$\text{SL}(2, \mathbb{F}_q)$, for which characters is the $G$-representation irreducible? Followup to here. Let $\mathbb{F}$ be a finite field with $q$ elements, and let $G = \text{SL}_2(\mathbb{F})$. The group $G$ acts linearly on the $2$-dimensional vector space $\mathbb{F}^2$ and fixes the origin $0$. Hence, $G$ acts on the set $X := \mathbb{F}^2 \setminus \{0\}$, the complement of the origin. For any group homomorphism $\chi: \mathbb{F}^\times \to S^1 \subset \mathbb{C}^\times$, in $\mathbb{C}\{X\}$, we define a subspace$$\mathbb{C}\{X\}^\chi := \{f \in \mathbb{C}\{X\} : f(z \cdot x) = \chi(z) \cdot f(x), \text{ for all }z \in \mathbb{F}^\times\}.$$We know that there is a vector space direct sum decomposition$$\mathbb{C}\{X\} = \oplus_{\chi \in \widehat{H}} \mathbb{C}\{X\}^\chi,$$where we put $H := \mathbb{F}^\times$.
My question is as follows. For which $\chi \in \widehat{H}$ is the $G$-representation $\mathbb{C}\{X\}^\chi$ irreducible?
 A: Since $G$ acts transitively on the set $X$ where the stabilizer of $(1,0)\in\mathbb F^2\setminus\{0\}$ is $N:=\{\begin{pmatrix}1&b\\0&1\end{pmatrix}:a\ne0\}$, the representation $\mathbb C\{X\}$ is $\mathrm{Ind}_N^G1$.
Now, the decomposition given is the decomposition
$$\mathrm{Ind}_N^G1=\bigoplus_{\chi\in H^\vee}\mathrm{Ind}_B^G\chi,$$
where $B:=\{\begin{pmatrix}a&b\\0&a^{-1}\end{pmatrix}:a\ne0\}$ is the Borel subgroup of $G$, and $\chi$ is viewed as a character of $B$ by inflating via $B\to B/N\cong H$. In other words, $\chi\begin{pmatrix}a&b\\0&a^{-1}\end{pmatrix}:=\chi(a)$.
Now, we can use Mackey theory to check when these representations are irreducible (and using the Bruhat decomposition $G=B\sqcup BwB$ with $w=\begin{pmatrix}&1\\1\end{pmatrix}$):
$$\begin{align*}\hom_G(\mathrm{Ind}_B^G\chi,\mathrm{Ind}_B^G\chi)&=\hom_B(\chi,\mathrm{Ind}_B^G\chi|_B)\\
&=\hom_B(\chi,\chi\oplus\mathrm{Ind}_{B\cap B^w}^B(\chi^{-1}))\\
&=\mathbb C\oplus\hom_B(\chi,\mathrm{Ind}_T^B(\chi^{-1}))\\
&=\mathbb C\oplus\hom_T(\chi,\chi^{-1}),\end{align*}$$
where $T\subset B$ is the maximal torus. Thus, $\mathbb C\{X\}^\chi=\mathrm{Ind}_B^G\chi$ is irreducible exactly when $\chi^2\ne1$, and otherwise it has length $2$. These are the so-called principal-series representations.
