TITLE: Reference request: Models of cuspidal representations of GL(n,k) where k is a finite field
QUESTION [8 upvotes]: Let $k=\mathbb{F}_q$ where $q$ is a prime power of odd cardinality.
Where could I find explicit models of all irreducible cuspidal (complex) representations of $GL_n(k)$ for $n\ge 3$?
I understand that the characters of such representations was constructed by J.A. Green “The characters of the finite general linear groups,” Trans. Amer. Math. Soc.,80, No. 2, 402–447 (1955).
If $n=2$, the cuspidal representations of $GL_2(k)$ can be constructed using Weil representations, which can be found in many online notes. But I could not find any explicit construction of cuspidal representations of $GL_n(k)$ when $n\ge 3$.

REPLY [4 votes]: Gel′fand S.I., Representations of the general linear group over a finite field, Lie groups and their representations (Proc. Summer School on Group Representations of the Bolya: J ́anos Math. Soc., Budapest, 1971), 119–132, Halsted, New York, 1975.  makes the action on the Kirilov model explicit.