The classification of finite simple groups shows that many (those of Lie type) are obtained as (projectivisations of) subgroups of some $GL_{n}(\mathbb{F}_{q})$.
Green first determined the character table of any $GL_{n}(\mathbb{F}_{q})$ in his influential 1955 paper, while others have since given more explicit constructions of certain `cuspidal' representations.
In this talk, I will introduce parabolic induction as a means of obtaining representations of $GL_{n}(\mathbb{F}_{q})$ from those of $GL_{m}(\mathbb{F}_{q})$ where $m<n$.
Finding the irreducible representations of any $GL_{n}(\mathbb{F}_{q})$ then becomes inductive on $n$ for fixed $q$, with the cuspidal representations serving as atoms for this process.
Harish-Chandra's philosophy of cusp forms reduces the problem to the following two steps:
- Find the cuspidal representations of any $GL_{n}(\mathbb{F}_{q})$
- Determine the irreducible components of any representation $\sigma_{1}\circ\dots\circ\sigma_{k}$ parabolically induced from cuspidals $\sigma_{i}$
The majority of my talk will then aim to address each of these points.