TITLE: Irreducible representations of $\operatorname{GL}_3(\mathbb{F}_q)$
QUESTION [5 upvotes]: I am trying to find all irreducible representations of $G = \operatorname{GL}_3(\mathbb{F}_q)$. I know that the order of $G$ is $(q^3-1)(q^3 - q)(q^3 - q^2)$ and the number of conjugacy classes is $q(q-1)(q+1)$.
My teacher suggested the following:
By Bushnell and Henniart's "The local Langlands conjecture for GL(2)", chapter 2, we know all irreducible representations of $\operatorname{GL}_2(\mathbb{F}_q)$.
Now look at $f: \operatorname{GL}_2(\mathbb{F}_q) \times \mathbb{F}_q^* \to B \subset \operatorname{GL}_3(\mathbb{F}_q)$
$$ (A, k) \mapsto \begin{pmatrix} A & 0 \\ 0 & k \\ \end{pmatrix} $$
Now for irreducible representations $\varphi$ of $\operatorname{GL}_2(\mathbb{F}_q)$ and characters $\chi$ of $\mathbb{F}_q^*$ look at $\operatorname{Ind}_P^G(\varphi \bigotimes \chi)$ where $P \subset G$ are all matrices of the form $$\begin{pmatrix} * & * & * \\ * & * & * \\ 0 & 0 & * \\ \end{pmatrix} $$ where we used the injection $B \to P$ and let $\varphi \bigotimes \chi$ work trivial on the upperright part.
Unfortunately, I do not know how to proceed from here. Does anybody know of a good source that would explain this approach (I recall my teacher using the words Levi-subgroups) or a better approach to constructing the irreducible representations of $G$?

REPLY [2 votes]: If you are willing to settle for the calculation of the characters of the irreducible complex representations of $\mathrm{GL}_n(F_q)$, then this is written down in section 6. of Chapter IV of Macdonald's book Symmetric functions and Hall polynomials, in an exposition parallel to (but more complicated than) the calculation of the characters of the symmetric groups.