The characters of the generalized Steinberg representations of finite general linear groups on the regular elliptic set

Let $k$ be a finite field, $k_{n}\vert k$ the degree $n$ extension of $k$, and $G=\operatorname{GL}_{n}(k)$ the general linear group with entries in $k$. This paper studies the ``generalized Steinberg" (GS) representations of $G$ and proves the equivalence of several different characterizations for this class of representations. As our main result we show that the union of the class of cuspidal and GS representations of $G$ is in natural one-one correspondence with the set of Galois orbits of characters of $k_{n}^{\times }$, the regular orbits of course corresponding to the cuspidal representations. Besides using Green's character formulas to define GS representations, we characterize GS representations by associating to them idempotents in certain commuting algebras corresponding to parabolic inductions and by showing that GS representations are the sole components of these induced representations which are ``generic" (have Whittaker vectors).