Some groups have simpler representation theories than others. For example, the representation theory of $S_n$ is simpler and more beautiful than the representation theory of $A_n$ – the characters of $S_n$ all have integer values, and are easily parametrized, while we saw that $A_4$ and $A_5$ have characters that do not extend to $S_4$ and $S_5$, and have more complicated descriptions using roots of unity (cube roots for $A_4$ and fifth roots for $A_5$). It is also true that the conjugacy classes of $S_n$ are easier to describe than the conjugacy classes of $A_n$. The best way to understand the conjugacy classes or the irreducible representations of $A_n$ is probably to first understand the conjugacy classes or representations of $S_n$, then understand how some conjugacy classes or representations split into two when restricted to the smaller group.
Similarly, the representation theory of $\operatorname{GL} (n, \mathbb{F}_q)$ is much simpler and more beautiful than the representation theory of $\operatorname{SL} (n, \mathbb{F}_q)$. This theory was completed in 1956 by J. A. Green in an important paper. We will use Mackey theory to construct one of the four families of irreducible representations when $n = 2$. Everything we do generalized beautifully to general $n$ in Green's theory.
The group $\operatorname{GL} (2, \mathbb{F}_q)$ has irreducible representations of degrees $1, q, q - 1$ and $q + 1$. There are $q - 1$ linear characters, which are the characters $\chi (\det (g))$ where $\chi$ is any linear character of $\mathbb{F}_q^{\times}$; this obviously is a character since $\det : \operatorname{GL} (2, \mathbb{F}_q) \longrightarrow \mathbb{F}_q^{\times}$ is a homomorphism.
There are also $q - 1$ irreducible characters of degree $q$. One irreducible character, called the Steinberg character can be obtained by taking the permutation character on $\mathbb{P}^1 (\mathbb{F}_q)$ and subtracting $1$. You already know this character, but we mention that the Steinberg character can be generalized to $\operatorname{GL} (n, \mathbb{F}_q)$. It is an irreducible character of degree $q^{\frac{1}{2} n (n - 1)}$ with remarkable properties. The remaining irreducible characters can be obtained by multiplying by the $q - 1$ linear characters.
The two families of degrees $q - 1$ and $q + 1$ are "twins.'' They are parametrized by characters of two important abelian subgroups called tori.
The first torus $T_{\operatorname{split}}$ is the diagonal subgroup of order $(q - 1)^2$ consisting of all matrices of the form \[ T_{\operatorname{split}} = \left\{ \left(\begin{array}{cc} t_1 & \\ & t_2 \end{array}\right) {\large{|}} t_1, t_2 \in \mathbb{F}_q^{\times} \right\} \] The characters of $T_{\operatorname{split}}$ are of the form $t = \left(\begin{array}{cc} t_1 & \\ & t_2 \end{array}\right) \longmapsto \psi (t) = \psi_1 (t_1) \psi_2 (t_2)$, and each such character parametrizes a representation that we will call $\pi (\psi_1, \psi_2)$. In order to construct it, we first extend the charater to a larger subgroup \[ B = \left\{ \left(\begin{array}{cc} t_1 & x\\ 0 & t_2 \end{array}\right) {\large{|}} t_1, t_2 \in \mathbb{F}_q^{\times}, x \in \mathbb{F}_q \right\} \] called the Borel subgroup, by the same formula: \[ \psi \left(\begin{array}{cc} t_1 & x\\ 0 & t_2 \end{array}\right) = \psi_1 (t_1) \psi_2 (t_2), \] and we define $\pi (\psi_1, \psi_2)$ to be the representation of $\operatorname{GL} (2, \mathbb{F}_q)$ induced from this. We will deduce its properties from Mackey's theorem as a typical application.
Exercise 4.4.1: Prove that the degree of $\pi (\psi_1, \psi_2)$ is $q + 1$.
The first step is to compute the double cosets $B \backslash G / B$ where $G = \operatorname{GL} (2, \mathbb{F}_q) .$
Lemma 4.4.1: (Bruhat decomposition.) There are two double cosets in $B \backslash G / B$. They are \[ \left\{ \left(\begin{array}{cc} a & b\\ c & d \end{array}\right) {\large{|}} c = 0 \right\} = B \left(\begin{array}{cc} 1 & 0\\ 0 & 1 \end{array}\right) B \] and \[ \left\{ \left(\begin{array}{cc} a & b\\ c & d \end{array}\right) {\large{|}} c \neq 0 \right\} = B \left(\begin{array}{cc} 0 & 1\\ 1 & 0 \end{array}\right) B. \]
Proof. (Click to Expand/Collapse)
Lemma 4.4.2: Let $\chi (\psi_1, \psi_2)$ and $\chi (\phi_1, \phi_2)$ be the characters of $\pi (\psi_1, \psi_2)$ and $\pi (\phi_1, \phi_2)$, respectively, where $\psi_i$ and $\phi_i$ are characters of $\mathbb{F}_q^{\times}$. Then \[ \left\langle \chi (\psi_1, \psi_2), \chi (\phi_1, \phi_2) \right\rangle = \left\{ \begin{array}{cc} 1 & \text{if $\phi_1 = \psi_1$ and $\phi_2 = \psi_2,$}\\ 0 & \text{otherwise;} \end{array} \right\} + \left\{ \begin{array}{cc} 1 & \text{if $\phi_1 = \psi_2$ and $\phi_2 = \psi_1,$}\\ 0 & \text{otherwise;} \end{array} \right\} . \]
Proof. (Click to Expand/Collapse)
| \[ \label{deltabconsistent} \Delta \left( \left(\begin{array}{cc} t_1 & x\\ 0 & t_2 \end{array}\right) g \left(\begin{array}{cc} u_1 & y\\ 0 & u_2 \end{array}\right) \right) = \phi_1 (t_1) \phi_2 (t_2) \Delta (g) \psi_1 (u_1) \psi_2 (u_2) . \] | (4.4.1) |
| \[ \label{twodeltav} \Delta \left( \left(\begin{array}{cc} 1 & 0\\ 0 & 1 \end{array}\right) \right), \hspace{2em} \Delta \left( \left(\begin{array}{cc} 0 & 1\\ 1 & 0 \end{array}\right) \right) \] | (4.4.2) |
First, we show that
| \[ \label{deltaicond} \Delta \left( \left(\begin{array}{cc} 1 & 0\\ 0 & 1 \end{array}\right) \right) = 0\qquad\text{unless}\qquad \phi_1 = \psi_1\quad\text{and}\quad \phi_2 = \psi_2 . \] | (4.4.3) |
Next we prove that
| \[ \label{deltawcond}\Delta \left( \left(\begin{array}{cc} 0 & 1\\ 1 & 0 \end{array}\right) \right) = 0\qquad\text{unless}\qquad\phi_1 = \psi_2\quad\text{and}\quad \phi_2 = \psi_1. \] | (4.4.4) |
To summarize, $\Delta$ is determined by its values (4.4.2). There are however necessary and sufficient conditions as to whether these values are nonzero. Taking these into account one obtains the Lemma.
Theorem 4.4.1: The representation $\pi (\psi_1, \psi_2)$ is irreducible unless $\psi_1 = \psi_2$. The representations $\pi (\psi_1, \psi_2)$ and $\pi (\phi_1, \phi_2)$, with $\psi_1 \neq \psi_2$ and $\phi_1 \neq \phi_2$ are isomorphic if and only if either $\psi_1 = \phi_1$ and $\psi_2 = \phi_2$ or $\psi_1 = \phi_2$ and $\psi_2 = \phi_1$.
Proof. (Click to Expand/Collapse)
This gives us $\frac{1}{2} (q - 1) (q - 2)$ irreducible representations of degree $q + 1$, parametrized by characters of $T_{\operatorname{split}}$, which is a group with $(q - 1)^2$ elements. The torus $T_{\operatorname{split}}$ has a "twin'' that we will call $T_{\operatorname{nonsplit}}$, which is an abelian group with $q^2 - 1$ elements. Of its $q^2 - 1$ linear characters, $q^2 - q$ characters correspond to irreduccible representations, which have degree $q - 1$. However, as with the $q + 1$ dimensional representations, two characters correspond to the same irreducible representation, so there are only $\frac{1}{2} (q^2 - q)$ irreducible of degree $q - 1$.
The torus $T_{\operatorname{nonsplit}}$ can be constructed as follows. There is a field $\mathbb{F}_{q^2}$ with $q^2$ elements, and it is a vector space of dimension 2 over $\mathbb{F}_q$. The group $\mathbb{F}_{q^2}^{\times}$ acts on $\mathbb{F}_{q^2}$ by multiplication, and choosing a basis $x_1, x_2$ of $\mathbb{F}_{q^2}$ as a vector space over $\mathbb{F}_q$, the elements of $\mathbb{F}_{q^2}^{\times}$ can be realized as matrices. Thus if $\alpha \in \mathbb{F}_{q^2}^{\times}$ we can write \begin{eqnarray*} \alpha x_1 = a x_1 + b x_2, & & \\ \alpha x_2 = c x_1 + d x_2, & & \end{eqnarray*} and we can take $T_{\operatorname{nonsplit}}$ to be the set of all such matrices $\left(\begin{array}{cc} a & b\\ c & d \end{array}\right)$. Of course, it depends on the choice of basis but only up to conjugacy.
The nature of the parametrization of irreducible representations of $\operatorname{GL} (2, \mathbb{F}_q)$ by linear characters of $T$, where $T$ is either $T_{\operatorname{split}}$ or $T_{\operatorname{nonsplit}}$ can be described as follows. If $g \in G = \operatorname{GL} (2, \mathbb{F}_q)$ has distinct eigenvalues, and if $\psi$ is a linear character of either $T_{\operatorname{split}}$ or $T_{\operatorname{nonsplit}}$ parametrizing the irreducible representation of $G$ with character $\chi$, then $\chi (g) = 0$ unless $g$ has a conjugate in $T$. If it does, it has two conjugates $\gamma_1$ and $\gamma_2$ in $T$. For example, if $T = T_{\operatorname{split}}$ and one conjugate is \[ \gamma_1 = \left(\begin{array}{cc} t_1 & \\ & t_2 \end{array}\right), \] the other is \[ \gamma_2 = \left(\begin{array}{cc} t_2 & \\ & t_1 \end{array}\right) . \] Again, if $T = T_{\operatorname{nonsplit}}$ and one conjugate is $\gamma_1$, the other is $\gamma_2 = \gamma_1^q$. Then \[ \chi (g) = \left\{ \begin{array}{ll} \psi (\gamma_1) + \psi (\gamma_2) & \text{if $T = T_{\operatorname{split}}$},\\ - \psi (\gamma_1) - \psi (\gamma_2) & \text{if $T = T_{\operatorname{nonsplit}}$} . \end{array} \right. \] This recipe only tells us the character values for most elements of $G$.
Although the analogy and relationship between the two classes of representations of $G$ is very deep, there is an important difference. The $q-1$-dimensional representations cannot directly be constructed by induction. One approach to constructing them is to take one of the generalized characters $\chi$ constructed by Green's Theorem 4.1.2 and write it as $\chi_1 - \chi_2$ where $\chi_1$ is one of the irreducibles of degree $q + 1$, and $\chi_2$ is one of the ones of degree $q - 1$. If $\chi_1$ is already constructed, this yields $\chi_2$. Another approach, due to Deligne and Lusztig, constructs the representations as acting on certain cohomology groups.