$\mathscr{C}^{\infty}_{c}(\mathbf{Q}_{p},\tau)$ the space of locally constant
compactly supported $\tau$-valued functions on $\mathbf{Q}_{p}$. We endow it
with a $\widetilde{B}$-action via the formula

(14) $\left[\left(\left(\begin{smallmatrix}a&b\\\
0&d\end{smallmatrix}\right),\zeta\right).L\right](x)=\tau\left(\left(\left(\begin{smallmatrix}d&0\\\
0&a\end{smallmatrix}\right),\zeta(a,d^{-1})\right)\right)L\left(\frac{b+xd}{a}\right),$

for all $\left(\begin{smallmatrix}a&b\\\ 0&d\end{smallmatrix}\right)\in B$,
$\zeta\in\mu_{2}$ and $L\in\mathscr{C}^{\infty}_{c}(\mathbf{Q}_{p},\tau)$.

If $\tau=\mathbf{1}$ is the trivial representation of $T$ viewed as a
$\widetilde{T}$-representation via inflation, we simply write
$\mathscr{C}^{\infty}_{c}(\mathbf{Q}_{p},k)$.

######  Lemma 4.23.

We have the chain of $\widetilde{B}$-equivariant isomorphisms

$\tilde{\pi}(\chi)^{\operatorname{bc}}\cong\mathscr{C}^{\infty}_{c}(\mathbf{Q}_{p},\tau_{\chi})\cong\mathscr{C}^{\infty}_{c}(\mathbf{Q}_{p},k)\otimes\tau_{\chi^{s}},$

where the first isomorphism is given by $f\mapsto\left[x\mapsto
f\left(\left(\begin{smallmatrix}0&1\\\
1&x\end{smallmatrix}\right),1\right)\right]$ and the second one is given by
$[x\mapsto L(x)v]\mapsfrom L\otimes v$ implicitly identifying
$\tau_{\chi^{s}}=\tau_{\chi}^{\tilde{s}}$.

###### Proof.

The equality $\widetilde{B}\tilde{s}\widetilde{B}=\widetilde{B}\tilde{s}U$
implies the first isomorphism. Here, note that the function $L$ attached to
$f$ is locally constant since so is $f$; it has compact support since $f$ is
locally constant around the identity element of $\widetilde{G}$. The last
isomorphism is clear. One checks that these are indeed
$\widetilde{B}$-equivariant. ∎

Define
$M_{0}=\bigoplus_{i,j=0,1}kF_{i,j}^{1}\subset\tilde{\pi}(\chi)^{\operatorname{bc}}\cap
M$.

######  Corollary 4.24.

The $\widetilde{B}$-representation $\tilde{\pi}(\chi)^{\operatorname{bc}}$ is
generated by $M_{0}$.

###### Proof.

Under the isomorphism
$\tilde{\pi}(\chi)^{\operatorname{bc}}\cong\mathscr{C}^{\infty}_{c}(\mathbf{Q}_{p},k)\otimes\tau_{\chi^{s}}$
of the previous lemma, the function $F_{i,j}^{1}$ corresponds to the element
$\mathbf{1}_{\mathbf{Z}_{p}}\otimes f_{i,j}$, where
$\mathbf{1}_{\mathbf{Z}_{p}}$ is the indicator function of $\mathbf{Z}_{p}$.
It suffices to show that $\mathscr{C}^{\infty}_{c}(\mathbf{Q}_{p},k)$ is
generated as a $B_{2}$-representation by this indicator function, which is
evident from the definition of the action (14). ∎

Recall that $X=\left(\begin{smallmatrix}1&1\\\
0&1\end{smallmatrix}\right)-1\in k\llbracket U\cap I\rrbracket\cong
k\llbracket X\rrbracket$ and $F=\left(\begin{smallmatrix}p&0\\\
0&1\end{smallmatrix}\right)$.

######  Definition 4.25.

Define $\mathcal{T}(M)^{\operatorname{bc}}\subset\mathcal{T}(M)$ to be the
$\widetilde{B}$-subrepresentation generated by $M_{0}$.

We also let $\mathcal{T}(M)^{\operatorname{bc}}_{0}=k\llbracket
X\rrbracket[F]M_{0}\subset\mathcal{T}(M)^{\operatorname{bc}}$ be the
$P^{+}$-subrepresentation generated by $M_{0}$; this is
$\widetilde{Z}$-stable.

We will prove that
$\mathcal{T}(M)^{\operatorname{bc}}=\mathcal{T}(M)^{\operatorname{bc}}_{0}[F^{-1}]$
is irreducible as a $\widetilde{B}$-representation. In order to do this, we
first show that $\mathcal{T}(M)^{\operatorname{bc}}_{0}$ is irreducible as a
$P^{+}\widetilde{Z}$-representation, for which we determine its $X$-torsion.

######  Lemma 4.26.

For all $i,j=0,1$, up to a non-zero scalar,

$\sum_{\lambda\in\mathbf{F}_{p}}(X+1)^{[\lambda]}FF_{i,j}^{1}=F_{i,j+1}^{1}.$

###### Proof.

On the subspace $\tilde{\pi}(\chi)^{I_{1}}$, the operator
$\sum_{\lambda\in\mathbf{F}_{p}}(X+1)^{[\lambda]}F$ is simply the Hecke
operator $T_{\Pi^{-1}s}=T_{\Pi^{-1}}T_{s}$, so the assertion follows from
Lemma 4.22. ∎

######  Corollary 4.27.

For all $i,j=0,1$, we have the following equalities of $X$-torsion subspaces
inside the space $\mathcal{T}(M)^{\operatorname{bc}}_{0}$.

  1. (i)

$\left(k\llbracket X\rrbracket[F^{2}]F_{i,j}^{1}\right)[X]=kF_{i,j}^{1}$;

  2. (ii)

$\left(k\llbracket X\rrbracket[F]F_{i,j}^{1}\right)[X]=kF_{i,j}^{1}\oplus
kF_{i,j+1}^{1}$;

  3. (iii)

$\mathcal{T}(M)^{\operatorname{bc}}_{0}[X]=M_{0}$.

###### Proof.

Applying Lemma 4.26 twice yields, up to a non-zero scalar,

(15)
$F_{i,j}^{1}=\sum_{\lambda,\mu\in\mathbf{F}_{p}}(X+1)^{[\lambda]+p[\mu]}F^{2}F_{i,j}^{1}.$

Iterating this $n$-times shows that $kF_{i,j}^{1}\subset\left(k\llbracket
X\rrbracket F^{2n}F_{i,j}^{1}\right)[X]\subset k\llbracket X\rrbracket
F^{2n}F_{i,j}^{1}$. The latter is a module over $k\llbracket
X\rrbracket/F^{2n}XF^{-2n}=k\llbracket X\rrbracket/X^{p^{2n}}$ generated by a
single element and thus has $1$-dimensional $X$-torsion, so the first
inclusion has to be an equality. Passing to the colimit over all
$n\in\mathbf{N}$, which is filtered by (15), finishes the proof of part (i).

For part (ii), note that Lemma 4.26 implies that $k\llbracket
X\rrbracket[F]F_{i,j}^{1}=k\llbracket
X\rrbracket[F^{2}]F_{i,j}^{1}+k\llbracket X\rrbracket[F^{2}]F_{i,j+1}^{1}$.
Applying part (i) now gives (ii).

Part (iii) can be deduced from (ii) just like (ii) itself was deduced from
(i). ∎

######  Proposition 4.28.

The $\widetilde{B}$-subrepresentation
$\mathcal{T}(M)^{\operatorname{bc}}\subset\mathcal{T}(M)$ is irreducible.

###### Proof.

We first claim that $\mathcal{T}(M)^{\operatorname{bc}}_{0}$ is irreducible as
a $P^{+}\widetilde{Z}$-representation. Indeed, by the previous corollary,
$\mathcal{T}(M)^{\operatorname{bc}}_{0}[X]=M_{0}$ and $H$ acts on the basis
elements $\\{F_{i,j}^{1}\\}_{i,j=0,1}$ via pairwise distinct characters, cf.
Lemma 4.22 (i). Thus, by smoothness, any non-zero
$P^{+}\widetilde{Z}$-subrepresentation of
$\mathcal{T}(M)^{\operatorname{bc}}_{0}$ must contain some $F_{i,j}^{1}$.
Using Lemma 4.26 and the action of $\tilde{\Pi}^{2}$, it must then contain
$M_{0}$ and must therefore be equal to
$\mathcal{T}(M)^{\operatorname{bc}}_{0}$. The proposition now follows from the
claim by writing
$\mathcal{T}(M)^{\operatorname{bc}}=\mathcal{T}(M)^{\operatorname{bc}}_{0}[F^{-1}]=\varinjlim_{n\geq
1}F^{-n}\mathcal{T}(M)^{\operatorname{bc}}_{0}$. ∎

######  Lemma 4.29.

The $\widetilde{B}$-representation
$\mathcal{T}(M)/\mathcal{T}(M)^{\operatorname{bc}}$ is equal to the image of
$M_{0}^{\perp}:=\bigoplus_{i,j=0,1}kF_{i,j}^{0}\subset
M\hookrightarrow\mathcal{T}(M)$ under the projection map.

###### Proof.

Let $Q$ denote the image of $M_{0}^{\perp}$ in the quotient
$\mathcal{T}(M)/\mathcal{T}(M)^{\operatorname{bc}}$. By definition,
$\mathcal{T}(M)$ is generated as a $\widetilde{G}$-representation by $M$.
Writing $\widetilde{G}=\widetilde{B}I_{1}\sqcup\widetilde{B}I_{1}\tilde{\Pi}$
and noting that $\tilde{\Pi}M=MT_{\Pi}^{-1}=M$, it follows from the inclusion
$M\subset\mathcal{T}(M)^{I_{1}}$ (in fact, $I_{1}$-stability of $M$ would be
enough) that $\mathcal{T}(M)$ is generated as a $\widetilde{B}$-representation
by $M=M_{0}\oplus M_{0}^{\perp}$. As
$M_{0}\subset\mathcal{T}(M)^{\operatorname{bc}}$, we deduce that
$\mathcal{T}(M)/\mathcal{T}(M)^{\operatorname{bc}}$ is generated as a
$\widetilde{B}$-representation by $Q$. It therefore suffices to prove that $Q$
is $\widetilde{B}$-stable.

It is certainly stable under the action of $B\cap I$ and
$\tilde{\Pi}^{2\mathbf{Z}}$, so it remains to show that $Q$ is $F^{\pm
1}$-stable. The subspace $M_{0}^{\perp}$ is stable under the Hecke operator
$T_{s\Pi}=T_{s}T_{\Pi}$ by Lemma 4.22. The operator $T_{s\Pi}$ is given by
$\sum_{\lambda\in\mathbf{F}_{p}}\left(\left(\begin{smallmatrix}1&0\\\
p[\lambda]&1\end{smallmatrix}\right),1\right)F^{-1}$, and we claim that for
all $m\in M_{0}^{\perp}$, the element

$F^{-1}m-mT_{s}T_{\Pi}=\sum_{\lambda\in\mathbf{F}_{p}^{\times}}\left(\begin{smallmatrix}1&0\\\
p[\lambda]&1\end{smallmatrix}\right)F^{-1}m\in\mathcal{T}(M)^{\operatorname{bc}}$

lies in $\mathcal{T}(M)^{\operatorname{bc}}$. Now for all
$\lambda\in\mathbf{F}_{p}^{\times}$, $\left(\begin{smallmatrix}1&0\\\
p[\lambda]&1\end{smallmatrix}\right)F^{-1}\in Bs(I_{1}\cap U)=B\Pi(I_{1}\cap
U)$. Since $M_{0}^{\perp}$ is $(I_{1}\cap U)$-stable and
$\mathcal{T}(M)^{\operatorname{bc}}$ is $\widetilde{B}$-stable, it is enough
to observe that
$\tilde{\Pi}M_{0}^{\perp}=M_{0}\subset\mathcal{T}(M)^{\operatorname{bc}}$,
which follows from Lemma 4.22. ∎

######  Corollary 4.30.

The canonical map
$\mathcal{T}(M)=\mathcal{T}(\mathcal{I}(\tilde{\pi}(\chi)))\to\tilde{\pi}(\chi)$
is an isomorphism.

###### Proof.

By construction, the canonical map of interest fits into a map of short exact
sequences

(16)
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{T}(M)^{\operatorname{bc}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{T}(M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{T}(M)/\mathcal{T}(M)^{\operatorname{bc}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\tilde{\pi}(\chi)^{\operatorname{bc}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\tilde{\pi}(\chi)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\tau_{\chi}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$

By Corollary 4.24 and Proposition 4.28, the left vertical arrow is an
isomorphism. The right vertical map is an isomorphism by the irreducibility of
its $4$-dimensional target and by Lemma 4.29. In conclusion, also the middle
arrow must be an isomorphism. ∎

We summarize the results of this section in the following theorem.

######  Theorem 4.31.

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

$\left\\{\begin{array}[]{c}\text{Principal right}\\\
\text{$\mathcal{H}$-modules}\end{array}\right\\}\cong\left\\{\begin{array}[]{c}\text{Genuine
principal series}\\\ \text{representations of
$\widetilde{G}$}\end{array}\right\\},$

where both sides are considered up to isomorphism.

######  Corollary 4.32.

Let $\pi$ be a genuine principal series of $\widetilde{G}$. Then each element
in $\pi/X\pi$ is killed by a polynomial in $k[F]$.

###### Proof.

Write $\pi=\mathcal{T}(M)$ for a principal $\mathcal{H}$-module $M$. It is
enough to show that the statement is true for the kernel and cokernel in the
top row of diagram (16). For the cokernel this is clear as it is finite
dimensional. For the kernel, write
$\mathcal{T}(M)^{\operatorname{bc}}=\mathcal{T}(M)^{\operatorname{bc}}_{0}[F^{-1}]$
and use Lemma 4.26, which implies that
$\mathcal{T}(M)^{\operatorname{bc}}_{0}=X\mathcal{T}(M)^{\operatorname{bc}}_{0}$.
∎

####  4.3.4. Extensions of principal series representations

In this section, we compute the extensions of two genuine principal series
representations. Quite surprisingly, this can be deduced from the already
existing results for principal series representations of the group
$G=\operatorname{GL}_{2}(\mathbf{Q}_{p})$. In what follows, the Ext-groups
appearing will be computed in the category of smooth $k$-representations of
$\widetilde{G}$-, resp. $G$.

We will use the same notation for a genuine principal series as in the last
section, i.e. given a smooth character $\chi\colon T_{2}\to k^{\times}$, we
put
$\tilde{\pi}(\chi)=\operatorname{Ind}^{\widetilde{G}}_{\widetilde{B}_{2}}(\chi\boxtimes\iota)$,
which up to isomorphism only depends on the restriction of $\chi$ to
$T^{\operatorname{sq}}$.

Given a smooth character $\chi\colon T\to k^{\times}$, we put

$\pi(\chi)=\operatorname{Ind}^{G}_{B}(\chi)\text{ and
}\tilde{\pi}(\chi)=\tilde{\pi}(\chi|_{T_{2}})=\operatorname{Ind}^{\widetilde{G}}_{\widetilde{B}_{2}}(\chi|_{T_{2}}\boxtimes\iota).$

######  Proposition 4.33 ([10, Proposition 4.3.15]).

Let $\chi,\chi^{\prime}\colon T\to k^{\times}$ be smooth characters.

  1. (i)

The space

$\operatorname{Ext}^{1}_{G}(\pi(\chi),\pi(\chi^{\prime}))=0$

vanishes unless $\chi^{\prime}=\chi$ or
$\chi^{\prime}=\chi^{s}\otimes(\omega\otimes\omega^{-1})$.

  2. (ii)

The map

$\operatorname{Ext}^{1}_{T}(\chi,\chi)\xrightarrow{\cong}\operatorname{Ext}^{1}_{G}(\pi(\chi),\pi(\chi)),$

induced by the exact functor $\operatorname{Ind}^{G}_{B}(-)$, is an
isomorphism.

  3. (iii)

If $\chi^{s}\otimes(\omega\otimes\omega^{-1})\neq\chi$, then

$\dim_{k}\operatorname{Ext}^{1}_{G}(\pi(\chi),\pi(\chi^{s}\otimes(\omega\otimes\omega^{-1})))=1.$

As in the genuine case, we let $\pi(\chi)^{\operatorname{bc}}\subset\pi(\chi)$
denote the $B$-invariant subspace consisting of those functions whose support
is contained in $BsB$ and we identify it $B$-equivariantly with
$\mathscr{C}^{\infty}_{c}(\mathbf{Q}_{p},k)\otimes\chi^{s}$. The key
ingredient for this section is the $\widetilde{B}$-equivariant isomorphism

(17)
$\tilde{\pi}(\chi)^{\operatorname{bc}}\cong\operatorname{Ind}^{\widetilde{B}}_{\widetilde{B}_{2}}(\pi(\chi)^{\operatorname{bc}}|_{B_{2}}\boxtimes\iota)$

resulting from Lemma 4.23.

Evaluation at the identity yields a short exact sequence

(18) $0\to\pi(\chi)^{\operatorname{bc}}\to\pi(\chi)\to\chi\to 0$

of $B$-representations.

######  Lemma 4.34.

Let $\chi,\chi^{\prime}\colon T\to k^{\times}$ be smooth characters.

  * (i)

Assume that $\chi^{\prime}\neq\chi$. The short exact sequence (18) induces a
natural isomorphism

$\operatorname{Ext}^{1}_{G}(\pi(\chi),\pi(\chi^{\prime}))\cong\operatorname{Ext}^{1}_{B}(\pi(\chi)^{\operatorname{bc}},\chi^{\prime}).$

  * (ii)

Assume that $\chi^{\prime}=\chi$ or
$\chi^{\prime}\neq\chi^{s}\otimes(\omega\otimes\omega^{-1})$. The canonical
map

$\operatorname{Ext}^{1}_{B}(\pi(\chi)^{\operatorname{bc}},\chi^{\prime})\hookrightarrow\operatorname{Ext}^{2}_{B}(\chi,\chi^{\prime}),$

induced by the sequence (18), is injective.

###### Proof.

By Shapiro’s Lemma we have
$\operatorname{Ext}^{1}_{G}(\pi(\chi),\pi(\chi^{\prime}))\cong\operatorname{Ext}^{1}_{G}(\pi(\chi)|_{B},\chi^{\prime}))$.
The short exact sequence (18) induces an exact sequence

(19)
$0\to\operatorname{Ext}^{1}_{B}(\chi,\chi^{\prime})\to\operatorname{Ext}^{1}_{B}(\pi(\chi)|_{B},\chi^{\prime})\to\operatorname{Ext}^{1}_{B}(\pi(\chi)^{\operatorname{bc}},\chi^{\prime})\to\operatorname{Ext}^{2}_{B}(\chi,\chi^{\prime}).$

The zero on the left-hand side holds because
$\operatorname{Hom}_{B}(\pi(\chi)^{\operatorname{bc}},\chi^{\prime})=0$ (as
both are irreducible, but non-isomorphic). By [10, Lemma 4.3.7], we have
$\operatorname{Ext}^{i}_{B}(\chi,\chi^{\prime})=\operatorname{Ext}^{i}_{T}(\chi,\chi^{\prime})$
for $0\leq i\leq 2$.

Assume that $\chi^{\prime}\neq\chi$. By [10, Lemma 4.3.10], we then have
$\operatorname{Ext}^{i}_{T}(\chi,\chi^{\prime})=0$ for all $i\geq 0$, which
gives the assertion in (i).

Assume now that $\chi^{\prime}=\chi$ or
$\chi^{\prime}\neq\chi^{s}\otimes(\omega\otimes\omega^{-1})$. By Proposition
4.33, the map
$\operatorname{Ext}^{1}_{B}(\chi,\chi^{\prime})\to\operatorname{Ext}^{1}_{B}(\pi(\chi)|_{B},\chi^{\prime})$
is surjective proving (ii). Indeed, if $\chi^{\prime}=\chi$, then it is even
isomorphism; if
$\chi\neq\chi^{\prime}\neq\chi^{s}\otimes(\omega\otimes\omega^{-1})$, then the
target is trivial. ∎

Recall from Notation 4.15 that for $i,j\in\mathbf{Z}/2\mathbf{Z}$ and a smooth
representation of a subgroup of $\widetilde{B}$, we put
$\pi[i,j]=\pi\otimes(\omega^{i\frac{p-1}{2}}\otimes\omega^{j\frac{p-1}{2}})$.
For the purpose of this section, we also define

$\pi(i,j)=\pi\otimes(\mu_{-1}^{i}\otimes\mu_{-1}^{j}),$

where, as introduced earlier in Secion 4.1, $\mu_{-1}$ is the smooth
unramified character mapping $p$ to $-1$.

######  Lemma 4.35.

Let $\chi\colon T^{\operatorname{sq}}\to k^{\times}$ be a smooth character
such that $\chi=\chi^{s}\otimes(\omega\otimes\omega^{-1})$ (as characters on
$T^{\operatorname{sq}}$). Then there exists a smooth character $\chi_{e}\colon
T\to k^{\times}$ such that $\chi_{e}|_{T^{\operatorname{sq}}}=\chi$ and
$\chi_{e}=\chi_{e}^{s}\otimes(\omega\otimes\omega^{-1})$ (as characters on
$T$).

###### Proof.

Let $\psi\colon T\to k^{\times}$ be any smooth character extending $\chi$,
i.e. $\psi|_{T^{\operatorname{sq}}}=\chi$. The assumption on $\chi$ means that
$\psi[i,j](a,b)=\psi^{s}\otimes(\omega\otimes\omega^{-1})$ for some (unique)
quadruple $(i,j,a,b)$ of elements in $\mathbf{Z}/2\mathbf{Z}$. One quickly
verifies that necessarily $i=j$ and $a=b$ as elements of
$\mathbf{Z}/2\mathbf{Z}$. Then $\chi_{e}:=\psi[0,j](0,b)$ does the job. ∎

######  Proposition 4.36.

Let $\chi,\chi^{\prime}\colon T\to k^{\times}$ be smooth characters.

  1. (i)

The space

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

vanishes unless
$\chi^{\prime}|_{T^{\operatorname{sq}}}=\chi|_{T^{\operatorname{sq}}}$ or
$\chi^{\prime}|_{T^{\operatorname{sq}}}=\chi^{s}\otimes(\omega\otimes\omega^{-1})|_{T^{\operatorname{sq}}}$.

  2. (ii)

The map

$\operatorname{Ext}^{1}_{T_{2}}(\chi|_{T_{2}},\chi|_{T_{2}})\xrightarrow{\cong}\operatorname{Ext}^{1}_{\widetilde{G}}(\tilde{\pi}(\chi),\tilde{\pi}(\chi)),$

induced by the exact functor
$\operatorname{Ind}^{\widetilde{G}}_{\widetilde{B}_{2}}(-\boxtimes\iota)$, is
an isomorphism.

  3. (iii)

If
$\chi^{s}\otimes(\omega\otimes\omega^{-1})|_{T^{\operatorname{sq}}}\neq\chi|_{T^{\operatorname{sq}}}$,
then

$\dim_{k}\operatorname{Ext}^{1}_{\widetilde{G}}(\tilde{\pi}(\chi),\tilde{\pi}(\chi^{s}\otimes(\omega\otimes\omega^{-1})))=1.$

###### Proof.

Assume first that
$\chi^{\prime}|_{T^{\operatorname{sq}}}\neq\chi|_{T^{\operatorname{sq}}}$,
i.e. $\chi^{\prime}\neq\chi[i,j](a,b)$ for any
$i,j,a,b\in\mathbf{Z}/2\mathbf{Z}$. Writing
$\tau_{\chi}|_{B_{2}}=\oplus_{i,j\in\mathbf{Z}/2\mathbf{Z}}\chi[i,j]$, we
deduce that

$\operatorname{Ext}^{1}_{\widetilde{G}}(\tilde{\pi}(\chi),\tilde{\pi}(\chi^{\prime}))\cong\operatorname{Ext}^{1}_{B_{2}}(\tilde{\pi}(\chi)^{\operatorname{bc}}|_{B_{2}},\chi^{\prime})$

just as in the proof of Lemma 4.34 (i). Restricting the isomorphism (17) to
$B_{2}$ and then moving the resultings twists (obtained from Mackey’s
decomposition and Lemma 4.14) over to $\chi^{\prime}$, Frobenius reciprocity
implies that the above Ext-space is isomorphic to

$\bigoplus_{i,j\in\mathbf{Z}/2\mathbf{Z}}\operatorname{Ext}^{1}_{B}(\pi(\chi)^{\operatorname{bc}},\operatorname{Ind}^{B}_{B_{2}}(\chi^{\prime}[i,j]))=\bigoplus_{i,j,a,b\in\mathbf{Z}/2\mathbf{Z}}\operatorname{Ext}^{1}_{B}(\pi(\chi)^{\operatorname{bc}},\chi^{\prime}[i,j](a,b)).$

By our assumption and Lemma 4.34 (i), this is isomorphic to

$\bigoplus_{i,j,a,b\in\mathbf{Z}/2\mathbf{Z}}\operatorname{Ext}^{1}_{G}(\pi(\chi),\pi(\chi^{\prime}[i,j](a,b))),$

which by Proposition 4.33 is non-zero (and then $1$-dimensional) if and only
if $\chi^{\prime}[i,j](a,b)=\chi^{s}\otimes(\omega\otimes\omega^{-1})$ for
some (necessarily unique) quadruple $(i,j,a,b)$ of elements in
$\mathbf{Z}/2\mathbf{Z}$, or equivalently if and only if
$\chi^{\prime}|_{T^{\operatorname{sq}}}=\chi^{s}\otimes(\omega\otimes\omega^{-1})|_{T^{\operatorname{sq}}}$.
This proves (i) and (iii).

To prove (ii), note that the statement only depends on the restriction of
$\chi$ to $T^{\operatorname{sq}}$. If
$\chi=\chi^{s}\otimes(\omega\otimes\omega^{-1})$ as characters of
$T^{\operatorname{sq}}$, Lemma 4.35 allows us to assume that this equality
holds on the full group $T$, which we will do. Using the evident analogue of
the long exact sequence (19), we reduce to showing the injectivity of the map

(20)
$\operatorname{Ext}^{1}_{B_{2}}(\tilde{\pi}(\chi)^{\operatorname{bc}}|_{B_{2}},\chi)\to\operatorname{Ext}^{2}_{B_{2}}(\tau_{\chi}|_{T_{2}},\chi).$

As before, the left-hand side is equal to
$\oplus_{i,j,a,b\in\mathbf{Z}/2\mathbf{Z}}\operatorname{Ext}^{1}_{B}(\pi(\chi)^{\operatorname{bc}},\chi[i,j](a,b))$.
The right-hand side is similarly equal to
$\oplus_{i,j,a,b\in\mathbf{Z}/2\mathbf{Z}}\operatorname{Ext}^{2}_{B}(\chi,\chi[i,j](a,b))$
and, by [10, Lemma 4.3.7, Lemma 4.3.10], this equals
$\operatorname{Ext}^{2}_{B}(\chi,\chi)$. By Proposition 4.33, the extension
space
$\operatorname{Ext}^{1}_{B}(\pi(\chi)^{\operatorname{bc}},\chi[i,j](a,b))$ is
trivial unless333Here we are using that we chose $\chi$ so that
$\chi=\chi^{s}\otimes(\omega\otimes\omega^{-1})$ if this happens to be true on
$T^{\operatorname{sq}}$. $i=j=a=b=0$ in $\mathbf{Z}/2\mathbf{Z}$. Thus, the
injectivity of (20) is equivalent to the injectivity of the connecting
homomorphism

$\operatorname{Ext}^{1}_{B}(\pi(\chi)^{\operatorname{bc}},\chi)\to\operatorname{Ext}^{2}_{B}(\chi,\chi).$

This is the content of Lemma 4.34 (ii). ∎

######  Remark 4.37.

Twisting induces an isomorphism
$\operatorname{Ext}^{1}_{T_{2}}(\chi|_{T_{2}},\chi|_{T_{2}})\cong\operatorname{Ext}^{1}_{T_{2}}(\mathbf{1},\mathbf{1})$,
which in turn can be identified with
$\operatorname{Hom}^{\operatorname{cts}}(T_{2},\overline{\mathbf{F}}_{p})$.
The latter contains the subspace of unramified homomorphisms, i.e. those
trivial on $T_{2}\cap K$, and the resulting inclusion induces a commutative
diagram

$\textstyle{\operatorname{Hom}^{\operatorname{cts}}(T_{2},k)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{\operatorname{Ext}^{1}_{T_{2}}(\chi|_{T_{2}},\chi|_{T_{2}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{\operatorname{Ext}^{1}_{\widetilde{G}}(\tilde{\pi}(\chi),\tilde{\pi}(\chi))}$$\textstyle{\operatorname{Hom}^{\operatorname{unr}}(T_{2},k)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{\operatorname{Ext}^{1}_{\mathcal{H}}(\mathcal{I}(\tilde{\pi}(\chi)),\mathcal{I}(\tilde{\pi}(\chi))).\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{T}}$

Looking only at those continuous homomorphisms killing the scalar matrix
$p^{2}$ yields the isomorphism
$\operatorname{Hom}_{p^{2}=0}^{\operatorname{unr}}(T_{2},k)\cong\operatorname{Ext}^{1}_{\mathcal{H}_{p^{2}=1}}(\mathcal{I}(\tilde{\pi}(\chi)),\mathcal{I}(\tilde{\pi}(\chi)))$.
In particular, the latter Ext-space is $1$-dimensional, cf. Proposition 3.49.

######  Remark 4.38.

Consider the diagram induced by evaluation at the identity,

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\tilde{\pi}(\chi)^{\operatorname{bc}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\tilde{\pi}(\chi)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\tau_{\chi}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi(\chi)^{\operatorname{bc}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi(\chi)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\chi\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0,}$

where the left vertical $B_{2}$-equivariant map is given by sending
$f\in\pi(\chi)^{\operatorname{bc}}$ to
$\tilde{f}\in\tilde{\pi}(\chi)^{\operatorname{bc}}$ uniquely determined by
$\tilde{f}\left(\left(\left(\begin{smallmatrix}0&1\\\
1&x\end{smallmatrix}\right),1\right)\right)=f\left(\left(\begin{smallmatrix}0&1\\\
1&x\end{smallmatrix}\right)\right)f_{0,0}$ for all $x\in\mathbf{Q}_{p}$, and
the right vertical $B_{2}$-equivariant map is given by mapping $1\in\chi$ to
the standard basis vector $f_{0,0}\in\tau_{\chi}$. By the isomorphism (17) and
the very definition of $\tau_{\chi}$, the $\widetilde{B}$-equivariant maps
obtained via Frobenius reciprocity from the outer vertical arrows are
isomorphisms. It is tempting to believe that this is also true for
$\tilde{\pi}(\chi)$ in a compatible way, i.e. such that the top sequence in
the above diagram is obtained from the lower one by applying the functor
$\operatorname{Ind}^{\widetilde{B}}_{\widetilde{B}_{2}}((-)\boxtimes\iota)$.
However, one may check that this diagram cannot be filled in, which also seems
to explain why the application of Lemma 4.35 in the proof of the previous
proposition is crucial.

###  4.4. Supersingular representations

######  Definition 4.39.

A genuine supersingular representation of $\widetilde{G}$ is a smooth
irreducible genuine $k$-representation of $\widetilde{G}$, which is not
isomorphic to a subquotient of a principal series.

######  Remark 4.40.

By Proposition 4.20, every genuine principal series is irreducible, so
removing “a subquotient of” gives an equivalent definition.

The goal of this section is to prove that the adjoint pair
$(\mathcal{T},\mathcal{I})$ induces a bijection between the set of isomorphism
classes of supersingular $\mathcal{H}$-modules and the set of isomorphism
classes of admissible genuine supersingular representations. We will later
show that any smooth irreducible representation of $\widetilde{G}$ is already
admissible, see Theorem 5.17. For showing that $\mathcal{T}(M)$ is
irreducible, for a supersingular Hecke module $M$, we will use a weight-
cycling argument, that has already been used for supersingular representations
of $G$, see [22, §4.1] or [15, §10].

####  4.4.1. Basic lemma

In a previous draft of this work we worked out the weight-cycling in the
specific case of interest (in dimension four). However, in a lecture [14] of
Florian Herzig at the Spring School on the mod-$p$ Langlands Correspondence,
which took place in Essen (online) in 2021, he formulated it more generally
(in dimension $n\geq 1$) and called the resulting statement “Basic Lemma”. In
this subsection, we simply reproduce his result.

Setup. Let $\pi$ be a smooth $k$-representation of the mirabolic subgroup $P$.
Suppose that we are given $k$-linearly independent $\Gamma$-eigenvectors
$v_{1},\ldots,v_{n}\in\pi[X]$ killed by $X$, satisfying

(21) $X^{s_{i}}F(v_{i})=c_{i}v_{i+1}\text{ for some $s_{i}\in\mathbf{Z}_{\geq
0}$ and $c_{i}\in k^{\times}$}$

for all $1\leq i\leq n$. Here and in what follows we understand the subscript
$i$ to be the unique element in $\\{1,\ldots,n\\}$ congruent to $i$ modulo
$n$. We define

$M=\bigoplus_{i=1}^{n}kv_{i}$

and let

$\pi_{M}=k\llbracket X\rrbracket[F]M\subset\pi.$

be the $k\llbracket X\rrbracket[F]$-submodule generated by $M$ in $\pi$.
Finally, for $1\leq i\leq n$, we put

$\pi_{i}=k\llbracket X\rrbracket[F^{n}]v_{i}\subset\pi_{M}.$

We wish to prove the following result.

######  Lemma 4.41 (Basic Lemma).

  1. (i)

For each $1\leq i\leq n$, the subspace $\pi_{i}\subset\pi$ is $\Gamma$-stable.

  2. (ii)

The subspace $\pi_{M}\subset\pi$ is $P^{+}$-stable.

  3. (iii)

For each $1\leq i\leq n$, $\pi_{i}[X]=kv_{i}$.

  4. (iv)

As $k\llbracket X\rrbracket[F^{n},\Gamma]$-modules,
$\pi_{M}=\bigoplus_{i=1}^{n}\pi_{i}$.

  5. (v)

If $s_{i}=0$ for all $1\leq i\leq n$, then $\pi_{i}=kv_{i}$.

  6. (vi)

If $s_{j}>0$ for some $1\leq j\leq n$, then
$\pi_{i}\cong\mathscr{C}^{\infty}(U\cap I,k)$ as $U\cap I$-representations and
the inclusion $kv_{i}\subset\pi_{i}$ is an injective envelope in the category
of smooth $U\cap I$-representations, for all $1\leq i\leq n$.

  7. (vii)

Assume that the following condition is satisfied: If for some $1\leq i\neq
j\leq n$, the vectors $v_{i},v_{j}$ have the same $\Gamma$-eigencharacter,
then $s_{i}\neq s_{j}$. Then $\pi_{M}$ is an irreducible
$P^{+}$-representation.

This is a more detailed version of the Basic Lemma in Herzig’s lecture alluded
to above. The last part on the irreducibility was for instance not contained
in Herzig’s version. However, the additional details are essentially just a
byproduct of the proof of the original version. Before jumping into the proof,
we make some preparations.

######  Definition 4.42.

For $1\leq i\leq n$, define

$e(i)=p^{n-1}s_{i}+p^{n-2}s_{i+1}+\ldots+p^{2}s_{i+n-3}+ps_{i+n-2}+s_{i+n-1}$

and, if $m\in\mathbf{Z}_{\geq 1}$, put

$e(i)_{m}=\sum_{j=0}^{m-1}p^{jn}e(i).$

We let

$c=\prod_{i=1}^{n}c_{i}$

be the product of the scalars appearing in (21).

######  Lemma 4.43.

Let $1\leq i\leq n$. The following hold true:

  1. (i)

$X^{p^{n-1}s_{i}+p^{n-2}s_{i+1}+\ldots+p^{2}s_{i+n-3}+ps_{i+n-2}}F^{n}(v_{i})=\left(\prod_{j\neq
i-1}c_{j}\right)F(v_{i-1})$;

  2. (ii)

for all $m\in\mathbf{Z}_{\geq 1}$, $X^{e(i)_{m}}F^{nm}(v_{i})=c^{m}v_{i}$.

###### Proof.

Part (i) follows from the relation $FX=X^{p}F$ and inductively applying the
equations (21) starting with $X^{s_{i}}F(v_{i})=c_{i}v_{i+1}$. The case of
$m=1$ in part (ii) now follows from this by applying $X^{s_{i-1}}$ to (i) and
using (21). Inductively applying the resulting relation gives the assertion
for an arbitrary $m\in\mathbf{Z}_{\geq 1}$. ∎

###### Proof of the Basic Lemma 4.41.

Part (i) and (ii) follow from the relations $\gamma F=F\gamma$ and $\gamma
X=((X+1)^{\gamma}-1)\gamma$ for all $\gamma\in\Gamma$. By Lemma 4.43 (i),
$F\pi_{i-1}\subset\pi_{i}$ and thus $\pi_{M}=\sum_{i}\pi_{i}$. In particular,
part (iv) will follow from part (iii) as the vectors $v_{1},\ldots,v_{n}$ are
linearly independent.

Fix $1\leq i\leq n$. For $m\in\mathbf{Z}_{\geq 1}$, let $\pi_{i,m}=k\llbracket
X\rrbracket F^{nm}(v_{i})$ be the $k\llbracket X\rrbracket$-submodule of
$\pi_{i}$ generated $F^{nm}(v_{i})$. Applying $F^{nm}$ to Lemma 4.43 (ii)
(with $m=1$), shows that $kv_{i}\subset\pi_{i,m}\subset\pi_{i,m+1}$. By Lemma
4.43 (ii), the $k\llbracket X\rrbracket$-linear map

(22) $\displaystyle k\llbracket X\rrbracket/(X^{e(i)_{m}+1})$
$\displaystyle\xrightarrow{\cong}\pi_{i,m}$ $\displaystyle f(X)$
$\displaystyle\mapsto c^{-m}f(X)F^{nm}(v_{i})$

is a well-defined isomorphism. Comparing $X$-torsion shows that $\pi_{i,m}[X]$
is $1$-dimensional, necessarily equal to $kv_{i}$. This proves (iii). The
isomorphisms (22) fit into a commutative diagram

(23) $\begin{gathered}\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
40.3697pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\pi_{i,m}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
10.051pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@hook{1}}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
122.12177pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
122.12177pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\pi_{i,m+1}}$}}}}}}}{\hbox{\kern-40.3697pt\raise-39.3372pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{k\llbracket
X\rrbracket/(X^{e(i)_{m}+1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.75pt\hbox{$\scriptstyle{\cong}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 0.0pt\raise-5.93555pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
40.3697pt\raise-39.3372pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@hook{1}}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
54.18146pt\raise-33.19554pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-3.14166pt\hbox{$\scriptstyle{\cdot X^{p^{nm}e(i)}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
88.3697pt\raise-39.3372pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
88.3697pt\raise-39.3372pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{k\llbracket
X\rrbracket/(X^{e(i+1)_{m+1}+1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces.}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
135.75053pt\raise-19.66861pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.75pt\hbox{$\scriptstyle{\cong}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
135.75053pt\raise-5.93555pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\ignorespaces\end{gathered}$

For $d\in\mathbf{Z}_{\geq 1}$, we identify the $k$-linear dual
$\left(k\llbracket X\rrbracket/(X^{d+1})\right)^{\vee}$ with $k\llbracket
X\rrbracket/(X^{d+1})$ via the $k\llbracket X\rrbracket$-linear map
corresponding to the projection $\operatorname{pr}_{d}\in\left(k\llbracket
X\rrbracket/(X^{d+1})\right)^{\vee}$ defined by
$\operatorname{pr}_{d}(X^{j})=\delta_{jd}$, the Kronecker delta, for $0\leq
j\leq d$. Under these identifications, the Pontryagin dual of the lower
horizontal map is simply the canonical projection. Passing to the inverse
limit, we obtain

$\pi_{i}^{\vee}=(\varinjlim_{m\geq 1}\pi_{i,m})^{\vee}=\varprojlim_{m\geq
1}\pi_{i,m}^{\vee}\cong\varprojlim_{m\geq 1}k\llbracket
X\rrbracket/(X^{e(i)_{m}+1}).$

If $s_{j}=0$ for all $1\leq j\leq n$, then $e(i)_{m}=0$, so we obtain
$\pi_{i}=kv_{i}$, which proves (v). If however $s_{j}$ is non-zero for some
$1\leq j\leq n$, then $e(i)_{m}$ tends to infinity as $m\to\infty$. In this
case we therefore obtain $\pi_{i}^{\vee}\cong k\llbracket X\rrbracket$ as
$k\llbracket X\rrbracket$-modules, or equivalently
$\pi_{i}\cong\mathscr{C}^{\infty}(U\cap I,k)$ as smooth $U\cap
I$-representations. In particular, $\pi_{i}$ is injective in the category of
smooth $U\cap I$-modules, and since $\pi_{i}[X]=kv_{i}$, we deduce that the
inclusion $kv_{i}\subset\pi_{i}$ is an injective envelope proving part (vi).

Assume now that the condition formulated in (vii) holds. Let
$\tau\subset\pi_{M}$ be a non-zero $P^{+}$-subrepresentation (or equivalently
a $k\llbracket X\rrbracket[F,\Gamma]$-submodule). Since
$\pi_{M}[X]=\oplus_{i=1}^{n}kv_{i}$ is a $\Gamma$-stable subspace, there
exists a subset $\\{i_{1},\ldots,i_{r}\\}\subset\\{1,\ldots,n\\}$, for some
$r\leq n$, such that

$\sum_{j=1}^{r}a_{i_{j}}v_{i_{j}}\in\tau\text{ for some $a_{i_{j}}\in
k^{\times}$,}$

where the $v_{i_{j}}$, for $1\leq j\leq r$, have the same
$\Gamma$-eigencharacter. After reordering we may assume that $s_{i_{j+1}}\geq
s_{i_{j}}$ for all $1\leq j\leq r-1$. Applying $X^{1+s_{i_{r-1}}}F(-)$ will
kill $v_{i_{j}}$ for $1\leq j\leq r-1$. Here we are using equation (21) and
the fact the $Xv_{j}=0$ for all $1\leq j\leq n$. Thus,
$a_{i_{r}}X^{1+s_{i_{r-1}}}F(v_{i_{r}})\in\tau$. Since $s_{i_{r-1}}\neq
s_{i_{r}}$ (and so $s_{i_{r-1}}<s_{i_{r}}$) by our assumption, applying
$X^{s_{i_{r}}-s_{i_{r-1}}-1}$ and using equation (21) once again, we see that
$v_{i_{r}+1}\in\tau$. Another application of the same equation now shows that
$\tau$ contains $v_{j}$ for all $1\leq j\leq n$ and must therefore be equal to
$\pi_{M}$. ∎

######  Corollary 4.44.

Assume that the following condition is satisfied: If for some $1\leq i\neq
j\leq n$, the vectors $v_{i},v_{j}$ have the same $\Gamma$-eigencharacter,
then $s_{i}\neq s_{j}$. Then the subspace

$\pi_{M}[F^{-1}]=\varinjlim_{m\geq 1}F^{-m}\pi_{M}\subset\pi$

is $P$-stable and -irreducible.

###### Proof.

By the Basic Lemma 4.41, the subspace $\pi_{M}\subset\pi$ is $P^{+}$-stable
and -irreducible. The equality $P=F^{-\mathbf{N}}P^{+}$ now implies the
assertion. ∎

####  4.4.2. From supersingular Hecke modules to supersingular representations

Let $M$ be a supersingular $\mathcal{H}(O)$-module for some orbit $O$.
Applying the Basic Lemma, we now prove that the $\mathcal{T}(M)$ is
irreducible as a $P$-representation (hence as a
$\widetilde{G}$-representation), and that the canonical map
$M\to\mathcal{I}(\mathcal{T}(M))$ is an isomorphism. Using Lemma 4.3, we may
and do assume that, in the non-square-regular case, $\mathbf{1}\in O$. We fix
a character $\chi\in O$, with $\chi=\mathbf{1}$ in the non-square-regular
case. By Definition 3.29, resp. 3.38, we can write

$M=\begin{cases}\operatorname{SS}(\chi,z)\text{ if $\chi$ is square-
regular}\\\ \operatorname{SS}(\lambda,z)\text{ if
$\chi=\mathbf{1}$}\end{cases}$

for some $z\in k^{\times}$ and $\lambda\in\\{0,-1\\}$. By Proposition 3.27,
resp. 3.36, we have

(24) $M=\bigoplus_{\psi\in\chi^{\tilde{\Pi}^{\mathbf{Z}}}}Me_{\psi}\text{ and
}\dim_{\overline{\mathbf{F}}_{p}}Me_{\psi}=1,$

where $\chi^{\tilde{\Pi}^{\mathbf{Z}}}$ denotes the $\Pi$-orbit of $\chi$ as
introduced in Definition 3.11.

It is explained in Remark 5.14 below that $\mathcal{T}(M)$ is non-zero, so
that the canonical map $M\hookrightarrow\mathcal{I}\mathcal{T}(M)$ is an
injection, via which we will view elements of $M$ as elements of
$\mathcal{T}(M)$.

######  Lemma 4.45.

For each $\psi\in\chi^{\tilde{\Pi}^{\mathbf{Z}}}$, the Hecke operator $T_{s}$
acts on $Me_{\psi}$ via a scalar. More precisely:

  1. (i)

If $O$ is square-regular, then $T_{s}=0$ on $M$.

  2. (ii)

If $O$ is non-square-regular, then $T_{s}$ acts on $Me_{\psi}$ via $\lambda$
(resp. $0$) if $\psi=\psi^{s}$ (resp. $\psi\neq\psi^{s}$).

###### Proof.

The decomposition (24) together with the relation
$e_{\psi}T_{s}=T_{s}e_{\psi^{s}}$ of Lemma 3.7 (iv) already implies (i). For
(ii), note that by applying the element $T_{\Pi}^{2}$ and using Lemma 3.9 (iv)
and (v), it suffices to treat the characters
$\psi=\mathbf{1},\mathbf{1}[0,1]$. Using part (v) once again and applying the
invertible element $T_{\Pi}$ from the right, the vanishing of
$Me_{\mathbf{1}[0,1]}T_{s}=Me_{\mathbf{1}}T_{\Pi}T_{s}$ is equivalent to the
vanishing of $Me_{\mathbf{1}}T_{\Pi}T_{s}T_{\Pi}$, which by parts (vi) and
(ix) of the cited lemma means that $T_{\Pi s\Pi}$, or equivalently
$S_{0,2}\in\mathcal{H}(\mathbf{1})$, kills $Me_{\mathbf{1}}$. This holds true
by the very definition of $M$, which also says that $T_{s}$, or equivalently
$S_{0,0}\in\mathcal{H}(\mathbf{1})$, acts on $Me_{\mathbf{1}}$ by the scalar
$\lambda$. ∎

Recall from Definition 4.9 that to a character $\psi\in\widehat{H}$ we
attached the semi-simple smooth $K$-representation $W(\psi)$, that is
irreducible if and only if $\psi\neq\psi^{s}$.

######  Corollary 4.46.

For each $\psi\in\chi^{\tilde{\Pi}^{\mathbf{Z}}}$, the
$\widetilde{K}$-subrepresentation
$\sigma(\psi):=\left\langle\widetilde{K}Me_{\psi}\right\rangle\subset\mathcal{T}(M)$
generated by $Me_{\psi}$ is irreducible. More precisely:

  1. (i)

If $O$ is square-regular, then $\sigma(\psi)=W(\psi)\boxtimes\iota$.

  2. (ii)

If $O$ is non-square-regular and $\psi\neq\psi^{s}$, then
$\sigma(\psi)=W(\psi)\boxtimes\iota$, while

$\sigma(\mathbf{1})=\begin{cases}\mathbf{1}\boxtimes\iota\text{ if
$\lambda=0$}\\\ \operatorname{Sym}^{p-1}(k^{2})\boxtimes\iota\text{ if
$\lambda=-1$}\end{cases}\text{ and
\phantom{m}}\sigma(\mathbf{1}[1,1])=\sigma(\mathbf{1})\otimes{\det}^{\frac{p-1}{2}}.$

###### Proof.

Frobenius reciprocity yields a $\widetilde{K}$-equivariant surjection
$\operatorname{Ind}^{\widetilde{K}}_{\widetilde{I}}(\psi\boxtimes\iota)\twoheadrightarrow\sigma(\psi)$
mapping the $I_{1}$-invariant function $\varphi$ with
$\operatorname{supp}(\varphi)=\widetilde{I}$ and $\varphi(1)=1$ to a fixed
element in $Me_{\psi}$. By Lemma 4.45, the operator $T_{s}$ acts on
$Me_{\psi}$ by a scalar. However, it does not act by a scalar on $\varphi$,
and so the surjection cannot be an isomorphism, i.e. the kernel is non-
trivial.

Assume that $\psi\neq\psi^{s}$. By Proposition 4.8, we have a short exact
sequence

$0\to
W(\psi^{s})\boxtimes\iota\to\operatorname{Ind}^{\widetilde{K}}_{\widetilde{I}}(\psi\boxtimes\iota)\to
W(\psi)\boxtimes\iota\to 0$

of $\widetilde{K}$-representation with the outer terms being irreducible.
Since this sequence is non-split, we deduce that $\sigma(\psi)\cong
W(\psi)\boxtimes\iota$.

Assume now that $\psi=\psi^{s}$. By Lemma 4.4, the element $\tilde{\Pi}^{2}$
defines a $\widetilde{K}$-equivariant isomorphism
$\mathcal{T}(M)\cong\mathcal{T}(M)\otimes\omega^{\frac{p-1}{2}}\circ{\det}$
restricting to an isomorphism
$\sigma(\mathbf{1}[1,1])\cong\sigma(\mathbf{1})\otimes{\det}^{\frac{p-1}{2}}$,
so it is enough to treat the case $\psi=\mathbf{1}$. By Proposition 4.8 once
again, we have
$\operatorname{Ind}^{\widetilde{K}}_{\widetilde{I}}(\mathbf{1}\boxtimes\iota)=\mathbf{1}\boxtimes\iota\oplus\operatorname{Sym}^{p-1}(k^{2})\boxtimes\iota$.
The $I_{1}$-invariants of the second summand are $1$-dimensional generated by
$\varphi T_{s}$. Hence, the kernel of
$\operatorname{Ind}^{\widetilde{K}}_{\widetilde{I}}(\mathbf{1}\boxtimes\iota)\twoheadrightarrow\sigma(\psi)$
meets $\operatorname{Sym}^{p-1}(k^{2})\boxtimes\iota$ trivially if and only if
$\varphi T_{s}$ does not lie in the kernel, i.e. if and only if $T_{s}$ does
not kill $Me_{\mathbf{1}}$, which by Lemma 4.45 just means that $\lambda=-1$.
∎

By Proposition 4.7, we may write, for each
$\psi\in\chi^{\tilde{\Pi}^{\mathbf{Z}}}$,

$\sigma(\psi)=\operatorname{Sym}^{r_{\psi}}(k^{2})\otimes{\det}^{b_{\psi}}$

for a unique $0\leq r_{\psi}\leq p-1$ and $0\leq b_{\psi}\leq p-2$.

######  Remark 4.47.

By Lemma 4.4, the element $\tilde{\Pi}^{2}$ defines an isomorphism
$\sigma(\psi[1,1])\cong\sigma(\psi)\otimes\omega^{\frac{p-1}{2}}\circ{\det}$,
i.e. $r_{\psi[1,1]}=r_{\psi}$ and $b_{\psi[1,1]}\equiv
b_{\psi}+\frac{p-1}{2}\bmod p-1$. It follows more generally from Lemma 4.46
that a single weight $\sigma(\psi)$ determines all the other ones, namely

$b_{\psi^{s}[1,0]}\equiv b_{\psi}+r_{\psi}\bmod p-1\text{ and
}r_{\psi^{s}[1,0]}=\begin{cases}\frac{p-1}{2}-r_{\psi}\text{ if $0\leq
r_{\psi}\leq\frac{p-1}{2}$}\\\ \frac{3(p-1)}{2}-r_{\psi}\text{ if
$\frac{p-1}{2}\leq r_{\psi}\leq p-1$.}\end{cases}$

######  Corollary 4.48.

For each $\psi\in\chi^{\tilde{\Pi}^{\mathbf{Z}}}$ and $v_{\psi}\in Me_{\psi}$,

$X^{r_{\psi^{s}[1,0]}}F(v_{\psi})=(-1)^{b_{\psi^{s}[1,0]}}r_{\psi^{s}[1,0]}!v_{\psi}T_{\Pi}^{-1}.$

###### Proof.

We may assume that $v_{\psi}\neq 0$. Since $F=\tilde{s}\tilde{\Pi}$ and
$\tilde{\Pi}$ acts on $M$ via the Hecke operator $T_{\Pi}^{-1}$, we need to
show that

(25)
$X^{r_{\psi^{s}[1,0]}}\tilde{s}(v_{\psi}T_{\Pi}^{-1})=(-1)^{b_{\psi^{s}[1,0]}}r_{\psi^{s}[1,0]}!v_{\psi}T_{\Pi}^{-1}.$

The relation $e_{\psi}T_{\Pi}^{-1}=T_{\Pi}^{-1}e_{\psi^{s}[1,0]}$ of Lemma 3.9
(i) implies that $0\neq v_{\psi}T_{\Pi}^{-1}\in
Me_{\psi^{s}[1,0]}\subset\sigma(\psi^{s}[1,0])^{I_{1}}$. By Lemma 4.46,
$\sigma(\psi^{s}[1,0])$ is a weight. Therefore, equation (25) is an explicit
computation in
$\sigma(\psi^{s}[1,0])=\operatorname{Sym}^{r_{\psi^{s}[1,0]}}(k^{2})\otimes{\det}^{b_{\psi^{s}[1,0]}}$,
that is for example carried out in [22, Lemma 3.2]. ∎

We now bring ourselves into the following setup: Fix $0\neq v_{1}\in
Me_{\chi}$ and put $v_{i}=v_{1}T_{\Pi}^{-i+1}$ for $1\leq i\leq 4$. We then
have

$M=\underbrace{kv_{1}}_{Me_{\chi}}\oplus\underbrace{kv_{2}}_{Me_{\chi^{s}[1,0]}}\oplus\underbrace{kv_{3}}_{Me_{\chi[1,1]}}\oplus\underbrace{kv_{4}}_{Me_{\chi^{s}[0,1]}}$

We further set

$s_{1}=r_{\chi^{s}[1,0]},\hskip 5.69046pts_{2}=r_{\chi[1,1]},\hskip
5.69046pts_{3}=r_{\chi^{s}[0,1]},\hskip 5.69046pts_{4}=r_{\chi}$

and

$\displaystyle c_{1}=(-1)^{b_{\chi^{s}[1,0]}}r_{\chi^{s}[1,0]}!,\hskip
8.5359pt$ $\displaystyle c_{2}=(-1)^{b_{\chi[1,1]}}r_{\chi[1,1]}!,$
$\displaystyle c_{3}=(-1)^{b_{\chi^{s}[0,1]}}r_{\chi^{s}[0,1]}!,\hskip
8.5359pt$ $\displaystyle
c_{4}=(-1)^{\frac{p-1}{2}}z^{-1}(-1)^{b_{\chi}}r_{\chi}!.$

By Corollary 4.48, we then have

$X^{s_{i}}F(v_{i})=c_{i}v_{i+1}\text{ for all $1\leq i\leq 4$.}$

Here, note that, by Lemma 3.9 (iv),
$T_{\Pi}^{-4}=(-1)^{\frac{p-1}{2}}T_{\Pi^{-4}}$, which acts on $M$ via
$(-1)^{\frac{p-1}{2}}z^{-1}$. By the Basic Lemma 4.41 (vii),

$\pi_{M}=k\llbracket X\rrbracket[F]M,$

is an irreducible $P^{+}$-subrepresentation of $\mathcal{T}(M)$: in the
square-regular case, the $\Gamma$-eigencharacters of the vectors
$v_{1},\ldots,v_{4}$ are pairwise distinct, while in case $\mathbf{1}\in O$,
the $\Gamma$-eigencharacters of $Me_{\mathbf{1}}$ and $Me_{\mathbf{1}[0,1]}$
(resp. $Me_{\mathbf{1}[1,1]}$ and $Me_{\mathbf{1}[1,0]}$) coincide, but the
corresponding $s_{i}$’s differ by $\frac{p-1}{2}$ (see Remark 4.47), so we can
indeed apply the Basic Lemma.

######  Proposition 4.49.

The equality $\mathcal{T}(M)=\pi_{M}[F^{-1}]$ holds true. In particular, the
restriction of $\mathcal{T}(M)$ to $P$ is irreducible.

###### Proof.

We first claim that $\pi_{M}$ contains the weights $\sigma(\psi)$ for all
$\psi\in\chi^{\tilde{\Pi}^{\mathbf{Z}}}$. The Iwahori decomposition and the
inclusion $(I\cap sUs)F\subset F(I\cap sUs)$ imply that $\pi_{M}$ is
$I$-stable. Now $\sigma(\psi)$, being a weight, is generated as a
$K$-representation by its $I_{1}$-invariants $Me_{\psi}$, which are contained
in $\pi_{M}$ by definition. Using the decomposition $K=I\sqcup IsI$, it
suffices to prove that $\tilde{s}Me_{\psi}\in\pi_{M}$. Writing
$\tilde{s}=F\tilde{\Pi}^{-1}$, we have

$\tilde{s}Me_{\psi}=F(Me_{\psi}T_{\Pi})=F(Me_{\psi^{s}[0,1]})\subset\pi_{M}.$

As already used a few times, the element $\tilde{\Pi}^{2}$ acting on
$\mathcal{T}(M)$ restricts to an isomorphism
$\sigma(\psi[1,1])\cong\sigma(\psi)\otimes{\det}^{\frac{p-1}{2}}$, which
implies that $M$ is stable under the action of $\widetilde{Z}$. Putting this
together with the previous claim, we obtain

(26)
$\mathcal{T}(M)\supset\pi_{M}[F^{-1}]=\left\langle\widetilde{B}.\sum_{\psi}\sigma(\psi)\right\rangle=\left\langle\widetilde{B}K.\sum_{\psi}\sigma(\psi)\right\rangle\supset\left\langle\widetilde{G}.M\right\rangle=\mathcal{T}(M),$

i.e. $\pi_{M}[F^{-1}]=\mathcal{T}(M)$. As explained prior to this proposition,
the condition of Corollary 4.44 is satisfied, whence
$\pi_{M}[F^{-1}]=\mathcal{T}(M)$ is irreducible as a $P$-representation. ∎

####  4.4.3. …and back

It remains to prove that the unit $M\to\mathcal{I}(\mathcal{T}(M))$ is an
isomorphism. We will use an analogue of the short exact sequence in [22,
Theorem 6.3], the proof of which we will closely follow.

For $1\leq i\leq 4$, let

$\pi_{i}=k\llbracket X\rrbracket[F^{4}]v_{i}$

as in the setup of Section 4.41. The Iwahori decomposition and the inclusion
$(I\cap sUs)F\subset F(I\cap sUs)$ imply that $\pi_{i}$ is
$\widetilde{I}$-stable, and $Z_{1}:=I_{1}\cap Z$ acts trivially on it. By part
(iii) and (vi) of the Basic Lemma 4.41,
$0\neq\pi_{i}^{I_{1}}\subset\pi_{i}^{I\cap U}=kv_{i}$ (hence
$\pi_{i}^{I_{1}}=kv_{i}$) and the inclusion $kv_{i}\subset\pi_{i}$ is an
injective envelope in the category of smooth $I\cap U$-representations. Thus,
$(\pi_{i}/kv_{i})^{I\cap U}=H^{1}(I\cap
U,kv_{i})\cong\operatorname{Hom}_{\operatorname{cts}}(\mathbf{Z}_{p},k)$ is
$1$-dimensional; in particular,
$(\pi_{i}/kv_{i})^{I_{1}}=(\pi_{i}/kv_{i})^{I\cap U}$.

Consider the short exact sequence

$0\to kv_{i}\to\pi_{i}\to\pi_{i}/kv_{i}\to 0$

of $\widetilde{I}/Z_{1}$-representations. The injection becomes an equality on
$I_{1}$-invariants, so we obtain an inclusion

$(\pi_{i}/kv_{i})^{I_{1}}\hookrightarrow H^{1}(I_{1}/Z_{1},kv_{i})\subset
H^{1}(I_{1}/Z_{1},M)\cong\operatorname{Hom}(I_{1}/Z_{1},M),$

whose $1$-dimensional image we denote by $\Delta_{i}$. Using the decomposition
$\pi_{M}=\oplus_{i=1}^{4}\pi_{i}$, we take the direct sum of the above
inclusions and call the resulting image $\Delta$:

$\Delta=\bigoplus_{i=1}^{4}\Delta_{i}=\operatorname{image}\left((\pi_{M}/M)^{I_{1}}\hookrightarrow
H^{1}(I_{1}/Z_{1},M)\cong\operatorname{Hom}(I_{1}/Z_{1},M)\right).$

The element $\tilde{\Pi}$ normalizes $I_{1}$ and its subgroup $Z_{1}$. Its
action on $M$ thus induces an action on
$H^{1}(I_{1}/Z_{1},M)\cong\operatorname{Hom}(I_{1}/Z_{1},M)$:

$[\tilde{\Pi}.f](x)=\tilde{\Pi}f(\tilde{\Pi}^{-1}x\tilde{\Pi})\text{ for all
$x\in I_{1}$ and $f\in\operatorname{Hom}(I_{1}/Z_{1},M)$.}$

In particular, $\tilde{\Pi}\Delta_{i}\subset H^{1}(I_{1}/Z_{1},kv_{i+1})$.
Note that $\tilde{\Pi}^{2}\Delta_{i}=\Delta_{i+2}$ since
$\tilde{\Pi}^{2}\pi_{i}=\pi_{i+2}$.

######  Lemma 4.50.

The intersection $\Delta\cap\tilde{\Pi}\Delta=0$ is trivial.

###### Proof.

We need to show that for each $1\leq i\leq 4$,
$\tilde{\Pi}\Delta_{i}\cap\Delta_{i+1}=0$. Acting by $\tilde{\Pi}$, we may
replace $i$ by $i+1$ and thus assume that the $H$-eigencharacter $\chi_{i}$ of
$v_{i}$ is regular, i.e. $\chi_{i}\neq\chi_{i}^{s}$. By [22, Proposition 5.2],
a $k$-basis of
$H^{1}(I_{1}/Z_{1},\overline{\mathbf{F}}_{p}v_{j})\cong\operatorname{Hom}(I_{1}/Z_{1},kv_{j})$,
for $1\leq j\leq 4$, is given by the homomorphisms
$\kappa^{u}_{j},\kappa^{\ell}_{j}$ defined by

$\kappa^{u}_{j}\left(\left(\begin{smallmatrix}a&b\\\
c&d\end{smallmatrix}\right)\right)=\omega(b)v_{j}\text{ and
}\kappa^{\ell}_{j}\left(\left(\begin{smallmatrix}a&b\\\
c&d\end{smallmatrix}\right)\right)=\omega(p^{-1}c)v_{j}\text{ for all
$\left(\begin{smallmatrix}a&b\\\ c&d\end{smallmatrix}\right)\in I_{1}$.}$

Since the eigencharacter of $v_{i}$ is regular, we have
$\Delta_{i}=k\kappa^{u}_{i}$, by Proposition 6.2 (i) in loc. cit. The cited
proposition also implies that
$\Delta_{i+1}=k(\kappa^{u}_{i+1}+b\kappa^{\ell}_{i+1})$ for some $b\in k$.
Noting that $\tilde{\Pi}\kappa^{u}_{i}$ is a non-zero scalar multiple of
$\kappa^{\ell}_{i+1}$, we obtain

$\tilde{\Pi}\Delta_{i}\cap\Delta_{i+1}=k\kappa^{\ell}_{i+1}\cap
k(\kappa^{u}_{i+1}+b\kappa^{\ell}_{i+1})=0,$

which finishes the proof. ∎

######  Proposition 4.51.

The $\langle\widetilde{I},\tilde{\Pi}\rangle$-equivariant sequence

$0\to
M\xrightarrow{\operatorname{diag}}\pi_{M}\oplus\tilde{\Pi}\pi_{M}\xrightarrow{(v,w)\mapsto
v-w}\mathcal{T}(M)\to 0$

is exact.

###### Proof.

With the previous preparations, the same proof as in [22, Theorem 6.3] works.
∎

######  Corollary 4.52.

The unit $M\to\mathcal{I}(\mathcal{T}(M))$ is an isomorphism.

###### Proof.

Since $M$ is irreducible, the unit is an inclusion, so it is enough to prove
that $\mathcal{I}(\mathcal{T}(M))$ has the same dimension as $M$. The equality
$M=\pi_{M}^{I_{1}}$ reduces us to proving that the sequence in Proposition
4.51 remains exact after taking $I_{1}$-invariants, i.e. we need that the map
$H^{1}(I_{1}/Z_{1},M)\to H^{1}(I_{1}/Z_{1},\pi_{M}\oplus\tilde{\Pi}\pi_{M})$
is injective. This amounts to showing that $W\cap\tilde{\Pi}W=0$, where $W$ is
the kernel of $H^{1}(I_{1}/Z_{1},M)\to H^{1}(I_{1},/Z_{1},\pi_{M})$. Since
$\pi_{M}^{I_{1}}=M$, the kernel $W$ is equal to the image of the injection
$(\pi_{M}/M)^{I_{1}}\hookrightarrow H^{1}(I_{1}/Z_{1},M)$, i.e. $W=\Delta$.
The corollary now follows from Lemma 4.50. ∎

######  Corollary 4.53.

The $K$-socle of $\mathcal{T}(M)$ is equal to
$\bigoplus_{\psi\in\chi^{\tilde{\Pi}^{\mathbf{Z}}}}\sigma(\psi)|_{K}$ with
$\sigma(\psi)$ as in Corollary 4.46.

###### Proof.

By Corollary 4.46, $\mathcal{T}(M)$ contains the pairwise distinct weights
$\sigma(\psi)$ for $\psi\in\chi^{\tilde{\Pi}^{\mathbf{Z}}}$, so we obtain an
inclusion
$\bigoplus_{\psi\in\chi^{\tilde{\Pi}^{\mathbf{Z}}}}\sigma(\psi)|_{K}\subset\operatorname{soc}_{K}(\mathcal{T}(M))$.
By Corollary 4.52 this is an equality on $I_{1}$-invariants, which implies the
assertion. ∎

######  Theorem 4.54.

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

$\left\\{\begin{array}[]{c}\text{Supersingular right}\\\
\text{$\mathcal{H}$-modules}\end{array}\right\\}\cong\left\\{\begin{array}[]{c}\text{Admissible
genuine supersingular}\\\ \text{representations of
$\widetilde{G}$},\end{array}\right\\}$

where both sides are considered up to isomorphism.

###### Proof.

Let $M$ be a supersingular $\mathcal{H}$-module. By Proposition 4.49, the
$\widetilde{G}$-representation $\mathcal{T}(M)$ is irreducible. If it was not
supersingular, then it would be isomorphic to a principal series, see Remark
4.40. By Theorem 4.31 and Corollary 4.52, this is impossible and we deduce
that $\mathcal{T}(M)$ is supersingular and that the assignment
$M\mapsto\mathcal{T}(M)$ is injective with inverse $\mathcal{I}$. Let now
$\pi$ be a genuine supersingular $\widetilde{G}$-representation. Then
$\mathcal{I}(\pi)$ is a non-zero finite dimensional $\mathcal{H}$-module and
hence contains some simple module $M$. By the classification results for
simple Hecke modules, $M$ is either principal or supersingular. In any case,
the attached $\widetilde{G}$-representation $\mathcal{T}(M)$ is irreducible
(as we just explained in the supersingular case and by Theorem 4.31 in the
principal case). By adjunction we obtain a non-trivial
$\widetilde{G}$-equivariant map $\mathcal{T}(M)\to\pi$, which has to be an
isomorphism as both sides are irreducible. Once again using Theorem 4.31, we
see that $M$ cannot be a principal module and therefore needs to be
supersingular. This proves surjectivitiy. ∎

######  Corollary 4.55.

Let $\pi_{1},\pi_{2}$ be genuine supersingular
$\widetilde{G}$-representations. Then the restriction functor from
$\widetilde{G}$ to $P$ induces an equality

$\operatorname{Hom}_{\widetilde{G}}(\pi_{1},\pi_{2})=\operatorname{Hom}_{P}(\pi_{1}|_{P},\pi_{2}|_{P}).$

In other words, $\pi_{1}\cong\pi_{2}$ as $\widetilde{G}$-representations if
and only if $\pi_{1}|_{P}\cong\pi_{2}|_{P}$ as $P$-representations.

###### Proof.

By Theorem 4.54, we may write $\pi_{i}=\mathcal{T}(M_{i})$ for a supersingular
$\mathcal{H}(O_{i})$-module $M_{i}$, for some orbit $O_{i}$, for $i=1,2$. By
Proposition 4.49, $\pi_{i}$ is then irreducible as a $P$-representation.
Assume now that $f\colon\pi_{1}|_{P}\cong\pi_{2}|_{P}$ is an isomorphism of
$P$-representations. Since $\pi_{2}=\pi_{M_{2}}[F^{-1}]$, we can choose $N\gg
0$ such that $F^{4N}f(M_{1})\subset\pi_{M_{2}}$. Using Lemma 4.43 (ii), we
then obtain $f(M_{1})\subset\pi_{M_{2}}$. Comparing $X$-torsion shows that
$f(M_{1})\subset M_{2}$, which is an equality by symmetry, i.e. $f$ restricts
to a $\Gamma$-equivariant isomorphism $f\colon M_{1}\cong M_{2}$.

We now check that $O_{1}=O_{2}$. The space $M_{i}$ is the direct sum of
$1$-dimensional $\Gamma$-stable subspaces. If $O_{i}$ is square-regular, then
the $\Gamma$-characters in $M_{i}$ appear with multiplicity one. If $O_{i}$ is
non-square-regular, then the $\Gamma$-characters in $M_{i}$ appear with
multiplicity two. This already implies that $O_{1}$ is square-regular if and
only if $O_{2}$ is square-regular. For $i=1,2$, let us now fix a character
$\chi_{i}\in O_{i}$, which we choose to satisfy $\chi_{i}=\chi_{i}^{s}$ in the
non-square-regular case. Let $O_{i}^{2}$ denote the set of characters on $H$
obtained by squaring those in $O_{i}$. Then $O_{i}^{2}$ determines $O_{i}$.
Concretely, $O_{i}^{2}=\\{\chi_{i}^{2},(\chi_{i}^{2})^{s}\\}$, which has
cardinality two if and only if $O_{i}$ is square-regular. We can recover
$O_{i}^{2}$ from $\\{\chi_{i}^{2}|_{\Gamma},(\chi_{i}^{2})^{s}|_{\Gamma}\\}$,
and so $O_{1}=O_{2}$. After possibly replacing $\chi_{2}$, we may assume that
$\chi_{1}=\chi_{2}$, which we denote by $\chi$. In the non-square-regular case
we may, after possibly twisting (Lemma 4.3), assume that $\chi=\mathbf{1}$.

For $i=1,2$ and $\psi\in\chi^{\tilde{\Pi}^{\mathbf{Z}}}$, let
$\sigma_{i}(\psi)\subset\pi_{i}$ denote the weight generated by
$M_{i}e_{\psi}$, see Lemma 4.46, and put
$r_{i}(\psi)=\dim_{k}\sigma_{i}(\psi)-1$. We claim that
$r_{1}(\psi)=r_{2}(\psi)$ for all $\psi\in\chi^{\tilde{\Pi}^{\mathbf{Z}}}$. By
Remark 4.47, it suffices to prove this for a single $\psi$. Choose $\psi$ such
that $r_{1}(\psi)\geq r_{1}(\psi^{\prime})$ for all
$\psi^{\prime}\in\chi^{\tilde{\Pi}^{\mathbf{Z}}}$. By symmetry, we may assume
that $r_{1}(\psi)\geq r_{2}(\psi)$. Assume by contradiction that
$r_{1}(\psi)>r_{2}(\psi)$. Looking at the $\Gamma$-eigencharacter, we must
have

$f(v_{\psi^{s}[0,1]})=\alpha w_{\psi^{s}[0,1]}+\beta w_{\psi^{s}}\text{ for
some $\alpha,\beta\in k$ with $\alpha\beta\neq 0$,}$

where we put $w_{\psi^{s}}:=0$ in the square-regular case. By the recursion
formula in Corollary 4.48, we have

(27) $X^{r_{1}(\psi)}F(v_{\psi^{s}[0,1]})=av_{\psi}\text{ for some $a\in
k^{\times}$.}$

Applying the $P$-equivariant isomorphism $f$ to this equation, we obtain

(28) $af(v_{\psi})=\alpha X^{r_{1}(\psi)}F(w_{\psi^{s}[0,1]})+\beta
X^{r_{1}(\psi)}F(w_{\psi^{s}}),$

where the first summand vanishes since $r_{1}(\psi)>r_{2}(\psi)$ (using the
analogue of equation (27) for $w_{\psi^{s}[0,1]}$ and the fact that
$Xw_{\psi}=0$). It follows that $\beta\neq 0$ and

(29) $X^{r_{1}(\psi)}F(w_{\psi^{s}})=\beta^{-1}af(v_{\psi})\neq 0.$

This already gives a contradiction in the square-regular case (as we put
$w_{\psi^{s}}=0$ in this case). It remains to treat the non-square-regular
case. Since the expression in (29) is non-zero, we deduce that
$r_{1}(\psi)\leq r_{2}(\psi[1,0])$; and since it is killed by $X$, we then
deduce that $r_{1}(\psi)=r_{2}(\psi[1,0])$ (both of these observations follow
from Corollary 4.48). Now either $\psi=\psi^{s}$ or $\psi[1,0]=\psi[1,0]^{s}$,
i.e. either $r_{i}(\psi)\in\\{0,p-1\\}$ (independent of $i$) or
$r_{i}(\psi[1,0])\in\\{0,p-1\\}$ (independent of $i$), but not both. Hence,
the equality $r_{1}(\psi)=r_{2}(\psi[1,0])$ is impossible. This proves the
claim.

Using the claim we obtain that $f(v_{\psi})=\gamma w_{\psi}$ for some
$\gamma\in k^{\times}$. Indeed, in the square-regular case, there is nothing
to prove. In the non-square-regular case, note that
$r_{2}(\psi[1,0])=r_{1}(\psi[1,0])<r_{1}(\psi)=r_{2}(\psi)$, so the second
summand in equation (28) vanishes, while the first one is a non-zero scalar
multiple of $w_{\psi}$.

For $i=1,2$, let us now write

$M_{i}=\begin{cases}\operatorname{SS}(\chi,z_{i})\text{ if $O$ is square-
regular}\\\ \operatorname{SS}(\lambda_{i},z_{i})\text{ if $\mathbf{1}\in
O$,}\end{cases}$

where $z_{i}\in k^{\times}$ and $\lambda_{i}\in\\{0,-1\\}$ as in Definition
3.29, resp. 3.38. It follows from Corollary 4.46 (ii) that
$\lambda_{1}=\lambda_{2}$ since $r_{1}(\chi)=r_{2}(\chi)$. In order to
conclude that $M_{1}=M_{2}$, it remains to show that $z_{1}=z_{2}$. By Lemma
4.43 (ii), we have

$X^{e(\psi)}F^{4}(v_{\psi})=(-1)^{\frac{p-1}{2}}z_{i}^{-1}\left(\prod_{\psi^{\prime}\in\chi^{\tilde{\Pi}^{\mathbf{Z}}}}r_{1}(\psi^{\prime})!\right)v_{\psi},$

where the exponent $e(\psi)$ only depends on $r_{1}(\psi)$ (which determines
all the other $r_{1}(\psi^{\prime})$). A similar equality also holds true for
$w_{\psi}$. Applying the $P$-equivariant map $f$ and using the equalities
$r_{1}(\psi^{\prime})=r_{2}(\psi^{\prime})$ for all
$\psi^{\prime}\in\chi^{\tilde{\Pi}^{\mathbf{Z}}}$ allows us to conclude
$z_{1}=z_{2}$. ∎

######  Corollary 4.56.

Let $\pi$ be a genuine supersingular $\widetilde{G}$-representation. Then each
element in $\pi/X\pi$ is killed by a power of $F$.

###### Proof.

Write $\pi=\mathcal{T}(M)$ for a supersingular $\mathcal{H}$-module $M$. By
Proposition 4.49, we have $\pi=\pi_{M}[F^{-1}]$ and the relations in Corollary
4.48 imply that $\pi_{M}=X\pi_{M}$, which finishes the proof. ∎

##  5\. Spherical Hecke algebras

In this section, we study the genuine mod-$p$ representation theory of
$\widetilde{G}$ via spherical Hecke algebras by following the approach of
Barthel-Livné as outlined in the introduction. The spherical Hecke algebra,
defined to be the endomorphism ring of the compact induction of some weight,
turns out to be a polynomial ring in a single operator $\widetilde{T}$ (not to
be confused with the metaplectic torus). Any smooth irreducible admissible
genuine $\widetilde{G}$-representation is a quotient of the cokernel of a
linear polynomial in $\widetilde{T}$, for a suitable weight. By following [6],
we show that these cokernels are of finite length and determine their Jordan-
Hölder decomposition. As an application, we prove that a smooth
$k$-representation of $\widetilde{G}$ is of finite length if and only if it is
finitely generated and admissible, extending the known result for $G$.

We set things up a bit more generally to also include characteristic $0$
coefficients as in [5] allowing us to show in the next section that in case
$k=\overline{\mathbf{F}}_{p}$ the aforementioned cokernels arise as the
mod-$p$ reduction of an invariant bounded lattice of a locally algebraic
$\overline{\mathbf{Q}}_{p}$-representations of $\widetilde{G}$

###  5.1. Computational preparations

Let $R$ be a $\mathbf{Z}_{p}$-algebra. We fix an integer $r\in\mathbf{Z}_{\geq
0}$ and define the $R$-linear representation $\operatorname{Sym}^{r}(R^{2})$
of $K$ just as in the case of $R=k$, Definition 4.6, i.e. the underlying
$R$-module is the space of homogeneous polynomials in $R[x,y]$ of degree $r$
with action defined by

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

Extend the $K$-action to an action of $KZ^{\operatorname{sq}}$ by letting the
scalar matrix $p^{2}$ act trivially, and denote the resulting representation
by $\underline{\operatorname{Sym}}^{r}(R^{2})$. By Lemma 2.6 (i), we have
$Z(\widetilde{G})=Z^{\operatorname{sq}}\times\mu_{2}$ and we write

$\tilde{\sigma}_{r}(R):=\underline{\operatorname{Sym}}^{r}(R^{2})\boxtimes\iota$

for the corresponding genuine $\widetilde{K}Z(\widetilde{G})$-representation.
We identify the spherical Hecke algebra
$\operatorname{End}_{\widetilde{G}}(\operatorname{c-Ind}^{\widetilde{G}}_{\widetilde{K}Z(\widetilde{G})}(\tilde{\sigma}_{r}(R)))$
with the algebra

(30)
$\biggl{\\{}\varphi\colon\widetilde{G}\to\operatorname{End}_{R}(\tilde{\sigma}_{r}(R)):\begin{array}[]{l}\diamond\hskip
2.84544pt\varphi(k_{2}gk_{2})=k_{2}\circ\varphi(g)\circ k_{1},\forall
g\in\widetilde{G},k_{1},k_{2}\in\widetilde{K}Z(\widetilde{G})\\\
\diamond\hskip 2.84544pt\operatorname{supp}(\varphi)\text{ is
compact}\end{array}\biggr{\\}},$

whose multiplication is given by convolution.

The following is an analogue of [5, Lemme 2.1.4.1] and its proof is similar.

######  Definition-Lemma 5.1.

There exists a unique function $\widetilde{\mathbf{T}}_{R}$ in the algebra
(30) with

  1. $\bullet$

$\operatorname{supp}(\widetilde{\mathbf{T}}_{R})=\widetilde{K}Z(\widetilde{G})\left(\left(\begin{smallmatrix}1&0\\\
0&p^{-2}\end{smallmatrix}\right),1\right)\widetilde{K}Z(\widetilde{G})$;

  2. $\bullet$

$\widetilde{\mathbf{T}}_{R}\left(\left(\left(\begin{smallmatrix}1&0\\\
0&p^{-2}\end{smallmatrix}\right),1\right)\right)(x^{r-i}y^{i})=p^{2(r-i)}x^{r-i}y^{i}$
for all $0\leq i\leq r$.

The corresponding endomorphism in
$\operatorname{End}_{\widetilde{G}}(\operatorname{c-Ind}^{\widetilde{G}}_{\widetilde{K}Z(\widetilde{G})}(\tilde{\sigma}_{r}(R)))$
is denoted by $\widetilde{T}_{R}$. If $R=k$, we simply write $\widetilde{T}$
instead of $\widetilde{T}_{k}$.

######  Proposition 5.2.

If $R$ is an $\mathbf{F}_{p}$-algebra and $0\leq r\leq p-1$, then the
$R$-algebra
$\operatorname{End}_{\widetilde{G}}(\operatorname{c-Ind}^{\widetilde{G}}_{\widetilde{K}Z(\widetilde{G})}(\tilde{\sigma}_{r}(R)))$
is a commutative polynomial ring in the operator $\widetilde{T}_{R}$.

###### Proof.

Since
$\operatorname{c-Ind}^{\widetilde{G}}_{\widetilde{K}Z(\widetilde{G})}(\tilde{\sigma}_{r}(\mathbf{F}_{p}))$
is finitely generated over $\mathbf{F}_{p}[\widetilde{G}]$, one may reduce to
the case $R=\mathbf{F}_{p}$, see also [23, Lemma 5.1]. Using the description
of the Hecke algebra given by (30) and the Cartan decomposition, the assertion
can be proved as in the case of $G=\operatorname{GL}_{2}(\mathbf{Q}_{p})$, see
[15, Theorem 21]. We only explain why a function supported on
$\left(\left(\begin{smallmatrix}p^{n}&0\\\
0&p^{m}\end{smallmatrix}\right),1\right)$, for $n\geq m\geq 0$, can only exist
if $n$ and $m$ are even. Let $F$ be such a function and put
$f=F\left(\left(\left(\begin{smallmatrix}p^{n}&0\\\
0&p^{m}\end{smallmatrix}\right),1\right)\right)\in\operatorname{End}_{\mathbf{F}_{p}}(\tilde{\sigma}_{r}(\mathbf{F}_{p}))$.
First assume that $n>m$. For $x\in\mathbf{Z}_{p}$, the equation

$\left(\left(\begin{smallmatrix}p^{n}&0\\\
0&p^{m}\end{smallmatrix}\right),1\right)\left(\left(\begin{smallmatrix}1&x\\\
0&1\end{smallmatrix}\right),1\right)=\left(\left(\begin{smallmatrix}1&p^{n-m}x\\\
0&1\end{smallmatrix}\right),1\right)\left(\left(\begin{smallmatrix}p^{n}&0\\\
0&p^{m}\end{smallmatrix}\right),1\right)$

implies that $f$ factors through the $1$-dimensional $U\cap K$-coinvariants
$\tilde{\sigma}_{r}(k)_{U\cap K}$, while the equation

$\left(\left(\begin{smallmatrix}1&0\\\
x&1\end{smallmatrix}\right),1\right)\left(\left(\begin{smallmatrix}p^{n}&0\\\
0&p^{m}\end{smallmatrix}\right),1\right)=\left(\left(\begin{smallmatrix}p^{n}&0\\\
0&p^{m}\end{smallmatrix}\right),1\right)\left(\left(\begin{smallmatrix}1&0\\\
p^{n-m}x&1\end{smallmatrix}\right),1\right)$

implies that $f$ takes values in the $1$-dimensional $sUs\cap K$-invariants
$\tilde{\sigma}_{r}(k)^{sUs\cap K}$. In conclusion, $f$ must be a scalar
multiple of the canonical map

$\tilde{\sigma}_{r}(\mathbf{F}_{p})\twoheadrightarrow\tilde{\sigma}_{r}(\mathbf{F}_{p})_{U\cap
K}\xleftarrow{\cong}\tilde{\sigma}_{r}(\mathbf{F}_{p})^{sUs\cap
K}\subset\tilde{\sigma}_{r}(\mathbf{F}_{p}),$

where the isomorphism is the projection. In particular, $f$ must be $T\cap
K$-equivariant. Now, by Example 2.3, for all
$\lambda,\mu\in\mathbf{F}_{p}^{\times}$,

$\left(\left(\begin{smallmatrix}p^{n}&0\\\
0&p^{m}\end{smallmatrix}\right),1\right)\left(\left(\begin{smallmatrix}[\lambda]&0\\\
0&[\mu]\end{smallmatrix}\right),1\right)=\left(\left(\begin{smallmatrix}[\lambda]&0\\\
0&[\mu]\end{smallmatrix}\right),\lambda^{m\frac{p-1}{2}}\mu^{n\frac{p-1}{2}}\right)\left(\left(\begin{smallmatrix}p^{n}&0\\\
0&p^{m}\end{smallmatrix}\right),1\right),$

and so the $T\cap K$-equivariance of $f$ can only hold true if $f=0$ or $m$
and $n$ are even.

If $n=m$, then Lemma 4.4 implies that $f$ defines a $K$-equivariant map
between $\tilde{\sigma}_{r}(k)$ and its twist by
$\omega^{n\frac{p-1}{2}}\circ{\det}$. Such a map can only be non-zero if the
twist is trivial, i.e. if $n$ is even. ∎

######  Remark 5.3.

If $R$ is a $\mathbf{Q}_{p}$-algebra and $r=0$, then the proposition is still
valid as follows for example from the Satake isomorphism [18, Theorem 10.1].

######  Definition 5.4.

Given a vector $v\in\tilde{\sigma}_{r}(R)$ and an element $g\in\widetilde{G}$,
denote by
$[g,v]\in\operatorname{c-Ind}^{\widetilde{G}}_{\widetilde{K}Z(\widetilde{G})}(\tilde{\sigma}_{r}(R))$
the unique function with support $\widetilde{K}Z(\widetilde{G})g^{-1}$ and
value $v$ at $g^{-1}$.

Next, we define an explicit set of representatives of
$\mathbf{Z}_{p}/p^{m}\mathbf{Z}_{p}$, for all $m\geq 0$, by letting
$J_{m}=\left\\{\sum_{j=0}^{m-1}p^{j}[\lambda_{j}]:\lambda_{j}\in\mathbf{F}_{p}\right\\}$
if $m\geq 1$ and setting $J_{0}=\\{0\\}$. Moreover, for $m\geq 0$ and
$\lambda\in J_{m}$, we put
$g^{0}_{m,\lambda}=\left(\begin{smallmatrix}p^{m}&\lambda\\\
0&1\end{smallmatrix}\right)$ and
$g^{1}_{m,\lambda}=\left(\begin{smallmatrix}1&0\\\
p\lambda&p^{m+1}\end{smallmatrix}\right).$ It follows from [5, Equation (1)]
that

(31) $\widetilde{G}=\bigsqcup_{m\geq 0,\lambda\in
J_{m}}\bigsqcup_{q,j=0,1}\widetilde{K}Z(\widetilde{G})(\tilde{g}^{q}_{m,\lambda})^{-1}\tilde{\Pi}^{-2j}.$

######  Definition 5.5.

For $m\geq 0$ and $q,j=0,1$, define the subspaces $S_{m}(R)\subset B_{m}(R)$
of
$\operatorname{c-Ind}^{\widetilde{G}}_{\widetilde{K}Z(\widetilde{G})}(\tilde{\sigma}_{r}(R))$
by

$S_{m}(R)=\left\\{f:\operatorname{supp}(f)\subset\bigsqcup_{\lambda\in
J_{m}}\bigsqcup_{q,j=0,1}\widetilde{K}Z(\widetilde{G})(\tilde{g}^{q}_{m,\lambda})^{-1}\tilde{\Pi}^{-2j}\right\\},$

and $B_{m}(R)=\sum_{n\leq m}S_{n}(R)$, respectively.

Note that $B_{m}(R)\subset B_{m+1}(R)$ for all $m\geq 0$. The decomposition
(31) implies that
$\operatorname{c-Ind}^{\widetilde{G}}_{\widetilde{K}Z(\widetilde{G})}(\tilde{\sigma}_{r}(R))=\sum_{m\geq
0}B_{m}(R)$. Given a function $f$ in this space, one may think of the minimal
$m\geq 0$ satisfying $f\in B_{m}(R)$ as the radius of the support of $f$.

The decomposition

$\widetilde{K}Z(\widetilde{G})\left(\left(\begin{smallmatrix}1&0\\\
0&p^{-2}\end{smallmatrix}\right),1\right)\widetilde{K}Z(\widetilde{G})=\bigsqcup_{\lambda\in
J_{2}}\widetilde{K}Z(\widetilde{G})(\tilde{g}^{0}_{2,\lambda})^{-1}\sqcup\bigsqcup_{\lambda\in
J_{1}}\widetilde{K}Z(\widetilde{G})(\tilde{g}^{1}_{1,\lambda})^{-1},$

which is quickly verified using $K=I\sqcup IsI$, implies that the endomorphism
$\widetilde{T}_{R}\in\operatorname{End}_{\widetilde{G}}\left(\operatorname{c-Ind}^{\widetilde{G}}_{\widetilde{K}Z(\widetilde{G})}(\tilde{\sigma}_{r}(R))\right)$
is explicitly given by

(32) $\widetilde{T}_{R}([1,v])=\sum_{\lambda\in
J_{2}}[\tilde{g}^{0}_{2,\lambda},\widetilde{\mathbf{T}}_{R}((\tilde{g}^{0}_{2,\lambda})^{-1})(v)]+\sum_{\mu\in
J_{1}}[\tilde{g}^{1}_{1,\mu},\widetilde{\mathbf{T}}_{R}((\tilde{g}^{1}_{1,\mu})^{-1})(v)],$

for all $v\in\tilde{\sigma}_{r}(R)$. Motivated by this description, we make
the following definition in analogy to [5, Lemme 2.2.1].

######  Definition 5.6.

Let $m\geq 0$, $\mu\in J_{m}$ and $v\in\tilde{\sigma}_{r}(R)$.

  1. (i)

Define $\widetilde{T}_{R}^{+}([\tilde{g}^{0}_{m,\mu},v])=\sum_{\lambda\in
J_{2}}[\tilde{g}^{0}_{m+2,\mu+p^{m}\lambda},\widetilde{\mathbf{T}}_{R}((\tilde{g}^{0}_{2,\lambda})^{-1})(v)]$.

  2. (ii)

Noting that $\tilde{g}^{1}_{m,\mu}=\tilde{\Pi}\tilde{g}^{0}_{m,\mu}\tilde{s}$,
set
$\widetilde{T}_{R}^{+}([\tilde{g}^{1}_{m,\mu},v])=\tilde{\Pi}\widetilde{T}_{R}^{+}([\tilde{g}^{0}_{m,\mu},\tilde{s}v])$.

  3. (iii)

For $q\in\\{0,1\\}$, put
$\widetilde{T}_{R}^{+}([\tilde{\Pi}^{2}\tilde{g}^{q}_{m,\mu},v])=\tilde{\Pi}^{2}\widetilde{T}_{R}^{+}([\tilde{g}^{q}_{m,\mu},v]).$

Extending $R$-linearly, cf. (31), yields the endomorphism
$\widetilde{T}^{+}_{R}$ on
$\operatorname{c-Ind}^{\widetilde{G}}_{\widetilde{K}Z(\widetilde{G})}(\tilde{\sigma}_{r}(R))$.

For $m\geq 0$, we thus have $\widetilde{T}^{+}_{R}(S_{m}(R))\subset
S_{m+2}(R)$ and one checks that the difference
$\widetilde{T}_{R}-\widetilde{T}^{+}_{R}$ preserves $B_{m}(R)$. We obtain:

######  Observation 5.7.

Let
$f\in\operatorname{c-Ind}^{\widetilde{G}}_{\widetilde{K}Z(\widetilde{G})}(\tilde{\sigma}_{r}(R))$.
If $\widetilde{T}_{R}(f)\in B_{m}(R)$ for some $m\geq 0$, then $f\in
B_{m-2}(R)$. In particular, the image of $\widetilde{T}_{R}$ intersects
$B_{1}(R)$ trivially, and for any $\lambda\in R$, the endomorphism
$\widetilde{T}_{R}-\lambda$ is not surjective.

###  5.2. Jordan-Hölder sequence

We now take $R=k$ and restrict $0\leq r\leq p-1$ so that the
$K$-representation $\tilde{\sigma}_{r}(k)$ is irreducible. Following the
exposition in [6, §6], we show that
$\operatorname{c-Ind}^{\widetilde{G}}_{\widetilde{K}Z(\widetilde{G})}(\tilde{\sigma}_{r}(k))/(\widetilde{T}-\lambda)$,
for $\lambda\in k$, has finite length as a $\widetilde{G}$-representation and
compute its Jordan-Hölder decomposition.

To ease notation let us put $\tilde{\sigma}_{r}:=\tilde{\sigma}_{r}(k)$ and
let $\chi=\omega^{r}\otimes\mathbf{1}\in\widehat{H}$ so that
$\chi\boxtimes\iota=\tilde{\sigma}_{r}^{I_{1}}$. We extend $\chi$ to a
character of $HZ^{\operatorname{sq}}$ by letting $p^{2}$ act trivially.

######  Proposition 5.8.

  1. (i)

If $\chi\neq\chi^{s}$, then

$\displaystyle\operatorname{c-Ind}^{\widetilde{G}}_{\widetilde{K}Z(\widetilde{G})}(\tilde{\sigma}_{r})$
$\displaystyle=\ker\left(T_{s}\colon\operatorname{c-Ind}^{\widetilde{G}}_{\widetilde{I}Z(\widetilde{G})}(\chi^{s}\boxtimes\iota)\to\operatorname{c-Ind}^{\widetilde{G}}_{\widetilde{I}Z(\widetilde{G})}(\chi\boxtimes\iota)\right)$
$\displaystyle=T_{s}\operatorname{c-Ind}^{\widetilde{G}}_{\widetilde{I}Z(\widetilde{G})}(\chi\boxtimes\iota).$

  2. (ii)

If $r=p-1$, then

$\displaystyle\operatorname{c-Ind}^{\widetilde{G}}_{\widetilde{K}Z(\widetilde{G})}(\tilde{\sigma}_{p-1})$
$\displaystyle=\ker\left(T_{s}+1\colon\operatorname{c-Ind}^{\widetilde{G}}_{\widetilde{I}Z(\widetilde{G})}(\mathbf{1}\boxtimes\iota)\to\operatorname{c-Ind}^{\widetilde{G}}_{\widetilde{I}Z(\widetilde{G})}(\mathbf{1}\boxtimes\iota)\right)$
$\displaystyle=T_{s}\operatorname{c-Ind}^{\widetilde{G}}_{\widetilde{I}Z(\widetilde{G})}(\mathbf{1}\boxtimes\iota).$

  3. (iii)

If $r=0$, then

$\displaystyle\operatorname{c-Ind}^{\widetilde{G}}_{\widetilde{K}Z(\widetilde{G})}(\tilde{\sigma}_{0})$
$\displaystyle=\ker\left(T_{s}\colon\operatorname{c-Ind}^{\widetilde{G}}_{\widetilde{I}Z(\widetilde{G})}(\mathbf{1}\boxtimes\iota)\to\operatorname{c-Ind}^{\widetilde{G}}_{\widetilde{I}Z(\widetilde{G})}(\mathbf{1}\boxtimes\iota)\right)$
$\displaystyle=(T_{s}+1)\operatorname{c-Ind}^{\widetilde{G}}_{\widetilde{I}Z(\widetilde{G})}(\mathbf{1}\boxtimes\iota).$

###### Proof.

Since the functor
$\operatorname{c-Ind}^{\widetilde{G}}_{\widetilde{K}Z(\widetilde{G})})(-)$ is
exact, it suffices to prove the equalities for $\widetilde{G}$ replaced by
$\widetilde{K}Z(\widetilde{G})\cong KZ^{\operatorname{sq}}\times\mu_{2}$. In
this case, the assertions follow from Proposition 4.8 and Lemma 3.7 (v). ∎

######  Corollary 5.9.

We have equalities of right $\mathcal{H}$-modules:

  1. (i)

If $\chi\neq\chi^{s}$ or $r=p-1$,

$\mathcal{I}\left(\operatorname{c-Ind}^{\widetilde{G}}_{\widetilde{K}Z(\widetilde{G})}(\tilde{\sigma}_{r})\right)=T_{s}e_{\chi}\mathcal{H}_{p^{2}=1}.$

  2. (ii)

If $r=0$,

$\mathcal{I}\left(\operatorname{c-Ind}^{\widetilde{G}}_{\widetilde{K}Z(\widetilde{G})}(\tilde{\sigma}_{0})\right)=(T_{s}+1)e_{\mathbf{1}}\mathcal{H}_{p^{2}=1}.$

###### Proof.

By Frobenius reciprocity,
$e_{\chi}\mathcal{H}_{p^{2}=1}\cong\mathcal{I}\left(\operatorname{c-Ind}^{\widetilde{G}}_{\widetilde{I}Z(\widetilde{G})}(\chi\boxtimes\iota)\right)$.
Taking $I_{1}$-invariants in Proposition 5.8 reduces us to proving the
equalities

  1. (i)

$\ker(T_{s}\colon e_{\chi^{s}}\mathcal{H}_{p^{2}=1}\to
e_{\chi}\mathcal{H}_{p^{2}=1})=T_{s}e_{\chi}\mathcal{H}_{p^{2}=1}$ for
$\chi\neq\chi^{s}$;

  2. (ii)

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

  3. (iii)

$\ker(T_{s}\colon e_{\mathbf{1}}\mathcal{H}_{p^{2}=1}\to
e_{\mathbf{1}}\mathcal{H}_{p^{2}=1})=(T_{s}+1)e_{\mathbf{1}}\mathcal{H}_{p^{2}=1}$.

This is the content of Lemma 3.39. ∎

######  Remark 5.10.

We will fix the identifications of the corollary so that the function
$[1,x^{r}]$ corresponds to $T_{s}e_{\chi}$ (resp. $(T_{s}+1)e_{\mathbf{1}}$).

Let now $O=O_{\chi}$ be the orbit of $\chi$. As in Section 3.5, we put

$\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 $O$ is non-square-regular},\end{cases}$

which defines a central element in $\mathcal{H}(O)$.

######  Lemma 5.11.

Under the identifications in Corollary 5.9, the action of the (restriction to
$I_{1}$-invariants of the) Hecke operator $\widetilde{T}$ is given by the
action of $\mathscr{Z}_{O}$.

###### Proof.

Since both $\widetilde{T}$ and $\mathscr{Z}_{O}$ are equivariant for the right
$\mathcal{H}$-action, it suffices to prove that
$\widetilde{T}([1,x^{r}])=\mathscr{Z}_{O}([1,x^{r}])$. The identifications of
Corollary 5.9 are induced by the isomorphism
$e_{\chi}\mathcal{H}_{p^{2}=1}\cong\operatorname{c-Ind}^{\widetilde{G}}_{\widetilde{I}Z(\widetilde{G})}(\chi\boxtimes\iota)$
obtained by Frobenius reciprocity, i.e. if
$\varphi\in\operatorname{c-Ind}^{\widetilde{G}}_{\widetilde{I}_{1}Z(\widetilde{G})}(\mathbf{1}\boxtimes\iota)$
denotes the unique function with support $\widetilde{I}_{1}Z(\widetilde{G})$
and value $1$ at the identity, then the image of an element $f\in
e_{\chi}\mathcal{H}_{p^{2}=1}$ is given by $f(\varphi)$.

Together with the relations
$(\tilde{g}^{0}_{2,\lambda})^{-1}=\tilde{s}\left(\left(\begin{smallmatrix}1&0\\\
0&p^{-2}\end{smallmatrix}\right),1\right)\left(\left(\begin{smallmatrix}0&1\\\
1&-\lambda\end{smallmatrix}\right),1\right)$ and
$(\tilde{g}^{1}_{1,\mu})^{-1}=\left(\left(\begin{smallmatrix}1&0\\\
0&p^{-2}\end{smallmatrix}\right),1\right)\left(\left(\begin{smallmatrix}1&0\\\
-p\mu&1\end{smallmatrix}\right),1\right)$, equation (32) implies that in
characteristic $p$,

$\widetilde{T}([1,x^{r}])=\sum_{\lambda\in
J_{2}}[\tilde{g}^{0}_{2,\lambda},x^{r}]+\sum_{\mu\in
J_{1}}[\tilde{g}^{1}_{1,\mu},p^{2r}x^{r}].$

Assume that $\chi\neq\chi^{s}$. The element $[1,x^{r}]$ is identified with the
element $T_{s}e_{\chi}\in T_{s}e_{\chi}\mathcal{H}_{p^{2}=1}$. By Lemma 3.7
(v), $T_{s}^{2}e_{\chi}=0$ and so the operator $\mathscr{Z}_{O}$ acts on
$T_{s}e_{\chi}$ via $(T_{s}T_{\Pi})^{2}$. By Lemma 3.9 (vii),
$(T_{\Pi}T_{s})^{2}=T_{(\Pi s)^{2}}$. The decomposition
$I_{1}\left(\begin{smallmatrix}1&0\\\
0&p^{2}\end{smallmatrix}\right)I_{1}=\sqcup_{\lambda\in J_{2}}I_{1}(\Pi
s)^{2}\left(\begin{smallmatrix}1&\lambda\\\ 0&1\end{smallmatrix}\right)$
implies that

(33) $(T_{\Pi}T_{s})^{2}e_{\chi}(\varphi)=\sum_{\lambda\in
J_{2}}\left(\begin{smallmatrix}1&\lambda\\\
0&1\end{smallmatrix}\right)\left(\left(\begin{smallmatrix}1&0\\\
0&p^{-2}\end{smallmatrix}\right),1\right)e_{\chi}(\varphi).$

Since $T_{\Pi^{4}}$ acts trivially, we may replace
$\left(\left(\begin{smallmatrix}1&0\\\
0&p^{-2}\end{smallmatrix}\right),1\right)$ by
$\left(\left(\begin{smallmatrix}p^{2}&0\\\
0&1\end{smallmatrix}\right),1\right)$. Applying the
$\widetilde{G}$-equivariant homomorphism $T_{s}$ to the above equation and
using that $T_{s}(T_{\Pi}T_{s})^{2}=(T_{s}T_{\Pi})^{2}T_{s}$, see Lemma 3.9
(viii), gives the assertion in case $\chi\neq\chi^{s}$.

Assume now that $r=p-1$ or $r=0$. By Lemma 3.9 (vi) and (ix), we have
$T_{\Pi}T_{s}T_{\Pi}=T_{\Pi s\Pi}$. The decomposition $I_{1}(\Pi
s\Pi)I_{1}=\sqcup_{\lambda\in J_{1}}I_{1}(\Pi
s\Pi)\left(\begin{smallmatrix}1&0\\\ p\lambda&1\end{smallmatrix}\right)$
yields

(34) $T_{\Pi s\Pi}e_{\mathbf{1}}(\varphi)=\sum_{\lambda\in
J_{1}}\left(\begin{smallmatrix}1&0\\\
p\lambda&1\end{smallmatrix}\right)\left(\left(\begin{smallmatrix}0&p^{-2}\\\
1&0\end{smallmatrix}\right),1\right)e_{\mathbf{1}}(\varphi).$

Applying the $\widetilde{G}$-equivariant map $T_{s}$ to it gives the formula

(35) $(T_{s}T_{\Pi})^{2}e_{\mathbf{1}}(\varphi)=\sum_{\lambda\in
J_{1}}\left(\begin{smallmatrix}1&0\\\
p\lambda&1\end{smallmatrix}\right)\left(\left(\begin{smallmatrix}0&p^{-2}\\\
1&0\end{smallmatrix}\right),1\right)T_{s}e_{\mathbf{1}}(\varphi).$

The equalities $T_{s}(T_{s}+1)e_{\mathbf{1}}=0$ of Lemma 3.7 (v) allows one to
deduce from (33), (34) and (35) that

$\mathscr{Z}_{O}T_{s}e_{\mathbf{1}}(\varphi)=\sum_{\lambda\in
J_{2}}\left(\begin{smallmatrix}1&\lambda\\\
0&1\end{smallmatrix}\right)\left(\left(\begin{smallmatrix}1&0\\\
0&p^{-2}\end{smallmatrix}\right),1\right)T_{s}e_{\mathbf{1}}(\varphi),$

and

$\displaystyle\mathscr{Z}_{O}(T_{s}+1)e_{\mathbf{1}}(\varphi)$
$\displaystyle=\sum_{\lambda\in J_{2}}\left(\begin{smallmatrix}1&\lambda\\\
0&1\end{smallmatrix}\right)\left(\left(\begin{smallmatrix}1&0\\\
0&p^{-2}\end{smallmatrix}\right),1\right)(T_{s}+1)e_{\mathbf{1}}(\varphi)$
$\displaystyle+\sum_{\lambda\in J_{1}}\left(\begin{smallmatrix}1&0\\\
p\lambda&1\end{smallmatrix}\right)\left(\left(\begin{smallmatrix}0&p^{-2}\\\
1&0\end{smallmatrix}\right),1\right)(T_{s}+1)e_{\mathbf{1}}(\varphi)$

Applying $T_{\Pi^{4}}=1$, the first equality settles the case $r=p-1$, while
the second equality covers the case $r=0$ by noting that $\tilde{s}$ fixes
$(T_{s}+1)e_{\mathbf{1}}(\varphi)$. ∎

Recall from Definition 3.41 that for each $\psi\in O$ and the element
$\lambda\in k$, we have attached a right $\mathcal{H}_{p^{2}=1}$-module
$M(\psi,\lambda)$ as well as the module $M(\lambda)$ in the non-regular case.
The previous corollary immediately gives:

######  Corollary 5.12.

As right $\mathcal{H}_{p^{2}=1}$-modules,

$\mathcal{I}\left(\operatorname{c-Ind}^{\widetilde{G}}_{\widetilde{K}Z(\widetilde{G})}(\tilde{\sigma}_{r})\right)/(\widetilde{T}-\lambda)=\begin{cases}M(\chi,\lambda)\text{
if $r\neq 0$}\\\ M(\lambda)\text{ if $r=0$.}\end{cases}$

We remind the reader that we parametrized the supersingular Hecke modules in
definitions 3.29 and 3.38, respectively.

######  Theorem 5.13.

  1. (i)

If $\lambda\neq 0$, then

$\operatorname{c-Ind}^{\widetilde{G}}_{\widetilde{K}Z(\widetilde{G})}(\tilde{\sigma}_{r})/(\widetilde{T}-\lambda)\cong\operatorname{Ind}^{\widetilde{G}}_{\widetilde{B}_{2}}(\psi\boxtimes\iota),$

where $\psi\colon T_{2}\to k^{\times}$ is the smooth character defined by the
requirements $\psi|_{H}=\chi^{s}$,
$\psi\left(\left(\begin{smallmatrix}p^{-2}&0\\\
0&1\end{smallmatrix}\right)\right)=\lambda$ and
$\psi\left(\left(\begin{smallmatrix}p^{-2}&0\\\
0&p^{-2}\end{smallmatrix}\right)\right)=1$.

  2. (ii)

Assume now that $\lambda=0$.

     * (a)

If $O$ is square-regular, then there is a non-split short exact sequence

$0\to\mathcal{T}(\operatorname{SS}(\chi[1,0],1))\to\operatorname{c-Ind}^{\widetilde{G}}_{\widetilde{K}Z(\widetilde{G})}(\tilde{\sigma}_{r})/(\widetilde{T})\to\mathcal{T}(\operatorname{SS}(\chi,1))\to
0.$

     * (b)

For $r=p-1$ and $r=0$, there are non-split short exact sequences

$0\to\mathcal{T}(\operatorname{SS}(0,1))\to\operatorname{c-Ind}^{\widetilde{G}}_{\widetilde{K}Z(\widetilde{G})}(\tilde{\sigma}_{p-1})/(\widetilde{T})\to\mathcal{T}(\operatorname{SS}(-1,1))\to
0$

and

$0\to\mathcal{T}(\operatorname{SS}(-1,1))\to\operatorname{c-Ind}^{\widetilde{G}}_{\widetilde{K}Z(\widetilde{G})}(\tilde{\sigma}_{0})/(\widetilde{T})\to\mathcal{T}(\operatorname{SS}(0,1))\to
0,$

respectively, while for $r=\frac{p-1}{2}$ there is a direct sum decomposition

$\operatorname{c-Ind}^{\widetilde{G}}_{\widetilde{K}Z(\widetilde{G})}(\tilde{\sigma}_{\frac{p-1}{2}})/(\widetilde{T})\cong\mathcal{T}(\operatorname{SS}(0,1))\oplus\mathcal{T}(\operatorname{SS}(-1,1)).$

###### Proof.

(i) Assume that $\bar{a}_{p}\neq 0$. By equation (11) (resp. (12)),
Proposition 3.43 (resp. 3.44) and Corollary 5.12, we have

$\mathcal{I}(\operatorname{Ind}^{\widetilde{G}}_{\widetilde{B}_{2}}(\psi\boxtimes\iota))\cong\mathcal{I}\left(\operatorname{c-Ind}^{\widetilde{G}}_{\widetilde{K}Z(\widetilde{G})}(\tilde{\sigma}_{r})\right)/(\widetilde{T}-\lambda).$

(In the square-regular case, use
$\operatorname{PS}(\chi^{s},x,0,z)=\operatorname{PS}(\chi,0,x,z)$, see Remark
3.30). The evident map from this object to the $I_{1}$-invariants of the
cokernel of $\widetilde{T}-\lambda$ is non-trivial by Observation 5.7 (as the
element $[1,x^{r}]$ is $I_{1}$-invariant and generates the compact induction
as a $\widetilde{G}$-representation). Hence, by adjunction, we obtain a non-
trivial $\widetilde{G}$-equivariant map

$\operatorname{Ind}^{\widetilde{G}}_{\widetilde{B}_{2}}(\psi\boxtimes\iota)=\mathcal{T}\mathcal{I}(\operatorname{Ind}^{\widetilde{G}}_{\widetilde{B}_{2}}(\psi\boxtimes\iota))\to\operatorname{c-Ind}^{\widetilde{G}}_{\widetilde{K}Z(\widetilde{G})}(\tilde{\sigma}_{r})/(\widetilde{T}-\lambda),$

where the first equality is due to Theorem 4.31. The source being irreducible,
this map is necessarily injective, while the right-hand side being generated
by the image of the $I_{1}$-invariant function $[1,x^{r}]$ implies that it is
surjective.

(ii) Assume now that $\lambda=0$. By Propositions 3.43 (resp. 3.44) and
Corollary 5.12, there are short exact sequences of the desired form if the
middle term is replaced by
$\mathcal{T}\left(\mathcal{I}\left(\operatorname{c-Ind}^{\widetilde{G}}_{\widetilde{K}Z(\widetilde{G})}(\tilde{\sigma}_{r})\right)/(\widetilde{T})\right)$.
For this we use the right exactness of the functor $\mathcal{T}$ and the fact
that $\mathcal{T}$ applied to a supersingular modules yields an irreducible
representation by Theorem 4.54. It remains to show that the canonical map

$\mathcal{T}\left(\mathcal{I}\left(\operatorname{c-Ind}^{\widetilde{G}}_{\widetilde{K}Z(\widetilde{G})}(\tilde{\sigma}_{r})\right)/(\widetilde{T})\right)\to\operatorname{c-Ind}^{\widetilde{G}}_{\widetilde{K}Z(\widetilde{G})}(\tilde{\sigma}_{r})/(\widetilde{T})$

is an isomorphism. The consideration of the element $[1,x^{r}]$ again shows
the surjectivity. By what we have just explained, it suffices to prove that
each supersingular subrepresentation (or equivalently its $I_{1}$-invariants)
of the source survives in the target.

In the square-regular case, the proof of Proposition 3.43 shows that the
element $T_{s}e_{\chi}T_{\Pi}T_{s}$ is a basis of
$\operatorname{SS}(\chi[1,0],1)(e_{\chi}+e_{\chi[1,0]})$. In the non-square-
regular case, the proof of Proposition 3.44 shows that the element
$T_{s}e_{\mathbf{1}}T_{\Pi}T_{s}T_{\Pi}$ is a basis element of
$\operatorname{SS}(0,1)e_{\mathbf{1}}\subset M(\mathbf{1},0)e_{\mathbf{1}}$,
while $(T_{s}+1)e_{\mathbf{1}}T_{\Pi}T_{s}T_{\Pi}$ is a basis of
$\operatorname{SS}(-1,1)e_{\mathbf{1}}\subset M(0)$; it also shows that in
case $\chi=\omega^{\frac{p-1}{2}}\otimes\mathbf{1}$ (i.e. $r=\frac{p-1}{2}$),
the element $T_{s}e_{\chi}T_{\Pi}T_{s}$ is a basis for the subspace
$\operatorname{SS}(-1,1)e_{\mathbf{1}}$, while $T_{s}e_{\chi}T_{\Pi}(T_{s}+1)$
is a basis element of the subspace $\operatorname{SS}(0,1)e_{\mathbf{1}}$. By
the Morita equivalences 3.19 and 3.31, respectively, it is enough to prove
that these basis elements survive in the cokernel of $\widetilde{T}$. We thus
need to show that

(36)
$\begin{cases}[1,x^{r}]T_{\Pi}T_{s}\notin\operatorname{Im}(\widetilde{T})\text{
if $0<r<p-1$}\\\
[1,x^{r}]T_{\Pi}T_{s}T_{\Pi}\notin\operatorname{Im}(\widetilde{T})\text{ if
$r=0,p-1$}\\\
[1,x^{r}]T_{\Pi}(T_{s}+1)\notin\operatorname{Im}(\widetilde{T})\text{ if
$r=\frac{p-1}{2}$.}\end{cases}$

Since these elements lie in $B_{1}(k)$, an application of Observation 5.7
finishes the proof. ∎

######  Remark 5.14.

The last part of the proof in particular shows that a supersingular
$\mathcal{H}_{p^{2}=1}$-module $M$ admits a non-trivial
$\mathcal{H}$-equivariant morphism to the non-zero (Observation 5.7) space
$\mathcal{I}\left(\operatorname{c-Ind}^{\widetilde{G}}_{\widetilde{K}Z(\widetilde{G})}(\tilde{\sigma})/(\widetilde{T})\right)$
for some weight $\tilde{\sigma}$. In particular, by adjunction,
$\mathcal{T}(M)$ is non-zero.

######  Definition 5.15.

For each $0\leq r\leq p-1$ with $r\neq\frac{p-1}{2}$, let $\tilde{\pi}(r)$ be
the unique irreducible (supersingular) quotient of
$\operatorname{c-Ind}^{\widetilde{G}}_{\widetilde{K}Z(\widetilde{G})}(\tilde{\sigma}_{r})/(\widetilde{T})$.
If $\eta\colon\mathbf{Q}_{p}^{\times}\to k^{\times}$ is a smooth character, we
put $\tilde{\pi}(r,\eta)=\tilde{\pi}(r)\otimes\eta\circ{\det}$.

######  Corollary 5.16.

Each genuine supersingular $\widetilde{G}$-representation is of the form
$\tilde{\pi}(r,\eta)$ for some $0\leq r\leq p-1$, $r\neq\frac{p-1}{2}$, and a
smooth character $\eta\colon\mathbf{Q}_{p}^{\times}\to k^{\times}$.

Moreover, $\tilde{\pi}(r)\cong\tilde{\pi}(r^{\prime},\eta)$ if and only if one
of the following holds

$\displaystyle r^{\prime}=r,\text{ $\eta_{\operatorname{nr}}^{4}=1$ and
$(\eta\eta_{\operatorname{nr}}^{-1})^{2}=1$;}$ $\displaystyle
r^{\prime}=\begin{cases}\frac{p-1}{2}-r\text{ if $0<r<\frac{p-1}{2}$}\\\
\frac{3}{2}(p-1)-r\text{ if $\frac{p-1}{2}<r<p-1$}\end{cases}\text{ and
\phantom{m} $\eta_{\operatorname{nr}}^{4}=1$,
$(\eta\eta_{\operatorname{nr}}^{-1}\omega^{-r})^{2}=1$,}$

where $\eta_{\operatorname{nr}}$ denotes the unramified part of $\eta$.

###### Proof.

Corollary 4.53 gives an explicit description of the socle of a genuine
supersingular representation. In particular, after twisting by a character
$\eta\circ\det$ it contains a weight $\tilde{\sigma}_{r}$ for some $0\leq
r\leq p-1$ with $r\neq\frac{p-1}{2}$. We may also assume that the central
element $p^{2}$ acts trivially. Since the weight appears with multiplicity
one, the operator $\widetilde{T}$ must act on the corresponding weight space
via a scalar. By Theorem 5.13 (i), this scalar must be $0$, i.e. the
supersingular representation is isomorphic to $\tilde{\pi}(r)$. This proves
the first part.

Any unramified character $\eta$ of order at most $4$ satisfies
$(\eta\circ\det)|_{Z^{\operatorname{sq}}}=1$ and so

$\operatorname{c-Ind}^{\widetilde{G}}_{\widetilde{K}Z(\widetilde{G})}(\tilde{\sigma}_{r})/(\widetilde{T})\otimes(\eta\circ{\det})\cong\operatorname{c-Ind}^{\widetilde{G}}_{\widetilde{K}Z(\widetilde{G})}(\tilde{\sigma}_{r})/(\widetilde{T}).$

By Lemma 4.4, the element $\tilde{\Pi}^{2}$ defines a
$\widetilde{G}$-equivariant isomorphism

$\operatorname{c-Ind}^{\widetilde{G}}_{\widetilde{K}Z(\widetilde{G})}(\tilde{\sigma}_{r})/(\widetilde{T})\cong\operatorname{c-Ind}^{\widetilde{G}}_{\widetilde{K}Z(\widetilde{G})}(\tilde{\sigma}_{r})/(\widetilde{T})\otimes(\mu_{-1}^{\frac{p-1}{2}}\omega^{\frac{p-1}{2}})\circ{\det}.$

Putting these two observations together, we obtain that
$\tilde{\pi}(r)\cong\tilde{\pi}(r,\eta)$ for $\eta$ satisfying
$\eta_{\operatorname{nr}}^{4}=1$ and
$(\eta\eta_{\operatorname{nr}}^{-1})^{2}=1$ (in fact this holds for the full
cokernel of $\widetilde{T}$). By Corollary 4.53 and Remark 4.47, we know that
$\tilde{\sigma}_{r^{\prime}}\otimes\omega^{r}\circ{\det}$ lies in the
$K$-socle of $\tilde{\pi}(r)$ and thus
$\tilde{\pi}(r)\cong\tilde{\pi}(r^{\prime})\otimes\omega^{r}\circ{\det}$. This
shows that the described intertwinings hold true.

That these exhaust all intertwinings can be deduced from looking at the
$K$-socle and the central character. ∎

###  5.3. Howe’s theorem

The goal of this section is to explain a proof of the following theorem.

######  Theorem 5.17.

A smooth $k$-representation of $\widetilde{G}$ is of finite length if and only
if it is finitely generated and admissible.

For the full subcategory $\operatorname{Mod}^{\operatorname{sm}}_{G}(k)$ of
$\operatorname{Mod}^{\operatorname{sm}}_{\widetilde{G}}(k)$ this is a
combination of [9, Theorem 2.3.8] and [2], so we will only focus on the
subcategory of genuine objects. The strategy is the same as that for the group
$G$, but we still need to justify certain auxiliary results.

Theorem 5.13 implies that the cokernels of linear polynomials in the Hecke
operator $\widetilde{T}$ are of finite length and the proof of [9, Theorem
2.3.8] then goes through to show that any finitely generated admissible object
of $\operatorname{Mod}^{\operatorname{sm}}_{\widetilde{G},\iota}(k)$ is indeed
of finite length.

For the converse, we need to show that a smooth irreducible genuine
representation of $\widetilde{G}$ is admissible. As a first step, we need to
check that the analogue of (the proof of) [1, Proposition 32] goes through to
prove the following result.

######  Proposition 5.18.

Let $\ell$ be a field of characteristic $p$. Let $\pi$ be a smooth irreducible
genuine $\ell$-representation of $\widetilde{G}$ admitting a central
character. Then $\pi$ is a quotient of
$\operatorname{c-Ind}^{\widetilde{G}}_{\widetilde{K}Z(\widetilde{G})}(\sigma\boxtimes\iota)/P(\widetilde{T})$
for some weight $\sigma$ and some irreducible polynomial
$P(\widetilde{T})\in\ell[\widetilde{T}]$.

The proof of this proposition requires two lemmas, the second of which is a
metaplectic version of [1, Proposition 18].

######  Lemma 5.19.

Assume that $\chi:=\sigma^{I_{1}}$ is square-regular. The Hecke operator
$T_{\Pi s}$ acts injectively on
$\operatorname{c-Ind}^{\widetilde{G}}_{\widetilde{K}Z(\widetilde{G})}(\sigma\boxtimes\iota)^{I_{1}}e_{\chi[1,0]}$.

###### Proof.

After twisting by a character, we may assume that
$\sigma=\operatorname{Sym}^{r}(\ell^{2})$, for some $0\leq r\leq p-1$, and the
central element $p^{2}$ acts trivially. By Corollary 5.9, we have
$\operatorname{c-Ind}^{\widetilde{G}}_{\widetilde{K}Z(\widetilde{G})}(\sigma\boxtimes\iota)^{I_{1}}e_{\chi[1,0]}=T_{s}e_{\chi}\mathcal{H}_{p^{2}=1}e_{\chi[1,0]}$.
For proving the desired injectivity, we may act by $T_{\Pi}$ from the left and
it suffices to prove injectivity of the map $T_{\Pi s}^{2}=T_{(\Pi s)^{2}}$.
In view of the isomorphisms in propositions 3.21 and 3.22, we need to show
that the element $Y\in\mathcal{H}(\chi[1,0])_{p^{2}=1}\cong\ell[X,Y]/(XY)$
acts injectively on $T_{\Pi s}\mathcal{H}(\chi[1,0],\chi)_{p^{2}=1}\cong
Y\ell[X,Y]/(XY)$, which is clear. We note that the results we used hold true
over an arbitrary field of characteristic $p$; alternatively, the lemma can be
checked after passing to an algebraic closure. ∎

######  Lemma 5.20.

Let $\psi\colon HZ^{\operatorname{sq}}\to k^{\times}$ be a smooth character.
Any non-zero subspace
$W\subset\operatorname{Hom}_{\widetilde{G}}(\operatorname{c-Ind}^{\widetilde{G}}_{\widetilde{I}Z(\widetilde{G})}(\psi\boxtimes\iota),\operatorname{c-Ind}^{\widetilde{G}}_{\widetilde{K}Z(\widetilde{G})}(\sigma\boxtimes\iota))$
stable under the action of
$\operatorname{End}_{\widetilde{G}}(\operatorname{c-Ind}^{\widetilde{G}}_{\widetilde{I}Z(\widetilde{G})}(\psi\boxtimes\iota))$
has finite codimension.

###### Proof.

After twisting we may assume that $\sigma=\operatorname{Sym}^{r}(\ell^{2})$,
for some $0\leq r\leq p-1$, and the central element $p^{2}$ acts trivially.
Write $\chi=\sigma^{I_{1}}$, viewed as a character of
$HZ^{\operatorname{sq}}$. The Hom-space in the statement is equal to the
$\mathcal{H}(\psi)_{p^{2}=1}$-module
$\operatorname{c-Ind}^{\widetilde{G}}_{\widetilde{K}Z(\widetilde{G})}(\sigma\boxtimes\iota)^{I_{1}}e_{\psi}$.

Assume that $\chi$ is square-regular. By Corollary 5.9 we then have
$\operatorname{c-Ind}^{\widetilde{G}}_{\widetilde{K}Z(\widetilde{G})}(\sigma\boxtimes\iota)^{I_{1}}e_{\psi}=T_{s}e_{\chi}\mathcal{H}_{p^{2}=1}e_{\psi}$.
Since the statement is empty if $\psi$ does not lie in the orbit of $\chi$, we
do from now on assume that $\psi\in O_{\chi}$. Acting by $T_{\Pi}$ from the
right, we may assume that $\psi=\chi[t,0]$ for $t=0,1$. If $t=0$, then acting
from the left by $T_{\Pi}$, we need to check that a non-zero
$\mathcal{H}(\chi)_{p^{2}=1}$-submodule of $T_{\Pi
s}\mathcal{H}(\chi)_{p^{2}=1}$ has finite codimension. But by propositions
3.21 and 3.22, this just means that a $\ell[X,Y]/(XY)$-submodule of $\ell[Y]$
(with $X=0$ on it) has finite codimension, i.e. that every non-zero ideal of
the polynomial ring $\ell[Y]$ has finite codimension. This is clear. The case
$t=1$ is treated similarly.

Assume now that $\chi$ is not square-regular, i.e. $r$ is a multiple of
$\frac{p-1}{2}$. Without loss of generality, we may assume that
$\psi=\mathbf{1}$.

If $r=p-1$, then by Corollary 5.9, we are considering a non-zero
$\mathcal{H}(\mathbf{1})_{p^{2}=1}$-submodule $W$ of
$T_{s}\mathcal{H}(\mathbf{1})_{p^{2}=1}$. Using the description of
$\mathcal{H}(\mathbf{1})_{p^{2}=1}$ as in Proposition 3.33, we decompose $W$
as

$WS_{0,0}\oplus W(S_{0,0}+1)\subset
S_{0,0}\mathcal{H}(\mathbf{1})_{p^{2}=1}S_{0,0}\oplus
S_{0,0}\mathcal{H}(\mathbf{1})_{p^{2}=1}(S_{0,0}+1)=S_{0,0}\mathcal{H}(\mathbf{1})_{p^{2}=1}.$

We note that each summand on the left-hand side is non-zero (act by
$S_{0,2}$). Now, as $\ell$-algebras, $\ell[Y]\cong
S_{0,0}\mathcal{H}(\mathbf{1})_{p^{2}=1}S_{0,0}$ via $1\mapsto-S_{0,0}$ and
$Y\mapsto S_{0,0}S_{0,2}S_{0,0}$, so we are done with the inclusion of the
first direct summand just like in the regular case. For the second one, note
that we again have
$\ell[Y]\cong(S_{0,0}+1)\mathcal{H}(\mathbf{1})_{p^{2}=1}(S_{0,0}+1)$, where
now $1\mapsto S_{0,0}+1$ and $Y\mapsto(S_{0,0}+1)S_{0,2}(S_{0,0}+1)$. As a
module over this ring, the summand
$S_{0,0}\mathcal{H}(\mathbf{1})_{p^{2}=1}(S_{0,0}+1)$ is generated by the
element $S_{0,0}S_{0,2}(S_{0,0}+1)$ and in particular any non-zero submodule
does indeed have finite codimension.

If $r=0$, then by Corollary 5.9, we are considering a non-zero
$\mathcal{H}(\mathbf{1})_{p^{2}=1}$-submodule $W$ of
$(S_{0,0}+1)\mathcal{H}(\mathbf{1})_{p^{2}=1}$. Interchanging $S_{0,0}+1$ and
$-S_{0,0}$ defines an automorphism of $\mathcal{H}(\mathbf{1})_{p^{2}=1}$,
which reduces to the previously treated case $r=p-1$.

In case $r=\frac{p-1}{2}$, we make use of Corollary 5.9 once again and act by
$T_{\Pi}$ from the left to reduce to showing that a non-zero
$\mathcal{H}(\mathbf{1})_{p^{2}=1}$-submodule $W$ of
$T_{\Pi}T_{s}T_{\Pi}\mathcal{H}(\mathbf{1})_{p^{2}=1}=S_{0,2}\mathcal{H}(\mathbf{1})_{p^{2}=1}$
has finite codimension (here we used that
$T_{\Pi}e_{\mathbf{1}}=e_{\chi}T_{\Pi}$, Lemma 3.9 (v)), which can now be
proved as for $r=p-1$ by decomposing the problem according to the idempotent
$-S_{0,0}$. ∎

###### Proof of Proposition 5.18.

Let $\pi$ be as in the statement. Picking a weight $\sigma\subset\pi|_{K}$ and
letting the central element $p^{2}$ act on it according to the central
character of $\pi$, we obtain a surjection
$\varphi\colon\operatorname{c-Ind}^{\widetilde{G}}_{\widetilde{K}Z(\widetilde{G})}(\sigma\boxtimes\iota)\twoheadrightarrow\pi$.
Since the compact induction is not irreducible (e.g. Theorem 5.13), the kernel
is non-zero and thus $\ker(\varphi)^{I_{1}}e_{\psi}\neq 0$ for some $\psi\in
O_{\chi}$. Either acting by $T_{\Pi}$ and using Lemma 3.9 (v) or acting by
$T_{\Pi s}$ and using Lemma 3.9 (v), (vii) and Lemma 3.7 (iv), we may assume
that $\psi=\chi$; here note that $T_{\Pi}$ (Lemma 3.9 (i)) and $T_{\Pi s}$
(Lemma 5.19) act injectively.

The inclusion $\chi=\sigma^{I_{1}}\subset\sigma$ gives rise to a surjective
map
$\operatorname{c-Ind}^{\widetilde{G}}_{\widetilde{I}Z(\widetilde{G})}(\chi\boxtimes\iota)\twoheadrightarrow\operatorname{c-Ind}^{\widetilde{G}}_{\widetilde{K}Z(\widetilde{G})}(\sigma\boxtimes\iota)$,
which in turn induces a commutative diagram

$\textstyle{\operatorname{Hom}_{\widetilde{G}}(\operatorname{c-Ind}^{\widetilde{G}}_{\widetilde{I}Z(\widetilde{G})}(\chi\boxtimes\iota),\operatorname{c-Ind}^{\widetilde{G}}_{\widetilde{K}Z(\widetilde{G})}(\sigma\boxtimes\iota))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Hom}_{\widetilde{G}}(\operatorname{c-Ind}^{\widetilde{G}}_{\widetilde{I}Z(\widetilde{G})}(\chi\boxtimes\iota),\pi)}$$\textstyle{\operatorname{End}_{\widetilde{G}}(\operatorname{c-Ind}^{\widetilde{G}}_{\widetilde{K}Z(\widetilde{G})}(\sigma\boxtimes\iota))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Hom}_{\widetilde{G}}(\operatorname{c-Ind}^{\widetilde{G}}_{\widetilde{K}Z(\widetilde{G})}(\sigma\boxtimes\iota),\pi)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

with injective vertical maps, where the horizontal maps are induced by
composing with $\varphi$. Lemma 5.20 implies that the top arrow has finite
image and thus the lower arrow does as well. But the image of the morphism at
the bottom is non-zero (it contains $\varphi$) and it is stable under the
action of the spherical Hecke algebra $\ell[\widetilde{T}]$ appearing as the
source of the arrow. Picking a non-zero element in this finite dimensional
$\ell[\widetilde{T}]$-submodule killed by some irreducible polynomial finishes
the proof. ∎

######  Proposition 5.21.

A smooth irreducible genuine $k$-representation of $\widetilde{G}$ is
admissible.

###### Proof.

As in [2], this can now be deduced from Proposition 5.18 and Theorem 5.13. ∎

##  6\. Relation to locally algebraic representations

Fix an algebraic closure $\overline{\mathbf{Q}}_{p}$ of $\mathbf{Q}_{p}$ with
ring of integers $\overline{\mathbf{Z}}_{p}$. Keeping the notation of the
previous section and putting $k=\overline{\mathbf{F}}_{p}$, we would like to
relate the smooth $\overline{\mathbf{F}}_{p}$-representations
$\operatorname{c-Ind}^{\widetilde{G}}_{\widetilde{K}Z(\widetilde{G})}(\tilde{\sigma}_{r}(\overline{\mathbf{F}}_{p}))/(\widetilde{T}-\bar{a}_{p})$,
for $\bar{a}_{p}\in\overline{\mathbf{F}}_{p}$ and $0\leq r\leq p-1$, to the
reduction modulo the maximal ideal in $\overline{\mathbf{Z}}_{p}$ of a
$\widetilde{G}$-stable bounded lattice of a locally algebraic
$\overline{\mathbf{Q}}_{p}$-representation (or equivalently its unitary
completion) of $\widetilde{G}$. Choosing a lift
$a_{p}\in\overline{\mathbf{Z}}_{p}$ of $\bar{a}_{p}$, a natural candidate to
consider is the representation

$\widetilde{\Pi}_{r,a_{p}}:=\operatorname{c-Ind}^{\widetilde{G}}_{\widetilde{K}Z(\widetilde{G})}(\tilde{\sigma}_{r}(\overline{\mathbf{Q}}_{p}))/(\widetilde{T}_{\overline{\mathbf{Q}}_{p}}-a_{p}).$

Although it will not be necessary for us, we first describe
$\widetilde{\Pi}_{r,a_{p}}$ in terms of (constituents of) locally algebraic
unramified principal series, which can be deduced from the corresponding
statement [5, Proposition 3.2.1] for the group $G$ by utilizing Savin’s local
Shimura correspondence.

###  6.1. Savin’s local Shimura correspondence

Let us denote by
$\mathcal{H}_{\iota}(\widetilde{G},\widetilde{I})=\operatorname{End}_{\widetilde{G}}(\operatorname{c-Ind}^{\widetilde{G}}_{\widetilde{I}}(\mathbf{1}\boxtimes\iota))$
the genuine Iwahori Hecke algebra, where $\mathbf{1}$ is the trivial
$\overline{\mathbf{Q}}_{p}^{\times}$-valuded character, and let
$\mathcal{H}(G,I)=\operatorname{End}_{G}(\operatorname{c-Ind}^{G}_{I}(\mathbf{1}))$
be the usual Iwahori Hecke algebra of $G$. The work [27, Proposition 3.1.2] of
Savin provides us with a description of
$\mathcal{H}_{\iota}(\widetilde{G},\widetilde{I})$ in terms of generators and
relations. Comparing with the description of $\mathcal{H}(G,I)$, see for
example [26], we obtain an explicit isomorphism

(37) $\mathcal{H}_{\iota}(\widetilde{G},I)\cong\mathcal{H}(G,I)$

of $\overline{\mathbf{Q}}_{p}$-algebras, see also [27, §4]. In representation
theoretic terms, this translates into an equivalence of categories

(38) $\left\\{\begin{array}[]{c}\text{Smooth genuine}\\\
\text{$\overline{\mathbf{Q}}_{p}$-representations of $\widetilde{G}$}\\\
\text{generated by their
$I$-invariants}\end{array}\right\\}\cong\left\\{\begin{array}[]{c}\text{Smooth
}\\\ \text{$\overline{\mathbf{Q}}_{p}$-representations of $G$}\\\
\text{generated by their $I$-invariants}\end{array}\right\\}.$

######  Lemma 6.1.

The equivalence (38) maps
$\operatorname{c-Ind}^{\widetilde{G}}_{\widetilde{K}Z(\widetilde{G})}(\mathbf{1}\boxtimes\iota)$
to $\operatorname{c-Ind}^{G}_{KZ}(\mathbf{1})$ and the endomorphism
$\widetilde{T}_{\overline{\mathbf{Q}}_{p}}$ of the former to the endomorphism
$p^{1/2}T_{\overline{\mathbf{Q}}_{p}}$ of the latter, where
$T_{\overline{\mathbf{Q}}_{p}}$ corresponds to the double coset
$KZ\left(\begin{smallmatrix}1&0\\\ 0&p^{-1}\end{smallmatrix}\right)KZ$.

###### Proof.

It is enough to check the assertions on $I$-invariants. Let
$\widetilde{T}_{s}\in\mathcal{H}_{\iota}(\widetilde{G},\widetilde{I})$ be the
function with support $\widetilde{I}\tilde{s}\widetilde{I}$ sending
$\tilde{s}$ to $1$ and let $T_{s}\in\mathcal{H}(G,I)$ be the indicator
function of $IsI$. For $n\geq m$, let
$\tilde{\mathbf{1}}_{2n,2m}\in\mathcal{H}_{\iota}(\widetilde{G},\widetilde{I})$
be the function with support
$\widetilde{I}\left(\left(\begin{smallmatrix}p^{2n}&0\\\
0&p^{2m}\end{smallmatrix}\right),1\right)\widetilde{I}$ and value $1$ at
$\left(\left(\begin{smallmatrix}p^{2n}&0\\\
0&p^{2m}\end{smallmatrix}\right),1\right)$, and let
$\mathbf{1}_{n,m}\in\mathcal{H}(G,I)$ be the indicator function of
$I\left(\begin{smallmatrix}p^{n}&0\\\ 0&p^{m}\end{smallmatrix}\right)I$. The
isomorphism (37) then maps $\widetilde{T}_{s}$ to $T_{s}$ and
$\tilde{\mathbf{1}}_{2n,2m}$ to $p^{\frac{n-m}{2}}\mathbf{1}_{n,m}$.

Considered as a subspace
$\operatorname{c-Ind}^{G}_{K}(\mathbf{1})^{I}\subset\operatorname{c-Ind}^{G}_{I}(\mathbf{1})^{I}\cong\mathcal{H}(G,I)$,
we have the equality
$\operatorname{c-Ind}^{G}_{K}(\mathbf{1})^{I}=(1+T_{s})\mathcal{H}(G,I)$ since
$1+T_{s}$ is the indicator function of $K$. One similarly has
$\operatorname{c-Ind}^{\widetilde{G}}_{\widetilde{K}}(\mathbf{1}\boxtimes\iota)^{I}=(1+\widetilde{T}_{s})\mathcal{H}_{\iota}(\widetilde{G},\widetilde{I})$.
This proves the first part. Concerning the Hecke operators, one may check that
$T_{\overline{\mathbf{Q}}_{p}}=p^{-1}(1+T_{s})\mathbf{1}_{0,-1}(1+T_{s})$ in
$\mathcal{H}(G,I)/(\mathbf{1}_{1,1}-1)$ and
$\widetilde{T}_{\overline{\mathbf{Q}}_{p}}=p^{-1}(1+\widetilde{T}_{s})\tilde{\mathbf{1}}_{0,-2}(1+\widetilde{T}_{s})$
in
$\mathcal{H}_{\iota}(\widetilde{G},\widetilde{I})/(\tilde{\mathbf{1}}_{2,2}-1)$.
∎

###  6.2. Genuine principal series in characteristic $0$

Fix an isomorphism $\overline{\mathbf{Q}}_{p}\cong\mathbf{C}$ and let
$|\cdot|\colon\mathbf{Q}_{p}\to\mathbf{Q}_{\geq 0}$ be the $p$-adic norm so
that $|p|=p^{-1}$. Via the fixed isomorphism, we will view fractional powers
of $|\cdot|$ as $\overline{\mathbf{Q}}_{p}^{\times}$-valued characters of
$\mathbf{Q}_{p}^{\times}$.

######  Definition 6.2.

  1. (i)

For two smooth characters $\chi_{1},\chi_{2}\colon
A\to\overline{\mathbf{Q}}_{p}^{\times}$, define the normalized principal
series

$\iota^{\widetilde{G}}_{\widetilde{B}_{2}}(\chi_{1}\otimes\chi_{2})=\operatorname{Ind}^{\widetilde{G}}_{\widetilde{B}_{2}}(\chi_{1}|\cdot|^{1/2}\otimes\chi_{2}|\cdot|^{-1/2}\boxtimes\iota),$

which only depends on the restriction of $\chi_{1},\chi_{2}$ to the subgroup
$S$ of squares in $\mathbf{Q}_{p}^{\times}$.

  2. (ii)

For two smooth characters
$\chi_{1},\chi_{2}\colon\mathbf{Q}_{p}^{\times}\to\overline{\mathbf{Q}}_{p}^{\times}$,
let

$\iota^{G}_{B}(\chi_{1}\otimes\chi_{2})=\operatorname{Ind}^{G}_{B}(\chi_{1}|\cdot|^{1/2}\otimes\chi_{2}|\cdot|^{-1/2})$

be the usual normalized principal series of $G$ attached to
$(\chi_{1},\chi_{2})$.

Given a character $\chi\colon A\to\overline{\mathbf{Q}}_{p}^{\times}$, denote
by
$\chi^{\prime}\colon\mathbf{Q}_{p}^{\times}\to\overline{\mathbf{Q}}_{p}^{\times}$
the character defined by

$\chi^{\prime}(x)=\chi(x^{2}),\text{ for all $x\in\mathbf{Q}_{p}^{\times}$.}$

######  Lemma 6.3.

Let $\chi_{1},\chi_{2}\colon A\to\overline{\mathbf{Q}}_{p}^{\times}$ be smooth
characters trivial on $T\cap K$. The equivalence (38) maps
$\iota^{\widetilde{G}}_{\widetilde{B}}(\chi_{1}\otimes\chi_{2})$ to
$\iota^{G}_{B}(\chi_{1}^{\prime}\otimes\chi_{2}^{\prime})$.

###### Proof.

First note that the genuine principal series must correspond to an
(unramified) principal series, say $\iota^{G}_{B}(\psi_{1}\otimes\psi_{2})$
for two smooth unramified characters $\psi_{1},\psi_{2}$. A basis for the
space $\iota^{G}_{B}(\psi_{1}\otimes\psi_{2})^{I}$ is given by the two
functions $\varphi,\varphi_{s}$ defined by the conditions
$\operatorname{supp}(\varphi)=BI$, $\operatorname{supp}(\varphi_{s})=BsI$ and
$\varphi(1)=\varphi_{s}(s)=1$. A basis for the space
$\iota^{\widetilde{G}}_{\widetilde{B}}(\chi_{1}\otimes\chi_{2})^{I}$ is given
by the two functions $\tilde{\varphi},\tilde{\varphi}_{s}$ defined similarly
but requiring that
$\tilde{\varphi}(1)=\tilde{\varphi}_{s}(\tilde{s})\in\operatorname{Ind}^{\widetilde{T}}_{\widetilde{T}_{2}}(\chi_{1}|\cdot|^{1/2}\otimes\chi_{2}|\cdot|^{-1/2}\boxtimes\iota)^{T\cap
K}$ is the standard basis vector $f_{0,0}$ as in Definition 4.18. Letting
$\mathbf{1}_{0,-1}\in\mathcal{H}(G,I)$ and
$\tilde{\mathbf{1}}_{0,-2}\in\mathcal{H}_{\iota}(\widetilde{G},\widetilde{I})$
be as in the proof of Lemma 6.1, one checks that with respect to the chosen
bases,

$\mathbf{1}_{0,-1}=\left(\begin{smallmatrix}p^{1/2}\psi_{2}(p)&p^{1/2}(p-1)\psi_{2}(p)\\\
0&p^{1/2}\psi_{1}(p)\end{smallmatrix}\right)\text{ and
}\tilde{\mathbf{1}}_{0,-2}=\left(\begin{smallmatrix}p\chi_{2}(p^{2})&p(p-1)\chi_{2}(p^{2})\\\
0&p\chi_{1}(p^{2})\end{smallmatrix}\right).$

Since the isomorphism (37) identifies $\tilde{\mathbf{1}}_{0,-2}$ with
$p^{1/2}\mathbf{1}_{0,-1}$, we deduce that
$\\{\psi_{1},\psi_{2}\\}=\\{\chi_{1}^{\prime},\chi_{2}^{\prime}\\}$. If
$\psi_{1}\psi_{2}^{-1}\neq|\cdot|^{\pm 1}$, then
$\iota^{G}_{B}(\psi_{1}\otimes\psi_{2})\cong\iota^{G}_{B}(\psi_{2}\otimes\psi_{1})$,
so we are done. Assume now by contradiction that
$\psi_{1}\psi_{2}^{-1}=|\cdot|$ and
$\chi_{1}^{\prime}(\chi_{2}^{\prime})^{-1}=|\cdot|^{-1}$. By [16, Theorem
I.2.9],
$\iota^{\widetilde{G}}_{\widetilde{B}}(\chi_{1}^{\prime}\otimes\chi_{2}^{\prime})$
contains a unique irreducible subrepresentation admitting a $K$-invariant
vector (and so the quotient cannot admit one), while
$\iota^{G}_{B}(\psi_{1}\otimes\psi_{2})$ is a non-split extensions of an
unramified character by the corresponding twist of the Steinberg
representation (not admitting a $K$-invariant vector). Since the equivalence
(38) preserves the property of having $K$-invariant vectors (the isomorphism
(37) maps the genuine indicator function of $\widetilde{K}$ to the indicator
function of $K$ as observered in the proof of Lemma 6.1), we obtain the
desired contradiction. ∎

######  Definition-Proposition 6.4.

Let $\chi_{1},\chi_{2}\colon A\to\overline{\mathbf{Q}}_{p}^{\times}$ be smooth
characters.

  1. (i)

If $\chi_{1}^{\prime}(\chi_{2}^{\prime})^{-1}\neq|\cdot|^{\pm 1}$, then
$\iota^{\widetilde{G}}_{\widetilde{B}_{2}}(\chi_{1}\otimes\chi_{2})$ is
irreducible.

Moreover, if $\psi_{1},\psi_{2}\colon A\to\overline{\mathbf{Q}}_{p}^{\times}$
is another pair of smooth characters, then
$\iota^{\widetilde{G}}_{\widetilde{B}_{2}}(\chi_{1}\otimes\chi_{2})\cong\iota^{\widetilde{G}}_{\widetilde{B}_{2}}(\psi_{1}\otimes\psi_{2})$
if and only if
$(\psi_{1}^{\prime},\psi_{2}^{\prime})=(\chi_{1}^{\prime},\chi_{2}^{\prime})$
or
$(\psi_{1}^{\prime},\psi_{2}^{\prime})=(\chi_{2}^{\prime},\chi_{1}^{\prime})$.

  2. (ii)

If $\chi_{1}^{\prime}(\chi_{2}^{\prime})^{-1}=|\cdot|^{-1}$, then there are
non-split short exact sequence

$0\to\tilde{\pi}(\chi_{1},\chi_{2})\to\iota^{\widetilde{G}}_{\widetilde{B}_{2}}(\chi_{1}\otimes\chi_{2})\to\tilde{\pi}(\chi_{2},\chi_{1})\to
0$

and

$0\to\tilde{\pi}(\chi_{2},\chi_{1})\to\iota^{\widetilde{G}}_{\widetilde{B}_{2}}(\chi_{2}\otimes\chi_{1})\to\tilde{\pi}(\chi_{1},\chi_{2})\to
0,$

with $\tilde{\pi}(\chi_{1},\chi_{2})$, $\tilde{\pi}(\chi_{2},\chi_{1})$
irreducible. Moreover, $\tilde{\pi}(\chi_{1},\chi_{2})^{K}\neq 0$ if and only
if $\chi_{2}^{\prime}$ is unramified, in which case
$\tilde{\pi}(\chi_{1},\chi_{2})$ and $\tilde{\pi}(\chi_{2},\chi_{1})$
correspond to $\chi_{2}^{\prime}|\cdot|^{-1/2}\circ\det$ and
$\operatorname{St}\otimes\chi_{2}^{\prime}|\cdot|^{-1/2}\circ\det$,
respectively, under the equivalence (38).

###### Proof.

(i) The irreducibility is [16, Corollary I.2.8], while the intertwinings
follow from the computation of the Jacquet module of a genuine principal
series, see Proposition I.2.1 of loc. cit.

(ii) After twisting, we may assume that $\chi_{2}^{\prime}$ is unramified, in
which case the assertion follows from the equivalence (38) and Lemma 6.3. ∎

For $y\in\overline{\mathbf{Q}}_{p}^{\times}$, let $\tilde{\mu}_{y}\colon
A\to\overline{\mathbf{Q}}_{p}^{\times}$ denote the unique character trivial on
$T\cap K$ and mapping $p^{2}$ to $y$.

######  Corollary 6.5.

Let $r\in\mathbf{Z}_{\geq 0}$ and $a_{p}\in\overline{\mathbf{Q}}_{p}$. Let
$\alpha,\beta\in\overline{\mathbf{Q}}_{p}$ be the roots of the polynomial
$X^{2}-a_{p}X+p^{2(r+1)}$.

  1. (i)

If $a_{p}\notin\\{\pm p^{r+1/2}(1+p)\\}$, then

$\widetilde{\Pi}_{r,a_{p}}\cong\operatorname{Sym}^{r}(\overline{\mathbf{Q}}_{p}^{2})\otimes\operatorname{Ind}^{\widetilde{G}}_{\widetilde{B}_{2}}(\tilde{\mu}_{\alpha^{-1}}\otimes\tilde{\mu}_{p^{2}\beta^{-1}}\boxtimes\iota).$

  2. (ii)

If $a_{p}\in\\{\pm p^{r+1/2}(1+p)\\}$, then there is a short exact sequence

$0\to\operatorname{Sym}^{r}(\overline{\mathbf{Q}}_{p}^{2})\otimes\tilde{\pi}(\tilde{\mu}_{\varepsilon},\tilde{\mu}_{\varepsilon}|\cdot|^{-1/2})\to\widetilde{\Pi}_{r,a_{p}}\to\operatorname{Sym}^{r}(\overline{\mathbf{Q}}_{p}^{2})\otimes\tilde{\pi}(\tilde{\mu}_{\varepsilon}|\cdot|^{-1/2},\tilde{\mu}_{\varepsilon})\to
0,$

where $\varepsilon=\frac{p^{-2r}a_{p}}{p(p+1)}$.

###### Proof.

The case $r=0$ follows from the equivalence (38), Lemma 6.1, Lemma 6.3 and the
corresponding statement for the group $G$, which is a special case of [5,
Proposition 3.2.1]. The general case is now a consequence of this by noting
that the isomorphism

$\displaystyle(\operatorname{Sym}^{r}(\overline{\mathbf{Q}}_{p}^{2})\otimes|\cdot|^{r/2}\circ\det)\otimes\operatorname{c-Ind}^{\widetilde{G}}_{\widetilde{K}Z(\widetilde{G})}(\mathbf{1}\boxtimes\iota)\cong\operatorname{c-Ind}^{\widetilde{G}}_{\widetilde{K}Z(\widetilde{G})}(\tilde{\sigma}_{r}(\overline{\mathbf{Q}}_{p}))$
$\displaystyle v\otimes f\mapsto[g\mapsto f(g)(g.v)]$

identifies $\operatorname{id}\otimes\widetilde{T}_{\overline{\mathbf{Q}}_{p}}$
with $p^{-r}\widetilde{T}_{\overline{\mathbf{Q}}_{p}}$. ∎

###  6.3. Mod-$p$ reduction

Let now $0\leq r\leq p-1$ and $a_{p}\in\overline{\mathbf{Z}}_{p}$.

######  Lemma 6.6.

Suppose that for $0\leq i\leq p-1$ we are given elements
$a_{i}\in\overline{\mathbf{Q}}_{p}$ such that
$\sum_{i=0}^{p-1}a_{i}[\lambda]^{i}\in\overline{\mathbf{Z}}_{p}$ for all
$\lambda\in\mathbf{F}_{p}$. Then $a_{i}\in\overline{\mathbf{Z}}_{p}$

###### Proof.

Choose $L/\mathbf{Q}_{p}$ finite large enough to contain all the $a_{i}$. Let
$\mathcal{O}_{L}$ be the ring of integers with uniformizer $\varpi$ and
residue field $k_{L}$. Let $N\gg 0$ such that
$b_{i}:=\varpi^{N}a_{i}\in\mathcal{O}_{L}$ for all $0\leq i\leq p-1$. By
assumption, $\sum_{i=0}^{p-1}b_{i}[\lambda]^{i}\in\varpi^{N}\mathcal{O}_{L}$
for all $\lambda\in\mathbf{F}_{p}$. We claim that
$b_{i}\in\varpi^{N}\mathcal{O}_{L}$. By induction, we may assume that $N=1$.
In this case, the mod-$\varpi$ reduction $\bar{f}(X)\in k_{L}[X]$ of the
polynomial $\sum_{i=0}^{p-1}b_{i}X^{i}\in\mathcal{O}_{L}[X]$ vanishes on the
set $\mathbf{F}_{p}$ of cardinality $p$, which forces $\bar{f}(X)=0$ for
degree reasons, i.e. $b_{i}\in\varpi\mathcal{O}_{L}$ as desired. ∎

By our restriction on $r$ being at most $p-1$, the following assertion can be
proved in the same way as Breuil shows Corollaire 4.1.2 in [5] by utilizing
the previous lemma, which allows to obtain appropriate versions of Proposition
3.3.3 and Théoréme 4.1.1 in loc. cit.

######  Proposition 6.7.

The $\widetilde{G}$-equivariant map

(39)
$\operatorname{c-Ind}^{\widetilde{G}}_{\widetilde{K}Z(\widetilde{G})}(\tilde{\sigma}_{r}(\overline{\mathbf{Z}}_{p}))/(\widetilde{T}_{\overline{\mathbf{Z}}_{p}}-a_{p})\hookrightarrow\operatorname{c-Ind}^{\widetilde{G}}_{\widetilde{K}Z(\widetilde{G})}(\tilde{\sigma}_{r}(\overline{\mathbf{Q}}_{p}))/(\widetilde{T}_{\overline{\mathbf{Q}}_{p}}-a_{p})=\widetilde{\Pi}_{r,a_{p}}$

is injective and defines a $\widetilde{G}$-stable bounded lattice in the
$\overline{\mathbf{Q}}_{p}[\widetilde{G}]$-module $\widetilde{\Pi}_{r,a_{p}}$,
i.e. a $\overline{\mathbf{Z}}_{p}[\widetilde{G}]$-submodule generating the
whole space over $\overline{\mathbf{Q}}_{p}$ but not containing a
$\overline{\mathbf{Q}}_{p}$-line.

######  Remark 6.8.

  1. (i)

We have

$\operatorname{c-Ind}^{\widetilde{G}}_{\widetilde{K}Z(\widetilde{G})}(\tilde{\sigma}_{r}(\overline{\mathbf{Z}}_{p}))/(\widetilde{T}_{\overline{\mathbf{Z}}_{p}}-a_{p})\otimes_{\overline{\mathbf{Z}}_{p}}\overline{\mathbf{F}}_{p}=\operatorname{c-Ind}^{\widetilde{G}}_{\widetilde{K}Z(\widetilde{G})}(\tilde{\sigma}_{r}(\overline{\mathbf{F}}_{p}))/(\widetilde{T}-\bar{a}_{p}),$

where $\bar{a}_{p}$ is the image of $a_{p}$ under the projection
$\overline{\mathbf{Z}}_{p}\twoheadrightarrow\overline{\mathbf{F}}_{p}$, so
from Theorem 5.13 we deduce a description of the (semi-simplified) mod-$p$
reduction of (any $\widetilde{G}$-stable bounded lattice in)
$\widetilde{\Pi}_{r,a_{p}}$.

  2. (ii)

In fact, Breuil allows $0\leq r\leq 2(p-1)$. But under the more restrictive
assumption $r\leq p-1$ only the $\widetilde{T}^{+}$-part of the Hecke
operator, see Definition 5.6, needs to be considered. We have not attempted to
investigate the more complicated cases $p\leq r\leq 2(p-1)$ in our setting.

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