This is the question from Serre's book #7.4. $G=\operatorname{SL}\_2(k)$, where $k$ is a finite field and $H\leq G$ such that $H$ consists of matrices $\begin{pmatrix} a & b \\ 0 & d \end{pmatrix}$. Let $w:k^\* \rightarrow \mathbb{C}^\*$ be a homomorphism and $\chi\_w$ be the character of degree 1 of $H$ defined by $\chi\_w \begin{pmatrix} a & b \\ c & d \end{pmatrix}=w(a)$. Show that the representation of $G$ induced by $\chi\_w$ is irreducible if $w^2\neq 1$.

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