Exercise 26, § 21

Let $q$ be a power of a prime number and F a finite field with $q$ elements; take G to be the group $\mathbf{G}\mathbf{L}(2,F)$.

a) Describe the conjugacy classes of G (distinguish four types of classes, containing, respectively, $1,q^2-1,q^2+q$, and $q^2-q$ elements).

b) The group G acts on the projective line P over F; consider the representation $\pi_P$ of G (Exercise 6). For any group homomorphism $\lambda : F^*\rightarrow K^*$, denote by $\delta_{\lambda}$ the representation of degree 1 associated with the homomorphism $\lambda \circ$ det from G to $K^*$, and set $\pi_{\lambda}=\pi_P\otimes \delta_{\lambda}$. Prove that $\pi_{\lambda}$ is irreducible, and calculate its character.

c) Let B be the subgroup of G consisting of the upper triangular matrices. For $\lambda , \mu$ in Hom(F$^*,K^*$), denote by $\beta_{\lambda ,\mu}$ the representation of B of degree 1 defined by $\beta_{\lambda ,\mu}((a_{ij})) =\lambda (a_{11})\mu(a_{22})$, and set $\pi_{\lambda ,\mu}=$ Ind$^G_B(\beta_{\lambda ,\mu})$. Prove that $\pi_{\lambda ,\mu}$ is irreducible for $\lambda \not=\mu$, while $\pi_{\lambda ,\lambda}$ is isomorphic to $\pi_P\oplus \delta_{\lambda}$ (apply Exercise 8). The representations $\pi_{\lambda ,\mu}$ and $\pi_{\lambda',\mu'}$ are isomorphic if and only if the subsets $\{\lambda , \mu\}$ and $\{\lambda ', \mu'\}$ of Hom(F$^*,K^*$) are equal.

d) Let $F'$ be an extension of F of degree 2; choosing a basis of $F'$ over F allows one to identify $F^{'*}$ with a subgroup of G = Aut$_F(F')$. Let $\varphi \in$ Hom(F$^{'*},K^*$); calculate Ind$^G_{F^{'*}}(\varphi )$, and prove that it is equal to Ind$^G_{F^{'*}}(\varphi^q)$. Denote by $\lambda$ the restriction of $\varphi$ to $F^*$, and set $\chi_{\varphi}=\chi_{\pi_P\otimes\pi_{\lambda ,1}}-\chi_{\pi_{\lambda ,1}}-$ Ind$^G_{F^{'*}}(\varphi )$. Prove that if $\varphi \not=\varphi^q$, then we have $\langle \chi_{\varphi}, \chi_{\varphi}\rangle = 1$ and $\chi_{\varphi}(1) =q-1$, so that $\chi_{\varphi}$ is the character of an irreducible representation $\pi_{\varphi}$ of dimension $q-1$. Prove that we thus obtain $^1_2q(q-1)$ nonisomorphic irreducible representations. e) Prove that the irreducible representations $\pi_{\lambda}$ for $\lambda \in$ Hom(F$^*,K^*$)$,\pi_{\lambda ,\mu}$ for a subset $\{\lambda , \mu\}$ of Hom(F$^*,K^*$) with two elements, and $\pi_{\varphi}$ for $\varphi \in$ Hom(F$^{'*},K^*$) with $\varphi \not=\varphi^q$ are pairwise nonisomorphic and that every irreducible representation of G is isomorphic to one of them.