde{T}$.

  2. (ii)

The canonical map

$\operatorname{c-Ind}^{\widetilde{G}}_{\widetilde{K}Z(\widetilde{G})}(\operatorname{Sym}^{r}(\overline{\mathbf{Z}}_{p}^{2})\boxtimes\iota)/(\widetilde{T}-a_{p})\hookrightarrow\operatorname{c-Ind}^{\widetilde{G}}_{\widetilde{K}Z(\widetilde{G})}(\operatorname{Sym}^{r}(\overline{\mathbf{Q}}_{p}^{2})\boxtimes\iota)/(\widetilde{T}-a_{p})$

is injective and defines a $\widetilde{G}$-stable bounded lattice.

  3. (iii)

Let $\bar{a}_{p}\in\overline{\mathbf{F}}_{p}$ denote the reduction of $a_{p}$
modulo the maximal ideal in $\overline{\mathbf{Z}}_{p}$.

     * •

If $\bar{a}_{p}$ is non-zero, then
$\operatorname{c-Ind}^{\widetilde{G}}_{\widetilde{K}Z(\widetilde{G})}(\operatorname{Sym}^{r}(\overline{\mathbf{F}}_{p}^{2})\boxtimes\iota)/(\widetilde{T}-\bar{a}_{p})$
is a principal series representation.

     * •

If $\bar{a}_{p}=0$, then
$\operatorname{c-Ind}^{\widetilde{G}}_{\widetilde{K}Z(\widetilde{G})}(\operatorname{Sym}^{r}(\overline{\mathbf{F}}_{p}^{2})\boxtimes\iota)/(\widetilde{T})$
is an extension of two distinct supersingular representations, which is split
if and only if $r=\frac{p-1}{2}$.

The pairs of genuine supersingular representations appearing in part (iii) are
precisely those alluded to above in terms of the classification of simple
$\mathcal{H}$-modules.

In Corollary 6.5, we also give a description of the locally algebraic
representation appearing in part (ii) of the theorem in terms of locally
algebraic irreducible $\overline{\mathbf{Q}}_{p}$-representations – in fact,
it will generically be a locally algebraic irreducible unramified principal
series. This is deduced from the corresponding statement [5] for the group $G$
by using Savin’s local Shimura correspondence [27] formulated in terms of
Iwahori Hecke algebras.

While we have not investigated the existence (let alone its mod-$p$ reduction)
of a $\widetilde{G}$-stable bounded lattice in such a locally algebraic
representation in case $p\leq r\leq 2(p-1)$, we still expect the metaplectic
version of [5, Proposition 5.3.4.1] to be true. We can prove a necessary
consequence of such a result, namely the non-vanishing of the Ext-space in
part (iii) of the following proposition, see Proposition 4.36. For two smooth
characters $\chi_{1},\chi_{2}$ of $A=p^{2\mathbf{Z}}\mathbf{Z}_{p}^{\times}$,
define the corresponding normalized principal series by
$\tilde{\pi}(\chi_{1},\chi_{2})=\operatorname{Ind}^{\widetilde{G}}_{\widetilde{B}_{2}}((\chi_{1}\otimes\chi_{2}\omega^{-1})\boxtimes\iota)$
only depending on the restriction of $\chi_{1},\chi_{2}$ to the subgroup $S$
of squares in $\mathbf{Q}_{p}^{\times}$, where
$\omega\colon\mathbf{Q}_{p}^{\times}\twoheadrightarrow\mathbf{F}_{p}^{\times}$
is the projection onto the torsion part.

######  Proposition 1.4.

Let $\chi_{1},\chi_{2}$ and $\chi_{1}^{\prime},\chi_{2}^{\prime}$ be two pairs
of smooth characters $A\to\overline{\mathbf{F}}_{p}^{\times}$.

  1. (i)

The space

$\operatorname{Ext}^{1}_{\widetilde{G}}(\tilde{\pi}(\chi_{1},\chi_{2}),\tilde{\pi}(\chi_{1}^{\prime},\chi_{2}^{\prime}))=0$

vanishes unless $(\chi^{\prime}_{1}|_{S},\chi^{\prime}_{2}|_{S})$ is equal to
$(\chi_{1}|_{S},\chi_{2}|_{S})$ or $(\chi_{2}|_{S},\chi_{1}|_{S})$.

  2. (ii)

The map

$\operatorname{Ext}^{1}_{T_{2}}(\chi_{1}\otimes\chi_{2}\omega^{-1},\chi_{1}\otimes\chi_{2}\omega^{-1})\xrightarrow{\cong}\operatorname{Ext}^{1}_{\widetilde{G}}(\tilde{\pi}(\chi_{1},\chi_{2}),\tilde{\pi}(\chi_{1},\chi_{2})),$

induced by the exact functor
$\operatorname{Ind}^{\widetilde{G}}_{\widetilde{B}_{2}}(-\boxtimes\iota)$, is
an isomorphism, where $T_{2}=\left(\begin{smallmatrix}A&0\\\
0&A\end{smallmatrix}\right)$.

  3. (iii)

If $\chi_{1}|_{S}\neq\chi_{2}|_{S}$, then

$\dim_{\overline{\mathbf{F}}_{p}}\operatorname{Ext}^{1}_{\widetilde{G}}(\tilde{\pi}(\chi_{1},\chi_{2}),\tilde{\pi}(\chi_{2},\chi_{1}))=1.$

This proposition is proved by comparing it to the situation for the group $G$,
where the statement is known, see for example [10].

Acknowledgements. This work is part of the author’s PhD thesis and he would
like to heartily thank his advisor Vytautas Paškūnas for suggesting the topic
at hand and for fruitful discussions during the past years. Special thanks go
to Florian Herzig for his lectures [14] at the Spring School on the mod-$p$
Langlands Correspondence, which took place in Essen (online) in April 2021;
Section 4.4.1 is more or less an excerpt of his material. This work was
partially funded by the DFG Graduiertenkolleg 2553.

###  1.2. Conventions and notations

Fix an odd prime number $p$ and an algebraically closed field $k$ of
characteristic $p$. Denote by $\mathbf{Q}_{p}$ the completion of the rationals
$\mathbf{Q}$ with respect to the $p$-adic valuation
$v\colon\mathbf{Q}_{p}\to\mathbf{Z}\cup\\{\infty\\}$. We use the standard
notations

$G=\operatorname{GL}_{2}(\mathbf{Q}_{p})\supset
B=\begin{pmatrix}\mathbf{Q}_{p}^{\times}&\mathbf{Q}_{p}\\\
0&\mathbf{Q}_{p}^{\times}\end{pmatrix}\supset
T=\begin{pmatrix}\mathbf{Q}_{p}^{\times}&0\\\
0&\mathbf{Q}_{p}^{\times}\end{pmatrix}\supset Z,$

where $Z\cong\mathbf{Q}_{p}^{\times}$ denotes the center of $G$, and we let
$K=\operatorname{GL}_{2}(\mathbf{Z}_{p})$ be the standard maximal compact open
subgroup of $G$. It will be convenient to also introduce the following
subgroups of $\mathbf{Q}_{p}^{\times}$:

$\mu_{2}=\\{\pm 1\\},\hskip
2.84544ptA=p^{2\mathbf{Z}}\mathbf{Z}_{p}^{\times}\supset
S=\\{x^{2}:x\in\mathbf{Q}_{p}^{\times}\\}.$

For a topological group $\mathscr{G}$, we denote by
$\operatorname{Mod}^{\operatorname{sm}}_{\mathscr{G}}(k)$ the category of
$k$-linear smooth representations of $\mathscr{G}$. Here, smooth means that
the stabilizer of each vector is open.

If $\mathscr{H}\subset\mathscr{G}$ is a closed subgroup and
$\pi\in\operatorname{Mod}^{\operatorname{sm}}_{\mathscr{H}}(k)$, we denote the
smooth induced representation by
$\operatorname{Ind}^{\mathscr{G}}_{\mathscr{H}}(\pi)$. It is right adjoint to
the restriction functor $(-)|_{\mathscr{H}}$, which we will refer to as
Frobenius reciprocity.

If $\mathscr{H}$ is an open subgroup, we denote the compact induction of $\pi$
by $\operatorname{c-Ind}^{\mathscr{G}}_{\mathscr{H}}(\pi)$. It is the
subrepresentation of $\operatorname{Ind}^{\mathscr{G}}_{\mathscr{H}}(\pi)$
consisting of those functions having finite support modulo $\mathscr{H}$, and
defines a left adjoint to the restriction functor $(-)|_{\mathscr{H}}$, which
we will also refer to as Frobenius reciprocity.

For
$\sigma_{1},\sigma_{2}\in\operatorname{Mod}^{\operatorname{sm}}_{\mathscr{H}}(k)$,
we will identify
$\operatorname{Hom}_{\mathscr{G}}(\operatorname{c-Ind}^{\mathscr{G}}_{\mathscr{H}}(\sigma_{1}),\operatorname{c-Ind}^{\mathscr{G}}_{\mathscr{H}}(\sigma_{2}))$
with the $k$-vector space

$\left\\{\varphi\colon\mathscr{G}\to\operatorname{Hom}_{k}(\sigma_{1},\sigma_{2}):\begin{array}[]{l}\diamond\hskip
2.84544pt\varphi(h_{2}gh_{1})=\sigma_{2}(h_{2})\varphi(g)\sigma_{1}(h_{1}),\forall
h_{1},h_{2}\in\mathscr{H},g\in\mathscr{G}\\\ \diamond\hskip
2.84544pt|\mathscr{H}\setminus\operatorname{supp}(\varphi)/\mathscr{H}|<\infty\end{array}\right\\}$

with left
$\operatorname{End}_{\mathscr{G}}(\operatorname{c-Ind}^{\mathscr{G}}_{\mathscr{H}}(\sigma_{2}))$-,
resp. right
$\operatorname{End}_{\mathscr{G}}(\operatorname{c-Ind}^{\mathscr{G}}_{\mathscr{H}}(\sigma_{1}))$-module
structure given by the convolution product

$(\varphi_{1}\varphi_{2})(g)=\sum_{x\in\mathscr{H}\setminus\mathscr{G}}\varphi_{1}(gx^{-1})\varphi_{2}(x),\text{
for all $g\in\mathscr{G}$.}$

For a subgroup $\mathscr{H}\subset\mathscr{G}$,
$\pi\in\operatorname{Mod}^{\operatorname{sm}}_{\mathscr{H}}(k)$ and
$g\in\mathscr{G}$, we denote by
$\pi^{g}=\pi(g(-)g^{-1})\in\operatorname{Mod}^{\operatorname{sm}}_{g^{-1}\mathscr{H}g}(k)$
the conjugated representation.

If $\mathscr{G}$ is abelian, we write
$\mathscr{G}^{\operatorname{sq}}=\\{g^{2}:g\in\mathscr{G}\\}$ for the subgroup
of squares.

For a ring $R$, we denote by $\operatorname{Mod}_{R}$ the category of right
$R$-modules and by $M_{n\times n}(R)$ the ring of $n\times n$-matrices with
entries in $R$, for $n\in\mathbf{Z}_{\geq 1}$.

##  2\. The metaplectic covering group

The metaplectic covering group is defined by an explicit cocycle relying on
the Hilbert symbol, which we briefly remind the reader of following [19, V.
§3].

###  2.1. Hilbert symbol

Let
$L=\mathbf{Q}_{p}(\sqrt{\smash[b]{\mathbf{Q}_{p}^{\times}}})=\mathbf{Q}_{p^{2}}(\sqrt{\smash[b]{p}})$
be the unique abelian extension of $\mathbf{Q}_{p}$ with norm group $S$. By
local class field theory and Kummer theory, respectively, we have isomorphisms
$\operatorname{Gal}(L/\mathbf{Q}_{p})\cong\mathbf{Q}_{p}^{\times}/S$ and
$\operatorname{Hom}(\operatorname{Gal}(L/\mathbf{Q}_{p}),\mu_{2})\cong\mathbf{Q}_{p}^{\times}/S$.
In particular, the evaluation map
$\operatorname{Gal}(L/\mathbf{Q}_{p})\times\operatorname{Hom}(\operatorname{Gal}(L/\mathbf{Q}_{p}),\mu_{2})\to\mu_{2}$
is identified with a non-degenerate bilinear map

$(-,-)\colon\mathbf{Q}_{p}^{\times}/S\times\mathbf{Q}_{p}^{\times}/S\to\mu_{2},$

which is called the Hilbert symbol. We will usually view it as a bilinear map
on $\mathbf{Q}_{p}^{\times}\times\mathbf{Q}_{p}^{\times}$ via inflation.

Denote by
$\omega\colon\mathbf{Q}_{p}^{\times}\twoheadrightarrow[\mathbf{F}_{p}^{\times}]\cong\mathbf{F}_{p}^{\times}$
the projection onto the torsion part, where $[-]$ denotes the Teichmüller
lift. By [19, V. Satz (3.4)], we then have

(1)
$(a,b)=\omega\left((-1)^{v(a)v(b)}\frac{b^{v(a)}}{a^{v(b)}}\right)^{\frac{p-1}{2}}\text{
for all $a,b\in\mathbf{Q}_{p}^{\times}$.}$

From this one deduces the following properties of the Hilbert symbol.

######  Lemma 2.1.

Let $a,a^{\prime},b,b^{\prime}\in\mathbf{Q}_{p}^{\times}$. Then

  1. (i)

$(aa^{\prime},b)=(a,b)(a^{\prime},b)$;

  2. (ii)

$(a,bb^{\prime})=(a,b)(a,b^{\prime})$;

  3. (iii)

$(a,b)^{-1}=(b,a)$;

  4. (iv)

$(a,1-a)=1$ and $(a,-a)=1$;

  5. (v)

$(a,x)=1$ for all $x\in\mathbf{Q}_{p}^{\times}$ if and only if $a\in S$;

  6. (vi)

$(a,b)=(b,a)$;

  7. (vii)

If $a,b\in A$, then $(a,b)=1$.

###### Proof.

Parts (i)-(v) are contained in [19, V. Satz (3.2)], while (vi) and (vii) are
immediate from (1) noting that an element in $\mu_{2}$ is self-inverse. ∎

###  2.2. Definition and first properties

The _metaplectic covering of $G$_ is the central topological extension

(2) $1\to\mu_{2}\to\widetilde{G}\to G\to 1$

defined by the two-cocycle $\sigma\colon G\times G\to\mu_{2}$,

(3)
$\sigma(g_{1},g_{2})=\left(\frac{\mathfrak{c}(g_{1}g_{2})}{\mathfrak{c}(g_{1})},\frac{\mathfrak{c}(g_{1}g_{2})}{\mathfrak{c}(g_{2})}\det(g_{1})\right)\text{
for all $(g_{1},g_{2})\in G\times G$},$

where $\mathfrak{c}\colon G\to\mathbf{Q}_{p}^{\times}$ is the function
$\mathfrak{c}\left(\left(\begin{smallmatrix}a&b\\\
c&d\end{smallmatrix}\right)\right)=\begin{cases}c\text{ if $c\neq 0$}\\\
d\text{ if $c=0$,}\end{cases}$ cf. [16, p. 41].

In other words, $\widetilde{G}=G\times\mu_{2}$ and the product of two elements
$(g_{i},\zeta_{i})$, $i=1,2$, is equal to
$(g_{1},\zeta_{1})(g_{2},\zeta_{2})=(g_{1}g_{2},\zeta_{1}\zeta_{2}\sigma(g_{1},g_{2}))$.

######  Notation 2.2.

Given a subset $D\subset G$, we denote by $\widetilde{D}$ its preimage in
$\widetilde{G}$ under the projection map $\widetilde{G}\to G$. For $g\in G$,
we put $\tilde{g}=(g,1)$.

######  Example 2.3.

The multiplication law on the covering group $\widetilde{T}$ of the torus $T$
is given by

$\left(\left(\begin{smallmatrix}x_{1}&0\\\
0&y_{1}\end{smallmatrix}\right),1\right)\left(\left(\begin{smallmatrix}x_{2}&0\\\
0&y_{2}\end{smallmatrix}\right),1\right)=\left(\left(\begin{smallmatrix}x_{1}x_{2}&0\\\
0&y_{1}y_{2}\end{smallmatrix}\right),(y_{2},x_{1})\right),$

for all $x_{i},y_{i}\in\mathbf{Q}_{p}^{\times}$, $i=1,2$. In particular, it is
not abelian, cf. Lemma 2.6 (ii) below. We also note that for all
$x,y\in\mathbf{Q}_{p}^{\times}$,

$\left(\left(\begin{smallmatrix}x&0\\\
0&y\end{smallmatrix}\right),1\right)^{-1}=\left(\left(\begin{smallmatrix}x^{-1}&0\\\
0&y^{-1}\end{smallmatrix}\right),(y,x)\right)\text{ and
}\left(\left(\begin{smallmatrix}0&x\\\
y&0\end{smallmatrix}\right),1\right)^{-1}=\left(\left(\begin{smallmatrix}0&y^{-1}\\\
x^{-1}&0\end{smallmatrix}\right),1\right).$

######  Lemma 2.4.

The metaplectic covering (2) splits over the subgroups
$K=\operatorname{GL}_{2}(\mathbf{Z}_{p})$,
$B_{2}=\left(\begin{smallmatrix}A&\mathbf{Q}_{p}\\\
0&A\end{smallmatrix}\right)$ and
$B_{S}=\left(\begin{smallmatrix}\mathbf{Q}_{p}^{\times}&\mathbf{Q}_{p}\\\
0&S\end{smallmatrix}\right)$. More precisely:

  1. (i)

The map $K\to\mu_{2}$ given by

$g=\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\mapsto\begin{cases}(c,d{\det(g)}^{-1})\text{ if }c\in
p\mathbf{Z}_{p}\setminus\\{0\\}\\\ 1\text{ otherwise}\end{cases}$

defines a splitting $K\times\mu_{2}\cong\widetilde{K}$.

  2. (ii)

We have equalities of groups $\widetilde{B}_{2}=B_{2}\times\mu_{2}$,
$\widetilde{B}_{S}=B_{S}\times\mu_{2}$.

###### Proof.

Part (i) is contained in [16, p. 43], while part (ii) follows from the cocycle
$\sigma$ being trivial when restricted to the subgroups $B_{2}$ and $B_{S}$,
respectively, which is a consequence of Lemma 2.1. ∎

######  Convention 2.5.

From now on we will fix the splitting on $K$ defined in part (i) of the lemma
and view subgroups of $K$ as subgroups of $\widetilde{K}\subset\widetilde{G}$.

######  Lemma 2.6.

We have the following equalities of groups describing the centers of
$\widetilde{G}$ and $\widetilde{T}$, respectively:

  1. (i)

$Z(\widetilde{G})=Z^{\operatorname{sq}}\times\mu_{2}$;

  2. (ii)

$Z(\widetilde{T})=T^{\operatorname{sq}}\times\mu_{2}$.

###### Proof.

Note that by Lemma 2.4 the groups on the right-hand side of the equation are
indeed subgroups of $\widetilde{G}$. Property (v) of the Hilbert symbol in
Lemma 2.1 implies that they are contained in the respective center. For
proving the converse inclusions, we first consider part (ii): Let
$(t,\zeta)\in Z(\widetilde{T})$. We may assume that $\zeta=1$ as $\mu_{2}$ is
a central subgroup. By the formula for the multiplication law in Example 2.3,
the commuting of $\tilde{t}$ with elements in $K\cap T$ forces the diagonal
entries of $t$ to lie in $A$, and the commuting with
$\left(\left(\begin{smallmatrix}p&0\\\ 0&1\end{smallmatrix}\right),1\right)$
and $\left(\left(\begin{smallmatrix}1&0\\\
0&p\end{smallmatrix}\right),1\right)$ then forces $t\in
T^{\operatorname{sq}}$, which finishes the proof of (ii). Part (i) now follows
from the inclusion $Z(\widetilde{G})\subset
Z(\widetilde{T})\cap\widetilde{Z}$. ∎

###  2.3. Genuine representations and further notations

Fix the non-trivial character
$\iota\colon\mu_{2}\hookrightarrow\mathbf{F}_{p}^{\times}\subset k^{\times}$.
Let $\mathscr{G}\subset G$ be a closed subgroup.

######  Definition 2.7.

We let
$\operatorname{Mod}^{\operatorname{sm}}_{\widetilde{\mathscr{G}},\iota}(k)$
denote the full subcategory of
$\operatorname{Mod}^{\operatorname{sm}}_{\widetilde{\mathscr{G}}}(\overline{\mathbf{F}}_{p})$
consisting of genuine representations, i.e. those objects on which $\mu_{2}$
acts via the non-trivial character $\iota$.

Since $p$ is prime to the order of the central subgroup $\mu_{2}$, we have a
decomposition of categories

$\operatorname{Mod}^{\operatorname{sm}}_{\widetilde{\mathscr{G}}}(k)=\operatorname{Mod}^{\operatorname{sm}}_{\widetilde{\mathscr{G}},\iota}(k)\times\operatorname{Mod}^{\operatorname{sm}}_{\mathscr{G}}(k),$

where the second factor is viewed as a full subcategory via inflation.

Recall that a smooth representation
$\pi\in\operatorname{Mod}^{\operatorname{sm}}_{\widetilde{\mathscr{G}}}(k)$ is
admissible if the subspace $\pi^{\mathscr{H}}$ of $\mathscr{H}$-fixed vectors
is finite dimensional for every open subgroups
$\mathscr{H}\subset\mathscr{G}$. Since the identity in $\mathscr{G}$ admits a
basis of open neighbourhood consisting of pro-$p$ groups and we are working
with mod-$p$ coefficients, admissibility can be checked for a single open
pro-$p$ subgroup $\mathscr{H}$, see [15, Proposition 36]. Examples of open
pro-$p$ subgroups of $K$ (and hence of $\widetilde{G}$) are given by the
congruence subgroup of level $m\geq 1$ and the pro-$p$ Iwahori subgroup:

$K_{m}=1+p^{m}M_{2\times 2}(\mathbf{Z}_{p}),\hskip
2.84544ptI_{1}=\begin{pmatrix}1+p\mathbf{Z}_{p}&\mathbf{Z}_{p}\\\
p\mathbf{Z}_{p}&1+p\mathbf{Z}_{p}\end{pmatrix}.$

Let $I=\begin{pmatrix}\mathbf{Z}_{p}^{\times}&\mathbf{Z}_{p}\\\
p\mathbf{Z}_{p}&\mathbf{Z}_{p}^{\times}\end{pmatrix}$ denote the Iwahori
subgroup and set $H=\begin{pmatrix}[\mathbf{F}_{p}^{\times}]&0\\\
0&[\mathbf{F}_{p}^{\times}]\end{pmatrix}$, we have the semi-direct product
decomposition $I=I_{1}\rtimes H$.

Finally, let us agree on the following notation: given a subgroup $C\subset G$
together with a splitting $\widetilde{C}\cong C\times\mu_{2}$, we denote the
image of an object $\pi$ under the induced equivalence
$\operatorname{Mod}^{\operatorname{sm}}_{C}(k)\cong\operatorname{Mod}^{\operatorname{sm}}_{\widetilde{C},\iota}(k)$
by $\pi\boxtimes\iota$.

##  3\. The pro-$p$ Iwahori Hecke algebra

In this section, we study the pro-$p$ Iwahori Hecke algebra by decomposing it
into smaller algebras which we then describe explicitly, enabling us to
determine the simple modules.

By Lemma 2.4 (i), the metaplectic covering splits over the maximal compact
open subgroup and thus also over the pro-$p$ Iwahori subgroup $I_{1}$. Since
$p$ is odd, any such splitting is unique and we in fact have an equality of
groups $\widetilde{I}_{1}=I_{1}\times\mu_{2}$ because
$1+p\mathbf{Z}_{p}\subset S$ so that the splitting in the cited lemma is
trivial on $I_{1}$. We will simply view the elements of $I_{1}$ as elements of
$\widetilde{I}_{1}$ and drop the $\mu_{2}$-component (which is $1$) from the
notation. Taking the trivial character $\mathbf{1}$ on $I_{1}$, we obtain the
genuine $k^{\times}$-valued character $\mathbf{1}\boxtimes\iota$ on
$\widetilde{I}_{1}$, which is just the inflation of $\iota$.

######  Definition 3.1.

The pro-$p$ Iwahori Hecke algebra $\mathcal{H}$ is the endomorphism ring of
the compact induction
$\operatorname{c-Ind}^{\widetilde{G}}_{\widetilde{I}_{1}}(\mathbf{1}\boxtimes\iota)$
in the category of $\widetilde{G}$-representations, i.e.
$\mathcal{H}=\operatorname{End}_{\widetilde{G}}(\operatorname{c-Ind}^{\widetilde{G}}_{\widetilde{I}_{1}}(\mathbf{1}\boxtimes\iota))$.

As agreed upon in our list of conventions and notations 1.2, we will identify
$\mathcal{H}$ with the $k$-algebra

$\left\\{\varphi\colon I_{1}\setminus\widetilde{G}/I_{1}\to
k:\begin{array}[]{l}\diamond\hskip
2.84544pt\varphi(g\zeta)=\zeta\varphi(g),\forall
g\in\widetilde{G},\zeta\in\mu_{2}\\\ \diamond\hskip
2.84544pt|\operatorname{supp}(\varphi)|<\infty\end{array}\right\\},$

the product being given by convolution.

###  3.1. Generators and relations

In this subsection, we prove some useful identities of Hecke operators by
following [21], which treats the case of $\operatorname{GL}_{2}$ over a local
field. We start by writing down a basis of $\mathcal{H}$ as a $k$-vector
space. Let $N_{G}(T)$ be the normalizer of $T$ in $G$. The following two
elements of $N_{G}(T)$ will play a prominent role in this work:

$s=\begin{pmatrix}0&1\\\ 1&0\end{pmatrix}\text{ and }\Pi=\begin{pmatrix}0&1\\\
p&0\end{pmatrix}.$

Putting $W_{1}:=N_{G}(T)/(I_{1}\cap T)$, we have the decomposition

(4) $\widetilde{G}=\bigsqcup_{w\in
W_{1}}\widetilde{I}_{1}\tilde{w}\widetilde{I}_{1},$

see also [21, Lemma 2.0.4]. The “genuine characteristic functions” of the
double cosets appearing in this decomposition will define a $k$-basis of
$\mathcal{H}$ – provided they are well-defined, which we now check.

######  Lemma 3.2.

For all $g\in I_{1}$ and $n\in N_{G}(T)$,
$\tilde{n}g(\tilde{n})^{-1}=(ngn^{-1},1)$. In particular, for all $n\in
N_{G}(T)$, $\mu_{2}\cap I_{1}\tilde{n}I_{1}(\tilde{n})^{-1}=\\{1\\}$.

###### Proof.

Recall that $N_{G}(T)=T\sqcup sT$. Let $n\in N_{G}(T)$ and write
$g=\left(\begin{smallmatrix}a&b\\\ c&d\end{smallmatrix}\right)\in I_{1}$. The
end of Example 2.3 gives a formula for the inverse $(\tilde{n})^{-1}$ of
$\tilde{n}=(n,1)$. Distinguishing between the cases $c=0$ and $c\neq 0$, if
$n\in T$, and the cases $b=0$ and $b\neq 0$, if $n\in sT$, the lemma follows
from the definition of the cocycle (3) and the identities of the Hilbert
symbol in Lemma 2.1, bearing in mind that $1+p\mathbf{Z}_{p}\subset S$. ∎

######  Definition 3.3.

For $w\in W_{1}$, let $T_{w}\in\mathcal{H}$ be the unique function with

$\operatorname{supp}(T_{w})=\widetilde{I}_{1}\tilde{w}\widetilde{I}_{1}\text{
and }T_{w}(\tilde{w})=1.$

If $n\in N_{G}(T)$, then we also write $T_{n}$ for the operator defined by the
class of $n$ in $W_{1}$.

######  Corollary 3.4.

The operators $T_{w}$, for $w\in W_{1}$, are well-defined and the collection
$\\{T_{w}\\}_{w\in W_{1}}$ is a $k$-basis of $\mathcal{H}$.

###### Proof.

For $t\in T\cap I_{1}$, we have $\tilde{g}\tilde{t}=\widetilde{gt}$ for all
$g\in G$, so the condition $T_{w}(\tilde{w})=1$ is independent of the choice
of a representative of $w$. By Lemma 3.2, the operator $T_{w}$ is then well-
defined. The decomposition (4) implies the assertion about the basis. ∎

######  Remark 3.5.

Let $n_{1},n_{2}\in N_{G}(T)$. Writing $I_{1}n_{2}I_{1}=\sqcup_{x\in
I_{1}/(I_{1}\cap n_{2}^{-1}I_{1}n_{2})}I_{1}n_{2}x$, the convolution product
of $T_{n_{1}}$ and $T_{n_{2}}$ reads

(5) $(T_{n_{1}}T_{n_{2}})(g)=\sum_{x\in I_{1}/(I_{1}\cap
n_{2}^{-1}I_{1}n_{2})}T_{n_{1}}(gx^{-1}(\tilde{n}_{2})^{-1})\text{ for all
$g\in\widetilde{G}$.}$

In particular, if $n_{2}$ or $n_{1}$ normalizes $I_{1}$, then
$T_{n_{1}}T_{n_{2}}=\zeta T_{n_{1}n_{2}}$, where $\zeta\in\mu_{2}$ is such
that $(n_{1}n_{2},1)(n_{2},1)^{-1}=(n_{1},\zeta)$.

######  Definition 3.6.

Let
$\widehat{H}=\operatorname{Hom}(H,\mathbf{F}_{p}^{\times})=\operatorname{Hom}(H,k^{\times})$
be the group of $k^{\times}$-valued characters of $H$. For
$\chi\in\widehat{H}$, define the Hecke operator

$e_{\chi}=\frac{1}{|H|}\sum_{h\in H}\chi(h)T_{h}.$

######  Lemma 3.7.

The following identities hold true for all characters
$\chi,\chi^{\prime}\in\widehat{H}$:

  1. (i)

$e_{\chi}^{2}=e_{\chi}$;

  2. (ii)

$e_{\chi}e_{\chi^{\prime}}=0$ if $\chi\neq\chi^{\prime}$;

  3. (iii)

$\sum_{\chi\in\widehat{H}}e_{\chi}=1$;

  4. (iv)

$T_{s}e_{\chi}=e_{\chi^{s}}T_{s}$;

  5. (v)

$T_{s}^{2}e_{\chi}=\begin{cases}0\text{ if }\chi\neq\chi^{s}\\\
-\chi(\left(\begin{smallmatrix}1&0\\\
0&-1\end{smallmatrix}\right))T_{s}e_{\chi}\text{ if
}\chi=\chi^{s}.\end{cases}$

###### Proof.

The identities can be checked in the endomorphism ring of the induction
$\operatorname{Ind}^{\widetilde{K}}_{\widetilde{I}_{1}}(\mathbf{1}\boxtimes\iota)$.
Via the splitting $\widetilde{K}\cong K\times\mu_{2}$, this is identified with
the endomorphism ring of $\operatorname{Ind}^{K}_{I_{1}}(\mathbf{1})$, so the
lemma follows from [21, Lemma 2.0.10, Lemma 2.0.12]. ∎

######  Definition 3.8.

Given a character $\chi\in\widehat{H}$ and a pair
$(i,j)\in\mathbf{Z}/2\mathbf{Z}$, define the twisted character
$\chi[i,j]\in\widehat{H}$ by

$\displaystyle\chi[i,j]\left(\left(\begin{smallmatrix}[\lambda]&0\\\
0&[\mu]\end{smallmatrix}\right)\right)=\chi\left(\left(\begin{smallmatrix}[\lambda]&0\\\
0&[\mu]\end{smallmatrix}\right)\right)(\lambda^{i}\mu^{j})^{\frac{p-1}{2}},\text{for
all $\lambda,\mu\in\mathbf{F}_{p}^{\times}$.}$

Recall that we put $\Pi=\left(\begin{smallmatrix}0&1\\\
p&0\end{smallmatrix}\right)\in N_{G}(T)$, which normalizes $I_{1}$. We will
make use of Remark 3.5 as well as the identities in Example 2.3 during the
proof of the following lemma.

######  Lemma 3.9.

Let $m\in\mathbf{Z}_{\geq 0}$. The following assertions hold true.

  1. (i)

$T_{\Pi}$ is invertible with inverse $T_{\Pi^{-1}}$;

  2. (ii)

$T_{\Pi}^{2}=T_{\Pi^{2}}$;

  3. (iii)

$T_{\Pi}^{4}=(-1)^{\frac{p-1}{2}}T_{\Pi^{4}}$ is central, and
$T_{n}T_{\Pi^{4}}=T_{n\Pi^{4}}=T_{\Pi^{4}n}$ for all $n\in N_{G}(T)$;

  4. (iv)

$T_{\Pi^{2}}T_{s}=T_{\Pi^{2}s}=T_{s\Pi^{2}}=(-1)^{\frac{p-1}{2}}T_{s}T_{\Pi^{2}}$;

  5. (v)

for all $\chi\in\widehat{H}$, $T_{\Pi}e_{\chi}=e_{\chi^{s}[1,0]}T_{\Pi}$;

  6. (vi)

$(T_{s}T_{\Pi})^{m}=T_{(s\Pi)^{m}}$;

  7. (vii)

$(T_{\Pi}T_{s})^{m}=T_{(\Pi s)^{m}}$;

  8. (viii)

$T_{(s\Pi)^{m}}T_{s}=T_{(s\Pi)^{m}s}=T_{s}T_{(\Pi s)^{m}}$;

  9. (ix)

$T_{\Pi}T_{(s\Pi)^{m}}=T_{\Pi(s\Pi)^{m}}=T_{(\Pi s)^{m}\Pi}=T_{(\Pi
s)^{m}}T_{\Pi}$;

###### Proof.

(i) This follows from $(\Pi,1)^{-1}=(\Pi^{-1},1)$.

(ii) This follows from $(\Pi^{2},1)(\Pi^{-1},1)=(1,1)$.

(iii) The first equality follows from
$(\Pi^{2},1)^{2}=(\Pi^{4},(p,p))=(\Pi^{4},(-1)^{\frac{p-1}{2}})$. This element
is central element in $\widetilde{G}$ by Lemma 2.6 (i), which implies the
other equalities.

(iv) The first equality follows from $(\Pi^{2}s,1)(s,1)=(\Pi^{2},1)$, the
second one is trivial, and the last equality follows from
$(s\Pi^{2},1)(\Pi^{2},1)^{-1}=(s\Pi^{2},1)(\Pi^{-2},(-1)^{\frac{p-1}{2}})=(s,(-1)^{\frac{p-1}{2}})$.

(v) Let $h=\left(\begin{smallmatrix}[\lambda]&0\\\
0&[\mu]\end{smallmatrix}\right)\in H$ and denote by $h^{s}=shs$ its conjugate
by $s$. The two equalities $(\Pi h,1)(h^{-1},1)=(\Pi,1)$ and
$(h^{s}\Pi,1)(\Pi^{-1},1)=(h^{s},\mu^{\frac{p-1}{2}})$ translate into

$T_{\Pi}T_{h}=T_{\Pi h}=T_{h^{s}\Pi}=\mu^{\frac{p-1}{2}}T_{h^{s}}T_{\Pi},$

which implies the assertion.

(vi) The case $m=1$ follows from the equality $(s\Pi,1)(\Pi^{-1},1)=(s,1)$. It
now suffices to prove $T_{(s\Pi)^{m}}T_{s\Pi}=T_{(s\Pi)^{m+1}}$ for all $m\geq
1$. Since the support of the product of two operators is contained in the
product of the supports (as follows from the definition of the convolution
product), the left-hand side has support contained in the preimage in
$\widetilde{G}$ of

$I_{1}\left(\begin{smallmatrix}p^{m}&0\\\
0&1\end{smallmatrix}\right)I_{1}\left(\begin{smallmatrix}p&0\\\
0&1\end{smallmatrix}\right)I_{1}=\bigcup_{\lambda\in\mathbf{F}_{p}}I_{1}\left(\begin{smallmatrix}p^{m}&0\\\
0&1\end{smallmatrix}\right)\left(\begin{smallmatrix}1&[\lambda]\\\
0&1\end{smallmatrix}\right)\left(\begin{smallmatrix}p&0\\\
0&1\end{smallmatrix}\right)I_{1}=I_{1}\left(\begin{smallmatrix}p^{m+1}&0\\\
0&1\end{smallmatrix}\right)I_{1}.$

The decomposition $I_{1}s\Pi
I_{1}=\sqcup_{\lambda\in\mathbf{F}_{p}}I_{1}s\Pi\left(\begin{smallmatrix}1&0\\\
p[\lambda]&1\end{smallmatrix}\right)$ yields that the evaluation of
$T_{(s\Pi)^{m}}T_{s\Pi}$ at $((s\Pi)^{m+1},1)$ is equal to

$\sum_{\lambda\in\mathbf{F}_{p}}T_{(s\Pi)^{m}}\left(((s\Pi)^{m+1},1)\left(\begin{smallmatrix}1&0\\\
-p[\lambda]&1\end{smallmatrix}\right)(s\Pi,1)^{-1}\right).$

If $\lambda\neq 0$, then $(s\Pi)^{m+1}\left(\begin{smallmatrix}1&0\\\
-p[\lambda]&1\end{smallmatrix}\right)(s\Pi)^{-1}$ does not lie in
$I_{1}(s\Pi)^{m}I_{1}$, so only the summand corresponding to $\lambda=0$
contributes and gives the value $1$.

(vii) This is analogous to (vi).

(viii) Since $I_{1}(s\Pi)^{m}I_{1}sI_{1}=I_{1}(s\Pi)^{m}sI_{1}$, it is enough
to evaluate at $((s\Pi)^{m}s,1)$. Now
$I_{1}sI_{1}=\sqcup_{\lambda\in\mathbf{F}_{p}}I_{1}s\left(\begin{smallmatrix}1&[\lambda]\\\
0&1\end{smallmatrix}\right)$, so

$T_{(s\Pi)^{m}}T_{s}(((s\Pi)^{m}s,1))=\sum_{\lambda\in\overline{\mathbf{F}}_{p}}T_{(s\Pi)^{m}}\left(((s\Pi)^{m}s,1)\left(\begin{smallmatrix}1&[\lambda]\\\
0&1\end{smallmatrix}\right)(s,1)\right).$

If $\lambda\neq 0$, then $(s\Pi)^{m}\left(\begin{smallmatrix}1&0\\\
[\lambda]&1\end{smallmatrix}\right)$ does not lie in $I_{1}(s\Pi)^{m}I_{1}$,
so only the summand corresponding to $\lambda=0$ contributes and gives the
value $1$. This proves the first equality. The second one is proved similarly.

(ix) The first identity follows from

$(\Pi(s\Pi)^{m},1)((s\Pi)^{-m},1)=(\Pi,(p^{-m},p)(-p^{m+1},p^{-m}))=(\Pi,1),$

the second one is trivial, the last one follows from $((\Pi
s)^{m}\Pi,1)(\Pi^{-1},1)=((\Pi s)^{m},1)$. ∎

######  Corollary 3.10.

The operator $T_{\Pi}$ restricts to a $\widetilde{G}$-equivariant isomorphism

$T_{\Pi}\colon\operatorname{c-Ind}^{\widetilde{G}}_{\widetilde{I}}(\chi\boxtimes\iota)\xrightarrow{\cong}\operatorname{c-Ind}^{\widetilde{G}}_{\widetilde{I}}(\chi^{s}[1,0]\boxtimes\iota),$

for each $\chi\in\widehat{H}$.

###### Proof.

By definition, $T_{\Pi}$ is a $\widetilde{G}$-equivariant endomorphism on
$\operatorname{c-Ind}^{\widetilde{G}}_{\widetilde{I}_{1}}(\mathbf{1}\boxtimes\iota)$.
The assertion follows from Lemma 3.9 (v) and (i), noting that
$\operatorname{c-Ind}^{\widetilde{G}}_{\widetilde{I}}(\chi\boxtimes\iota)=e_{\chi}\operatorname{c-Ind}^{\widetilde{G}}_{\widetilde{I}_{1}}(\mathbf{1}\boxtimes\iota)$.
∎

###  3.2. Decomposing the pro-$p$ Iwahori Hecke algebra

It follows from Lemma 3.7 that we have the decomposition of
$\widetilde{G}$-representations

(6)
$\operatorname{c-Ind}^{\widetilde{G}}_{\widetilde{I}_{1}}(\mathbf{1}\boxtimes\iota)=\bigoplus_{\chi\in\widehat{H}}e_{\chi}\operatorname{c-Ind}^{\widetilde{G}}_{\widetilde{I}_{1}}(\mathbf{1}\boxtimes\iota)=\bigoplus_{\chi\in\widehat{H}}\operatorname{c-Ind}^{\widetilde{G}}_{\widetilde{I}}(\chi\boxtimes\iota).$

As we will see below, there are non-zero intertwinings between two direct
summands if and only if the two respective characters are conjugate in the
following sense: The group $\widetilde{N_{G}(T)}$ acts on the set of genuine
characters of $\widetilde{H}$ by conjugation. Since $\mu_{2}$ is central, this
action descends to $N_{G}(T)$. Given $\chi\in\widehat{H}$ and $n\in N_{G}(T)$,
we let $\chi^{\tilde{n}}\in\widehat{H}$ be the unique character satisfying
$\chi^{\tilde{n}}\boxtimes\iota=(\chi\boxtimes\iota)^{\tilde{n}}$ as genuine
characters on $\widetilde{H}=H\times\mu_{2}$. This depends only on the image
of $n$ in $N_{G}(T)/(T\cap I)$ and defines an action of the latter group on
$\widehat{H}$. In particular, the subgroup generated by $\Pi$ acts on
$\widehat{H}$.

######  Definition 3.11.

Let $\chi\in\widehat{H}$.

  1. (i)

Define $O_{\chi}=\left\\{\chi^{\tilde{w}}:w\in N_{G}(T)/(T\cap I)\right\\}$ to
be the orbit of $\chi$. Denote the set of all orbits by $\mathcal{O}$.

  2. (ii)

The $\Pi$-orbit of $\chi$ is the orbit of $\chi$ under the action of the
subgroup generated by $\Pi$, i.e. it is equal to
$\chi^{\tilde{\Pi}^{\mathbf{Z}}}$.

Recall from Definition 3.8 that to $\chi$ and each pair
$(i,j)\in(\mathbf{Z}/2\mathbf{Z})^{2}$ we have associated a twisted character
$\chi[i,j]$.

######  Remark 3.12.

For $h=\left(\begin{smallmatrix}[\lambda]&0\\\
0&[\mu]\end{smallmatrix}\right)\in H$, one has
$\tilde{\Pi}\tilde{h}(\tilde{\Pi})^{-1}=(h^{s},\mu^{\frac{p-1}{2}})$, i.e. for
$\chi\in\widehat{H}$, $\chi^{\tilde{\Pi}}=\chi^{s}[0,1]$. Thus, the
$\Pi$-orbit of $\chi$ is equal to

$\chi^{\widetilde{\Pi}^{\mathbf{Z}}}=\left\\{\chi,\chi^{s}[0,1],\chi[1,1],\chi^{s}[1,0]\right\\}.$

######  Lemma 3.13.

Let $\chi\in\widehat{H}$.

  1. (i)

If $(\chi^{-1}\chi^{s})^{2}\neq 1$, then the map

$\displaystyle\mathbf{Z}/2\mathbf{Z}\times(\mathbf{Z}/2\mathbf{Z})^{2}$
$\displaystyle\cong O_{\chi}$ $\displaystyle(a,(i,j))$
$\displaystyle\mapsto\chi^{s^{a}}[i,j]$

is bijective.

  2. (ii)

If $(\chi^{-1}\chi^{s})^{2}=1$, then the map

$\displaystyle(\mathbf{Z}/2\mathbf{Z})^{2}$ $\displaystyle\cong O_{\chi}$
$\displaystyle(i,j)$ $\displaystyle\mapsto\chi[i,j]$

is bijective.

###### Proof.

The group $N_{G}(T)/(T\cap I)$ is generated by the elements $\Pi$ and $s$.
Remark 3.12 gives a description of the $\Pi$-orbit. Since
$(\chi\boxtimes\iota)^{\tilde{s}}=\chi^{s}\boxtimes\iota$, we deduce that the
map in (i) is surjective. Since we cannot have $\chi=\chi[i,j]$ for
$0\neq(i,j)\in(\mathbf{Z}/2\mathbf{Z})^{2}$, the map in (ii) is injective. If
additionally $(\chi^{-1}\chi^{s})^{2}\neq 1$, then we can neither have
$\chi=\chi^{s}[i,j]$ for any non-zero pair $(i,j)$, showing that the map in
(i) is also injective. If $(\chi^{-1}\chi^{s})^{2}=1$, we may choose
$t\in\mathbf{Z}/2\mathbf{Z}$ such that
$\chi^{-1}\chi^{s}\left(\left(\begin{smallmatrix}[\lambda]&0\\\
0&[\mu]\end{smallmatrix}\right)\right)=(\lambda^{-1}\mu)^{t\frac{p-1}{2}}$ for
all $\lambda,\mu\in\mathbf{F}_{p}^{\times}$, i.e. $\chi^{s}=\chi[t,t]$,
proving that the map in (ii) is surjective. ∎

######  Definition 3.14.

Let $\chi\in\widehat{H}$.

  1. (i)

We say that $\chi$ is square-regular if $(\chi^{-1}\chi^{s})^{2}\neq 1$.

  2. (ii)

We say that $\chi$ is non-square-regular if $(\chi^{-1}\chi^{s})^{2}=1$, i.e.
$\chi^{s}=\chi[t,t]$ for some $t\in\mathbf{Z}/2\mathbf{Z}$.

  3. (iii)

We say that an orbit $O\in\mathcal{O}$ is (non-)square-regular if one, or
equivalently every, character in it is so.

######  Definition 3.15.

  1. (i)

For an orbit $O\in\mathcal{O}$, define the Hecke algebra of $O$ by

$\mathcal{H}(O)=\operatorname{End}_{\widetilde{G}}(\bigoplus_{\chi\in
O}\operatorname{c-Ind}^{\widetilde{G}}_{\widetilde{I}}(\chi\boxtimes\iota)).$

  2. (ii)

For $\chi_{1},\chi_{2}\in\widehat{H}$, put

$\mathcal{H}(\chi_{1},\chi_{2})=\operatorname{End}_{\widetilde{G}}(\operatorname{c-Ind}^{\widetilde{G}}_{\widetilde{I}}(\chi_{1}\boxtimes\iota),\operatorname{c-Ind}^{\widetilde{G}}_{\widetilde{I}}(\chi_{2}\boxtimes\iota)).$

If $\chi_{1}=\chi_{2}$, we simply write $\mathcal{H}(\chi_{1})$.

######  Proposition 3.16.

The decomposition (6) induces a direct sum decomposition of $k$-algebras

$\mathcal{H}=\bigoplus_{O\in\mathcal{O}}\mathcal{H}(O).$

###### Proof.

We have to show that
$\operatorname{Hom}_{\widetilde{G}}(\operatorname{c-Ind}^{\widetilde{G}}_{\widetilde{I}}(\chi_{1}\boxtimes\iota),\operatorname{c-Ind}^{\widetilde{G}}_{\widetilde{I}}(\chi_{2}\boxtimes\iota))=0$
if $\chi_{1}$ and $\chi_{2}$ lie in distinct orbits. This can be proved as in
[30, p. 9]: For $n\in N_{G}(T)$, let $I_{n}=I\cap n^{-1}In$. Since
$I=I_{1}\rtimes H$, we have $I=I_{1}I_{n}$. Using the decomposition
$\widetilde{G}=\sqcup_{n\in N_{G}(T)/(T\cap
I)}\widetilde{I}\tilde{n}\widetilde{I}$, Mackey’s formula reads

$\operatorname{c-Ind}^{\widetilde{G}}_{\widetilde{I}}(\chi_{2}\boxtimes\iota)|_{\widetilde{I}}\cong\bigoplus_{n\in
N_{G}(T)/(T\cap
I)}\operatorname{Ind}^{\widetilde{I}}_{\widetilde{I}_{n}}((\chi_{2}\boxtimes\iota)^{\tilde{n}}).$

Therefore, using Frobenius reciprocity twice, the non-vanishing of the Hom-
space of interest is equivalent to
$(\chi_{1}\boxtimes\iota)|_{\widetilde{I}_{n}}=(\chi_{2}\boxtimes\iota)^{\tilde{n}}|_{\widetilde{I}_{n}}$
for some $n\in N_{G}(T)$. Smooth characters are trivial on the pro-$p$ group
$I_{1}$, so the equality $\widetilde{I}=I_{1}\widetilde{I}_{n}$ completes the
proof. ∎

We finish this section with two lemmas, the first of which will allow us to
reduce statements in the non-square-regular case to a consideration of the
trivial character, while the second one is a computational preliminary for the
upcoming sections, where we determine the Hecke algebra of an orbit
explicitly.

######  Lemma 3.17.

Let $\psi\colon\mathbf{Q}_{p}^{\times}\to k^{\times}$ be a smooth character.
Then the map

$\displaystyle\operatorname{c-Ind}^{\widetilde{G}}_{\widetilde{I}_{1}}(\mathbf{1}\boxtimes\iota)\otimes\psi\circ\det$
$\displaystyle\xrightarrow{\cong}\operatorname{c-Ind}^{\widetilde{G}}_{\widetilde{I}_{1}}(\mathbf{1}\boxtimes\iota)$
$\displaystyle f$
$\displaystyle\mapsto[(g,\zeta)\mapsto\psi(\det(g))f(g,\zeta)]$

is a $\widetilde{G}$-equivariant isomorphism. The resulting $k$-algebra
automorphism

$\operatorname{tw}_{\psi}\colon\mathcal{H}=\operatorname{End}_{\widetilde{G}}(\operatorname{c-Ind}^{\widetilde{G}}_{\widetilde{I}_{1}}(\mathbf{1}\boxtimes\iota)\otimes\psi\circ\det)\cong\mathcal{H},$

where the equality is induced by the functor $(-)\otimes\psi\circ\det$,
identifies $\mathcal{H}(O_{\chi})$ with $\mathcal{H}(O_{\chi\psi|_{H}})$.

###### Proof.

Checking that the map is well-defined $\widetilde{G}$-equivariant is straight
forward. Replacing $\psi$ by its inverse shows that the map is an isomorphism.
The rest is clear. ∎

######  Lemma 3.18.

Let $r,s\in\mathbf{Z}$ and let $\zeta\in\mu_{2}$.

  1. (i)

Let $x=\left(\begin{smallmatrix}a&b\\\ c&d\end{smallmatrix}\right)\in I$. Then
$y=\left(\begin{smallmatrix}a&p^{s-r}b\\\
p^{r-s}c&d\end{smallmatrix}\right)\in I$ if and only if $s-r+v(b)\geq 0$ and
$r-s+v(c)\geq 1$. In this case, the equality

$(x,\zeta)\left(\left(\begin{smallmatrix}p^{r}&0\\\
0&p^{s}\end{smallmatrix}\right),1\right)=\left(\left(\begin{smallmatrix}p^{r}&0\\\
0&p^{s}\end{smallmatrix}\right),1\right)(y,1)$

holds if and only if

$\zeta=\begin{cases}\omega(\det(x))^{r\frac{p-1}{2}}\text{ if }c\neq 0\\\
\omega\left(\det(x)^{s}d^{r-s}\right)^{\frac{p-1}{2}}\text{ if
}c=0.\end{cases}$

  2. (ii)

Let $x=\left(\begin{smallmatrix}a&b\\\ c&d\end{smallmatrix}\right)\in I$. Then
$y=\left(\begin{smallmatrix}d&p^{r-s}c\\\
p^{s-r}b&a\end{smallmatrix}\right)\in I$ if and only if $r-s+v(c)\geq 0$ and
$s-r+v(b)\geq 1$. In this case, the equality

$(x,\zeta)\left(\left(\begin{smallmatrix}0&p^{r}\\\
p^{s}&0\end{smallmatrix}\right),1\right)=\left(\left(\begin{smallmatrix}0&p^{r}\\\
p^{s}&0\end{smallmatrix}\right),1\right)(y,1)$

holds if and only if

$\zeta=\begin{cases}\omega\left(d^{v(b)+v(c)}\det(x)^{s-v(c)}\right)^{\frac{p-1}{2}}\text{
if }c\neq 0,b\neq 0\\\
\omega\left(d^{r-s+v(c)}\det(x)^{s-v(c)}\right)^{\frac{p-1}{2}}\text{ if
}c\neq 0,b=0\\\ \omega\left(d^{v(b)}\det(x)^{s}\right)^{\frac{p-1}{2}}\text{
if }c=0,b\neq 0\\\ \omega\left(d^{r-s}\det(x)^{s}\right)^{\frac{p-1}{2}}\text{
if }c=0,b=0.\end{cases}$

###### Proof.

This is straight forward. ∎

###  3.3. The Hecke algebra of a regular orbit

In this section, we fix a square-regular orbit $O$ and a character $\chi\in
O$. Define

$\mathcal{H}(\chi\oplus\chi[1,0])=\operatorname{End}_{\widetilde{G}}(\bigoplus_{t=0,1}\operatorname{c-Ind}^{\widetilde{G}}_{\widetilde{I}}(\chi[t,0]\boxtimes\iota))=\begin{pmatrix}\mathcal{H}(\chi)&\mathcal{H}(\chi[1,0],\chi)\\\
\mathcal{H}(\chi,\chi[1,0])&\mathcal{H}(\chi[1,0])\end{pmatrix}.$

By Lemma 3.13, Remark 3.12 and Corollary 3.10, we have an isomorphism of
algebras $\mathcal{H}(O)\cong M_{4\times 4}(\mathcal{H}(\chi\oplus\chi[1,0]))$
and so:

######  Corollary 3.19.

The functor

$\displaystyle\operatorname{Mod}_{\mathcal{H}(O)}$
$\displaystyle\xrightarrow{\sim}\operatorname{Mod}_{\mathcal{H}(\chi\oplus\chi[1,0])}$
$\displaystyle M$ $\displaystyle\mapsto M(e_{\chi}+e_{\chi[1,0]})$

is an equivalence of categories.

In order to understand the simple objects on the left-hand side of the
equivalence, it therefore suffices to study the algebra
$\mathcal{H}(\chi\oplus\chi[1,0])$ and its simple modules.

For $t_{1},t_{2}\in\mathbf{Z}/2\mathbf{Z}$, we view the Hecke module
$\mathcal{H}(\chi[t_{1},0],\chi[t_{2},0])$ as the space

$\left\\{\varphi\colon\widetilde{G}\to k:\begin{array}[]{l}\diamond\hskip
2.84544pt\varphi((x,\zeta_{x})g(y,\zeta_{y}))=(\chi[t_{2},0]\boxtimes\iota)((x,\zeta_{x}))\varphi(g)(\chi[t_{1},0]\boxtimes\iota)((y,\zeta_{y}))\\\
\phantom{\diamond}\text{ for all }g\in\widetilde{G},x,y\in
I,\zeta_{x},\zeta_{y}\in\mu_{2}\\\ \diamond\hskip
2.84544pt|\widetilde{I}\setminus\operatorname{supp}(\varphi)/\widetilde{I}|<\infty\end{array}\right\\}$

with left $\mathcal{H}(\chi[t_{2},0])$-, resp. right
$\mathcal{H}[t_{1},0])$-action given by the convolution product. As a
consequence of Lemma 3.18 and the regularity assumption on $\chi$, we obtain:

######  Corollary 3.20.

For $t_{1},t_{2}\in\mathbf{Z}/2\mathbf{Z}$,
$\mathcal{H}(\chi[t_{1},0],\chi[t_{2},0])$ has $k$-basis
$\\{T_{i,t_{1}-t_{2}+j}\\}_{i,j\in 2\mathbf{Z}}$, where $T_{i,t_{1}-t_{2}+j}$
is uniquely determined by

$\operatorname{supp}(T_{i,t_{1}-t_{2}+j})=\widetilde{I}\left(\left(\begin{smallmatrix}p^{i}&0\\\
0&p^{t_{1}-t_{2}+j}\end{smallmatrix}\right),1\right)\widetilde{I},\hskip
5.69046ptT_{i,t_{1}-t_{2}+j}\left(\left(\left(\begin{smallmatrix}p^{i}&0\\\
0&p^{t_{1}-t_{2}+j}\end{smallmatrix}\right),1\right)\right)=1.$

######  Proposition 3.21.

For each $t\in\mathbf{Z}/2\mathbf{Z}$, the map

$\displaystyle k[X,Y,Z^{\pm 1}]/(XY)$
$\displaystyle\xrightarrow{\cong}\mathcal{H}(\chi[t,0])$ $\displaystyle Z$
$\displaystyle\mapsto T_{2,2}$ $\displaystyle Z^{-1}$ $\displaystyle\mapsto
T_{-2,-2}$ $\displaystyle X$ $\displaystyle\mapsto T_{2,0}$ $\displaystyle Y$
$\displaystyle\mapsto T_{0,2}$

is a well-defined $k$-algebra isomorphism. In particular, the right-hand side
is commutative.

###### Proof.

The proposition follows from Corollary 3.20 and the three points

  1. (a)

$T_{2,2}^{M}T_{i,j}=T_{i+2M,j+2M}=T_{i,j}T_{2,2}^{M}$ for all $i,j\in
2\mathbf{Z}$ and $M\in\mathbf{Z}$;

  2. (b)

$T_{2,0}^{m}=T_{2m,0}$ and $T_{0,2}^{m}=T_{0,2m}$ for all
$m\in\mathbf{Z}_{\geq 0}$;

  3. (c)

$T_{2,0}T_{0,2}=0$;

which we quickly explain:

  1. (a)

follows from Lemma 3.9 (iii);

  2. (b)

follows from Lemma 3.9 (vi) and (vii);

  3. (c)

follows from Lemma 3.9 (vi), (vii) (applied to $m=1$) and Lemma 3.7 (v).

∎

Fix now integers $0\leq t_{1},t_{2}\leq 1$. We want to compute
$\mathcal{H}(\chi[t_{1},0],\chi[t_{2},0])$ as a left
$\mathcal{H}(\chi[t_{2},0])$\- and right $\mathcal{H}(\chi[t_{1},0])$-module,
respectively. In view of the previous proposition, we may assume that
$t_{1}\neq t_{2}$.

######  Proposition 3.22.

For $0\leq t_{1}\neq t_{2}\leq 1$, the maps of left
$\mathcal{H}(\chi[t_{2},0])$-, resp. right
$\mathcal{H}(\chi[t_{1},0])$-modules,

$\displaystyle\mathcal{H}(\chi[t_{2},0])/(T_{2,0})\oplus\mathcal{H}(\chi[t_{2},0])/(T_{0,2})$
$\displaystyle\xrightarrow{\cong}\mathcal{H}(\chi[t_{1},0],\chi[t_{2},0])$
$\displaystyle(1,0)$ $\displaystyle\mapsto T_{0,1}$ $\displaystyle(0,1)$
$\displaystyle\mapsto T_{2,1}$

and

$\displaystyle\mathcal{H}(\chi[t_{1},0])/(T_{2,0})\oplus\mathcal{H}(\chi[t_{1},0])/(T_{0,2})$
$\displaystyle\xrightarrow{\cong}\mathcal{H}(\chi[t_{1},0],\chi[t_{2},0])$
$\displaystyle(1,0)$ $\displaystyle\mapsto T_{0,1}$ $\displaystyle(0,1)$
$\displaystyle\mapsto T_{2,1}$

are well-defined isomorphisms.

###### Proof.

By Corollary 3.20, the space $\mathcal{H}(\chi[t_{1},0],\chi[t_{2},0])$ has
$k$-basis $\\{T_{i,1+j}\\}_{i,j\in 2\mathbf{Z}}$. The proposition now follows
from the identities, for $m\in\mathbf{Z}_{\geq 0}$,

  1. (a)

$T_{2,0}T_{0,1}=0=T_{0,1}T_{2,0}$;

  2. (b)

$T_{0,2}T_{2,1}=0=T_{2,1}T_{0,2}$;

  3. (c)

$T_{2,2}T_{i,1+j}=T_{i+2,1+j+2}=T_{i,1+j}T_{2,2}$ for all $i,j\in
2\mathbf{Z}$;

  4. (d)

$T_{0,2m}T_{0,1}=T_{0,1+2m}=T_{0,1}T_{0,2m}$;

  5. (e)

$T_{2m,0}T_{2,1}=T_{2(m+1),1}=T_{2,1}T_{2m,0}$.

To check these, write $T_{0,1+2m}=T_{(\Pi s)^{1+2m}}e_{\chi[t_{1},0]}$,
$T_{2(m+1),1}=T_{\Pi^{2}(s\Pi)^{2m+1}}e_{\chi[t_{1},0]}$ and, for
$t\in\\{t_{1},t_{2}\\}$, $T_{2,2}=T_{\Pi^{4}}e_{\chi[t,0]}$,
$T_{2m,0}=T_{(s\Pi)^{2m}}e_{\chi[t,0]}$, $T_{0,2m}=T_{(\Pi
s)^{2m}}e_{\chi[t,0]}$. The points (a)-(e) now follow from Lemma 3.9, and
Lemma 3.7 (v) to prove (a) and (b). ∎

######  Remark 3.23.

The points (a)-(e) show that the left action of $\mathcal{H}(\chi[t_{1},0])$
agrees with the right action of $\mathcal{H}(\chi[t_{2},0])$ on
$\mathcal{H}(\chi[t_{1},0],\chi[t_{2},0])$ under the isomorphism
$\mathcal{H}(\chi[t_{1},0])\cong\mathcal{H}(\chi[t_{2},0])$ resulting from
Proposition 3.21. Moreover, as a consequence of the proposition, we obtain
that $\mathcal{H}(\chi[1,0],\chi)\cong\mathcal{H}(\chi,\chi[1,0])$ as modules
over the commutative ring $\mathcal{H}(\chi)\cong\mathcal{H}(\chi[1,0])$.

Recall that we defined
$\mathcal{H}(\chi\oplus\chi[1,0])=\left(\begin{smallmatrix}\mathcal{H}(\chi)&\mathcal{H}(\chi[1,0],\chi)\\\
\mathcal{H}(\chi,\chi[1,0])&\mathcal{H}(\chi[1,0])\end{smallmatrix}\right)$.
By Proposition 3.21, the $k$-algebras on the diagonal may be identified with
$k[X,Y,Z^{\pm 1}]/(XY)$.

######  Corollary 3.24.

The center $Z(\mathcal{H}(\chi\oplus\chi[1,0]))$ of
$\mathcal{H}(\chi\oplus\chi[1,0])$ is isomorphic to the $k$-algebra
$k[X,Y,Z^{\pm 1}]/(XY)$ via the diagonal embedding.

###### Proof.

Considering elementary diagonal matrices, an element $b\in
Z(\mathcal{H}(\chi\oplus\chi[1,0]))$ needs to be diagonal, say
$b=\operatorname{diag}(P_{0},P_{1})$, where each $P_{t}\in k[X,Y,Z^{\pm
1}]/(XY)$ with $Z=T_{2,2},X=T_{2,0},Y=T_{0,2}\in\mathcal{H}(\chi[t,0])$.
Letting $\mathbf{T}_{0,1}=\left(\begin{smallmatrix}0&T_{0,1}\\\
0&0\end{smallmatrix}\right)$ and
$\mathbf{T}_{2,1}=\left(\begin{smallmatrix}0&T_{2,1}\\\
0&0\end{smallmatrix}\right)$, the equality
$b\mathbf{T}_{0,1}=\mathbf{T}_{0,1}b$ forces $P_{1}(0,Y,Z)=P_{0}(0,Y,Z)$,
while the equality $b\mathbf{T}_{2,1}=\mathbf{T}_{2,1}b$ forces
$P_{1}(X,0,Z)=P_{0}(X,0,Z)$. Since $XY=0$, this implies $P_{1}=P_{0}$. It
remains to show that such elements indeed lie in the center, which follows
from Remark 3.23. ∎

We now compute to which extend the off-diagonal entries contribute to the
diagonal.

######  Lemma 3.25.

Let $0\leq t_{1}\neq t_{2}\leq 1$. The image of the composition map

$\mathcal{H}(\chi[t_{2},0],\chi[t_{1},0])\otimes_{\mathcal{H}(\chi[t_{2},0])}\mathcal{H}(\chi[t_{1},0],\chi[t_{2},0])\to\mathcal{H}(\chi[t_{1},0])$

is equal to the ideal $(T_{2,0},T_{0,2})$ generated by $T_{2,0}$ and
$T_{0,2}$. More precisely,

  * •

$T_{0,1}\circ T_{2,1}=0=T_{2,1}\circ T_{0,1}$;

  * •

$T_{0,1}\circ T_{0,1}=T_{0,2}$;

  * •

$T_{2,1}\circ T_{2,1}=T_{4,2}=T_{2,2}T_{2,0}$.

###### Proof.

By Proposition 3.22, we have

$\mathcal{H}(\chi[t_{2},0],\chi[t_{1},0])=\mathcal{H}(\chi[t_{1},0])T_{0,1}+\mathcal{H}(\chi[t_{1},0])T_{2,1}$

and

$\mathcal{H}(\chi[t_{1},0],\chi[t_{2},0])=T_{0,1}\mathcal{H}(\chi[t_{1},0])+T_{2,1}\mathcal{H}(\chi[t_{1},0]).$

Since the composition map is $\mathcal{H}(\chi[t_{1},0])$-equivariant from
both sides, it is enough to compute how the generators compose, i.e. it is
enough to check the claimed identities. These follow from Lemma 3.9 and Lemma
3.7 (v) by writing the generators $T_{0,1}$ and $T_{2,1}$ as in the proof of
Proposition 3.22. ∎

####  3.3.1. Simple modules

In order to determine the simple modules, we first show that the center acts
via scalars in the coefficient field $k$ and then study those with a fixed
central character.

######  Lemma 3.26.

Any simple right $\mathcal{H}(\chi\oplus\chi[1,0])$-module admits a central
character.

###### Proof.

To ease notation, put $R=\mathcal{H}(\chi\oplus\chi[1,0])$, and identify the
center of $R$ with $k[X,Y,Z^{\pm 1}]/(XY)$ as in Corollary 3.24. Let $M$ be a
simple right $R$-module. The kernel of the central action map $k[X,Y,Z^{\pm
1}]/(XY)\to\operatorname{End}_{R}(M)$ must be a prime ideal. In particular, it
contains $X$ or $Y$ and we may without loss of generality assume that it
contains $X$. Thus, the action factors through the localized polynomial ring
$(k[Y])[Z^{\pm 1}]$ over the PID $k[Y]$. The prime ideals in this ring are

  1. (a)

$(0)$;

  2. (b)

$(g(Z))$ for an irreducible polynomial $g(Z)\in(k[Y])[Z^{\pm 1}]$;

  3. (c)

$(Y-y,Z-z)$ for $y\in k$ and $z\in k^{\times}$.

We need to exclude that the kernel of the map $(k[Y])[Z^{\pm
1}]\to\operatorname{End}_{R}(M)$ falls into (a) or (b). In case (a) and (b),
$M$ is naturally a $C=k(Y)$-vector space. Since $T_{2,0}=X=0$ on $M$, the last
relation in Lemma 3.25 implies that $\left(\begin{smallmatrix}0&T_{2,1}\\\
T_{2,1}&0\end{smallmatrix}\right)^{2}$ kills $M$. In particular, we may choose
some $0\neq v\in M$ with $v\left(\begin{smallmatrix}0&T_{2,1}\\\
T_{2,1}&0\end{smallmatrix}\right)=0$. We may assume that $ve_{\chi[t,0]}=v$
for either $t=0$ or $t=1$. By Lemma 3.25, the $C$-subvector space

(7) $C[Z^{\pm 1}]v+C[Z^{\pm 1}]v\left(\begin{smallmatrix}0&T_{0,1}\\\
T_{0,1}&0\end{smallmatrix}\right)\subset M.$

is stable under the action of $R$ and must therefore be equal to $M$. In
particular, $M$ is finitely generated as a module over $C[Z^{\pm 1}]$ and so
the $C[Z^{\pm 1}]$-module structure on $M$ cannot extend to $C(Z)$, which
excludes (a). In case (b), replacing $C$ by $k[Y]$ in (7) yields an $R$-stable
subspace and we deduce that $M$ is finitely generated over $k[Y]$, which is
impossible since $M$ is a $C$-vector space by assumption. ∎

Fix now a central character $\xi\colon k[X,Y,Z^{\pm 1}]/(XY)\to k$
corresponding to the $k$-rational point $(x,y,z)$ and put

$\mathcal{H}(\chi\oplus\chi[1,0])_{\xi}=\mathcal{H}(\chi\oplus\chi[1,0])\otimes_{Z(\mathcal{H}(\chi\oplus\chi[1,0])),\xi}k.$

######  Proposition 3.27.

Assume that $x=0=y$. Up to isomorphism, there exist precisely two simple right
$\mathcal{H}(\chi\oplus\chi[1,0])_{\xi}$-modules, namely the $1$-dimensional
$\mathcal{H}(\chi\oplus\chi[1,0])_{\xi}$-algebras

$\operatorname{pr}_{t}\colon\mathcal{H}(\chi\oplus\chi[1,0])_{\xi}\twoheadrightarrow\mathcal{H}(\chi[t,0])/(T_{2,0},T_{0,2},T_{2,2}-z)\cong
k,\hskip 2.84544pt\text{ for }t=0,1,$

induced by projecting to the diagonal entry in $\mathcal{H}(\chi[t,0])$.

###### Proof.

It follows from Lemma 3.25 and the assumption on $x$ and $y$ that the
projection maps $\operatorname{pr}_{t}$ are indeed $k$-algebra homomorphisms.
Since the respective target is $1$-dimensional over $k$, they define simple
$\mathcal{H}(\chi\oplus\chi[1,0])_{\xi}$-modules. Moreover, these are pairwise
non-isomorphic as the kernels are distinct two-sided maximal ideals.

For showing that these are the only simple right modules, we make use of
propositions 3.21 and 3.22 to identify

(8) $\mathcal{H}(\chi\oplus\chi[1,0])_{\xi}\cong\begin{pmatrix}k&k\oplus k\\\
k\oplus k&k\end{pmatrix}$

with multiplication $\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\begin{pmatrix}a^{\prime}&b^{\prime}\\\
c^{\prime}&d^{\prime}\end{pmatrix}=\begin{pmatrix}aa^{\prime}&ab^{\prime}+d^{\prime}b\\\
a^{\prime}c+dc^{\prime}&dd^{\prime}\end{pmatrix}$, where $k$ acts on $k\oplus
k$ via the usual componentwise scalar multiplication. Let now
$I\subset\mathcal{H}(\chi\oplus\chi[1,0])_{\xi}$ be a right ideal neither
contained in $\ker(\operatorname{pr}_{0})$ nor in
$\ker(\operatorname{pr}_{1})$. We claim that $I$ is the whole ring. Indeed, by
assumption $I$ contains elements $E_{0},E_{1}$, where the $(t+1)$-th entry on
the diagonal in $E_{t}$ is non-zero. By scaling we may assume that this non-
zero entry is equal to $1$. Since $I$ is a right ideal, we may further assume
that $E_{t}$ contains only one non-zero column for $t=0,1$. Hence,
$E_{0}+E_{1}$ is of the form $\begin{pmatrix}1&b\\\ c&1\end{pmatrix}$ for some
$b,c\in k\oplus k$. Since $\begin{pmatrix}1&b\\\
c&1\end{pmatrix}\begin{pmatrix}1&-b\\\ -c&1\end{pmatrix}=\begin{pmatrix}1&0\\\
0&1\end{pmatrix}$, we are done. ∎

######  Proposition 3.28.

Assume that $x\neq 0$ or $y\neq 0$. Up to isomorphism, there exists a unique
simple right $\mathcal{H}(\chi\oplus\chi[1,0])_{\xi}$-module, namely the
$2$-dimensional space $kv\oplus kw$ uniquely determined by $ve_{\chi}=v$ and
$we_{\chi[1,0]}=w$.

###### Proof.

Consider the idempotent $e_{\chi}=\begin{pmatrix}1&0\\\ 0&0\end{pmatrix}$ in
$\mathcal{H}(\chi\oplus\chi[1,0])_{\xi}$. Since $x$ or $y$ is non-zero, Lemma
3.25 implies that
$\mathcal{H}(\chi\oplus\chi[1,0])_{\xi}e_{\chi}\mathcal{H}(\chi\oplus\chi[1,0])_{\xi}=\mathcal{H}(\chi\oplus\chi[1,0])_{\xi}$,
i.e. $e_{\chi}$ is a full idempotent. Hence,
$\mathcal{H}(\chi\oplus\chi[1,0])_{\xi}$ is Morita equivalent to the field

$e_{\chi}\mathcal{H}(\chi\oplus\chi[1,0])_{\xi}e_{\chi}=\mathcal{H}(\chi)\otimes_{k[X,Y,Z^{\pm
1}]/(XY),\xi}k\cong k$

and therefore admits unique simple right module. The same works with
$e_{\chi}$ replaced by $e_{\chi[1,0]}$. Hence, if $M$ denotes the unique
simple right $\mathcal{H}(\chi\oplus\chi[1,0])_{\xi}$-module, we have
$M=Me_{\chi}\oplus Me_{\chi[1,0]}$ with each summand being $1$-dimensional. ∎

Under the isomorphism $Z(\mathcal{H}(O))\cong
Z(\mathcal{H}(\chi\oplus\chi[1,0]))\cong k[X,Y,Z^{\pm 1}]/(XY)$ resulting from
the Morita equivalence Corollary 3.19, the elements $X,Y,Z$ correspond to
$T_{(s\Pi)^{2}},T_{(\Pi s)^{2}},T_{\Pi^{4}}$, respectively.

######  Definition 3.29.

  1. (i)

We say that a simple right $\mathcal{H}(O)$-module is supersingular if it is
killed by the central elements $T_{(s\Pi)^{2}}$ and $T_{(\Pi s)^{2}}$.
Otherwise, we say that the module is principal.

  2. (ii)

For $\chi\in O$ and $z\in k^{\times}$, denote by $\operatorname{SS}(\chi,z)$
the unique supersingular $\mathcal{H}(O)$-module with
$\operatorname{SS}(\chi,z)e_{\chi}\neq 0$ and the central element
$T_{\Pi^{4}}$ acting by the scalar $z$.

  3. (iii)

For $\chi\in O$ and $(x,y,z)\in k^{3}$ with $xy=0\neq z$ and $x\neq 0$ or
$y\neq 0$, denote by $\operatorname{PS}(\chi,x,y,z)$ the unique principal
$\mathcal{H}(O)$-module with $\operatorname{PS}(\chi,x,y,z)e_{\chi}\neq 0$
such that $(T_{(s\Pi)^{2}},T_{(\Pi s)^{2}},T_{\Pi^{4}})$ act on
$\operatorname{PS}(\chi,x,y,z)e_{\chi}$ via the scalars $(x,y,z)$.

######  Remark 3.30.

The only intertwinings between the simple right $\mathcal{H}(O)$-modules are
$\operatorname{SS}(\chi,z)\cong\operatorname{SS}(\chi^{s}[1,0],z)$ and

$\operatorname{PS}(\chi[1,0],x,y,z)\cong\operatorname{PS}(\chi,x,y,z)\cong\operatorname{PS}(\chi^{s}[1,0],y,x,z)$

for $\chi\in O$. For the last isomorphism note that
$T_{\Pi}(T_{s}T_{\Pi})^{2}T_{\Pi}^{-1}=(T_{\Pi}T_{s})^{2}$ and
$T_{\Pi}(T_{\Pi}T_{s})^{2}T_{\Pi}^{-1}=(T_{s}T_{\Pi})^{2}$, which follows from
Lemma 3.9 (ii), (iv).

###  3.4. The Hecke algebra of a non-regular orbit

In this section, we fix a non-square-regular orbit $O$ and a character
$\chi\in O$. Possibly replacing $\chi$ by $\chi^{s}[0,1]$, we may assume that
$\chi=\chi^{s}$. Twisting appropriately as in Lemma 3.17, we may assume that
$\chi=\mathbf{1}$ is the trivial character. By Lemma 3.13 and Remark 3.12, the
orbit $O$ of agrees with the $\Pi$-orbit of $\mathbf{1}$ and thus, by
Corollary 3.10, we have an algebra isomorphism $\mathcal{H}(O)\cong M_{4\times
4}(\mathcal{H}(\mathbf{1}))$ and so:

######  Corollary 3.31.

The functor

$\displaystyle\operatorname{Mod}_{\mathcal{H}(O)}$
$\displaystyle\xrightarrow{\sim}\operatorname{Mod}_{\mathcal{H}(\mathbf{1})}$
$\displaystyle M$ $\displaystyle\mapsto Me_{\mathbf{1}}$

is an equivalence of categories.

Viewing $\mathcal{H}(\mathbf{1})$ as the $k$-algebra of genuine
bi-$I$-invariant functions $\widetilde{G}\to k$ endowed with the convolution
product, Lemma 3.18 yields:

######  Corollary 3.32.

The algebra $\mathcal{H}(\mathbf{1})$ has $k$-basis
$\\{T_{i,j},S_{i,j}\\}_{i,j\in 2\mathbf{Z}}$, where $T_{i,j}$ and $S_{i,j}$
are uniquely determined by

$\operatorname{supp}(T_{i,j})=\widetilde{I}\left(\left(\begin{smallmatrix}p^{i}&0\\\
0&p^{j}\end{smallmatrix}\right),1\right)\widetilde{I},\hskip
5.69046ptT_{i,j}\left(\left(\left(\begin{smallmatrix}p^{i}&0\\\
0&p^{j}\end{smallmatrix}\right),1\right)\right)=1$

and

$\operatorname{supp}(S_{i,j})=\widetilde{I}\left(\left(\begin{smallmatrix}0&p^{i}\\\
p^{j}&0\end{smallmatrix}\right),1\right)\widetilde{I},\hskip
5.69046ptS_{i,j}\left(\left(\left(\begin{smallmatrix}0&p^{i}\\\
p^{j}&0\end{smallmatrix}\right),1\right)\right)=1,$

respectively.

######  Proposition 3.33.

The $k$-algebra $\mathcal{H}(\mathbf{1})$ is generated by
$T_{2,2},T_{-2,-2},S_{0,0},S_{0,2}$, and the following is a full set of
relations: $T_{2,2}$ is central and invertible with inverse $T_{-2,-2}$,
$S_{0,0}^{2}=-S_{0,0}$ and $S_{0,2}^{2}=0$.

###### Proof.

Writing $T_{2,2}=T_{\Pi^{4}}e_{\mathbf{1}}$,
$T_{-2,-2}=T_{\Pi^{-4}}e_{\mathbf{1}}$, $S_{0,0}=T_{s}e_{\mathbf{1}}$ and
$S_{0,2}=T_{\Pi s\Pi}e_{\mathbf{1}}$, the relations follow from Lemma 3.9, and
Lemma 3.7 (v) to prove $S_{0,2}^{2}=0$. Furthermore, for all
$m\in\mathbf{Z}_{\geq 1}$,

  1. (a)

$(S_{0,0}S_{0,2})^{m}=T_{2m,0}$, $(S_{0,2}S_{0,0})^{m}=T_{0,2m}$;

  2. (b)

$T_{2m,0}S_{0,0}=S_{2m,0}$, $S_{0,2}T_{2m,0}=S_{0,2(m+1)}$;

  3. (c)

$T_{2,2}T_{i,j}=T_{i+2,j+2}$, $T_{2,2}S_{i,j}=S_{i+2,j+2}$ for all $i,j\in
2\mathbf{Z}$.

Indeed, (a) follows from (vi) and (vii), (b) follows from (viii) and (ix),
while (c) follows from (iii) of Lemma 3.9. Corollary 3.32 now finishes the
proof. ∎

######  Corollary 3.34.

The center of $\mathcal{H}(\mathbf{1})$ is a localized polynomial ring in two
variables, namely the $k$-algebra morphism

$\displaystyle k[\mathfrak{Z},Z^{\pm 1}]$
$\displaystyle\xrightarrow{\cong}Z(\mathcal{H}(\mathbf{1}))$
$\displaystyle\mathfrak{Z}$ $\displaystyle\mapsto
S_{0,0}S_{0,2}+S_{0,2}S_{0,0}+S_{0,2}$ $\displaystyle Z$ $\displaystyle\mapsto
T_{2,2}$

is a well-defined isomorphism.

###### Proof.

The explicit description of the algebra $\mathcal{H}(\mathbf{1})$ in the
previous proposition implies that the center is equal to the set of elements,
which are up to an integral power of $T_{2,2}$ of the form

$bT_{2,2}^{M}+\sum_{i\geq
1}a_{i}T_{2,2}^{N_{i}}(S_{0,0}S_{0,2})^{i}+\sum_{i\geq
1}a_{i}T_{2,2}^{N_{i}}(S_{0,2}S_{0,0})^{i}+\sum_{i\geq
0}a_{i+1}T_{2,2}^{N_{i+1}}(S_{0,2}S_{0,0})^{i}S_{0,2}$

for $b,a_{i}\in k$, $M\in\mathbf{Z}$ and $N_{i}\in\mathbf{Z}_{\geq 0}$. An
induction shows moreover that

$\mathfrak{Z}^{n}=(S_{0,0}S_{0,2})^{n}+(S_{0,2}S_{0,0})^{n}+(S_{0,2}S_{0,0})^{n-1}S_{0,2}$

for all $n\in\mathbf{Z}_{\geq 1}$, finishing the proof. ∎

####  3.4.1. Simple modules

######  Lemma 3.35.

Any simple right $\mathcal{H}(\mathbf{1})$-module is finite dimensional.

###### Proof.

Let $M$ be a simple right $\mathcal{H}(\mathbf{1})$-module. If $S_{0,0}+1$
kills this module, then $M$ descends to a module over the commutative finite
type $k$-algebra $\mathcal{H}(\mathbf{1})/(S_{0,0}+1)$ and we are done by
Hilbert’s Nullstellensatz. Hence, we may assume that there exists some non-
zero $v\in M(S_{0,0}+1)$. In particular, $vS_{0,0}=0$. Writing
$Z(\mathcal{H}(\mathbf{1}))=k[\mathfrak{Z},Z^{\pm 1}]$ as in Corollary 3.34,
one can now argue as in the square-reguler case, cf. proof of Lemma 3.26
replacing $v\left(\begin{smallmatrix}0&T_{0,1}\\\
T_{0,1}&0\end{smallmatrix}\right)$ by $vS_{0,2}$. ∎

######  Proposition 3.36.

The simple right $\mathcal{H}(\mathbf{1})$-modules are up to isomorphism as
follows:

  1. (SS)

For each $(\lambda,z)\in\\{0,-1\\}\times k^{\times}$, the character
$\mathcal{H}(\mathbf{1})\to k$, $T_{2,2}\mapsto z,S_{0,2}\mapsto
0,S_{0,0}\mapsto\lambda$.

  2. (PS)

For each $(\lambda,z)\in k^{\times}\times k^{\times}$, the $2$-dimensional
space $kv\oplus kw$ with basis $v,w$ and action determined by
$vT_{2,2}=zv,\hskip 2.84544ptvS_{0,2}=w,\hskip 2.84544ptwS_{0,0}=\lambda v.$

There are no isomorphisms between objects associated to different parameters.

###### Proof.

By the previous lemma, each simple right $\mathcal{H}(\mathbf{1})$-module $M$
is finite dimensional. The same poof used by Vignéras in [30, Theorem A.2] now
works here: If $\dim_{k}M>1$, then one shows that $S_{0,2}S_{0,0}$ cannot be
nilpotent on $M$, so one may choose an eigenvector $v\in M$ with non-zero
eigenvalue $\lambda\in k^{\times}$ and one has $M=kv\oplus kvS_{0,2}$. ∎

######  Remark 3.37.

The center $Z(\mathcal{H}(\mathbf{1}))\cong k[\mathfrak{Z},Z^{\pm 1}]$, see
Lemma 3.34, acts on the module of type (SS) (resp. (PS)) corresponding to the
parameter $(\lambda,z)$ via the central character corresponding to the
$k$-rational point $(0,z)$ (resp. $(\lambda,z)$).

Using the Morita equivalence Corollary 3.31, we identify the center of
$\mathcal{H}(O)$ with $Z(\mathcal{H}(\mathbf{1}))\cong k[\mathfrak{Z},Z^{\pm
1}]$.

######  Definition 3.38.

  1. (i)

We say that a simple right $\mathcal{H}(O)$-module is supersingular if the
center $k[\mathfrak{Z},Z^{\pm 1}]$ acts on it via a $k$-rational point of the
form $(0,z)$ for $z\in k^{\times}$. Otherwise, we say that the module is
principal.

  2. (ii)

For $(\lambda,z)\in\\{0,-1\\}\times k^{\times}$, denote by
$\operatorname{SS}(\lambda,z)$ the unique supersingular
$\mathcal{H}(O)$-module such that $\operatorname{SS}(\lambda,z)e_{\mathbf{1}}$
is equal to the character on $\mathcal{H}(\mathbf{1})$ corresponding to
$(\lambda,z)$ as in Proposition 3.36.

  3. (iii)

For $(\lambda,z)\in k^{\times}\times k^{\times}$, denote by
$\operatorname{PS}(\lambda,z)$ the unique right $\mathcal{H}(O)$-module such
that $\operatorname{PS}(\lambda,z)e_{\mathbf{1}}$ is equal to the
$2$-dimensional $\mathcal{H}(\mathbf{1})$-module corresponding to
$(\lambda,z)$ as in Proposition 3.36.

###  3.5. Extensions of simple modules

Using [6, §6] as a guideline, we compute the extensions of the simple modules
determined in the previous sections in the category of $\mathcal{H}$-modules
on which $T_{\Pi^{4}}$ acts via a fixed scalar. Applying Lemma 3.17 to an
appropriate unramified character, we may and do assume that $T_{\Pi^{4}}$ acts
trivially. We put

$\mathcal{H}_{p^{2}=1}=\mathcal{H}/(T_{\Pi^{4}}-1)$

and use a similar notation for the smaller algebras $\mathcal{H}(O)$,
$\mathcal{H}(\chi)$, etc. There can only be non-trivial extensions of two
simple modules if they descend to the Hecke algebra of the same orbit. We
therefore fix an orbit $O$ and we may and do without loss of generality assume
that $O=O_{\mathbf{1}}$ in the non-square-regular case. Define the element

$\mathcal{H}\ni\mathscr{Z}_{O}=\begin{cases}(T_{\Pi}T_{s})^{2}+(T_{s}T_{\Pi})^{2}\text{
if $O$ is square-regular}\\\
\sum_{i=0}^{3}T_{\Pi}^{-i}\left((T_{\Pi}T_{s})^{2}+(T_{s}T_{\Pi})^{2}+T_{\Pi}T_{s}T_{\Pi}\right)e_{\mathbf{1}}T_{\Pi}^{i}\text{
if $O=O_{\mathbf{1}}$.}\end{cases}$

which, when projected to $\mathcal{H}(O)$, is central by Corollary 3.24, resp.
3.34.

We start with two lemmas, the first of which we record for later reference.

######  Lemma 3.39.

We have equalities of right $\mathcal{H}_{p^{2}=1}$-modules:

  1. (i)

If $\chi\neq\chi^{s}$, then
$T_{s}e_{\chi}\mathcal{H}_{p^{2}=1}=\ker\left(T_{s}\colon
e_{\chi^{s}}\mathcal{H}_{p^{2}=1}\twoheadrightarrow
e_{\chi}T_{s}\mathcal{H}_{p^{2}=1}\right)$;

  2. (ii)

$T_{s}e_{\mathbf{1}}\mathcal{H}_{p^{2}=1}=\ker\left(T_{s}+1\colon
e_{\mathbf{1}}\mathcal{H}_{p^{2}=1}\twoheadrightarrow
e_{\mathbf{1}}(T_{s}+1)\mathcal{H}_{p^{2}=1}\right)$;

  3. (iii)

$(T_{s}+1)e_{\mathbf{1}}\mathcal{H}_{p^{2}=1}=\ker\left(T_{s}\colon
e_{\mathbf{1}}\mathcal{H}_{p^{2}=1}\twoheadrightarrow
e_{\mathbf{1}}T_{s}\mathcal{H}_{p^{2}=1}\right)$;

where the map $T_{s}$, resp. $T_{s}+1$, is given by acting from the left.

###### Proof.

It suffices to prove the equality after applying the idempotent cutting out
$\mathcal{H}(O)_{p^{2}=1}$. We distinguish between the square-regular and non-
square-regular case.

Assume that $O$ is square-regular. By the Morita equivalence of Corollary
3.19, it is enough to prove the statement after applying
$e:=e_{\chi}+e_{\chi[1,0]}$ from the right. Applying the invertible element
$T_{\Pi}$ from the left, the desired equality becomes equivalent to

$T_{\Pi}T_{s}e_{\chi}\mathcal{H}_{p^{2}=1}e=\ker(T_{s}T_{\Pi}^{-1}\colon
T_{\Pi}e_{\chi^{s}}\mathcal{H}_{p^{2}=1}e\to
e_{\chi}T_{s}\mathcal{H}_{p^{2}=1}e\subset\mathcal{H}(\chi\oplus\chi[1,0])_{p^{2}=1}).$

Using Lemma 3.9 (v), (vi) and (vii), this is equivalent to

$\displaystyle\left(\begin{smallmatrix}0&T_{0,1}\\\
T_{0,1}&0\end{smallmatrix}\right)\left(\begin{smallmatrix}\mathcal{H}(\chi)_{p^{2}=1}&\mathcal{H}(\chi[1,0],\chi)_{p^{2}=1}\\\
0&0\end{smallmatrix}\right)$
$\displaystyle=\ker\left(\left(\begin{smallmatrix}0&T_{0,-1}\\\
T_{0,-1}&0\end{smallmatrix}\right)\colon\left(\begin{smallmatrix}0&0\\\
\mathcal{H}(\chi,\chi[1,0])_{p^{2}=1}&\mathcal{H}(\chi[1,0])_{p^{2}=1}\end{smallmatrix}\right)\to\mathcal{H}(\chi\oplus\chi[1,0])_{p^{2}=1}\right)$
$\displaystyle=\ker\left(\left(\begin{smallmatrix}0&T_{2,1}\\\
T_{2,1}&0\end{smallmatrix}\right)\colon\left(\begin{smallmatrix}0&0\\\
\mathcal{H}(\chi,\chi[1,0])_{p^{2}=1}&\mathcal{H}(\chi[1,0])_{p^{2}=1}\end{smallmatrix}\right)\to\mathcal{H}(\chi\oplus\chi[1,0])_{p^{2}=1}\right).$

For the last equality we used that the central element $T_{2,2}$ is
invertible. The result now follows from Proposition 3.21, Proposition 3.22 and
Lemma 3.25.

Assume now that $O=O_{\mathbf{1}}$. By the Morita equivalence 3.31, it is
enough to prove the equalities after applying the idempotent $e_{\mathbf{1}}$.
In this case, (ii) and (iii) translate into

$S_{0,0}\mathcal{H}(\mathbf{1})_{p^{2}=1}=\ker\left(S_{0,0}+1\colon\mathcal{H}(\mathbf{1})_{p^{2}=1}\twoheadrightarrow(S_{0,0}+1)\mathcal{H}(\mathbf{1})_{p^{2}=1}\right)$

and

$(S_{0,0}+1)\mathcal{H}(\mathbf{1})_{p^{2}=1}=\ker\left(S_{0,0}\colon\mathcal{H}(\mathbf{1})_{p^{2}=1}\twoheadrightarrow
S_{0,0}\mathcal{H}(\mathbf{1})_{p^{2}=1}\right).$

For (i), note that by applying $T_{\Pi^{2}}$, we may replace $\chi$ by
$\chi[1,1]$ and so assume that $\chi=\mathbf{1}[1,0]$. Thus,
$e_{\chi}T_{\Pi}=T_{\Pi}e_{\mathbf{1}}$, see Lemma 3.9 (v). Acting by
$T_{\Pi}$ from the left, we find that (iii) is equivalent to

$\displaystyle T_{\Pi}T_{s}T_{\Pi}\mathcal{H}(\mathbf{1})_{p^{2}=1}$
$\displaystyle=\ker\left(T_{s}T_{\Pi}^{-1}\colon\mathcal{H}(\mathbf{1})_{p^{2}=1}\to
e_{\mathbf{1}[1,0]}T_{s}\mathcal{H}_{p^{2}=1}e_{\mathbf{1}}\right)$
$\displaystyle=\ker\left(T_{\Pi}^{-1}T_{s}T_{\Pi}^{-1}\colon\mathcal{H}(\mathbf{1})_{p^{2}=1}\twoheadrightarrow
T_{\Pi}^{-1}T_{s}T_{\Pi}^{-1}\mathcal{H}(\mathbf{1})_{p^{2}=1}\right),$

where in the last equality we used the injectivity of $T_{\Pi}^{-1}$. This
just means that

$\displaystyle S_{0,2}\mathcal{H}(\mathbf{1})_{p^{2}=1}$
$\displaystyle=\ker\left(T_{-2,-2}S_{0,2}\colon\mathcal{H}(\mathbf{1})_{p^{2}=1}\twoheadrightarrow
S_{0,2}\mathcal{H}(\mathbf{1})_{p^{2}=1}\right)$
$\displaystyle=\ker\left(S_{0,2}\colon\mathcal{H}(\mathbf{1})_{p^{2}=1}\twoheadrightarrow
S_{0,2}\mathcal{H}(\mathbf{1})_{p^{2}=1}\right)$

The assertions now follow from the description of $\mathcal{H}(\mathbf{1})$ in
Proposition 3.33. ∎

By similar arguments, one proves the following lemma.

######  Lemma 3.40.

For all $\lambda\in k$ and all $\chi\in O$, the endomorphism

$\mathscr{Z}_{O}-\lambda\colon e_{\chi}\mathcal{H}_{p^{2}=1}\to
e_{\chi}\mathcal{H}_{p^{2}=1}$

is injective.

######  Definition 3.41.

For $\lambda\in k$ and a character $\chi\in O$, define the
$\mathcal{H}_{p^{2}=1}$-modules $M(\chi,\lambda)$ by the short exact sequence

$0\to
T_{s}e_{\chi}\mathcal{H}_{p^{2}=1}\xrightarrow{\mathscr{Z}_{O}-\lambda}T_{s}e_{\chi}\mathcal{H}_{p^{2}=1}\to
M(\chi,\lambda)\to 0.$

For $O=O_{\mathbf{1}}$, we also define $M(\lambda)$ by

$0\to(T_{s}+1)e_{\mathbf{1}}\mathcal{H}_{p^{2}=1}\xrightarrow{\mathscr{Z}_{O}-\lambda}(T_{s}+1)e_{\mathbf{1}}\mathcal{H}_{p^{2}=1}\to
M(\lambda)\to 0.$

######  Remark 3.42.

Using the relations $e_{\chi}T_{\Pi}^{2}=T_{\Pi}^{2}e_{\chi[1,1]}$,
$T_{s}T_{\Pi}^{2}=(-1)^{\frac{p-1}{2}}T_{\Pi}^{2}T_{s}$ and the invertibility
of $T_{\Pi}$ (see Lemma 3.9), we obtain an isomorphism $T_{\Pi}^{2}\colon
T_{s}e_{\chi}\mathcal{H}_{p^{2}=1}\cong
T_{s}e_{\chi[1,1]}\mathcal{H}_{p^{2}=1}$ of right
$\mathcal{H}_{p^{2}=1}$-modules inducing an isomorphism on the quotients
$M(\chi,\lambda)\cong M(\chi[1,1],\lambda)$.

######  Proposition 3.43.

Assume that $O$ is square-regular. Let $\chi\in O$ and $\lambda\in k$.

  1. (i)

If $\lambda\neq 0$, then
$M(\chi,\lambda)\cong\operatorname{PS}(\chi,0,\lambda,1)$.

  2. (ii)

If $\lambda=0$, then there is a non-split short exact sequence

$0\to\operatorname{SS}(\chi[1,0],1)\to
M(\chi,0)\to\operatorname{SS}(\chi,1)\to 0$

of $\mathcal{H}_{p^{2}=1}$-modules.

###### Proof.

Put $e=e_{\chi}+e_{\chi[1,0]}$. The Morita equivalence 3.19 reduces to proving
the statement after applying $e$. Acting by $T_{\Pi}$ from the left defines an
isomorphism between $M(\chi,\lambda)$ and the cokernel of
$\mathscr{Z}_{O}-\lambda$ acting on
$T_{\Pi}T_{s}e_{\chi}\mathcal{H}_{p^{2}=1}$, which we denote by
$T_{\Pi}M(\chi,\lambda)$. It follows from propositions 3.22 and 3.21 that the
images of the elements $T_{\Pi}T_{s}e_{\chi}=\left(\begin{smallmatrix}0&0\\\
T_{0,1}&0\end{smallmatrix}\right)$, $T_{\Pi}T_{s}e_{\chi}T_{\Pi}T_{s}=T_{(\Pi
s)^{2}}e_{\chi[1,0]}=\left(\begin{smallmatrix}0&0\\\
0&T_{0,2}\end{smallmatrix}\right)$ form a basis of the cokernel of the map

$\left(\begin{smallmatrix}T_{2,0}+T_{0,2}-\lambda&0\\\
0&T_{2,0}+T_{0,2}-\lambda\end{smallmatrix}\right)\colon\left(\begin{smallmatrix}0&0\\\
T_{0,1}\mathcal{H}(\chi)_{p^{2}=1}&(T_{0,2})\end{smallmatrix}\right)\to\left(\begin{smallmatrix}0&0\\\
T_{0,1}\mathcal{H}(\chi)_{p^{2}=1}&(T_{0,2})\end{smallmatrix}\right),$

where $(T_{0,2})$ is the ideal generated by $T_{0,2}$ in
$\mathcal{H}(\chi[1,0])_{p^{2}=1}$. Moreover, the central elements
$(T_{2,0},T_{0,2},T_{2,2})$ act on the cokernel via the scalars
$(0,\lambda,1)$.

Part (i) follows from the uniqueness in Proposition 3.28.

For (ii), note that the $1$-dimensional subspace of $T_{\Pi}M(\chi,0)e$
spanned by the image of $\left(\begin{smallmatrix}0&0\\\
0&T_{0,2}\end{smallmatrix}\right)$ is
$\mathcal{H}(\chi\oplus\chi[1,0])$-stable and must therefore be isomorphic to
$\operatorname{SS}(\chi[1,0],0)e$. The cokernel of the resulting inclusion
$\operatorname{SS}(\chi[1,0])e\hookrightarrow T_{\Pi}M(\chi,0)e$ is
$1$-dimensional and $e_{\chi}$ acts trivially on it, so we obtain an exact
sequence of the desired form, by Proposition 3.27. It is non-split for if
$\alpha,\beta\in k$ such that $\alpha\left(\begin{smallmatrix}0&0\\\
T_{0,1}&0\end{smallmatrix}\right)+\beta\left(\begin{smallmatrix}0&0\\\
0&T_{0,2}\end{smallmatrix}\right)$ is fixed by $e_{\chi}$, then $\beta=0$, but
the $1$-dimensional subspace spanned by $\left(\begin{smallmatrix}0&0\\\
T_{0,1}&0\end{smallmatrix}\right)$ is not
$\mathcal{H}(\chi\oplus\chi[1,0])$-stable: $\left(\begin{smallmatrix}0&0\\\
T_{0,1}&0\end{smallmatrix}\right)\left(\begin{smallmatrix}0&T_{0,1}\\\
0&0\end{smallmatrix}\right)=\left(\begin{smallmatrix}0&0\\\
0&T_{0,2}\end{smallmatrix}\right)$, by Lemma 3.25. ∎

######  Proposition 3.44.

Assume that $O=O_{\mathbf{1}}$. Let $\chi\in O$ and $\lambda\in k$.

  1. (i)

If $\lambda\neq 0$, then
$M(\chi,\lambda)\cong\operatorname{PS}(\lambda,1)\cong M(\lambda)$.

  2. (ii)

If $\lambda=0$, then
$M(\chi,0)=\operatorname{SS}(0,1)\oplus\operatorname{SS}(-1,1)$ for
$\chi\neq\chi^{s}$, and for $\chi=\chi^{s}$ there are non-split exact
sequences

$0\to\operatorname{SS}(0,1)\to M(\chi,0)\to\operatorname{SS}(-1,1)\to 0$

and

$0\to\operatorname{SS}(-1,1)\to M(0)\to\operatorname{SS}(0,1)\to 0.$

###### Proof.

It is enough to prove the statements after applying $e_{\mathbf{1}}$. If
$\chi\neq\chi^{s}$, then, arguing as in the proof of Lemma 3.39,
$M(\chi,\lambda)$ is isomorphic to the cokernel of the endomorphism on
$S_{0,2}\mathcal{H}(\mathbf{1})_{p^{2}=1}$ defined by the central element
$\mathfrak{Z}-\lambda\in k[\mathfrak{Z},Z^{\pm 1}]\cong
Z(\mathcal{H}(\mathbf{1}))$ (see Corollary 3.34). Using the explicit
description of $\mathcal{H}(\mathbf{1})$ in Proposition 3.33, one checks that
a basis for this cokernel as a $k$-vector space is given by the images $v$ and
$w$ of the two elements $S_{0,2}$ and $S_{0,2}S_{0,0}$, respectively. By
construction, the center $k[\mathfrak{Z},Z^{\pm 1}]$ acts on it via the
rational point $(\lambda,z)$, which implies (i) by the uniqueness in
Proposition 3.36, see also Remark 3.37. If $\lambda=0$, then write the
cokernel as

$k(v+w)\oplus kw.$

Since $wS_{0,0}=-w$, $wS_{0,2}=0$, i.e. $kw=\operatorname{SS}(-1,1)$, and
$(v+w)S_{0,0}=0$, $(v+w)S_{0,2}=0$, i.e. $k(v+w)=\operatorname{SS}(0,1)$, this
settles the case $\chi\neq\chi^{s}$.

If $\chi=\chi^{s}$, we may assume that $\chi=\mathbf{1}$ by Remark 3.42. In
this case, a basis of $M(\chi,\lambda)$ (resp. $M(\lambda)$) is given by the
images $v$ and $w$ of $S_{0,0}$ and $S_{0,0}S_{0,2}$ (resp. $S_{0,0}+1$ and
$(S_{0,0}+1)S_{0,2}$). Again by uniqueness of the principal module with
central character corresponding to the rational point $(\lambda,z)$, we deduce
(i). Assume that $\lambda=0$, then for $M(\chi,\lambda)$, $vS_{0,0}=-v$,
$vS_{0,2}=w$ and $wS_{0,0}=0$, $wS_{0,2}=0$, so we obtain a non-split short
exact sequence

$0\to kw=\operatorname{SS}(0,1)e_{\mathbf{1}}\to
M(\chi,\lambda)e_{\mathbf{1}}\to\operatorname{SS}(-1,1)e_{\mathbf{1}}\to 0.$

For $M(0)$ on the other hand, we have $vS_{0,0}=0$, $vS_{0,2}=w$ and
$wS_{0,0}=-w$, $wS_{0,2}=0$, so we obtain a non-split short exact sequence

$0\to kw=\operatorname{SS}(-1,1)e_{\mathbf{1}}\to
M(0)e_{\mathbf{1}}\to\operatorname{SS}(0,1)e_{\mathbf{1}}\to 0,$

finishing the proof in case $\chi=\chi^{s}$. ∎

If a non-split extension between simple modules exists, then their central
characters have to coincide. In particular, there are no non-split extensions
neither of supersingular and principal modules nor of different principal
modules.

####  3.5.1. Extensions of supersingular modules

In order to compute the extensions of supersingular modules, we will
distinguish between the square-regular case and the non-square-regular case.

Square-regular case. Assume that $O$ is square-regular and write

$\mathcal{H}(\chi\oplus\chi[1,0])=\begin{pmatrix}\mathcal{H}(\chi)&\mathcal{H}(\chi[1,0],\chi)\\\
\mathcal{H}(\chi,\chi[1,0])&\mathcal{H}(\chi[1,0])\end{pmatrix}$

and identify

$\mathcal{H}(\chi\oplus\chi[1,0])_{\xi}\cong\begin{pmatrix}k&k\oplus k\\\
k\oplus k&\overline{\mathbf{F}}_{p}\end{pmatrix}$

as in (8), where $\xi$ is the central character corresponding to the singular
rational point $(0,0,1)$, which we may view as a central character of
$\mathcal{H}(O)$ via the Morita equivalence 3.19.

######  Definition 3.45.

For $\lambda,\mu\in k$, define $E_{\lambda,\mu}$ to be the unique
$\mathcal{H}(O)$-module so that $E_{\lambda,\mu}(e_{\chi}+e_{\chi[1,0]})$ is
the $2$-dimensional $\mathcal{H}(\chi\oplus\chi[1,0])_{\xi}$-module $kv\oplus
kw$ with

$v\left(\begin{smallmatrix}a&(b_{1},b_{2})\\\
(c_{1},c_{2})&d\end{smallmatrix}\right)=dv,\hskip
5.69046ptw\left(\begin{smallmatrix}a&(b_{1},b_{2})\\\
(c_{1},c_{2})&d\end{smallmatrix}\right)=aw+(\lambda b_{1}+\mu b_{2})v$

for all $\left(\begin{smallmatrix}a&(b_{1},b_{2})\\\
(c_{1},c_{2})&d\end{smallmatrix}\right)\in\mathcal{H}(\chi\oplus\chi[1,0])_{\xi}$.

######  Proposition 3.46.

  1. (i)

The map

$\displaystyle k^{2}$
$\displaystyle\xrightarrow{\cong}\operatorname{Ext}^{1}_{\mathcal{H}(O)}(\operatorname{SS}(\chi,1),\operatorname{SS}(\chi[1,0],1))$
$\displaystyle(\lambda,\mu)$ $\displaystyle\mapsto E_{\lambda,\mu}$

is a well-defined $k$-linear isomorphism.

  2. (ii)

The space
$\operatorname{Ext}^{1}_{\mathcal{H}(O)_{p^{2}=1}}(\operatorname{SS}(\chi,1),\operatorname{SS}(\chi,1))=0$
is trivial.

###### Proof.

(i) Since the two supersingular modules are non-isomorphic, any non-trivial
extension admits a central character, so the Ext-space in question is equal to

$\operatorname{Ext}^{1}_{\mathcal{H}(O)_{\xi}}(\operatorname{SS}(\chi,1),\operatorname{SS}(\chi[1,0],1))\cong\operatorname{Ext}^{1}_{\mathcal{H}(\chi\oplus\chi[1,0])_{\xi}}(\operatorname{SS}(\chi,1)e,\operatorname{SS}(\chi[1,0],1)e),$

where $e=e_{\chi}+e_{\chi[1,0]}$ and the isomorphism uses the Morita
equivalence 3.19. Write $E_{\lambda,\mu}e=kv\oplus kw$ as in its definition.
Let $\bar{w}$ denote the image of $w$ in $E_{\lambda,\mu}e/kv$, so that we
have a short exact sequence

$0\to kv\to E_{\lambda,\mu}e\to k\bar{w}\to 0$

It is now clear from the definition of $E_{\lambda,\mu}$ that the kernel
(resp. cokernel) is just $\operatorname{SS}(\chi[1,0],1)e$ (resp.
$\operatorname{SS}(\chi,1)e$) as
$\mathcal{H}(\chi\oplus\chi[1,0])_{\xi}$-modules, see also Proposition 3.27.

One verifies that the assignment is $k$-linear. If the extension
$E_{\lambda,\mu}e$ splits, then there exist $\alpha,\beta\in k$ such that
$\alpha v+\beta w$ is fixed by the idempotent $e_{\chi}$, which forces
$\alpha=0$ and so $\lambda,\mu=0$. This proves injectivity.

Conversely, let $0\to kv\to E\to k\bar{w}\to 0$ be an extension of the two
supersingular modules. Then the operator $e_{\chi}$ on $E$ is diagonalizable
with distinct eigenvalues, so we may choose an eigenvector $w$ with eigenvalue
$1$. Since $\left(\begin{smallmatrix}0&(1,0)\\\ 0&0\end{smallmatrix}\right)$
and $\left(\begin{smallmatrix}0&(0,1)\\\ 0&0\end{smallmatrix}\right)$ act
trivially on $k\bar{w}$, we can find $\lambda,\mu\in k$ so that they act on
$w$ via $\lambda v$ and $\mu v$, respectively. In other words,
$E=E_{\lambda,\mu}$.

(ii) We need to show that there are no non-trivial self-extensions of
$\operatorname{SS}(\chi,1)e$ in the category of
$\mathcal{H}(\chi\oplus\chi[1,0])_{p^{2}=1}$-modules. Let $E$ be such an
extension. The element $(e_{\chi}-1)^{2}=1-e_{\chi}$ kills $E$ and so for each
$f\in E$,

$f\left(\begin{smallmatrix}a&b\\\
c&d\end{smallmatrix}\right)=fe_{\chi}\left(\begin{smallmatrix}a&b\\\
c&d\end{smallmatrix}\right)=fe_{\chi}\left(\begin{smallmatrix}a&b\\\
c&d\end{smallmatrix}\right)e_{\chi}=f\left(\begin{smallmatrix}a&0\\\
0&0\end{smallmatrix}\right)\text{ for all $\left(\begin{smallmatrix}a&b\\\
c&d\end{smallmatrix}\right)\in\mathcal{H}(\chi\oplus\chi[1,0])_{p^{2}=1}$.}$

In particular, elements of the form $\left(\begin{smallmatrix}0&\ast\\\
0&0\end{smallmatrix}\right)$ or $\left(\begin{smallmatrix}0&0\\\
\ast&0\end{smallmatrix}\right)$ have to kill $E$. By Lemma 3.25, this implies
that also the elements of the form $\left(\begin{smallmatrix}a&0\\\
0&0\end{smallmatrix}\right)$ for $a\in(T_{2,0},T_{0,2})$ kill $E$. Since
$\mathcal{H}(\chi)_{p^{2}=1}=k\oplus(T_{2,0},T_{0,2})$ by Proposition 3.21, we
are done. ∎

######  Remark 3.47.

One may check that $M(\chi,0)\cong E_{1,0}$ and $M(\chi^{s}[1,0],0)\cong
E_{0,1}$.

Non-square-regular case. Assume that $O=O_{\mathbf{1}}$.

######  Proposition 3.48.

Let $M_{1},M_{2}$ be supersingular $\mathcal{H}(O)_{p^{2}=1}$-modules. Then

$\dim_{k}\operatorname{Ext}^{1}_{\mathcal{H}(O)_{p^{2}=1}}(M_{1},M_{2})=1.$

###### Proof.

It is enough to prove this in the category of
$\mathcal{H}(\mathbf{1})_{p^{2}=1}$-modules after applying $e_{\mathbf{1}}$.
Write $M_{2}=\operatorname{SS}(\lambda,1)$ for $\lambda\in\\{0,-1\\}$. Define
an extension

$0\to M_{1}e_{\mathbf{1}}\to E\to M_{2}e_{\mathbf{1}}\to 0$

by putting $E=kv\oplus kpw$ with $kv=M_{1}e_{\mathbf{1}}$ and
$wS_{0,0}=\lambda w$ and $wS_{0,2}=v$. This is a well-defined extension which
is non-split (as $S_{0,2}$ acts non-trivially on $E$). Conversely, given any
other such extension, say $F$, we may diagonalize the action of $S_{0,0}$
since $S_{0,0}(S_{0,0}+1)=0$. Choose a basis of eigenvectors $v,w$ with $v\in
M_{1}e_{\mathbf{1}}$. Then necessarily $wS_{0,0}=\lambda w$. Since $S_{0,2}$
acts trivially on $M_{2}e_{\mathbf{1}}$, there exists some $\mu\in k$ with
$wS_{0,2}=\mu v$. In conclusion, the extension $F$ is equal to the
$\mu$-scalar multiple of the extension $E$. ∎

####  3.5.2. Extensions of principal modules

In the square-regular case, we may write a principal module as
$M(\chi,\lambda)$ for a suitable $0\neq\lambda\in k$ and some $\chi\in O$, see
Proposition 3.43 and the intertwinings in Remark 3.30. In case
$O=O_{\mathbf{1}}$, we may write each principal module of
$\mathcal{H}(O)_{p^{2}=1}$ as $M(\lambda)$ for a suitable $\lambda\in
k^{\times}$, see Proposition 3.44. The following assertion can now either be
proved as in [6, Corollary 6.6], resp. [6, Corollary 6.7], or is also a
consequence of our later Ext-computations of principal series representations,
see Remark 4.37.

######  Proposition 3.49.

Let $\lambda\in k^{\times}$.

  1. (i)

If $O=O_{\chi}$ is square-regular, then the space
$\operatorname{Ext}^{1}_{\mathcal{H}(O)_{p^{2}=1}}(M(\chi,\lambda),M(\chi,\lambda))$
is $1$-dimensional with basis

$0\to
M(\chi,\lambda)\xrightarrow{\mathscr{Z}_{O}-\lambda}T_{s}e_{\chi}\mathcal{H}_{p^{2}=1}/(\mathscr{Z}_{O}-\lambda)^{2}\to
M(\chi,\lambda)\to 0.$

  2. (ii)

If $O=O_{\mathbf{1}}$, then the space
$\operatorname{Ext}^{1}_{\mathcal{H}(O)_{p^{2}=1}}(M(\lambda),M(\lambda))$ is
$1$-dimensional with basis given by the extension

$0\to
M(\lambda)\xrightarrow{\mathscr{Z}_{O}-\lambda}(T_{s}+1)e_{\mathbf{1}}\mathcal{H}_{p^{2}=1}/(\mathscr{Z}_{O}-\lambda)^{2}\to
M(\lambda)\to 0.$

##  4\. Genuine representation theory

In this part of the paper, we explain a complete classification result for the
smooth irreducible genuine $k$-representations of $\widetilde{G}$ as
introduced in Section 2.3. We do this by working out the strategy outlined in
the introduction. For
$\pi\in\operatorname{Mod}^{\operatorname{sm}}_{\widetilde{G},\iota}(k)$, the
$I_{1}$-invariants

(9)
$\pi^{I_{1}}\cong\operatorname{Hom}_{I_{1}}(\mathbf{1},\pi|_{I_{1}})\cong\operatorname{Hom}_{\widetilde{I}_{1}}(\mathbf{1}\boxtimes\iota,\pi|_{\widetilde{I}_{1}})\cong\operatorname{Hom}_{\widetilde{G}}(\operatorname{c-Ind}^{\widetilde{G}}_{\widetilde{I}_{1}}(\mathbf{1}\boxtimes\iota),\pi)$

carry a natural right action of the pro-$p$ Iwahori Hecke algebra
$\mathcal{H}$. We show that the adjoint pair $(\mathcal{T},\mathcal{I})$
defined by

$\displaystyle\mathcal{I}\colon\operatorname{Mod}^{\operatorname{sm}}_{\widetilde{G},\iota}(k)$
$\displaystyle\to\operatorname{Mod}_{\mathcal{H}}$ $\displaystyle\pi$
$\displaystyle\mapsto\mathcal{I}(\pi)=\pi^{I_{1}},$

and

$\displaystyle\hskip
52.63777pt\mathcal{T}\colon\operatorname{Mod}_{\mathcal{H}}$
$\displaystyle\to\operatorname{Mod}^{\operatorname{sm}}_{\widetilde{G},\iota}(k)$
$\displaystyle M$
$\displaystyle\mapsto\mathcal{T}(M)=M\otimes_{\mathcal{H}}\operatorname{c-Ind}^{\widetilde{G}}_{\widetilde{I}_{1}}(\mathbf{1}\boxtimes\iota),$

induces a bijection

$\left\\{\begin{array}[]{c}\text{Simple right}\\\
\text{$\mathcal{H}$-modules}\end{array}\right\\}\cong\left\\{\begin{array}[]{c}\text{Smooth
irreducible genuine}\\\ \text{$k$-representations of
$\widetilde{G}$}\end{array}\right\\},$

where both sides are considered up to isomorphism. The results of the previous
sections provide a classification of the left-hand side, and in particular
show that there are only two kinds of simple modules: principal and
supersingular modules. We will prove the desired bijection by considering the
two kinds of modules separately. Before we start, let us extend the list of
notations given by sections 1.2 and 2.3.

###  4.1. Preliminaries

We define subgroups of $B$ and $T$ by

$\displaystyle B_{2}=\begin{pmatrix}A&\mathbf{Q}_{p}\\\ 0&A\end{pmatrix}$
$\displaystyle\supset T_{2}$ $\displaystyle=\begin{pmatrix}A&0\\\
0&A\end{pmatrix},$ $\displaystyle U=\begin{pmatrix}1&\mathbf{Q}_{p}\\\
0&1\end{pmatrix}$ $\displaystyle
B_{S}=\begin{pmatrix}\mathbf{Q}_{p}^{\times}&\mathbf{Q}_{p}\\\
0&S\end{pmatrix}$ $\displaystyle\supset T_{S}$
$\displaystyle=\begin{pmatrix}\mathbf{Q}_{p}^{\times}&0\\\ 0&S\end{pmatrix}$
$\displaystyle P=\begin{pmatrix}\mathbf{Q}_{p}^{\times}&\mathbf{Q}_{p}\\\
0&1\end{pmatrix}$ $\displaystyle\supset P^{+}$
$\displaystyle=\begin{pmatrix}\mathbf{Z}_{p}\setminus\\{0\\}&\mathbf{Z}_{p}\\\
0&1\end{pmatrix}\supset\hskip 2.84544pt$
$\displaystyle\Gamma=\begin{pmatrix}\mathbf{Z}_{p}^{\times}&0\\\
0&1\end{pmatrix}$

We identify $k\llbracket U\cap I\rrbracket=k\llbracket X\rrbracket$, the power
series ring in the variable $X=\left(\begin{smallmatrix}1&1\\\
0&1\end{smallmatrix}\right)-1$. Following [8], we let
$F=\left(\begin{smallmatrix}p&0\\\ 0&1\end{smallmatrix}\right)$ act on $U\cap
I$ via conjugation: $F\left(\begin{smallmatrix}1&b\\\
0&1\end{smallmatrix}\right)F^{-1}=\left(\begin{smallmatrix}1&pb\\\
0&1\end{smallmatrix}\right)$ for all $b\in\mathbf{Z}_{p}$, inducing the flat
$k$-algebra endomorphism on $k\llbracket X\rrbracket$ given by $X\mapsto
X^{p}$, and we let $k\llbracket X\rrbracket[F]$ be the non-commutative
polynomial ring in $F$: $Ff(X)=F(f(X))F=f(X^{p})F$ for all $f(X)\in
k\llbracket X\rrbracket$. We also define $k\llbracket X\rrbracket[F,\Gamma]$
to be the twisted group ring so that the elements of $\Gamma$ commute with $F$
and $\gamma X=((X+1)^{\gamma}-1)\gamma$ for $\gamma\in\Gamma$.

If $M$ is a $k\llbracket X\rrbracket$-module, we denote the subspace of
$X$-torsion elements (i.e. those killed by $X$) in $M$ by $M[X]$.

The category of smooth $P^{+}$-representations is naturally a full subcategory
of the category of left $k\llbracket X\rrbracket[F,\Gamma]$-modules.

By Lemma 2.4, we have
$\widetilde{B}_{S}=B_{S}\times\mu_{2}\supset\widetilde{P}=P\times\mu_{2}$ and
$\widetilde{B}_{2}=B_{2}\times\mu_{2}$ as groups, which we will from now on
use without further mention. In particular, we will simply view $B_{S},P$ and
$B_{2}$ as subgroups of the corresponding covering.

Without further comment, we will make use of the following lemma.

######  Lemma 4.1.

Let $\pi\in\operatorname{Mod}^{\operatorname{sm}}_{\widetilde{G},\iota}(k)$
and $n\in N_{G}(T)$. Under the isomorphism (9), the action of the Hecke
operator $T_{n}$ on $\pi^{I_{1}}$ is given by

$vT_{n}=\sum_{x\in
I_{1}/(I_{1}\cap(n^{-1}I_{1}n))}x^{-1}(\tilde{n})^{-1}v,\text{ for all
$v\in\pi^{I_{1}}$.}$

###### Proof.

This follows from the definition, see also [21, Corollary 2.6]: Letting
$\varphi\in\operatorname{c-Ind}^{\widetilde{G}}_{\widetilde{I}_{1}}(\mathbf{1}\boxtimes\iota)$
be the unique function with support $\widetilde{I}_{1}$ and value $1$ at the
identity, the isomorphism
$\pi^{I_{1}}\cong\operatorname{Hom}_{\widetilde{G}}(\operatorname{c-Ind}^{\widetilde{G}}_{\widetilde{I}_{1}}(\mathbf{1}\boxtimes\iota),\pi)$
is given by $f(\varphi)\mapsfrom f$. Since
$\widetilde{I}_{1}\tilde{n}I_{1}=\sqcup_{x\in I_{1}/(I_{1}\cap
n^{-1}I_{1}n)}\widetilde{I}_{1}\tilde{n}x$, we have $T_{n}(\varphi)=\sum_{x\in
I_{1}/(I_{1}\cap n^{-1}I_{1}n)}x^{-1}(\tilde{n})^{-1}\varphi$. Applying the
$\widetilde{G}$-equivariant map $f$ gives the assertion. ∎

As in [21, Corollary 2.0.11], Lemma 3.7 has the following consequence.

######  Corollary 4.2.

Given $\pi\in\operatorname{Mod}^{\operatorname{sm}}_{\widetilde{G},\iota}(k)$,
the $I_{1}$-invariants
$\pi^{I_{1}}\in\operatorname{Mod}^{\operatorname{sm}}_{\widetilde{I},\iota}(k)$
decomposes as

$\pi^{I_{1}}=\bigoplus_{\chi\in\widehat{H}}\pi^{I_{1}}e_{\chi},$

and $\widetilde{I}$ acts on $\pi^{I_{1}}e_{\chi}$ via the character
$\chi\boxtimes\iota$.

In the study of the Hecke algebra of a non-square-regular orbit in Section
3.4, we used the $k$-algebra automorphism
$\operatorname{tw}\colon\mathcal{H}\cong\mathcal{H}$ of Lemma 3.17 induced by
twisting. We will also do this on the representation theoretic side and we now
check that this is harmless.

######  Lemma 4.3.

Let $\psi\colon\mathbf{Q}_{p}^{\times}\to k^{\times}$ be a smooth character.
Let $\operatorname{tw}_{\psi}\colon\mathcal{H}\cong\mathcal{H}$ be the
automorphism induced by twisting by $\psi\circ{\det}$. For each
$M\in\operatorname{Mod}_{\mathcal{H}}$, there is a $\widetilde{G}$-equivariant
isomorphism

$\mathcal{T}(M)\otimes\psi\circ{\det}\cong\mathcal{T}(\operatorname{tw}_{\psi}^{\ast}(M)),$

which is natural in $M$.

###### Proof.

Recall that $\operatorname{tw}_{\psi}$ is given by $h\mapsto
j(h\otimes\psi\circ\det)j^{-1}$ for $h\in\mathcal{H}$ , where
$j\colon\operatorname{c-Ind}^{\widetilde{G}}_{\widetilde{I}_{1}}(\mathbf{1}\boxtimes\iota)\otimes\psi\circ\det\cong\operatorname{c-Ind}^{\widetilde{G}}_{\widetilde{I}_{1}}(\mathbf{1}\boxtimes\iota)$
is the canonical isomorphism.

Using the adjoint pairs
$((-)\otimes\psi\circ\det,(-)\otimes\psi^{-1}\circ\det)$,
$(\mathcal{T},\mathcal{I})$ and
$(\operatorname{tw}_{\psi}^{\ast},\operatorname{tw}_{\psi,\ast})$, it
suffices, by Yoneda’s Lemma, to show that for each genuine
$\widetilde{G}$-representation $\pi$,

$\mathcal{I}(\pi\otimes\psi^{-1}\circ\det)\cong\operatorname{tw}_{\psi,\ast}\mathcal{I}(\pi)$

as $\mathcal{H}$-modules, natural in $\pi$. The left-hand side is equal to

$\displaystyle\operatorname{Hom}_{\widetilde{G}}(\operatorname{c-Ind}^{\widetilde{G}}_{\widetilde{I}_{1}}(\mathbf{1}\boxtimes\iota),\pi\otimes\psi^{-1}\circ\det)$
$\displaystyle=\operatorname{Hom}_{\widetilde{G}}(\operatorname{c-Ind}^{\widetilde{G}}_{\widetilde{I}_{1}}(\mathbf{1}\boxtimes\iota)\otimes\psi\circ\det,\pi)$
$\displaystyle\cong\operatorname{Hom}_{\widetilde{G}}(\operatorname{c-Ind}^{\widetilde{G}}_{\widetilde{I}_{1}}(\mathbf{1}\boxtimes\iota),\pi)$

where the $\mathcal{H}$-action in the second line is given by precomposition
with $h\otimes(\psi\circ\det)$ for $h\in\mathcal{H}$ and the isomorphism is
induced by $j$, so that the induced $\mathcal{H}$-action in the last line is
given by precomposition with $j(h\otimes\psi\circ\det)j^{-1}$ for
$h\in\mathcal{H}$, which proves the lemma. ∎

Speaking of twisting, the elements of $\widetilde{Z}$ define interesting
$\widetilde{G}$-equivariant isomorphisms. Besides the already introduced
smooth character
$\omega\colon\mathbf{Q}_{p}^{\times}\twoheadrightarrow[\mathbf{F}_{p}^{\times}]\cong\mathbf{F}_{p}^{\times}\subset
k^{\times}$, we denote the unramified character mapping $p$ to a fixed element
$y\in k^{\times}$ by $\mu_{y}$.

######  Lemma 4.4.

For all $\left(\left(\begin{smallmatrix}x&0\\\
0&x\end{smallmatrix}\right),\xi\right)\in\widetilde{Z}$ and
$(g,\zeta)\in\widetilde{G}$,

$\left(\left(\begin{smallmatrix}x&0\\\
0&x\end{smallmatrix}\right),\xi\right)(g,\zeta)\left(\left(\begin{smallmatrix}x&0\\\
0&x\end{smallmatrix}\right),\xi\right)^{-1}=\left(g,\zeta\left(\mu_{-1}^{v(x)}\omega^{v(x)}\mu_{\omega(x)}\right)^{\frac{p-1}{2}}(\det(g))\right).$

In particular, for
$\pi\in\operatorname{Mod}^{\operatorname{sm}}_{\widetilde{G},\iota}(k)$, the
map

$\left(\left(\begin{smallmatrix}x&0\\\
0&x\end{smallmatrix}\right),\xi\right)\colon\pi\xrightarrow{\cong}\pi\otimes\left(\mu_{-1}^{v(x)}\omega^{v(x)}\mu_{\omega(x)}\right)^{\frac{p-1}{2}}\circ{\det}$

is a $\widetilde{G}$-equivariant isomorphism.

###### Proof.

This is a straight forward computation using the definition of the cocycle (3)
defining $\widetilde{G}$. ∎

We finish the section with a quick observation.

######  Lemma 4.5.

There are no finite dimensional objects in
$\operatorname{Mod}^{\operatorname{sm}}_{\widetilde{G},\iota}(k)$.

###### Proof.

Let $\pi$ be a finite dimensional and smooth $\widetilde{G}$-representation,
which we may assume to be irreducible. Its kernel is an open subgroup of
$\widetilde{G}$. In particular, it contains a congruence subgroup $K_{m}$ for
some $m\geq 1$. Projecting to $G$ and conjugating by $F^{n}$ (resp.
$(sFs)^{n}$) for all $n\geq 0$, shows that the image of the kernel in $G$
contains $U$ as well as $sUs$ and thus
$\operatorname{SL}_{2}(\mathbf{Q}_{p})$. We conclude that each element of
$\widetilde{\operatorname{SL}_{2}(\mathbf{Q}_{p})}$ acts on $\pi$ by $\pm 1$
(hence by scalars). It follows that the restriction of $\pi$ to the subgroup
$P\cap T\cong\mathbf{Q}_{p}^{\times}$ is still irreducible and must therefore
be $1$-dimensional. However, the commutator subgroup of $\widetilde{G}$
contains $[\widetilde{T},\widetilde{T}]=\mu_{2}$ (cf. Lemma 2.6 (ii)),
contradicting genuinity. ∎

###  4.2. Weights

We remind the reader of the well-known classification of smooth irreducible
$k$-representations of $K$, so-called weights.

######  Definition 4.6.

For $r\in\mathbf{Z}_{\geq 0}$, define $\operatorname{Sym}^{r}(k^{2})$ to be
the space of homogeneous degree-$r$ polynomials in $k[x,y]$ with
$\operatorname{GL}_{2}(\mathbf{F}_{p})$-action defined by

$\left(\begin{smallmatrix}a&b\\\
c&d\end{smallmatrix}\right)P(x,y)=P(ax+cy,bx+dy),\text{ for all
$\left(\begin{smallmatrix}a&b\\\
c&d\end{smallmatrix}\right)\in\operatorname{GL}_{2}(\mathbf{F}_{p})$,
$P(x,y)\in\operatorname{Sym}^{r}(k^{2})$.}$

We will view it as representation of $K$ via inflation along the reduction
map.

######  Proposition 4.7.

The map

$\displaystyle\left\\{(a,b)\in\mathbf{Z}^{2}:\begin{array}[]{l}0\leq a-b\leq
p-1\\\ 0\leq b<p-1\end{array}\right\\}$
$\displaystyle\to\left\\{\begin{array}[]{c}\text{Smooth irreducible}\\\
\text{$k$-representations of $K$}\end{array}\right\\}$ $\displaystyle(a,b)$
$\displaystyle\mapsto\operatorname{Sym}^{a-b}(k^{2})\otimes{\det}^{b}$

is well-defined and bijective, where the right-hand side is considered up to
isomorphism.

###### Proof.

See for example [15, Proposition 14]. ∎

######  Proposition 4.8.

Given $(a,b)\in\mathbf{Z}^{2}$ with $0\leq a-b\leq p-1$ and $0\leq b<p-1$,
there is a short exact sequence of $K$-representations,

$0\to\operatorname{Sym}^{p-1-(a-b)}(k^{2})\otimes{\det}^{a}\to\operatorname{Ind}^{K}_{I}(\omega^{a}\otimes\omega^{b})\to\operatorname{Sym}^{a-b}(k^{2})\otimes{\det}^{b}\to
0.$

###### Proof.

This follows from Frobenius reciprocity and counting dimensions, see for
example [15, p. 23]. ∎

######  Definition 4.9.

Given a character $\psi\in\widehat{H}$, define the semi-simple
$K$-representation $W(\psi)$ by

$W(\psi)=\bigoplus_{\sigma:\sigma^{I_{1}}=\psi}\sigma,$

where the direct sum is over all smooth irreducible $K$-representations
$\sigma$ with $\sigma^{I_{1}}=\psi$.

######  Remark 4.10.

If $\psi\neq\psi^{s}$, then $W(\psi)$ is irreducible. Otherwise, $W(\psi)$
consists of the two constituents of $\operatorname{Ind}^{K}_{I}(\psi)$.

######  Definition 4.11.

The maximal semi-simple subrepresentation of a smooth $K$-representation $\pi$
will be denoted by $\operatorname{soc}_{K}(\pi)$ and is called the $K$-socle
of $\pi$.

###  4.3. Principal series representations

######  Definition 4.12.

A genuine principal series representation of $\widetilde{G}$ is a
representation of $\widetilde{G}$ isomorphic to the smooth parabolic induction
$\operatorname{Ind}^{\widetilde{G}}_{\widetilde{B}}(\tau)$ of some smooth
irreducible genuine $k$-representation $\tau$ of $\widetilde{T}$ viewed as a
representation of $\widetilde{B}$ via inflation.

In order to understand these objects, an investigation of the
$\widetilde{T}$-representations we are inducing is necessary, which is the
content of the next section.

####  4.3.1. Smooth irreducible genuine $\widetilde{T}$-representations

The classification result we are about to prove closely follows [18, Chapter
5], which deals with smooth genuine complex representations in a more general
setting, see especially Corollary 5.2 in loc. cit. It says that a smooth
irreducible genuine $\widetilde{T}$-representation is uniquely determined by
its central character. Before stating the result, we need two lemmas.

######  Lemma 4.13.

Any smooth irreducible genuine $k$-representation of $\widetilde{T}$ is finite
dimensional.

###### Proof.

The proof is inspired by [17, Lemma 2.4]. By Lemma 2.6,
$Z(\widetilde{T})=T^{\operatorname{sq}}\times\mu_{2}$. Filtering the inclusion
$Z(\widetilde{T})\subset\widetilde{T}$ by subgroups of successive index two,
and observing that, if $G_{1}\subset G_{2}$ is an inclusion of groups of index
two, then the restriction of an irreducible $G_{2}$-representation to $G_{1}$
is semi-simple of length at most two, the lemma reduces to the statement that
a smooth irreducible genuine $Z(\widetilde{T})$-representation, or
equivalently a smooth irreducible $T^{\operatorname{sq}}$-representation, is
finite dimensional. Modulo the pro-$p$ part, $T^{\operatorname{sq}}$ is a
finitely generated abelian group, so Hilbert’s Nullstellensatz implies that a
smooth irreducible $k$-representation of it is in fact $1$-dimensional. ∎

######  Lemma 4.14.

Let $i,j\in\mathbf{Z}$. For all $\left(\begin{smallmatrix}a&b\\\
0&d\end{smallmatrix}\right)\in B_{2}$,

$\left(\left(\begin{smallmatrix}p^{i}&0\\\
0&p^{j}\end{smallmatrix}\right),1\right)\left(\begin{smallmatrix}a&b\\\
0&d\end{smallmatrix}\right)\left(\left(\begin{smallmatrix}p^{i}&0\\\
0&p^{j}\end{smallmatrix}\right),1\right)^{-1}=\left(\left(\begin{smallmatrix}a&p^{i-j}b\\\
0&d\end{smallmatrix}\right),\omega(d)^{i\frac{p-1}{2}}\omega(a)^{j\frac{p-1}{2}}\right).$

###### Proof.

By Example 2.3, $\left(\left(\begin{smallmatrix}p^{i}&0\\\
0&p^{j}\end{smallmatrix}\right),1\right)^{-1}=\left(\left(\begin{smallmatrix}p^{-i}&0\\\
0&p^{-j}\end{smallmatrix}\right),(-1)^{ij\frac{p-1}{2}}\right)$. The lemma is
now a simple computation based on the definition of the cocycle (3) defining
$\widetilde{G}$. ∎

######  Notation 4.15.

For $i,j\in\mathbf{Z}$ and a smooth representation $\pi$ of a subgroup of
$\widetilde{B}$, denote by

$\pi[j,i]=\pi\otimes(\omega^{j\frac{p-1}{2}}\otimes\omega^{i\frac{p-1}{2}})$

the twist of $\pi$ by the character
$\omega^{j\frac{p-1}{2}}\otimes\omega^{i\frac{p-1}{2}}\colon\widetilde{B}\twoheadrightarrow
B\twoheadrightarrow T\to\mu_{2}\subset k^{\times}$ restricted to the subgroup
of consideration.

######  Definition-Proposition 4.16.

The map

$\displaystyle\left\\{\begin{array}[]{c}\text{Smooth characters}\\\
T^{\operatorname{sq}}\to k^{\times}\end{array}\right\\}$
$\displaystyle\xrightarrow{\cong}\left\\{\begin{array}[]{c}\text{Smooth
irreducible genuine}\\\ \text{$k$-representations of
$\widetilde{T}$}\end{array}\right\\}$ $\displaystyle\chi\colon
T^{\operatorname{sq}}\to k^{\times}$
$\displaystyle\mapsto\tau_{\chi}:=\operatorname{Ind}^{\widetilde{T}}_{\widetilde{T}_{2}}\left(\chi^{\prime}\boxtimes\iota\right)$

is a well-defined bijection, where $\chi^{\prime}$ is any extension of $\chi$
to $T_{2}$ and the right-hand side is considered up to isomorphism.

###### Proof.

Note that, since $k$ is divisible and so injective in the category of abelian
groups, any (smooth) character of $T^{\operatorname{sq}}$ extends to a
(smooth) character of $T_{2}$. Let now $\chi$ be a smooth character on
$T_{2}$. By Lemma 4.14 we have

(10)
$\operatorname{Ind}^{\widetilde{T}}_{\widetilde{T}_{2}}(\chi\boxtimes\iota)|_{T_{2}}=\bigoplus_{i,j\in\mathbf{Z}/2\mathbf{Z}}\chi[j,i],$

which is the direct sum over all distinct characters extending
$\chi|_{T^{\operatorname{sq}}}$. This proves irreducibility as well as the
independence of the chosen extension.

Conversely, by Lemma 4.13, any smooth irreducible genuine $k$-representation
$\tau$ of $\widetilde{T}$ is finite dimensional and thus admits a central
character, i.e. there exists a smooth character $\chi\colon T^{2}\to
k^{\times}$ with $\chi\subset\tau|_{T^{\operatorname{sq}}}$. Using Frobenius
reciprocity and that
$\operatorname{Ind}^{T_{2}}_{T^{\operatorname{sq}}}(\chi)$ is the direct sum
of the characters extending $\chi$ (since $T_{2}/T^{\operatorname{sq}}$ is
finite of order prime to $p$), we deduce that there exists a character
$\chi^{\prime}$ of $T_{2}$ extending $\chi$ with
$\chi^{\prime}\subset\tau|_{T_{2}}$. Using Frobenius reciprocity once again,
we obtain a non-trivial $\widetilde{T}$-equivariant morphism
$\operatorname{Ind}^{\widetilde{T}}_{\widetilde{T}_{2}}(\chi^{\prime}\boxtimes\iota)\to\tau$,
which has to be an isomorphism as both sides are irreducible. ∎

######  Remark 4.17.

  1. (i)

In the previous proposition, the subgroup $\widetilde{T}_{2}$ can be replaced
by any other maximal abelian subgroup of $\widetilde{T}$, for example by
$\widetilde{T}_{S}$.

  2. (ii)

A genuine principal series of $\widetilde{G}$ is of the form
$\operatorname{Ind}^{\widetilde{G}}_{\widetilde{B}_{2}}(\chi\boxtimes\iota)$
for a smooth character $\chi\colon T_{2}\to k^{\times}$ viewed as a character
of $B_{2}$ via inflation.

######  Definition 4.18.

For a smooth character $\chi\colon T_{2}\to k^{\times}$, define the standard
basis $\\{f_{i,j}\\}_{i,j=0,1}$ of $\tau_{\chi}$ by letting $f_{i,j}$ be the
unique function with

$\operatorname{supp}(f_{i,j})=\widetilde{T}_{2}\left(\left(\begin{smallmatrix}p^{i}&0\\\
0&p^{j}\end{smallmatrix}\right),1\right)\text{ and
}f_{i,j}\left(\left(\left(\begin{smallmatrix}p^{i}&0\\\
0&p^{j}\end{smallmatrix}\right),1\right)\right)=1.$

######  Lemma 4.19.

For all $i,j=0,1$, the following hold true.

  1. (i)

$f_{i,j}$ is an $H$-eigenvector with eigencharacter $\chi|_{H}[j,i]$;

  2. (ii)

$\left(\left(\begin{smallmatrix}1&0\\\
0&p^{-1}\end{smallmatrix}\right),1\right)f_{i,0}=(-1)^{i\frac{p-1}{2}}f_{i,1}$;

  3. (iii)

$\left(\left(\begin{smallmatrix}1&0\\\
0&p^{-1}\end{smallmatrix}\right),1\right)f_{i,1}=(-1)^{i\frac{p-1}{2}}\chi\left(\left(\begin{smallmatrix}1&0\\\
0&p^{-2}\end{smallmatrix}\right)\right)f_{i,0}$;

  4. (iv)

$\left(\left(\begin{smallmatrix}p^{-1}&0\\\
0&1\end{smallmatrix}\right),1\right)f_{0,j}=f_{1,j}$;

  5. (v)

$\left(\left(\begin{smallmatrix}p^{-1}&0\\\
0&1\end{smallmatrix}\right),1\right)f_{1,j}=\chi\left(\left(\begin{smallmatrix}p^{-2}&0\\\
0&1\end{smallmatrix}\right)\right)f_{0,j}$.

###### Proof.

Part (i) follows from Lemma 4.14. Part (ii) follows from the equation
$\left(\left(\begin{smallmatrix}p^{i}&0\\\
0&p\end{smallmatrix}\right)\right)\left(\left(\begin{smallmatrix}1&0\\\
0&p^{-1}\end{smallmatrix}\right),1\right)=\left(\left(\begin{smallmatrix}p^{i}&0\\\
0&1\end{smallmatrix}\right),(-1)^{i\frac{p-1}{2}}\right)$, see Example 2.3.
Part (iii) follows from (ii) since the element
$\left(\begin{smallmatrix}1&0\\\ 0&p^{-2}\end{smallmatrix}\right)$ is central
and $\tau_{\chi}$ has central character $\chi$. The remaining two parts are
proved analogously. ∎

####  4.3.2. Irreducibility of principal series representations

Let us fix a smooth character $\chi\colon T_{2}\to k^{\times}$ and denote by

$\tilde{\pi}(\chi)=\operatorname{Ind}^{\widetilde{G}}_{\widetilde{B}}(\tau_{\chi})$

the corresponding principal series. The goal of this section it to prove:

######  Proposition 4.20.

The genuine principal series $\tilde{\pi}(\chi)$ is irreducible as a
representation of $\widetilde{G}$. Moreover,
$\tilde{\pi}(\chi)\cong\tilde{\pi}(\psi)$ for another smooth character
$\psi\colon T_{2}\to k^{\times}$ if and only if
$\chi|_{T^{\operatorname{sq}}}=\psi|_{T^{\operatorname{sq}}}$.

The if-direction of the proposition trivially holds true as it does already
for $\tau_{\chi}$. In particular, to prove irreducibility, we may replace
$\chi$ by one of its twists $\chi[i,j]$. In the non-square-regular case, we
may and do therefore assume that $\chi|_{H}=\chi|_{H}^{s}$.

######  Definition 4.21.

For $i,j=0,1$, define elements
$F_{i,j}^{0},F_{i,j}^{1}\in\tilde{\pi}(\chi)^{I_{1}}$ uniquely determined by

$\operatorname{supp}(F_{i,j}^{0})=\widetilde{B}I_{1}\text{ and
}F_{i,j}^{0}\left(\left(\begin{smallmatrix}1&0\\\
0&1\end{smallmatrix}\right),1\right)=f_{i,j}$

and

$\operatorname{supp}(F_{i,j}^{1})=\widetilde{B}\tilde{s}I_{1}\text{ and
}F_{i,j}^{1}(\tilde{s})=f_{i,j}.$

The smooth character $\chi|_{T^{\operatorname{sq}}}$ being trivial on the
pro-$p$ group $T\cap I_{1}$ implies that these elements are well-defined.
Since $\\{f_{i,j}\\}_{i,j=0,1}$ is a $k$-basis of $\tau_{\chi}$, the
decomposition
$\widetilde{G}=\widetilde{B}I_{1}\sqcup\widetilde{B}\tilde{s}I_{1}$
immediately implies that $\\{F_{i,j}^{0},F_{i,j}^{1}\\}_{i,j=0,1}$ is a
$k$-basis of $\tilde{\pi}(\chi)^{I_{1}}$.

######  Lemma 4.22.

For all $i,j=0,1$, the following hold true.

  1. (i)

$F_{i,j}^{0}e_{\chi|_{H}[j,i]}=F_{i,j}^{0}$ and
$F_{i,j}^{1}e_{\chi|_{H}[j,i]^{s}}=F_{i,j}^{1}$;

  2. (ii)

$F_{i,0}^{0}T_{\Pi}=(-1)^{i\frac{p-1}{2}}F_{i,1}^{1}$ and
$F_{i,1}^{0}T_{\Pi}=(-1)^{i\frac{p-1}{2}}\chi\left(\left(\begin{smallmatrix}1&0\\\
0&p^{-2}\end{smallmatrix}\right)\right)F_{i,0}^{1}$;

  3. (iii)

$F^{1}_{0,j}T_{\Pi}=F_{1,j}^{0}$ and
$F^{1}_{1,j}T_{\Pi}=\chi\left(\left(\begin{smallmatrix}p^{-2}&0\\\
0&1\end{smallmatrix}\right)\right)F_{0,j}^{0}$;

  4. (iv)

$F^{0}_{i,j}T_{s}=F_{i,j}^{1}$.

###### Proof.

Parts (i) to (iii) follow from Lemma 4.19. For part (iv) recall that in the
non-square-regular case we agreed to choose $\chi$ so that
$\chi|_{H}=\chi|_{H}^{s}$. Part (i) together with Lemma 3.7 (iv) implies that
$F_{i,j}^{0}T_{s}$ is a linear combination of $F_{i,j}^{1}$ and $F_{j,i}^{0}$
(in the square-regular case, even just a scalar multiple of $F_{i,j}^{1}$). It
therefore suffices to evaluate at $\tilde{s}$ and $1$. Now,
$I_{1}sI_{1}=\sqcup_{\lambda\in\mathbf{F}_{p}}I_{1}s\left(\begin{smallmatrix}1&[\lambda]\\\
0&1\end{smallmatrix}\right)$, and so

$F_{i,j}^{0}T_{s}=\sum_{\lambda\in\mathbf{F}_{p}}\left(\begin{smallmatrix}1&[\lambda]\\\
0&1\end{smallmatrix}\right)\tilde{s}F_{i,j}^{0}.$

Since $\operatorname{supp}(F_{i,j}^{0})=\widetilde{B}I_{1}$, each summand
evaluated at $1$ gives $0$, while only the summand corresponding to
$\lambda=0$ contributes with value $1$ if one evaluates at $\tilde{s}$. This
concludes the proof. ∎

###### Proof of Proposition 4.20.

It is enough to prove that $\tilde{\pi}(\chi)^{I_{1}}$ is a simple right
$\mathcal{H}$-module. Indeed, this follows from $\tilde{\pi}(\chi)^{I_{1}}$
generating $\tilde{\pi}(\chi)$ as a $\widetilde{G}$-representation, which in
turn is a consequence of the fact222E.g. see [30, p. 24 (a)]; a shorter proof:
the Iwahori decomposition for the congruence subgroup $K_{m}$, for $m\geq 1$,
implies that $BK_{m}=BI_{1}\left(\begin{smallmatrix}p^{m-1}&0\\\
0&1\end{smallmatrix}\right)$. that each compact open subgroup of
$\widetilde{B}\setminus\widetilde{G}\cong B\setminus G$ is a finite disjoint
union of subsets of the form $B\setminus BI_{1}g$ for suitable $g\in G$. In
order to prove the irreducibility of $\tilde{\pi}(\chi)^{I_{1}}$ as an
$\mathcal{H}$-module, first note that
$\tilde{\pi}(\chi)^{I_{1}}e_{O_{\chi|_{H}}}=\tilde{\pi}(\chi)^{I_{1}}$, where
$e_{O_{\chi|_{H}}}\in\mathcal{H}$ is the idempotent cutting out
$\mathcal{H}(O_{\chi|_{H}})$. We distinguish two cases.

If $\chi|_{H}$ is square-regular, it suffices to prove that
$\tilde{\pi}(\chi)^{I_{1}}(e_{\chi|_{H}}+e_{\chi|_{H}[1,0]})$ is simple as a
module over $\mathcal{H}(\chi|_{H}\oplus\chi|_{H}[1,0])$, see Corollary 3.19.
By Lemma 4.22, we have

$\tilde{\pi}(\chi)^{I_{1}}(e_{\chi|_{H}}+e_{\chi|_{H}[1,0]})=kF_{0,0}^{0}+kF_{0,1}^{0},$

and the Hecke operators $(T_{(s\Pi)^{2}},T_{(\Pi s)^{2}},T_{\Pi^{4}})$ act on
this space via the scalars
$\left(\chi\left(\left(\begin{smallmatrix}p^{-2}&0\\\
0&1\end{smallmatrix}\right)\right),0,\chi\left(\left(\begin{smallmatrix}p^{-2}&0\\\
0&p^{-2}\end{smallmatrix}\right)\right)\right)$. The uniqueness in Proposition
3.28 now gives

(11)
$\tilde{\pi}(\chi)^{I_{1}}\cong\operatorname{PS}\left(\chi|_{H},\left(\chi\left(\left(\begin{smallmatrix}p^{-2}&0\\\
0&1\end{smallmatrix}\right)\right),0,\chi\left(\left(\begin{smallmatrix}p^{-2}&0\\\
0&p^{-2}\end{smallmatrix}\right)\right)\right)\right).$

Assume now that $\chi|_{H}=\chi|_{H}^{s}$. By twisting, we may even assume
that $\chi|_{H}=\mathbf{1}$. By Lemma 4.22, we have

$\tilde{\pi}(\chi)^{I_{1}}e_{\mathbf{1}}=kF_{0,0}^{0}+kF_{0,0}^{1}$

and the center $k[\mathfrak{Z},Z^{\pm 1}]$ of $\mathcal{H}(\mathbf{1})$
(Corollary 3.34) acts via the character corresponding to the $k$-rational
point $\left(\chi\left(\left(\begin{smallmatrix}p^{-2}&0\\\
0&1\end{smallmatrix}\right)\right),\chi\left(\left(\begin{smallmatrix}p^{-2}&0\\\
0&p^{-2}\end{smallmatrix}\right)\right)\right)$. Thus, by Proposition 3.36 and
the remark thereafter,

(12)
$\tilde{\pi}(\chi)^{I_{1}}=\operatorname{PS}\left(\chi\left(\left(\begin{smallmatrix}p^{-2}&0\\\
0&1\end{smallmatrix}\right)\right),\chi\left(\left(\begin{smallmatrix}p^{-2}&0\\\
0&p^{-2}\end{smallmatrix}\right)\right)\right).$

This concludes the proof of the irreducibility of $\tilde{\pi}(\chi)$.
Concerning the intertwinings: in the square-regular case, the isomorphism (11)
together with Remark 3.30 implies that
$\tilde{\pi}(\chi)\cong\tilde{\pi}(\psi)$ for another smooth character $\psi$
on $T_{2}$ only if $\psi|_{H}=\chi|_{H}[i,j]$ for some
$i,j\in\mathbf{Z}/2\mathbf{Z}$,
$\chi\left(\left(\begin{smallmatrix}p^{-2}&0\\\
0&1\end{smallmatrix}\right)\right)=\psi\left(\left(\begin{smallmatrix}p^{-2}&0\\\
0&1\end{smallmatrix}\right)\right)$ and
$\chi\left(\left(\begin{smallmatrix}p^{-2}&0\\\
0&p^{-2}\end{smallmatrix}\right)\right)=\psi\left(\left(\begin{smallmatrix}p^{-2}&0\\\
0&p^{-2}\end{smallmatrix}\right)\right)$, i.e. only if
$\chi|_{T^{\operatorname{sq}}}=\psi|_{T^{\operatorname{sq}}}$.

In the non-square-regular case, an isomorphism
$\tilde{\pi}(\chi)^{I_{1}}\cong\tilde{\pi}(\psi)^{I_{1}}$ of
$\mathcal{H}$-modules implies that $O_{\chi|_{H}}=O_{\psi|_{H}}$, i.e.
$\psi|_{H}=\chi|_{H}[i,j]$ for some $i,j\in\mathbf{Z}/2\mathbf{Z}$. From the
isomorphism (12) we can read off the values of $\chi$ and $\psi$ at
$\left(\begin{smallmatrix}p^{-2}&0\\\ 0&1\end{smallmatrix}\right)$ and
$\left(\begin{smallmatrix}p^{-2}&0\\\ 0&p^{-2}\end{smallmatrix}\right)$, so in
total $\chi|_{T^{\operatorname{sq}}}=\psi|_{T^{\operatorname{sq}}}$. ∎

####  4.3.3. From principal Hecke modules to principal series representations

The proof of the previous proposition shows that the assignment
$\pi\mapsto\mathcal{I}(\pi)$ defines a bijection between the isomorphism
classes of principal $\mathcal{H}$-modules and the isomorphism classes of
(irreducible) genuine principal series representations of $\widetilde{G}$. For
showing that the inverse is given by the functor $\mathcal{T}$, we need to
check that the canonical map

(13) $\mathcal{T}(\mathcal{I}(\tilde{\pi}(\chi)))\to\tilde{\pi}(\chi)$

is an isomorphism. To ease notation, put $M=\mathcal{I}(\tilde{\pi}(\chi))$.
After an unramified twist, we may and do also assume that
$\chi|_{H}=\mathbf{1}$ in the non-square-regular case, cf. Lemma 4.3. The map
(13) is non-trivial (by adjunction), so the source must be non-zero. In
particular, the unit $M\to\mathcal{I}\mathcal{T}(M)$ is an injection, via
which we will view $M=\tilde{\pi}(\chi)^{I_{1}}$ as a subspace of
$\mathcal{T}(M)$.

Denote by

$\tilde{\pi}(\chi)^{\operatorname{bc}}=\left\\{f\in\tilde{\pi}(\chi):\operatorname{supp}(f)\subset\widetilde{B}\tilde{s}\widetilde{B}\right\\}$

the $\widetilde{B}$-stable subspace consisting of those functions whose
support is contained in the big cell $\widetilde{B}\tilde{s}\widetilde{B}$. We
thus have an exact sequence of $\widetilde{B}$-representations

$0\to\tilde{\pi}(\chi)^{\operatorname{bc}}\to\tilde{\pi}(\chi)\to\tau_{\chi}\to
0$

induced by evaluation at the identity. In order to prove the canonical map
(13) to be an isomorphism, we will construct an analogue of this sequence for
the object $\mathcal{T}(M)$ and show that the canonical map induces an
isomorphism on the outer terms of the short exact sequences.

For a smooth $k$-representation $\tau$ of $\widetilde{T}$, denote by