Neither Etingof nor Fulton and Harris provide an actual construction for the cuspidal representations of $\mathrm{GL}_2(\mathbb{F}_q)$. Instead, they find the character that corresponds to it, without constructing the actual vector space and defining the action. I was looking for a (undergraduate) reference for their construction. Does such a reference exist? My professor suggested I look for such a reference, but admitted they couldn't recall an undergraduate appropriate one to recommend.
I have seen Deligne-Lusztig theory and cuspidal representations of $\mathrm{GL}_{2}(\mathbb{F}_{q})$, but I don't know what a cohomology is and know almost nothing about algebraic varieties.