  We define a new notion of cuspidality for representations of $\GL_n$ over a
finite quotient $\Oh_k$ of the ring of integers $\Oh$ of a non-Archimedean
local field $F$ using geometric and infinitesimal induction functors, which
involve automorphism groups $G_\lambda$ of torsion $\Oh$\nobreakdash-modules.
When $n$ is a prime, we show that this notion of cuspidality is equivalent to
strong cuspidality, which arises in the construction of supercuspidal
representations of $\GL_n(F)$. We show that strongly cuspidal representations
share many features of cuspidal representations of finite general linear
groups. In the function field case, we show that the construction of the
representations of $\GL_n(\Oh_k)$ for $k\geq 2$ for all $n$ is equivalent to
the construction of the representations of all the groups $G_\lambda$. A
functional equation for zeta functions for representations of $\GL_n(\Oh_k)$ is
established for representations which are not contained in an infinitesimally
induced representation. All the cuspidal representations for $\GL_4(\Oh_2)$ are
constructed. Not all these representations are strongly cuspidal.
