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% 29/11/2023 11AM

We have found:
\[ \Ind_B^G \mu_\theta (g) = \begin{cases}
  (q^2 - 1)\theta(\lambda) & [g]_G = \left[
  \begin{pmatrix}
    \lambda & 0 \\
    0 & \lambda
  \end{pmatrix}\right]_G \\
  -\theta(\lambda) & [g]_G = \left[
  \begin{pmatrix}
    \lambda & 1 \\
    0 & \lambda
  \end{pmatrix}\right]_G \\
  0 & \text{otherwise}
\end{cases} \]
and
\[ \Gip\langle \Ind_B^G \mu_\theta, \Ind_B^G
\mu_\theta \rangle_G = q .\]
Our next strategy is to induce characters from
$K$:
\begin{align*}
  \FF_{q^2}
  &\to M_2(\FF_q) \\
  \lambda + \mu\sqrt{\eps}
  &\mapsto
  \begin{pmatrix}
    \lambda & \eps\mu \\
    \mu & \lambda
  \end{pmatrix}
\end{align*}
induces an isomorphism of rings $\FF_{q^2}$ to $K
\cup \{0\}$. We will identify these rings. Under
this identification,
\begin{align*}
  \FF_q^\times
  &\leftrightarrow Z \\
  \lambda
  &\leftrightarrow
  \begin{pmatrix}
    \lambda & 0 \\
    0 & \lambda
  \end{pmatrix}
\end{align*}
Moreover
\[
\begin{pmatrix}
  \lambda & \eps\mu \\
  \mu & \lambda
\end{pmatrix}^q
=
\begin{pmatrix}
  \lambda & -\eps\mu \\
  -\mu & \lambda
\end{pmatrix}
\]
since $(\lambda + \mu\sqrt{\eps})^q = (\lambda -
\mu\sqrt{\eps})$.

We want to understand $\Ind_K^G \varphi$ for an
(\gls{irred}) \gls{char_rep} $\varphi$ of $K$.
First we consider the double cosets $K \leftquot G
/ K$ and then use Mackey to compute $\Gip\langle
\Ind_K^G \varphi, \Ind_K^G \varphi \rangle_G$. For
$k \in K$, $g \in G$,
\begin{align*}
  kgK = gK
  &\iff g^{-1} kg \in K \\
  &\iff g^{-1} kg \in \{k, k^q\}
\end{align*}
($[k]_G \cap K = \{k, k^q\}$). Writing
\[ t =
\begin{pmatrix}
  1 & 0 \\
  0 & -1
\end{pmatrix}
\]
we get
\[ t^{-1}
\begin{pmatrix}
  \lambda & \eps\mu \\
  \mu & \lambda
\end{pmatrix}
t =
\begin{pmatrix}
  \lambda & -\eps\mu \\
  -\mu & \lambda
\end{pmatrix}
\]
so $kgK = gK \iff g^{-1} kg = k$ or $(tg)^{-1}
k(tg) = k$. Furthermore since
\[ C_G \left(
\begin{pmatrix}
  \lambda & \eps\mu \\
  \mu & \lambda
\end{pmatrix}
\right) = \begin{cases}
  K & \text{if $\mu \neq 0$} \\
  G & \text{if $\mu = 0$}
\end{cases} \]
so $kgK = gK \iff gK \in \{K, tK\}$ or $k \in Z$.
It follows that
\[ |KgK| = \begin{cases}
  |K| & g \in K \cup tK \\
  \ub{\left| \frac{K}{2} \right| |K|}_{=(q^2 -
  1)(q + 1)} & \text{otherwise}
\end{cases} \]
so there are
\[ \frac{|G| - 2|K|}{\left|\frac{K}{2}\right||K|}
= \frac{\frac{|G|}{|K|} - 2}{\left| \frac{K}{2}
\right|} = \frac{q(q - 1) - 2}{q + 1} = q - 2 \]
double cosets of size $\left| \frac{K}{2} \right|
|K|$.

Now $K \cap \leftgrep{}^t K = K$, $K \cap
\leftgrep{}^g K = Z$ if $g \notin K \cup tK$. Thus
by Mackey,
\[ \Gip\langle \Ind_K^G \varphi, \Ind_K^G
\rangle_G = \langle \varphi, \varphi \rangle_K +
\langle \varphi, \leftgrep{}^t \varphi \rangle_K +
\sum_{g \in K \leftquot G / K - \{K, tK\}} \langle
\varphi|_Z, \leftgrep{}^g \varphi|_Z \rangle_Z .\]
Since $\leftgrep{}^g \varphi|_Z = \varphi|_Z$ for
all $g \in G$, $\leftgrep{}^t \varphi =
\varphi^q$. So if $\varphi$ has \gls{rep_deg} $1$.
\[ \Gip\langle \Ind_K^G \varphi, \Ind_K^G \varphi
\rangle_G = \begin{cases}
  q - 1 & \varphi \neq \varphi^q \\
  q & \varphi = \varphi^q
\end{cases} \]
Next we compute
\[ \Ind_K^G \varphi(g) = \begin{cases}
  q(q - 1) \varphi(\lambda) & g =
  \begin{pmatrix}
    \lambda & 0 \\
    0 & \lambda
  \end{pmatrix} \\
  \varphi(\alpha) + \varphi^q(\alpha) & g = \alpha
  \in \FF_{q^2} \setminus \FF_q \\
  0 & \text{otherwise}
\end{cases} \]
We can compute
\begin{align*}
  \Gip\langle \Ind_B^G \mu_\theta \Ind_K^G u
  \rangle_G
  &= \frac{1}{|G|} \left( \sum_{\lambda \in Z}
  (q^2 - 1) \ol{\theta(\lambda)} q(q - 1)
  \varphi(\lambda) + 0 \right) \\
  &= |Z| \langle \theta, \varphi|_Z \rangle_Z \\
  &= q - 1
  &&\text{if $\theta = \varphi|_Z$}
\end{align*}
If $\beta_\varphi = \Ind_B^G \mu_\theta = \Ind_B^G
\mu_\theta - \Ind_K^G \varphi$ for $\theta =
\varphi|_Z$.
\begin{align*}
  \Gip\langle \beta_\varphi, \beta_\varphi
  \rangle_G
  &= \Gip\langle \Ind_B^G \mu_\theta, \Ind_B^G
  \mu_\theta \rangle_G - 2\Gip\langle \Ind_B^G
  \mu_\theta, \Ind_K^G \varphi \rangle_G +
  \langle \Ind_K^G \varphi, \Ind_K^G \varphi
  \rangle \\
  &= q - 2(q - 1) + \begin{cases}
    q - 1 & \varphi \neq \varphi^q \\
    q & \varphi = \varphi^q
  \end{cases} \\
  &= \begin{cases}
    1 & \text{if $\varphi \neq \varphi^q$} \\
    2 & \text{if $\varphi = \varphi^q$}
  \end{cases}
\end{align*}
Also
\[ \beta_\varphi \left(
\begin{pmatrix}
  1 & 0 \\
  0 & 1
\end{pmatrix}
\right) = (q^2 - 1) - (q)(q - 1) = q - 1 > 0 \]
It follows that $\beta_\varphi$ is an \gls{irred}
\gls{char_rep} whenever $\varphi \neq \varphi^q$.
Since $\beta_\varphi = \beta_{\varphi^q}$,
$\varphi^{q^2} = \varphi$ and
\[ |\{\varphi : \varphi^q \neq \varphi\}| = q - 1 \]
We et
\[ \frac{(q^2 - 1) - (q - 1)}{2} = {q \choose 2} \]
\gls{irred} \glsref[char_rep]{characters} in this
way.
\begin{center}
  \begin{tabular}{c|c|c|c|c|c}
    & $\begin{pmatrix}\lambda & 0 \\ 0 &
    \lambda\end{pmatrix}$ & $\begin{pmatrix}
    \lambda & 1 \\ 0 & \lambda\end{pmatrix}$ &
    $\begin{pmatrix}\lambda & 0 \\ 0 & \mu\end{pmatrix}$
    & $\alpha, \alpha^q$ & \#\glsref[rep]{reps} \\
    \hline
    $\chichar_\theta$ & $\theta(\lambda)^2$ &
    $\theta(\lambda)^2$ &
    $\theta(\lambda)\theta(\mu)$ &
    $\theta(\alpha^{q + 1})$ & $q - 1$ \\
    $V_\theta$ & $q\theta(\lambda)^2$ & $0$ &
    $\theta(\lambda)\theta(\mu)$ &
    $-\theta(\alpha^{q + 1})$ & $q - 1$ \\
    $W_{\theta, \phi}$ & $(q + 1)\theta(\lambda
    \phi(\lambda)$ &
    $\theta(\lambda)\phi(\lambda)$ &
    $\theta(\lambda) \phi(\mu) + \phi(\lambda)
    \theta(\mu)$ & $0$ & ${q - 1 \choose 2}$ \\
    $\beta_\varphi$ & $(q - 1)\varphi(\lambda)$ &
    $-\varphi(\lambda)$ & $0$ & $-(\varphi +
    \varphi^q)(\alpha)$ & ${q \choose 2}$
  \end{tabular}
\end{center}
We have not compute the \glspl{rep} corresponding
to $\beta_\varphi$ explicitly. These are known as
\emph{discrete series representations}.

Drufield found these in $l$-adic \'etale cohomology groups
of an explicit algebraic curve $X / \FF_q$. They
can also be found as $p$-adic de Rham cohomology
groups over a similar space. These can be viewed
as generalisations of `functions on $X$'. This
work was generalised by Deligne-Lusztig for all
``finite groups of lie type''. Our computation
also allows us to compute the \gls{char_table} of
$\PGL_2(\FF_q) = \frac{\GL_2(\FF_q)}{Z}$ as its
\glspl{rep} are just \gls{irred} \glspl{rep} of
$G$ where $Z$ acts trivially, i.e.
$\chichar_\theta, V_\theta$ for $\theta^2 =
\indicator{}$, $W_{\theta, \theta^{-1}}$ for
$\theta \neq \theta^{-1}$ and $\beta_\varphi$ such
that $\varphi|_Z = \indicator{Z}$ (i.e.
$\varphi^{q + 1} = \indicator{}$ as well as
$\varphi^q \neq \varphi$). We can then also
compute the \gls{char_table} of $\PSL_2(\FF_q) =
\frac{\SL_2(\FF_q)}{(\pm I)}$ which has index $2$
in $\PGL_2(\FF_q)$. These groups $\PSL_2(\FF_q)$
are simple if $q \ge 5$ and this can be seen from
the \gls{char_table}.
