\section{Introduction}

Recently, the mod-$p$ representation theory of $p$-adic reductive groups has garnered a great deal of attention as a result of its roles in the mod-$p$ and $p$-adic Local Langlands Programs.  The expectation is that there exists a matching between (packets of) smooth mod-$p$ representations of a $p$-adic reductive group and certain Galois representations.  Representations of the group $\textnormal{GL}_2(\mathbb{Q}_p)$ have been widely studied and analyzed, and a (semisimple) mod-$p$ Local Langlands Correspondence has been established by Breuil (\cite{Br03}) based on the explicit determination of the irreducible mod-$p$ representations of $\textnormal{GL}_2(\mathbb{Q}_p)$.  Moreover, this correspondence is compatible with the $p$-adic Local Langlands Correspondence (cf. \cite{Br032}; see also \cite{Br04}, \cite{Co10}, \cite{Em10}, \cite{Ki09}, \cite{Ki10}, \cite{Pas10}).

The smooth irreducible mod-$p$ representations of $\textnormal{SL}_2(\mathbb{Q}_p)$ have recently been classified by Abdellatif in \cite{Ab12}, by examining restrictions of the irreducible representations of $\textnormal{GL}_2(\mathbb{Q}_p)$.  This allowed her to take the first steps towards a mod-$p$ Local Langlands Correspondence for $\textnormal{SL}_2(\mathbb{Q}_p)$.  In addition, her results are the first to consider a mod-$p$ Local Langlands Correspondence with $L$-packets.

Several aspects of the mod-$p$ representation theory of unitary groups have already been considered by Abdellatif in \cite{Ab11}, and by the author and Xu in \cite{KX12}.  In the present article, we utilize the work of Breuil and Abdellatif to investigate the smooth irreducible mod-$p$ representations of the unitary group $\textnormal{U}(1,1)(\mathbb{Q}_{p^2}/\mathbb{Q}_p)$, where $\mathbb{Q}_{p^2}$ denotes the unramified quadratic extension of the field of $p$-adic numbers $\mathbb{Q}_p$.  The irreducible subquotients of parabolically induced representations of $G$ have been classified by Abdellatif in \cite{Ab13}.  We shall be interested in representations which do not arise in this way, which we will refer to as \emph{supercuspidal} representations (we will comment on terminology at the end of this introduction).  These representations are the ones which are expected to play a central role in a potential Local Langlands Correspondence.  We now describe our method to classify these representations.

We start in a more general context.  Denote by $F$ a nonarchimedean local field of odd residual characteristic $p$, and $E$ an unramified quadratic extension.  Let $G$ be the group $\textnormal{U}(1,1)(E/F)$, $G_\textnormal{S} = \textnormal{SU}(1,1)(E/F)$ its derived subgroup, and $I_\textnormal{S}(1)$ the unique pro-$p$ Sylow subgroup of the standard Iwahori subgroup of $G_\textnormal{S}$.  The \emph{pro-$p$-Iwahori-Hecke algebra} $\mathcal{H}_{\ol{\mathbb{F}}_p}(G_\textnormal{S},I_\textnormal{S}(1))$ is the convolution algebra of compactly supported, $\ol{\mathbb{F}}_p$-valued functions on the double coset space $I_\textnormal{S}(1)\backslash G_\textnormal{S}/I_\textnormal{S}(1)$.  As $G_\textnormal{S}\cong \textnormal{SL}_2(F)$, the structure and properties of $\mathcal{H}_{\ol{\mathbb{F}}_p}(G_\textnormal{S},I_\textnormal{S}(1))$ are well-understood (cf. \cite{Ab11}, \cite{Vig05}).  In particular, a classification of finite-dimensional simple right modules for $\mathcal{H}_{\ol{\mathbb{F}}_p}(G_\textnormal{S},I_\textnormal{S}(1))$ is known (see \cite{Ab11}).  We review the necessary results in Chapter 3.

The results of \cite{Ab13} provide a classification of smooth irreducible nonsupercuspidal representations of any connected quasi-split reductive group of relative rank 1.  We make the computations explicit in Section 4, and obtain an explicit description of all irreducible nonsupercuspidal representations of $G$ (Theorem \ref{nonscclass}).  We then investigate the behavior of irreducible representations upon restriction to the derived subgroup $G_\textnormal{S}$.  From the classification of irreducible nonsupercuspidal representations of $G$ and $G_\textnormal{S}$, we obtain a result on the restrictions of supercuspidal representations of $G$ (Proposition \ref{nonsciffnonsc}).

Next, we specialize to the case where $F = \mathbb{Q}_p$ and $E = \mathbb{Q}_{p^2}$.  Under these assumptions, the smooth irreducible supercuspidal representations of $\textnormal{GL}_2(\mathbb{Q}_p)$ have been classified by Breuil in \cite{Br03}.  They are all of the form
$$\pi(r,0,\chi) = (\chi\circ\det)\otimes \frac{\textnormal{c-ind}_{\mathbb{Q}_p^\times\textnormal{GL}_2(\mathbb{Z}_p)}^{\textnormal{GL}_2(\mathbb{Q}_p)}(\textnormal{Sym}^r(\ol{\mathbb{F}}_p^2))}{(T_r)},$$
where $0\leq r \leq p - 1$, $\chi:\mathbb{Q}_p^\times\longrightarrow\ol{\mathbb{F}}_p^\times$ is a smooth character, and $T_r$ is a generator of a certain spherical Hecke algebra.  Results of Abdellatif (\cite{Ab12}) show that one can then obtain the irreducible supercuspidal representations of $\textnormal{SL}_2(\mathbb{Q}_p)$ by restriction; we have
$$\pi(r,0,\chi)|_{\textnormal{SL}_2(\mathbb{Q}_p)} \cong \pi_r~ \oplus~ \pi_{p - 1 - r},$$
where $\pi_r$ is an irreducible supercuspidal $\textnormal{SL}_2(\mathbb{Q}_p)$-subrepresentation of $\pi(r,0,1)$ generated by a distinguished pro-$p$-Iwarhori-fixed vector.  In light of the isomorphism $\textnormal{SL}_2(\mathbb{Q}_p)\cong\textnormal{SU}(1,1)(\mathbb{Q}_{p^2}/\mathbb{Q}_p) = G_\textnormal{S}$, we view the representations $\pi_r$ as smooth representations of $G_\textnormal{S}$.

Using a cohomological argument, we show in Section 5 that the representations $\pi_r$ lift to smooth irreducible representations of $G$, which we denote
$$\cusp{k}{r}$$
for $0\leq r \leq p - 1$ and $0\leq k < p + 1$ (see Definition \ref{defofsc}).  Moreover, we show that every smooth irreducible supercuspidal representation of $\textnormal{U}(1,1)(\mathbb{Q}_{p^2}/\mathbb{Q}_p)$ is of this form (Theorem \ref{scclass}), and thereby obtain a classification of all smooth irreducible representations (Corollary \ref{u11class}).  To conclude, we arrange the irreducible representations into sets called \emph{$L$-packets}, and determine the $L$-packets on $\textnormal{U}(1,1)(\mathbb{Q}_{p^2}/\mathbb{Q}_p)$ explicitly.

In the final section, we define the relevant Galois groups and $L$-groups attached to $G$.  Our definitions are adapted from the complex setting (see \cite{Ro90} for the classical definitions).  Thus, we are led to investigate Langlands parameters associated to the group $G$, that is, homomorphisms
$$\varphi:\textnormal{Gal}(\ol{\mathbb{Q}}_p/\mathbb{Q}_p)\longrightarrow {}^LG = \textnormal{GL}_2(\ol{\mathbb{F}}_p)\rtimes\textnormal{Gal}(\ol{\mathbb{Q}}_p/\mathbb{Q}_p),$$
such that the composition of $\varphi$ with the canonical projection ${}^LG\longrightarrow \textnormal{Gal}(\ol{\mathbb{Q}}_p/\mathbb{Q}_p)$ is the identity map of $\textnormal{Gal}(\ol{\mathbb{Q}}_p/\mathbb{Q}_p)$.  Our first (somewhat surprising) result in this direction is Proposition \ref{nostables}, which asserts that there do not exist any parameters $\varphi$ such that the Galois representation associated to $\varphi|_{\textnormal{Gal}(\ol{\mathbb{Q}}_p/\mathbb{Q}_{p^2})}$ is irreducible.

The aforementioned result suggests that the Langlands parameters associated to $G$ which are of interest are ``reducible'' in some sense.  We therefore consider two intermediate $L$-groups ${}^LJ$ and ${}^LT$, where $J$ is the unique \emph{elliptic endoscopic group} associated to $G$, and $T$ is the maximal torus of $G$.  The Langlands parameters we examine take the following form:
$$\varphi:\textnormal{Gal}(\ol{\mathbb{Q}}_p/\mathbb{Q}_p)\longrightarrow {}^LJ = (\ol{\mathbb{F}}_p^\times\times\ol{\mathbb{F}}_p^\times)\rtimes\textnormal{Gal}(\ol{\mathbb{Q}}_p/\mathbb{Q}_p) \ensuremath{\lhook\joinrel\relbar\joinrel\rightarrow} {}^LG$$
$$\psi:\textnormal{Gal}(\ol{\mathbb{Q}}_p/\mathbb{Q}_p)\longrightarrow {}^LT = (\ol{\mathbb{F}}_p^\times\times\ol{\mathbb{F}}_p^\times)\rtimes\textnormal{Gal}(\ol{\mathbb{Q}}_p/\mathbb{Q}_p) \ensuremath{\lhook\joinrel\relbar\joinrel\rightarrow} {}^LG$$
We stress that the action of $\textnormal{Gal}(\ol{\mathbb{Q}}_p/\mathbb{Q}_p)$ on $\ol{\mathbb{F}}_p^\times\times\ol{\mathbb{F}}_p^\times$ for the group ${}^LJ$ is \emph{not the same} as the action for the group ${}^LT$.
We classify all such parameters in Corollary \ref{u1u1llc} and Proposition \ref{paramsfromT}, and determine the possible equivalences among the parameters (Lemmas \ref{paramequiv}, \ref{paramequiv2}, and \ref{endvsnonend}).

In the complex setting, the parameters coming from ${}^LJ$ play a pivotal role in the representation theory of $G$ (cf. \cite{Ro90}).  In the characteristic $p$ setting, they remain of particular importance.  We denote the parameters factoring through ${}^LJ$ by $\varphi_{k,\ell}$, where $0\leq k,\ell < p + 1$ (see Definition \ref{endparams} for their precise description).  Using equivalences between the parameters $\varphi_{k,\ell}$ and the classification of $L$-packets, we obtain the following:
\begin{thmI}
Suppose $0\leq k,\ell < p + 1$ and $k\neq \ell$.  There exists a bijection between equivalence classes of Langlands parameters coming from the group ${}^LJ$ and $L$-packets of irreducible supercuspidal representations of the group $G = \textnormal{U}(1,1)(\mathbb{Q}_{p^2}/\mathbb{Q}_p)$, given by
$$\varphi_{k,\ell}\longleftrightarrow \{\cusp{\ell}{[k - \ell - 1]},~\cusp{k}{[\ell - k - 1]}\},$$
where $[k - \ell - 1]$ (resp. $[\ell - k - 1]$) denotes the unique integer between $0$ and $p - 1$ equivalent to $k - \ell - 1$ (resp. $\ell - k - 1$) modulo $p + 1$.  Moreover, this bijection is compatible with twisting by characters on both sides.
\end{thmI}

Finally, we extend this bijection to include semisimple nonsupercuspidal representations (Definition \ref{corr}), using the parameters coming from ${}^LT$.  This allows us to make explicit a case of \emph{endoscopic transfer} from irreducible representations of $J$ to semisimple $L$-packets on $G$ (see the remarks following Definition \ref{corr}).

\vspace{\baselineskip}

\noindent\textbf{Remark on Terminology.}  We briefly address our choice of nomenclature.  In the mod-$p$ representation theory of $p$-adic groups, the representations dubbed supersingular are the ones which play a crucial role.  The notion of supersingularity was introduced by Barthel and Livn\'{e} (\cite{BL94} and \cite{BL95}) in their classification of smooth irreducible mod-$p$ representations of $\textrm{GL}_2(F)$.  A smooth representation of $\textrm{GL}_2(F)$ is called \emph{supersingular} if a certain operator of a spherical Hecke algebra acts by zero, while it is called \emph{supercuspidal} if it is not a subquotient of a parabolically induced representation.  Theorems 33, 34, and Corollary 36(1) of \cite{BL94} show that a smooth representation of $\textrm{GL}_2(F)$ admitting a central character is supercuspidal if and only if it is supersingular.  In the present case, the study of spherical Hecke algebras for nonsplit groups has been initiated by Herzig in \cite{He11-2}, Henniart and Vign\'{e}ras in
\cite{HV11}, and Abdellatif in \cite{Ab11}.  Forthcoming work of Abe--Henniart--Herzig--Vign\'{e}ras \cite{AHHV} establishes the equivalence of these two notions for any quasi-split, connected, reductive group.  In anticipation of these results, we will make no distinction between supercuspidal and supersingular representations, and henceforth only use the term supercuspidal.

\vspace{\baselineskip}

\noindent\textbf{Acknowledgements.}   I would like to thank my advisor Rachel Ollivier, for many enlightening discussions throughout the course of working on this article, as well as her encouragement and advice.  I would also like to thank Ramla Abdellatif, Florian Herzig, and Shaun Stevens for many helpful comments and discussions.  During the preparation of this article, support was provided by NSF Grant DMS-0739400.

\vspace{\baselineskip}

\section{Notation}

\subsection{General Notation}\label{notation}
Fix an odd prime number $p$, and let $F$ be a nonarchimedean local field of residual characteristic $p$.  Denote by $\mathfrak{o}_F$ its ring of integers, and by $\mathfrak{p}_F$ the unique maximal ideal of $\mathfrak{o}_F$.  Fix a uniformizer $\varpi_F$ and let $k_F = \mathfrak{o}_F/\mathfrak{p}_F$ denote the (finite) residue field.  The field $k_F$ is a finite extension of $\mathbb{F}_p$ of size $q = p^f$.  We fix also a separable closure $\overline{F}$ of $F$, and let $k_{\ol{F}}$ denote its residue field.

Let $E$ denote the unique unramified extension of degree 2 in $\ol{F}$.  The ring of integers of $E$ is denoted $\mathfrak{o}_E$, and $\mathfrak{p}_E$ is its unique maximal ideal.  Since $E$ is unramified, we may and do take $\varpi_E = \varpi_F =: \varpi$ as our uniformizer.  The residue field of $E$ is $k_E = \mathfrak{o}_E/\mathfrak{p}_E$; it is a quadratic extension of $k_F$.  Let $\iota:k_{\ol{F}}\stackrel{\sim}{\longrightarrow} \ol{\mathbb{F}}_p$ denote a fixed isomorphism, and assume that every $\ol{\mathbb{F}}_p^\times$-valued character factors through $\iota$.  We identify $k_F$ and $k_E$ with $\mathbb{F}_q$ and $\mathbb{F}_{q^2}$, respectively, using the isomorphism $\iota$.  We shall denote by $\mathbb{Q}_p$ the field of $p$-adic numbers, and by $\mathbb{Q}_{p^2}$ its unique unramified quadratic extension in a fixed algebraic closure $\ol{\mathbb{Q}}_p$.

We write $E = F(\sqrt{\epsilon})$, where $\epsilon\in \mathfrak{o}_F^\times$ is some fixed but arbitrary nonsquare unit.  We let $x\longmapsto\ol{x}$ denote the nontrivial Galois automorphism of $E$ fixing $F$, and define $\textnormal{U}(1)(E/F)$ to be the kernel of the norm map
\begin{center}
 \begin{tabular}{rccl}
  $\textnormal{N}_{E/F}:$ & $E^\times$ & $\longrightarrow$ & $F^\times$\\
 & $x$ & $\longmapsto$ &  $x\ol{x}$.
 \end{tabular}
\end{center}
We denote by $\vartheta$ a fixed representative of the nontrivial coset in $\mathfrak{o}_E^\times/\mathfrak{o}_F^\times\textnormal{U}(1)(E/F)$.

Denote by $G$ the $F$-rational points of the algebraic group $\mathbf{U}(1,1)$.  We perform our computations using the following realization of $G$:  let $V$ denote a two-dimensional hyperbolic plane over $E$.  We identify $V$ with $E^2$ by a choice of basis, and for $\vec{x} = (x_1,x_2)^\top, \vec{y} = (y_1,y_2)^\top\in V$ the nondegenerate Hermitian form $\langle\cdot, \cdot\rangle$ is given by $$\langle\vec{x}, \vec{y}\rangle = \ol{x_1}{y_2} + \ol{x_2}{y_1}.$$ Letting $$s = \begin{pmatrix} 0 & 1\\ 1 & 0\end{pmatrix},$$ our form is represented by $\langle\vec{x},\vec{y}\rangle = \vec{x}^{\phantom{'}*}s\vec{y}$, where $m^*$ denotes the conjugate transpose of a matrix $m$ with coefficients in $E$.  With this notation, we have $$G = \{g\in \textrm{GL}_2(E): g^*sg = s\}.$$

Let $K$ denote the maximal compact subgroup of $G$ given by
$$K := \textrm{GL}_2(\mathfrak{o}_E)\cap G,$$
and let
$$K_1 :=\begin{pmatrix}1 + \mathfrak{p}_E & \mathfrak{p}_E \\ \mathfrak{p}_E & 1 + \mathfrak{p}_E\end{pmatrix}\cap G$$
denote its pro-$p$ radical.  The group $K$ is a representative of one of two conjugacy classes of maximal compact subgroups of $G$ (\cite{Ti79}, Sections 2.10 and 3.2).  We define
$$\Gamma := K/K_1 \cong \textnormal{U}(1,1)(\mathbb{F}_{q^2}/\mathbb{F}_{q}).$$
The Iwahori subgroup $I$ is defined as the preimage under the quotient map $K\longrightarrow \Gamma$ of the Borel subgroup of upper triangular matrices in $\Gamma$.  We denote by $I(1)$ the pro-$p$ radical of $I$, which is the preimage of the upper triangular unipotent elements of $\Gamma$.  Explicitly, we have
$$I := \begin{pmatrix}\mathfrak{o}_E^\times & \mathfrak{o}_E \\ \mathfrak{p}_E & \mathfrak{o}_E^\times\end{pmatrix}\cap G,\qquad I(1):=\begin{pmatrix}1 + \mathfrak{p}_E & \mathfrak{o}_E \\ \mathfrak{p}_E & 1 + \mathfrak{p}_E \end{pmatrix}\cap G.$$

Let $B$ denote the Borel subgroup of upper triangular elements of $G$, $U$ its unipotent radical, and $U^-$ the opposite unipotent.  The subgroups $U$ and $U^-$ are both isomorphic to the additive group $F$.  We define
$$u(x):=\begin{pmatrix}1 & x \\ 0 & 1\end{pmatrix},\qquad u^-(x):=\begin{pmatrix}1 & 0 \\ x & 1\end{pmatrix},$$
where $x\in E$ satisfies $x + \ol{x} = 0$.  We have $u(x)^{-1} = u(\ol{x}), u^-(x)^{-1} = u^-(\ol{x})$.

We let $G_\textrm{S}$ denote the derived subgroup of $G$.  For any subgroup $J$ of $G$, we let $J_\textnormal{S}$ denote its intersection with $G_\textnormal{S}$.  We have $G_\textrm{S} = \textnormal{SU}(1,1)(E/F)\cong \textnormal{SL}_2(F)$, the latter isomorphism given by conjugation by the element $$\begin{pmatrix}\sqrt{\epsilon} & 0 \\ 0 & 1 \end{pmatrix}\in \textnormal{GL}_2(E).$$  We shall exploit this isomorphism to give a classification of the smooth irreducible representations of $\textnormal{U}(1,1)(\mathbb{Q}_{p^2}/\mathbb{Q}_p)$.

\vspace{\baselineskip}
We define the following distinguished elements of $\textnormal{GL}_2(E)$:
\begin{center}
\begin{tabular}{cc}
 & \\
$\displaystyle{s:=\begin{pmatrix}0 &  1\\  1 & 0 \end{pmatrix}},$ &  $\displaystyle{s':=\begin{pmatrix} 0  & \varpi^{-1}\\ \varpi & 0 \end{pmatrix}},$\\
 & \\
$\displaystyle{n_s:=\begin{pmatrix}0 & -\sqrt{\epsilon}^{-1}\\ \sqrt{\epsilon}& 0\end{pmatrix}},$ &  $\displaystyle{n_{s'}:=\begin{pmatrix}0 & -\varpi^{-1}\sqrt{\epsilon}^{-1}\\\varpi\sqrt{\epsilon} & 0\end{pmatrix}},$\\
 & \\
$\displaystyle{\alpha := s's = \begin{pmatrix}\varpi^{-1} & 0\\  0 & \varpi\end{pmatrix}},$ & $\displaystyle{\alpha^{-1}:= ss' = \begin{pmatrix}\varpi & 0 \\ 0  & \varpi^{-1}\end{pmatrix}},$\\
 & \\
$\displaystyle{\beta := \begin{pmatrix}0 & 1 \\ \varpi & 0 \end{pmatrix}},$ & $\displaystyle{\theta := \begin{pmatrix}\vartheta & 0 \\ 0 & \ol{\vartheta}^{-1}\end{pmatrix}}.$\\
 &
\end{tabular}
\end{center}

\vspace{\baselineskip}
\subsection{Bruhat Decompositions}  The maximal torus $T$ of $G$ consists of all elements of the form
$$\begin{pmatrix}a & 0 \\ 0 & \ol{a}^{-1}\end{pmatrix},$$
with $a\in E^\times$.  Note that $T$ is not split over $F$.  The maximal torus of $G_\textnormal{S}$ is $T_\textnormal{S} = T\cap G_\textnormal{S}$; it consists of all elements of the form
$$\begin{pmatrix}a & 0 \\ 0 & a^{-1}\end{pmatrix},$$
with $a\in F^\times$, and is split over $F$.  The center $Z$ of $G$ is given by the subgroup of elements
$$\begin{pmatrix}a & 0 \\ 0 & a\end{pmatrix},$$
with $a\in \textnormal{U}(1)(E/F)$.

Let
$$T_0 := T\cap K,\qquad T_1:= T\cap K_1,\qquad H := T_0/T_1 \cong I/I(1) \cong \mathbb{F}_{q^2}^\times.$$
We will identify the characters of $H$ and those of $I/I(1)$.  We will also identify $\mathbb{F}_{q^2}$ with the image of the Teichm\"{u}ller lifting map $[\ \cdot\ ]:\mathbb{F}_{q^2}\longrightarrow \mathfrak{o}_E$ when convenient.

Let $N$ denote the normalizer of $T$ in $G$.  Then the affine Weyl group $W_{\textrm{aff}}$ is defined as $N/T_0$, and the finite Weyl group $W$ is defined as $N/T$.  The group $W_{\textrm{aff}}$ is a Coxeter group, generated by the classes of the two reflections $s$ and $s'$.  We have a decomposition $G = INI$, where two cosets $InI$ and $In'I$ are equal if and only if $n$ and $n'$ have the same image in $W_{\textrm{aff}}$.  This yields the Bruhat decomposition for the BN pair $(I,N)$ (cf. \cite{Ti79}, Section 3.3):
$$G = \bigsqcup_{w\in W_{\textrm{aff}}}IwI;$$
here we engage in the standard abuse of notation, letting $IwI$ denote $I\dot{w}I$ for any preimage $\dot{w}$ of $w$ in $N$.  We will take as our double coset representatives the elements $\alpha^n, n_s\alpha^n$, for $n\in\mathbb{Z}$.

Similarly, we have a Bruhat decomposition for the group $G_\textnormal{S}$:
$$G_\textnormal{S} = \bigsqcup_{w\in N_\textnormal{S}/T_{0,\textnormal{S}}} I_\textnormal{S} wI_\textnormal{S}.$$
In light of the isomorphism $N/T_0 \cong N_\textnormal{S}/T_{0,\textnormal{S}}$, we will take the elements $\alpha^n, n_s\alpha^n$, for $n\in\mathbb{Z}$ as our double coset representatives in $G_\textnormal{S}$.

\vspace{\baselineskip}

\section{Hecke Algebras}

In the course of determining the smooth irreducible representations of $\textnormal{U}(1,1)(\mathbb{Q}_{p^2}/\mathbb{Q}_p)$, we shall make essential use of the pro-$p$-Iwahori-Hecke algebra of the derived subgroup $\textnormal{SU}(1,1)(\mathbb{Q}_{p^2}/\mathbb{Q}_p)$.  We collect the relevant results here.

\vspace{\baselineskip}
\subsection{Preliminaries}  For this subsection only, let $G$ denote any locally profinite group.  We shall be interested in the category $\mathfrak{Rep}_C(G)$ of smooth representations of $G$ over $C$, where $C$ denotes an algebraically closed field.  We briefly recall some preliminary terminology.  Let $J$ be a closed subgroup of $G$, and let $(\sigma,V_\sigma)$ be a smooth $C$-representation of $J$ (meaning that stabilizers are open).  We denote by $\textnormal{ind}_J^{G}(\sigma)$ the space of functions $f:G\longrightarrow V_\sigma$ such that $f(jg) = \sigma(j)f(g)$ for $j\in J, g\in G$, and such that the action of $G$ given by right translation is smooth (meaning that there exists some open subgroup $J'$, depending on $f$, such that $f(gj') = f(g)$ for every $j'\in J', g\in G$).  We let $\textnormal{c-ind}_J^{G}(\sigma)$ denote the subspace of $\textnormal{ind}_J^{G}(\sigma)$ spanned by functions whose support in $J\backslash G$ is compact.  These functors are called \emph{induction} and \emph{compact
induction}, respectively.  We will mostly be concerned with the cases when $J$ is a compact open subgroup, or when $J = \mathbf{J}(F)$ (with $\mathbf{J}$ a parabolic subgroup of a reductive algebraic group $\mathbf{G}$ for which $G = \mathbf{G}(F)$).

\vspace{\baselineskip}
\subsection{Pro-$p$-Iwahori-Hecke Algebra}  Let $\pi$ be a smooth $C$-representation of the group $G_\textnormal{S} = \textnormal{SU}(1,1)(E/F)$.  Frobenius Reciprocity for compact induction gives $$\pi^{I_\textnormal{S}(1)}\cong \textrm{Hom}_{I_\textnormal{S}(1)}(1,\pi|_{I_\textnormal{S}(1)})\cong \textrm{Hom}_{G_\textnormal{S}}(\textrm{c-ind}_{I_\textnormal{S}(1)}^{G_\textnormal{S}}(1),\pi),$$ where $1$ denotes the trivial character of $I_\textnormal{S}(1)$.  The \emph{pro-$p$-Iwahori-Hecke algebra} $$\mathcal{H}_C(G_\textnormal{S},I_\textnormal{S}(1)) := \textnormal{End}_{G_\textnormal{S}}(\textnormal{c-ind}_{I_\textnormal{S}(1)}^{G_\textnormal{S}}(1))$$ is the algebra of ${G_\textnormal{S}}$-equivariant endomorphisms of the universal module $\textnormal{c-ind}_{I_\textnormal{S}(1)}^{G_\textnormal{S}}(1)$.  This algebra has a natural right action on $\textrm{Hom}_{G_\textnormal{S}}(\textrm{c-ind}_{I_\textnormal{S}(1)}^{G_\textnormal{S}}(1),\pi)$ by pre-composition, which induces a right action on $\pi^{I_\textnormal{S}(1)}$.  In this way, we obtain the functor of $I_\textnormal{S}(1)$-invariants, $\pi\longmapsto\pi^{I_\textnormal{S}(1)}$, from the category of smooth $C$-representations of ${G_\textnormal{S}}$ to the category of right $\mathcal{H}_C({G_\textnormal{S}},I_\textnormal{S}(1))$-modules.

By adjunction, we have a natural identification
$$\mathcal{H}_C({G_\textnormal{S}},I_\textnormal{S}(1)) \cong \textnormal{c-ind}_{I_\textnormal{S}(1)}^{G_\textnormal{S}}(1)^{I_\textnormal{S}(1)},$$
so we may view endomorphisms of $\textnormal{c-ind}_{I_\textnormal{S}(1)}^{G_\textnormal{S}}(1)$ as compactly supported functions on ${G_\textnormal{S}}$ which are $I_\textnormal{S}(1)$-biinvariant.  This leads to the following definition.

\begin{defn}
 Let $g\in {G_\textnormal{S}}$.  We let $\textnormal{T}_g\in \mathcal{H}_C({G_\textnormal{S}},I_\textnormal{S}(1))$ denote the endomorphism of $\textnormal{c-ind}_{I_\textnormal{S}(1)}^{G_\textnormal{S}}(1)$ corresponding by adjunction to the characteristic function of ${I_\textnormal{S}(1)gI_\textnormal{S}(1)}$; in particular, $\textnormal{T}_g$ maps the characteristic function of $I_\textnormal{S}(1)$ to the characteristic function of $I_\textnormal{S}(1)gI_\textnormal{S}(1)$.
\end{defn}

From this definition it is clear that $\textnormal{T}_g = \textnormal{T}_{g'}$ if and only if $I_\textnormal{S}(1)gI_\textnormal{S}(1) = I_\textnormal{S}(1)g'I_\textnormal{S}(1)$; moreover, since $W_{\textrm{aff}} = N_\textnormal{S}/T_{0,\textnormal{S}}$ is a set of representatives for the double coset space $I_\textnormal{S}\backslash {G_\textnormal{S}}/I_\textnormal{S}$, the group $N_\textnormal{S}/T_{1,\textnormal{S}}$ gives a set of representatives for $I_\textnormal{S}(1)\backslash {G_\textnormal{S}}/I_\textnormal{S}(1)$.  We therefore only consider the operators $\textnormal{T}_n$, where $n$ is a representative of a coset in $N_\textnormal{S}/T_{1,\textnormal{S}}$.  These operators give a basis for $\mathcal{H}_C({G_\textnormal{S}},I_\textnormal{S}(1))$ as a vector space over $C$.  Using the isomorphisms above, we see that if $\pi$ is a smooth $C$-representation of ${G_\textnormal{S}}$, $v\in \pi^{I_\textnormal{S}(1)}$, and $n\in N_\textnormal{S}$, then
\begin{equation}\label{act}
v\cdot \textnormal{T}_n = \sum_{u \in I_\textnormal{S}(1)\backslash I_\textnormal{S}(1)nI_\textnormal{S}(1)}u^{-1}.v = \sum_{u\in I_\textnormal{S}(1)/I_\textnormal{S}(1)\cap n^{-1}I_\textnormal{S}(1)n} un^{-1}.v.
\end{equation}
For more details, see \cite{Vig05}.

\vspace{\baselineskip}
\subsection{Decomposition with Idempotents} We assume henceforth that $C = \ol{\mathbb{F}}_p$.  Let $H_\textnormal{S} := T_{0,\textnormal{S}}/T_{1,\textnormal{S}} \cong \mathbb{F}_q^\times$, and denote by $\widehat{H_\textnormal{S}}$ the group of all $\ol{\mathbb{F}}_p^\times$-valued characters of $H_\textnormal{S}$.  For $r\in \mathbb{Z}$, we denote by $\chi_r$ the character of $H_\textnormal{S}$ given by $$\chi_r\begin{pmatrix}a & 0 \\ 0 & a^{-1}\end{pmatrix} = a^r,$$ where $a\in \mathbb{F}_q^\times$.  Every element of $\widehat{H_\textnormal{S}}$ is of the form $\chi_r$ with $0\leq r < q - 1$.

\begin{defn}
 For an $\ol{\mathbb{F}}_p$-character $\chi_r$ of $H_\textnormal{S}$, we define $$e_r = |H_\textnormal{S}|^{-1}\sum_{h\in H_\textnormal{S}}\chi_r(h)\textnormal{T}_h,$$ where $\textnormal{T}_h$ denotes the operator $\textnormal{T}_{t_0}$ for any preimage of $t_0$ of $h$ in $T_{0,\textnormal{S}}$.
\end{defn}

The following properties of the operators $e_r$ follow readily from the orthogonality relations of characters:
\begin{itemize}
 \item $e_re_r = e_r$;
 \item $e_re_{r'} = 0$ for $\chi_r \neq \chi_{r'}$;
 \item $\displaystyle{\textnormal{T}_{\textnormal{id}} = \sum_{r = 0}^{q-2} e_r}$.
\end{itemize}
Using these orthogonal idempotents $e_r$, we obtain the following structure theorem.  We note that the results of \cite{Vig05} apply in this setting, since the group $G_\textnormal{S}$ is \emph{split}.

\begin{thm}\label{gens}
The operators $\textnormal{T}_{n_s}$, $\textnormal{T}_{n_{s'}}$ and $e_r$, for $0\leq r < q-1$, generate $\mathcal{H}_{\ol{\mathbb{F}}_p}(G_\textnormal{S}, I_\textnormal{S}(1))$ as an algebra.
\end{thm}

\begin{proof}
 The claim follows from (the remark following) Theorem 1 of \cite{Vig05}, along with Fourier inversion on the elements $e_r$.
\end{proof}

In the study of Hecke modules over fields of characteristic $p$, the notion of supersingularity plays a prominent role.  We recall the definition here.

\begin{defn}[\cite{Vig05}, Definition 3]\label{defofssing}
 Let $M$ be a nonzero simple right $\mathcal{H}_{\ol{\mathbb{F}}_p}(G_\textnormal{S},I_\textnormal{S}(1))$-module which admits a central character.  We say $M$ is \emph{supersingular} if every element of the center of $\mathcal{H}_{\ol{\mathbb{F}}_p}(G_\textnormal{S},I_\textnormal{S}(1))$ which is of ``positive length'' acts by 0.  For the precise notion of ``positive length,'' see the discussion preceding Definition 2 (\emph{loc. cit.}).
\end{defn}

The finite-dimensional simple right $\mathcal{H}_{\ol{\mathbb{F}}_p}(G_\textnormal{S}, I_\textnormal{S}(1))$-modules have been classified in Chapitre 6 of \cite{Ab11}.  The supersingular modules take on a particularly simple form:

\begin{prop}\label{ssingmods}
 The supersingular $\mathcal{H}_{\ol{\mathbb{F}}_p}(G_\textnormal{S},I_\textnormal{S}(1))$-modules are one-dimensional.  They are given by:
\begin{center}
  \begin{tabular}{ccrclccclcccl}
   $M_{0}$ & : & $e_{0}$ & $\longmapsto$ & $1$, & & $\textnormal{T}_{n_s}$ & $\longmapsto$ & $\phantom{-}0$, & & $\textnormal{T}_{n_{s'}}$ & $\longmapsto$ & $-1$;\\
   $M_{q-1}$ & : & $e_{q-1}$ & $\longmapsto$ & $1$, & & $\textnormal{T}_{n_s}$ & $\longmapsto$ & $-1$, & & $\textnormal{T}_{n_{s'}}$ & $\longmapsto$ & $\phantom{-}0$;\\
   $M_{r}$ & : & $e_{r}$ & $\longmapsto$ & $1$, & & $\textnormal{T}_{n_s}$ & $\longmapsto$ & $\phantom{-}0$, & & $\textnormal{T}_{n_{s'}}$ & $\longmapsto$ & $\phantom{-}0$,
  \end{tabular}
 \end{center}
where $0 < r < q-1.$
\end{prop}

In order to make use of the machinery of Hecke modules, we will need a precise relationship between smooth representations of $\textnormal{SU}(1,1)(E/F)$ and $\mathcal{H}_{\ol{\mathbb{F}}_p}(G_\textnormal{S},I_\textnormal{S}(1))$-modules.  The following theorems provide us with the necessary link.

\begin{thm}[\cite{Ab11}, Corollaire 6.1.10 (i)]\label{bij1}
 The functor of $I_\textnormal{S}(1)$-invariants $\pi\longmapsto \pi^{I_\textnormal{S}(1)}$ induces a bijection between isomorphism classes of smooth, irreducible, nonsupercuspidal representations of the group $\textnormal{SU}(1,1)(E/F)$ and isomorphism classes of simple, finite-dimensional, nonsupersingular right $\mathcal{H}_{\ol{\mathbb{F}}_p}(G_\textnormal{S},I_\textnormal{S}(1))$-modules.
\end{thm}

\begin{thm}[\cite{Ab11}, Corollaire 6.1.10 (ii)]\label{bij2}
 The functor of $I_\textnormal{S}(1)$-invariants $\pi\longmapsto \pi^{I_\textnormal{S}(1)}$ induces a bijection between isomorphism classes of smooth, irreducible representations of the group $\textnormal{SU}(1,1)(\mathbb{Q}_{p^2}/\mathbb{Q}_p)$ and isomorphism classes of simple, finite-dimensional right $\mathcal{H}_{\ol{\mathbb{F}}_p}(G_\textnormal{S},I_\textnormal{S}(1))$-modules.  Under this bijection, supercuspidal representations correspond to supersingular modules.
\end{thm}

\vspace{\baselineskip}

\section{Nonsupercuspidal Representations}\label{secnonsc}
Let $\mathbb{G}$ be the $F$-rational points of a connected quasi-split reductive group of relative rank 1.  The smooth, irreducible, nonsupercuspidal representations of $\mathbb{G}$ have been classified by Abdellatif in \cite{Ab13}.  In particular, these results apply to the group $G = \textnormal{U}(1,1)(E/F)$.  We recall the results here.

\begin{thm}[\cite{Ab13}, Th\'eor\`eme 1.1]\label{ramla1}
 Let $\chi:T\longrightarrow \ol{\mathbb{F}}_p^\times$ be a smooth character of $T$, which we inflate to a smooth character of $B$.
\begin{enumerate}[(1)]
 \item As a $B$-module, the $\ol{\mathbb{F}}_p$-representation $\textnormal{ind}_B^G(\chi)|_{B}$ is of length 2, with irreducible subquotients given by the character $\chi$ and the representation $V_\chi$, consisting of elements of $\textnormal{ind}_B^G(\chi)$ which take the value $0$ at the identity.
 \item The following are equivalent:
\begin{enumerate}[(a)]
  \item The $B$-module $\textnormal{ind}_B^G(\chi)|_B$ is semisimple;
  \item The $G$-module $\textnormal{ind}_B^G(\chi)$ is reducible;
  \item The $B$-character $\chi$ extends to a character of $G$.
\end{enumerate}
 \item If $\chi$ extends to a character of $G$, then the $G$-module $\textnormal{ind}_B^G(\chi)$ admits as subquotients the representation $\chi$ (as a subrepresentation) and the representation $\chi\otimes\textnormal{St}_G$ (as a quotient).  Here $$\textnormal{St}_G := \textnormal{ind}_B^G(1_B)/1_G$$ denotes the Steinberg representation of $G$, and $1_B$ and $1_G$ denote the trivial characters of $B$ and $G$, respectively.  The short exact sequence
$$0 \longrightarrow \chi \longrightarrow \textnormal{ind}_B^G(\chi)\longrightarrow \chi\otimes\textnormal{St}_G\longrightarrow 0$$
does not split.
\end{enumerate}
\end{thm}

This theorem shows that the irreducible nonsupercuspidal representations divide into three families.  The next result demonstrates the lack of isomorphisms between these representations.

\begin{thm}[\cite{Ab13}, Th\'eor\`eme 1.2, Section 3.3]\label{ramla2}
There do not exist any isomorphisms between representations from distinct families.  If $\chi$ and $\chi'$ are two characters of $B$ which extend to $G$ (resp. do not extend to $G$) and there exists an isomorphism $\chi\otimes\textnormal{St}_G\cong\chi'\otimes\textnormal{St}_G$ (resp. $\textnormal{ind}_B^G(\chi)\cong \textnormal{ind}_B^G(\chi')$), then $\chi = \chi'$.
\end{thm}

We make Theorem \ref{ramla1} explicit.  For any finite extension $L$ of $F$, we let $\omega$ denote the character of $L^\times$ whose value at $\varpi_L$ is 1, and whose restriction to $\mathfrak{o}_L^\times$ is given by the composition
\begin{equation}\label{omega}
\omega:\mathfrak{o}_L^\times \stackrel{\mathfrak{r}_L}{\longrightarrow} k_L^\times \stackrel{\iota}{\longrightarrow} \ol{\mathbb{F}}_p^\times,
\end{equation}
where $\mathfrak{r}_L:\mathfrak{o}_L\longrightarrow k_L$ denotes the reduction modulo the maximal ideal.  For $\lambda\in \ol{\mathbb{F}}_p^\times$, we denote by $\mu_\lambda:L^\times\longrightarrow \ol{\mathbb{F}}_p^\times$ the unramified character taking the value $\lambda$ at $\varpi_L$.  In this notation, we have that any smooth character of $E^\times$ (resp. $F^\times$) is of the form $\mu_\lambda\omega^r$ for a unique $\lambda\in\ol{\mathbb{F}}_p^\times$ and a unique $0\leq r < q^2 - 1$ (resp. $0\leq r < q - 1$).  Likewise, any smooth character of $\textnormal{U}(1)(E/F)$ is of the form $\omega^r$ for a unique $0\leq r < q+1$.

Since $T\cong E^\times$, we will identify the smooth characters of $T$ (and $B$) with those of $E^\times$, by the formula
$$\mu_\lambda\omega^r\begin{pmatrix}a & b \\ 0 & \ol{a}^{-1}\end{pmatrix} := \mu_\lambda\omega^r(a).$$
Assume now that the character $\mu_\lambda\omega^r$ extends to a character of $G$.  This extension must therefore be trivial on the derived subgroup $G_\textnormal{S}$ and its maximal torus $T_\textnormal{S}$.  In particular, this implies
\begin{center}
\begin{tabular}{ccccc}
$\lambda$ &  = & $\mu_\lambda\omega^r(\varpi)$ & = & $1$,\\
$a^r$ & = & $\mu_\lambda\omega^r([a])$ & = & $1$,
\end{tabular}
\end{center}
where $a\in \mathbb{F}_q^\times$ is arbitrary.  Hence, we see that if the character $\mu_\lambda\omega^r$ extends to $G$, we must have $\lambda = 1$ and $r = (q - 1)m$ for $0\leq m < q + 1$.  The converse statement is easily verified, and combining Theorems \ref{ramla1} and \ref{ramla2}, we obtain the following theorem.

\begin{thm}\label{nonscclass}
 Let $\pi$ be a smooth, irreducible, nonsupercuspidal representation of the group $\textnormal{U}(1,1)(E/F)$.  Then $\pi$ is isomorphic to one and only one of the following representations:
\begin{itemize}
 \item the smooth $\ol{\mathbb{F}}_p$-characters $\omega^k\circ\det$, where $0\leq k < q + 1;$
 \item twists of the Steinberg representation $(\omega^k\circ\det)\otimes\textnormal{St}_G$, where $0\leq k < q + 1;$
 \item the principal series representations $\textnormal{ind}_B^G(\mu_\lambda\omega^r)$, where $\lambda\in\ol{\mathbb{F}}_p^\times$ and $0\leq r < q^2 - 1$ with $(r, \lambda)\neq ((q - 1)m,1)$.
\end{itemize}
\end{thm}

In addition to this classification, we will also need to know how the nonsupercuspidal representations behave upon restriction to $\textnormal{SU}(1,1)(E/F)$.

\begin{thm}\label{nonscres}  We have the following isomorphisms:
\begin{enumerate}[(a)]
 \item $\omega^k\circ\det|_{G_\textnormal{S}} \cong 1_{G_\textnormal{S}}$, where $1_{G_\textnormal{S}}$ denotes the trivial character of $G_\textnormal{S}$;
 \item $\textnormal{ind}_B^G(\mu_\lambda\omega^r)|_{G_\textnormal{S}} \cong \textnormal{ind}_{B_\textnormal{S}}^{G_\textnormal{S}}(\mu_\lambda\omega^{r'})$, where $r'$ denotes the unique integer such that $0\leq r' < q - 1$ and $r' \equiv r~ (\textnormal{mod}~ q - 1)$;
 \item $(\omega^k\circ\det)\otimes\textnormal{St}_G|_{G_\textnormal{S}} \cong \textnormal{St}_{G_\textnormal{S}}$, where $\textnormal{St}_{G_\textnormal{S}} := \textnormal{ind}_{B_\textnormal{S}}^{G_\textnormal{S}}(1_{B_\textnormal{S}})/1_{G_\textnormal{S}}$ is the Steinberg representation of $G_\textnormal{S}$.
\end{enumerate}
\end{thm}

\begin{proof}
 We proceed as in the proof of Th\'eor\`eme 2.16 of \cite{Ab12}.  Part (a) follows from the definition of $G_\textnormal{S}$.  For part (b), we note that $G = BG_\textnormal{S} = G_\textnormal{S} B$, and use the Mackey decomposition:
\begin{eqnarray*}
\textnormal{ind}_B^G(\mu_\lambda\omega^r)|_{G_\textnormal{S}} & \cong & \bigoplus_{g\in G_\textnormal{S} \backslash G/B} \textnormal{ind}_{gBg^{-1}\cap G_\textnormal{S}}^{G_\textnormal{S}}((\mu_\lambda\omega^r)^{g^{-1}}|_{gBg^{-1}\cap G_\textnormal{S}})\\
 & = & \textnormal{ind}_{B_\textnormal{S}}^{G_\textnormal{S}}(\mu_\lambda\omega^r|_{B_\textnormal{S}})\\
 & = & \textnormal{ind}_{B_\textnormal{S}}^{G_\textnormal{S}}(\mu_\lambda\omega^{r'}).
\end{eqnarray*}

This isomorphism gives $(\omega^{k}\circ\det)\otimes\textnormal{ind}_B^G(1_B)|_{G_\textnormal{S}} \cong \textnormal{ind}_B^G(\omega^{(1-q)k})|_{G_\textnormal{S}} \cong \textnormal{ind}_{B_\textnormal{S}}^{G_\textnormal{S}}(1_{B_\textnormal{S}})$.  Taking the quotient of $\textnormal{ind}_{B_\textnormal{S}}^{G_\textnormal{S}}(1_{B_\textnormal{S}})$ by the subrepresentation consisting of constant functions induces an injection $$(\omega^k\circ\det)\otimes\textnormal{St}_{G}|_{G_\textnormal{S}} = \textnormal{ind}_B^G(\omega^{(1-q)k})/(\omega^k\circ\det)|_{G_\textnormal{S}} \ensuremath{\lhook\joinrel\relbar\joinrel\rightarrow} \textnormal{ind}_{B_\textnormal{S}}^{G_\textnormal{S}}(1_{B_\textnormal{S}})/1_{G_\textnormal{S}} = \textnormal{St}_{G_\textnormal{S}}.$$
Proposition 2.7 (\emph{loc. cit.}) shows that $\textnormal{St}_{G_\textnormal{S}}$ is irreducible as a representation of $G_\textnormal{S}$, and therefore the injection is an isomorphism.  This proves (c).
\end{proof}

\begin{cor}\label{nonscres2}
 If $\pi$ is a smooth irreducible nonsupercuspidal representation of $\textnormal{U}(1,1)(E/F)$, then $\pi$ remains irreducible and nonsupercuspidal upon restriction to $\textnormal{SU}(1,1)(E/F)$.
\end{cor}

\begin{proof}
 This follows from the two preceding Theorems and Proposition 2.7 of \cite{Ab12}.
\end{proof}

\vspace{\baselineskip}

In determining the representations of $\textnormal{U}(1,1)(\mathbb{Q}_{p^2}/\mathbb{Q}_p)$, we shall need to know how supercuspidal representations behave upon restriction to $\textnormal{SU}(1,1)(\mathbb{Q}_{p^2}/\mathbb{Q}_p)$.  We record two results in this direction.

\begin{prop}\label{LLss}
 Let $\pi$ be a smooth irreducible representation of $G$.  Then $\pi$ admits a central character, and the restriction $\pi|_{G_\textnormal{S}}$ is semisimple of length at most 2.
\end{prop}

\begin{proof}
We proceed using a method outlined in Lemma 2.4 of \cite{LL79}.  We first note that, since $K_1$ is a pro-$p$ group, the vector space $\pi^{K_1}$ is nonzero, and has an action of the group $K$.  This action factors through $\Gamma$, and we let $\sigma\subset \pi^{K_1}$ be an irreducible $\Gamma$-subrepresentation (such a representation always exists since $\Gamma$ is finite).  As $\sigma$ is irreducible, the center of $K$ acts on $\sigma$ by a character.  The injection
$$\sigma\ensuremath{\lhook\joinrel\relbar\joinrel\rightarrow} \pi|_K$$
gives, by Frobenius Reciprocity, a $G$-equivariant surjection of $\textnormal{c-ind}_K^G(\sigma)$ onto $\pi$.  Since $K$ contains the center $Z$, we conclude that $\pi$ admits a central character.

The subgroup $ZG_\textnormal{S}$ is an index 2 subgroup of $G$; the choice of $\vartheta$ made in Subsection \ref{notation} implies that the matrix $\theta$ represents the nontrivial coset.  Consider first the restriction $\pi|_{ZG_\textnormal{S}}$, and suppose that it is reducible, with a nonzero proper subrepresentation $\tau$.  Let $V_\tau$ denote the underlying vector space of $\tau$.  It then follows that the space $V_\tau + \theta.V_\tau$ is nonzero and stable by $G$, so must be all of $\pi$, by irreducibility.  Likewise, the space $V_\tau \cap \theta.V_\tau$ is stable by $G$, and hence must be $0$.  Thus we see that
$$\pi|_{ZG_\textnormal{S}} \cong \tau\oplus\tau^\theta,$$
where $\tau^\theta$ denotes the representation with the same underlying space as $\tau$, with the action given by first conjugating an element of $ZG_\textnormal{S}$ by $\theta$.  Moreover, the same argument shows that $\tau$ must be irreducible as a representation of $ZG_\textnormal{S}$.  Since $\pi$ admits a central character, the representations $\tau$ and $\tau^\theta$ will remain irreducible upon further restricting to $G_\textnormal{S}$.  Therefore, we see that if $\pi$ is a smooth irreducible representation of $G$, the restriction $\pi|_{G_\textnormal{S}}$ is semisimple of length at most 2.
\end{proof}

\begin{prop}\label{nonsciffnonsc}
 Let $\pi$ be a smooth irreducible representation of $G$, and let $\tau \subset \pi|_{G_\textnormal{S}}$ be a nonzero irreducible $G_\textnormal{S}$-subrepresentation.  Then $\pi$ is supercuspidal if and only if $\tau$ is supercuspidal.
\end{prop}

\begin{proof}
 Suppose $\pi$ is not supercuspidal, so that $\pi$ is isomorphic to either a character of $G$, a twist of the Steinberg representation, or a principal series representation.  Theorem \ref{nonscres} implies that the restriction $\pi|_{G_\textnormal{S}}$ must be the trivial character of $G_\textnormal{S}$, the Steinberg representation of $G_\textnormal{S}$, or an irreducible principal series for $G_\textnormal{S}$, respectively.  Therefore $\tau$ cannot be supercuspidal.

 Suppose now that $\pi|_{G_\textnormal{S}}$ contains a nonsupercuspidal representation $\tau$.  If $\tau$ is the trivial character of $G_\textnormal{S}$, then Proposition \ref{LLss} implies $\pi$ is finite-dimensional.  This implies by smoothness and irreducibility that $\pi$ must be a character, and hence not supercuspidal.  We may therefore assume that $\tau$ is a quotient of a parabolically induced representation, that is to say, there exists $\chi:B_\textnormal{S}\longrightarrow \ol{\mathbb{F}}_p^\times$ such that $\tau$ is a quotient of $\textnormal{ind}_{B_\textnormal{S}}^{G_\textnormal{S}}(\chi)$.  Let $\widetilde{\chi}:ZB_\textnormal{S}\longrightarrow \ol{\mathbb{F}}_p^\times$ denote the character whose restriction to $B_\textnormal{S}$ is $\chi$, and whose restriction to $Z$ is the central character of $\pi$.  Mackey theory now implies that
$$\textnormal{Hom}_{ZG_\textnormal{S}}(\textnormal{ind}_{ZB_\textnormal{S}}^{ZG_\textnormal{S}}(\widetilde{\chi}), \pi|_{ZG_\textnormal{S}})\neq 0.$$
Using Frobenius Reciprocity and transitivity of induction, we obtain
\begin{eqnarray*}
 \textnormal{Hom}_{ZG_\textnormal{S}}(\textnormal{ind}_{ZB_\textnormal{S}}^{ZG_\textnormal{S}}(\widetilde{\chi}), \pi|_{ZG_\textnormal{S}}) & \cong & \textnormal{Hom}_{G}(\textnormal{ind}_{ZG_\textnormal{S}}^G(\textnormal{ind}_{ZB_\textnormal{S}}^{ZG_\textnormal{S}}(\widetilde{\chi})), \pi)\\
 & \cong & \textnormal{Hom}_{G}(\textnormal{ind}_{ZB_\textnormal{S}}^G(\widetilde{\chi}), \pi)\\
 & \cong & \textnormal{Hom}_{G}(\textnormal{ind}_{B}^G(\textnormal{ind}_{ZB_\textnormal{S}}^{B}(\widetilde{\chi})), \pi)\\
 & \cong & \textnormal{Hom}_{G}(\textnormal{ind}_{B}^G(\chi'\oplus\chi''), \pi),
\end{eqnarray*}
where $\chi'$ and $\chi''$ are the two extensions of $\widetilde{\chi}$ to $B$.  From this we see that $\pi$ is a quotient of a parabolically induced representation, and is therefore not supercuspidal.
\end{proof}

\vspace{\baselineskip}

\section{Supercuspidal Representations}

We suppose from this point onwards that $F = \mathbb{Q}_p, E = \mathbb{Q}_{p^2}$, and $\varpi = p$.  We let $\mathbb{Z}_p$ and $\mathbb{Z}_{p^2}$ denote the rings of integers of $\mathbb{Q}_p$ and $\mathbb{Q}_{p^2}$, respectively.

The supercuspidal representations of $\textnormal{GL}_2(\mathbb{Q}_p)$ and $\textnormal{SL}_2(\mathbb{Q}_p)$ have been classified by Breuil and Abdellatif, respectively (cf. \cite{Br03}, \cite{Ab12}).  We review their results here.

\vspace{\baselineskip}
\subsection{The Group $\textnormal{GL}_2(\mathbb{Q}_p)$}\label{gl2sc} Let $0\leq r \leq p - 1$ be an integer.  Denote by $\sigma_r = \textnormal{Sym}^r(\ol{\mathbb{F}}_p^2)$ the $\ol{\mathbb{F}}_p$-vector space of homogeneous polynomials in two variables of degree $r$, with an action of $\textnormal{GL}_2(\mathbb{Z}_p)$ given by
\begin{eqnarray*}
 \begin{pmatrix}a & b \\ c & d\end{pmatrix}.x^{r - i}y^i & = & (\mathfrak{r}_{\mathbb{Q}_p}(a)x + \mathfrak{r}_{\mathbb{Q}_p}(c)y)^{r - i}(\mathfrak{r}_{\mathbb{Q}_p}(b)x + \mathfrak{r}_{\mathbb{Q}_p}(d)y)^i,
\end{eqnarray*}
where $\mathfrak{r}_{\mathbb{Q}_p}:\mathbb{Z}_p\longrightarrow \mathbb{F}_p$ is the reduction map defined in Section \ref{secnonsc}.  We denote by $\mathbb{Q}_p^\times$ the center of $\textnormal{GL}_2(\mathbb{Q}_p)$, and extend the above action to $\mathbb{Q}_p^\times\textnormal{GL}_2(\mathbb{Z}_p)$ by letting $p\cdot \textnormal{id}$ act trivially.  Proposition 8 of \cite{BL94} shows that the algebra of $\textnormal{GL}_2(\mathbb{Q}_p)$-equivariant endomorphisms of the compactly induced representation $\textnormal{c-ind}_{\mathbb{Q}_p^\times\textnormal{GL}_2(\mathbb{Z}_p)}^{\textnormal{GL}_2(\mathbb{Q}_p)}(\sigma_r)$ is isomorphic to a polynomial algebra over $\ol{\mathbb{F}}_p$ in one variable, generated by an endomorphism denoted $T_r$.  For a smooth character $\chi$ of $\mathbb{Q}_p^\times$, we denote by $\pi(r,0,\chi)$ the representation of $\textnormal{GL}_2(\mathbb{Q}_p)$ afforded by the cokernel of the map $T_r$, twisted by $\chi$~:
$$\pi(r,0,\chi) := (\chi\circ\det)\otimes\frac{\textnormal{c-ind}_{\mathbb{Q}_p^\times\textnormal{GL}_2(\mathbb{Z}_p)}^{\textnormal{GL}_2(\mathbb{Q}_p)}(\sigma_r)}{(T_r)}~.$$

The necessary properties of the representations $\pi(r,0,\chi)$ are summarized in the following theorem, proved by Barthel-Livn\'{e} and Breuil.

\begin{thm}\label{BLandBr}
 In the following, $r$ denotes an integer $0\leq r \leq p - 1$ and $\chi:\mathbb{Q}_p^\times\longrightarrow\ol{\mathbb{F}}_p^\times$ a smooth character.
\begin{enumerate}[(a)]
 \item Every smooth irreducible supercuspidal representation of $\textnormal{GL}_2(\mathbb{Q}_p)$ is of the form $\pi(r,0,\chi)$.
 \item The only isomorphisms among the representations $\pi(r,0,\chi)$ are the following~:
\begin{eqnarray*}
 \pi(r,0,\chi) & \cong & \pi(r,0,\chi\mu_{-1})\\
 \pi(r,0,\chi) & \cong & \pi(p - 1 - r,0,\chi\omega^r)\\
 \pi(r,0,\chi) & \cong & \pi(p - 1 - r,0,\chi\mu_{-1}\omega^r).
\end{eqnarray*}
 \item Let $Iw(1)$ denote the standard upper pro-$p$-Iwahori subgroup of $\textnormal{GL}_2(\mathbb{Q}_p)$.  We have $$\pi(r,0,1)^{Iw(1)} = \ol{\mathbb{F}}_p\ol{[\textnormal{id}, x^r]} \oplus \ol{\mathbb{F}}_p\ol{[\beta,x^r]},$$ where, for $g\in \textnormal{GL}_2(\mathbb{Q}_p)$ and $v\in\sigma_r$, $\ol{[g,v]}$ denotes the image in $\pi(r,0,1)$ of the element $[g,v]$ of $\textnormal{c-ind}_{\mathbb{Q}_p^\times\textnormal{GL}_2(\mathbb{Z}_p)}^{\textnormal{GL}_2(\mathbb{Q}_p)}(\sigma_r)$ with support $\mathbb{Q}_p^\times\textnormal{GL}_2(\mathbb{Z}_p)g^{-1}$ and value $v$ at $g$.
\end{enumerate}
\end{thm}

\begin{proof}
 This follows from Theorems 33, 34 and Corollary 36 of \cite{BL94}, and Th\'eor\`eme 3.2.4 and Corollaires 4.1.1, 4.1.4, and 4.1.5 of \cite{Br03}.  We remark that the hypothesis of having a central character made in \cite{BL94}, \cite{BL95}, and \cite{Br03} may be omitted by \cite{Be11}.
\end{proof}

\vspace{\baselineskip}
\subsection{The Group $\textnormal{SL}_2(\mathbb{Q}_p)$} Consider now the group $\textnormal{SL}_2(\mathbb{Q}_p)$.  Th\'eor\`eme 3.36 of \cite{Ab12} implies that for any smooth irreducible representation $\sigma$ of $\textnormal{SL}_2(\mathbb{Q}_p)$, there exists a smooth irreducible representation $\Sigma$ of $\textnormal{GL}_2(\mathbb{Q}_p)$ such that $\sigma$ is a Jordan-H\"older factor of $\Sigma$.  Moreover, Corollaires 3.38 and 3.41 (\emph{loc. cit.}) imply that in order to classify supercuspidal representations of $\textnormal{SL}_2(\mathbb{Q}_p)$, it suffices to compute the restriction of the representations $\pi(r,0,1)$.  Let $\pi_{r,\infty}$ denote the $\textnormal{SL}_2(\mathbb{Q}_p)$-subrepresentation of $\pi(r,0,1)|_{\textnormal{SL}_2(\mathbb{Q}_p)}$ generated by $\ol{[\textnormal{id}, x^r]}$, and let $\pi_{r,0}$ denote the $\textnormal{SL}_2(\mathbb{Q}_p)$-subrepresentation of $\pi(r,0,1)|_{\textnormal{SL}_2(\mathbb{Q}_p)}$ generated by $\ol{[\beta,x^r]}$.

\begin{thm}\label{Ramla}
 In the following, $r$ denotes an integer $0\leq r \leq p - 1$.
\begin{enumerate}[(a)]
 \item We have $\pi(r,0,1)|_{\textnormal{SL}_2(\mathbb{Q}_p)} \cong \pi_{r,\infty} \oplus \pi_{r,0}$.
 \item The representations $\pi_{r,0}$ and $\pi_{r,\infty}$ are smooth, irreducible, admissible, and supercuspidal.
 \item Conversely, any smooth irreducible supercuspidal representation of $\textnormal{SL}_2(\mathbb{Q}_p)$ is isomorphic to one of the form $\pi_{r,0}$ or $\pi_{r,\infty}$.
 \item The only isomorphisms among the representations $\pi_{r,0}$ and $\pi_{r,\infty}$ are the following:
\begin{eqnarray*}
 \pi_{r,\infty} & \cong & \pi_{p - 1 - r,0}
\end{eqnarray*}
 \item Let $Iw_\textnormal{S}(1)$ denote the standard upper pro-$p$-Iwahori subgroup of $\textnormal{SL}_2(\mathbb{Q}_p)$.  We have
\begin{eqnarray*}
 \pi_{r,\infty}^{Iw_\textnormal{S}(1)} & = & \ol{\mathbb{F}}_p\ol{[\textnormal{id}, x^r]},\\
 \pi_{r,0}^{Iw_\textnormal{S}(1)} & = & \ol{\mathbb{F}}_p\ol{[\beta,x^r]}.
\end{eqnarray*}
\end{enumerate}
\end{thm}

\begin{proof}
 This follows from Propositions 4.5 and 4.7 and Corollaires 4.8 and 4.9 of \cite{Ab12}, and the comments following Corollaire 4.9.
\end{proof}

\begin{defn}
  We let $\pi_r$ denote the representation $\pi_{r,\infty}$.  In light of Theorem \ref{Ramla}, the representations $\pi_r$ with $0\leq r \leq p - 1$ are a full set of representatives for the set of supercuspidal representations of $\textnormal{SL}_2(\mathbb{Q}_p)$ (cf. \cite{Ab12}, Th\'eor\`eme 4.12).
\end{defn}

We shall henceforth view the representations $\pi_r$ as representations of $\textnormal{SU}(1,1)(\mathbb{Q}_{p^2}/\mathbb{Q}_p)$ via the isomorphism $\textnormal{SU}(1,1)(\mathbb{Q}_{p^2}/\mathbb{Q}_p)\cong\textnormal{SL}_2(\mathbb{Q}_p)$.  We may now be more precise about the structures of $\pi_r^{I_\textnormal{S}(1)}$ as Hecke modules.

\begin{lemma}
 The bijection of Theorem \ref{bij2} is given explicitly by $$\pi_{r}\longmapsto \pi_{r}^{I_\textnormal{S}(1)}\cong M_r,$$ where $M_r$ is the Hecke module defined in Proposition \ref{ssingmods}.
\end{lemma}

\begin{proof}
 This follows from Propositions 6.3.50, 6.3.52, and 6.3.54 of \cite{Ab11}.
\end{proof}

\vspace{\baselineskip}
\subsection{The Group $\textnormal{U}(1,1)(\mathbb{Q}_{p^2}/\mathbb{Q}_p)$}  We now proceed to examine the supercuspidal representions of $\textnormal{U}(1,1)(\mathbb{Q}_{p^2}/\mathbb{Q}_p)$.  For $n\geq 1$, let $\textnormal{U}(1)_n$ denote the group $\textnormal{U}(1)(\mathbb{Q}_{p^2}/\mathbb{Q}_p)\cap (1 + p^n\mathbb{Z}_{p^2})$.  We begin with a simple lemma.

\begin{lemma}\label{section}
 Let $n\geq 1$.  Any element $x \in 1 + p^n\mathbb{Z}_{p^2}$ can be written uniquely as $x = yz$, with $y\in 1 + p^n\mathbb{Z}_p$ and $z\in \textnormal{U}(1)_n$.  In other words, we have $$1 + p^n\mathbb{Z}_{p^2} \cong (1 + p^n\mathbb{Z}_p)\times\textnormal{U}(1)_n.$$
\end{lemma}

\begin{proof}
 Consider the norm map:
\begin{center}
\begin{tabular}{rccl}
$\textnormal{N}_{\mathbb{Q}_{p^2}/\mathbb{Q}_p}:$ & $\mathbb{Q}_{p^2}^\times$ &  $\longrightarrow$ &  $\mathbb{Q}_p^\times$\\
 & $x$ & $\longmapsto$ & $x\ol{x}$
\end{tabular}
\end{center}
 Since the extension $\mathbb{Q}_{p^2}/\mathbb{Q}_p$ is unramified, the norm map restricts to a surjection $$\textnormal{N}_{\mathbb{Q}_{p^2}/\mathbb{Q}_p}:1 + p^n\mathbb{Z}_{p^2}\longrightarrow 1 + p^n\mathbb{Z}_p$$ (\cite{Se68}, Chapitre V, Proposition 3(a)).  The kernel is precisely $\textnormal{U}(1)_n$, so we obtain an exact sequence $$1\longrightarrow \textnormal{U}(1)_n \longrightarrow 1 + p^n\mathbb{Z}_{p^2} \stackrel{\textnormal{N}_{\mathbb{Q}_{p^2}/\mathbb{Q}_p}}{\longrightarrow} 1 + p^n\mathbb{Z}_p \longrightarrow 1.$$  By Proposition 6(b) of Chapitre IV and Lemme 2 of Chapitre V (\emph{loc. cit.}), the map $x\longmapsto x^2$ is an automorphism of $1 + p^n\mathbb{Z}_p$ (since $p$ was assumed odd).  Therefore, the map $x\longmapsto x^{1/2}$ gives a section to the norm map in the short exact sequence above.  The result now follows.
\end{proof}

\begin{remark}
 The Lemma above and its proof are both valid for any unramified quadratic extension of nonarchimedean local fields of odd residual characteristic.
\end{remark}

Let $G_0$ denote the subgroup of $G$ generated by $G_\textnormal{S}$ and the central subgroup $$\left\{\begin{pmatrix}a & 0 \\ 0 & a \end{pmatrix}: a\in \textnormal{U}(1)_1\right\}.$$  Since $\textnormal{U}(1)_1$ is a pro-$p$ group, Lemma 3 of \cite{BL94} implies that the set of smooth, irreducible mod-$p$ representations of $G_0$ and $G_\textnormal{S}$ are in canonical bijection.  A more useful description of $G_0$ is given by the following Lemma.

\begin{lemma}\label{gzero}
 Denote by $\eta$ the (surjective) composite homomorphism $$G\stackrel{\det}{\longrightarrow}\textnormal{U}(1)(\mathbb{Q}_{p^2}/\mathbb{Q}_p)\stackrel{\mathfrak{r}_{\mathbb{Q}_{p^2}}}{\longrightarrow}\textnormal{U}(1)(\mathbb{F}_{p^2}/\mathbb{F}_p),$$  where $\mathfrak{r}_{\mathbb{Q}_{p^2}}$ is as in Section \ref{secnonsc}.  Then $\ker(\eta) = G_0$.
\end{lemma}

\begin{proof}
 Let $h_s:\mathbb{Q}_{p^2}^\times\longrightarrow T$ be the homomorphism given by $$h_s(a) = \begin{pmatrix}a & 0 \\ 0 & \ol{a}^{-1}\end{pmatrix}.$$  Every element $g$ of $G$ can be written (not necessarily uniquely) as $$g = h_s(x)h_s([a])g',$$ with $x\in 1 + p\mathbb{Z}_{p^2}, a\in\mathbb{F}_{p^2}^\times$ and $g'\in G_\textnormal{S}$.  Assume that $g\in \ker(\eta)$, so that $$a^{1 - p} \equiv 1\ (\textnormal{mod}\ 1 + p\mathbb{Z}_{p^2}).$$  This implies $a\in\mathbb{F}_p^\times$ and $h_s([a])\in G_\textnormal{S}$.  Hence, if $g$ is in $\ker(\eta)$, then it must be of the form $h_s(x)g'$ for $x\in 1 + p\mathbb{Z}_{p^2}$ and $g'\in G_\textnormal{S}$.  Lemma \ref{section} implies that $g\in G_0$, so $\ker(\eta)$ is a subgroup of $G_0$.  The reverse inclusion is easily verified.
\end{proof}

\vspace{\baselineskip}

By the Lemma above, we have $G/G_0\cong \textnormal{U}(1)(\mathbb{F}_{p^2}/\mathbb{F}_p)$.  Hilbert's Theorem 90 implies that $T_0G_0 = G$, so we may and do choose coset representatives $\{\delta_i\}_{i = 0}^p$ such that $\delta_i\in T_0$. In particular, each $\delta_i$ normalizes the subgroups $I(1)\cap U$, $I(1)\cap U^-$ and $I(1)\cap T_\textnormal{S}$.

Let $0\leq r \leq p - 1$, and let $\pi_r$ be a smooth irreducible supercuspidal representation of $G_\textnormal{S}$, inflated to $G_0$.  For $\delta_i$ a coset representative of $G/G_0$ as above, we let $\pi_r^{\delta_i}$ denote the representation with the same underlying space as $\pi_r$, with the action given by first conjugating an element of $G_0$ by $\delta_i$.  Since $\delta_i$ normalizes the subgroups $I(1)\cap U$, $I(1)\cap U^-$ and $I(1)\cap T_\textnormal{S}$, the Iwahori decomposition (cf. \cite{Ab13}, Lemme 2.2) implies we have $\pi_r^{I_\textnormal{S}(1)} = (\pi_r^{\delta_i})^{I_\textnormal{S}(1)}$ as vector spaces, and equation \eqref{act} shows that the actions of the operators $\textnormal{T}_{n_s}, \textnormal{T}_{n_{s'}}$ and $e_{r'}$ on these two spaces are the same.  By Theorem \ref{gens} we have $\pi_r^{I_\textnormal{S}(1)} \cong (\pi_r^{\delta_i})^{I_\textnormal{S}(1)}$ as $\mathcal{H}_{\ol{\mathbb{F}}_p}(G_\textnormal{S},I_\textnormal{S}(1))$-modules, and Theorem \ref{bij2} now implies that $\pi_r\cong \pi_r^{\delta_i}$ as $G_\textnormal{S}$-representations (and consequently as $G_0$-representations).  Therefore, we may lift $\pi_r$ to
a projective representation of
$G$.  Since
$$\textnormal{H}^2(G/G_0, \ol{\mathbb{F}}_p^\times) = \textnormal{H}^2(\textnormal{U}(1)(\mathbb{F}_{p^2}/\mathbb{F}_p), \ol{\mathbb{F}}_p^\times) = \widehat{\textnormal{H}}{}^0(\textnormal{U}(1)(\mathbb{F}_{p^2}/\mathbb{F}_p), \ol{\mathbb{F}}_p^\times)= 0,$$
this representation lifts to a genuine representation $\widetilde{\pi}_r$ of $G$.  For more details, see \cite{JMY06}.

It remains to determine the action of $G$ on the lift $\widetilde{\pi}_r$ of $\pi_r$.  Consider the homomorphism $h_s$ defined in the proof of Lemma \ref{gzero}.  Since the image under $h_s$ of $\textnormal{U}(1)_1$ acts trivially on $\widetilde{\pi}_r$, we have $\widetilde{\pi}_r^{I(1)} = \widetilde{\pi}_r^{I_\textnormal{S}(1)} = \ol{\mathbb{F}}_p v_r$, where $v_r = \ol{[\textnormal{id},x^r]}$.  As the elements $h_s([a])$ for $a\in \mathbb{F}_{p^2}^\times$ normalize $I(1)$, we have $$h_s([a]).v_r = a^mv_r,$$ for $0 \leq m < p^2 - 1$.  Since $a^{p + 1}\in \mathbb{F}_p^\times$ for $a\in \mathbb{F}_{p^2}^\times$, we have $h_s([a]^{p + 1})\in T_{0,\textnormal{S}}$ and $$a^{(p + 1)r}v_r = h_s([a]^{p+1}).v_r = a^{(p + 1)m}v_r,$$ by the action of $I_\textnormal{S}$ on $\ol{[\textnormal{id},x^r]}\in\pi_r^{I_\textnormal{S}(1)}$.  Thus, we must have $m = r + (1 - p)k$ for some $k\in \mathbb{Z}$.  This leads to the following definition.

\begin{defn}\label{defofsc}
 Let $0\leq r \leq p - 1$ and $0\leq k < p+1$.  We define the representation $(\omega^k\circ\det)\otimes\boldsymbol{\pi}_r$ of $G = \textnormal{U}(1,1)(\mathbb{Q}_{p^2}/\mathbb{Q}_p)$ by the following conditions:
\begin{itemize}
 \item $(\omega^k\circ\det)\otimes\boldsymbol{\pi}_r|_{G_\textnormal{S}} = \pi_r$;
 \item $((\omega^k\circ\det)\otimes\boldsymbol{\pi}_r)^{I(1)} = \pi_r^{I_\textnormal{S}(1)} = \ol{\mathbb{F}}_p v_r$ as vector spaces;
 \item $h_s([a]).v_r = a^{r + (1 - p)k}v_r$ for $a\in \mathbb{F}_{p^2}^\times$.
\end{itemize}
\end{defn}

\vspace{\baselineskip}

The last point defines the character that gives the action of $I$ on $((\omega^k\circ\det)\otimes\boldsymbol{\pi}_r)^{I(1)}$, and the preceding discussion ensures that the vector spaces $(\omega^k\circ\det)\otimes\boldsymbol{\pi}_r$ are bona fide representations of $G$.  We collect their properties in the following Proposition.

\begin{prop}\label{propsc}
 Let $0\leq r \leq p - 1$ and $0\leq k < p + 1$.
\begin{enumerate}[(a)]
 \item The representations $(\omega^k\circ\det)\otimes\boldsymbol{\pi}_r$ are smooth, irreducible, admissible, and supercuspidal representations of $G$.
 \item The representations $(\omega^k\circ\det)\otimes\boldsymbol{\pi}_r$ are pairwise nonisomorphic.
\end{enumerate}
\end{prop}

\begin{proof}
 (a) The claim about irreducibility follows from the fact that the restriction of the representation $(\omega^k\circ\det)\otimes\boldsymbol{\pi}_r$ to $G_\textnormal{S}$ is irreducible.  The space of $I(1)$-invariants is one-dimensional, and therefore $(\omega^k\circ\det)\otimes\boldsymbol{\pi}_r$ is admissible.  Proposition \ref{nonsciffnonsc} implies that the representations are supercuspidal.

 (b) Suppose that $$\varphi:(\omega^k\circ\det)\otimes\boldsymbol{\pi}_r\longrightarrow (\omega^{k'}\circ\det)\otimes\boldsymbol{\pi}_{r'}$$ is a $G$-equivariant isomorphism.  The map $\varphi$ then defines a $G_\textnormal{S}$-equivariant isomorphism, so we must have $r = r'$ by Theorem \ref{Ramla}.  The last point of Definition \ref{defofsc} now shows that we must have $k \equiv k'~(\textnormal{mod}~ p + 1)$, which implies $k = k'$ (since $0\leq k,k' < p + 1$).
\end{proof}

\begin{thm}\label{scclass}
 Let $\pi$ be a smooth irreducible supercuspidal representation of the group $G = \textnormal{U}(1,1)(\mathbb{Q}_{p^2}/\mathbb{Q}_p)$.  Then $\pi$ is isomorphic to a unique representation of the form $(\omega^k\circ\det)\otimes\boldsymbol{\pi}_r$ with $0\leq r \leq p - 1$ and $0\leq k < p + 1$.
\end{thm}

\begin{proof}
 Let $\pi$ be a smooth irreducible supercuspidal representation.  Propositions \ref{LLss} and \ref{nonsciffnonsc} imply that upon restriction to $G_\textnormal{S}$, we must have
$$\pi|_{G_\textnormal{S}} \cong \pi_r\quad \textnormal{or}\quad \pi|_{G_\textnormal{S}} \cong \pi_r \oplus \pi_r^\theta,$$
for some $0\leq r \leq p - 1$.  Assume the latter.  Since $\theta$ normalizes the subgroups $I(1)\cap U, I(1)\cap U^-$ and $I(1)\cap T_\textnormal{S}$, equation \eqref{act}, Theorem \ref{gens}, and the Iwahori decomposition imply that the Hecke modules $\pi_r^{I_\textnormal{S}(1)}$ and $(\pi_r^\theta)^{I_\textnormal{S}(1)}$ are isomorphic.  Theorem \ref{bij2} now implies that $\pi_r^\theta\cong \pi_r$ as $G_\textnormal{S}$-representations.  Let $v$ be a nonzero eigenvector for $T_0$ contained in $\pi^{I_\textnormal{S}(1)} \cong \pi_r^{I_\textnormal{S}(1)}\oplus (\pi_r^\theta)^{I_\textnormal{S}(1)}$.  The space $\langle G.v\rangle_{\ol{\mathbb{F}}_p}$ is stable by $G$, and therefore must be all of $\pi$.  This implies that $\pi|_{G_\textnormal{S}} \cong \pi_r$, a contradiction.

 We may therefore assume that $\pi|_{G_\textnormal{S}}\cong \pi_r$.  The discussion preceding Definition \ref{defofsc} then shows that there exists an integer $k$ such that $\pi\cong (\omega^k\circ\det)\otimes\boldsymbol{\pi}_r$.
\end{proof}

\begin{cor}\label{u11class}
 Let $\pi$ be a smooth irreducible representation of $G = \textnormal{U}(1,1)(\mathbb{Q}_{p^2}/\mathbb{Q}_p)$.  Then $\pi$ admits a central character and is admissible.  Moreover, $\pi$ is isomorphic to one and only one of the following representations:
\begin{itemize}
 \item the smooth $\ol{\mathbb{F}}_p$-characters $\omega^k\circ\det$, where $0\leq k < p + 1;$
 \item twists of the Steinberg representation $(\omega^k\circ\det)\otimes\textnormal{St}_G$, where $0\leq k < p + 1;$
 \item the principal series representations $\textnormal{ind}_B^G(\mu_\lambda\omega^r)$, where $\lambda\in\ol{\mathbb{F}}_p^\times$ and $0\leq r < p^2 - 1$ with $(r, \lambda)\neq ((p - 1)m, 1);$
\item the supercuspidal representations $(\omega^k\circ\det)\otimes\boldsymbol{\pi}_r$, where $0\leq r \leq p - 1$ and $0 \leq k < p + 1$.
\end{itemize}
\end{cor}

\begin{proof}
 If $\pi$ is not supercuspidal, then the result follows from Theorem \ref{nonscclass}, and if $\pi$ is supercuspidal it follows from Proposition \ref{propsc} and Theorem \ref{scclass}.  It only remains to prove that no supercuspidal representation is isomorphic to a nonsupercuspidal representation.  Assume this is the case; we then obtain a $G_\textnormal{S}$-equivariant isomorphism between a supercuspidal representation and a nonsupercuspidal representation, contradicting Corollaire 3.19 of \cite{Ab12}.
\end{proof}

\vspace{\baselineskip}
\subsection{$L$-packets}
We define the \emph{general unitary group} $\textnormal{GU}(1,1)(\mathbb{Q}_{p^2}/\mathbb{Q}_p)$ by $$\{g\in \textrm{GL}_2(\mathbb{Q}_{p^2}): g^*sg = \kappa s~\textnormal{for some}~ \kappa\in\mathbb{Q}_p^\times\}.$$  The association $g\longmapsto \kappa$ is in fact a character, and induces a surjective homomorphism $\textnormal{sim}:\textnormal{GU}(1,1)(\mathbb{Q}_{p^2}/\mathbb{Q}_p)\longrightarrow \mathbb{Q}_p^\times$.  We obtain a short exact sequence of groups
$$1\longrightarrow \textnormal{U}(1,1)(\mathbb{Q}_{p^2}/\mathbb{Q}_p)\longrightarrow \textnormal{GU}(1,1)(\mathbb{Q}_{p^2}/\mathbb{Q}_p)\stackrel{\textnormal{sim}}{\longrightarrow}\mathbb{Q}_p^\times \longrightarrow 1$$
which splits, and we have
\begin{equation}\label{gu(1,1)}
\textnormal{GU}(1,1)(\mathbb{Q}_{p^2}/\mathbb{Q}_p) = \textnormal{U}(1,1)(\mathbb{Q}_{p^2}/\mathbb{Q}_p)\left\{\begin{pmatrix}1 & 0 \\ 0 & a\end{pmatrix}:~ a\in\mathbb{Q}_p^\times\right\}.
\end{equation}
Since the group $G = \textnormal{U}(1,1)(\mathbb{Q}_{p^2}/\mathbb{Q}_p)$ is a normal subgroup of $\textnormal{GU}(1,1)(\mathbb{Q}_{p^2}/\mathbb{Q}_p)$, the latter group acts on $G$ by conjugation, and consequently acts on representations of $G$.  The following definition is adapted from the complex case (see Section 11.1 of \cite{Ro90}).

\begin{defn}
An \emph{$L$-packet of semisimple representations on $G = \textnormal{U}(1,1)(\mathbb{Q}_{p^2}/\mathbb{Q}_p)$} is a $\textnormal{GU}(1,1)(\mathbb{Q}_{p^2}/\mathbb{Q}_p)$-orbit of smooth semisimple representations of $G$.  An $L$-packet is called \emph{supercuspidal} if it consists entirely of irreducible supercuspidal representations.
\end{defn}

\begin{prop}\label{Lpackets}
 Let $\Pi$ be an $L$-packet of smooth irreducible representations on $G = \textnormal{U}(1,1)(\mathbb{Q}_{p^2}/\mathbb{Q}_p)$.  Then $\Pi$ has cardinality 1 if and only if it contains an irreducible nonsupercuspidal representation.  If $\Pi$ is a supercuspidal $L$-packet, then it is of the form $$\Pi = \{\cusp{k}{r},~ \cusp{k + r + 1}{p - 1 - r}\},$$ for some $0\leq r \leq p - 1,~ 0\leq k < p + 1$.
\end{prop}

\begin{proof}
 Assume first that $\Pi$ contains an irreducible nonsupercuspidal representation $\pi$, and let
$$t = \begin{pmatrix}1 & 0 \\ 0 & a \end{pmatrix}$$
with $a\in \mathbb{Z}_p^\times$.  Consider the representation $\pi^t$ obtained by conjugation by $t$.  Since
$$\pi^t|_{G_\textnormal{S}} = (\pi|_{G_\textnormal{S}})^{t}$$
and $t$ normalizes $I_\textnormal{S}(1)$, we have $((\pi|_{G_\textnormal{S}})^t)^{I_\textnormal{S}(1)} = (\pi|_{G_\textnormal{S}})^{I_\textnormal{S}(1)}$ as vector spaces.  Equation \eqref{act} implies that the action of $\mathcal{H}_{\ol{\mathbb{F}}_p}(G_\textnormal{S},I_\textnormal{S}(1))$ on these spaces is the same, and Theorem \ref{bij1} shows that $\pi^t|_{G_\textnormal{S}}\cong \pi|_{G_\textnormal{S}}$.  Theorems \ref{ramla2} and \ref{nonscres} now imply $\pi^t\cong (\omega^k\circ\det)\otimes\pi$ for some $0 \leq k < p+1$, and upon examining the action of the torus $T$, we conclude $\pi^t \cong \pi$.

 Consider now the element $$\beta s = \begin{pmatrix}1 & 0 \\ 0 & p\end{pmatrix}.$$  Proposition 2.8 of \cite{Ab12} implies $\pi^{\beta s}|_{G_\textnormal{S}}\cong \pi|_{G_\textnormal{S}}$, and we proceed as above to conclude $\pi^{\beta s}\cong \pi$.  These arguments imply that $\Pi = \{\pi\}$ by equation \eqref{gu(1,1)}.

 Assume now that $\Pi$ contains a supercuspidal representation $\pi = \cusp{k}{r}$, and let $t$ be as above.  The arguments of the first paragraph carry over (this time appealing to Theorem \ref{bij2} and Definition \ref{defofsc}), and we obtain $\pi^t\cong\pi$.

 To conclude the proof, we must compute $\pi^{\beta s}$; since $s\in\textnormal{U}(1,1)(\mathbb{Q}_{p^2}/\mathbb{Q}_p)$, we have $\pi^s\cong \pi$ and it suffices to determine $\pi^\beta$.  Corollaire 4.6 of \cite{Ab12} implies
$$\pi^\beta|_{G_\textnormal{S}}\cong \pi_r^\beta \cong \pi_{p - 1 - r},$$
and Theorem \ref{scclass} gives $\pi^\beta\cong \cusp{k'}{p - 1 - r}$ for some $k'$.  As the element $\beta$ normalizes $I(1)$, we have
$$\ol{\mathbb{F}}_p v_r = \pi^{I(1)} = (\pi^\beta)^{I(1)}$$
as vector spaces.  For $a\in \mathbb{F}_{p^2}^\times$, the action of $h_s([a])$ on $(\pi^\beta)^{I(1)}$ is given by
$$v_r\longmapsto a^{-pr - pk + k}v_r,$$
while the action of $h_s([a])$ on $(\cusp{k'}{p - 1 - r})^{I(1)}$ is given by
$$v_{p - 1 - r}\longmapsto a^{p - 1 - r + (1 - p)k'}v_{p - 1 - r}.$$
Since these characters coincide, we have that $(1 - p)k' - (1 - p) - r \equiv (1 - p)k - pr~(\textnormal{mod}~ p^2 - 1)$, which shows $k' \equiv r + k + 1~(\textnormal{mod}~ p + 1)$.  Hence
$$\Pi = \{\cusp{k}{r},~ \cusp{k + r + 1}{p - 1 - r}\},$$
which concludes the proof.
\end{proof}

\vspace{\baselineskip}

\section{Galois Groups and Representations}

In this section we recall the definitions associated to Galois representations attached to unitary groups.

\vspace{\baselineskip}
\subsection{Galois Groups}
Let $\mathcal{G}_{\mathbb{Q}_p} := \textrm{Gal}(\ol{\mathbb{Q}}_p/\mathbb{Q}_p)$ denote the absolute Galois group of $\mathbb{Q}_p$, and $\mathcal{I}_{\mathbb{Q}_p}$ the inertia subgroup of elements which act trivially on the residue field $k_{\ol{\mathbb{Q}}_p}$.  For any extension $F$ of $\mathbb{Q}_p$ contained in $\ol{\mathbb{Q}}_p$, we define $\mathcal{G}_F := \textrm{Gal}(\ol{\mathbb{Q}}_p/F)$.  Let $\mathbb{Q}_p^{ur}$ denote the maximal unramified extension of $\mathbb{Q}_p$ in $\ol{\mathbb{Q}}_p$; we may then realize the subgroup $\mathcal{I}_{\mathbb{Q}_p}$ as
$$\mathcal{I}_{\mathbb{Q}_p} = \textrm{Gal}(\ol{\mathbb{Q}}_p/\mathbb{Q}_p^{ur}).$$
This gives $\mathcal{G}_{\mathbb{Q}_p}/\mathcal{I}_{\mathbb{Q}_p} \cong \textrm{Gal}(\mathbb{Q}_p^{ur}/\mathbb{Q}_p)\cong \textrm{Gal}(\ol{\mathbb{F}}_p/\mathbb{F}_p)\cong \widehat{\mathbb{Z}}$, where the last isomorphism is given by sending the geometric Frobenius to 1.

For $n\geq 1$, we let $\mathbb{Q}_{p^n}$ denote the unique unramified extension of $\mathbb{Q}_p$ of degree $n$ contained in $\ol{\mathbb{Q}}_p$, and let
\begin{equation}\label{rec}
\iota_n:\mathbb{Q}_{p^n}^\times\longrightarrow \mathcal{G}_{\mathbb{Q}_{p^n}}^{ab}
\end{equation}
denote the reciprocity map of local class field theory, normalized so that uniformizers correspond to geometric Frobenius elements.  We shall denote by $\textnormal{Fr}_p$ a fixed element of $\mathcal{G}_{\mathbb{Q}_p}$ whose image in $\mathcal{G}_{\mathbb{Q}_p}^{ab}$ is equal to $\iota_1(p^{-1})$.

Using the injections $\iota_n$, we will identify the smooth $\ol{\mathbb{F}}_p$-characters of $\mathbb{Q}_{p^n}^\times$ and $\mathcal{G}_{\mathbb{Q}_{p^n}}$ in the following way.  We fix a compatible system $\{\sqrt[p^n-1]{p}\}_{n\geq 1}$ of $(p^n-1)^{\textnormal{th}}$ roots of $p$, and let $\omega_n:\mathcal{I}_{\mathbb{Q}_p}\longrightarrow \ol{\mathbb{F}}_p^\times$ denote the character given by
\begin{equation}\label{omegan}
\omega_n: h\longmapsto \iota\circ\mathfrak{r}_{\ol{\mathbb{Q}}_p}\left(\frac{h.\sqrt[p^n-1]{p}}{\sqrt[p^n-1]{p}}\right),
\end{equation}
where $h\in \mathcal{I}_{\mathbb{Q}_p}$ and $\mathfrak{r}_{\ol{\mathbb{Q}}_p}:\ol{\mathbb{Z}}_p\longrightarrow k_{\ol{\mathbb{Q}}_p}$ denotes the reduction modulo the maximal ideal.  Lemma 2.5 of \cite{Br07} shows that the character $\omega_n$ extends to a character of $\mathcal{G}_{\mathbb{Q}_{p^n}}$; we continue to denote by $\omega_n$ the extension which sends the element $\textnormal{Fr}_p^n$ to 1.

\begin{lemma}\label{firstgal}
 For $n\geq 1$, we have $\omega_n\circ\iota_n = \omega$, where $\omega$ is defined in equation \eqref{omega}.
\end{lemma}
\begin{proof}
 Let
$$(~\cdot~,*/\mathbb{Q}_{p^n}):\mathbb{Q}_{p^n}^\times\longrightarrow \mathcal{G}_{\mathbb{Q}_{p^n}}^{ab}$$
denote the norm residue symbol of the field $\mathbb{Q}_{p^n}$, constructed in Chapitre XIII of \cite{Se68}; with this notation we have $\iota_n(x) = (x^{-1},*/\mathbb{Q}_{p^n})$ for $x\in\mathbb{Q}_{p^n}^\times$.  We also let $\nu_n:\mathbb{Q}_{p^n}^\times\longrightarrow \mathbb{Z}$ denote the valuation of $\mathbb{Q}_{p^n}^\times$ normalized by $\nu_n(p) = 1$, and let
$$(~\cdot~,~ \cdot~)_{\nu_n}:\mathbb{Q}_{p^n}^\times\times\mathbb{Q}_{p^n}^\times\longrightarrow \boldsymbol{\mu}_{p^n - 1}(\mathbb{Q}_{p^n}^\times)$$
denote the Hilbert symbol, where $\boldsymbol{\mu}_{p^n - 1}(\mathbb{Q}_{p^n}^\times)$ denotes the group of $(p^n - 1)^{\textnormal{th}}$ roots of unity in $\mathbb{Q}_{p^n}^\times$ (cf. Chapitre XIV, \emph{loc. cit.}).

Denote by $\mathbb{Z}_{p^n}$ the ring of integers of $\mathbb{Q}_{p^n}$, and suppose $u\in\mathbb{Z}_{p^n}^\times$.  Propositions 6 and 8 of Chapitre XIV (\emph{loc. cit.}) imply
$$\frac{\iota_n(u).\sqrt[p^n - 1]{p}}{\sqrt[p^n - 1]{p}} = \frac{(u^{-1},*/\mathbb{Q}_{p^n}).\sqrt[p^n - 1]{p}}{\sqrt[p^n - 1]{p}} = (p,u^{-1})_{\nu_n} = \left[\mathfrak{r}_{\ol{\mathbb{Q}}_p}\left((-1)^{\nu_n(p)\nu_n(u^{-1})}\frac{p^{\nu_n(u^{-1})}}{u^{-\nu_n(p)}}\right)\right] = \left[\mathfrak{r}_{\ol{\mathbb{Q}}_p}(u)\right].$$
Applying $\iota\circ\mathfrak{r}_{\ol{\mathbb{Q}}_p}$ to both sides shows that $\omega_n\circ\iota_n(u) = \omega(u)$.

The functorial properties of the reciprocity maps $\iota_n$ imply that we have equalities
$$\omega_n\circ\iota_n(p^{-1}) = \omega_n\circ\textnormal{ver}\circ\iota_1(p^{-1}) = \omega_n(\textnormal{Fr}_p^n) = 1 = \omega(p^{-1}),$$
where $\textnormal{ver}:\mathcal{G}_{\mathbb{Q}_p}^{ab}\longrightarrow\mathcal{G}_{\mathbb{Q}_{p^n}}^{ab}$ denotes the transfer map.  The result now follows.
\end{proof}

\begin{lemma}\label{brlemma}
For $h\in\mathcal{I}_{\mathbb{Q}_p}$ and $n\geq 1$, we have $$\omega_n(\textnormal{Fr}_ph\textnormal{Fr}_p^{-1}) = \omega_n(h)^p.$$
\end{lemma}
\begin{proof}
 See the proof of Lemma 2.5 in \cite{Br07}.
\end{proof}

For $\lambda\in\ol{\mathbb{F}}_p^\times$ and $n\geq 1$, we let $\mu_{n,\lambda}:\mathcal{G}_{\mathbb{Q}_{p^n}}\longrightarrow \ol{\mathbb{F}}_p^\times$ denote the unramified character which is trivial on $\mathcal{I}_{\mathbb{Q}_{p}}$ and sends $\textnormal{Fr}_p^n$ to $\lambda$.

\begin{cor}
 Let $n\geq 1$.  Every smooth $\ol{\mathbb{F}}_p$-character of $\mathcal{G}_{\mathbb{Q}_{p^n}}$ is of the form $\mu_{n,\lambda}\omega_n^{r}$, where $\lambda\in\ol{\mathbb{F}}_p^\times$ and $0\leq r < p^n - 1$.  Moreover, the reciprocity maps $\iota_n$ induce a bijection between smooth $\ol{\mathbb{F}}_p$-characters of $\mathcal{G}_{\mathbb{Q}_{p^n}}$ and $\mathbb{Q}_{p^n}^\times$, given explicitly by $\mu_{n,\lambda}\omega_n^r\circ\iota_n = \mu_{\lambda^{-1}}\omega^r$.
\end{cor}

\begin{proof}
 This follows from Lemmas \ref{firstgal} and \ref{brlemma}.
\end{proof}

\vspace{\baselineskip}
\subsection{$L$-groups}  We now review the construction of $L$-groups for the group $\textnormal{U}(1,1)(\mathbb{Q}_{p^2}/\mathbb{Q}_p)$.  For general references on $L$-groups, the reader should consult \cite{Bo79}; for the specific case of unitary groups, see Appendix A of \cite{BC09}, \cite{Mi11}, or Section 1.8 of \cite{Ro90}.

Let $\widehat{G}$ denote the $\ol{\mathbb{F}}_p$-valued points of the dual group of $G = \textnormal{U}(1,1)(\mathbb{Q}_{p^2}/\mathbb{Q}_p)$; since $G$ splits over $\mathbb{Q}_{p^2}$, we have $\widehat{G} = \textrm{GL}_2(\ol{\mathbb{F}}_p)$.  Let $\alpha_1$ be the $\ol{\mathbb{F}}_p$-character of the diagonal maximal torus $\widehat{T}$ of $\textrm{GL}_2(\ol{\mathbb{F}}_p)$ defined by
\begin{center}
  $\alpha_1\begin{pmatrix}x & 0 \\ 0 & y\end{pmatrix}  =  xy^{-1}$
\end{center}
for $x,y\in \ol{\mathbb{F}}_p^\times$, and let $\widehat{B}$ be the Borel subgroup of upper triangular matrices containing $\widehat{T}$.  The root subgroup associated to $\alpha_1$ (with respect to $\widehat{B}$) is given by
$$\mathfrak{X}_{\alpha_1} = \left\{u(x)=\begin{pmatrix}1 & x \\ 0 & 1 \end{pmatrix}: x\in \ol{\mathbb{F}}_p\right\}.$$
Let $X_1 := u(1)$ be a basis for $\mathfrak{X}_{\alpha_1}$.  The discussion of \cite{Bo79}, Section 1, implies we have $\textrm{Aut}(\widehat{G}) \cong \textrm{Inn}(\widehat{G})\rtimes \Theta$, where $\Theta$ denotes the group of pinned automorphisms preserving the quadruple $\{\widehat{G}, \widehat{B}, \widehat{T}, X_1\}$.  The group $\Theta$ is of order 2, generated by the automorphism
$$g \longmapsto \Phi_2 (g^\top)^{-1}\Phi_2^{-1} = g\cdot\det(g)^{-1},$$ where $$\Phi_2 = \begin{pmatrix}\phantom{-}0 & 1\\ -1 & 0\end{pmatrix}.$$
Since the group $G$ splits over $\mathbb{Q}_{p^2}$, the discussion of Section 1 (\emph{loc. cit.}) implies we have an injection (indeed, an isomorphism) of $\textrm{Gal}(\mathbb{Q}_{p^2}/\mathbb{Q}_p)$ into $\Theta$.

\begin{defn}
 The \emph{$L$-group of $G$} is defined as the semidirect product $${}^LG = \widehat{G}\rtimes\mathcal{G}_{\mathbb{Q}_p} = \textnormal{GL}_2(\ol{\mathbb{F}}_p)\rtimes\mathcal{G}_{\mathbb{Q}_p},$$ with the action of $\mathcal{G}_{\mathbb{Q}_p}$ on $\widehat{G}$ given by
\begin{eqnarray*}
\textnormal{Fr}_pg\textnormal{Fr}_p^{-1} & = & \Phi_2(g^\top)^{-1}\Phi_2^{-1},\\
 hgh^{-1} & = & g,
\end{eqnarray*}
for $g\in \widehat{G}$, $h\in \mathcal{G}_{\mathbb{Q}_{p^2}}$.
\end{defn}

In addition to the group $G$, we shall also need unitary groups of lower rank.  In particular, we will need the endoscopic group associated to $G$.  For the general definition in the complex case, see Sections 4.2 and 4.6 of \cite{Ro90}.

\begin{defn}
\textnormal{(a)} The group $J := (\textnormal{U}(1)\times\textnormal{U}(1))(\mathbb{Q}_{p^2}/\mathbb{Q}_p)$ is the unique \emph{elliptic endoscopic group} associated to $G = \textnormal{U}(1,1)(\mathbb{Q}_{p^2}/\mathbb{Q}_p)$.

\noindent \textnormal{(b)} The \emph{$L$-group of $J$} is defined as the semidirect product $${}^LJ = \widehat{J}\rtimes\mathcal{G}_{\mathbb{Q}_p} = (\ol{\mathbb{F}}_p^\times\times\ol{\mathbb{F}}_p^\times)\rtimes\mathcal{G}_{\mathbb{Q}_p},$$ with the action of $\mathcal{G}_{\mathbb{Q}_p}$ on $\widehat{J}$ given by
\begin{eqnarray*}
\textnormal{Fr}_p(x,y)\textnormal{Fr}_p^{-1} & = & (x^{-1},y^{-1}),\\
 h(x,y)h^{-1} & = & (x,y),
\end{eqnarray*}
for $x,y\in \ol{\mathbb{F}}_p^\times$, $h\in \mathcal{G}_{\mathbb{Q}_{p^2}}$.

\noindent \textnormal{(c)} The \emph{$L$-group of $\textnormal{U}(1)(\mathbb{Q}_{p^2}/\mathbb{Q}_p)$} is defined as the semidirect product $${}^L\textnormal{U}(1) = \ol{\mathbb{F}}_p^\times\rtimes\mathcal{G}_{\mathbb{Q}_p},$$ with the action of $\mathcal{G}_{\mathbb{Q}_p}$ on $\ol{\mathbb{F}}_p^\times$ given by
\begin{eqnarray*}
\textnormal{Fr}_px\textnormal{Fr}_p^{-1} & = & x^{-1},\\
 hxh^{-1} & = & x,
\end{eqnarray*}
for $x\in \ol{\mathbb{F}}_p^\times$, $h\in \mathcal{G}_{\mathbb{Q}_{p^2}}$.
\end{defn}

\begin{prop}[\cite{Ro90}, Proposition 4.6.1]\label{Lembedding}
 There exists a homomorphism $$\xi:{}^LJ\ensuremath{\lhook\joinrel\relbar\joinrel\rightarrow} {}^LG,$$ which commutes with the projections to $\mathcal{G}_{\mathbb{Q}_p}$, given by
\begin{eqnarray*}
 (x,y) & \longmapsto & \begin{pmatrix}x & 0 \\ 0 & y\end{pmatrix},\\
 (1,1)\textnormal{Fr}_p & \longmapsto & \begin{pmatrix}0 & -1 \\ 1 & \phantom{-}0\end{pmatrix}\textnormal{Fr}_p,\\
 (1,1)h & \longmapsto & \begin{pmatrix}\mu_{2,-1}(h) & 0 \\ 0 & \mu_{2,-1}(h)\end{pmatrix}h,
\end{eqnarray*}
where $x,y\in\ol{\mathbb{F}}_p^\times$, $h\in\mathcal{G}_{\mathbb{Q}_{p^2}}$.
\end{prop}

\vspace{\baselineskip}
\subsection{Langlands Parameters for $\textnormal{U}(1)(\mathbb{Q}_{p^2}/\mathbb{Q}_p)$}  We begin by defining and investigating Langlands parameters in characteristic $p$ associated to $\textnormal{U}(1)(\mathbb{Q}_{p^2}/\mathbb{Q}_p)$.

\begin{defn}
A \emph{Langlands parameter} is a homomorphism
$$\varphi: \mathcal{G}_{\mathbb{Q}_p}\longrightarrow {}^L\textnormal{U}(1) = \ol{\mathbb{F}}_p^\times\rtimes \mathcal{G}_{\mathbb{Q}_p},$$
such that the composition of $\varphi$ with the canonical projection ${}^L\textnormal{U}(1)\longrightarrow \mathcal{G}_{\mathbb{Q}_p}$ is the identity map of $\mathcal{G}_{\mathbb{Q}_p}$.  We say two Langlands parameters are \emph{equivalent} if they are conjugate by an element of $\ol{\mathbb{F}}_p^\times$.
\end{defn}

With this definition, we come to our first result.

\begin{prop}\label{u1params}
 Let $\varphi:\mathcal{G}_{\mathbb{Q}_p}\longrightarrow {}^L\textnormal{U}(1)$ be a Langlands parameter.  Then there exists $0\leq k < p + 1$ such that, up to equivalence, $\varphi$ is of the following form:
\begin{center}
\begin{tabular}{ccc}
 $\textnormal{Fr}_p$ & $\longmapsto$ & $\textnormal{Fr}_p$\\
 $h$ & $\longmapsto$ & $\omega_2^{(1 - p)k}(h)h$,
\end{tabular}
\end{center}
where $h\in \mathcal{G}_{\mathbb{Q}_{p^2}}$.
\end{prop}

\begin{proof}
 The conjugation action of $\mathcal{G}_{\mathbb{Q}_p}$ on $\ol{\mathbb{F}}_p^\times$ shows that, up to equivalence, we have $\varphi(\textnormal{Fr}_p) = \textnormal{Fr}_p$.  It remains to determine the image of $\mathcal{G}_{\mathbb{Q}_{p^2}}$.  Since $\mathcal{G}_{\mathbb{Q}_{p^2}}$ acts trivially on $\ol{\mathbb{F}}_p^\times$, we see that $\varphi$ must be of the form
$$\varphi(h)  = \mu_{2,\lambda}\omega_2^{r}(h)h,$$
where $\lambda\in \ol{\mathbb{F}}_p^\times$, $0\leq r < p^2 - 1$, and $h\in\mathcal{G}_{\mathbb{Q}_{p^2}}$.  Examining the element $\varphi(\textnormal{Fr}_p^2)$, we see that
\begin{eqnarray*}
 \lambda\textnormal{Fr}_p^2 & = & \varphi(\textnormal{Fr}_p^2)\\
 & = & \varphi(\textnormal{Fr}_p)^2\\
 & = & \textnormal{Fr}_p^2,
\end{eqnarray*}
and thus $\lambda = 1$.

Lemma \ref{brlemma} now gives
\begin{eqnarray*}
 \omega_2^{pr}(h)\textnormal{Fr}_ph\textnormal{Fr}_p^{-1} & = & \varphi(\textnormal{Fr}_ph\textnormal{Fr}_p^{-1})\\
 & = & \varphi(\textnormal{Fr}_p)\varphi(h)\varphi(\textnormal{Fr}_p)^{-1}\\
 & = & \textnormal{Fr}_p\omega_2^{r}(h)h\textnormal{Fr}_p^{-1}\\
 & = & \omega_2^{-r}(h)\textnormal{Fr}_ph\textnormal{Fr}_p^{-1},
\end{eqnarray*}
which implies $r \equiv 0~(\textnormal{mod}~ p - 1)$.
\end{proof}

\begin{defn}
 Let $0\leq k < p + 1$.  We denote by $\eta_k:\mathcal{G}_{\mathbb{Q}_p}\longrightarrow {}^L\textnormal{U}(1)$ the Langlands parameter constructed in Proposition \ref{u1params}; explicitly, we have
\begin{center}
\begin{tabular}{ccc}
 $\eta_{k}(\textnormal{Fr}_p)$ & $=$ & $\textnormal{Fr}_p$,\\
 $\eta_{k}(h)$ & $=$ & $\omega_2^{(1 - p)k}h$,
\end{tabular}
\end{center}
where $h\in \mathcal{G}_{\mathbb{Q}_{p^2}}$.
\end{defn}

\begin{cor}\label{u1llc}
 There is a bijection between the Langlands parameters associated to the group $\textnormal{U}(1)(\mathbb{Q}_{p^2}/\mathbb{Q}_p)$ and smooth irreducible representations of $\textnormal{U}(1)(\mathbb{Q}_{p^2}/\mathbb{Q}_p)$, given explicitly by
$$\eta_k \longleftrightarrow \omega^k,$$
where $0\leq k < p + 1$, and $\omega$ is the character defined in equation \eqref{omega}.
\end{cor}

\vspace{\baselineskip}
\subsection{Langlands Parameters for $\textnormal{U}(1,1)(\mathbb{Q}_{p^2}/\mathbb{Q}_p)$} We now proceed to explore Langlands parameters in characteristic $p$ for the group $G = \textnormal{U}(1,1)(\mathbb{Q}_{p^2}/\mathbb{Q}_p)$.  For the analogous definitions in the complex setting, see \cite{Ro90}, \cite{Ro92}, and Appendix A of \cite{BC09}.
\begin{defn}\label{params}
 \textnormal{(a)} A \emph{Langlands parameter} is a homomorphism
$$\varphi: \mathcal{G}_{\mathbb{Q}_p}\longrightarrow {}^LG = \textnormal{GL}_2(\ol{\mathbb{F}}_p)\rtimes\mathcal{G}_{\mathbb{Q}_p},$$
such that the composition of $\varphi$ with the canonical projection ${}^LG\longrightarrow \mathcal{G}_{\mathbb{Q}_p}$ is the identity map of $\mathcal{G}_{\mathbb{Q}_p}$.  We say two Langlands parameters are \emph{equivalent} if they are conjugate by an element of $\widehat{G} = \textnormal{GL}_2(\ol{\mathbb{F}}_p)$.

\noindent \textnormal{(b)} Let $\varphi:\mathcal{G}_{\mathbb{Q}_p}\longrightarrow {}^LG$ be a Langlands parameter and let $0\leq k < p + 1$.  We define the \emph{twist of $\varphi$ by $\omega_2^{(1 - p)k}$}, denoted $\varphi\otimes\omega_2^{(1 - p)k}$, by
\begin{center}
\begin{tabular}{ccc}
 $\varphi\otimes\omega_2^{(1 - p)k}(h)$ & = & $\varphi(h)\begin{pmatrix}\omega_2^{(1 - p)k}(h) & 0 \\ 0 & \omega_2^{(1 - p)k}(h)\end{pmatrix}$,\\
 $\varphi\otimes\omega_2^{(1 - p)k}(\textnormal{Fr}_ph)$ & = & $\varphi(\textnormal{Fr}_ph)\begin{pmatrix}\omega_2^{(1 - p)k}(h) & 0 \\ 0 & \omega_2^{(1 - p)k}(h)\end{pmatrix}$,
\end{tabular}
\end{center}
where $h\in\mathcal{G}_{\mathbb{Q}_{p^2}}$.  One easily checks that this is well-defined and gives a bona fide Langlands parameter.
\end{defn}

\begin{defn}
 Let $\varphi:\mathcal{G}_{\mathbb{Q}_p}\longrightarrow {}^LG$ be a Langlands parameter.  Since the group $\mathcal{G}_{\mathbb{Q}_{p^2}}$ acts trivially on $\widehat{G}$, the restriction of $\varphi$ to $\mathcal{G}_{\mathbb{Q}_{p^2}}$ must be of the form $$\varphi(h) = \varphi_0(h)h,$$ where $h\in\mathcal{G}_{\mathbb{Q}_{p^2}}$ and $\varphi_0:\mathcal{G}_{\mathbb{Q}_{p^2}}\longrightarrow \widehat{G}$ is a \emph{homomorphism}.  As $\widehat{G} = \textnormal{GL}_2(\ol{\mathbb{F}}_p)$, $\varphi_0$ is a two-dimensional Galois representation; we call it the \emph{Galois representation associated to $\varphi$}.
\end{defn}

\begin{defn}
 Let $\varphi: \mathcal{G}_{\mathbb{Q}_p}\longrightarrow {}^LG$ be a Langlands parameter.  We say $\varphi$ is \emph{stable} if the associated Galois representation $\varphi_0:\mathcal{G}_{\mathbb{Q}_{p^2}}\longrightarrow \textnormal{GL}_2(\ol{\mathbb{F}}_p)$ is irreducible.
\end{defn}

\vspace{\baselineskip}

Our first result on Langlands parameters for $\textnormal{U}(1,1)(\mathbb{Q}_{p^2}/\mathbb{Q}_p)$ stands in stark contrast to the complex case (cf. \cite{Ro90}, Section 15.1).

\begin{prop}\label{nostables}
 There do not exist any stable parameters $\varphi:\mathcal{G}_{\mathbb{Q}_p}\longrightarrow {}^LG$.
\end{prop}

\begin{proof}
The inclusion $\textnormal{SL}_2(\mathbb{Q}_p)\ensuremath{\lhook\joinrel\relbar\joinrel\rightarrow} \textnormal{U}(1,1)(\mathbb{Q}_{p^2}/\mathbb{Q}_p)$ gives rise, by duality, to a homomorphism of $L$-groups, given explicitly by
\begin{eqnarray*}
 \imath :{}^LG = \textnormal{GL}_2(\ol{\mathbb{F}}_p)\rtimes \mathcal{G}_{\mathbb{Q}_p} & \longrightarrow & {}^L\textnormal{SL}_2 = \textnormal{PGL}_2(\ol{\mathbb{F}}_p)\times\mathcal{G}_{\mathbb{Q}_p}\\
 gh & \longmapsto & \llbracket g\rrbracket h,
\end{eqnarray*}
where $g\in \textnormal{GL}_2(\ol{\mathbb{F}}_p), h\in \mathcal{G}_{\mathbb{Q}_p}$, and $\llbracket g\rrbracket$ denotes the image in $\textnormal{PGL}_2(\ol{\mathbb{F}}_p)$ of the element $g$.  Given $\varphi:\mathcal{G}_{\mathbb{Q}_p} \longrightarrow {}^LG$, we consider the Langlands parameter $\imath\circ \varphi:\mathcal{G}_{\mathbb{Q}_p} \longrightarrow\textnormal{PGL}_2(\ol{\mathbb{F}}_p)\times\mathcal{G}_{\mathbb{Q}_p}$ and the associated Galois representation $(\imath\circ\varphi)_0:\mathcal{G}_{\mathbb{Q}_{p^2}}\longrightarrow \textnormal{PGL}_2(\ol{\mathbb{F}}_p)$.  It is clear that $\varphi_0$ is irreducible if and only if $(\imath\circ\varphi)_0$ is irreducible, and therefore it suffices to examine $\imath\circ\varphi$.

Now, since the group $\textnormal{SL}_2(\mathbb{Q}_p)$ is split, $\mathcal{G}_{\mathbb{Q}_p}$ acts trivially on $\widehat{\textnormal{SL}_2} = \textnormal{PGL}_2(\ol{\mathbb{F}}_p)$, and therefore $\imath\circ\varphi$ takes the form
$$\imath\circ\varphi(h) = \varphi'(h)h,$$
where $h\in \mathcal{G}_{\mathbb{Q}_p}$ and $\varphi':\mathcal{G}_{\mathbb{Q}_p}\longrightarrow \textnormal{PGL}_2(\ol{\mathbb{F}}_p)$ is a homomorphism.  Assume $\varphi'$ is irreducible.  By a theorem of Tate (see the proof of Theorem 4 (and its corollary) in \cite{Se77}), we have
$$\textnormal{H}^2(\mathcal{G}_{\mathbb{Q}_p},\ol{\mathbb{F}}_p^\times) = 0,$$
which implies that every projective representation has a lift to $\textnormal{GL}_2(\ol{\mathbb{F}}_p)$.  Hence, we may write $\varphi'(h) = \llbracket \widetilde{\varphi}'(h)\rrbracket$, where $\widetilde{\varphi}':\mathcal{G}_{\mathbb{Q}_p}\longrightarrow \textnormal{GL}_2(\ol{\mathbb{F}}_p)$ is an irreducible Galois representation.

It is well-known (\cite{Vig97}, Section 1.14) that every two-dimensional irreducible mod-$p$ representation of $\mathcal{G}_{\mathbb{Q}_p}$ is isomorphic to a representation of the form
$$\textnormal{ind}_{\mathcal{G}_{\mathbb{Q}_{p^2}}}^{\mathcal{G}_{\mathbb{Q}_p}}(\mu_{2,\lambda}\omega_2^m),$$
where $\lambda\in\ol{\mathbb{F}}_p^\times$, and $0\leq m < p^2 - 1$ satisfies $m\not\equiv pm~(\textnormal{mod}~ p^2 - 1)$.  By Mackey theory, we have
$$\widetilde{\varphi}'|_{\mathcal{G}_{\mathbb{Q}_{p^2}}} \cong \textnormal{ind}_{\mathcal{G}_{\mathbb{Q}_{p^2}}}^{\mathcal{G}_{\mathbb{Q}_p}}(\mu_{2,\lambda}\omega_2^m)|_{\mathcal{G}_{\mathbb{Q}_{p^2}}} \cong \mu_{2,\lambda}\omega_2^m\oplus\mu_{2,\lambda}\omega_2^{pm},$$
which implies that the original Langlands parameter $\varphi$ cannot be stable.
\end{proof}

\vspace{\baselineskip}
We have a completely analogous definition of Langlands parameters and equivalence for the endoscopic group $J$.  We may classify such parameters in a manner nearly identical to that of Proposition \ref{u1params}.

\begin{defn}\label{u1u1params}
 Let $0\leq k,\ell < p+1$.  We denote by $\eta_{k,\ell}:\mathcal{G}_{\mathbb{Q}_p}\longrightarrow {}^LJ$ the Langlands parameter defined by
\begin{center}
\begin{tabular}{ccc}
 $\eta_{k,\ell}(\textnormal{Fr}_p)$ & $=$ & $(1,~1)\textnormal{Fr}_p$,\\
 $\eta_{k,\ell}(h)$ & $=$ & $(\omega_2^{(1 - p)k}(h),~ \omega_2^{(1 - p)\ell}(h))h$,
\end{tabular}
\end{center}
where $h\in\mathcal{G}_{\mathbb{Q}_{p^2}}$.
\end{defn}

\begin{cor}\label{u1u1llc}
 The Langlands parameters $\eta_{k,\ell}$ are pairwise inequivalent, and correspond bijectively (via the correspondence of Corollary \ref{u1llc}) to the smooth irreducible representations of $J = (\textnormal{U}(1)\times\textnormal{U}(1))(\mathbb{Q}_{p^2}/\mathbb{Q}_p)$.  Explicitly, the bijection is given by
$$\eta_{k,\ell}\longleftrightarrow \omega^k\otimes\omega^\ell,$$
where $0\leq k, \ell < p + 1$.
\end{cor}

\vspace{\baselineskip}
Using the parameters $\eta_{k,\ell}$, we obtain the first nontrivial explicit examples of Langlands parameters for the group $G$.

\begin{defn}\label{endparams}
 Let $0\leq k,\ell < p+1$.  We denote by $\varphi_{k,\ell}:\mathcal{G}_{\mathbb{Q}_p}\longrightarrow {}^LG$ the Langlands parameter obtained by composing $\eta_{k,\ell}$ (of Definition \ref{u1u1params}) with $\xi$ (of Proposition \ref{Lembedding}).  Explicitly, we have
\begin{center}
\begin{tabular}{ccc}
 $\varphi_{k,\ell}(\textnormal{Fr}_p)$ & $=$ & $\begin{pmatrix}0 & -1\\ 1 & \phantom{-}0\end{pmatrix}\textnormal{Fr}_p$,\\
 $\varphi_{k,\ell}(h)$ & $=$ & $\begin{pmatrix}\mu_{2,-1}\omega_2^{(1 - p)k}(h) & 0\\ 0 & \mu_{2,-1}\omega_2^{(1 - p)\ell}(h)\end{pmatrix}h$,
\end{tabular}
\end{center}
for $h\in\mathcal{G}_{\mathbb{Q}_{p^2}}$.

We say $\varphi_{k,\ell}$ is \emph{regular} if $k\neq \ell$, and \emph{singular} otherwise.
\end{defn}

\begin{lemma}\label{paramequiv}
 Let $0\leq k, k', \ell, \ell' < p + 1$.  Then $\varphi_{k,\ell}$ is equivalent to $\varphi_{k',\ell'}$ if and only if the sets $\{k,\ell\}$ and $\{k',\ell'\}$ coincide.
\end{lemma}

\begin{proof}
 Conjugation by $\Phi_2\in \widehat{G}$ shows that $\varphi_{k,\ell}$ is equivalent to $\varphi_{\ell, k}$.  Assume conversely that $\varphi_{k,\ell}$ is equivalent to $\varphi_{k',\ell'}$.  Upon restricting to $\mathcal{G}_{\mathbb{Q}_{p^2}}$, we obtain
$$\mu_{2,-1}\omega_2^{(1 - p)k}\oplus\mu_{2,-1}\omega_2^{(1 - p)\ell}~\cong~ \mu_{2,-1}\omega_2^{(1 - p)k'}\oplus\mu_{2,-1}\omega_2^{(1 - p)\ell'},$$
 whence the result.
\end{proof}

\begin{cor}\label{scbij}
 There exists a bijection between $\widehat{G}$-equivalence classes of regular Langlands parameters coming from the endoscopic group $J = (\textnormal{U}(1)\times\textnormal{U}(1))(\mathbb{Q}_{p^2}/\mathbb{Q}_p)$ and $L$-packets of irreducible supercuspidal representations on the group $G = \textnormal{U}(1,1)(\mathbb{Q}_{p^2}/\mathbb{Q}_p)$, given by
$$\varphi_{k,\ell}\longleftrightarrow \{\cusp{\ell}{[k - \ell - 1]},~\cusp{k}{[\ell - k - 1]}\},$$
where $0\leq k,\ell < p + 1$, and where $[k - \ell - 1]$ (resp. $[\ell - k - 1]$) denotes the unique integer between $0$ and $p - 1$ equivalent to $k - \ell - 1$ (resp. $\ell - k - 1$) modulo $p + 1$.  Moreover, this bijection is compatible with twisting by characters on both sides (under the correspondence of Corollary \ref{u1llc}).
\end{cor}

\begin{proof}
Let $\Pi_{k,\ell}$ denote the $L$-packet on the right-hand side of the correspondence above.  Proposition \ref{Lpackets} and Lemma \ref{paramequiv} show that two $L$-packets of the form $\Pi_{k,\ell}$ and $\Pi_{k',\ell'}$ are identical if and only if $\{k,\ell\} = \{k', \ell'\}$, if and only if $\varphi_{k,\ell}$ is equivalent to $\varphi_{k', \ell'}$.
\end{proof}

\vspace{\baselineskip}

Our next task will be to extend the correspondence of the above corollary to nonsupercuspidal $L$-packets.  In doing so, we are led to consider Langlands parameters arising from a proper Levi subgroup of ${}^LG$.  We let
$${}^LT = \widehat{T}\rtimes \mathcal{G}_{\mathbb{Q}_p},$$
where the action of $\mathcal{G}_{\mathbb{Q}_p}$ on $\widehat{T}$ is the restriction of the action on $\widehat{G}$.

\begin{prop}\label{paramsfromT}
 Let $\varphi:\mathcal{G}_{\mathbb{Q}_p}\longrightarrow {}^LG$ be a Langlands parameter which factors through the group ${}^LT$, that is, such that $\varphi$ is the composition
$$\mathcal{G}_{\mathbb{Q}_p}\longrightarrow {}^LT\ensuremath{\lhook\joinrel\relbar\joinrel\rightarrow} {}^LG,$$
where the second arrow denotes the canonical inclusion.  Then there exist $0\leq r < p^2 - 1$ and $\lambda\in\ol{\mathbb{F}}_p^\times$ such that, up to equivalence, $\varphi$ is of the following form:
\begin{center}
\begin{tabular}{ccc}
 $\textnormal{Fr}_p$ & $\longmapsto$ & $\begin{pmatrix}1 & 0 \\ 0 & \lambda\end{pmatrix}\textnormal{Fr}_p$\\
 $h$ & $\longmapsto$ & $\begin{pmatrix}\mu_{2,\lambda^{-1}}\omega_2^r(h) & 0 \\ 0 & \mu_{2,\lambda}\omega_2^{-pr}(h) \end{pmatrix}h$,
\end{tabular}
\end{center}
where $h\in \mathcal{G}_{\mathbb{Q}_{p^2}}$.
\end{prop}

\begin{proof}
 We proceed as in the proof of Proposition \ref{u1params}.  Using the action of $\mathcal{G}_{\mathbb{Q}_p}$ on $\widehat{T}$ we may assume that, up to equivalence, we have
$$\varphi(\textnormal{Fr}_p) = \begin{pmatrix}1 & 0 \\ 0 & \lambda\end{pmatrix}\textnormal{Fr}_p$$
for some $\lambda\in \ol{\mathbb{F}}_p^\times$.  Let $\varphi_0:\mathcal{G}_{\mathbb{Q}_{p^2}}\longrightarrow \widehat{T}\ensuremath{\lhook\joinrel\relbar\joinrel\rightarrow} \widehat{G}$ be the Galois representation associated to $\varphi$, so that
$$\varphi(h) = \varphi_0(h)h = \begin{pmatrix}\mu_{2,\lambda_1}\omega_2^{r_1}(h) & 0 \\ 0 & \mu_{2,\lambda_2}\omega_2^{r_2}(h) \end{pmatrix}h,$$
where $0\leq r_i < p^2 - 1$, $\lambda_i\in\ol{\mathbb{F}}_p^\times$, and $h\in\mathcal{G}_{\mathbb{Q}_{p^2}}$.  In particular, for $h = \textnormal{Fr}_p^2$, we have
\begin{eqnarray*}
 \begin{pmatrix}\lambda_1 & 0 \\ 0 & \lambda_2\end{pmatrix}\textnormal{Fr}_p^2 & = & \varphi(\textnormal{Fr}_p^2)\\
 & = & \varphi(\textnormal{Fr}_p)^2\\
 & = & \begin{pmatrix}1 & 0 \\ 0 & \lambda\end{pmatrix}\textnormal{Fr}_p\begin{pmatrix}1 & 0 \\ 0 & \lambda\end{pmatrix}\textnormal{Fr}_p\\
 & = & \begin{pmatrix}\lambda^{-1} & 0 \\ 0 & \lambda\end{pmatrix}\textnormal{Fr}_p^2,
\end{eqnarray*}
and thus $\lambda_2 = \lambda_1^{-1} = \lambda$.

Again using Lemma \ref{brlemma}, we get, for $h\in \mathcal{G}_{\mathbb{Q}_{p^2}}$,
\begin{eqnarray*}
 \begin{pmatrix}\omega_2^{pr_1}(h) & 0 \\ 0 & \omega_2^{pr_2}(h)\end{pmatrix}\textnormal{Fr}_ph\textnormal{Fr}_p^{-1} & = & \varphi(\textnormal{Fr}_ph\textnormal{Fr}_p^{-1})\\
 & = & \varphi(\textnormal{Fr}_p)\varphi(h)\varphi(\textnormal{Fr}_p)^{-1}\\
 & = & \begin{pmatrix}1 & 0 \\ 0 & \lambda\end{pmatrix}\textnormal{Fr}_p\begin{pmatrix}\omega_2^{r_1}(h) & 0 \\ 0 & \omega_2^{r_2}(h)\end{pmatrix}h\textnormal{Fr}_p^{-1}\begin{pmatrix}1 & 0 \\ 0 & \lambda^{-1}\end{pmatrix}\\
 & = & \begin{pmatrix}\omega_2^{-r_2}(h) & 0 \\ 0 & \omega_2^{-r_1}(h)\end{pmatrix}\textnormal{Fr}_ph\textnormal{Fr}_p^{-1},
\end{eqnarray*}
which shows that $r_2 \equiv -pr_1~(\textnormal{mod}~ p^2 - 1)$.
\end{proof}

\begin{defn}
 Let $0\leq r < p^2 - 1$ and $\lambda\in\ol{\mathbb{F}}_p^\times$.  We denote by $\psi_{r,\lambda}:\mathcal{G}_{\mathbb{Q}_p}\longrightarrow {}^LG$ the Langlands parameter constructed in Proposition \ref{paramsfromT}; explicitly, we have
\begin{center}
\begin{tabular}{ccc}
 $\psi_{r,\lambda}(\textnormal{Fr}_p)$ & $=$ & $\begin{pmatrix}1 & 0\\ 0 & \lambda\end{pmatrix}\textnormal{Fr}_p$\\
 $\psi_{r,\lambda}(h)$ & $=$ & $\begin{pmatrix}\mu_{2,\lambda^{-1}}\omega_2^r(h) & 0 \\ 0 & \mu_{2,\lambda}\omega_2^{-pr}(h) \end{pmatrix}h$,
\end{tabular}
\end{center}
where $h\in\mathcal{G}_{\mathbb{Q}_{p^2}}$.
\end{defn}

\vspace{\baselineskip}
We shall also need more precise information about equivalence classes of the Langlands parameters $\varphi_{k,\ell}$ and $\psi_{r,\lambda}$.  This is the content of the following two Lemmas.

\begin{lemma}\label{paramequiv2}
  Let $0\leq r,r'<p^2 - 1$ and $\lambda, \lambda'\in \ol{\mathbb{F}}_p^\times$.  Then $\psi_{r,\lambda}$ is equivalent to $\psi_{r',\lambda'}$ if and only if $r' = r, \lambda' = \lambda$ or $r' \equiv -pr~(\textnormal{mod}~p^2 - 1), \lambda' = \lambda^{-1}.$
\end{lemma}

\begin{proof}
 Conjugation by the element
$$\begin{pmatrix}0 & 1 \\ -\lambda^{-1} & 0\end{pmatrix}$$
shows that $\psi_{r,\lambda}$ is equivalent to $\psi_{-pr,\lambda^{-1}}$.  Assume conversely that $\psi_{r,\lambda}$ is equivalent to $\psi_{r',\lambda'}$.  Upon restricting to $\mathcal{G}_{\mathbb{Q}_{p^2}}$, we obtain
$$\mu_{2,\lambda^{-1}}\omega_2^{r}\oplus\mu_{2,\lambda}\omega_2^{-pr}~\cong~\mu_{2,\lambda'^{-1}}\omega_2^{r'}\oplus\mu_{2,\lambda'}\omega_2^{-pr'},$$
which gives the desired result.
\end{proof}

\begin{lemma}\label{endvsnonend}
 Let $0\leq k,\ell < p+1,~~~$ $0\leq r < p^2 - 1$, and $\lambda\in\ol{\mathbb{F}}_p^\times$.  Then $\varphi_{k,\ell}$ is equivalent to $\psi_{r,\lambda}$ if and only if $k = \ell$, $r\equiv (1 - p)k~(\textnormal{mod}~p^2 - 1),$ and $\lambda = -1$.
\end{lemma}

\begin{proof}
Fix an element $d\in \ol{\mathbb{F}}_p^\times$ such that $d^2 = -1$, and set
$$g = \begin{pmatrix}1 & -d \\ 2^{-1} & 2^{-1}d\end{pmatrix}.$$
A bit of algebra verifies that $g\varphi_{k,k}(h)g^{-1} = \psi_{(1-p)k,-1}(h)$ for every $h\in\mathcal{G}_{\mathbb{Q}_p}$.

Assume conversely that $\varphi_{k,\ell}$ is equivalent to $\psi_{r,\lambda}$.  Upon restricting to $\mathcal{G}_{\mathbb{Q}_{p^2}}$, we obtain
$$\mu_{2,-1}\omega_2^{(1-p)k}\oplus\mu_{2,-1}\omega_2^{(1-p)\ell}~\cong~\mu_{2,\lambda^{-1}}\omega_2^r\oplus\mu_{2,\lambda}\omega_2^{-pr}.$$
This implies $\lambda = -1$, $r\equiv(1 - p)k~(\textnormal{mod}~p^2 - 1)$ and $k = \ell$, which gives the result.
\end{proof}

\vspace{\baselineskip}

\begin{defn}\label{corr}
 We define a \emph{``semisimple mod-$p$ correspondence for $G = \textnormal{U(1,1)}(\mathbb{Q}_{p^2}/\mathbb{Q}_p)$''} to be the following correspondence between certain $\widehat{G}$-equivalence classes of Langlands parameters over $\ol{\mathbb{F}}_p$ and certain isomorphism classes of semisimple $L$-packets on $\textnormal{U(1,1)}(\mathbb{Q}_{p^2}/\mathbb{Q}_p)$:
\begin{itemize}
 \item \emph{the supercuspidal case:}
Let $0\leq k,\ell<p + 1$ with $k\neq \ell$.
$$\boxed{\varphi_{k,\ell} \longleftrightarrow \left\{\cusp{\ell}{[k - \ell - 1]},~\cusp{k}{[\ell - k - 1]}\right\}}$$
\vspace{\baselineskip}
 \item \emph{the nonsupercuspidal case:}
Let $0\leq r \leq p - 1$, $\lambda\in\ol{\mathbb{F}}_p^\times$, and $0\leq k < p+1$.
\begin{itemize}
\item if $(r,\lambda)\neq (0,1),~(p-1,1)$:
$$\boxed{\psi_{r,\lambda}\otimes\omega_2^{(1-p)k} = \psi_{r + (1-p)k,\lambda}\longleftrightarrow \left\{(\omega^k\circ\det)\otimes\textnormal{ind}_B^G(\mu_{\lambda^{-1}}\omega^{-pr})~\oplus~(\omega^k\circ\det)\otimes\textnormal{ind}_B^G(\mu_{\lambda}\omega^{r})\right\}}$$
\item if $(r,\lambda) = (0,1)$:
$$\boxed{\psi_{0,1}\otimes\omega_2^{(1-p)k} = \psi_{(1-p)k,1}\longleftrightarrow \left\{\omega^k\circ\det~\oplus~(\omega^k\circ\det)\otimes\textnormal{St}_G~\oplus~\omega^k\circ\det~\oplus~(\omega^k\circ\det)\otimes\textnormal{St}_G\right\}}$$
\end{itemize}
\end{itemize}
\end{defn}

\vspace{2\baselineskip}

\subsection{Remarks}\begin{enumerate}
\item Corollary \ref{scbij} and Lemmas \ref{paramequiv} and \ref{paramequiv2} imply that the correspondence above is well-defined.

\vspace{\baselineskip}
\item We may state this correspondence more elegantly as follows.  For $0\leq r\leq p - 1$, the group $K$ acts irreducibly on the representation $\sigma_r$ defined in Subsection \ref{gl2sc}.  We let $\tau_{r,1}$ denote the endomorphism of $\textnormal{c-ind}_K^G(\sigma_r)$ which corresponds via Frobenius Reciprocity to the function with support $K\alpha^{-1}K$ and taking the value $U_r$ at $\alpha^{-1}$, where $U_r$ is the endomorphism of $\sigma_r$ given by
$$U_r.x^{r - i}y^i = \begin{cases}0 & \textnormal{if}~i\neq r,\\ y^r & \textnormal{if}~i = r.\end{cases}$$
The spherical Hecke algebra $\mathcal{H}_{\ol{\mathbb{F}}_p}(G,K,\sigma_r)$ of $G$-equivariant endomorphisms of the compactly induced representation $\textnormal{c-ind}_K^G(\sigma_r)$ is then isomorphic to a polynomial algebra over $\ol{\mathbb{F}}_p$ in one variable, generated by an endomorphism $\tau_r$.  Explicitly, we have
$$\tau_r = \begin{cases}\tau_{r,1}& \textnormal{if}~r \neq 0,\\ \tau_{r,1} + 1 & \textnormal{if}~ r = 0.\end{cases}$$
For $\lambda\in\ol{\mathbb{F}}_p^\times$, we define
$$\boldsymbol{\pi}(r,\lambda):=\frac{\textnormal{c-ind}_K^G(\sigma_r)}{(\tau_r - \lambda)}.$$
A simple argument shows that
$$\boldsymbol{\pi}(r,\lambda)|_{G_\textnormal{S}}\cong \pi_0(r,\lambda),$$
where $\pi_0(r,\lambda)$ denotes the representation of $\textnormal{SL}_2(\mathbb{Q}_p)$ (viewed as a representation of $G_\textnormal{S}$) defined in \cite{Ab12}, Section 3.4.  Using Th\'eor\`eme 3.18 (\emph{loc. cit.}) and the existence of certain $I(1)$-invariant elements of $\boldsymbol{\pi}(r,\lambda)$ (along with Theorem \ref{nonscres}), we deduce
$$\boldsymbol{\pi}(r,\lambda)\cong \begin{cases}\textnormal{ind}_B^G(\mu_{\lambda^{-1}}\omega^{-pr})& \textnormal{if}~(r,\lambda)\neq (0,1),\\ \textnormal{nonsplit extension of}~ 1_G~\textnormal{by}~\textnormal{St}_G& \textnormal{if}~(r,\lambda) = (0,1). \end{cases}$$

If we let $\pi^{\textnormal{ss}}$ denote the semisimplification of a smooth representation $\pi$ of $G$, we obtain
$$\boldsymbol{\pi}(r,\lambda)^\textnormal{ss} = \begin{cases}\textnormal{ind}_B^G(\mu_{\lambda^{-1}}\omega^{-pr}) & \textnormal{if}~(r,\lambda)\neq (0,1),~(p-1,1),\\ \omega^{p}\circ\det~\oplus~(\omega^{p}\circ\det)\otimes\textnormal{St}_G & \textnormal{if}~(r,\lambda) = (p - 1,1),\\ 1_G~\oplus~\textnormal{St}_G & \textnormal{if}~(r,\lambda) = (0,1).\end{cases}$$

\vspace{\baselineskip}

The correspondence of Definition \ref{corr} now takes the form:

\begin{defnI}\hfill\newline
\noindent $\bullet$ \emph{The supercuspidal case:}
Let $0\leq k,\ell<p + 1$ with $k\neq \ell$.
$$\boxed{\varphi_{k,\ell} \longleftrightarrow \left\{\cusp{\ell}{[k - \ell - 1]},~\cusp{k}{[\ell - k - 1]}\right\}}$$
\noindent $\bullet$ \emph{The nonsupercuspidal case:}
Let $0\leq r \leq p - 1$, $\lambda\in\ol{\mathbb{F}}_p^\times$, and $0\leq k < p+1$.
$$\boxed{\psi_{r,\lambda}\otimes\omega_2^{(1 - p)k} = \psi_{r + (1 - p)k,\lambda}\longleftrightarrow \{(\omega^k\circ\det)\otimes\boldsymbol{\pi}(r,\lambda)^{\textnormal{ss}}~\oplus~(\omega^{k + r + 1}\circ\det)\otimes\boldsymbol{\pi}(p - 1 - r,\lambda^{-1})^{\textnormal{ss}}\}}$$
\end{defnI}

\vspace{\baselineskip}
\item The correspondences of Corollary \ref{u1u1llc} and Definition \ref{corr}, along with the homomorphism $\xi$ of Proposition \ref{Lembedding}, imply that we have an \emph{endoscopic transfer map}
$$\widetilde{\xi}:\mathfrak{Rep}_{\ol{\mathbb{F}}_p}((\textnormal{U}(1)\times\textnormal{U}(1))(\mathbb{Q}_{p^2}/\mathbb{Q}_p))\longrightarrow \mathfrak{L}\textnormal{-}\mathfrak{pack}_{\ol{\mathbb{F}}_p}(\textnormal{U}(1,1)(\mathbb{Q}_{p^2}/\mathbb{Q}_p))$$
from the set of isomorphism classes of smooth irreducible representations of $(\textnormal{U}(1)\times\textnormal{U}(1))(\mathbb{Q}_{p^2}/\mathbb{Q}_p)$ to the set of isomorphism classes of $L$-packets of semisimple representations on $\textnormal{U}(1,1)(\mathbb{Q}_{p^2}/\mathbb{Q}_p)$.  Using Lemma \ref{endvsnonend}, the map $\widetilde{\xi}$ is given explicitly by
$$\widetilde{\xi}\left(\omega^k\otimes\omega^\ell\right) = \begin{cases} \left\{\cusp{\ell}{[k - \ell - 1]},~\cusp{k}{[\ell - k - 1]}\right\} & \textnormal{if}~ k \neq \ell,\\ \left\{\textnormal{ind}_B^G(\mu_{-1}\omega^{(1 - p)k})~\oplus~\textnormal{ind}_B^G(\mu_{-1}\omega^{(1 - p)k})\right\} & \textnormal{if}~k = \ell.\end{cases}$$
This bears a striking resemblence to the complex case (see Proposition 11.1.1 of \cite{Ro90}, especially points (c) and (e)).  Moreover, the equation
$$\widetilde{\xi}\left(1_J\right) = \widetilde{\xi}\left(\omega^0\otimes\omega^0\right) = \left\{\textnormal{ind}_B^G(\mu_{-1})~\oplus~\textnormal{ind}_B^G(\mu_{-1})\right\}$$
gives an example of \emph{transfer of unramified representations}, which may also be deduced from (a modified version of) the discussion in Section 2.7 of \cite{Mi11} (see also Theorem 4.4, \emph{loc. cit.}, and Section 4.5 of \cite{Ro90}).
\end{enumerate}