%! TEX root = RT.tex % vim: tw=50 % 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}.