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Combination Problem Understanding

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 $\operatorname{GL}_2(\mathbb{F}_q)$.

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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 $G$?

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This is the approach known as *parabolic induction*. It’s the standard way to approach the problem of classifying the representations of (the rational points of) algebraic groups. I suspect that you aren’t familiar with the theory of algebraic groups, so let me explicitly talk about general linear groups over finite fields. To simplify notation, let me fix a $q=p^n$ and write $G_n=GL_n(\mathbb{F}_q)$.

Inside $G_n$, there are natural occurances of subgroups $G_{n_1}\times\cdots\times G_{n_r}$, where $\sum_{i=1}^r n_i=n$, where the groups are the block-diagonal embeddings (and their $G_n$-conjugates, but we can just forget about this and assume that everything is in the standard block-diagonal form). These are *Levi subgroups*. In general, the idea of a Levi subgroup of a reductive group (something which behaves reasonably similarly to $GL_n$) is that it’s a subgroup with a smaller “semisimple rank”, which is somehow a measure of the expected complexity of the representation theory. So hopefully you classify the irreducible representations of these Levi subgroups — which all external tensor products of irreducible representations of the individual $G_{n_i}$ factors — and then try to build representations of $G_n$ from these.

There’s a natural way of doing this. To a Levi subgroup $M=G_{n_1}\times\cdots\times G_{n_r}$, there is a larger *parabolic subgroup* $P$ which contains $M$ as a quotient in a natural way — $P$ has a unipotent radical $N$, which is a certain normal subgroup, and $P=MN$. Then one can view an irreducible representation $\sigma$ of $M$ as an irreducible representation of $P$ by composing with the projection $P\rightarrow N$. The reason for doing this is that $P$ is now a large enough subgroup of $G_n$ that one can hope to describe the irreducible subquotients of the induced representation $Ind_P^G\ \sigma$. Explicitly in your case, the parabolic subgroup $P$ associated to $M=G_{n_1}\times\cdots\times G_{n_r}$ is the group of upper-triangular matrices obtained by “filling in” the $0$ entries above the block-diagonal of $M$. In principle, once you know the representations of $M$, you should be able to use character theory to describe the irreducible subquotients of this induced representation.

You might then hope that, having done this, you’ve obtained all representations of $G_n$, meaning that (after enough work) the classification boils down to understanding the representations of $\mathbb{F}_q^\times$. Unfortunately, this isn’t the case. There are *cuspidal representations*, which are by definition those representations which aren’t obtained by the process I just described. In fact, it’s really the classification of the cuspidal representations which is the difficult part of the classification.

I think that Dietrich’s link should be a good reference for this, although I can’t say that I’ve ever read it. Basically any approach to the construction of the representations of $G_n$ will proceed by parabolic induction (apart from case-specific methods relying on isomorphisms, e.g. writing $GL_2(\mathbf{F}_2)\simeq S_6$ and then classifying representations in this way). You’d be best off first trying to understand the classification of representations of $GL_2$, which will be simpler as there is a single Levi subgroup (up to conjugacy) which is abelian.

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.

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